Properties of Homogeneous Ellipsoids (2)

In addition to pulling from §53 of Chandrasekhar's EFE, here, we lean heavily on the papers by M. D. Weinberg & S. Tremaine (1983, ApJ, 271, 586) (hereafter, WT83) and by D. M. Christodoulou, D. Kazanas, I. Shlosman, & J. E. Tohline (1995, ApJ, 446, 472) (hereafter, Paper I).

Sequence-Defining Dimensionless Parameters

A Riemann sequence of S-type ellipsoids is defined by the value of the dimensionless parameter,

<math>~\equiv</math>

<math>~\frac{\zeta}{\Omega} = </math> constant,

[ EFE, §48, Eq. (31) ]
[ WT83, Eq. (5) ]
[ Paper I, Eq. (2.1) ]

where, <math>~\zeta</math> is the system's vorticity as measured in a frame rotating with angular velocity, <math>~\Omega</math>. Alternatively, we can use the dimensionless parameter,

<math>~\equiv</math>

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

[ EFE, §48, Eq. (40) ]
[ Paper I, Eq. (2.2) ]

or,

<math>~\Lambda</math>

<math>~\equiv</math>

<math>~-\biggl[\frac{ab}{a^2 + b^2} \biggr] \Omega f = -\Omega x \, .</math>

[ WT83, Eq. (4) ]

Conserved Quantities

Algebraic expressions for the conserved energy, <math>~E</math>, angular momentum, <math>~L</math>, and circulation, <math>~C</math>, are, respectively,

<math>~E</math>	<math>~=</math>	<math>~\frac{1}{2}v^2 + \frac{1}{2}(a^2 + b^2)(\Lambda^2 + \Omega^2) - 2ab\Lambda\Omega - 2I </math>
	<math>~\rightarrow</math>	<math>~\cancelto{0}{\frac{1}{2}v^2} + \frac{1}{2} [(a+bx)^2 + (b+ax)^2]\Omega^2 - 2I \, ,</math>

[ 1^st expression — EFE, §53, Eq. (239) ]
[ 2^nd expression — Paper I, Eq. (2.7) ]

where — see an accompanying discussion for the definitions of <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>,

[ 1^st expression — EFE, §53, Eq. (239) ]
[ 2^nd expression — Paper I, Eq. (2.8) ]

<math>~\frac{5L}{M}</math>	<math>~=</math>	<math>~(a^2 + b^2)\Omega - 2ab\Lambda</math>
	<math>~=</math>	<math>~ (a^2 + b^2 + 2abx)\Omega \, ;</math>

[ 1^st expression — EFE, §53, Eq. (240) ]
[ 2^nd expression — Paper I, Eq. (2.5) ]

<math>~\frac{5C}{M}</math>	<math>~=</math>	<math>~(a^2 + b^2)\Lambda - 2ab\Omega</math>
	<math>~=</math>	<math>~- [2ab + (a^2 + b^2)x ]\Omega \, .</math>

[ 1^st expression — EFE, §53, Eq. (241) ]
[ 2^nd expression — Paper I, Eq. (2.6) ]

Note that, based on the units chosen in Paper I, <math>~M = 5</math>, and <math>~abc = 15/4</math>.

Aside: Chandra's Notation

According to equation (107) in §21 of EFE, it appears as though,

And, according to equation (105) in §21 of EFE, it appears as though,

So, for example,

<math>~-\biggl[ \frac{A_1 - A_2}{a_1^2 - a_2^2} \biggr] \, ,</math>

and,

<math>~B_{12} </math>	<math>~=</math>	<math>~A_2 + a_1^2\biggl[ \frac{A_1 - A_2}{a_1^2 - a_2^2} \biggr] </math>
	<math>~=</math>	<math>~\frac{(a_1^2 - a_2^2)A_2 + a_1^2(A_1 - A_2)}{a_1^2 - a_2^2} </math>
	<math>~=</math>	<math>~\frac{a_1^2A_1 - a_2^2A_2 }{a_1^2 - a_2^2} \, .</math>

Free Energy Surface(s)

Scope

Consider a self-gravitating ellipsoid having the following properties:

