User:Tohline/VE/RiemannEllipsoids - Revision history

Tohline: /* Riemann S-Type Ellipsoids */

2020-09-08T23:02:43Z

Riemann S-Type Ellipsoids

← Older revision		Revision as of 23:02, 8 September 2020
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	</table>		</table>

	It has become customary to characterize each Riemann S-Type ellipsoid by the value of its equilibrium frequency ratio,		<span id="fDefined">It has become customary to characterize each Riemann S-Type ellipsoid by the value of its equilibrium frequency ratio, </span>
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Summary */

2020-08-21T03:00:09Z

Summary

← Older revision		Revision as of 03:00, 21 August 2020
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	==Summary==		==Summary==

	~~Here~~ we employ the 2<sup>nd</sup>-order tensor virial equation (TVE),		Drawing from our [[User:Tohline/VE#Virial_Equations_.28Rotating_Frame.29\|accompanying discussion of virial equations as viewed from a rotating frame of reference]], here we employ the 2<sup>nd</sup>-order tensor virial equation (TVE),
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
	<tr>		<tr>

Tohline: /* Specific Case of a2 > a1 > a3 */

2020-08-17T21:14:33Z

Specific Case of a2 > a1 > a3

← Older revision		Revision as of 21:14, 17 August 2020
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	</table>		</table>
	</div>		</div>

			===Oblate Spheroids [a<sub>2</sub> = a<sub>1</sub> > a<sub>3</sub>]===

			Starting with the case of <math>~a_2 > a_1 > a_3 > 0</math> and setting <math>~a_2 = a_1</math>, we recognize, first, that <math>~k = 0</math>. Hence, we have,

			<table align="center" border=0 cellpadding="3">
			<tr>
			<td align="right">
			<math>
			~A_3
			</math>
			</td>
			<td align="center">
			<math>
			~=
			</math>
			</td>
			<td align="left">
			<math>
			~2\biggl[ \frac{ \sin\theta - (a_3/a_1)E(\theta,0)}{\sin^3\theta} \biggr] \, ,
			</math>
			</td>
			</tr>
			</table>

