3Dconfigurations/DescriptionOfRiemannTypeI - Revision history

Jet53man: /* Basic Relations */

2022-03-30T16:48:37Z

Basic Relations

← Older revision		Revision as of 12:48, 30 March 2022
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	</table>		</table>

	<font color="red">REMINDER:</font>  For a given choice of the eccentricity, there are two viable solutions … the ''direct'' configuration and its ''adjoint.'' In the context of ~~Maclaurin spheroids~~, this pair of solutions arises from the choice of the sign <math>(\pm)</math> in the expression for <math>\Omega_3</math>; in the context of Type I Riemann ellipsoids, the pair arises from the choice of the sign <math>(\mp)</math> in the <font color="red">STEP #3</font> determination of <math>\beta</math> and <math>\gamma</math>. In both physical contexts, the ''direct'' (Jacobi-like) solution results from selecting the ''inferior'' sign while the ''adjoint'' (Dedekind-like) solution results from selecting the ''superior'' sign.		<font color="red">REMINDER:</font>  For a given choice of the eccentricity, there are two viable solutions … the ''direct'' configuration and its ''adjoint.'' In the context of [[ThreeDimensionalConfigurations/RiemannStype#Expressions_Supplied_by_EFE\|Riemann S-type ellipsoids]], this pair of solutions arises from the choice of the sign <math>(\pm)</math> in the expression for <math>\Omega_3</math>; in the context of Type I Riemann ellipsoids (<i>i.e.</i>, here) the pair arises from the choice of the sign <math>(\mp)</math> in the <font color="red">STEP #3</font> determination of <math>\beta</math> and <math>\gamma</math>. In both physical contexts, the ''direct'' (Jacobi-like) solution results from selecting the ''inferior'' sign while the ''adjoint'' (Dedekind-like) solution results from selecting the ''superior'' sign.

	===Frequency Ratio===		===Frequency Ratio===

Jet53man: /* Frequency Ratio */

2022-03-24T21:48:29Z

Frequency Ratio

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    <td align="left" width="50%">Analogous to Figure 5 from &sect;32, p. 79 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]; shows how the square of the normalized rotation frequency varies with eccentricity, <math>e = (1 - a_3/a_1)^{1 / 2},</math> along the (black-dotted) Maclaurin sequence and along the Jacobi sequence (series of purple circular markers).</td>
    <td align="left">&nbsp;</td>
 </tr>
 </table>

Jet53man: /* Frequency Ratio */

2022-03-24T21:00:02Z

Frequency Ratio

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 </table>
-In an [[Apps/MaclaurinSpheroidSequence#Maclaurin_Spheroid_Sequence|accompanying discussion of the Maclaurin spheroid sequence]], a number of different plots have been used to display how various physical parameters vary along the sequence.  The solid curve that appears in Figure 1 of that discussion has been redrawn as a dotted curve in the left-hand panel of Figure 1 of ''this'' chapter (immediately below); it shows how <math>\Omega^2_\mathrm{Mc}</math> varies with the spheroid's eccentricity, <math>e</math>.  The small green square identifies the configuration along the sequence that exhibits the largest (dimensionless) frequency
+In an [[Apps/MaclaurinSpheroidSequence#Maclaurin_Spheroid_Sequence|accompanying discussion of the Maclaurin spheroid sequence]], a number of different plots have been used to display how various physical parameters vary along the sequence.  The solid curve that appears in Figure 1 of that discussion has been redrawn as a black-dotted curve in the left-hand panel of Figure 1 of ''this'' chapter (immediately below); it shows how <math>\Omega^2_\mathrm{Mc}</math> varies with the spheroid's eccentricity, <math>e</math>.  The small solid-green square marker identifies the location along the sequence where the system with the maximum angular velocity resides:
-<table border="1" align="center" cellpadding="2">
+<table border="1" align="center" cellpadding="8" width="80%">
 <tr>
    <td align="center" colspan="2">
@@ Line 516: / Line 534: @@
 [[File:f_McA.png|400px|center|FrequencyRatio]]
 </td>
 </tr>
 </table>

Jet53man: /* Frequency Ratio */

2022-03-24T20:33:17Z

Frequency Ratio

← Older revision		Revision as of 16:33, 24 March 2022
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	<table border="0" align="center" cellpadding="2"><tr>		In an [[Apps/MaclaurinSpheroidSequence#Maclaurin_Spheroid_Sequence\|accompanying discussion of the Maclaurin spheroid sequence]], a number of different plots have been used to display how various physical parameters vary along the sequence. The solid curve that appears in Figure 1 of that discussion has been redrawn as a dotted curve in the left-hand panel of Figure 1 of ''this'' chapter (immediately below); it shows how <math>\Omega^2_\mathrm{Mc}</math> varies with the spheroid's eccentricity, <math>e</math>. The small green square identifies the configuration along the sequence that exhibits the largest (dimensionless) frequency

			<table border="1" align="center" cellpadding="2">
			<tr>
			<td align="center" colspan="2">
			<b>Figure 1:   Parameter Variations Along the Maclaurin Spheroid Sequence</b>
			</td>
			</tr>
			<tr>
	<td align="center">		<td align="center">
	[[File:JacobiSequenceTooA.png\|400px\|center\|JacobiSequenceToo]]		[[File:JacobiSequenceTooA.png\|400px\|center\|JacobiSequenceToo]]
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	[[File:f_McA.png\|400px\|center\|FrequencyRatio]]		[[File:f_McA.png\|400px\|center\|FrequencyRatio]]
	</td>		</td>
	</tr></table>		</tr>
			</table>

	==Example Equilibrium Models==		==Example Equilibrium Models==

Jet53man: /* Frequency Ratio */

2022-03-23T21:30:03Z

Frequency Ratio

← Older revision		Revision as of 17:30, 23 March 2022
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	</tr>		</tr>
	</table>		</table>

			<table border="0" align="center" cellpadding="2"><tr>
			<td align="center">
			[[File:JacobiSequenceTooA.png\|400px\|center\|JacobiSequenceToo]]
			</td>
			<td align="center">
			[[File:f_McA.png\|400px\|center\|FrequencyRatio]]
			</td>
			</tr></table>

	==Example Equilibrium Models==		==Example Equilibrium Models==

Jet53man: /* Maclaurin Spheroid Limit */

2022-03-23T20:18:34Z

Maclaurin Spheroid Limit

← Older revision		Revision as of 16:18, 23 March 2022
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	<table border="0" align="center" cellpadding="8">		<table border="0" align="center" cellpadding="8">
	<tr>		<tr>
	<td align="right"><math>\Omega_\mathrm{Mc}^2 \equiv \frac{\omega_0^2}{\pi G \rho}</math></td>		<td align="right"><math>\Omega_\mathrm{Mc}^2 \equiv \frac{\omega_0^2}{\pi G \rho} = 2e^2 B_{13} </math></td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left"><math>2(3-2e^2)(1 - e^2)^{1 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{6(1-e^2)}{e^2} \, .</math></td>		<td align="left">
			<math>
			2(3-2e^2)(1 - e^2)^{1 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{6(1-e^2)}{e^2} \, .</math>
			</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
	[<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], §32, p. 78, Eq. (6)		[<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], §32, pp. 77-78, Eqs. (4) & (6)
	</td>		</td>
	</tr>		</tr>
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	<table border="0" align="center" cellpadding="8">		<table border="0" align="center" cellpadding="8">
	</tr>		</tr>
	<td align="right"><math>\mathcal{H}</math></td>		<td align="right"><math>\mathcal{H} \equiv \frac{16B_{13}}{\Omega^2_\mathrm{Mc}}</math></td>
	<td align="center"><math>~~\equiv~~</math></td>		<td align="center"><math>=</math></td>
	<td align="left"><math>\frac{~~16B_{13}~~}{~~\Omega~~^~~2_\mathrm{Mc~~}~~}\, ,~~</math></td>		<td align="left">
			<math>
			\frac{8}{e^2}
			</math>
			</td>
	</tr>		</tr>
	</table>		</table>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>f = \frac{\zeta_3}{\Omega_3}</math>		<math>f_\mathrm{Mc} = \biggl(\frac{\zeta_3}{\Omega_3}\biggr)_\mathrm{Mc}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>\frac{4}{f +2}</math>		<math>\frac{4}{f_\mathrm{Mc} +2}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>\Rightarrow ~~~ f</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>1 \pm \biggl[ 1 + \frac{8}{e^2} \biggr]^{1 / 2}</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>\Rightarrow ~~~ \frac{4e}{f_\mathrm{Mc} +2}</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>e \pm ( 8 + e^2 )^{1 / 2}</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>\Rightarrow ~~~ f_\mathrm{Mc}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>4\biggl[ 1 \pm (1 + ~~\mathcal{H}~~ )^{1 / 2} \biggr]^{-1} -2 \, .</math>		<math>4e\biggl[ e \pm ( 8 + e^2 )^{1 / 2} \biggr]^{-1} - 2 \, .</math>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Maclaurin Spheroid Limit */

