User:Tohline/Apps/MaclaurinSpheroids - Revision history

Tohline: /* Gravitational Potential */

2020-08-24T21:19:34Z

Gravitational Potential

← Older revision		Revision as of 21:19, 24 August 2020
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	</table>		</table>

			====Example Equi-gravitational-potential Contours====
	~~<table border~~=~~"1" cellpadding~~=~~"10" align~~=~~"center" width~~=~~"80%"><tr><td align~~=~~"left">~~
	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 [[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 …
	<table border="0" align="center" width="80%">		<table border="0" align="center" width="80%">
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	</table>		</table>

	In the meridional <math>~(\varpi, z)</math> plane, the surface of this oblate-spheroidal configuration — identified ~~here, in Figure 1,~~ by the thick black ~~solid~~ curve — is defined by the expression,		In the meridional <math>~(\varpi, z)</math> plane, the surface of this oblate-spheroidal configuration — identified by the thick, solid-black curve below, in Figure 1 — is defined by the expression,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	Therefore, for the example model parameters specified above, our selection of equipotential surfaces to plot should be guided by the following constraints.		Therefore, for the example model parameters specified above, our selection of equipotential surfaces to plot should be guided by the following constraints.
	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8" width="80%">
	<tr>		<tr>
	<td align="center" colspan="2">Equipotential Contour Lies …</td>		<td align="center" colspan="2">Equipotential Contour Lies …</td>
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	</tr>		</tr>
	<tr><td align="left" colspan="2">		<tr><td align="left" colspan="2">
	<div align="center">'''Meridional Plane Cross-section'''<br />		<div align="center">'''Figure 1:   Meridional Plane Cross-section'''<br />
	[[File:MacAtJacBifurcationJustGravity01.png\|550px\|center\|Maclaurin Spheroid Cross-section at Jacobi Bifurcation]]</div>		[[File:MacAtJacBifurcationJustGravity01.png\|550px\|center\|Maclaurin Spheroid Cross-section at Jacobi Bifurcation]]</div>
	''Solid black curve'': Surface of oblate spheroid having a<sub>3</sub>/a<sub>1</sub> = 0.582724. ''Dashed curves'': Equi-gravitational-potential contours plotted in increments of <math>~\Delta\phi_\mathrm{choice} = 0.075</math>; specifically, <math>~\phi_\mathrm{choice}</math> = 0.029 (black), 0.104 (dark blue), 0.179 (red), 0.254 (light blue), 0.329 (green), 0.404 (purple), and 0.479 (orange).		''Solid black curve'': Surface of oblate spheroid having a<sub>3</sub>/a<sub>1</sub> = 0.582724. ''Dashed curves'': Equi-gravitational-potential contours plotted in increments of <math>~\Delta\phi_\mathrm{choice} = 0.075</math>; specifically, <math>~\phi_\mathrm{choice}</math> = 0.029 (black), 0.104 (dark blue), 0.179 (red), 0.254 (light blue), 0.329 (green), 0.404 (purple), and 0.479 (orange).
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	</li>		</li>
	</ol>		</ol>
	~~</td></tr></table>~~

