Editing SSC/Dynamics/FreeFall (section)

==Analytic Model of the Collapse of a Uniform-Density Sphere==
Our primary objective in this chapter is to show how a mathematical model of the free-fall collapse of a uniform-density sphere &#8212; specifically, the model presented in the introduction of [http://adsabs.harvard.edu/abs/1965ApJ...142.1431L LMS65], as [[#Free-Fall_Collapse|reprinted above]] &#8212; can be derived from our standard, principal set of governing equations.  As it turns out, much of the calculus underpinning this derivation can be developed in the context of a simpler physical model, namely, the radial free-fall of a massless particle in a point-mass potential.  This is not a continuum fluids problem nor a self-gravitating problem, so the continuity equation and the Poisson equation are superfluous.  But the nonlinear dynamics associated with free-fall in a time-invariant, point-mass potential mirrors the dynamics embodied by the Euler equation in the context of the collapse of a self-gravitating, homogeneous fluid sphere.  So we begin by discussing this simpler, "single particle in a point-mass potential" problem. This is followed by a derivation of the model of the free-fall collapse of a "uniform-density sphere" presented by LMS65, as viewed from an inertial frame of reference.  Then, using techniques described by [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; Weber (1980)], we show how the same model can be derived as a ''similarity solution'' when viewed from a non-inertial, spherically symmetric coordinate system whose radial coordinate is undergoing an appropriately accelerated collapse.

===Single Particle in a Point-Mass Potential===

Suppose we examine the free-fall of a single (massless) particle, located a distance <math>~|\vec{r}|</math> from an immovable point-like object of mass, <math>~M</math>.  The particle will feel a distance-dependent acceleration due to a gradient in the gravitational potential of the form,
<div align="center">
<math>~\frac{d\Phi}{dr} = \frac{GM}{r^2} \, ,</math>
</div>
and the Euler equation, as just derived, serves to describe the particle's governing equation of motion, namely,
<div align="center">
<math>~\ddot{r} =  - \frac{GM}{r^2} \, ,</math>
</div>
where we have used dots to denote differentiation with respect to time (see also equation 1 from LMS65, [[#Free-Fall_Collapse|reprinted above]]).  If we multiply this equation through by <math>~2\dot{r} = 2dr/dt</math>, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~2\dot{r} \frac{d\dot{r}}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2GM}{r^2} \cdot \frac{dr}{dt} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ d(\dot{r}^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2GM \cdot d(r^{-1}) \, ,</math>
  </td>
</tr>
</table>
</div>
which integrates once to give,
<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}{r} - k \, , </math>
  </td>
</tr>
</table>
</div>
<span id="RoleOfIntegrationConstant">where, as an integration constant, <math>~k</math> is independent of time.  </span>

<table border="1" width="75%" align="center" cellpadding="10">
<tr><th align="center">
Role of Integration Constant
</th></tr>
<tr><td align="left">
Within the context of this particular physical problem, the constant, <math>~k</math>, should be used to specify the initial velocity, <math>~v_i</math>, of the particle that begins its collapse from the radial position, <math>~r_i</math>.  Specifically,
<div align="center">
<math>~k = \frac{2GM}{r_i} - v_i^2 \, .</math>
</div>
Without this explicit specification, it should nevertheless be clear that, in order to ensure that <math>~\dot{r}^2</math> is positive &#8212; and, hence, <math>~\dot{r}</math> is real &#8212; the constant must be restricted to values,
<div align="center">
<math>~k \leq \frac{2GM}{r_i}  \, .</math>
</div>

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

Taking the square root of both sides of our derived "kinetic energy" equation, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dr}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \biggl[ \frac{2GM}{r} - k \biggr]^{1/2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ dt </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \pm \biggl[ \frac{2GM}{r} - k \biggr]^{-1/2} dr \, .</math>
  </td>
</tr>
</table>
</div>

