Editing SSC/Stability/Isothermal (section)

==Groundwork==
===Equilibrium Model===

====Review====
In an [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|accompanying discussion]], while reviewing the original derivations of {{ Ebert55 }} and {{ Bonnor56 }}, we have detailed the equilibrium properties of pressure-truncated isothermal spheres.  These properties have been expressed in terms of the ''isothermal Lane-Emden function'', <math>~\psi(\xi)</math>, which provides a solution to the governing,
<div align="center">
<table border="1" cellpadding="8" align="center">
<tr><td align="center">
Isothermal Lane-Emden Equation <p></p>
{{ Math/EQ_SSLaneEmden02 }}
</td></tr>
</table>
</div>

A parallel presentation of these details can be found in &sect;2 &#8212; specifically, equations (2.4) through (2.10) &#8212; of {{ Yabushita68 }}.  Each of Yabushita's key mathematical expressions can be mapped to ours via the variable substitutions presented here in Table 1.

<div align="center">
<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="7">
<font size="+1"><b>Table 1:</b></font>&nbsp; Mapping Between Our Notation and that employed by {{ Yabushita68 }}
  </td>
</tr>
<tr>
  <td align="right">Yabushita's (1968) Notation:</td>
  <td align="center" width="8%"><math>~x</math></td>
  <td align="center" width="8%"><math>~\psi</math></td>
  <td align="center" width="8%"><math>~\mu</math></td>
  <td align="center" width="8%"><math>~M</math></td>
  <td align="center" width="8%"><math>~x_0</math></td>
  <td align="center" width="8%"><math>~p_0</math></td>
</tr>
<tr>
  <td align="right">[[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Our Notation]]:</td>
  <td align="center"><math>~\xi</math></td>
  <td align="center"><math>~-\psi</math></td>
  <td align="center"><math>~\bar\mu</math></td>
  <td align="center"><math>~M_{\xi_e}</math></td>
  <td align="center"><math>~\xi_e</math></td>
  <td align="center"><math>~P_e</math></td>
</tr>
</table>
</div>

For example, given the system's sound speed, <math>~c_s</math>, and total mass, <math>~M_{\xi_e}</math>, the expression from [[SSC/Structure/BonnorEbert#Pressure|our presentation]] that shows how the bounding external pressure, <math>~P_e</math>, depends on the dimensionless Lane-Emden function, <math>~\psi</math>, is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_e</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{c_s^8}{4\pi G^3 M_{\xi_e}^2} \biggr) ~\xi_e^4 \biggl(\frac{d\psi}{d\xi}\biggr)^2_e e^{-\psi_e}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \xi_e^2 \biggl(\frac{d\psi}{d\xi}\biggr)_e e^{-(1/2)\psi_e}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{c_s^4}\biggl[ G^3 M_{\xi_e}^2 ~(4\pi P_e)\biggr]^{1 / 2} \, ,</math>
  </td>
</tr>
</table>
</div>

which &#8212; see the boxed-in excerpt that follows &#8212; exactly matches equation (2.9) of {{ Yabushita68 }}, after recalling that the system's sound speed is related to its temperature via the relation,
<div align="center">
<math>c_s^2 = \frac{\Re T}{\bar{\mu}} \, .</math>
</div>

And, [[SSC/Structure/BonnorEbert#Radius|our expression]] for the truncated configuration's equilibrium radius is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{GM_{\xi_e}}{c_s^2} \biggl[ \xi \biggl(\frac{d\psi}{d\xi}\biggr) \biggr]_e^{-1}</math>
  </td>
</tr>
</table>
</div>
which &#8212; see the boxed-in excerpt that follows &#8212; matches equation (2.10) of {{ Yabushita68 }}.


