User:Tohline/ThreeDimensionalConfigurations/RiemannStype

Riemann S-type Ellipsoids

General Coefficient Expressions

As has been detailed in an accompanying chapter, the gravitational potential anywhere inside or on the surface, <math>~(a_1,a_2,a_3) ~\leftrightarrow~(a,b,c)</math>, of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:

<math> ~A_1 </math>	<math> ~= </math>	<math>~2\biggl(\frac{b}{a}\biggr)\biggl(\frac{c}{a}\biggr) \biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, , </math>
<math> ~A_3 </math>	<math> ~= </math>	<math> ~2\biggl(\frac{b}{a}\biggr) \biggl[ \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, , </math>
<math> ~A_2 </math>	<math> ~= </math>	<math>~2 - (A_1+A_3) \, ,</math>

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,

<math>~\theta = \cos^{-1} \biggl(\frac{c}{a} \biggr)</math>	and	<math>~k = \biggl[\frac{1 - (b/a)^2}{1 - (c/a)^2} \biggr]^{1/2} \, .</math>
[ EFE, Chapter 3, §17, Eq. (32) ]

TEST (part 1) Notation: Use <math>~\phi</math> in place of <math>~\theta</math>.
<math>~\frac{b}{a}</math>	<math>~\frac{c}{a}</math>	<math>~\phi</math>		<math>~k</math>		Numerical Recipes		<math>~A_1</math>	<math>~A_2</math>	<math>~A_3</math>
		(deg)	(rad)	(deg)	(rad)	<math>~F(\phi,k) </math>	<math>~E(\phi,k) </math>
0.9	0.641	50.13357253	0.874995907	32.53852919	0.567904468	0.909025949	0.843118048	0.521450273	0.595131012	0.883418715

Equilibrium Conditions for Riemann S-type Ellipsoids

Pulling from Chapter 7 — specifically, §48 — of Chandrasekhar's EFE, we understand that the semi-axis ratios, <math>~(\tfrac{b}{a}, \tfrac{c}{a})</math> associated with Riemann S-type ellipsoids are given by the roots of the equation,

<math>~ \biggl[ \frac{a^2 b^2}{a^2 + b^2} \biggr] f \biggl( \frac{\Omega^2}{\pi G \rho} \biggr) </math>	<math>~=</math>	<math>~a^2 b^2 A_{12} - c^2 A_3 \, ,</math>
[ EFE, §48, Eq. (34) ]

and the associated value of the square of the equilibrium configuration's angular velocity is,

<math>~\biggl[ 1 + \frac{a^2 b^2 \cdot f^2}{(a^2 + b^2)^2} \biggr] \frac{\Omega^2}{\pi G \rho}</math>	<math>~=</math>	<math>~2B_{12} \, ,</math>
[ EFE, §48, Eq. (33) ]

where,

<math>~A_{12}</math>	<math>~\equiv</math>	<math>~-\frac{A_1-A_2}{(a^2 - b^2)} \, ,</math>
[ EFE, §21, Eq. (107) ]
<math>~B_{12}</math>	<math>~\equiv</math>	<math>~A_2 - a^2A_{12} \, .</math>
[ EFE, §21, Eq. (105) ]

(Notice that if we set <math>~f \rightarrow 0</math>, this pair of expressions simplifies to the pair we have provided in a separate discussion of the equilibrium conditions for Jacobi ellipsoids.) Following Chandrasekhar's lead and eliminating <math>~\Omega^2</math> between these two expressions, we obtain,

<math>~0</math>	<math>~=</math>	<math>~ \biggl[ \frac{a^2 b^2}{(a^2 + b^2)^2} \biggr] f^2 + \biggl[ \frac{2a^2 b^2 B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} \biggr]\frac{f}{a^2 + b^2} + 1 \, . </math>
[ EFE, §48, Eq. (35) ]

