Apps/MaclaurinSpheroids - Revision history

Jet53man: /* Gravitational Potential */

2022-03-23T21:17:36Z

Gravitational Potential

← Older revision		Revision as of 17:17, 23 March 2022
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	<td align="center">		<td align="center">
	<math>~\frac{\pi}{2} - \biggl(\frac{a_3}{a_1}\biggr) +\frac{\pi}{4}\biggl(\frac{a_3}{a_1}\biggr)^2		<math>~\frac{\pi}{2} - \biggl(\frac{a_3}{a_1}\biggr) +\frac{\pi}{4}\biggl(\frac{a_3}{a_1}\biggr)^2
	- \mathcal{O}\biggl(\frac{a_3}{a_1}\biggr)</math>		- \mathcal{O}\biggl(\frac{a_3^3}{a_1^3}\biggr)</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>\frac{\pi}{2} \biggl( \frac{a_3}{a_1}\biggr) - 2\biggl(\frac{a_3}{a_1}\biggr)^2+ \mathcal{O}\biggl(\frac{a_3}{a_1}\biggr)</math>		<math>\frac{\pi}{2} \biggl( \frac{a_3}{a_1}\biggr) - 2\biggl(\frac{a_3}{a_1}\biggr)^2+ \mathcal{O}\biggl(\frac{a_3^3}{a_1^3}\biggr)</math>
	</td>		</td>
	</tr>		</tr>
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	<td align="center">		<td align="center">
	<math> 2 - \pi \biggl( \frac{a_3}{a_1}\biggr) + 4\biggl(\frac{a_3}{a_1}\biggr)^2		<math> 2 - \pi \biggl( \frac{a_3}{a_1}\biggr) + 4\biggl(\frac{a_3}{a_1}\biggr)^2
	- \mathcal{O}\biggl(\frac{a_3}{a_1}\biggr)</math>		- \mathcal{O}\biggl(\frac{a_3^3}{a_1^3}\biggr)</math>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Gravitational Potential */

2022-03-23T17:17:29Z

Gravitational Potential

← Older revision		Revision as of 13:17, 23 March 2022
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	<tr>		<tr>
	<td colspan="3" align="center">		<td colspan="3" align="center">
	'''Table 2:'''  [[Appendix/Ramblings/PowerSeriesExpressions#Maclaurin_Spheroid_Index_Symbols\|Limiting Values]]		'''Table 1:'''  [[Appendix/Ramblings/PowerSeriesExpressions#Maclaurin_Spheroid_Index_Symbols\|Limiting Values]]
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Gravitational Potential */

2022-03-23T17:16:28Z

Gravitational Potential

← Older revision		Revision as of 13:16, 23 March 2022
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	<span id="LimitingValues">Note that these three expressions</span> have the following values in the limit of a sphere <math>~(e=0)</math> or in the limit of an infinitesimally thin disk <math>~(e=1)</math>:		<span id="LimitingValues">Note that these three expressions</span> have the following values in the limit of a sphere <math>~(e=0)</math> or in the limit of an infinitesimally thin disk <math>~(e=1)</math>:


