User:Tohline/ThreeDimensionalConfigurations/JacobiEllipsoids
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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, <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:
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<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> |
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<math> ~A_3 </math> |
<math> ~= </math> |
<math> ~\biggl(\frac{b}{a}\biggr) \biggl[ \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, , </math> |
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<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,
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<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,
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<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,
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<math>~\frac{\Omega^2}{\pi G \rho}</math> |
<math>~=</math> |
<math>~2B_{12} \, ,</math> |
| [ EFE, §39, Eq. (5) ] | ||
where,
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<math>~A_{12}</math> |
<math>~\equiv</math> |
<math>~-\frac{A_1-A_2}{(a^2 - b^2)} \, ,</math> |
| [ EFE, §21, Eq. (107) ] | ||
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<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:
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Roots of the Governing Relation
To simplify notation, here we will set,
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<math>~x \equiv \frac{b}{a}</math> |
and |
<math>~y \equiv \frac{c}{a} \, ,</math> |
in which case the governing relation is,
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<math>~f_J</math> |
<math>~=</math> |
<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 value of <math>~x</math> that defines the root. Using this approach, the Newton-Raphson technique requires specification of, not only the function, <math>~f_J</math>, but also its first derivative,
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<math>~f_J^'</math> |
<math>~=</math> |
<math>~\frac{df}{dx} \, .</math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |