Editing SSC/Stability/n3PolytropeLAWE (section)

==Numerical Integration==

===From the Core to the Surface===

Here we use the [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward|finite-difference algorithm described separately]] to integrate the discretized LAWE from the center of the polytropic configuration, outward to its surface, which in this case &#8212; see, for example, [[SSC/Structure/Polytropes#Horedt2004|p. 77 of Horedt (2004)]] &#8212; is located at the polytropic-coordinate location,
<div align="center">
<math>~\xi_\mathrm{max} = 6.89684862 \, .</math>
</div>
It is assumed, at the outset, that we have in hand an appropriately discretized description of the unperturbed, equilibrium properties of an <math>~n=3</math> polytrope; specifically, at each radial grid line, we have tabulated values of the radial coordinate, <math>~0 \le \xi_i \le \xi_\mathrm{max}</math>, the Lane-Emden function, <math>~\theta_i</math>, and its first radial derivative, <math>~\theta_i'</math>.

The algorithm is as follows (substitute <math>~n=3</math> everywhere):
<ul>
  <li>Establish an equally spaced radial-coordinate grid containing <math>~N</math> grid zones (and, accordingly, <math>~N+1</math> grid ''lines''), in which case the grid-spacing parameter, <math>~\Delta_\xi \equiv \xi_\mathrm{max}/N</math>. </li>
  <li>Specify a value of the adiabatic exponent, <math>~\gamma</math>, which, in turn, determines the value of the parameter, <math>~\alpha \equiv (3-4/\gamma) \, .</math></li>
  <li>Choose a value for the (square of the) dimensionless oscillation frequency, <math>~\sigma_c^2</math>,  which we will accomplish by assigning a value to the parameter, <p></p>
<div align="center">
<math>~\mathfrak{F} \equiv \frac{\sigma_c^2}{\gamma} - 2\alpha \, .</math>
</div>
  </li>
  <li>Set the eigenfunction to unity at the center <math>~(\xi_0 = 0)</math> of the configuration, that is, set <math>~x_0 = 1</math>.</li>
  <li>Determine the value of the eigenfunction at the first grid ''line'' away from the center &#8212; having coordinate location, <math>~\xi_1 = \Delta_\xi </math> &#8212; via the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement|derived power-series expression]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
x_1 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x_0 \biggl[ 1 - \frac{\Delta_\xi^2 (n+1) \mathfrak{F}}{60} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
  </li>
  <li>At all other grid lines, <math>~i=2,N</math>, determine the value of the eigenfunction, <math>~x_i</math>, via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_i \biggl[2\theta_{i-1} +\frac{4\Delta_\xi \theta_{i-1}}{\xi_{i-1}} - \Delta_\xi (n+1)(- \theta^')_{i-1}\biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x_{i-1}\biggl\{4\theta_{i-1} - \frac{\Delta_\xi^2(n+1)}{3}\biggl[ \mathfrak{F}+2\alpha  - 
2\alpha \biggl(- \frac{3\theta^'}{\xi}\biggr)_{i-1} \biggr]  \biggr\} 
+ x_{i-2} \biggl[\frac{4\Delta_\xi \theta_{i-1}}{\xi_{i-1}} - \Delta_\xi (n+1)(- \theta^')_{i-1} - 2\theta_{i-1}\biggr] 
\, .</math>
  </td>
</tr>
</table>
</div>
  </li>
</ul>

We divided our model into <math>~N = 200</math> radial zones and, using this algorithm, integrated the LAWE from the center of the configuration to the surface, for <math>~\alpha = 0.4</math>, and approximately 40 different chosen values of the frequency parameter across the range, <math>~-0.7 \le \mathfrak{F} \le + 0.3</math>.  The radial displacement functions resulting from these integrations are presented in Figure 2 as an animation sequence.  The specified value of <math>~\mathfrak{F}</math> is displayed at the top of each animation frame, and the resulting displacement function, <math>~x(r/R)</math>, is traced by the small, red circular markers in each frame.

<div align="center" id="SchwarzschildMovie">
<table border="0" cellpadding="5" align="center">
<tr>
  <th colspan="1" align="center"><font size="+1">Figure 2:</font>&nbsp; Numerically Determined Eigenfunctions for Various <math>~\mathfrak{F}</math></th>
  <td align="center" rowspan="2">
<table border="1" cellpadding="5">
<tr>
  <th align="center" colspan="3"><font size="+1">Table 2</font></th>
</tr>
<tr>
  <td align="center" rowspan="2">Mode</td>
  <td align="center" colspan="1">Match Schwarzschild</td>
  <td align="center" colspan="1">Match B.C.</td>
</tr>
<tr>
  <td align="center"><math>~\mathfrak{F}</math></td>
  <td align="center"><math>~\mathfrak{F}</math></td>
</tr>
<tr>
  <td align="center">0</td>
  <td align="center">-0.644131578154</td>
  <td align="center">-0.644131577959</td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center">-0.47013976423</td>
  <td align="center">-0.47013975308</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">-0.2121284391</td>
  <td align="center">-0.2121282667</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">+0.1202565375</td>
  <td align="center">+0.120257856</td>
</tr>
</table>
</td>
</tr>
<tr>
<td align="center" bgcolor="white">
[[File:Schwarzschild1941movie.gif|Eigenfunctions for Standard Model]]
</td>
</tr>
</table>
</div>