	<table align="center" border=1 cellpadding="8">		<table align="center" border=1 cellpadding="8">
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	</td>		</td>
	<td align="center">		<td align="center">
	<b><math>\frac{c}{a} \rightarrow 0</math></b>		<b><math>\frac{a_3}{a_1} \rightarrow 0</math></b>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~\frac{\pi}{2} - \biggl(\frac{c}{a}\biggr) +\frac{\pi}{4}\biggl(\frac{c}{a}\biggr)^2		<math>~\frac{\pi}{2} - \biggl(\frac{a_3}{a_1}\biggr) +\frac{\pi}{4}\biggl(\frac{a_3}{a_1}\biggr)^2
	- \mathcal{O}\biggl(\frac{~~c^3~~}{~~a^3~~}\biggr)</math>		- \mathcal{O}\biggl(\frac{a_3}{a_1}\biggr)</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>\frac{\pi}{2} \biggl( \frac{c}{a}\biggr) - 2\biggl(\frac{c}{a}\biggr)^2+ \mathcal{O}\biggl(\frac{~~c^3~~}{~~a^3~~}\biggr)</math>		<math>\frac{\pi}{2} \biggl( \frac{a_3}{a_1}\biggr) - 2\biggl(\frac{a_3}{a_1}\biggr)^2+ \mathcal{O}\biggl(\frac{a_3}{a_1}\biggr)</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math> 2 - \pi \biggl( \frac{c}{a}\biggr) + 4\biggl(\frac{c}{a}\biggr)^2		<math> 2 - \pi \biggl( \frac{a_3}{a_1}\biggr) + 4\biggl(\frac{a_3}{a_1}\biggr)^2
	- \mathcal{O}\biggl(\frac{~~c^3~~}{~~a^3~~}\biggr)</math>		- \mathcal{O}\biggl(\frac{a_3}{a_1}\biggr)</math>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Gravitational Potential */

2022-03-23T17:10:04Z

Gravitational Potential

← Older revision		Revision as of 13:10, 23 March 2022
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	<span id="LimitingValues">Note that these three expressions</span> have the following values in the limit of a sphere <math>~(e=0)</math> or in the limit of an infinitesimally thin disk <math>~(e=1)</math>:		<span id="LimitingValues">Note that these three expressions</span> have the following values in the limit of a sphere <math>~(e=0)</math> or in the limit of an infinitesimally thin disk <math>~(e=1)</math>:

	<table align="center" border=1 cellpadding="3">
			<table align="center" border=1 cellpadding="8">
	<tr>		<tr>
	<td colspan="3" align="center">		<td colspan="3" align="center">
	Limiting Values~~<br>~~		'''Table 2:'''  [[Appendix/Ramblings/PowerSeriesExpressions#Maclaurin_Spheroid_Index_Symbols\|Limiting Values]]
	~~(for oblate spheroids)~~
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="center">		<td align="center">
	<b><math>~e=0</math></b>		<b><math>e \rightarrow 0</math></b>
	</td>		</td>
	<td align="center">		<td align="center">
	<b><math>~~~e=1~~</math></b>		<b><math>\frac{c}{a} \rightarrow 0</math></b>
	</td>		</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">		<td align="center">
	<b><math>~~~A_1~~</math></b>		<b><math>~\frac{\sin^{-1}e}{e}</math></b>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>\frac{2}{3}</math>		<math>~1 + \frac{e^2}{6} + \mathcal{O}\biggl(e^4\biggr)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~0</math>		<math>~\frac{\pi}{2} - \biggl(\frac{c}{a}\biggr) +\frac{\pi}{4}\biggl(\frac{c}{a}\biggr)^2
			- \mathcal{O}\biggl(\frac{c^3}{a^3}\biggr)</math>
	</td>		</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">		<td align="center">
	<b><math>~~~A_3~~</math></b>		<b><math>A_1</math></b>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>\frac{2}{3}</math>		<math>\frac{2}{3}\biggl[1 - \frac{e^2}{5} - \mathcal{O}\biggl(e^4\biggr)\biggr]</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~2</math>		<math>\frac{\pi}{2} \biggl( \frac{c}{a}\biggr) - 2\biggl(\frac{c}{a}\biggr)^2+ \mathcal{O}\biggl(\frac{c^3}{a^3}\biggr)</math>
	</td>		</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">		<td align="center">
	<b><math>~~~I_\mathrm{BT}~~</math></b>		<b><math>A_3</math></b>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~2</math>		<math>\frac{2}{3}\biggl[1 + \frac{2e^2}{5} + \mathcal{O}\biggl(e^4\biggr)\biggr]</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~0</math>		<math> 2 - \pi \biggl( \frac{c}{a}\biggr) + 4\biggl(\frac{c}{a}\biggr)^2
			- \mathcal{O}\biggl(\frac{c^3}{a^3}\biggr)</math>
			</td>
			</tr>
			<tr>
			<td align="center">
			<b><math>I_\mathrm{BT}</math></b>
			</td>
			<td align="center">
			<math>2</math>
			</td>
			<td align="center">
			<math>0</math>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Gravitational Potential */

