jetwiki - User contributions [en]

Appendix/References

2021-11-03T21:35:22Z

174.64.14.12: /* LRS93b */

__FORCETOC__


=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
<ol type="1">
<li>Dyson1893</li>
<li>Dyson1893full</li>
<li>Dyson1893figure     <math>\Leftarrow</math>     This is the template highlighted below.</li>
<li>BAC84hereafter</li>
</ol>

If you want to see which of our MediaWiki chapters includes a link to — and, therefore, likely discusses — a particular journal article, step through the following options:
<ol type="1">
<li>Under the left-hand column, click on Tools/Special_pages;</li>
<li>Click on Page_tools/What_links_here;</li>
<li>In the input-box provided next to "Page," type the desired "Template" name. For example, type in:   <font color="Darkgreen">Template:Dyson1893full</font></li>
<li>Note that when a figure from a particular journal article is displayed in one or more of our MediaWiki chapters, it will usually be accompanied by a figure heading that cites the article. Hence, typing in, for example, <font color="Darkgreen">Template:Dyson1893figure</font> should identify all of the MediaWiki chapters that display a figure from Dyson (1893).</li>
</ol>
==18<sup>th</sup> Century==

==19<sup>th</sup> Century==

===Clausius1870===

[[Special:WhatLinksHere/Template:Clausius1870full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Clausius1870figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Clausius1870figure }}</div>

===Dyson1893===

[[Special:WhatLinksHere/Template:Dyson1893full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Dyson1893figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
{{ Dyson1893full }}<br />
{{ Dyson1893 }}<br />
<div align="center">{{ Dyson1893figure }}</div>

==20<sup>th</sup> Century==

===Decade0===

===Decade1===
====Eddington18====
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[[Special:WhatLinksHere/Template:Eddington18figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Eddington18figure }}</div>

===Decade2===
====Miller29====


[[Special:WhatLinksHere/Template:Miller29full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Miller29figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Miller29figure }}</div>
<div id="Miller29acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 64):<br /><font color="darkgreen">"I am greatly indebted to Professor Eddington for his valuable suggestions and help."</font>
</td></tr>
</table>
</div>

===Decade3===
====Milne30====
[[Special:WhatLinksHere/Template:Milne30full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Milne30figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Milne30figure }}</div>

===Decade4===
====LP41====
[[Special:WhatLinksHere/Template:LP41full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LP41figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ LP41figure }}</div>

====Schwarzschild41====
[[Special:WhatLinksHere/Template:Schwarzschild41full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Schwarzschild41figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Schwarzschild41figure }}</div>
<div id="Schwarzschild41acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 252):<br /><font color="darkgreen">"I am indebted to Miss Lillian Feinstein for her most helpful co-operation in the work with the punched-card machines."</font>
</td></tr>
</table>
</div>

====Cowling41====

[[Special:WhatLinksHere/Template:Cowling41full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Cowling41figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Cowling41figure }}</div>

===Decade5===

====Chatterji51====


[[Special:WhatLinksHere/Template:Chatterji51full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Chatterji51figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Chatterji51figure }}</div>

====Chatterji52====


[[Special:WhatLinksHere/Template:Chatterji52full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Chatterji52figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Chatterji52figure }}</div>

====CF53====
[[Special:WhatLinksHere/Template:CF53full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:CF53figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ CF53figure }}</div>

====MS56====
[[Special:WhatLinksHere/Template:MS56full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MS56figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ MS56figure }}</div>
<div id="MS56acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 513):<br /><font color="darkgreen">"One of us (L. M.) wishes to record with thanks the generous financial aid of the Commonwealth Fund of New York, and the warm Hospitality of the Princeton University Observatory."</font>
</td></tr>
</table>
</div>

===Decade6===
====PG61====


[[Special:WhatLinksHere/Template: PG61full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template: PG61figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ PG61figure }}</div>
<div id="PG61acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====Hunter62====


[[Special:WhatLinksHere/Template:Hunter62full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Hunter62figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Hunter62figure }}</div>
<div id="Hunter62acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 608):<br /><font color="darkgreen">"I am very grateful to Dr. L. Mestel for introducing me to this problem and for his interest and advice throughout."<br />

"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
</td></tr>
</table>
</div>

====LB62====


[[Special:WhatLinksHere/Template:LB62full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LB62figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LB62figure }}</div>
<div id="LB62acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====LMS65====


[[Special:WhatLinksHere/Template:LMS65full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LMS65figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LMS65figure }}</div>
<div id="LMS65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
</td></tr>
</table>
</div>

====Mestel65====


[[Special:WhatLinksHere/Template:Mestel65full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Mestel65figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Mestel65figure }}</div>
<div id="Mestel65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====HRW66====



[[Special:WhatLinksHere/Template:HRW66full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:HRW66figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ HRW66figure }}</div>
<div id="HRW66acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 551):<br /><font color="darkgreen">"This research was partially supported by the National Science Foundation (Grant GP-975); the authors are glad to acknowledge its assistance. We are also grateful to Professor S. Chandrasekhar for helpful comments, and to Mr. Jackson, of the Yerkes Observatory, for the radii of polytropes of indices 3.0 and 3.25 (see Table 4)."</font>
</td></tr>
</table>
</div>

===Decade7===

====Horedt70====
[[Special:WhatLinksHere/Template:Horedt70full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Horedt70figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Horedt70figure }}</div>
<div id="Horedt70acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 85):<br /><font color="darkgreen">"The author is greatly indebted to Dr M. Penston for his precious support and to an anonymous referee."</font>
</td></tr>
</table>
</div>

====VH74====

{{ VH74full }}<br />
{{ VH74 }}<br />
{{ VH74hereafter }}<br />

[[Special:WhatLinksHere/Template:VH74full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:VH74figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ VH74figure }}</div>

====Weber76====
[[Special:WhatLinksHere/Template:Weber76full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Weber76figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Weber76figure }}</div>
<div id="Weber76acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 123):<br /><font color="darkgreen">"I am indebted to Professor Frank Shu for suggesting the problem, providing advice and encouragement, and suggesting changes in the original draft of the paper. I also wish to acknowledge the tenure of an NSF Graduate Fellowship."</font>
</td></tr>
</table>
</div>

====Collins78====
[[Special:WhatLinksHere/Template:Collins78full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Collins78figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Collins78figure }}</div>

===Decade8===
====GW80====

[[Special:WhatLinksHere/Template:GW80full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:GW80figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ GW80figure }}</div>
<div id="GW80acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 996):<br /><font color="darkgreen">"We are grateful for comments by Doug Keeley and Frank Shu. We also acknowledge Glen Hermannsfeldt for excellent assistance with the computing. Stephen V. Weber was supported by a Chaim Weizmann postdoctoral fellowship. This work was supported by NSF grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7821453 AST78-21453] and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7680801 AST76-80801] A02."</font>
</td></tr>
</table>
</div>

====Kimura81b====

[[Special:WhatLinksHere/Template:Kimura81bfull|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Kimura81bfigure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Kimura81bfigure }}</div>
<div id="Kimura81backnowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 308):<br /><font color="darkgreen">"A portion of the work reported here formed part of the author's doctoral thesis, submitted to the University of Tokyo. The author is indebted to Professors W. Unno and S. Aoki for helpful discussions. Thanks are also due to Mrs. K. Sakurai for her assistance in preparing the manuscript."</font>
</td></tr>
</table>
</div>

====Whitworth81====
[[Special:WhatLinksHere/Template:Whitworth81full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Whitworth81figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Whitworth81figure }}</div>
<div id="Whitworth81acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====Tohline81====
[[Special:WhatLinksHere/Template:Tohline81full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Tohline81figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Tohline81figure }}</div>
<div id="Tohline81acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 726):<br /><font color="darkgreen">"Because Leon Mestel's (1965; {{Mestel65hereafter}}) paper has not been specifically referenced in the body of this paper, I must acknowledge its importance here. It was from Professor Mestel's work that I first gained a clear glimpse of how this problem could be attacked analytically. It is a pleasure also to acknowledge conversations with Alan Boss from which I first began to appreciate the fact that an analysis akin to the one presented in this paper was desperately needed. I thank Alan also for furnishing me with his data prior to its publication."<br />"This work was performed under the auspices of the [https://en.wikipedia.org/wiki/Energy_Research_and_Development_Administration USERDA]."</font>
</td></tr>
</table>
</div>

====Stahler83====
[[Special:WhatLinksHere/Template:Stahler83full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Stahler83figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Stahler83figure }}</div>
<div id="Stahler83acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 180):<br /><font color="darkgreen">"I wish to thank Martin Duncan, Mark Haugan, and Saul Teukolsky for useful discussions. This research was supported in part by National Science Foundation grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8116370 AST81-16370]."</font>
</td></tr>
</table>
</div>

====BAC84====

{{ BAC84full }}<br />
{{ BAC84 }}<br />
{{ BAC84hereafter }}<br />

[[Special:WhatLinksHere/Template:BAC84full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:BAC84figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ BAC84figure }}</div>
<div id="BAC84acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 844):<br /><font color="darkgreen">"We thank Willy Fowler, Lee Lindblom, Wolfgang Ober, Bob Wagoner, and Stan Woosley for useful discussions. In particular, Stan Woosley often emphasized the many ways rotation could transform simplicity into complexity. This research was supported by grants NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8022876 AST-80-22876] at Chicago, NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7923243 AST 79-23243] at Berkeley, and NSF PHY 81-19387 at Stanford, and by the [https://en.wikipedia.org/wiki/Science_and_Engineering_Research_Council SERC] at Cambridge."</font>
</td></tr>
</table>
</div>

====MF85b====



[[Special:WhatLinksHere/Template:MF85bfull|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MF85bfigure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ MF85bfigure }}</div>
<div id="MF85backnowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====Tohline85====
[[Special:WhatLinksHere/Template:Tohline85full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Tohline85figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Tohline85figure }}</div>
<div id="Tohline85acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 187):<br /><font color="darkgreen">"Frank Shu and Hugh Van Horn have made extremely enlightening comments to me in the course of this investigation. I thank them for their interest and support. I am indebted to Mort Roberts for allowing me to pull my thoughts together in the friendly, invigorating environment of NRAO, Charlottesville, and to E. F. Zganjar, A. U. Landolt, and C. L. Perry for doing all the dirty work at LSU while I was gone for the summer. This work was supported by the National Science Foundation through grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8217744 AST-8217744]."</font>
</td></tr>
</table>
</div>

====FWW86====
[[Special:WhatLinksHere/Template:FWW86full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:FWW86figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ FWW86figure }}</div>
<div id="FWW86acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 685):<br /><font color="darkgreen">"The authors gratefully acknowledge many stimulating and informative conversations with Willy Fowler on the subject of supermassive stars. This research has been supported by the National Science Foundation under grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8108509 AST81-08509] A01 and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8418185 AST 84-18185] and, at [https://en.wikipedia.org/wiki/Lawrence_Livermore_National_Laboratory Livermore], by the Department of Energy under W-7405-EN6-48</font> [probably should be ENG, rather than EN6]."
</td></tr>
</table>
</div>

====Horedt86====

[[Special:WhatLinksHere/Template:Horedt86full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Horedt86figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Horedt86figure }}</div>
<div id="Horedt86acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

===Decade9===
====LRS93b====

[[Special:WhatLinksHere/Template:LRS93bfull|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LRS93bfigure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LRS93bfigure }}</div>
<div id="LRS93backnowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 248):<br /><font color="darkgreen">"This work was supported by NSF grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=9015451 AST 90-15451] and NASA grant NAGW-2364 to Cornell University. Partial support was also provided by NASA through grant HF-1037.01-92A awarded by the Space Telescope Science Institute which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA under contract NAS5-26555. Computations were performed on the Cornell National Supercomputer Facility, a resource of the Center for Theory and Simulation in Science and Engineering at Cornell University, which receives major funding from the NSF and from the IBM Corporation, with additional support from New York State and members of its Corporate Research Institute."</font>
</td></tr>
</table>
</div>

==21<sup>st</sup> Century==

===MTF2002===
[[Special:WhatLinksHere/Template:MTF2002full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MTF2002figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ MTF2002figure }}</div>

===ZEUS-MP2006===
[[Special:WhatLinksHere/Template:ZEUS-MP2006full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:ZEUS-MP2006figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ ZEUS-MP2006figure }}</div>

===PK2007===

[[Special:WhatLinksHere/Template:PK2007full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:PK2007figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ PK2007figure }}</div>

===MT2012===
[[Special:WhatLinksHere/Template:MT2012full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MT2012figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">
{{ MT2012figure }}
</div>

=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Appendix/References

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=Journal Articles=
==Explanation==

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"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
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<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

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<tr><td align="left">Acknowledgements (p. 551):<br /><font color="darkgreen">"This research was partially supported by the National Science Foundation (Grant GP-975); the authors are glad to acknowledge its assistance. We are also grateful to Professor S. Chandrasekhar for helpful comments, and to Mr. Jackson, of the Yerkes Observatory, for the radii of polytropes of indices 3.0 and 3.25 (see Table 4)."</font>
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<tr><td align="left">Acknowledgements (p. 996):<br /><font color="darkgreen">"We are grateful for comments by Doug Keeley and Frank Shu. We also acknowledge Glen Hermannsfeldt for excellent assistance with the computing. Stephen V. Weber was supported by a Chaim Weizmann postdoctoral fellowship. This work was supported by NSF grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7821453 AST78-21453] and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7680801 AST76-80801] A02."</font>
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<tr><td align="left">Acknowledgements (p. 726):<br /><font color="darkgreen">"Because Leon Mestel's (1965; {{Mestel65hereafter}}) paper has not been specifically referenced in the body of this paper, I must acknowledge its importance here. It was from Professor Mestel's work that I first gained a clear glimpse of how this problem could be attacked analytically. It is a pleasure also to acknowledge conversations with Alan Boss from which I first began to appreciate the fact that an analysis akin to the one presented in this paper was desperately needed. I thank Alan also for furnishing me with his data prior to its publication."<br />"This work was performed under the auspices of the [https://en.wikipedia.org/wiki/Energy_Research_and_Development_Administration USERDA]."</font>
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====Stahler83====
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<tr><td align="left">Acknowledgements (p. 180):<br /><font color="darkgreen">"I wish to thank Martin Duncan, Mark Haugan, and Saul Teukolsky for useful discussions. This research was supported in part by National Science Foundation grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8116370 AST81-16370]."</font>
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====BAC84====

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<div align="center">{{ BAC84figure }}</div>
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<tr><td align="left">Acknowledgements (p. 844):<br /><font color="darkgreen">"We thank Willy Fowler, Lee Lindblom, Wolfgang Ober, Bob Wagoner, and Stan Woosley for useful discussions. In particular, Stan Woosley often emphasized the many ways rotation could transform simplicity into complexity. This research was supported by grants NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8022876 AST-80-22876] at Chicago, NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7923243 AST 79-23243] at Berkeley, and NSF PHY 81-19387 at Stanford, and by the [https://en.wikipedia.org/wiki/Science_and_Engineering_Research_Council SERC] at Cambridge."</font>
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<tr><td align="left">Acknowledgements (p. 187):<br /><font color="darkgreen">"Frank Shu and Hugh Van Horn have made extremely enlightening comments to me in the course of this investigation. I thank them for their interest and support. I am indebted to Mort Roberts for allowing me to pull my thoughts together in the friendly, invigorating environment of NRAO, Charlottesville, and to E. F. Zganjar, A. U. Landolt, and C. L. Perry for doing all the dirty work at LSU while I was gone for the summer. This work was supported by the National Science Foundation through grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8217744 AST-8217744]."</font>
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====FWW86====
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<tr><td align="left">Acknowledgements (p. 685):<br /><font color="darkgreen">"The authors gratefully acknowledge many stimulating and informative conversations with Willy Fowler on the subject of supermassive stars. This research has been supported by the National Science Foundation under grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8108509 AST81-08509] A01 and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8418185 AST 84-18185] and, at [https://en.wikipedia.org/wiki/Lawrence_Livermore_National_Laboratory Livermore], by the Department of Energy under W-7405-EN6-48</font> [probably should be ENG, rather than EN6]."
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====Horedt86====

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===Decade9===
====LRS93b====

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<tr><td align="left">Acknowledgements (p. 248):<br /><font color="darkgreen">"This work was supported by NSF grant AST 90-15451 and NASA grant NAGW-2364 to Cornell University. Partial support was also provided by NASA through grant HF-1037.01-92A awarded by the Space Telescope Science Institute which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA under contract NAS5-26555. Computations were performed on the Cornell National Supercomputer Facility, a resource of the Center for Theory and Simulation in Science and Engineering at Cornell University, which receives major funding from the NSF and from the IBM Corporation, with additional support from New York State and members of its Corporate Research Institute."</font>
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==21<sup>st</sup> Century==

===MTF2002===
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=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
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=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
<ol type="1">
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If you want to see which of our MediaWiki chapters includes a link to — and, therefore, likely discusses — a particular journal article, step through the following options:
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==18<sup>th</sup> Century==

==19<sup>th</sup> Century==

===Clausius1870===

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==20<sup>th</sup> Century==

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<tr><td align="left">Acknowledgements (p. 64):<br /><font color="darkgreen">"I am greatly indebted to Professor Eddington for his valuable suggestions and help."</font>
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===Decade3===
====Milne30====
[[Special:WhatLinksHere/Template:Milne30full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Milne30figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Milne30figure }}</div>

===Decade4===
====LP41====
[[Special:WhatLinksHere/Template:LP41full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LP41figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ LP41figure }}</div>

====Schwarzschild41====
[[Special:WhatLinksHere/Template:Schwarzschild41full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Schwarzschild41figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Schwarzschild41figure }}</div>
<div id="Schwarzschild41acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 252):<br /><font color="darkgreen">"I am indebted to Miss Lillian Feinstein for her most helpful co-operation in the work with the punched-card machines."</font>
</td></tr>
</table>
</div>

====Cowling41====

[[Special:WhatLinksHere/Template:Cowling41full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Cowling41figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Cowling41figure }}</div>

===Decade5===

====Chatterji51====


[[Special:WhatLinksHere/Template:Chatterji51full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Chatterji51figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Chatterji51figure }}</div>

====Chatterji52====


[[Special:WhatLinksHere/Template:Chatterji52full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Chatterji52figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Chatterji52figure }}</div>

====CF53====
[[Special:WhatLinksHere/Template:CF53full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:CF53figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ CF53figure }}</div>

====MS56====
[[Special:WhatLinksHere/Template:MS56full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MS56figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ MS56figure }}</div>
<div id="MS56acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 513):<br /><font color="darkgreen">"One of us (L. M.) wishes to record with thanks the generous financial aid of the Commonwealth Fund of New York, and the warm Hospitality of the Princeton University Observatory."</font>
</td></tr>
</table>
</div>

===Decade6===
====PG61====


[[Special:WhatLinksHere/Template: PG61full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template: PG61figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ PG61figure }}</div>
<div id="PG61acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====Hunter62====


[[Special:WhatLinksHere/Template:Hunter62full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Hunter62figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Hunter62figure }}</div>
<div id="Hunter62acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 608):<br /><font color="darkgreen">"I am very grateful to Dr. L. Mestel for introducing me to this problem and for his interest and advice throughout."<br />

"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
</td></tr>
</table>
</div>

====LB62====


[[Special:WhatLinksHere/Template:LB62full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LB62figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LB62figure }}</div>
<div id="LB62acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====LMS65====


[[Special:WhatLinksHere/Template:LMS65full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LMS65figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LMS65figure }}</div>
<div id="LMS65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
</td></tr>
</table>
</div>

====Mestel65====


[[Special:WhatLinksHere/Template:Mestel65full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Mestel65figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Mestel65figure }}</div>
<div id="Mestel65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====HRW66====



[[Special:WhatLinksHere/Template:HRW66full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:HRW66figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ HRW66figure }}</div>
<div id="HRW66acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 551):<br /><font color="darkgreen">"This research was partially supported by the National Science Foundation (Grant GP-975); the authors are glad to acknowledge its assistance. We are also grateful to Professor S. Chandrasekhar for helpful comments, and to Mr. Jackson, of the Yerkes Observatory, for the radii of polytropes of indices 3.0 and 3.25 (see Table 4)."</font>
</td></tr>
</table>
</div>

===Decade7===

====Horedt70====
[[Special:WhatLinksHere/Template:Horedt70full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Horedt70figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Horedt70figure }}</div>
<div id="Horedt70acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 85):<br /><font color="darkgreen">"The author is greatly indebted to Dr M. Penston for his precious support and to an anonymous referee."</font>
</td></tr>
</table>
</div>

====VH74====

{{ VH74full }}<br />
{{ VH74 }}<br />
{{ VH74hereafter }}<br />

[[Special:WhatLinksHere/Template:VH74full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:VH74figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ VH74figure }}</div>

====Weber76====
[[Special:WhatLinksHere/Template:Weber76full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Weber76figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Weber76figure }}</div>
<div id="Weber76acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 123):<br /><font color="darkgreen">"I am indebted to Professor Frank Shu for suggesting the problem, providing advice and encouragement, and suggesting changes in the original draft of the paper. I also wish to acknowledge the tenure of an NSF Graduate Fellowship."</font>
</td></tr>
</table>
</div>

====Collins78====
[[Special:WhatLinksHere/Template:Collins78full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Collins78figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Collins78figure }}</div>

===Decade8===
====GW80====

[[Special:WhatLinksHere/Template:GW80full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:GW80figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ GW80figure }}</div>
<div id="GW80acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 996):<br /><font color="darkgreen">"We are grateful for comments by Doug Keeley and Frank Shu. We also acknowledge Glen Hermannsfeldt for excellent assistance with the computing. Stephen V. Weber was supported by a Chaim Weizmann postdoctoral fellowship. This work was supported by NSF grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7821453 AST78-21453] and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7680801 AST76-80801] A02."</font>
</td></tr>
</table>
</div>

====Kimura81b====

[[Special:WhatLinksHere/Template:Kimura81bfull|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Kimura81bfigure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Kimura81bfigure }}</div>
<div id="Kimura81backnowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 308):<br /><font color="darkgreen">"A portion of the work reported here formed part of the author's doctoral thesis, submitted to the University of Tokyo. The author is indebted to Professors W. Unno and S. Aoki for helpful discussions. Thanks are also due to Mrs. K. Sakurai for her assistance in preparing the manuscript."</font>
</td></tr>
</table>
</div>

====Whitworth81====
[[Special:WhatLinksHere/Template:Whitworth81full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Whitworth81figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Whitworth81figure }}</div>
<div id="Whitworth81acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====Tohline81====
[[Special:WhatLinksHere/Template:Tohline81full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Tohline81figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Tohline81figure }}</div>
<div id="Tohline81acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 726):<br /><font color="darkgreen">"Because Leon Mestel's (1965; {{Mestel65hereafter}}) paper has not been specifically referenced in the body of this paper, I must acknowledge its importance here. It was from Professor Mestel's work that I first gained a clear glimpse of how this problem could be attacked analytically. It is a pleasure also to acknowledge conversations with Alan Boss from which I first began to appreciate the fact that an analysis akin to the one presented in this paper was desperately needed. I thank Alan also for furnishing me with his data prior to its publication."<br />"This work was performed under the auspices of the [https://en.wikipedia.org/wiki/Energy_Research_and_Development_Administration USERDA]."</font>
</td></tr>
</table>
</div>

====Stahler83====
[[Special:WhatLinksHere/Template:Stahler83full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Stahler83figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Stahler83figure }}</div>
<div id="Stahler83acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 180):<br /><font color="darkgreen">"I wish to thank Martin Duncan, Mark Haugan, and Saul Teukolsky for useful discussions. This research was supported in part by National Science Foundation grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8116370 AST81-16370]."</font>
</td></tr>
</table>
</div>

====BAC84====

{{ BAC84full }}<br />
{{ BAC84 }}<br />
{{ BAC84hereafter }}<br />

[[Special:WhatLinksHere/Template:BAC84full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:BAC84figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ BAC84figure }}</div>
<div id="BAC84acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 844):<br /><font color="darkgreen">"We thank Willy Fowler, Lee Lindblom, Wolfgang Ober, Bob Wagoner, and Stan Woosley for useful discussions. In particular, Stan Woosley often emphasized the many ways rotation could transform simplicity into complexity. This research was supported by grants NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8022876 AST-80-22876] at Chicago, NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7923243 AST 79-23243] at Berkeley, and NSF PHY 81-19387 at Stanford, and by the [https://en.wikipedia.org/wiki/Science_and_Engineering_Research_Council SERC] at Cambridge."</font>
</td></tr>
</table>
</div>

====MF85b====



[[Special:WhatLinksHere/Template:MF85bfull|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MF85bfigure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ MF85bfigure }}</div>
<div id="MF85backnowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====Tohline85====
[[Special:WhatLinksHere/Template:Tohline85full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Tohline85figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Tohline85figure }}</div>
<div id="Tohline85acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 187):<br /><font color="darkgreen">"Frank Shu and Hugh Van Horn have made extremely enlightening comments to me in the course of this investigation. I thank them for their interest and support. I am indebted to Mort Roberts for allowing me to pull my thoughts together in the friendly, invigorating environment of NRAO, Charlottesville, and to E. F. Zganjar, A. U. Landolt, and C. L. Perry for doing all the dirty work at LSU while I was gone for the summer. This work was supported by the National Science Foundation through grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8217744 AST-8217744]."</font>
</td></tr>
</table>
</div>

====FWW86====
[[Special:WhatLinksHere/Template:FWW86full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:FWW86figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ FWW86figure }}</div>
<div id="FWW86acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 685):<br /><font color="darkgreen">"The authors gratefully acknowledge many stimulating and informative conversations with Willy Fowler on the subject of supermassive stars. This research has been supported by the National Science Foundation under grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8108509 AST81-08509] A01 and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8418185 AST 84-18185] and, at [https://en.wikipedia.org/wiki/Lawrence_Livermore_National_Laboratory Livermore], by the Department of Energy under W-7405-EN6-48</font> [probably should be ENG, rather than EN6]."
</td></tr>
</table>
</div>

====Horedt86====

[[Special:WhatLinksHere/Template:Horedt86full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Horedt86figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Horedt86figure }}</div>
<div id="Horedt86acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

===Decade9===
====LRS93b====

[[Special:WhatLinksHere/Template:LRS93bfull|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LRS93bfigure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LRS93bfigure }}</div>

==21<sup>st</sup> Century==

===MTF2002===
[[Special:WhatLinksHere/Template:MTF2002full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MTF2002figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ MTF2002figure }}</div>

===ZEUS-MP2006===
[[Special:WhatLinksHere/Template:ZEUS-MP2006full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:ZEUS-MP2006figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ ZEUS-MP2006figure }}</div>

===PK2007===

[[Special:WhatLinksHere/Template:PK2007full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:PK2007figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ PK2007figure }}</div>

===MT2012===
[[Special:WhatLinksHere/Template:MT2012full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MT2012figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">
{{ MT2012figure }}
</div>

=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Appendix/References

2021-11-03T21:21:39Z

174.64.14.12: /* Whitworth81 */

__FORCETOC__


=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
<ol type="1">
<li>Dyson1893</li>
<li>Dyson1893full</li>
<li>Dyson1893figure     <math>\Leftarrow</math>     This is the template highlighted below.</li>
<li>BAC84hereafter</li>
</ol>

If you want to see which of our MediaWiki chapters includes a link to — and, therefore, likely discusses — a particular journal article, step through the following options:
<ol type="1">
<li>Under the left-hand column, click on Tools/Special_pages;</li>
<li>Click on Page_tools/What_links_here;</li>
<li>In the input-box provided next to "Page," type the desired "Template" name. For example, type in:   <font color="Darkgreen">Template:Dyson1893full</font></li>
<li>Note that when a figure from a particular journal article is displayed in one or more of our MediaWiki chapters, it will usually be accompanied by a figure heading that cites the article. Hence, typing in, for example, <font color="Darkgreen">Template:Dyson1893figure</font> should identify all of the MediaWiki chapters that display a figure from Dyson (1893).</li>
</ol>
==18<sup>th</sup> Century==

==19<sup>th</sup> Century==

===Clausius1870===

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<div align="center">{{ Clausius1870figure }}</div>

===Dyson1893===

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==20<sup>th</sup> Century==

===Decade0===

===Decade1===
====Eddington18====
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<div align="center">{{ Eddington18figure }}</div>

===Decade2===
====Miller29====


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<div align="center">{{ Miller29figure }}</div>
<div id="Miller29acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 64):<br /><font color="darkgreen">"I am greatly indebted to Professor Eddington for his valuable suggestions and help."</font>
</td></tr>
</table>
</div>

===Decade3===
====Milne30====
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===Decade4===
====LP41====
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====Schwarzschild41====
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<div align="center">{{ Schwarzschild41figure }}</div>
<div id="Schwarzschild41acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 252):<br /><font color="darkgreen">"I am indebted to Miss Lillian Feinstein for her most helpful co-operation in the work with the punched-card machines."</font>
</td></tr>
</table>
</div>

====Cowling41====

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<div align="center">{{ Cowling41figure }}</div>

===Decade5===

====Chatterji51====


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====Chatterji52====


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<div align="center">{{ Chatterji52figure }}</div>

====CF53====
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[[Special:WhatLinksHere/Template:CF53figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ CF53figure }}</div>

====MS56====
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[[Special:WhatLinksHere/Template:MS56figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ MS56figure }}</div>
<div id="MS56acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 513):<br /><font color="darkgreen">"One of us (L. M.) wishes to record with thanks the generous financial aid of the Commonwealth Fund of New York, and the warm Hospitality of the Princeton University Observatory."</font>
</td></tr>
</table>
</div>

===Decade6===
====PG61====


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[[Special:WhatLinksHere/Template: PG61figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ PG61figure }}</div>
<div id="PG61acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====Hunter62====


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[[Special:WhatLinksHere/Template:Hunter62figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Hunter62figure }}</div>
<div id="Hunter62acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 608):<br /><font color="darkgreen">"I am very grateful to Dr. L. Mestel for introducing me to this problem and for his interest and advice throughout."<br />

"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
</td></tr>
</table>
</div>

====LB62====


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[[Special:WhatLinksHere/Template:LB62figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LB62figure }}</div>
<div id="LB62acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====LMS65====


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[[Special:WhatLinksHere/Template:LMS65figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LMS65figure }}</div>
<div id="LMS65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
</td></tr>
</table>
</div>

====Mestel65====


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<div align="center">{{ Mestel65figure }}</div>
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<tr><td align="left">Acknowledgements:  n/a
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</div>

====HRW66====



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<div align="center">{{ HRW66figure }}</div>
<div id="HRW66acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 551):<br /><font color="darkgreen">"This research was partially supported by the National Science Foundation (Grant GP-975); the authors are glad to acknowledge its assistance. We are also grateful to Professor S. Chandrasekhar for helpful comments, and to Mr. Jackson, of the Yerkes Observatory, for the radii of polytropes of indices 3.0 and 3.25 (see Table 4)."</font>
</td></tr>
</table>
</div>

===Decade7===

====Horedt70====
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<div align="center">{{ Horedt70figure }}</div>
<div id="Horedt70acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 85):<br /><font color="darkgreen">"The author is greatly indebted to Dr M. Penston for his precious support and to an anonymous referee."</font>
</td></tr>
</table>
</div>

====VH74====

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<div align="center">{{ VH74figure }}</div>

====Weber76====
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<div align="center">{{ Weber76figure }}</div>
<div id="Weber76acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 123):<br /><font color="darkgreen">"I am indebted to Professor Frank Shu for suggesting the problem, providing advice and encouragement, and suggesting changes in the original draft of the paper. I also wish to acknowledge the tenure of an NSF Graduate Fellowship."</font>
</td></tr>
</table>
</div>

====Collins78====
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<div align="center">{{ Collins78figure }}</div>

===Decade8===
====GW80====

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<div align="center">{{ GW80figure }}</div>
<div id="GW80acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 996):<br /><font color="darkgreen">"We are grateful for comments by Doug Keeley and Frank Shu. We also acknowledge Glen Hermannsfeldt for excellent assistance with the computing. Stephen V. Weber was supported by a Chaim Weizmann postdoctoral fellowship. This work was supported by NSF grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7821453 AST78-21453] and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7680801 AST76-80801] A02."</font>
</td></tr>
</table>
</div>

====Kimura81b====

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<div align="center">{{ Kimura81bfigure }}</div>
<div id="Kimura81backnowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 308):<br /><font color="darkgreen">"A portion of the work reported here formed part of the author's doctoral thesis, submitted to the University of Tokyo. The author is indebted to Professors W. Unno and S. Aoki for helpful discussions. Thanks are also due to Mrs. K. Sakurai for her assistance in preparing the manuscript."</font>
</td></tr>
</table>
</div>

====Whitworth81====
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<tr><td align="left">Acknowledgements:  n/a
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====Tohline81====
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<div align="center">{{ Tohline81figure }}</div>
<div id="Tohline81acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 726):<br /><font color="darkgreen">"Because Leon Mestel's (1965; {{Mestel65hereafter}}) paper has not been specifically referenced in the body of this paper, I must acknowledge its importance here. It was from Professor Mestel's work that I first gained a clear glimpse of how this problem could be attacked analytically. It is a pleasure also to acknowledge conversations with Alan Boss from which I first began to appreciate the fact that an analysis akin to the one presented in this paper was desperately needed. I thank Alan also for furnishing me with his data prior to its publication."<br />"This work was performed under the auspices of the [https://en.wikipedia.org/wiki/Energy_Research_and_Development_Administration USERDA]."</font>
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====Stahler83====
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<div align="center">{{ Stahler83figure }}</div>
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<tr><td align="left">Acknowledgements (p. 180):<br /><font color="darkgreen">"I wish to thank Martin Duncan, Mark Haugan, and Saul Teukolsky for useful discussions. This research was supported in part by National Science Foundation grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8116370 AST81-16370]."</font>
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</table>
</div>

====BAC84====

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<div align="center">{{ BAC84figure }}</div>
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<tr><td align="left">Acknowledgements (p. 844):<br /><font color="darkgreen">"We thank Willy Fowler, Lee Lindblom, Wolfgang Ober, Bob Wagoner, and Stan Woosley for useful discussions. In particular, Stan Woosley often emphasized the many ways rotation could transform simplicity into complexity. This research was supported by grants NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8022876 AST-80-22876] at Chicago, NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7923243 AST 79-23243] at Berkeley, and NSF PHY 81-19387 at Stanford, and by the [https://en.wikipedia.org/wiki/Science_and_Engineering_Research_Council SERC] at Cambridge."</font>
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====MF85b====



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<tr><td align="left">Acknowledgements:  n/a
</td></tr>
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</div>

====Tohline85====
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<div id="Tohline85acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 187):<br /><font color="darkgreen">"Frank Shu and Hugh Van Horn have made extremely enlightening comments to me in the course of this investigation. I thank them for their interest and support. I am indebted to Mort Roberts for allowing me to pull my thoughts together in the friendly, invigorating environment of NRAO, Charlottesville, and to E. F. Zganjar, A. U. Landolt, and C. L. Perry for doing all the dirty work at LSU while I was gone for the summer. This work was supported by the National Science Foundation through grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8217744 AST-8217744]."</font>
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</table>
</div>

====FWW86====
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<div id="FWW86acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 685):<br /><font color="darkgreen">"The authors gratefully acknowledge many stimulating and informative conversations with Willy Fowler on the subject of supermassive stars. This research has been supported by the National Science Foundation under grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8108509 AST81-08509] A01 and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8418185 AST 84-18185] and, at [https://en.wikipedia.org/wiki/Lawrence_Livermore_National_Laboratory Livermore], by the Department of Energy under W-7405-EN6-48</font> [probably should be ENG, rather than EN6]."
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====Horedt86====

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==21<sup>st</sup> Century==

===MTF2002===
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=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Appendix/References

2021-11-03T21:20:56Z

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=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
<ol type="1">
<li>Dyson1893</li>
<li>Dyson1893full</li>
<li>Dyson1893figure     <math>\Leftarrow</math>     This is the template highlighted below.</li>
<li>BAC84hereafter</li>
</ol>

If you want to see which of our MediaWiki chapters includes a link to — and, therefore, likely discusses — a particular journal article, step through the following options:
<ol type="1">
<li>Under the left-hand column, click on Tools/Special_pages;</li>
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<li>In the input-box provided next to "Page," type the desired "Template" name. For example, type in:   <font color="Darkgreen">Template:Dyson1893full</font></li>
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</ol>
==18<sup>th</sup> Century==

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===Clausius1870===

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<tr><td align="left">Acknowledgements (p. 64):<br /><font color="darkgreen">"I am greatly indebted to Professor Eddington for his valuable suggestions and help."</font>
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<tr><td align="left">Acknowledgements (p. 252):<br /><font color="darkgreen">"I am indebted to Miss Lillian Feinstein for her most helpful co-operation in the work with the punched-card machines."</font>
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<tr><td align="left">Acknowledgements (p. 513):<br /><font color="darkgreen">"One of us (L. M.) wishes to record with thanks the generous financial aid of the Commonwealth Fund of New York, and the warm Hospitality of the Princeton University Observatory."</font>
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<tr><td align="left">Acknowledgements (p. 608):<br /><font color="darkgreen">"I am very grateful to Dr. L. Mestel for introducing me to this problem and for his interest and advice throughout."<br />

"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
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<div align="center">{{ LB62figure }}</div>
<div id="LB62acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====LMS65====


[[Special:WhatLinksHere/Template:LMS65full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LMS65figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LMS65figure }}</div>
<div id="LMS65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
</td></tr>
</table>
</div>

====Mestel65====


[[Special:WhatLinksHere/Template:Mestel65full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Mestel65figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Mestel65figure }}</div>
<div id="Mestel65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====HRW66====



[[Special:WhatLinksHere/Template:HRW66full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:HRW66figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ HRW66figure }}</div>
<div id="HRW66acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 551):<br /><font color="darkgreen">"This research was partially supported by the National Science Foundation (Grant GP-975); the authors are glad to acknowledge its assistance. We are also grateful to Professor S. Chandrasekhar for helpful comments, and to Mr. Jackson, of the Yerkes Observatory, for the radii of polytropes of indices 3.0 and 3.25 (see Table 4)."</font>
</td></tr>
</table>
</div>

===Decade7===

====Horedt70====
[[Special:WhatLinksHere/Template:Horedt70full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Horedt70figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Horedt70figure }}</div>
<div id="Horedt70acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 85):<br /><font color="darkgreen">"The author is greatly indebted to Dr M. Penston for his precious support and to an anonymous referee."</font>
</td></tr>
</table>
</div>

====VH74====

{{ VH74full }}<br />
{{ VH74 }}<br />
{{ VH74hereafter }}<br />

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[[Special:WhatLinksHere/Template:VH74figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ VH74figure }}</div>

====Weber76====
[[Special:WhatLinksHere/Template:Weber76full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Weber76figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Weber76figure }}</div>
<div id="Weber76acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 123):<br /><font color="darkgreen">"I am indebted to Professor Frank Shu for suggesting the problem, providing advice and encouragement, and suggesting changes in the original draft of the paper. I also wish to acknowledge the tenure of an NSF Graduate Fellowship."</font>
</td></tr>
</table>
</div>

====Collins78====
[[Special:WhatLinksHere/Template:Collins78full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Collins78figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Collins78figure }}</div>

===Decade8===
====GW80====

[[Special:WhatLinksHere/Template:GW80full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:GW80figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ GW80figure }}</div>
<div id="GW80acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 996):<br /><font color="darkgreen">"We are grateful for comments by Doug Keeley and Frank Shu. We also acknowledge Glen Hermannsfeldt for excellent assistance with the computing. Stephen V. Weber was supported by a Chaim Weizmann postdoctoral fellowship. This work was supported by NSF grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7821453 AST78-21453] and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7680801 AST76-80801] A02."</font>
</td></tr>
</table>
</div>

====Kimura81b====

[[Special:WhatLinksHere/Template:Kimura81bfull|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Kimura81bfigure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Kimura81bfigure }}</div>
<div id="Kimura81backnowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 308):<br /><font color="darkgreen">"A portion of the work reported here formed part of the author's doctoral thesis, submitted to the University of Tokyo. The author is indebted to Professors W. Unno and S. Aoki for helpful discussions. Thanks are also due to Mrs. K. Sakurai for her assistance in preparing the manuscript."</font>
</td></tr>
</table>
</div>

====Whitworth81====
[[Special:WhatLinksHere/Template:Whitworth81full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Whitworth81figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Whitworth81figure }}</div>

====Tohline81====
[[Special:WhatLinksHere/Template:Tohline81full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Tohline81figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Tohline81figure }}</div>
<div id="Tohline81acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 726):<br /><font color="darkgreen">"Because Leon Mestel's (1965; {{Mestel65hereafter}}) paper has not been specifically referenced in the body of this paper, I must acknowledge its importance here. It was from Professor Mestel's work that I first gained a clear glimpse of how this problem could be attacked analytically. It is a pleasure also to acknowledge conversations with Alan Boss from which I first began to appreciate the fact that an analysis akin to the one presented in this paper was desperately needed. I thank Alan also for furnishing me with his data prior to its publication."<br />"This work was performed under the auspices of the [https://en.wikipedia.org/wiki/Energy_Research_and_Development_Administration USERDA]."</font>
</td></tr>
</table>
</div>

====Stahler83====
[[Special:WhatLinksHere/Template:Stahler83full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Stahler83figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Stahler83figure }}</div>
<div id="Stahler83acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 180):<br /><font color="darkgreen">"I wish to thank Martin Duncan, Mark Haugan, and Saul Teukolsky for useful discussions. This research was supported in part by National Science Foundation grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8116370 AST81-16370]."</font>
</td></tr>
</table>
</div>

====BAC84====

{{ BAC84full }}<br />
{{ BAC84 }}<br />
{{ BAC84hereafter }}<br />

[[Special:WhatLinksHere/Template:BAC84full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:BAC84figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ BAC84figure }}</div>
<div id="BAC84acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 844):<br /><font color="darkgreen">"We thank Willy Fowler, Lee Lindblom, Wolfgang Ober, Bob Wagoner, and Stan Woosley for useful discussions. In particular, Stan Woosley often emphasized the many ways rotation could transform simplicity into complexity. This research was supported by grants NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8022876 AST-80-22876] at Chicago, NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7923243 AST 79-23243] at Berkeley, and NSF PHY 81-19387 at Stanford, and by the [https://en.wikipedia.org/wiki/Science_and_Engineering_Research_Council SERC] at Cambridge."</font>
</td></tr>
</table>
</div>

====MF85b====



[[Special:WhatLinksHere/Template:MF85bfull|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MF85bfigure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ MF85bfigure }}</div>
<div id="MF85backnowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====Tohline85====
[[Special:WhatLinksHere/Template:Tohline85full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Tohline85figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Tohline85figure }}</div>
<div id="Tohline85acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 187):<br /><font color="darkgreen">"Frank Shu and Hugh Van Horn have made extremely enlightening comments to me in the course of this investigation. I thank them for their interest and support. I am indebted to Mort Roberts for allowing me to pull my thoughts together in the friendly, invigorating environment of NRAO, Charlottesville, and to E. F. Zganjar, A. U. Landolt, and C. L. Perry for doing all the dirty work at LSU while I was gone for the summer. This work was supported by the National Science Foundation through grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8217744 AST-8217744]."</font>
</td></tr>
</table>
</div>

====FWW86====
[[Special:WhatLinksHere/Template:FWW86full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:FWW86figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ FWW86figure }}</div>
<div id="FWW86acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 685):<br /><font color="darkgreen">"The authors gratefully acknowledge many stimulating and informative conversations with Willy Fowler on the subject of supermassive stars. This research has been supported by the National Science Foundation under grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8108509 AST81-08509] A01 and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8418185 AST 84-18185] and, at [https://en.wikipedia.org/wiki/Lawrence_Livermore_National_Laboratory Livermore], by the Department of Energy under W-7405-EN6-48</font> [probably should be ENG, rather than EN6]."
</td></tr>
</table>
</div>

====Horedt86====

[[Special:WhatLinksHere/Template:Horedt86full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Horedt86figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Horedt86figure }}</div>

===Decade9===
====LRS93b====

[[Special:WhatLinksHere/Template:LRS93bfull|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LRS93bfigure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LRS93bfigure }}</div>

==21<sup>st</sup> Century==

===MTF2002===
[[Special:WhatLinksHere/Template:MTF2002full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MTF2002figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ MTF2002figure }}</div>

===ZEUS-MP2006===
[[Special:WhatLinksHere/Template:ZEUS-MP2006full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:ZEUS-MP2006figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ ZEUS-MP2006figure }}</div>

===PK2007===

[[Special:WhatLinksHere/Template:PK2007full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:PK2007figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ PK2007figure }}</div>

===MT2012===
[[Special:WhatLinksHere/Template:MT2012full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MT2012figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">
{{ MT2012figure }}
</div>

=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Appendix/References

2021-11-03T21:20:32Z

174.64.14.12: /* LB62 */

__FORCETOC__


=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
<ol type="1">
<li>Dyson1893</li>
<li>Dyson1893full</li>
<li>Dyson1893figure     <math>\Leftarrow</math>     This is the template highlighted below.</li>
<li>BAC84hereafter</li>
</ol>

If you want to see which of our MediaWiki chapters includes a link to — and, therefore, likely discusses — a particular journal article, step through the following options:
<ol type="1">
<li>Under the left-hand column, click on Tools/Special_pages;</li>
<li>Click on Page_tools/What_links_here;</li>
<li>In the input-box provided next to "Page," type the desired "Template" name. For example, type in:   <font color="Darkgreen">Template:Dyson1893full</font></li>
<li>Note that when a figure from a particular journal article is displayed in one or more of our MediaWiki chapters, it will usually be accompanied by a figure heading that cites the article. Hence, typing in, for example, <font color="Darkgreen">Template:Dyson1893figure</font> should identify all of the MediaWiki chapters that display a figure from Dyson (1893).</li>
</ol>
==18<sup>th</sup> Century==

==19<sup>th</sup> Century==

===Clausius1870===

[[Special:WhatLinksHere/Template:Clausius1870full|What chapter(s) cite this article?]]<br />
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<div align="center">{{ Clausius1870figure }}</div>

===Dyson1893===

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{{ Dyson1893full }}<br />
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==20<sup>th</sup> Century==

===Decade0===

===Decade1===
====Eddington18====
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<div align="center">{{ Eddington18figure }}</div>

===Decade2===
====Miller29====


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[[Special:WhatLinksHere/Template:Miller29figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Miller29figure }}</div>
<div id="Miller29acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 64):<br /><font color="darkgreen">"I am greatly indebted to Professor Eddington for his valuable suggestions and help."</font>
</td></tr>
</table>
</div>

===Decade3===
====Milne30====
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<div align="center">{{ Milne30figure }}</div>

===Decade4===
====LP41====
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<div align="center">{{ LP41figure }}</div>

====Schwarzschild41====
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<div align="center">{{ Schwarzschild41figure }}</div>
<div id="Schwarzschild41acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 252):<br /><font color="darkgreen">"I am indebted to Miss Lillian Feinstein for her most helpful co-operation in the work with the punched-card machines."</font>
</td></tr>
</table>
</div>

====Cowling41====

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<div align="center">{{ Cowling41figure }}</div>

===Decade5===

====Chatterji51====


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<div align="center">{{ Chatterji51figure }}</div>

====Chatterji52====


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<div align="center">{{ Chatterji52figure }}</div>

====CF53====
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[[Special:WhatLinksHere/Template:CF53figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ CF53figure }}</div>

====MS56====
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[[Special:WhatLinksHere/Template:MS56figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ MS56figure }}</div>
<div id="MS56acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 513):<br /><font color="darkgreen">"One of us (L. M.) wishes to record with thanks the generous financial aid of the Commonwealth Fund of New York, and the warm Hospitality of the Princeton University Observatory."</font>
</td></tr>
</table>
</div>

===Decade6===
====PG61====


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[[Special:WhatLinksHere/Template: PG61figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ PG61figure }}</div>
<div id="PG61acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====Hunter62====


[[Special:WhatLinksHere/Template:Hunter62full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Hunter62figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Hunter62figure }}</div>
<div id="Hunter62acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 608):<br /><font color="darkgreen">"I am very grateful to Dr. L. Mestel for introducing me to this problem and for his interest and advice throughout."<br />

"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
</td></tr>
</table>
</div>

====LB62====


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[[Special:WhatLinksHere/Template:LB62figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LB62figure }}</div>
<div id="LB62acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====LMS65====


[[Special:WhatLinksHere/Template:LMS65full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LMS65figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LMS65figure }}</div>
<div id="LMS65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
</td></tr>
</table>
</div>

====Mestel65====


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<div align="center">{{ Mestel65figure }}</div>

====HRW66====



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[[Special:WhatLinksHere/Template:HRW66figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ HRW66figure }}</div>
<div id="HRW66acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 551):<br /><font color="darkgreen">"This research was partially supported by the National Science Foundation (Grant GP-975); the authors are glad to acknowledge its assistance. We are also grateful to Professor S. Chandrasekhar for helpful comments, and to Mr. Jackson, of the Yerkes Observatory, for the radii of polytropes of indices 3.0 and 3.25 (see Table 4)."</font>
</td></tr>
</table>
</div>

===Decade7===

====Horedt70====
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<div align="center">{{ Horedt70figure }}</div>
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<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 85):<br /><font color="darkgreen">"The author is greatly indebted to Dr M. Penston for his precious support and to an anonymous referee."</font>
</td></tr>
</table>
</div>

====VH74====

{{ VH74full }}<br />
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<div align="center">{{ VH74figure }}</div>

====Weber76====
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<div align="center">{{ Weber76figure }}</div>
<div id="Weber76acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 123):<br /><font color="darkgreen">"I am indebted to Professor Frank Shu for suggesting the problem, providing advice and encouragement, and suggesting changes in the original draft of the paper. I also wish to acknowledge the tenure of an NSF Graduate Fellowship."</font>
</td></tr>
</table>
</div>

====Collins78====
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<div align="center">{{ Collins78figure }}</div>

===Decade8===
====GW80====

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[[Special:WhatLinksHere/Template:GW80figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ GW80figure }}</div>
<div id="GW80acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 996):<br /><font color="darkgreen">"We are grateful for comments by Doug Keeley and Frank Shu. We also acknowledge Glen Hermannsfeldt for excellent assistance with the computing. Stephen V. Weber was supported by a Chaim Weizmann postdoctoral fellowship. This work was supported by NSF grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7821453 AST78-21453] and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7680801 AST76-80801] A02."</font>
</td></tr>
</table>
</div>

====Kimura81b====

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<div align="center">{{ Kimura81bfigure }}</div>
<div id="Kimura81backnowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 308):<br /><font color="darkgreen">"A portion of the work reported here formed part of the author's doctoral thesis, submitted to the University of Tokyo. The author is indebted to Professors W. Unno and S. Aoki for helpful discussions. Thanks are also due to Mrs. K. Sakurai for her assistance in preparing the manuscript."</font>
</td></tr>
</table>
</div>

====Whitworth81====
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<div align="center">{{ Whitworth81figure }}</div>

====Tohline81====
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[[Special:WhatLinksHere/Template:Tohline81figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

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<div id="Tohline81acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 726):<br /><font color="darkgreen">"Because Leon Mestel's (1965; {{Mestel65hereafter}}) paper has not been specifically referenced in the body of this paper, I must acknowledge its importance here. It was from Professor Mestel's work that I first gained a clear glimpse of how this problem could be attacked analytically. It is a pleasure also to acknowledge conversations with Alan Boss from which I first began to appreciate the fact that an analysis akin to the one presented in this paper was desperately needed. I thank Alan also for furnishing me with his data prior to its publication."<br />"This work was performed under the auspices of the [https://en.wikipedia.org/wiki/Energy_Research_and_Development_Administration USERDA]."</font>
</td></tr>
</table>
</div>

====Stahler83====
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<div align="center">{{ Stahler83figure }}</div>
<div id="Stahler83acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 180):<br /><font color="darkgreen">"I wish to thank Martin Duncan, Mark Haugan, and Saul Teukolsky for useful discussions. This research was supported in part by National Science Foundation grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8116370 AST81-16370]."</font>
</td></tr>
</table>
</div>

====BAC84====

{{ BAC84full }}<br />
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[[Special:WhatLinksHere/Template:BAC84figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ BAC84figure }}</div>
<div id="BAC84acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 844):<br /><font color="darkgreen">"We thank Willy Fowler, Lee Lindblom, Wolfgang Ober, Bob Wagoner, and Stan Woosley for useful discussions. In particular, Stan Woosley often emphasized the many ways rotation could transform simplicity into complexity. This research was supported by grants NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8022876 AST-80-22876] at Chicago, NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7923243 AST 79-23243] at Berkeley, and NSF PHY 81-19387 at Stanford, and by the [https://en.wikipedia.org/wiki/Science_and_Engineering_Research_Council SERC] at Cambridge."</font>
</td></tr>
</table>
</div>

====MF85b====



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<div align="center">{{ MF85bfigure }}</div>
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<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
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</div>

====Tohline85====
[[Special:WhatLinksHere/Template:Tohline85full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Tohline85figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Tohline85figure }}</div>
<div id="Tohline85acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 187):<br /><font color="darkgreen">"Frank Shu and Hugh Van Horn have made extremely enlightening comments to me in the course of this investigation. I thank them for their interest and support. I am indebted to Mort Roberts for allowing me to pull my thoughts together in the friendly, invigorating environment of NRAO, Charlottesville, and to E. F. Zganjar, A. U. Landolt, and C. L. Perry for doing all the dirty work at LSU while I was gone for the summer. This work was supported by the National Science Foundation through grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8217744 AST-8217744]."</font>
</td></tr>
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</div>

====FWW86====
[[Special:WhatLinksHere/Template:FWW86full|What chapter(s) cite this article?]]<br />
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<div align="center">{{ FWW86figure }}</div>
<div id="FWW86acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 685):<br /><font color="darkgreen">"The authors gratefully acknowledge many stimulating and informative conversations with Willy Fowler on the subject of supermassive stars. This research has been supported by the National Science Foundation under grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8108509 AST81-08509] A01 and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8418185 AST 84-18185] and, at [https://en.wikipedia.org/wiki/Lawrence_Livermore_National_Laboratory Livermore], by the Department of Energy under W-7405-EN6-48</font> [probably should be ENG, rather than EN6]."
</td></tr>
</table>
</div>

====Horedt86====

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<div align="center">{{ Horedt86figure }}</div>

===Decade9===
====LRS93b====

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<div align="center">{{ LRS93bfigure }}</div>

==21<sup>st</sup> Century==

===MTF2002===
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===ZEUS-MP2006===
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<div align="center">{{ ZEUS-MP2006figure }}</div>

===PK2007===

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[[Special:WhatLinksHere/Template:PK2007figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ PK2007figure }}</div>

===MT2012===
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[[Special:WhatLinksHere/Template:MT2012figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

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{{ MT2012figure }}
</div>

=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Appendix/References

2021-11-03T21:20:03Z

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=Journal Articles=
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"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
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<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

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<tr><td align="left">Acknowledgements (p. 996):<br /><font color="darkgreen">"We are grateful for comments by Doug Keeley and Frank Shu. We also acknowledge Glen Hermannsfeldt for excellent assistance with the computing. Stephen V. Weber was supported by a Chaim Weizmann postdoctoral fellowship. This work was supported by NSF grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7821453 AST78-21453] and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7680801 AST76-80801] A02."</font>
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<tr><td align="left">Acknowledgements (p. 308):<br /><font color="darkgreen">"A portion of the work reported here formed part of the author's doctoral thesis, submitted to the University of Tokyo. The author is indebted to Professors W. Unno and S. Aoki for helpful discussions. Thanks are also due to Mrs. K. Sakurai for her assistance in preparing the manuscript."</font>
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<tr><td align="left">Acknowledgements (p. 726):<br /><font color="darkgreen">"Because Leon Mestel's (1965; {{Mestel65hereafter}}) paper has not been specifically referenced in the body of this paper, I must acknowledge its importance here. It was from Professor Mestel's work that I first gained a clear glimpse of how this problem could be attacked analytically. It is a pleasure also to acknowledge conversations with Alan Boss from which I first began to appreciate the fact that an analysis akin to the one presented in this paper was desperately needed. I thank Alan also for furnishing me with his data prior to its publication."<br />"This work was performed under the auspices of the [https://en.wikipedia.org/wiki/Energy_Research_and_Development_Administration USERDA]."</font>
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<tr><td align="left">Acknowledgements (p. 180):<br /><font color="darkgreen">"I wish to thank Martin Duncan, Mark Haugan, and Saul Teukolsky for useful discussions. This research was supported in part by National Science Foundation grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8116370 AST81-16370]."</font>
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<tr><td align="left">Acknowledgements (p. 844):<br /><font color="darkgreen">"We thank Willy Fowler, Lee Lindblom, Wolfgang Ober, Bob Wagoner, and Stan Woosley for useful discussions. In particular, Stan Woosley often emphasized the many ways rotation could transform simplicity into complexity. This research was supported by grants NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8022876 AST-80-22876] at Chicago, NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7923243 AST 79-23243] at Berkeley, and NSF PHY 81-19387 at Stanford, and by the [https://en.wikipedia.org/wiki/Science_and_Engineering_Research_Council SERC] at Cambridge."</font>
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<tr><td align="left">Acknowledgements (p. 187):<br /><font color="darkgreen">"Frank Shu and Hugh Van Horn have made extremely enlightening comments to me in the course of this investigation. I thank them for their interest and support. I am indebted to Mort Roberts for allowing me to pull my thoughts together in the friendly, invigorating environment of NRAO, Charlottesville, and to E. F. Zganjar, A. U. Landolt, and C. L. Perry for doing all the dirty work at LSU while I was gone for the summer. This work was supported by the National Science Foundation through grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8217744 AST-8217744]."</font>
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<tr><td align="left">Acknowledgements (p. 685):<br /><font color="darkgreen">"The authors gratefully acknowledge many stimulating and informative conversations with Willy Fowler on the subject of supermassive stars. This research has been supported by the National Science Foundation under grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8108509 AST81-08509] A01 and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8418185 AST 84-18185] and, at [https://en.wikipedia.org/wiki/Lawrence_Livermore_National_Laboratory Livermore], by the Department of Energy under W-7405-EN6-48</font> [probably should be ENG, rather than EN6]."
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==21<sup>st</sup> Century==

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=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
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=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
<ol type="1">
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<li>Dyson1893figure     <math>\Leftarrow</math>     This is the template highlighted below.</li>
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If you want to see which of our MediaWiki chapters includes a link to — and, therefore, likely discusses — a particular journal article, step through the following options:
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==18<sup>th</sup> Century==

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<tr><td align="left">Acknowledgements (p. 64):<br /><font color="darkgreen">"I am greatly indebted to Professor Eddington for his valuable suggestions and help."</font>
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[[Special:WhatLinksHere/Template:LP41figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ LP41figure }}</div>

====Schwarzschild41====
[[Special:WhatLinksHere/Template:Schwarzschild41full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Schwarzschild41figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Schwarzschild41figure }}</div>
<div id="Schwarzschild41acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 252):<br /><font color="darkgreen">"I am indebted to Miss Lillian Feinstein for her most helpful co-operation in the work with the punched-card machines."</font>
</td></tr>
</table>
</div>

====Cowling41====

[[Special:WhatLinksHere/Template:Cowling41full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Cowling41figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Cowling41figure }}</div>

===Decade5===

====Chatterji51====


[[Special:WhatLinksHere/Template:Chatterji51full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Chatterji51figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Chatterji51figure }}</div>

====Chatterji52====


[[Special:WhatLinksHere/Template:Chatterji52full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Chatterji52figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Chatterji52figure }}</div>

====CF53====
[[Special:WhatLinksHere/Template:CF53full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:CF53figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ CF53figure }}</div>

====MS56====
[[Special:WhatLinksHere/Template:MS56full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MS56figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ MS56figure }}</div>
<div id="MS56acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 513):<br /><font color="darkgreen">"One of us (L. M.) wishes to record with thanks the generous financial aid of the Commonwealth Fund of New York, and the warm Hospitality of the Princeton University Observatory."</font>
</td></tr>
</table>
</div>

===Decade6===
====PG61====


[[Special:WhatLinksHere/Template: PG61full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template: PG61figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ PG61figure }}</div>

====Hunter62====


[[Special:WhatLinksHere/Template:Hunter62full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Hunter62figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Hunter62figure }}</div>
<div id="Hunter62acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 608):<br /><font color="darkgreen">"I am very grateful to Dr. L. Mestel for introducing me to this problem and for his interest and advice throughout."<br />

"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
</td></tr>
</table>
</div>

====LB62====


[[Special:WhatLinksHere/Template:LB62full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LB62figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LB62figure }}</div>

====LMS65====


[[Special:WhatLinksHere/Template:LMS65full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LMS65figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LMS65figure }}</div>
<div id="LMS65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
</td></tr>
</table>
</div>

====Mestel65====


[[Special:WhatLinksHere/Template:Mestel65full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Mestel65figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Mestel65figure }}</div>

====HRW66====



[[Special:WhatLinksHere/Template:HRW66full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:HRW66figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ HRW66figure }}</div>
<div id="HRW66acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 551):<br /><font color="darkgreen">"This research was partially supported by the National Science Foundation (Grant GP-975); the authors are glad to acknowledge its assistance. We are also grateful to Professor S. Chandrasekhar for helpful comments, and to Mr. Jackson, of the Yerkes Observatory, for the radii of polytropes of indices 3.0 and 3.25 (see Table 4)."</font>
</td></tr>
</table>
</div>

===Decade7===

====Horedt70====
[[Special:WhatLinksHere/Template:Horedt70full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Horedt70figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Horedt70figure }}</div>
<div id="Horedt70acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 85):<br /><font color="darkgreen">"The author is greatly indebted to Dr M. Penston for his precious support and to an anonymous referee."</font>
</td></tr>
</table>
</div>

====VH74====

{{ VH74full }}<br />
{{ VH74 }}<br />
{{ VH74hereafter }}<br />

[[Special:WhatLinksHere/Template:VH74full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:VH74figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ VH74figure }}</div>

====Weber76====
[[Special:WhatLinksHere/Template:Weber76full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Weber76figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Weber76figure }}</div>
<div id="Weber76acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 123):<br /><font color="darkgreen">"I am indebted to Professor Frank Shu for suggesting the problem, providing advice and encouragement, and suggesting changes in the original draft of the paper. I also wish to acknowledge the tenure of an NSF Graduate Fellowship."</font>
</td></tr>
</table>
</div>

====Collins78====
[[Special:WhatLinksHere/Template:Collins78full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Collins78figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Collins78figure }}</div>

===Decade8===
====GW80====

[[Special:WhatLinksHere/Template:GW80full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:GW80figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ GW80figure }}</div>
<div id="GW80acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 996):<br /><font color="darkgreen">"We are grateful for comments by Doug Keeley and Frank Shu. We also acknowledge Glen Hermannsfeldt for excellent assistance with the computing. Stephen V. Weber was supported by a Chaim Weizmann postdoctoral fellowship. This work was supported by NSF grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7821453 AST78-21453] and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7680801 AST76-80801] A02."</font>
</td></tr>
</table>
</div>

====Kimura81b====

[[Special:WhatLinksHere/Template:Kimura81bfull|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Kimura81bfigure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Kimura81bfigure }}</div>
<div id="Kimura81backnowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 308):<br /><font color="darkgreen">"A portion of the work reported here formed part of the author's doctoral thesis, submitted to the University of Tokyo. The author is indebted to Professors W. Unno and S. Aoki for helpful discussions. Thanks are also due to Mrs. K. Sakurai for her assistance in preparing the manuscript."</font>
</td></tr>
</table>
</div>

====Whitworth81====
[[Special:WhatLinksHere/Template:Whitworth81full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Whitworth81figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Whitworth81figure }}</div>

====Tohline81====
[[Special:WhatLinksHere/Template:Tohline81full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Tohline81figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Tohline81figure }}</div>
<div id="Tohline81acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 726):<br /><font color="darkgreen">"Because Leon Mestel's (1965; {{Mestel65hereafter}}) paper has not been specifically referenced in the body of this paper, I must acknowledge its importance here. It was from Professor Mestel's work that I first gained a clear glimpse of how this problem could be attacked analytically. It is a pleasure also to acknowledge conversations with Alan Boss from which I first began to appreciate the fact that an analysis akin to the one presented in this paper was desperately needed. I thank Alan also for furnishing me with his data prior to its publication."<br />"This work was performed under the auspices of the [https://en.wikipedia.org/wiki/Energy_Research_and_Development_Administration USERDA]."</font>
</td></tr>
</table>
</div>

====Stahler83====
[[Special:WhatLinksHere/Template:Stahler83full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Stahler83figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Stahler83figure }}</div>
<div id="Stahler83acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 180):<br /><font color="darkgreen">"I wish to thank Martin Duncan, Mark Haugan, and Saul Teukolsky for useful discussions. This research was supported in part by National Science Foundation grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8116370 AST81-16370]."</font>
</td></tr>
</table>
</div>

====BAC84====

{{ BAC84full }}<br />
{{ BAC84 }}<br />
{{ BAC84hereafter }}<br />

[[Special:WhatLinksHere/Template:BAC84full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:BAC84figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ BAC84figure }}</div>
<div id="BAC84acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 844):<br /><font color="darkgreen">"We thank Willy Fowler, Lee Lindblom, Wolfgang Ober, Bob Wagoner, and Stan Woosley for useful discussions. In particular, Stan Woosley often emphasized the many ways rotation could transform simplicity into complexity. This research was supported by grants NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8022876 AST-80-22876] at Chicago, NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7923243 AST 79-23243] at Berkeley, and NSF PHY 81-19387 at Stanford, and by the [https://en.wikipedia.org/wiki/Science_and_Engineering_Research_Council SERC] at Cambridge."</font>
</td></tr>
</table>
</div>

====MF85b====



[[Special:WhatLinksHere/Template:MF85bfull|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MF85bfigure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ MF85bfigure }}</div>
<div id="MF85backnowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements:  n/a
</td></tr>
</table>
</div>

====Tohline85====
[[Special:WhatLinksHere/Template:Tohline85full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Tohline85figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Tohline85figure }}</div>
<div id="Tohline85acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 187):<br /><font color="darkgreen">"Frank Shu and Hugh Van Horn have made extremely enlightening comments to me in the course of this investigation. I thank them for their interest and support. I am indebted to Mort Roberts for allowing me to pull my thoughts together in the friendly, invigorating environment of NRAO, Charlottesville, and to E. F. Zganjar, A. U. Landolt, and C. L. Perry for doing all the dirty work at LSU while I was gone for the summer. This work was supported by the National Science Foundation through grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8217744 AST-8217744]."</font>
</td></tr>
</table>
</div>

====FWW86====
[[Special:WhatLinksHere/Template:FWW86full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:FWW86figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ FWW86figure }}</div>
<div id="FWW86acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 685):<br /><font color="darkgreen">"The authors gratefully acknowledge many stimulating and informative conversations with Willy Fowler on the subject of supermassive stars. This research has been supported by the National Science Foundation under grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8108509 AST81-08509] A01 and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8418185 AST 84-18185] and, at [https://en.wikipedia.org/wiki/Lawrence_Livermore_National_Laboratory Livermore], by the Department of Energy under W-7405-EN6-48</font> [probably should be ENG, rather than EN6]."
</td></tr>
</table>
</div>

====Horedt86====

[[Special:WhatLinksHere/Template:Horedt86full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Horedt86figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Horedt86figure }}</div>

===Decade9===
====LRS93b====

[[Special:WhatLinksHere/Template:LRS93bfull|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LRS93bfigure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LRS93bfigure }}</div>

==21<sup>st</sup> Century==

===MTF2002===
[[Special:WhatLinksHere/Template:MTF2002full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MTF2002figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ MTF2002figure }}</div>

===ZEUS-MP2006===
[[Special:WhatLinksHere/Template:ZEUS-MP2006full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:ZEUS-MP2006figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ ZEUS-MP2006figure }}</div>

===PK2007===

[[Special:WhatLinksHere/Template:PK2007full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:PK2007figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ PK2007figure }}</div>

===MT2012===
[[Special:WhatLinksHere/Template:MT2012full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MT2012figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">
{{ MT2012figure }}
</div>

=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Appendix/References

2021-11-03T20:53:20Z

174.64.14.12: /* Tohline85 */

__FORCETOC__


=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
<ol type="1">
<li>Dyson1893</li>
<li>Dyson1893full</li>
<li>Dyson1893figure     <math>\Leftarrow</math>     This is the template highlighted below.</li>
<li>BAC84hereafter</li>
</ol>

If you want to see which of our MediaWiki chapters includes a link to — and, therefore, likely discusses — a particular journal article, step through the following options:
<ol type="1">
<li>Under the left-hand column, click on Tools/Special_pages;</li>
<li>Click on Page_tools/What_links_here;</li>
<li>In the input-box provided next to "Page," type the desired "Template" name. For example, type in:   <font color="Darkgreen">Template:Dyson1893full</font></li>
<li>Note that when a figure from a particular journal article is displayed in one or more of our MediaWiki chapters, it will usually be accompanied by a figure heading that cites the article. Hence, typing in, for example, <font color="Darkgreen">Template:Dyson1893figure</font> should identify all of the MediaWiki chapters that display a figure from Dyson (1893).</li>
</ol>
==18<sup>th</sup> Century==

==19<sup>th</sup> Century==

===Clausius1870===

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==20<sup>th</sup> Century==

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===Decade1===
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====Miller29====


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<div align="center">{{ Miller29figure }}</div>
<div id="Miller29acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 64):<br /><font color="darkgreen">"I am greatly indebted to Professor Eddington for his valuable suggestions and help."</font>
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===Decade3===
====Milne30====
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===Decade4===
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<div align="center">{{ Schwarzschild41figure }}</div>
<div id="Schwarzschild41acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 252):<br /><font color="darkgreen">"I am indebted to Miss Lillian Feinstein for her most helpful co-operation in the work with the punched-card machines."</font>
</td></tr>
</table>
</div>

====Cowling41====

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===Decade5===

====Chatterji51====


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====Chatterji52====


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====CF53====
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====MS56====
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<div align="center">{{ MS56figure }}</div>
<div id="MS56acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 513):<br /><font color="darkgreen">"One of us (L. M.) wishes to record with thanks the generous financial aid of the Commonwealth Fund of New York, and the warm Hospitality of the Princeton University Observatory."</font>
</td></tr>
</table>
</div>

===Decade6===
====PG61====


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<div align="center">{{ PG61figure }}</div>

====Hunter62====


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<div align="center">{{ Hunter62figure }}</div>
<div id="Hunter62acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 608):<br /><font color="darkgreen">"I am very grateful to Dr. L. Mestel for introducing me to this problem and for his interest and advice throughout."<br />

"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
</td></tr>
</table>
</div>

====LB62====


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<div align="center">{{ LB62figure }}</div>

====LMS65====


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<div align="center">{{ LMS65figure }}</div>
<div id="LMS65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
</td></tr>
</table>
</div>

====Mestel65====


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<div align="center">{{ Mestel65figure }}</div>

====HRW66====



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<div align="center">{{ HRW66figure }}</div>
<div id="HRW66acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 551):<br /><font color="darkgreen">"This research was partially supported by the National Science Foundation (Grant GP-975); the authors are glad to acknowledge its assistance. We are also grateful to Professor S. Chandrasekhar for helpful comments, and to Mr. Jackson, of the Yerkes Observatory, for the radii of polytropes of indices 3.0 and 3.25 (see Table 4)."</font>
</td></tr>
</table>
</div>

===Decade7===

====Horedt70====
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<div align="center">{{ Horedt70figure }}</div>
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<tr><td align="left">Acknowledgements (p. 85):<br /><font color="darkgreen">"The author is greatly indebted to Dr M. Penston for his precious support and to an anonymous referee."</font>
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</div>

====VH74====

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====Weber76====
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<div align="center">{{ Weber76figure }}</div>
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<tr><td align="left">Acknowledgements (p. 123):<br /><font color="darkgreen">"I am indebted to Professor Frank Shu for suggesting the problem, providing advice and encouragement, and suggesting changes in the original draft of the paper. I also wish to acknowledge the tenure of an NSF Graduate Fellowship."</font>
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</table>
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====Collins78====
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<div align="center">{{ Collins78figure }}</div>

===Decade8===
====GW80====

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<div align="center">{{ GW80figure }}</div>
<div id="GW80acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 996):<br /><font color="darkgreen">"We are grateful for comments by Doug Keeley and Frank Shu. We also acknowledge Glen Hermannsfeldt for excellent assistance with the computing. Stephen V. Weber was supported by a Chaim Weizmann postdoctoral fellowship. This work was supported by NSF grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7821453 AST78-21453] and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7680801 AST76-80801] A02."</font>
</td></tr>
</table>
</div>

====Kimura81b====

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<div align="center">{{ Kimura81bfigure }}</div>
<div id="Kimura81backnowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 308):<br /><font color="darkgreen">"A portion of the work reported here formed part of the author's doctoral thesis, submitted to the University of Tokyo. The author is indebted to Professors W. Unno and S. Aoki for helpful discussions. Thanks are also due to Mrs. K. Sakurai for her assistance in preparing the manuscript."</font>
</td></tr>
</table>
</div>

====Whitworth81====
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====Tohline81====
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<div align="center">{{ Tohline81figure }}</div>
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<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 726):<br /><font color="darkgreen">"Because Leon Mestel's (1965; {{Mestel65hereafter}}) paper has not been specifically referenced in the body of this paper, I must acknowledge its importance here. It was from Professor Mestel's work that I first gained a clear glimpse of how this problem could be attacked analytically. It is a pleasure also to acknowledge conversations with Alan Boss from which I first began to appreciate the fact that an analysis akin to the one presented in this paper was desperately needed. I thank Alan also for furnishing me with his data prior to its publication."<br />"This work was performed under the auspices of the [https://en.wikipedia.org/wiki/Energy_Research_and_Development_Administration USERDA]."</font>
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====Stahler83====
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<tr><td align="left">Acknowledgements (p. 180):<br /><font color="darkgreen">"I wish to thank Martin Duncan, Mark Haugan, and Saul Teukolsky for useful discussions. This research was supported in part by National Science Foundation grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8116370 AST81-16370]."</font>
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====BAC84====

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<div align="center">{{ BAC84figure }}</div>
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<tr><td align="left">Acknowledgements (p. 844):<br /><font color="darkgreen">"We thank Willy Fowler, Lee Lindblom, Wolfgang Ober, Bob Wagoner, and Stan Woosley for useful discussions. In particular, Stan Woosley often emphasized the many ways rotation could transform simplicity into complexity. This research was supported by grants NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8022876 AST-80-22876] at Chicago, NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7923243 AST 79-23243] at Berkeley, and NSF PHY 81-19387 at Stanford, and by the [https://en.wikipedia.org/wiki/Science_and_Engineering_Research_Council SERC] at Cambridge."</font>
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<tr><td align="left">Acknowledgements:  n/a
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====Tohline85====
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<tr><td align="left">Acknowledgements (p. 187):<br /><font color="darkgreen">"Frank Shu and Hugh Van Horn have made extremely enlightening comments to me in the course of this investigation. I thank them for their interest and support. I am indebted to Mort Roberts for allowing me to pull my thoughts together in the friendly, invigorating environment of NRAO, Charlottesville, and to E. F. Zganjar, A. U. Landolt, and C. L. Perry for doing all the dirty work at LSU while I was gone for the summer. This work was supported by the National Science Foundation through grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8217744 AST-8217744]."</font>
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====FWW86====
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=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

