Editing SSC/Perturbations (section)

==The Eigenvalue Problem==
Here we adopt a notation and presentation very similar to what can be found in &sect;38 of [<b>[[Appendix/References#KE94|<font color="red">KW94</font>]]</b>].  In particular, we will use <math>~m</math> rather than the more cumbersome <math>M_r</math> to tag each (Lagrangian) mass shell, both initially and at all later times.  As is customary in perturbation studies throughout the field of physics, we will assume that the pressure <math>~P(m,t)</math>, density <math>\rho(m,t)</math>, and radial position <math>r(m,t)</math> of each mass shell at any time {{Math/VAR_Time01}} can be written in the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_0(m) + P_1(m,t) = P_0(m) \biggl[1 + p(m) e^{i\omega t}  \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0(m) + \rho_1(m,t) = \rho_0(m) \biggl[1 + d(m) e^{i\omega t}  \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_0(m) + r_1(m,t) = r_0(m) \biggl[1 + x(m) e^{i\omega t}  \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>

where the subscript "1" denotes the variation of any variable away from its initial value (subscript 0) as drawn from the derived '''structure''' of the selected initial equilibrium model.  These expressions encompass the hypothesis that, when perturbations away from the initial equilibrium state are sufficiently small &#8212; that is, <math>~|p|</math>, <math>~|d|</math>, and <math>~|x|</math> all <math>~\ll 1</math> &#8212; the perturbation can be treated as a product of functions that are separable in <math>~m</math> and {{Math/VAR_Time01}}, and that in general the time-dependent component can be represented by an exponential with an imaginary argument.  The task is to solve a linearized version of the coupled set of key relations for the "eigenfunctions" <math>~p_i(m)</math>, <math>~d_i(m)</math>, and <math>~x_i(m)</math> associated with various characteristic "eigenfrequencies" <math>~\omega_i</math> of the underlying equilibrium model.

===Linearizing the Key Equations===

====Adiabatic form of the First Law of Thermodynamics====

Plugging the perturbed expressions for <math>~P(m,t)</math> and <math>~\rho(m,t)</math> into the adiabatic form of the First Law of Thermodynamics, we obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(i\omega) \rho_0(m) \biggl[1 + d(m) e^{i\omega t} \biggr]P_0(m) p(m) e^{i\omega t} - \gamma_\mathrm{g}(i\omega) P_0(m) \biggl[1 + p(m) e^{i\omega t}  \biggr] \rho_0(m)  d(m) e^{i\omega t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~
(i\omega) \rho_0(m)P_0(m) e^{i\omega t} \biggl\{\biggl[1 + d(m) e^{i\omega t} \biggr] p(m)  - \gamma_\mathrm{g} \biggl[1 + p(m) e^{i\omega t}  \biggr]   d(m)  \biggr\}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>

Because we are seeking solutions that will be satisfied throughout the configuration &#8212; that is, for all mass shells <math>~m</math> &#8212; the expression inside the curly brackets must be zero. Hence,
<div align="center">
<math>
~p(m) - \gamma_\mathrm{g} d(m) + (1 - \gamma_\mathrm{g} ) d(m)p(m) e^{i\omega t} =0  \, .
</math>
</div>
Also, because we are only examining deviations from the initial equilibrium state in which <math>~|d(m)|</math> and <math>~|p(m)|</math> are both <math>\ll 1</math>, then the third term on the left-hand-side of this equation, which contains a product of these two small quantities, must be much smaller than the first two terms.  As is standard in perturbation theory throughout physics, for our stability analysis, we will drop this "quadradic" term and keep only terms that are linear in the small quantities.  This leads to the following algebraic relationship between <math>~d(m)</math> and <math>~p(m)</math>:
<div align="center">
<math>
~p = \gamma_\mathrm{g} d  \, .
</math>
</div>

