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# Jacobi Ellipsoids

## General Coefficient Expressions

As has been detailed in an accompanying chapter, the gravitational potential anywhere inside or on the surface, $~(a_1,a_2,a_3) ~\leftrightarrow~(a,b,c)$, of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:

 $~A_1$ $~=$ $~2\biggl(\frac{b}{a}\biggr)\biggl(\frac{c}{a}\biggr) \biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,$ $~A_3$ $~=$ $~\biggl(\frac{b}{a}\biggr) \biggl[ \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,$ $~A_2$ $~=$ $~2 - (A_1+A_3) \, ,$

where, $~F(\theta,k)$ and $~E(\theta,k)$ are incomplete elliptic integrals of the first and second kind, respectively, with arguments,

 $~\theta = \cos^{-1} \biggl(\frac{c}{a} \biggr)$ and $~k = \biggl[\frac{1 - (b/a)^2}{1 - (c/a)^2} \biggr]^{1/2} \, .$ [ 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, $~(\tfrac{b}{a},\tfrac{c}{a})$ associated with Jacobi ellipsoids are given by the roots of the equation,

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

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

 $~\frac{\Omega^2}{\pi G \rho}$ $~=$ $~2B_{12} \, ,$ [ EFE, §39, Eq. (5) ]

where,

 $~A_{12}$ $~\equiv$ $~-\frac{A_1-A_2}{(a^2 - b^2)} \, ,$ [ EFE, §21, Eq. (107) ] $~B_{12}$ $~\equiv$ $~A_2 - a^2A_{12} \, .$ [ EFE, §21, Eq. (105) ]

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

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

## Roots of the Governing Relation

To simplify notation, here we will set,

 $~x \equiv \frac{b}{a}$ and $~y \equiv \frac{c}{a} \, ,$

in which case the governing relation is,

 $~f_J$ $~=$ $~\frac{x^2}{1-x^2} \biggl[ 2(1-A_1)-A_3\biggr]-y^2 A_3 =0 \, .$

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

 $~f_J^'$ $~=$ $~\frac{df_J}{dx} \, .$

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

 $~f_J^'$ $~=$ $~ \biggl[ 2(1-A_1)-A_3\biggr]\biggl[ \frac{2x}{(1-x^2)^2} \biggr] +\frac{x^2}{1-x^2} \biggl[ 2(1-A_1^')-A_3^'\biggr] -y^2 A_3^' \, ,$

where, given that $~\theta$ does not depend on $~x$,

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

Now, according to online Wolfram documentation,