As has been shown by [http://adsabs.harvard.edu/abs/1934QJMat...5...73M McCrea &amp; Milne (1934)], this function can be integrated in closed form to give an analytic prescription for <math>~t(r)</math>.  Equation (17) in the McCrea &amp; Milne (1934) paper presents the function to be integrated as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~t</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\int\limits_0^\theta \frac{\theta^{1/2} d\theta}{(\alpha + A\theta)^{1/2}} \, ,</math>
  </td>
</tr>
</table>
</div>
which can be straightforwardly obtained from our expression after adopting the positive root of the "kinetic energy" equation and setting,
<div align="center">
<math>~r \rightarrow \theta \, ,</math> 
&nbsp; &nbsp; &nbsp; 
<math>~2GM \rightarrow \alpha \, ,</math> 
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;
<math>~k \rightarrow A' = -A \, .</math> 
</div>
The results obtained assuming three different ranges/values for the constant, <math>~k</math>, are presented at the end of &sect;4 of [http://adsabs.harvard.edu/abs/1934QJMat...5...73M McCrea &amp; Milne (1934)] and are reprinted here in an effort to fully acknowledge this early contribution.
<div align="center" id="McCreaMilne1934Solution">
<table border="1" align="center" cellpadding="5">
<tr>
  <th align="center" colspan="1">
W. H. McCrea and E. A. Milne

[http://adsabs.harvard.edu/abs/1934QJMat...5...73M (1934) ''Quarterly Journal of Mathematics Oxford'', 5, 73]
  </th>
</tr>
<tr>
  <th align="center" colspan="1">
[[File:McCreaMilne1934Solution.png|500px|McCrea &amp; Milne (1934) Time Solution]]
  </th>
</tr>
</table>
</div>

In the following subsections, we will rederive these algebraic, <math>~t(r)</math> solutions in the context of three separate, physically  interesting scenarios, all of which involve infall, so we will adopt the velocity root having only the negative sign. 

====Falling from rest at a finite distance &hellip;====
In this case, we set <math>~v_i = 0</math> in the definition of <math>~k</math>, so,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dr}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- ~\biggl[\frac{2GM}{r} - \frac{2GM}{r_i}\biggr]^{1/2} =
\biggl(\frac{2GM}{r_i}\biggr)^{1/2} \biggl[\frac{r_i}{r}-1  \biggr]^{1/2}
\, , </math>
  </td>
</tr>
</table>
</div>

and the relevant expression to be integrated is,
<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{2GM}{r_i} \biggr)^{-1/2} \biggl[ \biggl( \frac{r_i}{r} \biggr) - 1 \biggr]^{-1/2} dr  \, .</math>
  </td>
</tr>
</table>
</div>

Following [http://adsabs.harvard.edu/abs/1965ApJ...142.1431L LMS65], we see that this equation can be straightforwardly integrated by first making the substitution,
<div align="center">
<math>~\cos^2\zeta \equiv \frac{r}{r_i}  \, ,</math>
</div>
which also means,
<div align="center">
<math>~dr = - 2r_i \sin\zeta \cos\zeta d\zeta \, .</math>
</div>

The relevant integral is, therefore,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\int_0^t dt </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ \biggl(\frac{2r_i^3}{GM} \biggr)^{1/2} \int_0^\zeta \cos^2\zeta d\zeta \, ,</math>
  </td>
</tr>
</table>
</div>
where the limits of integration have been set to ensure that <math>~r/r_i = 1</math> at time <math>~t=0</math>.  After integration, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ t </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{2r_i^3}{GM} \biggr)^{1/2} \biggl[ \frac{\zeta}{2} + \frac{1}{4}\sin(2\zeta) \biggr]  \, .</math>
  </td>
</tr>
</table>
</div>
The physically relevant portion of this formally periodic solution is the interval in time from when <math>~r/r_i = 1 ~ (\zeta = 0)</math> to when <math>~r/r_i \rightarrow 0</math> for the first time <math>~(\zeta = \pi/2)</math>.  The particle's free-fall comes to an end at the time associated with <math>~\zeta = \pi/2</math>, that is, at the so-called "free-fall time,"
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tau_\mathrm{ff} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{2r_i^3}{GM} \biggr)^{1/2} \biggl[ \frac{\zeta}{2} + \frac{1}{4}\sin(2\zeta) \biggr]_{\zeta=\pi/2}  
= \biggl(\frac{\pi^2 r_i^3}{8GM} \biggr)^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>

