Editing SSC/Stability/GammaVariation (section)

===Comments on Uniform-Density Configurations===

According to [[#The_Sterne37_Solution|Sterne's stability analysis]], the square of the oscillation frequency, <math>\omega^2/(4\pi G \rho_c)</math>, of the fundamental mode is negative for all values of <math>\gamma_\mathrm{g} < \tfrac{4}{3}</math>.  All models with <math>\gamma_\mathrm{g} < \tfrac{4}{3}</math> are therefore dynamically unstable toward collapse with a radial-displacement eigenfunction given by that of the fundamental mode.  We appreciate as well that all models with <math>\gamma_\mathrm{g}  < \tfrac{2}{5}</math> are (also) dynamically unstable toward collapse with a radial-displacement eigenfunction given by the 1<sup>st</sup> overtone mode.

At the same time, an examination of each model's [[#Entropy_Distribution|entropy distribution]] indicates that models with <math>\gamma_\mathrm{g} > 1</math> are unstable toward convection throughout their entire volume.  Hence, we identify the following model regimes:

<table border="1" align="center" cellpadding="8" width="80%">

<tr>
  <td align="center"><math>\infty \ge \gamma_\mathrm{g} > \tfrac{4}{3}</math></td>
  <td align="left">Dynamically stable against collapse, but unstable toward convection throughout.</td>
</tr>

<tr>
  <td align="center"><math>\tfrac{4}{3} > \gamma_\mathrm{g} > 1</math></td>
  <td align="left">Unstable toward convection throughout and, simultaneously dynamically unstable toward collapse with the eigenfunction provided by the fundamental mode. (All other radial ''overtone'' modes are dynamically stable against collapse.)</td>
</tr>

<tr>
  <td align="center"><math>1 > \gamma_\mathrm{g} > \tfrac{2}{5}</math></td>
  <td align="left">Stable against convection, but dynamically unstable toward collapse with the eigenfunction provided by the fundamental mode. (All other radial ''overtone'' modes are dynamically stable against collapse.)</td>
</tr>

<tr>
  <td align="center"><math>\tfrac{2}{5} > \gamma_\mathrm{g} </math></td>
  <td align="left">Stable against convection, but dynamically unstable ''simultaneously'' toward collapse due to the fundamental and 1<sup>st</sup> overtone modes.</td>
</tr>
</table>