Editing Apps/GoldreichWeber80 (section)

===Imposed Constraints===
{{ GW80 }} specifically choose to examine the spherically symmetric collapse of a <math>~\gamma = 4/3</math> fluid.  With this choice of adiabatic index, the equation of state becomes,
<div align="center">
<math>~H = 4 \kappa \rho^{1/3} \, .</math>
</div>
And because a strictly radial flow-field exhibits no vorticity (i.e., <math>\vec\zeta = 0</math>), the Euler equation can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\partial v_r}{\partial t} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ \nabla_r \biggl[ H + \Phi + \frac{1}{2}v_r^2 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
{{ GW80 }} also realize that, because the flow is vorticity free, the velocity can be obtained from a stream function, <math>~\psi</math>, via the relation,
<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>
Hence, the continuity equation becomes,
<div 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>~-~ \nabla_r^2 \psi \, ,</math>
  </td>
</tr>
</table>
</div>

and the Euler equation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\partial }{\partial t} \biggl[ \nabla_r \psi \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ \nabla_r \biggl[ H + \Phi + \frac{1}{2}(\nabla_r \psi)^2 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

Since we are, up to this point in the discussion, still referencing the inertial-frame radial coordinate, the <math>~\nabla_r</math> operator can be moved outside of the partial time-derivative on the lefthand side of this equation to give,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\nabla_r \biggl[ \frac{\partial \psi}{\partial t} + H + \Phi + \frac{1}{2}(\nabla_r \psi)^2 \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
This means that the terms inside the square brackets must sum to a constant that is independent of spatial position.  Following the lead of {{ GW80 }}, this "integration constant" will be incorporated into the potential, in which case we have,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\partial \psi}{\partial t} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ \biggl[ H + \Phi + \frac{1}{2} ( \nabla_r \psi )^2 \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
which matches equation (5) of {{ GW80 }}.

Now, because it is more readily integrable, we ultimately would like to work with a differential equation that contains the total, rather than partial, time derivative of <math>~\psi</math>.  So we will take this opportunity to shift from an Eulerian representation of the Euler equation to a Lagrangian representation, invoking the same (familiar to fluid dynamicists) operator transformation as we have used in our [[PGE/Euler#Eulerian_Representation|general discussion of the Euler equation]], namely,
<div align="center" id="TimeDerivativeTransformation">
<math>~\frac{\partial\psi}{\partial t} ~~ \rightarrow ~~ \frac{d\psi}{dt} - \vec{v}\cdot \nabla\psi \, .</math>
</div>
In the context of the {{ GW80 }} model, we are dealing with a one-dimension (spherically symmetric), radial flow, so,
<div align="center">
<math>\vec{v}\cdot \nabla\psi = v_r \nabla_r \psi \, .</math>
</div>
But, given that we have adopted a stream-function representation of the flow in which <math>~v_r = \nabla_r\psi</math>, we appreciate that this term can either be written as <math>~v_r^2</math> or <math>~(\nabla_r\psi)^2</math>.  We choose the latter representation, so the Euler equation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d\psi}{dt} - (\nabla_r\psi)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ \biggl[ H + \Phi + \frac{1}{2} ( \nabla_r \psi )^2 \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
or, combining like terms on the left and right,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d\psi}{dt} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} ( \nabla_r \psi )^2 - H - \Phi  \, .</math>
  </td>
</tr>
</table>
</div>