	==Equilibrium Structure==		==Equilibrium Structure==

Tohline: /* Gravitational Potential */

2020-08-24T21:06:51Z

Gravitational Potential

← Older revision		Revision as of 21:06, 24 August 2020
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	<td align="center" colspan="1"><math>~\biggl[ \frac{\phi_\mathrm{choice} - 0.32878}{ 0.18711 } \biggr]^{1 / 2} \le \varpi \le (\phi_\mathrm{choice}/0.51589)^{1 / 2}</math></td>		<td align="center" colspan="1"><math>~\biggl[ \frac{\phi_\mathrm{choice} - 0.32878}{ 0.18711 } \biggr]^{1 / 2} \le \varpi \le (\phi_\mathrm{choice}/0.51589)^{1 / 2}</math></td>
	</tr>		</tr>
	<tr><td align="~~center~~" colspan="2">		<tr><td align="left" colspan="2">
	'''Meridional Plane Cross-section'''<br />		<div align="center">'''Meridional Plane Cross-section'''<br />
	[[File:MacAtJacBifurcationJustGravity01.png\|~~450px~~\|center\|Maclaurin Spheroid Cross-section at Jacobi Bifurcation]]		[[File:MacAtJacBifurcationJustGravity01.png\|550px\|center\|Maclaurin Spheroid Cross-section at Jacobi Bifurcation]]</div>
	</~~td></tr>~~		''Solid black curve'': Surface of oblate spheroid having a<sub>3</sub>/a<sub>1</sub> = 0.582724. ''Dashed curves'': Equi-gravitational-potential contours plotted in increments of <math>~\Delta\phi_\mathrm{choice} = 0.075</math>; specifically, <math>~\phi_\mathrm{choice}</math> = 0.029 (black), 0.104 (dark blue), 0.179 (red), 0.254 (light blue), 0.329 (green), 0.404 (purple), and 0.479 (orange).
	~~<tr>~~
	~~<td align="left" colspan="2"~~>
	''Solid black curve'': Surface of oblate spheroid having a<sub>3</sub>/a<sub>1</sub> = 0.582724. ''Dashed curves'': Equi-gravitational-potential contours plotted in increments of <math>~\Delta\phi_\mathrm{choice} = 0.075</math>; specifically, <math>~\phi_\mathrm{choice}</math> = 0.029 (black), 0.104 (dark blue), 0.179 (red), 0.254 (light blue), 0.329 (green), 0.4 (purple), and 0.479 (orange).
	</td>		</td>
	</tr>		</tr>

Tohline: /* Gravitational Potential */

2020-08-24T21:00:55Z

Gravitational Potential

← Older revision		Revision as of 21:00, 24 August 2020
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	<td align="center" colspan="1"><math>~0 \le \varpi \le (\phi_\mathrm{choice}/0.51589)^{1 / 2}</math></td>		<td align="center" colspan="1"><math>~0 \le \varpi \le (\phi_\mathrm{choice}/0.51589)^{1 / 2}</math></td>
	<td align="center" colspan="1"><math>~\biggl[ \frac{\phi_\mathrm{choice} - 0.32878}{ 0.18711 } \biggr]^{1 / 2} \le \varpi \le (\phi_\mathrm{choice}/0.51589)^{1 / 2}</math></td>		<td align="center" colspan="1"><math>~\biggl[ \frac{\phi_\mathrm{choice} - 0.32878}{ 0.18711 } \biggr]^{1 / 2} \le \varpi \le (\phi_\mathrm{choice}/0.51589)^{1 / 2}</math></td>
			</tr>
			<tr><td align="center" colspan="2">
			'''Meridional Plane Cross-section'''<br />
			[[File:MacAtJacBifurcationJustGravity01.png\|450px\|center\|Maclaurin Spheroid Cross-section at Jacobi Bifurcation]]
			</td></tr>
			<tr>
			<td align="left" colspan="2">
			''Solid black curve'': Surface of oblate spheroid having a<sub>3</sub>/a<sub>1</sub> = 0.582724. ''Dashed curves'': Equi-gravitational-potential contours plotted in increments of <math>~\Delta\phi_\mathrm{choice} = 0.075</math>; specifically, <math>~\phi_\mathrm{choice}</math> = 0.029 (black), 0.104 (dark blue), 0.179 (red), 0.254 (light blue), 0.329 (green), 0.4 (purple), and 0.479 (orange).
			</td>
	</tr>		</tr>
	</table>		</table>

Tohline: /* Gravitational Potential */

2020-08-24T17:15:18Z

Gravitational Potential

← Older revision		Revision as of 17:15, 24 August 2020
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	<math>~\biggl[ \frac{\phi_\mathrm{choice} - a_3^2 A_3}{ A_1 - a_3^2 A_3 } \biggr]^{1 / 2} \, .</math>		<math>~\biggl[ \frac{\phi_\mathrm{choice} - a_3^2 A_3}{ A_1 - a_3^2 A_3 } \biggr]^{1 / 2} \, .</math>
	</td>		</td>
			</tr>
			</table>