<div align="center" id="Yabushita68">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Equations extracted<sup>&dagger;</sup> from p. 110 of <br />{{ Yabushita68figure }}
</td></tr>
<tr>
  <td align="left">
<!-- [[File:Yabushita68Eqns.png|600px|center|Yabushita (1968)]] -->

<table border="0" align="center" cellpadding="5" width="100%">
<tr>
  <td align="right" width="50%">
<math>
x^2 \frac{d\psi}{dx} e^{1 /  2)\psi}\biggr|_{x=x_0}
</math>
  </td>
  <td align="center" width="5%"><math>=</math></td>
  <td align="left">
<math>
- \biggl(\frac{\mu}{\Re T}\biggr)^2 M G^{3 / 2} (4\pi p_0)^{1 / 2}
</math>
  </td>
  <td align="right" width="5%">(2.9)</td>
</tr>

<tr>
  <td align="right" width="50%">
<math>
R \equiv \ell_0 x_0
</math>
  </td>
  <td align="center" width="5%"><math>=</math></td>
  <td align="left">
<math>
- \frac{M G \mu}{\Re T} \biggl(x_0 \frac{d\psi}{dx}\biggr|_{x_0} \biggr)^{-1}
</math>
  </td>
  <td align="right" width="5%">(2.10)</td>
</tr>
</table>

  </td>
</tr>
<tr><td align="left">
<sup>&dagger;</sup>Layout of equations has been modified from the original publication.
</td></tr>
</table>
</div>

====P-V Diagram====

As we have discussed in a [[SSC/Structure/BonnorEbert#P-V_Diagram|separate chapter that focuses on the structural properties of pressure-truncated Isothermal spheres]], {{ Bonnor56 }} examined the ''sequence'' of equilibrium models that is generated by varying the truncation radius over the range, <math>0 < \xi_e < \infty</math>.  In a diagram that shows how <math>P_e(\xi_e)</math> varies with equilibrium volume, <math>V(\xi_e) \propto R^3</math>, Bonnor noticed that there is a pressure above which no equilibrium configurations exist.  The original P-V diagram published by [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)] has been reproduced here, in the left-hand panel of our Figure 1.  The right-hand panel of Figure 1 shows the same equilibrium sequence, as generated from our [[SSC/Structure/IsothermalSphere#Our_Numerical_Integration|numerical integration of the isothermal Lane-Emden equation]]; we have adopted [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Whitworth's normalizations]], <math>P_\mathrm{rf}</math> and <math>V_\mathrm{rf}</math>.


<div align="center">
<b>Figure 1: &nbsp; Bonnor's P-V Diagram</b> (see [[SSC/Structure/BonnorEbert#Fig1|accompanying discussion]] for details)

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center">
Figure extracted from p. 355 of <br />{{ Bonnor56figure }}
<!--
[http://adsabs.harvard.edu/abs/1956MNRAS.116..351B W. B. Bonnor (1956)]<p></p>
"''Boyle's Law and Gravitational Instability''"<p></p>
MNRAS, vol. 116, pp. 351 - 359 &copy; Royal Astronomical Society
-->
  </td>
  <td align="left">
The {{ Bonnor56 }} equilibrium sequence, but as generated from our [[SSC/Structure/IsothermalSphere#Our_Numerical_Integration|numerical integration of the isothermal Lane-Emden equation]]; in our plot, we have adopted normalizations, <math>P_\mathrm{rf}</math> and <math>V_\mathrm{rf}</math>, from {{ Whitworth81 }}.
  </td>
</tr>
<tr>
  <td align="center">
[[File:Bonnor1956Fig1reproducedB.png|400px|center|Bonnor (1956, MNRAS, 116, 351)]]
  </td>
  <td align="center">
[[File:IsothermalEquilSequence.png|800px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
  </td>
</tr>
</table>
</div>

As is [[SSC/Stability/Isothermal#From_the_Analysis_of_Taff_and_Van_Horn_.281974.29|discussed below]], in separate studies, {{ Yabushita68 }} and {{ TVH74 }} examined the lowest-order modes of radial oscillations that arise in pressure-truncated isothermal spheres.  In both studies, numerical techniques were used to solve the eigenvalue problem associated with the [[#IsothermalLAWE|isothermal LAWE]], as derived below.  The nine individual equilibrium models that were studied by {{ TVH74hereafter }} are identified by the nine small, filled circular markers along the sequence that has been displayed in the right-hand panel of our Figure 1; as labeled, they correspond to models with <math>~\xi_e</math> = 2, 3, 4, 5, 6, 7, 8, 9, and 10.  Earlier, Yabushita studied the oscillation modes of three of these same configurations; specifically, the models with <math>~\xi_e</math> = 6, 7, and 8, which straddle the location along the equilibrium sequence of the model associated with the pressure maximum (the [[SSC/FreeEnergy/EquilibriumSequenceInstabilities#Instabilities_Associated_with_Equilibrium_Sequence_Turning_Points|turning point]] labeled "A" in Bonnor's P-V diagram).