2022-03-19T17:06:26Z

Gravitational Potential

← Older revision		Revision as of 13:06, 19 March 2022
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	</table>		</table>

	Note that these three expressions have the following values in the limit of a sphere <math>~(e=0)</math> or in the limit of an infinitesimally thin disk <math>~(e=1)</math>:		<span id="LimitingValues">Note that these three expressions</span> have the following values in the limit of a sphere <math>~(e=0)</math> or in the limit of an infinitesimally thin disk <math>~(e=1)</math>:

	<table align="center" border=1 cellpadding="3">		<table align="center" border=1 cellpadding="3">

Jet53man: /* Maclaurin Spheroids */

2021-11-14T18:06:45Z

Maclaurin Spheroids

← Older revision		Revision as of 14:06, 14 November 2021
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	{\| class="MaclaurinSpheroids" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| class="MaclaurinSpheroids" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
	\|-		\|-
	! style="height: ~~150px~~; width: ~~150px~~; background-color:#ffff99;" \|[[H_BookTiledMenu#Spheroidal_.26_Spheroidal-Like\|<b>Uniform-Density<br />Maclaurin<br />Spheroids</b>]]		! style="height: 125px; width: 125px; background-color:#ffff99;" \|[[H_BookTiledMenu#Spheroidal_.26_Spheroidal-Like\|<b>Uniform-Density<br />Maclaurin<br />Spheroids</b>]]
	\|}		\|}

Jet53man at 23:52, 12 September 2021

2021-09-12T23:52:03Z

← Older revision		Revision as of 19:52, 12 September 2021
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	{\| class="MaclaurinSpheroids" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| class="MaclaurinSpheroids" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
	\|-		\|-
	! style="height: 150px; width: 150px; background-color:#ffff99;" \|[[H_BookTiledMenu#~~Two~~-~~Dimensional_Configurations_.28Axisymmetric.29~~\|<b>Uniform-Density<br />Maclaurin<br />Spheroids</b>]]		! style="height: 150px; width: 150px; background-color:#ffff99;" \|[[H_BookTiledMenu#Spheroidal_.26_Spheroidal-Like\|<b>Uniform-Density<br />Maclaurin<br />Spheroids</b>]]
	\|}		\|}

	There is no particular reason why one should guess ahead of time that the equilibrium properties of ''any'' rotating, self-gravitating configuration should be describable in terms of analytic functions. As luck would have it, however, the gravitational potential at the surface of and inside an homogeneous spheroid is expressible analytically. (The potential is constant on concentric spheroidal surfaces that generally have a different axis ratio from the spheroidal mass distribution.) Furthermore, the gradient of the gravitational potential is separable in cylindrical coordinates, proving to be a simple ''linear'' function of both <math>\varpi</math> and <math>~z</math>.		There is no particular reason why one should guess ahead of time that the equilibrium properties of ''any'' rotating, self-gravitating configuration should be describable in terms of analytic functions. As luck would have it, however, the gravitational potential at the surface of and inside an homogeneous spheroid is expressible analytically. (The potential is constant on concentric spheroidal surfaces that generally have a different axis ratio from the spheroidal mass distribution.) Furthermore, the gradient of the gravitational potential is separable in cylindrical coordinates, proving to be a simple ''linear'' function of both <math>\varpi</math> and <math>~z</math>.