<span id="Parametric">In summary,</span> then, the solution, <math>~r(t)</math>, to this simplified but dynamically relevant problem is provided by the following pair of analytically prescribable parametric relations (see also equations 2 &amp; 3 from LMS65, [[#Free-Fall_Collapse|reprinted above]]):

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Parametric &nbsp;<math>~r(t)</math>&nbsp; Solution
</th></tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~ \frac{r}{r_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>
</table>
</td></tr>
</table>
We note, as well, that the radially directed velocity is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~v_r = \frac{dr}{dt} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \biggl(\frac{2GM}{r_i} \biggr)^{1/2} \biggl[ \frac{1}{\cos^2\zeta}  - 1 \biggr]^{1/2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \biggl(\frac{2GM}{r_i} \biggr)^{1/2} \tan\zeta \, , </math>
  </td>
</tr>
</table>
</div>
which formally becomes infinite in magnitude when <math>~\zeta \rightarrow \pi/2</math>, that is, when <math>~t \rightarrow \tau_\mathrm{ff}</math>.

It is worth demonstrating that the parametric solution derived here is identical to the solution published by [http://adsabs.harvard.edu/abs/1934QJMat...5...73M McCrea &amp; Milne (1934)] &#8212;  [[#McCreaMilne1934Solution|reprinted above]] &#8212; for the case, <math>~k > 0</math>, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~t</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\theta^{1/2} (\alpha - A' \theta)^{1/2}}{A'} + \frac{\alpha}{(A')^{3/2}} \sin^{-1} 
\biggl( \frac{A' \theta}{\alpha} \biggr)^{1/2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\alpha}{(A')^{3/2}} \biggl[ - \biggl(\frac{A' \theta}{\alpha} \biggr)^{1/2} \biggl(1 - \frac{A' \theta}{\alpha} \biggr)^{1/2} + \sin^{-1} 
\biggl( \frac{A' \theta}{\alpha} \biggr)^{1/2} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
After reversing the substitutions detailed above, that is, after setting,
<div align="center">
<math>~\theta \rightarrow r \, ,</math> 
&nbsp; &nbsp; &nbsp; 
<math>~ \alpha \rightarrow 2GM \, ,</math> 
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;
<math>~A' \rightarrow k   \, ,</math> 
</div>
and remembering that, for this particular model example, we have set <math>~v_i = 0 ~\Rightarrow ~ k = 2GM/r_i</math>, the key dimensionless ratio in the McCrea &amp; Milne expression becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{A' \theta}{\alpha} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{r}{r_i} = \cos^2\zeta \, ,</math>
  </td>
</tr>
</table>
</div>
and the pre-factor on the righthand side becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[\frac{\alpha^2}{(A')^3} \biggr]^{1/2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{r_i^3}{2GM} \biggr]^{1/2} = \frac{2}{\pi} \tau_\mathrm{ff} \, .</math>
  </td>
</tr>
</table>
</div>

Hence, the McCrea &amp; Milne solution becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<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[  -\cos\zeta \sin\zeta + \sin^{-1} (\cos\zeta)
\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{\pi} \biggl[  -\frac{1}{2}\sin(2\zeta) + \biggl(\frac{\pi}{2} - \zeta \biggr)
\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 1 - \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>
</table>
</div>
We see that, by shifting the defined zero-point in time in this last expression such that <math>~t \rightarrow (\tau_\mathrm{ff} - t)</math>, which also reverses the ''sign'' on time, we have exact agreement between the solution that we have derived &#8212; designed to match the one published by {{ LMS65 }} &#8212; and the result for <math>k > 0</math> that was published by McCrea &amp; Milne in 1934.

