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>

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Acceleration at the Pole

Prolate Spheroids

In our above review, for consistency, we assumed that the longest axis of the ellipsoid was aligned with the <math>~x</math>-axis in all cases — for prolate spheroids as well as for oblate spheroids and for the more generic, triaxial ellipsoids. In this discussion, in order to better align with the operational features of a standard cylindrical coordinate system, we will orient the prolate-spheroidal configuration such that its major axis and, hence, its axis of symmetry aligns with the <math>~z</math>-axis while the center of the spheroid remains at the center of the (cylindrical) coordinate grid. In this case, the surface will be defined by the ellipse,

<math>~\frac{\varpi^2}{a_3^2} + \frac{z^2}{a_1^2} = 1 ~~~~\Rightarrow ~~~~ \varpi = a_3\sqrt{1-z^2/a_1^2} \, ,</math>

and the gravitational potential will be given by the expression,

<math> ~\Phi(\vec{x}) = -\pi G \rho \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 z^2 + A_3 \varpi^2 \biggr) \biggr]. </math>

The magnitude of the gravitational acceleration at the pole <math>~(\varpi, z) = (0, a_1)</math> of this prolate spheroid can be obtained from the gravitational potential via the expression,

<math>~\mathcal{A} \equiv \biggl|- \frac{\partial \Phi}{\partial z}\biggr|_{a_1}</math>

<math>~=</math>

<math>~2\pi G \rho A_1 a_1 \, ,</math>

where, as above,

<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>

We should also be able to derive this expression for <math>~\mathcal{A}</math> by integrating the <math>~z</math>-component of the differential acceleration over the mass distribution, that is,

<math>~\mathcal{A}</math>	<math>~=</math>	<math>~\int \biggl[ \frac{G }{r^2} \cdot \frac{(a_1-z)}{r} \biggr] dm = \int \biggl[ \frac{(a_1-z)G }{r^3} \biggr] 2\pi \varpi d\varpi dz</math>
	<math>~=</math>	<math>~2\pi G\rho \int^{a_1}_{-a_1} (a_1-z)dz \int_0^{a_3\sqrt{1-z^2/a_1^2}} [\varpi^2+(z-a_1)^2]^{-3/2}\varpi d\varpi \, ,</math>

where the distance, <math>~r</math>, has been measured from the pole, that is,

<math>~r^2 = \varpi^2 + (z-a_1)^2 \, .</math>

Performing the integral over <math>~\varpi</math> gives,

<math>~\mathcal{A}</math>	<math>~=</math>	<math>~2\pi G\rho \int^{a_1}_{-a_1} (a_1-z)dz \biggl\{ -[\varpi^2+(z-a_1)^2]^{-1/2} \biggr\}_0^{a_3\sqrt{1-z^2/a_1^2}} </math>
	<math>~=</math>	<math>~2\pi G\rho \int^{a_1}_{-a_1} (a_1-z)dz \biggl\{ \frac{1}{z - a_1} -\biggl[ a_3^2 \biggl(1-\frac{z^2}{a_1^2} \biggr) + a_1^2\biggl(1-\frac{z}{a_1}\biggr)^2 \biggr]^{-1/2} \biggr\} </math>
	<math>~=</math>	<math>~ - 2\pi G\rho a_1 \int^{1}_{-1} d\zeta \biggl\{ \frac{1-\zeta}{1-\zeta } - (1-\zeta)\biggl[ \biggl(\frac{a_3}{a_1}\biggr)^2 \biggl(1-\zeta^2 \biggr) + \biggl(1-\zeta\biggr)^2 \biggr]^{-1/2} \biggr\} </math>
	<math>~=</math>	<math>~ - 2\pi G\rho a_1 \int^{1}_{-1} d\zeta \biggl\{ 1 - (1-\zeta) [ (2-e^2) - 2\zeta + e^2\zeta^2 ]^{-1/2} \biggr\} \, , </math>

where, <math>~\zeta\equiv z/a_1</math>. For later reference, we will identify the expression inside the curly braces as the function, <math>~\mathcal{Z}</math>; specifically,

<math>~\mathcal{Z}</math>	<math>~\equiv</math>	<math>~1 - (1-\zeta) [ (2-e^2) - 2\zeta + e^2\zeta^2 ]^{-1/2} </math>
	<math>~=</math>	<math>~1 - \frac{1}{\sqrt{X}} + \frac{\zeta}{\sqrt{X}} \, ,</math>

where, in an effort to line up with notation found in integral tables, in this last expression we have used the notation, <math>~X \equiv a + b\zeta + c\zeta^2</math> and, in our case,

<math>a \equiv (2-e^2)\, ,</math>       <math>b \equiv -2\, ,</math>       and       <math>c \equiv e^2\, .</math>

We find that,

<math>~\int_{-1}^1 \mathcal{Z} d\zeta</math>	<math>~=</math>	<math>~\zeta\biggr|_{-1}^{1} + \biggl\{ \frac{\sqrt{X}}{c} \biggr\}_{-1}^1 -\biggl[1 + \frac{b}{2c} \biggr]\int_{-1}^1 \frac{d\zeta}{\sqrt{X}} </math>
	<math>~=</math>	<math>~2 + \biggl\{ \frac{\sqrt{(2-e^2) -2\zeta + e^2\zeta^2}}{e^2} \biggr\}_{-1}^1 -\biggl[1 - \frac{1}{e^2} \biggr] \biggl\{ \frac{1}{\sqrt{c}} \ln \biggl[2\sqrt{cX} + 2c\zeta + b \biggr] \biggr\}_{-1}^1 </math>
	<math>~=</math>	<math>~2 + \biggl\{ \frac{\sqrt{(2-e^2) -2 + e^2}}{e^2} \biggr\} - \biggl\{ \frac{\sqrt{(2-e^2) +2 + e^2}}{e^2} \biggr\} -\biggl[1 - \frac{1}{e^2} \biggr] \biggl\{ \frac{1}{e} \ln \biggl[2\sqrt{e^2[(2-e^2) -2\zeta + e^2\zeta^2]} + 2e^2\zeta - 2 \biggr] \biggr\}_{-1}^1 </math>
	<math>~=</math>	<math>~2 - \frac{2}{e^2} -\biggl[\frac{e^2-1}{e^3} \biggr] \biggl\{ \ln \biggl[2e^2 - 2 \biggr] - \ln \biggl[4e - 2e^2 - 2 \biggr] \biggr\} </math>
	<math>~=</math>	<math>~2\biggl[\frac{e^2 - 1}{e^2}\biggr] -\biggl[\frac{e^2-1}{e^3} \biggr] \biggl\{ \ln \biggl[-2(1-e^2) \biggr] - \ln \biggl[-2(1-e)^2\biggr] \biggr\} </math>
	<math>~=</math>	<math>~- \biggl\{ \biggl[\frac{1-e^2}{e^3} \biggr] \ln \biggl[\frac{1+e}{1-e} \biggr] -2\biggl[\frac{1-e^2 }{e^2}\biggr] \biggr\} \, . </math>

With the exception of the leading sign, this expression exactly matches the function, <math>~A_1</math>, that has been specified for prolate spheroids. Hooray!

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