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=Radial Oscillations of Pressure-Truncated Isothermal Spheres=
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<font size="-1">[[H_BookTiledMenu#Stability_Analysis|<b>via<br />Direct<br />Numerical<br />Integration</b>]]</font>
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==Overview==
Both {{ Ebert55full }} and {{ Bonnor56full }} showed that, when presented in a pressure-volume diagram, the equilibrium sequence of pressure-truncated isothermal spheres displays an astrophysical interesting ''turning point''.  This turning point signifies that there is a critical external pressure above which no equilibrium configurations exist.  As we have summarized in a [[SSC/Structure/BonnorEbert#Following_Bonnor.27s_Presentation|related discussion]], {{ Bonnor56 }} furthermore showed that this turning point lies along the equilibrium sequence approximately halfway between configurations that have a truncation radius of <math>\xi_e = 6</math> and <math>\xi_e = 7</math>, and that it is identified by the model whose isothermal Lane-Emden function, <math>\psi(\xi)</math>, exhibits the property,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[e^{\psi} \biggl( \frac{d\psi}{d\xi}\biggr)^2\biggr]_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2 \, ,</math>
  </td>
</tr>
</table>
</div>
''at its surface''.  At the time, it was left unanswered whether this turning point is only significant in the context of ''secular'' evolution along the equilibrium sequence, or whether it is associated with the onset of a dynamical instability.  As far as we have been able to determine, {{ Yabushita68full }} was the first to point out that, in principle, the relative [dynamical] stability of this "turning point" model can be ascertained by using linear stability analysis techniques to examine the sign of <math>~\lambda_0^2</math>, which is the square of the eigenfrequency of the configuration's fundamental radial mode of oscillation.

The stability analysis presented by {{ Yabushita68 }} showed that the marginally [dynamically] unstable model does lie between <math>~\xi_e = 6</math> and <math>~\xi_e = 7</math>.  This was confirmed by the follow-up stability analysis published by {{ TVH74full }}.  As is demonstrated in subsequent subsections of this chapter, we have reproduced in detail the subset of the {{ TVH74 }} results that are directly related to this stability question.  Extending the application of these standard linear stability analysis techniques, we have [[#Some_Results|furthermore determined]] that the marginally unstable model has a truncation radius of,

<div align="center">
<math>\xi_e \approx 6.4510534 \, ,</math>
</div>
and that, for this specific model,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[e^{\psi} \biggl( \frac{d\psi}{d\xi}\biggr)^2\biggr]_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>1.9999588 \, .</math>
  </td>
</tr>
</table>
</div>
Hence, to a high degree of precision, it appears as though the equilibrium configuration that sits at the limiting-pressure turning point can also be identified as the model along the equilibrium sequence that separates dynamically stable configurations from dynamically unstable ones.

==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>

==Previously Published Eigenvalues and Eigenfunctions==
===From the Analysis of Taff &amp; van Horn (1974)===
The left-hand column of Composite Display 1, below, contains a pair of images that present properties of the eigenvectors that resulted from the {{ TVH74full }} analysis of radial oscillations in pressure-truncated isothermal spheres, assuming that the configurations remain isothermal &#8212; that is, adopting an adiabatic exponent of <math>~\Gamma_1 = 1</math> &#8212; during the oscillations.  In the image titled, "Table I", that we have extracted from their paper, the first column of numbers identifies values of nine adopted truncation radii in the range, <math>~2 \le x_0 \le 10</math>, while the second column lists the corresponding value of <math>~\lambda_0^2</math> that were determined via their numerical analysis.  For three values of the truncation radius &#8212; <math>~x_0 = 6, 7, \And 8</math> &#8212; the third column lists the values of <math>~\lambda_0^2</math> that had been previously reported by {{ Yabushita68full }}.

In the right-hand column of Composite Display 1, we have detailed some results from our own [[#Our_Numerical_Integration|numerical analysis]] of the same set of nine configurations that were studied by {{ TVH74 }}.  In the second column of our table that has the heading, "Fundamental Mode," we have listed the value of <math>~\mathfrak{F}</math> that was required in our analysis in order to generate an eigenfunction whose logarithmic derivative at each configuration's surface was precisely negative three (to five significant digits).  In order to facilitate quantitative comparison with the work of {{ TVH74 }}, the third column of our table lists for each model the corresponding value of <math>\lambda_0^2</math>; recalling that <math>\gamma = 1</math> and <math>\alpha = -1</math>, each of these values was determined via the relation,
<div align="center">
<math>~\lambda_0^2 = \frac{\gamma(\mathfrak{F}+2\alpha)}{6} = \frac{(\mathfrak{F}-2)}{6} \, .</math>
</div>
The agreement between our numerically determined fundamental-mode eigenvalues (highlighted by pink, rectangular boxes) and the ones reported by {{ TVH74 }} is excellent, across the board.  Notice that <math>\lambda_0^2</math> is negative for the models having <math>x_0 = 7, 8, 9, \And 10</math>, which indicates that these four models are dynamically unstable.


<div align="center" id="Yabushita68">
'''<font size="+1>Composite Display 1:</font>&nbsp; Select Eigenfrequencies for Pressure-Truncated Isothermal Spheres'''

<table border="1" cellpadding="5" width="75%">
<tr>
  <td align="center">
Data extracted from Tables I &amp; II (p. 429) of<br />{{ TVH74figure }}
  </td>
  <td align="center">
From [[#Our_Numerical_Integration|Our Analysis]]<p></p><math>~\alpha = -1</math>&nbsp;&nbsp;<math>~(\gamma=1)</math><p></p>
B.C.:&nbsp;&nbsp; <math>~\frac{d\ln x}{d\ln\xi} = -3.00000</math>
  </td>
</tr>

<tr>
  <td align="center" rowspan="1">
<!-- [[File:TaffAndVanHorn1974Table1.png|center|Taff &amp; Van Horn (1974)]] -->
Fundamental mode eigenvalues <math>\lambda_0^2</math> vs. <math>x_0</math> <math>(\Gamma_1=1)</math>
  </td>
  <td align="center">Fundamental Mode</td>
</tr>
<tr>
  <td align="center">
<table border="0" align="center" cellpadding="6">

<tr>
  <td align="center" rowspan="1"><math>x_0</math>
<br />&nbsp; <hr /></td>
  <td align="center"><math>\lambda_0^2</math> <br />from p. 429 of {{ TVH74hereafter }}<hr /></td>
  <td align="center"><math>\lambda_0^2</math> <br />from p. 116 of {{ Yabushita688hereafter }}<hr /></td>
</tr>

<tr>
  <td align="center"><math>2</math></td>
  <td align="center"><math>1.822~~</math></td>
  <td align="center">---</td>
</tr>

<tr>
  <td align="center"><math>3</math></td>
  <td align="center"><math>0.5860~</math></td>
  <td align="center">---</td>
</tr>

<tr>
  <td align="center"><math>4</math></td>
  <td align="center"><math>0.2161~</math></td>
  <td align="center">---</td>
</tr>

<tr>
  <td align="center"><math>5</math></td>
  <td align="center"><math>0.07607</math></td>
  <td align="center">---</td>
</tr>

<tr>
  <td align="center"><math>6</math></td>
  <td align="center"><math>0.01543</math></td>
  <td align="center"><math>+~0.01614</math></td>
</tr>

<tr>
  <td align="center"><math>7</math></td>
  <td align="center"><math>-~0.01316</math></td>
  <td align="center"><math>-~0.01274</math></td>
</tr>

<tr>
  <td align="center"><math>8</math></td>
  <td align="center"><math>-~0.02742</math></td>
  <td align="center"><math>-~0.02710</math></td>
</tr>

<tr>
  <td align="center"><math>9</math></td>
  <td align="center"><math>-~0.03477</math></td>
  <td align="center">---</td>
</tr>

<tr>
  <td align="center"><math>10</math></td>
  <td align="center"><math>-~0.03862</math></td>
  <td align="center">---</td>
</tr>
</table>
  </td>
  <td align="center">