	If the spheroid is uniformly rotating, this behavior conspires nicely with the behavior of the [[~~User:Tohline/~~PGE/RotatingFrame#Centrifugal_and_Coriolis_Accelerations\|centrifugal acceleration]] — which also will be a linear function of <math>\varpi</math> — to permit an analytic (and integrable) prescription of the pressure gradient. Not surprisingly, it resembles the functional form of the pressure gradient that is required to balance the gravitational force in [[~~User:Tohline/~~SSC/Structure/UniformDensity\|uniform-density spheres]].		If the spheroid is uniformly rotating, this behavior conspires nicely with the behavior of the [[PGE/RotatingFrame#Centrifugal_and_Coriolis_Accelerations\|centrifugal acceleration]] — which also will be a linear function of <math>\varpi</math> — to permit an analytic (and integrable) prescription of the pressure gradient. Not surprisingly, it resembles the functional form of the pressure gradient that is required to balance the gravitational force in [[SSC/Structure/UniformDensity\|uniform-density spheres]].

	As a consequence of this good fortune, the equilibrium structure of a uniformly rotating, uniform-density <math>~(n = 0)</math>, axisymmetric configuration can be shown to be precisely an oblate spheroid whose internal properties are describable in terms of analytic expressions. As we show [[~~User:Tohline/~~Apps/MaclaurinSpheroids/GoogleBooks#Interpreting_Maclaurin.27s_Key_Concluding_Theorem\|in an accompanying discussion]], the angular velocity that is required to keep a self-gravitating spheroid of a specified eccentricity in equilibrium can be obtained from a theorem — we will refer to it as [[~~User:Tohline/~~Apps/MaclaurinSpheroids/GoogleBooks#MaclaurinTheorem\|Maclaurin's Theorem]] — that was derived from purely geometric arguments over 270 years ago by Colin Maclaurin (1742) in ''A Treatise of Fluxions.'' This result has been enumerated in many subsequent publications (e.g., Tassoul 1978; Chandrasekhar 1987). It should be appreciated that both [https://books.google.com/books?id=dCAOAAAAQAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Volume I] and [https://books.google.com/books?id=xfQ7AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Volume II] of Maclaurin's (1742) ''Treatise'' can now be accessed online via Google Books. Selected excerpts from these two volumes are shown [[~~User:Tohline/~~Apps/MaclaurinSpheroids/GoogleBooks#Excerpts_from_A_Treatise_of_Fluxions\|in our accompanying discussion]].		As a consequence of this good fortune, the equilibrium structure of a uniformly rotating, uniform-density <math>~(n = 0)</math>, axisymmetric configuration can be shown to be precisely an oblate spheroid whose internal properties are describable in terms of analytic expressions. As we show [[Apps/MaclaurinSpheroids/GoogleBooks#Interpreting_Maclaurin.27s_Key_Concluding_Theorem\|in an accompanying discussion]], the angular velocity that is required to keep a self-gravitating spheroid of a specified eccentricity in equilibrium can be obtained from a theorem — we will refer to it as [[Apps/MaclaurinSpheroids/GoogleBooks#MaclaurinTheorem\|Maclaurin's Theorem]] — that was derived from purely geometric arguments over 270 years ago by Colin Maclaurin (1742) in ''A Treatise of Fluxions.'' This result has been enumerated in many subsequent publications (e.g., Tassoul 1978; Chandrasekhar 1987). It should be appreciated that both [https://books.google.com/books?id=dCAOAAAAQAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Volume I] and [https://books.google.com/books?id=xfQ7AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Volume II] of Maclaurin's (1742) ''Treatise'' can now be accessed online via Google Books. Selected excerpts from these two volumes are shown [[Apps/MaclaurinSpheroids/GoogleBooks#Excerpts_from_A_Treatise_of_Fluxions\|in our accompanying discussion]].

