Editing Aps/MaclaurinSpheroidFreeFall (section)

==Gravitational Potential==
Here, our specific interest is in modeling the free-fall collapse of a uniform-density spheroid. Ignoring, for the moment, the time-dependent nature of this problem, we appreciate from [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Gravitational_Potential|a separate, detailed derivation]] that the gravitational potential inside (or on the surface) of an homogeneous, triaxial ellipsoid with semi-axes <math>(x, y, z) = (a_1, a_2, a_3)</math> is given, to within an arbitrary additive constant, by the expression,

<div align="center">
<math>
~\Phi(\vec{x}) = \pi G \rho \biggl[ A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr],
</math>
</div>
where the three, spatially independent coefficients, <math>A_1, A_2,</math> and <math>A_3</math> are functions of the chosen lengths of the three semi-axes.  When deriving mathematical expressions for each of the three <math>A_i</math> coefficients, in our [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Derivation_of_Expressions_for_Ai|accompanying discussion]] we have found it useful to initially attach a subscript, <math>(\ell, m, ~\mathrm{or}~ s)</math>  &#8212; indicating whether the coefficient is associated with the (largest, medium-length, or smallest) semi-axis &#8212; before specifying how, for a given problem, <math>(\ell, m, s)</math> are appropriately associated with the three <math>(x, y, z)</math> coordinate axes.  

===Oblate Spheroids===
For example, for an oblate-spheroidal mass distribution, by definition the "largest" and "medium-length" semi-axes are equal to one another.  Hence, <math>a_\ell = a_m</math> and, according to [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#When_am_.3D_a.E2.84.93|our associated derivation]],

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>A_\ell = A_m</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-
\frac{(1-e^2)}{e^2} + 
~\frac{(1-e^2)^{1 / 2}}{ e^3 } \biggl[
\sin^{-1}e
\biggr] \, ,
</math>
  </td>
</tr>
</table>

where, <math>e \equiv (1 - a_s^2/a_\ell^2)^{1 / 2}</math>; and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>A_s</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{e^2} - 
~\frac{2(1-e^2)^{1 / 2}}{ e^3 } \biggl[
\sin^{-1}e
\biggr] \, .
</math>
  </td>
</tr>
</table>
Conventionally, the <math>z</math>-axis is aligned with the symmetry (in this case, shortest) axis of the mass distribution, so we set <math>A_1 = A_2 = A_\ell</math>, and <math>A_3 = A_s</math>.  Therefore &#8212; see also our [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Oblate_Spheroids_.7F.27.22.60UNIQ--postMath-00000039-QINU.60.22.27.7F|parallel discussion]] &#8212; we appreciate that, for oblate-spheroidal mass distributions,

<div align="center">
<math>
\Phi(\vec{x}) = \pi G \rho \biggl[A_\ell \varpi^2 + A_s z^2 \biggr].
</math>
</div>
These same coefficient expressions may also be found in, for example: &nbsp;Chapter 3, Eq. (36) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]; &sect;4.5, Eqs. (48) &amp; (49) of [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>]; and the first column of Table 2-1 (p. 57) of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>].  Note that, as we have pointed out in a [[Apps/MaclaurinSpheroids#Gravitational_Potential|separate discussion of Maclaurin Spheroids]], the expressions for <math>A_\ell</math> and <math>A_s</math> have the following values in the limit of a sphere <math>(e=0)</math> or in the limit of an infinitesimally thin disk <math>(e=1)</math>:


