Editing SSC/Dynamics/FreeFall (section)

====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>