====Entropy Conservation====
If, instead, we simply demand that the specific entropy remain constant in time, then the expression that relates <math>P</math> to <math>\rho</math> is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{P}{\rho^{\gamma_g}}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\exp\biggl[ \frac{s(\gamma_g - 1)}{\mathfrak{R}/\bar\mu}\biggr]\, .
</math>
  </td>
</tr>
</table>

Plugging the perturbed expressions for <math>P(m,t)</math> and <math>\rho(m,t)</math> into this expression gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\exp\biggl[ \frac{s(\gamma_g - 1)}{\mathfrak{R}/\bar\mu}\biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ P_0(m) \biggl[1 + p(m) e^{i\omega t}  \biggr] \biggr\}
\biggl\{\rho_0(m) \biggl[1 + d(m) e^{i\omega t}  \biggr] \biggr\}^{-\gamma_g}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>P_0(m) \rho_0(m)^{-\gamma_g} 
\biggl[1 + p(m) e^{i\omega t}  \biggr]
\biggl[1 + d(m) e^{i\omega t}  \biggr]^{-\gamma_g} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[1 + p(m) e^{i\omega t}  \biggr]
\biggl[1 + d(m) e^{i\omega t}  \biggr]^{-\gamma_g} \, .
</math>
  </td>
</tr>
</table>
Now, according to the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]],

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>(1 \pm x)^n</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 ~\pm ~nx + \biggl[\frac{n(n-1)}{2!}\biggr]x^2
~\pm~ \biggl[\frac{n(n-1)(n-2)}{3!}\biggr]x^3
+ \biggl[\frac{n(n-1)(n-2)(n-3)}{4!}\biggr]x^4
~~\pm ~~ \cdots
</math>
&nbsp; &nbsp;&nbsp; for <math>~(x^2 < 1)</math>
  </td>
</tr>
</table>

Hence, we see that,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[ 1 + d(m) e^{i\omega t}\biggr]^{-\gamma_g}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
1 ~- ~\gamma_g d(m) e^{i\omega t} 
+ \biggl[\frac{\gamma_g(\gamma_g+1)}{2!}\biggr]\biggl[d(m) e^{i\omega t}\biggr]^2 \, .
</math>
  </td>
</tr>
</table>
Dropping terms of higher order than unity, the relationship between <math>p(m)</math> and <math>d(m)</math> becomes,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>1</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math> 
\biggl[1 + p(m) e^{i\omega t}  \biggr]
\biggl[1 ~- ~\gamma_g d(m) e^{i\omega t} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
1 + p(m) e^{i\omega t} 
~- ~\gamma_g d(m) e^{i\omega t} 
~- ~\gamma_g \cancelto{\mathrm{small}}{d(m)p(m)} e^{2i\omega t}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ p(m)</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>  
\gamma_g d(m)  \, .
</math>
  </td>
</tr>
</table>

This is identical to the algebraic relationship between <math>p(m)</math> and <math>d(m)</math> that has been derived, [[#Adiabatic_form_of_the_First_Law_of_Thermodynamics|immediately above]].

