User:Tohline/ThreeDimensionalConfigurations/RiemannStype - Revision history

Tohline: /* Example Mach Surface */

2020-06-14T22:18:25Z

Example Mach Surface

← Older revision		Revision as of 22:18, 14 June 2020
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{ (\Omega_f^2+\lambda^2) }{2} \biggl[ x^2 + y^2\biggr]		\frac{ (\Omega_f^2+\lambda^2) }{2(\pi G \rho)} \biggl[ \biggl(\frac{x}{a}\biggr)^2 + \biggl( \frac{y}{a}\biggr)^2\biggr]
	- \Omega_f \lambda \biggl[ \biggl(\frac{b}{a}\biggr) x^2 + \biggl( \frac{a}{b} \biggr)y^2 \biggr]		- \frac{\Omega_f \lambda}{(\pi G \rho)} \biggl[ \biggl(\frac{b}{a}\biggr) \biggl( \frac{x}{a}\biggr)^2 + \biggl( \frac{a}{b} \biggr) \biggl( \frac{y}{a}\biggr)^2 \biggr]
	-~~\pi G \rho~~ \biggl[\biggl(~~A_1~~ x^2 + A_2 y^2 +A_3 z^2 ~~\biggr)~~ -~~A_3~~ c^2 \biggr]		- \biggl[A_1 \biggl( \frac{x}{a}\biggr)^2 + A_2 \biggl(\frac{y}{a}\biggr)^2 +A_3 \biggl( \frac{z^2 - c^2}{a^2}\biggr) \biggr] \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>






	Drawing from equation (7) of [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou (2006)] and from a [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Gravitational_Potential\|separate discussion of gravitational potential of homogeneous ellipsoids]], we see that the effective potential is,		Drawing from equation (7) of [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou (2006)] and from a [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Gravitational_Potential\|separate discussion of gravitational potential of homogeneous ellipsoids]], we see that the effective potential is,

Tohline: /* Example Mach Surface */

2020-06-14T22:11:51Z

Example Mach Surface

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	In an [[User:Tohline/Apps/MaclaurinSpheroids#Equilibrium_Structure\|accompanying discussion]] of axisymmetric configurations, we have recognized that, at any point inside the configuration, the square of the sound speed is given approximately by the enthalpy where,		In an [[User:Tohline/Apps/MaclaurinSpheroids#Equilibrium_Structure\|accompanying discussion]] of axisymmetric configurations, we have recognized that, at any point inside the configuration, the square of the sound speed is given approximately by the enthalpy where,
	<table border="0" ~~align~~="~~center~~" ~~cellpadding~~="8">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~c^2 \sim H(x, y, z)</math>		<math>~
			c^2 \sim H(x, y, z)
			</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>~=</math>
	</td>		</td>
			<td align="left">
			<math>~\frac{P(x, y, z)}{\rho} =
			C_B - \Phi_\mathrm{eff}(x, y, z)
			</math>
			</td>
			</tr>

			<tr>
	<td align="right">		<td align="right">
	~~<math>~\frac{P(x, y, z)}{\rho} </math>~~
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>~=</math>
	</td>		</td>
			<td align="left">
			<math>~
			C_B - \biggl[
			\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)
			\biggr]
			</math>
			</td>
			</tr>

			<tr>
	<td align="right">		<td align="right">
	<math>~C_B - \~~Phi_~~\mathrm{~~eff~~}(x, y, z) \~~, .~~</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			C_B +
			\pi G \rho \biggl[ I_\mathrm{BT} a^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
			+ \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>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-\pi G \rho \biggl[ I_\mathrm{BT} a^2 - A_3 c^2 \biggr] +
			\pi G \rho \biggl[ I_\mathrm{BT} a^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
			+ \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>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\pi G \rho \biggl[A_3 c^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
			+ \frac{ (\Omega_f^2+\lambda^2) }{2} \biggl[ x^2 + y^2\biggr]
			- \Omega_f \lambda \biggl[ \biggl(\frac{b}{a}\biggr) x^2 + \biggl( \frac{a}{b} \biggr)y^2 \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~ \frac{c^2}{a^2 (\pi G\rho)}</math>
			</td>
			<td align="center">
			<math>~\sim</math>
			</td>
			<td align="left">
			<math>~
			\frac{ (\Omega_f^2+\lambda^2) }{2} \biggl[ x^2 + y^2\biggr]
			- \Omega_f \lambda \biggl[ \biggl(\frac{b}{a}\biggr) x^2 + \biggl( \frac{a}{b} \biggr)y^2 \biggr]
			-\pi G \rho \biggl[\biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) -A_3 c^2 \biggr]
			</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>