<table border="0" align="center" cellpadding="6">
<tr>
  <td align="center"><math>~\xi_0</math></td>
  <td align="center"><math>~\mathfrak{F}</math></td>
  <td align="center"><math>~\lambda_0^2 = \frac{\gamma(\mathfrak{F}+2\alpha)}{6}</math></td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">12.92907</td>
  <td align="center" bgcolor="pink">+1.821512</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">5.51614</td>
  <td align="center" bgcolor="pink">+0.586023</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="center">3.29671</td>
  <td align="center" bgcolor="pink">+0.216118</td>
</tr>
<tr>
  <td align="center">5</td>
  <td align="center">2.456412</td>
  <td align="center" bgcolor="pink">+0.0760687</td>
</tr>
<tr>
  <td align="center">6</td>
  <td align="center">2.092651</td>
  <td align="center" bgcolor="pink">+0.0154418</td>
</tr>
<tr>
  <td align="center">7</td>
  <td align="center">1.921062</td>
  <td align="center" bgcolor="pink">-0.013156</td>
</tr>
<tr>
  <td align="center">8</td>
  <td align="center">1.8354928</td>
  <td align="center" bgcolor="pink">-0.027418</td>
</tr>
<tr>
  <td align="center">9</td>
  <td align="center">1.791388</td>
  <td align="center" bgcolor="pink">-0.034769</td>
</tr>
<tr>
  <td align="center">10</td>
  <td align="center">1.7683435</td>
  <td align="center" bgcolor="pink">-0.038609</td>
</tr>
</table>

  </td>
</tr>
<tr>
  <td align="center" rowspan="1">
Fundamental and First Harmonic Eigenfunctions and Eigenvalues <math>(\Gamma_1=1)</math>
  </td>
  <td align="center">First Harmonic</td>
</tr>
<tr>
  <td align="left" rowspan="1">
<!--[[File:TaffAndVanHorn1974Table2.png|center|Taff &amp; Van Horn (1974)]] -->
<table border="0" align="center" cellpadding="6">

<tr>
  <td align="center"><math>x_0</math><hr /></td>
  <td align="center"><math>\rho_c/<\rho></math><hr /></td>
  <td align="center"><math>\lambda_0^2</math><hr /></td>
  <td align="center"><math>\xi_0(0)</math><hr /></td>
  <td align="center"><math>x_\mathrm{pk}</math><hr /></td>
  <td align="center"><math>\xi_0(x_\mathrm{pk})</math><hr /></td>
  <td align="center"><math>\lambda_1^2</math><hr /></td>
  <td align="center"><math>\xi_1(0)</math><hr /></td>
  <td align="center"><math>x_{1,1}</math><hr /></td>
</tr>

<tr>
  <td align="center"><math>3</math></td>
  <td align="center"><math>1.934</math></td>
  <td align="center"><math>0.5860</math></td>
  <td align="center"><math>2.970</math></td>
  <td align="center">----</td>
  <td align="center">----</td>
  <td align="center" bgcolor="pink"><math>3.766</math></td>
  <td align="center"><math>-8.567</math></td>
  <td align="center"><math>2.07</math></td>
</tr>

<tr>
  <td align="center"><math>6</math></td>
  <td align="center"><math>5.018</math></td>
  <td align="center"><math>0.01543</math></td>
  <td align="center"><math>3.552</math></td>
  <td align="center">----</td>
  <td align="center">----</td>
  <td align="center" bgcolor="pink"><math>0.7178</math></td>
  <td align="center"><math>-5.561</math></td>
  <td align="center"><math>3.85</math></td>
</tr>

<tr>
  <td align="center"><math>9</math></td>
  <td align="center"><math>10.72</math></td>
  <td align="center"><math>-~0.03477</math></td>
  <td align="center"><math>5.311</math></td>
  <td align="center">----</td>
  <td align="center">----</td>
  <td align="center" bgcolor="pink"><math>0.2444</math></td>
  <td align="center"><math>-~4.685</math></td>
  <td align="center"><math>5.29</math></td>
</tr>
</table>

  </td>
  <td align="center">
<table border="0" align="center" cellpadding="6">
<tr>
  <td align="center"><math>~\xi_1</math></td>
  <td align="center"><math>~\mathfrak{F}</math></td>
  <td align="center"><math>~\lambda_1^2 = \frac{\gamma(\mathfrak{F}+2\alpha)}{6}</math></td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">56.87349</td>
  <td align="center" bgcolor="white">+9.14558</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">24.58903</td>
  <td align="center" bgcolor="pink">+3.76484</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="center">13.640525</td>
  <td align="center" bgcolor="white">+1.94009</td>
</tr>
<tr>
  <td align="center">5</td>
  <td align="center">8.798197</td>
  <td align="center" bgcolor="white">+1.13303</td>
</tr>
<tr>
  <td align="center">6</td>
  <td align="center">6.306545</td>
  <td align="center" bgcolor="pink">+0.71776</td>
</tr>
<tr>
  <td align="center">7</td>
  <td align="center">4.88991</td>
  <td align="center" bgcolor="white">+0.48165</td>
</tr>
<tr>
  <td align="center">8</td>
  <td align="center">4.024628</td>
  <td align="center" bgcolor="white">+0.33744</td>
</tr>
<tr>
  <td align="center">9</td>
  <td align="center">3.46662</td>
  <td align="center" bgcolor="pink">+0.24444</td>
</tr>
<tr>
  <td align="center">10</td>
  <td align="center">3.091117</td>
  <td align="center" bgcolor="white">+0.18185</td>
</tr>
</table>
  </td>
</tr>
</table>
</div>


For three of their models &#8212; the ones having <math>x_0 = 3, 6, \And ~9</math> &#8212; {{ TVH74 }} also determined an eigenvalue for the first harmonic mode of oscillation; see the seventh column of numbers, labeled <math>\lambda_1^2</math>, that appears in the digital image of their "Table II" that we have extracted from their paper and presented in Composite Display 1, above. Via our own [[#Our_Numerical_Integration|numerical analysis]], we have determined an eigenvalue for the first harmonic mode of oscillation in all nine of the models.  These results are presented in the right-hand panel of Composite Display 1, under the heading, "First Harmonic."  Again, the agreement between our numerically determined eigenvalues (see the three highlighted by pink, rectangular boxes) and the ones reported by {{ TVH74hereafter }} is excellent.

<span id="CompositeDisplay2">The right-hand column of Composite Display 2, below,</span> presents a pair of animations that display how our numerically derived displacement function, <math>x(\xi)</math>, varies with radius &#8212; from the center of the isothermal sphere, out to <math>\xi = 9</math> &#8212; for a variety of values of the square of the eigenfrequency, <math>\lambda_0^2</math>; each frame of both animations is tagged by the relevant value of <math>\lambda_0^2</math>.  The segment of the <math>x(\xi)</math> curve that has been drawn in blue identifies the ''eigenfunction'' that corresponds to the specified value of the eigenfrequency.  In each frame, the radial location at which the blue segment terminates simultaneously identifies the truncation radius of the relevant isothermal sphere, and the radius at which the boundary condition, <math>d\ln x/d\ln\xi = -3</math>, has been enforced.  In every frame, the <math>x(\xi)</math> function has been normalized such that the displacement amplitude is unity at the truncated configuration's surface.  


<div align="center" id="Yabushita68">
'''<font size="+1>Composite Display 2:</font>&nbsp; Select Eigenfunctions for Pressure-Truncated Isothermal Spheres'''

<table border="1" cellpadding="5" width="75%">
<tr>
  <td colspan="1" align="center">
Figure &amp; caption extracted from p. 430 of <br />{{ TVH74figure }}
</td>
  <td align="center">
From [[#Our_Numerical_Integration|Our Analysis]]<p></p><math>~\alpha = -1</math>&nbsp;&nbsp;<math>~(\gamma=1)</math><p></p>
B.C.:&nbsp;&nbsp; <math>~\frac{d\ln x}{d\ln\xi} = -3.00000</math>
  </td>
</tr>
<tr>
  <td align="left">
[[File:TaffAndVanHorn1974Fig1.png|600px|center|Taff &amp; Van Horn (1974)]]
  </td>
  <td align="left">
[[File:TaffVanHorn1974Harmonic.gif|450px|First Harmonic mode animation]]<p></p>
[[File:TaffVanHorn1974Fundamental.gif|450px|Fundamental mode animation]]
  </td>
</tr>
</table>
</div>

The blue curve segments in the bottom animation identify ''fundamental-mode'' eigenfunctions; most significantly, no radial nodes appear between the center and the surface of the truncated configuration.  The ones that terminate at <math>\xi_e = 3, 6, \And ~9</math> appear to be identical to the three fundamental-mode eigenfunctions with corresponding values of the truncation radius that appear in Figure 1a of {{ TVH74 }} &#8212; see the bottom half of the image that has been extracted from the {{ TVH74hereafter }} publication and presented here in the left-hand column of our Composite Display 2.  

The blue curve segments in the ''top'' animation identify "first harmonic" eigenfunctions; for each of the displayed eigenfunctions, one radial node exists between the center and the surface of the truncated configuration.  The ones that terminate at <math>\xi_e = 3, 6, \And ~9</math>, appear to be identical to the three first harmonic eigenfunctions with corresponding values of the truncation radius that appear in Figure 1b of {{ TVH74 }} &#8212; see the top half of the image that has been extracted from the {{ TVH74hereafter }} publication and presented here in the left-hand column of our Composite Display 2.