====Continuity Equation====

Adopting the same approach, we will now "linearize" each term in the continuity equation:
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
~\frac{d\rho}{dt}  
</math>
  <td align="center">
<math>
\rightarrow
</math>
  </td>
  <td align="left">
<math>
(i\omega)\rho_0 d~e^{i\omega t}
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
\frac{\rho}{r}  
</math>
  </td>
  <td align="center">
<math>
\rightarrow
</math>
  </td>
  <td align="left">
<math>
\frac{\rho_0}{r_0} \biggl[1 + d e^{i\omega t}  \biggr] \biggl[1 + x e^{i\omega t}  \biggr]^{-1} 
\approx  \frac{\rho_0}{r_0}  \biggl[1 + d ~e^{i\omega t}  \biggr]\biggl[1 - x~ e^{i\omega t}  \biggr] \approx 
 \frac{\rho_0}{r_0} \biggl[1 + (d - x) ~e^{i\omega t}  \biggr]
</math>
</tr>
<tr>
  <td align="right">
<math>
\frac{dr}{dt}
</math>
  </td>
  <td align="center">
<math>
\rightarrow
</math>
  </td>
  <td align="left">
<math>
(i\omega) r_0 x~e^{i\omega t}
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
~\rho^2 r^2 
</math>
  </td>
  <td align="center">
<math>
\rightarrow
</math>
  </td>
  <td align="left">
<math>
\rho_0^2 r_0^2 \biggl[1 + d e^{i\omega t}  \biggr]^2 \biggl[1 + x e^{i\omega t}  \biggr]^2 
\approx  \rho_0^2 r_0^2 \biggl[1 + 2d ~e^{i\omega t}  \biggr]\biggl[1 + 2x~ e^{i\omega t}  \biggr] \approx 
\rho_0^2 r_0^2 \biggl[1 + 2(d + x) ~e^{i\omega t}  \biggr]
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
\frac{d}{dm}\biggl(\frac{dr}{dt}\biggr)
</math>
  </td>
  <td align="center">
<math>
\approx
</math>
  </td>
  <td align="left">
<math>
\frac{d}{dm}\biggl[(i\omega) r_0 x~e^{i\omega t}\biggr] = (i\omega) e^{i\omega t} \biggl[x\frac{dr_0}{dm} + r_0\frac{dx}{dm} \biggr] = (i\omega) e^{i\omega t} \biggl[\frac{x}{4\pi r_0^2 \rho_0} + r_0\frac{dx}{dm} \biggr]
</math>
  </td>
</tr>
</table>

In the last step of this last expression we have made use of the fact that, in the initial, unperturbed equilibrium model, <math>dr_0/dm = 1/(4\pi r_0^2 \rho_0)</math>.  Combining all of these terms and linearizing the combined expression further, the linearized continuity equation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(i\omega)\rho_0 d~e^{i\omega t}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~- \frac{2\rho_0}{r_0} \biggl[1 + (d - x) ~e^{i\omega t}  \biggr] (i\omega) r_0 x~e^{i\omega t} - 4\pi \rho_0^2 r_0^2 \biggl[1 + 2(d + x) ~e^{i\omega t}  \biggr](i\omega) e^{i\omega t} \biggl[\frac{x}{4\pi r_0^2 \rho_0} + r_0\frac{dx}{dm} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \rho_0 d </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~- 3\rho_0 x - 4\pi \rho_0^2 r_0^3 \frac{dx}{dm}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 4\pi \rho_0 r_0^3 \frac{dx}{dm}  </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~- 3 x - d \, ,</math>
  </td>
</tr>
</table>
<table border="1" cellpadding="8"><tr><td align="center">
NOTE: &nbsp; An alternate approach to deriving this expression can be found in an [[SSC/Structure/BiPolytropes/51RenormaizePart2#STEP4|accompanying <font color="red">ASIDE</font>]].
</td></tr></table>
</div>

or,
<div align="center">
<math>
r_0 \frac{dx}{dr_0} \approx - 3 x - d ,
</math>
</div>

where, to obtain this last expression, we have switched back from differentiation with respect to <math>~m</math> to differentiation with respect to <math>~r_0</math>.

====Euler + Poisson Equations====
Finally, linearizing each term in the combined "Euler + Poisson" equation gives:

