Editing SSC/Perturbations (section)

===Summary Set of Linearized Equations===
In summary, the following three linearized equations describe the physical relationship between the three dimensionless perturbation amplitudes <math>~p(r_0)</math>, <math>~d(r_0)</math> and <math>~x(r_0)</math>, for various characteristic eigenfrequencies, <math>~\omega</math>:

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<font color="#770000">'''Linearized'''</font><br />
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
<math>
r_0 \frac{dx}{dr_0} = - 3 x - d ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:Euler"><font color="#770000">'''Euler + Poisson Equations'''</font></span><br />
<math>
\frac{P_0}{\rho_0} \frac{dp}{dr_0}  = (4x + p)g_0 + \omega^2 r_0 x ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
<math>
p = \gamma_\mathrm{g} d  \, .
</math>
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It is customary to combine these three relations to obtain a single, second-order ODE in terms of the fractional displacement, <math>~x</math> as follows.  Using the third expression to replace <math>~d</math> by <math>~p</math> in the first expression, then differentiating the first expression with respect to <math>~r_0</math> generates,
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<math>~\frac{d}{dr_0} \biggl[ r_0 \frac{dx}{dr_0}\biggr]</math>
  </td>
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<math>~=</math>
  </td>
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<math>~- \frac{d}{dr_0}\biggl[ 3 x + \frac{p}{\gamma_\mathrm{g}} \biggr]</math>
  </td>
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<math>~\Rightarrow ~~~~~ r_0 \frac{d^2x}{dr_0^2} + 4 \frac{dx}{dr_0}</math>
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<math>~=</math>
  </td>
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<math>~- \frac{1}{\gamma_\mathrm{g}} \frac{dp}{dr_0} \, .</math>
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Similarly, replacing <math>~p</math> by <math>~d</math> in the second expression, then using the first expression to eliminate <math>~d</math> gives,
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<math>~\frac{P_0}{\rho_0} \frac{dp}{dr_0}</math>
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<math>~=</math>
  </td>
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<math>~\biggl[4x + \gamma_\mathrm{g}\biggl( -3x -r_0\frac{dx}{dr_0} \biggr) \biggr] g_0 + \omega^2 r_0 x </math>
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<math>~\Rightarrow ~~~~~ \frac{1}{\gamma_\mathrm{g}} \frac{dp}{dr_0}</math>
  </td>
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<math>~=</math>
  </td>
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<math>~- \frac{dx}{dr_0}  \biggl(\frac{r_0 g_0 \rho_0}{P_0}\biggr) + \biggl[\omega^2 + 
(4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr] \biggl(\frac{r_0 \rho_0}{\gamma_\mathrm{g} P_0} \biggr) x \, .</math>
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Finally, then, combining these two expressions gives the desired 2<sup>nd</sup>-order ODE, which we will henceforth refer to as the,

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<font color="#770000">'''LAWE: &nbsp; Linear Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
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After deriving this equation in his review article titled, ''Pulsating Stars'' (see [[#Review_Article_by_Cox_.281974.29|related information, below]]), 
[http://adsabs.harvard.edu/abs/1974RPPh...37..563C Cox (1974)] offers the following immediate observation:  If the motion following a perturbation were homolgous &#8212; that is, if <math>~x \equiv \delta r/r_0</math> were constant in space &#8212; then the terms involving the first and second derivatives of <math>~x</math> would vanish.  These terms therefore arise solely from differential expansion or compression of the configuration's layers.  For the idealized case of homologous oscillations, the governing linearized equation becomes,
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<math>~\omega^2 = (3\gamma_g - 4) \frac{g_0}{r_0} \, .</math>
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The physical interpretation of this relation is that stable oscillatory motion <math>~(\omega^2 > 0)</math> is possible only for <math>~\gamma_g > 4/3 \, .</math>  Otherwise, for <math>~\gamma_g < 4/3 \, ,</math> "the motion will be aperiodic on a time scale which is seen to be of the general order of magnitude of the &hellip; [[SSC/FreeFall#Free-Fall_Collapse|free-fall time]] &hellip; [and] this behaviour corresponds to dynamical instability &hellip;"