			Therefore, for the example model parameters specified above, our selection of equipotential surfaces to plot should be guided by the following constraints.
			<table border="1" align="center" cellpadding="8">
			<tr>
			<td align="center" colspan="2">Equipotential Contour Lies …</td>
			</tr>
			<tr>
			<td align="center" colspan="1" width="50%">Entirely Inside Spheroid's Surface</td>
			<td align="center" colspan="1">Partially Outside Spheroid's Surface</td>
			</tr>
			<tr>
			<td align="center" colspan="1"><math>~0 < \phi_\mathrm{choice} < 0.32878</math></td>
			<td align="center" colspan="1"><math>~0.32878 < \phi_\mathrm{choice} < 0.51589</math></td>
			</tr>
			<tr>
			<td align="center" colspan="1"><math>~0 \le \varpi \le (\phi_\mathrm{choice}/0.51589)^{1 / 2}</math></td>
			<td align="center" colspan="1"><math>~\biggl[ \frac{\phi_\mathrm{choice} - 0.32878}{ 0.18711 } \biggr]^{1 / 2} \le \varpi \le (\phi_\mathrm{choice}/0.51589)^{1 / 2}</math></td>
	</tr>		</tr>
	</table>		</table>

Tohline: /* Gravitational Potential */

2020-08-24T01:03:42Z

Gravitational Potential

← Older revision		Revision as of 01:03, 24 August 2020
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow~~~\varpi^2 </math>		<math>~\Rightarrow~~~\varpi </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{\phi_\mathrm{choice} - a_3^2 A_3}{ A_1 - a_3^2 A_3 } \, .</math>		<math>~\biggl[ \frac{\phi_\mathrm{choice} - a_3^2 A_3}{ A_1 - a_3^2 A_3 } \biggr]^{1 / 2} \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Gravitational Potential */

2020-08-24T01:02:35Z

Gravitational Potential

← Older revision		Revision as of 01:02, 24 August 2020
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	<td align="center"><math>~\frac{a_3}{a_1} = 0.582724 \, ,</math></td>		<td align="center"><math>~\frac{a_3}{a_1} = 0.582724 \, ,</math></td>
	<td align="center"><math>~e = 0.81267 \, ,</math></td>		<td align="center"><math>~e = 0.81267 \, ,</math></td>
			<td align="center"> </td>
			</tr>
			<tr>
	<td align="center"><math>~A_1 = A_2 = 0.51589042 \, ,</math></td>		<td align="center"><math>~A_1 = A_2 = 0.51589042 \, ,</math></td>
	<td align="center"><math>~A_3 = 0.96821916 \, ,</math></td>		<td align="center"><math>~A_3 = 0.96821916 \, ,</math></td>
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	</tr>		</tr>
	</table>		</table>
	Throughout the interior of this configuration, each associated <math>~\Phi_\mathrm{~~grav~~}</math> = constant, equipotential surface is defined by the expression,		Throughout the interior of this configuration, each associated <math>~\Phi_\mathrm{eff}</math> = constant, equipotential surface is defined by the expression,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\phi_\mathrm{choice} \equiv \frac{\Phi_\mathrm{~~grav~~}}{\pi G \rho} + I_\mathrm{BT}a_1^2 </math>		<math>~\phi_\mathrm{choice} \equiv \frac{\Phi_\mathrm{eff}}{\pi G \rho} + I_\mathrm{BT}a_1^2 </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</table>		</table>

			----

			'''No Rotation'''

			When we do not consider the effects of rotation and plot, instead, just the equi-gravitational-potential surfaces, then
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~z </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\pm \frac{1}{\sqrt{A_3}} \biggl[ \phi_\mathrm{choice} - A_1 \varpi^2\biggr]^{1 / 2} \, . </math>
			</td>
			</tr>
			</table>

			Because we know that the <math>~\Phi_\mathrm{grav}</math> = constant surfaces are all less flattened than the configuration itself, we should expect that the largest value of the potential that will arise inside — actually, on the surface of — the flattened spheroidal configuration will be found at <math>~(\varpi, z) = (1, 0)</math>, that is, when,
			<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>~\pm \frac{1}{\sqrt{A_3}} \biggl[ \phi_\mathrm{choice} - A_1 \biggr]^{1 / 2} \, . </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~\phi_\mathrm{choice}\biggr\|_\mathrm{max}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ A_1 \, . </math>
			</td>
			</tr>
			</table>
			So we will plot various equipotential surfaces having, <math>~0 < \phi_\mathrm{choice} < A_1 \, ,</math> recognizing that they will each cut through the equatorial plane <math>~(z = 0)</math> at the radial coordinate given by,
			<div align="center">
			<math>~\varpi = \sqrt{\phi_\mathrm{choice}/A_1} \, .</math>
			</div>
			Next, we recognize that the largest equipotential surface that fits entirely within the surface of the oblate spheroidal configuration has the value of the potential that is found on the symmetry axis and at the pole of the spheroid, that is, at <math>~(\varpi, z) = (0, a_3) \, .</math> For this case we find,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\phi_\mathrm{choice}\biggr\|_\mathrm{mid}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~a_3^2 A_3 \, .</math>
			</td>
			</tr>
			</table>
			Hence, all equipotential surfaces having <math>~0 < \phi_\mathrm{choice} \le a_3^2 A_3</math> will lie entirely within the spheroid. But equipotential surfaces having <math>~a_3^2 A_3 < \phi_\mathrm{choice} \le A_1</math> will cut through the surface of the spheroid at the value of <math>~\varpi</math> where "the two values of z<sup>2</sup> match," that is, where,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~a_3^2(1-\varpi^2)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{A_3} \biggl[ \phi_\mathrm{choice} - A_1 \varpi^2\biggr] </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~\varpi^2 </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{\phi_\mathrm{choice} - a_3^2 A_3}{ A_1 - a_3^2 A_3 } \, .</math>
			</td>
			</tr>
			</table>