===From Yabushita's (1992) Analysis===

In the portion (&sect;5) of his analysis that is focused on the stability of pressure-truncated polytropic spheres, {{ Yabushita92 }} examined the eigenvalue problem governed by the following wave equation:
<!--
<div align="center" id="HRW66excerpt">
<table border="2" cellpadding="10">
<tr>
  <th align="center">
Radial Pulsation Equation Extracted<sup>&dagger;</sup> from p. 182 of [http://adsabs.harvard.edu/abs/1992Ap%26SS.193..173Y S. Yabushita (1992)]<p></p>
"''Similarity Between the Structure and Stability of Isothermal and Polytropic Gas Spheres''"<p></p>
Astrophysics and Space Science, vol. 193, pp. 173-183 &copy; [http://www.springer.com/astronomy/astrophysics+and+astroparticles/journal/10509 Springer]
  </th>
<tr>
  <td>
[[File:Yabushita1992WaveEquation2.png|650px|center|Yabushita (1992)]]
  </td>
</tr>
<tr><td align="left">
<sup>&dagger;</sup>Equations and text displayed here exactly as it appears in the original publication.
</td></tr>
</table>
</div>
-->

<div align="center" id="Yabushita68">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Radial pulsation equation extracted<sup>&dagger;</sup> from p. 182 of <br />{{ Yabushita92figure }}
</td></tr>
<tr>
  <td align="left">
<!-- 
[[File:Yabushita1992WaveEquation2.png|650px|center|Yabushita (1992)]]
-->
[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline:  In Yabushita's Eq. (5.3), the (2/x) term inside the square brackets should be (4/x).]]
<table border="0" align="center" cellpadding="5" width="100%">
<tr>
  <td align="right" width="50%">
<math>
\frac{d^2 h}{dx^2} + \biggl[ \frac{2}{x} + (n+1) \frac{1}{\theta} \frac{d\theta}{dx} \biggr]\frac{dh}{dx}
+ (3-n)\frac{1}{x} \frac{1}{\theta} \frac{d\theta}{dx}~ h
</math>
  </td>
  <td align="center" width="5%"><math>=</math></td>
  <td align="left"><math>0 \, ;</math></td>
  <td align="right" width="5%">(5.3)</td>
</tr>
</table>
while the boundary condition may be expressed in the form

<table border="0" align="center" cellpadding="5" width="100%">
<tr>
  <td align="right" width="50%">
<math>
\frac{dh}{dx} + \frac{3h}{x}
</math>
  </td>
  <td align="center" width="5%"><math>=</math></td>
  <td align="left"><math>0 \, ,</math> &nbsp; &nbsp; &nbsp; <math>x = x_0\, .</math></td>
  <td align="right" width="5%">(5.4)</td>
</tr>
</table>

  </td>
</tr>
<tr><td align="left">
<sup>&dagger;</sup>Equations and text displayed here exactly as it appears in the original publication.
</td></tr>
</table>
</div>

Let's examine the overlap between this pair of governing relations and the ones employed by {{ HRW66full }}, hereafter {{ HRW66hereafter }}.  If we replace the variable <math>~X</math> with <math>~h</math>, set <math>~\gamma = (n+1)/n</math>, and set the dimensionless eigenfrequency, <math>~s</math>, to zero in the [[SSC/Stability/Polytropes#HRW66excerpt|radial pulsation equation employed by HRW66]], 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>~
\frac{d^2 h}{dx^2} + \biggl[\frac{4}{x} + (n+1) \frac{\theta^'}{\theta} \biggr] \frac{dh}{dx} + (n+1)\biggl[ 3 - \frac{4n}{(n+1)} \biggr] \biggl[ \frac{\theta^' h}{\theta x} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2 h}{dx^2} + \biggl[\frac{4}{x} + (n+1) \frac{\theta^'}{\theta} \biggr] \frac{dh}{dx} + (3-n) \biggl[ \frac{\theta^' h}{\theta x} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
This matches equation (5.3) of {{ Yabushita92 }} &#8212; see the above boxed-in image &#8212; except the <math>(4/x)</math> term appears as <math>(2/x)</math> in Yabushita's article; giving the benefit of the doubt, <font color="red">this is most likely a typographical error</font> in {{ Yabushita92 }}.  According to {{ HRW66hereafter }}, the corresponding central boundary condition is, 
<div align="center">
<math>\frac{dh}{dx} = 0</math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>x=0 \, .</math>
</div>
While &#8212; after changing the sign on the right-hand side of {{ HRW66hereafter }}'s equation (58) as argued in our [[SSC/Perturbations#ChristyCox|accompanying discussion]] in order to align with the separate derivations presented by [http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C Christy (1965)] and [http://adsabs.harvard.edu/abs/1967IAUS...28....3C Cox (1967)]  &#8212; the corresponding boundary condition at the surface is, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dh}{dx}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{h}{x} \biggr[  3 - \frac{4}{\gamma}  + \cancelto{0}{\frac{x s^2}{\gamma q}} \biggr]</math>
  </td>
  <td align="left" colspan="2">
&nbsp;
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{n-3}{n+1} \biggl(\frac{h}{x} \biggr) \, .</math>
  </td>
  <td align="left">
&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~x = x_0 \, .</math>
  </td>
</tr>
</table>
</div>
This surface boundary condition, which has been used by the astrophysics community in the context of ''isolated'' polytropic configurations, is different from the one displayed as equation (5.4) of {{ Yabushita92 }}.  The surface boundary condition chosen by Yabushita &#8212; effectively,
<div align="center">
<math>~\frac{d \ln h}{d\ln x} = -3 \, ,</math>
</div>
&#8212; does seem to be more appropriate in the context of a study of the stability of ''pressure-truncated'' polytropes because, as argued by {{ LP41full }} and as reviewed in our [[SSC/Perturbations#Set_the_Surface_Pressure_Fluctuation_to_Zero|accompanying discussion]], it ensures that the pressure fluctuation ''at the surface'' is zero.  It is worth noting that Yabushita's surface boundary condition matches the surface boundary condition chosen by {{ TVH74 }} in their study of pressure-truncated ''isothermal'' spheres; in their words (see p. 428 of their article):  &nbsp; [Setting the surface logarithmic derivative to negative 3] <font color="green">expresses the condition that the pressure at the perturbed surface always remain[s] equal to the confining pressure exerted by the external medium in which the [pressure-truncated] sphere must be embedded</font>.

==Our Numerical Integration==
===Algorithm===

Here we show how a relatively simple, finite-difference algorithm can be developed to numerically integrate the governing LAWE from the center of the isothermal configuration, outward to its surface, which we will mark by the radial location, <math>~\xi_\mathrm{surf}</math>.  

Following a [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#Integrating_Outward_Through_the_Core|parallel discussion]], in preparation for discretization on a discrete radial grid, <math>~0 \le \xi_i \le \xi_\mathrm{surf}</math>, the [[#Taff_and_Van_Horn_.281974.29|above-defined]] LAWE for an isothermal may be written as,

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

<tr>
  <td align="right">
<math>~x_i''</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  - \biggl[4 
- \xi_i \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] \frac{x_i'}{\xi_i} 
- \biggl[ \frac{\sigma_c^2}{6\gamma}  - \frac{\alpha}{\xi} \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr]  x_i  \, .
</math>
  </td>
</tr>
</table>
</div>

Now, using the [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#General_Approach|general finite-difference approach described separately]], we make the substitutions,

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

<tr>
  <td align="right">
<math>~x_i'</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{x_+ - x_-}{2 \Delta_\xi}   \, ;
</math>
  </td>
</tr>
</table>
</div>
and,