	Drawing from equation (7) of [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou (2006)] and from a [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Gravitational_Potential\|separate discussion of gravitational potential of homogeneous ellipsoids]], we see that the effective potential is,		Drawing from equation (7) of [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou (2006)] and from a [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Gravitational_Potential\|separate discussion of gravitational potential of homogeneous ellipsoids]], we see that the effective potential is,

Tohline: /* Relate EFE to Ou(2006) */

2020-06-14T20:16:31Z

Relate EFE to Ou(2006)

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	===Relate EFE to Ou(2006)===		===Relate EFE to Ou(2006)===

	As we have already acknowledged, according to [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou (2006)], at any coordinate position inside or on the surface of the ellipsoid, <math>~(x, y)</math>, the three components of the velocity as viewed from a frame of rotation that is spinning at the equilibrium configuration's frequency, <math>~\~~Omega~~</math>, are,		As we have already acknowledged, according to [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou (2006)], at any coordinate position inside or on the surface of the ellipsoid, <math>~(x, y)</math>, the three components of the velocity as viewed from a frame of rotation that is spinning at the equilibrium configuration's frequency, <math>~\Omega_f</math>, are,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\vec{v}</math>		<math>~{\vec{v}}_\mathrm{rot}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</tr>		</tr>
	</table>		</table>
	which, gratifyingly agrees with Ou's equation (17).		which, gratifyingly agrees with Ou's equation (17). It is worth noting as well that, when viewed from the inertial reference frame, the velocity field is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\vec{v}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~{\vec{v}}_\mathrm{rot} + \vec\Omega_f \times \vec{x} \, .</math>
			</td>
			</tr>
			</table>

			[https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract Chandrasekhar (1965)], p. 892, §II, Eq. (13)
			</div>
			Broken down into its Cartesian components, this is
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\vec{v}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\boldsymbol{\hat\imath} \biggl[\lambda \biggl( \frac{a}{b} \biggr) - \Omega_f \biggr]y
			+
			\boldsymbol{\hat\jmath} \biggl[- \lambda \biggl( \frac{b}{a} \biggr) + \Omega_f \biggr]x
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\boldsymbol{\hat\imath} \biggl[- \biggl( \frac{a b}{a^2 + b^2} \biggr)\zeta \biggl( \frac{a}{b} \biggr) - \Omega_f \biggr]y
			+
			\boldsymbol{\hat\jmath} \biggl[\biggl( \frac{a b}{a^2 + b^2} \biggr)\zeta\biggl( \frac{b}{a} \biggr) + \Omega_f \biggr]x
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-\boldsymbol{\hat\imath} \biggl[ \biggl( \frac{a^2}{a^2 + b^2} \biggr) f + 1 \biggr] \Omega_fy
			+
			\boldsymbol{\hat\jmath} \biggl[\biggl( \frac{b^2}{a^2 + b^2} \biggr) f + 1 \biggr] \Omega_fx \, .
			</math>
			</td>
			</tr>
			</table>

			[https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract Chandrasekhar (1965)], p. 892, §II, Eq. (14)
			</div>

	===Summary===		===Summary===

Tohline: /* Equilibrium Conditions for Riemann S-type Ellipsoids */

2020-06-14T19:46:14Z

Equilibrium Conditions for Riemann S-type Ellipsoids

← Older revision		Revision as of 19:46, 14 June 2020
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	==Equilibrium Conditions for Riemann S-type Ellipsoids==		==Equilibrium Conditions for Riemann S-type Ellipsoids==

			We begin this section by quoting from the first paragraph in §II, p. 892 of [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract Chandrasekhar (1965)]. "<font color="darkgreen">The problem that is to be considered … is that of a homogeneous mass, rotating uniformly with an angular velocity <math>\vec\Omega_f</math>, with internal motions having a uniform vorticity <math>~\vec\zeta</math> in the direction of <math>~\Omega_f</math> and in the frame of reference rotating with the angular velocity <math>~\vec\Omega_f</math>.</font>" As did Chandrasekhar, we will find it useful to refer to the ratio of these highlighted frequencies as the key model 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_f} \, .</math>
			</td>
			</tr>
			</table>

			[https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract Chandrasekhar (1965)], p. 892, §II, Eq. (15)
			</div>

	===Based on Virial Equilibrium===		===Based on Virial Equilibrium===

Tohline: /* Relate EFE to Ou(2006) */

2020-06-14T18:00:32Z

Relate EFE to Ou(2006)