====Falling from rest at infinity &hellip;====
In this case, we set <math>~k= 0</math>, so the relevant expression to be integrated is,
<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{2GM}{r} \biggr]^{-1/2} dr = - (2GM)^{-1/2} r^{1/2} dr \, .</math>
  </td>
</tr>
</table>
</div>
Upon integration, this gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~t + C_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -  \frac{2}{3}(2GM)^{-1/2} r^{3/2} \, ,</math>
  </td>
</tr>
</table>
</div>
where, <math>~C_0</math> is an integration constant.  In this case, it is useful to simply let <math>~t=0</math> mark the time at which <math>~r = 0</math> &#8212; hence, also, <math>~C_0 = 0</math> &#8212; so at all earlier times (<math>~t</math> intrinsically negative) we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- t </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl( \frac{2r^3}{9GM} \biggr)^{1/2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ r </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl( \frac{9}{2} \cdot GMt^2 \biggr)^{1/3} \, .</math>
  </td>
</tr>
</table>
</div>


It is straightforward to demonstrate that this derived solution is identical to the solution published by [http://adsabs.harvard.edu/abs/1934QJMat...5...73M McCrea &amp; Milne (1934)] &#8212;  [[#McCreaMilne1934Solution|reprinted above]] &#8212; for the case, <math>~k = 0</math>, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~t</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{2}{3} \frac{\theta^{3/2}}{\alpha^{1/2}} 
= \biggl( \frac{4\theta^3}{9\alpha} \biggr)^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>
After reversing the substitutions detailed above, that is, after setting,
<div align="center">
<math>~\theta \rightarrow r \, ,</math> 
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<math>~ \alpha \rightarrow 2GM \, ,</math> 
</div>
the McCrea &amp; Milne solution becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~t </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl( \frac{2r^3}{9GM} \biggr)^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>
Aside from reversing the ''sign'' on time, we have exact agreement between the solution that we have derived and the result for <math>~k = 0</math> that was published by McCrea &amp; Milne in 1934.


====Falling from a finite distance with an initially nonzero velocity &hellip;====
Here, we examine the case in which <math>~0 < r_i < \infty</math> and <math>~0 < v_i^2 < GM/r_i</math>, in which case, the constant <math>~k</math> is a nonzero, positive number.  The relevant expression to be integrated is,

<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>~ - k^{-1/2}\biggl[ \frac{a}{r} - 1 \biggr]^{-1/2} dr \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~ a \equiv \frac{2GM}{k} \, .</math>
</div>
Using [http://integrals.wolfram.com/index.jsp?expr=-%28a%2Fx-1%29%5E%28-1%2F2%29&random=false Wolfram Mathematica's online integrator], we find,

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

<tr>
  <td align="right">
<math>~- \int \biggl[ \frac{a}{r} - 1 \biggr]^{-1/2} dr</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r ( ar^{-1} -1 )^{1/2} + \frac{a}{2} \tan^{-1} \biggl[ \frac{(2r-a)(ar^{-1} - 1)^{1/2}}{2(r-a)} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

Hence, we find,

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

<tr>
  <td align="right">
<math>~k^{1/2}(t + C_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r ( ar^{-1} -1 )^{1/2} + \frac{a}{2} \tan^{-1} \biggl[ \frac{(2r-a)(ar^{-1} - 1)^{1/2}}{2(r-a)} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r ( ar^{-1} -1 )^{1/2} + \frac{a}{2} \tan^{-1} \biggl[ \frac{(ar^{-1}-2)(ar^{-1} - 1)^{1/2}}{2(ar^{-1}-1)} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r ( ar^{-1} -1 )^{1/2} + \frac{a}{2} \tan^{-1} \biggl[ \frac{(ar^{-1}-2)}{2(ar^{-1}-1)^{1/2}} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r k^{-1/2}( akr^{-1} -k )^{1/2} + \frac{a}{2} \tan^{-1} \biggl[ \frac{(akr^{-1}-2k)}{2k^{1/2}(akr^{-1}-k)^{1/2}} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

Let's determine the constant, <math>~C_0</math>.  When <math>~t = 0</math>, we can write,

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

<tr>
  <td align="right">
<math>~[akr^{-1} - k]_{t=0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2GM}{r_i} - \biggl[\frac{2GM}{r_i} - v_i^2  \biggr] = v_i^2
\, .</math>
  </td>
</tr>
</table>
</div>
Hence,

