User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids

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,

<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>

Evaluation of Coefficients

The integrals defining <math>A_i</math> and <math>I_\mathrm{BT}</math> can be evaluated in terms of the incomplete elliptic integral of the first kind,

<math> F(\theta,k) \equiv \int_0^\theta \frac{d\theta '}{\sqrt{1 - k^2 \sin^2\theta '}} ~~ , </math>

and/or the incomplete elliptic integral of the second kind,

<math> E(\theta,k) \equiv \int_0^\theta {\sqrt{1 - k^2 \sin^2\theta '}}~d\theta ' ~~ , </math>

where, for our particular problem,

<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 the integrals can be evaluated 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 <math>(a_1 > a_2 > a_3)</math>

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>,

<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>
<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>
<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>
<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>

Oblate Spheroids <math>(a_1 = a_2 > a_3)</math>

If the longest axis (<math>a_1</math>) and the intermediate axis (<math>a_2</math>) of the ellipsoid are equal to one another, then an equatorial cross-section of the object presents a circle of radius <math>a_1</math> and the object is referred to as an oblate spheroid. For homogeneous oblate spheroids, evaluation of the integrals defining <math>A_i</math> and <math>I_\mathrm{BT}</math> gives,

<math> A_1 </math>	<math> = </math>	<math> \frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} ~~; </math>
<math> A_2 </math>	<math> = </math>	<math> A_1 ~~; </math>
<math> A_3 </math>	<math> = </math>	<math> \frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1/2} ~~; </math>
<math> I_\mathrm{BT} </math>	<math> = </math>	<math> 2A_1 + A_3 (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] ~~, </math>

where the eccentricity,

<math> e \equiv \biggl[1 - \biggl(\frac{a_3}{a_1}\biggr)^2 \biggr]^{1/2} ~~. </math>

Prolate Spheroids <math>(a_1 > a_2 = a_3)</math>

If the shortest axis (<math>a_3</math>) and the intermediate axis (<math>a_2</math>) of the ellipsoid are equal to one another, then a cross-section in the <math>x-y</math> plane of the object presents a circle of radius <math>a_3</math> and the object is referred to as a prolate spheroid. For homogeneous prolate spheroids, evaluation of the integrals defining <math>A_i</math> and <math>I_\mathrm{BT}</math> gives,

<math> A_1 </math>	<math> = </math>	<math> \ln\biggl[ \frac{1+e}{1-e} \biggr] \frac{(1-e^2)}{e^3} - \frac{2(1-e^2)}{e^2} ~~; </math>
<math> A_2 </math>	<math> = </math>	<math> \frac{1}{e^2} - \ln\biggl[ \frac{1+e}{1-e} \biggr]\frac{(1-e^2)}{2e^3} ~~; </math>
<math> A_3 </math>	<math> = </math>	<math> A_2 ~~; </math>
<math> I_\mathrm{BT} </math>	<math> = </math>	<math> \ln\biggl[ \frac{1+e}{1-e} \biggr]\frac{(1-e^2)^{1/2}}{e} ~~, </math>

where, again, the eccentricity,

<math> e \equiv \biggl[1 - \biggl(\frac{a_3}{a_1}\biggr)^2 \biggr]^{1/2} ~~. </math>

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