<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\frac{d^2r}{dt^2}  
</math>
  </td>
  <td align="center">
<math>
\rightarrow
</math>
  </td>
  <td align="left">
<math>
\frac{d}{dt}\biggl[(i\omega) r_0 x~e^{i\omega t}\biggr] = - \omega^2 r_0 x~e^{i\omega t}
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
r^2 \frac{dP}{dm}  
</math>
  </td>
  <td align="center">
<math>
\rightarrow
</math>
  </td>
  <td align="left">
<math>
r_0^2 \biggl[1 + x~ e^{i\omega t}  \biggr]^2 \biggl\{\frac{dP_0}{dm} \biggl[1 + p~ e^{i\omega t}  \biggr]  + P_0~e^{i\omega t} \frac{dp}{dm}   \biggr\} \approx r_0^2 \frac{dP_0}{dm} \biggl[1 + (2x+p)~ e^{i\omega t}  \biggr] + P_0 r_0^2~e^{i\omega t} \frac{dp}{dm}
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
\frac{Gm}{r^2}  
</math>
  </td>
  <td align="center">
<math>
\rightarrow
</math>
  </td>
  <td align="left">
<math>
\frac{Gm}{ r_0^2} \biggl[1 + x~ e^{i\omega t}  \biggr]^{-2} \approx \frac{Gm}{ r_0^2} 
\biggl[1 -2 x~ e^{i\omega t}  \biggr] \, .
</math>
  </td>
</tr>
</table>

Hence, the combined linearized relation is,
<div align="center">
<math>
- \omega^2 r_0 x~e^{i\omega t} \approx -4\pi \biggl\{r_0^2 \frac{dP_0}{dm} \biggl[1 + (2x+p)~ e^{i\omega t}  \biggr] + P_0 r_0^2~e^{i\omega t} \frac{dp}{dm} \biggr\} - \frac{Gm}{ r_0^2} \biggl[1 -2 x~ e^{i\omega t}  \biggr]
</math><br />
<math>
\Rightarrow ~~~~~ e^{i\omega t} \biggl\{(2x + p)4\pi r_0^2 \frac{dP_0}{dm}-2x \frac{Gm}{r_0^2} + 4\pi P_0 r_0^2 \frac{dp}{dm} -\omega^2 r_0 x  \biggr\} \approx - 4\pi r_0^2 \frac{dP_0}{dm} - \frac{Gm}{r_0^2}
</math><br />
<math>
\Rightarrow ~~~~~  4\pi P_0 r_0^2 \frac{dp}{dm}  \approx (4x + p)g_0 + \omega^2 r_0 x \, ,
</math><br />
</div>
<span id="g0">where,</span> in order to obtain this last expression we have made use of the fact that, in the unperturbed equilibrium configuration,
<div align="center" id="g0Defined">
<math> 
g_0(m) \equiv \frac{Gm}{r_0^2} = - 4\pi r_0^2 \frac{dP_0}{dm} = - \frac{1}{\rho_0} \frac{dP_0}{dr_0} \, .
</math><br />
</div>
Switching back from differentiation with respect to <math>~m</math> to differentiation with respect to <math>~r_0</math>, the "Euler + Poisson" combined linearized relation can alternatively be written as,
<div align="center">
<math>
\frac{P_0}{\rho_0} \frac{dp}{dr_0}  \approx (4x + p)g_0 + \omega^2 r_0 x .
</math><br />
</div>

===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>:

<div align="center">
<table border="1" cellpadding="10">
<tr><td align="center">
<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>
</td></tr>
</table>
</div>

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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{dr_0} \biggl[ r_0 \frac{dx}{dr_0}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{d}{dr_0}\biggl[ 3 x + \frac{p}{\gamma_\mathrm{g}} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ r_0 \frac{d^2x}{dr_0^2} + 4 \frac{dx}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{\gamma_\mathrm{g}} \frac{dp}{dr_0} \, .</math>
  </td>
</tr>
</table>
</div>

Similarly, replacing <math>~p</math> by <math>~d</math> in the second expression, then using the first expression to eliminate <math>~d</math> gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{P_0}{\rho_0} \frac{dp}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[4x + \gamma_\mathrm{g}\biggl( -3x -r_0\frac{dx}{dr_0} \biggr) \biggr] g_0 + \omega^2 r_0 x </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ \frac{1}{\gamma_\mathrm{g}} \frac{dp}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>

