Appendix/Ramblings/TurningPoints - Revision history

Jet53man: /* References */

2022-05-09T21:37:08Z

References

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	<li>Chapter 8 (pp. 189-241) of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>]</li>		<li>Chapter 8 (pp. 189-241) of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>]</li>
	<li>Chapter 16 (pp. 448-469) of [<b>[[Appendix/References#T78\|<font color="red">T78</font>]]</b>]</li>		<li>Chapter 16 (pp. 448-469) of [<b>[[Appendix/References#T78\|<font color="red">T78</font>]]</b>]</li>
	<li>~~[http://adsabs.harvard.edu/abs/1993ApJ...406L..63L Lai~~, ~~Rasio, & Shapiro (1993a, ApJL, 406, L63)] —~~ ''Hydrodynamic Instability and Coalescence of Close Binary Systems''~~</li>~~		<li>{{ LRS93afull }}, ''Hydrodynamic Instability and Coalescence of Close Binary Systems.''
	<li>~~<b>LRS93b</b> —~~ {{ LRS93bfull }} ~~—~~ ''Ellipsoidal Figures of Equilibrium: Compressible Models''</li>		<li>{{ LRS93bfull }}, ''Ellipsoidal Figures of Equilibrium: Compressible Models.''</li>
	<li>Other LRS papers:		<li>Other LRS papers:
	<ul>		<ul>
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	{{ ~~LRS93bfigure~~ }}		Reproduction of Fig. 1 (p. L64) from<br />
			{{ LRS93afigure }}
	</td>		</td>
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Jet53man: /* References */

2022-05-09T17:27:55Z

References

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	<ul>		<ul>
	<li>~~[http://rsta.royalsocietypublishing.org/content/206/402-412/161 G. H. Darwin (1906)]~~ ''On the Figure and Stability of a Liquid Satellite''~~, Philosophical Transactions of the Royal Society A, 206, 161.~~</li>		<li>{{ Darwin06full }}, ''On the Figure and Stability of a Liquid Satellite.''</li>
	<li>[http://adsabs.harvard.edu/abs/1919pcsd.book.....J J. H. Jeans (1919)] ''Problems of Cosmogony and Stellar Dynamics'' -- see especially beginning in Ch. III (p. 46)</li>		<li>[http://adsabs.harvard.edu/abs/1919pcsd.book.....J J. H. Jeans (1919)] ''Problems of Cosmogony and Stellar Dynamics'' -- see especially beginning in Ch. III (p. 46)</li>
	<li>[http://adsabs.harvard.edu/abs/1959cbs..book.....K Z. Kopal (1959)], ''Close Binary Systems'' -- see especially, Ch. III (p. 125); but not particularly useful.</li>		<li>[http://adsabs.harvard.edu/abs/1959cbs..book.....K Z. Kopal (1959)], ''Close Binary Systems'' -- see especially, Ch. III (p. 125); but not particularly useful.</li>

Jet53man: /* Preface */

2022-05-05T23:03:48Z

Preface

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	</ol>		</ol>
	</li>		</li>
	<li>[https://www.google.com/books/edition/Mémoires_de_la_section_des_sciences/x3gVAAAAQAAJ?hl=en&gbpv=0 ~~Volume 2~~], pp. </li>		<li>[https://www.google.com/books/edition/Mémoires_de_la_section_des_sciences/x3gVAAAAQAAJ?hl=en&gbpv=0 <b>Tome Second (1851 - 1854):</b>]
			<ol type="a">
			<li>(1851, pp. 21 - 32) — ''Mémoire sur la figure d'une masse fluide, soumise à 'attraction d'un point éloigné.</li>
			</ol>
			</li>
	</ol>		</ol>
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	Also found the following (presumably related) article by Éd. Roche via a "Google Books" search:		Also found the following (presumably related) article by Éd. Roche via a "Google Books" search:
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Jet53man: /* Preface */

2022-05-05T22:56:06Z

Preface

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	----		----
	<div align="center">[[File:RocheReferences.png\|600px\|Roche References]]</div>		<div align="center">[[File:RocheReferences.png\|600px\|Roche References]]</div>
	<br />
	~~Finally~~ found the following (presumably related) article by Éd. Roche via a "Google Books" search:		The following articles by Édouard Roche have been found via "Google Book" searches:
			<ol>
			<li>[https://www.google.com/books/edition/_/mGpFAAAAcAAJ?hl=en&sa=X&ved=2ahUKEwiW3Oz_rMn3AhXnomoFHbAcBHMQ7_IDegQICxAC <b>Tome Premier (1847 - 1850):</b>]
			<ol type="a">
			<li>(1848, pp. 113 - 124) — ''Calcul de l'Inégalité parallactique''</li>
			<li>(1849, pp. 245 - 262) — ''Mémoire sur la figure d'une masse fluide, soumise à 'attraction d'un point éloigné. (1<sup>re</sup> Partie.)</li>
			<li>(1850, pp. 333 - 348) — ''Mémoire sur la figure d'une masse fluide, soumise à 'attraction d'un point éloigné. (2<sup>e</sup> Partie.)</li>
			</ol>
			</li>
			<li>[https://www.google.com/books/edition/Mémoires_de_la_section_des_sciences/x3gVAAAAQAAJ?hl=en&gbpv=0 Volume 2], pp. </li>
			</ol>

			Also found the following (presumably related) article by Éd. Roche via a "Google Books" search:
	<ul>		<ul>
	<li>		<li>

Jet53man: /* Preface */

2022-05-05T22:28:24Z

Preface

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	<div align="center">[[File:RocheReferences.png\|600px\|Roche References]]</div>		<div align="center">[[File:RocheReferences.png\|600px\|Roche References]]</div>
	<br />		<br />
	Finally found ~~this~~ article by Éd. Roche via a [https://www.google.com/books/edition/Mémoires_de_la_section_des_sciences/x3gVAAAAQAAJ?hl=en&gbpv=1&dq=d%27une+Masse+Fluide+Soumise&pg=PA21&printsec=frontcover ~~Google search~~]~~; see~~ pp. 21 - 32.		Finally found the following (presumably related) article by Éd. Roche via a "Google Books" search:
			<ul>
			<li>
			Académie des Sciences et Lettres de Montpellier<br />
			Mémoires de la Section des Section des Sciences.<br />
			<b>Tome Second (1851 - 1854)</b><br />
			[https://www.google.com/books/edition/Mémoires_de_la_section_des_sciences/x3gVAAAAQAAJ?hl=en&gbpv=1&dq=d%27une+Masse+Fluide+Soumise&pg=PA21&printsec=frontcover Édouard Roche — ''Mémoire sur la figure d'une masse fluide, soumise à l'attraction d'un point éloigné] (pp. 21 - 32)
			</li>
			</ul>
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Jet53man: /* Preface */

2022-05-05T22:09:01Z

Preface

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	"The laborious computations necessary for obtaining numerical results were carried out, partly at least, by graphical methods. Verification of the calculations, which as far as I know have never been repeated, forms part of the work of the present paper. The distance from a spherical planet which has been called 'Roche's limit' is expressed by the number of planetary radii in the radius vector of the nearest possible infinitesimal liquid satellite, of the same density as the planet, revolving so as always to present the same aspect to the planet. Our moon, if it were homogeneous, would have the form of one of Roche's ellipsoids; but its present radius vector is of course far greater than the limit. Roche assigned to the limit in question the numerical value 2.44; it the present paper I show that the true value is 2.455, and the closeness of the agreement with the previously accepted value affords a remarkable testimony to the accuracy with which he must have drawn his figures."		"The laborious computations necessary for obtaining numerical results were carried out, partly at least, by graphical methods. Verification of the calculations, which as far as I know have never been repeated, forms part of the work of the present paper. The distance from a spherical planet which has been called 'Roche's limit' is expressed by the number of planetary radii in the radius vector of the nearest possible infinitesimal liquid satellite, of the same density as the planet, revolving so as always to present the same aspect to the planet. Our moon, if it were homogeneous, would have the form of one of Roche's ellipsoids; but its present radius vector is of course far greater than the limit. Roche assigned to the limit in question the numerical value 2.44; it the present paper I show that the true value is 2.455, and the closeness of the agreement with the previously accepted value affords a remarkable testimony to the accuracy with which he must have drawn his figures."

	<sup>*</sup>''La figure d'une masse fluide soumise à l'attraction d'un point éloigné,'' 'Acad. des Sci. de Montpellier,' vol. 1, 1847-50, p. 243.		<sup>*</sup>''La figure d'une masse fluide soumise à l'attraction d'un point éloigné,'' 'Acad. des Sci. de Montpellier,' vol. 1, 1847-50, p. 243.
			</font>
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	<div align="center">[[File:RocheReferences.png\|600px\|Roche References]]</div>		<div align="center">[[File:RocheReferences.png\|600px\|Roche References]]</div>
	</~~font~~>		<br />
			Finally found this article by Éd. Roche via a [https://www.google.com/books/edition/Mémoires_de_la_section_des_sciences/x3gVAAAAQAAJ?hl=en&gbpv=1&dq=d%27une+Masse+Fluide+Soumise&pg=PA21&printsec=frontcover Google search]; see pp. 21 - 32.
	</td>		</td>
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Jet53man: /* Preface */