====Other Properties====
<span id="IsothermalVariables">Also, as has been summarized in our [[SSC/Structure/BonnorEbert#P-V_Diagram|accompanying discussion]] of the equilibrium properties of pressure-truncated isothermal spheres, we have,</span>

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

<tr>
  <td align="right">
<math>~r_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \xi \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P_0 = c_s^2 \rho_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(c_s^2 \rho_c) e^{-\psi} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~M_r  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr]  \, .</math>
  </td>
</tr>
</table>
</div>

Hence, for isothermal configurations,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~g_0 \equiv \frac{GM_r}{r_0^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~G\biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr]
\biggl[ \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \xi\biggr]^{-2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_s^2  
\biggl( \frac{4\pi G \rho_c}{c_s^2} \biggr)^{1 / 2} 
\biggl( \frac{d\psi}{d\xi} \biggr) \, .
</math>
  </td>
</tr>
</table>
</div>

===Linearized Wave Equation===
In our [[SSC/Perturbations#2ndOrderODE|introductory discussion of techniques that facilitate linear stability analyses]], we derived what we now repeatedly refer to as the "key" form of the
<div align="center" id="2ndOrderODE">
<font color="#770000">'''LAWE: &nbsp; Linear Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}

[<b>[[Appendix/References#P00|<font color="red">P00</font>]]</b>], Vol. II, &sect;3.7.1, p. 174, Eq. (3.144)
</div>
Here we review two published articles that have presented a partial analysis of radial modes of oscillation in pressure-truncated isothermal spheres.  The analyses presented in both of these papers, effectively, employ this key wave equation, but the authors of these articles present it in different forms.

====Yabushita (1968)====
The linearized wave equation that {{ Yabushita68full }} used to examine the radial pulsation modes of pressure-truncated isothermal spheres is displayed in the following, boxed-in image: 

<div align="center" id="Yabushita68">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Equation extracted from p. 111 of <br />{{ Yabushita68figure }}
</td></tr>
<tr>
  <td align="left">
<!-- [[File:Yabushita68WaveEq.png|500px|center|Yabushita (1968)]] -->

<table border="0" align="center" cellpadding="5" width="100%">
<tr>
  <td align="right" width="65%">
<math>
\frac{\partial^2}{\partial t^2} \delta\rho
- \nabla^2\delta p - 8\pi G \bar\rho \delta\rho
-\nabla\bar\rho \cdot \nabla\delta\Phi - \nabla\delta\rho\cdot \nabla\Phi
</math>
  </td>
  <td align="center" width="5%"><math>=</math></td>
  <td align="left">
<math>
0 \, .
</math>
  </td>
  <td align="right" width="5%">(2.12)</td>
</tr>
</table>

  </td>
</tr>
</table>
</div>

This equation can be obtained straightforwardly through a strategic combination of three of the following four linearized principal governing equations that we have derived in our [[SSC/StabilityEulerianPerspective#Summary_and_Combinations|accompanying, broad introductory discussion]] of linear stability analyses, namely,

<div align="center">
<table border="1" cellpadding="15">
<tr><td align="center">
<font color="#770000">'''Linearized'''</font><br />
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
<math>
\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v} + \vec{v}\cdot \nabla\rho_0 = 0 ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br />
<math>
\frac{\partial \vec{v}}{\partial t}  = - \nabla\Phi_1 - \frac{1}{\rho_0} \nabla P_1 + \frac{\rho_1}{\rho_0^2} \nabla P_0  \, ,
</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_1 = \biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1\, ,
</math>

<font color="#770000">'''Linearized'''</font><br />
<font color="#770000">'''Poisson Equation'''</font><br />