Finally, then, combining these two expressions gives the desired 2<sup>nd</sup>-order ODE, which we will henceforth refer to as the,

<div align="center" id="2ndOrderODE">
<font color="#770000">'''LAWE: &nbsp; Linear Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
</div>


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,
<div align="center">
<math>~\omega^2 = (3\gamma_g - 4) \frac{g_0}{r_0} \, .</math>
</div>
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;"

===Boundary Conditions===
Two boundary conditions must accompany the derived, 2<sup>nd</sup>-order ODE.  It is customary to establish one of these conditions at the center of the spherically symmetric configuration, and the other at the surface.

====Inner Boundary====
In order for the solution to be physically reasonable, the eigenfunction, <math>~x(r_0)</math>, must be "regular" at the center of the configuration.  This demand will be met if the function's first derivative goes to zero at the center, that is, if,
<div align="center">
<math>~\frac{dx}{dr_0} = 0</math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = 0 \, .</math>
</div> 

====Outer Boundary====

=====Set the Surface Pressure Fluctuation to Zero=====
Using the [[#PGE:AdiabaticFirstLaw|above "Linearized Adiabatic Form of the First Law of Thermodynamics"]] to replace the fractional density variation, <math>~d</math>, in favor of the fractional pressure variation, <math>~p</math> in the [[#Continuity|above "Linearized Equation of Continuity"]], gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~p</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\gamma_g \biggl( 3 x +  r_0 \frac{dx}{dr_0} \biggr) \, .</math>
  </td>
</tr>
</table>
</div>

[http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux &amp; Pekeris (1941]; see [[#Ledoux_and_Pekeris_.281941.29|additional discussion below]]) suggest that an adequate outer boundary condition is provided by setting the fractional pressure fluctuation, <math>~p</math>, to zero at the surface.  Leaning on this just-derived relation, therefore, they recommend (see their equation 4) imposing the following surface boundary constraint on the fractional radial variation, <math>~x</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~-\gamma_g \biggl( 3 x +  r_0 \frac{dx}{dr_0} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, .</math>
  </td>
</tr>
</table>
</div>

=====Ensure Finite-Amplitude Fluctuations=====
Here we follow the discussion provided by [http://adsabs.harvard.edu/abs/1967IAUS...28....3C J. P. Cox (1967)]; specifically, the relevant discussion begins in the middle of p. 21, in association with Cox's equation (3.7).  Text drawn directly from [http://adsabs.harvard.edu/abs/1967IAUS...28....3C J. P. Cox (1967)] is presented here in green.  


<font color="green">At the surface <math>~(r_0 = R)</math> of our</font> oscillating, spherically symmetric configuration <font color="green">we must require, in general, that all relative pulsation variables &hellip; be finite.  The specific surface condition can be obtained most generally from the</font> [[#Summary_Set_of_Linearized_Equations|above derived]],
<div align="center">
<font color="#770000">'''Linearized Euler + Poisson Equations'''</font><br />
<math>
\frac{P_0}{\rho_0} \frac{dp}{dr_0}  = (4x + p)g_0 + \omega^2 r_0 x .
</math>
</div>
Multiplying through by <math>~(R\rho_0/P_0)</math> and remembering that 
<div align="center">
<math>g_0 = - \frac{1}{\rho_0} \frac{dP_0}{dr_0} \, ,</math>
</div>
this relation assumes the form of equation (3.7) in [http://adsabs.harvard.edu/abs/1967IAUS...28....3C J. P. Cox (1967)], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R \frac{dp}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{R}{P_0} \frac{dP_0}{dr_0}\biggl[(4x + p) + \frac{\omega^2 r_0 x}{g_0} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R}{\lambda_p}\biggl[(4x + p) + \frac{\omega^2 r_0 x}{g_0} \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
where, as is highlighted by Cox in association with his equation (2.42'),
<div align="center">
<math>~\lambda_p \equiv -\biggl(\frac{d\ln P_0}{dr_0} \biggr)^{-1} \, ,</math>
</div>
<font color="green">is the (equilibrium) pressure scale height</font> of the configuration.  (Note that, the [https://en.wikipedia.org/wiki/Scale_height pressure scale height] is often represented by the variable, <math>~H</math>, instead of <math>~\lambda_p</math>.)


