Editing SSC/Dynamics/FreeFall (section)

====Discussion====
Upon further assessment of the term on its righthand side, this last expression can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
a^2 \ddot{a}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\lambda G\biggl( \frac{4\pi}{3}\rho_0 a_i^3 \biggr) = ~-~\lambda GM_i \, , 
</math>
  </td>
</tr>
</table>
</div>
where, <math>~M_i</math> is independent of time and is the mass associated with the initial scale length, <math>~a_i</math>.  Except that <math>~a</math> appears in place of <math>~R</math>, we see that this identically matches the equation of motion for the collapsing, uniform-density sphere [[#Uniform-Density_Sphere|presented above]] if we set <math>~\lambda = 1</math>.  Therefore, for a system of any initial size, <math>~a_i</math>, that collapses from rest with an initial (uniform) density, <math>~\rho_0</math>, this equation can be straightforwardly integrated twice using the above sequence of steps to give the following parametric relationship between time and the system's instantaneous scale length at that time:

<table border="1" cellpadding="10" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~ \frac{a}{a_i} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \cos^2\zeta </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \frac{t}{\tau_\mathrm{ff}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{2}{\pi} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta) \biggr] </math>
  </td>
</tr>

<tr>
<td align="left" colspan="3">
where, &nbsp; &nbsp; <math>\tau_\mathrm{ff} \equiv \biggl[ \frac{3\pi}{32G\rho_0} \biggr]^{1/2} \, .</math>
</td>
</tr>
</table>
</td></tr>
</table>
 

Also, when we plug our derived functional expression for the dimensionless gravitational potential, <math>~\sigma</math>, into the dimensionless Poisson equation, we find,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla_\mathfrak{x}^2 \biggl[ \frac{\lambda}{6}\mathfrak{x}^2 \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~f </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\biggl( \frac{\lambda}{6} \biggr) \frac{1}{\mathfrak{x}^2}
\frac{d}{d\mathfrak{x}}\biggl[ \mathfrak{x}^2 \frac{d}{d\mathfrak{x}}\mathfrak{x}^2 \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~f </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\lambda</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~f \, ,</math>
  </td>
</tr>
</table>
</div>
so the proper physical solution is, <math>~f = 1</math>, as expected.  This means that, to within an additive constant, the gravitational potential will depend inversely on the time-dependent scale length, <math>~a</math>, and quadratically on the dimensionless &#8212; and time ''independent'' &#8212; radial coordinate, <math>\mathfrak{x}</math>, via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G \rho_c a^2 \biggl( \frac{1}{6} ~\mathfrak{x}^2 \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{2\pi G \rho_0 a_i^3}{3 a} \biggr)\mathfrak{x}^2 
= \biggl( \frac{ G M_i}{2 a} \biggr)\mathfrak{x}^2 \, .</math>
  </td>
</tr>
</table>
</div>

This expression for the gravitational potential looks a bit peculiar because it is zero at the center of the configuration and is otherwise everywhere positive.  Customarily, a constant is subtracted from this function in order to ensure that it is everywhere negative and properly normalized to the expected value at the surface of the sphere.  For example, in a [[SSC/Structure/UniformDensity#Summary|separate discussion of the internal properties of isolated, uniform-density spheres]] that are in hydrostatic balance, the derived gravitational potential is,

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

<tr>
  <td align="right">
<math>~\Phi(r)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{3G M}{2R} \biggl[ 1 - \frac{1}{3} \biggl(\frac{r}{R} \biggr)^2 \biggr]  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{G M}{2R} \biggl(\frac{r}{R} \biggr)^2 - \frac{3G M}{2R} \, .  </math>
  </td>
</tr>
</table>
</div>
We could subtract the quantity, <math>~[3GM_i/(2a)]</math>, from our derived expression for the potential of a free-falling homogeneous sphere in order for it to reflect this more familiar normalization, but this doesn't make a lot of sense because the quantity being subtracted &#8212; while constant in space &#8212; varies with time.