<math>
\nabla^2 \Phi_1 = 4\pi G \rho_1\, .
</math>
</td></tr>
</table>
</div>

Taking the partial time-derivative of the linearized equation of continuity gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>- \nabla\cdot \frac{\partial \vec{v}}{\partial t} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\rho_0}\frac{\partial^2 \rho_1}{\partial t^2}  + \frac{\nabla\rho_0}{\rho_0} \cdot \frac{\partial\vec{v}}{\partial t}  \, ;</math>
  </td>
</tr>
</table>
</div>
and, taking the divergence of the linearized Euler equation gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>-\nabla\cdot \frac{\partial \vec{v}}{\partial t} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\nabla^2 \Phi_1 + \nabla\cdot \biggl[\frac{1}{\rho_0} \nabla P_1\biggr]  - \nabla \cdot \biggl[ \frac{\rho_1}{\rho_0^2} \nabla P_0 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Combining the two, then making two substitutions using (1) the linearized Poisson equation and (2) the linearized Euler equation, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\partial^2 \rho_1}{\partial t^2}  + \nabla\rho_0 \cdot \frac{\partial\vec{v}}{\partial t} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \nabla^2 \Phi_1 + \rho_0 \nabla\cdot \biggl[\frac{1}{\rho_0} \nabla P_1\biggr]  - \rho_0\nabla \cdot \biggl[ \frac{\rho_1}{\rho_0^2} \nabla P_0 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
\frac{\partial^2 \rho_1}{\partial t^2}  + \nabla\rho_0 \cdot \biggl[ - \nabla\Phi_1 - \frac{1}{\rho_0} \nabla P_1 + \frac{\rho_1}{\rho_0^2} \nabla P_0 \biggr] 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi G \rho_0 \rho_1 + \nabla^2 P_1  + \rho_0 \nabla P_1 \cdot \nabla \biggl(\frac{1}{\rho_0} \biggr)  
- \rho_0\nabla \cdot \biggl[ \frac{\rho_1}{\rho_0^2} \nabla P_0 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Rearranging terms, and using the replacement ''equilibrium'' relation, <math>\nabla P_0 = - \rho_0\nabla\Phi_0</math>, gives,

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

<tr>
  <td align="right">
<math>
\frac{\partial^2 \rho_1}{\partial t^2}  - \nabla^2 P_1 - 4\pi G \rho_0 \rho_1  - \nabla\rho_0\cdot\nabla\Phi_1 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\nabla\rho_0}{\rho_0} \cdot \biggl[  \nabla P_1 + \rho_1 \nabla \Phi_0 \biggr] 
+ \rho_0 \nabla P_1 \cdot \nabla \biggl(\frac{1}{\rho_0} \biggr)  
+ \rho_0\nabla \cdot \biggl[ \frac{\rho_1}{\rho_0} \nabla \Phi_0 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\nabla\rho_0}{\rho_0} \cdot \biggl[  \nabla P_1 \biggr] + \frac{\rho_1}{\rho_0} \biggl[ \nabla\rho_0\cdot \nabla \Phi_0 \biggr]
- \frac{1}{\rho_0} \nabla P_1 \cdot \nabla \rho_0  
+ \rho_0 \nabla \Phi_0 \cdot \nabla \biggl[ \frac{\rho_1}{\rho_0} \biggr] 
+ \rho_1\nabla^2  \Phi_0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\rho_1}{\rho_0} \biggl[ \nabla\rho_0\cdot \nabla \Phi_0 \biggr] - \frac{\rho_1}{\rho_0} \biggl[ \nabla \Phi_0 \cdot \nabla\rho_0\biggr]
+ \nabla \Phi_0 \cdot \nabla  \rho_1
+ 4\pi G \rho_0 \rho_1 
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\frac{\partial^2 \rho_1}{\partial t^2}  - \nabla^2 P_1 - 8\pi G \rho_0 \rho_1  - \nabla\rho_0\cdot\nabla\Phi_1 
- \nabla \Phi_0 \cdot \nabla  \rho_1
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, .</math>
  </td>
</tr>
</table>
</div>
This is identical to equation (2.12) of {{ Yabushita68 }}.  Letting <math>t \rightarrow (4\pi G \rho_c)^{-1 / 2} \tau</math> and noting that <math>\rho_c = P_c/c_s^2</math>, we also have,

