Editing SSC/Virial/PolytropesSummary (section)

===Physical Meaning of Parameter &eta;<sub>ad</sub>===

As defined in our above discussion, <math>~\eta_\mathrm{ad}</math> is the ratio of the two terms that are summed together in the definition of the structural form factor, <math>\tilde\mathfrak{f}_A</math>.  It is worth pointing out what physical quantities are associated with these two terms.  

At any radial location within a polytropic configuration, the [[SSC/Structure/Polytropes#Lane-Emden_Equation|Lane-Emden function]], <math>\theta</math>, is defined in terms of a ratio of the local density to the configuration's central density, specifically,
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<math>\theta \equiv \biggl(\frac{\rho}{\rho_c} \biggr)^{1/n} \, .</math>
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Remembering that, at any location within the configuration, the pressure is related to the density via the polytropic equation of state,
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<math>P = K\rho^{(n+1)/n} \, ,</math>
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we see that,
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<math>\frac{P}{P_c} = \theta^{n+1} \, .</math>
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Hence, the quantity, <math>\tilde\theta^{n+1}</math>, which appears as the second term in our definition of <math>\tilde\mathfrak{f}_A</math>, is the ratio, <math>(P/P_c)_{\tilde\xi}</math>, evaluated at the surface of the truncated polytropic sphere.  But, by construction, the pressure at this location equals the pressure of the external medium in which the polytrope is embedded, so we can write,
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<math>\tilde\theta^{n+1} = \frac{P_e}{P_c}  \, .</math>
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In our [[SSC/Virial/Polytropes#Strategy2|accompanying detailed analysis]], we have employed the virial theorem expression to demonstrate that the first term in our definition of <math>~\tilde\mathfrak{f}_A</math> provides a measure the configuration's normalized central pressure.  Specifically, we show that,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{4\pi}{3} \biggr) \frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>[3 (n+1) (\tilde\theta^')^2]^{-1} \, .</math>
  </td>
</tr>
</table>
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We conclude, therefore, that quite generally,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>(5-n) \tilde\mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3}{4\pi} \biggr) \frac{G M_\mathrm{limit}^2}{P_c R_\mathrm{eq}^4}  + (5-n) \frac{P_e}{P_c} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3}{4\pi} \biggr) \frac{G M_\mathrm{limit}^2}{P_c R_\mathrm{eq}^4} \biggl[1  + \eta_\mathrm{ad} \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
and that,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\eta_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{4\pi (5-n)}{3} \biggr] \frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \, .</math>
  </td>
</tr>
</table>
</div>