For a given <math>~f</math>, this last expression determines the ratios of the axes of the ellipsoids that are compatible with equilibrium; and the value of <math>~\Omega^2</math>, that is to be associated with a particular solution of this last expression, then follows from either one of the first two expressions. For convenience of evaluation and for greater clarity, let's rewrite this last (quadratic) equation in the form,

<math>~0</math>

<math>~=</math>

<math>~ \alpha f^2 + \beta f + 1 \, , </math>

in which case the pair of solutions is,

<math>~f</math>

<math>~=</math>

<math>~ \frac{1}{2\alpha}\biggr\{ - \beta \pm \biggl[ \beta^2 - 4\alpha \biggr]^{1 / 2} \biggr\} \, ; </math>

and the corresponding values of the angular velocity are provided by the expression,

<math>~\omega \equiv \frac{\Omega}{\sqrt{G\rho}}</math>

<math>~=</math>

<math>~ \pm \biggl\{ 2 \pi B_{12} \biggl[ 1 + \alpha f^2 \biggr]^{-1} \biggr\}^{1 / 2} </math>

TEST (part 2)
<math>~\frac{b}{a}</math>	<math>~\frac{c}{a}</math>	<math>~a^2 A_{12}</math>	<math>~ B_{12}</math>	<math>~\alpha \equiv \frac{(b/a)^2}{[ 1 + (b/a)^2]^2} </math>	<math>~\beta \equiv \biggl[ \frac{2 B_{12}}{(c/b)^2 A_3 - a^2 A_{12}} \biggr]\frac{1}{1 + (b/a)^2} </math>	Direct		Adjoint
						<math>~f </math>	<math>~\omega = \frac{\Omega}{\sqrt{G\rho}} </math>	<math>~f^\dagger </math>	<math>~\omega^\dagger = \frac{\Omega^\dagger}{\sqrt{G\rho}} </math>
0.9	0.641	0.387793362	0.207337649	0.247245200	3.797483556	- 0.268008879	± 1.131374734	-15.09117122	± 0.150771618

Now, according to 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,

<math>~\vec{v}</math>

<math>~=</math>

<math>~\lambda \biggl( \frac{ay}{b} , - \frac{bx}{a} , 0 \biggr) \, ,</math>

where, <math>~\lambda</math> is an overall scale factor. But, according to §48 of EFE, we see that,

<math>~\vec{u}</math>

<math>~=</math>

<math>~\biggl( Q_1 y , Q_2 x , 0 \biggr) \, ,</math>

where,

<math>~Q_1</math>

<math>~\equiv</math>

<math>~- \biggl[ \frac{a^2}{a^2 + b^2} \biggr]\zeta</math>

and,

<math>~Q_2</math>

<math>~\equiv</math>

<math>~+ \biggl[ \frac{b^2}{a^2 + b^2} \biggr]\zeta \, ,</math>

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,

<math>~\lambda \biggl( \frac{a}{b} \biggr) </math>

<math>~=</math>

<math>~- \biggl[ \frac{a^2}{a^2 + b^2} \biggr]\zeta</math>

and,

<math>~- \lambda \biggl( \frac{b}{a} \biggr) </math>

<math>~=</math>

<math>~+ \biggl[ \frac{b^2}{a^2 + b^2} \biggr]\zeta </math>

<math>~\Rightarrow ~~~ \lambda </math>

<math>~=</math>

<math>~- \biggl[ \frac{a b}{a^2 + b^2} \biggr]\zeta = - \biggl[ \frac{b}{a} + \frac{a}{b} \biggr]^{-1} \zeta\, ,</math>

which, gratifyingly agrees with Ou's equation (17).