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

<tr>
  <td align="right">
<math>0 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi G \rho_c^2 ~\biggl\{
\frac{\partial^2}{\partial \tau^2}\biggl(\frac{\rho_1}{\rho_c}\biggr)  - \frac{1}{4\pi G \rho_c^2}\nabla^2 P_1 - 2 \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{\rho_1}{\rho_c}\biggr)
- \frac{1}{4\pi G \rho_c^2} \nabla\rho_0\cdot\nabla\Phi_1 
- \frac{1}{4\pi G \rho_c^2} \nabla \Phi_0 \cdot \nabla  \rho_1
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial^2}{\partial \tau^2}\biggl(\frac{\rho_1}{\rho_c}\biggr)  - \frac{c_s^2}{4\pi G \rho_c}\nabla^2 \biggl(\frac{P_1}{P_c}\biggr) - 2\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{\rho_1}{\rho_c}\biggr)  
- \frac{c_s^2}{4\pi G \rho_c} \nabla\biggl( \frac{\rho_0}{\rho_c}\biggr) \cdot \nabla \biggl( \frac{\Phi_1 }{c_s^2}\biggr)
- \frac{c_s^2}{4\pi G \rho_c} \nabla \biggl( \frac{\Phi_0}{c_s^2}\biggr) \cdot \nabla \biggl(\frac{ \rho_1}{\rho_c} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial^2}{\partial \tau^2}\biggl(\frac{\rho_1}{\rho_c}\biggr)  - \nabla_\xi^2 \biggl(\frac{P_1}{P_c}\biggr) - 2\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{\rho_1}{\rho_c}\biggr)  
- \nabla_\xi\biggl( \frac{\rho_0}{\rho_c}\biggr) \cdot \nabla_\xi \biggl( \frac{\Phi_1 }{c_s^2}\biggr)
- \nabla_\xi \biggl( \frac{\Phi_0}{c_s^2}\biggr) \cdot \nabla_\xi \biggl(\frac{ \rho_1}{\rho_c} \biggr) \, ,
</math>
  </td>
</tr>
</table>
</div>
where, in this last step, we have switched from the radial coordinate, <math>r</math>, to the dimensionless coordinate, <math>\xi \equiv r (4\pi G\rho_c/c_s^2)^{1 / 2}</math>.  This matches equation (2.15) of {{ Yabushita68 }}.  

Now, in principle, we can rewrite this linearized wave equation entirely in terms of the density perturbation by recognizing that, from the isothermal equation of state,
<div align="center">
<math>\frac{P_1}{P_c} = \frac{\rho_1}{\rho_c}</math>;
</div>
and from the Poisson equation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla^2\Phi_1 = 4\pi G \rho_1</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>\Rightarrow</math> &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>\nabla_\xi^2 \biggl( \frac{\Phi_1}{c_s^2} \biggr) = \frac{\rho_1}{\rho_c} \, .</math>
  </td>
</tr>
</table>
</div>
<span id="gDefinition">But this does not work directly because our</span> just-derived, governing linearized wave equation contains the term, <math>\nabla_\xi \Phi_1</math>, rather than, <math>\nabla_\xi^2 \Phi_1</math>.  Instead, following the lead of [http://adsabs.harvard.edu/abs/1957ZA.....42..263E Ebert (1957)], {{ Yabushita68 }} rewrites all of the perturbed variables in terms of a new function, <math>g(\xi)</math>, defined such that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>g e^{i\omega t}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\xi^2 \frac{d}{d\xi}\biggl( \frac{\Phi_1}{c_s^2}\biggr)</math>
  </td>
</tr>
</table>
</div>
in which case, from the Poisson equation, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\rho_1}{\rho_c} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\xi^2} \frac{d}{d\xi} \biggl[\xi^2 \frac{d}{d\xi}\biggl( \frac{\Phi_1}{c_s^2} \biggr)\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\xi^2} \frac{dg}{d\xi} ~e^{i\omega t} \, .</math>
  </td>
</tr>
</table>
</div>
This matches both equation (12) of [http://adsabs.harvard.edu/abs/1957ZA.....42..263E Ebert (1957)], and equation (2.17) of {{ Yabushita68 }}.  The governing wave equation therefore becomes,

