Editing SSC/Perturbations (section)

===Schwarzschild (1941)===
In the same volume of ''The Astrophysical Journal'', [http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild (1941)] published work on "Overtone Pulsations for the Standard [Stellar] Model."  The following excerpt has been drawn from the first page of this article.

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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
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[[File:Schwarzschild1941.png|700px|center|Schwarzschild (1941, ApJ, 94, 245)]]
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&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.
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The similarity between Schwarzschild's "pulsation equation" and the [[#2ndOrderODE|governing 2nd-order ODE that we have derived]] is immediately apparent; for example, the eigenfrequency, <math>~\omega</math>, is the same in both, 
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<math>~\xi_1 \leftrightarrow x</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
<math>\xi_0 \leftrightarrow r_0 \, .</math> 
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But the two equations are not exactly the same.  To show this, we begin by comparing the last term on the lefthand-side in both expressions and presume that Schwarzschild's <math>~u</math> is related to the state variables in our equation as,
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<math>u = \frac{\gamma_g P_0}{\rho_0} \, .</math>
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Restricting the discussion to only polytropic equations of state &#8212; that is, <math>~P_0 = K\rho_0^{(n+1)/n}</math> &#8212; we can write,
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<math>~u</math>
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<math>~=</math>
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<math>~\gamma_g K\rho_0^{1/n} \, ,</math>
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which means that,
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<math>~\frac{du}{d\xi_0}</math>
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<math>~=</math>
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<math>~\biggl( \frac{\gamma_g K}{n} \biggr) \rho_0^{(1-n)/n} \frac{d\rho_0}{d\xi_0} 
= \biggl( \frac{\gamma_g}{n+1} \biggr) \frac{1}{\rho_0} \frac{dP_0}{d\xi_0} = - \biggl( \frac{\gamma_g}{n+1} \biggr)  g_0 \, ,</math>
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where the last step results from recalling that, by [[#g0Defined|our definition above]], the unperturbed gravitational acceleration is,
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<math>g_0 = - \frac{1}{\rho_0} \frac{dP_0}{dr_0} \, .</math>
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Hence, when considering polytropic configurations, the following substitutions are appropriate between the two equations:
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<math>\frac{du}{d\xi_0} \leftrightarrow - \biggl( \frac{\gamma_g}{n+1} \biggr) g_0 </math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
<math>\frac{1}{u} \frac{du}{d\xi_0} \leftrightarrow - \frac{1}{(n+1)} \frac{g_0 \rho_0}{P_0} \, .</math>
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Making these substitutions into Schwarzschild's pulsation equation gives,

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\frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \frac{4}{(n+1)}\biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} + \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 - \biggl( \frac{4a\gamma_g}{n+1} \biggr)\frac{g_0}{r_0} \biggr]  x = 0 \, .
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Appreciating that, in Schwarzschild's expression, <math>~a \equiv - (4-3\gamma_g)/\gamma_g</math>, we see that our expression matches his if and only if <math>~n=3</math>.  This is, indeed, precisely the "standard model" that Schwarzschild was considering.

In a separate chapter that discusses the [[SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres|radial oscillations of n =3 polytropes]], we provide a much more in-depth review of this published investigation by Schwarzshild.