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 EFE and summarized in Table 2-2 (p. 57) of BT87, 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>

[ EFE, Chapter 3, Eq. (40)^1,2 ]
[ BT87, Chapter 2, Table 2-2 ]

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>

[ EFE, Chapter 3, Eqs. (18), (15 & 22)¹, & (8), respectively ]
[ BT87, Chapter 2, Table 2-2 ]

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>

[ EFE, Chapter 3, Eq. (32) ]

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>

[ EFE, Chapter 3, Eqs. (33), (34) & (35) ]

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>

[ EFE, Chapter 3, Eq. (36) ]
[ T78, §4.5, Eqs. (48) & (49) ]

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

[ EFE, Chapter 3, Eq. (38) ]

where, again, the eccentricity,

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

Footnotes

1. In EFE this equation is written in terms of a variable <math>I</math> instead of <math>I_\mathrm{BT}</math> as defined here. The two variables are related to one another straightforwardly through the expression, <math>I = I_\mathrm{BT} a_1^2</math>.
2. Throughout EFE, Chandrasekhar adopts a sign convention for the scalar gravitational potential that is opposite to the sign convention being used here.

© 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