	==Adopted (Internal) Velocity Field==		==Adopted (Internal) Velocity Field==

Tohline: /* General Coefficient Expressions */

2020-08-17T20:49:11Z

General Coefficient Expressions

← Older revision		Revision as of 20:49, 17 August 2020
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	<td align="right">		<td align="right">
	<math>		<math>
	~~~\frac{~~A_1~~}{a_1 a_3 a_2}~~		~A_1
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{2}{a_1^3}		<math>~2 \biggl( \frac{a_2}{a_1} \biggr)\biggl( \frac{a_3}{a_1} \biggr)
	\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,		\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
	</math>		</math>
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	<td align="right">		<td align="right">
	<math>		<math>
	~~~\frac{~~A_2~~}{a_1 a_3 a_2}~~		~A_2
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	~\frac{2}{a_1^3} \biggl[ \frac{(a_3/~~a_2~~) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,		~2 \biggl( \frac{a_3}{a_1} \biggr) \biggl[ \frac{(a_3/a_1) \sin\theta - (a_2/a_1)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
	</math>		</math>
	</td>		</td>
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	<td align="right">		<td align="right">
	<math>		<math>
	~~~\frac{~~A_3~~}{a_1 a_3 a_2}~~ = ~~\frac{~~2 - (A_1 + A_2)~~}{a_1 a_3 a_2}~~		~A_3 = 2 - (A_1 + A_2)
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{ 2}{a_1^3 }		\frac{2a_2 a_3}{a_1^2}
	\biggl[ \frac{		\biggl[ \frac{
	E(\theta, k)		E(\theta, k)
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	</table>		</table>
	</div>		</div>
			NOTE: All ''irrotational'' ellipsoids belong to this category of configurations.

	===Specific Case of a<sub>2</sub> > a<sub>1</sub> > a<sub>3</sub>===		===Specific Case of a<sub>2</sub> > a<sub>1</sub> > a<sub>3</sub>===

Tohline: /* Specific Case of a1 > a3 > a2 */

2020-08-17T20:34:38Z

Specific Case of a1 > a3 > a2

← Older revision		Revision as of 20:34, 17 August 2020
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	===Specific Case of a<sub>1</sub> > a<sub>3</sub> > a<sub>2</sub>===		===Specific Case of a<sub>1</sub> > a<sub>3</sub> > a<sub>2</sub>===

	When we discuss configurations in which <math>~a_1 > a_3 > a_2 > 0</math> — these are usually referred to in EFE as prolate S-Type Riemann ellipsoids — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_s, a_s)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_m, a_m)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,		When we discuss configurations in which <math>~a_1 > a_3 > a_2 > 0</math> — these are usually referred to in [[User:Tohline/Appendix/References#EFE\|EFE]] as prolate S-Type Riemann ellipsoids — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_s, a_s)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_m, a_m)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
	<div align="center">		<div align="center">
	<table align="center" border=0 cellpadding="3">		<table align="center" border=0 cellpadding="3">
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	</td>		</td>
	</tr>		</tr>
			<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §48d, footnote to Table VII (p. 143)</font> ]</td></tr>
	</table>		</table>
	</div>		</div>

Tohline: /* Specific Case of a1 > a3 > a2 */

2020-08-17T20:30:41Z

Specific Case of a1 > a3 > a2

← Older revision		Revision as of 20:30, 17 August 2020
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	<td align="right">		<td align="right">
	<math>		<math>
	~\frac{~~A_\ell~~}{~~a_\ell a_m a_s~~}		~\frac{A_1}{a_1 a_3 a_2}
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{2}{~~a_\ell~~^3}		<math>~\frac{2}{a_1^3}
	\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,		\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
	</math>		</math>
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	<td align="right">		<td align="right">
	<math>		<math>
	~\frac{~~A_s~~}{~~a_\ell a_m a_s~~}		~\frac{A_2}{a_1 a_3 a_2}
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	~\frac{2}{~~a_\ell~~^3} \biggl[ \frac{(~~a_m~~/~~a_s~~) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,		~\frac{2}{a_1^3} \biggl[ \frac{(a_3/a_2) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
	</math>		</math>
	</td>		</td>
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	<td align="right">		<td align="right">
	<math>		<math>
	~\frac{~~A_m~~}{~~a_\ell a_m a_s~~} = \frac{2 - (~~A_\ell~~ + ~~A_s~~)}{~~a_\ell a_m a_s~~}		~\frac{A_3}{a_1 a_3 a_2} = \frac{2 - (A_1 + A_2)}{a_1 a_3 a_2}
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{ 2}{~~a_\ell~~^3 }		\frac{ 2}{a_1^3 }
	\biggl[ \frac{		\biggl[ \frac{
	E(\theta, k)		E(\theta, k)
	-~(1-k^2)		-~(1-k^2)
	F(\theta, k)		F(\theta, k)
	-~(~~a_s~~/~~a_m~~)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}		-~(a_2/a_3)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
	\biggr] \, ,		\biggr] \, ,
	</math>		</math>
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	</table>		</table>
	</div>		</div>
	where, ~~<math>~F(\theta,k)</math> and <math>~E(\theta,k)</math> are~~ incomplete elliptic integrals of the first and second kind~~, respectively, with arguments~~,		where, the arguments of the incomplete elliptic integrals of the first and second kind are,
	<div align="center">		<div align="center">
	<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>~\theta = \cos^{-1} \biggl(\frac{~~a_s~~}{~~a_\ell~~} \biggr)</math>		<math>~\theta = \cos^{-1} \biggl(\frac{a_2}{a_1} \biggr)</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~k = \biggl[\frac{1 - (~~a_m~~/~~a_\ell~~)^2}{1 - (~~a_s~~/~~a_\ell~~)^2} \biggr]^{1/2} \, .</math>		<math>~k = \biggl[\frac{1 - (a_3/a_1)^2}{1 - (a_2/a_1)^2} \biggr]^{1/2} \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* General Coefficient Expressions */

2020-08-17T20:22:29Z

General Coefficient Expressions

← Older revision		Revision as of 20:22, 17 August 2020