Models Examined by Ou (2006)

Table 1 (see below) lists a subset of the Riemann S-type ellipsoids that were studied by Ou (2006); properties of various so-called Direct configurations can be found in Ou's Table 1, while properties of various Adjoint configurations can be found in his Table 5. Each row of our Table 1 was constructed as follows:

• The pair of axis ratios <math>~(\tfrac{b}{a}, \tfrac{c}{a} )</math> associated with one of Ou's (2006) uniform-density, incompressible <math>~(n=0)</math> ellipsoid models (columns 1 and 2 from Ou's Table 1) has been copied into columns 1 and 2 of our table.
• Properties of Direct Configurations …
• The pair of parameter values <math>~(\omega_\mathrm{analytic}, \lambda_\mathrm{analytic})</math> that is required in order for this to be an equilibrium configuration — as specified by the above set of analytical expressions from EFE — is copied from, respectively, columns 11 and 13 of Ou's Table 1 into columns 3 and 4 of our table; in our table, the "analytic" subscript has been dropped from the column headings.
• The value of the equilibrium configuration's vorticity, <math>~\zeta</math> — see column 5 of our table — has been determined from the expression,
  

  <math>~\zeta = - \biggl[ \frac{1 + (b/a)^2}{b/a} \biggr] \lambda \, .</math>

• Column 6 of our table lists the value of the frequency ratio, <math>~f \equiv \zeta/\omega</math>.
• Properties of Adjoint Configurations [in order to distinguish from Direct configuration properties, a superscript † has been attached to each parameter name] …
• As listed in column 7 of our Table, the "spin" angular velocity of the adjoint equilibrium configuration has been determined from the vorticity of the direct configuration via the relation,
  

  <math>~\omega^\dagger = \zeta \biggl[\frac{b/a}{1 + (b/a)^2}\biggr] \, .</math>

• As listed in column 10 of our Table, the ratio <math>~(f^\dagger)</math> of the vorticity to the angular velocity in the adjoint equilibrium configuration has been determined from the same ratio <math>~(f)</math> in the direct configuration via the relation,
  

  <math>~f^\dagger = \frac{1}{f} \biggl\{ \frac{[1 + (b/a)^2]^2}{(b/a)^2} \biggr\} \, .</math>

• As indicated, the value of the vorticity in the adjoint equilibrium configuration (column 9 of our table) has been determined from a product of <math>~\omega^\dagger</math> and <math>~f^\dagger</math>.
• As listed in column 8 of our table, the value of the parameter, <math>~\lambda^\dagger</math>, has been determined from the vorticity in the adjoint equilibrium configuration via the relation,
  

  <math>~\lambda^\dagger = -~ \zeta^\dagger \biggl[ \frac{b}{a} + \frac{a}{b}\biggr]^{-1} \, .</math>

Divider1

Table 1:   Example Riemann S-type Ellipsoids [Cells with a pink background contain numbers copied directly from Table 1 of Ou (2006)] [Cells with a yellow background contain numbers drawn from Table IV (p. 103) of EFE]
<math>~\frac{b}{a}</math>	<math>~\frac{c}{a}</math>	Properties of Direct Configurations				Properties of Adjoint Configurations
		<math>~\omega = \frac{\Omega}{\sqrt{G \rho}}</math>	<math>~\lambda</math>	<math>~\zeta </math>	<math>~f \equiv \frac{\zeta}{\omega}</math>	<math>~\omega^\dagger </math>	<math>~\lambda^\dagger </math>	<math>~\zeta^\dagger = \omega^\dagger f^\dagger</math>	<math>~f^\dagger </math>
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
0.90	0.795	1.14704	0.43181	-0.86842	-0.75709	-0.43181	-1.14704	+2.30682	-5.3422
	0.641	1.13137	0.15077	- 0.30322	- 0.26801	- 0.15077	-1.13137	2.27531	- 15.0913
	0.590	1.10661	0.06406	-0.12883	-0.11642	-0.06406	-1.10661	+2.22552	-34.7411
	0.564	1.09034	0.02033	-0.04089	-0.03750	-0.02033	-1.09034	+2.19279	-107.86
	0.538	1.07148	- 0.02324	+0.04674	+0.04362	+0.02324	- 1.07148	+2.15487	+92.722
	0.487	1.02639	- 0.10880	+0.21881	+0.21318	+0.10880	-1.02639	+2.06418	+18.972
	0.333	0.79257	- 0.39224	+0.78884	+0.99529	+0.39224	-0.79257	+1.59395	+4.06370
0.28	0.256	0.80944	0.03668	-0.14127	-0.17453	-0.03668	-0.80944	+3.11750	-84.992
	0.245083	0.796512a	0.0	0.0	0.0	0.0	…	…	<math>~\infty</math>
	0.231	0.77651	- 0.04714	+0.18156	+0.23381	+0.04714	-0.77651	+2.99067	+63.442
	0.205	0.72853	- 0.13511	+0.52037	+0.71427	+0.13511	-0.72853	+2.80588	+20.7674
aAccording to Table IV (p. 103) of EFE, the square of the angular velocity of this Jacobi ellipsoid is, <math>~\Omega^2/(\pi G\rho) = 0.201946</math>; from this value, we find that, <math>~\omega = \sqrt{\pi} \cdot \sqrt{0.201946} = 0.796512</math>.