			----

			'''With Rotation'''
	This expression is only applicable to our physical problem under the following conditions:		This expression is only applicable to our physical problem under the following conditions:
	<ol>		<ol>

Tohline: /* Gravitational Potential */

2020-08-23T21:52:12Z

Gravitational Potential

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	<ol>		<ol>
	<li>		<li>
	The argument of the square root must not be negative, that is,<math>~\varpi</math> must be confined to the range,		The argument of the square root must not be negative, that is, <math>~\varpi</math> must be confined to the range,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</tr>		</tr>
	</table>		</table>
			Note that, in turn, in order to ensure that the argument of ''this'' square root is not negative, we should only explore rotation rates for which <math>~\omega_0^2/(2\pi G \rho) \le A_1 \, .</math>
			</li>
			<li>
			In order that our equipotential surface be relevant only to the ''interior'' of our configuration, for every allowed value of <math>~\varpi ,</math> the value of <math>~z</math> corresponding to the potential surface must be less than or equal to the value of <math>~z</math> at the surface of the configuration. That is,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~a_3^2(1-\varpi^2) </math>
			</td>
			<td align="center">
			<math>~\le</math>
			</td>
			<td align="left">
			<math>~\frac{1}{A_3} \biggl[ \phi_\mathrm{choice} - \biggl( A_1 - \frac{\omega_0^2}{2\pi G \rho}\biggr) \varpi^2\biggr]</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~\biggl( A_1 - \frac{\omega_0^2}{2\pi G \rho} - a_3^2 A_3^2 \biggr) \varpi^2 </math>
			</td>
			<td align="center">
			<math>~\le</math>
			</td>
			<td align="left">
			<math>~\phi_\mathrm{choice} - a_3^2 A_3^2 </math>
			</td>
			</tr>
			</table>
	</li>		</li>
	</ol>		</ol>

Tohline: /* Gravitational Potential */

2020-08-23T21:19:02Z

Gravitational Potential

← Older revision		Revision as of 21:19, 23 August 2020
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	</table>		</table>


			<table border="1" cellpadding="10" align="center" width="80%"><tr><td align="left">
	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 [[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 …
	<table border="0" align="center" width="80%">		<table border="0" align="center" width="80%">
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	<math>~\pm a_3 [1 - \varpi^2]^{1 / 2} \, ,</math>		<math>~\pm a_3 [1 - \varpi^2]^{1 / 2} \, ,</math>
	</td>		</td>
	<td align="right">        for <math>~0 \le \varpi \le 1 \, .</math></td>		<td align="right">        for <math>~0 \le \| \varpi \| \le 1 \, .</math></td>
	</tr>		</tr>
	</table>		</table>
	~~Each~~ associated <math>~\Phi_\mathrm{grav}</math> = constant, equipotential surface is defined by the expression,		Throughout the interior of this configuration, each associated <math>~\Phi_\mathrm{grav}</math> = constant, equipotential surface is defined by the expression,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<td align="right">		<td align="right">
	<math>~\phi_\mathrm{choice} \equiv \frac{\Phi_\mathrm{grav}}{\pi G \rho} + I_\mathrm{BT}a_1^2 </math>		<math>~\phi_\mathrm{choice} \equiv \frac{\Phi_\mathrm{grav}}{\pi G \rho} + I_\mathrm{BT}a_1^2 </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left" colspan="1">
			<math>~\biggl( A_1 - \frac{\omega_0^2}{2\pi G \rho}\biggr) \varpi^2 + A_3 z^2 </math>
			</td>
			</tr>
			</table>

