Editing SSC/Perturbations (section)

===Ledoux and Pekeris (1941)===
Historically, by the 1940s, the [[#2ndOrderODE|expression just derived]] was a relatively familiar one to astrophysicists.  For example, the opening paragraph of {{ LP41full }} reads:
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Paragraph extracted from &sect;1 of<br />{{ LP41figure }}<br /> &copy; American Astronomical Society
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<!--[[File:LedouxPekeris1941.jpg|600px|center|Ledoux &amp; Pekeris (1941, ApJ, 94, 124)]]-->
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"The differential equation which governs the adiabatic and radial oscillations of a gaseous star is</font>
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Xr~\frac{d^2\xi}{dr^2} + \frac{d\xi}{dr}\biggl[4X + r~\frac{dX}{dr}\biggr]
+ \xi\biggl[\biggl(\sigma^2 + 4~\frac{Gm(r)}{r^3}\biggr)r\rho + 3~\frac{dX}{dr}  \biggr]
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  <td align="center" width="3%"><math>=</math></td>
  <td align="left"><math>0 \, ,</math></td>
  <td align="right" width="10%">(1)</td>
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<font color="darkgreen">where <math>\xi</math> denotes the ratio <math>\delta r/r</math> of the radial displacement to the radius.  <math>X = \Gamma_1 P, P</math> being the sum of the gaseous pressure <math>p_G</math> and the pressure of radiation <math>p_R</math>.  The quantity <math>\Gamma_1</math> is the adiabatic exponent defined by <math>\delta P/P = \Gamma_1 \delta\rho/\rho</math> and has the well-known value,</font>
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  <td align="right"><math>
\Gamma_1
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  <td align="center" width="3%"><math>=</math></td>
  <td align="left" width="50%"><math>\beta + \frac{(4-3\beta)^2(\gamma - 1)}{\beta + 12(\gamma - 1)(1 - \beta)} \, ,</math></td>
  <td align="right" width="10%">(2)</td>
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<font color="darkgreen">where <math>\beta = p_G/P</math> and <math>\gamma = c_P/c_V, c_p</math> and <math>c_V</math> denoting, respectively, the specific heats at constant pressure and constant volume for the matter; <math>\sigma</math> is equal to <math>2\pi \nu = 2\pi/\tau, \nu</math> being the frequency and <math>\tau</math> the period of oscillation; <math>\rho</math> is the density at <math>r</math> and <math>m(r)</math> is the mass interior to <math>r</math>.</font>
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If we divide their equation (1) through by <math>~Xr = \Gamma_1 P r</math> and recognize that,
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\frac{dX}{dr} = \frac{dX}{dm}\frac{dm}{dr} = - \Gamma_1 g_0 \rho \, ,
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we obtain,

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\frac{d^2\xi}{dr^2} + \biggl[ \frac{4}{r} - \frac{g_0 \rho}{P} \biggr] \frac{d\xi}{dr} +\frac{\rho}{\Gamma_1 P} \biggl[ \sigma^2 + (4 - 3\Gamma_1) \frac{g_0}{r} \biggr] \xi = 0 \, .
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This is clearly the [[#2ndOrderODE|same <math>2^\mathrm{nd}</math>-order, ordinary differential equation as the one we have derived]], but with a [[SR#Adiabatic_Exponent|more general definition of the adiabatic exponent]] that allows consideration of a situation where the total pressure is a sum of both gas and radiation pressure.