Editing SSC/Dynamics/FreeFall (section)

===Uniform-Density Sphere===

Now, let's consider the (pressure-free) collapse, from rest, of a uniform-density sphere of total mass <math>~M_\mathrm{tot}</math> and radius, <math>~R(t)</math>.  If we use a subscript "0" to label the radius of the sphere at time <math>~t=0</math>, then the initial mass-density throughout the sphere is,
<div align="center">
<math>~\rho_0 = \frac{3M_\mathrm{tot}}{4\pi R_0^3} \, .</math>
</div>
If we not only assume that the total mass of this configuration remains constant but that all of the mass ''remains fully enclosed within the surface'' of radius, <math>~R(t)</math>, throughout the collapse (the validity of this second assumption will be critically assessed shortly), then at all points across the surface of the configuration, the acceleration will be given &#8212; analogous to the single-particle case, above &#8212; by,
<div align="center">
<math>~\frac{d\Phi}{dR} = \frac{GM_\mathrm{tot}}{R^2} \, ,</math>
</div>
and the equation of motion for the surface is, as before,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ddot{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{GM_\mathrm{tot}}{R^2} \, .</math>
  </td>
</tr>
</table>
</div>

As in the single-particle case, above, this 2<sup>nd</sup>-order ODE can be integrated once to generate a "kinetic energy" equation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\dot{R}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2GM_\mathrm{tot}}{R} - k(R_i, v_i) \, , </math>
  </td>
</tr>
</table>
</div>

and integrated a second time to give the following parametric relationship between the sphere's radius, and 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{R}{R_0} </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{\pi^2 R_0^3}{8GM_\mathrm{tot}} \biggr)^{1/2} 
= \biggl[ \frac{3\pi}{32G\rho_0} \biggr]^{1/2}</math>
</td>
</tr>
</table>
</td></tr>
</table>

It is important to notice, from this result, that the timescale for collapse, <math>~\tau_\mathrm{ff}</math>, depends only on the density of the configuration in its initial state.  It is important to realize, as well, that the derived parametric solution that gives the ratio <math>~R/R_0</math> as a function of time applies for ''all positions within'' the sphere.  In this more general way of interpreting the solution, <math>~R</math> represents ''any'' radial position, <math>~R_0</math> represents the value of ''that'' <math>~R</math> at time <math>~t=0</math>, and the relevant mass is the mass interior to that position, <math>~M_R</math>, rather than the configuration's total mass.  This works because, for spherically symmetric configurations, the acceleration only depends on the mass ''interior'' to each position.  The ultimate result is that the free-fall collapse of an initially uniform-density sphere proceeds homologously.  This happens because, independent of <math>~R</math>, the timescale for collapse only depends on <math>~\rho_0</math> and, by design, <math>~\rho_0</math> is independent of <math>~R</math>.  This is just a restatement of the behavior emphasized by [http://adsabs.harvard.edu/abs/1965ApJ...142.1431L LMS65], as [[#Free-Fall_Collapse|reprinted above]]:  "&hellip; a uniform sphere contracts homologously, and so stays uniform."

Because the pressure-free collapse of an initially uniform-density sphere proceeds in an homologous fashion, the mass interior to any radial shell remains constant.   This fully justifies the assumption of constant mass that was made earlier in this derivation. 

<span id="Velocity">The expression for the time-dependent velocity that was obtained, above, in the context of a [[#Falling_from_rest_at_a_finite_distance_.E2.80.A6| particle falling from rest at a finite distance]] can also be generalized here to the case of a collapsing uniform-density sphere.</span>  A radial shell initially at any position, <math>~R_i \le R_0</math>, within the sphere will enclose a mass, <math>M_i = 4\pi \rho_0 R_i^3/3</math>.  Hence the radially directed velocity of that shell at any time, <math>~t</math> (specified via the parameter, <math>~\zeta</math>), will be,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~v_r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \biggl(\frac{2GM_i}{R_i} \biggr)^{1/2} \tan\zeta 
= - R_i \biggl[ \biggl(\frac{8\pi G\rho_0}{3} \biggr)^{1/2} \tan\zeta \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  
- R \biggl[ \biggl(\frac{8\pi G\rho_0}{3} \biggr)^{1/2} \frac{\sin\zeta}{\cos^3\zeta} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Because everything inside the square brackets of this last expression is independent of space, the expression tells us that, at any time during the collapse, the radially directed velocity is linearly proportional to the radial coordinate of the shell.  

Knowing the velocity field, we can use the [[#Assembling_the_Key_Relations|continuity equation]] to determine the variation with time of the configuration's density.  Specifically, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln\rho}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \nabla\cdot \vec{v} = - \frac{1}{R^2} \frac{d}{dR} \biggl( R^2 v_r \biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ \biggl(\frac{8\pi G\rho_0}{3} \biggr)^{1/2} \frac{\sin\zeta}{\cos^3\zeta} \biggr] 
\frac{1}{R^2} \frac{d}{dR} \biggl( R^3 \biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 3\biggl[ \biggl(\frac{8\pi G\rho_0}{3} \biggr)^{1/2} \frac{\sin\zeta}{\cos^3\zeta} \biggr] 
\, ,</math>
  </td>
</tr>
</table>
</div>
so we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d\ln\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3\pi}{2\tau_\mathrm{ff}} \biggl(\frac{\sin\zeta}{\cos^3\zeta}\biggr) dt \, . </math>
  </td>
</tr>
</table>
</div>
But, from the function, <math>~t(\zeta)</math>, we deduce that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~dt</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{2\tau_\mathrm{ff}}{\pi} \biggr) d[\zeta + \sin\zeta\cos\zeta]
= \biggl( \frac{4\tau_\mathrm{ff}}{\pi} \biggr) \cos^2\zeta ~d\zeta \, .</math>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d\ln\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~6\tan\zeta ~d\zeta </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 6 d\ln(\cos\zeta) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~d\ln(\cos^{-6}\zeta) \, ,</math>
  </td>
</tr>
</table>
</div>
which, upon integration, gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ln\rho -~ \mathrm{constant}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ln(\cos^{-6}\zeta) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ln\biggl(\frac{R}{R_0} \biggr)^{-3} \, .</math>
  </td>
</tr>
</table>
</div>
Because <math>~\rho \rightarrow \rho_0</math> when <math>~R \rightarrow R_0</math>, the constant of integration must be <math>~\ln\rho_0</math>, giving us, finally,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho}{\rho_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{R}{R_0} \biggr)^{-3} \, .</math>
  </td>
</tr>
</table>
</div>

<span id="PressureFreeSummary">Finally we note that, given this quantified relationship between <math>~\rho</math> and <math>~R</math> along with an appreciation that the governing equation of motion applies to any radial position within the homogeneous sphere, we can rewrite the "acceleration" and "kinetic energy" equations as,</span>

<table border="1" cellpadding="10" align="center">
<tr>
  <td align="center">
Pressure-Free Collapse of an Homogeneous Sphere
  </td>
</tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~ \ddot{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{GM_R}{R^2} = -\frac{4}{3}\pi G \rho R \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{R}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2GM_R}{R} - k(R_i, v_i) = \frac{8}{3}\pi G \rho R^2 - k(R_i, v_i) \, .</math>
  </td>
</tr>
</table>
</td></tr>
</table>

We will refer back to this pair of dynamical equations in our [[#Relationship_to_Relativistic_Cosmologies|briefly discuss relativistic cosmologies, below]].