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

<tr>
  <td align="right">
<math>~C_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r_i k^{-1}v_i + \biggl(\frac{a}{2k^{1/2}} \biggr) \tan^{-1} \biggl[ \frac{(v_i^2-k)}{2k^{1/2}v_i} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
GMk^{-3/2} \biggl\{
2[\eta(1-\eta)]^{1/2} + \tan^{-1} \biggl[ \biggl( \eta - \frac{1}{2} \biggr) [ \eta(1-\eta)]^{-1/2} \biggr]
\biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, in this last expression,
<div align="center">
<math>\eta \equiv \frac{v_i^2 r_i}{2GM} \, .</math>
</div>
(This last expression needs to be checked for errors, as it has been rather hastily derived.)

====Relationship to Kepler's 3<sup>rd</sup> Law====

[[File:CommentButton02.png|right|100px|Note from J. E. Tohline:  I was first made aware of this relationship, as a graduate student, while listening to John Faulkner give a lecture on the free-fall problem to a class of undergraduates at UCSC.]]It is useful to note a relationship between [http://astro.physics.uiowa.edu/ITU/glossary/keplers-third-law/ Kepler's 3<sup>rd</sup> law] and the free-fall problem, as introduced [[#Falling_from_rest_at_a_finite_distance_.E2.80.A6|here in the context of the motion of a single (massless) particle that falls from rest]] toward a point-like object of mass, <math>~M</math>.  According to Kepler's 3<sup>rd</sup> law, when a massless particle orbits a point-like object of mass, <math>~M</math>, the particle's orbital period, <math>~P_\mathrm{orb}</math>, is related to the semi-major axis, <math>~a_\mathrm{orb}</math>, of its elliptical orbit via the algebraic expression,
<div align="center">
<math>~P^2_\mathrm{orb} = \frac{4\pi a_\mathrm{orb}^3}{GM} \, .</math>
</div>
This relation works for orbits of any eccentricity, 
<div align="center">
<math>~e \equiv \biggl[ 1 - \frac{b_\mathrm{orb}^2}{a_\mathrm{orb}^2} \biggr]^{1/2} \, ,</math>
</div>
where, <math>~b_\mathrm{orb}</math> is the semi-minor axis of the orbit, the extremes being:  <math>~e = 0 ~(b_\mathrm{orb} = a_\mathrm{orb})</math> for a circular orbit, and <math>~e = 1 ~(b_\mathrm{orb} = 0)</math> for a purely radially directed (in-fall) trajectory.  

In the context of our current discussion, it should be clear that a particle that "free-falls" from rest at an initial distance, <math>~r_i</math>, from a point mass object will follow a trajectory synonymous with a Keplerian orbit having eccentricity, <math>~e=1</math>.  The particle's initial position coincides with the apo-center of this orbit and the point mass object is located at the peri-center (as well as at one focus) of the orbit, so the semi-major axis is <math>~a_\mathrm{orb} = r_i/2</math>.  We also recognize that the particle will move from the apo-center to the peri-center of its orbit &#8212; completing its "free-fall" onto the point-mass object &#8212; in a time, <math>~\tau = P_\mathrm{orb}/2</math>.  From Kepler's 3<sup>rd</sup> law, we therefore deduce that,
<div align="center">
<math>~\tau = \frac{1}{2} \biggl[ \frac{4\pi (r_i/2)^3}{GM} \biggr]^{1/2} = \biggl( \frac{\pi r_i^3}{8GM} \biggr)^{1/2} \, ,</math>
</div>
which precisely matches the free-fall time, <math>~\tau_\mathrm{ff}</math>, derived above.

===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]].

