Editing SSC/Dynamics/FreeFall (section)

===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.