<font color="green">Since <math>~R/\lambda_p \gg 1</math> at the photosphere for most stars (and <math>~R/\lambda_p = \infty</math> if <math>~P/\rho</math> is assumed to vanish at the surface), a reasonable surface boundary condition would be</font> to force the terms inside the square brackets on the right-hand side of this expression to sum to zero, that is, for the pressure fluctuation to obey the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~p</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl(4 + \frac{\omega^2 r_0 }{g_0} \biggr)x </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl( 4 + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) x</math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, .</math>
  </td>
</tr>
</table>
</div>
<font color="green">This boundary condition prevents <math>~dp/dr_0</math> from having a large (or infinite) value at <math>~r_0 = R</math> and also requires that <math>~p</math> be finite at <math>~r_0 = R</math> even if <math>~P_0 = 0</math> here.</font>  This is the surface boundary condition specified in two key review articles on this subject &#8212; one by [http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C R. F. Christy (1965]; see [[#Review_Article_by_Christy_.281966.29|discussion below]]) and another, almost a decade later, by [http://adsabs.harvard.edu/abs/1974RPPh...37..563C J. P. Cox (1974]; see [[#Review_Article_by_Cox_.281974.29|additional discussion below]]).  

<span id="ChristyCox">
[http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C Christy (1965)] and [http://adsabs.harvard.edu/abs/1967IAUS...28....3C Cox (1967)] also point out that, by calling upon the [[#PGE:AdiabaticFirstLaw|above "Linearized Adiabatic Form of the First Law of Thermodynamics"]] to replace the fractional pressure variation, <math>~p</math>, in favor of the fractional density variation, <math>~d</math>; then using the [[#Continuity|above "Linearized Equation of Continuity"]] to replace <math>~d</math> in favor of the fractional radial displacement, <math>~x</math>, the same boundary condition may be written as,</span>

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \gamma_g d</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( 4 + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) x</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ 3 x +  r_0 \frac{dx}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( 4 + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) \frac{x}{\gamma_g}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ r_0 \frac{d\ln x}{dr_0}</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>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, ,</math>
  </td>
</tr>
</table>
</div>

which, as [http://adsabs.harvard.edu/abs/1967IAUS...28....3C Cox (1967)] summarizes (see following his equation 3.9), <font color="green">gives the logarithmic slope <math>~d\ln x/dr_0</math> of the relative pulsation amplitude <math>~x</math> at <math>~r_0 = R</math> in terms of <math>~\gamma_g</math> and the dimensionless frequency <math>~\omega^2R^3/GM_\mathrm{tot}</math> and assures that both <math>~x</math> and <math>~dx/dr_0</math> are finite at <math>~r_0 = R</math>.</font>  This is the boundary condition <font color="green">conventionally used in connection with the adiabatic wave equation</font>.

=====Ignore Atmospheric Inertia=====
Here we echo the discussion presented in &sect;38.1 of [<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], where an alternative boundary condition at the surface of our spherically symmetric, oscillating configuration is recommended; text drawn verbatim from this reference is shown here in green.  <font color="green">We simplify the atmosphere by assuming its mass <math>~m_a</math> to be comprised in a thin layer at <math>~r_0 = R(t)</math>, which follows the changing <math>~R</math> during the oscillations and provides the outer boundary condition at each moment by its weight</font>.  This amounts to ignoring the inertia of this very thin, nearly massless layer and is accomplished, in practice, by setting to zero the second time-derivative on the left-hand side of the [[#PGE:Euler|above "Euler + Poisson Equation"]]. Hence, the pressure, <math>~P_b</math>, at the base of the outermost (atmospheric) layer of the oscillating configuration is described, at all times, by,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cancelto{0}{\frac{d^2 r}{dt^2}}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~- 4\pi r^2 \frac{dP}{dM_r} - \frac{GM_r}{r^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ \frac{GM_\mathrm{tot}}{4\pi R^4}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~- \frac{dP}{dM_r} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~- \frac{\Delta P}{\Delta M_r} = - \frac{(0 - P_b)}{m_a} = \frac{P_b}{m_a} \, .</math>
  </td>
</tr>
</table>
</div>

