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<math>~- \biggl[ \frac{a b}{a^2 + b^2} \biggr]\zeta \, .</math>
<math>~- \biggl[ \frac{a b}{a^2 + b^2} \biggr]\zeta = - \biggl[ \frac{b}{a} + \frac{a}{b} \biggr]^{-1} \zeta\, ,</math>
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which, gratifyingly agrees with Ou's equation (17).


=See Also=
=See Also=

Revision as of 04:10, 19 August 2019


Riemann S-type Ellipsoids

Whitworth's (1981) Isothermal Free-Energy Surface
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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 + \biggl( \frac{a^2 b^2}{a^2 + b^2} \biggr) f^2 \biggr] \frac{\Omega^2}{\pi G \rho}</math>

<math>~=</math>

<math>~2B_{12} \, ,</math>

[ EFE, §39, Eq. (5) ]

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>~\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 vorticity. The transformation from EFE's notation to the one used by Ou is, then,

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

<math>~\equiv</math>

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

      and,      

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

<math>~\equiv</math>

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

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

<math>~\equiv</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).

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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