Editing SSC/Perturbations (section)

===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>