User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids
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Properties of Homogeneous Ellipsoids
Gravitational Potential
The Defining Integral Expressions
As has been shown in a separate discussion (not yet typed!), the acceleration due to the gravitational attraction of a distribution of mass <math>~\rho</math><math>(\vec{x})</math> can be derived from the gradient of a scalar potential <math>~\Phi</math><math>(\vec{x})</math> defined as follows:
<math> \Phi(\vec{x}) \equiv - \int \frac{G \rho(\vec{x}')}{|\vec{x}' - \vec{x}|} d^3 x' . </math>
As has been explicitly demonstrated in Chapter 3 of Chandrasekhar (1987) and summarized in Table 2-2 (p. 57) of Binney & Tremaine (1987), for an homogeneous ellipsoid this volume integral can be evaluated analytically in closed form. Specifically, at an internal point or on the surface of an homogeneous ellipsoid with semi-axes <math>(x,y,z) = (a_1,a_2,a_3)</math>,
<math>
\Phi(\vec{x}) = -\pi G \rho \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr],
</math>
where,
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<math> A_i </math> |
<math> \equiv </math> |
<math> a_1 a_2 a_3 \int_0^\infty \frac{du}{\Delta (a_i^2 + u )} , </math> |
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<math> I_\mathrm{BT} </math> |
<math> \equiv </math> |
<math> \frac{a_2 a_3}{a_1} \int_0^\infty \frac{du}{\Delta} = A_1 + A_2\biggl(\frac{a_2}{a_1}\biggr)^2+ A_3\biggl(\frac{a_3}{a_1}\biggr)^2 , </math> |
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<math> \Delta </math> |
<math> \equiv </math> |
<math> \biggl[ (a_1^2 + u)(a_2^2 + u)(a_3^2 + u) \biggr]^{1/2} . </math> |
Evaluation of Coefficients
The integrals defining <math>A_i</math> and <math>I_\mathrm{BT}</math> can be expressed in terms of standard incomplete elliptic integrals of the first kind, <math>F(\theta,k)</math>, and second kind, <math>E(\theta,k)</math>, where,
<math>
\theta \equiv \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr) ,
</math>
<math> k \equiv \biggl[\frac{(a_1^2 - a_2^2)}{(a_1^2 - a_3^2)} \biggr]^{1/2} , </math>
or in terms of more elementary functions if either <math>a_2 = a_1</math> (oblate spheroids) or <math>a_3 = a_2</math> (prolate spheroids).
Triaxial Configurations
If the three principal axes of the configuration are unequal in length and related to one another such that <math>a_1 > a_2 > a_3 </math>,
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<math> A_1 </math> |
<math> = </math> |
<math> \frac{2a_2 a_3}{a_1^2} \biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] ~~; </math> |
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<math> A_2 </math> |
<math> = </math> |
<math> \frac{2a_2 a_3}{a_1^2} \biggl[ \frac{E(\theta,k) - (1-k^2)F(\theta,k) - (a_3/a_2)k^2\sin\theta}{k^2 (1-k^2) \sin^3\theta}\biggr] ~~; </math> |
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<math> A_3 </math> |
<math> = </math> |
<math> \frac{2a_2 a_3}{a_1^2} \biggl[ \frac{(a_2/a_3) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] ~~; </math> |
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<math> I_\mathrm{BT} </math> |
<math> = </math> |
<math> \frac{2a_2 a_3}{a_1^2} \biggl[ \frac{F(\theta,k)}{\sin\theta} \biggr] ~~. </math> |
See Also
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