Editing SSC/Stability/Isothermal (section)

==Overview==
Both {{ Ebert55full }} and {{ Bonnor56full }} showed that, when presented in a pressure-volume diagram, the equilibrium sequence of pressure-truncated isothermal spheres displays an astrophysical interesting ''turning point''.  This turning point signifies that there is a critical external pressure above which no equilibrium configurations exist.  As we have summarized in a [[SSC/Structure/BonnorEbert#Following_Bonnor.27s_Presentation|related discussion]], {{ Bonnor56 }} furthermore showed that this turning point lies along the equilibrium sequence approximately halfway between configurations that have a truncation radius of <math>\xi_e = 6</math> and <math>\xi_e = 7</math>, and that it is identified by the model whose isothermal Lane-Emden function, <math>\psi(\xi)</math>, exhibits the property,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[e^{\psi} \biggl( \frac{d\psi}{d\xi}\biggr)^2\biggr]_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2 \, ,</math>
  </td>
</tr>
</table>
</div>
''at its surface''.  At the time, it was left unanswered whether this turning point is only significant in the context of ''secular'' evolution along the equilibrium sequence, or whether it is associated with the onset of a dynamical instability.  As far as we have been able to determine, {{ Yabushita68full }} was the first to point out that, in principle, the relative [dynamical] stability of this "turning point" model can be ascertained by using linear stability analysis techniques to examine the sign of <math>~\lambda_0^2</math>, which is the square of the eigenfrequency of the configuration's fundamental radial mode of oscillation.

The stability analysis presented by {{ Yabushita68 }} showed that the marginally [dynamically] unstable model does lie between <math>~\xi_e = 6</math> and <math>~\xi_e = 7</math>.  This was confirmed by the follow-up stability analysis published by {{ TVH74full }}.  As is demonstrated in subsequent subsections of this chapter, we have reproduced in detail the subset of the {{ TVH74 }} results that are directly related to this stability question.  Extending the application of these standard linear stability analysis techniques, we have [[#Some_Results|furthermore determined]] that the marginally unstable model has a truncation radius of,

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<math>\xi_e \approx 6.4510534 \, ,</math>
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and that, for this specific model,
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<tr>
  <td align="right">
<math>\biggl[e^{\psi} \biggl( \frac{d\psi}{d\xi}\biggr)^2\biggr]_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>1.9999588 \, .</math>
  </td>
</tr>
</table>
</div>
Hence, to a high degree of precision, it appears as though the equilibrium configuration that sits at the limiting-pressure turning point can also be identified as the model along the equilibrium sequence that separates dynamically stable configurations from dynamically unstable ones.