			(Notice that, written in this manner, <math>~\phi_\mathrm{choice}</math> assumes its minimum value (zero) when <math>~(\varpi, z) = (0, 0)</math>, that is, at the center of the configuration.) This means that,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~z </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl[ A_1 - \frac{\omega_0^2}{2\pi G \rho}\biggr] \varpi^2 ~~+ A_3 z~~^2 </math>		<math>~\pm \frac{1}{\sqrt{A_3}} \biggl[ \phi_\mathrm{choice} - \biggl( A_1 - \frac{\omega_0^2}{2\pi G \rho}\biggr) \varpi^2\biggr]^{1 / 2} \, . </math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

			This expression is only applicable to our physical problem under the following conditions:
			<ol>
			<li>
			The argument of the square root must not be negative, that is,<math>~\varpi</math> must be confined to the range,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~0 \le \| \varpi \|</math>
			</td>
			<td align="center">
			<math>~\le</math>
			</td>
			<td align="left">
			<math>~\biggl[ \phi_\mathrm{choice}\biggl( A_1 - \frac{\omega_0^2}{2\pi G \rho}\biggr)^{-1 }\biggr]^{1 / 2} \, .</math>
			</td>
			</tr>
			</table>

			</li>
			</ol>
			</td></tr></table>

	==Equilibrium Structure==		==Equilibrium Structure==

Tohline: /* Gravitational Potential */

2020-08-23T20:25:16Z

Gravitational Potential

← Older revision		Revision as of 20:25, 23 August 2020
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	</tr>		</tr>
	</table>		</table>
	~~The corresponding~~ equipotential ~~surfaces — depicted in Figure 1 by the set of dashed meridional-plane curves — are~~ defined by the expression,		Each associated <math>~\Phi_\mathrm{grav}</math> = constant, equipotential surface is defined by the expression,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\phi_\mathrm{choice} \equiv \frac{\Phi_\mathrm{~~choice~~}}{\pi G \rho} </math>		<math>~\phi_\mathrm{choice} \equiv \frac{\Phi_\mathrm{grav}}{\pi G \rho} + I_\mathrm{BT}a_1^2 </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \biggl[ ~~I_\mathrm{BT} a_1^2 - \biggl(~~A_1 ~~\varpi^2 + A_3 z^2 \biggr) \biggr]~~ - ~~\frac{1}{2}\varpi^2 \biggl(~~ \frac{\omega_0^2}{\pi G \rho}\biggr)</math>		<math>~\biggl[ A_1 - \frac{\omega_0^2}{2\pi G \rho}\biggr] \varpi^2 + A_3 z^2 </math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>


	==Equilibrium Structure==		==Equilibrium Structure==

Tohline: /* Maclaurin Spheroids (axisymmetric structure) */

2020-08-23T19:52:22Z

Maclaurin Spheroids (axisymmetric structure)

← Older revision		Revision as of 19:52, 23 August 2020
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			In the meridional <math>~(\varpi, z)</math> plane, the surface of this oblate-spheroidal configuration — identified here, in Figure 1, by the thick black solid curve — is defined by the expression,
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			<math>~\frac{\varpi^2}{a_1^2} + \frac{z^2}{a_3^2}</math>
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			<math>~=</math>
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			<math>~1 </math>
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			<math>~\Rightarrow ~~~ z</math>
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			<math>~=</math>
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			<math>~\pm a_3 [1 - \varpi^2]^{1 / 2} \, ,</math>
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			<td align="right">        for <math>~0 \le \varpi \le 1 \, .</math></td>
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			The corresponding equipotential surfaces — depicted in Figure 1 by the set of dashed meridional-plane curves — are defined by the expression,

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			<math>~\phi_\mathrm{choice} \equiv \frac{\Phi_\mathrm{choice}}{\pi G \rho} </math>
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			<math>~=</math>
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			<math>~- \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 \varpi^2 + A_3 z^2 \biggr) \biggr] - \frac{1}{2}\varpi^2 \biggl( \frac{\omega_0^2}{\pi G \rho}\biggr)</math>
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	==Equilibrium Structure==		==Equilibrium Structure==
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	=Related Wikipedia Discussions=		=Related Wikipedia Discussions=