Editing SSC/Stability/GammaVariation (section)

==Example Fundamental Modes for Isolated Configurations==
For an isolated polytrope whose surface does not extend to infinity &#8212; that is, for <math>0 < n < 5</math> &#8212; the eigenvector for the fundamental mode of radial oscillation depends on the specification of a single parameter: <math>\gamma</math>.  Then, for virtually any choice of the square of the radial oscillation frequency, <math>\sigma_c^2</math>, the governing polytropic LAWE can be integrated (usually, numerically) to obtain the radial-displacement, <math>x(\xi)</math>, that is consistent with that choice of <math>\sigma_c^2</math>.  While this function, <math>x(\xi)</math>, satisfies the LAWE, its slope at the surface of the polytrope usually will not satisfy the physically relevant boundary condition.  Other "guesses" for <math>\sigma_c^2</math> must be made until the <math>x(\xi)</math> function satisfies the proper boundary condition; the result provides the eigenfrequency and eigenfunction (together, the eigenvector) that are associated with the specified value of <math>\gamma</math>.

As an example, consider specifying <math>\gamma = \tfrac{5}{3}</math> for an isolated, <math>n = 1</math> polytrope. The following table records the value of the square of the eigenfrequency that has been independently determined by three different research groups:  1.155 by {{ Chatterji51 }}; 1.1499 by {{ HRW66 }}; and 1.1492896    [[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope|herein]].  Also  for comparison, the corresponding ''eigenfunction''  obtained from two of these investigations has been displayed graphically [[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope|herein]].

Not unexpectedly, when a different value of <math>\gamma</math> is specified, the result is a different radial oscillation eigenfrequency along with a different eigenfunction.  However, as was first demonstrated by {{ Sterne37 }}, for an <math>n = 0</math> (uniform-density) polytrope, even though the eigenfrequency varies with the choice of <math>\gamma</math>, the radial displacement ''eigenfunction'' is identically the same for all chosen <math>\gamma</math>.

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="6">Published Fundamental-Mode Oscillation Frequencies</td>
</tr>

<tr>
  <td align="center"><math>n</math></td>
  <td align="center"><math>\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>\gamma</math></td>
  <td align="center"><math>\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c}</math>
  <td align="center">Publication</td>
  <td align="center">Relevant<br />JETohlineWiki<br />Chapter</td>
</tr>

<tr>
  <td align="center"><math>0</math></td>
  <td align="center">1</td>
  <td align="center">Any <math>\gamma</math></td>
  <td align="center"><math>6(\gamma - 4/3)</math></td>
  <td align="left"><sup>c</sup>{{ Sterne37 }}</td>
  <td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|here]]</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{5}{3}</math></td>
  <td align="center"><math>2</math></td>
  <td align="left"><sup>b</sup>{{ HRW66 }}</td>
  <td align="center">[[SSC/Stability/Polytropes/Pt3#Tables|here]]</td>
</tr>

<tr>
  <td align="center" bgcolor="lightgrey" colspan="6">&nbsp; </td>
</tr>

<tr>
  <td align="center"><math>1</math></td>
  <td align="center"><math>\frac{\pi^2}{3}</math></td>
  <td align="center"><math>\frac{5}{3}</math></td>
  <td align="center"><math>1.155</math></td>
  <td align="left"><sup>d</sup>{{ Chatterji51 }}</td>
  <td align="center">[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope|here]]</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{5}{3}</math></td>
  <td align="center"><math>1.1499</math></td>
  <td align="left"><sup>b</sup>{{ HRW66 }}</td>
  <td align="center">[[SSC/Stability/Polytropes/Pt3#Tables|here]]</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{5}{3}</math></td>
  <td align="center"><math>1.1492896</math></td>
  <td align="center" colspan="2">[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Isolated_n_=_1_Polytrope|Our imposed surface B.C.]]</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{20}{13}</math></td>
  <td align="center"><math>0.715</math></td>
  <td align="left"><sup>d</sup>{{ Chatterji51 }}</td>
  <td align="center">n/a</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{10}{7}</math></td>
  <td align="center"><math>0.334</math></td>
  <td align="left"><sup>d</sup>{{ Chatterji51 }}</td>
  <td align="center">n/a</td>
</tr>

<tr>
  <td align="center" bgcolor="lightgrey" colspan="6">&nbsp; </td>
</tr>

<tr>
  <td align="center"><math>3</math></td>
  <td align="right">54.18248</td>
  <td align="center"><math>\frac{5}{3}</math></td>
  <td align="center"><math>0.34175</math></td>
  <td align="left"><sup>a</sup>{{ Schwarzschild41 }}</td>
  <td align="center">[[SSC/Stability/n3PolytropeLAWE#Schwarzschild_(1941)|here]]</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{5}{3}</math></td>
  <td align="center"><math>0.34161</math></td>
  <td align="left"><sup>b</sup>{{ HRW66 }}</td>
  <td align="center">[[SSC/Stability/Polytropes/Pt3#Tables|here]]</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{20}{13}</math></td>
  <td align="center"><math>0.23979</math></td>
  <td align="left"><sup>a</sup>{{ Schwarzschild41 }}</td>
  <td align="center">n/a</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{10}{7}</math></td>
  <td align="center"><math>0.12604</math></td>
  <td align="left"><sup>a</sup>{{ Schwarzschild41 }}</td>
  <td align="center">n/a</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\frac{4}{3}</math></td>
  <td align="center"><math>0.0</math></td>
  <td align="left"><sup>a</sup>{{ Schwarzschild41 }}</td>
  <td align="center">n/a</td>
</tr>

<tr>
  <td align="left" colspan="6">
NOTES:<br />
<ol type="a">
<li>&nbsp;<math>\sigma_c^2 = \tfrac{3}{2}\gamma \omega^2_\mathrm{Sch}</math></li>
<li>&nbsp;<math>\omega^2 = s^2_\mathrm{HRW66}</math></li>
<li>&nbsp;<math>\omega^2 = n^2_\mathrm{Sterne37}</math></li>
<li>&nbsp;<math>\sigma_c^2 = 3\gamma \omega^2_\mathrm{Chatterji}</math></li>
</ol>
  </td>
</tr>
</table>