Editing Apps/GoldreichWeber80 (section)

====Length====

In their investigation, {{ GW80 }} chose the same length scale for normalization that is used in deriving the [[SSC/Structure/Polytropes#Lane-Emden_equation|Lane-Emden equation]], which governs the hydrostatic structure of a polytrope of index {{ Math/MP_PolytropicIndex }}, that is,
<div align="center">
<math>
a_\mathrm{n} \equiv \biggl[\frac{1}{4\pi G}~ \biggl( \frac{H_c}{\rho_c} \biggr)\biggr]^{1/2}  \, ,
</math>
</div>
where the subscript, "c", denotes central values.  In this case <math>~(n = 3)</math>, substitution of the equation of state expression for <math>~H_c</math> leads to,
<div align="center">
<math>
a = \rho_c^{-1/3} \biggl(\frac{\kappa}{\pi G}\biggr)^{1/2} \, .
</math>
</div>
''Most significantly'', {{ GW80 }} (see their equation 6) allow the normalizing scale length to vary with time in order for the governing equations to accommodate a self-similar dynamical solution.  In doing this, they effectively adopted an ''accelerating'' coordinate system with a time-dependent dimensionless radial coordinate,
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<math>~\vec\mathfrak{x} \equiv \frac{1}{a(t)} \vec{r} \, .</math>
</div>

This, in turn, will mean that either the central density varies with time, or the specific entropy of all fluid elements (captured by the value of <math>~\kappa</math>) varies with time, or both.  In practice, {{ GW80 }} assume that <math>~\kappa</math> is held fixed, so the time-variation in the scale length, <math>~a</math>, reflects a time-varying central density; specifically,
<div align="center">
<math>
\rho_c = \biggl(\frac{\kappa}{\pi G}\biggr)^{3/2} [a(t)]^{-3} \, .
</math>
</div>

Given the newly adopted dimensionless radial coordinate, the following replacements for the spatial operators should be made, as appropriate, throughout the set of governing equations:
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<math>~\nabla_r ~\rightarrow~ a^{-1} \nabla_\mathfrak{x}</math>&nbsp; &nbsp; &nbsp; &nbsp;
and
&nbsp; &nbsp; &nbsp; &nbsp;<math>~\nabla_r^2 ~\rightarrow~ a^{-2} \nabla_\mathfrak{x}^2 \, .</math>
</div>
Specifically, 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>