Editing Apps/GoldreichWeber80 (section)

====Mass-Density and Speed====

Next, {{ GW80 }} (see their equation 10) choose to normalize the density by the central density, specifically defining a dimensionless function,
<div align="center">
<math>f \equiv \biggl( \frac{\rho}{\rho_c} \biggr)^{1/3} \, ,</math>
</div>
which, in order to successfully identify a similarity solution, may be a function of space but not of time.  Keeping in mind that <math>~n = 3</math>, this is also in line with the formulation and evaluation of the [[SSC/Structure/Polytropes#Lane-Emden_equation|Lane-Emden equation]], where the primary ''dependent'' structural variable is the dimensionless polytropic enthalpy, 
<div align="center">
<math>\Theta_H \equiv \biggl( \frac{\rho}{\rho_c} \biggr)^{1/n} \, .</math>
</div>

Also, {{ GW80 }} (see their equation 11) normalize the gravitational potential to the square of the central sound speed,
<div align="center">
<math>c_s^2 = \frac{\gamma P_c}{\rho_c} = \frac{4}{3} \kappa \rho_c^{1/3} 
= \frac{4}{3}\biggl(\frac{\kappa^3}{\pi G}\biggr)^{1/2} [a(t)]^{-1}  \, .</math>
</div>
Specifically, their 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}{c_s^2} = \biggl[ \frac{3}{4} \biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} a(t) \biggr] \Phi \, ,</math>
  </td>
</tr>
</table>
</div>
and the similarly normalized enthalpy may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{H}{c_s^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{3}{4} \biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} a(t) \biggr] 4\kappa \rho^{1/3} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3 \biggl( \frac{\rho}{\rho_c} \biggr)^{1/3} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3f \, .</math>
  </td>
</tr>

</table>
</div>
With these additional scalings, our continuity equation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\cancelto{0}{\frac{d\ln f^3}{dt}}  +  \frac{d\ln \rho_c}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ a^{-2} \nabla_\mathfrak{x}^2 \psi  \, ,</math>
  </td>
</tr>
</table>
</div>

where the first term on the lefthand side has been set to zero because, as stated above, <math>~f</math> may be a function of space but not of time; our Euler equation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\biggl[ \frac{3}{4} \biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} a(t) \biggr]  
\biggl[ \frac{d\psi}{dt} - \frac{1}{2a^2} ( \nabla_\mathfrak{x} \psi )^2 \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 3 f - \sigma  \, ;</math>
  </td>
</tr>
</table>
</div>

and the Poisson equation becomes,
<div align="center">
<math>\nabla_\mathfrak{x}^2 \sigma = 3f^3 \, .</math>
</div>