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

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\omega^2 \biggl[\frac{1}{\xi^2} \frac{dg}{d\xi} \biggr]  - \frac{1}{\xi^2}\frac{d}{d\xi} \biggl[ \xi^2 \frac{d}{d\xi}\biggl( \frac{1}{\xi^2} \frac{dg}{d\xi} \biggr) \biggr]
- 2\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl[ \frac{1}{\xi^2} \frac{dg}{d\xi} \biggr]   
- \nabla_\xi\biggl( \frac{\rho_0}{\rho_c}\biggr) \cdot \frac{g}{\xi^2}
- \nabla_\xi \biggl( \frac{\Phi_0}{c_s^2}\biggr) \cdot \frac{d}{d\xi} \biggl[ \frac{1}{\xi^2} \frac{dg}{d\xi}\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
Finally, inserting the background, equilibrium structural profiles, in particular,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\rho_0}{\rho_c} = e^{-\psi}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>\frac{\Phi_0}{c_s^2} = \psi \, ,</math>
  </td>
</tr>
</table>
</div>
[<font color="red">NOTE: </font> Here, and throughout this H_Book, our adopted sign convention for <math>\psi</math> is opposite that adopted by {{ Yabushita68 }}; see is equations (2.5) and (2.6)] we have,

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

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\omega^2 \biggl[\frac{1}{\xi^2} \frac{dg}{d\xi} \biggr]  
- \frac{1}{\xi^2}\frac{d}{d\xi} \biggl[ \xi^2 \frac{d}{d\xi}\biggl( \frac{1}{\xi^2} \frac{dg}{d\xi} \biggr) \biggr]
+ e^{-\psi } \biggl[ - \frac{2}{\xi^2} \frac{dg}{d\xi}    + \frac{g}{\xi^2}\frac{d\psi}{d\xi} \biggr]
- \frac{d\psi}{d\xi} \cdot \frac{d}{d\xi} \biggl[ \frac{1}{\xi^2} \frac{dg}{d\xi}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\xi^2} \biggl\{
-\omega^2 \biggl[\frac{dg}{d\xi} \biggr]  
- \frac{d}{d\xi} \biggl[ \xi^2 \frac{d}{d\xi}\biggl( \frac{1}{\xi^2} \frac{dg}{d\xi} \biggr) \biggr]
+ e^{-\psi} \biggl[ -2\frac{dg}{d\xi}    + g \frac{d\psi}{d\xi} \biggr]
- \xi^2 \frac{d\psi}{d\xi} \cdot \frac{d}{d\xi} \biggl[ \frac{1}{\xi^2} \frac{dg}{d\xi}\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\xi^2} \biggl\{
-\omega^2 \biggl[\frac{dg}{d\xi} \biggr]  + \frac{2}{\xi} \frac{d^2g}{d\xi^2} - \frac{2}{\xi^2} \frac{dg}{d\xi} - \frac{d^3g}{d\xi^3} 
+ e^{-\psi} \biggl[ -2\frac{dg}{d\xi}    + g \frac{d\psi}{d\xi} \biggr]
- \frac{d\psi}{d\xi} \biggl[ - \frac{2}{\xi} \frac{dg}{d\xi} + \frac{d^2g}{d\xi^2}\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\omega^2 ~\frac{dg}{d\xi}  + \frac{d^3g}{d\xi^3}  
+  \frac{d^2g}{d\xi^2}\biggl[- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggl] 
+ \frac{dg}{d\xi} \biggl[ \frac{2}{\xi^2} - \frac{2}{\xi} \cdot \frac{d\psi}{d\xi} + 2e^{-\psi} \biggr]
- g \biggl[ e^{-\psi} \frac{d\psi}{d\xi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{d^3g}{d\xi^3} - B_1(\xi) \frac{d^2 g}{d\xi^2} + [B_2(\xi) + \omega^2] \frac{dg}{d\xi} + B_3(\xi) g \, ,</math>
  </td>
</tr>
</table>
</div>
where,