2022-03-23T19:39:02Z

Maclaurin Spheroid Limit

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 </tr>
 </table>
-is constant.  Let's see if it proves useful to identify the same types of model sequences in the context of Type I Riemann Ellipsoids.  Adopting the parameter,
+is constant.  Below, we will examine how such sequences behave across the domain of Type I Riemann Ellipsoids.  In anticipation of this discussion, here we examine how <math>f</math> varies along the ''limiting'' Maclaurin spheroid sequence.
 <table border="0" align="center" cellpadding="8">
 </tr>
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    </td>
    <td align="left">
-<math>1 \pm  [1 + \mathcal{H} ]^{1 / 2} \, .</math>
+<math>1 \pm  (1 + \mathcal{H} )^{1 / 2} \, .</math>
    </td>
 </tr>
@@ Line 436: / Line 438: @@
    </td>
    <td align="left">
-<math>2\biggl[\frac{\Omega_\mathrm{Mc}}{\Omega_3} - 1 \biggr] </math>
+<math>2\biggl[\frac{\Omega_\mathrm{Mc}}{\Omega_3} - 1 \biggr] \, .</math>
    </td>
 </tr>

Jet53man: /* Maclaurin Spheroid Limit */

2022-03-23T17:55:20Z

Maclaurin Spheroid Limit

← Older revision		Revision as of 13:55, 23 March 2022
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	</tr>		</tr>
	</table>		</table>
	is constant. Let's see if it proves useful to identify the same types of model sequences in the context of Type I Riemann Ellipsoids.		is constant. Let's see if it proves useful to identify the same types of model sequences in the context of Type I Riemann Ellipsoids. Adopting the parameter,
			<table border="0" align="center" cellpadding="8">
			</tr>
			<td align="right"><math>\mathcal{H}</math></td>
			<td align="center"><math>\equiv</math></td>
			<td align="left"><math>\frac{16B_{13}}{\Omega^2_\mathrm{Mc}}\, ,</math></td>
			</tr>
			</table>
			we have the relation,
			<table border="0" cellpadding="5" align="center">


	~~<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">~~
	~~Note that, [[Apps/MaclaurinSpheroids#LimitingValues\|in the limit of an infinitesimally thin disk]] <math>(e \rightarrow 1)</math>, <math>A_1 = 0</math> and <math>A_3 = 2</math>. Hence,~~
	~~<table border="0" align="center" cellpadding="8">~~
	<tr>		<tr>
	<td align="right"><math>\Omega_\mathrm{Mc}~~^2\biggr\|_{e = 1~~}</math></td>		<td align="right">
	<td align="center"><math>=</math></td>		<math>\frac{2\Omega_3}{\Omega_\mathrm{Mc}}</math>
	<td align="left"><math>~~2(3-2e^2)(~~1 ~~- e^2)^{~~1 ~~/ 2}~~ \~~cdot \frac{\sin^~~{-1} ~~e}{e~~^~~3} - \frac~~{6(1~~-e^2)}{e^~~2} \, .</math></td>		</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>1 \pm [1 + \mathcal{H} ]^{1 / 2} \, .</math>
			</td>
	</tr>		</tr>
			</table>
			But we also see that,
			<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">~~ ~~</td>		<td align="right">
	<td align="center"><math>=</math></td>		<math>f = \frac{\zeta_3}{\Omega_3}</math>
	<td align="left"><math>2~~(1 - e^2)^{1 / 2}~~ \~~cdot~~ \frac{\~~sin^~~{-1} e}{~~e^3~~} - ~~\frac{6(~~1~~-e^2)}{e^2}~~ \~~, .~~</math></td>		</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>2\biggl[\frac{\Omega_\mathrm{Mc}}{\Omega_3} - 1 \biggr] </math>
			</td>
	</tr>		</tr>
	</table>		</table>