	==Properties of Uniform-Density Spheroids==		==Properties of Uniform-Density Spheroids==
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	===Gravitational Potential===		===Gravitational Potential===
	In an accompanying discussion entitled, ''[[~~User:Tohline/~~ThreeDimensionalConfigurations/HomogeneousEllipsoids#Properties_of_Homogeneous_Ellipsoids\|Properties of Homogeneous Ellipsoids]]'', an expression is given for the gravitational potential <math>\Phi(\vec{x})</math> at an internal point or on the surface of an homogeneous ellipsoid with semi-axes <math>~(x,y,z) = (a_1,a_2,a_3)</math>. For an [[~~User:Tohline/~~ThreeDimensionalConfigurations/HomogeneousEllipsoids#oblate\|homogeneous, oblate spheroid]] in which <math>~a_1 = a_2 \geq a_3</math>, this analytic expression defining the potential reduces to the form,		In an accompanying discussion entitled, ''[[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Properties_of_Homogeneous_Ellipsoids\|Properties of Homogeneous Ellipsoids]]'', an expression is given for the gravitational potential <math>\Phi(\vec{x})</math> at an internal point or on the surface of an homogeneous ellipsoid with semi-axes <math>~(x,y,z) = (a_1,a_2,a_3)</math>. For an [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#oblate\|homogeneous, oblate spheroid]] in which <math>~a_1 = a_2 \geq a_3</math>, this analytic expression defining the potential reduces to the form,

	<div align="center">		<div align="center">
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	</math>		</math>
	</p>		</p>
	[<b>[[~~User:Tohline/~~Appendix/References#ST83\|<font color="red">ST83</font>]]</b>], §7.3, p. 169, Eq. (7.3.1)		[<b>[[Appendix/References#ST83\|<font color="red">ST83</font>]]</b>], §7.3, p. 169, Eq. (7.3.1)
	</div>		</div>

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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
	[<b>[[~~User:Tohline/~~Appendix/References#ST83\|<font color="red">ST83</font>]]</b>], §7.3, p. 170, Eqs. (7.3.8)		[<b>[[Appendix/References#ST83\|<font color="red">ST83</font>]]</b>], §7.3, p. 170, Eqs. (7.3.8)
	</td>		</td>
	</tr>		</tr>
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	====Example Equi-gravitational-potential Contours====		====Example Equi-gravitational-potential Contours====
	As an example, let's examine the gravitational potential everywhere inside (and on the surface) of the oblate spheroid whose properties are presented in the first row of model data in [[~~User:Tohline/~~ThreeDimensionalConfigurations/HomogeneousEllipsoids#Table1\|Table 1 of our accompanying discussion of the properties of homogeneous ellipsoids]]. That is, let's examine a model with <math>~a_1 = 1.0</math> and …		As an example, let's examine the gravitational potential everywhere inside (and on the surface) of the oblate spheroid whose properties are presented in the first row of model data in [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Table1\|Table 1 of our accompanying discussion of the properties of homogeneous ellipsoids]]. That is, let's examine a model with <math>~a_1 = 1.0</math> and …
	<table border="0" align="center" width="80%">		<table border="0" align="center" width="80%">
	<tr>		<tr>
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	==Equilibrium Structure==		==Equilibrium Structure==