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

<tr>
  <td align="right">
<math>~
x_i'' 
</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{x_+ - 2x_i + x_-}{\Delta_\xi^2} \, ,</math>
  </td>
</tr>
</table>
</div>
which will provide an approximate expression for <math>~x_+ \equiv x_{i+1}</math>, given the values of <math>~x_- \equiv x_{i-1}</math> and <math>~x_i</math>.  Specifically, if the center of the configuration is denoted by the grid index, <math>~i=1</math>, then for zones, <math>~i = 3 \rightarrow N</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x_+ - 2x_i + x_-}{\Delta_\xi^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  - \biggl[4 
- \xi_i \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] \frac{1}{\xi_i} \biggl[ \frac{x_+ - x_-}{2 \Delta_\xi}\biggr]
- \biggl[ \frac{\sigma_c^2}{6\gamma}  - \frac{\alpha}{\xi} \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr]  x_i  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_+\biggl\{
\frac{1 }{\Delta_\xi^2} + \biggl[4 - \xi_i \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] \frac{1}{\xi_i} \biggl[ \frac{1}{2 \Delta_\xi}\biggr] \biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  
x_- \biggl\{ \biggl[4 - \xi_i \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] \frac{1}{\xi_i} \biggl[ \frac{1}{2 \Delta_\xi}\biggr]
- \frac{1 }{\Delta_\xi^2} \biggr\}
+ x_i \biggl\{ \frac{2}{\Delta_\xi^2} 
- \biggl[ \frac{\sigma_c^2}{6\gamma}  - \frac{\alpha}{\xi} \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr]  \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_+\biggl\{
2 + \biggl[4 - \xi_i \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] \frac{\Delta_\xi}{\xi_i}  \biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  
x_- \biggl\{ \biggl[4 - \xi_i \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr] \frac{\Delta_\xi}{\xi_i} - 2 \biggr\}
+ x_i \biggl\{ 4 
- 2\Delta_\xi^2 \biggl[ \frac{\sigma_c^2}{6\gamma}  - \frac{\alpha}{\xi} \biggl(\frac{d\psi }{d\xi}\biggr)_i \biggr]  \biggr\} 
</math>
  </td>
</tr>
</table>
</div>

At the very center <math>~(\xi_1=0)</math>, we will set <math>~x_1=1</math>; then at the first radial grid line off the center <math>~(\xi_2 = \Delta_\xi)</math>, we will call upon the [[Appendix/Ramblings/PowerSeriesExpressions#IsothermalDisplacement|derived series expansion]] to set,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1-\frac{\mathfrak{F}\Delta_\xi^2}{60} \, .</math>
  </td>
</tr>
</table>
</div>

===Some Results===

When examining ''isothermal'' <math>(\gamma_\mathrm{g} = 1)</math> radial oscillations in pressure-truncated isothermal spheres, the marginally unstable equilibrium configuration should be the one along the equilibrium sequence for which the fundamental-mode eigenfrequency is precisely zero &#8212; that is, <math>\mathfrak{F} = -2\alpha = 2</math> &#8212; and for which the fundamental-mode eigenfunction displays <math>d\ln x/d\ln\xi = -3</math> at the surface of the truncated sphere.  Via the numerical modeling, just described, we have determined that this marginally unstable configuration has the following properties.

<div align="center">
<table border="1" align="center" cellpadding="5">
<tr>
  <th colspan="2" align="center">Numerically Determined Properties<br />of Marginally Unstable Model</th>
</tr>
<tr>
  <td align="center"><math>\xi_\mathrm{surf}</math></td>
  <td align="center">6.4510534</td>
</tr>
<tr>
  <td align="center"><math>\psi_\mathrm{surf}</math></td>
  <td align="center">2.642199</td>
</tr>
<tr>
  <td align="center"><math>\frac{d\psi}{d\xi}\biggr|_\mathrm{surf}</math></td>
  <td align="center">0.3773673</td>
</tr>
<tr>
  <td align="center"><math>e^{-\psi_\mathrm{surf}}</math></td>
  <td align="center">0.0712045</td>
</tr>
<tr>
  <td align="center"><math>\biggl[ e^\psi \biggl( \frac{d\psi}{d\xi} \biggr)^2\biggr]_\mathrm{surf}</math></td>
 <td align="center">1.9999588</td>
</tr>
<tr>
  <td align="center"><math>\mathfrak{F}</math></td>
 <td align="center">2</td>
</tr>
<tr>
  <td align="center"><math>\frac{d\ln x}{d\ln\xi}\biggr|_\mathrm{surf}</math></td>
 <td align="center">-3.0000000</td>
</tr>
<tr>
  <td align="center"><math>\frac{\xi^2}{x} \cdot \frac{d^2x}{d\xi^2}\biggr|_\mathrm{surf}</math></td>
 <td align="center">2.262333</td>
</tr>
</table>
</div>

==Implications==

===Playing Around===
Our own analysis of this problem decidedly validates the results published by earlier authors.  In particular, it appears as though the {{ Yabushita688hereafter }} conjecture is correct; specifically, a careful LAWE analysis seems to indicate that &#8212; if we consider only isothermal fluctuations, that is, <math>\gamma_\mathrm{g}=1</math> &#8212; the dynamical instability sets in at the point along the equilibrium sequence where the pressure maximum arises.  Let's see if we can put this conjecture on an even firmer foundation by manipulating the LAWE analytically.

When <math>\gamma_\mathrm{g} = 1</math>, the LAWE for pressure-truncated isothermal spheres is,
<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^2x}{d\xi^2} + \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} +
\frac{1 }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr)   \biggr]  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\xi^2}{x} \cdot \frac{d^2x}{d\xi^2} + \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{d\ln x}{d\ln\xi} 
+  \xi^2\biggl[\frac{\sigma_c^2}{6} +
\frac{1 }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr)   \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d}{d\ln \xi}\biggl[ \frac{d\ln x}{d\ln\xi}\biggr] + \biggl[\frac{d\ln x}{d\ln\xi}-1 \biggr] \frac{d\ln x}{d\ln\xi}
+ \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{d\ln x}{d\ln\xi} 
+  \xi^2\biggl[\frac{\sigma_c^2}{6} +
\frac{1 }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr)   \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{dy_\xi}{d\ln \xi} + \biggl[y_\xi-1 \biggr] y_\xi
+ \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] y_\xi 
+  \xi^2\biggl[\frac{\sigma_c^2}{6} +
\frac{1 }{\xi} \biggl( \frac{d\psi}{d\xi} \biggr)   \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where, we have introduced the shorthand notation,
<div align="center">
<math>y_\xi \equiv \frac{d\ln x}{d\ln\xi} \, .</math>
</div>

Now, precisely ''at'' the onset of dynamical instability, we should find that, <math>\sigma_c^2 = 0</math>.  We also know, from the work of {{ Bonnor56 }}, that the equilibrium configuration at the pressure maximum is identified by the model whose Lane-Emden function ''at its surface'' exhibits the property,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[e^{\psi} \biggl( \frac{d\psi}{d\xi}\biggr)^2\biggr]_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{d\psi}{d\xi}\biggr)_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ 2^{1 / 2} e^{-\psi/2} \biggr]_\mathrm{surf} \, .</math>
  </td>
</tr>
</table>
</div>

Therefore, ''at the surface'' the condition expected from the LAWE is,
<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>\biggl\{
\xi \cdot \frac{dy_\xi}{d\xi} + (y_\xi-1 ) y_\xi
+ \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] y_\xi 
+  \xi \biggl( \frac{d\psi}{d\xi} \biggr) 
\biggr\}_\mathrm{surf}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{
\xi \cdot \frac{dy_\xi}{d\xi} + (y_\xi + 3 ) y_\xi 
+  \xi (1-y_\xi)\biggl( \frac{d\psi}{d\xi} \biggr) 
\biggr\}_\mathrm{surf}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{
\xi \cdot \frac{dy_\xi}{d\xi} + (y_\xi + 3 ) y_\xi 
+  2^{1 / 2}\xi (1-y_\xi)e^{-\psi/2} 
\biggr\}_\mathrm{surf} \, .
</math>
  </td>
</tr>
</table>
</div>

Let's see what this implies if, at the surface of the configuration, the logarithmic derivative of the fundamental-mode eigenfunction has the value,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>[y_\xi]_\mathrm{surf} = \frac{d\ln x}{d\ln\xi} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-3 \, .</math>
  </td>
</tr>
</table>
</div>

In this case, 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>\biggl\{
\frac{dy_\xi}{d\xi}  
+  2^{5 / 2}e^{-\psi/2} 
\biggr\}_\mathrm{surf} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{
\frac{dy_\xi}{d\xi}  
+  4 \cdot \frac{d\psi}{d\xi} 
\biggr\}_\mathrm{surf} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{
\frac{d}{d\xi}  \biggl[y_\xi + 4\psi \biggr]
\biggr\}_\mathrm{surf} \, ,
</math>
  </td>
</tr>
</table>
</div>
<!--
Not sure what to make of this!
-->

which means that, at the surface,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>y_\xi + 4\psi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>b_0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{d\ln x}{d\ln\xi} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>b_0 - 4\psi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ b_0 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\psi_\mathrm{surf} - 3 \, .</math>
  </td>
</tr>
</table>
</div>
[NOTE:  We have deduced empirically that, <math>b_0 \approx 7.568796</math>.]  