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

<tr>
  <td align="right">
<math>B_1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{2}{\xi} - \frac{d\psi}{d\xi} \biggl] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2\biggl[ \frac{1}{\xi^2} - \frac{1}{\xi} \cdot \frac{d\psi}{d\xi} + e^{-\psi} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B_3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \biggl[ e^{-\psi} \frac{d\psi}{d\xi} \biggr]</math>
  </td>
</tr>
</table>
</div>
Taking into account that our sign convention on <math>~\psi</math> is opposite to that adopted by  [http://adsabs.harvard.edu/abs/1957ZA.....42..263E Ebert (1957)], this last form of the governing wave equation matches his eqs. (13) and (14) when his parameter, <math>\alpha</math>, is set to unity (isothermal condition) and the variable substitution, <math>\lambda \leftrightarrow i\omega</math>, is made.

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

<tr>
  <td align="right">
<math>\frac{d}{d\xi} \biggl[ \frac{dg}{d\xi}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggr) \biggl]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d^2g}{d\xi^2}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggl)
+
\biggl(\frac{2}{\xi^2} +\frac{d^2\psi}{d\xi^2} \biggr) \frac{dg}{d\xi}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
\frac{d^2g}{d\xi^2}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggl)
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d}{d\xi} \biggl[ \frac{dg}{d\xi}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggr) \biggl]
-
\frac{dg}{d\xi}\biggl(\frac{2}{\xi^2} +\frac{d^2\psi}{d\xi^2} \biggr)  \, ,
</math>
  </td>
</tr>
</table>
</div>
we can rewrite the governing wave equation as,

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

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{d}{d\xi} \biggl\{
\omega^2 g + \frac{d^2g}{d\xi^2} 
+  \frac{dg}{d\xi}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggr) 
\biggr\}
+ \frac{dg}{d\xi} \biggl[ \frac{2}{\xi^2} - \frac{2}{\xi} \cdot \frac{d\psi}{d\xi} + 2e^{-\psi} \biggr]
- \frac{dg}{d\xi}\biggl(\frac{2}{\xi^2} +\frac{d^2\psi}{d\xi^2} \biggr)  
+ g \frac{d}{d\xi} \biggl(e^{-\psi} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{d}{d\xi} \biggl\{
\frac{d^2g}{d\xi^2} 
+  \frac{dg}{d\xi}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggr) +
\omega^2 g \biggr\}
+ \frac{dg}{d\xi} \biggl[ - \frac{2}{\xi} \cdot \frac{d\psi}{d\xi} + 2e^{-\psi} - \frac{d^2\psi}{d\xi^2} \biggr]
+ \frac{d}{d\xi} \biggl(ge^{-\psi} \biggr) - e^{-\psi} \frac{dg}{d\xi} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{d}{d\xi} \biggl\{
\frac{d^2g}{d\xi^2} 
+  \frac{dg}{d\xi}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggr) +
g(e^{-\psi} + \omega^2) \biggr\}
+ \frac{dg}{d\xi} \biggl[ - \frac{2}{\xi} \cdot \frac{d\psi}{d\xi} + e^{-\psi} - \frac{d^2\psi}{d\xi^2} \biggr]
\, .
</math>
  </td>
</tr>
</table>
</div>

Finally, we recognize that the last term in this expression drops out because, according to the isothermal Lane-Emden equation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>-\frac{d^2\psi}{d\xi^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2}{\xi} \cdot \frac{d\psi}{d\xi} - e^{-\psi}  \, .</math>
  </td>
</tr>
</table>
</div>
So, the governing wave equation becomes,

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

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{d}{d\xi} \biggl\{
\frac{d^2g}{d\xi^2} 
+  \frac{dg}{d\xi}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggr) +
g(e^{-\psi} + \omega^2) \biggr\}
\, ,
</math>
  </td>
</tr>
</table>
</div>
which can be integrated once to give, what we will refer to as the,
<div align="center" id="Yabushita68LAWE">
<font color="maroon"><b>Yabushita68 Isothermal LAWE</b></font><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math> 
\frac{d^2g}{d\xi^2} +  \frac{dg}{d\xi}\biggl(- \frac{2}{\xi} +\frac{d\psi}{d\xi} \biggr) + g e^{-\psi}  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
C_0 - g \omega^2
\, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>C_0</math> is the integration constant.  Once again, taking into account the different adopted sign on <math>\psi</math>, acknowledging the variable substitution, <math>\lambda \leftrightarrow i\omega</math>, and considering only an isothermal equation of state <math>(\gamma = 1)</math>, we recognize that this is precisely the same form of the governing wave equation that appears as equation (2.19) of {{ Yabushita68 }}.