Appreciating that <math>~R</math> and <math>~P_b</math> are the only time-varying quantities in this expression, perturbing then linearizing the expression gives (at <math>~r_0 = R</math>),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(P_b)_0[1 + p e^{i\omega t}]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{GM_\mathrm{tot} m_a}{4\pi R_0^4}\biggl[ 1 + x e^{i\omega t} \biggr]^{-4}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ 1 + p e^{i\omega t}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~1 -4 x e^{i\omega t} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ p + 4x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, .</math>
  </td>
</tr>
</table>
</div>

This is identical to the boundary condition presented as equation (38.12) in [<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>].  Combining this condition with the [[#Summary_Set_of_Linearized_Equations|above "linearized adiabatic form of the first law of thermodynamics"]] allows us to write, as well,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \gamma_g d + 4x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, ;</math>
  </td>
</tr>
</table>
</div>
and, in combination with the  [[#Summary_Set_of_Linearized_Equations|above "linearized continuity equation"]], to also conclude that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ r_0 \frac{dx}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-3x + \frac{4x}{\gamma_g}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~~~ \biggl( \frac{r_0}{x} \biggr) \frac{dx}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\gamma_g} (4-3\gamma_g )</math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, .</math>
  </td>
</tr>
</table>
</div>
This matches the boundary condition, as presented in equation (38.13) of [<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>].

=====Implications=====
'''First Case:'''  If the surface pressure fluctuation is set to zero, we have just deduced that, at the surface of the configuration,
<div align="center">
<math>~3 x +  r_0 \frac{dx}{dr_0} =0 \, .</math>
</div>
This implies that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x}{d\ln r_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-3</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ d\ln x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-3d\ln r_0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \ln x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ln C_0 - 3\ln r_0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{C_0}{r_0^3} \, .</math>
  </td>
</tr>
</table>
</div>

It is customary to normalize the radial eigenfunction, <math>~x</math>, in such a way that it goes to unity at the surface.  Therefore, in order to satisfy this "first case" boundary condition, at the surface of the oscillating configuration, the eigenfunction must display the behavior,
<div align="center">
<math>~x = \biggl( \frac{R}{r_0}\biggr)^{3} \, .</math>
</div>

'''Second Case:'''  If, instead, we insist that the ''first derivative'' of the surface pressure fluctuation be zero, then, as we have just deduced, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( 4 - 3\gamma_g + \frac{\omega^2 r_0}{g_0}\biggr) \frac{x}{\gamma_g}-r_0 \frac{dx}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, .</math>
  </td>
</tr>
</table>
</div>
But, rearranging terms in the full ''[[#2ndOrderODE|linear adiabatic wave equation]]'', we see that, throughout the entire structure (including the surface),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ 
\biggl(\frac{g_0 \rho_0 }{P_0 r_0} \biggr)\biggl[\biggl(4 - 3\gamma_\mathrm{g}  + \frac{\omega^2 r_0}{g_0} \biggr)  \frac{x}{\gamma_\mathrm{g}}
- r_0 \frac{dx}{dr_0}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{d^2x}{dr_0^2} + \biggl(\frac{4}{r_0}\biggr)\frac{dx}{dr_0} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Since, according to the "second case" surface boundary condition, the term inside the square brackets on the left-hand side of this expression must be zero at the surface, it must also be true that the term inside the square brackets on the right-hand side is zero.  That is, the "second case" boundary condition will be satisfied if, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2x}{dr_0^2} + \biggl(\frac{4}{r_0}\biggr)\frac{dx}{dr_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~\frac{1}{r_0^4} \cdot \frac{d}{dr_0}\biggl[ r_0^4\frac{dx}{dr_0} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~r_0^4\frac{dx}{dr_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~C_0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~dx </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~C_0 r_0^{-4} dr_0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~C_1 - \biggl(\frac{C_0}{3} \biggr) r_0^{-3} </math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, .</math>
  </td>
</tr>
</table>
</div>