<!--  INCORRECT TO ASSUME THAT THE "SURFACE CONDITION" RELATION CAN BE DIFFERENTIATED OR INTEGRATED ...
Let's go back to Bonnor's condition at the surface and see if we can integrate that relation once.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e^{\psi/2}d\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{1 / 2} d\xi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{1}{2} e^{\psi/2}d\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{-1 / 2} d\xi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ e^{\psi/2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{-1 / 2} \xi + a_0</math>
  </td>
</tr>
</table>
</div>
[NOTE:  We have deduced empirically that, <math>~a_0 \approx -0.8140441</math>.]  

Returning to the [[SSC/Structure/IsothermalSphere#Chandrasekhar|isothermal Lane-Emden equation]], we know as well that,
<div align="center">
{{ Math/EQ_SSLaneEmden02 }}
</div>
So, when combined with the desired condition,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{d\psi}{d\xi}\biggr)_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 2^{1 / 2} e^{-\psi/2} \biggr]_\mathrm{surf} \, ,</math>
  </td>
</tr>
</table>
</div>
we expect that, at the surface,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{1 / 2} \biggl[ 2\xi e^{-\psi/2} - \frac{\xi^2}{2} e^{-\psi/2} \frac{d\psi}{d\xi}\biggr]_\mathrm{surf}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 2^{3 / 2}\xi e^{-\psi/2} - \xi^2 e^{-\psi} \biggr]_\mathrm{surf}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 2 e^{-\psi} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2^{3 / 2}}{\xi} e^{-\psi/2} \biggr]_\mathrm{surf}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ e^{-\psi/2} \biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2^{1 / 2}}{\xi} \biggr]_\mathrm{surf} \, .</math>
  </td>
</tr>
</table>
</div>
Given this relation, we can derive expressions for both of the above integration constants &#8212; <math>~a_0</math> and <math>~b_0</math> &#8212; in terms of <math>~\xi_\mathrm{surf}</math>.  Specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[e^{\psi/2} - \frac{\xi}{\sqrt{2}}  \biggr]_\mathrm{surf}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\sqrt{2}\biggl(\frac{\xi}{4-\xi}\biggr) - \frac{\xi}{\sqrt{2}}  \biggr]_\mathrm{surf}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{-1 / 2}\biggl[ \frac{2\xi - \xi(4-\xi)}{4-\xi}  \biggr]_\mathrm{surf}</math>
  </td>
</tr>
</table>
</div>
END DELETION DUE TO IMPROPER DIFFERENTIATION AND INTEGRATION -->

===Yabushita (1974)===
====Review====
From our [[SSC/Structure/IsothermalSphere#Summary|summary of the equilibrium properties of isothermal spheres]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r = R_0 \xi \, ;</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{\rho}{\rho_c} = e^{-\psi} \, ;</math>
  </td>
  <td align="center">
&nbsp; &nbsp; and &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\mathfrak{M}(\xi) \equiv \frac{M_r}{M_0} = \xi^2 \frac{d\psi}{d\xi} \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R_0 \equiv \biggl[\frac{c_s^2}{4\pi G \rho_c}\biggr]^{1 / 2} \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; and &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~M_0 \equiv \biggl[ \frac{c_s^6}{4\pi G^3 \rho_c} \biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>
</div>
The relativistic version of these expressions is provided as equation (2.1) of {{ Yabushita74 }}.  Tracking back, directly to the statement of,

<div align="center">
<span id="HydrostaticBalance"><font color="#770000">'''Mass Conservation'''</font></span><br />

{{Math/EQ_SSmassConservation01}}
</div>
we straightforwardly recognize that (see also {{ Yabushita74hereafter }}'s equation 2.2),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\mathfrak{M}}{d\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{R_0}{M_0} \biggl[4\pi R_0^2 \rho_c \biggr]\xi^2 e^{-\psi} = \xi^2 e^{-\psi} \, .
</math>
  </td>
</tr>
</table>
</div>
This makes it quite clear, therefore, that both the LHS and the RHS of the
<div align="center">
<table border="0" cellpadding="8" align="center">
<tr><td align="center">
<font color="maroon"><b>Isothermal Lane-Emden Equation</b></font> <p></p>
{{ Math/EQ_SSLaneEmden02 }}
</td></tr>
</table>
</div>
are valid functional expressions for the quantity, <math>~\xi^{-2} (d\mathfrak{M}/d\xi)</math>.  Differentiating again, we also see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2\mathfrak{M}}{d\xi^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\xi e^{-\psi} -\xi^2 e^{-\psi} \frac{d\psi}{d\xi}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\xi e^{-\psi} - \mathfrak{M} e^{-\psi} \, .</math>
  </td>
</tr>
</table>
</div>

====Stability Analysis====
Let's see if we can understand the steps that were taken by {{ Yabushita74full }} in deriving an analytically defined, isothermal displacement function.  He begins by stating that the linearized equation governing radial oscillations of a ''relativistic'' isothermal sphere is (see his equation 3.1),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{d^2 f}{d\xi^2} + \biggl( - \frac{2}{\xi} + \frac{d\psi}{d\xi} + q\xi e^{\lambda-\psi} \biggr) \frac{df}{d\xi} 
+ e^{\lambda - \psi}\biggl[ 1 + \frac{2q\xi}{(1+q)} \cdot \frac{d\psi}{d\xi} \biggr] f 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\sigma_Y^2 c^4}{4\pi G \rho_c(1+q)} e^{\lambda - \nu} f
\, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>e^{-\lambda}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 - \frac{2qM(\xi)}{(1+q)\xi}  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\nu</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2q\psi(\xi)}{(1+q)} \, .</math>
  </td>
</tr>
</table>
</div>
In the ''Newtonian'' limit, we know that, <math>q \rightarrow 0</math> and <math>c^2 \rightarrow 1</math>, in which case <math>e^{-\lambda} \rightarrow 1</math>, <math>\nu \rightarrow 0</math>, so this governing LAWE becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{d^2 f}{d\xi^2} + \biggl( - \frac{2}{\xi} + \frac{d\psi}{d\xi} \biggr) \frac{df}{d\xi} + e^{-\psi} f 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\sigma_Y^2}{4\pi G \rho_c} \biggr] f
\, .
</math>
  </td>
</tr>
</table>
</div>
Gratifyingly, this is identical to, what we have labeled earlier as, the [[#Yabushita68LAWE|Yabushita68 Isothermal LAWE]], with <math>f</math> being used instead of <math>g</math> to represent the unknown, radially dependent amplitude of the perturbation; with the integration constant, <math>C_0</math>, set to zero; and with,
<div align="center">
<math>\omega^2 ~\rightarrow ~ -\frac{\sigma_Y^2}{4\pi G \rho_c} \, .</math>
</div>

Following {{ Yabushita74 }}, let's rewrite this linearized wave equation as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\mathcal{L}(f)
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\sigma_Y^2}{4\pi G \rho_c} \biggr] f
\, ,
</math>
  </td>
</tr>
</table>
</div>
where the differential operator, <math>\mathcal{L}</math>, is defined by the left-hand side of the equation, which is identical to the left-hand side of the [[#Yabushita68LAWE|Yabushita68 Isothermal LAWE]].  {{ Yabushita74 }} noticed, first, that when this operator acts on the mass function, <math>\mathfrak{M}(\xi)</math>, the result is (see his equation 3.2, with <math>q = 0</math>),
<div align="center">
<table border="0" cellpadding="5" align="center">

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\xi^2 e^{-\psi} \frac{d\psi}{d\xi} = \biggl( \frac{d\mathfrak{M}}{d\xi} \biggr)\frac{d\psi}{d\xi}  \, ,</math>
  </td>
</tr>
</table>
</div>
where, drawing from the isothermal Lane-Emden equation, we have employed the substitution,
<div align="center">
<math>\frac{d^2\psi}{d\xi^2} = e^{-\psi} - \frac{2}{\xi} \frac{d\psi}{d\xi} \, .</math>
</div>