===Homologous Collapse in an Accelerated Reference Frame===

As we have shown, the free-fall collapse from rest of an initially uniform-density sphere occurs in an homologous fashion:  During the collapse, the system maintains the same radial density (specifically, uniform density) profile; at all times the magnitude of the radial velocity of each spherical shell of material is linearly proportional to the shell's distance from the center; and all mass shells hit the center at precisely the same time, that is, at <math>~t = \tau_\mathrm{ff}</math>.  This evolutionary behavior is reminiscent of the behavior that is displayed by the self-similar model that Goldreich &amp; Weber [http://adsabs.harvard.edu/abs/1980ApJ...238..991G (1980, ApJ, 238, 991)] developed to describe the near-homologous collapse of stellar cores; an [[Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores|accompanying chapter]] contains our review of this work.  Two key differences are that, in the Goldreich &amp; Weber work, the underlying density distribution resembles that of an <math>~n = 3</math> polytrope, rather than an <math>~n=0</math> (''i.e.,'' uniform density) polytrope; and the dynamical equations incorporate a noninertial, radially collapsing coordinate system.  Here we investigate what might be learned by mapping the classic free-fall problem onto a Goldreich &amp; Weber-type noninertial coordinate frame.

====Adaptation from Goldreich &amp; Weber (1980)====
We begin with the set of governing equations, derived by [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; Weber (1980)], that result from expressing the vorticity-free velocity flow-field, <math>~\vec{v}</math>, in terms of a stream function, <math>~\psi</math>, viz.,
<div align="center">
<math>~\vec{v} = \nabla\psi ~~~~~\Rightarrow~~~~~v_r = \nabla_r\psi </math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
<math>~\nabla\cdot \vec{v} = \nabla_r^2 \psi \, ;</math>
</div>
and from adopting a dimensionless radial coordinate that is defined by normalizing the inertial coordinate vector, <math>~\vec{r}</math>, to a time-varying length, <math>~a(t)</math>, viz.,
<div align="center">
<math>~\vec\mathfrak{x} \equiv \frac{1}{a(t)} \vec{r} \, .</math>
</div>
As is described in detail in [[Apps/GoldreichWeber80#GoverningWithStreamFunction|an accompanying discussion]], the continuity equation, the Euler equation, and the Poisson equation become, respectively,

<div align="center" id="GoverningWithStreamFunction">
<table border="1" align="center" cellpadding="10" width="55%">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{1}{\rho} \frac{d\rho}{dt} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ a^{-2} \nabla_\mathfrak{x}^2 \psi  \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{d\psi}{dt} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} a^{-2} ( \nabla_\mathfrak{x} \psi )^2 - H - \Phi  \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~a^{-2}\nabla_\mathfrak{x}^2 \Phi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G \rho \, .</math>
  </td>
</tr>
</table>

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

Because Goldreich &amp; Weber were modeling the collapse of a stellar core that is initially in (or nearly in) hydrostatic balance and obeys a <math>~\gamma = 4/3</math> gas law, they supplemented this set of dynamical equations with an <math>~n=3</math>, [[SR#Barotropic_Structure|polytropic equation of state]],
<div align="center">
<math>~H = 4\kappa \rho^{1/3} \, ,</math>
</div>
to relate the key state variables to one another.  Here, in our study of free-fall collapse, it is appropriate for us to simply set <math>~H = 0</math>, not only initially but at all times.
 

