User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids

Properties of Homogeneous Ellipsoids

Gravitational Potential

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,

<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>
<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>
<math> \Delta </math>	<math> \equiv </math>	<math> \biggl[ (a_1^2 + u)(a_2^2 + u)(a_3^2 + u) \biggr]^{1/2} . </math>

As is detailed below, the integrals defining <math>A_i</math> and <math>I_\mathrm{BT}</math> can be expressed in terms of standard incomplete elliptic integrals, 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).

See Also

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation