Editing SSC/Perturbations (section)

===Eddington (1926)===

[[File:Eddington1930Cover.png|thumb|right|200px|Eddington (1926)]]To our knowledge, a derivation of [[#2ndOrderODE|this governing 2nd-order ODE]] was first presented by A. S. Eddington (1926) in a book titled, ''The Internal Constitution of the Stars''.  (This entire book has been digitally scanned and is now [https://www.google.com/books/edition/The_Internal_Constitution_of_the_Stars/N6UNAQAAIAAJ?hl=en&gbpv=1&printsec=frontcover available online].)  The derived expression, which appears on p. 188 as equation (127.6) of Eddington's book, is presented in the following, framed image.


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Pulsation equation extracted from &sect;127 (p. 188) of<br />[https://archive.org/details/TheInternalConstitutionOfTheStars A. S. Eddington (1926)]<br />
''The Internal Constitution of the Stars''<br />Cambridge:  Cambridge University Press
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<!--[[File:Eddington1930.png|600px|center|Eddington (1926)]]-->
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  <td align="right" width="60%"><math>
\frac{d^2\xi_1}{d\xi_0^2} + \frac{4-\mu}{\xi_0} \frac{d\xi_1}{d\xi_0} + \biggl\{ \frac{n^2\rho_0}{\gamma P_0} - \biggl(3 - \frac{4}{\gamma}\biggr)
\frac{\mu}{\xi_0^2} \biggr\}\xi_1
  </math></td>
  <td align="center" width="3%"><math>=</math></td>
  <td align="left"><math>0</math></td>
  <td align="right" width="10%">(127.6)</td>
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  <td align="left" colspan="4">where &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;<math>\mu = g_0\rho_0\xi_0/P_0 \, .</math></td>
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The similarity between Eddington's expression and the [[#2ndOrderODE|governing 2nd-order ODE that we have derived]] is immediately apparent.  Specifically, simply after inserting Eddington's definition of his composite variable, <math>\mu</math>, and making the substitutions,  
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<math>\xi_1 \rightarrow x \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
<math>\xi_0 \rightarrow r_0 \, ,</math> 
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
<math>n^2 \rightarrow \omega^2 \, ,</math> 
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Eddington's pulsation equation becomes,
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<math>
\frac{d^2 x}{dr_0^2} + \biggl[ \frac{4}{r_0} - \frac{g_0 \rho_0}{P_0} \biggr] \frac{dx}{dr_0}
+ \frac{\rho_0}{\gamma P_0} \biggl\{\omega^2 + \biggl(4 - 3\gamma\biggr) \frac{g_0 }{r_0}  \biggr\}x
</math>
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<math>=</math>
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<math>0 \, ,</math>
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which exactly matches [[#2ndOrderODE|our derived governing relation]].