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	where, <math>~\lambda</math> is an overall scale factor. But, according to §48 of EFE, we see that,		where, <math>~\lambda</math> is an overall scale factor. But, according to §48 of EFE, we see that,
			<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</tr>		</tr>
	</table>		</table>

			[https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract Chandrasekhar (1965)], p. 892, §II, Eq. (9)
			</div>
	where,		where,
			<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</tr>		</tr>
	</table>		</table>

			[https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract Chandrasekhar (1965)], p. 892, §II, Eq. (10)
			</div>
	and <math>~\zeta</math> is the scalar magnitude of the vorticity vector, <math>~\vec\zeta</math>. The transformation from EFE's notation to the one used by Ou is, then,		and <math>~\zeta</math> is the scalar magnitude of the vorticity vector, <math>~\vec\zeta</math>. The transformation from EFE's notation to the one used by Ou is, then,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Our Parameter Determinations */

2020-06-13T20:54:33Z

Our Parameter Determinations

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	<li>''Dedekind'' sequence — a smooth curve that lies precisely on top of the ''Jacobi'' sequence. Each configuration along this sequence is ''adjoint'' to a model on the ''Jacobi'' sequence that shares its (b/a, c/a) axis-ratio pair. All ellipsoidal figures along this sequence have <math>~1/f = \Omega_f/\zeta = 0</math> and are therefore stationary as viewed from the ''inertial'' frame; the angular momentum of each configuration is stored in its internal motion (vorticity).</li>		<li>''Dedekind'' sequence — a smooth curve that lies precisely on top of the ''Jacobi'' sequence. Each configuration along this sequence is ''adjoint'' to a model on the ''Jacobi'' sequence that shares its (b/a, c/a) axis-ratio pair. All ellipsoidal figures along this sequence have <math>~1/f = \Omega_f/\zeta = 0</math> and are therefore stationary as viewed from the ''inertial'' frame; the angular momentum of each configuration is stored in its internal motion (vorticity).</li>
	<li>The X = -1 ''self-adjoint'' sequence — At every point along this sequence, the value of the key frequency ratio, <math>~\zeta/\Omega_f</math>, in the ''adjoint'' configuration <math>~(f_+)</math> is identical to the value of the frequency ratio in the ''direct'' configuration <math>~(f_-)</math>; specifically, <math>~f_+ = f_- = -(a^2+b^2)/(ab)</math>. The data identifying the location of the small, solid-black markers along this sequence have been drawn from §48, Table VI of [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>].</li>		<li>The X = -1 ''self-adjoint'' sequence — At every point along this sequence, the value of the key frequency ratio, <math>~\zeta/\Omega_f</math>, in the ''adjoint'' configuration <math>~(f_+)</math> is identical to the value of the frequency ratio in the ''direct'' configuration <math>~(f_-)</math>; specifically, <math>~f_+ = f_- = -(a^2+b^2)/(ab)</math>. The data identifying the location of the small, solid-black markers along this sequence have been drawn from §48, Table VI of [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>].</li>
	<li>The X = +1 ''self-adjoint'' sequence — At every point along this sequence, the value of the key frequency ratio, <math>~\zeta/\Omega_f</math>, in the ''adjoint'' configuration <math>~(f_+)</math> is identical to the value of the frequency ratio in the ''direct'' configuration <math>~(f_-)</math>; specifically, <math>~f_+ = f_- = -(a^2+b^2)/(ab)</math>. The data identifying the location of the small, solid-black markers along this sequence have been drawn from §48, Table VI of [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>].</li>		<li>The X = +1 ''self-adjoint'' sequence — At every point along this sequence, the value of the key frequency ratio, <math>~\zeta/\Omega_f</math>, in the ''adjoint'' configuration <math>~(f_+)</math> is identical to the value of the frequency ratio in the ''direct'' configuration <math>~(f_-)</math>; specifically, <math>~f_+ = f_- = +(a^2+b^2)/(ab)</math>. The data identifying the location of the small, solid-black markers along this sequence have been drawn from §48, Table VI of [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>].</li>
	</ul>		</ul>

Tohline: /* Our Parameter Determinations */

2020-06-13T20:51:29Z

Our Parameter Determinations

← Older revision		Revision as of 20:51, 13 June 2020
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	</ul>		</ul>