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=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
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If you want to see which of our MediaWiki chapters includes a link to — and, therefore, likely discusses — a particular journal article, step through the following options:
<ol type="1">
<li>Under the left-hand column, click on Tools/Special_pages;</li>
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==18<sup>th</sup> Century==

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<tr><td align="left">Acknowledgements (p. 64):<br /><font color="darkgreen">"I am greatly indebted to Professor Eddington for his valuable suggestions and help."</font>
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<tr><td align="left">Acknowledgements (p. 513):<br /><font color="darkgreen">"One of us (L. M.) wishes to record with thanks the generous financial aid of the Commonwealth Fund of New York, and the warm Hospitality of the Princeton University Observatory."</font>
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<tr><td align="left">Acknowledgements (p. 608):<br /><font color="darkgreen">"I am very grateful to Dr. L. Mestel for introducing me to this problem and for his interest and advice throughout."<br />

"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
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<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
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<tr><td align="left">Acknowledgements (p. 551):<br /><font color="darkgreen">"This research was partially supported by the National Science Foundation (Grant GP-975); the authors are glad to acknowledge its assistance. We are also grateful to Professor S. Chandrasekhar for helpful comments, and to Mr. Jackson, of the Yerkes Observatory, for the radii of polytropes of indices 3.0 and 3.25 (see Table 4)."</font>
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====Weber76====
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<div id="Weber76acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 123):<br /><font color="darkgreen">"I am indebted to Professor Frank Shu for suggesting the problem, providing advice and encouragement, and suggesting changes in the original draft of the paper. I also wish to acknowledge the tenure of an NSF Graduate Fellowship."</font>
</td></tr>
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====Collins78====
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===Decade8===
====GW80====

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<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 996):<br /><font color="darkgreen">"We are grateful for comments by Doug Keeley and Frank Shu. We also acknowledge Glen Hermannsfeldt for excellent assistance with the computing. Stephen V. Weber was supported by a Chaim Weizmann postdoctoral fellowship. This work was supported by NSF grants [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7821453 AST78-21453] and [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7680801 AST76-80801] A02."</font>
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====Kimura81b====

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<div id="Kimura81backnowledgement">
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<tr><td align="left">Acknowledgements (p. 308):<br /><font color="darkgreen">"A portion of the work reported here formed part of the author's doctoral thesis, submitted to the University of Tokyo. The author is indebted to Professors W. Unno and S. Aoki for helpful discussions. Thanks are also due to Mrs. K. Sakurai for her assistance in preparing the manuscript."</font>
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====Whitworth81====
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====Tohline81====
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<div id="Tohline81acknowledgement">
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<tr><td align="left">Acknowledgements (p. 726):<br /><font color="darkgreen">"Because Leon Mestel's (1965; {{Mestel65hereafter}}) paper has not been specifically referenced in the body of this paper, I must acknowledge its importance here. It was from Professor Mestel's work that I first gained a clear glimpse of how this problem could be attacked analytically. It is a pleasure also to acknowledge conversations with Alan Boss from which I first began to appreciate the fact that an analysis akin to the one presented in this paper was desperately needed. I thank Alan also for furnishing me with his data prior to its publication."<br />"This work was performed under the auspices of the [https://en.wikipedia.org/wiki/Energy_Research_and_Development_Administration USERDA]."</font>
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====Stahler83====
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<tr><td align="left">Acknowledgements (p. 180):<br /><font color="darkgreen">"I wish to thank Martin Duncan, Mark Haugan, and Saul Teukolsky for useful discussions. This research was supported in part by National Science Foundation grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8116370 AST81-16370]."</font>
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====BAC84====

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<tr><td align="left">Acknowledgements (p. 844):<br /><font color="darkgreen">"We thank Willy Fowler, Lee Lindblom, Wolfgang Ober, Bob Wagoner, and Stan Woosley for useful discussions. In particular, Stan Woosley often emphasized the many ways rotation could transform simplicity into complexity. This research was supported by grants NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8022876 AST-80-22876] at Chicago, NSF [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=7923243 AST 79-23243] at Berkeley, and NSF PHY 81-19387 at Stanford, and by the [https://en.wikipedia.org/wiki/Science_and_Engineering_Research_Council SERC] at Cambridge."</font>
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====MF85b====



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====Tohline85====
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<div id="Tohline85acknowledgement">
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<tr><td align="left">Acknowledgements (p. 187):<br /><font color="darkgreen">"Frank She and Hugh Van Horn have made extremely enlightening comments to me in the course of this investigation. I thank them for their interest and support. I am indebted to Mort Roberts for allowing me to pull my thoughts together in the friendly, invigorating environment of NRAO, Charlottesville, and to E. F. Zganjar, A. U. Landolt, and C. L. Perry for doing all the dirty work at LSU while I was gone for the summer. This work was supported by the National Science Foundation through grant [https://www.nsf.gov/awardsearch/simpleSearchResult?queryText=8217744 AST-8217744]."</font>
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====FWW86====
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====Horedt86====

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===Decade9===
====LRS93b====

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==21<sup>st</sup> Century==

===MTF2002===
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===ZEUS-MP2006===
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===PK2007===

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===MT2012===
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=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
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Appendix/References

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=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
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==18<sup>th</sup> Century==

==19<sup>th</sup> Century==

===Clausius1870===

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==20<sup>th</sup> Century==

===Decade0===

===Decade1===
====Eddington18====
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<div align="center">{{ Eddington18figure }}</div>

===Decade2===
====Miller29====


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[[Special:WhatLinksHere/Template:Miller29figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Miller29figure }}</div>
<div id="Miller29acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 64):<br /><font color="darkgreen">"I am greatly indebted to Professor Eddington for his valuable suggestions and help."</font>
</td></tr>
</table>
</div>

===Decade3===
====Milne30====
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<div align="center">{{ Milne30figure }}</div>

===Decade4===
====LP41====
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[[Special:WhatLinksHere/Template:LP41figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ LP41figure }}</div>

====Schwarzschild41====
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[[Special:WhatLinksHere/Template:Schwarzschild41figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Schwarzschild41figure }}</div>
<div id="Schwarzschild41acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 252):<br /><font color="darkgreen">"I am indebted to Miss Lillian Feinstein for her most helpful co-operation in the work with the punched-card machines."</font>
</td></tr>
</table>
</div>

====Cowling41====

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<div align="center">{{ Cowling41figure }}</div>

===Decade5===

====Chatterji51====


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<div align="center">{{ Chatterji51figure }}</div>

====Chatterji52====


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[[Special:WhatLinksHere/Template:Chatterji52figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Chatterji52figure }}</div>

====CF53====
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[[Special:WhatLinksHere/Template:CF53figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ CF53figure }}</div>

====MS56====
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[[Special:WhatLinksHere/Template:MS56figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ MS56figure }}</div>
<div id="MS56acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 513):<br /><font color="darkgreen">"One of us (L. M.) wishes to record with thanks the generous financial aid of the Commonwealth Fund of New York, and the warm Hospitality of the Princeton University Observatory."</font>
</td></tr>
</table>
</div>

===Decade6===
====PG61====


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[[Special:WhatLinksHere/Template: PG61figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ PG61figure }}</div>

====Hunter62====


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[[Special:WhatLinksHere/Template:Hunter62figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Hunter62figure }}</div>
<div id="Hunter62acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 608):<br /><font color="darkgreen">"I am very grateful to Dr. L. Mestel for introducing me to this problem and for his interest and advice throughout."<br />

"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
</td></tr>
</table>
</div>

====LB62====


[[Special:WhatLinksHere/Template:LB62full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LB62figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LB62figure }}</div>

====LMS65====


[[Special:WhatLinksHere/Template:LMS65full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:LMS65figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ LMS65figure }}</div>
<div id="LMS65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
</td></tr>
</table>
</div>

====Mestel65====


[[Special:WhatLinksHere/Template:Mestel65full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:Mestel65figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Mestel65figure }}</div>

====HRW66====



[[Special:WhatLinksHere/Template:HRW66full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:HRW66figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ HRW66figure }}</div>
<div id="HRW66acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 551):<br /><font color="darkgreen">"This research was partially supported by the National Science Foundation (Grant GP-975); the authors are glad to acknowledge its assistance. We are also grateful to Professor S. Chandrasekhar for helpful comments, and to Mr. Jackson, of the Yerkes Observatory, for the radii of polytropes of indices 3.0 and 3.25 (see Table 4)."</font>
</td></tr>
</table>
</div>

===Decade7===

====Horedt70====
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<div align="center">{{ Horedt70figure }}</div>

====VH74====

{{ VH74full }}<br />
{{ VH74 }}<br />
{{ VH74hereafter }}<br />

[[Special:WhatLinksHere/Template:VH74full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:VH74figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ VH74figure }}</div>

====Weber76====
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[[Special:WhatLinksHere/Template:Weber76figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Weber76figure }}</div>

====Collins78====
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[[Special:WhatLinksHere/Template:Collins78figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Collins78figure }}</div>

===Decade8===
====GW80====

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[[Special:WhatLinksHere/Template:GW80figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ GW80figure }}</div>

====Kimura81b====

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[[Special:WhatLinksHere/Template:Kimura81bfigure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ Kimura81bfigure }}</div>

====Whitworth81====
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[[Special:WhatLinksHere/Template:Whitworth81figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Whitworth81figure }}</div>

====Tohline81====
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<div align="center">{{ Tohline81figure }}</div>

====Stahler83====
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<div align="center">{{ Stahler83figure }}</div>

====BAC84====

{{ BAC84full }}<br />
{{ BAC84 }}<br />
{{ BAC84hereafter }}<br />

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[[Special:WhatLinksHere/Template:BAC84figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ BAC84figure }}</div>

====MF85b====



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[[Special:WhatLinksHere/Template:MF85bfigure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ MF85bfigure }}</div>

====Tohline85====
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[[Special:WhatLinksHere/Template:Tohline85figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Tohline85figure }}</div>

====FWW86====
[[Special:WhatLinksHere/Template:FWW86full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:FWW86figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ FWW86figure }}</div>

====Horedt86====

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[[Special:WhatLinksHere/Template:Horedt86figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ Horedt86figure }}</div>

===Decade9===
====LRS93b====

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<div align="center">{{ LRS93bfigure }}</div>

==21<sup>st</sup> Century==

===MTF2002===
[[Special:WhatLinksHere/Template:MTF2002full|What chapter(s) cite this article?]]<br />
[[Special:WhatLinksHere/Template:MTF2002figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ MTF2002figure }}</div>

===ZEUS-MP2006===
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[[Special:WhatLinksHere/Template:ZEUS-MP2006figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">{{ ZEUS-MP2006figure }}</div>

===PK2007===

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[[Special:WhatLinksHere/Template:PK2007figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />
<div align="center">{{ PK2007figure }}</div>

===MT2012===
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[[Special:WhatLinksHere/Template:MT2012figure|In what chapter(s), if any, does a digital image (e.g., a figure or table) from this article appear?]]<br />

<div align="center">
{{ MT2012figure }}
</div>

=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Appendix/References

2021-11-02T22:06:57Z

174.64.14.12: /* Miller29 */

__FORCETOC__


=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
<ol type="1">
<li>Dyson1893</li>
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<li>Dyson1893figure     <math>\Leftarrow</math>     This is the template highlighted below.</li>
<li>BAC84hereafter</li>
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If you want to see which of our MediaWiki chapters includes a link to — and, therefore, likely discusses — a particular journal article, step through the following options:
<ol type="1">
<li>Under the left-hand column, click on Tools/Special_pages;</li>
<li>Click on Page_tools/What_links_here;</li>
<li>In the input-box provided next to "Page," type the desired "Template" name. For example, type in:   <font color="Darkgreen">Template:Dyson1893full</font></li>
<li>Note that when a figure from a particular journal article is displayed in one or more of our MediaWiki chapters, it will usually be accompanied by a figure heading that cites the article. Hence, typing in, for example, <font color="Darkgreen">Template:Dyson1893figure</font> should identify all of the MediaWiki chapters that display a figure from Dyson (1893).</li>
</ol>
==18<sup>th</sup> Century==

==19<sup>th</sup> Century==

===Clausius1870===

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<div id="Miller29acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 64):<br /><font color="darkgreen">"I am greatly indebted to Professor Eddington for his valuable suggestions and help."</font>
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<div id="Schwarzschild41acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 252):<br /><font color="darkgreen">"I am indebted to Miss Lillian Feinstein for her most helpful co-operation in the work with the punched-card machines."</font>
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====Cowling41====

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<tr><td align="left">Acknowledgements (p. 513):<br /><font color="darkgreen">"One of us (L. M.) wishes to record with thanks the generous financial aid of the Commonwealth Fund of New York, and the warm Hospitality of the Princeton University Observatory."</font>
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<tr><td align="left">Acknowledgements (p. 608):<br /><font color="darkgreen">"I am very grateful to Dr. L. Mestel for introducing me to this problem and for his interest and advice throughout."<br />

"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
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====LB62====


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<div align="center">{{ LMS65figure }}</div>
<div id="LMS65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
</td></tr>
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</div>

====Mestel65====


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====FWW86====
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====LRS93b====

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==21<sup>st</sup> Century==

===MTF2002===
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===ZEUS-MP2006===
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===PK2007===

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===MT2012===
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</div>

=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Appendix/References

2021-11-02T21:58:41Z

174.64.14.12: /* Schwarzschild41 */

__FORCETOC__


=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
<ol type="1">
<li>Dyson1893</li>
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<li>Dyson1893figure     <math>\Leftarrow</math>     This is the template highlighted below.</li>
<li>BAC84hereafter</li>
</ol>

If you want to see which of our MediaWiki chapters includes a link to — and, therefore, likely discusses — a particular journal article, step through the following options:
<ol type="1">
<li>Under the left-hand column, click on Tools/Special_pages;</li>
<li>Click on Page_tools/What_links_here;</li>
<li>In the input-box provided next to "Page," type the desired "Template" name. For example, type in:   <font color="Darkgreen">Template:Dyson1893full</font></li>
<li>Note that when a figure from a particular journal article is displayed in one or more of our MediaWiki chapters, it will usually be accompanied by a figure heading that cites the article. Hence, typing in, for example, <font color="Darkgreen">Template:Dyson1893figure</font> should identify all of the MediaWiki chapters that display a figure from Dyson (1893).</li>
</ol>
==18<sup>th</sup> Century==

==19<sup>th</sup> Century==

===Clausius1870===

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<div id="Schwarzschild41acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 252):<br /><font color="darkgreen">"I am indebted to Miss Lillian Feinstein for her most helpful co-operation in the work with the punched-card machines."</font>
</td></tr>
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<tr><td align="left">Acknowledgements (p. 513):<br /><font color="darkgreen">"One of us (L. M.) wishes to record with thanks the generous financial aid of the Commonwealth Fund of New York, and the warm Hospitality of the Princeton University Observatory."</font>
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<tr><td align="left">Acknowledgements (p. 608):<br /><font color="darkgreen">"I am very grateful to Dr. L. Mestel for introducing me to this problem and for his interest and advice throughout."<br />

"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
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<div align="center">{{ LMS65figure }}</div>
<div id="LMS65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
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</div>

====Mestel65====


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==21<sup>st</sup> Century==

===MTF2002===
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===PK2007===

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===MT2012===
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=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Appendix/References

2021-11-02T21:50:13Z

174.64.14.12: /* MS56 */

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=Journal Articles=
==Explanation==

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<tr><td align="left">Acknowledgements (p. 513):<br /><font color="darkgreen">"One of us (L. M.) wishes to record with thanks the generous financial aid of the Commonwealth Fund of New York, and the warm Hospitality of the Princeton University Observatory."</font>
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<tr><td align="left">Acknowledgements (p. 608):<br /><font color="darkgreen">"I am very grateful to Dr. L. Mestel for introducing me to this problem and for his interest and advice throughout."<br />

"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
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<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

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==21<sup>st</sup> Century==

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=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Appendix/References

2021-11-02T21:44:47Z

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<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

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==21<sup>st</sup> Century==

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=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Appendix/References

2021-11-02T21:44:21Z

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=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
<ol type="1">
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<li>Dyson1893figure     <math>\Leftarrow</math>     This is the template highlighted below.</li>
<li>BAC84hereafter</li>
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If you want to see which of our MediaWiki chapters includes a link to — and, therefore, likely discusses — a particular journal article, step through the following options:
<ol type="1">
<li>Under the left-hand column, click on Tools/Special_pages;</li>
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<li>In the input-box provided next to "Page," type the desired "Template" name. For example, type in:   <font color="Darkgreen">Template:Dyson1893full</font></li>
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<tr><td align="left">Acknowledgement (p. 608):<br /><font color="darkgreen">"I am very grateful to Dr. L. Mestel for introducing me to this problem and for his interest and advice throughout."<br />

"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
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<tr><td align="left">Acknowledgements (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
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===Decade8===
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====Whitworth81====
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====Tohline81====
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====Stahler83====
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====FWW86====
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===Decade9===
====LRS93b====

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==21<sup>st</sup> Century==

===MTF2002===
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===ZEUS-MP2006===
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===PK2007===

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=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Appendix/References

2021-11-02T21:42:38Z

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=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
<ol type="1">
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<li>Dyson1893figure     <math>\Leftarrow</math>     This is the template highlighted below.</li>
<li>BAC84hereafter</li>
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If you want to see which of our MediaWiki chapters includes a link to — and, therefore, likely discusses — a particular journal article, step through the following options:
<ol type="1">
<li>Under the left-hand column, click on Tools/Special_pages;</li>
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<li>In the input-box provided next to "Page," type the desired "Template" name. For example, type in:   <font color="Darkgreen">Template:Dyson1893full</font></li>
<li>Note that when a figure from a particular journal article is displayed in one or more of our MediaWiki chapters, it will usually be accompanied by a figure heading that cites the article. Hence, typing in, for example, <font color="Darkgreen">Template:Dyson1893figure</font> should identify all of the MediaWiki chapters that display a figure from Dyson (1893).</li>
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<tr><td align="left">Acknowledgement (p. 608):<br /><font color="darkgreen">"I am very grateful to Dr. L. Mestel for introducing me to this problem and for his interest and advice throughout."<br />

"This research was supported by the U.S. Air Force under contract AF 49(638)-708, monitored by the Air Force Office of Scientific Research of the Air Research and Development Command."</font>
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<div id="LMS65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgement (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
</td></tr>
</table>
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====Whitworth81====
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====Stahler83====
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====FWW86====
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==21<sup>st</sup> Century==

===MTF2002===
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===PK2007===

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=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Appendix/References

2021-11-02T21:35:59Z

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=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
<ol type="1">
<li>Dyson1893</li>
<li>Dyson1893full</li>
<li>Dyson1893figure     <math>\Leftarrow</math>     This is the template highlighted below.</li>
<li>BAC84hereafter</li>
</ol>

If you want to see which of our MediaWiki chapters includes a link to — and, therefore, likely discusses — a particular journal article, step through the following options:
<ol type="1">
<li>Under the left-hand column, click on Tools/Special_pages;</li>
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<li>In the input-box provided next to "Page," type the desired "Template" name. For example, type in:   <font color="Darkgreen">Template:Dyson1893full</font></li>
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</ol>
==18<sup>th</sup> Century==

==19<sup>th</sup> Century==

===Clausius1870===

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===Decade5===

====Chatterji51====


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====Chatterji52====


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====CF53====
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====MS56====
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===Decade6===
====PG61====


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====Hunter62====


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====LB62====


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====LMS65====


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<div id="LMS65acknowledgement">
<table border="0" align="center" width="80%">
<tr><td align="left">Acknowledgement (p. 1446):<br /><font color="darkgreen">"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
</td></tr>
</table>
</div>

====Mestel65====


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====HRW66====



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===Decade7===

====Horedt70====
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====VH74====

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====Weber76====
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====Collins78====
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===Decade8===
====GW80====

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====Kimura81b====

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====Whitworth81====
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====Tohline81====
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====Stahler83====
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====BAC84====

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====MF85b====



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====Tohline85====
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====FWW86====
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====Horedt86====

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===Decade9===
====LRS93b====

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==21<sup>st</sup> Century==

===MTF2002===
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===ZEUS-MP2006===
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===PK2007===

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===MT2012===
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=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Appendix/References

2021-11-02T21:31:34Z

174.64.14.12: /* LMS65 */

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=Journal Articles=
==Explanation==

Three or four templates are available for each individual journal-article reference. For example, for the Dyson(1893) publication:
<ol type="1">
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If you want to see which of our MediaWiki chapters includes a link to — and, therefore, likely discusses — a particular journal article, step through the following options:
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==18<sup>th</sup> Century==

==19<sup>th</sup> Century==

===Clausius1870===

[[Special:WhatLinksHere/Template:Clausius1870full|What chapter(s) cite this article?]]<br />
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===Dyson1893===

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==20<sup>th</sup> Century==

===Decade0===

===Decade1===
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====Schwarzschild41====
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====Cowling41====

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====CF53====
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====MS56====
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====Hunter62====


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====LB62====


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====LMS65====


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<div id="LMS65acknowledgement">
<table border="0" align="center" width="75%">
<tr><td align="left"><font color="darkgreen">
"One of us (L. M.) wishes to thank Professor C. C. Lin for arranging a visit to the Department of Mathematics, Massachusetts Institute of Technology as research associate during July and August, 1964. The cosmogonical importance of the present problem first arose during discussions with Professor E. E. Salpeter during his visit to Cambridge, England, in 1961. The authors are grateful to the Computation Center at Massachusetts Institute of Technology for the use of its facilities."<br />

"This work is supported in part by the National Aeronautics and space Administration through the Center for Space Research at Massachusetts Institute of Technology."</font>
</td></tr>
</table>
</div>

====Mestel65====


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====HRW66====



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===Decade7===

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====VH74====

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====Weber76====
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====Collins78====
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===Decade8===
====GW80====

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====Kimura81b====

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====Whitworth81====
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====Tohline81====
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====Stahler83====
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====BAC84====

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====MF85b====



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====Tohline85====
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====FWW86====
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====Horedt86====

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===Decade9===
====LRS93b====

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==21<sup>st</sup> Century==

===MTF2002===
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<div align="center">{{ MTF2002figure }}</div>

===ZEUS-MP2006===
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===PK2007===

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===MT2012===
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</div>

=Books=
==Key Parallel References==

The following seven references serve as excellent supplemental printed texts to this Wiki-based H_Book because the fundamental physics and astrophysics concepts that are covered in these texts have significant overlap with the concepts we are discussing. The degree to which these references provide discussions that are parallel to ours is illustrated in our accompanying [[Appendix/EquationTemplates|key equations]] appendix.

* [<b><font color="red">BT87</font></b>] <span id="BT87">'''Binney, J. & Tremaine, S.''' 1987,</span> Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

* [<b><font color="red">C67</font></b>] <span id="C67">'''Chandrasekhar, S.''' 1967 (originally, 1939),</span> An Introduction to the Study of Stellar Structure (New York: Dover)

* [<b><font color="red">H87</font></b>] <span id="H87">'''Huang, K.''' 1987 (originally 1963),</span> Statistical Mechanics (New York: John Wiley & Sons)

* [<b><font color="red">KW94</font></b>] <span id="KW94">'''Kippenhahn, R. & Weigert, A.''' 1994,</span> Stellar Structure and Evolution (New York: Springer-Verlag)

* [<b><font color="red">LL75</font></b>] <span id="LL75">'''Laundau, L. D. & Lifshitz, E. M.''' 1975</span> (originally, 1959), Fluid Mechanics (New York: Pergamon Press)

* [<b><font color="red">P00</font></b>] <span id="P00">'''Padmanabhan, T.''' 2000,</span> Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)

* [<b><font color="red">ST83</font></b>] <span id="ST83">[http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983],</span> Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA

==Other References (Listed Chronologically)==

* [<b><font color="red">Lamb32</font></b>] <span id="Lamb32">'''Lamb, Horace''' 1932 (originally, 1879; we're referencing a 1945 reprint of), 6<sup>th</sup> Edition,</span> Hydrodynamics (New York: Dover)

* [<b><font color="red">MF53</font></b>] <span id="MF53">'''Morse, Philip M. & Feshbach, H.''' 1953,</span> Methods of Theoretical Physics: Parts I and II (New York: McGraw-Hill Book Company)

* [<b><font color="red">Clayton68</font></b>] <span id="Clayton68">'''Clayton, Donald D.''' 1968,</span> Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill Book Company)

* [<b><font color="red">EFE</font></b>] <span id="EFE">'''Chandrasekhar, S.''' 1987 (originally, 1969),</span> Ellipsoidal Figures of Equilibrium (New York: Dover)

* [<b><font color="red">CRC</font></b>] <span id="CRC">'''Selby, Samuel M.''' 1971 (19<sup>th</sup> edition),</span> CRC Standard Mathematical Tables (Cleveland, Ohio: The Chemical Rubber Co.)

* [<b><font color="red">T78</font></b>] <span id="T78">'''Tassoul, Jean-Louis''' 1978,</span> Theory of Rotating Stars (Princeton, NJ: Princeton Univ. Press)

* [<b><font color="red">Shu92</font></b>] <span id="Shu92">'''Shu, Frank H.''' 1992,</span> The Physics of Astrophysics, Volume I: Radiation & Volume II: Gas Dynamics (Mill Valey, California: University Science Books)

* [<b><font color="red">HK94</font></b>] <span id="HK94">'''Hansen, C. J. & Kawaler, S. D.''' 1994,</span> Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer-Verlag)

* [<b><font color="red">Maeder09</font></b>] <span id="Maeder09">'''Maeder, André''' 2009,</span> Physics, Formation and Evolution of Rotating Stars (Berlin: Springer)

* [<b><font color="red">Choudhuri10</font></b>] <span id="Choudhuri10">'''Choudhuri, Arnab Rai''' 2010,</span> Astrophysics for Physicists (Cambridge: Cambridge University Press)

=Appendix of EFE=

The presentation of the subject in this book (Ellipsoidal Figures of Equilibrium) is based on the following papers:
==Setup==
* [Publication I] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240: ''The virial theorem in hydromagnetics''
* [Publication II] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500: ''The virial tensor and its application to self-gravitating fluids''
* [Publication III] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..662C/abstract S. Chandrasekhar (1961)], ApJ, 134, 662: ''A Theorem on rotating polytropes''
<br />
* [Publication IV] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..238C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 238: ''On super-potentials in the theory of Newtonian gravitation''
* [Publication VII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1032C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1032: ''On the superpotentials in the theory of Newtonian gravitation. II. Tensors of higher rank''
* [Publication VIII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1037C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1037: ''The potentials and the superpotentials of homogeneous ellipsoids''
* [Publication XXXV] [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..293C/abstract S. Chandrasekhar (1968)], ApJ, 152, 293: ''The virial equations of the fourth order''

==Spheroidal & Ellipsoidal Sequences==
* [Publication V] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 135, 248: ''On the oscillations and the stability of rotating gaseous masses''
* [Publication IX] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1048C/abstract S. Chandrasekhar (1962)], ApJ, 136, 1048: ''On the point of bifurcation along the sequence of the Jacobi ellipsoids''
* [Publication X] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1069: ''On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model''
* [Publication XI] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1082: ''On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes''
* [Publication XIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1142C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1142: ''On the stability of the Jacobi ellipsoids''
* [Publication XIV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1162: ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [Publication XV] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1172C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 137, 1172: ''The equilibrium and the stability of the Jeans spheroids''
* [Publication XVI] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1185C/abstract S. Chandrasekhar (1963)], ApJ, 137, 1185: ''The points of bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences''
* [Publication XXIII] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar & N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 323: ''On the ellipsoidal figures of equilibrium of homogeneous masses''
* [Publication XXIV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...141.1043C/abstract S. Chandrasekhar (1965)], ApJ, 141, 1043: ''The equilibrium and the stability of the Dedekind ellipsoids''
* [Publication XXV] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890: ''The equilibrium and the stability of the Riemann ellipsoids. I''
* [Publication XXVIII] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842: ''The equilibrium and the stability of the Riemann ellipsoids. II''
* [Publication XXIX] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878: ''On Riemann's criterion for the stability of liquid ellipsoids''
* [Publication XXXVII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [Publication XXXVIII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145: ''The finite-amplitude oscillations of the Maclaurin spheroids''

==Binary Systems==
* [Publication XIX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The equilibrium and stability of the Roche ellipsoids''
* [Publication XX] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214: ''On the principle of the exchange of stabilities. I. The Roche ellipsoids''
* [Publication XXI] [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..599C/abstract S. Chandrasekhar (1964)], ApJ, 140, 599: ''The equilibrium and the stability of the Darwin ellipsoids''
* [Publication XXXI] S. Chandrasekhar (1969), [https://books.google.com/books/about/Publications_of_the_Ramanujan_Institute.html?id=KiFLnwEACAAJ Publications of the Ramanujan Institute], 1, 213 - 222: ''The effect of viscous dissipation on the stability of the Roche ellipsoid''
* [Publication XXXIX] [https://ui.adsabs.harvard.edu/abs/1968ApJ...153..511A/abstract M. L. Aizenman (1968)], ApJ, 153, 511: ''The equilibrium and the stability of the Roche-Riemann ellipsoids''
* [Publication XL] [https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1419C/abstract S. Chandrasekhar (1969)], ApJ, 157, 1419: ''The stability of the congruent Darwin ellipsoids''

==Effects of General Relativity==
* [Publication XXVI] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1513C/abstract S. Chandrasekhar (1965)], ApJ, 142, 1513: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem''
* [Publication XXX] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract S. Chandrasekhar (1967)], ApJ, 147, 334: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids''
* [Publication XXXI] [https://ui.adsabs.harvard.edu/abs/1967ApJ...147..383C/abstract S. Chandrasekhar (1967)], ApJ, 147, 383: ''Virial relations for uniformly rotating fluid masses in general relativity''
* [Publication XXXII] [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..621C/abstract S. Chandrasekhar (1967)], ApJ, 148, 621: ''The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. III. The deformed figures of the Jacobi ellipsoids''

==Other==
* [Publication VI] [https://books.google.com/books/about/Proceedings.html?id=GyZPQwAACAAJ S. Chandrasekhar (1962)], Proc. 4th U. S. Nat. Congress of Applied Mechanics, pp. 9 - 14: ''An approach to the theory of the equilibrium and the stability of rotating masses via the virial theorem and its extensions''
* [Publication XII] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar & N. R. Lebovitz (1962)], ApJ, 136, 1105: ''On the occurrence of multiple frequencies and beats in the β Canis Majoris stars''
* [Publication XVII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & N. R. Lebovitz (1963)], ApJ, 138, 185: ''Non-radial oscillations and the convective instability of gaseous masses''
* [Publication XVIII] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..185C/abstract S. Chandrasekhar & P. H. Roberts (1963)], ApJ, 138, 801: ''The ellipticity of a slowly rotating configuration''
* [Publication XXII] [http://inspirehep.net/record/1364947?ln=en S. Chandrasekhar (1964)], Lectures in Theoretical Physics, Vol. VI, Boulder 1963 (Boulder: University of Colorado Press), pp. 1 - 72: ''The higher order virial equations and their applications to the equilibrium and stability of rotating configurations''
* [Publication XXVII] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: ''The Riemann ellipsoids''
* [Publication XXXIII] [https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203 S. Chandrasekhar (1967)], Communications on Pure and Applied Mathematics, 20, 251: ''Ellipsoidal figures of equilibrium — an historical account''
* [Publication XXXIV] [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''

=See Also=
* [https://www.lib.uchicago.edu/e/scrc/findingaids/view.php?eadid=ICU.SPCL.CHANDRASEKHAR Guide to the Subrahmanyan Chandrasekhar Papers 1913 - 2011] (University of Chicago Library)

 <br />
{{ SGFfooter }}

Aps/MaclaurinSpheroidFreeFall

2021-11-02T21:14:51Z

174.64.14.12: /* Simplified Governing Relations */

__FORCETOC__

=Free-Fall Collapse of an Homogeneous Spheroid=
{| class="FreeFallSpheroid" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#ffff99;" |[[H_BookTiledMenu#Nonlinear_Dynamical_Evolution_2|<b>Free-Fall<br />Collapse<br />of an<br />Homogeneous<br />Spheroid</b>]]
|}
<table border="0" cellpadding="3" align="center" width="80%">
<tr><td align="left">
<font color="darkgreen">
"What is the form of the collapse under gravitational forces of a uniformly rotating spheroidal gas cloud? In the special case where initially the gas is absolutely cold and of uniform density within the spheroid, we show that the collapse proceeds through a series of uniform, uniformly rotating spheroids until a disk is formed."
</font>
</td></tr>
<tr><td align="right">
— Drawn from {{ LB62full }}

</td></tr></table>

==Simplified Governing Relations==

When studying the dynamical evolution of strictly axisymmetric configurations, it proves useful to write the spatial operators in our overarching set of [[PGE|principal governing equations]] in terms of cylindrical coordinates, <math>(\varpi, \varphi, z)</math>, and to simplify the individual equations as described in our [[AxisymmetricConfigurations/PGE#Governing_Equations_.28CYL..29|accompanying discussion]]. The resulting set of simplified governing relations is …

<div align="center">
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

<math>\frac{d\rho}{dt} + \frac{\rho}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \varpi \dot\varpi \biggr]
+ \rho \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0 </math><br />

<span id="PGE:Euler">
<font color="#770000">'''Euler Equation'''</font>
</span><br />

<math>
{\hat{e}}_\varpi \biggl[ \frac{d \dot\varpi}{dt} - \frac{j^2}{\varpi^3} \biggr] + {\hat{e}}_z \biggl[ \frac{d \dot{z}}{dt} \biggr] = -
{\hat{e}}_\varpi \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] - {\hat{e}}_z \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr]
</math><br />
where, the specific angular momentum,     <math>
j(\varpi,z) \equiv \varpi^2 \dot\varphi = \mathrm{constant} ~(\mathrm{i.e.,}~\mathrm{independent~of~time})
</math><br />

<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

{{Math/EQ_FirstLaw02}}

<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<math>
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} = 4\pi G \rho .
</math><br />
</div>

Here, our specific interest is in modeling the free-fall collapse of a uniform-density spheroid. This study is closely tied to our [[SSC/Dynamics/FreeFall#Free-Fall|separate discussion of the free-fall collapse of uniform-density ''spheres'']]. For example, by definition, an element of fluid is in "free fall" if its motion in a gravitational field is unimpeded by pressure gradients. The most straightforward way to illustrate how such a system evolves is to set <math>P = 0</math> in all of the governing equations. In doing this, the continuity equation and the Poisson equation remain unchanged; the equation formulated by the first law of thermodynamics becomes irrelevant; and the two components of the Euler equation become,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right"><math>\hat{\mathbf{e}}_\varpi</math>:</td>
<td align="right">
<math>\frac{d\dot{\varpi}}{dt} - \frac{j^2}{\varpi^3}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{\partial\Phi}{\partial\varpi} \, ,</math>
</td>
</tr>

<tr>
<td align="right"><math>\hat{\mathbf{e}}_z</math>:</td>
<td align="right">
<math>\frac{d\dot{z}}{dt} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{\partial\Phi}{\partial z} \, .</math>
</td>
</tr>
</table>

=Key References=

* {{ LB62full }}: ''On the gravitational collapse of a cold rotating gas cloud'' <br />NOTE … according to the [https://ui.adsabs.harvard.edu/abs/1962PCPS...58..709L/abstract new ADS listing], the authors associated with this paper should be, D. Lynden-Bell & C. T. C. Wall ([https://en.wikipedia.org/wiki/C._T._C._Wall Charles Terence Clegg "Terry" Wall]); however, the archived article, itself, lists Lynden-Bell as the sole author while indicating that the paper was simply being ''communicated'' by Wall.

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline: In §II of this "1964" article, Lynden-Bell references his 1962 article with an incorrect year (Lynden-Bell 1963); within his list of REFERENCES, the year (1962) is correct, but the journal volume is incorrectly identified as 50 (it should be vol. 58).]] <br />

* [https://ui.adsabs.harvard.edu/abs/1964ApJ...139.1195L/abstract D. Lynden-Bell (1964)], ApJ, 139, 1195 - 1216: ''On Large-Scale Instabilities during Gravitational Collapse and the Evolution of Shrinking Maclaurin Spheroids''
* Classic paper by C. C. Lin, Leon Mestel, and Frank Shu [https://ui.adsabs.harvard.edu/abs/1965ApJ...142.1431L/abstract (1965, ApJ, 142, 1431 - 1446)] titled, "The Gravitational Collapse of a Uniform Spheroid."

=See Also=

{{ SGFfooter }}

Appendix/References

2021-10-25T20:26:31Z

174.64.14.12: /* Decade5 */

SSC/Index

2021-10-14T21:02:21Z

174.64.14.12: /* Free Energy & Virial Analysis */

__FORCETOC__


=Spherically Symmetric Configurations (SSC) Index=
==Detailed Force Balance==
<ol type="A">
<li>
<font color="orange">SGF Menu Tile:</font>   [[SSCpt1/PGE|One-Dimensional PGEs]]<br />
SGF link:   '''SSCpt1/PGE'''<br />
File Headline:   PGE for Spherically Symmetric Configurations<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/PGE User:Tohline/SphericallySymmetricConfigurations/PGE]
<ol>
<li>
SGF link:   '''SSCpt2/SolutionStrategies'''<br />
File Headline:   Spherically Symmetric Configurations (Part II)<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies]
</li>

<li>
SGF link:   '''SSCpt2/Stability'''<br />
File Headline:   Spherically Symmetric Configurations (Stability - Part II)<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SSC/Perturbations User:Tohline/SSC/Perturbations]
</li>
</ol>
</li>
<li>
<font color="orange">SGF Menu Tile:</font>   [[SSCpt2/IntroductorySummary|Hydrostatic Balance Equation]]<br />
SGF link:   '''SSCpt2/IntroductorySummary'''<br />
File Headline:   Hydrostatic Balance Equation<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/IntroductorySummary#Applications User:Tohline/SphericallySymmetricConfigurations/IntroductorySummary]
</li>
</ol>

==Free Energy & Virial Analysis==
<ol type="A">
<li>
<font color="orange">SGF Menu Tile:</font>   [[VE|Global Energy Considerations]]<br />
SGF link:   '''VE'''<br />
File Headline:   Global Energy Considerations<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/VE#Global_Energy_Considerations User:Tohline/VE]
<ol>
<li>
SGF link:   [[PGE/RotatingFrame|'''PGE/RotatingFrame''']]<br />
File Headline:   Rotating Reference Frame<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/PGE/RotatingFrame User:Tohline/PGE/RotatingFrame]
</li>
</ol>
</li>

<li>
<font color="orange">SGF Menu Tile:</font>   [[SSCpt1/Virial|Free-Energy of Spherical Systems]]<br />
SGF link:   '''SSCpt1/Virial'''<br />
File Headline:   Virial Equilibrium of Spherically Symmetric Configurations<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/Virial User:Tohline/SphericallySymmetricConfigurations/Virial]
<ol>
<li>
SGF link:   [[SSC/Virial/Polytropes|'''SSC/Virial/Polytropes''']]<br />
File Headline:   Virial Equilibrium of Adiabatic Spheres<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SSC/Virial/Polytropes User:Tohline/SSC/Virial/Polytropes]
</li>

<li>
SGF link:   [[SSC/BipolytropeGeneralizationVersion2|'''SSC/BipolytropeGeneralizationVersion2''']]<br />
File Headline:   Bipolytrope Generalization<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SSC/BipolytropeGeneralization_Version2 User:Tohline/SSC/BipolytropeGeneralization_Version2]
</li>

<li>
SGF link:   [[AxisymmetricConfigurations/SolutionStrategies|'''AxisymmetricConfigurations/SolutionStrategies''']]<br />
File Headline:   Axisymmetric Configurations (Solution Strategies)<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/AxisymmetricConfigurations/SolutionStrategies User:Tohline/AxisymmetricConfigurations/SolutionStrategies]<br />
<font color="darkgreen">NOTE: This chapter contains "Simple Rotation Profiles"</font>
</li>

<li>
SGF link:   [[SSC/Structure/PolytropesEmbedded|'''SSC/Structure/PolytropesEmbedded''']]<br />
File Headline:   Embedded Polytropic Spheres<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SSC/Structure/PolytropesEmbedded User:Tohline/SSC/Structure/PolytropesEmbedded]
</li>

<li>
SGF link:   [[SSC/Structure/Polytropes|'''SSC/Structure/Polytropes''']]<br />
File Headline:   Polytropic Spheres<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SSC/Structure/Polytropes User:Tohline/SSC/Structure/Polytropes]
</li>

<li>
SGF link:   [[SSC/Virial/PolytropesSummary|'''SSC/Virial/PolytropesSummary''']]<br />
File Headline:   Virial Equilibrium of Adiabatic Spheres (Summary)<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SSC/Virial/PolytropesSummary User:Tohline/SSC/Virial/PolytropesSummary]
</li>

<li>
SGF link:   [[SSC/Structure/BiPolytropes/Analytic51|'''SSC/Structure/BiPolytropes/Analytic51''']]<br />
File Headline:   BiPolytrope with n<sub>c</sub> = 5 and n<sub>e</sub> = 1<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SSC/Structure/BiPolytropes/Analytic51 User:Tohline/SSC/Structure/BiPolytropes/Analytic51]
</li>

<li>
SGF link:   [[SSC/Virial/Isothermal|'''SSC/Virial/Isothermal''']]<br />
File Headline:   Virial Equilibrium of Isothermal Spheres<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SSC/Virial/Isothermal User:Tohline/SSC/Virial/Isothermal]
</li>

<li>
SGF link:   [[SSC/Structure/LimitingMasses|'''SSC/Structure/LimitingMasses''']]<br />
File Headline:   Mass Upper Limits<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SSC/Structure/LimitingMasses User:Tohline/SSC/Structure/LimitingMasses]
</li>
</ol>
</li>

<li>
<font color="orange">SGF Menu Tile:</font>   [[SSCpt1/Virial/FormFactors|Structural Form Factors]]<br />
SGF link:   '''SSCpt1/Virial/FormFactors'''<br />
File Headline:   Structural Form Factors<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SSC/Virial/FormFactors User:Tohline/SSC/Virial/FormFactors]
<ol>
<li>SGF link:   [[SSC/Structure/PolytropesEmbedded|'''SSC/Structure/PolytropesEmbedded''']]</li>

<li>
SGF link:   [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain|'''SSC/Virial/PolytropesEmbedded/SecondEffortAgain''']]<br />
File Headline:   Virial Equilibrium of Adiabatic Spheres (Summary)<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain]
</li>

<li>SGF link:   [[SSC/Structure/Polytropes|'''SSC/Structure/Polytropes''']]<br /></li>

<li>
SGF link:   [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain|'''SSC/Virial/PolytropesEmbedded/FirstEffortAgain''']]<br />
File Headline:   Virial Equilibrium of Adiabatic Spheres<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain]
</li>

<li>SGF link:   [[SSC/Virial/PolytropesSummary|'''SSC/Virial/PolytropesSummary''']]</li>
<li>SGF link:   [[SSC/Structure/BiPolytropes/Analytic51|'''SSC/Structure/BiPolytropes/Analytic51''']]</li>
</ol>
</li>

<li>
<font color="orange">SGF Menu Tile:</font>   File:FreeNRGpressureRadiusIsothermal.png; [[SSCpt1/Virial/PolytropesEmbeddedOutline|Whitworth's (1981) Isothermal Free-Energy Surface]]<br />
SGF link:   '''SSCpt1/Virial/PolytropesEmbeddedOutline'''<br />
File Headline:   Virial Equilibrium of Embedded Polytropic Spheres<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SSC/Virial/PolytropesEmbeddedOutline User:Tohline/SSC/Virial/PolytropesEmbeddedOutline]
<ol>
<li>SGF link:   [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain|'''SSC/Virial/PolytropesEmbedded/SecondEffortAgain''']]</li>

<li>
SGF link:   [[SSC/Structure/PolytropesASIDE1|'''SSC/Structure/PolytropesASIDE1''']]<br />
File Headline:   ASIDE: Whitworth's Scaling<br />
(old) VisTrails link:   [https://www.vistrails.org/index.php/User:Tohline/SSC/Structure/PolytropesASIDE1 User:Tohline/SSC/Structure/PolytropesASIDE1]
</li>
<li>SGF link:   [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain|'''SSC/Virial/PolytropesEmbedded/FirstEffortAgain''']]</li>

<li>SGF link:   [[SSC/Virial/Polytropes|'''SSC/Virial/Polytropes''']]</li>
<li>SGF link:   [[SSC/Structure/PolytropesEmbedded|'''SSC/Structure/PolytropesEmbedded''']]</li>
<li>SGF link:   [[SSC/Virial/PolytropesSummary|'''SSC/Virial/PolytropesSummary''']]</li>
</ol>
</li>
</ol>

{{ SGFfooter }}

ThreeDimensionalConfigurations/Challenges

2021-10-02T22:05:40Z

174.64.14.12: /* Ou's (2006) Detailed Force Balance */

__FORCETOC__

=Challenges Constructing Ellipsoidal-Like Configurations=
First, let's review the three different approaches that we have described for constructing Riemann S-type ellipsoids. Then let's see how these relate to the technique that has been used to construct infinitesimally thin, nonaxisymmetric disks.

==Riemann S-type Ellipsoids==

Usually, the density, <math>~\rho</math>, and the pair of axis ratios, <math>~b/a</math> and <math>~c/a</math>, are specified. Then, the Poisson equation is solved to obtain <math>~\Phi_\mathrm{grav}</math> in terms of <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>. The aim, then, is to determine the value of the central enthalpy, <math>~H_0</math> — alternatively, the thermal energy density, <math>~\Pi</math> — and the two parameters, <math>~\Omega_f</math> and <math>~\lambda</math>, that determine the magnitude of the velocity flow-field. Keep in mind that, as viewed from a frame of reference that is spinning with the ellipsoid (at angular frequency, <math>~\Omega_f</math>), the adopted (rotating-frame) velocity field is,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\lambda \biggl[ \boldsymbol{\hat\imath} \biggl( \frac{a}{b}\biggr) y - \boldsymbol{\hat\jmath} \biggl( \frac{b}{a} \biggr) x \biggr] \, .</math>
</td>
</tr>
</table>
Hence, the inertial-frame velocity is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\bold{v}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\bold{u} + \bold{\hat{e}}_\varphi \Omega_f \varpi \, .</math>
</td>
</tr>
</table>

While we will fundamentally rely on the <math>~(\Omega_f, \lambda)</math> parameter pair to define the velocity flow-field, in discussions of Riemann S-type ellipsoids it is customary to also refer to the following two additional parameters: The (rotating-frame) vorticity,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\boldsymbol{\zeta} \equiv \boldsymbol{\nabla \times}\bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\boldsymbol{\hat\imath} \biggl[ \frac{\partial u_z}{\partial y} - \frac{\partial u_y}{\partial z} \biggr]
+ \boldsymbol{\hat\jmath} \biggl[ \frac{\partial u_x}{\partial z} - \frac{\partial u_z}{\partial x} \biggr]
+ \bold{\hat{k}} \biggl[ \frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\bold{\hat{k}} \biggl[ - \lambda \biggl(\frac{b}{a} + \frac{a}{b}\biggr) \biggr] \, ;</math>
</td>
</tr>
</table>
and the dimensionless frequency ratio,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{ \zeta}{\Omega_f} \, .</math>
</td>
</tr>
</table>

===2<sup>nd</sup>-Order TVE Expressions===

As we have discussed in detail in an [[VE/RiemannEllipsoids#Riemann_S-Type_Ellipsoids|accompanying chapter]], the three diagonal elements of the <math>~(3 \times 3)</math> 2<sup>nd</sup>-order tensor virial equation are sufficient to determine the equilibrium values of <math>~\Pi</math>, <math>~\Omega_3</math>, and <math>~\zeta_3</math>.

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="2">Indices</td>
<td align="center" rowspan="2">2<sup>nd</sup>-Order TVE Expressions that are Relevant to Riemann S-Type Ellipsoids</td>
</tr>
<tr>
<td align="center" width="5%"><math>~i</math></td>
<td align="center" width="5%"><math>~j</math></td>
</tr>
<tr>
<td align="center"><math>~1</math></td>
<td align="center"><math>~1</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi
+\biggl\{
\Omega_3^2
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
~-~(2\pi G\rho) A_1
\biggr\} a^2
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~2</math></td>
<td align="center"><math>~2</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi
+ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ \biggl\{
\Omega_3^2
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~-~( 2\pi G \rho) A_2
\biggr\}b^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~3</math></td>
<td align="center"><math>~3</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi
- (2\pi G \rho)A_3 c^2
</math>
</td>
</tr>
</table>

</td>
</tr>
</table>

The <math>~(i, j) = (3, 3)</math> element gives <math>~\Pi</math> directly in terms of known parameters. The <math>~(1, 1)</math> and <math>~(2, 2)</math> elements can then be combined in a couple of different ways to obtain a coupled set of expressions that define <math>~\Omega_3</math> and <math>~f \equiv \zeta_3/\Omega_3</math>.

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\biggl[ \frac{b^2 a^2}{b^2+a^2}\biggr] f \Omega_3^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2}{ b^2 - a^2} - A_3 c^2\biggr] \, ;
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (34)</font> ]</td></tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\Omega_3^2 \biggl\{1
+ \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] f^2 \biggr\}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2\pi G\rho}{ (a^2-b^2) }
\biggl[
A_1 a^2
- A_2 b^2
\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (33)</font> ]</td></tr>
</table>

</td></tr></table>

===Ou's (2006) Detailed Force Balance===

In a separate [[ThreeDimensionalConfigurations/RiemannStype#Based_on_Detailed_Force_Balance|accompanying chapter]], we have described in detail how [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou(2006)] used, essentially, the HSCF technique to solve the detailed force-balance equations. Beginning with the,

<div align="center">
<font color="#770000">'''Eulerian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr]_{rot} + ({\vec{v}}_{rot}\cdot \nabla) {\vec{v}}_{rot}= - \frac{1}{\rho} \nabla P - \nabla \Phi_\mathrm{grav}
- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} \, ,</math> </div>

it can be shown that, for the velocity fields associated with all Riemann S-type ellipsoids,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~({\vec{v}}_{rot}\cdot \nabla) {\vec{v}}_{rot}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\nabla \biggl[ \frac{1}{2} \lambda^2(x^2 + y^2) \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
+\nabla\biggl[\frac{1}{2} \Omega_f^2 (x^2 + y^2) \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~- 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \nabla\biggl[ \Omega_f \lambda\biggl( \frac{b}{a} x^2 + \frac{a}{b}y^2 \biggr) \biggr] \, .
</math>
</td>
</tr>
</table>

<font color="orange">Hence, within</font> each steady-state <font color="orange"> configuration the following Bernoulli's function must be uniform in space:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
H + \Phi_\mathrm{grav} - \frac{1}{2} \Omega_f^2(x^2 + y^2)
- \frac{1}{2} \lambda^2(x^2 + y^2)
+ \Omega_f \lambda \biggl(\frac{b}{a}x^2 + \frac{a}{b}y^2 \biggr)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
C_B \, ,
</math>
</td>
</tr>
</table>

[https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou(2006)], p. 550, §2, Eq. (6)
</div>

<font color="orange">where <math>~C_B</math> is a constant.</font> So, at the surface of the ellipsoid (where the enthalpy ''H = 0'') on each of its three principal axes, the equilibrium conditions demanded by the expression for detailed force balance become, respectively:

<ol type="I">
<li>On the x-axis, where (x, y, z) = (a, 0, 0):
<br />
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~2\biggl[ \frac{C_B}{a^2} + (\pi G\rho)I_\mathrm{BT} \biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(2\pi G \rho) A_1 - \Omega_f^2 - \lambda^2 + 2\Omega_f \lambda \biggl(\frac{b}{a} \biggr)
</math>
</td>
</tr>
</table>
</li>
<li>On the y-axis, where (x, y, z) = (0, b, 0):
<br />
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~2\biggl[ \frac{C_B}{a^2} + (\pi G\rho)I_\mathrm{BT} \biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(2\pi G \rho) A_2 \biggl( \frac{b^2}{a^2}\biggr)
- \Omega_f^2 \biggl( \frac{b^2}{a^2} \biggr)
- \lambda^2\biggl( \frac{b^2}{a^2} \biggr)
+ 2\Omega_f \lambda \biggl(\frac{b}{a}\biggr)
</math>
</td>
</tr>
</table>
</li>
<li>On the z-axis, where (x, y, z) = (0, 0, c):
<br />
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Rightarrow ~~~ 2 \biggl[ \frac{C_B}{a^2} + (\pi G\rho)I_\mathrm{BT}\biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(2\pi G \rho) A_3 \biggl( \frac{c^2}{a^2}\biggr)
</math>
</td>
</tr>
</table>
</li>
</ol>

This third expression can be used to replace the left-hand-side of the first and second expressions. Then via some additional algebraic manipulation, the first and second expressions can be combined to provide the desired solutions for the parameter pair, <math>~(\Omega_f, \lambda)</math>, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\Omega_f^2}{(\pi G \rho)}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2} \biggl[M + \sqrt{ M^2 - 4N^2} \biggr] \, ,</math>
</td>
<td align="center">      and      </td>
<td align="right">
<math>~\frac{\lambda^2}{(\pi G \rho)}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2} \biggl[M - \sqrt{ M^2 - 4N^2} \biggr] \, ,</math>
</td>
</tr>
</table>
[https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou(2006)], p. 551, §2, Eqs. (15) & (16)
</div>
where,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~M</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
2\biggl[ A_1 - A_2 \biggl( \frac{b^2}{a^2}\biggr) \biggr]\biggl[ \frac{a^2}{a^2 - b^2} \biggr] \, ,</math>     and,
</td>
</tr>

<tr>
<td align="right">
<math>~N</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\frac{1}{a b ( a^2 - b^2 )}\biggl[ A_3 ( a^2 - b^2 )c^2 - (A_2 - A_1) a^2 b^2 \biggr] \, .
</math>
</td>
</tr>
</table>

===Hybrid Scheme===

In a separate chapter we have detailed the [[Appendix/Ramblings/Hybrid_Scheme_Implications#Hybrid_Scheme|hybrid scheme]]. For steady-state configurations, the three components of the combined Euler + Continuity equations give,

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
<div align="center">'''Hybrid Scheme Summary for ''Steady-State'' Configurations'''</div>

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right"><math>~\boldsymbol{\hat{k}:}</math></td>
<td align="right">
<math>~
\bold\nabla \cdot (\rho v_z \bold{u})
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\bold{\hat{k}} \cdot (\rho \bold{a}) \, ;</math>
</td>
</tr>

<tr>
<td align="right"><math>~\bold{\hat{e}_\varpi:}</math></td>
<td align="right">
<math>~
\bold\nabla \cdot (\rho v_\varpi \bold{u})
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\bold{\hat{e}}_\varpi \cdot (\rho \bold{a}) + \frac{v_\varphi^2}{\varpi} \, ;</math>
</td>
</tr>

<tr>
<td align="right"><math>~\bold{\hat{e}_\varphi:}</math></td>
<td align="right">
<math>~
\bold\nabla \cdot (\rho \varpi v_\varphi \bold{u})
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\bold{\hat{e}}_\varphi \cdot (\rho \varpi \bold{a}) \, .</math>
</td>
</tr>
</table>

</td></tr></table>

In this context, the vector acceleration that drives the fluid flow is, simply,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\bold{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-\nabla(H + \Phi_\mathrm{grav} ) \, .</math>
</td>
</tr>
</table>

Then, for the velocity flow-patterns in Riemann S-type ellipsoids, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla \cdot (\rho v_z \bold{u})</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0</math>           (because <math>~v_z = 0</math>);
</td>
</tr>

<tr>
<td align="right">
<math>~\nabla \cdot (\rho v_\varpi \bold{u})</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\lambda^2}{\varpi^3} \biggl[\frac{a}{b} - \frac{b}{a} \biggr] \biggl\{
y^4 \biggl(\frac{a}{b}\biggr)
-
x^4 \biggl(\frac{b}{a}\biggr)
\biggr\}\rho \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\nabla \cdot (\rho \varpi v_\varphi \bold{u})</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \lambda xy \Omega_f \biggl[\frac{a}{b} - \frac{b}{a} \biggr]\rho \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\varpi v_\varphi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-
\biggl[ \lambda \biggl(\frac{b}{a}\biggr) - \Omega_f\biggr]x^2
-
\biggl[ \lambda \biggl(\frac{a}{b}\biggr) - \Omega_f\biggr]y^2
\, .
</math>
</td>
</tr>
</table>

<font color="red">'''Vertical Component:'''</font>   Given that <math>~\bold{\hat{k}}\cdot (\rho \bold{a}) = 0</math>, we deduce that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H_0 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\pi G \rho c^2 A_3 \, . </math>
</td>
</tr>
</table>

<font color="red">'''Azimuthal Component:'''</font>   Irrespective of the <math>~(x, y, z)</math> location of each fluid element, this component requires,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
- a b \lambda \Omega_f
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi G \rho \biggl[ \frac{( A_1 - A_2 )a^2b^2}{b^2 - a^2} - c^2 A_3
\biggr] \, .
</math>
</td>
</tr>
</table>

<font color="red">'''Radial Component:'''</font>   After inserting the "azimuthal component" relation and marching through a fair amount of algebraic manipulation, we find that Irrespective of the <math>~(x, y, z)</math> location of each fluid element, this component requires,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~
\frac{2\pi G \rho }{(a^2 - b^2) }
\biggl[ A_1 a^2 - A_2 b^2 \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \lambda^2 + \Omega_f^2\biggr] \, .
</math>
</td>
</tr>
</table>

==Compressible Structures==

Here we draw heavily on the published work of [http://adsabs.harvard.edu/abs/1996ApJS..105..181K Korycansky & Papaloizou] (1996, ApJS, 105, 181; hereafter KP96) that we have reviewed in a [[Apps/Korycansky_Papaloizou_1996#Korycansky_and_Papaloizou_.281996.29|separate chapter]], and on the doctoral dissertation of [https://digitalcommons.lsu.edu/gradschool_disstheses/6650/ Saied W. Andalib (1998)].

===Part I===
Returning to the above-mentioned,

<div align="center">
<font color="#770000">'''Eulerian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\frac{\partial \bold{u}}{\partial t} + (\bold{u}\cdot \nabla) \bold{u} = - \frac{1}{\rho} \nabla P - \nabla \Phi_\mathrm{grav}
- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) - 2{\vec{\Omega}}_f \times \bold{u} \, ,</math> </div>

we next note — as we have done in our [[PGE/Euler#in_terms_of_the_vorticity:|broader discussion of the Euler equation]] — that,
<div align="center">
<math>
(\bold{u} \cdot\nabla)\bold{u} = \frac{1}{2}\nabla(\bold{u} \cdot \bold{u}) - \bold{u} \times(\nabla\times\bold{u})
= \frac{1}{2}\nabla(u^2) + \boldsymbol\zeta \times \bold{u} ,
</math>
</div>
where, as above, <math>\boldsymbol\zeta \equiv \nabla\times \bold{u}</math> is the [https://en.wikipedia.org/wiki/Vorticity vorticity]. Making this substitution, we obtain what is essentially equation (7) of [http://adsabs.harvard.edu/abs/1996ApJS..105..181K KP96], that is, the,

<div align="center">
Euler Equation<br />
written <font color="#770000">'''in terms of the Vorticity'''</font> and<br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\frac{\partial \bold{u}}{\partial t} + (\boldsymbol\zeta+2{\vec\Omega}_f) \times {\bold{u}}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi + \frac{1}{2}u^2 - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr]</math> .
</div>
Hence, in steady-state, the Euler equation becomes,
<div align="center">
<math>
\nabla F_B + \vec{A} = 0 ,
</math>
</div>
where, the scalar "Bernoulli" function,
<div align="center">
<math>
F_B \equiv \frac{1}{2}u^2 + H + \Phi - \frac{1}{2}|\Omega\hat{k} \times \vec{x}|^2 ;
</math>
</div>
and,
<div align="center">
<math>
\vec{A} \equiv ({\boldsymbol\zeta}+2{\vec\Omega}_f) \times {\bold{u}} .
</math>
</div>

For later use …
<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
<ol>
<li><font color="red">Curl of steady-state Euler equation:</font>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\nabla F_B + \bold{A}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\nabla \times \biggl[ \nabla F_B + \bold{A} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\nabla \times \bold{A} \, .</math>
</td>
</tr>
</table>
This last step is justified because the [https://en.wikipedia.org/wiki/Vector_calculus_identities#Curl_of_gradient_is_zero curl of any gradient is zero].
</li>
<li>
<font color="red">KP96 only deal with two-dimensional motion in the equatorial plane and, hence, there is no vertical motion:</font><br />
Hence, <math>~\bold{u}</math> lies in the equatorial plane; both <math>~\vec\zeta</math> and <math>~\vec\Omega_f</math> only have z-components; and, <math>~\bold{A}</math> is perpendicular to both <math>~\vec\Omega_f</math> and <math>~\bold{u}</math>. Also, given that <math>~\bold{A}</math> necessarily lies in the equatorial plane, its curl will only have a z-component, that is,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\nabla \times \bold{A} = 0</math>
</td>
<td align="center">
      <math>~\Leftrightarrow</math>     
</td>
<td align="left">
<math>~[\nabla \times \bold{A}]_z = 0 \, .</math>
</td>
</tr>
</table>
</li>
<li>
<font color="red">"Dot" <math>~\bold{u}</math> into the steady-state Euler equation:</font>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\nabla F_B + \bold{A}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\bold{u} \cdot \biggl[ \nabla F_B + \bold{A} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\bold{u} \cdot \nabla F_B \, .</math>
</td>
</tr>
</table>
This last step is justified because <math>~\bold{A}</math> is necessarily always perpendicular to <math>~\bold{u}</math>.
</li>
</ol>
</td></tr></table>

----

We will leave discussion of the Euler equation, for the moment, and instead look at the continuity equation. As viewed from the rotating frame of reference,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\partial (\rho \bold{u})}{\partial t} + \nabla\cdot (\rho\bold{u})</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, .</math>
</td>
</tr>
</table>

If we are able to write the momentum density (in the rotating frame) in terms of a stream-function, <math>~\Psi</math>, such that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho\bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\nabla \times (\boldsymbol{\hat{k}} \Psi)
=
\boldsymbol{\hat\imath} \biggl[ \frac{\partial \Psi}{\partial y} \biggr]
- \boldsymbol{\hat\jmath} \biggl[ \frac{\partial \Psi}{\partial x}\biggr] \, ,</math>
</td>
</tr>
</table>
then satisfying the steady-state continuity equation is guaranteed because the [https://en.wikipedia.org/wiki/Vector_calculus_identities#Divergence_of_curl_is_zero divergence of a curl is always zero]. Note, as well, that when written in terms of this stream-function, the z-component of the vorticity will be,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\zeta_z </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial }{\partial x}\biggl[- \frac{1}{\rho} \frac{\partial \Psi}{\partial x} \biggr] - \frac{\partial }{\partial y} \biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial y} \biggr] \, .
</math>
</td>
</tr>
</table>

Note that the steady-state continuity equation may be rewritten in the form,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla\cdot \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \bold{u} \cdot \nabla [ \ln \rho] \, .
</math>
</td>
</tr>
</table>

----

It can also be shown that the condition, <math>~[\nabla \times \bold{A}]_z = 0</math> can be rewritten as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla\cdot \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \bold{u} \cdot \nabla [ \ln(2\Omega_f + \zeta_z] \, .
</math>
</td>
</tr>
</table>

By combining these last two expressions, we appreciate that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\bold{u} \cdot \nabla \ln \biggl[ \frac{(2\Omega_f + \zeta_z)}{\rho} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, .</math>
</td>
</tr>
</table>
This means that, in the steady-state flow whose spatial structure we are seeking, the velocity vector, <math>~\bold{u}</math> (and also the momentum density vector, <math>~\rho \bold{u}</math>) must everywhere be tangent to contours of constant ''vortensity'', <math>~[(2\Omega_f + \zeta_z)/\rho]</math>.

We need a function <math>~g(\Psi) </math> such that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~g(\Psi) </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho} \biggl\{
\zeta_z + 2\Omega_f
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho} \biggl\{
2\Omega_f
-
\frac{\partial }{\partial x}\biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial x} \biggr]
- \frac{\partial }{\partial y} \biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial y} \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>

Let's try, <math>~\Psi = \rho^2</math>, and
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\rho_c \biggl\{1 - \biggl[ \frac{y^2}{b^2} + \frac{x^2}{a^2}\biggr]\biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~\frac{\partial^2 \rho}{\partial x^2} = -\frac{\partial}{\partial x}\biggl\{ \frac{2\rho_c x}{a^2}\biggr\} = - \frac{2\rho_c}{a^2}</math>
</td>
<td align="center">
    and,    
</td>
<td align="left">
<math>~\frac{\partial^2 \rho}{\partial y^2} = - \frac{\partial}{\partial y}\biggl\{ \frac{2\rho_c y}{b^2} \biggr\} = - \frac{2\rho_c}{b^2} \, .</math>
</td>
</tr>
</table>

Then,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~g(\Psi) </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho} \biggl\{
2\Omega_f
-
\frac{\partial }{\partial x}\biggl[\frac{1}{\rho} \frac{\partial \rho^2}{\partial x} \biggr]
- \frac{\partial }{\partial y} \biggl[\frac{1}{\rho} \frac{\partial \rho^2}{\partial y} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho} \biggl\{
2\Omega_f
-
2 \biggl[ \frac{\partial^2 \rho}{\partial x^2} \biggr]
- 2 \biggl[\frac{\partial^2 \rho}{\partial y^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho} \biggl\{
2\Omega_f
+
4\rho_c \biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>

Hence,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~g(\Psi) </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
2\Omega_f
+
4\rho_c \biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr]
\biggr\} \Psi^{-1 / 2} \, .
</math>
</td>
</tr>
</table>

Next, given that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{dF_B}{d\Psi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- g(\Psi) \, ,</math>
</td>
</tr>
</table>
we conclude that, to within an additive constant,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~ F_B(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \int g(\Psi) d\Psi
= - g_0 \int \Psi^{-1 / 2} d\Psi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- 2g_0 \Psi^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- 2g_0 \rho \, ,
</math>
</td>
</tr>

</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~g_0</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\biggl\{ 2\Omega_f + 4\rho_c \biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr] \biggr\} \, .
</math>
</td>
</tr>
</table>

Here's what to do:

Given <math>~g(\Psi)</math>, write out the functional forms of <math>~\bold{A}</math> and <math>~F_B(\Psi)</math>. Then see if <math>~\nabla F_B = - \bold{A}</math>.

===Part II===

Consider a steady-state configuration that is the compressible analog of a Riemann S-type ellipsoid; even better, give the configuration a "peanut" shape rather than the shape of an ellipsoid. As viewed from a frame that is spinning with the configuration's overall angular velocity, <math>~\vec\Omega_f = \boldsymbol{\hat{k}} \Omega_f</math>, generally we expect the configuration's internal (and surface) flow to be represented by a set of nested stream-lines and at every <math>~(x, y)</math> location the fluid's velocity (and its momentum-density vector) will be tangent to the stream-line that runs through that point. It is customary to represent the stream-function by a scalar quantity, <math>~\Psi(x, y)</math>, appreciating that each stream-line will be defined by a curve, <math>~\Psi = \mathrm{constant}</math>; also, the local spatial gradient of <math>~\Psi(x,y)</math> will be perpendicular to the local stream-line, hence it will be perpendicular to the local velocity vector as well. If we specifically introduce the stream-function via the relation,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\rho\bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\nabla \times (\boldsymbol{\hat{k}} \Psi)
=
\boldsymbol{\hat\imath} \biggl[ \frac{\partial \Psi}{\partial y} \biggr]
- \boldsymbol{\hat\jmath} \biggl[ \frac{\partial \Psi}{\partial x}\biggr] \, ,</math>
</td>
</tr>
</table>

it will display all of the just-described attributes and we are also guaranteed that the steady-state continuity equation will be satisfied everywhere, because the divergence of a curl is always zero.

We also have demonstrated that the vector, <math>~\bold{A}</math>, has the right properties if,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\bold{u} \cdot \nabla \ln \biggl[ \frac{(2\Omega_f + \zeta_z)}{\rho} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, .</math>
</td>
</tr>
</table>

This means that, at every location in the plane of the fluid flow, the gradient of the ''vortensity'' also must be perpendicular to the velocity vector. This constraint can be immediately satisfied if we simply demand that the vortensity be a function of the stream-function, <math>~\Psi</math>, that is, we need,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{(2\Omega_f + \zeta_z)}{\rho}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~g(\Psi) \, .</math>
</td>
</tr>
</table>
In other words, the scalar vortensity is constant along each stream-line. And, once we have determined the mathematical expression for this function, we will know that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\bold{A}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~[ \boldsymbol{\hat{k}} g(\Psi) ] \times \rho\bold{u} \, ;</math>
</td>
</tr>
</table>
Furthermore, we should be able to determine the mathematical expression for <math>~F_B(x,y)</math> because,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{dF_B}{d\Psi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- g(\Psi) \, .</math>
</td>
</tr>
</table>
As a check, we should find that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla F_B + \bold{A}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, .</math>
</td>
</tr>
</table>

===Part III===

Here we closely follow Chapter 4 of [https://digitalcommons.lsu.edu/gradschool_disstheses/6650/ Saied W. Andalib (1998)].

From §4.1 (p. 80): "<font color="orange">Euler's equation, the equation of continuity, the Poisson equation and the equation of state … govern the dynamics and evolution of these equilibrium configurations.</font>"

====Equation of Continuity====
In steady state, <math>~\partial (\rho\bold{u})/\partial t = 0</math>. Hence the rotating-frame-based continuity equation becomes,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla \cdot (\rho\bold{u})</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, .</math>
</td>
</tr>
</table>
If we insist that the momentum-density vector be expressible in terms of the curl of a vector — for example,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho\bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\nabla \times \boldsymbol{\mathfrak{J}} \, ,</math>
</td>
</tr>
<tr><td align="center" colspan="3">[https://digitalcommons.lsu.edu/gradschool_disstheses/6650/ Saied W. Andalib (1998)], §4.1, p. 80, Eq. (4.1)
</td></tr>
</table>

then satisfying this steady-state continuity equation is guaranteed because the [https://en.wikipedia.org/wiki/Vector_calculus_identities#Divergence_of_curl_is_zero divergence of a curl is always zero]. "<font color="orange">The task</font> of satisfying the steady-state equation of continuity <font color="orange">then shifts to identifying an appropriate expression for the vector potential, <math>~\boldsymbol{\mathfrak{J}} \, .</math></font>" In the most general case, in terms of this vector potential the three Cartesian components of the momentum-density vector are,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho u_x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial \mathfrak{J}_z}{\partial y}
-
\frac{\partial \mathfrak{J}_y}{\partial z} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\rho u_y</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial \mathfrak{J}_x}{\partial z}
-
\frac{\partial \mathfrak{J}_z}{\partial x} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\rho u_z</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial \mathfrak{J}_y}{\partial x}
-
\frac{\partial \mathfrak{J}_x}{\partial y} \, .
</math>
</td>
</tr>
</table>
Here, <font color="red">we will follow Andalib's lead and only look for fluid flows with no vertical motions</font>. That is to say, we will set <math>~\rho u_z = 0</math>, in which case this last expression establishes the constraint,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\partial \mathfrak{J}_y}{\partial x}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial \mathfrak{J}_x}{\partial y} \, .
</math>
</td>
</tr>
</table>
"<font color="orange">A general solution to this equation can be found only if there exists a scalar function <math>~\Gamma(x, y, z)</math> such that …</font>"

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{J}_y = \frac{\partial \Gamma}{\partial y}</math>
</td>
<td align="center">
      and,      
</td>
<td align="left">
<math>~\mathfrak{J}_x = \frac{\partial \Gamma}{\partial x} \, ;</math>
</td>
</tr>
</table>
note that this adopted functional behavior works because the constraint becomes,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\partial^2 \Gamma}{\partial x \partial y}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\partial^2 \Gamma}{\partial y \partial x} \, .</math>
</td>
</tr>
</table>

Hence, the expressions for the x- and y-components of the momentum-density vector may be rewritten, respectively, as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho u_x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial \mathfrak{J}_z}{\partial y}
-
\frac{\partial}{\partial z} \biggl[ \frac{\partial \Gamma}{\partial y} \biggr]
=
+ \frac{\partial}{\partial y}\biggl[ \mathfrak{J}_z - \frac{\partial\Gamma}{\partial z} \biggr]
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\rho u_y</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial }{\partial z}\biggl[ \frac{\partial \Gamma}{\partial x} \biggr]
-
\frac{\partial \mathfrak{J}_z}{\partial x}
=
- \frac{\partial}{\partial x}\biggl[ \mathfrak{J}_z - \frac{\partial\Gamma}{\partial z} \biggr]
\, .
</math>
</td>
</tr>
</table>

If we again <font color="red">follow Andalib's lead and only look for models in which the x-y-plane flow is independent of the vertical coordinate, z</font>, then, <font color="orange"><math>~\mathfrak{J}_z</math> and <math>~\partial\Gamma/\partial z</math> must be functions of x and y only. Therefore, <math>~\mathfrak{J}_z</math> is independent of z and <math>~\Gamma</math> is at most linear in z.</font> Now, rather than focusing on the determination of <math>~\Gamma(x,y)</math>, we can just as well define the scalar function,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Psi(x,y)</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\mathfrak{J}_z - \frac{\partial\Gamma}{\partial z} \, ,
</math>
</td>
</tr>
</table>
in which case "<font color="orange">… the components of the momentum density may be written as:</font>"
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\boldsymbol{\hat\imath} \frac{\partial \Psi}{\partial y} - \boldsymbol{\hat\jmath} \frac{\partial \Psi}{\partial x} \, .
</math>
</td>
</tr>
</table>
It is straightforward to demonstrate that this expression for the momentum-density vector does satisfy the steady-state continuity equation. "<font color="orange">The function <math>~\Psi(x, y)</math> will serve a similar role as the velocity potential for incompressible fluids.</font>"

====Related Useful Expressions====

Given that, by our design, the fluid motion will be confined to the x-y-plane, the fluid vorticity will have only a z-component; that is,
<div align="center">
<math>~\vec\zeta = \nabla \times \bold{u} = \boldsymbol{\hat{k}} \zeta_z</math>,
</div>
where,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\zeta_z </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y} \, .
</math>
</td>
</tr>
</table>

And when it is written in terms of <math>~\Psi(x,y)</math>, this z-component of the vorticity will be obtained from the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\zeta_z </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial }{\partial x}\biggl[- \frac{1}{\rho} \frac{\partial \Psi}{\partial x} \biggr] - \frac{\partial }{\partial y} \biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial y} \biggr] \, .
</math>
</td>
</tr>
</table>
<span id="LHS">This is useful to know because</span>, in the Euler equation (see immediately below) we will encounter a term that involves the cross product of the vector, <math>~(\vec\zeta + 2\vec\Omega) </math>, with the rotating-frame-based velocity vector. Appreciating as well that the vector, <math>~\vec\Omega = \boldsymbol{\hat{k}} \Omega_f</math>, only has a nonzero z-component, we recognize that this term may be written as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(\vec\zeta + 2\vec\Omega) \times \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\boldsymbol{\hat{k}} \biggl[ \frac{ (\zeta_z + 2\Omega_f)}{\rho} \biggr] \times \biggl[ \boldsymbol{\hat\imath} \frac{\partial \Psi}{\partial y} - \boldsymbol{\hat\jmath} \frac{\partial \Psi}{\partial x} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{ (\zeta_z + 2\Omega_f)}{\rho} \biggr] \biggl[ \boldsymbol{\hat\jmath} \frac{\partial \Psi}{\partial y} + \boldsymbol{\hat\imath} \frac{\partial \Psi}{\partial x} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{ (\zeta_z + 2\Omega_f)}{\rho} \biggr] \nabla\Psi \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[https://digitalcommons.lsu.edu/gradschool_disstheses/6650/ Saied W. Andalib (1998)], §4.2, p. 83, Eq. (4.13)</td></tr>
</table>

====Euler Equation====
We begin with the,

<div align="center">
Euler Equation<br />
written <font color="#770000">'''in terms of the Vorticity'''</font> and<br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\frac{\partial \bold{u}}{\partial t} + (\boldsymbol\zeta+2{\vec\Omega}_f) \times {\bold{u}}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi + \frac{1}{2}u^2 - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr]</math> .
</div>

Next, we rewrite this expression to incorporate the following three realizations:
<ul>
<li>For a barotropic fluid, the term involving the pressure gradient can be replaced with a term involving the enthalpy via the relation, <math>~\nabla H = \nabla P/\rho</math>.</li>
<li>The expression for the centrifugal potential can be rewritten as, <math>~\tfrac{1}{2}|\vec\Omega_f \times \vec{x}|^2 = \tfrac{1}{2}\Omega_f^2 (x^2 + y^2)</math>.</li>
<li>In steady state, <math>~\partial \bold{u}/\partial t = 0</math>.</li>
</ul>
This means that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ (\boldsymbol\zeta+2{\vec\Omega}_f) \times {\bold{u}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>- \nabla \biggl[H + \Phi_\mathrm{grav} + \frac{1}{2}u^2 - \frac{1}{2}\Omega_f^2 (x^2 + y^2) \biggr] \, .</math>
</td>
</tr>
</table>

If the term on the left-hand-side of this equation can be expressed in terms of the gradient of a scalar function, then it can be readily grouped with all the other terms on the right-hand-side, which already are in the gradient form.