====Taff and Van Horn (1974)====
Drawing on the expressions for the radial profiles of various physical variables in equilibrium isothermal spheres, [[#IsothermalVariables|as provided above]], our more familiar, "key" form of the wave equation can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\xi^2}{r_0^2}\biggl\{
\frac{d^2x}{d\xi^2} + \biggl[4 - \biggl(\frac{r_0 g_0 \rho_0}{P_0}\biggr) \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} 
+ \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)\biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr]  x\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\xi^2}{r_0^2}\biggl\{
\frac{d^2x}{d\xi^2} + \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} 
+ \frac{1}{4\pi G \rho_c \gamma_\mathrm{g}} \biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})
\frac{4\pi G \rho_c }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr)   \biggr]  x\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi G \rho_c}{\gamma_\mathrm{g} c_s^2} \biggl\{
\gamma_\mathrm{g}\frac{d^2x}{d\xi^2} + \gamma_\mathrm{g}\biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} 
+  \biggl[\frac{\sigma_c^2}{6} - (3\gamma_\mathrm{g} - 4)~
\frac{1 }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr)   \biggr]  x\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Aside from the leading (constant) coefficient, this expression is identical to the linearized wave equation that {{ TVH74full }} used to examine the radial pulsation modes of pressure-truncated isothermal spheres; their governing relation is displayed in the following, boxed-in expression:

<div align="center" id="TVH74">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Equation extracted from p. 427 of<br />{{ TVH74figure }} 
</td></tr>
<tr>
  <td align="left">
<!-- [[File:TaffAndVanHornEq1.png|500px|center|Taff &amp; Van Horn (1974)]] -->
<div align="center"><math>
\Gamma_1 \frac{d^2\xi}{dx^2} + \Gamma_1\frac{d\xi}{dx}\biggl[ \frac{4}{x} - \frac{d\psi}{dx}\biggr]
+ \xi\biggl[ \lambda^2 - \frac{(3\Gamma_1-4)}{x} \frac{d\psi}{dx} \biggr] = 0 \, .
</math></div>
  </td>
</tr>
</table>
</div>

This equation &#8212; in the following, slightly rewritten form &#8212; can be found among our selected set of [[Appendix/EquationTemplates#Stability:__Radial_Pulsation|''key equations'' associated with the study of radial pulsation]], and will henceforth be referred to as the,
<div align="center" id="IsothermalLAWE">
<font color="maroon"><b>Isothermal LAWE</b></font><br />
{{ Math/EQ_RadialPulsation03 }}
</div>

A mapping between our expression and the one copied directly from {{ TVH74 }} is facilitated by the variable mapping provided here in Table 2; note, in particular, that the roles of the two variables, <math>x</math> and <math>\xi</math> are swapped.

<div align="center">
<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="6">
<font size="+1"><b>Table 2:</b></font>&nbsp; Mapping between our notation and that employed by {{ TVH74 }}
  </td>
</tr>
<tr>
  <td align="right">Taff &amp; van Horn's Notation:</td>
  <td align="center" width="8%"><math>x</math></td>
  <td align="center" width="8%"><math>\xi</math></td>
  <td align="center" width="8%"><math>\psi</math></td>
  <td align="center" width="8%"><math>\Gamma_1</math></td>
  <td align="center" width="20%"><math>\lambda^2</math></td>
</tr>
<tr>
  <td align="right">Our Notation:</td>
  <td align="center"><math>\xi</math></td>
  <td align="center"><math>x</math></td>
  <td align="center"><math>\psi</math></td>
  <td align="center"><math>\gamma_\mathrm{g}</math></td>
  <td align="center"><math>\sigma_c^2/6</math></td>
</tr>
</table>
</div>