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Table 2:   Example Riemann S-type Ellipsoids
<math>~\frac{b}{a}</math>	<math>~\frac{c}{a}</math>	Properties of Direct Configurations				Properties of Adjoint Configurations
		<math>~\omega = \frac{\Omega}{\sqrt{G \rho}}</math>	<math>~\lambda</math>	<math>~\zeta </math>	<math>~f \equiv \frac{\zeta}{\Omega}</math>	<math>~\omega^\dagger </math>	<math>~\lambda^\dagger </math>	<math>~\zeta^\dagger = \omega^\dagger f^\dagger</math>	<math>~f^\dagger </math>
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
0.90	0.795	+1.147036091720	+0.431809451699	-0.868416786194	-0.757096320116	-0.431809460593	-1.147036104571	+2.306817054749	-5.342210487323
	0.641	+1.131374738327	+0.150771621841	-0.303218483925	-0.268008886644	-0.150771621877	-1.131374730590	+2.275320291519	-15.091170863305
	0.590	+1.106612583610	+0.064060198174	-0.128832176328	-0.116420305902	-0.064060197762	-1.106612576964	+2.225520849228	-34.741086358509
	0.564	+1.090339840378	+0.020334563779	-0.040895067155	-0.037506716440	-0.020334563809	-1.090339837153	+2.192794561386	-107.8358300897
	0.538	+1.071485625744	-0.023236834336	+0.046731855720	+0.043614077664	+0.023236835120	-1.071485656401	+2.154876708984	+92.735376233270
	0.487	+1.026387311947	-0.108799837242	+0.218808561563	+0.213183225210	+0.108799835209	-1.026387320039	+2.064178943634	+18.972261524065
	0.333	+0.792566980901	-0.392440787995	+0.789242029190	+0.995804846843	+0.392440793882	-0.792566979129	+1.593940258026	+4.061606964516
0.28	0.256	+0.809436834686	+0.036676037913	±0.141255140305	±0.174510396110	±0.036676038521	±0.809436833116	±3.117488145828	±85.000678306244
	0.245083	0.796512a	0.0	0.0	0.0	0.0	…	…	<math>~\infty</math>
	0.231	+0.776514825339	-0.047142035397	±0.181564182043	±0.233819345828	±0.047142037070	±0.776514835457	±2.990691423416	±63.440011724689
	0.205	+0.728526018042	-0.135108121071	±0.520359277725	±0.714263156392	±0.135108125079	±0.728526039364	±2.805866003036	±20.767558718483
aAccording to Table IV (p. 103) of EFE, the square of the angular velocity of this Jacobi ellipsoid is, <math>~\Omega^2/(\pi G\rho) = 0.201946</math>; from this value, we find that, <math>~\omega = \sqrt{\pi} \cdot \sqrt{0.201946} = 0.796512</math>.

See Also

• Properties of Maclaurin Spheroids
• Excerpts from Maclaurin's (1742) A Treatise of Fluxions

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation