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) ]

If we rewrite the expression for the system's free energy in terms of <math>~L</math> (and x) instead of <math>~\Omega</math> (and x), we have,

<math>~\frac{1}{2} \biggl(\frac{5L}{M}\biggr)^2 \frac{(a+bx)^2 + (b+ax)^2}{(a^2 + b^2 + 2abx)^2} - 2I \, ,</math>

[ Paper I, Eq. (3.4) ]

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\biggl(\frac{b}{a}, \frac{c}{a}, \Omega, x\biggr)</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, replacing <math>~\Omega</math> in favor of <math>~L</math>, we have,

<math>~E\biggl(\frac{b}{a}, \frac{c}{a}, L, x\biggr)</math>	<math>~=</math>	<math>~\frac{L^2}{2a^2} \biggl[ \biggl(1+\frac{b}{a}\cdot x \biggr)^2 + \biggl(\frac{b}{a}+x \biggr)^2 \biggr] \biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2 + 2\biggl(\frac{b}{a}\biggr)x \biggr]^{-2} - 2I </math>
	<math>~=</math>	<math>~\frac{L^2}{2} \biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{-2/3} \biggl[ \biggl(1+\frac{b}{a}\cdot x \biggr)^2 + \biggl(\frac{b}{a}+x \biggr)^2 \biggr] \biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2 + 2\biggl(\frac{b}{a}\biggr)x \biggr]^{-2} </math>
		<math>~- 2\biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{2/3} \biggl[A_1 + A_2\biggl(\frac{b}{a}\biggr)^2 + A_3\biggl(\frac{c}{a}\biggr)^2 \biggr]\, .</math>

Adopted Evolutionary Constraints

Conserve Only L

Let's fix the total angular momentum, <math>~L</math>, of a triaxial configuration and examine how its free energy varies as we allow it to contort through different triaxial shapes — that is, as its pair of axis ratios varies, always maintaining <math>~\tfrac{b}{a} < 1</math> — and as we vary <math>~x</math>, which characterizes the fraction of angular momentum that is stored in internal spin versus overall figure rotation. The desired free-energy function, <math>~E(\tfrac{b}{a},\tfrac{c}{a}, x)|_L</math>, has just been defined, but visualizing its behavior is difficult because, in this situation, the free energy is a three-dimensional "surface" lying in the four-dimensional domain, <math>~(\tfrac{b}{a},\tfrac{c}{a}, x, E_L)</math>. But, acknowledging that we are primarily interested in identifying extrema of this free-energy function, the discussion presented in §3.2 of Christodoulou et al. (1995) shows us how to reduce the dimensionality of this problem by one. There, it is shown that, as long as <math>~\tfrac{b}{a} \ne 1</math>, extrema exist in the <math>~x</math>-coordinate direction — that is, <math>~\partial E_L/\partial x = 0</math> — only if <math>~x = 0.</math> For a given choice of <math>~L</math>, therefore, the relevant free-energy "surface" is defined by the expression,

<math>~E\biggl(\frac{b}{a}, \frac{c}{a}\biggr)\biggr|_L</math>

<math>~\frac{L^2}{2} \biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{-2/3} \biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2\biggr]^{-1} - 2\biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{2/3} \biggl[A_1 + A_2\biggl(\frac{b}{a}\biggr)^2 + A_3\biggl(\frac{c}{a}\biggr)^2 \biggr]\, .</math>

Figure 3 of Christodoulou et al. (1995) presents a contour plot of this <math>~E_L</math> function for the specific case of <math>~L = 4.71488</math>, which, for reference, is the total angular momentum of an equilibrium Maclaurin spheroid having an eccentricity, <math>~e = 0.85</math>. We have reprinted this black-and-white contour plot in the left panel of row (i) of our Figure 1 montage; note, however, that we have flipped the plot horizontally and rotated it by 90° in order that the orientation of the axis pair, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math>, conform to the orientation used by Chandrasekhar (1965).

Figure 1: 3D Free-Energy Surface & Its Projection onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> Plane