Editing ThreeDimensionalConfigurations/HomogeneousEllipsoids (section)

==Gravitational Potential==

===The Defining Integral Expressions===

As has been shown in a separate discussion titled, [[PGE/PoissonOrigin#Origin_of_the_Poisson_Equation|"Origin of the Poisson Equation,"]] the acceleration due to the gravitational attraction of a distribution of mass {{Math/VAR_Density01}} <math>(\vec{x})</math> can be derived from the gradient of a scalar potential {{Math/VAR_NewtonianPotential01}} <math>(\vec{x})</math> defined as follows:

<div align="center">
<math>
\Phi(\vec{x}) \equiv - \int \frac{G \rho(\vec{x}')}{|\vec{x}' - \vec{x}|} d^3 x' .
</math>
</div>

As has been explicitly demonstrated in Chapter 3 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] and summarized in Table 2-2 (p. 57) of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], 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>,

<div align="center">
<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><br />

[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eq. (40)</font><sup>1,2</sup> <br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], <font color="#00CC00">Chapter 2, Table 2-2</font>
</div>

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 15 August 2020:  This integral definition of A_i also appears as Eq. (5) of &sect;10.2 (p. 234) of T78, but it contains an error &#8212; in the denominator on the right-hand-side, a_1 appears instead of a_i.]]where,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
A_i
</math>
  </td>
  <td align="center">
<math>
\equiv
</math>
  </td>
  <td align="left">
<math>
a_1 a_2 a_3 \int_0^\infty \frac{du}{\Delta (a_i^2 + u )} ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>I_\mathrm{BT}</math>  </td>
  <td align="center"><math>\equiv</math>  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Delta
</math>
  </td>
  <td align="center">
<math>
\equiv
</math>
  </td>
  <td align="left">
<math>
\biggl[ (a_1^2 + u)(a_2^2 + u)(a_3^2 + u)  \biggr]^{1/2} .
</math>
  </td>
</tr>
</table>

<div align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eqs. (18), (15 &amp; 22)</font><sup>1</sup><font color="#00CC00">, &amp; (8)</font>, respectively<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], <font color="#00CC00">Chapter 2, Table 2-2</font> 
</div>

This definite-integral definition of <math>A_i</math> may also be found in:
* [<b>[[Appendix/References#Lamb32|<font color="red">Lamb32</font>]]</b>]: as Eq. (6) in &sect;114 (p. 153); and as Eq. (5) in &sect;373 (p. 700).
* [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>]: as Eq. (5) in &sect;10.2 (p. 234), but note that there is an error in the denominator of the right-hand-side &#8212; <math>a_1</math> appears instead of <math>a_i</math>.

===Evaluation of Coefficients===

As is [[#Derivation_of_Expressions_for_Ai|detailed below]], the integrals defining <math>A_i</math> and <math>I_\mathrm{BT}</math> can be evaluated in terms of the [http://en.wikipedia.org/wiki/Elliptic_integral#Incomplete_elliptic_integral_of_the_first_kind incomplete elliptic integral of the first kind],

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

and/or the [http://en.wikipedia.org/wiki/Elliptic_integral#Incomplete_elliptic_integral_of_the_second_kind incomplete elliptic integral of the second kind],

<div align="center">
<math>
E(\theta,k) \equiv \int_0^\theta {\sqrt{1 - k^2 \sin^2\theta '}}d\theta ' \, ,
</math>
</div> 

where, for our particular problem,

<div align="center">
<math>
\theta \equiv \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr) \, ,
</math><br />

<math>
k \equiv \biggl[\frac{a_1^2 - a_2^2}{a_1^2 - a_3^2} \biggr]^{1/2} = \biggl[\frac{1 - (a_2/a_1)^2}{1 - (a_3/a_1)^2} \biggr]^{1 / 2} \, ,
</math><br />

[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eq. (32)</font> 

</div>

or the integrals can be evaluated in terms of more elementary functions if either <math>a_2 = a_1</math> ([[#Oblate_Spheroids|oblate spheroids]]) or <math>a_3 = a_2</math> ([[#Prolate_Spheroids|prolate spheroids]]).