	~~</td></tr></table>~~

	==Example Equilibrium Models==		==Example Equilibrium Models==

Jet53man: /* Maclaurin Spheroid Limit */

2022-03-23T17:31:55Z

Maclaurin Spheroid Limit

@@ Line 279: / Line 279: @@
 ==Maclaurin Spheroid Limit==
 In Chapter 7, &sect;51(c) (pp. 165 - 166) of [[Appendix/References#EFE|[<font color="red">EFE</font>] ]], Chandrasekhar shows that <font color="orange">"&hellip; the ''entire'' [[Apps/MaclaurinSpheroidSequence|Maclaurin sequence]] can be considered as limiting <math>(a_2/a_1 \rightarrow 1)</math> forms of the Riemann Ellipsoids of type I."</font>  First, we [[Apps/MaclaurinSpheroidSequence#Equilibrium_Angular_Velocity|recall]] that, as viewed from the inertial frame, each Maclaurin spheroid of eccentricity, <math>e = (1 - a_3^2/a_1^2)^{1 / 2}</math>, rotates uniformly with angular velocity,
 <table border="0" align="center" cellpadding="8">
@@ Line 388: / Line 391: @@
 <font color="red">REMINDER:</font>&nbsp; For a given choice of the eccentricity, there are two viable solutions &hellip; the ''direct'' configuration and its ''adjoint.''  In the context of Maclaurin spheroids, this pair of solutions arises from the choice of the sign <math>(\pm)</math> in the  expression for <math>\Omega_3</math>; in the context of Type I Riemann ellipsoids, the pair arises from the choice of the sign <math>(\mp)</math> in the <font color="red">STEP #3</font> determination of <math>\beta</math> and <math>\gamma</math>.  In both physical contexts, the ''direct'' (Jacobi-like) solution results from selecting the ''inferior'' sign while the ''adjoint'' (Dedekind-like) solution results from selecting the ''superior'' sign.
 <table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">

Jet53man: /* Maclaurin Spheroid Limit */

2022-03-19T17:15:02Z

Maclaurin Spheroid Limit

← Older revision		Revision as of 13:15, 19 March 2022
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	<font color="red">REMINDER:</font>  For a given choice of the eccentricity, there are two viable solutions … the ''direct'' configuration and its ''adjoint.'' In the context of Maclaurin spheroids, this pair of solutions arises from the choice of the sign <math>(\pm)</math> in the expression for <math>\Omega_3</math>; in the context of Type I Riemann ellipsoids, the pair arises from the choice of the sign <math>(\mp)</math> in the <font color="red">STEP #3</font> determination of <math>\beta</math> and <math>\gamma</math>. In both physical contexts, the ''direct'' (Jacobi-like) solution results from selecting the ''inferior'' sign while the ''adjoint'' (Dedekind-like) solution results from selecting the ''superior'' sign.		<font color="red">REMINDER:</font>  For a given choice of the eccentricity, there are two viable solutions … the ''direct'' configuration and its ''adjoint.'' In the context of Maclaurin spheroids, this pair of solutions arises from the choice of the sign <math>(\pm)</math> in the expression for <math>\Omega_3</math>; in the context of Type I Riemann ellipsoids, the pair arises from the choice of the sign <math>(\mp)</math> in the <font color="red">STEP #3</font> determination of <math>\beta</math> and <math>\gamma</math>. In both physical contexts, the ''direct'' (Jacobi-like) solution results from selecting the ''inferior'' sign while the ''adjoint'' (Dedekind-like) solution results from selecting the ''superior'' sign.

			<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
			Note that, [[Apps/MaclaurinSpheroids#LimitingValues\|in the limit of an infinitesimally thin disk]] <math>(e \rightarrow 1)</math>, <math>A_1 = 0</math> and <math>A_3 = 2</math>. Hence,
			<table border="0" align="center" cellpadding="8">
			<tr>
			<td align="right"><math>\Omega_\mathrm{Mc}^2\biggr\|_{e = 1}</math></td>
			<td align="center"><math>=</math></td>
			<td align="left"><math>2(3-2e^2)(1 - e^2)^{1 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{6(1-e^2)}{e^2} \, .</math></td>
			</tr>
			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left"><math>2(1 - e^2)^{1 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{6(1-e^2)}{e^2} \, .</math></td>
			</tr>
			</table>

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

	==Example Equilibrium Models==		==Example Equilibrium Models==