Editing Apps/SMS (section)

====For any value of the ratio of specific heats====
From equation (2) of {{ LP41full }} &#8212; see, for example, [[SSC/Perturbations#Ledoux_and_Pekeris_.281941.29|our brief summary of this work]] &#8212; or, equally well, from equation (131) in Chapter II of [[Appendix/References#C67|<b>[<font color="red">C67</font>]</b>]], we see that when the total pressure is of the form being considered here, a general expression for the adiabatic exponent,
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<math>\Gamma_1 \equiv \frac{d\ln P}{d\ln\rho} \, ,</math>
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is,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~\Gamma_1</math>
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<math>~=</math>
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<math>~\beta + \frac{(\gamma-1)(4-3\beta)^2}{\beta + 12(\gamma-1) (1 - \beta)} \, ,</math>
  </td>
</tr>
</table>
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where, <math>~\gamma</math> is the ratio of specific heats associated with the ideal-gas component of the equation of state.  Notice that <math>~\beta = 1 </math> represents a situation where there is no radiation pressure.  In this limit the expression simplifies to,
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  <td align="right">
<math>~\Gamma_1\biggr|_{\beta=1}</math>
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<math>~=</math>
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<math>~\gamma \, ,</math>
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</tr>
</table>
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which makes sense.  On the other hand, setting <math>~\beta = 0</math> represents the other extreme, where there is no ideal-gas contribution to the pressure.  In this case, we have,
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<tr>
  <td align="right">
<math>~\Gamma_1\biggr|_{\beta=0}</math>
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  <td align="center">
<math>~=</math>
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<math>~\frac{16(\gamma-1)}{12(\gamma-1) } = \frac{4}{3} \, .</math>
  </td>
</tr>
</table>
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