2022-05-05T18:47:01Z

Preface

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	<sup>*</sup>''La figure d'une masse fluide soumise à l'attraction d'un point éloigné,'' 'Acad. des Sci. de Montpellier,' vol. 1, 1847-50, p. 243.		<sup>*</sup>''La figure d'une masse fluide soumise à l'attraction d'un point éloigné,'' 'Acad. des Sci. de Montpellier,' vol. 1, 1847-50, p. 243.
			----
			<div align="center">[[File:RocheReferences.png\|600px\|Roche References]]</div>
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Jet53man: /* Schönberg-Chandrasekhar Mass */

2022-05-05T18:05:49Z

Schönberg-Chandrasekhar Mass

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	<tr>		<tr>
	<td align="center" colspan="2">		<td align="center" colspan="1" width="33%">
	~~Figures~~ from ~~(left) [http:~~//~~adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)]~~		Reproduction of Figure 1 from<br />{{ HC41figure }}
	~~and (~~center~~) [http:~~/~~/adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]~~		</td>
			<td align="center" colspan="1" width="33%">
			Reproduction of Figure 1 from<br />{{ SC42figure }}
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	<td align="center" colspan="2">		<td align="center" colspan="1" width="33%">
	~~Figures~~ from ~~(left) [http:~~//~~adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)]~~		Reproduction of Figure 1 from<br />{{ HC41figure }}
	~~and (~~center~~) [http:~~/~~/adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]~~		</td>
			<td align="center" colspan="1" width="33%">
			Reproduction of Figure 1 from<br />{{ SC42figure }}
	</td>		</td>
	<td align="center">		<td align="center">

Jet53man: /* Bonnor-Ebert Sphere */

2022-05-05T17:58:46Z

Bonnor-Ebert Sphere

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	<table border="1" cellpadding="5">		<table border="1" cellpadding="5">
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	Reproduction of Figure 2 from<br />{{ Ebert55figure }}		Reproduction of Figure 2 from<br />{{ Ebert55figure }}
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	Reproduction of Figure 1 from<br />{{ Bonnor56figure }}		Reproduction of Figure 1 from<br />{{ Bonnor56figure }}
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	<table border="1">		<table border="1">
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	~~Figures~~ from ~~(left) [http:~~//~~adsabs.harvard.edu~~/~~abs/1955ZA.....37..217E Ebert (1955)] and (right) [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)]~~		Reproduction of Figure 2 from<br />{{ Ebert55figure }}
			</td>
			<td align="center" colspan="1">
			Reproduction of Figure 1 from<br />{{ Bonnor56figure }}
	</td>		</td>
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Jet53man: /* Bonnor-Ebert Sphere */

2022-05-05T17:50:19Z

Bonnor-Ebert Sphere

← Older revision		Revision as of 13:50, 5 May 2022
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	==Bonnor-Ebert Sphere==		==Bonnor-Ebert Sphere==
	In the mid-1950s, {{ Ebert55full }} and {{ Bonnor56full }} independently realized that an equilibrium isothermal gas cloud can be constructed with a finite radius by embedding it in a hot, tenuous external medium. As has been described in an [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|accompanying chapter]], the relevant mathematical model is constructed by chopping off the ''isolated'' isothermal sphere at some finite radius — call it, <math>\xi_e</math> — and imposing an externally applied pressure, <math>P_e</math>, that is equal to the pressure of the isothermal gas at the specified edge of the truncated sphere. By varying <math>\xi_e</math>, a ''sequence'' of equilibrium models can be constructed, as illustrated in Figure 1.		In the mid-1950s, {{ Ebert55full }} and {{ Bonnor56full }} independently realized that an equilibrium isothermal gas cloud can be constructed with a finite radius by embedding it in a hot, tenuous external medium. As has been described in an [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|accompanying chapter]], the relevant mathematical model is constructed by chopping off the ''isolated'' isothermal sphere at some finite radius — call it, <math>\xi_e</math> — and imposing an externally applied pressure, <math>P_e</math>, that is equal to the pressure of the isothermal gas at the specified edge of the truncated sphere. By varying <math>\xi_e</math>, a ''sequence'' of equilibrium models can be constructed, as illustrated in Figure 1. (See also our [[SSC/Stability/InstabilityOnsetOverview#Fig1\|more extensive discussion of isothermal equilibrium sequences]].)

	<div align="center">		<div align="center">