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) ]

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.

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

where,

<math>~\equiv</math>

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

and,

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

and,

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

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

<math>~\Rightarrow ~~~ \lambda </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 ellipsoid axis ratios <math>~(\tfrac{b}{a}, \tfrac{c}{a} )</math> associated with one of Ou's (2006) <math>~n=0</math> models (columns 1 and 2 from Ou's Table 1) is 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 expression from EFE — is copied from, respectively, columns 11 and 13 of Ou's Table 1 into columns 3 and 4 of our table.

Table 1: Example Riemann S-type Ellipsoids
<math>~\frac{b}{a}</math>	<math>~\frac{c}{a}</math>	Properties of Direct Configurations				Properties of Adjoint Configurations
<math>~\frac{b}{a}</math>	<math>~\frac{c}{a}</math>	<math>~\omega_a = \frac{\Omega_a}{\pi G \rho}</math>	<math>~\lambda_a</math>	<math>~\zeta = - \biggl[ \frac{1 + (b/a)^2}{b/a} \biggr] \lambda_a</math>	<math>~f \equiv \zeta/\Omega_a</math>	<math>~\omega_a^\dagger = \zeta_a \biggl[\frac{b/a}{1 + (b/a)^2}\biggr]</math>	<math>~\lambda^\dagger = - \zeta^\dagger \biggl[ \frac{b}{a} + \frac{a}{b}\biggr]^{-1}</math>	<math>~\zeta^\dagger = \omega^\dagger f^\dagger</math>	<math>~f^\dagger = \frac{1}{f_a} \biggl\{ \frac{[1 + (b/a)^2]^2}{(b/a)^2} \biggr\}</math>
0.90	0.795	1.14704	0.43181
0.90	0.641	1.13137	0.15077	- 0.30322	- 0.26801	- 0.15077	-1.13137	2.27531	- 15.0913

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Contents

Riemann S-type Ellipsoids

General Coefficient Expressions

Equilibrium Conditions for Riemann S-type Ellipsoids

Models Examined by Ou (2006)

See Also

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