<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
<div align="center">
'''Striving for Gradient Form'''
</div>

As we have already demonstrated [[#LHS|above]], the term on the left-hand-side of the Euler equation can be rewritten as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(\vec\zeta + 2\vec\Omega) \times \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{ (\zeta_z + 2\Omega_f)}{\rho} \biggr] \nabla\Psi \, .</math>
</td>
</tr>
</table>

If the term inside the square brackets on the right-hand-side were a constant — that is, independent of position — then it could immediately be moved inside the gradient operator and we will have accomplished our objective. But, while <math>~\Omega_f</math> ''is'' constant "<font color="orange">… generally the vorticity and density are both functions of <math>~x</math> and <math>~y</math>.</font>" As Andalib has explained, "<font color="orange">The expression … can be cast in the form of a gradient only if</font>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ \frac{ (\zeta_z + 2\Omega_f)}{\rho} \biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~g(\Psi) \, ,</math>
</td>
</tr>
</table>
<font color="orange">where <math>~g</math> is an arbitrary function.</font>" Specifically in this case, the term on the left-hand-side of the Euler equation may be written as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(\vec\zeta + 2\vec\Omega) \times \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\nabla F_B(\Psi) </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{dF_B(\Psi)}{d\Psi} \biggr] \nabla \Psi \, .</math>
</td>
</tr>
</table>
That is, we accomplish our objective by recognizing that the sought-after function, <math>~F_B(\Psi)</math>, is obtained from <math>~g(\Psi)</math> via the relation,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{dF_B(\Psi)}{d\Psi} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~g(\Psi) \, .</math>
</td>
</tr>
</table>

----

For example, try …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~F_B(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~C_0 \Psi + \frac{1}{2} C_1 \Psi^2 </math>
</td>
</tr>
<tr><td align="center" colspan="3">[https://digitalcommons.lsu.edu/gradschool_disstheses/6650/ Saied W. Andalib (1998)], §4.3, p. 85, Eq. (4.22)</td></tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ g(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~C_0 + C_1 \Psi \, .</math>
</td>
</tr>
</table>

This means that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ \frac{ (\zeta_z + 2\Omega_f)}{\rho} \biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~~C_0 + C_1 \Psi</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
2\Omega_f - C_0 \rho
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~~C_1 \rho \Psi - \zeta_z </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~~C_1 \rho \Psi + \frac{\partial }{\partial x}\biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial x} \biggr] + \frac{\partial }{\partial y} \biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial y} \biggr] \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[https://digitalcommons.lsu.edu/gradschool_disstheses/6650/ Saied W. Andalib (1998)], §4.3, p. 85, Eq. (4.24)</td></tr>
</table>

</td></tr></table>

<span id="AndalibBernoulli">Having transformed the left-hand-side term</span> into the gradient of the scalar function, <math>~F_B(\Psi)</math>, the Euler equation can now be written as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\nabla \biggl[H + \Phi_\mathrm{grav} + F_B(\Psi) + \frac{1}{2}u^2 - \frac{1}{2}\Omega_f^2 (x^2 + y^2) \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ 0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~
H + \Phi_\mathrm{grav} + F_B(\Psi) + \frac{1}{2}u^2 - \frac{1}{2}\Omega_f^2 (x^2 + y^2)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>C_B \, , </math>
</td>
</tr>
</table>
where we will refer to <math>~C_B</math> as the Bernoulli constant.

====Strategy====

<font color="red">'''STEP 0:'''</font>  Choose the pair of model-sequence parameters, <math>~(C_0, C_1)</math>, that are associated with the function, <math>~F_B(\Psi)</math>. Hold these fixed during iterations.

<font color="red">'''STEP 1:'''</font>  Guess a density distribution, <math>~\rho(x,y)</math>. For example, pick the equatorial-plane (uniform) density distribution of a Riemann S-type ellipsoid with an equatorial-plane axis-ratio, <math>~b/a</math> and meridional-plane axis-ratio, <math>~c/a</math>; use the same <math>~b/a</math> ratio to define two points on the configuration's surface throughout the iteration cycle.

<font color="red">'''STEP 2:'''</font>  Given <math>~\rho(x,y)</math>, solve the Poisson equation to obtain, <math>~\Phi_\mathrm{grav}(x,y)</math>. In the first iteration, this should exactly match the <math>~A_1, A_2, A_3</math> values associated with the chosen Riemann S-type ellipsoid.

<font color="red">'''STEP 3:'''</font>  Guess a value of <math>~\Omega_f</math> — perhaps the spin-frequency associated with your "initial guess" Riemann ellipsoid — then solve the following two-dimensional, elliptic PDE to obtain <math>~\Psi(x,y)</math> …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
2\Omega_f - C_0 \rho
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~~C_1 \rho \Psi + \frac{\partial }{\partial x}\biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial x} \biggr] + \frac{\partial }{\partial y} \biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial y} \biggr] \, .</math>
</td>
</tr>
</table>
<table border="1" cellpadding="10" width="70%" align="center"><tr><td align="left">
<div align="center">'''Boundary Condition'''</div>

Moving along various rays from the center of the configuration, outward, the surface is determined by the location along each ray where <math>~H(x,y)</math> goes to zero for the first time. We set <math>~\Psi = 0</math> at these various surface locations. At each of these locations, the velocity vector must be tangent to the surface. This requirement also, then, sets the value of <math>~\partial \Psi/\partial y</math> and <math>~\partial \Psi/\partial x</math> at each location.
</td></tr></table>

<font color="red">'''STEP 4:'''</font>  Determine (rotating-frame) velocity from knowledge of <math>~\Psi(x,y)</math> and <math>~\rho(x,y)</math>.
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho}\biggl\{ \boldsymbol{\hat\imath} \biggl[ \frac{\partial \Psi}{\partial y} \biggr]
- \boldsymbol{\hat\jmath} \biggl[ \frac{\partial \Psi}{\partial x}\biggr] \biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ u^2 = \bold{u} \cdot \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho^2}\biggl\{ \biggl[ \frac{\partial \Psi}{\partial y} \biggr]^2
+ \biggl[ \frac{\partial \Psi}{\partial x}\biggr]^2 \biggr\} \, .</math>
</td>
</tr>
</table>

<font color="red">'''STEP 5:'''</font>  Using the "scalar Euler equation,"

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
H + \Phi_\mathrm{grav} + C_0 \Psi + \frac{1}{2} C_1 \Psi^2 + \frac{1}{2}u^2 - \frac{1}{2}\Omega_f^2 (x^2 + y^2)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>C_B \, , </math>
</td>
</tr>
<tr><td align="center" colspan="3">[https://digitalcommons.lsu.edu/gradschool_disstheses/6650/ Saied W. Andalib (1998)], §4.3, p. 85, Eq. (4.23)</td></tr>
</table>
<ul>
<li>Set <math>~H = 0</math> at two different points on the surface of the configuration — usually at <math>~(x,y) = (a,0)</math> and <math>~(x,y) = (0,b)</math> — to determine values of the two constants, <math>~\Omega_f^2</math> and <math>~C_B</math>.</li>
<li>At all points inside the configuration, determine <math>~H(x,y)</math>.</li>
</ul>
<font color="red">'''STEP 6:'''</font>  Use the barotropic equation of state to determine the "new" mass-density distribution from the knowledge of the enthalpy, <math>~H(x,y)</math>.

==Compare==

===Incompressible===
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\lambda \biggl[ \boldsymbol{\hat\imath} \biggl( \frac{a}{b}\biggr) y - \boldsymbol{\hat\jmath} \biggl( \frac{b}{a} \biggr) x \biggr] \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ u^2 \equiv \bold{u}\cdot \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\lambda^2 \biggl[ b^2 \biggl(\frac{x^2}{a^2}\biggr) + a^2 \biggl(\frac{y^2}{b^2}\biggr) \biggr] \, .
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ \boldsymbol{\zeta} \equiv \boldsymbol{\nabla \times}\bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\boldsymbol{\hat\imath} \biggl[ \frac{\partial u_z}{\partial y} - \frac{\partial u_y}{\partial z} \biggr]
+ \boldsymbol{\hat\jmath} \biggl[ \frac{\partial u_x}{\partial z} - \frac{\partial u_z}{\partial x} \biggr]
+ \bold{\hat{k}} \biggl[ \frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\bold{\hat{k}} \biggl[ - \lambda \biggl(\frac{b}{a} + \frac{a}{b}\biggr) \biggr] \, .</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~
\boldsymbol{\zeta} \times \bold{u}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \lambda^2 \biggl(\frac{b}{a} + \frac{a}{b}\biggr) \biggl[ \boldsymbol{\hat\imath} \biggl(\frac{b}{a}\biggr)x + \boldsymbol{\hat\jmath} \biggl(\frac{a}{b}\biggr)y \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \nabla \biggl\{
\frac{\lambda^2(a^2 + b^2)}{2} \biggl[ \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>

Returning to the above-mentioned,

<div align="center">
<font color="#770000">'''Eulerian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\frac{\partial \bold{u}}{\partial t} + (\bold{u}\cdot \nabla) \bold{u} = - \frac{1}{\rho} \nabla P - \nabla \Phi_\mathrm{grav}
- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) - 2{\vec{\Omega}}_f \times \bold{u} \, ,</math> </div>

we next note — as we have done in our [[PGE/Euler#in_terms_of_the_vorticity:|broader discussion of the Euler equation]] — that,
<div align="center">
<math>
(\bold{u} \cdot\nabla)\bold{u} = \frac{1}{2}\nabla(\bold{u} \cdot \bold{u}) - \bold{u} \times(\nabla\times\bold{u})
= \frac{1}{2}\nabla(u^2) + \boldsymbol\zeta \times \bold{u} \, .
</math>
</div>
Making this substitution, we obtain
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\frac{\partial \bold{u}}{\partial t} + \biggl[ \frac{1}{2}\nabla(u^2) + \boldsymbol\zeta \times \bold{u} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{\rho} \nabla P - \nabla \Phi_\mathrm{grav}
- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) - 2{\vec{\Omega}}_f \times \bold{u}
</math>
</td>
</tr>
</table>

it can be shown that, for the velocity fields associated with all Riemann S-type ellipsoids,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~({\vec{v}}_{rot}\cdot \nabla) {\vec{v}}_{rot}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\nabla \biggl[ \frac{1}{2} \lambda^2(x^2 + y^2) \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
+\nabla\biggl[\frac{1}{2} \Omega_f^2 (x^2 + y^2) \biggr] \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~- 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \nabla\biggl[ \Omega_f \lambda\biggl( \frac{b}{a} x^2 + \frac{a}{b}y^2 \biggr) \biggr] \, .
</math>
</td>
</tr>
</table>

As a check,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(\bold{u} \cdot\nabla)\bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}\nabla(u^2) + \boldsymbol\zeta \times \bold{u} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\nabla \biggl\{\frac{\lambda^2}{2} \biggl[ b^2 \biggl(\frac{x^2}{a^2}\biggr) + a^2 \biggl(\frac{y^2}{b^2}\biggr) \biggr]
- \frac{\lambda^2(a^2 + b^2)}{2} \biggl[ \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right" colspan="3">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\nabla \frac{\lambda^2}{2} \biggl\{ -(x^2 + y^2)
\biggr\} \, .
</math>
</td>
</tr>
</table>
Yes! This expression matches the one that appears just a few lines earlier in this discussion.

Now, let's switch the order of terms in the steady-state Euler equation to permit an easier comparison with our attempt to develop the compressible models.
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\cancel{\frac{\partial \bold{u}}{\partial t}} + \biggl[ \frac{1}{2}\nabla(u^2) + \boldsymbol\zeta \times \bold{u} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{\rho} \nabla P - \nabla \Phi_\mathrm{grav}
- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) - 2{\vec{\Omega}}_f \times \bold{u}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~
\Rightarrow~~~ 0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \nabla \biggl[ H + \Phi_\mathrm{grav} + \frac{1}{2} u^2 - \frac{1}{2} \Omega_f^2 (x^2 + y^2) \biggr]
- (\boldsymbol\zeta + 2{\vec{\Omega}}_f )\times \bold{u} \, .
</math>
</td>
</tr>
</table>

===Compressible===

Now in our [[#AndalibBernoulli|above discussion of Andalib's work]], the steady-state form of the Euler equation was formulated as,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\nabla \biggl[H + \Phi_\mathrm{grav} + F_B(\Psi) + \frac{1}{2}u^2 - \frac{1}{2}\Omega_f^2 (x^2 + y^2) \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ 0 \, .</math>
</td>
</tr>
</table>
It is easy to appreciate that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla F_B(\Psi) </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(\boldsymbol\zeta + 2{\vec{\Omega}}_f )\times \bold{u} \, .</math>
</td>
</tr>
</table>
As we have shown, in the case of the incompressible (Riemann S-type ellipsoid) models,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(\boldsymbol\zeta + 2{\vec{\Omega}}_f )\times \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\nabla \biggl\{
\frac{\lambda^2(a^2 + b^2)}{2} \biggl[ \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr]
+
\biggl[ \Omega_f \lambda\biggl( \frac{b}{a} x^2 + \frac{a}{b}y^2 \biggr) \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>

If we attempt to directly relate these two expressions, we must acknowledge that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~F_B(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\lambda^2(a^2 + b^2)}{2} \biggl[ \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr]
+
\biggl[ \Omega_f \lambda\biggl( \frac{b}{a} x^2 + \frac{a}{b}y^2 \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{x^2}{a^2} \biggl[\frac{1}{2}\lambda^2(a^2 + b^2) + \Omega_f \lambda a b \biggr]
+
\frac{y^2}{b^2} \biggl[ \frac{1}{2}\lambda^2(a^2 + b^2) + \Omega_f \lambda a b \biggr] \, .
</math>
</td>
</tr>
</table>

As we have discussed above, Andalib (1998) found that some interesting model sequences could be constructed if he adopted the functional form,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~F_B(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~C_0 \Psi + \frac{1}{2} C_1 \Psi^2 \, .</math>
</td>
</tr>
</table>
Evidently, the incompressible (uniform-density) Riemann S-type ellisoids can be retrieved from our derived compressible-model formalism if we set, <math>~C_1 = 0</math>, and,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Psi(x,y)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{x^2}{a^2} + \frac{y^2}{b^2} \, ,</math>
</td>
</tr>
</table>
with,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~C_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{1}{2}\lambda^2(a^2 + b^2) + \Omega_f \lambda a b \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{\zeta_z}{2}\biggl( \frac{a^2 b^2}{a^2 + b^2}\biggr)\biggl[ 2\Omega_f - \zeta_z \biggr] \, .</math>
</td>
</tr>
</table>

<font color="red">Afterthought:</font>  Because we want <math>~\Psi(x,y)</math> to go to zero at the surface, it likely will be better to set,
<div align="center">
<math>~\Psi \equiv 1 - \biggl[ \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr] \, ,</math>
</div>
then adjust the sign of <math>~C_0</math> and add a constant (zeroth-order term) to the definition of <math>~F_B(\Psi)</math>.

===Trial #1===

Restricting our discussion to nonaxisymmetric, thin disks, let's assume <math>~\rho</math> is uniform throughout the configuration and that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Psi(x,y)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Psi_0 \biggl[1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr) \biggr] \, .
</math>
</td>
</tr>
</table>

This means that,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\frac{ \partial \Psi}{\partial x} = - \Psi_0 \biggl( \frac{2x}{a^2}\biggr)</math>
</td>
<td align="center">
      and,      
</td>
<td align="left">
<math>~\frac{ \partial \Psi}{\partial y} = - \Psi_0 \biggl( \frac{2y}{b^2}\biggr) \, .</math>
</td>
</tr>
</table>
The momentum density vector is governed by the relation,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\rho\bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\boldsymbol{\hat\imath} \biggl[ \frac{\partial \Psi}{\partial y} \biggr]
- \boldsymbol{\hat\jmath} \biggl[ \frac{\partial \Psi}{\partial x}\biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \Psi_0 \biggl\{-
\boldsymbol{\hat\imath} \biggl[ \frac{2y}{b^2} \biggr]
+ \boldsymbol{\hat\jmath} \biggl[ \frac{2x}{a^2}\biggr]
\biggr\} \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[<math>~\Psi</math> has units of "density × length<sup>2</sup> per time"]</td>
</table>
First of all, let's see if the steady-state continuity equation is satisfied:
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla \cdot (\rho \bold{u})</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial (\rho u_x)}{\partial x} + \frac{\partial (\rho u_y)}{\partial y}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\rho \biggl\{
\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\Psi_0 \rho \biggl\{
\frac{\partial }{\partial x}\biggl[ - \frac{2y}{b^2} \biggr] + \frac{\partial }{\partial y}\biggl[ \frac{2x}{a^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, .</math>      <font color="red">Q.E.D.</font>
</td>
</tr>
</table>

Next, let's determine the z-component of the vorticity and the vortensity:
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\zeta_z</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\Psi_0 \biggl\{
\frac{\partial }{\partial x}\biggl[ \frac{2x}{\rho a^2} \biggr] + \frac{\partial }{\partial y}\biggl[ \frac{2y}{\rho b^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2\Psi_0}{\rho} \biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr]
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ \frac{(\zeta_z + 2\Omega_f)}{\rho}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{\rho} \biggl\{
\frac{2\Psi_0}{\rho} \biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr]
+ 2\Omega_f \biggr\}\, .
</math>
</td>
</tr>
</table>
This means that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
(\vec\zeta + 2\vec\Omega) \times \bold{u}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{ \frac{(\zeta_z + 2\Omega_f)}{\rho} \biggr\} \boldsymbol{\hat{k}} \times (\rho \bold{u} )</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\Psi_0 \biggl\{ \frac{(\zeta_z + 2\Omega_f)}{\rho} \biggr\} \boldsymbol{\hat{k}} \times \biggl\{-
\boldsymbol{\hat\imath} \biggl[ \frac{2y}{b^2} \biggr]
+ \boldsymbol{\hat\jmath} \biggl[ \frac{2x}{a^2}\biggr]
\biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-2 \Psi_0 \biggl[ \frac{(\zeta_z + 2\Omega_f)}{\rho} \biggr] \biggl[
\boldsymbol{\hat\imath} \biggl( \frac{x}{a^2}\biggr)
+ \boldsymbol{\hat\jmath} \biggl( \frac{y}{b^2} \biggr)
\biggr] \, .</math>
</td>
</tr>
</table>
But this entire process was designed to ensure that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
(\vec\zeta + 2\vec\Omega) \times \bold{u}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\nabla F_B(\Psi) \, ,</math>
</td>
</tr>
</table>
where, <math>~F_B(\Psi) \equiv C_0 \Psi</math>. Let's see if we get the same expression …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla F_B(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \boldsymbol{\hat\imath} \frac{\partial}{\partial x} + \boldsymbol{\hat\jmath} \frac{\partial}{\partial y} \biggr]C_0 \Psi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
C_0 \Psi_0\biggl[ \boldsymbol{\hat\imath} \frac{\partial}{\partial x} + \boldsymbol{\hat\jmath} \frac{\partial}{\partial y} \biggr]
\biggl[1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- 2C_0 \Psi_0\biggl[ \boldsymbol{\hat\imath} \biggl( \frac{x}{a^2} \biggr) + \boldsymbol{\hat\jmath} \biggl( \frac{y}{b^2} \biggr) \biggr] \, .
</math>
</td>
</tr>
</table>
This is indeed the same expression as above if we set,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~C_0 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{(\zeta_z + 2\Omega_f)}{\rho} \, .</math>
</td>
</tr>
</table>
<font color="red">Hooray!!</font>

Finally, let's make sure that the elliptic PDE "constraint" equation is satisfied.

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
2\Omega_f
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~C_0 \rho+~C_1 \rho \Psi + \frac{\partial }{\partial x}\biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial x} \biggr] + \frac{\partial }{\partial y} \biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial y} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(\zeta_z + 2\Omega_f)+~\cancel{C_1 \rho \Psi} - \frac{\partial }{\partial x}\biggl[\frac{\Psi_0}{\rho} \frac{2x}{a^2} \biggr] - \frac{\partial }{\partial y} \biggl[\frac{\Psi_0}{\rho} \frac{2y}{b^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\zeta_z + 2\Omega_f - \biggl[\frac{\Psi_0}{\rho} \frac{2}{a^2} \biggr] - \biggl[\frac{\Psi_0}{\rho} \frac{2}{b^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\zeta_z + 2\Omega_f - \frac{2\Psi_0}{\rho}\biggl[\frac{1}{a^2} + \frac{1}{b^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2\Omega_f \, .</math>       <font color="red">Yes!</font>
</td>
</tr>
</table>

===Trial #2===

Still restricting our discussion to nonaxisymmetric, thin disks, let's try, <math>~\Psi = \Psi_0 (\rho/\rho_c)^2</math>, and
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\rho_c \biggl\{1 - \biggl[ \frac{y^2}{b^2} + \frac{x^2}{a^2}\biggr]\biggr\} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~\frac{\partial^2 \rho}{\partial x^2} = -\frac{\partial}{\partial x}\biggl\{ \frac{2\rho_c x}{a^2}\biggr\} = - \frac{2\rho_c}{a^2}</math>
</td>
<td align="center">
    and,    
</td>
<td align="left">
<math>~\frac{\partial^2 \rho}{\partial y^2} = - \frac{\partial}{\partial y}\biggl\{ \frac{2\rho_c y}{b^2} \biggr\} = - \frac{2\rho_c}{b^2} \, .</math>
</td>
</tr>
</table>

<table border="1" align="center" width="90%" cellpadding="10"><tr><td align="left">
<font color="red">'''IMPORTANT NOTE''' (by Tohline on 22 September 2020):</font>    As I have come to appreciate today — after studying the relevant sections of both [[Appendix/References#EFE|EFE]] and [[Appendix/References#BT87|BT87]] — this is an example of a heterogeneous density distribution whose gravitational potential has an analytic prescription. As is discussed in a [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Inhomogeneous_Ellipsoids_Leading_to_Ferrers_Potentials| separate chapter]], the potential that it generates is sometimes referred to as a ''Ferrers'' potential, for the exponent, n = 1.

In our [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#GravFor1|accompanying discussion]] we find that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{ \Phi_\mathrm{grav}(\bold{x})}{(-\pi G\rho_c)} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2} I_\mathrm{BT} a_1^2
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr)
~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
~+ \frac{1}{6} \biggl(3A_{11}x^4 + 3A_{22}y^4 + 3A_{33}z^4 \biggr)
\, ,
</math>
</td>
</tr>
</table>

where,
<table border="0" align="center" cellpadding="10" width="80%">
<tr>
<td align="center" width="50%">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="3">for <math>~i \ne j</math></td></tr>
<tr>
<td align="right">
<math>~A_{ij}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~-\frac{A_i-A_j}{(a_i^2 - a_j^2)} </math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">§21, Eq. (107)</font> ]</td></tr>
</table>

</td>
<td align="center" width="50%">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="3">for <math>~i = j</math></td></tr>
<tr>
<td align="right">
<math>~2A_{ii} + \sum_{\ell = 1}^3 A_{i\ell}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{a_i} </math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">§21, Eq. (109)</font> ]</td></tr>
</table>

</td>
</tr>
</table>
More specifically, in the three cases where the indices, <math>~i=j</math>,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3A_{11}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2}{a_1^2} - (A_{12} + A_{13}) \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~3A_{22}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2}{a_2^2} - (A_{21} + A_{23}) \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~3A_{33}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2}{a_3^2} - (A_{31} + A_{32}) \, .
</math>
</td>
</tr>
</table>

</td></tr></table>

This means that,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\frac{ \partial \Psi}{\partial x} = \biggl( \frac{\Psi_0}{\rho_c^2}\biggr) 2\rho \frac{ \partial \rho}{\partial x} = -2\rho \biggl( \frac{2\rho_c x}{a^2} \biggr)\biggl( \frac{\Psi_0}{\rho_c^2}\biggr)</math>
</td>
<td align="center">
      and,      
</td>
<td align="left">
<math>~\frac{ \partial \Psi}{\partial y} = \biggl( \frac{\Psi_0}{\rho_c^2}\biggr)2\rho \frac{ \partial \rho}{\partial y} = -2\rho \biggl( \frac{2\rho_c y}{b^2} \biggr)\biggl( \frac{\Psi_0}{\rho_c^2}\biggr) \, .</math>
</td>
</tr>
</table>

<span id="ConstraintTrial2"> </span>
<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
<div align="center">'''The Elliptic PDE Constraint Equation'''</div>

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
2\Omega_f
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~C_0 \rho+~\cancel{C_1 \rho \Psi} + \frac{\partial }{\partial x}\biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial x} \biggr] + \frac{\partial }{\partial y} \biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial y} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~C_0 \rho
- \frac{\partial }{\partial x}\biggl[ \biggl( \frac{4 x}{a^2} \biggr)\biggl( \frac{\Psi_0}{\rho_c}\biggr) \biggr]
- \frac{\partial }{\partial y} \biggl[ \biggl( \frac{4 y}{b^2} \biggr)\biggl( \frac{\Psi_0}{\rho_c}\biggr) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~C_0 \rho - \frac{4\Psi_0}{\rho_c}\biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr]
</math>
</td>
</tr>
</table>
</td></tr></table>

The momentum density vector is governed by the relation,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\rho\bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\boldsymbol{\hat\imath} \biggl[ \frac{\partial \Psi}{\partial y} \biggr]
- \boldsymbol{\hat\jmath} \biggl[ \frac{\partial \Psi}{\partial x}\biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \boldsymbol{\hat\imath} \biggl[ 2\rho \biggl( \frac{2\rho_c y}{b^2} \biggr)\biggl( \frac{\Psi_0}{\rho_c^2}\biggr) \biggr]
+ \boldsymbol{\hat\jmath} \biggl[ 2\rho \biggl( \frac{2\rho_c x}{a^2} \biggr)\biggl( \frac{\Psi_0}{\rho_c^2}\biggr)\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\rho \biggl\{
- \boldsymbol{\hat\imath} \biggl[ \frac{4\Psi_0 y}{\rho_c b^2} \biggr]
+ \boldsymbol{\hat\jmath} \biggl[ \frac{4\Psi_0 x}{\rho_c a^2} \biggr]
\biggr\}
\, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[<math>~\Psi</math> has units of "density × length<sup>2</sup> per time"]</td>
</table>

As above, let's see if the steady-state continuity equation is satisfied:
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla \cdot (\rho \bold{u})</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial (\rho u_x)}{\partial x} + \frac{\partial (\rho u_y)}{\partial y}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{\partial }{\partial x}\biggl[ \frac{4\Psi_0 y \rho}{\rho_c b^2} \biggr] + \frac{\partial }{\partial y}\biggl[ \frac{4\Psi_0 x \rho }{\rho_c a^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\Psi_0}{\rho_c} \biggl\{
- \frac{y}{b^2} \cdot \frac{\partial \rho}{\partial x} + \frac{ x }{ a^2} \cdot \frac{\partial \rho}{\partial y}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\Psi_0}{\rho_c} \biggl\{
\frac{y}{b^2} \biggl[ \frac{2\rho_c x}{a^2} \biggr] - \frac{ x }{ a^2} \biggl[ \frac{2 \rho_c y}{b^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, .</math>      <font color="red">Yes!</font>
</td>
</tr>
</table>

Next, as above, let's determine the z-component of the vorticity and the vortensity:
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\zeta_z</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{ 4\Psi_0}{\rho_c} \biggl\{
\frac{\partial }{\partial x}\biggl[ \frac{x}{a^2} \biggr] + \frac{\partial }{\partial y}\biggl[ \frac{y}{b^2 } \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{ 4\Psi_0}{\rho_c} \biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr]
\, .
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ \frac{(\zeta_z + 2\Omega_f)}{\rho}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{\rho} \biggl\{
\frac{ 4\Psi_0}{\rho_c} \biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr]
+ 2\Omega_f \biggr\}\, .
</math>
</td>
</tr>
</table>
This means that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
(\vec\zeta + 2\vec\Omega) \times \bold{u}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{ \frac{(\zeta_z + 2\Omega_f)}{\rho} \biggr\} \boldsymbol{\hat{k}} \times (\rho \bold{u} )</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{ 4\Psi_0 (\zeta_z + 2\Omega_f)}{\rho_c} \boldsymbol{\hat{k}} \times
\biggl\{
- \boldsymbol{\hat\imath} \biggl[ \frac{y}{b^2} \biggr]
+ \boldsymbol{\hat\jmath} \biggl[ \frac{x}{a^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{ 4\Psi_0 (\zeta_z + 2\Omega_f)}{\rho_c}
\biggl\{
\boldsymbol{\hat\imath} \biggl[ \frac{x}{a^2} \biggr]
+ \boldsymbol{\hat\jmath} \biggl[ \frac{y}{b^2} \biggr]
\biggr\}
</math>
</td>
</tr>
</table>

Now, let's examine the gradient of <math>~F_B(\Psi)</math>.
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla F_B(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \boldsymbol{\hat\imath} \frac{\partial}{\partial x} + \boldsymbol{\hat\jmath} \frac{\partial}{\partial y} \biggr]C_0 \Psi</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{ C_0 \Psi_0}{\rho_c^2} \biggl[ \boldsymbol{\hat\imath} \frac{\partial \rho^2}{\partial x} + \boldsymbol{\hat\jmath} \frac{\partial \rho^2}{\partial y} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{ 2C_0 \rho \Psi_0}{\rho_c^2} \biggl\{
-\boldsymbol{\hat\imath} \biggl[ \frac{2 \rho_c x}{a^2} \biggr]
- \boldsymbol{\hat\jmath} \biggl[ \frac{2 \rho_c y}{b^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{ 4 C_0 \rho \Psi_0}{\rho_c} \biggl\{
\boldsymbol{\hat\imath} \biggl[ \frac{x}{a^2} \biggr]
+ \boldsymbol{\hat\jmath} \biggl[ \frac{y}{b^2} \biggr]
\biggr\} \, ,
</math>
</td>
</tr>
</table>
which is identical to the immediately preceding expression if we set,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~C_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{(\zeta_z + 2\Omega_f)}{\rho} \, .
</math>
</td>
</tr>
</table>

Continuing with the [[#ConstraintTrial2|above examination of the elliptic PDE "constraint" equation]], we find that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~2\Omega_f</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~C_0 \rho - \zeta_z
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(2\Omega_f + \zeta_z) - \zeta_z
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2\Omega_f \, .</math>      <font color="red">Hooray!</font>
</td>
</tr>
</table>

===Trial #3===

If the density distribution has been specified, then <math>~\Psi</math> is the "stream-function" from which all rotating-frame velocities are determined. Specifically,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho} \biggl[
\boldsymbol{\hat\imath} \frac{\partial \Psi}{\partial y} - \boldsymbol{\hat\jmath} \frac{\partial \Psi}{\partial x}
\biggr] \, .
</math>
</td>
</tr>
</table>
Most importantly, as has been [[#Related_Useful_Expressions|detailed above]], the term on the left-hand-side of the steady-state Euler equation becomes,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(\vec\zeta + 2\vec\Omega) \times \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{ (\zeta_z + 2\Omega_f)}{\rho} \biggr] \nabla\Psi \, ,</math>
</td>
</tr>
<tr><td align="center" colspan="3">[https://digitalcommons.lsu.edu/gradschool_disstheses/6650/ Saied W. Andalib (1998)], §4.2, p. 83, Eq. (4.13)</td></tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\zeta_z </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\frac{\partial }{\partial x}\biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial x} \biggr] - \frac{\partial }{\partial y} \biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial y} \biggr] \, .
</math>
</td>
</tr>
</table>

Still restricting our discussion to infinitesimally thin, nonaxisymmetric disks, let's assume that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\rho_c \biggl[1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr) \biggr] </math>
</td>
</tr>
</table>

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Rightarrow~~~\frac{\partial \rho}{\partial x} = -\biggl( \frac{2\rho_c x}{a^2} \biggr)</math>
</td>
<td align="center">
    and,    
</td>
<td align="left">
<math>~\frac{\partial \rho}{\partial y} = -\biggl( \frac{2\rho_c y}{b^2} \biggr) \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~\frac{\partial^2 \rho}{\partial x^2} = -\frac{\partial}{\partial x}\biggl\{ \frac{2\rho_c x}{a^2}\biggr\} = - \frac{2\rho_c}{a^2}</math>
</td>
<td align="center">
    and,    
</td>
<td align="left">
<math>~\frac{\partial^2 \rho}{\partial y^2} = - \frac{\partial}{\partial y}\biggl\{ \frac{2\rho_c y}{b^2} \biggr\} = - \frac{2\rho_c}{b^2} \, .</math>
</td>
</tr>
</table>
And, let's assume that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Psi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\Psi_0 \biggl( \frac{\rho}{\rho_c} \biggr)^q \, ,</math>
</td>
</tr>
</table>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Rightarrow~~~ \frac{\partial\Psi}{\partial x} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\rho^{q-1}\biggl(\frac{q \Psi_0}{\rho_c^q}\biggr) \frac{\partial\rho}{\partial x}
=
- \rho^{q-1}\biggl(\frac{q \Psi_0}{\rho_c^q}\biggr) \biggl(\frac{2\rho_c x}{a^2} \biggr)
=
- \biggl( \frac{\rho}{\rho_c}\biggr)^{q-1} \biggl(\frac{2q \Psi_0x}{a^2} \biggr) \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
and, similarly,       <math>~\frac{\partial\Psi}{\partial y} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl( \frac{\rho}{\rho_c}\biggr)^{q-1} \biggl(\frac{2q \Psi_0 y}{b^2} \biggr) \, .
</math>
</td>
</tr>
</table>

This means that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~
\frac{1}{\rho} \biggl\{
\boldsymbol{\hat\imath} \biggl[ \biggl( \frac{\rho}{\rho_c}\biggr)^{q-1} \biggl(\frac{2q \Psi_0 y}{b^2} \biggr) \biggr]
- \boldsymbol{\hat\jmath} \biggl[ \biggl( \frac{\rho}{\rho_c}\biggr)^{q-1} \biggl(\frac{2q \Psi_0x}{a^2} \biggr) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~\bold{u}\cdot \bold{u} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho^2} \biggl\{
\biggl[ \biggl( \frac{\rho}{\rho_c}\biggr)^{q-1} \biggl(\frac{2q \Psi_0 y}{b^2} \biggr) \biggr]^2
+ \biggl[ \biggl( \frac{\rho}{\rho_c}\biggr)^{q-1} \biggl(\frac{2q \Psi_0x}{a^2} \biggr) \biggr]^2
\biggr\} \, .
</math>
</td>
</tr>
</table>

It also means that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\zeta_z </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial }{\partial x}\biggl[\biggl( \frac{\rho}{\rho_c}\biggr)^{q-2} \biggl(\frac{2q \Psi_0 x}{\rho_c a^2} \biggr) \biggr]
+
\frac{\partial }{\partial y} \biggl[\biggl( \frac{\rho}{\rho_c}\biggr)^{q-2} \biggl(\frac{2q \Psi_0 y}{\rho_c b^2} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho}{\rho_c}\biggr)^{q-2} \biggl(\frac{2q \Psi_0 }{\rho_c a^2} \biggr)
~+ ~
\biggl(\frac{2q \Psi_0 x}{\rho_c a^2} \biggr) \frac{\partial }{\partial x} \biggl( \frac{\rho}{\rho_c}\biggr)^{q-2}
~+ ~
\biggl( \frac{\rho}{\rho_c}\biggr)^{q-2} \biggl(\frac{2q \Psi_0 }{\rho_c b^2} \biggr)
~+ ~
\biggl(\frac{2q \Psi_0 y}{\rho_c b^2} \biggr) \frac{\partial }{\partial y} \biggl( \frac{\rho}{\rho_c}\biggr)^{q-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho}{\rho_c}\biggr)^{q-2} \biggl(\frac{2q \Psi_0 }{\rho_c } \biggr) \biggl[ \frac{1}{a^2} + \frac{1}{b^2}\biggr]
~+ ~
\biggl(\frac{2q \Psi_0 x}{\rho_c a^2} \biggr) \biggl[ (q-2) \rho_c^{2-q} \rho^{q-3} \frac{\partial \rho}{\partial x}\biggr]
~+ ~
\biggl(\frac{2q \Psi_0 y}{\rho_c b^2} \biggr) \biggl[ (q-2) \rho_c^{2-q} \rho^{q-3} \frac{\partial \rho}{\partial y}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\rho}{\rho_c}\biggr)^{q-2} \biggl(\frac{2q \Psi_0 }{\rho_c } \biggr) \biggl[ \frac{1}{a^2} + \frac{1}{b^2}\biggr]
~- ~
\biggl(\frac{2q \Psi_0 x}{\rho_c^2 a^2} \biggr) \biggl[ (q-2) \biggl(\frac{\rho}{\rho_c}\biggr)^{q-3} \frac{2\rho_c x}{a^2}\biggr]
~- ~
\biggl(\frac{2q \Psi_0 y}{\rho_c^2 b^2} \biggr) \biggl[ (q-2) \biggl(\frac{\rho}{\rho_c}\biggr)^{q-3} \frac{2\rho_c y}{b^2}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{q\Psi_0}{\rho_c} \biggl\{
2\biggl( \frac{\rho}{\rho_c}\biggr)^{q-2} \biggl[ \frac{1}{a^2} + \frac{1}{b^2}\biggr]
~- ~
\biggl(\frac{2 x}{ a^2} \biggr) \biggl[ (q-2) \biggl(\frac{\rho}{\rho_c}\biggr)^{q-3} \frac{2x}{a^2}\biggr]
~- ~
\biggl(\frac{2 y}{ b^2} \biggr) \biggl[ (q-2) \biggl(\frac{\rho}{\rho_c}\biggr)^{q-3} \frac{2 y}{b^2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{q\Psi_0}{\rho_c} \biggl(\frac{\rho}{\rho_c}\biggr)^{q-3} \biggl[
2\biggl( \frac{\rho}{\rho_c}\biggr) \biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)
~- ~ 4(q-2) \biggl(\frac{x^2}{ a^4} \biggr)
~- ~ 4(q-2) \biggl(\frac{y^2}{ b^4} \biggr) \biggr]
</math>
</td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(\vec\zeta + 2\vec\Omega) \times \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \boldsymbol{\hat\imath} \frac{\partial \Psi}{\partial x} + \boldsymbol{\hat\jmath} \frac{\partial \Psi}{\partial y} \biggr]
\biggl\{
\frac{ 2\Omega_f}{\rho_c}\biggl( \frac{\rho}{\rho_c}\biggr)^{-1}
+
\biggl( \frac{\rho}{\rho_c}\biggr)^{-1} \frac{ \zeta_z }{\rho_c}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~
\biggl[ \boldsymbol{\hat\imath} \biggl( \frac{\rho}{\rho_c}\biggr)^{q-1} \biggl(\frac{2q \Psi_0 x}{a^2} \biggr) + \boldsymbol{\hat\jmath} \biggl( \frac{\rho}{\rho_c}\biggr)^{q-1} \biggl(\frac{2q \Psi_0 y}{b^2} \biggr) \biggr]
\biggl\{
\frac{ 2\Omega_f}{\rho_c}\biggl( \frac{\rho}{\rho_c}\biggr)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+
\frac{q\Psi_0}{\rho_c^2} \biggl(\frac{\rho}{\rho_c}\biggr)^{q-4} \biggl[
2\biggl( \frac{\rho}{\rho_c}\biggr) \biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)
~- ~ 4(q-2) \biggl(\frac{x^2}{ a^4} \biggr)
~- ~ 4(q-2) \biggl(\frac{y^2}{ b^4} \biggr) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~2q\Psi_0
\biggl[ \boldsymbol{\hat\imath} \biggl(\frac{x}{a^2} \biggr) + \boldsymbol{\hat\jmath} \biggl(\frac{y}{b^2} \biggr) \biggr]
\biggl( \frac{\rho}{\rho_c}\biggr)^{q-2} \biggl\{
\frac{ 2\Omega_f}{\rho_c}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+
\frac{q\Psi_0}{\rho_c^2} \biggl(\frac{\rho}{\rho_c}\biggr)^{q-3} \biggl[
2\biggl( \frac{\rho}{\rho_c}\biggr) \biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)
~- ~ 4(q-2) \biggl(\frac{x^2}{ a^4} \biggr)
~- ~ 4(q-2) \biggl(\frac{y^2}{ b^4} \biggr) \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>

====Exponent q = 2====
Notice that, if <math>~q = 2</math>,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ (\vec\zeta + 2\vec\Omega) \times \bold{u} \biggr]_{q=2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\frac{8\Psi_0}{\rho_c}\biggl\{ \Omega_f + \frac{2\Psi_0}{\rho_c} \biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr) \biggr\}
\biggl[ \boldsymbol{\hat\imath} \biggl(\frac{x}{a^2} \biggr) + \boldsymbol{\hat\jmath} \biggl(\frac{y}{b^2} \biggr) \biggr]
\, .
</math>
</td>
</tr>
</table>

Now, if we choose a function,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~F_B(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{D_1}{2} \Psi^{1 / 2}
= \frac{D_1}{2} \Psi_0^{1 / 2} \biggl(\frac{\rho}{\rho_c}\biggr)
= \frac{D_1}{2} \Psi_0^{1 / 2} \biggl[1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr) \biggr]
\, ,</math>
</td>
</tr>
</table>
we obtain,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla F_B(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\frac{D_1}{2} \Psi_0^{1 / 2} \cdot \nabla \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~D_1 \Psi_0^{1 / 2}
\biggl[ \boldsymbol{\hat\imath} \biggl(\frac{x}{a^2} \biggr) + \boldsymbol{\hat\jmath} \biggl(\frac{y}{b^2} \biggr) \biggr]
\, .
</math>
</td>
</tr>
</table>
This is consistent with the elliptic PDE constraint if,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~D_1 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8\Psi_0^{1 / 2}}{\rho_c}\biggl\{ \Omega_f + \frac{2\Psi_0}{\rho_c} \biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr) \biggr\} \, .</math>
</td>
</tr>
</table>

Also if <math>~q = 2</math>, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho} \biggl\{
\boldsymbol{\hat\imath} \frac{\partial \Psi}{\partial y} - \boldsymbol{\hat\jmath} \frac{\partial \Psi}{\partial x}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho} \biggl\{
-\boldsymbol{\hat\imath} \biggl( \frac{\rho}{\rho_c}\biggr) \biggl(\frac{2q \Psi_0 y}{b^2} \biggr)
+
\boldsymbol{\hat\jmath} \biggl( \frac{\rho}{\rho_c}\biggr) \biggl(\frac{2q \Psi_0 x}{a^2} \biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
-\boldsymbol{\hat\imath} \biggl(\frac{4 \Psi_0 y}{\rho_c b^2} \biggr)
+
\boldsymbol{\hat\jmath} \biggl(\frac{4 \Psi_0 x}{\rho_c a^2} \biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \bold{u}\cdot \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{4 \Psi_0 y}{\rho_c b^2} \biggr)^2
+
\biggl(\frac{4 \Psi_0 x}{\rho_c a^2} \biggr)^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{4 \Psi_0}{\rho_c}\biggr)^2 \biggl[ \frac{x^2}{a^4} + \frac{y^2}{b^4} \biggr]
\, .
</math>
</td>
</tr>
</table>

Keep in mind that, [[#AndalibBernoulli|as discussed above]], we are trying to satisfy the scalar Bernoulli relation,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~C_B - H - \Phi_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
F_B(\Psi) + \frac{1}{2}u^2 - \frac{1}{2}\Omega_f^2 (x^2 + y^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{4\Psi_0}{\rho_c}\biggl\{ \Omega_f + \frac{2\Psi_0}{\rho_c} \biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr) \biggr\}\biggl[1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr) \biggr]
+ \frac{1}{2}\biggl( \frac{4 \Psi_0}{\rho_c}\biggr)^2 \biggl[ \frac{x^2}{a^4} + \frac{y^2}{b^4} \biggr]
- \frac{1}{2}\Omega_f^2 (x^2 + y^2) \, .
</math>
</td>
</tr>
</table>
The right-hand-side of this expression does not appear to be rich enough to balance the gravitational potential (on the left-hand-side) which, as [[#Trial_.232|detailed above]], contains <math>~x^2 y^2</math> and <math>~x^4</math> and <math>~y^4</math> terms.

====Exponent q = 3====
Alternatively, if <math>~q = 3</math>,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ (\vec\zeta + 2\vec\Omega) \times \bold{u} \biggr]_{q=3}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~6\Psi_0
\biggl[ \boldsymbol{\hat\imath} \biggl(\frac{x}{a^2} \biggr) + \boldsymbol{\hat\jmath} \biggl(\frac{y}{b^2} \biggr) \biggr]
\biggl( \frac{\rho}{\rho_c}\biggr) \biggl\{
\frac{ 2\Omega_f}{\rho_c}
+
\frac{3\Psi_0}{\rho_c^2} \biggl[
2\biggl( \frac{\rho}{\rho_c}\biggr) \biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)
~- ~ 4 \biggl(\frac{x^2}{ a^4} \biggr)
~- ~ 4 \biggl(\frac{y^2}{ b^4} \biggr) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~ 6\Psi_0
\biggl\{
\frac{ 2\Omega_f}{\rho_c} \biggl( \frac{\rho}{\rho_c}\biggr)
-
\frac{12\Psi_0}{\rho_c^2} \biggl[
\frac{x^2}{ a^4}
+ \frac{y^2}{ b^4} \biggr]\biggl( \frac{\rho}{\rho_c}\biggr)
+
\frac{6\Psi_0}{\rho_c^2}
\biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)
\biggl( \frac{\rho}{\rho_c}\biggr)^2
\biggr\} \boldsymbol{\hat{f}} \, ,
</math>
</td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\boldsymbol{\hat{f}}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\biggl[ \boldsymbol{\hat\imath} \biggl(\frac{x}{a^2} \biggr) + \boldsymbol{\hat\jmath} \biggl(\frac{y}{b^2} \biggr) \biggr] \, .
</math>
</td>
</tr>
</table>

If we choose a function,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~F_B(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
D_1\Psi^{2/3} + D_2 \Psi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
D_1\Psi_0^{2/3} \biggl(\frac{\rho}{\rho_c}\biggr)^2
+ D_2 \Psi_0 \biggl(\frac{\rho}{\rho_c}\biggr)^3 \, ,
</math>
</td>
</tr>
</table>

we obtain,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla F_B(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\biggl[ 2D_1 \Psi_0^{2/3} \biggl(\frac{\rho}{\rho_c}\biggr) + 3D_2\Psi_0 \biggl( \frac{\rho}{\rho_c}\biggr)^2\biggr]
\cdot \nabla \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\biggl\{
2D_1 \Psi_0^{2/3} \biggl[1 - \biggl(\frac{x^2}{a^2} + \frac{y^2}{b^2}\biggr) \biggr]
+ 3D_2\Psi_0 \biggl[1 - \biggl(\frac{x^2}{a^2} + \frac{y^2}{b^2}\biggr) \biggr]^2
\biggr\}
\cdot \biggl[ \boldsymbol{\hat\imath} \biggl(\frac{2x}{a^2} \biggr) + \boldsymbol{\hat\jmath} \biggl(\frac{2y}{b^2} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\biggl\{
2D_1 \Psi_0^{2/3} \biggl[1 - \biggl(\frac{x^2}{a^2} + \frac{y^2}{b^2}\biggr) \biggr]
+ 3D_2\Psi_0 \biggl[1 - 2\biggl(\frac{x^2}{a^2} + \frac{y^2}{b^2}\biggr) + \biggl(\frac{x^2}{a^2} + \frac{y^2}{b^2}\biggr)^2 \biggr]
\biggr\}
\cdot \biggl[ \boldsymbol{\hat\imath} \biggl(\frac{2x}{a^2} \biggr) + \boldsymbol{\hat\jmath} \biggl(\frac{2y}{b^2} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\biggl\{ \biggl(2D_1 \Psi_0^{2/3} + 3D_2\Psi_0 \biggr) -
(2D_1 \Psi_0^{2/3} +6D_2\Psi_0)\biggl(\frac{x^2}{a^2} + \frac{y^2}{b^2}\biggr)
+ 3D_2\Psi_0 \biggl(\frac{x^2}{a^2} + \frac{y^2}{b^2}\biggr)^2
\biggr\}
\boldsymbol{\hat{f}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\biggl\{ \biggl(2D_1 \Psi_0^{2/3} + 3D_2\Psi_0 \biggr) -
(2D_1 \Psi_0^{2/3} +6D_2\Psi_0)\biggl(\frac{x^2}{a^2} + \frac{y^2}{b^2}\biggr)
+ 3D_2\Psi_0
\biggl[ \frac{x^4}{a^4} + 2\biggl( \frac{x^2 y^2}{a^2 b^2} \biggr) + \frac{y^4}{b^4}\biggr]
\biggr\}
\boldsymbol{\hat{f}} \, .
</math>
</td>
</tr>
</table>

----

Let's reorganize and expand the terms in both of these expressions in order to ascertain whether or not they can be matched. First …

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ (\vec\zeta + 2\vec\Omega) \times \bold{u} \biggr]_{q=3}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~ 6\Psi_0
\biggl\{
\frac{ 2\Omega_f}{\rho_c} \biggl[ 1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr) \biggr]
-
\frac{12\Psi_0}{\rho_c^2} \biggl[
\frac{x^2}{ a^4}
+ \frac{y^2}{ b^4} \biggr] \biggl[ 1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+
\frac{6\Psi_0}{\rho_c^2}
\biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)
\biggl[ 1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr) \biggr]^2
\biggr\} \boldsymbol{\hat{f}} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~ 6\Psi_0
\biggl\{
\frac{ 2\Omega_f}{\rho_c} - \frac{ 2\Omega_f}{\rho_c} \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr)
-
\frac{12\Psi_0}{\rho_c^2} \biggl[ \frac{x^2}{ a^4} + \frac{y^2}{ b^4} \biggr]
+
\frac{12\Psi_0}{\rho_c^2} \biggl[ \frac{x^2}{ a^4} + \frac{y^2}{ b^4} \biggr] \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+
\frac{6\Psi_0}{\rho_c^2}
\biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)
\biggl[ 1 - 2\biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr) + \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr)^2 \biggr]
\biggr\} \boldsymbol{\hat{f}} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~ 6\Psi_0
\biggl\{
\biggl[ \frac{ 2\Omega_f}{\rho_c} +\frac{6\Psi_0}{\rho_c^2} \cdot \frac{(a^2 + b^2)}{a^2b^2} \biggr]
- \biggl[ \frac{ 2\Omega_f}{\rho_c} + \frac{12\Psi_0}{\rho_c^2}\biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)\biggr]\biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
-
\frac{12\Psi_0}{\rho_c^2} \biggl( \frac{x^2}{ a^4} + \frac{y^2}{ b^4} \biggr)
+
\frac{12\Psi_0}{\rho_c^2}
\biggl( \frac{x^4}{ a^6} + \frac{x^2 y^2}{ a^4 b^2} + \frac{x^2 y^2}{ a^2 b^4} + \frac{y^4}{ b^6}\biggr)
+ \frac{6\Psi_0}{\rho_c^2}\biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)\biggl( \frac{x^4}{a^4} + \frac{2x^2 y^2}{a^2 b^2}+ \frac{y^4}{b^4} \biggr)
\biggr\} \boldsymbol{\hat{f}} \, ,
</math>
</td>
</tr>
</table>

In order for the zeroth-order terms to match, we need,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~2D_1 \Psi_0^{2/3} + 3D_2\Psi_0 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
6\Psi_0 \biggl\{ \frac{ 2\Omega_f}{\rho_c} +\frac{6\Psi_0}{\rho_c^2} \cdot \frac{(a^2 + b^2)}{a^2b^2} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ 2D_1 \Psi_0^{2/3} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
3\Psi_0 \biggl\{ \frac{ 4\Omega_f \rho_c a^2 b^2}{\rho_c^2 a^2 b^2} +\frac{12\Psi_0(a^2 + b^2)}{\rho_c^2 a^2 b^2} \biggr\}
- 3D_2\Psi_0
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
3\Psi_0 \biggl\{ \frac{ 4\Omega_f \rho_c a^2 b^2}{\rho_c^2 a^2 b^2} +\frac{12\Psi_0(a^2 + b^2)}{\rho_c^2 a^2 b^2} -D_2\biggr\} \, .
</math>
</td>
</tr>
</table>
Then we also need,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ 6D_2\Psi_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~6\Psi_0 \biggl[ \frac{ 2\Omega_f}{\rho_c} + \frac{12\Psi_0}{\rho_c^2}\biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)\biggr]
- 2D_1 \Psi_0^{2/3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~6\Psi_0 \biggl[ \frac{ 2\Omega_f}{\rho_c} + \frac{12\Psi_0}{\rho_c^2}\biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)\biggr]
-
3\Psi_0 \biggl\{ \frac{ 4\Omega_f \rho_c a^2 b^2}{\rho_c^2 a^2 b^2} +\frac{12\Psi_0(a^2 + b^2)}{\rho_c^2 a^2 b^2} -D_2\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ 3D_2\Psi_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~6\Psi_0 \biggl[ \frac{ 2\Omega_f \rho_c a^2b^2}{\rho_c^2 a^2 b^2} + \frac{12\Psi_0}{\rho_c^2}\frac{(a^2 + b^2)}{a^2b^2} \biggr]
-
6\Psi_0 \biggl[ \frac{ 2\Omega_f \rho_c a^2 b^2}{\rho_c^2 a^2 b^2} +\frac{6\Psi_0(a^2 + b^2)}{\rho_c^2 a^2 b^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~6\Psi_0 \biggl[ \frac{6\Psi_0(a^2 + b^2)}{\rho_c^2a^2b^2}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ D_2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{12\Psi_0(a^2 + b^2)}{\rho_c^2a^2b^2} \, .
</math>
</td>
</tr>
</table>

Finally, we need,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3D_2\Psi_0
\biggl[ \frac{x^4}{a^4} + 2\biggl( \frac{x^2 y^2}{a^2 b^2} \biggr) + \frac{y^4}{b^4}\biggr] \cdot \frac{1}{6\Psi_0}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-
\frac{12\Psi_0}{\rho_c^2} \biggl( \frac{x^2}{ a^4} + \frac{y^2}{ b^4} \biggr)
+
\frac{12\Psi_0}{\rho_c^2}
\biggl( \frac{x^4}{ a^6} + \frac{x^2 y^2}{ a^4 b^2} + \frac{x^2 y^2}{ a^2 b^4} + \frac{y^4}{ b^6}\biggr)
+ \frac{6\Psi_0}{\rho_c^2}\biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)\biggl( \frac{x^4}{a^4} + \frac{2x^2 y^2}{a^2 b^2}+ \frac{y^4}{b^4} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{(a^2 + b^2)}{a^2b^2}
\biggl[ \frac{x^4}{a^4} + 2\biggl( \frac{x^2 y^2}{a^2 b^2} \biggr) + \frac{y^4}{b^4}\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- 2 \biggl( \frac{x^2}{ a^4} + \frac{y^2}{ b^4} \biggr)
+ 2 \biggl( \frac{x^4}{ a^6} + \frac{x^2 y^2}{ a^4 b^2} + \frac{x^2 y^2}{ a^2 b^4} + \frac{y^4}{ b^6}\biggr)
+ \frac{(a^2 + b^2)}{a^2b^2} \biggl( \frac{x^4}{a^4} + \frac{2x^2 y^2}{a^2 b^2}+ \frac{y^4}{b^4} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\biggl( \frac{x^2}{ a^4} + \frac{y^2}{ b^4} \biggr) </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{x^4}{ a^6} + \frac{x^2 y^2}{ a^4 b^2} + \frac{x^2 y^2}{ a^2 b^4} + \frac{y^4}{ b^6}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\frac{x^2}{ a^4} + \frac{y^2}{ b^4} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{x^4}{ a^6} + \frac{y^4}{ b^6} + \frac{x^2 y^2(a^2 + b^2) }{a^4 b^4}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
x^4 b^6 + y^4 a^6
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
a^2 b^2 [x^2 b^4 - (a^2+b^2)x^2 y^2 + y^2a^4]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
a^2 b^2 [x^4 b^4 - (a^2+b^2)x^2 y^2 + y^4a^4]
+a^2b^2[x^2b^4 + y^2a^4]
-a^2b^2[x^4b^4 + y^4a^4]
</math>
</td>
</tr>
</table>

----

Keeping in mind that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Psi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\Psi_0 \biggl( \frac{\rho}{\rho_c} \biggr)^q \, ,</math>
</td>
</tr>
</table>
and that, after setting <math>~q = 3</math>, we have chosen,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~F_B(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
D_1\Psi^{2/3} + D_2 \Psi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
D_1\Psi_0^{2/3} \biggl(\frac{\rho}{\rho_c}\biggr)^2
+ D_2 \Psi_0 \biggl(\frac{\rho}{\rho_c}\biggr)^3 \, ,
</math>
</td>
</tr>
</table>

let's try again.

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ (\vec\zeta + 2\vec\Omega) \times \bold{u} \biggr]_{q=3} - \nabla F_B(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~ 6\Psi_0
\biggl\{
\frac{ 2\Omega_f}{\rho_c} \biggl( \frac{\rho}{\rho_c}\biggr)
-
\frac{12\Psi_0}{\rho_c^2} \biggl[
\frac{x^2}{ a^4}
+ \frac{y^2}{ b^4} \biggr]\biggl( \frac{\rho}{\rho_c}\biggr)
+
\frac{6\Psi_0}{\rho_c^2}
\biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)
\biggl( \frac{\rho}{\rho_c}\biggr)^2
\biggr\} \cdot \boldsymbol{\hat{f}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+~\biggl[ 2D_1 \Psi_0^{2/3} \biggl(\frac{\rho}{\rho_c}\biggr) + 3D_2\Psi_0 \biggl( \frac{\rho}{\rho_c}\biggr)^2\biggr]
\cdot \nabla \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~ 3\Psi_0
\biggl\{
\frac{ 2\Omega_f}{\rho_c}
-
\frac{12\Psi_0}{\rho_c^2} \biggl[
\frac{x^2}{ a^4}
+ \frac{y^2}{ b^4} \biggr]
+
\frac{6\Psi_0}{\rho_c^2}
\biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)
\biggl( \frac{\rho}{\rho_c}\biggr)
\biggr\} \cdot 2\biggl(\frac{\rho}{\rho_c}\biggr)\boldsymbol{\hat{f}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+~\biggl[ 2D_1 \Psi_0^{2/3} + 3D_2\Psi_0 \biggl( \frac{\rho}{\rho_c}\biggr)\biggr]
\cdot 2\biggl(\frac{\rho}{\rho_c}\biggr)\boldsymbol{\hat{f}} \, .
</math>
</td>
</tr>
</table>
Now, set …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~2D_1 \Psi_0^{2/3}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{6\Psi_0 \Omega_f}{\rho_c}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ D_1 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3\Psi_0^{1 / 3} \Omega_f}{\rho_c} \, ;</math>
</td>
</tr>
</table>
and, set …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3D_2 \Psi_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{18 \Psi_0^2(a^2 + b^2)}{\rho_c^2 a^2 b^2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ D_2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{6 \Psi_0(a^2 + b^2)}{\rho_c^2 a^2 b^2} \, .
</math>
</td>
</tr>
</table>
We then have,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ (\vec\zeta + 2\vec\Omega) \times \bold{u} \biggr]_{q=3} - \nabla F_B(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~ 3\Psi_0
\biggl\{
-
\frac{12\Psi_0}{\rho_c^2} \biggl[
\frac{x^2}{ a^4}
+ \frac{y^2}{ b^4} \biggr]
\biggr\} \cdot 2\biggl(\frac{\rho}{\rho_c}\biggr)\boldsymbol{\hat{f}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{72\Psi_0^2}{\rho_c^2} \biggl[
\frac{x^2}{ a^4}
+ \frac{y^2}{ b^4} \biggr]
\cdot \biggl(\frac{\rho}{\rho_c}\biggr)\boldsymbol{\hat{f}} \, .
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(\bold{u}\cdot \bold{u}) \nabla \biggl(\frac{\rho}{\rho_c}\biggr) \, .
</math>
</td>
</tr>
</table>
<font color="red">'''VERY INTERESTING!''' (29 September 2020)</font>

====Exponent q = 4====

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ (\vec\zeta + 2\vec\Omega) \times \bold{u} \biggr]_{q=4}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~8\Psi_0
\biggl[ \boldsymbol{\hat\imath} \biggl(\frac{x}{a^2} \biggr) + \boldsymbol{\hat\jmath} \biggl(\frac{y}{b^2} \biggr) \biggr]
\biggl( \frac{\rho}{\rho_c}\biggr)^{2} \biggl\{
\frac{ 2\Omega_f}{\rho_c}
+
\frac{8\Psi_0}{\rho_c^2} \biggl(\frac{\rho}{\rho_c}\biggr) \biggl[
\biggl( \frac{\rho}{\rho_c}\biggr) \biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)
~- ~ 4 \biggl(\frac{x^2}{ a^4} \biggr)
~- ~ 4 \biggl(\frac{y^2}{ b^4} \biggr) \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>

===Trial #4===
We begin with the,

<div align="center">
Euler Equation<br />
written <font color="#770000">'''in terms of the Vorticity'''</font> and<br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\frac{\partial \bold{u}}{\partial t} + (\boldsymbol\zeta+2{\vec\Omega}_f) \times {\bold{u}}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi + \frac{1}{2}u^2 - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr]</math> .
</div>

Next, we rewrite this expression to incorporate the following three realizations:
<ul>
<li>For a barotropic fluid, the term involving the pressure gradient can be replaced with a term involving the enthalpy via the relation, <math>~\nabla H = \nabla P/\rho</math>.</li>
<li>The expression for the centrifugal potential can be rewritten as, <math>~\tfrac{1}{2}|\vec\Omega_f \times \vec{x}|^2 = \tfrac{1}{2}\Omega_f^2 (x^2 + y^2)</math>.</li>
<li>In steady state, <math>~\partial \bold{u}/\partial t = 0</math>.</li>
</ul>
This means that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ (\boldsymbol\zeta+2{\vec\Omega}_f) \times {\bold{u}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>- \nabla \biggl[H + \Phi_\mathrm{grav} + \frac{1}{2}u^2 - \frac{1}{2}\Omega_f^2 (x^2 + y^2) \biggr] \, .</math>
</td>
</tr>
</table>

If the term on the left-hand-side of this equation can be expressed in terms of the gradient of a scalar function, then it can be readily grouped with all the other terms on the right-hand-side, which already are in the gradient form.

Building on the insight that we have gained from the [[#Exponent_q_.3D_3|above examination of systems for which the exponent, q = 3]], let's change the <math>~\tfrac{1}{2}\nabla u^2</math> term on the RHS to <math>~\tfrac{1}{2}\nabla (\rho u)^2</math> then reexamine the LHS.
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ [ (\boldsymbol\zeta+2{\vec\Omega}_f) \times {\bold{u}} ]_{q=3}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>- \nabla \biggl[H + \Phi_\mathrm{grav} - \frac{1}{2}\Omega_f^2 (x^2 + y^2) \biggr] - \nabla \biggl[\frac{1}{2}u^2\biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \nabla \biggl[H + \Phi_\mathrm{grav} - \frac{1}{2}\Omega_f^2 (x^2 + y^2) \biggr]
- \biggl(\frac{\rho}{\rho_c}\biggr)^{-2} \biggl\{ \biggl(\frac{\rho}{\rho_c}\biggr)^{2} \nabla \biggl[\frac{1}{2}u^2\biggr]\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \nabla \biggl[H + \Phi_\mathrm{grav} - \frac{1}{2}\Omega_f^2 (x^2 + y^2) \biggr]
- \biggl(\frac{\rho}{\rho_c}\biggr)^{-2} \biggl\{
\nabla \biggl[\frac{1}{2}\biggl(\frac{\rho}{\rho_c}\biggr)^{2} u^2\biggr]
- \frac{1}{2}u^2\nabla \biggl[\biggl(\frac{\rho}{\rho_c}\biggr)^{2}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ [ (\boldsymbol\zeta+2{\vec\Omega}_f) \times {\bold{u}} ]_{q=3} - \nabla F_B</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
- \nabla \biggl[H + \Phi_\mathrm{grav} + F_B - \frac{1}{2}\Omega_f^2 (x^2 + y^2) \biggr]
- \biggl(\frac{\rho}{\rho_c}\biggr)^{-2} \biggl\{
\nabla \biggl[\frac{1}{2}\biggl(\frac{\rho}{\rho_c}\biggr)^{2} u^2\biggr]
- \frac{1}{2}u^2\nabla \biggl[\biggl(\frac{\rho}{\rho_c}\biggr)^{2}\biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>
Now, rewriting the LHS gives,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~[ (\boldsymbol\zeta+2{\vec\Omega}_f) \times {\bold{u}} ]_{q=3} - \nabla F_B</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~ 3\Psi_0
\biggl\{
\frac{ 2\Omega_f}{\rho_c}
-
\frac{12\Psi_0}{\rho_c^2} \biggl[
\frac{x^2}{ a^4}
+ \frac{y^2}{ b^4} \biggr]
+
\frac{6\Psi_0}{\rho_c^2}
\biggl( \frac{1}{a^2} + \frac{1}{b^2}\biggr)
\biggl( \frac{\rho}{\rho_c}\biggr)
\biggr\} \cdot 2\biggl( \frac{\rho}{\rho_c}\biggr)\boldsymbol{\hat{f}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+~\biggl\{
2D_1 \Psi_0^{2/3}
+ 3D_2\Psi_0 \biggl( \frac{\rho}{\rho_c}\biggr)
\biggr\} \cdot 2\biggl( \frac{\rho}{\rho_c}\biggr)\boldsymbol{\hat{f}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3\Psi_0
\biggl\{
\frac{12\Psi_0}{\rho_c^2} \biggl[
\frac{x^2}{ a^4}
+ \frac{y^2}{ b^4} \biggr]
\biggr\} \cdot 2\biggl( \frac{\rho}{\rho_c}\biggr)\boldsymbol{\hat{f}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~
\frac{36\Psi_0^2}{\rho_c^2} \biggl[
\frac{x^2}{ a^4}
+ \frac{y^2}{ b^4} \biggr]
\cdot \nabla \biggl(\frac{\rho}{\rho_c}\biggr)^2 \, ,
</math>
</td>
</tr>
</table>
where we have set,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~D_1 = \frac{3\Psi_0^{1 / 3} \Omega_f}{\rho_c}</math>
</td>
<td align="center">
      and,      
</td>
<td align="left">
<math>~D_2 = \frac{6\Psi_0 (a^2 + b^2)}{\rho_c^2 a^2 b^2} \, .</math>
</td>
</tr>
</table>

Notice that when the exponent, <math>~q=3</math>, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~[ \bold{u}\cdot \bold{u} ]_{q=3} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho^2} \biggl\{
\biggl[ \biggl( \frac{\rho}{\rho_c}\biggr)^{q-1} \biggl(\frac{2q \Psi_0 y}{b^2} \biggr) \biggr]^2
+ \biggl[ \biggl( \frac{\rho}{\rho_c}\biggr)^{q-1} \biggl(\frac{2q \Psi_0x}{a^2} \biggr) \biggr]^2
\biggr\}_{q=3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho^2} \biggl\{
\biggl[ \biggl( \frac{\rho}{\rho_c}\biggr)^{2} \biggl(\frac{6 \Psi_0 y}{b^2} \biggr) \biggr]^2
+ \biggl[ \biggl( \frac{\rho}{\rho_c}\biggr)^{2} \biggl(\frac{6 \Psi_0x}{a^2} \biggr) \biggr]^2
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{36\Psi_0^2}{\rho_c^2}
\biggl( \frac{\rho}{\rho_c}\biggr)^2
\biggl[ \frac{x^2}{a^4} + \frac{ y^2}{b^4} \biggr] \, .
</math>
</td>
</tr>
</table>

Hence,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~[ (\boldsymbol\zeta+2{\vec\Omega}_f) \times {\bold{u}} ]_{q=3} - \nabla F_B</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~
\biggl( \frac{\rho}{\rho_c}\biggr)^{-2} u^2
\cdot \nabla \biggl(\frac{\rho}{\rho_c}\biggr)^2 \, .
</math>
</td>
</tr>
</table>

===Trial#5===

Let's return to the above-mentioned,

<div align="center">
<font color="#770000">'''Eulerian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\frac{\partial \bold{u}}{\partial t} + (\bold{u}\cdot \nabla) \bold{u} = - \frac{1}{\rho} \nabla P - \nabla \Phi_\mathrm{grav}
- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) - 2{\vec{\Omega}}_f \times \bold{u} \, .</math> </div>

In steady state, this can be rewritten as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(\bold{u}\cdot \nabla) \bold{u} + 2{\vec{\Omega}}_f \times \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \nabla \biggl[H + \Phi_\mathrm{grav} - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr] \, .
</math>
</td>
</tr>
</table>
Let's focus on the left-hand-side, which is expressed entirely in terms of the rotating-frame velocity, <math>~\bold{u}</math>, and the (constant) angular frequency of rotation of the coordinate frame, <math>~\Omega_f</math>.
Rewriting the LHS, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
'''LHS:'''     <math>~2{\vec{\Omega}}_f \times \bold{u} + (\bold{u}\cdot \nabla) \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2\Omega_f \biggl[ \boldsymbol{\hat\jmath} u_x - \boldsymbol{\hat\imath} u_y\biggr]
+
\biggl[ u_x \frac{\partial}{\partial x} + u_y \frac{\partial}{\partial y} \biggr] \biggl[\boldsymbol{\hat\imath} u_x + \boldsymbol{\hat\jmath} u_y \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2\Omega_f \biggl[ \boldsymbol{\hat\jmath} u_x - \boldsymbol{\hat\imath} u_y\biggr]
+
\boldsymbol{\hat\imath}\biggl[ u_x \frac{\partial u_x}{\partial x} + u_y \frac{\partial u_x}{\partial y} \biggr]
+
\boldsymbol{\hat\jmath} \biggl[ u_x \frac{\partial u_y}{\partial x} + u_y \frac{\partial u_y}{\partial y} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2\Omega_f \biggl[ \boldsymbol{\hat\jmath} u_x - \boldsymbol{\hat\imath} u_y\biggr]
+ \biggl[\boldsymbol{\hat\imath} u_x \frac{\partial u_x}{\partial x} + \boldsymbol{\hat\jmath} u_y \frac{\partial u_y}{\partial y} \biggr]
+
\biggl[ \boldsymbol{\hat\imath}u_y \frac{\partial u_x}{\partial y}
+
\boldsymbol{\hat\jmath} u_x \frac{\partial u_y}{\partial x} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2\Omega_f \biggl[ \boldsymbol{\hat\jmath} u_x - \boldsymbol{\hat\imath} u_y\biggr]
+ \frac{1}{2}\biggl\{ \biggl[\boldsymbol{\hat\imath} \frac{\partial u_x^2}{\partial x} + \boldsymbol{\hat\jmath} \frac{\partial u_y^2}{\partial y} \biggr]
+ \biggl[\boldsymbol{\hat\imath} \frac{\partial u_y^2}{\partial x} + \boldsymbol{\hat\jmath} \frac{\partial u_x^2}{\partial y} \biggr] \biggr\}
+
\biggl[ \boldsymbol{\hat\imath}u_y \frac{\partial u_x}{\partial y}
+
\boldsymbol{\hat\jmath} u_x \frac{\partial u_y}{\partial x} \biggr]
- \frac{1}{2}\biggl[\boldsymbol{\hat\imath} \frac{\partial u_y^2}{\partial x} + \boldsymbol{\hat\jmath} \frac{\partial u_x^2}{\partial y} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2\Omega_f \biggl[ \boldsymbol{\hat\jmath} u_x - \boldsymbol{\hat\imath} u_y\biggr]
+ \frac{1}{2}\nabla u^2
+
\boldsymbol{\hat\imath} u_y \biggl[ \frac{\partial u_x}{\partial y} - \frac{\partial u_y}{\partial x} \biggr]
+ \boldsymbol{\hat\jmath} u_x \biggl[ \frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y} \biggr]
\, .
</math>
</td>
</tr>
</table>

Next, to the extent possible, let's express the LHS in terms of the dimensionless mass density,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\sigma \equiv \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2}\biggr) \, .</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
We will assume that the stream-function,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Psi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\Psi_0 \sigma^q \, ,</math>
</td>
</tr>
</table>
in which case,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho_c \sigma \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\boldsymbol{\hat\imath} \frac{\partial\Psi}{\partial y} - \boldsymbol{\hat\jmath} \frac{\partial\Psi}{\partial x}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ \rho_c \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{q}{(q-1)} \biggl[ \boldsymbol{\hat\imath} \frac{\partial\sigma^{q-1}}{\partial y} - \boldsymbol{\hat\jmath} \frac{\partial\sigma^{q-1}}{\partial x} \biggr] \, .</math>
</td>
</tr>
</table>
That is,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho_c u_x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{q}{(q-1)} \biggl( \frac{\partial\sigma^{q-1}}{\partial y} \biggr) </math>
</td>
<td align="center">      and,      </td>
<td align="right">
<math>~\rho_c u_y</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{q}{(q-1)} \biggl( \frac{\partial\sigma^{q-1}}{\partial x} \biggr) \, .</math>
</td>
</tr>
</table>

</td></tr></table>

The first term on the LHS becomes,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
2\Omega_f \biggl[ \boldsymbol{\hat\jmath} u_x - \boldsymbol{\hat\imath} u_y\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2\Omega_f}{\rho_c} \cdot \frac{q}{q-1}
\biggl[ \boldsymbol{\hat\jmath} \biggl( \frac{\partial\sigma^{q-1}}{\partial y} \biggr)
+ \boldsymbol{\hat\imath} \biggl( \frac{\partial\sigma^{q-1}}{\partial x} \biggr)\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2\Omega_f}{\rho_c} \cdot \frac{q}{q-1} \nabla (\sigma^{q-1}) \, .
</math>
</td>
</tr>
</table>

The third term on the LHS becomes,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\boldsymbol{\hat\imath} u_y \biggl[ \frac{\partial u_x}{\partial y} - \frac{\partial u_y}{\partial x} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\boldsymbol{\hat\imath} \biggl( \frac{u_y}{\rho_c} \biggr) \biggl\{
\frac{\partial }{\partial y} \biggl[ \frac{q}{(q-1)} \biggl( \frac{\partial\sigma^{q-1}}{\partial y} \biggr) \biggr]
+
\frac{\partial }{\partial x} \biggl[ \frac{q}{(q-1)} \biggl( \frac{\partial\sigma^{q-1}}{\partial x} \biggr) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{q}{(q-1)} \cdot \boldsymbol{\hat\imath} \biggl( \frac{u_y}{\rho_c} \biggr) \biggl\{ \nabla^2 \sigma^{q-1}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\biggl[ \frac{q}{\rho_c(q-1)}\biggr]^2 \biggl\{ \nabla^2 \sigma^{q-1} \cdot \boldsymbol{\hat\imath} \biggl( \frac{\partial\sigma^{q-1}}{\partial x} \biggr)
\biggr\}
</math>
</td>
</tr>
</table>

Similarly, the fourth term on the LHS becomes,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\boldsymbol{\hat\jmath} u_x \biggl[ \frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\boldsymbol{\hat\jmath} \biggl( \frac{u_x}{\rho_c} \biggr) \biggl\{
\frac{\partial }{\partial x} \biggl[ \frac{q}{(q-1)} \biggl( \frac{\partial\sigma^{q-1}}{\partial x} \biggr) \biggr]
+
\frac{\partial }{\partial y} \biggl[ \frac{q}{(q-1)} \biggl( \frac{\partial\sigma^{q-1}}{\partial y} \biggr) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~ \frac{q}{(q-1)} \cdot \boldsymbol{\hat\jmath} \biggl( \frac{u_x}{\rho_c} \biggr) \biggl\{
\nabla^2 \sigma^{q-1}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~ \biggl[ \frac{q}{\rho_c(q-1)}\biggr]^2\biggl\{ \nabla^2 \sigma^{q-1} \cdot \boldsymbol{\hat\jmath}
\biggl( \frac{\partial \sigma^{q-1}}{\partial y} \biggr)
\biggr\} \, .
</math>
</td>
</tr>
</table>

<table border="1" width="80%" cellpadding="10" align="center"><tr><td align="left">
Note that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\partial \sigma^{q-1}}{\partial x_i}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(q-1)\sigma^{q-2} \frac{\partial \sigma}{\partial x_i}
=
- (q-1)\sigma^{q-2} \biggl( \frac{2x_i}{a_i^2} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
(\rho_c u)^2 = \rho_c^2(u_x^2 + u_y^2)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{q^2}{(q-1)^2}\biggl\{
\biggl[ - (q-1)\sigma^{q-2} \biggl( \frac{2x}{a^2} \biggr) \biggr]^2
+
\biggl[ - (q-1)\sigma^{q-2} \biggl( \frac{2y}{b^2} \biggr) \biggr]^2
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4q^2 \sigma^{2(q-2)}
\biggl[\frac{x^2}{a^4} +\frac{y^2}{b^4} \biggr] \, .
</math>
</td>
</tr>
</table>
</td></tr></table>

Hence,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla^2 \sigma^{q-1}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial}{\partial x} \biggl[ \frac{\partial \sigma^{q-1}}{\partial x}\biggr]
+
\frac{\partial}{\partial y} \biggl[ \frac{\partial \sigma^{q-1}}{\partial y}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- (q-1)\biggl\{
\frac{\partial}{\partial x} \biggl[ \sigma^{q-2} \biggl( \frac{2x}{a^2} \biggr)\biggr]
+
\frac{\partial}{\partial y} \biggl[ \sigma^{q-2} \biggl( \frac{2y}{b^2} \biggr)\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- (q-1)\biggl\{
\sigma^{q-2} \frac{\partial}{\partial x} \biggl[ \biggl( \frac{2x}{a^2} \biggr)\biggr] +
\biggl( \frac{2x}{a^2} \biggr)\frac{\partial}{\partial x} \biggl[ \sigma^{q-2} \biggr]
+
\sigma^{q-2}\frac{\partial}{\partial y} \biggl[ \biggl( \frac{2y}{b^2} \biggr)\biggr] +
\biggl( \frac{2y}{b^2} \biggr)\frac{\partial}{\partial y} \biggl[ \sigma^{q-2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- (q-1)\biggl\{
\biggl[ \frac{2}{a^2} + \frac{2}{b^2} \biggr] \sigma^{q-2} +
\frac{2x(q-2)}{a^2} \biggl[ \sigma^{q-3} \frac{\partial \sigma}{\partial x}\biggr]
+
\frac{2y(q-2)}{b^2} \biggl[ \sigma^{q-3} \frac{\partial \sigma}{\partial y}\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- (q-1)
\biggl\{ \biggl[ \frac{2}{a^2} + \frac{2}{b^2} \biggr] \sigma^{q-2} -
2 (q-2)\sigma^{q-3} \biggl[ \frac{2x^2}{a^4} +\frac{2y^2}{b^4} \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>
So, when they are added together, the third and fourth terms give,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\boldsymbol{\hat\imath} u_y \biggl[ \frac{\partial u_x}{\partial y} - \frac{\partial u_y}{\partial x} \biggr]
+ \boldsymbol{\hat\jmath} u_x \biggl[ \frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\biggl[ \frac{q}{\rho_c(q-1)}\biggr]^2 \nabla^2 \sigma^{q-1} \biggl\{ \boldsymbol{\hat\imath} \biggl( \frac{\partial\sigma^{q-1}}{\partial x} \biggr)
+~ \boldsymbol{\hat\jmath}
\biggl( \frac{\partial \sigma^{q-1}}{\partial y} \biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{2q^2}{\rho_c^2(q-1)}\biggr] \biggl\{ \biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr] \sigma^{q-2} -
2 (q-2)\sigma^{q-3} \biggl[ \frac{x^2}{a^4} +\frac{y^2}{b^4} \biggr]
\biggr\}
\biggl\{ \boldsymbol{\hat\imath} \biggl( \frac{\partial\sigma^{q-1}}{\partial x} \biggr)
+~ \boldsymbol{\hat\jmath}
\biggl( \frac{\partial \sigma^{q-1}}{\partial y} \biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{2q^2}{\rho_c^2(q-1)}\biggr]\biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr] \biggl\{ \sigma^{q-2}
\biggr\}
\biggl\{ \boldsymbol{\hat\imath} \biggl( \frac{\partial\sigma^{q-1}}{\partial x} \biggr)
+~ \boldsymbol{\hat\jmath}
\biggl( \frac{\partial \sigma^{q-1}}{\partial y} \biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
-~\biggl[ \frac{4(q-2)q^2}{\rho_c^2(q-1)}\biggr]
\biggl\{ \biggl[\frac{(\rho_c u)^2}{4q^2}\biggr] \sigma^{1-q} \biggr\}\biggl\{ \boldsymbol{\hat\imath} \biggl( \frac{\partial\sigma^{q-1}}{\partial x} \biggr)
+~ \boldsymbol{\hat\jmath}
\biggl( \frac{\partial \sigma^{q-1}}{\partial y} \biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2q^2}{\rho_c^2(2q-3)} \biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr] \nabla \sigma^{2q-3}
-~\frac{(q-2)(\rho_c u)^2}{\rho_c^2}
\nabla\ln(\sigma) \, .
</math>
</td>
</tr>
</table>

Hence,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
'''LHS:'''     <math>~2{\vec{\Omega}}_f \times \bold{u} + (\bold{u}\cdot \nabla) \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2q\Omega_f}{\rho_c(q-1)} \nabla (\sigma^{q-1})
+ \frac{1}{2}\nabla u^2
+ \frac{2q^2}{\rho_c^2(2q-3)} \biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr] \nabla \sigma^{2q-3}
-~(q-2)u^2
\nabla\ln(\sigma) \, .
</math>
</td>
</tr>
</table>

<table border="1" width="80%" cellpadding="10" align="center"><tr><td align="left">
Note that,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\frac{1}{2}\nabla u^2
-~(q-2)u^2
\nabla\ln(\sigma)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2} \biggl\{ \nabla u^2
-~u^2
\nabla\ln[\sigma^{2(q-2)}]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{u^2}{2} \biggl\{ \nabla \ln [u^2]
-~
\nabla\ln[\sigma^{2(q-2)}]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{u^2}{2} \biggl\{ \nabla \ln \biggl[ \frac{u^2}{ \sigma^{2(q-2)} } \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{u^2}{2} \biggl\{ \nabla \ln \biggl[ \biggl( \frac{2q}{ \rho_c } \biggr)^2 \biggl( \frac{x^2}{a^4} + \frac{y^2}{b^4} \biggr)\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2q^2}{\rho_c^2} \sigma^{2(q-2)}
\biggl[\frac{x^2}{a^4} +\frac{y^2}{b^4} \biggr]
\biggl\{ \nabla \ln \biggl[ \biggl( \frac{2q}{ \rho_c } \biggr)^2 \biggl( \frac{x^2}{a^4} + \frac{y^2}{b^4} \biggr)\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\sigma^{2(q-2)}}{2}
\biggl\{ \nabla \biggl[ \biggl( \frac{2q}{ \rho_c } \biggr)^2 \biggl( \frac{x^2}{a^4} + \frac{y^2}{b^4} \biggr)\biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>
Or,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\frac{1}{2}\nabla u^2
+~(2-q)u^2
\nabla\ln(\sigma)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2} \biggl\{ \nabla u^2
+~u^2
\nabla\ln[\sigma^{2(2-q)}]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{1}{2}\biggr)\sigma^{-2(2-q)} \biggl\{ \sigma^{2(2-q)}\nabla u^2
+~u^2
\nabla[\sigma^{2(2-q)}]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{1}{2}\biggr)\sigma^{2(q-2)} \biggl\{
\nabla[u^2 \sigma^{2(2-q)}]
\biggr\} \, .
</math>
</td>
</tr>
</table>

</td></tr></table>

====Exponent q = 2====
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
'''LHS:'''     <math>~[2{\vec{\Omega}}_f \times \bold{u} + (\bold{u}\cdot \nabla) \bold{u}]_{q=2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{4\Omega_f}{\rho_c} \nabla \sigma
+ \frac{1}{2}\nabla u^2
+ \frac{8}{\rho_c^2} \biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr] \nabla \sigma
\, .
</math>
</td>
</tr>
</table>

====Exponent q = 3====
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
'''LHS:'''     <math>~[2{\vec{\Omega}}_f \times \bold{u} + (\bold{u}\cdot \nabla) \bold{u} ]_{q=3}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\Omega_f}{\rho_c} \nabla \sigma^{2}
+ \frac{1}{2}\nabla u^2
+ \frac{6}{\rho_c^2} \biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr] \nabla \sigma^{3}
-~u^2
\nabla\ln(\sigma) \, .
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\Omega_f}{\rho_c} \nabla \sigma^{2}
+ \frac{6}{\rho_c^2} \biggl[ \frac{1}{a^2} + \frac{1}{b^2} \biggr] \nabla \sigma^{3}
+ u^2\nabla \ln\biggl(\frac{u}{\sigma}\biggr)
\, .
</math>
</td>
</tr>
</table>

===Trial #6===

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
'''LHS:'''     <math>~2{\vec{\Omega}}_f \times \bold{u} + (\bold{u}\cdot \nabla) \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2\Omega_f \biggl[ \boldsymbol{\hat\jmath} u_x - \boldsymbol{\hat\imath} u_y\biggr]
+ \frac{1}{2}\nabla u^2
+
\boldsymbol{\hat\imath} u_y \biggl[ \frac{\partial u_x}{\partial y} - \frac{\partial u_y}{\partial x} \biggr]
+ \boldsymbol{\hat\jmath} u_x \biggl[ \frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y} \biggr]
\, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
We will assume that the stream-function,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Psi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\Psi_0 (\alpha - \sigma )^q </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{\partial\Psi}{\partial x_i}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~\Psi_0 q(\alpha - \sigma )^{q-1} \frac{\partial\sigma}{\partial x_i} \, ,</math>
</td>
</tr>
</table>
in which case,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho_c \sigma \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\boldsymbol{\hat\imath} \frac{\partial\Psi}{\partial y} - \boldsymbol{\hat\jmath} \frac{\partial\Psi}{\partial x}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\boldsymbol{\hat\imath} \Psi_0 q(\alpha - \sigma )^{q-1} \frac{\partial\sigma}{\partial y}
+
\boldsymbol{\hat\jmath}~\Psi_0 q(\alpha - \sigma )^{q-1} \frac{\partial\sigma}{\partial x}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ \rho_c \bold{u}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\boldsymbol{\hat\imath} \biggl[ \frac{\Psi_0 q(\alpha - \sigma )^{q-1} }{\sigma} \biggr]\frac{\partial\sigma}{\partial y}
+
\boldsymbol{\hat\jmath}~\biggl[ \frac{\Psi_0 q(\alpha - \sigma )^{q-1}}{\sigma}\biggr] \frac{\partial\sigma}{\partial x}
\, .</math>
</td>
</tr>
</table>
That is,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho_c u_x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~\biggl[ \frac{\Psi_0 q(\alpha - \sigma )^{q-1} }{\sigma} \biggr]\frac{2y}{b^2} </math>
</td>
<td align="center">      and,      </td>
<td align="right">
<math>~\rho_c u_y</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl[ \frac{\Psi_0 q(\alpha - \sigma )^{q-1}}{\sigma}\biggr] \frac{2x}{a^2} \, .</math>
</td>
</tr>
</table>

</td></tr></table>

The first term on the LHS becomes,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
2\Omega_f \biggl[ \boldsymbol{\hat\jmath} u_x - \boldsymbol{\hat\imath} u_y\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2\Omega_f}{\rho_c} \biggl[- \frac{\Psi_0 q(\alpha - \sigma )^{q-1}}{\sigma}\biggr]
\biggl[\boldsymbol{\hat\jmath} \biggl( \frac{2y}{b^2} \biggr)
+ \boldsymbol{\hat\imath} \biggl( \frac{2x}{a^2} \biggr)\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
+ \frac{2\Omega_f}{\rho_c} \biggl[ \frac{\Psi_0 q(\alpha - \sigma )^{q-1}}{\sigma}\biggr]
\nabla \sigma \, .
</math>
</td>
</tr>
</table>

===Trial #7===

====Uncluttered Setup====

Let's simply look at the vortensity expression as defined in [[#Part_II|Part II, above]], namely,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~g(\Psi)</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{(2\Omega_f + \zeta_z)}{\rho_c\sigma} \, ,</math>
</td>
</tr>
</table>
and recognize that we are ultimately interested in the function, <math>~F_B(\Psi)</math>, defined such that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{dF_B(\Psi)}{d\Psi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-g(\Psi) \, .</math>
</td>
</tr>
</table>

We start with the expression for the z-component of the vorticity,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\zeta_z </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\frac{\partial }{\partial x}\biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial x} \biggr] - \frac{\partial }{\partial y} \biggl[\frac{1}{\rho} \frac{\partial \Psi}{\partial y} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\biggl\{ \frac{\partial }{\partial x}\biggl[ \biggl( \frac{\Psi_0}{\rho_c \sigma} \biggr) \frac{\partial \sigma^q}{\partial x} \biggr]
+ \frac{\partial }{\partial y} \biggl[ \biggl( \frac{\Psi_0}{\rho_c\sigma} \biggr) \frac{\partial \sigma^q}{\partial y} \biggr]
\biggr\}
\, .
</math>
</td>
</tr>
</table>

Next, appreciate that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{1}{\sigma} \frac{\partial \sigma^q}{\partial x_i}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
q \sigma^{q-2} \frac{\partial \sigma}{\partial x_i}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{q}{q-1} \biggr) \frac{\partial \sigma^{q-1}}{\partial x_i} \, .
</math>
</td>
</tr>
</table>

Hence,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\zeta_z </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~ \frac{q\Psi_0}{\rho_c (q-1) } \biggl\{ \frac{\partial }{\partial x}\biggl[ \frac{\partial \sigma^{q-1}}{\partial x} \biggr]
+ \frac{\partial }{\partial y} \biggl[ \frac{\partial \sigma^{q-1}}{\partial y} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~ \biggl[ \frac{q\Psi_0}{\rho_c (q-1) } \biggr] \nabla^2\sigma^{q-1}
\, .
</math>
</td>
</tr>
</table>

So, the vortensity is,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~g(\sigma) = \frac{(2\Omega_f + \zeta_z )}{\rho_c\sigma}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{2\Omega_f}{\rho_c}\biggr)\sigma^{-1}
-
\biggl[ \frac{q\Psi_0}{\rho_c^2 (q-1) } \biggr] \sigma^{-1} \nabla^2\sigma^{q-1} \, .
</math>
</td>
</tr>
</table>

Let's switch to the stream-function via the ''assumed'' relation,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\sigma</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{\Psi}{\Psi_0}\biggr)^{1/q} \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ g(\Psi) </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{2\Omega_f}{\rho_c}\biggr)\biggl( \frac{\Psi}{\Psi_0}\biggr)^{-1/q}
-
\biggl[ \frac{q\Psi_0}{\rho_c^2 (q-1) } \biggr] \biggl( \frac{\Psi}{\Psi_0}\biggr)^{-1/q} \nabla^2\biggl( \frac{\Psi}{\Psi_0}\biggr)^{(q-1)/q} \, .
</math>
</td>
</tr>
</table>

This expression gives the vortensity in what appears to be the desired form — that is, expressed strictly in terms of the stream function, <math>~\Psi</math> — for a wide range of values of the exponent, <math>~q</math>. [<font color="red">'''CAUTION:'''</font>   the <math>~\nabla^2</math> operator is an exception.] It is not yet (13 October 2020) clear to me how — or if — the second term on the right-hand-side of this expression can be integrated to give <math>~F(\Psi)</math>. But the first term can be obtained from,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~F_{1^\mathrm{st}}(\Psi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\biggl(\frac{2\Omega_f}{\rho_c}\biggr) \frac{q}{(q-1)} \biggl( \frac{\Psi}{\Psi_0}\biggr)^{(q-1)/q}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{dF_{1^\mathrm{st}}(\Psi)}{d\Psi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\biggl(\frac{2\Omega_f}{\rho_c}\biggr) \biggl( \frac{\Psi}{\Psi_0}\biggr)^{-1/q} \, .
</math>
</td>
</tr>
</table>

====T5 Coordinates====
Let's evaluate the <math>~\nabla^2</math> operator by expressing it and its argument in terms of [[Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates|T5 Coordinates]]. Note that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\lambda_1</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~(x^2 + \kappa^2 y^2)^{1 / 2} \, ,</math>
</td>
</tr>
</table>
where, <math>~\kappa \equiv a/b</math>. The specified density distribution is, therefore,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\sigma \equiv \frac{\rho}{\rho_c}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~1 - \frac{\lambda_1^2}{a^2} \, ,</math>
</td>
</tr>
</table>
and the stream-function is,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\Psi}{\Psi_0}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\sigma^q = \biggl[ 1 - \frac{\lambda_1^2}{a^2} \biggr]^q \, .</math>
</td>
</tr>
</table>

The relevant [[Appendix/Ramblings/EllipticCylinderCoordinates#Laplacian|T5-Coordinate System Laplacian]] is,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla^2 \mathfrak{F}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{1}{\lambda_1^2 \ell^2} \biggr]
\biggl[ \frac{\partial^2 \mathfrak{F}}{\partial \lambda_1^2}\biggr]
-
\biggl[ \frac{1}{\lambda_1^3 \ell^2} \biggr]
\frac{\partial \mathfrak{F}}{\partial \lambda_1}
+
\biggl[ \frac{(1 + \kappa^2)}{\lambda_1 } \biggr]
\frac{\partial \mathfrak{F}}{\partial \lambda_1} \, ,
</math>
</td>
</tr>
</table>
where,
<div align="center">
<math>~\ell \equiv (x^2 + \kappa^4 y^2)^{- 1 / 2} \, ,</math>
</div>
and in the present context,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{F}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{\Psi}{\Psi_0}\biggr)^{(q-1)/q} =
\biggl[ 1 - \frac{\lambda_1^2}{a^2} \biggr]^{q-1} \, .
</math>
</td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\partial \mathfrak{F}}{\partial \lambda_1}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2(q-1)\lambda_1}{a^2}
\biggl[ 1 - \frac{\lambda_1^2}{a^2} \biggr]^{q-2}
=
- \biggl[ \frac{2(q-1)\lambda_1}{a^2}\biggr]\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-2)/q} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\partial^2 \mathfrak{F}}{\partial \lambda_1^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2(q-1)}{a^2}
\biggl[ 1 - \frac{\lambda_1^2}{a^2} \biggr]^{q-2}
+ \frac{2(q-1)\lambda_1}{a^2}
\biggl[ 1 - \frac{\lambda_1^2}{a^2} \biggr]^{q-3} \biggl[ \frac{2(q-2) \lambda_1 }{ a^2 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2(q-1)}{a^2}
\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-2)/q}
+ \frac{4(q-1)(q-2)\lambda_1^2}{a^4}
\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-3)/q}
\, .
</math>
</td>
</tr>
</table>
And,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla^2 \mathfrak{F}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{1}{\lambda_1^2 \ell^2} \biggr]
\biggl\{ - \frac{2(q-1)}{a^2}
\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-2)/q}
+ \frac{4(q-1)(q-2)\lambda_1^2}{a^4}
\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-3)/q}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
-
\biggl[ \frac{1}{\lambda_1^3 \ell^2} \biggr]
\biggl\{
- \biggl[ \frac{2(q-1)\lambda_1}{a^2}\biggr]\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-2)/q}
\biggr\}
+
\biggl[ \frac{(1 + \kappa^2)}{\lambda_1 } \biggr]
\biggl\{
- \biggl[ \frac{2(q-1)\lambda_1}{a^2}\biggr]\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-2)/q}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{4(q-1)(q-2)}{a^4\ell^2}
\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-3)/q}
-
\biggl[ \frac{2(q-1)(1 + \kappa^2)}{a^2}\biggr]\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-2)/q} \, .
</math>
</td>
</tr>
</table>

====All Together====

Putting this all together, we obtain,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ g(\Psi) </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{2\Omega_f}{\rho_c}\biggr)\biggl( \frac{\Psi}{\Psi_0}\biggr)^{-1/q}
-
\biggl[ \frac{q\Psi_0}{\rho_c^2 (q-1) } \biggr] \biggl( \frac{\Psi}{\Psi_0}\biggr)^{-1/q}
\biggl\{
\frac{4(q-1)(q-2)}{a^4\ell^2}
\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-3)/q}
-
\biggl[ \frac{2(q-1)(1 + \kappa^2)}{a^2}\biggr]\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-2)/q}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl(\frac{2\Omega_f}{\rho_c}\biggr)\biggl( \frac{\Psi}{\Psi_0}\biggr)^{-1/q}
+
\biggl[ \frac{2q(1 + \kappa^2)\Psi_0}{\rho_c^2 a^2} \biggr] \biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-3)/q}
-
\biggl[ \frac{4 q(q-2) \Psi_0}{\rho_c^2 a^4} \biggr] (x^2 + \kappa^4 y^2)
\biggl( \frac{\Psi}{\Psi_0} \biggr)^{(q-4)/q} \, .
</math>
</td>
</tr>
</table>

Note that, for the [[Appendix/Ramblings/EllipticCylinderCoordinates#Invert_Coordinate_Mapping|specific ''example'' case of]] <math>~\kappa^2 = 2</math>,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ \frac{1}{\ell^2} \biggr]_{\kappa^2=2} = (x^2 + 4y^2)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
x^2 \Lambda
=
\frac{\Lambda \lambda_2^2}{2} \biggl[ \Lambda - 1 \biggr]
=
2\lambda_1^2 \biggl[ \frac{\Lambda}{\Lambda + 1} \biggr]
\, .
</math>
</td>
</tr>
</table>
where,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Lambda</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\biggl[ 1 + \frac{4\lambda_1^2}{\lambda_2^2} \biggr]^{1 / 2} \, .</math>
</td>
</tr>
</table>

=See Also=

* [[ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations|Construction Challenges (Pt. 1)]]
* [[ThreeDimensionalConfigurations/ChallengesPt2|Construction Challenges (Pt. 2)]]
* [[ThreeDimensionalConfigurations/ChallengesPt3|Construction Challenges (Pt. 3)]]
* [[ThreeDimensionalConfigurations/ChallengesPt4|Construction Challenges (Pt. 4)]]
* [[ThreeDimensionalConfigurations/ChallengesPt5|Construction Challenges (Pt. 5)]]
* Related discussions of models viewed from a rotating reference frame:
** [[PGE/RotatingFrame#Rotating_Reference_Frame|PGE]]
** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[Apps/RiemannEllipsoids_Compressible|Compressible Riemann Ellipsoids]].

{{ SGFfooter }}

VE/RiemannEllipsoids

2021-09-30T23:05:31Z

174.64.14.12: Created page with "__FORCETOC__  =Steady-State 2<sup>nd</sup>-Order Tensor Virial Equations= ==Summary== Drawing from our User:Tohline/VE#Virial_Equatio..."

__FORCETOC__


=Steady-State 2<sup>nd</sup>-Order Tensor Virial Equations=

==Summary==

Drawing from our [[User:Tohline/VE#Virial_Equations_.28Rotating_Frame.29|accompanying discussion of virial equations as viewed from a rotating frame of reference]], here we employ the 2<sup>nd</sup>-order tensor virial equation (TVE),
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi
+ \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx \, ,
</math>
</td>
</tr>
</table>
to determine the equilibrium conditions of uniform-density <math>~(\rho)</math> ellipsoids that have semi-axes, <math>~(a_1, a_2, a_3) \leftrightarrow (a, b, c),</math> and an internal velocity field, <math>~\vec{u}</math> (as [[#Adopted_.28Internal.29_Velocity_Field|prescribed below]]), that preserves this specified ellipsoidal shape, as viewed from a frame of reference that is rotating with angular velocity, <math>~\vec\Omega</math>. Because each of the indices, <math>~i</math> and <math>~j</math>, run from 1 to 3, inclusive, this TVE appears to provide nine equilibrium constraints; and once the values of the density and the three semi-axes are specified, there appear to be seven unknowns: <math>~\Pi</math> and the three pairs of velocity-field components <math>~(\Omega_1, \zeta_1)</math>, <math>~(\Omega_2, \zeta_2)</math>, <math>~(\Omega_3, \zeta_3).</math> In practice, however, only five constraints are relevant/independent because, as is encapsulated in …
<table border="0" width="60%" align="center" cellpadding="10"><tr><td align="left">
<div align="center"><font color="maroon">'''Riemann's Fundamental Theorem'''</font></div>
<font color="darkgreen">… non-trivial solutions are obtained only if no more than two of the three pairs of velocity-field components are different from zero.</font>
</td></tr></table>
<span id="SummaryTable">Following EFE</span>, we will set <math>~\Omega_1 = \zeta_1 = 0</math>, in which case the only applicable TVE constraint relations are the five identified in the following table of equations.

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="2">Indices</td>
<td align="center" rowspan="2">Each Associated 2<sup>nd</sup>-Order TVE Expression</td>
</tr>
<tr>
<td align="center" width="5%"><math>~i</math></td>
<td align="center" width="5%"><math>~j</math></td>
</tr>
<tr>
<td align="center"><math>~1</math></td>
<td align="center"><math>~1</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi
+\biggl\{
( \Omega_2^2 + \Omega_3^2)
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2
~-~(2\pi G\rho) A_1
\biggr\} a^2
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
+ \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 c^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~2</math></td>
<td align="center"><math>~2</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi
+ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ \biggl\{
\Omega_3^2
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~-~( 2\pi G \rho) A_2
\biggr\}b^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~3</math></td>
<td align="center"><math>~3</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi
+ \biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 a^2
+ \biggl\{
\Omega_2^2 + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2
- (2\pi G \rho)A_3
\biggr\}c^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~2</math></td>
<td align="center"><math>~3</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
\biggr\} \Omega_2\Omega_3c^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~3</math></td>
<td align="center"><math>~2</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
\biggr\} \Omega_2 \Omega_3b^2
</math>
</td>
</tr>
</table>

</td>
</tr>
</table>

==General Coefficient Expressions==

In the context of our discussion of configurations that are triaxial ellipsoids, we begin by adopting the <math>~(\ell, m, s)</math> subscript notation to identify which semi-axis length is the (largest, medium-length, smallest). As has been detailed in an [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Derivation_of_Expressions_for_Ai|accompanying chapter]], the gravitational potential anywhere inside or on the surface of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~\frac{A_\ell}{a_\ell a_m a_s}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~\frac{2}{a_\ell^3}
\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{A_s}{a_\ell a_m a_s}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\frac{2}{a_\ell^3} \biggl[ \frac{(a_m/a_s) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{A_m}{a_\ell a_m a_s} = \frac{2 - (A_\ell + A_s)}{a_\ell a_m a_s}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~
\frac{ 2}{a_\ell^3 }
\biggl[ \frac{
E(\theta, k)
-~(1-k^2)
F(\theta, k)
-~(a_s/a_m)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
</td>
</tr>

</table>
</div>
where, <math>~F(\theta,k)</math> and <math>~E(\theta,k)</math> are incomplete elliptic integrals of the first and second kind, respectively, with arguments,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_s}{a_\ell} \biggr)</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~k = \biggl[\frac{1 - (a_m/a_\ell)^2}{1 - (a_s/a_\ell)^2} \biggr]^{1/2} \, .</math>
</td>
</tr>
</table>
</div>

===Specific Case of a<sub>1</sub> > a<sub>2</sub> > a<sub>3</sub>===

When we discuss configurations in which <math>~a_1 > a_2 > a_3 > 0</math> — such as Jacobi, Dedekind, or ''most'' Riemann S-Type ellipsoids — we must adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_m, a_m)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_s, a_s)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~A_1
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~2\biggl(\frac{a_2}{a_1}\biggr)\biggl(\frac{a_3}{a_1}\biggr)
\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_3
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~2\biggl(\frac{a_2}{a_1}\biggr) \biggl[ \frac{(a_2/a_1) \sin\theta - (a_3/a_1)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_2
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~2 - (A_1+A_3) \, ,</math>
</td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr)</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~k = \biggl[\frac{1 - (a_2/a_1)^2}{1 - (a_3/a_1)^2} \biggr]^{1/2} \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §17, Eq. (32)</font> ]</td></tr>
</table>
</div>

===Specific Case of a<sub>1</sub> > a<sub>3</sub> > a<sub>2</sub>===

When we discuss configurations in which <math>~a_1 > a_3 > a_2 > 0</math> — these are usually referred to in [[User:Tohline/Appendix/References#EFE|EFE]] as prolate S-Type Riemann ellipsoids — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_s, a_s)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_m, a_m)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~A_1
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~2 \biggl( \frac{a_2}{a_1} \biggr)\biggl( \frac{a_3}{a_1} \biggr)
\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_2
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~2 \biggl( \frac{a_3}{a_1} \biggr) \biggl[ \frac{(a_3/a_1) \sin\theta - (a_2/a_1)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_3 = 2 - (A_1 + A_2)
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~
\frac{2a_2 a_3}{a_1^2}
\biggl[ \frac{
E(\theta, k)
-~(1-k^2)
F(\theta, k)
-~(a_2/a_3)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
</td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals of the first and second kind are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_2}{a_1} \biggr)</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~k = \biggl[\frac{1 - (a_3/a_1)^2}{1 - (a_2/a_1)^2} \biggr]^{1/2} \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48d, footnote to Table VII (p. 143)</font> ]</td></tr>
</table>
</div>
NOTE: All ''irrotational'' ellipsoids belong to this category of configurations.

===Specific Case of a<sub>2</sub> > a<sub>1</sub> > a<sub>3</sub>===

When we discuss configurations in which <math>~a_2 > a_1 > a_3 > 0</math> — for example, ''most'' Riemann ellipsoids of Types I, II, & III — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_m, a_m)</math>, <math>~(A_2, a_2) \leftrightarrow (A_\ell, a_\ell)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_s, a_s)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~A_2
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~2 \biggl( \frac{a_1}{a_2} \biggr)\biggl( \frac{a_3}{a_2} \biggr)
\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_3
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~2\biggl( \frac{a_1}{a_2}\biggr) \biggl[ \frac{(a_1/a_2) \sin\theta - (a_3/a_2)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~A_1 = 2 - (A_2 + A_3)
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>~
\frac{ 2a_1 a_3}{a_2^2 }
\biggl[ \frac{
E(\theta, k)
-~(1-k^2)
F(\theta, k)
-~(a_3/a_1)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
</td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_3}{a_2} \biggr)</math>
</td>
<td align="center">
      and      
</td>
<td align="left">
<math>~k = \biggl[\frac{1 - (a_1/a_2)^2}{1 - (a_3/a_2)^2} \biggr]^{1/2} \, .</math>
</td>
</tr>
</table>
</div>

===Oblate Spheroids [a<sub>2</sub> = a<sub>1</sub> > a<sub>3</sub>]===

Starting with the case of <math>~a_2 > a_1 > a_3 > 0</math> and setting <math>~a_2 = a_1</math>, we recognize, first, that <math>~k = 0</math>. Hence, we have,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>
~A_3
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~2\biggl[ \frac{ \sin\theta - (a_3/a_1)E(\theta,0)}{\sin^3\theta} \biggr] \, ,
</math>
</td>
</tr>
</table>

==Adopted (Internal) Velocity Field==

EFE (p. 130) states that the … <font color="#007700">kinematical requirement, that the motion <math>~(\vec{u})</math>, associated with <math>~\vec{\zeta}</math>, preserves the ellipsoidal boundary, leads to the following expressions for its components:</font>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~u_1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[ \frac{a_1^2}{a_1^2 + a_2^2}\biggr] \zeta_3 x_2 + \biggl[ \frac{a_1^2}{a_1^2+a_3^2}\biggr] \zeta_2 x_3 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~u_2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 x_3 + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x_1 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~u_3</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x_1 + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 x_2 \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (1)</font> ]</td></tr>
</table>

==Equilibrium Expressions==
[<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b> §11(b), p. 22] <font color="#007700">Under conditions of a stationary state, [the tensor virial equation] gives,</font>
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \delta_{ij}\Pi \, .</math>
</td>
</tr>
</table>
</div>
<font color="#007700">[This] provides six integral relations which must obtain whenever the conditions are stationary</font>.

When viewing the (generally ellipsoidal) configuration from a rotating frame of reference, the 2<sup>nd</sup>-order TVE takes on the more general form:
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi
+ \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx
\, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 2, §12, Eq. (64)</font> ]</td></tr>
</table>

EFE (p. 57) also shows that … <font color="#007700">The potential energy tensor … for a homogeneous ellipsoid is given by</font>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\mathfrak{W}_{ij}}{\pi G\rho}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-2A_i I_{ij} \, ,</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §22, Eq. (128)</font> ]</td></tr>
</table>
<font color="#007700">where</font>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~I_{ij}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §22, Eq. (129)</font> ]</td></tr>
</table>

<font color="#007700">is the moment of inertia tensor.</font> Expressions for all nine components of the kinetic energy tensor, <math>~\mathfrak{T}_{ij}</math> are derived in [[#Appendix_E:_.C2.A0_Kinetic_Energy_Components|Appendix E]], below; and expressions for each of the six Coriolis components can be found in [[#Appendix_B:_.C2.A0Coriolis_Component_u1x2|Appendices B, C, & D]].

===The Three Diagonal Elements===
For <math>~i = j = 1</math>, we have,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi
+ \Omega^2 I_{11} - \Omega_1\Omega_k I_{k1} + 2\epsilon_{1lm}\Omega_m \int_V \rho u_lx_1 d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi + \Omega^2 I_{11}
- \Omega_1^2I_{11}
+ 2 \Omega_3 \int_V \rho u_2x_1 ~d^3x
- 2\Omega_2 \int_V \rho u_3x_1 ~d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi
+( \Omega_2^2 + \Omega_3^2) I_{11}
+ 2 \Omega_3\rho \int_V u_2x ~d^3x
- 2\Omega_2\rho \int_V u_3 x~ d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 I_{22}
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 I_{33}
~-~(2\pi G\rho) A_1 I_{11} + \Pi
+( \Omega_2^2 + \Omega_3^2) I_{11}
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 I_{11}
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2 I_{11}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Pi
+ \biggl\{
( \Omega_2^2 + \Omega_3^2)
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2
~-~(2\pi G\rho) A_1
\biggr\} I_{11}
+
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 I_{22}
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 I_{33}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ -\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
( \Omega_2^2 + \Omega_3^2)
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2
~-~(2\pi G\rho) A_1
\biggr\} a^2
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
+ \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 c^2 \, .
</math>
</td>
</tr>
</table>

Once we choose the values of the (semi) axis lengths <math>~(a, b, c)</math> of an ellipsoid — from which the value of <math>~A_1</math> can be immediately determined — along with a specification of <math>~\rho</math>, this equation has the following five unknowns: <math>~\Pi, \Omega_2, \Omega_3, \zeta_2, \zeta_3</math>. Similarly, for <math>~i = j = 2</math>,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{22} + \mathfrak{W}_{22} + \Pi
+ \Omega^2 I_{22} - \Omega_2\Omega_k I_{k2} + 2\epsilon_{2lm}\Omega_m \int_V \rho u_lx_2 d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{22} + \mathfrak{W}_{22} + \Pi
+ (\Omega_1^2 + \Omega_3^2) I_{22} + 2\Omega_1 \rho \int_V u_3 y ~d^3x
- 2\Omega_3 \rho \int_V u_1 y ~d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 I_{33}
+
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 I_{11}
~-~( 2\pi G \rho) A_2 {I}_{22}
+ \Pi
+ (\Omega_1^2 + \Omega_3^2) I_{22}
+ 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1 I_{22}
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3 I_{22}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Pi
+
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 I_{11}
+ \biggl\{
(\Omega_1^2 + \Omega_3^2)
+ 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~-~( 2\pi G \rho) A_2
\biggr\}{I}_{22}
+ \biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 I_{33}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~-\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ \biggl\{
(\Omega_1^2 + \Omega_3^2)
+ 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~-~( 2\pi G \rho) A_2
\biggr\}b^2
+ \biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 c^2 \, .
</math>
</td>
</tr>
</table>

This gives us a second equation, but an additional pair of (for a total of seven) unknowns: <math>~\Omega_1, \zeta_1</math>. For the third diagonal element — that is, for <math>~i=j=3</math> — we have,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{33} + \mathfrak{W}_{33} + \Pi
+ \Omega^2 I_{33} - \Omega_3\Omega_k I_{k3} + 2\epsilon_{3lm}\Omega_m \int_V \rho u_lx_3 ~d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{33} + \mathfrak{W}_{33} + \Pi
+ (\Omega_1^2 + \Omega_2^2) I_{33} + 2\Omega_2 \rho \int_V u_1 z ~d^3x
- 2\Omega_1 \rho \int_V u_2 z ~d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 I_{11}
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 I_{22}
- (2\pi G \rho)A_3 I_{33} + \Pi
+ (\Omega_1^2 + \Omega_2^2) I_{33} + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2 I_{33}
+ 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1 I_{33}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Pi
+
\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 I_{11}
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 I_{22}
+ \biggl\{
(\Omega_1^2 + \Omega_2^2) + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2
+ 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1
- (2\pi G \rho)A_3
\biggr\}I_{33}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ -\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 a^2
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 b^2
+ \biggl\{
(\Omega_1^2 + \Omega_2^2) + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2
+ 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1
- (2\pi G \rho)A_3
\biggr\}c^2 \, .
</math>
</td>
</tr>
</table>
This gives us three equations ''vs.'' seven unknowns.

===Off-Diagonal Elements===

Notice that the off-diagonal components of both <math>~I_{ij}</math> and <math>~\mathfrak{W}_{ij}</math> are zero. Hence, the equilibrium expression that is dictated by each off-diagonal component of the 2<sup>nd</sup>-order TVE is,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j d^3x
\, .
</math>
</td>
</tr>
</table>

For example — as is explicitly illustrated on p. 130 of EFE — for <math>~i=2</math> and <math>~j=3</math>,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} + 2\Omega_1 \cancelto{0}{\int_V \rho u_3x_3 d^3x}
- 2\Omega_3 \int_V \rho u_1x_3 d^3x \, ,
</math>
</td>
</tr>
<tr><td align="center" colspan="4">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (3)</font> ]</td></tr>
</table>
whereas for <math>~i=3</math> and <math>~j=2</math>,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 d^3x
- 2\Omega_1 \cancelto{0}{\int_V \rho u_2 x_2 d^3x}
\, .
</math>
</td>
</tr>
<tr><td align="center" colspan="4">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (4)</font> ]</td></tr>
</table>

<table border="1" cellpadding="8" align="center" width="80%"><tr><td align="left">
Given our adoption of a uniform-density configuration whose surface has a precisely ellipsoidal shape and, along with it, our adoption of the above specific prescription for the internal velocity field, <math>~\vec{u}</math>, we recognize that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int_V \rho u_i x_j d^3x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0</math>
</td>
<td align="right">      if    <math>~i = j \, .</math>
</tr>
<tr><td align="center" colspan="4">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (5)</font> ]</td></tr>
</table>
This has allowed us to set to zero one of the integrals in each of these last two expressions. In what follows, we will benefit from recognizing, as well, that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{T}_{32} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathfrak{T}_{23}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2} \int_V \rho v_2 v_3 d^3x \, .</math>
</td>
</tr>
</table>
</td></tr></table>

Our first off-diagonal element is, then,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33}
- 2\Omega_3 \rho \int_V u_1 z d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- ~
\biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 a^2
- \Omega_2\Omega_3 c^2
- 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_3 \zeta_2 c^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
\Omega_2\Omega_3
+ \biggl[ \frac{\zeta_2 a^2}{a^2 + c^2 }\biggr] \biggl[ 2\Omega_3 + \frac{\zeta_3 b^2}{b^2+a^2}\biggr]
\biggr\} c^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
\biggr\} \Omega_2\Omega_3c^2 \, .
</math>
</td>
</tr>
</table>

The second is,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \rho \int_V u_1 y d^3x
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- ~
\biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 a^2
- \Omega_3 \Omega_2 b^2
- 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\Omega_2 \zeta_3 b^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
\Omega_2 \Omega_3
+ \biggl[ \frac{\zeta_3 a^2}{a^2+b^2}\biggr] \biggl[2\Omega_2 + \frac{\zeta_2 c^2}{c^2 + a^2}\biggr]
\biggr\} b^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
\biggr\} \Omega_2 \Omega_3b^2 \, .
</math>
</td>
</tr>
</table>

===How Solution is Obtained ===
Adding this pair of governing expressions we obtain,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33}
- 2\Omega_3 \int_V \rho u_1x_3 dx \biggr]
+
\biggl[2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx
\biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4 \mathfrak{T}_{23} - \Omega_2\Omega_3(I_{22}+ I_{33} )
+
2 \int_V \rho u_1 (\Omega_2 x_2 - \Omega_3 x_3) dx \, ;
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (6)</font> ]</td></tr>
</table>
and subtracting the pair gives,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33}
- 2\Omega_3 \int_V \rho u_1x_3 dx \biggr]
-
\biggl[2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2\Omega_3 (I_{22} - I_{33} )
- 2 \int_V \rho u_1 ( \Omega_2 x_2 + \Omega_3 x_3) dx \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (7)</font> ]</td></tr>
</table>

=Various Degrees of Simplification=

==Riemann Ellipsoids of Types I, II, & III==
In this, most general, case, the two vectors <math>~\vec{\Omega}</math> and <math>~\vec\zeta</math> are not parallel to any of the principal axes of the ellipsoid, and they are not aligned with each other, but they both lie in the <math>~y-z</math>-plane — that is to say, <math>~(\Omega_1, \zeta_1) = (0, 0)</math>. For a given specified density <math>~(\rho)</math> and choice of the three semi-axes <math>~(a_1, a_2, a_3) \leftrightarrow (a, b, c)</math>, all five of the expressions displayed in our above [[#SummaryTable|''Summary Table'']] must be used in order to determine the equilibrium configuration's associated values of the five unknowns: <math>~\Pi, (\Omega_2, \zeta_2), (\Omega_3, \zeta_3)</math>. Here we show how these five unknowns can be derived from the five constraint equations, closely following the analysis that is presented in §47 (pp. 129 - 132) of [ [[User:Tohline/Appendix/References#EFE|EFE]] ].

===Constraints Due to Off-Diagonal Elements===
We begin by subtracting the constraint equation provided by the first off-diagonal element <math>~(i, j) = (2, 3)</math> from the constraint equation provided by the second off-diagonal element <math>~(i, j) = (3, 2) </math>. This gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl\{
1
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
\biggr\} \Omega_2\Omega_3c^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{
1
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
\biggr\} \Omega_2 \Omega_3b^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ 1 + \frac{\zeta_3}{2 \Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr] \biggl[1 + \frac{\zeta_2}{2\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr]
+ \frac{\zeta_2}{\Omega_2} \cdot \frac{\zeta_3}{\Omega_3} \biggl[ \frac{a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ \frac{b^2}{b^2+a^2} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr]
+ \frac{\zeta_3}{\Omega_3} \cdot \frac{\zeta_2}{\Omega_2} \biggl[ \frac{a^2 b^2}{a^2+b^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
c^2 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (11)</font> ]</td></tr>
</table>

Adding the two instead gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
1
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
\biggr\} \Omega_2\Omega_3c^2
+
\biggl\{
1
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
\biggr\} \Omega_2 \Omega_3b^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2 + c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2 b^2}{a^2+b^2}\biggr] \biggl[2 + \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2 + c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2a^2 c^2}{a^2 + c^2 }\biggr]
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2a^2 b^2}{a^2+b^2}\biggr]
+ \frac{\zeta_2}{\Omega_2} \cdot \frac{\zeta_3}{\Omega_3} \biggl[ \frac{2a^2 b^2 c^2}{(a^2 + c^2)( b^2+a^2 ) }\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (10)</font> ]</td></tr>
</table>

The first of these relations cleanly gives an expression for the frequency ratio, <math>~\zeta_3/\Omega_3</math>, in terms of the ''other'' frequency ratio, <math>~\zeta_2/\Omega_2</math>. This allows us to rewrite the second relation in terms of the ratio, <math>~\zeta_2/\Omega_2</math>, alone. We obtain,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2 + c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2a^2 c^2}{a^2 + c^2 }\biggr]
+ \biggl\{ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2a^2 b^2}{a^2+b^2}\biggr] \biggr\}
+ \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{(a^2 + c^2) } \biggr] \cdot \biggl\{ \frac{\zeta_3}{\Omega_3} \biggl[ \frac{2a^2 b^2 }{( b^2+a^2 ) }\biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2 + c^2
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2a^2 c^2}{a^2 + c^2 }\biggr]
+ \biggl\{ c^2 - b^2 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] \biggr\}
+ \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{(a^2 + c^2) } \biggr] \cdot \biggl\{ c^2 - b^2 + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2c^2
+ \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{a^2 + c^2 } \biggr] (4a^2 + c^2 - b^2 )
+ \biggl\{ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{c^2}{a^2 + c^2 }\biggr] \biggr\}^22a^2
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
ASIDE:   Alternatively, given that,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\frac{\zeta_2}{\Omega_2}\biggl[ \frac{c^2}{a^2 + c^2 }\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2a^2}\biggl[ b^2 - c^2+ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]
</math>
</td>
</tr>
</table>
the quadratic equation that governs the value of the frequency ratio, <math>~\zeta_3/\Omega_3</math> is …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4 a^2 c^2
+ \biggl[ b^2 - c^2+ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr] (4a^2 + c^2 - b^2 )
+ \biggl[ b^2 - c^2+ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4 a^2 c^2
+ \biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr] (4a^2 + c^2 - b^2 )
+ ( b^2 - c^2) (4a^2 + c^2 - b^2 )
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ (b^2 - c^2)^2 + 2(b^2 - c^2) \biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]
+ \biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4 a^2 c^2
+ ( b^2 - c^2) (4a^2 + c^2 - b^2 )
+ (b^2 - c^2)^2
+ \biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]
\biggl[ (4a^2 + c^2 - b^2 )+ 2(b^2 - c^2) \biggr]
+ \biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]^2
+ \biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]
(4a^2 + b^2 - c^2 )
+ 4 a^2 b^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ 0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \biggl(\frac{\zeta_3}{\Omega_3} \biggr)^2 \frac{a^2 b^2}{(a^2+b^2)^2}\biggr]
+ \frac{1}{2}\biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{1}{a^2+b^2}\biggr) \biggr]
(4a^2 + b^2 - c^2 )
+ 1 \, .
</math>
</td>
</tr>
</table>

Now, in our discussion of Riemann S-Type ellipsoids, there is also a quadratic equation that governs the equilibrium frequency ratio, <math>~f \equiv \zeta_3/\Omega_3</math>. It is, specifically,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2 b^2}{(a^2 + b^2)^2} \biggr] f^2
+ \biggl[ \frac{2a^2 b^2 B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} \biggr]\frac{f}{a^2 + b^2} + 1 \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§48, Eq. (35)</font> ]</td></tr>
</table>
Notice that the first and third terms of this quadratic equation exactly match the first and third terms of the quadratic equation, which we have just derived, that governs the same frequency ratio in Riemann ellipsoids of Types I, II & III. Does the second term match? That is, is the coefficient of the linear term the same in both quadratic relations? Well, …
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \frac{2a^2 b^2 B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2a^2 b^2 \biggl[ c^2 A_3 + a^2 b^2 \biggl( \frac{A_1 - A_2}{a^2-b^2} \biggr)\biggr]^{-1} \biggl[A_2 + a^2\biggl( \frac{A_1 - A_2}{a^2-b^2} \biggr)\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2a^2 b^2 \biggl[ c^2 A_3(a^2 - b^2) + a^2 b^2 (A_1 - A_2) \biggr]^{-1} \biggl[a^2 A_1 - b^2 A_2\biggr] \, .
</math>
</td>
</tr>
</table>

Even appreciating that we can make the substitution, <math>~A_3 = (2 - A_1 - A_2)</math>, I don't see any way that this coefficient expression can be manipulated to match the associated coefficient in the other expression, namely, <math>~(4a^2 + b^2 - c^2)/2</math>.

</td></tr></table>

This is a quadratic equation whose solution gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~4a^2 \cdot \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{a^2 + c^2 } \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- (4a^2 + c^2 - b^2 ) \pm \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2} \, .
</math>
</td>
</tr>
</table>

For the other frequency ratio we therefore find,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
2\biggl\{ b^2 -c^2
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr] \biggr\}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2 \cdot \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- (4a^2 + c^2 - b^2 ) \pm \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~ \Rightarrow ~~~
4a^2 \cdot \frac{\zeta_3}{\Omega_3}\biggl[ \frac{b^2}{a^2+b^2}\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- (4a^2 + b^2 -c^2) \pm \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2}
</math>
</td>
</tr>
</table>
<span id="OffDiagonal"> </span>
<table border="1" width="80%" align="center" cellpadding="10"><tr><td align="left">
<div align="center">'''SUMMARY:   Riemann Ellipsoids of Types I, II, & III'''</div>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\beta \equiv~ -~\frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{a^2 + c^2 } \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{4a^2}\biggl\{ (4a^2 -b^2 + c^2 ) \mp \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2} \biggr\} \, ;
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (16)</font> ]</td></tr>

<tr>
<td align="right">
<math>~
\gamma \equiv~-~\frac{\zeta_3}{\Omega_3}\biggl[ \frac{b^2}{a^2+b^2}\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{4a^2}\biggl\{ (4a^2 + b^2 -c^2) \mp \biggl[ (4a^2 + c^2 - b^2 )^2 - 16a^2 c^2 \biggr]^{1 / 2} \biggr\} \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, Eq. (17)</font> ]</td></tr>
</table>

As is emphasized in [[User:Tohline/Appendix/References#EFE|EFE]] (Chapter 7, §47, p. 131) "<font color="darkgreen">… the signs in front of the radicals, in the two expressions, go together.</font> Furthermore, "<font color="darkgreen">the two roots … correspond to the fact that, consistent with Dedekind's theorem, two states of internal motions are compatible with the same external figure.</font>"

----

As has also been pointed out in [[User:Tohline/Appendix/References#EFE|EFE]] (Chapter 7, §51, p. 158), from the steps that have led to the development and solution of the above pair of quadratic equations we can demonstrate that the following relations also hold:

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\beta^2 - 2\beta + \frac{c^2}{a^2} = \biggl[ \frac{c^2 - b^2}{2a^2}\biggr]\beta \, ,</math>
</td>
<td align="center">
       
</td>
<td align="left">
<math>~\gamma^2 -2\gamma + \frac{b^2}{a^2} = \biggl[ \frac{b^2 - c^2}{2a^2} \biggr]\gamma</math>
</td>
</tr>

<tr>
<td align="right">
<math>~1 - 2\beta + \biggl(\frac{a^2}{c^2}\biggr)\beta^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{4a^2 - b^2 - 3c^2}{2c^2}\biggr]\beta \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~1 - 2\gamma + \biggl(\frac{a^2}{b^2}\biggr)\gamma^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]\gamma \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eqs. (161) - (163)</font> ]</td></tr>
</table>

</td></tr></table>

===Constraints Due to Diagonal Elements===

Next, to simplify manipulations, let's replace the frequency ratios by these newly defined — and ''known'' — parameters, <math>~\beta</math> and <math>~\gamma</math>, in the three diagonal-element expressions that are written out in our above [[#SummaryTable|Summary Table]].

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="2">Indices</td>
<td align="center" rowspan="2">Rewritten Diagonal-Element Expressions</td>
</tr>
<tr>
<td align="center" width="5%"><math>~i</math></td>
<td align="center" width="5%"><math>~j</math></td>
</tr>
<tr>
<td align="center"><math>~1</math></td>
<td align="center"><math>~1</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\biggl\{
( \Omega_2^2 + \Omega_3^2)
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]
+ 2 \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \biggl[ - \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr]
~-~(2\pi G\rho) A_1
\biggr\} a^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]^2 b^2
- \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \biggl[ - \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr]^2 c^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
- \Omega_2^2 - \Omega_3^2
+ 2 \Omega_3^2 \gamma
+ 2 \Omega_2^2 \beta
~+~(2\pi G\rho) A_1
\biggr\} a^2
- \biggl( \frac{a^4}{b^2}\biggr) \Omega_3^2\gamma^2
- \biggl( \frac{a^4}{c^2}\biggr) \Omega_2^2 \beta^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{ \Omega_2^2 \biggl[2 \beta - 1 - \biggl( \frac{a^2}{c^2}\biggr) \beta^2 \biggr]
+ \Omega_3^2 \biggl[ 2 \gamma - 1 - \biggl( \frac{a^2}{b^2}\biggr)\gamma^2 \biggr]
~+~(2\pi G\rho) A_1
\biggr\}a^2
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (158)</font> ]</td></tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~2</math></td>
<td align="center"><math>~2</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]^2 a^2
- \biggl\{
\Omega_3^2
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]
~-~( 2\pi G \rho) A_2
\biggr\}b^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma^2 a^2
- \Omega_3^2 b^2
+ 2 a^2 \Omega_3^2 \gamma
~+~( 2\pi G \rho) b^2A_2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~a^2 \Omega_3^2 \biggl[\gamma^2 - 2\gamma
+ \biggl( \frac{b^2}{a^2}\biggr) \biggr]
~+~( 2\pi G \rho) b^2A_2
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (159)</font> ]</td></tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~3</math></td>
<td align="center"><math>~3</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \biggl[ - \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr]^2 a^2
- \biggl\{
\Omega_2^2 + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \biggl[ - \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr]
- (2\pi G \rho)A_3
\biggr\}c^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \Omega_2^2 \beta^2 a^2
-\Omega_2^2c^2 + 2 a^2\Omega_2^2 \beta
+ (2\pi G \rho)c^2 A_3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~a^2\Omega_2^2\biggl[
\beta^2 - 2 \beta
+ \biggl( \frac{c^2}{a^2}\biggr) \biggr]
+ (2\pi G \rho)c^2 A_3
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (160)</font> ]</td></tr>
</table>

</td>
</tr>
</table>

Using the <math>~(i, j) = (3, 3)</math> element to preplace <math>~\Pi</math> in the other two expressions, we obtain,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\biggl[
\beta^2 - 2 \beta
+ \biggl( \frac{c^2}{a^2}\biggr) \biggr]
+
\Omega_2^2 \biggl[2 \beta - 1 - \biggl( \frac{a^2}{c^2}\biggr) \beta^2 \biggr]
+ \Omega_3^2 \biggl[ 2 \gamma - 1 - \biggl( \frac{a^2}{b^2}\biggr)\gamma^2 \biggr]
~+~2\pi G\rho \biggl[ A_1 - \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr] \, ;
</math>
</td>
</tr>
</table>

and,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\Omega_2^2\biggl[
\beta^2 - 2 \beta
+ \biggl( \frac{c^2}{a^2}\biggr) \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_3^2 \biggl[\gamma^2 - 2\gamma
+ \biggl( \frac{b^2}{a^2}\biggr) \biggr]
~+~2\pi G \rho \biggl[\biggl( \frac{c^2}{a^2}\biggr) A_3 - \biggl( \frac{b^2}{a^2} \biggr)A_2 \biggr] \, .
</math>
</td>
</tr>
</table>

Inserting the [[#OffDiagonal|various relations highlighted above]], these two expressions may be rewritten as,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2
\biggl[ \frac{c^2 - b^2}{2a^2}\biggr]\beta
~-~\Omega_2^2 \biggl[ \frac{4a^2 - b^2 - 3c^2}{2c^2}\biggr]\beta
~-~\Omega_3^2 \biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]\gamma
~+~2\pi G\rho \biggl[ A_1 - \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl[ \frac{c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)}{2a^2c^2}
\biggr]
~-~\Omega_3^2 \gamma \biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]
~+~2\pi G\rho \biggl[ A_1 - \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr] \, ;
</math>
</td>
</tr>
</table>

<span id="Temporary">and,</span>

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~\Omega_2^2
\biggl[ \frac{c^2 - b^2}{2a^2}\biggr]\beta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_3^2
\biggl[ \frac{b^2 - c^2}{2a^2} \biggr]\gamma~+~2\pi G \rho \biggl[\biggl( \frac{c^2}{a^2}\biggr) A_3 - \biggl( \frac{b^2}{a^2} \biggr)A_2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \Omega_2^2 \beta
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
~-~\Omega_3^2 \gamma
~+~4\pi G \rho \biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ -~\Omega_3^2 \gamma
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2 \beta~-~4\pi G \rho \biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr] \, .
</math>
</td>
</tr>
</table>
Together, then,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl[ \frac{c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)}{2a^2c^2}
\biggr]
~+~2\pi G\rho \biggl[ A_1 - \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr]
~+~\biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]\biggl\{
\Omega_2^2 \beta~-~4\pi G \rho \biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl[ \frac{c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)}{2a^2c^2}
~+~\frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]
~+~2\pi G\rho \biggl[ \frac{a^2A_1 - c^2 A_3}{a^2} \biggr]
~-~2\pi G \rho \biggl[ \frac{4a^2 - c^2 - 3b^2}{b^2}\biggr]
\biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl\{ \frac{b^2[ c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)] + (4a^2 - c^2 - 3b^2)a^2 c^2}{2a^2 b^2 c^2}
\biggr\}
~+~2\pi G\rho \biggl\{
\biggl[ \frac{a^2A_1 - c^2 A_3}{a^2} \biggr]
~-~\biggl[ \frac{4a^2 - c^2 - 3b^2}{b^2}\biggr]
\biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 - b^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta
\biggl\{ \frac{b^2 c^2(c^2 - b^2) + a^2 b^2(c^2 - b^2 ) + ( c^2 - b^2)a^2 c^2 + a^2 (4a^2 -2b^2 - 2c^2 )(c^2 - b^2 ) }{2a^2 b^2 c^2}
\biggr\}
~+~2\pi G\rho
\biggl[ \frac{b^2(a^2A_1 - c^2 A_3) + a^2(3b^2-4a^2 + c^2)B_{23} }{a^2b^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_2^2\beta \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 }{2a^2 b^2 }
\biggr]
~+~2\pi G\rho
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{a^2b^2} \biggr] \, ,
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (170)</font> ]</td></tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~B_{23}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{A_2 b^2 - A_3 c^2}{b^2 - c^2} \biggr] \, .</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §21, Eqs. (105) & (107)</font> ]</td></tr>
</table>

Similarly, given that ([[#Temporary|see just above]]),

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Omega_2^2 \beta \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
~+~4\pi G \rho \biggl[ \frac{ c^2 A_3 - b^2 A_2}{c^2 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
~+~4\pi G \rho \biggl[ \frac{ (c^2 - b^2)B_{23} }{c^2 } \biggr] \, ,
</math>
</td>
</tr>
</table>
we have,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
~+~4\pi G \rho \biggl[ \frac{ (c^2 - b^2)B_{23} }{c^2 } \biggr]
\biggr\}
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 }{2a^2 b^2 }
\biggr]
~+~2\pi G\rho
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{a^2b^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{b^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 }{2a^2 c^2 }
\biggr]
~+~2\pi G \rho \biggl\{
\biggl[ \frac{ [4a^4 - a^2 (b^2 + c^2) + b^2 c^2 ](c^2 - b^2)B_{23} }{a^2 b^2 c^2 }
\biggr]
~+~
\biggl[ \frac{a^2c^2 (3b^2-4a^2 + c^2)B_{23} + b^2c^2(a^2A_1 - c^2 A_3) }{a^2b^2 c^2} \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{b^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 }{2a^2 c^2 }
\biggr]
~+~2\pi G \rho
\biggl[ \frac{ a^2( b^2 + 3c^2 - 4a^2 ) B_{23} +c^2(a^2A_1 - b^2 A_2) }{a^2 c^2 }
\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §51, Eq. (171)</font> ]</td></tr>
</table>

Finally, looking back at the <math>~(i, j) = (3, 3)</math> constraint and recognizing that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~-~
\Omega_2^2\beta (c^2 - b^2)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi G\rho c^2
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 } \biggr] \, ,
</math>
</td>
</tr>
</table>

we find,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~2\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~2a^2\Omega_2^2\biggl[
\beta^2 - 2 \beta
+ \biggl( \frac{c^2}{a^2}\biggr) \biggr]
+ (4\pi G \rho)c^2 A_3
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(4\pi G \rho)c^2 A_3
-~
(c^2 - b^2 )\Omega_2^2 \beta
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(4\pi G \rho)c^2 A_3
+~
4\pi G\rho c^2
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\pi G \rho c^2 \biggl\{ A_3
+~
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) }{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 } \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>

==Riemann S-Type Ellipsoids==
In this case, we assume that <math>~\vec{\Omega}</math> and <math>~\vec\zeta</math> are aligned with each other and, as well, are aligned with the <math>~z</math>-axis; that is to say, in addition to setting <math>~(\Omega_1, \zeta_1) = (0, 0)</math> we also set <math>~(\Omega_2, \zeta_2) = (0, 0)</math>. So, there are only three unknowns — <math>~\Pi, (\Omega_3, \zeta_3)</math> — and they can be determined by ignoring off-axis expressions and simultaneously solving the ''diagonal element'' expressions displayed in our above [[#SummaryTable|''Summary Table'']]. Furthermore, two of the three diagonal-element expressions can be simplified because we are setting <math>~(\Omega_2, \zeta_2) = (0, 0)</math>. The three relevant equilibrium constraints are:

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="2">Indices</td>
<td align="center" rowspan="2">2<sup>nd</sup>-Order TVE Expressions that are Relevant to Riemann S-Type Ellipsoids</td>
</tr>
<tr>
<td align="center" width="5%"><math>~i</math></td>
<td align="center" width="5%"><math>~j</math></td>
</tr>
<tr>
<td align="center"><math>~1</math></td>
<td align="center"><math>~1</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi
+\biggl\{
\Omega_3^2
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
~-~(2\pi G\rho) A_1
\biggr\} a^2
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~2</math></td>
<td align="center"><math>~2</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi
+ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ \biggl\{
\Omega_3^2
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~-~( 2\pi G \rho) A_2
\biggr\}b^2
</math>
</td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center"><math>~3</math></td>
<td align="center"><math>~3</math></td>
<td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi
- (2\pi G \rho)A_3 c^2
</math>
</td>
</tr>
</table>

</td>
</tr>
</table>

The <math>~(i, j) = (3, 3)</math> component expression immediately identifies the value of one of the unknowns, namely,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{2^3\pi^2}{3\cdot 5} \biggr) G \rho^2A_3 a b c^3 \, .
</math>
</td>
</tr>
</table>

From the remaining pair of diagonal-element expressions, we therefore have,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
a^2 \Omega_3^2
+ 2 \biggl[ \frac{b^2a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 ) \, ,
</math>
</td>
</tr>
</table>

and,
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+
b^2 \Omega_3^2
+ 2 \biggl[ \frac{a^2b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) \, .
</math>
</td>
</tr>
</table>
Multiplying the first of these two expressions through by <math>~b^2</math> and the second through by <math>~a^2</math>, then subtracting the second from the first gives,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
b^2\biggl\{
2 \biggl[ \frac{b^2a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 ) \biggr\}
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
-~
a^2\biggl\{
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ 2 \biggl[ \frac{a^2b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl\{
2 \biggl[ \frac{b^4 a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 )b^2 \biggr\}
~-~
\biggl\{
2 \biggl[ \frac{a^4 b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) a^2
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{b^2 a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi G\rho \biggl[ \frac{(A_3 c^2 - A_2 b^2) a^2 ~-~(A_3 c^2 - A_1 a^2 )b^2}{ b^2 - a^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2}{ b^2 - a^2} - A_3 c^2\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (30)</font> ]</td></tr>
</table>
Note that — as EFE has done and as we have recorded in a [[User:Tohline/ThreeDimensionalConfigurations/JacobiEllipsoids#Equilibrium_Conditions_for_Jacobi_Ellipsoids|related discussion]] — the first term on the right-hand-side of this last expression can be expressed more compactly in terms of the coefficient, <math>~A_{12}</math>.

Alternatively, dividing the first expression through by <math>~a^2</math> and the second by <math>~b^2</math>, then adding the pair of expressions gives,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~
0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Omega_3^2
+ 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3
+ \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2
~+~(2\pi G\rho)(A_3 c^2 - A_1 a^2 )\frac{1}{a^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~+~
\biggl[ \frac{a^2 b^2}{(b^2+a^2)^2}\biggr] \zeta_3^2
+
\Omega_3^2
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) \frac{1}{b^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2\Omega_3^2 + 2 \Omega_3 \zeta_3
+ 2\biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2
~+~2\pi G\rho \biggl[ \frac{A_3 c^2 - A_1 a^2 }{a^2} + \frac{A_3c^2 - A_2 b^2}{b^2}\biggr] \, .
</math>
</td>
</tr>
</table>

If we divide through by 2, then replace the product, <math>~\Omega_3\zeta_3</math>, in this expression by the relation derived immediately above, we have,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>~
\Omega_3^2
+ \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
~-~\pi G\rho \biggl[ \frac{b^2 (A_3 c^2 - A_1 a^2) + a^2(A_3c^2 - A_2 b^2 ) }{a^2b^2} \biggr]
~-~
\pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2 - A_3 c^2(b^2 - a^2)}{ b^2 - a^2} \biggr]\biggl[ \frac{b^2+a^2}{b^2 a^2}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\pi G\rho}{ a^2b^2(a^2-b^2) }
\biggl\{ [ b^2 (A_3 c^2 - A_1 a^2) + a^2(A_3c^2 - A_2 b^2 )](b^2-a^2)
~+~
[ (A_1 - A_2)a^2b^2 - A_3 c^2(b^2 - a^2) ](b^2+a^2)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\pi G\rho}{ a^2b^2(a^2-b^2) }
\biggl\{ [ - A_1 a^2 b^2 - A_2 a^2 b^2 ](b^2-a^2)
~+~
(A_1 - A_2)a^2b^2 (b^2+a^2)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2\pi G\rho}{ (a^2-b^2) }
\biggl[
A_1 a^2
- A_2 b^2
\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (29)</font> ]</td></tr>
</table>

<span id="fDefined">It has become customary to characterize each Riemann S-Type ellipsoid by the value of its equilibrium frequency ratio, </span>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{\zeta_3}{\Omega_3} \, ,</math>
</td>
</tr>
</table>
in which case the relevant pair of constraint equations becomes,

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\biggl[ \frac{b^2 a^2}{b^2+a^2}\biggr] f \Omega_3^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2}{ b^2 - a^2} - A_3 c^2\biggr] \, ;
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (34)</font> ]</td></tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\Omega_3^2 \biggl\{1
+ \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] f^2 \biggr\}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2\pi G\rho}{ (a^2-b^2) }
\biggl[
A_1 a^2
- A_2 b^2
\biggr] \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (33)</font> ]</td></tr>
</table>

</td></tr></table>

These two equations can straightforwardly be combined to generate a quadratic equation for the frequency ratio, <math>~f</math>. Then, once the value of <math>~f</math> has been determined, either expression can be used to determine the corresponding equilibrium value for <math>~\Omega_3</math> in the unit of <math>~(\pi G \rho)^{1 / 2}</math>. The fact that the value of <math>~f</math> is determined from the solution of a quadratic equation underscores the realization that, for a given specification of the ellipsoidal geometry <math>~(a, b, c)</math>, if an equilibrium exists — ''i.e.,'' if the solution for <math>~f</math> is real rather than imaginary — then two equally valid, and usually different (''i.e.,'' non-degenerate), values of <math>~f</math> will be realized. This means that two different underlying flows — one ''direct'' and the other ''adjoint'' — will sustain the shape of the ellipsoidal configuration, as viewed from a frame that is rotating about the <math>~z</math>-axis with frequency, <math>~\Omega_3</math>.

==Jacobi and Dedekind Ellipsoids==
Describe …

==Maclaurin Spheroids==

Describe …

=Appendices:  Various Integrals Over Ellipsoid Volume=
Throughout this set of appendices, we work with a uniform-density ellipsoid whose surface is defined by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \, .
</math>
</td>
</tr>
</table>

==Appendix A:  Volume==

Here we seek to find the volume of the ellipsoid via the ''Cartesian'' integral expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint dx ~dy ~dz \, .
</math>
</td>
</tr>
</table>

===Preliminaries===
First, we will integrate over <math>~x</math> and specify the integration limits via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x_\ell</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
a\biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} \, ;
</math>
</td>
</tr>
</table>

second, we will integrate over <math>~z</math> and specify the integration limits via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~z_\ell</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
c\biggl[ 1 - \frac{y^2}{b^2} \biggr]^{1 / 2} \, ;
</math>
</td>
</tr>
</table>
third, we will integrate over <math>~y</math> and set the limits of integration as <math>~\pm b</math>.

===Carry Out the Integration===
Following thestrategy that has just been outlined, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iint dy ~dz \int_{-x_\ell}^{+x_\ell} dx
=
\iint dy ~dz \biggl[ x \biggr]_{-x_\ell}^{+x_\ell}
=
2\int dy \int x_\ell ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2a\int dy \int \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} dz
=
\frac{2a}{c} \int dy \int_{-z_\ell}^{+z_\ell} \biggl[ z_\ell^2- z^2 \biggr]^{1 / 2} dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{c} \int \frac{dy}{2} \biggl[ z\sqrt{ z_\ell^2- z^2 } + z_\ell^2 \sin^{-1} \biggl( \frac{z}{|z_\ell |} \biggr) \biggr]_{-z_\ell}^{+z_\ell}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{c} \int \biggl[ z_\ell \cancelto{0}{\sqrt{ z_\ell^2- z_\ell^2 }} + z_\ell^2 \sin^{-1} \biggl(1\biggr) \biggr] dy
=
\frac{2a}{c} \int \biggl[ \frac{\pi}{2} z_\ell^2 \biggr] dy
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi a c \int_{-b}^{+b} \biggl( 1 - \frac{y^2}{b^2} \biggr) dy
=
\pi a c \biggl[ y - \frac{y^3}{3b^2} \biggr]_{-b}^{+b}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{4\pi}{3} \cdot a b c\, .
</math>
</td>
</tr>
</table>

==Appendix B:  Coriolis Component u<sub>1</sub>x<sub>2</sub>==
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\iiint [u_1 y] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} y ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \iiint y^2 ~dx ~dy ~dz
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \iiint yz ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int dz \int_{-x_\ell}^{+x_\ell} dx
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z ~dz \int_{-x_\ell}^{+x_\ell} dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{2a^2}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int x_\ell dz
+
\biggl[ \frac{2a^2}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z~x_\ell ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} dz
+
\biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z~\biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int_{-z_\ell}^{+z_\ell} \biggl[ z_\ell^2- z^2 \biggr]^{1 / 2} dz
~+~
\frac{1}{c} \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int_{-z_\ell}^{+z_\ell} z~\biggl[ z_\ell^2 - z^2 \biggr]^{1 / 2} ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \cdot \frac{1}{2} \biggl\{ z \sqrt{z_\ell^2 - z^2} + z_\ell^2 \sin^{-1}\biggl(\frac{z}{|z_\ell |}\biggr) \biggr\}_{-z_\ell}^{+z_\ell}
~-~
\frac{1}{c} \biggl[ \frac{2a^3}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \cdot \frac{1}{3} \biggl\{ \biggl[ z_\ell^2 - z^2 \biggr]^{3 / 2} \biggr\}_{-z_\ell}^{+z_\ell}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{c}\biggl[ \frac{2a^3}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \cdot \frac{1}{2} \biggl\{ z_\ell^2 \sin^{-1}\biggl(\frac{z}{|z_\ell |}\biggr) \biggr\}_{-z_\ell}^{+z_\ell}
=
- \pi a~c\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \int_{-b}^b y^2 \biggl[1 - \frac{y^2}{b^2} \biggr] dy
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \pi ac\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{y^3}{3} - \frac{y^5}{5b^2} \biggr]_{-b}^{+b}
=
- 2\pi a b^3 c\biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{2}{15} \biggr]
=
- \frac{4\pi abc}{3} \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \biggl[\frac{b^2}{5} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{I_{22}}{\rho} \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, p. 130, Eq. (9a)</font> ]</td></tr>
</table>

==Appendix C:  Coriolis Component u<sub>1</sub>x<sub>3</sub>==

Here we will additionally make use of the integration limits,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~y_\ell^2</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~b^2 \biggl(1 - \frac{z^2}{c^2}\biggr) \, .</math>
</td>
</tr>
</table>
Integration over the relevant Coriolis component gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\iiint [u_1 z] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} z ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-
\biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3\iiint \cancelto{0}{y z ~dx ~dy ~dz}
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \iiint z^2 ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int dy \int_{-x_\ell}^{+x_\ell} dx
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2a\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int dy \biggl\{ \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \int_{-y_\ell}^{+y_\ell} \biggl[ y_\ell^2 - y^2 \biggr]^{1 / 2} dy
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int z^2 dz \cdot \frac{1}{2}\biggl\{ y \sqrt{y_\ell^2 - y^2} + y_\ell^2 \sin^{-1}\biggr( \frac{y}{|y_\ell |} \biggr)\biggr\}_{-y_\ell}^{+y_\ell}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{b}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int_{-c}^c z^2 \biggl\{ \frac{\pi}{2} y_\ell^2 \biggr\} dz
=
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int_{-c}^c z^2 \biggl\{ 1 - \frac{z^2}{c^2} \biggr\} dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{z^3}{3} - \frac{z^5}{5c^2} \biggr\}_{-c}^{+c}
=
\pi a b \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{1}{3} - \frac{1}{5} \biggr\}2c^3
=
\frac{4 \pi a b c}{3}\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \biggl\{\frac{c^2}{5} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~+
~\frac{I_{33}}{\rho} \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, p. 130, Eq. (9b)</font> ]</td></tr>
</table>

==Appendix D:   The Other Four Coriolis Components ==

It follows that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\iiint [u_2 x] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \cancelto{0}{\biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 z} + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x \biggr\} x ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
+~\frac{I_{11}}{\rho}\biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\iiint [u_2 z] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 z + \cancelto{0}{\biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x} \biggr\} z ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\frac{I_{33}}{\rho} \biggl[\frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\iiint [u_3 x] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x + \cancelto{0}{\biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 y} \biggr\} x ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-~\frac{I_{11}}{\rho} \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\iiint [u_3 y] ~dx ~dy ~dz</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\iiint \biggl\{ - \cancelto{0}{\biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x} + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 y \biggr\} y ~dx ~dy ~dz
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
+~\frac{I_{22}}{\rho} \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 \, .
</math>
</td>
</tr>
</table>

==Appendix E:   Kinetic Energy Components ==

===Diagonal Elements===
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{11} = \int_V u_1 u_1 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_1^2] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\}^2 ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint \biggl\{
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 y^2
- 2\cancelto{0}{\biggl[ \frac{a^2}{a^2 + b^2}\biggr]
\biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \zeta_3} yz
+ \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 z^2
\biggr\} ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 \iiint y^2 ~dx ~dy ~dz
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2\iiint z^2 ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 \biggl[ \frac{I_{22}}{\rho} \biggr]
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 \biggl[ \frac{I_{33}}{\rho} \biggr] \, .</math>
</td>
</tr>
</table>
Similarly,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{22} = \int_V u_2 u_2 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_2^2] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint \biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x \biggr\}^2 ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2 \biggl[ \frac{I_{33}}{\rho} \biggr]
+
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 \biggl[ \frac{I_{11}}{\rho} \biggr] \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{33} = \int_V u_3 u_3 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_2^2] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint \biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}^2 ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2 \biggl[ \frac{I_{11}}{\rho} \biggr]
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 \biggl[ \frac{I_{22}}{\rho} \biggr] \, .</math>
</td>
</tr>
</table>

===Off-Diagonal Elements===
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{23} = \int_V u_2 u_3 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_2 u_3] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint
\biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x\biggr\}
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint
\biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z \biggr\}
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~+ \iiint
\biggl\{\biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x\biggr\}
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~\iiint
\biggl\{\biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 \biggr\}
x^2~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- ~\frac{I_{11}}{\rho}
\biggl[ \frac{b^2}{b^2+a^2}\biggr] \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3
</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, §47, p. 130, Eq. (8)</font> ]</td></tr>
</table>
Similarly,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{12} = \int_V u_1 u_2 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_1 u_2] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint
\biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\}
\biggl\{ - \biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 z + \biggl[ \frac{b^2}{b^2+a^2}\biggr] \zeta_3 x \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~
\biggl[ \frac{a^2}{a^2+c^2}\biggr]
\biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \zeta_2
\iiint z^2~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-~ \frac{I_{33}}{\rho}
\biggl[ \frac{a^2}{a^2+c^2}\biggr]
\biggl[ \frac{b^2}{b^2 + c^2}\biggr] \zeta_1 \zeta_2
\, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\biggl( \frac{2}{\rho}\biggr)\mathfrak{T}_{31} = \int_V u_3 u_1 d^3x </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint [u_3 u_1] ~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\iiint
\biggl\{ - \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 x + \biggl[ \frac{c^2}{c^2+b^2}\biggr] \zeta_1 y \biggr\}
\biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\}
~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -~
\biggl[ \frac{c^2}{c^2+b^2}\biggr]
\biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_1\zeta_3
\iiint y^2~dx ~dy ~dz</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -~ \frac{I_{22}}{\rho}
\biggl[ \frac{c^2}{c^2+b^2}\biggr]
\biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_1\zeta_3 \, .
</math>
</td>
</tr>
</table>

And, finally,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{T}_{32}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathfrak{T}_{23} \, ;</math>
</td>
<td align="center">     </td>
<td align="right">
<math>~\mathfrak{T}_{21}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathfrak{T}_{12} \, ;</math>
</td>
<td align="center">      and,     </td>
<td align="right">
<math>~\mathfrak{T}_{13}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mathfrak{T}_{31} \, .</math>
</td>
</tr>
</table>

=See Also=
* [[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|Properties of Maclaurin Spheroids]]
* [[Apps/MaclaurinSpheroids/GoogleBooks#Excerpts_from_A_Treatise_of_Fluxions|Excerpts from Maclaurin's (1742) ''A Treatise of Fluxions'']]

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174.64.14.12: Created page with "  =Properties of Homogeneous Ellipsoids (2)= In addition to pulling..."

=Properties of Homogeneous Ellipsoids (2)=

In addition to pulling from §53 of [[Appendix/References#EFE|Chandrasekhar's EFE]], here, we lean heavily on the papers by [https://ui.adsabs.harvard.edu/abs/1983ApJ...271..586W/abstract M. D. Weinberg & S. Tremaine (1983, ApJ, 271, 586)] (hereafter, WT83) and by [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract D. M. Christodoulou, D. Kazanas, I. Shlosman, & J. E. Tohline (1995a, ApJ, 446, 472)] (hereafter, Paper I).

==Sequence-Defining Dimensionless Parameters==

A Riemann sequence of ''S''-type ellipsoids is defined by the value of the dimensionless parameter,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{\zeta}{\Omega} = </math> constant,
</td>
</tr>
</table>

[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§48, Eq. (31)</font> ]<br />
[ [https://ui.adsabs.harvard.edu/abs/1983ApJ...271..586W/abstract WT83], <font color="#00CC00">Eq. (5)</font> ]<br />
[ [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <font color="#00CC00">Eq. (2.1)</font> ]<br />
</div>
where, <math>~\zeta</math> is the system's vorticity as measured in a frame rotating with angular velocity, <math>~\Omega</math>. Alternatively, we can use the dimensionless parameter,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\biggl[\frac{ab}{a^2 + b^2} \biggr]f \, ,</math>
</td>
</tr>
</table>
[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§48, Eq. (40)</font> ]<br />
[ [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <font color="#00CC00">Eq. (2.2)</font> ]<br />
</div>
or,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Lambda</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~-\biggl[\frac{ab}{a^2 + b^2} \biggr] \Omega f = -\Omega x \, .</math>
</td>
</tr>
</table>
[ [https://ui.adsabs.harvard.edu/abs/1983ApJ...271..586W/abstract WT83], <font color="#00CC00">Eq. (4)</font> ]<br />
</div>

==Conserved Quantities==
Algebraic expressions for the conserved energy, <math>~E</math>, angular momentum, <math>~L</math>, and circulation, <math>~C</math>, are, respectively,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~E</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}v^2 + \frac{1}{2}(a^2 + b^2)(\Lambda^2 + \Omega^2) - 2ab\Lambda\Omega - 2I </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~\rightarrow</math>
</td>
<td align="left">
<math>~\cancelto{0}{\frac{1}{2}v^2} + \frac{1}{2} [(a+bx)^2 + (b+ax)^2]\Omega^2 - 2I \, ,</math>
</td>
</tr>
</table>

[ 1<sup>st</sup> expression — [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§53, Eq. (239)</font> ]<br />
[ 2<sup>nd</sup> expression — [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <font color="#00CC00">Eq. (2.7)</font> ]<br />
</div>
where — see an [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Triaxial_Configurations|accompanying discussion]] for the definitions of <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~I</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~A_1a^2 + A_2b^2 + A_3c^2 \, ;</math>
</td>
</tr>
</table>

[ 1<sup>st</sup> expression — [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§53, Eq. (239)</font> ]<br />
[ 2<sup>nd</sup> expression — [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <font color="#00CC00">Eq. (2.8)</font> ]<br />
</div>

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{5L}{M}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(a^2 + b^2)\Omega - 2ab\Lambda</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ (a^2 + b^2 + 2abx)\Omega \, ;</math>
</td>
</tr>
</table>

[ 1<sup>st</sup> expression — [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§53, Eq. (240)</font> ]<br />
[ 2<sup>nd</sup> expression — [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <font color="#00CC00">Eq. (2.5)</font> ]<br />
</div>

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{5C}{M}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(a^2 + b^2)\Lambda - 2ab\Omega</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- [2ab + (a^2 + b^2)x ]\Omega \, .</math>
</td>
</tr>
</table>

[ 1<sup>st</sup> expression — [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§53, Eq. (241)</font> ]<br />
[ 2<sup>nd</sup> expression — [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <font color="#00CC00">Eq. (2.6)</font> ]<br />
</div>

If we rewrite the expression for the system's free energy in terms of <math>~L</math> (and ''x'') instead of <math>~\Omega</math> (and ''x''), we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~E</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2} \biggl(\frac{5L}{M}\biggr)^2 \frac{(a+bx)^2 + (b+ax)^2}{(a^2 + b^2 + 2abx)^2} - 2I \, ,</math>
</td>
</tr>
</table>

[ [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <font color="#00CC00">Eq. (3.4)</font> ]<br />
</div>

Note that, based on the units chosen in [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <math>~M = 5</math>, and <math>~abc = 15/4</math>.

=Aside: Chandra's Notation=
According to equation (107) in §21 of EFE, it appears as though,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~A_i - A_j</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- (a_i^2 - a_j^2)A_{ij} \, .</math>
</td>
</tr>
</table>
</div>
And, according to equation (105) in §21 of EFE, it appears as though,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~B_{ij}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~A_j - a_i^2A_{ij} \, .</math>
</td>
</tr>
</table>
</div>

So, for example,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~A_{12} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-\biggl[ \frac{A_1 - A_2}{a_1^2 - a_2^2} \biggr] \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~B_{12} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~A_2 + a_1^2\biggl[ \frac{A_1 - A_2}{a_1^2 - a_2^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{(a_1^2 - a_2^2)A_2 + a_1^2(A_1 - A_2)}{a_1^2 - a_2^2} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{a_1^2A_1 - a_2^2A_2 }{a_1^2 - a_2^2} \, .</math>
</td>
</tr>
</table>
</div>

=Free Energy Surface(s)=

==Scope==

Consider a self-gravitating ellipsoid having the following properties:
<ul>
<li>Semi-axis lengths, <math>~(x,y,z)_\mathrm{surface} = (a,b,c)</math>, and corresponding volume, <math>~4\pi/(3abc)</math>  ; and consider only the situations <math>0 \le b/a \le 1</math> and <math>0 \le c/a \le 1</math>  ;</li>
<li>Total mass, <math>~M</math>  ;</li>
<li>Uniform density, <math>~\rho = (3 M)/(4\pi abc) </math>  ;</li>
<li>Figure is spinning about its ''c'' axis with angular velocity, <math>~\Omega</math>  ;</li>
<li>Internal, steady-state flow exhibiting the following characteristics:</li>
<ul>
<li>No vertical (''z'') motion;</li>
<li>Elliptical (''x-y'' plane) streamlines everywhere having an ellipticity that matches that of the overall figure, that is, <math>~e = (1-b^2/a^2)^{1/2}</math>  ;</li>
<li>The velocity components, <math>~v_x</math> and <math>~v_y</math>, are linear in the coordinate and, overall, characterized by the magnitude of the vorticity, <math>~\zeta</math>  .</li>
</ul>
</ul>

Such a configuration is uniquely specified by the choice of six key parameters:   <math>~a</math>, <math>~b</math>, <math>~c</math>, <math>~M</math>, <math>~\Omega</math>, and <math>~\zeta</math>  .

==Free Energy of Incompressible, Constant Mass Systems==
We are interested, here, in examining how the free energy of such a system will vary as it is allowed to "evolve" as an ''incompressible'' fluid — ''i.e.,'' holding <math>~\rho</math> fixed — through different ellipsoidal shapes while conserving its total mass. Following [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], we choose to set <math>~M = 5</math> — which removes mass from the list of unspecified key parameters — and we choose to set <math>~\rho = \pi^{-1}</math>, which is then reflected in a specification of the semi-axis, <math>~a</math>, in terms of the pair of dimensionless axis ratios, <math>~b/a</math> and <math>~c/a</math>, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~a^3</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3Ma^2}{4\pi(bc)\rho} = \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1}\, .</math>
</td>
</tr>
</table>
</div>

Moving forward, then, a unique ellipsoidal configuration is identified via the specification of ''four'', rather than six, key parameters —   <math>~b/a</math>, <math>~c/a</math>, <math>~\Omega</math>, and <math>~x</math>   — and the free energy of that configuration is given by the expression,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~E\biggl(\frac{b}{a}, \frac{c}{a}, \Omega, x\biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{a^2}{2} \biggl[\biggl(1+\frac{b}{a} \cdot x\biggr)^2 + \biggl(\frac{b}{a}+x\biggr)^2\biggr]\Omega^2 - 2I </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{2/3} \biggl\{\frac{1}{2}
\biggl[\biggl(1+\frac{b}{a} \cdot x\biggr)^2 + \biggl(\frac{b}{a}+x\biggr)^2\biggr]\Omega^2 - \frac{2I}{a^2}\biggr\} \, ,</math>
</td>
</tr>
</table>
</div>

where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\biggl[\frac{(b/a)}{1 + (b/a)^2} \biggr]\frac{\zeta}{\Omega} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{I}{a^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[A_1 + A_2\biggl(\frac{b}{a}\biggr)^2 + A_3\biggl(\frac{c}{a}\biggr)^2 \biggr] \, ,</math>
</td>
</tr>
</table>
</div>

and the functional behavior of the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>, are given by the expressions provided in an [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Evaluation_of_Coefficients|accompanying discussion]].

<span id="E_Lexpression">Alternatively,</span> replacing <math>~\Omega</math> in favor of <math>~L</math>, we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~E\biggl(\frac{b}{a}, \frac{c}{a}, L, x\biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{L^2}{2a^2} \biggl[ \biggl(1+\frac{b}{a}\cdot x \biggr)^2
+ \biggl(\frac{b}{a}+x \biggr)^2 \biggr] \biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2 + 2\biggl(\frac{b}{a}\biggr)x \biggr]^{-2} - 2I </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{L^2}{2} \biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{-2/3}
\biggl[ \biggl(1+\frac{b}{a}\cdot x \biggr)^2 + \biggl(\frac{b}{a}+x \biggr)^2 \biggr]
\biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2 + 2\biggl(\frac{b}{a}\biggr)x \biggr]^{-2} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~- 2\biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{2/3}
\biggl[A_1 + A_2\biggl(\frac{b}{a}\biggr)^2 + A_3\biggl(\frac{c}{a}\biggr)^2 \biggr]\, .</math>
</td>
</tr>
</table>
</div>



==Adopted Evolutionary Constraints==

===Conserve Only L===
Let's fix the total angular momentum, <math>~L</math>, of a triaxial configuration and examine how the configuration's free energy varies as we allow it to contort through different triaxial shapes — that is, as its pair of axis ratios varies, always maintaining <math>~\tfrac{b}{a} < 1</math> — and as we vary <math>~x</math>, which characterizes the fraction of angular momentum that is stored in internal spin versus overall figure rotation. The desired free-energy function, <math>~E(\tfrac{b}{a},\tfrac{c}{a}, x)|_L</math>, has [[#E_Lexpression|just been defined]], but visualizing its behavior is difficult because, in this situation, the free energy is a warped, ''three-dimensional'' surface draped across the four-dimensional domain, <math>~(\tfrac{b}{a},\tfrac{c}{a}, x, E_L)</math>.

Acknowledging that we are primarily interested in identifying extrema of this free-energy function, the discussion presented in §3.2 of [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] shows us how to reduce the dimensionality of this problem by one. There, it is shown that, as long as <math>~\tfrac{b}{a} \ne 1</math>, extrema exist in the <math>~x</math>-coordinate direction — that is, <math>~\partial E_L/\partial x = 0</math> — only if <math>~x = 0.</math> For a given choice of <math>~L</math>, therefore, the relevant ''two-dimensional'' free-energy surface is defined by the expression,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~E\biggl(\frac{b}{a}, \frac{c}{a}, x=0\biggr)\biggr|_L</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{L^2}{2} \biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{-2/3}
\biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2\biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- 2\biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{2/3}
\biggl[A_1 + A_2\biggl(\frac{b}{a}\biggr)^2 + A_3\biggl(\frac{c}{a}\biggr)^2 \biggr]\, .</math>
</td>
</tr>
</table>
</div>

Figure 3 of [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] presents a black-and-white contour plot of this <math>~E_L</math> function for the specific case of <math>~L = 4.71488</math>, which, for reference, is the total angular momentum of an equilibrium [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|Maclaurin spheroid]] having an eccentricity, <math>~e = 0.85</math> (see [[#Table1|Table 1, below]]). We have digitally extracted this black-and-white contour plot from p. 477 of the (PDF-formatted) [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] publication and have reprinted it as the left-hand panel of our Figure 1. Note that we have flipped the plot horizontally and rotated it by 90° so that the orientation of the axis pair, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math>, conforms with the orientation of a related, information-rich diagram presented by [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract Chandrasekhar (1965)] — see also our [[User:Tohline/ThreeDimensionalConfigurations/JacobiEllipsoids#Sequence_Plots|accompanying discussion of equilibrium sequence plots]].

<div align="center" id="Figure1">
<table border="0" cellpadding="5" align="center">
<tr>
<th align="center" colspan="1"><font size="+1">Figure 1:</font> Free-Energy Surface Projected onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> Plane
</th>
</tr>
<tr><td align="center">

<table border="1" cellpadding="5" align="center">
<tr>
<td align="center" colspan="1">
[[File:JacobiPaperIFig3flipped.png|240px|Christodoulou1995Fig3 Flipped]]
</td>
<td align="center" colspan="1">
[[File:VisTrailsFig3e.png|240px|Both 2D contour plots overlaid]]
</td>
<td align="center" colspan="1">
[[File:VisTrailsFig3d.png|240px|Our 2D colored contour plot]]
</td>
</tr>
<tr>
<td align="left" colspan="3">
All three contour plots show how the free-energy, <math>~E_L</math>, varies across the <math>~(\tfrac{b}{a}, \tfrac{c}{a})</math> domain for the specific case of <math>~L = 4.71488</math>. Horizontal axis is <math>~0 \le \tfrac{b}{a} \le 1</math> and vertical axis is <math>~0 \le \tfrac{c}{a} \le 1</math>.
</td>
</tr>
<tr>
<td align="center" width="240px"><b>Left-hand Panel:</b><br />Black-and-white contour plot<br />
extracted from p. 477 of [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I]<br />
"''Phase-Transition Theory of Instabilities. I. Second-Harmonic Instability and Bifurcation Points''"<p></p>
ApJ, vol. 446, pp. 472-484 © [http://aas.org/ AAS]
</td>
<td align="center" width="240px"><b>Middle Panel:</b><br />Black-and-white contour plot digitally overlaid on color contour plot.</td>
<td align="center" width="240px"><b>Right-hand Panel:</b><br />Color contour plot<br />created here as a projection of the free-energy surface shown in Fig. 2.</td>
</tr>
</table>

</td></tr>
</table>
</div>

In our Figure 2, this same <math>~E_L</math> function has been displayed as a warped, two-dimensional free-energy surface draped across the three-dimensional <math>~(\tfrac{b}{a},\tfrac{c}{a},E)</math> domain, where depth as well as color has been used to tag energy values. The two-dimensional, colored contour plot presented in the right-hand panel of our Figure 1 results from the projection of this free-energy surface onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> plane; it reproduces in quantitative detail the black-and-white contour plot that we have extracted from [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I]. In an effort to (qualitatively) illustrate this agreement, we have digitally "pasted" the black-and-white contour plot from [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] onto our colored contour plot and presented the combined image in the middle panel of our Figure 1.

Our Figure 2 image of the free-energy surface helps illuminate the description of this surface that appears in the caption of Fig. 3 from [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I]. Quoting from that figure caption:  "The [equilibrium] Maclaurin spheroid sits on a saddle point <math>~[(\tfrac{b}{a},\tfrac{c}{a}) = (1.0,0.52678); E_0 = -7.81842]</math>, while a global minimum with <math>~E_0 = -7.83300</math> exists at <math>~(\tfrac{b}{a},\tfrac{c}{a}) = (0.588,0.428)</math>."

<div align="center" id="Figure2">
<table border="0" cellpadding="5" align="center">
<tr>
<th align="center" colspan="1"><font size="+1">Figure 2:</font> Free-Energy Surface
</th>
</tr>
<tr><td align="center">

<table border="1" cellpadding="0" align="center">
<tr>
<td align="center" colspan="1" bgcolor="#CCFFFF">
[[File:VistrailsFig3b.png|600px|Christodoulou1995Fig3 Flipped]]
</td>
</tr>
</table>

</td></tr>
</table>
</div>

===Animation===

The animation sequence presented, below, as Figure 3 displays the warped free-energy surface (right) in conjunction with its projection onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> plane (left) for configurations having nineteen different total angular momentum values, <math>~L</math>, as detailed in column 5 of Table 1. The four-digit number that tags each frame of this animation sequence identifies the eccentricity (column 1 of Table 1) of the Maclaurin spheroid that is associated with each selected value of <math>~L</math>. In each frame of the animation, the equilibrium configuration associated with that Maclaurin spheroid is identified by the extremum of the free energy that appears along the right-hand edge <math>~(\tfrac{b}{a} = 1)</math> of the warped surface. For values of <math>~e < 0.81267</math> — corresponding to <math>~L < 4.23296</math> — the Maclaurin spheroid (marked by a small white circle/sphere) sits at the location of the absolute minimum of the free-energy surface and the configuration is stable. But for all larger values of the eccentricity/angular momentum, the Maclaurin spheroid (marked by a small dark-blue circle/sphere) is associated with a ''saddle point'' of the free-energy surface — that is, the configuration is in equilibrium, but it is (secularly) unstable — and the absolute energy minimum shifts off-axis to the location of a Jacobi ellipsoid (marked by a small white circle/sphere) having the same total angular momentum. As [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] points out, evolution from the unstable axisymmetric equilibrium configuration to the stable triaxial configuration occurs along the narrow valley/canyon connecting the two extrema of the free energy.

<div align="center" id="Figure3">
<table border="0" cellpadding="5" align="center">
<tr>
<th align="center" colspan="1"><font size="+1">Figure 3:</font> Animation
</th>
</tr>
<tr><td align="center">

<table border="1" cellpadding="0" align="center">
<tr>
<td align="center" colspan="1" bgcolor="#CCFFFF">
[[File:JacobiMaclaurin2.gif|640px|Animation related to Fig. 3 from Christodoulou1995]]
</td>
</tr>
</table>

</td></tr>
</table>
</div>



<div align="center" id="Table1">
<table border="1" cellpadding="5" align="center" width="500px">
<tr>
<td align="center" colspan="8"><b><font size="+1">Table 1:</font>  Parameter Values Associated with Each Frame
of the Figure 3 Animation</b><br />
(parameter values associated with Figures 1 & 2 are highlighted in pink)
</td>
</tr>
<tr>
<td align="center" colspan="4">Maclaurin Spheroid</td>
<td align="center" rowspan="2"><math>~L^\dagger</math></td>
<td align="center" colspan="3">Jacobi Ellipsoids</td>
</tr>
<tr>
<td align="center"><math>~e</math></td>
<td align="center"><math>~\frac{c}{a}</math></td>
<td align="center"><math>~E_L</math></td>
<td align="center"><math>~E_\mathrm{plot}^\ddagger</math></td>
<td align="center"><math>~\frac{b}{a}</math></td>
<td align="center"><math>~\frac{c}{a}</math></td>
<td align="center"><math>~E_\mathrm{Jac}</math></td>
</tr>
<tr>
<td align="center">0.650</td>
<td align="center">0.7599342</td>
<td align="center">-8.9018255</td>
<td align="center">0.0</td>
<td align="center">2.8270256</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
</tr>
<tr>
<td align="center">0.675</td>
<td align="center">0.7378177</td>
<td align="center">-8.8165100</td>
<td align="center">0.0</td>
<td align="center">2.9985043</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
</tr>
<tr>
<td align="center">0.700</td>
<td align="center">0.7141428</td>
<td align="center">-8.7216343</td>
<td align="center">0.0</td>
<td align="center">3.1820090</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
</tr>
<tr>
<td align="center">0.725</td>
<td align="center">0.6887489</td>
<td align="center">-8.6155943</td>
<td align="center">0.0</td>
<td align="center">3.3796768</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
</tr>
<tr>
<td align="center">0.750</td>
<td align="center">0.66143783</td>
<td align="center">-8.4963506</td>
<td align="center">0.0</td>
<td align="center">3.5942337</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
</tr>
<tr>
<td align="center">0.775</td>
<td align="center">0.6319612</td>
<td align="center">-8.3612566</td>
<td align="center">0.0</td>
<td align="center">3.8292360</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
</tr>
<tr>
<td align="center">0.790</td>
<td align="center">0.6131068</td>
<td align="center">-8.2711758</td>
<td align="center">0.0</td>
<td align="center">3.9819677</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
</tr>
<tr>
<td align="center">0.795</td>
<td align="center">0.6066094</td>
<td align="center">-8.2394436</td>
<td align="center">0.0</td>
<td align="center">4.0351072</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
</tr>
<tr>
<td align="center">0.800</td>
<td align="center">0.6000000</td>
<td align="center">-8.2067933</td>
<td align="center">0.0</td>
<td align="center">4.0894508</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
</tr>
<tr>
<td align="center">0.805</td>
<td align="center">0.5932748</td>
<td align="center">-8.1731817</td>
<td align="center">0.0</td>
<td align="center">4.1450581</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
</tr>
<tr>
<td align="center">0.810</td>
<td align="center">0.5864299</td>
<td align="center">-8.1385621</td>
<td align="center">0.0</td>
<td align="center">4.2019932</td>
<td align="center">---</td>
<td align="center">---</td>
<td align="center">---</td>
</tr>
<tr>
<td align="center">0.815</td>
<td align="center">0.5794610</td>
<td align="center">-8.1028846</td>
<td align="center">0.0064</td>
<td align="center">4.2603252</td>
<td align="center">0.880967</td>
<td align="center">0.545588</td>
<td align="center">-8.102934</td>
</tr>
<tr>
<td align="center">0.820</td>
<td align="center">0.5723635</td>
<td align="center">-8.0660955</td>
<td align="center">0.0524</td>
<td align="center">4.3201286</td>
<td align="center">0.797543</td>
<td align="center">0.516311</td>
<td align="center">-8.066596</td>
</tr>
<tr>
<td align="center">0.825</td>
<td align="center">0.5651327</td>
<td align="center">-8.0281369</td>
<td align="center">0.1116</td>
<td align="center">4.3814839</td>
<td align="center">0.744298</td>
<td align="center">0.496028</td>
<td align="center">-8.029578</td>
</tr>
<tr>
<td align="center">0.830</td>
<td align="center">0.5577634</td>
<td align="center">-7.9889461</td>
<td align="center">0.1665</td>
<td align="center">4.4444785</td>
<td align="center">0.702967</td>
<td align="center">0.479341</td>
<td align="center">-7.991848</td>
</tr>
<tr>
<td align="center">0.835</td>
<td align="center">0.5502499</td>
<td align="center">-7.9484555</td>
<td align="center">0.2140</td>
<td align="center">4.5092074</td>
<td align="center">0.668439</td>
<td align="center">0.464724</td>
<td align="center">-7.953367</td>
</tr>
<tr>
<td align="center">0.840</td>
<td align="center">0.5425864</td>
<td align="center">-7.9065917</td>
<td align="center">0.2551</td>
<td align="center">4.5757737</td>
<td align="center">0.638420</td>
<td align="center">0.451485</td>
<td align="center">-7.914095</td>
</tr>
<tr>
<td align="center">0.845</td>
<td align="center">0.5347663</td>
<td align="center">-7.8632747</td>
<td align="center">0.2912</td>
<td align="center">4.6442903</td>
<td align="center">0.611646</td>
<td align="center">0.439241</td>
<td align="center">-7.873990</td>
</tr>
<tr>
<td align="center" bgcolor="pink">0.850</td>
<td align="center" bgcolor="pink">0.5267827</td>
<td align="center" bgcolor="pink">-7.8184175</td>
<td align="center">0.3232</td>
<td align="center" bgcolor="pink">4.7148806</td>
<td align="center" bgcolor="pink">0.587337</td>
<td align="center" bgcolor="pink">0.427750</td>
<td align="center" bgcolor="pink">-7.833003</td>
</tr>
<tr>
<td align="left" colspan="8"><sup>&dagger;</sup>Here, the units of angular momentum are as used in [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I]. In order to convert to units of <math>~L</math> as used in EFE (see, for example, Table I, in Chapter 5, §32), multiply by <math>~[2^2/(3\cdot 5^{10})]^{1/6} = 0.0717585</math>.
</tr>
<tr>
<td align="left" colspan="8"><sup>&Dagger;</sup><math>~E_\mathrm{plot}</math> is a normalized value of <math>~E_L</math> that has been used for plotting purposes. It's definition is: <math>E_\mathrm{plot} = 0.25*\biggl\{ \log_{10}\biggl[0.0001 + \frac{(E_L + |E_\mathrm{Jac}|)}{|E_\mathrm{Jac}|} \biggr] + 4\biggr\}</math>
</tr>
</table>
</div>

<pre>
PROPERTIES OF VARIOUS MACLAURIN SPHEROIDS

eccenL covera omega2 ellChandra L

0.815 5.7946096D-01 3.7625539D-01 3.0571438D-01 4.2603252D+00
0.820 5.7236352D-01 3.8058727D-01 3.1000578D-01 4.3201286D+00
0.825 5.6513273D-01 3.8489420D-01 3.1440854D-01 4.3814839D+00
0.830 5.5776339D-01 3.8917054D-01 3.1892894D-01 4.4444785D+00
0.835 5.5024994D-01 3.9341001D-01 3.2357378D-01 4.5092074D+00
0.840 5.4258640D-01 3.9760569D-01 3.2835048D-01 4.5757737D+00
0.845 5.3476630D-01 4.0174986D-01 3.3326712D-01 4.6442903D+00
0.850 5.2678269D-01 4.0583395D-01 3.3833257D-01 4.7148806D+00
0.855 5.1862800D-01 4.0984835D-01 3.4355656D-01 4.7876802D+00
0.860 5.1029403D-01 4.1378236D-01 3.4894980D-01 4.8628384D+00
0.865 5.0177186D-01 4.1762394D-01 3.5452411D-01 4.9405200D+00
0.870 4.9305172D-01 4.2135955D-01 3.6029264D-01 5.0209081D+00
0.875 4.8412292D-01 4.2497391D-01 3.6626999D-01 5.1042063D+00
0.880 4.7497368D-01 4.2844972D-01 3.7247248D-01 5.1906420D+00
0.885 4.6559102D-01 4.3176729D-01 3.7891846D-01 5.2804708D+00
0.890 4.5596052D-01 4.3490417D-01 3.8562861D-01 5.3739810D+00
0.895 4.4606614D-01 4.3783459D-01 3.9262639D-01 5.4714996D+00
0.900 4.3588989D-01 4.4052888D-01 3.9993856D-01 5.5733994D+00
0.905 4.2541157D-01 4.4295266D-01 4.0759585D-01 5.6801086D+00
0.910 4.1460825D-01 4.4506586D-01 4.1563375D-01 5.7921218D+00
0.915 4.0345384D-01 4.4682147D-01 4.2409362D-01 5.9100155D+00
0.920 3.9191836D-01 4.4816395D-01 4.3302405D-01 6.0344667D+00
0.925 3.7996710D-01 4.4902713D-01 4.4248265D-01 6.1662784D+00
0.930 3.6755952D-01 4.4933139D-01 4.5253852D-01 6.3064134D+00
0.935 3.5464771D-01 4.4897998D-01 4.6327550D-01 6.4560401D+00
0.940 3.4117444D-01 4.4785386D-01 4.7479681D-01 6.6165969D+00
0.945 3.2707033D-01 4.4580450D-01 4.8723156D-01 6.7898831D+00
0.950 3.1224990D-01 4.4264348D-01 5.0074442D-01 6.9781934D+00
</pre>

<pre>
PROPERTIES OF JACOBI ELLIPSOIDS THAT HAVE THE SAME ANGULAR MOMENTA (L) AS THE ABOVE MACLAURIN SPHEROIDS

e b/a c/a A1 A2 A3 omega2 a L_C L energy

0.815 0.880967 0.545588 0.474189 0.557354 0.968456 0.371826 1.983364 0.305714 4.260325 -8.102934
0.820 0.797543 0.516311 0.441622 0.589410 0.968968 0.366634 2.088279 0.310006 4.320129 -8.066596
0.825 0.744298 0.496028 0.419233 0.611283 0.969484 0.361394 2.165672 0.314409 4.381484 -8.029578
0.830 0.702967 0.479341 0.400927 0.629069 0.970004 0.356104 2.232633 0.318929 4.444479 -7.991848
0.835 0.668439 0.464724 0.384983 0.644489 0.970527 0.350761 2.293990 0.323574 4.509207 -7.953367
0.840 0.638420 0.451485 0.370620 0.658325 0.971055 0.345362 2.351945 0.328350 4.575774 -7.914095
0.845 0.611646 0.439241 0.357403 0.671009 0.971588 0.339905 2.407740 0.333267 4.644290 -7.873990
0.850 0.587337 0.427750 0.345063 0.682812 0.972126 0.334386 2.462170 0.338333 4.714881 -7.833003
0.855 0.564969 0.416851 0.333417 0.693915 0.972668 0.328802 2.515795 0.343557 4.787680 -7.791082
0.860 0.544173 0.406427 0.322334 0.704450 0.973216 0.323150 2.569038 0.348950 4.862838 -7.748172
0.865 0.524676 0.396390 0.311717 0.714514 0.973770 0.317425 2.622239 0.354524 4.940520 -7.704210
0.870 0.506269 0.386673 0.301490 0.724181 0.974329 0.311624 2.675686 0.360293 5.020908 -7.659127
0.875 0.488788 0.377221 0.291594 0.733511 0.974895 0.305741 2.729638 0.366270 5.104206 -7.612848
0.880 0.472100 0.367988 0.281979 0.742554 0.975467 0.299772 2.784332 0.372472 5.190642 -7.565289
0.885 0.456097 0.358937 0.272604 0.751348 0.976047 0.293710 2.840003 0.378918 5.280471 -7.516357
0.890 0.440687 0.350033 0.263435 0.759930 0.976635 0.287549 2.896880 0.385629 5.373981 -7.465947
0.895 0.425792 0.341249 0.254440 0.768329 0.977230 0.281283 2.955205 0.392626 5.471500 -7.413941
0.900 0.411344 0.332556 0.245593 0.776572 0.977835 0.274902 3.015230 0.399939 5.573399 -7.360207
0.905 0.397284 0.323929 0.236868 0.784683 0.978449 0.268398 3.077231 0.407596 5.680109 -7.304592
0.910 0.383556 0.315345 0.228242 0.792685 0.979073 0.261762 3.141512 0.415634 5.792122 -7.246923
0.915 0.370112 0.306780 0.219694 0.800598 0.979708 0.254980 3.208416 0.424094 5.910016 -7.187000
0.920 0.356903 0.298209 0.211202 0.808442 0.980356 0.248041 3.278337 0.433024 6.034467 -7.124589
0.925 0.343885 0.289608 0.202744 0.816240 0.981017 0.240927 3.351732 0.442483 6.166278 -7.059417
0.930 0.331013 0.280950 0.194298 0.824010 0.981692 0.233621 3.429144 0.452539 6.306413 -6.991160
0.935 0.318242 0.272206 0.185842 0.831774 0.982384 0.226102 3.511226 0.463276 6.456040 -6.919428
0.940 0.305523 0.263344 0.177348 0.839558 0.983094 0.218342 3.598775 0.474797 6.616597 -6.843746
0.945 0.292805 0.254325 0.168790 0.847385 0.983825 0.210310 3.692789 0.487232 6.789883 -6.763530
0.950 0.280029 0.245105 0.160133 0.855288 0.984579 0.201966 3.794537 0.500744 6.978193 -6.678040
</pre>

=See Also=
* [[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|Properties of Maclaurin Spheroids]]
* [[Apps/MaclaurinSpheroids/GoogleBooks#Excerpts_from_A_Treatise_of_Fluxions|Excerpts from Maclaurin's (1742) ''A Treatise of Fluxions'']]

{{SGFfooter}}

ThreeDimensionalConfigurations/BinaryFission

2021-09-30T22:57:33Z

174.64.14.12: Created page with "__FORCETOC__ =Fission Hypothesis of Binary Star Formation= <table border="0" cellpadding="3" align="center" width="60%"> <tr><td align="left"> <font color="darkgreen"> "This..."

__FORCETOC__

=Fission Hypothesis of Binary Star Formation=
<table border="0" cellpadding="3" align="center" width="60%">
<tr><td align="left">
<font color="darkgreen">
"This paper is concerned with one of the oldest hypotheses in stellar-evolution theory, namely, that a binary system originates from the splitting of a single star into two components due to rotational instability during contraction of the star. In spite of many attempts, no satisfactory theory of this process has been developed …"</font>
</td></tr>
<tr><td align="right">
— Drawn from §I of [https://ui.adsabs.harvard.edu/abs/1966ApJ...143..111R/abstract Ian W. Roxburgh (1966)]
</td></tr></table>

==Potentially Useful References==

* [https://ui.adsabs.harvard.edu/abs/1965RSPSA.286....1C/abstract S. Chandrasekhar (1965)], Proc. Roy. Soc. London. Series A, Mathematical and Physical Sciences, 286, pp. 1 - 26: ''The Stability of a Rotating Liquid Drop''
* [https://ui.adsabs.harvard.edu/abs/1966ApJ...143..111R/abstract I. W. Roxburgh (1966)], ApJ, 143, 111: ''On the Fission Theory of the Origin of Binary Stars''
* [https://trove.nla.gov.au/work/21340423?selectedversion=NBD179107 J. P. Ostriker (1970)], Stellar Rotation: Proceedings of the IAU Colloquium held at the Ohio State University, Columbus, OH, U.S.A., September 8-11, 1969, edited by Arne Slettebak, p. 147: <font color="maroon">''Fission and the Origin of Binary Stars''</font>
* [https://ui.adsabs.harvard.edu/abs/1972ApJ...175..171L/abstract N. R. Lebovitz (1972)], ApJ, 175, 171: <font color="maroon">''On the Fission Theory of Binary Stars''</font>
* §IVa of [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract P. Bodenheimer & J. P. Ostriker (1973)], ApJ, 180, 159 [Part VIII]
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%"> </td><td align="left">
<font color="green">The ancient problem of whether a rotating, contracting star will, if it has sufficient angular momentum, subdivide and become a binary system must now be reexamined. The existence of generalized zero-viscosity polytropic sequences which do not terminate and which do reach the points of dynamical over stability … answers several of the objections that have been raised against the fission hypothesis. The new situation was reviewed by [https://trove.nla.gov.au/work/21340423?selectedversion=NBD179107 J. P. Ostriker (1970)] in a discussion which anticipates the results presented in this paper. A point of view is presented by [https://ui.adsabs.harvard.edu/abs/1972ApJ...175..171L/abstract N. R. Lebovitz (1972)] which envisions a different evolution but predicts fission at the same critical value of</font> <math>~T/|W|</math>.
</td></tr></table>
* [https://ui.adsabs.harvard.edu/abs/1984Ap%26SS..99...71H/abstract I. Hachisu & Y. Eriguchi (1984)], Astrophysics & Space Sciences, 99, 71: ''Fission Sequence and Equilibrium Models of Rigidity [sic] Rotating Polytropes''
* [http://www.phys.lsu.edu/astro/movie_captions/fission.html Webpage summary of Cazes' simulations]
* [https://inis.iaea.org/search/search.aspx?orig_q=RN:17071096 R. H. Durisen & J. E. Tohline (1985)], ''Protostars and Planets II'', pp. 534 - 575, Univ. of Arizona Press: <font color="maroon"> ''Fission of Rapidly Rotating Fluid Systems''</font>
* [http://articles.adsabs.harvard.edu//full/1986ormo.conf..487D/0000487.000.html Durisen & Gingold (198x)], <font color="maroon">''Numerical Simulations of Fission''</font>
* [https://ui.adsabs.harvard.edu/abs/1987GApFD..38...15L/abstract N. R. Lebovitz (1987)], Geophysical and Astrophysical Fluid Dynamics, 38, 15: <font color="maroon">''Binary Fission via Inviscid Trajectories''</font>
<ul>
<li>Phase-Transition Theory of Instabilities:</li>
<ol type="I">
<li>[https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract D. M. Christodoulou, D. Kazanas, I. Shlosman & J. E. Tohline (1995a)], ApJ, 446, 472: ''Second-Harmonic Instability and Bifurcation Points''</li>
<li>[https://ui.adsabs.harvard.edu/abs/1995ApJ...446..485C/abstract D. M. Christodoulou, D. Kazanas, I. Shlosman & J. E. Tohline (1995b)], ApJ, 446, 485: ''Fourth-Harmonic Bifurcations and λ-Transitions''</li>
<li>[https://ui.adsabs.harvard.edu/abs/1995ApJ...446..500C/abstract D. M. Christodoulou, D. Kazanas, I. Shlosman & J. E. Tohline (1995c)], ApJ, 446, 500: ''The Third-Harmonic Bifurcation on the Jacobi Sequence and the Fission Problem''</li>
<li>[https://ui.adsabs.harvard.edu/abs/1995ApJ...446..510C/abstract D. M. Christodoulou, D. Kazanas, I. Shlosman & J. E. Tohline (1995d)], ApJ, 446, 510: ''Critical Points on the Maclaurin Sequence and Nonlinear Fission Processes''</li>
</ol>
</ul>
* [https://ui.adsabs.harvard.edu/abs/2001IAUS..200...23B/abstract I. A. Bonnell (2001)], Proceedings of IAU Symposium 200, held 10 - 15 April, 2000 in Potsdam, Germany, edited by Hans Zinnecker and Robert D. Mathieu, p. 23: ''The Formation of Close Binary Stars''
* [https://ui.adsabs.harvard.edu/abs/2002ARA%26A..40..349T/abstract J. T. Tohline (2002)], ARAA, 40, 349: ''The Origin of Binary Stars''

==Illustration==

<div align="center">
<table border="1" cellpadding="5">
<tr>
<td align="center">
'''Figure 1'''
<br>
[[File:SkylabFission.jpg|300px|Droplet Fission]]
<br><br>
YouTube video:
<br>
[http://www.youtube.com/watch?v=61dH_CS_oqA Skylab Drop Dynamics Experiment (1975)]
</td>
<td align="center" rowspan="1" colspan="2">
'''Figure 2'''
<br>
[[File:BrownScriven1980_Fig5.jpg|300px|Theoretical Model]]<br>
[http://adsabs.harvard.edu/abs/1980RSPSA.371..331B Brown & Scriven (1980)]
<br>
(Journal of Fluid Mechanics, 276, 389)
</td>
<td align="center" rowspan="2">
'''Figure 4'''
<br>
[[File:Wang1994_Fig10.jpg|350px|USML-1 Experiment]]<br>
[http://dx.doi.org/10.1017/S0022112094002612 Wang, Anilkumar, Lee & Lin (1994)]
<br>
(Proc. Roy. Soc. London, 371, 331)
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2">
'''<font color="black">Figure 3</font>'''
<br>
[[File:HachisuEriguchi1984.jpg|300px|Hachisu & Eriguchi scenario]]<br>
[http://adsabs.harvard.edu/abs/1984Ap%26SS..99...71H Hachisu & Eriguchi (1984)]
<br>
(Astrophysics and Space Science, 99, 71)
</td>
<td align="center" valign="top" colspan="1" bgcolor="black">
'''<font color="white">Figure 5</font>'''
<br>
[[File:LSU_Stable.animated.gif|100px|Kimberly New simulation]]<br>
[http://adsabs.harvard.edu/abs/1997ApJ...490..311N Figure 10 from New & Tohline (1997)]<br>
<font color="white">(The Astrophysical Journal, 490, 311)</font>
</td>
</tr>
</table>
</div>

=Related Discussions=
==Fission in Nuclear Physics==
The nuclear physics community also draws an analogy between the fission of a rotating fluid drop and the spontaneous fission of atomic nuclei; see, for example, the figure associated with the [http://en.wikipedia.org/wiki/Nuclear_fission#Energetics Wikipedia discussion of the energetics of nuclear fission].

See also [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967b)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''. The first few lines of the Appendix of this paper state …
<table align="center" width="80%"><tr><td align="left">
<font color="green">
"The virial method has recently been extended to phenomena of interest in the theory of nuclear fission ([https://ui.adsabs.harvard.edu/abs/1967JMP.....8...98R/abstract Rosenkilde 1967a)]</font>, Journal of Mathematical Physics, 8, 98 <font color="green">… For a rotating, charged, inviscid liquid drop held together by surface tension, there exist axisymmetric figures of equilibrium which are nearly spheroids. Moreover, prolate as well as oblate figures are possible. If these figures are assumed to be spheroids, then the investigation (based on the virial method) of their stability closely resembles the corresponding investigation for the self-gravitating Maclaurin spheroids …"
</font>
</td></tr></table>

==Drop Dynamics Experiments==
[On '''<font color="red">1 January 2014</font>''', J. E. Tohline wrote ...] As I was putting this chapter together, I had difficulty documenting the various drop dynamics experiments that have been conducted by astronauts in various Earth-orbiting (zero <math>g</math>) environments. Here is the relevant information that I have found, to date.
===Skylab===
Experiments showing the ''fission'' of liquid drops were evidently conducted during the Skylab 2, Skylab 3, and Skylab 4 missions (circa 1973-1974).
* As has been documented in a short film review written by Howard Voss and published in the [http://dx.doi.org/10.1119/1.10227 American Journal of Physics (44/10, 1021, Oct 1976)], film footage from a variety of Skylab experiments was produced by NASA, edited by Thomas Campbell & Robert Fuller, and, beginning in 1976, distributed as 12 [http://en.wikipedia.org/wiki/Super_8_film Super 8] film loops by the [http://www.aapt.org/ American Association of Physics Teachers] (AAPT).
* As is documented in [http://web.physics.ucsb.edu/~lecturedemonstrations/Linked%20files/Media%20library/Skylab%20guide%20(videodisc).pdf A Teacher's Guide for the Skylab Physics Videodisc] the content of all 12 Super 8 film loops was made available for distribution in [http://en.wikipedia.org/wiki/Videodisc Videodisc] format in 1987 through the [http://www.aapt.org/ AAPT].
* The YouTube video referenced in and linked to the caption of Figure 1, above, is the digitized version of the Skylab film loop that illustrates fission of a water droplet.

<div align="center">
<table border="1" width="65%" cellpadding="8" cellspacing="3">
<tr>
<td align="left" colspan="2">
<font size="-1">According to the [http://web.physics.ucsb.edu/~lecturedemonstrations/Linked%20files/Media%20library/Skylab%20guide%20(videodisc).pdf Teacher's Guide] mentioned above, the activities shown in the above-referenced films were carried out by three teams of Skylab Astronauts:</font>
</td>
</tr>
<tr>
<td align="center">
[[File:SkylabAstronauts.jpg|500px|center|Skylab Astronauts]]
</td>
<td align="center">
<font size="-1">
[[File:Skylab_2_Kerwin3.jpg|thumb|Kerwin blows water droplet from a straw]]
<br>
[http://en.wikipedia.org/wiki/Skylab_2 Skylab 2] (First Team)
</font>
</td>
</tr>
</table>
</div>

===Space Shuttle Flights===
Experiments illustrating the dynamical behavior of liquid drops were conducted during several space shuttle missions. Some experiments were performed inside the European Space Agency's "spacelab module" and others were performed with the aid of a "Drop Physics Module (DPM)" inside the United States Microgravity Laboratory (USML), each being a "portable" laboratory that was housed in the shuttle's payload bay.
* [[File:Sts51b_patch.jpg|100px|right|Spacelab 3]]Wang, Trinh, Croonquist, and Elleman (1986; [http://adsabs.harvard.edu/abs/1986PhRvL..56..452W Physical Review Letters, 56, 452]) report results from a controlled drop dynamics experiment that was conducted during the "Spacelab III mission" (see the final acknowledgement paragraph of their paper), which took place during shuttle flight [http://en.wikipedia.org/wiki/STS-51-B STS-51-B (29 April - 6 May 1985)]. Taylor Wang — one of the authors of this ''PRL'' publication — flew as one of the seven members of the space shuttle crew, specifically as Payload Specialist 2. His narrated account of some of the experimental activities is available, beginning at 7 minutes 32 seconds into the [http://www.nss.org/resources/library/shuttlevideos/shuttle17.htm STS-51B "Post Flight Presentation" video].
* [[File:fluid_drop.jpg|100px|right|frame|USML-1 Droplet Fission]]Another mission — [http://www.nasa.gov/mission_pages/shuttle/shuttlemissions/archives/sts-50.html USML-1 during shuttle flight STS-50] — took place in early 1992. According to [http://www.jpl.nasa.gov/releases/95/release_1995_9571.html information provided by NASA/JPL's public information office], "… the transition of rotating liquid drops into a 'dog-bone,' or two-lobed shape, was studied in detail …" As can be seen, beginning at 2 minutes 28 seconds into the [http://www.nss.org/resources/library/shuttlevideos/shuttle48.htm STS-50 "Post Flight Presentation" video], Eugene Trinh flew was one of the members of this shuttle mission who conducted these DMP experiments. Detailed results from DPM experiments during the USML-1 mission have been published in the Journal of Fluid Mechanics: T. G. Wang, A. V. Anilkumar, C. P. Lee and K. C. Lin (1994). ''Bifurcation of rotating liquid drops: results from USML-1 experiments in Space.'' [http://dx.doi.org/10.1017/S0022112094002612 Journal of Fluid Mechanics, 276, pp 389-403]
* I think that the three-frame black & white image shown here on the right presents a result from mission USML-1. That is how this image is referenced in an [http://www.phys.lsu.edu/astro/movie_captions/fission.html online discussion of fission] that I put together about a decade ago.
* Yet another mission — [http://www.nasa.gov/mission_pages/shuttle/shuttlemissions/archives/sts-73.html USML-2 during shuttle flight STS-73] — took place in the fall of 1995. Some additional drop dynamics experiments were conducted during this mission — see, for example, about 7 minutes and 40 seconds into the [http://www.nss.org/resources/library/shuttlevideos/shuttle72.htm STS-73 "Post Flight Presentation" video]. Lee, Anilkumar, Hmelo, & Wang (1998; [http://adsabs.harvard.edu/abs/1998JFM...354...43L Journal of Fluid Mechanics, 354, 43]) report results from these controlled drop dynamics experiments.

===International Space Station===
* See the two "Gallery of Fluid Motions" mpg movies that accompany the preprint by [http://arxiv.org/abs/1210.4073v1 Ueno et al. (2012)].

==Online References==
* [http://www.phys.lsu.edu/astro/movie_captions/fission.html The Fission Mechanism for Binary Star Formation]
* [http://www.phys.lsu.edu/~tohline/fission.movies.html Fission Simulations at LSU]
* T. G. Wang, A. V. Anilkumar, C. P. Lee and K. C. Lin (1994). ''Bifurcation of rotating liquid drops: results from USML-1 experiments in Space.'' [http://dx.doi.org/10.1017/S0022112094002612 Journal of Fluid Mechanics, 276, pp 389-403]
* [http://trs-new.jpl.nasa.gov/dspace/bitstream/2014/18294/1/99-1767.pdf Ohsaka & Trinh (19xx)]

{{SGFfooter}}

ThreeDimensionalConfigurations/CAREs

2021-09-30T22:55:21Z

174.64.14.12: Created page with "__FORCETOC__  =New Thoughts Regarding the Formation of Binary Stars= It is Februa..."

__FORCETOC__

=New Thoughts Regarding the Formation of Binary Stars=

It is February, 2020 and I have begun to develop some new ideas regarding the manner by which binary stars might form from initially equilibrium configurations. These new thoughts have been sparked by my recent experience playing with — and, in particular, employing COLLADA to help build visually insightful models of — Type I Riemann ellipsoids.

==Learning from Classical Fission Hypothesis==

According to the classical fission hypothesis as described in particular by Lebovitz, relying especially on equilibrium models of incompressible fluids, an initially axisymmetric configuration (Maclaurin spheroid) can spontaneously deform into an ellipsoidal configuration when the Jacobi sequence bifurcates from the Maclaurin sequence. (Models along the Jacobi sequence are examples of Riemann S-type ellipsoids.) Upon further contraction/cooling along the Jacobi sequence, the equilibrium configuration eventually encounters an m = 3 instability; or, alternatively, it may have an opportunity to deform gradually into a peanut-shaped, "contact" binary system.

Using self-consistent-field techniques, numerous groups have been able to numerically construct compressible analogs of Maclaurin spheroids. Most have employed polytropic equations of state having a variety of different index values; generally speaking, highly flattened configurations can be constructed only if they are differentially rotating.

Over the years, especially through collaborations with Dick Durisen, Harold Williams, and John Cazes, nonlinear hydrodynamic techniques have been used to model the spontaneous development of nonaxisymmetric structure — almost exclusively, m = 2 bar-like distortions — in equilibrium models that are compressible analogs of Maclaurin spheroids. Mostly we have examined models that are ''dynamically'' unstable, which means that we have chosen models that have a T/|W| that is significantly higher than what is necessary to encounter the Jacobi-sequence bifurcation. One key exception is the modeling performed by Shangli Ou in collaboration with Lee Lindblom; we introduced a post-Newtonian radiation-reaction term into the simulation and were able to watch lower T/|W| models deform and evolve to the Dedekind sequence, which is adjoint sequence to the Jacobi sequence.

While evolving high T/|W| models that are dynamically unstable to the bar-mode, John Cazes demonstrated that the initially axisymmetric configuration evolves to a quasi-steady-state ellipsoidal configuration that, in many respects, can be described as a compressible analog of a Riemann S-type ellipsoid (CARE). [Note: He obtained qualitatively similar evolutionary results whether his initial Maclaurin-like model had an n' = 3/2 angular velocity profile, or had uniform vortensity.] This relatively long-lived bar-like configuration is differentially rotating (see relevant movie) and exhibits a "violin Mach surface" where the fluid transitions from supersonic to subsonic regions. The mild shock fronts associated with the violin Mach surface necessarily introduce dissipation, and therefore each CARE is, strictly speaking, cannot be steady-state. Two especially relevant elements of Cazes' simulations are the following:

<ol>
<li>
In Chandrasekhar's EFE, there is a chapter/subsection that discusses the nonlinear evolution of models from the Maclaurin sequence to Riemann S-type configurations. Evidently in the mid-to-late 60s, one researcher published a paper in which he "numerically integrated" a few such evolutions to illustrate how evolution toward a specific Riemann model might occur. It appears as though, using energy minima arguments, Christodoulou has presented a similar evolution; he did this, in part, in an effort to perhaps quantitatively understand Cazes' simulation results.
</li>
<li>
Cazes slowly "cooled" one of his CARE models and discovered that the bar became even more elongated over time and, eventually, encountered a "radial" oscillation which sloshed the fluid back and forth between a centrally condensed configuration to a configuration having a pair of off-axis density maxima — presenting the appearance of a common-envelope binary. The flow-field (see movie) showed some common-envelope flow, but also displayed circulation about the off-axis density maxima. This is the closest we have come to actually ''seeing'' an event that loosely can be described as binary formation.
</li>
</ol>

==Andalib's Work==
<ol>
<li>
Can we develop an SCF technique that let's us generate a wide range of CAREs? This may be impossible, given that the Cazes models exhibit a violin Mach surface and, therefore, do not actually represent steady-state structures.
</li>
<li>
What should the 3D velocity field look like? The Cazes CAREs appear to exhibit an angular-velocity profile that is independent of the vertical (z) coordianate.
</li>
<li>
Using an approach analogous to that used by [[User:Tohline/Apps/Korycansky_Papaloizou_1996#Korycansky_and_Papaloizou_.281996.29|Korycansy & Papaloizou (1996)]], Andalib was able to generate two-dimensional compressible equilibrium structures that have a wide variety of nonaxisymmetric (bar-like ''and'' binary like) shapes and internal velocity distributions; they all have uniform vortensity. Some look quite similar to the flows found in Cazes quasi-steady-state CAREs, including the common-envelope binary flow. How do we extend this work to 3D structures? I developed some on-line notes that are helpful, but incomplete; this was done following a discussion in the spring of 2010 that I had with David and Eric at BYU. I tried, for example, enforcing no motion in the vertical (z) direction, but the remaining constraint equations were still pretty daunting.
</li>
</ol>

==New Idea==

Perhaps we should move away from Riemann S-type ellipsoids and attempt instead to develop an SCF technique that can be used to construct a variety of Type I Riemann ellipsoids. This might be useful because:
<ol>
<li>
The velocity flow-field is not constrained to be independent of z. (Although a related constraint appears to be in place.) This would lead to significant redesign of my above-mentioned "Incomplete" on-line notes. This state of affairs might demand that each system evolve on a viscous time scale toward a flow field that has no z-component.
</li>
<li>
In the Type I Riemann ellipsoids, the steady-state flow-field already shows a pair of off-axis circulations — analogous to what is seen in a binary system — even though the underlying density distribution is uniform. This flow would presumably get "locked in" as the configuration cooled and encourage/necessarily imply the development of off-axis density maxima develop.
</li>
<li>
Given that the spin-axis of the interflow is not aligned with the spin-axis of the ellipsoidal figure, one can imagine that a binary system formed from such a configuration would have a wide range of interesting properties; the spin and orbital angular momentum axes would be different from one another, for example. This reminds me of work that Peter Bodenheimer did in the late 70s — see [https://ui.adsabs.harvard.edu/abs/1978ApJ...224..488B/abstract P. Bodenheimer (1978, ApJ, 224, 448)] — when he used fragmentation of "rings' to estimate how multiple multiple star systems might divide up the angular momentum.
</li>
</ol>

=See Also=
<p> </p>
{| class="ChallengesA" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"
|-
! style="height: 150px; width: 150px; background-color:white; border-right: 2px dashed black; " |[[ThreeDimensionalConfigurations/Challenges|<b>Construction<br />Challenges</b>]]<br />(Pt. 1)
|
! style="height: 150px; width: 150px; background-color:white; border-right: 2px dashed black; " |[[ThreeDimensionalConfigurations/ChallengesPt2|<b>Construction<br />Challenges</b>]]<br />(Pt. 2)
|
! style="height: 150px; width: 150px; background-color:white; border-right: 2px dashed black; " |[[ThreeDimensionalConfigurations/ChallengesPt3|<b>Construction<br />Challenges</b>]]<br />(Pt. 3)
|
! style="height: 150px; width: 150px; background-color:white; border-right: 2px dashed black; " |[[ThreeDimensionalConfigurations/ChallengesPt4|<b>Construction<br />Challenges</b>]]<br />(Pt. 4)
|
! style="height: 150px; width: 150px; background-color:white; " |[[ThreeDimensionalConfigurations/ChallengesPt5|<b>Construction<br />Challenges</b>]]<br />(Pt. 5)
|}
<p> </p>

{{SGFfooter}}

ThreeDimensionalConfigurations/FerrersPotential

2021-09-30T22:52:32Z

174.64.14.12: Created page with "  =Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids..."

=Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids=

In an [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids|accompanying chapter]] titled, ''Properties of Homogeneous Ellipsoids (1),'' we have shown how analytic expressions may be derived for the gravitational potential inside of uniform-density ellipsoids. In that discussion, we largely followed the derivations of [[User:Tohline/Appendix/References#EFE|EFE]]. In the latter part of the nineteenth-century, [https://babel.hathitrust.org/cgi/pt?id=uc1.$b417536&view=1up&seq=15 N. M. Ferrers, (1877, Quarterly Journal of Pure and Applied Mathematics, 14, 1 - 22)] showed that very similar analytic expressions can be derived for ellipsoids that have certain, specific inhomogeneous mass distributions. Here we specifically discuss the case of configurations that exhibit concentric ellipsoidal iso-density surfaces of the form,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\rho</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \rho_c \biggl[ 1 - \biggl( \frac{x^2}{a_1^2} + \frac{y^2}{a_2^2} + \frac{z^2}{a_3^2}\biggr) \biggr] \, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" width="90%" cellpadding="10">
<tr><td align="center">SUMMARY — copied from [[User:Tohline/ThreeDimensionalConfigurations/Challenges#Trial_.232|accompanying, ''Trial #2'' Discussion]]</td></tr>

<tr><td align="left">
After studying the relevant sections of both [[User:Tohline/Appendix/References#EFE|EFE]] and [[User:Tohline/Appendix/References#BT87|BT87]] — this is an example of a heterogeneous density distribution whose gravitational potential has an analytic prescription. As is discussed in a [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Inhomogeneous_Ellipsoids_Leading_to_Ferrers_Potentials| separate chapter]], the potential that it generates is sometimes referred to as a ''Ferrers'' potential, for the exponent, n = 1.

In our [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#GravFor1|accompanying discussion]] we find that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{ \Phi_\mathrm{grav}(\bold{x})}{(-\pi G\rho_c)} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2} I_\mathrm{BT} a_1^2
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr)
~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
~+ \frac{1}{6} \biggl(3A_{11}x^4 + 3A_{22}y^4 + 3A_{33}z^4 \biggr)
\, ,
</math>
</td>
</tr>
</table>

where,
<table border="0" align="center" cellpadding="10" width="80%">
<tr>
<td align="center" width="50%">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="3">for <math>~i \ne j</math></td></tr>
<tr>
<td align="right">
<math>~A_{ij}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~-\frac{A_i-A_j}{(a_i^2 - a_j^2)} </math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§21, Eq. (107)</font> ]</td></tr>
</table>

</td>
<td align="center" width="50%">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="3">for <math>~i = j</math></td></tr>
<tr>
<td align="right">
<math>~2A_{ii} + \sum_{\ell = 1}^3 A_{i\ell}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{a_i} </math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§21, Eq. (109)</font> ]</td></tr>
</table>

</td>
</tr>
</table>
More specifically, in the three cases where the indices, <math>~i=j</math>,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~3A_{11}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2}{a_1^2} - (A_{12} + A_{13}) \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~3A_{22}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2}{a_2^2} - (A_{21} + A_{23}) \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~3A_{33}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2}{a_3^2} - (A_{31} + A_{32}) \, .
</math>
</td>
</tr>
</table>

</td></tr></table>

==Derivation==

<table border="1" cellpadding="8" width="80%" align="center">
<tr><td align="left">
Other references to Ferrers Potential:
<ul>
<li>[https://babel.hathitrust.org/cgi/pt?id=uc1.$b417536&view=1up&seq=15 Ferrers, N. M. (1877, Quarterly Journal of Pure and Applied Mathematics, 14, 1)] … from [[User:Tohline/Appendix/References#BT87|BT87]] References (p. 711)</li>
<li>[https://docs.galpy.org/en/v1.6.0/reference/potentialferrers.html Galpy methods]</li>
<li>[https://www.mdpi.com/2075-4434/5/4/101 Lucas Antonio Caritá, Irapuan Rodriguez, & I. Puerari (2017)] <i>Explicit Second Partial Derivatives of the Ferrers Potential</i></li>
<li>[https://www.researchgate.net/publication/225744594_Anisotropic_and_inhomogeneous_S-type_Riemann_ellipsoids_inside_spheroidal_halos_II Martin G. Abrahamyan (2006, Astrophysics, 49(3), 306-319)], ''Anisotropic and inhomogeneous S-type Riemann ellipsoids inside spheroidal halos. II''</li>
</ul>
</td>
</tr>
</table>

Following §2.3.2 (beginning on p. 60) of [[User:Tohline/Appendix/References#BT87|BT87]], let's examine ''inhomogeneous'' configurations whose isodensity surfaces (including the surface, itself) are defined by triaxial ellipsoids on which the Cartesian coordinates <math>~(x_1, x_2, x_3)</math> satisfy the condition that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~m^2</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~a_1^2 \sum_{i=1}^{3} \frac{x_i^2}{a_i^2} \, ,</math>
</td>
</tr>
<tr><td align="center" colspan="3">
[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §20, p. 50, Eq. (75)</font> ]<br />
[ [[User:Tohline/Appendix/References#BT87|BT87]], <font color="#00CC00">§2.3.2, p. 61, Eq. (2-97)</font> ]
</td></tr>
</table>
be constant. <span id="DensitySpecification">More specifically,</span> let's consider the case (related to the so-called ''Ferrers'' potentials) in which the configuration's density distribution is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho(m^2)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\rho_c \biggl[1 - \frac{m^2}{a_1^2}\biggr]^n </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\rho_c \biggl[1 -
\sum_{i=1}^{3} \frac{x_i^2}{a_i^2} \biggr]^n
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\rho_c \biggl[1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}\biggr) \biggr]^n
\, .</math>
</td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
<font color="red">'''NOTE:'''</font>     In our [[User:Tohline/ThreeDimensionalConfigurations/Challenges#Trial_.232|accompanying discussion]] of compressible analogues of Riemann S-type ellipsoids, we have discovered that — at least in the context of infinitesimally thin, nonaxisymmetric disks — this heterogeneous density profile can be nicely paired with an analytically expressible stream function, at least for the case where the integer exponent is, n = 1.
</td></tr></table>

According to Theorem 13 of [[User:Tohline/Appendix/References#EFE|EFE]] — see his Chapter 3, §20 (p. 53) — the potential at any point inside a triaxial ellipsoid with this specific density distribution is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Phi_\mathrm{grav}(\bold{x})</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{\pi G \rho_c a_1 a_2 a_3}{(n+1)} \int_0^\infty \frac{ du}{\Delta } Q^{n+1} \, ,
</math>
</td>
</tr>
<tr><td align="center" colspan="3">
[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §20, p. 53, Eq. (101)</font> ]
</td></tr>
</table>
where, <math>~\Delta</math> has the same definition as [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Derivation_of_Expressions_for_Ai|above]], and,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~Q</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u } \, .
</math>
</td>
</tr>
</table>

For purposes of illustration, in what follows we will assume that, <math>~a_1 > a_2 > a_3</math>.

===The Case Where n = 0===

When <math>~n = 0</math>, we have a uniform-density configuration, and the "interior" potential will be given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Phi_\mathrm{grav}(\bold{x})</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \pi G \rho_c a_1 a_2 a_3 \int_0^\infty \frac{ du}{\Delta } \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \pi G \rho_c a_1 a_2 a_3 \biggl\{
\int_0^\infty \frac{ du}{\Delta }
- \int_0^\infty \frac{ du}{\Delta } \biggl( \frac{x^2}{ a_1^2 + u } \biggr)
- \int_0^\infty \frac{ du}{\Delta } \biggl( \frac{y^2}{ a_2^2 + u } \biggr)
- \int_0^\infty \frac{ du}{\Delta } \biggl( \frac{z^2}{ a_3^2 + u } \biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \pi G \rho_c a_1 a_2 a_3 \biggl\{
\int_0^\infty \frac{ du}{\Delta }
~ - ~x^2 \int_0^\infty \frac{ du}{\Delta (a_1^2 + u) }
~ - ~y^2 \int_0^\infty \frac{ du}{\Delta (a_2^2 + u) }
~ - ~ \int_0^\infty \frac{ du}{\Delta (a_3^2 + u) }
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\pi G \rho_c \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr] \, .</math>
</td>
</tr>
</table>

As a check, let's see if this scalar potential satisfies the differential form of the
<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{User:Tohline/Math/EQ_Poisson01}}
</div>
<span id="SumTo2">Given that,</span>
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\sum_{\ell = 1}^3 A_\ell</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2 \, ,</math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§21, Eq. (108)</font> ]</td></tr>
</table>

we find,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla^2\Phi_\mathrm{grav} = \biggl[\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\biggr]\Phi_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
+ 2\pi G \rho_c (A_1 + A_2 + A_3) = 4\pi G\rho_c \, .
</math>
</td>
</tr>
</table>
Q.E.D.

===The Case Where n = 1===

When <math>~n = 1</math>, we have a specific heterogeneous density configuration, and the "interior" potential will be given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{ \Phi_\mathrm{grav}(\bold{x})}{(-\pi G\rho_c)} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2} a_1 a_2 a_3 \int_0^\infty \frac{ du}{\Delta } \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u } \biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2} a_1 a_2 a_3 \biggl\{
\int_0^\infty \frac{ du}{\Delta } \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u } \biggr]
~- ~ x^2 \int_0^\infty \frac{ du}{\Delta (a_1^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
~- ~y^2 \int_0^\infty \frac{ du}{\Delta (a_2^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u } \biggr]
~ - ~z^2 \int_0^\infty \frac{ du}{\Delta (a_3^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u } \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>

The first definite-integral expression inside the curly braces is, to within a leading factor of <math>~\tfrac{1}{2}</math>, identical to the entire expression for the normalized potential that was derived in the case where n = 0. That is, we can write,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{ \Phi_\mathrm{grav}(\bold{x})}{(-\pi G\rho_c)} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2} \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
~- \frac{1}{2} a_1 a_2 a_3 \biggl\{ ~ x^2 \int_0^\infty \frac{ du}{\Delta (a_1^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
~+~y^2 \int_0^\infty \frac{ du}{\Delta (a_2^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u } \biggr]
~ + ~z^2 \int_0^\infty \frac{ du}{\Delta (a_3^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u } \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>
Then, from §22, p. 56 of [[User:Tohline/Appendix/References#EFE|EFE]], we see that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~a_1 a_2 a_3 \int_0^\infty \frac{ du}{\Delta (a_i^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u } \biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( A_i - \sum_{\ell=1}^3 A_{i\ell} x_\ell^2 \biggr) \, .
</math>
</td>
</tr>
<tr><td align="center" colspan="3">
[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, §22, p. 53, Eq. (125)</font> ]
</td></tr>
</table>

<span id="GravFor1">Applying this result to each of the other three definite integrals gives us,</span>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{ \Phi_\mathrm{grav}(\bold{x})}{(-\pi G\rho_c)} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2} \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
~- \frac{x^2}{2} \biggl( A_1 - \sum_{\ell=1}^3 A_{1\ell} x_\ell^2 \biggr)
~- \frac{y^2}{2} \biggl( A_2 - \sum_{\ell=1}^3 A_{2\ell} x_\ell^2 \biggr)
~- \frac{z^2}{2} \biggl( A_3 - \sum_{\ell=1}^3 A_{3\ell} x_\ell^2 \biggr) \, .
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2} \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
~- \frac{x^2}{2} \biggl[ A_1 - \biggl( A_{11}x^2 + A_{12}y^2 + A_{13}z^2 \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
~- \frac{y^2}{2} \biggl[ A_2 - \biggl( A_{21}x^2 + A_{22}y^2 + A_{23}z^2 \biggr) \biggr]
~- \frac{z^2}{2} \biggl[ A_3 - \biggl( A_{31}x^2 + A_{32}y^2 + A_{33}z^2 \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2} I_\mathrm{BT} a_1^2
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr)
~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
~+ \frac{1}{2} \biggl(A_{11}x^4 + A_{22}y^4 + A_{33}z^4 \biggr)
\, ,
</math>
</td>
</tr>
</table>

where,
<table border="1" align="center" cellpadding="10" width="80%">
<tr>
<td align="center" width="50%">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="3">for <math>~i \ne j</math></td></tr>
<tr>
<td align="right">
<math>~A_{ij}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~-\frac{A_i-A_j}{(a_i^2 - a_j^2)} </math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§21, Eq. (107)</font> ]</td></tr>
</table>

</td>
<td align="center" width="50%">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="3">for <math>~i = j</math></td></tr>
<tr>
<td align="right">
<math>~2A_{ii} + \sum_{\ell = 1}^3 A_{i\ell}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{a_i} </math>
</td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§21, Eq. (109)</font> ]</td></tr>
</table>

</td>
</tr>
</table>

and we have made use of the symmetry relation, <math>~A_{ij} = A_{ji}</math>. Again, as a check, let's see if this scalar potential satisfies the differential form of the
<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{User:Tohline/Math/EQ_Poisson01}}
</div>
We find,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla^2 \biggl[ \frac{\Phi_\mathrm{grav}}{-2\pi G \rho_c} \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2}\biggl[\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\biggr]
\biggl[- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr)
~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
~+ \frac{1}{2} \biggl(A_{11}x^4 + A_{22}y^4 + A_{33}z^4 \biggr)
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial}{\partial x} \biggl[- A_1 x ~+ A_{12} x y^2 + A_{13} x z^2 ~+ A_{11}x^3 \biggr]
+\frac{\partial}{\partial y} \biggl[- A_2 y ~+ A_{12} x^2y + A_{23} y z^2~+ A_{22}y^3 \biggr]
+ \frac{\partial}{\partial z}\biggl[- A_3 z ~+ A_{13} x^2z + A_{23} y^2z~+ A_{33}z^3 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[- A_1 + A_{12} y^2 + A_{13} z^2 ~+ 3A_{11}x^2 \biggr]
+ \biggl[- A_2 + A_{12} x^2 + A_{23} z^2~+ 3A_{22}y^2 \biggr]
+ \biggl[- A_3 + A_{13} x^2 + A_{23} y^2~+ 3A_{33}z^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ - (A_1 + A_2 + A_3) + x^2(3A_{11} + A_{12} + A_{13}) + y^2( 3A_{22} + A_{12} + A_{23}) + z^2( 3A_{33} + A_{13} + A_{23})\, .
</math>
</td>
</tr>
</table>

In addition to recognizing, as [[#SumTo2|stated above]], that <math>~(A_1 + A_2 + A_3) = 2</math>, and making explicit use of the relation,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~2A_{ii} + \sum_{\ell = 1}^3 A_{i\ell}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{a_i} \, ,</math>
</td>
</tr>
</table>
this last expression can be simplified to discover that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\nabla^2 \biggl[ \frac{\Phi_\mathrm{grav}}{-2\pi G \rho_c} \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ - (2) + \frac{2x^2}{a_1^2} + \frac{2y^2}{a_2^2} + \frac{2z^2}{a_3^2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \nabla^2 \Phi_\mathrm{grav}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ 4\pi G \rho_c \biggl[ 1 - \biggl( \frac{x^2}{a_1^2} + \frac{y^2}{a_2^2} + \frac{z^2}{a_3^2}\biggr) \biggr] \, .
</math>
</td>
</tr>
</table>
This does indeed demonstrate that the derived gravitational potential is consistent with [[#DensitySpecification|our selected mass distribution in the case where n = 1]], namely,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\rho</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \rho_c \biggl[ 1 - \biggl( \frac{x^2}{a_1^2} + \frac{y^2}{a_2^2} + \frac{z^2}{a_3^2}\biggr) \biggr] \, .
</math>
</td>
</tr>
</table>

Q.E.D.