Semi-axis lengths, <math>~(x,y,z)_\mathrm{surface} = (a,b,c)</math>, and corresponding volume, <math>~4\pi/(3abc)</math> ; and consider only the situations <math>0 \le b/a \le 1</math> and <math>0 \le c/a \le 1</math> ;
Total mass, <math>~M</math> ;
Uniform density, <math>~\rho = (3 M)/(4\pi abc) </math> ;
Figure is spinning about its c axis with angular velocity, <math>~\Omega</math> ;
Internal, steady-state flow exhibiting the following characteristics:

No vertical (z) motion;
Elliptical (x-y plane) streamlines everywhere having an ellipticity that matches that of the overall figure, that is, <math>~e = (1-b^2/a^2)^{1/2}</math> ;
The velocity components, <math>~v_x</math> and <math>~v_y</math>, are linear in the coordinate and, overall, characterized by the magnitude of the vorticity, <math>~\zeta</math> .

Such a configuration is uniquely specified by the choice of six key parameters: <math>~a</math>, <math>~b</math>, <math>~c</math>, <math>~M</math>, <math>~\Omega</math>, and <math>~\zeta</math> .

Free Energy of Incompressible, Constant Mass Systems

We are interested, here, in examining how the free energy of such a system will vary as it is allowed to "evolve" as an incompressible fluid — i.e., holding <math>~\rho</math> fixed — through different ellipsoidal shapes while conserving its total mass. Following Paper I, we choose to set <math>~M = 5</math> — which removes mass from the list of unspecified key parameters — and we choose to set <math>~\rho = \pi^{-1}</math>, which is then reflected in a specification of the semi-axis, <math>~a</math>, in terms of the pair of dimensionless axis ratios, <math>~b/a</math> and <math>~c/a</math>, namely,

<math>~\frac{3Ma^2}{4\pi(bc)\rho} = \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1}\, .</math>

Moving forward, then, a unique ellipsoidal configuration is identified via the specification of four, rather than six, key parameters — <math>~b/a</math>, <math>~c/a</math>, <math>~\Omega</math>, and <math>~x</math> — and the free energy of that configuration is given by the expression,

<math>~E</math>	<math>~=</math>	<math>~\frac{a^2}{2} \biggl[\biggl(1+\frac{b}{a} \cdot x\biggr)^2 + \biggl(\frac{b}{a}+x\biggr)^2\biggr]\Omega^2 - 2I </math>
	<math>~=</math>	<math>~\biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{2/3} \biggl\{\frac{1}{2} \biggl[\biggl(1+\frac{b}{a} \cdot x\biggr)^2 + \biggl(\frac{b}{a}+x\biggr)^2\biggr]\Omega^2 - \frac{2I}{a^2}\biggr\} \, ,</math>

where,

<math>~x</math>	<math>~\equiv</math>	<math>~\biggl[\frac{(b/a)}{1 + (b/a)^2} \biggr]\frac{\zeta}{\Omega} \, ,</math>
<math>~\frac{I}{a^2}</math>	<math>~=</math>	<math>~\biggl[A_1 + A_2\biggl(\frac{b}{a}\biggr)^2 + A_3\biggl(\frac{c}{a}\biggr)^2 \biggr] \, ,</math>

and the functional behavior of the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>, are given by the expressions provided in an accompanying discussion.

Alternatively, we can write,

<math>~E</math>	<math>~=</math>	<math>~\frac{1}{2}(a^2 + b^2)(\Lambda^2 + \Omega^2) - 2ab\Lambda\Omega - 2I </math>
	<math>~=</math>	<math>~a^2 \biggl\{ \frac{\Omega^2}{2}\biggl[1 + \biggl(\frac{b}{a}\biggr)^2\biggr]\biggl(\frac{\Lambda^2}{\Omega^2} + 1\biggr) - 2\Omega^2\biggl(\frac{b}{a}\biggr)\frac{\Lambda}{\Omega} - \frac{2I}{a^2} \biggl\}</math>
	<math>~=</math>	<math>~a^2 \biggl\{ \frac{\Omega^2}{2}\biggl[1 + \biggl(\frac{b}{a}\biggr)^2\biggr] \biggl[ \frac{(b/a)^2f^2}{(1 + b^2/a^2)^2} + 1\biggr] + \biggl(\frac{2\Omega^2 f}{1 + b^2/a^2} \biggr) - \frac{2I}{a^2} \biggl\}</math>