	Riemann S-type ellipsoids all lie between or on the two (self-adjoint) curves marked "X = -1" and "X = +1" in the EFE Diagram. The yellow circular markers in the diagram shown here, on the left, identify four Riemann S-type ellipsoids that were examined by Ou (2006) and that we have also chosen to use as examples.		Riemann S-type ellipsoids all lie between or on the two (self-adjoint) curves marked "X = -1" and "X = +1" in the EFE Diagram. The yellow circular markers in the diagram shown here, on the left, identify four Riemann S-type ellipsoids that were examined by [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou (2006)] and that we have also chosen to use as examples.
	</td>		</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="left">		<td align="left">
	[[File:EFEdiagram4.png\|left\|500px\|EFE Diagram ~~with~~ example Ou(2006) ~~models identified~~]]		[[File:EFEdiagram4.png\|left\|500px\|EFE Diagram identifying example models from Ou (2006)]]
	</td>		</td>
	</tr>		</tr>

Tohline: /* Our Parameter Determinations */

2020-06-13T20:47:53Z

Our Parameter Determinations

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	</table>		</table>

			<span id="Fig2"> </span>
	<table border="1" cellpadding="5" width="90%" align="center">		<table border="1" cellpadding="5" width="90%" align="center">
	<tr><td align="center" colspan="2">'''Figure 2:  EFE Diagram'''</td~~></tr>~~		<tr><td align="center" colspan="1">'''Figure 2:  EFE Diagram'''</td>
	~~<tr~~>		<td align="left" rowspan="2">
	<td align="left">
	~~[[File:EFEdiagram4.png\|left\|500px\|EFE Diagram with example Ou(2006) models identified]]~~
	~~</td>~~
	~~<td align~~="~~left~~">
	In the context of our broad discussion of ellipsoidal figures of equilibrium, the label "EFE Diagram" refers to a two-dimensional parameter space defined by the pair of axis ratios (b/a, c/a), ''usually'' covering the ranges, 0 ≤ b/a ≤ 1 and 0 ≤ c/a ≤ 1. The classic/original version of this diagram appears as Figure 2 on p. 902 of [http://adsabs.harvard.edu/abs/1965ApJ...142..890C S. Chandrasekhar (1965, ApJ, vol. 142, pp. 890-921)]; a somewhat less cluttered version appears on p. 147 of Chandrasekhar's [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>].		In the context of our broad discussion of ellipsoidal figures of equilibrium, the label "EFE Diagram" refers to a two-dimensional parameter space defined by the pair of axis ratios (b/a, c/a), ''usually'' covering the ranges, 0 ≤ b/a ≤ 1 and 0 ≤ c/a ≤ 1. The classic/original version of this diagram appears as Figure 2 on p. 902 of [http://adsabs.harvard.edu/abs/1965ApJ...142..890C S. Chandrasekhar (1965, ApJ, vol. 142, pp. 890-921)]; a somewhat less cluttered version appears on p. 147 of Chandrasekhar's [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>].

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	Riemann S-type ellipsoids all lie between or on the two (self-adjoint) curves marked "X = -1" and "X = +1" in the EFE Diagram. The yellow circular markers in the diagram shown here, on the left, identify four Riemann S-type ellipsoids that were examined by Ou (2006) and that we have also chosen to use as examples.		Riemann S-type ellipsoids all lie between or on the two (self-adjoint) curves marked "X = -1" and "X = +1" in the EFE Diagram. The yellow circular markers in the diagram shown here, on the left, identify four Riemann S-type ellipsoids that were examined by Ou (2006) and that we have also chosen to use as examples.
			</td>
			</tr>
			<tr>
			<td align="left">
			[[File:EFEdiagram4.png\|left\|500px\|EFE Diagram with example Ou(2006) models identified]]
	</td>		</td>
	</tr>		</tr>

Tohline: /* Our Parameter Determinations */

2020-06-13T20:43:13Z

Our Parameter Determinations

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	<tr>		<tr>
	<td align="left">		<td align="left">
	[[File:EFEdiagram4.png\|left\|~~450px~~\|EFE Diagram with example Ou(2006) models identified]]		[[File:EFEdiagram4.png\|left\|500px\|EFE Diagram with example Ou(2006) models identified]]
	</td>		</td>
	<td align="left">		<td align="left">