=See Also=
<ul>
<li>Our [[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates#Speculation6|''Speculation6'' ]] effort to develop a "Concentric Ellipsoidal (T6) Coordinate System."</li>
<li>[[User:Tohline/ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations|Challenges Constructing Ellipsoidal-Like Configurations]]</li>
<li>[[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Properties_of_Homogeneous_Ellipsoids_.281.29|Properties of Homogeneous Ellipsoids (1)]] — The Gravitational Potential (A<sub>i</sub> Coefficients)</li>
</ul>

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174.64.14.12: /* Ellipsoidal & Ellipsoidal-Like */

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==Spherically Symmetric Configurations==
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{| class="Chap3C" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"
|-
! style="height: 150px; width: 150px; background-color:lightgreen; border-right:2px solid black;" |[[SSCpt2/IntroductorySummary#Applications|<b>Hydrostatic<br />Balance<br />Equation</b>]]
|
! style="height: 150px; width: 310px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ Template:Math/EQ_SShydrostaticBalance01 }}</div>
|
! style="height: 150px; width: 150px;" |[[SSCpt2/SolutionStrategies#Solution_Strategies|Solution<br />Strategies]]
|}

{| class="Chap3D" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; " |[[SSC/Structure/UniformDensity#Isolated_Uniform-Density_Sphere|<b>Uniform-Density<br />Sphere</b>]]
|}
<p> </p>
{| class="Chap3E" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[SSC/Structure/IsothermalSphere#Isothermal_Sphere|<b>Isothermal<br />Sphere</b>]]
|
! style="height: 150px; width: 310px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ Math/EQ_SSLaneEmden02 }}</div>
|
! style="height: 150px; width: 150px;" |[[SSC/Structure/IsothermalSphere#Our_Numerical_Integration|via<br />Direct<br />Numerical<br />Integration]]
|}

{| class="Chap3F" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[SSC/Structure/Polytropes#Polytropic_Spheres|<b>Isolated<br />Polytropes</b>]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[SSC/Structure/Lane1870#Lane.27s_1870_Work|<b>Lane<br />(1870)</b>]]
|
! style="height: 150px; width: 310px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ Math/EQ_SSLaneEmden01 }}</div>
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SSC/Structure/Polytropes#Known_Analytic_Solutions|Known<br />Analytic<br />Solutions]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SSC/Structure/Polytropes#Straight_Numerical_Integration|via<br />Direct<br />Numerical<br />Integration]]
|
! style="height: 150px; width: 150px; " |[[SSC/Structure/Polytropes#HSCF_Technique|via<br />Self-Consistent<br />Field (SCF)<br />Technique]]
|}

<span id="MoreModels"> </span>
{| class="Chap3G" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |[[SSC/Structure/WhiteDwarfs#White_Dwarfs|<b>Zero-Temperature<br />White Dwarf</b>]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee;" |[[SSC/Structure/WhiteDwarfs#Chandrasekhar_mass|Chandrasekhar<br />Limiting<br />Mass<br />(1935)]]
|}

{| class="Chap3H" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px; "|[[SSC/Structure/Polytropes/VirialSummary|Virial Equilibrium<br />of<br />Pressure-Truncated<br />Polytropes]]
|}
{| class="Chap3I" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |<b>Pressure-Truncated<br />Configurations</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Bonnor-Ebert<br />(Isothermal)<br />Spheres<br />(1955 - 56)]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|Embedded<br />Polytropes]]
|
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|Equilibrium<br />Sequence<br />Turning-Points]]<br /><font color="green">&hearts;</font>
|
! style="height: 150px; width: 150px; border-right:2px dashed black; " |[[File:MassVsRadiusCombined02.png|130px|link=SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Equilibrium sequences of Pressure-Truncated Polytropes]]
|
! style="height: 150px; width: 150px; " |[[Appendix/Ramblings/TurningPoints#Turning_Points|Turning-Points<br />(Broader Context)]]
|}

{| class="Chap3J" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px; "|[[SSC/Structure/BiPolytropes/FreeEnergy51#Free_Energy_of_BiPolytrope_with|Free Energy<br />of<br />Bipolytropes]]<br /><br />(n<sub>c</sub>, n<sub>e</sub>) = (5, 1)
|}
{| class="Chap3K" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[SSC/Structure/BiPolytropes#BiPolytropes|<b>Composite<br />Polytropes</b>]]<br />(Bipolytropes)
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SSC/Structure/BiPolytropes/Analytic1.53#BiPolytrope_with__and_ne_.3D_3|Milne<br />(1930)]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass|Schönberg-<br />Chandrasekhar<br />Mass<br />(1942)]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5|Murphy (1983)<br /><br /><br />Analytic]]<br />(n<sub>c</sub>, n<sub>e</sub>) = (1, 5)
|
! style="height: 150px; width: 150px; border-right:2px dashed black; " |[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|Eggleton, Faulkner<br />& Cannon (1998)<br /><br />Analytic]]<br />(n<sub>c</sub>, n<sub>e</sub>) = (5, 1)
|
! style="height: 150px; width: 150px; " |[[Image:TurningPoints51Bipolytropes.png|150px|link=SSC/Stability/BiPolytropes#Planned_Approach|Equilibrium sequences of (5, 1) Bipolytropes]]
|}
<br />

===Stability Analysis===

{| class="Chap4A" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:darkgrey;"
|-
! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">1D STABILITY</font>
|}

<font color="darkgreen"><span id="BKB74pt1">Three different approaches are used in the study of hydrodynamical stability of stars</span> and other gravitating objects …  
<ul><li>The first approach is based on the use of the equations of small oscillations. In that case the problem is reduced to a search for the solution of the boundary-value problem of the Stourme-Liuville type for the linearised system of equations of small oscillations. The solutions consist of a set of eigenfrequencies and eigenfunctions.</font> The following set of menu tiles include links to chapters where this approach has been applied to: (a) uniform-density configurations, (b) pressure-truncated isothermal spheres, (c) an isolated n = 3 polytrope, (d) pressure-truncated n = 5 configurations, and (e) bipolytropes having <math>(n_c, n_e) = (1, 5)</math>.</li>

<li>Second, one can derive <font color="darkgreen">a variational principle from the equations of small oscillations.</font> Below, an appropriately labeled (purple) menu tile links to a chapter in which the foundation for this approach is developed.  <font color="darkgreen">With the aid of the variational principle, the problem is reduced to the search of the best trial functions; this leads to approximate eigenvalues of oscillations. In spite of the simplifications introduced by the use of the variational principle and by not solving the equations of motion exactly, the problem still remains complicated …</font> One menu tile, below, links to a chapter in which an analytic (''exact'') demonstration of the variational principle's utility is provided in the context pressure-truncated n = 5 polytropes.</li>

<li>The third approach is what we have already referred to as a free-energy — and associated virial theorem — analysis. <font color="darkgreen">When this method is used, it is not necessary to use the equations of small oscillations but, instead, the functional expression for the total energy of the momentarily stationary (but not necessarily in equilibrium) star is sufficient. The condition that the first variation of the energy vanishes, determines the state of equilibrium of the star and the positiveness of a second variation indicates stability.</font> </li>
</ul>

<span id="BKB74pt2"><font color="darkgreen">If one wants to know from a stability analysis the answer to only one question — whether the model is stable or not — then the most straightforward procedure is to use the third, static method … For the application of this method, one needs to construct only equilibrium, stationary models, with no further calculation. Generally the static analysis gives no information about the shape of the modes of oscillation, but, in the vicinity of critical points, where instability sets in, this method makes it possible to find the eigenfunction of the mode which becomes unstable at the critical point.</font></span> Generally in what follows, this will be referred to as the "B-KB74 conjecture;" a menu tile carrying this label is linked to a chapter in which this approach is used to analyze the onset of a dynamical instability along the equilibrium sequence of pressure-truncated n = 5 polytropes.
<div align="right">--- Text in ''green'' taken directly from [http://adsabs.harvard.edu/abs/1974A%26A....31..391B G. S. Bisnovatyi-Kogan & S. I. Blinnikov (1974)]; B-KB74, for short.</div>
 <br />
 <br />

{| class="Chap4B" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[File:SSC_SynopsisImage2.png|150px|link=SSC/SynopsisStyleSheet#Stability|Synopsis: Stability of Spherical Structures]]
|
! style="height: 150px; width: 150px; background-color:#9390DB;"|[[SSC/VariationalPrinciple#Ledoux.27s_Variational_Principle|Variational<br />Principle]]
|}
{| class="Chap4C" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:lightgreen; border-right:2px solid black;" |<b>Radial<br />Pulsation<br />Equation</b>
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|Example<br />Derivations<br />&<br />Statement of<br />Eigenvalue<br />Problem]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[SSC/PerspectiveReconciliation#Reconciling_Eulerian_versus_Lagrangian_Perspectives|(poor attempt at)<br />Reconciliation]]
|
! style="height: 150px; width: 150px;" |[[SSC/SoundWaves#Sound_Waves|Relationship<br />to<br />Sound Waves]]
|}
<p> </p>
{| class="Chap4D" style="margin-right: auto; margin-left: 230px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[File:ImageOfDerivations06GoodJeansBonnor.png|120px|thumb|center|Jeans (1928) or Bonnor (1957)]]
|
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[File:ImageOfDerivations07GoodLedouxWalraven.png|120px|thumb|center|Ledoux & Walraven (1958)]]
|
! style="height: 150px; width: 150px; " |[[File:ImageOfDerivations08GoodRosseland.png|120px|thumb|center|Rosseland (1969)]]
|}
<p> </p>
{| class="Chap4E" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |<b>Uniform-Density<br />Configurations</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black; " |[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Sterne's<br />Analytic Sol'n<br />of Eigenvalue<br />Problem<br />(1937)]]
|
! style="height: 150px; width: 150px; " |[[File:Sterne1937SolutionPlot1.png|150px|link=SSC/Stability/UniformDensity#Properties_of_Eigenfunction_Solutions|Sterne's (1937) Solution to the Eigenvalue Problem for Uniform-Density Spheres]]
|}
<p> </p>
{| class="Chap4F" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[SSC/Stability/Isothermal|<b>Pressure-Truncated<br />Isothermal<br />Spheres</b>]]
|
! style="height: 150px; width: 620px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ Math/EQ_RadialPulsation03 }}</div>
|
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[SSC/Stability/Isothermal#Our_Numerical_Integration|via<br />Direct<br />Numerical<br />Integration]]
|
! style="height: 150px; width: 310px;" |[[File:TaffVanHorn1974Fundamental.gif|400px|Fundamental-Mode Eigenvectors]]
|}
<span id="MoreStabilityAnalyses"> </span>

{| class="Chap5A" style="margin-right: auto; margin-left: 230px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black; " |[[SSC/Stability/InstabilityOnsetOverview#Yabushita.27s_Insight_Regarding_Stability|Yabushita's<br />Analytic Sol'n for<br />Marginally Unstable<br />Configurations<br />(1974)]]
|
! style="height: 150px; width: 310px;"|<table border="0" cellpadding="2" align="center">
<tr>
<td align="center">
<math>\sigma_c^2 = 0 \, , ~~~~\gamma_\mathrm{g} = 1</math>
</td>
</tr>
<tr>
<td align="center">
 and  
</td>
</tr>
<tr>
<td align="center">
<math>x = 1 - \biggl( \frac{1}{\xi e^{-\psi}}\biggr) \frac{d\psi}{d\xi} </math>
</td>
</tr>
</table>
|}
<p> </p>
{| class="Chap5B" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres|<b>Polytropes</b>]]
|
! style="height: 150px; width: 620px; text-align:center; border-right:2px dashed black;" |<div align="center">{{ Math/EQ_RadialPulsation02 }}</div>
|
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres|Isolated<br />n = 3<br />Polytrope]]
|
! style="height: 150px; width: 150px; " |[[File:Schwarzschild1941movie.gif|300px|link=SSC/Stability/n3PolytropeLAWE#SchwarzschildMovie|Schwarzschild's Modal Analysis]]
|}
<br />
{| class="Chap5C" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px;"|[[Appendix/Ramblings/NonlinarOscillation#Radial_Oscillations_in_Pressure-Truncated_n_.3D_5_Polytropes|''Exact''<br />Demonstration<br />of<br />B-KB74<br />Conjecture]]
|
! style="height: 150px; width: 150px; border-left:2px dashed black;"|[[SSC/VariationalPrinciple#Directly_to_n_.3D_5_Polytropic_Configurations|''Exact''<br />Demonstration<br />of<br />Variational<br />Principle]]
|}
{| class="Chap5D" style="margin-right: auto; margin-left: 230px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_Spheres|Pressure-Truncated<br />n = 5<br />Configurations]]
|
! style="height: 250px; width: 500px; " |[[File:N5Truncated2.gif|600px|n5 Truncated Movie]]
|}
<br />
{| class="Chap5E" style="margin-right: auto; margin-left: 230px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black; " |[[SSC/Stability/InstabilityOnsetOverview#Polytropic_Stability|Our (2017)<br />Analytic Sol'n for<br />Marginally Unstable<br />Configurations<br /><font color="green">&hearts;</font>]]
|
! style="height: 150px; width: 465px;"|<table border="0" cellpadding="2" align="center">
<tr>
<td align="center">
<math>~\sigma_c^2 = 0 \, , ~~~~\gamma_\mathrm{g} = (n+1)/n</math>
</td>
</tr>
<tr>
<td align="center">
 and  
</td>
</tr>
<tr>
<td align="center">
<math>~x = \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] </math>
</td>
</tr>
</table>
|}
<p> </p>
<br />
{| class="Chap5F" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px;"|[[SSC/StabilityConjecture/Bipolytrope51|B-KB74<br />Conjecture<br />RE: Bipolytrope]]<br /><br />(n<sub>c</sub>, n<sub>e</sub>) = (5, 1)
|}
{| class="Chap5G" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |<b>BiPolytropes</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black; " |[[SSC/Stability/MurphyFiedler85|Murphy & Fiedler<br />(1985b)]]<br /><br />(n<sub>c</sub>, n<sub>e</sub>) = (1,5)
|
! style="height: 150px; width: 150px; " |[[SSC/Stability/BiPolytropes/HeadScratching|Our<br />Broader<br />Analysis]]
|}

===Nonlinear Dynamical Evolution===

{| class="Chap6A" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:darkgrey;"
|-
! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">1D DYNAMICS</font>
|}
<p> </p>
{| class="Chap6B" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99;;" |[[SSC/Dynamics/FreeFall#Free-Fall|<b>Free-Fall<br />Collapse</b>]]
|}
<p> </p>
{| class="Chap6C" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black;" |[[SSC/Dynamics/IsothermalCollapse#Collapse_of_Isothermal_Spheres|<b>Collapse of<br />Isothermal<br />Spheres</b>]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[http://adsabs.harvard.edu/abs/1993ApJ...416..303F via<br/>Direct<br />Numerical<br />Integration]
|
! style="height: 150px; width: 150px;" |[[SSC/Dynamics/IsothermalSimilaritySolution#Similarity_Solution|Similarity<br />Solution]]
|}
<p> </p>
{| class="Chap6D" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99;" |[[Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores|<b>Collapse of<br />an Isolated<br />n = 3<br />Polytrope</b>]]
|}
<p> </p>

==Two-Dimensional Configurations (Axisymmetric)==
{| class="Chap7A" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:navy;"
|-
! style="height: 50px; width: 800px; background-color:lightgrey;"|<font color="navy" size="+2">(Initially) Axisymmetric Configurations</font>
|}
<p> </p>
{| class="Chap7B" style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color: black;"
|-
! style="height: 150px; width: 150px; background-color:black;" |[[AxisymmetricConfigurations/Storyline|<font color="white">Storyline</font>]]
|}
<p> </p>
{| class="Chap7C" style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color: black;"
|-
! style="height: 150px; width: 150px; background-color:lightgreen;" |[[AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29|PGEs<br />for<br />Axisymmetric<br />Systems]]
|}

===Axisymmetric Equilibrium Structures===

{| class="Chap8A" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:darkgrey;"
|-
! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">2D STRUCTURE</font>
|}
<p> </p>
{| class="Chap8B" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"
|-
! style="height: 150px; width: 150px; background-color:lightgreen; border-right:2px solid black;" |[[AxisymmetricConfigurations/Equilibria|<b>Constructing<br />Steady-State<br />Axisymmetric<br />Configurations</b>]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[2DStructure/AxisymmetricInstabilities|Axisymmetric<br />Instabilities<br />to Avoid]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''Simple''<br />Rotation<br />Profiles]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[AxisymmetricConfigurations/HSCF|Hachisu Self-Consistent-Field<br />[HSCF]<br />Technique]]
|
! style="height: 150px; width: 150px; " |[[AxisymmetricConfigurations/SolvingPE#Common_Theme:_Determining_the_Gravitational_Potential_for_Axisymmetric_Mass_Distributions|Solving the<br />Poisson Equation]]
|}
<p> </p>
{| class="Chap8C" style="margin-right: auto; margin-left: 230px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[2DStructure/UsingTC#Common_Theme:_Determining_the_Gravitational_Potential_for_Axisymmetric_Mass_Distributions|Using<br />Toroidal Coordinates<br />to Determine the<br />Gravitational<br /> Potential]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[File:Apollonian_myway4.png|150px|link=2DStructure/ToroidalCoordinateIntegrationLimits#Mapping_from_Cylindrical_to_Toroidal_Coordinates|Apollonian Circles]]
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[2DStructure/TCsimplification#Common_Theme:_Determining_the_Gravitational_Potential_for_Axisymmetric_Mass_Distributions|Attempt at<br />Simplification<br /><br /><font color="green" size="+2">&hearts;</font>]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black;" |[[Apps/WongAP#Common_Theme:_Determining_the_Gravitational_Potential_for_Axisymmetric_Mass_Distributions|Wong's<br />Analytic Potential<br />(1973)]]
|
! style="height: 150px; width: 150px; background-color:#D0FFFF;" |[[File:MovieWongN4.gif|130px|link=Apps/DysonWongTori#The_Coulomb_Potential|n = 3 contribution to potential]]
|}

====Spheroidal & Spheroidal-Like====
<br />
{| class="Chap9A" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |[[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|<b>Uniform-Density<br />(Maclaurin)<br />Spheroids</b>]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black;" |[[Apps/MaclaurinSpheroids/GoogleBooks#Excerpts_from_A_Treatise_of_Fluxions|Maclaurin's<br />Original Text<br />&<br />Analysis<br />(1742)]]
|
! style="height: 200px; width: 200px; border-right:2px dashed black;" |[[File:Maclaurin01.gif|282px|link=Apps/MaclaurinSpheroids/GoogleBooks#Prolate_Spheroid|Our Construction of Maclaurin's Figure 291Pt2]]
|
! style="height: 150px; width: 150px; background-color:maroon;" |[[Apps/MaclaurinSpheroidSequence|<font color="white">Maclaurin<br />Spheroid<br />Sequence</font>]]
|}
<p> </p>
{| class="Chap9B" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |<b>Rotationally<br />Flattened<br />Isothermal<br />Configurations</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[Apps/HayashiNaritaMiyama82#Rotationally_Flattened_Isothermal_Structures|Hayashi, Narita<br /> & Miyama's<br />Analytic Sol'n<br />(1982)]]
|
! style="height: 150px; width: 150px;" |[[Apps/ReviewStahler83|Review of<br /> Stahler's (1983)<br />Sol'n Technique]]
|}
<p> </p>
{| class="Chap9C" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |<b>Rotationally<br />Flattened<br />Polytropes</b>
|
! style="height: 150px; width: 150px;" |[[Apps/RotatingPolytropes|Example<br />Equilibria]]
|}
<p> </p>
{| class="Chap9D" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |<b>Rotationally<br />Flattened<br />White Dwarfs</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[Apps/OstrikerBodenheimerLyndenBell66|Ostriker<br />Bodenheimer<br />& Lynden-Bell<br />(1966)]]
|
! style="height: 150px; width: 150px;" |[[Apps/RotatingWhiteDwarfs|Example<br />Equilibria]]
|}

====Toroidal & Toroidal-Like====
 <br />

{| class="Chap10B" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |<b>Massless<br />Polytropic<br />Configurations</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black;" |[[Apps/PapaloizouPringleTori#Massless_Polytropic_Tori|Papaloizou-Pringle<br />Tori<br />(1984)]]
|
! style="height: 150px; width: 250px; background-color:#D0FFFF;" |[[File:TorusMovie1.gif|250px|link=Apps/PapaloizouPringleTori#Boundary_Conditions|Pivoting PP Torus]]
|}
<p> </p>
{| class="Chap10C" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |<b>Self-Gravitating<br />Incompressible<br />Configurations</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dashed black;" |[[Apps/DysonPotential|Dyson<br />(1893)]]
|
! style="height: 150px; width: 150px; " |[[Apps/DWT#Common_Theme:_Determining_the_Gravitational_Potential_for_Axisymmetric_Mass_Distributions|Dyson-Wong<br />Tori]]
|}
<p> </p>
{| class="Chap10D" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |<b>Self-Gravitating<br />Compressible<br />Configurations</b>
|
! style="height: 150px; width: 150px;" |[[Apps/Ostriker64|Ostriker<br />(1964)]]
|}
<p> </p>

===Stability Analysis===
{| class="Chap11A" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:darkgrey;"
|-
! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">2D STABILITY</font>
|}
<p> </p>
====Sheroidal & Spheroidal-Like====
{| class="Chap11B" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |<b>Linear<br />Analysis<br /> of<br />Bar-Mode<br />Instability</b>
|
! style="height: 150px; width: 150px; border-right:2px dashed black;" |[[Apps/RotatingPolytropes/BarmodeIncompressible|Bifurcation<br />from<br />Maclaurin<br />Sequence]]
|
! style="height: 150px; width: 150px;" |[[Apps/RotatingPolytropes/BarmodeEigenvector|Traditional<br />Analyses]]
|}
<p> </p>
* [https://ui.adsabs.harvard.edu/abs/1949ApJ...109..149C/abstract T. G. Cowling & R. A. Newing (1949)], ApJ, 109, 149: ''The Oscillations of a Rotating Star''
* [https://ui.adsabs.harvard.edu/abs/1965ApJ...141..210C/abstract M. J. Clement (1965)], ApJ, 141, 210: ''The Radial and Non-Radial Oscillations of Slowly Rotating Gaseous Masses''
* [https://ui.adsabs.harvard.edu/abs/1963ApJ...137..777R/abstract P. H. Roberts & K. Stewartson (1963)], ApJ, 137, 777: ''On the Stability of a Maclaurin spheroid of small viscosity''
* [https://ui.adsabs.harvard.edu/abs/1967ApJ...148..825R/abstract C. E. Rosenkilde (1967)], ApJ, 148, 825: ''The tensor virial-theorem including viscous stress and the oscillations of a Maclaurin spheroid''
* [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..267C/abstract S. Chandrasekhar & N. R. Lebovitz (1968)], ApJ, 152, 267: ''The Pulsations and the Dynamical Stability of Gaseous Masses in Uniform Rotation''
* [https://ui.adsabs.harvard.edu/abs/1977ApJ...213..497H/abstract C. Hunter (1977)], ApJ, 213, 497: ''On Secular Stability, Secular Instability, and Points of Bifurcation of Rotating Gaseous Masses''
* [https://ui.adsabs.harvard.edu/abs/1985ApJ...294..474I/abstract J. N. Imamura, J. L. Friedman & R. H. Durisen (1985)], ApJ, 294, 474: ''Secular stability limits for rotating polytropic stars''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%"> </td><td align="left">
<font color="green">The equilibrium models are calculated using the polytrope version (Bodenheimer & Ostriker 1973) of the Ostriker and Mark (1968) self-consistent field (SCF) code … the equilibrium models rotate on cylinders and are completely specified by <math>~n</math>, the total angular momentum, and the specific angular momentum distribution <math>~j(m_\varpi)</math>. Here <math>~m_\varpi</math> is the mass contained within a cylinder of radius <math>~\varpi</math> centered on the rotation axis. The angular momentum distribution is prescribed in several ways: (1) imposing strict uniform rotation; (2) using the same <math>~j(m_\varpi)</math> as that of a uniformly rotating spherical polybrope of index <math>~n^'</math> (see Bodenheimer and Ostriker 1973); and (3) using <math>~j(m_\varpi) \propto m_\varpi</math>, which we refer to as <math>~n^' = L</math>, <math>~L</math> for "linear."</font>
</td></tr></table>
* [https://ui.adsabs.harvard.edu/abs/1990ApJ...355..226I/abstract J. R. Ipser & L. Lindblom (1990)], ApJ, 355, 226: ''The Oscillations of Rapidly Rotating Newtonian Stellar Models''
* [https://ui.adsabs.harvard.edu/abs/1991ApJ...373..213I/abstract J. R. Ipser & L. Lindblom (1991)], ApJ, 373, 213: ''The Oscillations of Rapidly Rotating Newtonian Stellar Models. II. Dissipative Effects''
* [https://ui.adsabs.harvard.edu/abs/2000ApJ...528..946I/abstract J. N. Imamura, J. L. Friedman & R. H. Durisen (2000)], ApJ, 528, 946: ''Nonaxisymmetric Dynamic Instabilities of Rotating Polytropes. II. Torques, Bars, and Mode Saturation with Applications to Protostars and Fizzlers''
* [https://ui.adsabs.harvard.edu/abs/2003MNRAS.343..619S/abstract M. Shibata, S. Karino, & Y. Eriguchi (2003)], MNRAS, 343, 619 - 626: ''Dynamical bar-mode instability of differentially rotating stars: effects of equations of state and velocity profiles''
* [https://ui.adsabs.harvard.edu/abs/2019ApJ...877....9H/abstract G. P. Horedt (2019)], ApJ, 877, 9: ''On the Instability of Polytropic Maclaurin and Roche ellipsoids''
<p> </p>

====Toroidal & Toroidal-Like====
{| class="Chap11C" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"

|-
! style="height: 150px; width: 150px; background-color:lightgreen; " |[[Apps/PapaloizouPringle84#Formulation_of_Eigenvalue_Problem|<b>Defining the<br />Eigenvalue Problem</b>]]
|}
<p> </p>
{| class="Chap11D" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |<b>(Massless)<br />Papaloizou-Pringle<br />Tori</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black;" |[[Apps/ImamuraHadleyCollaboration#Analytic_Solution|Analytic Analysis<br />by<br />Blaes<br />(1985)]]
|
! style="height: 150px; width: 150px; " |[[File:N1.5j2_Combinedsmall.png|450px|center|link=Apps/ImamuraHadleyCollaboration#Plots_of_a_Few_Example_Eigenvectors|j2 Eigenfunction from Blaes85|]]
|}
<p> </p>
{| class="Chap11E" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |<b>Self-Gravitating<br />Polytropic<br />Rings</b>
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right:2px dotted black;" |[https://ui.adsabs.harvard.edu/abs/1990ApJ...361..394T/abstract Numerical Analysis<br />by<br />Tohline & Hachisu<br />(1990)]
|
! style="height: 150px; width: 150px; background-color:black; border-right:2px dashed black; " |[[File:Minitorus.animated.gif|150px|center|PP torus instability]]
|
! style="height: 150px; width: 150px;" |[[Apps/WoodwardTohlineHachisu94#Online_Movies|Thick<br />Accretion<br />Disks]]<br />(WTH94)
|}
<p> </p>
===Nonlinear Dynamical Evolution===

====Sheroidal & Spheroidal-Like====

{| class="Chap12A" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:darkgrey;"
|-
! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">2D DYNAMICS</font>
|}
<p> </p>

{| class="Chap12B" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99;" |[[Aps/MaclaurinSpheroidFreeFall|<b>Free-Fall<br />Collapse<br />of an<br />Homogeneous<br />Spheroid</b>]]
|}
<p> </p>
{| class="Chap12C" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right:2px solid black; " |<b>Nonlinear<br />Development of<br />Bar-Mode</b>
|
! style="height: 150px; width: 150px; border-right:2px dotted black;" |[[Apps/RotatingPolytropes/BarmodeLinearTimeDependent|Initially<br />Axisymmetric<br /> & Differentially<br />Rotating<br />Polytropes]]
|
! style="height: 150px; width: 150px; background-color: black;" |[[File:Dissertation.fig3cropped.png|112px|link=Apps/RotatingPolytropes/BarmodeLinearTimeDependent|Cazes Model A Simulation]]
|}
<p> </p>

==Two-Dimensional Configurations (Nonaxisymmetric Disks)==
{| class="Chap13A" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:navy;"
|-
! style="height: 50px; width: 800px; background-color:lightgrey;"|<font color="navy" size="+2">Infinitesimally Thin, Nonaxisymmetric Disks</font>
|}
<p> </p>

{| class="Chap13B" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:darkgrey;"
|-
! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">2D STRUCTURE</font>
|}
<p> </p>
{| class="Chap13C" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"

|-
! style="height: 150px; width: 150px; background-color:lightgreen; " |[[Apps/Korycansky_Papaloizou_1996#Korycansky_and_Papaloizou_.281996.29|<b>Constructing<br />Infinitesimally Thin<br />Nonaxisymmetric<br />Disks</b>]]
|}
<p> </p>

==Three-Dimensional Configurations==
{| class="Chap14A" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:navy;"
|-
! style="height: 50px; width: 800px; background-color:lightgrey;"|<font color="navy" size="+2">(Initially) Three-Dimensional Configurations</font>
|}
<p> </p>

===Equilibrium Structures===
{| class="Chap14B" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:darkgrey;"
|-
! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">3D STRUCTURE</font>
|}


<table border="0" cellpadding="3" align="center" width="60%">
<tr><td align="left">
Special numerical techniques must be developed <font color="darkgreen">"to build three-dimensional compressible equilibrium models with complicated flows."</font> To date … <font color="darkgreen">"techniques have only been developed to build compressible equilibrium models of nonaxisymmetric configurations for a few systems with simplified rotational profiles, e.g., rigidly rotating systems ([https://ui.adsabs.harvard.edu/abs/1984PASJ...36..239H/abstract Hachisu & Eriguchi 1984]; [https://ui.adsabs.harvard.edu/abs/1986ApJS...62..461H/abstract Hachisu 1986)], irrotational systems ([https://ui.adsabs.harvard.edu/abs/1998ApJS..118..563U/abstract Uryū & Eriguchi 1998]), and configurations that are stationary in the inertial frame ([https://ui.adsabs.harvard.edu/abs/1996MNRAS.282..653U/abstract Uryū & Eriguchi 1996])."</font>
</td></tr>
<tr><td align="right">
— Drawn from §1 of [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou (2006)], ApJ, 639, 549
</td></tr></table>
====Ellipsoidal & Ellipsoidal-Like====

{| class="Chap15A" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"

|-
! style="height: 150px; width: 150px; background-color:lightgreen; " |[https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract <b>Constructing<br />Ellipsoidal<br /> & Ellipsoidal-Like<br />Configurations</b>]
|}
<p> </p>
{| class="Chap15B" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px; background-color:#9390DB;"|[[VE/RiemannEllipsoids|Steady-State<br />2<sup>nd</sup>-Order<br />Tensor Virial<br />Equations]]
|}
{| class="Chap15C" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right: 2px solid black; " |<b>Uniform-Density<br />Incompressible<br />Ellipsoids</b>
|
! style="height: 150px; width: 150px; background-color:white; border-right: 2px dashed black; " |[[ThreeDimensionalConfigurations/HomogeneousEllipsoids|<b>The<br />Gravitational<br />Potential</b>]]<br />(A<sub>i</sub> coefficients)
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right: 2px dashed black; " |[[ThreeDimensionalConfigurations/JacobiEllipsoids#Jacobi_Ellipsoids|<b>Jacobi<br />Ellipsoids</b>]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right: 2px dashed black;" |[[ThreeDimensionalConfigurations/RiemannStype#Riemann_S-type_Ellipsoids|<b>Riemann<br />S-Type<br />Ellipsoids</b>]]
|
! style="height: 150px; width: 150px; background-color:#ffeeee; border-right: 2px dashed black;" |[[ThreeDimensionalConfigurations/RiemannTypeI|<b>Type I<br />Riemann<br />Ellipsoids</b>]]
|
! style="height: 150px; width: 150px; background-color:white;" |[[ThreeDimensionalConfigurations/MeetsCOLLADAandOculusRiftS|<b>Riemann<br />meets<br />COLLADA<br />&<br /> Oculus Rift S</b>]]
|}
<p> </p>

{| class="Chap15D" style="margin-right: auto; margin-left: 550px; border-style: solid; border-width: 3px; border-color: black;"
|-
! style="height: 150px; width: 150px; background-color:white; border-right: 2px dashed black; " |[https://www.aimspress.com/article/10.3934/math.2019.2.215/fulltext.html <b>A Gauge Theory<br />of<br />Riemann Ellipsoids</b>]
|
! style="height: 150px; width: 150px; background-color:white; " |[https://ui.adsabs.harvard.edu/abs/2020PhRvL.124e2501S/abstract <b>Nuclear<br />Wobbling Motion</b>]
|}
<p> </p>

{| class="Chap16A" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right: 2px solid black; " |<b>Compressible<br />Analogs of<br />Riemann Ellipsoids</b>
|
! style="height: 150px; width: 150px; background-color:pink; border-right: 2px dashed black;" |[[ThreeDimensionalConfigurations/FerrersPotential|<b>Ferrers<br />Potential<br />(1877)</b>]]
|
! style="height: 150px; width: 150px; background-color:white;" |[[ThreeDimensionalConfigurations/CAREs|<b>Thoughts<br />&<br />Challenges</b>]]
|}
<p> </p>
* [https://ui.adsabs.harvard.edu/abs/1985Ap.....23..654K/abstract B. P. Kondrat'ev (1985)], Astrophysics, 23, 654: ''Irrotational and zero angular momentum ellipsoids in the Dirichlet problem''
* [https://ui.adsabs.harvard.edu/abs/1993ApJS...88..205L/abstract D. Lai, F. A. Rasio & S. L. Shapiro (1993)], ApJS, 88, 205: ''Ellipsoidal Figures of Equilibrium: Compressible models''
====Binary Systems====

* [https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..539C/abstract S. Chandrasekhar (1933)], MNRAS, 93, 539: ''The equilibrium of distorted polytropes. IV. the rotational and the tidal distortions as functions of the density distribution''
* [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The Equilibrium and the Stability of the Roche Ellipsoids''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%"> </td><td align="left">
<font color="green">Roche's problem is concerned with the equilibrium and the stability of rotating homogeneous masses which are, further, distorted by the constant tidal action of an attendant rigid spherical mass.</font>
</td></tr></table>

===Stability Analysis===
{| class="Chap17A" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:darkgrey;"
|-
! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">3D STABILITY</font>
|}

====Ellipsoidal & Ellipsoidal-Like====

<p> </p>

====Binary Systems====

* [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182: ''The Equilibrium and the Stability of the Roche Ellipsoids''
* [https://ui.adsabs.harvard.edu/abs/2019ApJ...877....9H/abstract G. P. Horedt (2019)], ApJ, 877, 9: ''On the Instability of Polytropic Maclaurin and Roche Ellipsoids''

===Nonlinear Evolution===
{| class="Chap18A" width=100% style="margin-right: auto; margin-left: 0px; border-style: solid; border-width: 3px; border-color:darkgrey;"
|-
! style="height: 30px; width: 800px; background-color:lightgrey;"|<font color="white" size="+1">3D DYNAMICS</font>
|}
<p> </p>
{| class="Chap18B" style="float:right; margin-left: 150px; border-style: solid; border-width: 3px; border-color:black;"
|-
! style="height: 150px; width: 150px; background-color:#D0FFFF;"|[[File:JacobiMaclaurin2.gif|300px|link=ThreeDimensionalConfigurations/EFE_Energies#Animation|Animation related to Fig. 3 from Christodoulou1995]]
|
! style="height: 150px; width: 150px; background-color:#9390DB; border-left:2px solid black;"|[[ThreeDimensionalConfigurations/EFE_Energies#Properties_of_Homogeneous_Ellipsoids_.282.29|Free-Energy<br />Evolution<br />from the Maclaurin<br />to the Jacobi<br />Sequence]]
|}
{| class="Chap18C" style="margin-right: auto; margin-left: 75px; border-style: solid; border-width: 3px; border-color: black;"
|-
! style="height: 150px; width: 150px; background-color:#ffff99; border-right: 2px solid black; " |[[ThreeDimensionalConfigurations/BinaryFission#Fission_Hypothesis_of_Binary_Star_Formation|<b>Fission<br />Hypothesis</b>]]
|
! style="height: 150px; width: 150px; background-color:white;" |[http://www.phys.lsu.edu/~tohline/fission.movies.html <b>"Fission"<br />Simulations<br />at LSU</b>]
|}
<p> </p>

====Secular====
* [https://ui.adsabs.harvard.edu/abs/1971ApJ...170..143F/abstract M. Fujimoto (1971)], ApJ, 170, 143: ''Nonlinear Motions of Rotating Gaseous Ellipsoids''
* [https://ui.adsabs.harvard.edu/abs/1973ApJ...181..513P/abstract W. H. Press & S. A. Teukolsky (1973)], ApJ, 181, 513: ''On the Evolution of the Secularly Unstable, Viscous Maclaurin Spheroids''
* [https://ui.adsabs.harvard.edu/abs/1977ApJ...213..193D/abstract S. L. Detweiler & L. Lindblom (1977)], ApJ, 213, 193: ''On the evolution of the homogeneous ellipsoidal figures.''

====Dynamical====

=See Also=

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Appendix/Permissions

2021-09-23T20:08:41Z

174.64.14.12: /* Dyson1893 */

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=Permissions=
==Dyson1893==
<table border="0" cellpadding="2" align="left" width="100%">
<tr>
<td align="right" colspan="2" width="10%"><font size="-1">🔵</font></td>
<td align="left" colspan="2">
Wiki chapter titled:   [[Apps/DysonPotential#Evaluation|Dyson (1893)]]
</td>
</tr>
<tr>
<td align="right" colspan="2" width="10%"> </td>
<td align="left" colspan="2">
<table border="1" align="left" width="90%" cellpadding="3">
<tr>
<td align="right" width="15%">Author(s):</td>
<td align="left">F. W. (Frank Watson) Dyson</td>
</tr>
<tr>
<td align="right">Title:</td>
<td align="left">''II.   The Potential of an Anchor Ring''</td>
</tr>
<tr>
<td align="right">Reference:</td>
<td align="left">[http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893, Philosophical Transactions of the Royal Society of London. A., 184, 43 - 95)]</td>
</tr>
<tr>
<td align="right">DOI:</td>
<td align="left">[https://doi.org/10.1098/rsta.1893.0002 https://doi.org/10.1098/rsta.1893.0002]</td>
</tr>
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<td align="left">Royal Society Publishing</td>
</tr>
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<td align="right">Information Entry:</td>
<td align="left">2021/09/15</td>
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<td align="right">Relevant Permission:</td>
<td align="left">"[https://royalsociety.org/journals/permissions/ You do not need to seek permissions for re-use of material over 70 years old for up to 5 articles or figures — re-use is only subject to acknowledgement.]"</td>
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 <br />

=See Also=

{{ SGFfooter }}

SSC/Structure/UniformDensity

2021-09-23T00:06:59Z

174.64.14.12: /* Summary */

=Isolated Uniform-Density Sphere=

{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#ffff99;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Uniform-Density<br />Sphere</b>]]</font>
|}
Here we derive the interior structural properties of an isolated uniform-density sphere using all three [[SSCpt2/SolutionStrategies#Solution_Strategies|solution strategies]]. While deriving essentially the same solution three different ways might seem like a bit of overkill, this approach proves to be instructive because (a) it forces us to examine the structural behavior of a number of different physical parameters, and (b) it illustrates how to work through the different solution strategies for one model whose structure can in fact be derived analytically using any of the techniques. As we shall see when studying other self-gravitating configurations, the three strategies are not always equally fruitful.
 <br />
 <br />
 <br />
==Solution Technique 1==

Adopting [[SSCpt2/SolutionStrategies#Technique_1|solution technique #1]], we need to solve the integro-differential equation,

<div align="center">
{{Math/EQ_SShydrostaticBalance01}}
</div>
appreciating that,
<div align="center">
<math> M_r \equiv \int_0^r 4\pi r^2 \rho dr </math> .
</div>
For a uniform-density configuration, {{Math/VAR_Density01}} = <math>~\rho_c</math> = constant, so the density can be pulled outside the mass integral and the integral can be completed immediately to give,
<div align="center">
<math> M_r = \frac{4\pi}{3}\rho_c r^3 </math> .
</div>
Hence, the differential equation describing hydrostatic balance becomes,
<div align="center">
<math> \frac{dP}{dr} = - \frac{4\pi G}{3} \rho_c^2 r </math> .
</div>
Integrating this from the center of the configuration — where <math>~r=0</math> and <math>~P = P_c</math> — out to an arbitrary radius <math>~r</math> that is still inside the configuration, we obtain,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \int_{P_c}^P dP </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4\pi G}{3} \rho_c^2 \int_0^r r dr </math>
</td>
</tr>

<tr>
<td align="right">
<math>~ \Rightarrow ~~~ P </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~P_c - \frac{2\pi G}{3} \rho_c^2 r^2 \, .</math>
</td>
</tr>
</table>

We expect the pressure to drop to zero at the surface of our spherical configuration — that is, at <math>~r=R</math> — so the central pressure must be,

<div align="center">
<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr)</math> ,
</div>

where <math>~M</math> is the total mass of the configuration. Finally, then, we have,

<div align="center">
<math>P(r) = P_c\biggl[1 - \biggl(\frac{r}{R}\biggr)^2 \biggr] </math> .
</div>

==Solution Technique 3==

Adopting [[SSCpt2/SolutionStrategies#Technique_3|solution technique #3]], we need to solve the ''algebraic'' expression,
<div align="center">
<math>~H + \Phi = C_\mathrm{B}</math> .
</div>
in conjunction with the Poisson equation,
<div align="center">
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) = 4\pi G \rho </math> .
</div>
Appreciating that, as shown above, for a uniform density ({{Math/VAR_Density01}} = <math>\rho_c</math> = constant) configuration,
<div align="center">
<math> M_r = \int_0^r 4\pi r^2 \rho dr = \frac{4\pi}{3}\rho_c r^3 </math> ,
</div>
we can integrate the Poisson equation once to give,
<div align="center">
<math> \frac{d\Phi}{dr} = \frac{4\pi G}{3} \rho_c r </math> ,
</div>
everywhere inside the configuration. Integrating this expression from any point inside the configuration to the surface, we find that,
<div align="center">
<math> \int_{\Phi(r)}^{\Phi_\mathrm{surf}} d\Phi = \frac{4\pi G}{3} \rho_c \int_r^R r dr </math> <br />
<math>\Rightarrow ~~~~~ \Phi_\mathrm{surf} - \Phi(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math>
</div>
Turning to the above algebraic condition, we will adopt the convention that {{Math/VAR_Enthalpy01}} is set to zero at the surface of a barotropic configuration, in which case the constant, <math>C_\mathrm{B}</math>, must be,
<div align="center">
<math>~C_\mathrm{B} = (H + \Phi)_\mathrm{surf} = \Phi_\mathrm{surf}</math> .
</div>
Therefore, everywhere inside the configuration {{Math/VAR_Enthalpy01}} must be given by the expression,
<div align="center">
<math>~H(r) = \Phi_\mathrm{surf} - \Phi(r)</math> .
</div>
Matching this with our solution of the Poisson equation, we conclude that, throughout the configuration,
<div align="center">
<math> H(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>
Comparing this result with the result we obtained using solution technique #1, it is clear that throughout a uniform-density, self-gravitating sphere,
<div align="center">
<math>\frac{P}{H} = \rho</math> .
</div>

==Solution Technique 2==

Adopting [[SSCpt2/SolutionStrategies#Technique_2|solution technique #2]], we need to solve the following single, <math>2^\mathrm{nd}</math>-order ODE:
<div align="center">
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d H}{dr} \biggr) = - 4\pi G \rho </math> .
</div>
Appreciating again that, for a uniform density ({{Math/VAR_Density01}} = <math>\rho_c</math> = constant) configuration,
<div align="center">
<math> M_r = \int_0^r 4\pi r^2 \rho dr = \frac{4\pi}{3}\rho_c r^3 </math> ,
</div>
we can integrate the <math>2^\mathrm{nd}</math>-order ODE once to give,
<div align="center">
<math> \frac{dH}{dr} = -\frac{4\pi G}{3} \rho_c r </math> ,
</div>
everywhere inside the configuration. Integrating this expression from any point inside the configuration to the surface — where, again, we adopt the convention that {{Math/VAR_Enthalpy01}} = 0 — we find that,
<div align="center">
<math> \int_{H(r)}^{0} dH = - \frac{4\pi G}{3} \rho_c \int_r^R r dr </math> <br />
<math>\Rightarrow ~~~~~ H(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
</div>

<font color="darkblue">
==Summary==
</font>

From the above derivations, we can describe the properties of a uniform-density, self-gravitating sphere as follows:
* <font color="red">Mass</font>:
: Given the density, <math>\rho_c</math>, and the radius, <math>R</math>, of the configuration, the total mass is,
<div align="center">
<math>M = \frac{4\pi}{3} \rho_c R^3 </math> ;
</div>

: and, expressed as a function of <math>M</math>, the mass that lies interior to radius <math>r</math> is,
<div align="center">
<math>\frac{M_r}{M} = \biggl(\frac{r}{R} \biggr)^3</math> .
</div>
* <font color="red">Pressure</font>:
: Given values for the pair of model parameters <math>( \rho_c , R )</math>, or <math>( M , R )</math>, or <math>( \rho_c , M )</math>, the central pressure of the configuration is,
<div align="center">
<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr) = \biggl[ \frac{\pi}{6} G^3 \rho_c^4 M^2 \biggr]^{1/3}</math> ;<br />
[http://astrowww.phys.uvic.ca/~tatum/celmechs/celm5.pdf J. B. Tatum (2021)] Celestial Mechanics class notes (UVic), §5.13, p. 45, Eq. (5.13.4)
</div>

: and, expressed in terms of the central pressure <math>P_c</math>, the variation with radius of the pressure is,
<div align="center">
<math>P(r) = P_c \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>

* <span id="UniformSphereEnthalpy"><font color="red">Enthalpy</font>:</span>
: Throughout the configuration, the enthalpy is given by the relation,
<div align="center">
<math>H(r) = \frac{P(r)}{ \rho_c} = \frac{GM}{2R} \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>

* <span id="UniformSpherePotential"><font color="red">Gravitational potential</font>: </span>
: Throughout the configuration — that is, for all <math>r \leq R</math> — the gravitational potential is given by the relation,
<div align="center">
<math>\Phi_\mathrm{surf} - \Phi(r) = H(r) = \frac{G M}{2R} \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
</div>
: Outside of this spherical configuration— that is, for all <math>r \geq R</math> — the potential should behave like a point mass potential, that is,
<div align="center">
<math>\Phi(r) = - \frac{GM}{r} </math> .
</div>
: Matching these two expressions at the surface of the configuration, that is, setting <math>\Phi_\mathrm{surf} = - GM/R</math>, we have what is generally considered the properly normalized prescription for the gravitational potential inside a uniform-density, spherically symmetric configuration:
<div align="center">
<math>\Phi(r) = - \frac{G M}{R} \biggl\{ 1 + \frac{1}{2}\biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] \biggr\} = - \frac{3G M}{2R} \biggl[ 1 - \frac{1}{3} \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .<br />
[http://astrowww.phys.uvic.ca/~tatum/celmechs/celm5.pdf J. B. Tatum (2021)] Celestial Mechanics class notes (UVic), §5.8.9, p. 36, Eq. (5.8.23)
</div>

* <font color="red">Mass-Radius relationship</font>:
: We see that, for a given value of <math>\rho_c</math>, the relationship between the configuration's total mass and radius is,
<div align="center">
<math>M \propto R^3 ~~~~~\mathrm{or}~~~~~R \propto M^{1/3} </math> .
</div>

* <font color="red">Central- to Mean-Density Ratio</font>:
: Because this is a uniform-density structure, the ratio of its central density to its mean density is unity, that is,
<div align="center">
<math>\frac{\rho_c}{\bar{\rho}} = 1 </math> .
</div>

=Uniform-Density Sphere Embedded in an External Medium=
For the ''isolated'' uniform-density sphere, discussed above, the surface of the configuration was identified as the radial location where the pressure drops to zero. Here we embed the sphere in a hot, tenuous medium that exerts a confining "external" pressure, <math>~P_e</math>, and ask how the configuration's equilibrium radius, <math>~R_e</math>, changes in response to this applied external pressure, for a given (fixed) total mass and central pressure, <math>~P_c</math>.

Following [[SSC/Structure/UniformDensity#Solution_Technique_1|solution technique #1]], the derivation remains the same up through the integration of the hydrostatic balance equation to obtain the relation,
<div align="center">
<math>P = P_c - \frac{2\pi G}{3} \rho_c^2 r^2 \, .</math>
</div>
Now we set <math>~P = P_e</math> at the surface of our spherical configuration — that is, at <math>~r=R_e</math> — so we can write,
<div align="center">
<math>P_c - P_e = \frac{2\pi G}{3} \rho_c^2 R_e^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R_e^4} \biggr)</math>

<math>\Rightarrow ~~~~~ P_c \biggl( 1 - \frac{P_e}{P_c} \biggr) = \frac{3G}{8\pi}\biggl( \frac{M^2}{R_e^4} \biggr) \, ,</math>
</div>
where <math>M</math> is the total mass of the configuration. Solving for the equilibrium radius, we have,
<div align="center">
<math> R_e = \biggl[ \biggl( \frac{3}{2^3\pi} \biggr) \frac{G M^2}{P_c} \biggl( 1 - \frac{P_e}{P_c} \biggr)^{-1} \biggr]^{1/4} \, .</math>
</div>
As it should, when the ratio <math>~P_e/P_c \rightarrow 0</math>, this relation reduces to the one obtained, above, for the isolated uniform-density sphere, namely,
<div align="center">
<math> R_e^4 = \biggl( \frac{3}{8\pi} \biggr) \frac{G M^2}{P_c} \, .</math>
</div>

{{ SGFfooter }}

SSC/Structure/UniformDensity

2021-09-23T00:06:08Z

174.64.14.12: /* Solution Technique 1 */

=Isolated Uniform-Density Sphere=

{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#ffff99;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Uniform-Density<br />Sphere</b>]]</font>
|}
Here we derive the interior structural properties of an isolated uniform-density sphere using all three [[SSCpt2/SolutionStrategies#Solution_Strategies|solution strategies]]. While deriving essentially the same solution three different ways might seem like a bit of overkill, this approach proves to be instructive because (a) it forces us to examine the structural behavior of a number of different physical parameters, and (b) it illustrates how to work through the different solution strategies for one model whose structure can in fact be derived analytically using any of the techniques. As we shall see when studying other self-gravitating configurations, the three strategies are not always equally fruitful.
 <br />
 <br />
 <br />
==Solution Technique 1==

Adopting [[SSCpt2/SolutionStrategies#Technique_1|solution technique #1]], we need to solve the integro-differential equation,

<div align="center">
{{Math/EQ_SShydrostaticBalance01}}
</div>
appreciating that,
<div align="center">
<math> M_r \equiv \int_0^r 4\pi r^2 \rho dr </math> .
</div>
For a uniform-density configuration, {{Math/VAR_Density01}} = <math>~\rho_c</math> = constant, so the density can be pulled outside the mass integral and the integral can be completed immediately to give,
<div align="center">
<math> M_r = \frac{4\pi}{3}\rho_c r^3 </math> .
</div>
Hence, the differential equation describing hydrostatic balance becomes,
<div align="center">
<math> \frac{dP}{dr} = - \frac{4\pi G}{3} \rho_c^2 r </math> .
</div>
Integrating this from the center of the configuration — where <math>~r=0</math> and <math>~P = P_c</math> — out to an arbitrary radius <math>~r</math> that is still inside the configuration, we obtain,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \int_{P_c}^P dP </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4\pi G}{3} \rho_c^2 \int_0^r r dr </math>
</td>
</tr>

<tr>
<td align="right">
<math>~ \Rightarrow ~~~ P </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~P_c - \frac{2\pi G}{3} \rho_c^2 r^2 \, .</math>
</td>
</tr>
</table>

We expect the pressure to drop to zero at the surface of our spherical configuration — that is, at <math>~r=R</math> — so the central pressure must be,

<div align="center">
<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr)</math> ,
</div>

where <math>~M</math> is the total mass of the configuration. Finally, then, we have,

<div align="center">
<math>P(r) = P_c\biggl[1 - \biggl(\frac{r}{R}\biggr)^2 \biggr] </math> .
</div>

==Solution Technique 3==

Adopting [[SSCpt2/SolutionStrategies#Technique_3|solution technique #3]], we need to solve the ''algebraic'' expression,
<div align="center">
<math>~H + \Phi = C_\mathrm{B}</math> .
</div>
in conjunction with the Poisson equation,
<div align="center">
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) = 4\pi G \rho </math> .
</div>
Appreciating that, as shown above, for a uniform density ({{Math/VAR_Density01}} = <math>\rho_c</math> = constant) configuration,
<div align="center">
<math> M_r = \int_0^r 4\pi r^2 \rho dr = \frac{4\pi}{3}\rho_c r^3 </math> ,
</div>
we can integrate the Poisson equation once to give,
<div align="center">
<math> \frac{d\Phi}{dr} = \frac{4\pi G}{3} \rho_c r </math> ,
</div>
everywhere inside the configuration. Integrating this expression from any point inside the configuration to the surface, we find that,
<div align="center">
<math> \int_{\Phi(r)}^{\Phi_\mathrm{surf}} d\Phi = \frac{4\pi G}{3} \rho_c \int_r^R r dr </math> <br />
<math>\Rightarrow ~~~~~ \Phi_\mathrm{surf} - \Phi(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math>
</div>
Turning to the above algebraic condition, we will adopt the convention that {{Math/VAR_Enthalpy01}} is set to zero at the surface of a barotropic configuration, in which case the constant, <math>C_\mathrm{B}</math>, must be,
<div align="center">
<math>~C_\mathrm{B} = (H + \Phi)_\mathrm{surf} = \Phi_\mathrm{surf}</math> .
</div>
Therefore, everywhere inside the configuration {{Math/VAR_Enthalpy01}} must be given by the expression,
<div align="center">
<math>~H(r) = \Phi_\mathrm{surf} - \Phi(r)</math> .
</div>
Matching this with our solution of the Poisson equation, we conclude that, throughout the configuration,
<div align="center">
<math> H(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>
Comparing this result with the result we obtained using solution technique #1, it is clear that throughout a uniform-density, self-gravitating sphere,
<div align="center">
<math>\frac{P}{H} = \rho</math> .
</div>

==Solution Technique 2==

Adopting [[SSCpt2/SolutionStrategies#Technique_2|solution technique #2]], we need to solve the following single, <math>2^\mathrm{nd}</math>-order ODE:
<div align="center">
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d H}{dr} \biggr) = - 4\pi G \rho </math> .
</div>
Appreciating again that, for a uniform density ({{Math/VAR_Density01}} = <math>\rho_c</math> = constant) configuration,
<div align="center">
<math> M_r = \int_0^r 4\pi r^2 \rho dr = \frac{4\pi}{3}\rho_c r^3 </math> ,
</div>
we can integrate the <math>2^\mathrm{nd}</math>-order ODE once to give,
<div align="center">
<math> \frac{dH}{dr} = -\frac{4\pi G}{3} \rho_c r </math> ,
</div>
everywhere inside the configuration. Integrating this expression from any point inside the configuration to the surface — where, again, we adopt the convention that {{Math/VAR_Enthalpy01}} = 0 — we find that,
<div align="center">
<math> \int_{H(r)}^{0} dH = - \frac{4\pi G}{3} \rho_c \int_r^R r dr </math> <br />
<math>\Rightarrow ~~~~~ H(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
</div>

<font color="darkblue">
==Summary==
</font>

From the above derivations, we can describe the properties of a uniform-density, self-gravitating sphere as follows:
* <font color="red">Mass</font>:
: Given the density, <math>\rho_c</math>, and the radius, <math>R</math>, of the configuration, the total mass is,
<div align="center">
<math>M = \frac{4\pi}{3} \rho_c R^3 </math> ;
</div>

: and, expressed as a function of <math>M</math>, the mass that lies interior to radius <math>r</math> is,
<div align="center">
<math>\frac{M_r}{M} = \biggl(\frac{r}{R} \biggr)^3</math> .
</div>
* <font color="red">Pressure</font>:
: Given values for the pair of model parameters <math>( \rho_c , R )</math>, or <math>( M , R )</math>, or <math>( \rho_c , M )</math>, the central pressure of the configuration is,
<div align="center">
<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr) = \biggl[ \frac{\pi}{6} G^3 \rho_c^4 M^2 \biggr]^{1/3}</math> ;
</div>

: and, expressed in terms of the central pressure <math>P_c</math>, the variation with radius of the pressure is,
<div align="center">
<math>P(r) = P_c \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>

* <span id="UniformSphereEnthalpy"><font color="red">Enthalpy</font>:</span>
: Throughout the configuration, the enthalpy is given by the relation,
<div align="center">
<math>H(r) = \frac{P(r)}{ \rho_c} = \frac{GM}{2R} \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>

* <span id="UniformSpherePotential"><font color="red">Gravitational potential</font>: </span>
: Throughout the configuration — that is, for all <math>r \leq R</math> — the gravitational potential is given by the relation,
<div align="center">
<math>\Phi_\mathrm{surf} - \Phi(r) = H(r) = \frac{G M}{2R} \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
</div>
: Outside of this spherical configuration— that is, for all <math>r \geq R</math> — the potential should behave like a point mass potential, that is,
<div align="center">
<math>\Phi(r) = - \frac{GM}{r} </math> .
</div>
: Matching these two expressions at the surface of the configuration, that is, setting <math>\Phi_\mathrm{surf} = - GM/R</math>, we have what is generally considered the properly normalized prescription for the gravitational potential inside a uniform-density, spherically symmetric configuration:
<div align="center">
<math>\Phi(r) = - \frac{G M}{R} \biggl\{ 1 + \frac{1}{2}\biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] \biggr\} = - \frac{3G M}{2R} \biggl[ 1 - \frac{1}{3} \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .<br />
[http://astrowww.phys.uvic.ca/~tatum/celmechs/celm5.pdf J. B. Tatum (2021)] Celestial Mechanics class notes (UVic), §5.8.9, p. 36, Eq. (5.8.23)
</div>

* <font color="red">Mass-Radius relationship</font>:
: We see that, for a given value of <math>\rho_c</math>, the relationship between the configuration's total mass and radius is,
<div align="center">
<math>M \propto R^3 ~~~~~\mathrm{or}~~~~~R \propto M^{1/3} </math> .
</div>

* <font color="red">Central- to Mean-Density Ratio</font>:
: Because this is a uniform-density structure, the ratio of its central density to its mean density is unity, that is,
<div align="center">
<math>\frac{\rho_c}{\bar{\rho}} = 1 </math> .
</div>

=Uniform-Density Sphere Embedded in an External Medium=
For the ''isolated'' uniform-density sphere, discussed above, the surface of the configuration was identified as the radial location where the pressure drops to zero. Here we embed the sphere in a hot, tenuous medium that exerts a confining "external" pressure, <math>~P_e</math>, and ask how the configuration's equilibrium radius, <math>~R_e</math>, changes in response to this applied external pressure, for a given (fixed) total mass and central pressure, <math>~P_c</math>.

Following [[SSC/Structure/UniformDensity#Solution_Technique_1|solution technique #1]], the derivation remains the same up through the integration of the hydrostatic balance equation to obtain the relation,
<div align="center">
<math>P = P_c - \frac{2\pi G}{3} \rho_c^2 r^2 \, .</math>
</div>
Now we set <math>~P = P_e</math> at the surface of our spherical configuration — that is, at <math>~r=R_e</math> — so we can write,
<div align="center">
<math>P_c - P_e = \frac{2\pi G}{3} \rho_c^2 R_e^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R_e^4} \biggr)</math>

<math>\Rightarrow ~~~~~ P_c \biggl( 1 - \frac{P_e}{P_c} \biggr) = \frac{3G}{8\pi}\biggl( \frac{M^2}{R_e^4} \biggr) \, ,</math>
</div>
where <math>M</math> is the total mass of the configuration. Solving for the equilibrium radius, we have,
<div align="center">
<math> R_e = \biggl[ \biggl( \frac{3}{2^3\pi} \biggr) \frac{G M^2}{P_c} \biggl( 1 - \frac{P_e}{P_c} \biggr)^{-1} \biggr]^{1/4} \, .</math>
</div>
As it should, when the ratio <math>~P_e/P_c \rightarrow 0</math>, this relation reduces to the one obtained, above, for the isolated uniform-density sphere, namely,
<div align="center">
<math> R_e^4 = \biggl( \frac{3}{8\pi} \biggr) \frac{G M^2}{P_c} \, .</math>
</div>

{{ SGFfooter }}

SSC/Structure/UniformDensity

2021-09-23T00:03:44Z

174.64.14.12: /* Solution Technique 1 */

=Isolated Uniform-Density Sphere=

{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#ffff99;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Uniform-Density<br />Sphere</b>]]</font>
|}
Here we derive the interior structural properties of an isolated uniform-density sphere using all three [[SSCpt2/SolutionStrategies#Solution_Strategies|solution strategies]]. While deriving essentially the same solution three different ways might seem like a bit of overkill, this approach proves to be instructive because (a) it forces us to examine the structural behavior of a number of different physical parameters, and (b) it illustrates how to work through the different solution strategies for one model whose structure can in fact be derived analytically using any of the techniques. As we shall see when studying other self-gravitating configurations, the three strategies are not always equally fruitful.
 <br />
 <br />
 <br />
==Solution Technique 1==

Adopting [[SSCpt2/SolutionStrategies#Technique_1|solution technique #1]], we need to solve the integro-differential equation,

<div align="center">
{{Math/EQ_SShydrostaticBalance01}}
</div>
appreciating that,
<div align="center">
<math> M_r \equiv \int_0^r 4\pi r^2 \rho dr </math> .
</div>
For a uniform-density configuration, {{Math/VAR_Density01}} = <math>~\rho_c</math> = constant, so the density can be pulled outside the mass integral and the integral can be completed immediately to give,
<div align="center">
<math> M_r = \frac{4\pi}{3}\rho_c r^3 </math> .
</div>
Hence, the differential equation describing hydrostatic balance becomes,
<div align="center">
<math> \frac{dP}{dr} = - \frac{4\pi G}{3} \rho_c^2 r </math> .
</div>
Integrating this from the center of the configuration — where <math>~r=0</math> and <math>~P = P_c</math> — out to an arbitrary radius <math>~r</math> that is still inside the configuration, we obtain,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \int_{P_c}^P dP </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4\pi G}{3} \rho_c^2 \int_0^r r dr </math>
</td>
</tr>

<tr>
<td align="right">
<math>~ \Rightarrow ~~~ P </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~P_c - \frac{2\pi G}{3} \rho_c^2 r^2 \, .</math>
</td>
</tr>
</table>

We expect the pressure to drop to zero at the surface of our spherical configuration — that is, at <math>~r=R</math> — so the central pressure must be,

<div align="center">
<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr)</math> ,<br />
[http://astrowww.phys.uvic.ca/~tatum/celmechs/celm5.pdf J. B. Tatum (2021)] Celestial Mechanics class notes (UVic), §5.13, p. 45, Eq. (5.13.4)
</div>

where <math>~M</math> is the total mass of the configuration. Finally, then, we have,

<div align="center">
<math>P(r) = P_c\biggl[1 - \biggl(\frac{r}{R}\biggr)^2 \biggr] </math> .
</div>

==Solution Technique 3==

Adopting [[SSCpt2/SolutionStrategies#Technique_3|solution technique #3]], we need to solve the ''algebraic'' expression,
<div align="center">
<math>~H + \Phi = C_\mathrm{B}</math> .
</div>
in conjunction with the Poisson equation,
<div align="center">
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) = 4\pi G \rho </math> .
</div>
Appreciating that, as shown above, for a uniform density ({{Math/VAR_Density01}} = <math>\rho_c</math> = constant) configuration,
<div align="center">
<math> M_r = \int_0^r 4\pi r^2 \rho dr = \frac{4\pi}{3}\rho_c r^3 </math> ,
</div>
we can integrate the Poisson equation once to give,
<div align="center">
<math> \frac{d\Phi}{dr} = \frac{4\pi G}{3} \rho_c r </math> ,
</div>
everywhere inside the configuration. Integrating this expression from any point inside the configuration to the surface, we find that,
<div align="center">
<math> \int_{\Phi(r)}^{\Phi_\mathrm{surf}} d\Phi = \frac{4\pi G}{3} \rho_c \int_r^R r dr </math> <br />
<math>\Rightarrow ~~~~~ \Phi_\mathrm{surf} - \Phi(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math>
</div>
Turning to the above algebraic condition, we will adopt the convention that {{Math/VAR_Enthalpy01}} is set to zero at the surface of a barotropic configuration, in which case the constant, <math>C_\mathrm{B}</math>, must be,
<div align="center">
<math>~C_\mathrm{B} = (H + \Phi)_\mathrm{surf} = \Phi_\mathrm{surf}</math> .
</div>
Therefore, everywhere inside the configuration {{Math/VAR_Enthalpy01}} must be given by the expression,
<div align="center">
<math>~H(r) = \Phi_\mathrm{surf} - \Phi(r)</math> .
</div>
Matching this with our solution of the Poisson equation, we conclude that, throughout the configuration,
<div align="center">
<math> H(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>
Comparing this result with the result we obtained using solution technique #1, it is clear that throughout a uniform-density, self-gravitating sphere,
<div align="center">
<math>\frac{P}{H} = \rho</math> .
</div>

==Solution Technique 2==

Adopting [[SSCpt2/SolutionStrategies#Technique_2|solution technique #2]], we need to solve the following single, <math>2^\mathrm{nd}</math>-order ODE:
<div align="center">
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d H}{dr} \biggr) = - 4\pi G \rho </math> .
</div>
Appreciating again that, for a uniform density ({{Math/VAR_Density01}} = <math>\rho_c</math> = constant) configuration,
<div align="center">
<math> M_r = \int_0^r 4\pi r^2 \rho dr = \frac{4\pi}{3}\rho_c r^3 </math> ,
</div>
we can integrate the <math>2^\mathrm{nd}</math>-order ODE once to give,
<div align="center">
<math> \frac{dH}{dr} = -\frac{4\pi G}{3} \rho_c r </math> ,
</div>
everywhere inside the configuration. Integrating this expression from any point inside the configuration to the surface — where, again, we adopt the convention that {{Math/VAR_Enthalpy01}} = 0 — we find that,
<div align="center">
<math> \int_{H(r)}^{0} dH = - \frac{4\pi G}{3} \rho_c \int_r^R r dr </math> <br />
<math>\Rightarrow ~~~~~ H(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
</div>

<font color="darkblue">
==Summary==
</font>

From the above derivations, we can describe the properties of a uniform-density, self-gravitating sphere as follows:
* <font color="red">Mass</font>:
: Given the density, <math>\rho_c</math>, and the radius, <math>R</math>, of the configuration, the total mass is,
<div align="center">
<math>M = \frac{4\pi}{3} \rho_c R^3 </math> ;
</div>

: and, expressed as a function of <math>M</math>, the mass that lies interior to radius <math>r</math> is,
<div align="center">
<math>\frac{M_r}{M} = \biggl(\frac{r}{R} \biggr)^3</math> .
</div>
* <font color="red">Pressure</font>:
: Given values for the pair of model parameters <math>( \rho_c , R )</math>, or <math>( M , R )</math>, or <math>( \rho_c , M )</math>, the central pressure of the configuration is,
<div align="center">
<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr) = \biggl[ \frac{\pi}{6} G^3 \rho_c^4 M^2 \biggr]^{1/3}</math> ;
</div>

: and, expressed in terms of the central pressure <math>P_c</math>, the variation with radius of the pressure is,
<div align="center">
<math>P(r) = P_c \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>

* <span id="UniformSphereEnthalpy"><font color="red">Enthalpy</font>:</span>
: Throughout the configuration, the enthalpy is given by the relation,
<div align="center">
<math>H(r) = \frac{P(r)}{ \rho_c} = \frac{GM}{2R} \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>

* <span id="UniformSpherePotential"><font color="red">Gravitational potential</font>: </span>
: Throughout the configuration — that is, for all <math>r \leq R</math> — the gravitational potential is given by the relation,
<div align="center">
<math>\Phi_\mathrm{surf} - \Phi(r) = H(r) = \frac{G M}{2R} \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
</div>
: Outside of this spherical configuration— that is, for all <math>r \geq R</math> — the potential should behave like a point mass potential, that is,
<div align="center">
<math>\Phi(r) = - \frac{GM}{r} </math> .
</div>
: Matching these two expressions at the surface of the configuration, that is, setting <math>\Phi_\mathrm{surf} = - GM/R</math>, we have what is generally considered the properly normalized prescription for the gravitational potential inside a uniform-density, spherically symmetric configuration:
<div align="center">
<math>\Phi(r) = - \frac{G M}{R} \biggl\{ 1 + \frac{1}{2}\biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] \biggr\} = - \frac{3G M}{2R} \biggl[ 1 - \frac{1}{3} \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .<br />
[http://astrowww.phys.uvic.ca/~tatum/celmechs/celm5.pdf J. B. Tatum (2021)] Celestial Mechanics class notes (UVic), §5.8.9, p. 36, Eq. (5.8.23)
</div>

* <font color="red">Mass-Radius relationship</font>:
: We see that, for a given value of <math>\rho_c</math>, the relationship between the configuration's total mass and radius is,
<div align="center">
<math>M \propto R^3 ~~~~~\mathrm{or}~~~~~R \propto M^{1/3} </math> .
</div>

* <font color="red">Central- to Mean-Density Ratio</font>:
: Because this is a uniform-density structure, the ratio of its central density to its mean density is unity, that is,
<div align="center">
<math>\frac{\rho_c}{\bar{\rho}} = 1 </math> .
</div>

=Uniform-Density Sphere Embedded in an External Medium=
For the ''isolated'' uniform-density sphere, discussed above, the surface of the configuration was identified as the radial location where the pressure drops to zero. Here we embed the sphere in a hot, tenuous medium that exerts a confining "external" pressure, <math>~P_e</math>, and ask how the configuration's equilibrium radius, <math>~R_e</math>, changes in response to this applied external pressure, for a given (fixed) total mass and central pressure, <math>~P_c</math>.

Following [[SSC/Structure/UniformDensity#Solution_Technique_1|solution technique #1]], the derivation remains the same up through the integration of the hydrostatic balance equation to obtain the relation,
<div align="center">
<math>P = P_c - \frac{2\pi G}{3} \rho_c^2 r^2 \, .</math>
</div>
Now we set <math>~P = P_e</math> at the surface of our spherical configuration — that is, at <math>~r=R_e</math> — so we can write,
<div align="center">
<math>P_c - P_e = \frac{2\pi G}{3} \rho_c^2 R_e^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R_e^4} \biggr)</math>

<math>\Rightarrow ~~~~~ P_c \biggl( 1 - \frac{P_e}{P_c} \biggr) = \frac{3G}{8\pi}\biggl( \frac{M^2}{R_e^4} \biggr) \, ,</math>
</div>
where <math>M</math> is the total mass of the configuration. Solving for the equilibrium radius, we have,
<div align="center">
<math> R_e = \biggl[ \biggl( \frac{3}{2^3\pi} \biggr) \frac{G M^2}{P_c} \biggl( 1 - \frac{P_e}{P_c} \biggr)^{-1} \biggr]^{1/4} \, .</math>
</div>
As it should, when the ratio <math>~P_e/P_c \rightarrow 0</math>, this relation reduces to the one obtained, above, for the isolated uniform-density sphere, namely,
<div align="center">
<math> R_e^4 = \biggl( \frac{3}{8\pi} \biggr) \frac{G M^2}{P_c} \, .</math>
</div>

{{ SGFfooter }}

AxisymmetricConfigurations/PoissonEq

2021-09-23T00:00:05Z

174.64.14.12: /* Spherical Harmonics and Associated Legendre Functions */

__FORCETOC__

=Solving the (Multi-dimensional) Poisson Equation Numerically=

==Overview==

The set of [[PGE#Principal_Governing_Equations|Principal Governing Equations]] that serves as the foundation of our study of the structure, stability, and dynamical evolution of self-gravitating fluids contains an equation of motion (the ''Euler'' equation) that includes an acceleration due to local gradients in the (Newtonian) gravitational potential, <math>\Phi</math>. As has been pointed out in an [[PGE/PoissonOrigin#Origin_of_the_Poisson_Equation|accompanying chapter that discusses the origin of the Poisson equation]], the mathematical definition of this acceleration is fundamentally drawn from Isaac Newton's inverse-square law of gravitation, but takes into account that our fluid systems are not ensembles of point-mass sources but, rather, are represented by a continuous ''distribution'' of mass via the function, <math>\rho(\vec{x},t)</math>. As indicated, in our study, <math>\rho</math> may depend on time as well as space. The acceleration felt at any point in space may be obtained by integrating over the accelerations exerted by each differential mass element. Alternatively — and more commonly — as has been explicitly demonstrated in, respectively, [[PGE/PoissonOrigin#Step_1|Step 1]] and [[PGE/PoissonOrigin#Step_3|Step 3]] of the same accompanying chapter, at any point in time the spatial variation of the gravitational potential, <math>\Phi(\vec{x})</math>, is determined from <math>~\rho(\vec{x})</math> via either an ''integral'' or a ''differential'' equation as follows:

<div align="center">
<table border="1" cellpadding="8" align="center" width="80%">
<tr><th align="center" colspan="2"><font size="+0">Table 1:  Poisson Equation</font></th></tr>
<tr>
<th align="center">Integral Representation</th>
<th align="center">Differential Representation </th>
</tr>
<tr>
<td align="center">
<table border="0" align="center">
<tr>
<td align="right">
<math>\Phi(\vec{x})</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>-G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
</td>
</tr>
</table>
</td>
<td align="center">
{{ Math/EQ_Poisson01 }}
</td>
</tr>
</table>
</div>

While it is possible in some restricted situations to determine analytic expressions for the matched pair of functions, <math>\Phi </math> and <math>\rho</math>, that satisfy the Poisson equation, modeling the vast majority of interesting astrophysical problems requires the develop of a numerical scheme to solve the Poisson equation. In what follows, our aim is twofold: (a) To recount — in a reasonable amount of detail — the steps that we have taken over the past, approximately forty years to develop more and more accurate and efficient ways to solve the Poisson equation in full three-dimensional generality; and (b) to list/summarize alternative techniques that have been successfully employed by other research groups over the years.

<table border="0" align="center" cellpadding="5" width="100%">
<tr>
<td align="center">
<table border="1"><tr><td align="center" bgcolor="red">     </td></tr></table>
</td>
<td align="left" colspan="2">Dimensionality …</td>
<td align="left">2D   or   3D</td>
</tr>

<tr>
<td align="center">
<table border="1"><tr><td align="center" bgcolor="lightgreen">     </td></tr></table>
</td>
<td align="left" colspan="2">Computational Mesh Used for ''Differential Representation'' …</td>
<td align="left">[Car]tesian   [Cyl]indrical   [Sph]erical   Gridless ([Lag]rangian)</td>
</tr>

<tr>
<td align="center">
<table border="1"><tr><td align="center" bgcolor="yellow">     </td></tr></table>
</td>
<td align="left" colspan="2">Green's Function for ''Integral Representation'' …</td>
<td align="left">[Sph]erical   [Tor]oidal   Gridless ([Lag]rangian)</td>
</tr>
</table>

For each chosen problem, a research group must decide/specify, at a minimum:   (1) The ''dimensionality'' of the problem; that is, whether the study will be restricted to 2D (e.g., axisymmetric) systems or whether the problem will be tackled in its full 3D complexity. (2) Whether the gravitational potential ''inside'' the mass distribution will be determined by solving the ''integral representation'' or the ''differential representation'' of the Poisson equation and, if the latter, in what coordinate frame (e.g., cylindrical or spherical) the differential operator and the computational mesh will be based. (Note that even when the interior solution is obtained by evaluating the ''differential representation'' of the Poisson equation, the ''integral representation'' will likely be employed to evaluate the potential on a boundary that lies outside the mass distribution.) (3) Which Green's function representation of the term, <math>~|\vec{x}^{~'} - \vec{x}|^{-1}</math>, will be used if/when the ''integral representation'' is evaluated. As the numerical techniques employed by each research group are introduced, below, a small red/green/yellow boxed icon has been interlaced with the text in an effort to highlight, up front, which choices the group has made.

==Our Approach — Initially Heavily Influenced by Black & Bodenheimer (1975)==

<table border="0" cellpadding="5" width="100%" align="center"><tr><td align="left">
<table border="1" cellpadding="5" align="left"><tr>
<td align="center" bgcolor="red">  2D  </td>
<td align="center" bgcolor="lightgreen">  Cyl  </td>
<td align="center" bgcolor="yellow">  Sph  </td>
</tr></table>
</td></tr></table>
[http://adsabs.harvard.edu/abs/1975ApJ...199..619B D. C. Black & P. Bodenheimer (1975, ApJ, 199, 619 - 632)] published a detailed description of the numerical techniques that they implemented in an effort to model, in two dimensions, the dynamical collapse of ''axisymmetric'' interstellar gas clouds. Among the techniques was their method of solving, in cylindrical coordinates, the ''differential representation'' of the two-dimensional (2D) Poisson equation. Values of the potential along the outer boundaries of their cylindrical mesh were obtained by employing a spherical-harmonic expansion of the Green's function to evaluate the ''integral representation'' of the Poisson equation.