<table align="center" border=1 cellpadding="8">
<tr>
  <td colspan="3" align="center">
'''Table 1:''' &nbsp;[[Appendix/Ramblings/PowerSeriesExpressions#Maclaurin_Spheroid_Index_Symbols|Limiting Values]] (for oblate spheroids)
  </td>
</tr>
<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<b><math>e \rightarrow 0</math></b>
  </td>
  <td align="center">
<b><math>\frac{a_3}{a_1} \rightarrow 0</math></b>
  </td>
</tr>
<tr>
  <td align="center">
<b><math>~\frac{\sin^{-1}e}{e}</math></b>
  </td>
  <td align="center">
<math>~1 + \frac{e^2}{6} + \mathcal{O}\biggl(e^4\biggr)</math>
  </td>
  <td align="center">
<math>~\frac{\pi}{2} - \biggl(\frac{a_3}{a_1}\biggr) +\frac{\pi}{4}\biggl(\frac{a_3}{a_1}\biggr)^2
- \mathcal{O}\biggl(\frac{a_3^3}{a_1^3}\biggr)</math>
  </td>
</tr>
<tr>
  <td align="center">
<b><math>A_\ell</math></b>
  </td>
  <td align="center">
<math>\frac{2}{3}\biggl[1 - \frac{e^2}{5} - \mathcal{O}\biggl(e^4\biggr)\biggr]</math>
  </td>
  <td align="center">
<math>\frac{\pi}{2} \biggl( \frac{a_3}{a_1}\biggr) - 2\biggl(\frac{a_3}{a_1}\biggr)^2+ \mathcal{O}\biggl(\frac{a_3^3}{a_1^3}\biggr)</math>
  </td>
</tr>
<tr>
  <td align="center">
<b><math>A_s</math></b>
  </td>
  <td align="center">
<math>\frac{2}{3}\biggl[1 + \frac{2e^2}{5} + \mathcal{O}\biggl(e^4\biggr)\biggr]</math>
  </td>
  <td align="center">
<math> 2 - \pi \biggl( \frac{a_3}{a_1}\biggr) + 4\biggl(\frac{a_3}{a_1}\biggr)^2
- \mathcal{O}\biggl(\frac{a_3^3}{a_1^3}\biggr)</math>
  </td>
</tr>
</table>

===Prolate Spheroids===
For a prolate-spheroidal mass distribution, by definition the "smallest" and "medium-length" semi-axes are equal to one another.  Hence, <math>a_m = a_s</math> and, according to [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#When_am_.3D_as|our associated derivation]],

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>A_\ell</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{ (1-e^2)}{e^3} \cdot 
\ln \biggl[ \frac{1+e}{ 1-e }   \biggr]
- \frac{2(1-e^2)}{e^2 } 
\, ,
</math>
  </td>
</tr>
</table>

where, as above, <math>e \equiv (1 - a_s^2/a_\ell^2)^{1 / 2}</math>; and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>A_s = A_m</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{e^2 } 
- \frac{ (1-e^2)}{2e^3} 
\cdot
\ln \biggl( \frac{1+e}{ 1-e }   \biggr) \, .
</math>
  </td>
</tr>
</table>
If the symmetry (in this case, longest) axis of this prolate-spheroidal mass distribution is aligned with the <math>z</math>-axis of the coordinate system, then we should set <math>A_1 = A_2 = A_s</math>, and <math>A_3 = A_\ell</math>.  This means that the expression for the gravitational potential is,

<div align="center">
<math>
\Phi(\vec{x}) = \pi G \rho \biggl[A_s \varpi^2 + A_\ell z^2 \biggr].
</math>
</div>
These same coefficient expressions may also be found in, for example, the second column of Table 2-1 (p. 57) of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>].


<font color="red">NOTE:</font>&nbsp; If, following [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], we instead align the longest (and, in this case, symmetry) axis of the prolate mass distribution with the <math>x</math>-axis, then <math>A_2 = A_3 = A_s</math> and <math>A_1 = A_\ell</math>.  This matches the coefficient expressions presented in our [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Prolate_Spheroids_.7F.27.22.60UNIQ--postMath-0000004C-QINU.60.22.27.7F|parallel discussion]] of the potential inside and on the surface of a prolate-spheroidal mass distribution.

===Consider a Time-Varying Eccentricity===

If the eccentricity of an homogeneous spheroid varies with time &#8212; that is, if <math>e \rightarrow e(t)</math> &#8212; while it remains homogeneous, the result will be a potential of the form,

<div align="center">
<math>
\Phi(\vec{x}, t) = A(t) \varpi^2 + C(t) z^2 \, ,
</math>
</div>

whether the spheroid is oblate or the spheroid is prolate.