Editing Aps/MaclaurinSpheroidFreeFall (section)

==Lynden-Bell's (1962) Insight==
Let's examine the analysis by {{ LB62full }} &#8212; hereafter, {{ LB62hereafter }} &#8212; of the "Gravitational Collapse of a Cold Rotating Gas Cloud."

===Motion of a Single Particle===
Consider a particle that, at time <math>t=0</math>, is at position <math>(\varpi_0, \phi_0, z_0)</math> and is moving about the <math>z</math>-axis with velocity, <math>\varpi_0\Omega ~\Rightarrow~ j_0 = \varpi_0^2 \Omega</math>.  Consider furthermore that its acceleration is subject to the force arising from an axisymmetric gravitational potential of the form,

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

[This is the gravitational potential adopted by {{ LB62hereafter }} &#8212; see his equation (1) &#8212; except he adopted a different sign convention to ours.  He would therefore have also attached a sign to the gradient of the potential that is the opposite of the sign that appears on the right-hand side of our Euler equation expression.]  In this case, the two components of the Euler equation that govern the particle's motion are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>\hat{\mathbf{e}}_\varpi</math>:</td>
  <td align="right">
<math>\ddot\varpi - \frac{j_0^2}{\varpi^3}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 2A\varpi    \, ,</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\hat{\mathbf{e}}_z</math>:</td>
  <td align="right">
<math>\ddot{z} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 2Cz    \, ,</math>
  </td>
</tr>
</table>
where we have adopted the familiar shorthand notation, <math>d\dot\varpi/dt \rightarrow \ddot\varpi</math> and <math>d\dot{z}/dt \rightarrow \ddot{z}</math>.  If we divide the first of these relations by <math>\varpi_0</math> and the second by <math>z_0</math>, then adopt the dimensionless variables, <math>R \equiv \varpi/\varpi_0</math> and <math>Z \equiv z/z_0</math>, we can write,
<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\frac{\ddot\varpi}{\varpi_0}  - \frac{j_0^2}{\varpi_0 \varpi^3}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align=left">
<math>- 2A \biggl( \frac{\varpi}{\varpi_0} \biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \ddot{R}  - \frac{\Omega^2}{R^3}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align=left">
<math>- 2A R \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ LB62hereafter }}, p. 710, Eq. (10)
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\ddot{z}}{z_0} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 2C \biggl(\frac{z}{z_0}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \ddot{Z} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 2C Z \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ LB62hereafter }}, p. 710, Eq. (11)
  </td>
</tr>
</table>
Finally, we use <math>\varphi(t)</math> to represent the particle's time-varying angular-coordinate position ''relative to'' its initial position &#8212; that is, we adopt the definition, <math>\varphi(t) \equiv \phi(t) - \phi_0</math>. Then, conservation of angular momentum implies that, at any moment, the particle's rotation frequency about the symmetry axis will be,
<table border="0" cellpadding="0" align="center">

<tr>
  <td align="right">
<math>\dot\varphi = \dot\phi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\Omega}{R^2} \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ LB62hereafter }}, p. 710, Eq. (9)
  </td>
</tr>
</table>
This governing set of evolutionary equations has been set up such that at time, <math>t=0</math>: <math>R = 1</math>, <math>Z = 1</math>, <math>\varphi = 0</math>, <math>\dot{R} = 0</math>, <math>\dot{Z} = 0</math>, and <math>\dot\varphi = \Omega</math>. With this set of initial conditions in hand, along with an appropriate specification of the two time-dependent coefficients, <math>A(t)</math> and <math>C(t)</math>, the set of governing relations can be integrated (numerically) to give <math>R(t), \varphi(t)</math>, and <math>Z(t)</math>.  This is the result that {{ LB62 }} established for the motion of one particle.

===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>