As is acknowledged in the last paragraph of their paper, Black & Bodenheimer were introduced "to the pitfalls of two-dimensional hydrodynamics" by Drs. J. LeBlanc and J. Wilson who, at the time, were both staff scientists at the Lawrence Livermore Laboratory. In 1976, having completed my formal graduate-level course work in the astronomy program at UC, Santa Cruz, Peter Bodenheimer (on the faculty of Lick Observatory and UC, Santa Cruz) asked me if I would be interested in working with him and David Black (a planetary scientist at NASA, Ames Research Center) on the development of a fully three-dimensional (3D) hydrodynamics code to study, not only the collapse, but also the fragmentation of self-gravitating gas clouds. I jumped at the opportunity. As a result, an effort to model the process of spontaneous cloud fragmentation became the focus of my doctoral dissertation research.

<table border="0" cellpadding="5" width="100%" align="center"><tr><td align="left">
<table border="1" cellpadding="5" align="left"><tr>
<td align="center" bgcolor="red">  3D  </td>
<td align="center" bgcolor="lightgreen">  Cyl  </td>
<td align="center" bgcolor="yellow">  Sph  </td>
</tr></table>
</td></tr></table>
Borrowing from the [http://adsabs.harvard.edu/abs/1975ApJ...199..619B Black & Bodenheimer (1975)] work, I decided to solve the set of 3D [[PGE#Principal_Governing_Equations|principal governing equations]] on a cylindrical coordinate <math>~(\varpi, \phi, z)</math> mesh:
* whose outermost radial and vertical boundaries were placed entirely outside of the mass distribution (by way of illustration, see the green-dashed lines in Figure 1);
* that exhibited reflection symmetry through the <math>~(z = 0)</math> equatorial plane ;
* that allowed for non-uniform (logarithmically stretched) grid spacing in both the radial and vertical directions; and,
* (extending the Black & Bodenheimer work from 2D to 3D) with strictly uniform spacing in the azimuthal-coordinate direction.

----

<table border="0" align="center" width="85%">
<tr><td align="left">
<table border="0" cellpadding="10" width="200px" align="right">
<tr><td align="center">
<table border="1" cellpadding="5" align="center">
<tr><th align="center">Figure 1</th></tr>
<tr><td align="center">[[File:HSCF_MeridionalPlaneGrid04.png|center|185px|Schematic of grid and mass distribution]]</td></tr>
</table>
</td></tr></table>
<span id="MeshChoice"><font color="darkblue">'''Mesh Choice''':</font> In retrospect,</span> I am still comfortable with the choice that was made, at the time, of a cylindrical coordinate mesh with uniform spacing in the azimuthal-coordinate direction. When expressed in terms of cylindrical coordinates — as opposed to cartesian coordinates, for example — the azimuthal component of the Euler equation offers a natural means by which the conservation of angular momentum can be monitored, if not enforced. And, by enabling the implementation of [https://en.wikipedia.org/wiki/Fast_Fourier_transform FFTs] — hence, providing the ability to rapidly transform functions, like <math>~\Phi(\phi)</math> and <math>~\rho(\phi)</math>, back and forth between ''real'' space and ''Fourier'' space — a uniform descritization of the azimuthal grid facilitated the development of an efficient 3D Poisson solver as well as tools to straightforwardly analyze the behavior — e.g., the exponential growth — of individual nonaxisymmetric modes.
</td></tr>
</table>

----

Also, following the lead of [http://adsabs.harvard.edu/abs/1975ApJ...199..619B Black & Bodenheimer (1975)], a hybrid scheme was developed to solve the Poisson equation. As defined in Table 1 above, its ''integral representation'' was evaluated at grid locations along the outermost radial and vertical boundaries — illustrated by the dashed green lines displayed in Figure 1 — by integrating over the "pink" mass distribution lying inside of these grid boundaries. Separately, the ''differential'' form of the Poisson equation was used to evaluate the potential at all ''interior'' grid locations; this differential expression was supplemented by the implementation of [https://en.wikipedia.org/wiki/Neumann_boundary_condition ''Neumann'' boundary conditions] (reflection symmetry) along the equatorial plane, and by using the values of the potential just determined along the outermost grid boundaries to provide [https://en.wikipedia.org/wiki/Dirichlet_boundary_condition ''Dirichlet'' boundary conditions] along those grid boundaries.

===Determining Values of the Potential on the Mesh Boundary===
Let's determine the potential, <math>~\Phi_B</math>, at all points along the ''boundary'' of the cylindrical coordinate mesh by evaluating Table 1's ''integral representation'' of the Poisson equation.

====Using a Spherical Harmonic Expansion====
=====Full Three-Dimensional Generality=====

Following the lead of [http://adsabs.harvard.edu/abs/1975ApJ...199..619B Black & Bodenheimer (1975)], we will insert into this integral relation the Green's function expression for <math>~|\vec{x}^{~'}- \vec{x} |^{-1} </math> as given in terms of ''Spherical Harmonics'', <math>~Y_{\ell m}</math>, which in turn can be written in terms of ''Associated Legendre Functions.'' [[#Ylm|Table 2, below]], provides the primary details.

Written in the context of a spherical coordinate system we have,

<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B(r,\theta,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-G \int \frac{1}{|\vec{x}^{~'} - \vec{x}|} ~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-G \int
\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{4\pi}{2\ell+1} \biggl[ \frac{r_<^\ell}{r_>^{\ell+1}} \biggr] Y_{\ell m}^*(\theta^', \phi^') Y_{\ell m}(\theta,\phi)
~\rho(r^', \theta^', \phi^') d^3x^'
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-4\pi G
\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{Y_{\ell m}(\theta,\phi)}{(2\ell+1)} \biggl[ \frac{1}{r^{\ell+1}}\int_0^r (r^')^\ell Y_{\ell m}^*(\theta^', \phi^')
~\rho(r^', \theta^', \phi^') d^3x^'
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
        <math>~
+ r^\ell \int_r^\infty (r^')^{-(\ell+1)} Y_{\ell m}^*(\theta^', \phi^')
~\rho(r^', \theta^', \phi^') d^3x^' \biggr] \, .
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 66, Eq. (2-122)
</td>
</tr>
</table>

If the distance from the origin, <math>~r</math>, of a boundary point (''i.e.'', any point lying along the dashed green lines in Figure 1) is greater than the distance from the origin, <math>~r^'</math>, of ''all'' of the (pink) mass elements, the second integral can be completely ignored. We have, then, an expression that will henceforth be referred to as,

<div align="center" id="PotentialA">
<table border="0" align="center">
<tr>
<td align="center" colspan="3">
<font color="#770000">'''Form A of the Boundary Potential'''</font><br />
</td>
</tr>
<tr>
<td align="right">
<math>~ \Phi_B(r,\theta,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-4\pi G
\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{Y_{\ell m}(\theta,\phi)}{(2\ell+1)} \biggl[ \frac{1}{r^{\ell+1}}\int_0^r (r^')^\ell Y_{\ell m}^*(\theta^', \phi^')
~\rho(r^', \theta^', \phi^') d^3x^' \biggr] \, .
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], p. 137, Eq. (4.2)<br />
[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], p. 238, Eqs. (7.53) - (7.54)
</td>
</tr>
</table>
</div>

Rewriting this expression for <math>~\Phi_B</math> in terms of cylindrical coordinates — which aligns with our chosen grid coordinate system — and admitting that in practice our summation over the index, <math>~\ell</math>, cannot extend to infinity, we have,

<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B(\varpi, \phi, z)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-4\pi G
\sum_{\ell=0}^{\ell_\mathrm{max}} \sum_{m=-\ell}^{+\ell} \frac{Y_{\ell m}}{(2\ell+1)} \biggl[ \varpi^2 + z^2 \biggr]^{-(\ell+1)/2} \int Y_{\ell m}^* \biggl[ (\varpi^')^2 + (z^')^2 \biggr]^{\ell/2}
~\rho(\varpi^', \phi^', z^') d^3x^' \, .
</math>
</td>
</tr>
</table>

Note that, as a consequence of assuming that our configurations have equatorial-plane symmetry, the weighted integral over the mass distribution necessarily goes to zero anytime the sum of the two indexes, <math>~(\ell + m)</math>, is an odd number. This is because, in each of these situations — again, see [[#Ylm|Table 2, below]] for details — the <math>~Y_{\ell m}</math> includes an overall factor of <math>~\cos\theta</math>, which necessarily switches signs between the two hemispheres. After setting <math>~\ell_\mathrm{max}=4</math> — and dropping all terms in the summation for which the index sum, <math>~(\ell + m)</math>, is odd — this expression becomes precisely the relation that was used to determine the ''boundary'' values of the gravitational potential in our earliest set of simulations; see, for example, [http://adsabs.harvard.edu/abs/1978PhDT.........6T Tohline (1978)], [http://adsabs.harvard.edu/abs/1980ApJ...235..866T Tohline (1980)], and [http://adsabs.harvard.edu/abs/1980ApJ...242..209B Bodenheimer, Tohline, & Black (1980)].

=====Simplification for 2D, Axisymmetric Systems=====
It is easy to show that this last expression for <math>~\Phi_B</math> — which has been used in our 3D simulations — is a ''generalization'' of the expression for <math>~\Phi_B</math> that was employed by [http://adsabs.harvard.edu/abs/1975ApJ...199..619B Black & Bodenheimer (1975)] for 2D, axisymmetric simulations. In axisymmetric systems, by definition, physical variables exhibit no variation in the azimuthal coordinate direction. Hence, in the expression for <math>~\Phi_B</math>:
* the azimuthal coordinate, <math>~\phi</math>, need not appear explicitly as an independent variable;
* the index, <math>~m</math>, must be set to zero, so there is no summation over this index; and,
* every surviving spherical harmonic can be written more simply in terms of a Legendre function, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~Y_{\ell m} \rightarrow Y_{\ell 0}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\sqrt{\frac{(2\ell+1 )}{4\pi}} P_\ell(\chi) \, ,</math>
</td>
<td align="center">        where,</td>
<td align="left">
<math>~\chi \equiv \frac{z}{(\varpi^2 + z^2)^{1 / 2}} \, .</math>
</td>
</tr>
</table>
</div>
Note that the argument, <math>~\chi</math>, is still the spherical-coordinate expression, <math>~\cos\theta</math>, but here it has been written in terms of cylindrical coordinates. We have, therefore,

<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B(\varpi, z)\biggr|_{2D}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- G
\sum_{\ell=0}^{\ell_\mathrm{max}} P_\ell(\chi) \biggl[ \varpi^2 + z^2 \biggr]^{-(\ell+1)/2} \int P_\ell(\chi^') \biggl[ (\varpi^')^2 + (z^')^2 \biggr]^{\ell/2}
~\rho(\varpi^', z^') d^3x^' \, ,
</math>
</td>
</tr>
</table>
where, <math>~d^3x^' = 2\pi \varpi^' d\varpi^' dz^'</math>. This is precisely the same as equation (5) from [http://adsabs.harvard.edu/abs/1975ApJ...199..619B Black & Bodenheimer (1975)]; see also, equations (8) and (9) in [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)].

====Using Toroidal Functions====

=====In 3D=====
<table border="0" cellpadding="5" width="100%" align="center"><tr><td align="left">
<table border="1" cellpadding="5" align="left"><tr>
<td align="center" bgcolor="red">  3D  </td>
<td align="center" bgcolor="lightgreen">  Cyl  </td>
<td align="center" bgcolor="yellow">  Tor  </td>
</tr></table>
</td></tr></table>
NOTE: Throughout this chapter subsection, text that appears in a dark green font has been taken ''verbatim'' from [http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl & J. E. Tohline (1999)].

[[#MeshChoice|As mentioned above]], from the beginning of my research activities — following the lead of Black & Bodenheimer — it has seemed reasonable to me that numerical simulations of time-evolving, rotationally flattened fluid systems should be carried out on a cylindrical, rather than cartesian, coordinate mesh. When modeling rotationally flattened configurations, a cylindrical mesh has even seemed preferable to a ''spherical'' coordinate mesh because the "top" grid boundary (horizontal green-dashed line segment in [[#MeshChoice|Figure 1]]) can straightforwardly be dropped to a <math>~z</math>-coordinate location that is smaller than the <math>~\varpi</math>-coordinate location of the "side" grid boundary (vertical green-dashed line segment in [[#MeshChoice|Figure 1]]), thereby reducing the number of grid cells — and, correspondingly reducing the cost of modeling the less interesting, ''vacuum'' region — outside of the fluid system. [See, however, [[#Boss_.281980.29|Boss (1980)]] for an alternate point of view.] At the same time, however, it has not seemed reasonable to determine the values of the potential along the (cylindrical-grid) boundary by adopting a Green's function that is expressed in terms ''spherical harmonics''.

Over a period of approximately twenty years, off and on, my research group considered <font color="darkgreen">whether it might be advantageous in our numerical simulations to cast the Green's function in a cylindrical coordinate system. The "familiar" expression for the cylindrical Green's function expansion can be found in variety of references (see [<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>]; [https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)]</font>), and for convenience is [[#Familiar_Expression_for_the_Cylindrical_Green.27s_Function_Expansion|repeated below]]. <font color="darkgreen">It is expressible in terms of an infinite sum over the azimuthal quantum number <math>~m</math> and an infinite integral over products of Bessel functions of various orders multiplied by an exponential function. </font> Note that [http://adsabs.harvard.edu/abs/1985ApJ...290...75V J. V. Villumsen (1985, ApJ, 290, 75 - 85)] attempted to solve the potential problem in this manner; <font color="darkgreen">he presents a technique in which each infinite integral over products of Bessel functions is evaluated numerically using a Gauss-Legendre integrator … He then emphasizes the obvious problem that, because of the infinite integrals involved, a calculation of the potential via this straightforward application of the familiar cylindrical Green's function expansion is numerically much more difficult than a calculation of the potential using a ''spherical'' Green's function expansion.</font>

<font color="red"><b>Eureka!</b></font> Via his dogged efforts and an extraordinarily in-depth investigation of this problem, [[Appendix/Ramblings/CCGF#Compact_Cylindrical_Green_Function_.28CCGF.29|in 1999 Howard S. Cohl discovered]] that, in cylindrical coordinates, the relevant Green's function can be written in a much more compact and much more practical form. Specifically, he realized that,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=-\infty}^{\infty} e^{im(\phi - \phi^')}Q_{m- 1 / 2}(\chi) \, ,
</math>
</td>
</tr>
</table>
where,<br />
<div align="center"><math>~\chi \equiv \frac{\varpi^2 + (\varpi^')^2 + (z - z^')^2}{2\varpi \varpi^'} \, ,</math><br /><br />
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl & J. E. Tohline (1999)], p. 88, Eqs. (15) & (16)<br />
See also: [http://adsabs.harvard.edu/abs/2007AmJPh..75..724S Selvaggi, Salon & Chari (2007)] §II, eq. (5)<br />
and the [https://dlmf.nist.gov/14.19#ii DLMF's definition of Toroidal Functions], <math>~Q_{m - 1 / 2}^{0}</math>
</div>
and, <math>~Q_{m - 1 / 2}</math> is the zero-order, half-(odd)integer degree, Llegendre function of the second kind — also referred to as a ''toroidal'' function of zeroth order; see [[#Toroidal_Functions|additional details, below]]. Hence, anywhere along the boundary of our cylindrical-coordinate mesh, a valid expression for the gravitational potential is,

<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B(\varpi,\phi,z)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-G \int \frac{1}{|\vec{x}^{~'} - \vec{x}|} ~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-G \int \biggl\{
\frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=-\infty}^{\infty} e^{im(\phi - \phi^')}Q_{m- 1 / 2}(\chi)
\biggr\}~
\rho(\varpi^',\phi^',z^') \varpi^' d\varpi^' d\phi^' dz^'
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{G}{\pi \sqrt{\varpi}} \int \rho(\varpi^',\phi^',z^') \sqrt{\varpi^'} d\varpi^' d\phi^' dz^'
\sum_{m=0}^{\infty} \epsilon_m \cos[m(\phi - \phi^')] Q_{m- 1 / 2}(\chi) \, ,
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl & J. E. Tohline (1999)], p. 88, Eq. (18)
</td>
</tr>
</table>
where, <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0=1</math> and <math>~\epsilon_m = 2</math> for <math>~m\ge 1</math>. Following this discovery, most of my research group's 3D numerical modeling of self-gravitating fluids has been conducted using ''Toroidal functions'' instead of ''Spherical Harmonics'' to evaluate the boundary potential on our cylindrical-coordinate meshes; see, for example, [http://adsabs.harvard.edu/abs/2002ApJS..138..121M P. M. Motl, J. E. Tohline & J. Frank (2002)]; [http://adsabs.harvard.edu/abs/2005ApJ...625L.119O C. D. Ott, S. Ou, J. E. Tohline & A. Burrows (2005)]; [http://adsabs.harvard.edu/abs/2006ApJ...643..381D M. C. R. D'Souza, P. M. Motl, J. E. Tohline, & J. Frank (2006)]; and [http://adsabs.harvard.edu/abs/2012ApJS..199...35M D. C. Marcello & J. E. Tohline (2012)].

=====For Axisymmetric Systems=====

As was done, [[#Simplification_for_2D.2C_Axisymmetric_Systems|above]], in the context of our discussion of a spherical-harmonics-based expression for the boundary potential, let's consider how this toroidal-function-based expression for the boundary potential can be simplified when examining 2D (axisymmetric) rather than fully 3D systems. In axisymmetric systems, by definition, physical variables exhibit no variation in the azimuthal coordinate direction. Hence, in the expression for <math>~\Phi_B</math>:
* the azimuthal coordinate, <math>~\phi</math>, need not appear explicitly as an independent variable, so the integral over this angular coordinate immediately gives, <math>~2\pi</math>;
* the index, <math>~m</math>, must be set to zero, so there is no summation over this index; and,
* drawing from the [[#Toroidal_Functions|additional details provided in Table 5, below]] the single surviving toroidal function is, <math>~Q_{-1 / 2}</math>, which can be rewritten in terms of the complete elliptic integral of the first kind.

As a result of these simplifications, anywhere along the boundary of our cylindrical-coordinate mesh, a valid expression for the ''axisymmetric'' gravitational potential is,

<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B(\varpi,z)\biggr|_{2D}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{2G}{\sqrt{\varpi}} \int\limits_{\varpi^'} \int\limits_{z^'} d\varpi^' dz^' \rho(\varpi^',z^') \sqrt{\varpi^'}
Q_{- 1 / 2}(\chi)
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{2G}{\sqrt{\varpi}} \int\limits_{\varpi^'} \int\limits_{z^'} d\varpi^' dz^' \rho(\varpi^',z^') \sqrt{\varpi^'}
\mu K(\mu) \, ,
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl & J. E. Tohline (1999)], p. 89, Eqs. (31) & (32)
</td>
</tr>
</table>
where,
<table border="0" align="center">
<tr>
<td align="right">
<math>~\mu </math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\biggl[ \frac{2}{1+\chi} \biggr]^{1 / 2} =
\biggl[ \frac{4\varpi \varpi^'}{(\varpi + \varpi^')^2 + (z - z^')^2} \biggr]^{1 / 2}
\, .
</math>
</td>
</tr>
</table>
Actually, this expression for the potential is not only valid along the outer boundaries of the computational mesh, but ''anywhere'' inside or outside of the mass distribution.

===Extension from 2D to 3D===

An extension from 2D to 3D was accomplished by using separate, finite Fourier series expansion to represent the azimuthal dependence of <math>~\rho</math> and <math>~\Phi</math>. This allowed us to decouple the ''differential'' expression for the Poisson equation into a finite number of independent Fourier modes, and to straightforwardly solve a finite set of 2D Helmholtz equations, instead of solving a single 2D Poisson equation.

==Intermediate==

<table border="1" width="95%" align="center" cellpadding="10"><tr><td align="left">
Much of the material enclosed in this ''text box'' has been drawn from §3.1 of Jackson (1975).

In spherical coordinates, <math>~(r,\theta,\phi)</math>, the Poisson equation takes the form,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~4\pi G \rho(r,\theta,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{r} \frac{\partial^2}{\partial r^2} (r\Phi)
+ \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial\theta} \biggl( \sin\theta \frac{\partial\Phi}{\partial\theta}\biggr)
+ \frac{1}{r^2 \sin^2\theta} \frac{\partial^2\Phi}{\partial\phi^2} \, .
</math>
</td>
</tr>
</table>
As it turns out, the potential — which, generally, is a function of all three coordinates — can be written as the product of three functions, each of which is a function of only one coordinate. To demonstrate this, specifically suppose that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Phi(r,\theta,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{U( r )}{r} P(\theta)Q(\phi) \, .
</math>
</td>
</tr>
</table>
Then the differential form of the Poisson equation can be rewritten as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~4\pi G \rho(r,\theta,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{PQ}{r} \frac{d^2U}{dr^2}
+ \frac{UQ}{r^3\sin\theta} \frac{d}{d\theta}\biggl(\sin\theta \frac{dP}{d\theta}\biggr)
+ \frac{UP}{r^3\sin^2\theta} \frac{d^2Q}{d\phi^2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{r^3 \sin^2\theta}{ UPQ} \biggr] 4\pi G \rho(r,\theta,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{r^3 \sin^2\theta}{ UPQ} \biggr] \biggl\{
\frac{PQ}{r} \frac{d^2U}{dr^2}
+ \frac{UQ}{r^3\sin\theta} \frac{d}{d\theta}\biggl(\sin\theta \frac{dP}{d\theta}\biggr)
+ \frac{UP}{r^3\sin^2\theta} \frac{d^2Q}{d\phi^2}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{r^2\sin^2\theta}{U} \biggr] \frac{d^2U}{dr^2}
+ \frac{\sin\theta}{P} \frac{d}{d\theta}\biggl(\sin\theta \frac{dP}{d\theta}\biggr)
+ \frac{1}{Q} \frac{d^2Q}{d\phi^2} \, .
</math>
</td>
</tr>
</table>
Setting, <math>~Q = e^{\pm im\phi}</math>, gives,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \biggl[ \frac{r^3 \sin^2\theta}{ UP e^{\pm i m\phi }} \biggr] 4\pi G \rho(r,\theta,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{r^2\sin^2\theta}{U} \biggr] \frac{d^2U}{dr^2}
+ \frac{\sin\theta}{P} \frac{d}{d\theta}\biggl(\sin\theta \frac{dP}{d\theta}\biggr)
-m^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{r^3 }{ UP e^{\pm i m\phi}} \biggr] 4\pi G \rho(r,\theta,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{r^2 }{U} \biggr) \frac{d^2U}{dr^2}
+ \biggl[ \frac{1}{P\sin\theta} \frac{d}{d\theta}\biggl(\sin\theta \frac{dP}{d\theta}\biggr)
- \frac{m^2}{\sin^2\theta} \biggr] \, .
</math>
</td>
</tr>
</table>

Now, if for each selected pair of the integer indexes, <math>~(\ell, m)</math>, the function <math>~P</math> is identified as the associated Legendre function, <math>~P_\ell^m</math>, as defined above in Table 2, then the set of terms inside the square brackets on the right-hand-side of this last expression can be set equal to <math>~[-\ell(\ell + 1)]</math> because every associated Legendre function satisfies the differential equation,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \frac{1}{P_\ell^m \sin\theta} \frac{d}{d\theta}\biggl(\sin\theta \frac{dP_\ell^m}{d\theta}\biggr) + \ell(\ell+1) - \frac{m^2}{\sin^2\theta}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
0 \, .
</math>
</td>
</tr>
</table>

Rewriting our adopted ''separable'' expression for the potential as the sum over all possible <math>~(\ell,m)</math>-component solutions, that is,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Phi(r,\theta,\phi) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell [\Phi(r,\theta,\phi)]_{\ell m}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell \frac{U_{\ell m}( r )}{r} P_\ell^m(\theta)Q_m(\phi) \, ,
</math>
</td>
</tr>
</table>

a solution to the Poisson equation then imposes the requirement that, for each <math>~(\ell,m)</math> pair,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\biggl( \frac{r^2 }{U_{\ell m}} \biggr) \frac{d^2U_{\ell m} }{dr^2} - \ell(\ell+1)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl[ \frac{r^3 }{ U_{\ell m} P_\ell^m e^{\pm i m\phi}} \biggr] 4\pi G [\rho(r,\theta\phi)]_{\ell m} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
\frac{1}{r} \frac{d^2U_{\ell m} }{dr^2} - \biggl[ \frac{\ell(\ell+1)}{r^2} \biggr]\frac{U_{\ell m} }{r}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl[ \frac{1}{ P_\ell^m e^{\pm i m\phi}} \biggr] 4\pi G [\rho(r,\theta\phi)]_{\ell m} \, .</math>
</td>
</tr>
</table>

</td></tr></table>

==Other Approaches==

===Deupree (1974)===

<table border="0" cellpadding="5" width="100%" align="center"><tr><td align="left">
<table border="1" cellpadding="5" align="left"><tr>
<td align="center" bgcolor="red">  2D  </td>
<td align="center" bgcolor="lightgreen">  Sph  </td>
<td align="center" bgcolor="yellow">  Tor  </td>
</tr></table>
</td></tr></table>

<table border="0" cellpadding="3" align="center" width="75%">
<tr><td align="left">
<font color="#009999">
"The fully nonlinear, nonradial, nonadiabatic calculation of stellar oscillations has not as yet been attempted by anyone, to the best of the author's knowledge. Only Deupree (1974b, 1975) so far seems to have taken some steps in this direction. He has carried out numerical calculations of the nonlinear axisymmetric oscillations in the adiabatic ([http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree 1974b]) and nonadiabatic ([http://adsabs.harvard.edu/abs/1975ApJ...198..419D Deupree 1975]) approximations, with apparently encouraging results."</font>
</td></tr>
<tr><td align="right">
— Drawn from §3.5 of the review article by [http://adsabs.harvard.edu/abs/1976ARA%26A..14..247C J. P. Cox (1976, ARAA, 14, 247 - 273)]
</td></tr></table>

Apparently, [http://adsabs.harvard.edu/abs/1974ApJ...194..393D R. G. Deupree (1974, ApJ, 194, 393 - 402)] was the first astronomer to employ self-gravitating, numerical hydrodynamic techniques to model ''nonlinear, nonradial stellar pulsations.'' He chose to carry out his simulations on a spherical coordinate mesh with his earliest simulations being restricted to the examination of 2D (axisymmetric) configurations. The outer boundary of his computational grid was identified as (see his §IIb) <font color="darkgreen">a spherical surface completely exterior to the star.</font>

====His Derived Expression for the Boundary Potential====
In designing an algorithm to solve the Poisson equation, [http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] considered <font color="darkgreen">the major difficulty [to be] the evaluation of the potential boundary conditions.</font> He decided to determine the boundary potential via an evaluation of the ''integral representation'' of the Poisson equation; <font color="darkgreen">once the boundary conditions [were] specified, the potential inside the star [was] evaluated by [employing] the [http://adsabs.harvard.edu/abs/1964ApJ...139..306H Henyey method]</font> to solve the ''differential representation'' of the Poisson equation. The assumption of azimuthal symmetry meant that, for example, the density was specified at various meridional-plane locations, <math>~\rho(r,\theta)</math>, with <font color="darkgreen">each zone [being] considered as a uniform density circular ring.</font> He argued that <font color="darkgreen">the potential of [each such] ring at any point on the spherical boundary [could] easily be evaluated by</font> treating it as an infinitesimally thin ring of mass, <math>~\delta M = 2\pi a \rho ~\delta A</math> — where, <math>~\delta A</math> <font color="darkgreen">is the cross-sectional area of the [grid] zone, and <math>~a</math> is the radius of [that] ring</font> — then drawing upon the analytic analysis described by [https://archive.org/details/foundationsofpot033485mbp Kellogg (1929)] or, equivalently, by [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential MacMillan (1958; originally 1930)]. Then, <font color="darkgreen">the total potential at the boundary is the sum of the potentials from all of the [meridional-plane] zones.</font>

----

<table border="0" align="center" cellpadding="5" width="80%"><tr><td align="left">
Note that, over the years, a number of other research groups have adopted this same approach to evaluate the gravitational potential of axisymmetric mass distributions — that is, summing up the potential contributions due to an ensemble of "infinitesimally thin rings." But none appear to have recognized Deupree's earlier, pioneering investigation. See, for example, the brief reviews that we have written regarding the related investigations by: [[#Stahler_.281983.29|Stahler (1983)]], [[Apps/DysonWongTori#Bannikova_et_al._.282011.29|Bannikova et al. (2011)]], and [[Apps/DysonWongTori#Fukushima_.282016.29|Fukushima (2016)]].
</td></tr></table>

----

More specifically, equation (16) from [http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] states that the differential contribution to the boundary potential due to each infinitesimally thin ring is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\delta\Phi_B</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2G(2\pi a\rho ~\delta A)}{\pi p} \int_0^{\pi/2} \frac{d\psi}{[\cos^2\psi + (q/p)^2\sin^2\psi]^{1 / 2}} \, ,
</math>
</td>
</tr>
</table>
</div>

where, <math>~q</math> is the shortest distance between the ring and the boundary point, and <math>~p</math> is the longest distance. (Note that, in order to match our conventions, we have inserted the negative sign along with a leading factor of <math>~G</math>.) Referencing our accompanying [[Apps/DysonWongTori#DeupreeReference|detailed analysis of the potential due to a thin ring]], and adopting the variable mappings, <math>~p \leftrightarrow \rho_1</math> and <math>~q \leftrightarrow \rho_2</math>, we see that Deupree's expression is indeed identical to the expression for the potential derived by [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential MacMillan (1958)].

Recognizing, [[Apps/DysonWongTori#RingPotential|as did MacMillan]], that the definite integral in this expression is related to the complete elliptic integral of the first kind, and introducing the ratio of lengths, <math>~c \equiv p/q</math>, Deupree's expression for the (differential contribution to the) potential can be rewritten as,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\delta\Phi_B</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{2G (2\pi a\rho ~\delta A) c}{\pi p} \int_0^{\pi/2} \frac{d\psi}{ \sqrt{1 - k^2 \sin^2\psi }} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[ \frac{2G (2\pi a\rho ~\delta A) c}{\pi p} \biggr] K(k) \, ,</math>
</td>
</tr>
</table>
where, <math>~K(k)</math> is the [http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html complete elliptic integral of the first kind] for the modulus, <math>~k = \sqrt{1-c^2}</math>.

====Comparison With Other Related Derivations====
Now, given that Deupree chose to construct and evolve his models using a spherical coordinate system, he would have specified the relevant lengths, <math>~p</math> and <math>~q</math>, and each ring's differential cross-section, <math>~\delta A</math>, in terms of spherical coordinates. In an effort to more clearly illustrate the connection between Deupree's expression for the (differential contribution to the) boundary potential and the expression for the boundary potential that we have [[#For_Axisymmetric_Systems|derived above for axisymmetric systems]], we will insert expressions for these terms that apply, instead, to a cylindrical-coordinate mesh.

Following the same line of reasoning as has been presented in our [[Apps/DysonWongTori#CylindricalLocation|accompanying discussion of MacMillan's work]], if the meridional-plane locations of the infinitesimally thin ring and the desired point on the boundary are, respectively, <math>~(\varpi^',z^')</math> and <math>~(\varpi,z)</math>, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~k = \sqrt{1 - c^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl[\frac{4\varpi \varpi^'}{(\varpi + \varpi^')^2 + (z - z^')^2 }\biggr]^{1 / 2} \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{c}{p}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[(\varpi + \varpi^')^2 + (z - z^')^2 \biggr]^{- 1 / 2} = \frac{k}{\sqrt{4\varpi \varpi^'}} \, .</math>
</td>
</tr>
</table>
</div>
Hence — after acknowledging that, in cylindrical coordinates, the radius of each "infinitesimally thin ring" is, <math>~a = \varpi^'</math>, and the differential cross-section of each ring is, <math>~\delta A = \delta\varpi^' \delta z^'</math> — Deupree's expression for the (differential contribution to the) potential may be rewritten as,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\delta\Phi_B(\varpi,z)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{G (2\pi a \rho ~\delta A) }{\pi } \biggl[ \frac{k }{\sqrt{\varpi \varpi^'}} \biggr] K(k) </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{2G (\delta M) }{\pi } \cdot \frac{K(k) }{ \sqrt{(\varpi + a)^2 + (z - z^')^2 }} \, , </math>
</td>
</tr>
</table>
or it may be rewritten as,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\delta\Phi_B(\varpi,z)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{G (2\pi a \rho ~\delta A) }{\pi } \biggl[ \frac{k }{\sqrt{\varpi \varpi^'}} \biggr] K(k) </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{2G }{\sqrt{\varpi}} \biggl[ \delta\varpi^' \delta z^' \rho(\varpi^', z^') \sqrt{\varpi^'} k K(k) \biggr] \, .</math>
</td>
</tr>
</table>

Notice that the first of these two rewritten expressions aligns perfectly with our "[[Appendix/Equation_templates#Other_Equations_with_Assigned_Templates|key equation]]" that gives the gravitational potential of an axisymmetric torus in the thin ring (TR) approximation, namely,
<table border="0" align="center" cellpadding="10"><tr><td align="center">
{{ Math/EQ TRApproximation }}
</td>
<td align="center" rowspan="2">[[File:FlatColorContoursCropped.png|225px|link=Apps/DysonWongTori#ThinRingContours]]</td>
</tr></table>
(See our [[Apps/DysonWongTori#ThinRingContours|accompanying discussion]] for more information on the meridional-plane contour plot that is displayed to the right of this equation.) Next, referring back to the expression that was [[#For_Axisymmetric_Systems|derived above for axisymmetric systems from a toroidal-function-based Green's function]], namely,

<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B(\varpi,z)\biggr|_{2D}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{2G}{\sqrt{\varpi}} \int\limits_{\varpi^'} \int\limits_{z^'} d\varpi^' dz^' \rho(\varpi^',z^') \sqrt{\varpi^'}
\mu K(\mu) \, ,
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl & J. E. Tohline (1999)], p. 89, Eqs. (31) & (32)
</td>
</tr>
</table>
where,
<table border="0" align="center">
<tr>
<td align="right">
<math>~\mu </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{4\varpi \varpi^'}{(\varpi + \varpi^')^2 + (z - z^')^2} \biggr]^{1 / 2}
\, ,
</math>
</td>
</tr>
</table>
we see that the second of these rewritten expressions for Deupree's <math>~\delta\Phi_B</math> aligns perfectly with our derived expression for the differential contribution to the potential of any axisymmetric mass distribution. It is therefore fair to say that the expression that Deupree used to determine the gravitational potential along the boundary of his modeled configurations is derivable from a 3D Green's function that is written in terms of ''toroidal functions''.

===Cook (1977)===

<table border="0" cellpadding="5" width="100%" align="center"><tr><td align="left">
<table border="1" cellpadding="5" align="left"><tr>
<td align="center" bgcolor="red">  3D  </td>
<td align="center" bgcolor="lightgreen">  Cyl  </td>
<td align="center" bgcolor="yellow">  Cyl  </td>
</tr></table>
</td></tr></table>

Key results from [https://www.osti.gov/servlets/purl/7294335 T. L. Cook's (1977) doctoral dissertation] — titled, ''Three-Dimensional Dynamics of Protostellar Evolution'' — were published as [http://adsabs.harvard.edu/abs/1978ApJ...225.1005C T. L. Cook & F. H. Harlow (1978, ApJ, 225, 1005 - 1020)]. Cook's three-dimensional hydrodynamic simulations were conducted on a cylindrical-coordinate mesh. While few details of the technique used to solve the Poisson equation are provided in the ApJ article, §II.B of [https://www.osti.gov/servlets/purl/7294335 Cook's (1977) dissertation] explains that values of the gravitational potential across the ''interior'' regions of the mesh were obtained by solving the ''differential representation'' of the Poisson equation, subject to the boundary conditions at each boundary point, calculated by "performing a numerical integration over all mass points" using the ''integral representation'' of the Poisson equation, namely,
<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B(\varpi,\phi,z)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-G \int \frac{1}{|\vec{x}^{~'} - \vec{x}|} ~\rho(\vec{x}^{~'}) d^3x^' \, ,
</math>
</td>
</tr>
</table>
with,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\frac{1}{|\vec{x}^{~'} - \vec{x}|}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \varpi^2 + (\varpi^')^2 - 2\varpi \varpi^' \cos|\phi - \phi^'| + (z - z^')^2 \biggr]^{- 1 / 2} \, .
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[https://www.osti.gov/servlets/purl/7294335 Cook (1977)], p. 15, Eq. (II-16)<br />
See also: [http://adsabs.harvard.edu/abs/2007AmJPh..75..724S Selvaggi, Salon & Chari (2007)] §II, eq. (2)
</td>
</tr>
</table>

(Note that the leading factor of the gravitational constant, <math>~G</math>, does not appear explicitly in Cook's equation II-16, although it should.) Presumably the coordinate locations of the boundary cells, <math>~(\varpi,\phi,z)</math>, were in no case coincident with the coordinate locations of any of the ''interior'' grid cells, <math>~(\varpi^',\phi^',z^')</math>, so there was no danger that the integration would encounter a singularity as a consequence of the distance, <math>~|\vec{x}^{~'} - \vec{x}|</math>, being zero.

===Norman and Wilson (1978)===

<table border="0" cellpadding="5" width="100%" align="center"><tr><td align="left">
<table border="1" cellpadding="5" align="left"><tr>
<td align="center" bgcolor="red">  3D  </td>
<td align="center" bgcolor="lightgreen">  Cyl  </td>
<td align="center" bgcolor="yellow">  Sph  </td>
</tr></table>
</td></tr></table>

Following the discussion presented in §4.1 of [https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], it is sometimes useful to rewrite [[#PotentialA|''Form A'' of the boundary potential]] as,
<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B(r,\theta,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-4\pi G
\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{q_{\ell m}}{r^{\ell+1}} \frac{Y_{\ell m}(\theta,\phi)}{(2\ell+1)} \, ,
</math>
</td>
</tr>
</table>

<span id="MultipoleMoments">where we have introduced what is commonly referred to as the,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="center" colspan="3">
<font color="#770000">'''Multipole Moments of the Mass Distribution'''</font>
</td>
</tr>
<tr>
<td align="right">
<math>~q_{\ell m}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\int (r^')^\ell Y_{\ell m}^*(\theta^', \phi^')
~\rho(r^', \theta^', \phi^') d^3x^'
\, .
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], p. 137, Eq. (4.3)
</td>
</tr>
</table>

When written explicitly in terms of cartesian coordinates — see [[#Ylm|Table 2, below]], for each of the relevant <math>~Y_{\ell m}</math> expressions — the first few of these moments have the functional representations [[#Multipole_Moments_of_the_Mass_Distribution|derived below]]. For the cases that correspond to positive values of the index, <math>~m</math>, the set of ''multipole moment'' expressions that have been included in our [[#qlm|Table 3 summary]] exactly matches the set of expressions presented as equations (4.4), (4.5), and (4.6) in [https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)]. (Jackson's equation 4.7 explains how to map these expressions to the cases corresponding to negative values of the index, <math>~m</math>.)

<span id="PhiSeriesExpansion">Let's now look at various terms in the summed expression for the boundary potential</span> with each term expressing the contribution for a separate value of the index, <math>~\ell</math>. Isolating the first three terms from all the rest, for example, we have,

<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B(r,\theta,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \Phi_B(r,\theta,\phi) \biggr|_{\ell=0} +~ \Phi_B(r,\theta,\phi)\biggr|_{\ell=1} + ~\Phi_B(r,\theta,\phi)\biggr|_{\ell=2}
-~4\pi G
\sum_{\ell=3}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{q_{\ell m}}{r^{\ell+1}} \frac{Y_{\ell m}(\theta,\phi)}{(2\ell+1)} \, .
</math>
</td>
</tr>
</table>
We recognize, first, that as we consider boundary points that lie farther and farther away from the mass distribution, the magnitude of the first <math>(\ell = 0)</math> term drops off as <math>~r^{-1}</math> — the expected behavior of the potential outside of a point mass; the second <math>~(\ell=1)</math> term drops off as <math>~r^{-3}</math>; and the third <math>~(\ell=2)</math> term drops off as <math>~r^{-5}</math>. Below, we have evaluated in more detail the behavior of these first three terms. This evaluation gives us the,
<table border="0" align="center">
<tr>
<td align="center" colspan="3">
<font color="#770000">'''Boundary Potential Written in Terms of Multipole Moments '''</font>
</td>
</tr>
<tr>
<td align="right">
<math>~ \Phi_B(r,\theta,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
~-~~\frac{GM}{r}
~~- ~~\frac{GM}{r^3} \biggl[ \vec{x} \cdot \vec{x}_\mathrm{com} \biggr]
~~-~~\frac{G}{2r^5} \sum_{i=1}^3 \sum_{j=1}^3 Q_{i,j} \biggl[ x_i x_j \biggr]
~~-~~4\pi G \sum_{\ell=3}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{q_{\ell m}}{r^{\ell+1}} \frac{Y_{\ell m}(\theta,\phi)}{(2\ell+1)} \, ,
</math>
</td>
</tr>
</table>
where, as defined below, <math>~\vec{x}_\mathrm{com}</math> is the [[#Second_Term|center-of-mass location]], and <math>~Q_{i,j} </math> is the [[#QuadrupoleMomentTensor|traceless quadrupole moment tensor]].

If a modeled mass-density distribution, <math>~\rho( \vec{x}^{~'})</math>, has been configured such that the center-of-mass of the system coincides with the origin of the coordinate system — that is, if it has been configured such that <math>~\vec{x}_\mathrm{com} = 0</math> — then the second term in this series can be set to zero. If, in addition, all terms having <math>~\ell \ge 3</math> are ignored because their values drop off rapidly with distance — specifically, inverse distance to the <math>~(2\ell + 1)</math> power — then a reasonably good approximation for the potential on the boundary of the modeled system is given by the expression,
<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
-\frac{GM}{r}
~-\frac{G}{2r^5} \sum_{i=1}^3 \sum_{j=1}^3 Q_{i,j} \biggl[ x_i x_j \biggr] \, .
</math>
</td>
</tr>
</table>

This is precisely the relation that was adopted by [http://adsabs.harvard.edu/abs/1978ApJ...224..497N M. L. Norman & J. R. Wilson (1978, ApJ, 224, 497 - 511)] when, in the context of star formation, they modeled the ''Fragmentation of Isothermal Rings'' — see specifically their equation (18).

===Boss (1980)===

<table border="0" cellpadding="5" width="100%" align="center"><tr><td align="left">
<table border="1" cellpadding="5" align="left"><tr>
<td align="center" bgcolor="red">  3D  </td>
<td align="center" bgcolor="lightgreen">  Sph  </td>
<td align="center" bgcolor="yellow">  Sph  </td>
</tr></table>
</td></tr></table>

Building upon equations (5) and (7) from [http://adsabs.harvard.edu/abs/1980ApJ...236..619B A. P. Boss (1980, ApJ, 236, 619 - 627)] — see also Jackson's equation (3.58) — we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Phi(r,\theta,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\sum_{\ell=0}^\infty \sum_{m = -\ell}^{\ell} \Phi_{\ell m} ( r ) Y_{\ell m}(\theta,\phi) \, ,
</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho(r,\theta,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\sum_{\ell=0}^\infty \sum_{m = -\ell}^{\ell} \rho_{\ell m} ( r ) Y_{\ell m}(\theta,\phi) \, ,
</math>
</td>
</tr>
</table>
</div>
where the coefficients in the series expansion of <math>~\rho</math> can each be obtained from the known spatial density distribution via the integral expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho_{\ell m}(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int d\Omega Y_{\ell m}^*(\theta,\phi) \rho(r,\theta,\phi) \, ,
</math>
</td>
</tr>
</table>
</div>
and the differential solid angle is,
<div align="center">
<math>~d\Omega \equiv \sin\theta d\theta d\phi \, .</math>
</div>

===Stahler (1983)===

<table border="0" cellpadding="5" width="100%" align="center"><tr><td align="left">
<table border="1" cellpadding="5" align="left"><tr>
<td align="center" bgcolor="red">  2D  </td>
<td align="center" bgcolor="yellow">  Tor  </td>
</tr></table>
</td></tr></table>

[http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] used a [[AxisymmetricConfigurations/HSCF#Introduction|self-consistent-field]] technique to construct equilibrium sequences of rotationally flattened, isothermal gas clouds. At each iteration step, the method that he adopted to evaluate the gravitational potential along the outer boundary of his computational mesh was essentially the same as the method used by [http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] — see our [[#Deupree_.281974.29|above description]] — to model the time-dependent behavior of pulsating stars. It appears as though Stahler was unaware of Deupree's earlier development of this method to evaluate the boundary potential, as Deupree's (1974) paper is not among Stahler's list of references.


In describing [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler's (1983a)] method, we will first draw upon our "[[Appendix/Equation_templates#Other_Equations_with_Assigned_Templates|key equation]]" that gives the gravitational potential of an axisymmetric torus in the thin ring (TR) approximation, namely,
<table border="0" align="center" cellpadding="10"><tr><td align="center">
{{ Math/EQ TRApproximation }}
</td>
<td align="center" rowspan="2">[[File:FlatColorContoursCropped.png|225px|link=Apps/DysonWongTori#ThinRingContours]]</td>
</tr></table>
(See our [[Apps/DysonWongTori#ThinRingContours|accompanying discussion]] for more information on the meridional-plane contour plot that is displayed to the right of this equation.) Stahler has argued that a reasonably good approximation to the gravitational potential due to any extended axisymmetric mass distribution can be obtained by adding up the contributions due to many ''thin rings'' — <math>~\delta M(\varpi^', z^')</math> being the appropriate differential mass contributed by each ring element — positioned at various meridional coordinate locations throughout the mass distribution. According to his independent derivation, the differential contribution to the potential, <math>~\delta\Phi_g(\varpi, z)</math>, due to each differential mass element is (see his equation 11 and the explanatory text that follows it):
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\delta\Phi_g(\varpi,z)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\frac{2G}{\pi \varpi^'}\biggr] \frac{\delta M}{[(\alpha + 1)^2 + \beta^2]^{1 / 2}}
\times K\biggl\{ \biggl[ \frac{4\alpha}{(\alpha+1)^2 + \beta^2} \biggr]^{1 / 2} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[\frac{2G}{\pi }\biggr] \frac{\delta M}{[(\varpi + \varpi^')^2 + (z^' - z)^2]^{1 / 2}}
\times K\biggl\{ \biggl[ \frac{4\varpi^' \varpi}{(\varpi +\varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Stahler's expression for each ''thin ring'' contribution is clearly the same as our "key equation" expression for <math>~\Phi_\mathrm{TR}</math> if the individual ring being considered cuts through the meridional plane at <math>~(\varpi^', z^') = (a, 0)</math>.

In the context of our [[#For_Axisymmetric_Systems|above discussion of a Green's function expression written in terms of toroidal functions]], we have shown that the exact integral expression for the gravitational potential due to any axisymmetric mass-density distribution, <math>~\rho(\varpi^', z^')</math>, is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Phi(\varpi, z)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{2G}{\varpi^{1 / 2}} \int\int (\varpi^')^{1 / 2} \mu K(\mu) \rho(\varpi^', z^') d\varpi^' dz^' \, ,</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mu^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{4\varpi^' \varpi}{(\varpi + \varpi^')^2 + (z^' - z)^2} \biggr] \, .
</math>
</td>
</tr>
</table>
</div>
Recognizing that, for axisymmetric structures, the differential mass element is, <math>~dM^' = 2\pi \rho(\varpi^', z^') \varpi^' d\varpi^' dz^'</math>, this integral expression may be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Phi(\varpi, z)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{2G}{\varpi^{1 / 2}} \int\int (\varpi^')^{1 / 2} \mu K(\mu) \biggl[ \frac{dM^'}{2\pi \varpi^'} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{G}{\pi} \int\int \biggl[ \frac{1}{\varpi^'\varpi}\biggr]^{1 / 2} \biggl[\frac{4\varpi^' \varpi}{(\varpi + \varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} K(\mu) dM^' </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{2G}{\pi} \int\int \biggl[\frac{1}{(\varpi + \varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} K(\mu) dM^' \, .</math>
</td>
</tr>
</table>
</div>
We see that our expression for the differential contribution to the potential exactly matches [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler's (1983a)]. It is therefore fair to say that Stahler's expression for the gravitational potential is derivable from a 3D Green's function that is written in terms of toroidal functions. We note that, in his study of axisymmetric systems, Stahler made the decision to evaluate the gravitational potential both inside as well as outside of the mass distribution using the same integral expression.

==Constructing Two-Dimensional, Axisymmetric Structures==
As has been explained in [[AxisymmetricConfigurations/SolutionStrategies#Technique|an accompanying discussion]], our objective is to solve an ''algebraic'' expression for hydrostatic balance,
<div align="center">
<math>~H + \Phi + \Psi = C_0</math> ,
</div>

in conjunction with the Poisson equation in a form that is appropriate for two-dimensional, axisymmetric systems, namely,
<div align="center">
<math>~
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} = 4\pi G \rho .
</math>
</div>

===Steps to Follow===
<table border="0" width="33%" align="right" cellpadding="5"><tr><td align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">
Annotation of<br /> Figure 1 from [http://adsabs.harvard.edu/abs/1986ApJS...61..479H I. Hachisu (1986)]<p></p>
</td></tr>
<tr><td>
[[File:HSCF_MeridionalPlaneGrid03.png|center|250px|HSCF Meridional Plane]]</td></tr>
<tr><td align="left">One quadrant of a meridional-plane cross-section (pink) through: (a) spheroidal structure; (b) toroidal structure. A green, dashed rectangular grid boundary is also illustrated.</td></tr></table>
</td></tr></table>
<ol>
<li>'''Choose a particular [[SR#Barotropic_Structure|barotropic equation of state]].'''   More specifically, functionally define the density-enthalpy relationship, <math>~\rho(H)</math>, and identify what value, <math>~H_\mathrm{surface}</math>, the enthalpy will have at the surface of your configuration. For example, if a ''polytropic'' equation of state is adopted, <math>~H_\mathrm{surface} = 0</math> is a physically reasonable prescription.</li>
<li>Choosing from, for example, a list of astrophysically relevant ''[[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|simple rotation profiles]],'' '''specify the corresponding functional form of the centrifugal potential''', <math>~\Psi(\varpi)</math>, that will define the radial distribution of specific angular momentum in your equilibrium configuration. If the choice is uniform rotation, then <math>~\Psi = - \varpi^2 \omega_0^2/2 \, ,</math> where <math>~\omega_0</math> is a constant to be determined.</li>
<li>On your chosen computational lattice — for example, on a cylindrical-coordinate mesh — '''identify two boundary points''', A and B, that will lie on the surface of your equilibrium configuration. These two points should remain fixed in space during the HSCF iteration cycle and ultimately will confine the volume and define the geometry of the derived equilibrium object. Note that, by definition, the enthalpy at these two points is, <math>~H_A = H_B = H_\mathrm{surface}</math>.</li>
<li>Throughout the volume of your computational lattice, ''' ''guess'' a trial distribution of the mass density,''' <math>~\rho(\varpi,z)</math>, such that no material falls outside a volume defined by the two boundary points, A and B, that were identified in Step #3. Usually an initially uniform density distribution will suffice to start the SCF iteration.</li>
<li>Via ''some'' accurate numerical algorithm, '''solve the Poisson equation''' to determine the gravitational potential, <math>~\Phi(\varpi,z)</math>, throughout the computational lattice corresponding to the trial mass-density distribution that was specified in Step #4 (or in Step #9).</li>
<li>From the gravitational potential determined in Step #5, '''identify the values of <math>~\Phi_A</math> and <math>~\Phi_B</math>''' at the two boundary points that were selected in Step #3.</li>
<li>From the "known" values of the enthalpy (Step #3) and the gravitational potential (Step #6) at the two selected surface boundary points A and B, '''determine the values of the constants, <math>~C_0</math> and <math>~\omega_0</math>,''' that appear in the algebraic equation that defines hydrostatic equilibrium.</li>
<li>From the most recently determined values of the gravitational potential, <math>~\Phi(\varpi,z)</math> (Step #5), and the values of the two constants, <math>~C_0</math> and <math>~\omega_0</math> just determined (Step #7), '''determine the enthalpy distribution throughout the computational lattice.'''</li>
<li>From <math>~H(\varpi,z)</math> and the selected barotropic equation of state (Step #1), '''calculate an "improved guess" of the density distribution,''' <math>~\rho(\varpi,z)</math>, throughout the computational lattice.</li>
<li>'''Has the model converged to a satisfactory equilibrium solution?''' (Usually a satisfactory solution has been achieved when the derived model parameters — for example, the values of <math>~C_0</math> and <math>~\omega_0</math> — change very little between successive iterations and the viral error is sufficiently small.)
<ul>
<li>If the answer is, "NO":   Repeat steps 5 through 10.</li>
<li>If the answer is, "YES":   Stop iteration.</li>
</ul>
</li>
</ol>

=Special Functions & Other Broadly Used Representations=

==Spherical Harmonics and Associated Legendre Functions==

<div align="center" id="Ylm">
<table border="1" cellpadding="8" align="center" width="80%">
<tr>
<th align="center" colspan="2"><font size="+0">Table 2: <br />Green's Function in Terms of Associated Legendre Functions,
<math>~P_\ell^m(\cos\theta)</math>, <br />and the Spherical Harmonics, <math>~Y_{\ell m}(\theta,\phi)</math></font></th>
</tr>
<tr>
<td align="left" colspan="2">
<table border="0" align="center">
<tr>
<td align="right">
<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'} |}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{4\pi}{2\ell+1} \biggl[ \frac{r_<^\ell}{r_>^{\ell+1}} \biggr] Y_{\ell m}^*(\theta^', \phi^') Y_{\ell m}(\theta,\phi)</math>
</td>
</tr>
</table>
<div align="center">
[https://archive.org/details/ClassicalElectrodynamics2nd J. D. Jackson (1975, 2<sup>nd</sup> Edition)], Eq. (3.70)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 656, Eq. (1C-31)
</div>
</td>

</tr>
<tr>
<td align="left" rowspan="2">
Note:<br />
<div align="center"><math>~Y_{\ell m}(\theta,\phi) = \biggl[ \frac{(2\ell + 1 )(\ell - m)!}{4\pi(\ell + m)!} \biggr]^{1 / 2} P_\ell^m(\cos\theta)e^{im\phi}</math><br /><br />
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], Eq. (3.53)<br />
[https://dlmf.nist.gov/14.30#i DLMF] §14.30i<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 655, Eq. (1C-27)<br /><br />
<math>~Y_{\ell,-m}(\theta,\phi) = (-1)^{m}Y_{\ell m}^*(\theta,\phi)</math><br />
</div>

----

<div align="center"><math>~P_{\ell}^m(x) = (-1)^m (1-x^2)^{m/2} ~ \frac{d^m}{dx^m} P_\ell(x)</math><br />
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], Eq. (3.49)<br />
[https://dlmf.nist.gov/14.6#i DLMF] §14.6i<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 655, Eq. (1C-20)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], p. 481, Eq. (C.5)
</div>

----

<div align="center"><math>~P_{\ell}(x) = \frac{1}{2^\ell \ell !} ~ \frac{d^\ell}{dx^\ell} (x^2 - 1)^\ell</math><br />
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], Eq. (3.16)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 655, Eq. (1C-21)
</div>
</td>
<th align="center">Leading Legendre Functions
<div align="center">
[https://archive.org/details/FieldTheoryHandbookMoonAndSpencer P. Moon & D. E. Spencer (1971)], p. 205<br />
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], Eq. (3.15)<br />
[https://dlmf.nist.gov/18.5#iv DLMF] §18.5iv<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 655, Eq. (1C-25)
</div>
</th>
</tr>
<tr>
<td align="center" rowspan="1">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~P_0(x)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~1</math>
</td>
</tr>
<tr>
<td align="right">
<math>~P_1(x)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~x</math>
</td>
</tr>
<tr>
<td align="right">
<math>~P_2(x)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\tfrac{1}{2}(3x^2 - 1)</math>
</td>
</tr>
<tr>
<td align="right">
<math>~P_3(x)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\tfrac{1}{2}(5x^3 - 3x)</math>
</td>
</tr>
<tr>
<td align="right">
<math>~P_4(x)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\tfrac{1}{8}(35x^4 - 30x^2 + 3)</math>
</td>
</tr>
<tr>
<td align="right">
<math>~P_5(x)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\tfrac{1}{8}(63x^5 - 70x^3 + 15x)</math>
</td>
</tr>
<tr>
<td align="right">
<math>~P_6(x)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\tfrac{1}{16}(231x^6 - 315x^4 + 105x^2 - 5)</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[http://astrowww.phys.uvic.ca/~tatum/celmechs/celm5.pdf J. B. Tatum (2021)] Celestial Mechanics class notes (UVic), §5.11, p. 41, Eq. (5.11.8)
</td>
</tr>
</table>
</td>
</tr>

<tr>
<th align="center" colspan="2">Leading Spherical Harmonics
<div align="center">
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], §3.5<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 656, Eq. (1C-33)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], p. 482, Eqs. (C.7) - (C.16)
</div>
</th>
</tr>
<tr>
<td align="left" rowspan="1">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~Y_{00} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{\sqrt{4\pi}}</math>
</td>
</tr>
<tr>
<td align="right">
<math>~Y_{1,\pm 1} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mp \sqrt{\frac{3}{8\pi}} ~\sin\theta ~e^{\pm i\phi}</math>
</td>
</tr>
<tr>
<td align="right">
<math>~Y_{10} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\sqrt{\frac{3}{4\pi}} ~\cos\theta </math>
</td>
</tr>
<tr>
<td align="right">
<math>~Y_{2,\pm 2} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{4} \sqrt{\frac{15}{2\pi}} ~\sin^2\theta ~e^{\pm 2i\phi}</math>
</td>
</tr>
<tr>
<td align="right">
<math>~Y_{2, \pm 1} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mp \sqrt{\frac{15}{8\pi}} ~\sin\theta \cos\theta ~e^{\pm i\phi}</math>
</td>
</tr>
<tr>
<td align="right">
<math>~Y_{20} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\sqrt{\frac{5}{4\pi}} ~(\tfrac{3}{2}\cos^2\theta - \tfrac{1}{2})</math>
</td>
</tr>
</table>

</td>
<td align="left" rowspan="1">

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~Y_{3,\pm 3} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mp \frac{1}{4} \sqrt{\frac{35}{4\pi}} ~\sin^3\theta ~e^{\pm 3i\phi}</math>
</td>
</tr>
<tr>
<td align="right">
<math>~Y_{3, \pm 2} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{4}\sqrt{\frac{105}{2\pi}} ~\sin^2\theta \cos\theta ~e^{\pm 2i\phi}</math>
</td>
</tr>
<tr>
<td align="right">
<math>~Y_{3, \pm 1} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\mp \frac{1}{4}\sqrt{\frac{21}{4\pi}} ~\sin\theta (5\cos^2\theta - 1) ~e^{\pm i\phi}</math>
</td>
</tr>
<tr>
<td align="right">
<math>~Y_{30} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\sqrt{\frac{7}{4\pi}} ~(\tfrac{5}{2}\cos^3\theta - \tfrac{3}{2}\cos\theta) </math>
</td>
</tr>
</table>

----

Note that, for all cases where <math>~m=0</math> :<br />
<div align="center"><math>~Y_{\ell 0}(\theta,\phi) = \sqrt{\frac{(2\ell + 1 )}{4\pi} } ~P_\ell(\cos\theta)</math><br />
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], Eq. (3.57)
</div>
</td>
</tr>
</table>
</div>

==Multipole Expansions==

===Mass Multipole Moments===

As an extension of the [[#MultipoleMoments|above discussion of]]
<table border="0" cellpadding="5" align="center">

<tr>
<td align="center" colspan="3">
<font color="#770000">'''Multipole Moments of the Mass Distribution'''</font>
</td>
</tr>
<tr>
<td align="right">
<math>~q_{\ell m}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\int (r^')^\ell Y_{\ell m}^*(\theta^', \phi^')
~\rho(r^', \theta^', \phi^') d^3x^'
\, ,
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], p. 137, Eq. (4.3)
</td>
</tr>
</table>
here we evaluate a set of the leading order mass-multipole moments in terms of their cartesian-coordinate representations.

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~q_{00}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{\sqrt{4\pi}}
\int \rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~q_{11}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int r^' \biggl[ Y_{11}^* \biggr]~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- ~\int r^' \biggl[ \sqrt{\frac{3}{8\pi}}\sin\theta ~\biggl(\cos\phi^' - i\sin\phi^' \biggr) \biggr]~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \sqrt{\frac{3}{8\pi}}~\int r^' \biggl[ \frac{\sqrt{(x^')^2+(y^')^2}}{r^'}
~\biggl(\frac{x^'}{\sqrt{(x^')^2+(y^')^2}} - \frac{iy^'}{\sqrt{(x^')^2+(y^')^2}} \biggr)\biggr]
~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \sqrt{\frac{3}{8\pi}}~\int (x^' - iy^')
~\rho(\vec{x}^{~'}) d^3x^'
\, ;
</math>
</td>
</tr>
<tr>
<td align="right">
<math>~q_{1,- 1}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
+ \sqrt{\frac{3}{8\pi}}~\int (x^' + iy^')
~\rho(\vec{x}^{~'}) d^3x^'
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~q_{10}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int r^' \biggl[ Y_{10}^* \biggr]~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int r^' \biggl[ \sqrt{\frac{3}{4\pi}}~\cos\theta^' \biggr]~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\sqrt{\frac{3}{4\pi}} \int z^' \rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~q_{22}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int (r^')^2 \biggl[ Y_{22}^*(\theta^', \phi^') \biggr]~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int (r^')^2 \biggl\{ \frac{1}{4}\sqrt{\frac{15}{2\pi}} ~\sin^2\theta^' ~\biggl[\cos(2\phi^') - i\sin(2\phi^')\biggr] \biggr\}~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{4}\sqrt{\frac{15}{2\pi}}
\int (r^')^2 \biggl\{ \frac{(x^')^2+(y^')^2}{(r^')^2} ~\biggl[\frac{ (x^')^2 - (y^')^2 -
2i ( x^' y^' )}{(x^')^2+(y^')^2} \biggr] \biggr\}~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{4}\sqrt{\frac{15}{2\pi}}
\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
</td>
</tr>
<tr>
<td align="right">
<math>~q_{2,-2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{4}\sqrt{\frac{15}{2\pi}}
\int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~q_{21}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int (r^')^2 \biggl[ Y_{21}^*(\theta^', \phi^') \biggr]~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \sqrt{\frac{15}{8\pi}} \int (r^')^2 \biggl[ \sin\theta^' \cos\theta^' (\cos\phi^' -i\sin\phi^') \biggr]~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \sqrt{\frac{15}{8\pi}} \int (r^')^2
\biggl[ \frac{z^' \sqrt{(x^')^2+(y^')^2}}{(r^')^2}
~\biggl(\frac{x^'}{\sqrt{(x^')^2+(y^')^2}} - \frac{iy^'}{\sqrt{(x^')^2+(y^')^2}} \biggr)\biggr]
~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \sqrt{\frac{15}{8\pi}} \int z^'(x^' - iy^')
~\rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
</td>
</tr>
<tr>
<td align="right">
<math>~q_{2,-1}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
+ \sqrt{\frac{15}{8\pi}} \int z^'(x^' + iy^')
~\rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~q_{20}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int (r^')^2 \biggl[ Y_{20}^*(\theta^', \phi^') \biggr]~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int (r^')^2 \biggl[ \sqrt{\frac{5}{16\pi}} (3\cos^2\theta^' - 1) \biggr]~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\sqrt{\frac{5}{16\pi}}
\int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \, .
</math>
</td>
</tr>
</table>
</div>

<div align="center" id="qlm">
<table border="1" align="center" cellpadding="8" width="80%">
<tr><th>Table 3:   Summary of Cartesian Expressions for Leading Mass-Multipole Moments</th></tr>

<tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~q_{0, 0}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{\sqrt{4\pi}}
\int \rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~q_{1,\pm 1}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\mp \sqrt{\frac{3}{8\pi}}~\int (x^' \mp iy^')
~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~q_{1,0}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\sqrt{\frac{3}{4\pi}} \int z^' \rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~q_{2,\pm 2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{4}\sqrt{\frac{15}{2\pi}}
\int (x^' \mp iy^')^2~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~q_{2,\pm 1}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\mp \sqrt{\frac{15}{8\pi}} \int z^'(x^' \mp iy^')
~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~q_{2,0}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\sqrt{\frac{5}{16\pi}}
\int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^'
</math>
</td>
</tr>

</table>

</td></tr></table>
</div>

===Contributions to the Gravitational Potential in Terms of Multipole Moments===

Expanding on the [[#PhiSeriesExpansion|above discussion of the gravitational potential, written as a series expansion in terms of mass-multipole moments and spherical harmonics]],
<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B(r,\theta,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \Phi_B(r,\theta,\phi) \biggr|_{\ell=0} +~ \Phi_B(r,\theta,\phi)\biggr|_{\ell=1} + ~\Phi_B(r,\theta,\phi)\biggr|_{\ell=2}
-~4\pi G
\sum_{\ell=3}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{q_{\ell m}}{r^{\ell+1}} \frac{Y_{\ell m}(\theta,\phi)}{(2\ell+1)} \, ,
</math>
</td>
</tr>
</table>
here we evaluate the first three, leading order terms.

====First Term====
First (term with <math>~\ell = 0</math>):
<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B(r,\theta,\phi)\biggr|_{\ell = 0}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-4\pi G
\biggl[ q_{00} \biggr] \frac{ Y_{00}(\theta,\phi)}{r}
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-4\pi G
\biggl[ \frac{1}{\sqrt{4\pi}}
\int \rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{1}{\sqrt{4\pi}~r}
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{GM}{r} \, .
</math>
</td>
</tr>
</table>

====Second Term====
Second (terms with <math>~\ell = 1</math>):
<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B(r,\theta,\phi)\biggr|_{\ell = 1}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{4\pi G}{3}
\sum_{m=-1}^{+1} \biggl[ q_{1 m}\biggr] \frac{Y_{1 m}(\theta,\phi)}{r^2}
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{4\pi G}{3r^2}
\biggl\{
\biggl[ q_{1,- 1}\biggr] Y_{1, -1}(\theta,\phi)
+ \biggl[ q_{1 0}\biggr] Y_{1 0}(\theta,\phi)
+ \biggl[ q_{1 1}\biggr] Y_{1 1}(\theta,\phi)
\biggr\}
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{4\pi G}{3r^2}
\biggl\{
\biggl[ + \frac{3}{8\pi}~\int (x^' + iy^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{(x - iy) }{r}
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ \frac{3}{4\pi} \int z^' \rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{z}{r}
+ \biggl[ \frac{3}{8\pi}~\int (x^' - iy^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{(x + iy) }{r}
\biggr\}
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{G}{r^3}
\biggl\{
\biggl[ \frac{1}{2}~\int (x^' + iy^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] (x - iy)
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ \int z^' \rho(\vec{x}^{~'}) d^3x^' \biggr] z
+ \biggl[ \frac{1}{2}~\int (x^' - iy^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] (x + iy)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{G}{r^3}
\biggl\{
\biggl[ \int (x^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] (x )
+ \biggl[ \int (y^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] (y )
+ \biggl[ \int z^' \rho(\vec{x}^{~'}) d^3x^' \biggr] z
\biggr\}
</math>
</td>
</tr>
<tr>
<td align="right">
<math>~</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM}{r^3} \biggl[ \vec{x} \cdot \vec{x}_\mathrm{com} \biggr]</math>
</td>
</tr>
<tr>
<td align="right">
[https://en.wikipedia.org/wiki/Center_of_mass#A_continuous_volume where]:
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\vec{x}_\mathrm{com} \equiv \frac{1}{M} \int ~\vec{x}^{~'} \rho(\vec{x}^{~'}) d^3x^'\, .
</math>
</td>
</tr>
</table>

====Third Term====

Third (terms with <math>~\ell = 2</math>):
<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B(r,\theta,\phi)\biggr|_{\ell = 2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{4\pi G}{5r^3} \biggl\{
\biggl[ q_{2, -2} \biggr] Y_{2,-2} + \biggl[ q_{2, 2} \biggr] Y_{2,2}
+ \biggl[ q_{2,-1} \biggr] Y_{2, -1} + \biggl[ q_{2,1} \biggr] Y_{2, 1}
+ \biggl[ q_{20} \biggr] Y_{20} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{4\pi G}{5r^3} \biggl\{
\biggl[ \frac{1}{4}\sqrt{\frac{15}{2\pi}}
\int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{1}{4}\sqrt{\frac{15}{2\pi}} \sin^2\theta e^{-2i\phi}
+ \biggl[\frac{1}{4}\sqrt{\frac{15}{2\pi}}
\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{1}{4}\sqrt{\frac{15}{2\pi}} \sin^2\theta e^{2i\phi}
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ \sqrt{\frac{15}{8\pi}} \int z^'(x^' + iy^') ~\rho(\vec{x}^{~'}) d^3x^' \biggr] \sqrt{\frac{15}{8\pi}} \sin\theta \cos\theta e^{-i\phi}
+ \biggl[ \sqrt{\frac{15}{8\pi}} \int z^'(x^' - iy^') ~\rho(\vec{x}^{~'}) d^3x^' \biggr] \sqrt{\frac{15}{8\pi}} \sin\theta \cos\theta e^{i\phi}
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ \sqrt{\frac{5}{16\pi}}
\int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr] \sqrt{\frac{5}{16\pi}} (3\cos^2\theta - 1)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{4\pi G}{5r^3} \biggl\{
\frac{3\cdot 5}{2^5 \pi} \biggl(\frac{\varpi}{r}\biggr)^2
\biggl[ \int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[ \cos(2\phi) - i\sin(2\phi) \biggr]
+ \frac{3\cdot 5}{2^5 \pi} \biggl(\frac{\varpi}{r}\biggr)^2
\biggl[\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[ \cos(2\phi) + i\sin(2\phi) \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{3\cdot 5}{2^3\pi} \biggl( \frac{\varpi z}{r^2} \biggr) \biggl[\int z^'(x^' + iy^') ~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[ \cos\phi - i\sin\phi \biggr]
+ \frac{3\cdot 5}{2^3\pi} \biggl( \frac{\varpi z}{r^2} \biggr) \biggl[ \int z^'(x^' - iy^') ~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[ \cos\phi + i\sin\phi \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{5}{2^4\pi} \biggl[ \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl( \frac{3z^2 - r^2}{r^2} \biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{4G}{r^5} \biggl\{
\frac{3}{2^5 }
\biggl[ \int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[ x^2 - y^2 - 2i xy\biggr]
+ \frac{3}{2^5 }
\biggl[\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[ x^2 - y^2 + 2i xy \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{3z}{2^3} \biggl[\int z^'(x^' + iy^') ~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[x - i y \biggr]
+ \frac{3z}{2^3} \biggl[ \int z^'(x^' - iy^') ~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[x + i y \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl( \frac{3z^2 - r^2}{2^4} \biggr) \biggl[ \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{G}{2r^5} \biggl\{
\frac{3( x^2 - y^2 )}{2^2 }
\biggl[ \int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' + \int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ \frac{3i xy}{2 }
\biggl[\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' - \int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ 3xz \biggl[\int z^'(x^' + iy^') ~\rho(\vec{x}^{~'}) d^3x^' + \int z^'(x^' - iy^') ~\rho(\vec{x}^{~'}) d^3x^'\biggr]
+ 3i yz\biggl[ \int z^'(x^' - iy^') ~\rho(\vec{x}^{~'}) d^3x^' - \int z^'(x^' + iy^') ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl( \frac{3z^2 - r^2}{2} \biggr) \biggl[ \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{G}{2r^5} \biggl\{
\frac{3( x^2 - y^2 )}{2^2 }
\biggl[ \int [2(x^')^2 - 2(y^')^2]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ \frac{3i xy}{2 }
\biggl[\int -4i x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ 3xz \biggl[\int 2x^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 3i yz\biggl[ \int -2iy^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ \biggl( \frac{3z^2 - r^2}{2} \biggr) \biggl[ \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{G}{2r^5} \biggl\{
\frac{3( x^2 - y^2 )}{2}
\biggl[ \int [(x^')^2 - (y^')^2]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 6 xy \biggl[\int x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ 6xz \biggl[\int x^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 6yz\biggl[ \int y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ \biggl( \frac{3z^2 - r^2}{2} \biggr) \biggl[ \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
\biggr\} \, .
</math>
</td>
</tr>
</table>

Rearranging terms, we have,

<table border="0" align="center">
<tr>
<td align="right">
<math>~ \Phi_B(r,\theta,\phi)\biggr|_{\ell = 2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 2xz \biggl[\int 3x^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+\frac{3( x^2 - y^2 )}{2}
\biggl[ \int [(x^')^2 - (y^')^2]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ \biggl( \frac{3z^2 - r^2}{2} \biggr) \biggl[ \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 2xz \biggl[\int 3x^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+\frac{3( x^2 )}{2}
\biggl[ \int [(x^')^2 - (y^')^2]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
- \biggl( \frac{x^2}{2} \biggr) \biggl[ \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
-\frac{3( y^2 )}{2}
\biggl[ \int [(x^')^2 - (y^')^2]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
- \biggl( \frac{y^2}{2} \biggr) \biggl[ \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ z^2 \biggl[ \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 2xz \biggl[\int 3x^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ z^2 \biggl[ \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+\frac{( x^2 )}{2}
\biggl[ \int [3(x^')^2 - 3(y^')^2-3(z^')^2 + (r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+\frac{( y^2 )}{2}
\biggl[ \int [3(x^')^2 - 3(y^')^2 -3(z^')^2 + (r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 2xz \biggl[\int 3x^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ z^2 \biggl[ \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+\frac{( x^2 )}{2}
\biggl[ \int [3(x^')^2 - 3(y^')^2-3(z^')^2 + 3(r^')^2 - 3(r^')^2 + (r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+\frac{( y^2 )}{2}
\biggl[ \int [-3(x^')^2 + 3(y^')^2 -3(z^')^2 + 3(r^')^2 - 3(r^')^2 + (r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 2xz \biggl[\int 3x^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ z^2 \biggl[ \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+\frac{( x^2 )}{2}
\biggl[ \int [6(x^')^2 - 2(r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+\frac{( y^2 )}{2}
\biggl[ \int [6(y^')^2 - 2(r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 2xz \biggl[\int 3x^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ x^2
\biggl[ \int [3(x^')^2 - (r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ y^2
\biggl[ \int [3(y^')^2 - 2(r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
+ z^2 \biggl[ \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{G}{2r^5} \sum_{i=1}^3 \sum_{j=1}^3 Q_{i,j} \biggl[ x_i x_j \biggr] \, ,
</math>
</td>
</tr>
</table>
<span id="QuadrupoleMomentTensor">where,</span> following [https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], we have introduced the,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="center" colspan="3">
<font color="#770000">'''Traceless Quadrupole Moment Tensor'''</font>
</td>
</tr>
<tr>
<td align="right">
<math>~Q_{i,j}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\int [ 3(x_i^') (x_j^') - (r^')^2 \delta_{ij} ] \rho(\vec{x}^{~'}) d^3x^' \, .
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], p. 138, Eq. (4.9)<br />
[http://adsabs.harvard.edu/abs/1978ApJ...224..497N M. L. Norman & J. R. Wilson (1978)], p. 501, Eq. (19)<br />
</td>
</tr>
</table>

==Familiar Expression for the Cylindrical Green's Function Expansion==
<div align="center" id="CylindricalGreenFunction">
<table border="1" cellpadding="8" align="center" width="80%">
<tr>
<th align="center"><font size="+0">Table 4:  Cylindrical-Coordinate Green's Function Expansion</font></th>
</tr>
<tr>
<td align="left">
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\sum_{m=-\infty}^{\infty} e^{im(\phi - \phi^')} \int_0^\infty dk ~J_m(k\varpi) ~J_m(k\varpi^') e^{-k(z_> - z_<)}
</math>
</td>
</tr>
</table>
where <math>~J_m</math> is an order <math>~m</math> Bessel function of the first kind.<br />
<div align="center">
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], exercise [3.14]<br />
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl & J. E. Tohline (1999)], p. 88, Eq. (3)
</div>
</td>
</tr>
</table>
</div>

==Toroidal Functions==


<div align="center" id="Toroidal">
<table border="1" cellpadding="8" align="center" width="80%">
<tr>
<th align="center"><font size="+0">Table 5:  Green's Function in Terms of<br />Zero Order, Half-(Odd)Integer Degree, Associated Legendre Functions of the Second Kind, <math>~Q^0_{m-1 / 2}(\chi)</math><br />(also referred to as Toroidal Functions)</font></th>
</tr>
<tr>
<td align="left">
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=-\infty}^{\infty} e^{im(\phi - \phi^')}Q_{m- 1 / 2}(\chi)
</math>
</td>
</tr>
</table>
where:<br />
<div align="center"><math>~\chi \equiv \frac{\varpi^2 + (\varpi^')^2 + (z - z^')^2}{2\varpi \varpi^'}</math><br /><br />
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl & J. E. Tohline (1999)], p. 88, Eqs. (15) & (16)<br />
See also the [https://dlmf.nist.gov/14.19#ii DLMF's definition of Toroidal Functions], <math>~Q_{m - 1 / 2}^{0}</math>
</div>
</td>
</tr>

<tr>
<td align="left">
Note that, according to, for example, equation (8.731.5) of Gradshteyn & Ryzhik (1994),
<div align="center">
<math>~Q^0_{-m - 1 / 2}(\chi) = Q^0_{m- 1 / 2}(\chi) \, .</math>
</div>
Hence, the Green's function can straightforwardly be rewritten in terms of a simpler summation over just ''non-negative'' values of the index, <math>~m</math>.
</td>
</tr>

<tr>
<td align="left">
Referencing equations (8.13.3) and (8.13.7), respectively, of Abramowitz & Stegun (1965), we see that for the smallest two values of the ''non-negative'' index, <math>~m</math>, the function, <math>~Q_{m- 1 / 2}(\chi)</math>, can be rewritten in terms of, the more familiar, complete elliptic integrals of the first and second kind. Specifically,
<table border="0" cellpadding="1" align="center" width="100%">

<tr>
<td align="left" colspan="3">
for <math>~m = 0</math>,
</td>
</tr>
<tr>
<td align="right">
<math>~Q_{m- 1 / 2}(\chi) ~\rightarrow~ Q_{-1 / 2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\mu K(\mu) \, ,
</math>
</td>
</tr>
<tr>
<td align="left" colspan="3">
and, for <math>~m = 1</math>,
</td>
</tr>

<tr>
<td align="right">
<math>~Q_{m- 1 / 2}(\chi) ~\rightarrow~ Q_{+ 1 / 2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\chi \mu K(\mu) - (1+\chi)\mu E(\mu) \, ,
</math>
</td>
</tr>
<tr>
<td align="left" colspan="3">
where,
</td>
</tr>
<tr>
<td align="right">
<math>~\mu \equiv \biggl[ \frac{2}{1+\chi} \biggr]^{1 / 2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{4\varpi \varpi^'}{(\varpi + \varpi^')^2 + (z - z^')^2} \biggr]^{1 / 2}
\, .
</math>
</td>
</tr>
</table>
----

<div align="center">
Excerpt from p. 337 of [https://books.google.com/books?id=MtU8uP7XMvoC&printsec=frontcover&dq=Abramowitz+and+stegun&hl=en&sa=X&ved=0ahUKEwialra5xNbaAhWKna0KHcLAASAQ6AEILDAA#v=onepage&q=Abramowitz%20and%20stegun&f=false M. Abramowitz & I. A. Stegun (1995)]

</div>
[[File:AbramowitzStegun ToroidalFunctions2.png|center|700px|Abramowitz & Stegun (1965)]]
</td>
</tr>

<tr>
<td align="left">
Finally, equation (8.5.3) from Abramowitz & Stegun (1965) or equation (8.832.4) of Gradshteyn & Ryzhik (1994) — also see equation (2) of [http://adsabs.harvard.edu/abs/2000JCoPh.161..204G Gil, Segura & Temme (2000)] — provide the recurrence relation for all other values of the index, <math>~m</math>. Specifically, for all <math>~m \ge 2</math>,
<div align="center">
<math>~Q_{m - 1 / 2}(\chi) = 4\biggl[\frac{m-1}{2m-1}\biggr] \chi Q_{m- 3 / 2}(\chi)
- \biggl[ \frac{2m-3}{2m-1}\biggr] Q_{m- 5 / 2}(\chi) \, .</math>
</div>
----

<div align="center">
Excerpt from p. 490 of [https://dl-acm-org.libezp.lib.lsu.edu/citation.cfm?id=365474&picked=prox W. Guatschi (1965, Communications of the ACM, vol. 8, issue 8, 488 - 492)]
</div>
[[File:ToroidalRecurrenceRelation.png|center|500px|Guatschi (1965, Communications of the ACM, 8, 488 - 492)]]
</td>
</tr>
</table>
</div>

=Related Discussions=

==Toroidal Functions==

* [http://adsabs.harvard.edu/abs/2007AmJPh..75..724S J. Selvaggi, S. Salon, & M. V. K. Chari (2007, American Journal of Physics, 75, 724 - 727)] — ''An Application of Toroidal Functions in Electrostatics''
* [http://adsabs.harvard.edu/abs/2000JCoPh.161..204G A. Gil, J. Segura & N. M. Temme (2000, Journal of Computational Physics, 161, 204 - 217)] — ''Computing Toroidal Functions for Wide Ranges of the Parameters''
* [http://adsabs.harvard.edu/abs/2000CoPhC.124..104S J. Segura & A. Gil (2000, Computer Physics Communications, 124, 104 - 122)] — '' Evaluation of Toroidal Harmonics''
* [http://adsabs.harvard.edu/abs/1997JMP....38.3679B J. W. Bates (1997, Journal of Mathematical Physics, vol. 38, issue 7, 3679-3691)] — ''On Toroidal Green's Functions''
* [https://dl-acm-org.libezp.lib.lsu.edu/citation.cfm?id=365474&picked=prox W. Guatschi (1965, Communications of the ACM, 8, 488 - 492)] — ''Algorithms (Algorithm 259): Legendre Functions for Arguments Larger than One''
<table border="0" align="center" width="92%">
<tr><td align="left">
Note that, in this Guatschi (1965) article, a description of the "<b>procedure</b> <i>toroidal</i>" begins on p. 490, near the bottom of the right-hand column.
</td></tr>
</table>
* [https://www.jstor.org/stable/2369515?seq=1#page_scan_tab_contents A. B. Basset (1893, American Journal of Mathematics, vol. 15, No. 4, pp. 287 - 302)] — ''On Toroidal Functions''
* [http://www.jstor.org/stable/109363?seq=1#page_scan_tab_contents W. M. Hicks (1881, Philosophical Transactions of the Royal Society of London, vol. 172, pp. 609 - 652)] — ''On Toroidal Functions''

==Reviews==

* [http://adsabs.harvard.edu/abs/1969ARA%26A...7..665S P. A. Strittmatter (1969, Annual Review of Astronomy and Astrophysics, 7, 665 - 684)] — ''Stellar Rotation''
* [http://adsabs.harvard.edu/abs/1967ARA%26A...5..465L N. R. Lebovitz (1967, Annual Review of Astronomy and Astrophysics, 5, 465 - 480)] — ''Rotating Fluid Masses''

==Solution Methods==

* [http://adsabs.harvard.edu/abs/1985A%26A...146..260E Y. Eriguchi & E. Mueller (1985, A&A, 146, 260 - 268)] — ''A General Computational Method for Obtaining Equilibria of Self-Gravitating and Rotating Gases''
* [http://adsabs.harvard.edu/abs/1983ApJ...268..155S S. W. Stahler (1983, ApJ, 268, 155 - 184)] — ''The Equilibria of Rotating, Isothermal Clouds. I. - Method of Solution''
* [http://adsabs.harvard.edu/abs/1978PASJ...30..507E Y. Eriguchi (1978, PASJ, 30, 507 - 518)] — ''Hydrostatic Equilibria of Rotating Polytropes''
* [http://adsabs.harvard.edu/abs/1975SvA....19..151B S. I. Blinnikov (1975, Soviet Astronomy, 19, 151 - 156)] — ''Self-Consistent Field Method in the Theory of Rotating Stars''
* [http://adsabs.harvard.edu/abs/1974ApJ...194..709C M. J. Clement (1974, ApJ, 194, 709 - 714)] — ''On the Solution of Poisson's Equation for Rapidly Rotating Stars''
* [http://adsabs.harvard.edu/abs/1970ApJ...161..579J S. Jackson (1970, ApJ, 161, 579 - 585)] — ''Rapidly Rotating Stars. V. The Coupling of the Henyey and the Self-Consistent Methods''
* [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & J. W.-K. Mark (1968, ApJ, 151, 1075 - 1088)] — ''Rapidly Rotating Stars. I. The Self-Consistent-Field Method''
* [http://adsabs.harvard.edu/abs/1964ApJ...140..552J R. A. James (1964, ApJ, 140, 552 - 582)] — ''The Structure and Stability of Rotating Gas Masses''

==Early Eriguchi Applications==

* [http://adsabs.harvard.edu/abs/1985A%26A...147..161E Y. Eriguchi & E. Mueller (1985, A&A, 147, 161 - 168)] — ''Equilibrium Models of Differentially Rotating Polytropes and the Collapse of Rotating Stellar Cores''
* [http://adsabs.harvard.edu/abs/1984PASJ...36..497H I. Hachisu & Y. Eriguchi (1984, PASJ, 36, 497 - 503)] — ''Bifurcation Points on the Maclaurin Sequence''
* [http://adsabs.harvard.edu/abs/1984PASJ...36..259H I. Hachisu & Y. Eriguchi (1984, PASJ, 36, 259 - 276)] — ''Binary Fluid Star''
* [http://adsabs.harvard.edu/abs/1984PASJ...36..239H I. Hachisu & Y. Eriguchi (1984, PASJ, 36, 239 - 257)] — ''Fission of Dumbbell Equilibrium and Binary State of Rapidly Rotating Polytropes''
* [http://adsabs.harvard.edu/abs/1983MNRAS.204..583H I. Hachisu & Y. Eriguchi (1983, MNRAS, 204, 583 - 589)] — ''Bifurcations and Phase Transitions of Self-Gravitating and Uniformly Rotating Fluid''
* [http://adsabs.harvard.edu/abs/1982PThPh..68..206H I. Hachisu & Y. Eriguchi (1982, Prog. Theor. Phys., 68, 206 - 221)] — ''Bifurcation and Fission of Three Dimensional, Rigidly Rotating and Self-Gravitating Polytropes''
* [http://adsabs.harvard.edu/abs/1982PThPh..68..191H I. Hachisu, Y. Eriguchi, & D. Sugimoto (1982, Prog. Theor. Phys., 68, 191 - 205)] — ''Rapidly Rotating Polytropes and Concave Hamburger Equilibrium''
* [http://adsabs.harvard.edu/abs/1981PThPh..65.1870E Y. Eriguchi & D. Sugimoto (1981, Prog. Theor. Phys., 65, 1870 - 1875)] — ''Another Equilibrium Sequence of Self-Gravitating and Rotating Incompressible Fluid''
* [http://adsabs.harvard.edu/abs/1980PThPh..63.1957F T. Fukushima, Y. Eriguchi, D. Sugimoto, G. S. Bisnovatyi-Kogan (1980, Prog. Theor. Phys., 63, 1957 - 1970)] — ''Concave Hamburger Equilibrium of Rotating Bodies''

==Other Example Applications==

* [http://adsabs.harvard.edu/abs/1985A%26A...152..325M E. Mueller & Y. Eriguchi (1985, A&A, 152, 325 - 335)] — ''Equilibrium Models of Differentially Rotating, Completely Catalyzed, Zero-Temperature Configurations With Central Densities Intermediate to White Dwarf and Neutron Star Densities''
* [http://adsabs.harvard.edu/abs/1981ApJ...250..362I J. R. Ipser & R. A. Managan (1981, ApJ, 250, 362 - 372)] — ''On the Existence and Structure of Inhomogeneous Analogs of the Dedekind and Jacobi Ellipsoids''
* [http://adsabs.harvard.edu/abs/1977ApJ...211..568C C. T. Cunningham (1977, ApJ, 211, 568 - 578)] — ''Rapidly Rotating Spheroids of Polytropic Index n = 1''
* [http://adsabs.harvard.edu/abs/1975ApJ...199..179D R. H. Durisen (1975, ApJ, 199, 179 - 183)] — ''Upper Mass Limits for Stable Rotating White Dwarfs''
* [http://adsabs.harvard.edu/abs/1973ApJ...180..159B P. Bodenheimer & J. P. Ostriker (1973, ApJ, 180, 159 - 170)] — ''Rapidly Rotating Stars. VIII. Zero-Viscosity Polytropic Sequences'
* [http://adsabs.harvard.edu/abs/1971ApJ...167..153B P. Bodenheimer (1971, ApJ, 167, 153 - 163)] — ''Rapidly Rotating Stars. VII. Effects of Angular Momentum on Upper-Main-Sequence Models''
* [http://adsabs.harvard.edu/abs/1970ApJ...161.1101B P. Bodenheimer & J. P. Ostriker (1970, ApJ, 161, 1101 - 1113)] — ''Rapidly Rotating Stars. VI. Pre-Main-Sequence Evolution of Massive Stars''
* R. Kippenhahn & H.-C. Thomas (1970) in [http://adsabs.harvard.edu/abs/1970IAUCo...4.....S Proceedings of the 4th IAU Colloquium], held at the Ohio State University, Columbus, Ohio, September 8 - 11, 1969, Dordrecht: Riedel Publishing Co., edited by A. Slettebak — ''Stellar Rotation'' ==> Purchase proceedings from [http://www.springer.com/us/book/9789027701565 Springer], from [https://trove.nla.gov.au/work/21340423?selectedversion=NBD179107 Australia], or from [https://books.google.com/books/about/Stellar_Rotation.html?id=4NnsCAAAQBAJ Google]
** In the introductory section of his paper, [http://adsabs.harvard.edu/abs/1970ApJ...161..579J S. Jackson (1970)] references this article by Kippenhahn & Thomas in the context of uniformly rotating, and therefore only mildly distorted, structures.
* [http://adsabs.harvard.edu/abs/1969ApJ...155..987O J. P. Ostriker & J. L. Tassoul (1969, ApJ, 155, 987 - )] — ''On the Oscillations and Stability of Rotating Stellar Models. II. Rapidly Rotating White Dwarfs''
* [http://adsabs.harvard.edu/abs/1969ApJ...156.1051C M. J. Clement (1969, ApJ, 156, 1051 - 1068)] — ''Differential Rotation in Stars on the Upper Main Sequence''
* [http://adsabs.harvard.edu/abs/1968ApJ...154..627M J. W.-K. Mark (1968, ApJ, 154, 627 - )] — ''Rapidly Rotating Stars. III. Massive Main-Sequence Stars''
* [http://adsabs.harvard.edu/abs/1968ApJ...151.1089O J. P. Ostriker & P. Bodenheimer (1968, ApJ, 151, 1089 - )] — ''Rapidly Rotating Stars. II. Massive White Dwarfs''
* [http://adsabs.harvard.edu/abs/1968ApJ...151..203F J. Faulkner, I. W. Roxburgh, & P. A. Strittmatter (1968, ApJ, 151, 203 - 216)] — ''Uniformly Rotating Main-Sequence Stars''
* [http://adsabs.harvard.edu/abs/1965ApJ...142..208S R. Stoeckly (1965, ApJ, 142, 208 - 228)] — ''Polytropic Models with Fast, Non-Uniform Rotation''
** In the introductory section of his paper, [http://adsabs.harvard.edu/abs/1970ApJ...161..579J S. Jackson (1970)] states that a differentially rotating polytropic structure with a rotationally induced ''extreme distortion'' was first illustrated in this article by Stoeckly.

==Henyey Technique for Nonrotating Stars==
* [http://adsabs.harvard.edu/abs/1959ApJ...129..628H L. G. Henyey, L. Wilets, K. H. Böhm, R. Lelevier, & R. D. Levee (1959, ApJ, 129, 628 - )] — ''A Method for Automatic Computation of Stellar Evolution''
* [http://adsabs.harvard.edu/abs/1964ApJ...139..306H L. G. Henyey, J. E. Forbes, & N. L. Gould (1964, ApJ, 139, 306 - )] — ''A New Method of Automatic Computation of Stellar Evolution''

=See Also=
<ul>
<li>Common Theme:
<ul>
<li>
Menu Tile:  [[AxisymmetricConfigurations/SolvingPE|Solving the (Axisymmetric) Poisson Equation]]
   <math>\Rightarrow</math>     Points principally to: [[AxisymmetricConfigurations/PoissonEq|AxisymmetricConfigurations/PoissonEq]]
</li>
<li>
Menu Tile:  [[2DStructure/UsingTC|Using Toroidal Coordinates to Determine the Gravitational Potential]]
   <math>\Rightarrow</math>     Points principally to: [[2DStructure/ToroidalGreenFunction|2DStructure/ToroidalGreenFunction]]
</li>
<li>
Menu Tile:  [[2DStructure/TCsimplification|Attempt at Simplification]]
   <math>\Rightarrow</math>     Points principally to: [[2DStructure/ToroidalCoordinates|2DStructure/ToroidalCoordinates]]
</li>
<li>
Menu Tile:  [[Apps/WongAP|Wong's Analytic Potential (1973)]]
   <math>\Rightarrow</math>     Points principally to: [[Apps/Wong1973Potential|Apps/Wong1973Potential]]
</li>
<li>
Menu Tile:  [[Apps/DWT|Dyson-Wong Tori]] (Thin Ring Approximation)
   <math>\Rightarrow</math>     Points principally to: [[Apps/DysonWongTori|Apps/DysonWongTori]]
</li>
</ul>
</li>
<li>
(Apollonian Circles) [[2DStructure/ToroidalCoordinateIntegrationLimits|Toroidal-Coordinates Integration Limits]]
</li>
<li>
[[Appendix/Ramblings/ToroidalCoordinates#Toroidal_Configurations_and_Related_Coordinate_Systems|Determining the Relationships between Toroidal Coordinates and other Coordinate Systems]]
</li>
</ul>

{{ SGFfooter }}

SSC/Structure/UniformDensity

2021-09-22T23:54:12Z

174.64.14.12: /* Summary */

=Isolated Uniform-Density Sphere=

{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#ffff99;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Uniform-Density<br />Sphere</b>]]</font>
|}
Here we derive the interior structural properties of an isolated uniform-density sphere using all three [[SSCpt2/SolutionStrategies#Solution_Strategies|solution strategies]]. While deriving essentially the same solution three different ways might seem like a bit of overkill, this approach proves to be instructive because (a) it forces us to examine the structural behavior of a number of different physical parameters, and (b) it illustrates how to work through the different solution strategies for one model whose structure can in fact be derived analytically using any of the techniques. As we shall see when studying other self-gravitating configurations, the three strategies are not always equally fruitful.
 <br />
 <br />
 <br />
==Solution Technique 1==

Adopting [[SSCpt2/SolutionStrategies#Technique_1|solution technique #1]], we need to solve the integro-differential equation,

<div align="center">
{{Math/EQ_SShydrostaticBalance01}}
</div>
appreciating that,
<div align="center">
<math> M_r \equiv \int_0^r 4\pi r^2 \rho dr </math> .
</div>
For a uniform-density configuration, {{Math/VAR_Density01}} = <math>~\rho_c</math> = constant, so the density can be pulled outside the mass integral and the integral can be completed immediately to give,
<div align="center">
<math> M_r = \frac{4\pi}{3}\rho_c r^3 </math> .
</div>
Hence, the differential equation describing hydrostatic balance becomes,
<div align="center">
<math> \frac{dP}{dr} = - \frac{4\pi G}{3} \rho_c^2 r </math> .
</div>
Integrating this from the center of the configuration — where <math>~r=0</math> and <math>~P = P_c</math> — out to an arbitrary radius <math>~r</math> that is still inside the configuration, we obtain,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \int_{P_c}^P dP </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4\pi G}{3} \rho_c^2 \int_0^r r dr </math>
</td>
</tr>

<tr>
<td align="right">
<math>~ \Rightarrow ~~~ P </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~P_c - \frac{2\pi G}{3} \rho_c^2 r^2 \, .</math>
</td>
</tr>
</table>

We expect the pressure to drop to zero at the surface of our spherical configuration — that is, at <math>~r=R</math> — so the central pressure must be,

<div align="center">
<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr)</math> ,
</div>

where <math>~M</math> is the total mass of the configuration. Finally, then, we have,

<div align="center">
<math>P(r) = P_c\biggl[1 - \biggl(\frac{r}{R}\biggr)^2 \biggr] </math> .
</div>

==Solution Technique 3==

Adopting [[SSCpt2/SolutionStrategies#Technique_3|solution technique #3]], we need to solve the ''algebraic'' expression,
<div align="center">
<math>~H + \Phi = C_\mathrm{B}</math> .
</div>
in conjunction with the Poisson equation,
<div align="center">
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) = 4\pi G \rho </math> .
</div>
Appreciating that, as shown above, for a uniform density ({{Math/VAR_Density01}} = <math>\rho_c</math> = constant) configuration,
<div align="center">
<math> M_r = \int_0^r 4\pi r^2 \rho dr = \frac{4\pi}{3}\rho_c r^3 </math> ,
</div>
we can integrate the Poisson equation once to give,
<div align="center">
<math> \frac{d\Phi}{dr} = \frac{4\pi G}{3} \rho_c r </math> ,
</div>
everywhere inside the configuration. Integrating this expression from any point inside the configuration to the surface, we find that,
<div align="center">
<math> \int_{\Phi(r)}^{\Phi_\mathrm{surf}} d\Phi = \frac{4\pi G}{3} \rho_c \int_r^R r dr </math> <br />
<math>\Rightarrow ~~~~~ \Phi_\mathrm{surf} - \Phi(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math>
</div>
Turning to the above algebraic condition, we will adopt the convention that {{Math/VAR_Enthalpy01}} is set to zero at the surface of a barotropic configuration, in which case the constant, <math>C_\mathrm{B}</math>, must be,
<div align="center">
<math>~C_\mathrm{B} = (H + \Phi)_\mathrm{surf} = \Phi_\mathrm{surf}</math> .
</div>
Therefore, everywhere inside the configuration {{Math/VAR_Enthalpy01}} must be given by the expression,
<div align="center">
<math>~H(r) = \Phi_\mathrm{surf} - \Phi(r)</math> .
</div>
Matching this with our solution of the Poisson equation, we conclude that, throughout the configuration,
<div align="center">
<math> H(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>
Comparing this result with the result we obtained using solution technique #1, it is clear that throughout a uniform-density, self-gravitating sphere,
<div align="center">
<math>\frac{P}{H} = \rho</math> .
</div>

==Solution Technique 2==

Adopting [[SSCpt2/SolutionStrategies#Technique_2|solution technique #2]], we need to solve the following single, <math>2^\mathrm{nd}</math>-order ODE:
<div align="center">
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d H}{dr} \biggr) = - 4\pi G \rho </math> .
</div>
Appreciating again that, for a uniform density ({{Math/VAR_Density01}} = <math>\rho_c</math> = constant) configuration,
<div align="center">
<math> M_r = \int_0^r 4\pi r^2 \rho dr = \frac{4\pi}{3}\rho_c r^3 </math> ,
</div>
we can integrate the <math>2^\mathrm{nd}</math>-order ODE once to give,
<div align="center">
<math> \frac{dH}{dr} = -\frac{4\pi G}{3} \rho_c r </math> ,
</div>
everywhere inside the configuration. Integrating this expression from any point inside the configuration to the surface — where, again, we adopt the convention that {{Math/VAR_Enthalpy01}} = 0 — we find that,
<div align="center">
<math> \int_{H(r)}^{0} dH = - \frac{4\pi G}{3} \rho_c \int_r^R r dr </math> <br />
<math>\Rightarrow ~~~~~ H(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
</div>

<font color="darkblue">
==Summary==
</font>

From the above derivations, we can describe the properties of a uniform-density, self-gravitating sphere as follows:
* <font color="red">Mass</font>:
: Given the density, <math>\rho_c</math>, and the radius, <math>R</math>, of the configuration, the total mass is,
<div align="center">
<math>M = \frac{4\pi}{3} \rho_c R^3 </math> ;
</div>

: and, expressed as a function of <math>M</math>, the mass that lies interior to radius <math>r</math> is,
<div align="center">
<math>\frac{M_r}{M} = \biggl(\frac{r}{R} \biggr)^3</math> .
</div>
* <font color="red">Pressure</font>:
: Given values for the pair of model parameters <math>( \rho_c , R )</math>, or <math>( M , R )</math>, or <math>( \rho_c , M )</math>, the central pressure of the configuration is,
<div align="center">
<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr) = \biggl[ \frac{\pi}{6} G^3 \rho_c^4 M^2 \biggr]^{1/3}</math> ;
</div>

: and, expressed in terms of the central pressure <math>P_c</math>, the variation with radius of the pressure is,
<div align="center">
<math>P(r) = P_c \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>

* <span id="UniformSphereEnthalpy"><font color="red">Enthalpy</font>:</span>
: Throughout the configuration, the enthalpy is given by the relation,
<div align="center">
<math>H(r) = \frac{P(r)}{ \rho_c} = \frac{GM}{2R} \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
</div>

* <span id="UniformSpherePotential"><font color="red">Gravitational potential</font>: </span>
: Throughout the configuration — that is, for all <math>r \leq R</math> — the gravitational potential is given by the relation,
<div align="center">
<math>\Phi_\mathrm{surf} - \Phi(r) = H(r) = \frac{G M}{2R} \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
</div>
: Outside of this spherical configuration— that is, for all <math>r \geq R</math> — the potential should behave like a point mass potential, that is,
<div align="center">
<math>\Phi(r) = - \frac{GM}{r} </math> .
</div>
: Matching these two expressions at the surface of the configuration, that is, setting <math>\Phi_\mathrm{surf} = - GM/R</math>, we have what is generally considered the properly normalized prescription for the gravitational potential inside a uniform-density, spherically symmetric configuration:
<div align="center">
<math>\Phi(r) = - \frac{G M}{R} \biggl\{ 1 + \frac{1}{2}\biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] \biggr\} = - \frac{3G M}{2R} \biggl[ 1 - \frac{1}{3} \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .<br />
[http://astrowww.phys.uvic.ca/~tatum/celmechs/celm5.pdf J. B. Tatum (2021)] Celestial Mechanics class notes (UVic), §5.8.9, p. 36, Eq. (5.8.23)
</div>

* <font color="red">Mass-Radius relationship</font>:
: We see that, for a given value of <math>\rho_c</math>, the relationship between the configuration's total mass and radius is,
<div align="center">
<math>M \propto R^3 ~~~~~\mathrm{or}~~~~~R \propto M^{1/3} </math> .
</div>

* <font color="red">Central- to Mean-Density Ratio</font>:
: Because this is a uniform-density structure, the ratio of its central density to its mean density is unity, that is,
<div align="center">
<math>\frac{\rho_c}{\bar{\rho}} = 1 </math> .
</div>

=Uniform-Density Sphere Embedded in an External Medium=
For the ''isolated'' uniform-density sphere, discussed above, the surface of the configuration was identified as the radial location where the pressure drops to zero. Here we embed the sphere in a hot, tenuous medium that exerts a confining "external" pressure, <math>~P_e</math>, and ask how the configuration's equilibrium radius, <math>~R_e</math>, changes in response to this applied external pressure, for a given (fixed) total mass and central pressure, <math>~P_c</math>.

Following [[SSC/Structure/UniformDensity#Solution_Technique_1|solution technique #1]], the derivation remains the same up through the integration of the hydrostatic balance equation to obtain the relation,
<div align="center">
<math>P = P_c - \frac{2\pi G}{3} \rho_c^2 r^2 \, .</math>
</div>
Now we set <math>~P = P_e</math> at the surface of our spherical configuration — that is, at <math>~r=R_e</math> — so we can write,
<div align="center">
<math>P_c - P_e = \frac{2\pi G}{3} \rho_c^2 R_e^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R_e^4} \biggr)</math>

<math>\Rightarrow ~~~~~ P_c \biggl( 1 - \frac{P_e}{P_c} \biggr) = \frac{3G}{8\pi}\biggl( \frac{M^2}{R_e^4} \biggr) \, ,</math>
</div>
where <math>M</math> is the total mass of the configuration. Solving for the equilibrium radius, we have,
<div align="center">
<math> R_e = \biggl[ \biggl( \frac{3}{2^3\pi} \biggr) \frac{G M^2}{P_c} \biggl( 1 - \frac{P_e}{P_c} \biggr)^{-1} \biggr]^{1/4} \, .</math>
</div>
As it should, when the ratio <math>~P_e/P_c \rightarrow 0</math>, this relation reduces to the one obtained, above, for the isolated uniform-density sphere, namely,
<div align="center">
<math> R_e^4 = \biggl( \frac{3}{8\pi} \biggr) \frac{G M^2}{P_c} \, .</math>
</div>

{{ SGFfooter }}

H BookTiledMenu

2021-09-21T21:09:42Z

174.64.14.12: /* Sheroidal & Spheroidal-Like */

Aps/MaclaurinSpheroidFreeFall

2021-09-21T21:05:43Z

174.64.14.12: /* Key References */

Aps/MaclaurinSpheroidFreeFall

2021-09-21T21:02:41Z

174.64.14.12:

Aps/MaclaurinSpheroidFreeFall

2021-09-21T20:56:46Z

174.64.14.12: /* See Also */

Appendix/FormatRecommendations

2021-09-21T20:47:14Z

174.64.14.12: /* Format Recommendations */

__FORCETOC__ 

=Format Recommendations=

==Equations==
===Simplest Form===
Example extracted from Wiki chapter titled:  [[Apps/DysonPotential#Comparison_With_Thin_Ring_Approximation|Dyson (1893)]]

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{\pi}{GM}\biggr] \Phi_\mathrm{TR}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{2K(k)}{R_1} \, ,</math>
</td>
</tr>
</table>

Raw text used to generate this simple equation:
<pre>

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{\pi}{GM}\biggr] \Phi_\mathrm{TR}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{2K(k)}{R_1} \, ,</math>
</td>
</tr>
</table>

</pre>

===With Title and References===
Example extracted from Wiki chapter titled:  [[Apps/Ostriker64#Gravitational_Potential|Ostriker (1964)]]

<div align="center" id="GravitationalPotential">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="3">
<font color="#770000">'''Scalar Gravitational Potential'''</font>
</td>
</tr>
<tr>
<td align="right">
<math>\Phi(\vec{x})</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math> -G \iiint \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-3)<br />
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], §10, p. 17, Eq. (11)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], §4.2, p. 77, Eq. (12)
</td>
</tr>
</table>
</div>

Raw text used to generate this expression ensemble:
<pre>

<div align="center" id="GravitationalPotential">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="3">
<font color="#770000">'''Scalar Gravitational Potential'''</font>
</td>
</tr>
<tr>
<td align="right">
<math>\Phi(\vec{x})</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math> -G \iiint \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-3)<br />
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], §10, p. 17, Eq. (11)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], §4.2, p. 77, Eq. (12)
</td>
</tr>
</table>
</div>

</pre>

===Multiple Lines===
Example extracted from Wiki chapter titled:  [[Apps/DysonPotential#Proof|Dyson (1893)]]

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{\pi}{GM}\biggr] \Phi_\mathrm{TR}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{2}{R_1} \biggl[(1+k_1)K(k_1) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{2K(\mu)}{R_1} \biggl[1+\frac{R_1-R}{R_1+R} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{4K(\mu)}{R_1+R} \, .</math>
</td>
</tr>
</table>

Raw text used to generate this multi-line expression:
<pre>

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{\pi}{GM}\biggr] \Phi_\mathrm{TR}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{2}{R_1} \biggl[(1+k_1)K(k_1) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{2K(\mu)}{R_1} \biggl[1+\frac{R_1-R}{R_1+R} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{4K(\mu)}{R_1+R} \, .</math>
</td>
</tr>
</table>

</pre>

==Darkgreen Quotation Inset==

Example extracted from Wiki chapter titled:  [[Apps/RotatingPolytropes/BarmodeLinearTimeDependent|Onset of Bar-mode Instability …]]

<table border="0" cellpadding="3" align="center" width="80%">
<tr><td align="left">
<font color="darkgreen">
"… the onset of instability is not very sensitive to the compressibility or angular momentum distribution of the polytrope when the models are parameterized by T/|W|</font> — [in particular, the m = 2 barmode becomes unstable at T/|W| &sim; 0.26 - 0.28. ] <font color="darkgreen">The polytrope eigenfunctions are … qualitatively different from the Maclaurin eigenfunctions in one respect: they develop strong spiral arms. The spiral arms are stronger for more compressible polytropes and for polytropes whose angular momentum distributions deviate significantly from those of the Maclaurin spheroids."
</font>
</td></tr>
<tr><td align="right">
— Drawn from [https://ui.adsabs.harvard.edu/abs/1998ApJ...497..370T/abstract Toman, Imamura, Pickett & Durisen (1998)], ApJ, 497, 370
</td></tr></table>

Raw text used to generate this example quotation inset:
<pre>

<table border="0" cellpadding="3" align="center" width="80%">
<tr><td align="left">
<font color="darkgreen">
"… the onset of instability is not very sensitive to the compressibility or angular momentum distribution of the polytrope when the models are parameterized by T/|W|</font> — [in particular, the m = 2 barmode becomes unstable at T/|W| &sim; 0.26 - 0.28. ] <font color="darkgreen">The polytrope eigenfunctions are … qualitatively different from the Maclaurin eigenfunctions in one respect: they develop strong spiral arms. The spiral arms are stronger for more compressible polytropes and for polytropes whose angular momentum distributions deviate significantly from those of the Maclaurin spheroids."
</font>
</td></tr>
<tr><td align="right">
— Drawn from [https://ui.adsabs.harvard.edu/abs/1998ApJ...497..370T/abstract Toman, Imamura, Pickett & Durisen (1998)], ApJ, 497, 370
</td></tr></table>

</pre>

==Pink Comment Balloon==
Example extracted from Wiki chapter titled: [[Apps/MaclaurinSpheroids#Apply_Boundary_Conditions|Maclaurin Spheroids]]


[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline: In Tassoul (1978), the leading coefficient in the expression for the pressure — and, hence, the central pressure — is too large by a factor of 2.]]We know from our [[SR#Barotropic_Structure|separate discussion of supplemental, barotropic equations of state]] that, for a uniform-density, <math>~n = 0</math> polytropic configuration, the pressure is related to the enthalpy via the expression, <math>~P = H\rho</math>. Hence, we conclude that,
<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>
P(\varpi,z)
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\pi G \rho^2 a_1^2 A_3 (1-e^2)\biggl[1 - \biggl( \frac{\varpi}{a_1} \biggr)^2 - \biggl( \frac{z}{a_3} \biggr)^2
\biggr]
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], §4.5, p. 86, Eq. (51)<br />
[<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>], §7.3, p. 172, Eqs. (7.3.16) & (7.3.17)
</td>
</tr>
</table>


Raw text used to generate this illustration of the pink comment balloon:

<pre>

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline: In Tassoul (1978), the leading coefficient in the expression for the pressure — and, hence, the central pressure — is too large by a factor of 2.]]We know from our [[SR#Barotropic_Structure|separate discussion of supplemental, barotropic equations of state]] that, for a uniform-density, <math>~n = 0</math> polytropic configuration, the pressure is related to the enthalpy via the expression, <math>~P = H\rho</math>. Hence, we conclude that,
<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>
P(\varpi,z)
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\pi G \rho^2 a_1^2 A_3 (1-e^2)\biggl[1 - \biggl( \frac{\varpi}{a_1} \biggr)^2 - \biggl( \frac{z}{a_3} \biggr)^2
\biggr]
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], §4.5, p. 86, Eq. (51)<br />
[<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>], §7.3, p. 172, Eqs. (7.3.16) & (7.3.17)
</td>
</tr>
</table>

</pre>

==Wikitable Overflow==
Example extracted from Wiki chapter titled:  [[Appendix/Ramblings/ToroidalCoordinates#Examples|Toroidal Configurations and Related Coordinate Systems]]

<div id="Example2" style="width: 85%; height: 15em; overflow: auto;">
<table align="center" border="1" cellpadding="5">
<tr><th align="center" colspan="10">Example 2</th></tr>
<tr>
<td align="center" colspan="2" width="25%"><math>~\varpi_t</math></td>
<td align="center" colspan="2" width="25%"><math>~r_t</math></td>
<td align="center" colspan="2" width="25%"><math>~Z_0</math></td>
<td align="center" colspan="2"><math>~a</math></td>
<td align="center" colspan="2"><math>~\Kappa</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~\tfrac{3}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{1}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{3}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{1}{3}</math></td>
<td align="center" colspan="2"><math>~(\tfrac{5}{12})^2</math></td>
</tr>
<tr>
<th colspan="10" align="center">Torus Intersection Points</th>
</tr>
<tr>
<td align="center" colspan="2" rowspan="2"><math>~\xi_1</math></td>
<td align="center" colspan="1" rowspan="2"><math>~\beta</math></td>
<td align="center" colspan="1" rowspan="2"><math>~\ell</math></td>
<td align="center" colspan="3" bgcolor="yellow">Intersection #1 (''superior'' sign)</td>
<td align="center" colspan="3" bgcolor="yellow">Intersection #2 (''inferior'' sign)</td>
</tr>
<tr>
<td align="center"><math>~\xi_2</math>
<td align="center"><math>~\varpi_i</math>
<td align="center"><math>~z_i</math>
<td align="center"><math>~\xi_2</math>
<td align="center"><math>~\varpi_i</math>
<td align="center"><math>~z_i</math>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.1927843</math></td>
<td align="center" colspan="1"><math>~+0.138485</math></td>
<td align="center" colspan="1"><math>~1.000000</math></td>
<td align="center" colspan="1"><math>~0.885198</math></td>
<td align="center" colspan="1"><math>~0.704606</math></td>
<td align="center" colspan="1"><math>~0.245844</math></td>
<td align="center" colspan="3">Degenerate Coordinate Values</td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.176</math></td>
<td align="center" colspan="1"><math>~+0.116568</math></td>
<td align="center" colspan="1"><math>~0.981258</math></td>
<td align="center" colspan="1"><math>~0.922142</math></td>
<td align="center" colspan="1"><math>~0.812595</math></td>
<td align="center" colspan="1"><math>~0.242037</math></td>
<td align="center" colspan="1"><math>~0.841611</math></td>
<td align="center" colspan="1"><math>~0.616896</math></td>
<td align="center" colspan="1"><math>~0.211621</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.160</math></td>
<td align="center" colspan="1"><math>~+0.092267</math></td>
<td align="center" colspan="1"><math>~0.962725</math></td>
<td align="center" colspan="1"><math>~0.933386</math></td>
<td align="center" colspan="1"><math>~0.864726</math></td>
<td align="center" colspan="1"><math>~0.222121</math></td>
<td align="center" colspan="1"><math>~0.824945</math></td>
<td align="center" colspan="1"><math>~0.584858</math></td>
<td align="center" colspan="1"><math>~0.187691</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.144</math></td>
<td align="center" colspan="1"><math>~+0.063705</math></td>
<td align="center" colspan="1"><math>~0.943871</math></td>
<td align="center" colspan="1"><math>~0.940238</math></td>
<td align="center" colspan="1"><math>~0.908969</math></td>
<td align="center" colspan="1"><math>~0.192948</math></td>
<td align="center" colspan="1"><math>~0.813713</math></td>
<td align="center" colspan="1"><math>~0.560766</math></td>
<td align="center" colspan="1"><math>~0.163372</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.127</math></td>
<td align="center" colspan="1"><math>~+0.027202</math></td>
<td align="center" colspan="1"><math>~0.924221</math></td>
<td align="center" colspan="1"><math>~0.944608</math></td>
<td align="center" colspan="1"><math>~0.949856</math></td>
<td align="center" colspan="1"><math>~0.150191</math></td>
<td align="center" colspan="1"><math>~0.806047</math></td>
<td align="center" colspan="1"><math>~0.539788</math></td>
<td align="center" colspan="1"><math>~0.135318</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.111</math></td>
<td align="center" colspan="1"><math>~-0.015045</math></td>
<td align="center" colspan="1"><math>~0.907444</math></td>
<td align="center" colspan="1"><math>~0.946487</math></td>
<td align="center" colspan="1"><math>~0.980806</math></td>
<td align="center" colspan="1"><math>~0.096065</math></td>
<td align="center" colspan="1"><math>~0.802617</math></td>
<td align="center" colspan="1"><math>~0.523232</math></td>
<td align="center" colspan="1"><math>~0.105244</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.094</math></td>
<td align="center" colspan="1"><math>~-0.071947</math></td>
<td align="center" colspan="1"><math>~0.894425</math></td>
<td align="center" colspan="1"><math>~0.945995</math></td>
<td align="center" colspan="1"><math>~0.999208</math></td>
<td align="center" colspan="1"><math>~0.019887</math></td>
<td align="center" colspan="1"><math>~0.803522</math></td>
<td align="center" colspan="1"><math>~0.509118</math></td>
<td align="center" colspan="1"><math>~0.066901</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.078</math></td>
<td align="center" colspan="1"><math>~-0.142539</math></td>
<td align="center" colspan="1"><math>~0.892548</math></td>
<td align="center" colspan="1"><math>~0.942353</math></td>
<td align="center" colspan="1"><math>~0.989322</math></td>
<td align="center" colspan="1"><math>~-0.072283</math></td>
<td align="center" colspan="1"><math>~0.810056</math></td>
<td align="center" colspan="1"><math>~0.500846</math></td>
<td align="center" colspan="1"><math>~0.020554</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.061</math></td>
<td align="center" colspan="1"><math>~-0.247448</math></td>
<td align="center" colspan="1"><math>~0.916366</math></td>
<td align="center" colspan="1"><math>~0.932024</math></td>
<td align="center" colspan="1"><math>~0.916375</math></td>
<td align="center" colspan="1"><math>~-0.186599</math></td>
<td align="center" colspan="1"><math>~0.827074</math></td>
<td align="center" colspan="1"><math>~0.505248</math></td>
<td align="center" colspan="1"><math>~-0.050956</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.0449467</math></td>
<td align="center" colspan="1"><math>~-0.398902</math></td>
<td align="center" colspan="1"><math>~1.000000</math></td>
<td align="center" colspan="1"><math>~0.885198</math></td>
<td align="center" colspan="1"><math>~0.632605</math></td>
<td align="center" colspan="1"><math>~-0.220722</math></td>
<td align="center" colspan="3">Degenerate Coordinate Values</td>
</tr>
</table>
</div>


In order to see how this overflow command works, place your mouse cursor anywhere inside the displayed table, then scroll down/up. Here is the raw text illustrating how to handle "Wikitable Overflow":
<pre>

<div id="Example2" style="width: 85%; height: 15em; overflow: auto;">
<table align="center" border="1" cellpadding="5">
<tr><th align="center" colspan="10">Example 2</th></tr>
<tr>
<td align="center" colspan="2" width="25%"><math>~\varpi_t</math></td>
<td align="center" colspan="2" width="25%"><math>~r_t</math></td>
<td align="center" colspan="2" width="25%"><math>~Z_0</math></td>
<td align="center" colspan="2"><math>~a</math></td>
<td align="center" colspan="2"><math>~\Kappa</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~\tfrac{3}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{1}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{3}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{1}{3}</math></td>
<td align="center" colspan="2"><math>~(\tfrac{5}{12})^2</math></td>
</tr>
<tr>
<th colspan="10" align="center">Torus Intersection Points</th>
</tr>
<tr>
<td align="center" colspan="2" rowspan="2"><math>~\xi_1</math></td>
<td align="center" colspan="1" rowspan="2"><math>~\beta</math></td>
<td align="center" colspan="1" rowspan="2"><math>~\ell</math></td>
<td align="center" colspan="3" bgcolor="yellow">Intersection #1 (''superior'' sign)</td>
<td align="center" colspan="3" bgcolor="yellow">Intersection #2 (''inferior'' sign)</td>
</tr>
<tr>
<td align="center"><math>~\xi_2</math>
<td align="center"><math>~\varpi_i</math>
<td align="center"><math>~z_i</math>
<td align="center"><math>~\xi_2</math>
<td align="center"><math>~\varpi_i</math>
<td align="center"><math>~z_i</math>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.1927843</math></td>
<td align="center" colspan="1"><math>~+0.138485</math></td>
<td align="center" colspan="1"><math>~1.000000</math></td>
<td align="center" colspan="1"><math>~0.885198</math></td>
<td align="center" colspan="1"><math>~0.704606</math></td>
<td align="center" colspan="1"><math>~0.245844</math></td>
<td align="center" colspan="3">Degenerate Coordinate Values</td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.176</math></td>
<td align="center" colspan="1"><math>~+0.116568</math></td>
<td align="center" colspan="1"><math>~0.981258</math></td>
<td align="center" colspan="1"><math>~0.922142</math></td>
<td align="center" colspan="1"><math>~0.812595</math></td>
<td align="center" colspan="1"><math>~0.242037</math></td>
<td align="center" colspan="1"><math>~0.841611</math></td>
<td align="center" colspan="1"><math>~0.616896</math></td>
<td align="center" colspan="1"><math>~0.211621</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.160</math></td>
<td align="center" colspan="1"><math>~+0.092267</math></td>
<td align="center" colspan="1"><math>~0.962725</math></td>
<td align="center" colspan="1"><math>~0.933386</math></td>
<td align="center" colspan="1"><math>~0.864726</math></td>
<td align="center" colspan="1"><math>~0.222121</math></td>
<td align="center" colspan="1"><math>~0.824945</math></td>
<td align="center" colspan="1"><math>~0.584858</math></td>
<td align="center" colspan="1"><math>~0.187691</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.144</math></td>
<td align="center" colspan="1"><math>~+0.063705</math></td>
<td align="center" colspan="1"><math>~0.943871</math></td>
<td align="center" colspan="1"><math>~0.940238</math></td>
<td align="center" colspan="1"><math>~0.908969</math></td>
<td align="center" colspan="1"><math>~0.192948</math></td>
<td align="center" colspan="1"><math>~0.813713</math></td>
<td align="center" colspan="1"><math>~0.560766</math></td>
<td align="center" colspan="1"><math>~0.163372</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.127</math></td>
<td align="center" colspan="1"><math>~+0.027202</math></td>
<td align="center" colspan="1"><math>~0.924221</math></td>
<td align="center" colspan="1"><math>~0.944608</math></td>
<td align="center" colspan="1"><math>~0.949856</math></td>
<td align="center" colspan="1"><math>~0.150191</math></td>
<td align="center" colspan="1"><math>~0.806047</math></td>
<td align="center" colspan="1"><math>~0.539788</math></td>
<td align="center" colspan="1"><math>~0.135318</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.111</math></td>
<td align="center" colspan="1"><math>~-0.015045</math></td>
<td align="center" colspan="1"><math>~0.907444</math></td>
<td align="center" colspan="1"><math>~0.946487</math></td>
<td align="center" colspan="1"><math>~0.980806</math></td>
<td align="center" colspan="1"><math>~0.096065</math></td>
<td align="center" colspan="1"><math>~0.802617</math></td>
<td align="center" colspan="1"><math>~0.523232</math></td>
<td align="center" colspan="1"><math>~0.105244</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.094</math></td>
<td align="center" colspan="1"><math>~-0.071947</math></td>
<td align="center" colspan="1"><math>~0.894425</math></td>
<td align="center" colspan="1"><math>~0.945995</math></td>
<td align="center" colspan="1"><math>~0.999208</math></td>
<td align="center" colspan="1"><math>~0.019887</math></td>
<td align="center" colspan="1"><math>~0.803522</math></td>
<td align="center" colspan="1"><math>~0.509118</math></td>
<td align="center" colspan="1"><math>~0.066901</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.078</math></td>
<td align="center" colspan="1"><math>~-0.142539</math></td>
<td align="center" colspan="1"><math>~0.892548</math></td>
<td align="center" colspan="1"><math>~0.942353</math></td>
<td align="center" colspan="1"><math>~0.989322</math></td>
<td align="center" colspan="1"><math>~-0.072283</math></td>
<td align="center" colspan="1"><math>~0.810056</math></td>
<td align="center" colspan="1"><math>~0.500846</math></td>
<td align="center" colspan="1"><math>~0.020554</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.061</math></td>
<td align="center" colspan="1"><math>~-0.247448</math></td>
<td align="center" colspan="1"><math>~0.916366</math></td>
<td align="center" colspan="1"><math>~0.932024</math></td>
<td align="center" colspan="1"><math>~0.916375</math></td>
<td align="center" colspan="1"><math>~-0.186599</math></td>
<td align="center" colspan="1"><math>~0.827074</math></td>
<td align="center" colspan="1"><math>~0.505248</math></td>
<td align="center" colspan="1"><math>~-0.050956</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.0449467</math></td>
<td align="center" colspan="1"><math>~-0.398902</math></td>
<td align="center" colspan="1"><math>~1.000000</math></td>
<td align="center" colspan="1"><math>~0.885198</math></td>
<td align="center" colspan="1"><math>~0.632605</math></td>
<td align="center" colspan="1"><math>~-0.220722</math></td>
<td align="center" colspan="3">Degenerate Coordinate Values</td>
</tr>
</table>
</div>

</pre>

==Permissions==
Here is an example layout that we have adopted to provide a record of ''[[Appendix/Permissions|Permissions]]'' that have been granted to us by other authors and/or publishers to reproduce figures (and/or digitally clipped images of other material) from previously published (usually journal) articles.

<table border="0" cellpadding="2" align="left" width="100%">
<tr>
<td align="right" colspan="2" width="10%"><font size="-1">🔵</font></td>
<td align="left" colspan="2">
Wiki chapter titled:   [[Apps/DysonPotential#Evaluation|Dyson (1893)]]
</td>
</tr>
<tr>
<td align="right" colspan="2" width="10%"> </td>
<td align="left" colspan="2">
<table border="1" align="left" width="90%" cellpadding="3">
<tr>
<td align="right" width="15%">Author(s):</td>
<td align="left">F. W. (Frank Watson) Dyson</td>
</tr>
<tr>
<td align="right">Title:</td>
<td align="left">''II.   The Potential of an Anchor Ring''</td>
</tr>
<tr>
<td align="right">Reference:</td>
<td align="left">[http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893, Philosophical Transactions of the Royal Society of London. A., 184, 43 - 95)]</td>
</tr>
<tr>
<td align="right">DOI:</td>
<td align="left">[https://doi.org/10.1098/rsta.1893.0002 https://doi.org/10.1098/rsta.1893.0002]</td>
</tr>
<tr>
<td align="right">(Scanned Images) Copyright:</td>
<td align="left">© 2017, Royal Society</td>
</tr>
<tr>
<td align="right">Publisher:</td>
<td align="left">Royal Society Publishing</td>
</tr>
<tr>
<td align="right">(Apparently)<br />Relevant Permission:</td>
<td align="left">"[https://royalsociety.org/journals/permissions/ You do not need to seek permissions for re-use of material over 70 years old for up to 5 articles or figures — re-use is only subject to acknowledgement.]"</td>
</tr>
<tr>
<td align="right">Information Entry:</td>
<td align="left">2021/09/15</td>
</tr>
</table>
</td>
</tr>
</table>
 <br />

Here is the raw text that has been typed into our MediaWiki editor in order to generate this example ''Permissions'' layout. Generally speaking the statements of permission that we have received from various authors/publishers have been grouped in our [[Appendix/Permissions|Permissions Appendix]], but this raw text can be cut (from here) and pasted into any other MediaWiki-formatted chapter to serve as a template of this adopted ''Permissions'' format.

<pre>

<table border="0" cellpadding="2" align="left" width="100%">
<tr>
<td align="right" colspan="2" width="10%"><font size="-1">&#x1f535;</font></td>
<td align="left" colspan="2">
Wiki chapter titled: &nbsp; [[Apps/DysonPotential#Evaluation|Dyson (1893)]]
</td>
</tr>
<tr>
<td align="right" colspan="2" width="10%"> </td>
<td align="left" colspan="2">
<table border="1" align="left" width="90%" cellpadding="3">
<tr>
<td align="right" width="15%">Author(s):</td>
<td align="left">F. W. (Frank Watson) Dyson</td>
</tr>
<tr>
<td align="right">Title:</td>
<td align="left">''II.   The Potential of an Anchor Ring''</td>
</tr>
<tr>
<td align="right">Reference:</td>
<td align="left">[http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893, Philosophical Transactions of the Royal Society of London. A., 184, 43 - 95)]</td>
</tr>
<tr>
<td align="right">DOI:</td>
<td align="left">[https://doi.org/10.1098/rsta.1893.0002 https://doi.org/10.1098/rsta.1893.0002]</td>
</tr>
<tr>
<td align="right">(Scanned Images) Copyright:</td>
<td align="left">&copy; 2017, Royal Society</td>
</tr>
<tr>
<td align="right">Publisher:</td>
<td align="left">Royal Society Publishing</td>
</tr>
<tr>
<td align="right">(Apparently)<br />Relevant Permission:</td>
<td align="left">"[https://royalsociety.org/journals/permissions/ You do not need to seek permissions for re-use of material over 70 years old for up to 5 articles or figures &#8212; re-use is only subject to acknowledgement.]"</td>
</tr>
<tr>
<td align="right">Information Entry:</td>
<td align="left">2021/09/15</td>
</tr>
</table>
</td>
</tr>
</table>

</pre>

=See Also=

{{ SGFfooter }}

Appendix/FormatRecommendations

2021-09-21T20:39:08Z

174.64.14.12: /* Format Recommendations */

__FORCETOC__ 

=Format Recommendations=

==Equations==
===Simplest Form===
Example extracted from Wiki chapter titled:  [[Apps/DysonPotential#Comparison_With_Thin_Ring_Approximation|Dyson (1893)]]

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{\pi}{GM}\biggr] \Phi_\mathrm{TR}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{2K(k)}{R_1} \, ,</math>
</td>
</tr>
</table>

Raw text used to generate this simple equation:
<pre>

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{\pi}{GM}\biggr] \Phi_\mathrm{TR}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{2K(k)}{R_1} \, ,</math>
</td>
</tr>
</table>

</pre>

===With Title and References===
Example extracted from Wiki chapter titled:  [[Apps/Ostriker64#Gravitational_Potential|Ostriker (1964)]]

<div align="center" id="GravitationalPotential">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="3">
<font color="#770000">'''Scalar Gravitational Potential'''</font>
</td>
</tr>
<tr>
<td align="right">
<math>\Phi(\vec{x})</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math> -G \iiint \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-3)<br />
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], §10, p. 17, Eq. (11)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], §4.2, p. 77, Eq. (12)
</td>
</tr>
</table>
</div>

Raw text used to generate this expression ensemble:
<pre>

<div align="center" id="GravitationalPotential">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="3">
<font color="#770000">'''Scalar Gravitational Potential'''</font>
</td>
</tr>
<tr>
<td align="right">
<math>\Phi(\vec{x})</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math> -G \iiint \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-3)<br />
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], §10, p. 17, Eq. (11)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], §4.2, p. 77, Eq. (12)
</td>
</tr>
</table>
</div>

</pre>

===Multiple Lines===
Example extracted from Wiki chapter titled:  [[Apps/DysonPotential#Proof|Dyson (1893)]]

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{\pi}{GM}\biggr] \Phi_\mathrm{TR}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{2}{R_1} \biggl[(1+k_1)K(k_1) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{2K(\mu)}{R_1} \biggl[1+\frac{R_1-R}{R_1+R} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{4K(\mu)}{R_1+R} \, .</math>
</td>
</tr>
</table>

Raw text used to generate this multi-line expression:
<pre>

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{\pi}{GM}\biggr] \Phi_\mathrm{TR}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{2}{R_1} \biggl[(1+k_1)K(k_1) \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{2K(\mu)}{R_1} \biggl[1+\frac{R_1-R}{R_1+R} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>- \frac{4K(\mu)}{R_1+R} \, .</math>
</td>
</tr>
</table>

</pre>

==Pink Comment Balloon==
Example extracted from Wiki chapter titled: [[Apps/MaclaurinSpheroids#Apply_Boundary_Conditions|Maclaurin Spheroids]]


[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline: In Tassoul (1978), the leading coefficient in the expression for the pressure — and, hence, the central pressure — is too large by a factor of 2.]]We know from our [[SR#Barotropic_Structure|separate discussion of supplemental, barotropic equations of state]] that, for a uniform-density, <math>~n = 0</math> polytropic configuration, the pressure is related to the enthalpy via the expression, <math>~P = H\rho</math>. Hence, we conclude that,
<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>
P(\varpi,z)
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\pi G \rho^2 a_1^2 A_3 (1-e^2)\biggl[1 - \biggl( \frac{\varpi}{a_1} \biggr)^2 - \biggl( \frac{z}{a_3} \biggr)^2
\biggr]
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], §4.5, p. 86, Eq. (51)<br />
[<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>], §7.3, p. 172, Eqs. (7.3.16) & (7.3.17)
</td>
</tr>
</table>


Raw text used to generate this illustration of the pink comment balloon:

<pre>

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline: In Tassoul (1978), the leading coefficient in the expression for the pressure — and, hence, the central pressure — is too large by a factor of 2.]]We know from our [[SR#Barotropic_Structure|separate discussion of supplemental, barotropic equations of state]] that, for a uniform-density, <math>~n = 0</math> polytropic configuration, the pressure is related to the enthalpy via the expression, <math>~P = H\rho</math>. Hence, we conclude that,
<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>
P(\varpi,z)
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\pi G \rho^2 a_1^2 A_3 (1-e^2)\biggl[1 - \biggl( \frac{\varpi}{a_1} \biggr)^2 - \biggl( \frac{z}{a_3} \biggr)^2
\biggr]
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], §4.5, p. 86, Eq. (51)<br />
[<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>], §7.3, p. 172, Eqs. (7.3.16) & (7.3.17)
</td>
</tr>
</table>

</pre>

==Wikitable Overflow==
Example extracted from Wiki chapter titled:  [[Appendix/Ramblings/ToroidalCoordinates#Examples|Toroidal Configurations and Related Coordinate Systems]]

<div id="Example2" style="width: 85%; height: 15em; overflow: auto;">
<table align="center" border="1" cellpadding="5">
<tr><th align="center" colspan="10">Example 2</th></tr>
<tr>
<td align="center" colspan="2" width="25%"><math>~\varpi_t</math></td>
<td align="center" colspan="2" width="25%"><math>~r_t</math></td>
<td align="center" colspan="2" width="25%"><math>~Z_0</math></td>
<td align="center" colspan="2"><math>~a</math></td>
<td align="center" colspan="2"><math>~\Kappa</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~\tfrac{3}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{1}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{3}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{1}{3}</math></td>
<td align="center" colspan="2"><math>~(\tfrac{5}{12})^2</math></td>
</tr>
<tr>
<th colspan="10" align="center">Torus Intersection Points</th>
</tr>
<tr>
<td align="center" colspan="2" rowspan="2"><math>~\xi_1</math></td>
<td align="center" colspan="1" rowspan="2"><math>~\beta</math></td>
<td align="center" colspan="1" rowspan="2"><math>~\ell</math></td>
<td align="center" colspan="3" bgcolor="yellow">Intersection #1 (''superior'' sign)</td>
<td align="center" colspan="3" bgcolor="yellow">Intersection #2 (''inferior'' sign)</td>
</tr>
<tr>
<td align="center"><math>~\xi_2</math>
<td align="center"><math>~\varpi_i</math>
<td align="center"><math>~z_i</math>
<td align="center"><math>~\xi_2</math>
<td align="center"><math>~\varpi_i</math>
<td align="center"><math>~z_i</math>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.1927843</math></td>
<td align="center" colspan="1"><math>~+0.138485</math></td>
<td align="center" colspan="1"><math>~1.000000</math></td>
<td align="center" colspan="1"><math>~0.885198</math></td>
<td align="center" colspan="1"><math>~0.704606</math></td>
<td align="center" colspan="1"><math>~0.245844</math></td>
<td align="center" colspan="3">Degenerate Coordinate Values</td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.176</math></td>
<td align="center" colspan="1"><math>~+0.116568</math></td>
<td align="center" colspan="1"><math>~0.981258</math></td>
<td align="center" colspan="1"><math>~0.922142</math></td>
<td align="center" colspan="1"><math>~0.812595</math></td>
<td align="center" colspan="1"><math>~0.242037</math></td>
<td align="center" colspan="1"><math>~0.841611</math></td>
<td align="center" colspan="1"><math>~0.616896</math></td>
<td align="center" colspan="1"><math>~0.211621</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.160</math></td>
<td align="center" colspan="1"><math>~+0.092267</math></td>
<td align="center" colspan="1"><math>~0.962725</math></td>
<td align="center" colspan="1"><math>~0.933386</math></td>
<td align="center" colspan="1"><math>~0.864726</math></td>
<td align="center" colspan="1"><math>~0.222121</math></td>
<td align="center" colspan="1"><math>~0.824945</math></td>
<td align="center" colspan="1"><math>~0.584858</math></td>
<td align="center" colspan="1"><math>~0.187691</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.144</math></td>
<td align="center" colspan="1"><math>~+0.063705</math></td>
<td align="center" colspan="1"><math>~0.943871</math></td>
<td align="center" colspan="1"><math>~0.940238</math></td>
<td align="center" colspan="1"><math>~0.908969</math></td>
<td align="center" colspan="1"><math>~0.192948</math></td>
<td align="center" colspan="1"><math>~0.813713</math></td>
<td align="center" colspan="1"><math>~0.560766</math></td>
<td align="center" colspan="1"><math>~0.163372</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.127</math></td>
<td align="center" colspan="1"><math>~+0.027202</math></td>
<td align="center" colspan="1"><math>~0.924221</math></td>
<td align="center" colspan="1"><math>~0.944608</math></td>
<td align="center" colspan="1"><math>~0.949856</math></td>
<td align="center" colspan="1"><math>~0.150191</math></td>
<td align="center" colspan="1"><math>~0.806047</math></td>
<td align="center" colspan="1"><math>~0.539788</math></td>
<td align="center" colspan="1"><math>~0.135318</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.111</math></td>
<td align="center" colspan="1"><math>~-0.015045</math></td>
<td align="center" colspan="1"><math>~0.907444</math></td>
<td align="center" colspan="1"><math>~0.946487</math></td>
<td align="center" colspan="1"><math>~0.980806</math></td>
<td align="center" colspan="1"><math>~0.096065</math></td>
<td align="center" colspan="1"><math>~0.802617</math></td>
<td align="center" colspan="1"><math>~0.523232</math></td>
<td align="center" colspan="1"><math>~0.105244</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.094</math></td>
<td align="center" colspan="1"><math>~-0.071947</math></td>
<td align="center" colspan="1"><math>~0.894425</math></td>
<td align="center" colspan="1"><math>~0.945995</math></td>
<td align="center" colspan="1"><math>~0.999208</math></td>
<td align="center" colspan="1"><math>~0.019887</math></td>
<td align="center" colspan="1"><math>~0.803522</math></td>
<td align="center" colspan="1"><math>~0.509118</math></td>
<td align="center" colspan="1"><math>~0.066901</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.078</math></td>
<td align="center" colspan="1"><math>~-0.142539</math></td>
<td align="center" colspan="1"><math>~0.892548</math></td>
<td align="center" colspan="1"><math>~0.942353</math></td>
<td align="center" colspan="1"><math>~0.989322</math></td>
<td align="center" colspan="1"><math>~-0.072283</math></td>
<td align="center" colspan="1"><math>~0.810056</math></td>
<td align="center" colspan="1"><math>~0.500846</math></td>
<td align="center" colspan="1"><math>~0.020554</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.061</math></td>
<td align="center" colspan="1"><math>~-0.247448</math></td>
<td align="center" colspan="1"><math>~0.916366</math></td>
<td align="center" colspan="1"><math>~0.932024</math></td>
<td align="center" colspan="1"><math>~0.916375</math></td>
<td align="center" colspan="1"><math>~-0.186599</math></td>
<td align="center" colspan="1"><math>~0.827074</math></td>
<td align="center" colspan="1"><math>~0.505248</math></td>
<td align="center" colspan="1"><math>~-0.050956</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.0449467</math></td>
<td align="center" colspan="1"><math>~-0.398902</math></td>
<td align="center" colspan="1"><math>~1.000000</math></td>
<td align="center" colspan="1"><math>~0.885198</math></td>
<td align="center" colspan="1"><math>~0.632605</math></td>
<td align="center" colspan="1"><math>~-0.220722</math></td>
<td align="center" colspan="3">Degenerate Coordinate Values</td>
</tr>
</table>
</div>


In order to see how this overflow command works, place your mouse cursor anywhere inside the displayed table, then scroll down/up. Here is the raw text illustrating how to handle "Wikitable Overflow":
<pre>

<div id="Example2" style="width: 85%; height: 15em; overflow: auto;">
<table align="center" border="1" cellpadding="5">
<tr><th align="center" colspan="10">Example 2</th></tr>
<tr>
<td align="center" colspan="2" width="25%"><math>~\varpi_t</math></td>
<td align="center" colspan="2" width="25%"><math>~r_t</math></td>
<td align="center" colspan="2" width="25%"><math>~Z_0</math></td>
<td align="center" colspan="2"><math>~a</math></td>
<td align="center" colspan="2"><math>~\Kappa</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~\tfrac{3}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{1}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{3}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{1}{3}</math></td>
<td align="center" colspan="2"><math>~(\tfrac{5}{12})^2</math></td>
</tr>
<tr>
<th colspan="10" align="center">Torus Intersection Points</th>
</tr>
<tr>
<td align="center" colspan="2" rowspan="2"><math>~\xi_1</math></td>
<td align="center" colspan="1" rowspan="2"><math>~\beta</math></td>
<td align="center" colspan="1" rowspan="2"><math>~\ell</math></td>
<td align="center" colspan="3" bgcolor="yellow">Intersection #1 (''superior'' sign)</td>
<td align="center" colspan="3" bgcolor="yellow">Intersection #2 (''inferior'' sign)</td>
</tr>
<tr>
<td align="center"><math>~\xi_2</math>
<td align="center"><math>~\varpi_i</math>
<td align="center"><math>~z_i</math>
<td align="center"><math>~\xi_2</math>
<td align="center"><math>~\varpi_i</math>
<td align="center"><math>~z_i</math>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.1927843</math></td>
<td align="center" colspan="1"><math>~+0.138485</math></td>
<td align="center" colspan="1"><math>~1.000000</math></td>
<td align="center" colspan="1"><math>~0.885198</math></td>
<td align="center" colspan="1"><math>~0.704606</math></td>
<td align="center" colspan="1"><math>~0.245844</math></td>
<td align="center" colspan="3">Degenerate Coordinate Values</td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.176</math></td>
<td align="center" colspan="1"><math>~+0.116568</math></td>
<td align="center" colspan="1"><math>~0.981258</math></td>
<td align="center" colspan="1"><math>~0.922142</math></td>
<td align="center" colspan="1"><math>~0.812595</math></td>
<td align="center" colspan="1"><math>~0.242037</math></td>
<td align="center" colspan="1"><math>~0.841611</math></td>
<td align="center" colspan="1"><math>~0.616896</math></td>
<td align="center" colspan="1"><math>~0.211621</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.160</math></td>
<td align="center" colspan="1"><math>~+0.092267</math></td>
<td align="center" colspan="1"><math>~0.962725</math></td>
<td align="center" colspan="1"><math>~0.933386</math></td>
<td align="center" colspan="1"><math>~0.864726</math></td>
<td align="center" colspan="1"><math>~0.222121</math></td>
<td align="center" colspan="1"><math>~0.824945</math></td>
<td align="center" colspan="1"><math>~0.584858</math></td>
<td align="center" colspan="1"><math>~0.187691</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.144</math></td>
<td align="center" colspan="1"><math>~+0.063705</math></td>
<td align="center" colspan="1"><math>~0.943871</math></td>
<td align="center" colspan="1"><math>~0.940238</math></td>
<td align="center" colspan="1"><math>~0.908969</math></td>
<td align="center" colspan="1"><math>~0.192948</math></td>
<td align="center" colspan="1"><math>~0.813713</math></td>
<td align="center" colspan="1"><math>~0.560766</math></td>
<td align="center" colspan="1"><math>~0.163372</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.127</math></td>
<td align="center" colspan="1"><math>~+0.027202</math></td>
<td align="center" colspan="1"><math>~0.924221</math></td>
<td align="center" colspan="1"><math>~0.944608</math></td>
<td align="center" colspan="1"><math>~0.949856</math></td>
<td align="center" colspan="1"><math>~0.150191</math></td>
<td align="center" colspan="1"><math>~0.806047</math></td>
<td align="center" colspan="1"><math>~0.539788</math></td>
<td align="center" colspan="1"><math>~0.135318</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.111</math></td>
<td align="center" colspan="1"><math>~-0.015045</math></td>
<td align="center" colspan="1"><math>~0.907444</math></td>
<td align="center" colspan="1"><math>~0.946487</math></td>
<td align="center" colspan="1"><math>~0.980806</math></td>
<td align="center" colspan="1"><math>~0.096065</math></td>
<td align="center" colspan="1"><math>~0.802617</math></td>
<td align="center" colspan="1"><math>~0.523232</math></td>
<td align="center" colspan="1"><math>~0.105244</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.094</math></td>
<td align="center" colspan="1"><math>~-0.071947</math></td>
<td align="center" colspan="1"><math>~0.894425</math></td>
<td align="center" colspan="1"><math>~0.945995</math></td>
<td align="center" colspan="1"><math>~0.999208</math></td>
<td align="center" colspan="1"><math>~0.019887</math></td>
<td align="center" colspan="1"><math>~0.803522</math></td>
<td align="center" colspan="1"><math>~0.509118</math></td>
<td align="center" colspan="1"><math>~0.066901</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.078</math></td>
<td align="center" colspan="1"><math>~-0.142539</math></td>
<td align="center" colspan="1"><math>~0.892548</math></td>
<td align="center" colspan="1"><math>~0.942353</math></td>
<td align="center" colspan="1"><math>~0.989322</math></td>
<td align="center" colspan="1"><math>~-0.072283</math></td>
<td align="center" colspan="1"><math>~0.810056</math></td>
<td align="center" colspan="1"><math>~0.500846</math></td>
<td align="center" colspan="1"><math>~0.020554</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.061</math></td>
<td align="center" colspan="1"><math>~-0.247448</math></td>
<td align="center" colspan="1"><math>~0.916366</math></td>
<td align="center" colspan="1"><math>~0.932024</math></td>
<td align="center" colspan="1"><math>~0.916375</math></td>
<td align="center" colspan="1"><math>~-0.186599</math></td>
<td align="center" colspan="1"><math>~0.827074</math></td>
<td align="center" colspan="1"><math>~0.505248</math></td>
<td align="center" colspan="1"><math>~-0.050956</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.0449467</math></td>
<td align="center" colspan="1"><math>~-0.398902</math></td>
<td align="center" colspan="1"><math>~1.000000</math></td>
<td align="center" colspan="1"><math>~0.885198</math></td>
<td align="center" colspan="1"><math>~0.632605</math></td>
<td align="center" colspan="1"><math>~-0.220722</math></td>
<td align="center" colspan="3">Degenerate Coordinate Values</td>
</tr>
</table>
</div>

</pre>

==Permissions==
Here is an example layout that we have adopted to provide a record of ''[[Appendix/Permissions|Permissions]]'' that have been granted to us by other authors and/or publishers to reproduce figures (and/or digitally clipped images of other material) from previously published (usually journal) articles.

<table border="0" cellpadding="2" align="left" width="100%">
<tr>
<td align="right" colspan="2" width="10%"><font size="-1">🔵</font></td>
<td align="left" colspan="2">
Wiki chapter titled:   [[Apps/DysonPotential#Evaluation|Dyson (1893)]]
</td>
</tr>
<tr>
<td align="right" colspan="2" width="10%"> </td>
<td align="left" colspan="2">
<table border="1" align="left" width="90%" cellpadding="3">
<tr>
<td align="right" width="15%">Author(s):</td>
<td align="left">F. W. (Frank Watson) Dyson</td>
</tr>
<tr>
<td align="right">Title:</td>
<td align="left">''II.   The Potential of an Anchor Ring''</td>
</tr>
<tr>
<td align="right">Reference:</td>
<td align="left">[http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893, Philosophical Transactions of the Royal Society of London. A., 184, 43 - 95)]</td>
</tr>
<tr>
<td align="right">DOI:</td>
<td align="left">[https://doi.org/10.1098/rsta.1893.0002 https://doi.org/10.1098/rsta.1893.0002]</td>
</tr>
<tr>
<td align="right">(Scanned Images) Copyright:</td>
<td align="left">© 2017, Royal Society</td>
</tr>
<tr>
<td align="right">Publisher:</td>
<td align="left">Royal Society Publishing</td>
</tr>
<tr>
<td align="right">(Apparently)<br />Relevant Permission:</td>
<td align="left">"[https://royalsociety.org/journals/permissions/ You do not need to seek permissions for re-use of material over 70 years old for up to 5 articles or figures — re-use is only subject to acknowledgement.]"</td>
</tr>
<tr>
<td align="right">Information Entry:</td>
<td align="left">2021/09/15</td>
</tr>
</table>
</td>
</tr>
</table>
 <br />

Here is the raw text that has been typed into our MediaWiki editor in order to generate this example ''Permissions'' layout. Generally speaking the statements of permission that we have received from various authors/publishers have been grouped in our [[Appendix/Permissions|Permissions Appendix]], but this raw text can be cut (from here) and pasted into any other MediaWiki-formatted chapter to serve as a template of this adopted ''Permissions'' format.

<pre>

<table border="0" cellpadding="2" align="left" width="100%">
<tr>
<td align="right" colspan="2" width="10%"><font size="-1">&#x1f535;</font></td>
<td align="left" colspan="2">
Wiki chapter titled: &nbsp; [[Apps/DysonPotential#Evaluation|Dyson (1893)]]
</td>
</tr>
<tr>
<td align="right" colspan="2" width="10%"> </td>
<td align="left" colspan="2">
<table border="1" align="left" width="90%" cellpadding="3">
<tr>
<td align="right" width="15%">Author(s):</td>
<td align="left">F. W. (Frank Watson) Dyson</td>
</tr>
<tr>
<td align="right">Title:</td>
<td align="left">''II.   The Potential of an Anchor Ring''</td>
</tr>
<tr>
<td align="right">Reference:</td>
<td align="left">[http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893, Philosophical Transactions of the Royal Society of London. A., 184, 43 - 95)]</td>
</tr>
<tr>
<td align="right">DOI:</td>
<td align="left">[https://doi.org/10.1098/rsta.1893.0002 https://doi.org/10.1098/rsta.1893.0002]</td>
</tr>
<tr>
<td align="right">(Scanned Images) Copyright:</td>
<td align="left">&copy; 2017, Royal Society</td>
</tr>
<tr>
<td align="right">Publisher:</td>
<td align="left">Royal Society Publishing</td>
</tr>
<tr>
<td align="right">(Apparently)<br />Relevant Permission:</td>
<td align="left">"[https://royalsociety.org/journals/permissions/ You do not need to seek permissions for re-use of material over 70 years old for up to 5 articles or figures &#8212; re-use is only subject to acknowledgement.]"</td>
</tr>
<tr>
<td align="right">Information Entry:</td>
<td align="left">2021/09/15</td>
</tr>
</table>
</td>
</tr>
</table>

</pre>

=See Also=

{{ SGFfooter }}

Aps/MaclaurinSpheroidFreeFall

2021-09-21T20:15:28Z

174.64.14.12:

Aps/MaclaurinSpheroidFreeFall

2021-09-21T20:04:51Z

174.64.14.12: /* Free-Fall Collapse of an Homogeneous Spheroid */