Editing Aps/MaclaurinSpheroidFreeFall (section)

===Evolution of the Spheroid===
Following his examination of the motion of an individual particle, {{ LB62hereafter }} recognized that, <font color="darkgreen">"<math>R(t), \varphi(t)</math>, and <math>Z(t)</math> are all independent of <math>\varpi_0, \phi_0</math>, and <math>z_0</math> because</font> [none of the three evolutionary equations] <font color="darkgreen">nor the above initial conditions mention them."</font>  You only need to integrate the coupled set of governing relations once then &#8212; assuming that the functions, <math>A(t)</math> and <math>C(t)</math>, are the same in all cases &#8212; the time-dependent coordinates of any particle are given by <math>(\varpi_0 R, \phi_0 + \varphi, z_0 Z)</math>, where <math>(\varpi_0 , \phi_0, z_0)</math> are the initial coordinates of that particle.  <font color="darkgreen">"Thus the result of the motion is merely a change of scales."</font>

Consider then, as did {{ LB62hereafter }}, the evolution of a spheroid that is initially uniformly filled with free particles and whose only motion, initially, is uniform rotation, <math>\Omega</math>, about the z-axis.  As {{ LB62hereafter }} puts it, since the motion of each particle can be described merely via a change of scales: <font color="darkgreen">"&hellip; the distribution of the particles remains uniform, and the boundary remains spheroidal"</font>; and, while the angular frequency of each particle, <math>\dot\phi</math>, varies with time, <font color="darkgreen">"&hellip; since <math>\dot\phi = \dot\varphi</math> the rotation remains uniform in space."</font>

It should be clear, as well, that the eccentricity of the evolving spheroid will vary with time.  Specifically in the case of an oblate spheroid, the time-dependent semi-axes are <math>(a_\ell R(t),a_\ell R(t), a_s Z(t))</math>; hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>e(t)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 1 - \biggl( \frac{a_s Z}{a_\ell R} \biggr)^2 \biggr]^{1 / 2}
=
\biggl[ 1 - ( 1 - e_0^2 )\biggl( \frac{Z}{R} \biggr)^2 \biggr]^{1 / 2} \, ,
</math>
  </td>
</tr>
</table>
where, <math>e_0 = (1 - a_s^2/a_\ell^2)^{1 / 2}</math> is the eccentricity of the spheroid initially, and the time-variation enters via the pair of functions, <math>Z(t)</math> and <math>R(t)</math>.  In the case of an prolate spheroid, the time-dependent semi-axes are <math>(a_s R(t),a_s R(t), a_\ell Z(t))</math>; hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>e(t)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 1 - \biggl( \frac{a_s R}{a_\ell Z} \biggr)^2 \biggr]^{1 / 2}
=
\biggl[ 1 - ( 1 - e_0^2 )\biggl( \frac{R}{Z} \biggr)^2 \biggr]^{1 / 2} \, .
</math>
  </td>
</tr>
</table>
<span id="Table1">In turn,</span> the time-dependent behavior of the coefficients in the expression for the gravitational potential, <math>A(t)</math> and <math>C(t)</math>, is drawn from <math>e(t)</math> as detailed in Table 1, immediately below.

<table border="1" align="center" cellpadding="3">
<tr>
  <td align="center" colspan="3">
Table 1: &nbsp; Time-Dependent Coefficients of the Gravitational Potential<br />
<div align="center"><math>\Phi(\vec{x}, t) = A(t) \varpi^2 + C(t) z^2\, ,</math></div>
where, it is understood that the eccentricity of the spheroid, <math>e(t)</math>, varies with time.
  </td>
</tr>
<tr>
  <td align="center">Oblate Spheroid</td>
  <td align="center">Prolate Spheroid</td>
</tr>
<tr>
  <td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{A(t)}{\pi G \rho(t)}</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>

<tr>
  <td align="right">
<math>\frac{C(t)}{\pi G \rho(t)}</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>
<tr>
  <td align="left" colspan="3">where,</td>
</tr>

<tr>
  <td align="right">
<math>e(t)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 1 - ( 1 - e_0^2 )\biggl[ \frac{Z(t)}{R(t)} \biggr]^2 \biggr\}^{1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\rho(t)}{\rho_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{R^2(t)Z(t)} \, .
</math>
  </td>
</tr>
</table>
  </td>
  <td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{A(t)}{\pi G \rho(t)}</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>
<tr>
  <td align="right">
<math>\frac{C(t)}{\pi G \rho(t)}</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>
<tr>
  <td align="left" colspan="3">where,</td>
</tr>

<tr>
  <td align="right">
<math>e(t)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 1 - ( 1 - e_0^2 )\biggl[ \frac{R(t)}{Z(t)} \biggr]^2 \biggr\}^{1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\rho(t)}{\rho_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{R^2(t)Z(t)} \, .
</math>
  </td>
</tr>
</table>

  </td>
</tr>
</table>