Following the lead of [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; Weber (1980)] &#8212; again, see [[Apps/GoldreichWeber80#Homologous_Solution|our accompanying discussion]] &#8212; we adopt a stream function of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}a \dot{a} \mathfrak{x}^2 \, ,</math>
  </td>
</tr>
</table>
</div>
which, when acted upon by the various relevant operators, gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla_\mathfrak{x}\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a \dot{a} \mathfrak{x} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\nabla^2_\mathfrak{x}\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{1}{2}a \dot{a} \biggr) \frac{1}{\mathfrak{x}^2} \frac{d}{d\mathfrak{x}} \biggl[\mathfrak{x}^2 \frac{d}{d\mathfrak{x}}  \mathfrak{x}^2 \biggr]
= 3 a \dot{a} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{d\psi}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathfrak{x}^2 \biggl[ \frac{1}{2}\dot{a}^2 + \frac{1}{2}a\ddot{a} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Hence, the radial velocity profile is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~v_r = a^{-1}\nabla_\mathfrak{x} \psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\dot{a}\mathfrak{x} \, ;
</math>
  </td>
</tr>
</table>
</div>
and the continuity equation gives,
<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>-~ \frac{3\dot{a}}{a} \, .</math>
  </td>
</tr>
</table>
</div>
Because we are hoping to identify a similarity solution, it will be advantageous to rewrite the mass density as a product of two functions:  One that depends only on time, <math>~\rho_c(t)</math>, and one that reflects spatial variations, <math>f(\mathfrak{x})</math>.  Specifically, we will write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\rho  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_c f \, .</math>
  </td>
</tr>
</table>
</div>
(Because, here, we are modeling the homologous collapse of a uniform-density sphere, this step isn't formally necessary.  Ultimately, for example, we expect to find that <math>~f=1</math>, reflecting the system's spatial homogeneity. But rewriting the density in this fashion will make the analogy with Goldreich &amp; Weber's (1980) derivation clearer.)  Plugging this new expression into the continuity equation gives,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\cancelto{0}{\frac{d\ln f}{dt}} + \frac{d\ln \rho_c}{dt} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{d\ln a^3}{dt}  \, ,</math>
  </td>
</tr>
</table>
</div>
which means that the product, <math>~a^3 \rho_c</math>, is independent of time.  Hence, if <math>~a_i</math> and <math>~\rho_0</math> are, respectively, the system's scale length and density initially, we can write,
<div align="center">
<math>\frac{\rho_c}{\rho_0} = \biggl( \frac{a}{a_i}\biggr)^{-3} \, .</math>
</div>


As written, each term in the Euler equation has units of velocity-squared.  Goldreich &amp; Weber (1980) chose to normalize the Euler equation by dividing through by the square of the (time-varying) sound speed.  This is not a good choice in our examination of the free-fall problem because we are altogether ignoring the effects of pressure.  Instead, an appropriate normalization would seem to be,   
<div align="center">
<math>v_\mathrm{norm}^2 \equiv 4\pi G\rho_c a^2  \, .</math>
</div>
Adopting this normalization, the dimensionless gravitational potential is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\Phi }{4\pi G\rho_c a^2} \, ,</math>
  </td>
</tr>
</table>
</div>

and (remembering to set <math>~H = 0</math>) the Euler equation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~ - \sigma </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{1}{4\pi G\rho_c a^2} \biggr)
\biggl\{ \mathfrak{x}^2 \biggl[ \frac{1}{2}\dot{a}^2 + \frac{1}{2}a\ddot{a} \biggr]  - 
\frac{1}{2} ( \dot{a} \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{a}{4\pi G\rho_0 a_i^3 } \biggr)~ \mathfrak{x}^2 \biggl[\frac{1}{2} ( a \ddot{a} )\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{\sigma}{\mathfrak{x}^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl(\frac{4\tau_\mathrm{ff}^2}{3\pi^2 a_i^3} \biggr) a^2 \ddot{a}  \, ;
</math>
  </td>
</tr>

</table>
</div>
and the dimensionless Poisson equation is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a^{-2} \nabla_\mathfrak{x}^2 [ 4\pi G \rho_c a^2 \sigma] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G \rho_c f  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\nabla_\mathfrak{x}^2 \sigma </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~f \, . </math>
  </td>
</tr>
</table>
</div>

As was argued by [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich &amp; Weber (1980)], because everything on the lefthand side of the scaled Euler equation depends only on the dimensionless spatial coordinate, <math>~\mathfrak{x}</math>, while everything on the righthand side depends only on time &#8212; via the parameter, <math>~a(t)</math> &#8212; both expressions must equal the same (dimensionless) constant.  If, following Goldreich &amp; Weber, we call this constant, <math>~\lambda/6</math>, the terms on the lefthand side lead us to conclude that, to within an additive constant, the dimensionless gravitational potential is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\lambda}{6} ~\mathfrak{x}^2  \, .</math>
  </td>
</tr>
</table>
</div>
From the terms on the righthand side we conclude, furthermore, that the differential equation governing the time-dependent variation of the scale length, <math>~a</math>, is,
<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>~-~\frac{\lambda}{6}\biggl(\frac{3\pi^2 a_i^3}{4\tau_\mathrm{ff}^2 } \biggr) \, .
</math>
  </td>
</tr>
</table>
</div>

====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.