Again, given that it is customary to normalize the radial eigenfunction, <math>~x</math>, such that it goes to unity at the surface, the eigenfunction must display the behavior,
<div align="center">
<math>~x = 1 + \frac{C_0}{3}\biggl(\frac{1}{R^3} - \frac{1}{r_0^3} \biggr) \, ,</math>
</div>
at the surface of the oscillating configuration in order to satisfy this "second case" boundary condition.

'''Third Case:'''  If we follow the lead of [<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>] and choose to establish a surface boundary condition that effectively ignores the inertia of the configuration's atmosphere, then, as we have just determined,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x}{d \ln r_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 3 + \frac{4}{\gamma_g} </math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, .</math>
  </td>
</tr>
</table>
</div>
As in the "first case" discussed above, this constraint leads to a power-law <math>~x(r_0)</math> behavior at the surface.  Specifically, this "third case" boundary condition &#8212; along with the convention that <math>~x \rightarrow 1</math> at the surface &#8212; demands an eigenfunction whose behavior at the surface is,
<div align="center">
<math>~x = \biggl( \frac{R}{r_0}\biggr)^{3-4/\gamma_g} \, .</math>
</div>

===What Makes This an Eigenvalue Problem?===
Our own study of radial oscillations in spherically symmetric, self-gravitating fluids has fostered the following appreciation.  Generally, with knowledge only of the boundary condition at the center of the configuration and an associated power-series expansion that provides a description of the displacement function, <math>x(r_0)</math>, near the center &#8212; see, for example, the expansion that is appropriate for [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement|polytropic]] or for [[Appendix/Ramblings/PowerSeriesExpressions#IsothermalDisplacement|isothermal]] spheres &#8212; the [[#2ndOrderODE|LAWE]] can be straightforwardly integrated (numerically) from the center, outward to (or at least very ''near'' to) the surface for practically ''any'' specified value of the (square of the) oscillation frequency, <math>~\omega^2</math>.  As a result, integration from the center, outward, can very naturally generate a ''continuous spectrum'' of displacement functions, if the integration is unconstrained by specification of an outer boundary condition.  This is illustrated, for example, by Figure 2 in our [[SSC/Stability/n3PolytropeLAWE#Figure2|discussion of oscillating n = 3 polytropes]].  Typically, at low frequencies, the displacement function exhibits no, or only a few, radial nodes; but the number of radial nodes that reside within the volume of the configuration steadily grows as the specified frequency increases.

The continuous spectrum is transformed into a discrete spectrum of oscillation ''modes'' when the outward integration is forced to generate a displacement function whose behavior ''at the configuration's surface'' matches a specific surface boundary condition.  For example, as [[#Implications|described above]], if the aim is to ensure that there are no pressure fluctuations at the surface throughout an oscillation, then the only physically relevant displacement functions are the ones whose logarithmic radial derivative at the surface is negative three.  And each of these relevant displacement functions &#8212; now, an ''eigenfunction'' &#8212; will be associated with a ''discrete'' value the oscillation frequency &#8212; the associated ''eigenfrequency.''  Understanding this interplay between a derived solution to the LAWE and the imposed boundary condition helps clarify why this is an eigenvalue problem.