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	When we discuss configurations in which <math>~a_1 > a_3 > a_2 > 0</math> — these are usually referred to in EFE as prolate S-Type Riemann ellipsoids — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_s, a_s)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_m, a_m)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,		When we discuss configurations in which <math>~a_1 > a_3 > a_2 > 0</math> — these are usually referred to in EFE as prolate S-Type Riemann ellipsoids — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_s, a_s)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_m, a_m)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
			<div align="center">
			<table align="center" border=0 cellpadding="3">
			<tr>
			<td align="right">
			<math>
			~\frac{A_\ell}{a_\ell a_m a_s}
			</math>
			</td>
			<td align="center">
			<math>
			~=
			</math>
			</td>
			<td align="left">
			<math>~\frac{2}{a_\ell^3}
			\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>
			~\frac{A_s}{a_\ell a_m a_s}
			</math>
			</td>
			<td align="center">
			<math>
			~=
			</math>
			</td>
			<td align="left">
			<math>
			~\frac{2}{a_\ell^3} \biggl[ \frac{(a_m/a_s) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>
			~\frac{A_m}{a_\ell a_m a_s} = \frac{2 - (A_\ell + A_s)}{a_\ell a_m a_s}
			</math>
			</td>
			<td align="center">
			<math>
			~=
			</math>
			</td>
			<td align="left">
			<math>~
			\frac{ 2}{a_\ell^3 }
			\biggl[ \frac{
			E(\theta, k)
			-~(1-k^2)
			F(\theta, k)
			-~(a_s/a_m)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
			\biggr] \, ,
			</math>
			</td>
			</tr>

			</table>
			</div>
			where, <math>~F(\theta,k)</math> and <math>~E(\theta,k)</math> are incomplete elliptic integrals of the first and second kind, respectively, with arguments,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\theta = \cos^{-1} \biggl(\frac{a_s}{a_\ell} \biggr)</math>
			</td>
			<td align="center">
			and
			</td>
			<td align="left">
			<math>~k = \biggl[\frac{1 - (a_m/a_\ell)^2}{1 - (a_s/a_\ell)^2} \biggr]^{1/2} \, .</math>
			</td>
			</tr>
			</table>
			</div>

	===Specific Case of a<sub>2</sub> > a<sub>1</sub> > a<sub>3</sub>===		===Specific Case of a<sub>2</sub> > a<sub>1</sub> > a<sub>3</sub>===

Tohline: /* Specific Case of a2 > a1 > a3 */

2020-08-17T20:21:37Z

Specific Case of a2 > a1 > a3

← Older revision		Revision as of 20:21, 17 August 2020
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	</table>		</table>
	</div>		</div>

			===Specific Case of a<sub>1</sub> > a<sub>3</sub> > a<sub>2</sub>===

			When we discuss configurations in which <math>~a_1 > a_3 > a_2 > 0</math> — these are usually referred to in EFE as prolate S-Type Riemann ellipsoids — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_s, a_s)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_m, a_m)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,

	===Specific Case of a<sub>2</sub> > a<sub>1</sub> > a<sub>3</sub>===		===Specific Case of a<sub>2</sub> > a<sub>1</sub> > a<sub>3</sub>===

	When we discuss configurations in which <math>~a_2 > a_1 > a_3 > 0</math> — for example, ~~prolate S-Type Riemann ellipsoids and~~ ''most'' Riemann ellipsoids of Types I, II, & III — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_m, a_m)</math>, <math>~(A_2, a_2) \leftrightarrow (A_\ell, a_\ell)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_s, a_s)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,		When we discuss configurations in which <math>~a_2 > a_1 > a_3 > 0</math> — for example, ''most'' Riemann ellipsoids of Types I, II, & III — we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_m, a_m)</math>, <math>~(A_2, a_2) \leftrightarrow (A_\ell, a_\ell)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_s, a_s)</math>. This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
	<div align="center">		<div align="center">
	<table align="center" border=0 cellpadding="3">		<table align="center" border=0 cellpadding="3">

Tohline: /* Specific Case of a2 > a1 > a3 */

2020-08-17T01:33:49Z

Specific Case of a2 > a1 > a3

← Older revision		Revision as of 01:33, 17 August 2020
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{ 2}{a_2^3 }		\frac{ 2a_1 a_3}{a_2^2 }
	\biggl[ \frac{		\biggl[ \frac{
	E(\theta, k)		E(\theta, k)
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	F(\theta, k)		F(\theta, k)
	-~(a_3/a_1)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}		-~(a_3/a_1)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
	\biggr]~~a_2 a_1 a_3~~ \, ,		\biggr] \, ,
	</math>		</math>
	</td>		</td>

Tohline: /* General Coefficient Expressions */

2020-08-17T01:29:43Z

General Coefficient Expressions

← Older revision		Revision as of 01:29, 17 August 2020
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	<td align="right">		<td align="right">
	<math>		<math>
	~~~\frac{~~A_2~~}{a_2 a_1 a_3}~~		~A_2
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{2}{a_2^3}		<math>~2 \biggl( \frac{a_1}{a_2} \biggr)\biggl( \frac{a_3}{a_2} \biggr)
	\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,		\biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
	</math>		</math>
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	<td align="right">		<td align="right">
	<math>		<math>
	~~~\frac{~~A_3~~}{a_2 a_1 a_3}~~		~A_3
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	~\frac{2}{a_2^3} \biggl[ \frac{(a_1/~~a_3~~) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,		~2\biggl( \frac{a_1}{a_2}\biggr) \biggl[ \frac{(a_1/a_2) \sin\theta - (a_3/a_2)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
	</math>		</math>
	</td>		</td>
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	<td align="right">		<td align="right">
	<math>		<math>
	~~~\frac{~~A_1~~}{a_2 a_1 a_3}~~ = ~~\frac{~~2 - (A_2 + A_3)~~}{a_2 a_1 a_3}~~		~A_1 = 2 - (A_2 + A_3)
	</math>		</math>
	</td>		</td>
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	F(\theta, k)		F(\theta, k)
	-~(a_3/a_1)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}		-~(a_3/a_1)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
	\biggr] \, ,		\biggr]a_2 a_1 a_3 \, ,
	</math>		</math>
	</td>		</td>