Editing SSC/Stability/n3PolytropeLAWE (section)

===Schwarzschild (1941)===
We can directly compare our results with [http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild's (1941)] published work on "Overtone Pulsations for the Standard [Stellar] Model."  To begin with, it is straightforward to demonstrate that the last form of the LAWE, provided above, matches equation (2) from [[SSC/Perturbations#Schwarzschild_.281941.29|Schwarzschild (1941)]], if <math>~n</math> is set to 3 &#8212; see the boxed-in excerpt, immediately below.  Note as well that Schwarzschild's dimensionless oscillation frequency &#8212; defined in his equation (1) and which we will label, <math>~\omega_\mathrm{Sch}</math> &#8212; is related to our dimensionless frequency via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma_c^2</math>
  </td>
  <td align="center">
<math>~~\leftrightarrow~~</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{3\gamma_g}{2} \biggr) \omega_\mathrm{Sch}^2 \, .</math>
  </td>
</tr>
</table>
</div>
[http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild (1941)] numerically integrated the LAWE for <math>~n=3</math> polytropic spheres to find eigenvectors (''i.e.,'' the spatially discrete eigenfunction and corresponding eigenfrequency) for five separate oscillation modes (the fundamental mode, plus the 1<sup>st</sup>, 2<sup>nd</sup>, 3<sup>rd</sup>, and 4<sup>th</sup> overtones) for models having four different adopted adiabatic indexes <math>~\gamma_g = \tfrac{4}{3}, \tfrac{10}{7}, \tfrac{20}{13}, \tfrac{5}{3})</math>.  


<div align="center">
<table border="1" cellpadding="5">
<tr><td align="center">
Paragraph extracted from [http://adsabs.harvard.edu/abs/1941ApJ....94..245S M. Schwarzschild (1941)]<p></p>
"''Overtone Pulsations for the Standard Model''"<p></p>
ApJ, vol. 94, pp. 245 - 252 &copy; American Astronomical Society
</td></tr>
<tr><td>
[[File:Schwarzschild1941.png|700px|center|Schwarzschild (1941, ApJ, 94, 245)]]
</td></tr>
<tr><td align="left">
&nbsp; &nbsp; &nbsp; <sup>3</sup>A. S. Eddington (1930), [https://archive.org/details/TheInternalConstitutionOfTheStars ''The Internal Constitution of the Stars''], pp. 188 and 192.
</td></tr>
</table>
</div>

Drawing from our [[SSC/Perturbations#Ensure_Finite-Amplitude_Fluctuations|discussion of the historical treatment of boundary conditions]], we presume that Schwarzschild imposed the following constraint at the surface:

<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>~\frac{1}{\gamma_g} \biggl( 4 - 3\gamma_g + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{3\omega^2 R^3}{4\pi G \gamma \bar\rho} - \alpha \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \biggl[ \frac{\sigma_c^2}{\gamma } \biggl(\frac{\rho_c}{\bar\rho}\biggr) -2 \alpha \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \biggl\{ \biggl[ \mathfrak{F} + 2\alpha \biggr] \biggl(\frac{\rho_c}{\bar\rho}\biggr) -2 \alpha \biggr\} \, .</math>
  </td>
</tr>
</table>
</div>

Recognizing from an [[SSC/Structure/Polytropes#Horedt2004|accompanying tabulation]] that, for <math>~n=3</math> polytropes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho_c}{\bar\rho}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~54.18248 \, ,</math>
  </td>
</tr>
</table>
</div>
we presume that the surface boundary condition imposed by Schwarzschild was,

<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>

Our Table 1 catalogs the eigenfrequencies that Schwarzschild determined (drawn from ''his'' Table 1) for these twenty different models/modes.

<table border="1" cellpadding="8" align="center">
<tr>
  <th colspan="6" align="center">
<font size="+1">Table 1:</font> &nbsp; From Table 1 of [http://adsabs.harvard.edu/abs/1941ApJ....94..245S M. Schwarzschild (1941)]
  </th>
</tr>
<tr>
  <td align="center" rowspan="2">Mode</td>
  <td align="center"><math>~\alpha = 0.0</math><p></p><math>~(\gamma_g = 4/3)</math></td>
  <td align="center"><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"><math>~\alpha = 0.6</math><p></p><math>~(\gamma_g = 5/3)</math></td>
</tr>
<tr>
  <td align="center"><math>~\omega_\mathrm{Sch}^2</math>
  <td align="center"><math>~\omega_\mathrm{Sch}^2</math>
  <td align="center"><math>~\omega_\mathrm{Sch}^2</math>
  <td align="center"><math>~\mathfrak{F} = \biggl[\frac{3\omega_\mathrm{Sch}^2}{2} - 2\alpha \biggr]</math>
  <td align="center"><math>~\omega_\mathrm{Sch}^2</math>
</tr>
<tr>
  <td align="center">0</td>
  <td align="center">0.00000</td>
  <td align="center">0.05882</td>
  <td align="center">0.10391</td>
  <td align="center" bgcolor="pink">-0.64414</td>
  <td align="center">0.13670</td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center">0.16643</td>
  <td align="center">0.19139</td>
  <td align="center">0.21998</td>
  <td align="center" bgcolor="pink">-0.47003</td>
  <td align="center">0.25090</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">0.3392</td>
  <td align="center">0.3648</td>
  <td align="center">0.3920</td>
  <td align="center" bgcolor="pink">-0.2120</td>
  <td align="center">0.4209</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">0.5600</td>
  <td align="center">0.5863</td>
  <td align="center">0.6136</td>
  <td align="center" bgcolor="pink">+0.1204</td>
  <td align="center">0.6420</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="center">0.8283</td>
  <td align="center">0.8554</td>
  <td align="center">0.8832</td>
  <td align="center" bgcolor="pink">+0.5248</td>
  <td align="center">0.9117</td>
</tr>
</table>

[http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild (1941)] also documented the radial structure of the eigenfunction that is associated with each of these twenty model/mode eigenfrequencies.  Each column of his Table 4, except the first, presents numerical values of the amplitude of a specific model/mode at 84 discrete radial locations throughout the n = 3 polytrope; the first column of the table lists the corresponding radial coordinate, <math>~\xi</math>.  Focusing on the model that he analyzed assuming <math>~\alpha = 0.4</math>, we have typed his five columns of data into an Excel spreadsheet and have used this data to generate the pair of plots displayed, below, in Figure 1.  The left-hand panel displays the eigenfunction amplitude versus radius, <math>~x(\xi)</math>, for the fundamental mode as well as for the first four overtones; it essentially replicates Figure 1 from [http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild (1941)].  The right-hand panel displays the same data, but as a semi-log plot; specifically, it displays <math>~y(\xi)</math>, where,
<div align="center">
<math>~y \equiv \frac{1}{2} \log_{10}[x^2 + 10^{-8}] \, .</math>
</div>
Each sharp valley in this semi-log plot highlights the location of a node in the corresponding eigenfunction, that is, it identifies where <math>~x(\xi)</math> crosses through zero.

<span id="Fig1">&nbsp;</span>
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center"><font size="+1">Figure 1:</font>&nbsp;
Schwarzschild's Eigenfunctions for an n = 3 Polytrope with <math>\alpha = 0.4 ~(\gamma = 20/13)</math>
  </th>
</tr>
<tr>
<td align="center">[[File:Schwarzschild1941Combined3.png|800 px|Schwarzschild (1941) eigenfunctions]]</td>
</tr>
</table>