Tohline: /* Our Parameter Determinations */

2020-06-13T20:42:38Z

Our Parameter Determinations

← Older revision		Revision as of 20:42, 13 June 2020
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	<li>''Jacobi'' sequence — the smooth curve that runs through the set of small, dark-blue, diamond-shaped markers; the data identifying the location of these markers have been drawn from §39, Table IV of [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>]. The small red circular markers lie along this same sequence; their locations are taken from our own determinations, as detailed in [[User:Tohline/ThreeDimensionalConfigurations/JacobiEllipsoids#Table2\|Table 2]] of our accompanying discussion of Jacobi ellipsoids. All of the models along this sequence have <math>~f \equiv \zeta/\Omega_f = 0</math> and are therefore solid-body rotators, that is, there is no internal motion when the configuration is viewed from a frame that is rotating with frequency, <math>~\Omega_f</math>.</li>		<li>''Jacobi'' sequence — the smooth curve that runs through the set of small, dark-blue, diamond-shaped markers; the data identifying the location of these markers have been drawn from §39, Table IV of [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>]. The small red circular markers lie along this same sequence; their locations are taken from our own determinations, as detailed in [[User:Tohline/ThreeDimensionalConfigurations/JacobiEllipsoids#Table2\|Table 2]] of our accompanying discussion of Jacobi ellipsoids. All of the models along this sequence have <math>~f \equiv \zeta/\Omega_f = 0</math> and are therefore solid-body rotators, that is, there is no internal motion when the configuration is viewed from a frame that is rotating with frequency, <math>~\Omega_f</math>.</li>
	<li>''Dedekind'' sequence — a smooth curve that lies precisely on top of the ''Jacobi'' sequence. Each configuration along this sequence is ''adjoint'' to a model on the ''Jacobi'' sequence that shares its (b/a, c/a) axis-ratio pair. All ellipsoidal figures along this sequence have <math>~1/f = \Omega_f/\zeta = 0</math> and are therefore stationary as viewed from the ''inertial'' frame; the angular momentum of each configuration is stored in its internal motion (vorticity).</li>		<li>''Dedekind'' sequence — a smooth curve that lies precisely on top of the ''Jacobi'' sequence. Each configuration along this sequence is ''adjoint'' to a model on the ''Jacobi'' sequence that shares its (b/a, c/a) axis-ratio pair. All ellipsoidal figures along this sequence have <math>~1/f = \Omega_f/\zeta = 0</math> and are therefore stationary as viewed from the ''inertial'' frame; the angular momentum of each configuration is stored in its internal motion (vorticity).</li>
	<li>The X = -1 ''self-adjoint'' sequence — At every point along this sequence, the value of the key frequency ratio in the ''adjoint'' configuration (f<~~sub~~>+</~~sub~~>) is identical to the value of the frequency ratio in the ''direct'' configuration (f<~~sub~~>-</~~sub~~>); specifically, f<~~sub~~>+~~</sub>~~ = ~~f<sub>~~-</~~sub~~> = &~~zeta;/&Omega~~;<~~sub~~>f<~~/sub>~~ = ~~+(a<sup~~>2</~~sup~~>~~+b<sup>2~~</~~sup~~>~~)/(ab)~~.</li>		<li>The X = -1 ''self-adjoint'' sequence — At every point along this sequence, the value of the key frequency ratio, <math>~\zeta/\Omega_f</math>, in the ''adjoint'' configuration <math>~(f_+)</math> is identical to the value of the frequency ratio in the ''direct'' configuration <math>~(f_-)</math>; specifically, <math>~f_+ = f_- = -(a^2+b^2)/(ab)</math>. The data identifying the location of the small, solid-black markers along this sequence have been drawn from §48, Table VI of [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>].</li>
	<li>The X = +1 ''self-adjoint'' sequence —</li>		<li>The X = +1 ''self-adjoint'' sequence — At every point along this sequence, the value of the key frequency ratio, <math>~\zeta/\Omega_f</math>, in the ''adjoint'' configuration <math>~(f_+)</math> is identical to the value of the frequency ratio in the ''direct'' configuration <math>~(f_-)</math>; specifically, <math>~f_+ = f_- = -(a^2+b^2)/(ab)</math>. The data identifying the location of the small, solid-black markers along this sequence have been drawn from §48, Table VI of [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>].</li>
	</ul>		</ul>

			Riemann S-type ellipsoids all lie between or on the two (self-adjoint) curves marked "X = -1" and "X = +1" in the EFE Diagram. The yellow circular markers in the diagram shown here, on the left, identify four Riemann S-type ellipsoids that were examined by Ou (2006) and that we have also chosen to use as examples.
	~~It is useful to appreciate at the outset, for example, that~~ Riemann S-type ellipsoids ~~only populate a subset of the EFE Diagram's entire parameter space. More specifically, they~~ all lie between or on the two (self-adjoint) curves marked "X = -1" and "X = +1" in the EFE Diagram ~~that is~~ shown here, on the ~~right. Keeping this in mind~~, ~~we summarize here a sequence of steps that should be taken in order to construct and thereby quantitatively detail all of the physical properties that are associated with any~~ Riemann S-type ~~ellipsoid~~ that ~~lies in this allowed region of the EFE Diagram~~.

	</td>		</td>