Then he noticed, separately, that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathcal{L}(\xi^3 e^{-\psi})</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d^2 }{d\xi^2}\biggl[ \xi^3 e^{-\psi} \biggr] + \biggl( - \frac{2}{\xi} + \frac{d\psi}{d\xi} \biggr) \frac{d}{d\xi}\biggl[ \xi^3 e^{-\psi} \biggr] + e^{-\psi} \biggl[ \xi^3 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[ 3\xi^2 e^{-\psi} - \xi^3 e^{-\psi} \frac{d\psi}{d\xi} \biggr] 
+ \biggl( - \frac{2}{\xi} + \frac{d\psi}{d\xi} \biggr) \biggl[ 3\xi^2 e^{-\psi} - \xi^3 e^{-\psi} \frac{d\psi}{d\xi} \biggr] 
+ \xi^3 e^{-2\psi}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 6\xi e^{-\psi} - 3\xi^2 e^{-\psi} \frac{d\psi}{d\xi} \biggr] 
+ \biggl[ -3\xi^2 e^{-\psi} \frac{d\psi}{d\xi} + \xi^3 e^{-\psi} \biggl(\frac{d\psi}{d\xi}\biggr)^2 - \xi^3 e^{-\psi} \frac{d^2\psi}{d\xi^2} \biggr] 
+ \biggl[ - 6\xi e^{-\psi} + 2\xi^2 e^{-\psi} \frac{d\psi}{d\xi} \biggr] 
+ \biggl[ 3\xi^2 e^{-\psi} \frac{d\psi}{d\xi}  - \xi^3 e^{-\psi} \biggl( \frac{d\psi}{d\xi} \biggr)^2 \biggr] 
+ \xi^3 e^{-2\psi}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\xi^3 e^{-2\psi}-\xi^2 e^{-\psi} \frac{d\psi}{d\xi} - \xi^3 e^{-\psi} \frac{d^2\psi}{d\xi^2} \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\xi^3 e^{-2\psi}-\xi^2 e^{-\psi} \frac{d\psi}{d\xi} - \xi^3 e^{-\psi} \biggl[e^{-\psi} - \frac{2}{\xi} \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>
\xi^2 e^{-\psi} \frac{d\psi}{d\xi} \, ,
</math>
  </td>
</tr>
</table>
</div>
which is identical to <math>\mathcal{L}(\mathfrak{M})</math>.  Hence &#8212; employing the function name, <math>g</math>, in preference to <math>f</math> &#8212; a perturbation amplitude given by the function displayed in the following boxed-in image,

<div align="center">
<table border="1" cellpadding="5">
<tr><td align="center">
Discovery equation, extracted from<br />{{ Yabushita74figure }} 
</td></tr>
<tr><td>
<!-- [[File:Yabushita1974Solution.png|500px|center|Yabushita (1974, MNRAS, 167, 95)]] -->
<table border="0" align="center" width="100%" cellpadding="8">
  <tr>
<td align="center"><math>g \equiv M - \xi^3 e^{-\psi}</math></td>
<td align="center" width="5%">(3.6)</td>
  </tr>
</table>
</td></tr>
</table>
</div>

is a solution to 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>

if the integration constant, <math>C_0</math>, and the square of the oscillation frequency, <math>\omega^2</math>, are both set to zero.

===Yabushita (1975)===
====Evolving from 1968 and 1974, to 1975====
As we have [[#Yabushita_.281968.29|discussed, above]], {{ Yabushita68 }} couched his stability analysis in terms of the search for a solution to, what we have labeled as, the [[#Yabushita68LAWE|Yabushita68 Isothermal LAWE]].  In taking this approach, he introduced (a rather unconventional) [[#gDefinition|perturbation variable]],
<div align="center">
<math>g \equiv \xi^2 \frac{d}{d\xi}\biggl( \frac{\delta\Phi}{c_s^2}\biggr) \, .</math>
</div>
He pointed out that, defined in this way, the density variation, <math>\delta\rho/\rho_c</math>, can be determined from <math>g</math> via the differential relation,
<div align="center">
<math>\frac{\delta \rho}{\rho_c} = \frac{1}{\xi^2} \frac{dg}{d\xi} \, .</math>
</div>
It was in this context that {{ Yabushita74 }} discovered that the eigenvector for the fundamental mode of radial oscillation in the configuration that has the [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Bonnor-Ebert limiting mass]], can be specified analytically.  Specifically, the eigenfrequency is zero &#8212; that is, the configuration is marginally [dynamically] unstable &#8212; and the eigenfunction is, precisely,
<div align="center">
<math>g_Y = M - \xi^3 e^{-\psi} = \xi^2 \frac{d\psi}{d\xi} - \xi^3 e^{-\psi} \, .</math>
</div>

{{ Yabushita75full }}, on the other hand, adopted a more conventional approach and couched his discussions of radial oscillations and stability in terms of the search for solutions to, what we have referred to as, the (standard) [[#IsothermalLAWE|Isothermal LAWE]], involving the radial displacement function, <math>x \equiv \delta r/r_0</math>.  In order to facilitate a transition from his earlier work, we demonstrate, here, the relationship between the perturbation variable, <math>g</math>, and the displacement function, <math>x</math>.

Following the [[SSC/PerspectiveReconciliation#Approach_by_Ledoux_and_Walraven|derivation of the linearized wave equation]] presented, as by {{ LW58full }} &#8212; see, also, equation <font color="green">&#x2460;</font> in an [https://www.vistrails.org/images/ImageOfDerivations07GoodLedouxWalraven.png associated derivation diagram] &#8212; the linearized continuity equation may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\delta\rho}{\rho_c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{r^2}\frac{d}{d r} \biggl[ r^3 \biggl(\frac{\rho}{\rho_c}\biggr) \frac{\delta r}{r_0} \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{d}{d \xi} \biggl[ \xi^3 \biggl(\frac{\rho}{\rho_c}\biggr) x \biggr]\, .</math>
  </td>
</tr>
</table>
</div>
Hence, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\xi^2} \frac{dg}{d\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{\xi^2}\frac{d}{d \xi} \biggl[ \xi^3 \biggl(\frac{\rho}{\rho_c}\biggr) x \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ g</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \xi^3 \biggl(\frac{\rho}{\rho_c}\biggr) x </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{g}{\xi^3}~ e^{\psi} \, . </math>
  </td>
</tr>
</table>
</div>

So, in terms of the radial displacement function, the eigenfunction for the marginally unstable configuration that has the Bonnor-Ebert limiting mass is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_Y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{g_Y}{\xi^3}~ e^{\psi} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[\frac{1}{\xi} \frac{d\psi}{d\xi} - e^{-\psi}\biggr]e^{\psi} \, .</math>
  </td>
</tr>
</table>
</div>

====Verification of Analytic Solution====
[http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y Yabushita (1975)] states that the radial eigenfunction for the marginally unstable isothermal configuration is (see his equation 5.6),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\psi '}{\xi} - e^{-\psi}\biggr] e^{\psi} \, .</math>
  </td>
</tr>
</table>
</div>
As we have just demonstrated, this expression for the displacement function is, indeed, equivalent to the expression that was derived by Yabushita (1974) for the perturbation function, <math>~g_Y</math>.  (The overall sign has been reversed on the expression, but this leading factor of negative one is arbitrary.)   In order to see if this, indeed, satisfies the

<div align="center">
<font color="maroon"><b>Isothermal LAWE </b></font> with <math>~\sigma_c^2=0</math><br />
<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^2x}{d\xi^2} + \biggl[4 - \xi \psi ' \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} 
+  \biggl[
\frac{\psi ' }{\xi} \biggr]  x \, ,
</math>
  </td>
</tr>
</table>
</div>
we need to evaluate the first and second derivatives of the eigenfunction.  We also need to keep in mind that from the isothermal Lane-Emden equation, we know that,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \psi '' </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~e^{-\psi} - \frac{2 \psi '}{\xi} \, .</math>
  </td>
</tr>
</table>
</div>
So, the first derivative of the eigenfunction is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dx}{d\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d}{d\xi} \biggl\{ \biggl[ \frac{\psi '}{\xi} - e^{-\psi}\biggr] e^{\psi}  \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\psi '}{\xi} - e^{-\psi}\biggr] \psi ' e^{\psi}  
+ \biggl[ \frac{\psi ''}{\xi} - \frac{\psi '}{\xi^2} + \psi ' e^{-\psi}\biggr] e^{\psi}  
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{(\psi ')^2}{\xi}  - \frac{3\psi '}{\xi^2}\biggr]e^{\psi} + \frac{1}{\xi}  
\, .
</math>
  </td>
</tr>
</table>
</div>
And the second derivative is,

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

<tr>
  <td align="right">
<math>\frac{d^2x}{d\xi^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\xi}\biggl\{\biggl[\frac{(\psi ')^2}{\xi}  - \frac{3\psi '}{\xi^2}\biggr]e^{\psi} + \frac{1}{\xi}  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{(\psi ')^2}{\xi}  - \frac{3\psi '}{\xi^2}\biggr] \psi ' e^{\psi}  
+e^{\psi} \biggl\{\biggl[\frac{2\psi ' \psi ''}{\xi} - \frac{(\psi ')^2}{\xi^2} \biggr] 
-\biggl[ \frac{3\psi ''}{\xi^2} - \frac{6\psi '}{\xi^3}\biggr] \biggr\}
- \frac{1}{\xi^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~e^{\psi}\biggl\{ 
\frac{(\psi ')^3}{\xi}  - \frac{4 (\psi ')^2}{\xi^2}   
+ \frac{6\psi '}{\xi^3}  
+ \biggl[ \frac{2\psi '}{\xi}  - \frac{3}{\xi^2} \biggr]\psi ''\biggr\}
- \frac{1}{\xi^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~e^{\psi}\biggl\{ 
\frac{(\psi ')^3}{\xi}  - \frac{4 (\psi ')^2}{\xi^2}   
+ \frac{6\psi '}{\xi^3}  
- \biggl[ \frac{2\psi '}{\xi}  - \frac{3}{\xi^2} \biggr] \frac{2 \psi '}{\xi}   
\biggr\}
+ \frac{2\psi '}{\xi}  - \frac{4}{\xi^2}    
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{\psi}\biggl\{ 
\frac{(\psi ')^3}{\xi}  - \frac{8 (\psi ')^2}{\xi^2}   + \frac{12\psi '}{\xi^3}  
\biggr\}+ \frac{2\psi '}{\xi}  - \frac{4}{\xi^2}   \, . 
</math>
  </td>
</tr>
</table>
</div>