<span id="triaxial">&nbsp;</span>
====Triaxial Configurations (a<sub>1</sub> > a<sub>2</sub> > a<sub>3</sub>)====

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>, 

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
A_1
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2} \biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_2
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_3
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
<math>
I_\mathrm{BT}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{2a_2 a_3}{a_1^2} \biggl[ \frac{F(\theta,k)}{\sin\theta} \biggr] \, .
</math>
  </td>
</tr>

</table>

<div align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eqs. (33), (34) &amp; (35)</font> 
</div>

Notice that there is no need to specify the actual value of <math>a_1</math> in any of these expressions, as they each can be written in terms of the pair of axis ''ratios'', <math>a_2/a_1</math> and <math>a_3/a_1</math>.  As a sanity check, let's see if these three expressions can be related to one another in the manner described by equation (108) in &sect;21 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], namely, 
<div align="center">
<math>\sum_{\ell=1}^3 A_\ell = 2 \, .</math>
</div>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{a_1^2}{2a_2 a_3} \biggl[A_1 + A_3 + A_2\biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} 
+ \frac{(a_2/a_3) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>+ \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}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{k^2(1-k^2)\sin^3\theta} \biggl\{(1-k^2)F(\theta,k) - (1-k^2)E(\theta,k) 
+ k^2(a_2/a_3) \sin\theta </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>- k^2E(\theta,k)  + E(\theta,k) - (1-k^2)F(\theta,k) - (a_3/a_2)k^2\sin\theta\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{(1-k^2)\sin^2\theta} \biggl[ \frac{a_2}{a_3} - \frac{a_3}{a_2} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{a_1^2}{a_2 a_3} \, .</math>
  </td>
</tr>
</table>

Q.E.D.

<span id="oblate">&nbsp;</span>

====Oblate Spheroids (a<sub>1</sub> = a<sub>2</sub> > a<sub>3</sub>)====

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, 

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
A_1
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{e^2} \biggl[  \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} ~~;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_2
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
A_1  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>A_3</math>  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
\frac{2}{e^2} \biggl[  (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1 / 2} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>I_\mathrm{BT}</math>  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
2A_1 + A_3 (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] \, ,
</math>
  </td>
</tr>

</table>

<div align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eq. (36)</font><br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], <font color="#00CC00">&sect;4.5, Eqs. (48) &amp; (49)</font>
</div>
where the eccentricity,
<div align="center">
<math>
e \equiv \biggl[1 - \biggl(\frac{a_3}{a_1}\biggr)^2  \biggr]^{1 / 2} \, .
</math>
</div>

<span id="prolate">&nbsp;</span>

====Prolate Spheroids (a<sub>1</sub> > a<sub>2</sub> = a<sub>3</sub>)====

If the shortest axis <math>(a_3)</math> and the intermediate axis <math>(a_2)</math> of the ellipsoid are equal to one another &#8212; and the symmetry (longest, <math>a_1</math>) axis is aligned with the <math>x</math>-axis &#8212; then a cross-section in the <math>y-z</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, 

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
A_1
</math>
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
\ln\biggl[ \frac{1+e}{1-e} \biggr] \frac{(1-e^2)}{e^3} - \frac{2(1-e^2)}{e^2} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_2
</math>
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
\frac{1}{e^2} - \ln\biggl[ \frac{1+e}{1-e} \biggr]\frac{(1-e^2)}{2e^3}  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_3
</math>
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>
A_2 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
I_\mathrm{BT}
</math>
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>~
A_1 + 2(1-e^2)A_2 = \ln\biggl[ \frac{1+e}{1-e} \biggr]\frac{(1-e^2)}{e} \, ,
</math>
  </td>
</tr>
</table>
<div align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, Eq. (38)</font>
</div>

where, again, the eccentricity,
<div align="center">
<math>
e \equiv \biggl[1 - \biggl(\frac{a_3}{a_1}\biggr)^2  \biggr]^{1/2} \, .
</math>
</div>


<font color="red">NOTE:</font>&nbsp; If, instead, the longest (and, in this case, symmetry) axis of the prolate mass distribution is aligned with the <math>z</math>-axis &#8212; in which case, <math>a_1 = a_2 < a_3</math> &#8212; then, <math>e = (1 - a_1^2/a_3^2)^{1 / 2}</math> and the mathematical expressions for the <math>A_i</math> coefficients must be altered; they are essentially "swapped."  This modified set of coefficient expressions can be found in a [[Aps/MaclaurinSpheroidFreeFall#Prolate_Spheroids|parallel discussion]] of the potential inside and on the surface of prolate-spheroidal mass distributions, as well as in the second column of Table 2-1 (p. 57) of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>].