	===Governing Relations===		===Governing Relations===
	To obtain the equilibrium structure of Maclaurin spheroids, we will adopt the [[~~User:Tohline/~~AxisymmetricConfigurations/SolutionStrategies#Technique\|technique outlined earlier for determining the structure of axisymmetric configurations]]. Specifically, the algebraic expression,		To obtain the equilibrium structure of Maclaurin spheroids, we will adopt the [[AxisymmetricConfigurations/SolutionStrategies#Technique\|technique outlined earlier for determining the structure of axisymmetric configurations]]. Specifically, the algebraic expression,
	<div align="center">		<div align="center">
	<math>~H + \Phi_\mathrm{eff} = C_\mathrm{B}</math> ,		<math>~H + \Phi_\mathrm{eff} = C_\mathrm{B}</math> ,
	</div>		</div>
	must be solved in conjunction with the [[~~User:Tohline/~~AxisymmetricConfigurations/PGE#Governing_Equations\|Poisson equation written in cylindrical coordinates for axisymmetric configurations]], namely,		must be solved in conjunction with the [[AxisymmetricConfigurations/PGE#Governing_Equations\|Poisson equation written in cylindrical coordinates for axisymmetric configurations]], namely,
	<div align="center">		<div align="center">
	<math>		<math>
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	===Expression for Effective Potential===		===Expression for Effective Potential===
	For any value of the eccentricity, <math>~e</math>, the [[~~User:Tohline/~~Apps/MaclaurinSpheroids#Gravitational_Potential\|above expression for the gravitational potential]] satisfies this two-dimensional Poisson equation. Furthermore, an algebraic expression defining the centrifugal potential inside a uniformly rotating configuration can be drawn from our accompanying table that summarizes the properties of various [[~~User:Tohline/~~AxisymmetricConfigurations/SolutionStrategies#SRPtable\|simple rotation profiles]]. Together, these relations give us the relevant expression for the effective potential, namely,		For any value of the eccentricity, <math>~e</math>, the [[Apps/MaclaurinSpheroids#Gravitational_Potential\|above expression for the gravitational potential]] satisfies this two-dimensional Poisson equation. Furthermore, an algebraic expression defining the centrifugal potential inside a uniformly rotating configuration can be drawn from our accompanying table that summarizes the properties of various [[AxisymmetricConfigurations/SolutionStrategies#SRPtable\|simple rotation profiles]]. Together, these relations give us the relevant expression for the effective potential, namely,
	<div align="center">		<div align="center">
	<math>		<math>
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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
	[<b>[[~~User:Tohline/~~Appendix/References#T78\|<font color="red">T78</font>]]</b>], §4.5, p. 86, Eq. (52)<br />		[<b>[[Appendix/References#T78\|<font color="red">T78</font>]]</b>], §4.5, p. 86, Eq. (52)<br />
	[<b>[[~~User:Tohline/~~Appendix/References#ST83\|<font color="red">ST83</font>]]</b>], §7.3, p. 172, Eq. (7.3.18)		[<b>[[Appendix/References#ST83\|<font color="red">ST83</font>]]</b>], §7.3, p. 172, Eq. (7.3.18)
	</td>		</td>
	</tr>		</tr>
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	</table>		</table>

	[[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 [[~~User:Tohline/~~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,		[[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">		<table align="center" border="0" cellpadding="5">
	<tr>		<tr>
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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
	[<b>[[~~User:Tohline/~~Appendix/References#T78\|<font color="red">T78</font>]]</b>], §4.5, p. 86, Eq. (51)<br />		[<b>[[Appendix/References#T78\|<font color="red">T78</font>]]</b>], §4.5, p. 86, Eq. (51)<br />
	[<b>[[~~User:Tohline/~~Appendix/References#ST83\|<font color="red">ST83</font>]]</b>], §7.3, p. 172, Eqs. (7.3.16) & (7.3.17)		[<b>[[Appendix/References#ST83\|<font color="red">ST83</font>]]</b>], §7.3, p. 172, Eqs. (7.3.16) & (7.3.17)
	</td>		</td>
	</tr>		</tr>

Jet53man at 23:46, 12 September 2021

2021-09-12T23:46:19Z

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2021-09-12T23:44:00Z

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=Maclaurin Spheroids=
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! style="height: 150px; width: 150px; background-color:#ffff99;" |[[H_BookTiledMenu#Two-Dimensional_Configurations_.28Axisymmetric.29|Uniform-Density Maclaurin Spheroids]]
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When constructing rotating equilibrium configurations that obey a [[SR#Time-Independent_Problems|barotropic equation of state]], keep in mind that certain physical variable profiles should be avoided because they will lead to structures that are unstable toward the dynamical development of shape-distorting or ''convective''-type motions. Here are a few well-known examples.
  
  
  
  
  
  

=See Also=

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Apps/MaclaurinSpheroids - Revision history

Jet53man: /* Gravitational Potential */

Jet53man: /* Gravitational Potential */

Jet53man: /* Gravitational Potential */

Jet53man: /* Gravitational Potential */

Jet53man: /* Gravitational Potential */

Jet53man: /* Maclaurin Spheroids */

Jet53man at 23:52, 12 September 2021

Jet53man at 23:46, 12 September 2021

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