Plugging these expressions into the isothermal LAWE, we have,

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

<tr>
  <td align="right">
<math>~
\frac{d^2x}{d\xi^2} + \biggl[4 - \xi \psi ' \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} +  \biggl[\frac{\psi ' }{\xi} \biggr]  x 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{\psi}\biggl\{ 
\frac{(\psi ')^3}{\xi}  - \frac{8 (\psi ')^2}{\xi^2}   + \frac{12\psi '}{\xi^3}  
\biggr\}+ \frac{2\psi '}{\xi}  - \frac{4}{\xi^2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[\frac{4}{\xi} - \psi ' \biggr] \biggl\{ \biggl[ \frac{(\psi ')^2}{\xi}  - \frac{3\psi '}{\xi^2}\biggr]e^{\psi} + \frac{1}{\xi}\biggr\}
+  \frac{\psi ' }{\xi}  \biggl\{ \biggl[ \frac{\psi '}{\xi} - e^{-\psi}\biggr] e^{\psi}  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{\psi}\biggl\{
\frac{(\psi ')^3}{\xi}  - \frac{8 (\psi ')^2}{\xi^2}   + \frac{12\psi '}{\xi^3} + \biggl(\frac{\psi '}{\xi} \biggr)^2   
+ \biggl[\frac{4}{\xi} - \psi ' \biggr] \cdot\biggl[ \frac{(\psi ')^2}{\xi}  - \frac{3\psi '}{\xi^2}\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{\xi}\biggl[\frac{4}{\xi} - \psi ' \biggr]  
- \frac{\psi ' }{\xi}  
+ \frac{2\psi '}{\xi}  - \frac{4}{\xi^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{\psi}\biggl\{
\frac{(\psi ')^3}{\xi}  - \frac{8 (\psi ')^2}{\xi^2}   + \frac{12\psi '}{\xi^3} + \biggl(\frac{\psi '}{\xi} \biggr)^2   
+ \frac{4(\psi ')^2}{\xi^2}  - \frac{12\psi '}{\xi^3} 
- \frac{(\psi ')^3}{\xi}  + \frac{3(\psi ')^2}{\xi^2} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0 \, .
</math>
  </td>
</tr>
</table>
</div>
Amazing!!

Actually, we should have started with the simpler version of the expression for the displacement function, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{\psi ' e^\psi}{\xi} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{dx}{d\xi} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl\{ \frac{\psi '' e^\psi}{\xi} + \frac{(\psi ')^2 e^\psi}{\xi}  - \frac{\psi ' e^\psi}{\xi^2} \biggr\}</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl\{ \frac{1}{\xi}  + \frac{(\psi ')^2 e^\psi}{\xi}  - \frac{3\psi ' e^\psi}{\xi^2} \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ - \frac{d\ln x}{d\ln\xi} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{x}\biggl\{ 1  + (\psi ')^2 e^\psi  - \frac{3\psi ' e^\psi}{\xi} \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{x}\biggl[ (\psi ')^2 e^\psi  + 3x-2\biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Hence, if at the surface we impose a boundary condition, <math>~d\ln x/d\ln\xi = -3</math>, then this also means that, at the surface,

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

<tr>
  <td align="right">
<math>~3x_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ (\psi ')^2 e^\psi  + 3x-2\biggr]_\mathrm{surf} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \biggl[ (\psi ')^2 e^\psi  \biggr]_\mathrm{surf} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2 \, .</math>
  </td>
</tr>
</table>
</div>
And this is precisely the condition that Bonnor recognized was associated with the critical turning point along the equilibrium sequence.  Hence, we can now also precisely associate the marginally (dynamically) unstable configuration as being the configuration at the turning point.

<span id="CentralValue">For the record,</span> it would also be good to know what the limiting value of the Yabushita displacement function is at the center of the isothermal configuration.  Drawing on some of our already-developed [[Appendix/Ramblings/PowerSeriesExpressions#Approximate_Power-Series_Expressions|power-series expressions]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{\psi ' e^\psi}{\xi} </math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~1 - \frac{1}{\xi} 
\biggl[1 + \biggl(\frac{\xi^2}{6} - \frac{\xi^4}{120}\biggr) + \frac{1}{2} \biggl(\frac{\xi^2}{6} - \frac{\xi^4}{120}\biggr)^2  \biggr]
\frac{d}{d\xi}\biggl[ \frac{\xi^2}{6} - \frac{\xi^4}{120} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~1 -  
\biggl[1 + \frac{\xi^2}{6}  \biggr]
\biggl[ \frac{1}{3} - \frac{\xi^2}{30} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~1 -  
\biggl[ \frac{1}{3} - \frac{\xi^2}{30} \biggr]
+\frac{\xi^2}{18} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{2}{3} + \frac{2^2\xi^2}{3^2 \cdot 5} \, .
</math>
  </td>
</tr>
</table>
</div>

==Try to Generalize==
===Derivation of Polytropic Displacement Function===
<table border="0" cellpadding="8" align="right">
<tr><td align="center">iPhone snapshot of "whiteboard" Eureka! moment.</td></tr>
<tr><td align="center">
[[File:EurekaWhiteBoardMarch2017sm2.png|center|420px|Whiteboard Eureka moment! (18 March 2017)]]
</td></tr>
</table>
Now, let's turn our attention to the,
<div align="center">
<font color="maroon"><b>Polytropic LAWE</b></font><br />
{{ Math/EQ_RadialPulsation02 }}
</div>

Based on the analytic expression that Yabushita (1975) derived for the isothermal case, plus some "white board" poking around, the following general polytropic displacement function looks promising:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^' }{\xi \theta^{n}} \, .</math>
  </td>
</tr>
</table>
</div>

Keeping in mind that, according to the [[SSC/Structure/Lane1870#LaneEmden|polytropic Lane-Emden equation]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta^{''}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \theta^n - \frac{2\theta^'}{\xi} \, ,</math>
  </td>
</tr>
</table>
</div>

the first derivative is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dx}{d\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{n-3}{n-1}\biggr) \biggl\{
\frac{\theta^{''} }{\xi \theta^{n}} - \frac{n(\theta^')^2 }{\xi \theta^{n+1}} - \frac{\theta^' }{\xi^2 \theta^{n}} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{3-n}{n-1}\biggr) \biggl\{
\frac{1}{\xi \theta^{n}}\biggl[\theta^n + \frac{2\theta^'}{\xi}  \biggr] + \frac{n(\theta^')^2 }{\xi \theta^{n+1}} + \frac{\theta^' }{\xi^2 \theta^{n}} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{3-n}{n-1}\biggr) \biggl\{ \frac{1}{\xi} 
+ \frac{n(\theta^')^2 }{\xi \theta^{n+1}} + \frac{3\theta^' }{\xi^2 \theta^{n}} \biggr\} 
\, ;
</math>
  </td>
</tr>
</table>
</div>
and the second derivative is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d^2 x}{d\xi^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{3-n}{n-1}\biggr) \biggl\{ 
-\frac{1}{\xi^2} 
+ \frac{2n(\theta^')\theta^{''} }{\xi \theta^{n+1}} 
- \frac{n(\theta^')^2 }{\xi^2 \theta^{n+1}} 
- (n+1) \frac{n(\theta^')^3 }{\xi \theta^{n+2}} 
+ \frac{3\theta^{''} }{\xi^2 \theta^{n}} 
- \frac{6\theta^' }{\xi^3 \theta^{n}} 
- \frac{3n (\theta^')^2 }{\xi^2 \theta^{n+1}} 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{n-3}{n-1}\biggr) \biggl\{ 
\frac{1}{\xi^2} 
+ \bigg[ \frac{2n(\theta^')}{\xi \theta^{n+1}} + \frac{3 }{\xi^2 \theta^{n}} \biggr] \biggl[ \theta^n + \frac{2\theta^'}{\xi} \biggr]
+ (n+1) \frac{n(\theta^')^3 }{\xi \theta^{n+2}} 
+ \frac{6\theta^' }{\xi^3 \theta^{n}} 
+ \frac{4n (\theta^')^2 }{\xi^2 \theta^{n+1}} 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{n-3}{n-1}\biggr) \biggl\{ 
\frac{1}{\xi^2} 
+ \frac{2n(\theta^')}{\xi \theta} + \frac{3 }{\xi^2 } 
+ \frac{4n(\theta^')^2}{\xi^2 \theta^{n+1}} + \frac{6\theta^' }{\xi^3 \theta^{n}} 
+ (n+1) \frac{n(\theta^')^3 }{\xi \theta^{n+2}} 
+ \frac{6\theta^' }{\xi^3 \theta^{n}} 
+ \frac{4n (\theta^')^2 }{\xi^2 \theta^{n+1}} 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{n-3}{n-1}\biggr) \biggl\{ \frac{4}{\xi^2} + \frac{2n(\theta^')}{\xi \theta} 
+ \frac{12\theta^' }{\xi^3 \theta^{n}}+ \frac{8n(\theta^')^2}{\xi^2 \theta^{n+1}}  
+ (n+1) \frac{n(\theta^')^3 }{\xi \theta^{n+2}} \biggr\} 
\, .
</math>
  </td>
</tr>
</table>
</div>
<span id="ForceSigmaZero">Hence, setting</span> <math>~\sigma_c^2 = 0</math> in the polytropic LAWE gives,

