Jacobi 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 Jacobi Ellipsoids

Pulling from Chapter 6 — specifically, §39 — of Chandrasekhar's EFE, we understand that the semi-axis ratios, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> associated with Jacobi ellipsoids are given by the roots of the equation,

<math>~a^2 b^2 A_{12}</math>	<math>~=</math>	<math>~c^2 A_3 \, ,</math>
[ EFE, §39, Eq. (4) ]

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

<math>~\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) ]

Taken together, we see that, written in terms of the two primary coefficients, <math>~A_1</math> and <math>~A_3</math>, the pair of defining relations for Jacobi ellipsoids is:

<math>~f_J</math>	<math>~\equiv</math>	<math>~\biggl(\frac{b}{a}\biggr)^2 \biggl[ \frac{2(1-A_1)-A_3}{1 - (b/a)^2} \biggr]-\biggl(\frac{c}{a}\biggr)^2 A_3 =0 </math>
and
<math>~\frac{\Omega^2}{\pi G \rho}</math>	<math>~=</math>	<math>~2\biggl\{2 - (A_1+A_3) - \biggl[ \frac{2(1-A_1)-A_3}{1 - (b/a)^2} \biggr] \biggr\}</math>

Roots of the Governing Relation

To simplify notation, here we will set,

<math>~x \equiv \frac{b}{a}</math>

and

<math>~y \equiv \frac{c}{a} \, ,</math>

in which case the governing relation is,

<math>~\frac{x^2}{1-x^2} \biggl[ 2(1-A_1)-A_3\biggr]-y^2 A_3 =0 \, .</math>

Our plan is to employ the Newton-Raphson method to find the root(s) of the <math>~f_J = 0</math> relation, typically holding <math>~y</math> fixed and using the Newton-Raphson technique to identify the corresponding "root" value of <math>~x</math>. Using this approach, the Newton-Raphson technique requires specification of, not only the function, <math>~f_J</math>, but also its first derivative,

Let's determine the requisite expression, using a prime superscript to indicate differentiation with respect to <math>~x</math>.

<math>~ \biggl[ 2(1-A_1)-A_3\biggr]\biggl[ \frac{2x}{(1-x^2)^2} \biggr] +\frac{x^2}{1-x^2} \biggl[ 2(1-A_1^')-A_3^'\biggr] -y^2 A_3^' \, , </math>

where, given that <math>~\theta</math> does not depend on <math>~x</math>,

<math> ~A_1^' </math>	<math> ~= </math>	<math>~\frac{2y}{\sin^3\theta} \cdot \frac{d}{dx}\biggl\{ \frac{x}{k^2} \biggl[ F(\theta,k) - E(\theta,k) \biggr] \biggr\} </math>
	<math> ~= </math>	<math>~\frac{2y}{k^3 \sin^3\theta} \cdot \biggl\{ [ F - E ] [1 - 2xk^' ] +xk [ F^' - E^' ]\biggr\} \, , </math>
<math> ~A_3^' </math>	<math> ~= </math>	<math> ~\frac{1}{\sin^3\theta} \cdot \frac{d}{dx}\biggl\{ \frac{x}{(1-k^2)} \biggl[ x \sin\theta - yE(\theta,k)\biggr] \biggr\} </math>
	<math> ~= </math>	<math> ~\frac{1}{(1-k^2)^2\sin^3\theta} \biggl\{ \biggl[ x \sin\theta - yE\biggr]\biggl[ (1-k^2) +2xkk^' \biggr] + x(1-k^2) \biggl[ \sin\theta - yE^'\biggr] \biggr\}\, , </math>
<math>~k^'</math>	<math>~=</math>	<math>~ \frac{d}{dx}\biggl[\frac{1 - x^2}{1 - y^2} \biggr]^{1/2} = \frac{-x}{(1 - x^2)^{1/2}(1 - y^2)^{1/2}} \, , </math>
<math>~F^'</math>	<math>~=</math>	<math>~ \frac{\partial F(\theta,k)}{\partial k} \cdot k^' \, , </math>
<math>~E^'</math>	<math>~=</math>	<math>~ \frac{\partial E(\theta,k)}{\partial k} \cdot k^' \, . </math>

Now, according to online WolframResearch documentation — see, in particular, the subsection titled, "Representations of Derivatives" —

<math>~\frac{\partial F(z|m)}{\partial m}</math>

and,

<math>~\frac{\partial E(z|m)}{\partial m}</math>

where, <math>~z~\leftrightarrow~\theta</math>, and,

<math>~m \equiv k^2 ~~~~\Rightarrow~~~~\frac{dm}{dk} = 2k \ .</math>

Hence, we have,

<math>~F^'</math>	<math>~=</math>	<math>~ \biggl[\frac{\partial F(z\|m)}{\partial m} \cdot \frac{dm}{dk}\biggr] k^' </math>
	<math>~=</math>	<math>~ \biggl[ \frac{E(\theta,k)}{2(1-k^2)k^2} - \frac{F(\theta,k)}{2k^2} - \frac{\sin(2\theta)}{4(1-k^2)\sqrt{1-k^2\sin^2\theta}} \biggr] 2kk^' \, , </math>
<math>~E^'</math>	<math>~=</math>	<math>~ \biggl[ \frac{\partial E(z\|m)}{\partial m} \cdot \frac{dm}{dk}\biggr] k^' </math>
	<math>~=</math>	<math>~ \biggl[ E(\theta,k) - F(\theta,k) \biggr] \frac{k^'}{k} \, . </math>

This, then, gives us all of the expressions necessary to specify the derivative, <math>~f_J^'</math> analytically.

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Contents

Jacobi Ellipsoids

General Coefficient Expressions

Equilibrium Conditions for Jacobi Ellipsoids

Roots of the Governing Relation

See Also

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