Each frame of the Figure 2 animation also displays, as smooth solid curves, the radial eigenfunctions that Schwarzschild (1941) obtained for the fundamental mode (blue curve) and the first three overtone modes (green, purple, &amp; orange curves, repectively) for his model with <math>~\alpha = 0.4</math>.  These are the same curves that appear in the left-hand panel of Figure 1, but here the displacement amplitude has been renormalized such that <math>~x_0 = 1</math>, and, along the horizontal axis, the radial location is marked in terms of the fractional radius, <math>~r/R \equiv \xi/\xi_\mathrm{max}</math>.  In our examination of this model, as we approached each specific value of a modal eigenfrequency identified by Schwarzschild &#8212; see the frequencies highlighted in pink in our Table 1 &#8212; we fine-tuned our choice of the eigenfrequency in order to find a displacement function whose surface ''amplitude'' matched, to a high level of precision, the surface amplitude associated with Schwarzschild's corresponding published eigenfunction.  The column of our Table 2 whose heading is "Match Schwarzschild" identifies &#8212; to at least 10 digits precision &#8212; the frequency choice that was required in order for these surface ''amplitudes'' to match in each case.  

It is gratifying to see that our resulting frequencies match well the values published by Schwarzschild (as highlighted in pink, above).  But this does not satisfactorily explain why, among the entire range of displacement functions displayed (in red) in the Figure 2 animation, Schwarzschild labeled these specific ones as the eigenmodes.  As we shall now demonstrate, his eigenmode identifications resulted from the imposition of a specific, physically justified constraint on the ''slope'', rather than the value, of the displacement function at the surface of the configuration.  (See also our separate brief answer to the question, "[[SSC/Perturbations#What_Makes_This_an_Eigenvalue_Problem.3F|What makes this an eigenvalue problem?]]".)

===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:

<div align="center">
<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>~=</math>
  </td>
  <td align="left">
<math>~27.09124 ( \mathfrak{F} + 2\alpha )  - \alpha  \, .</math>
  </td>
</tr>
</table>
</div>

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,
<div align="center">
<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>
  </td>
  <td align="left">
<math>~\frac{\xi_\mathrm{max}}{x_N} \biggl[ \frac{x_{N+1}-x_{N-1}}{2\Delta_\xi} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <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>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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.

===Our Results===

<table border="1" cellpadding="8" align="center">
<tr>
  <th colspan="9" align="center">
<font size="+1">Table 3:</font> &nbsp; Our Results (to be compared w/ Table 1, above)
  </th>
</tr>
<tr>
  <td align="center" rowspan="2">Mode</td>
  <td align="center" colspan="2"><math>~\alpha = 0.0</math><p></p><math>~(\gamma_g = 4/3)</math></td>
  <td align="center" colspan="2"><math>~\alpha = 0.2</math><p></p><math>~(\gamma_g = 10/7)</math></td>
  <td align="center" colspan="2"><math>~\alpha = 0.4</math><p></p><math>~(\gamma_g = 20/13)</math></td>
  <td align="center" colspan="2"><math>~\alpha = 0.6</math><p></p><math>~(\gamma_g = 5/3)</math></td>
</tr>
<tr>
  <td align="center"><math>~\mathfrak{F}</math>
  <td align="center"><math>~\omega_\mathrm{Sch}^2 = \frac{2}{3}\biggl(\mathfrak{F}+2\alpha  \biggr)</math>
  <td align="center"><math>~\mathfrak{F}</math>
  <td align="center"><math>~\omega_\mathrm{Sch}^2</math>
  <td align="center"><math>~\mathfrak{F} </math>
  <td align="center"><math>~\omega_\mathrm{Sch}^2</math>
  <td align="center"><math>~\mathfrak{F}</math>
  <td align="center"><math>~\omega_\mathrm{Sch}^2</math>
</tr>
<tr>
  <td align="center">0</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">-0.311782342981</td>
  <td align="center">0.058812</td>
  <td align="center" bgcolor="pink">-0.644131577959</td>
  <td align="center">0.103912</td>
  <td align="center">---</td>
  <td align="center">---</td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center">+0.24946512002</td>
  <td align="center">0.166310</td>
  <td align="center">-0.113086698932</td>
  <td align="center">0.191276</td>
  <td align="center" bgcolor="pink">-0.47013975308</td>
  <td align="center">0.219907</td>
  <td align="center">---</td>
  <td align="center">---</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">+0.50882623652</td>
  <td align="center">0.339217</td>
  <td align="center">+0.14705874055</td>
  <td align="center">0.364706</td>
  <td align="center" bgcolor="pink">-0.2121282667</td>
  <td align="center">0.391914</td>
  <td align="center">---</td>
  <td align="center">---</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">+0.83977118</td>
  <td align="center">0.559847</td>
  <td align="center">+0.479241829</td>
  <td align="center">0.586161</td>
  <td align="center" bgcolor="pink">+0.120257856</td>
  <td align="center">0.613505</td>
  <td align="center">---</td>
  <td align="center">---</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="center">+1.24253191</td>
  <td align="center">0.828355</td>
  <td align="center">+0.8832297</td>
  <td align="center">0.855486</td>
  <td align="center" bgcolor="pink">+0.52498863</td>
  <td align="center">0.883326</td>
  <td align="center">---</td>
  <td align="center">---</td>
</tr>
</table>