Editing SSC/Stability/n3PolytropeLAWE (section)

=Truncated n = 3 Polytropes=

We understand that, for ''isolated'' <math>~n=3</math> polytropic spheres, the value of the adiabatic exponent for which the configuration is marginally unstable is <math>~\gamma_g = 4/3</math>, which is equivalent to, <math>~\alpha \equiv (3-4/\gamma_g) = 0</math>.  This critical condition is identified by examining when the oscillation frequency of the ''fundamental'' mode goes to zero.  Let's use our numerical integration tool to determine what this critical value of the adiabatic exponent is for ''truncated'', <math>~n=3</math> polytropes.  We will accomplish this as follows:

* At various truncation radii, <math>~0 < \xi_\mathrm{surf}/\xi_\mathrm{max} < 1</math> &hellip;
** Force <math>~\mathfrak{F} = -2\alpha</math>;
** Iterate on the choice of <math>~\alpha</math> until the displacement function with no radial nodes (''i.e.,'' the ''fundamental'' mode) satisfies the surface boundary condition of <math>~(d\ln x/d\ln \xi)_\mathrm{surf} = -3</math>, to a desired level of accuracy.
The following table shows the values of <math>~\alpha_\mathrm{crit}</math> &#8212; and associated values of <math>~\gamma_\mathrm{crit}</math> &#8212; that we obtained for nine different values of <math>~\xi_\mathrm{surf}/\xi_\mathrm{max}</math>; in each case, iterations were continued until the desired surface boundary condition was satisfied to six significant digits.

<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="6">Pressure-Truncated n = 3 Polytropes<br /><math>~N_\mathrm{zones} = 200</math>
</tr>
<tr>
  <td align="center">Edge Zone</td>
  <td align="center"><math>~\frac{\xi_\mathrm{surf}}{\xi_\mathrm{max}}</math></td>
  <td align="center"><math>~\alpha_\mathrm{crit}</math></td>
  <td align="center"><math>~\mathfrak{F}</math></td>
  <td align="center">Surface B.C</td>
  <td align="center"><math>~\gamma_\mathrm{crit}</math></td>
</tr>
<tr>
  <td align="center">200</td>
  <td align="center">1.00</td>
  <td align="center">0</td>
  <td align="center">0</td>
  <td align="center">---</td>
  <td align="center"><math>~\tfrac{4}{3}</math></td>
</tr>
<tr>
  <td align="center">180</td>
  <td align="center">0.90</td>
  <td align="center">-0.000201541</td>
  <td align="center"><math>~-2\alpha_\mathrm{crit}</math></td>
  <td align="center">-3.00000</td>
  <td align="center">1.333244</td>
</tr>
<tr>
  <td align="center">160</td>
  <td align="center">0.8</td>
  <td align="center">-0.00327575</td>
  <td align="center"><math>~-2\alpha_\mathrm{crit}</math></td>
  <td align="center">-3.00000</td>
  <td align="center">1.331879</td>
</tr>
<tr>
  <td align="center">150</td>
  <td align="center">0.75</td>
  <td align="center">-0.00808603</td>
  <td align="center"><math>~-2\alpha_\mathrm{crit}</math></td>
  <td align="center">-3.00000</td>
  <td align="center">1.329749</td>
</tr>
<tr>
  <td align="center">120</td>
  <td align="center">0.60</td>
  <td align="center">-0.0576031</td>
  <td align="center"><math>~-2\alpha_\mathrm{crit}</math></td>
  <td align="center">-3.00000</td>
  <td align="center">1.308214</td>
</tr>
<tr>
  <td align="center">100</td>
  <td align="center">0.5</td>
  <td align="center">-0.159111</td>
  <td align="center"><math>~-2\alpha_\mathrm{crit}</math></td>
  <td align="center">-3.00000</td>
  <td align="center">1.266179</td>
</tr>
<tr>
  <td align="center">80</td>
  <td align="center">0.40</td>
  <td align="center">-0.405712</td>
  <td align="center"><math>~-2\alpha_\mathrm{crit}</math></td>
  <td align="center">-3.00000</td>
  <td align="center">1.174497</td>
</tr>
<tr>
  <td align="center">50</td>
  <td align="center">0.25</td>
  <td align="center">-1.74909</td>
  <td align="center"><math>~-2\alpha_\mathrm{crit}</math></td>
  <td align="center">-3.00000</td>
  <td align="center">0.842266</td>
</tr>
<tr>
  <td align="center">20</td>
  <td align="center">0.10</td>
  <td align="center">-14.6648</td>
  <td align="center"><math>~-2\alpha_\mathrm{crit}</math></td>
  <td align="center">-3.00000</td>
  <td align="center">0.226439</td>
</tr>
</table>
</div>