Editing SSC/Stability/n3PolytropeLAWE (section)

===Surface Boundary Condition===

As was stated, [[#Schwarzschild_.281941.29|above]], we presume that as Schwarzschild searched for natural modes of oscillation in isolated, <math>~n=3</math> polytropes, he imposed the following boundary condition at the surface of the configuration:

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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x}{d\ln \xi}\biggr|_\mathrm{surface}</math>
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  <td align="center">
<math>~=</math>
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  <td align="left">
<math>~27.09124 ( \mathfrak{F} + 2\alpha )  - \alpha  \, .</math>
  </td>
</tr>
</table>
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In order to duplicate his findings, then, we need to fine tune our specification of the oscillation frequency such that the resulting displacement function presents this behavior at the surface of our model. A finite-difference expression of this logarithmic derivative that is consistent with the above-described finite-difference algorithm, is,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x}{d\ln \xi} \biggr|_\mathrm{surface}</math>
  </td>
  <td align="center">
<math>~\approx</math>
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  <td align="left">
<math>~\frac{\xi_\mathrm{max}}{x_N} \biggl[ \frac{x_{N+1}-x_{N-1}}{2\Delta_\xi} \biggr] \, .</math>
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</table>
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Everything is known here, except for the quantity, <math>~x_{N+1}</math>, which can be evaluated using the last expression in our algorithm one more time to, in effect, evaluate the eigenfunction just outside the surface.  That is, we obtain <math>~x_{N+1}</math> and, in turn, obtain a value for the logarithmic derivative at the surface, via the expression,
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  <td align="right">
<math>~x_{N+1} \biggl[2\theta_{N} +\frac{4\Delta_\xi \theta_{N}}{\xi_\mathrm{max}} - \Delta_\xi (n+1)(- \theta^')_{N}\biggr] </math>
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<math>~=</math>
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<math>~
x_{N}\biggl\{4\theta_{N} - \frac{\Delta_\xi^2(n+1)}{3}\biggl[ \mathfrak{F}+2\alpha  - 
2\alpha \biggl(- \frac{3\theta^'}{\xi}\biggr)_{N} \biggr]  \biggr\} 
+ x_{N-1} \biggl[\frac{4\Delta_\xi \theta_{N}}{\xi_\mathrm{max}} - \Delta_\xi (n+1)(- \theta^')_{N} - 2\theta_{N}\biggr] 
\, .</math>
  </td>
</tr>
</table>
</div>

We added to our numerical algorithm a step that evaluates, in this manner, the logarithmic derivative of the displacement function at the surface of our polytropic configuration.  The eigenfrequencies that generated displacement functions with this surface behavior are listed for four separate modes in the column of Table 2 titled, "Match B.C."  In every case the values agree to at least five decimal places with the "Match Schwarzschild" eigenfrequencies.  We conclude, therefore, that it ''was'' the implementation of this surface boundary condition that permitted Schwarzschild to quantitatively identify the properties of the eigenvectors associated with natural radial modes of oscillation in <math>~n=3</math> polytropes.