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

<tr>
  <td align="right">
LAWE
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{d\xi^2} + \biggl[ 4 + (n+1) \biggl(\frac{\xi \theta^'}{\theta} \biggr) \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi}  
+ \alpha (n+1) \biggl(\frac{\xi \theta^'}{\theta} \biggr)  \frac{x}{\xi^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{n-3}{n-1}\biggr) \biggl\{ \frac{4}{\xi^2} + \frac{2n(\theta^')}{\xi \theta} 
+ \frac{12\theta^' }{\xi^3 \theta^{n}}+ \frac{8n(\theta^')^2}{\xi^2 \theta^{n+1}}  
+ (n+1) \frac{n(\theta^')^3 }{\xi \theta^{n+2}} \biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl(\frac{n-3}{n-1}\biggr) \biggl[ 4 + (n+1) \biggl(\frac{\xi \theta^'}{\theta} \biggr) \biggr] \biggl[ \frac{1}{\xi^2} 
+ \frac{n(\theta^')^2 }{\xi^2 \theta^{n+1}} + \frac{3\theta^' }{\xi^3 \theta^{n}} \biggr]
+ \alpha (n+1) \biggl(\frac{\theta^'}{\xi \theta} \biggr) \biggl[ 1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^' }{\xi \theta^{n}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\alpha (n+1) \biggl(\frac{\theta^'}{\xi \theta} \biggr) + 
\biggl(\frac{n-3}{n-1}\biggr) \biggl\{ \frac{4}{\xi^2} + \frac{2n(\theta^')}{\xi \theta} 
+ \frac{12\theta^' }{\xi^3 \theta^{n}}+ \frac{8n(\theta^')^2}{\xi^2 \theta^{n+1}}  
+ (n+1) \frac{n(\theta^')^3 }{\xi \theta^{n+2}}  
- \frac{4}{\xi^2} 
- \frac{4n(\theta^')^2 }{\xi^2 \theta^{n+1}} - \frac{12\theta^' }{\xi^3 \theta^{n}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-  (n+1) \biggl(\frac{ \theta^'}{\xi\theta} \biggr) 
-  (n+1)  \biggl[  \frac{n(\theta^')^3 }{\xi \theta^{n+2}}  \biggr]
- (n+1) \biggl[  \frac{3 (\theta^')^2 }{\xi^2 \theta^{n+1}} \biggr]
+ \alpha (n+1) \biggl[ \frac{(\theta^')^2 }{\xi^2 \theta^{n+1}} \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\alpha (n+1) \biggl(\frac{\theta^'}{\xi \theta} \biggr) + 
\biggl(\frac{n-3}{n-1}\biggr) \biggl\{ (n-1) \biggl(\frac{ \theta^'}{\xi\theta} \biggr) 
+ \frac{(\theta^')^2}{\xi^2 \theta^{n+1}}  \biggl[4n - 3(n+1)  + \alpha (n+1) \biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[ \alpha (n+1) +  (n-3) ] \biggl[ \biggl(\frac{ \theta^'}{\xi\theta} \biggr) 
+ \biggl(\frac{n-3}{n-1}\biggr) \frac{(\theta^')^2}{\xi^2 \theta^{n+1}}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[ \alpha (n+1) +  (n-3) ] \biggl[ 1
+ \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^'}{\xi \theta^{n}}  \biggr] \biggl(\frac{ \theta^'}{\xi\theta} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- [ \alpha (n+1) +  (n-3) ] \biggl(\frac{x}{\xi^2} \biggr) Q \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, if <math>~\gamma = (n+1)/n</math>, then, <math>~\alpha = (3-4/\gamma) = (3-n)/(n+1)</math>, and the polytropic LAWE is satisfied.  <font color="red">'''HOOOORAAAAY!''' (18 March 2017)</font>

===Behavior of First Logarithmic Derivative===

<span id="LogarithmicDerivative">For the record,</span> let's develop an expression for the logarithmic derivative of this displacement function.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x}{d\ln \xi} = \frac{\xi}{x} \cdot \frac{dx}{d\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{3-n}{n-1}\biggr) \biggl[ 1 
+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr] 
\biggl[ 1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^' }{\xi \theta^{n}}  \biggr]^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (n-3) \biggl[ 1 
+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr] 
\biggl[ (n-1) + (n-3) \frac{\theta^' }{\xi \theta^{n}}  \biggr]^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[ (n-1) + (n-3) \frac{\theta^' }{\xi \theta^{n}}  \biggr] \frac{d\ln x}{d\ln \xi} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (n-3) \biggl[ 1 
+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[ (n-1)  \biggr] \frac{d\ln x}{d\ln \xi} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (n-3) \biggl\{ \biggl[ 1 
+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr]  
+ \biggl[ \frac{\theta^' }{\xi \theta^{n}}  \biggr] \frac{d\ln x}{d\ln \xi} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (n-3) \biggl\{ \biggl[ 1 
+ \frac{n(\theta^')^2 }{\theta^{n+1}}  \biggr]  
+ \biggl[ \frac{\theta^' }{\xi \theta^{n}}  \biggr] \biggl[3 + \frac{d\ln x}{d\ln \xi}\biggr] \biggr\}
</math>
  </td>
</tr>
</table>
</div>

Now, as we have [[#From_Yabushita.27s_.281992.29_Analysis|discussed above]], [http://adsabs.harvard.edu/abs/1992Ap%26SS.193..173Y Yabushita (1992)] has argued that for pressure-truncated configurations, the appropriate surface boundary condition should be,
<div align="center">
<math>\frac{d\ln x}{d\ln\xi} = -3 \, .</math>
</div>
If we adopt this surface boundary condition, then this just-derived expression becomes,

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

<tr>
  <td align="right">
<math>~3 (n-1)   </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(n-3) \biggl[ 1 
+ \frac{n(\theta^')^2 }{\theta^{n+1}}  \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
(n-3) \biggl[ \frac{n(\theta^')^2 }{\theta^{n+1}}  \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2n 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\frac{(\theta^')^2 }{\theta^{n+1}} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2 }{(n-3) } \, .
</math>
  </td>
</tr>
</table>
</div>
According to [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Derivation|Horedt (1970)]], this is precisely the functional condition that pinpoints the <math>~P_e</math>-max turning point(s) along the equilibrium sequence in a <math>~P-V</math> diagram, for all pressure-truncated, polytropic spheres with <math>~n \ge 3</math>.  And, according to [[SSC/Structure/PolytropesEmbedded#Other_Limits|Kimura (1981)]], this is precisely the functional condition that pinpoints the maximum mass turning point(s) along equlibrium sequences as displayed in a mass-radius diagram.

<!-- CHRISTY OR COX BOUNDARY CONDITIONS ...
On the other hand, as we have [[SSC/Perturbations#ChristyCox|discussed separately]] &#8212; see also, [[#From_Yabushita.27s_.281992.29_Analysis|above]] &#8212; in order to align with the separate derivations presented by [http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C Christy (1965)] and [http://adsabs.harvard.edu/abs/1967IAUS...28....3C Cox (1967)]  &#8212; the corresponding boundary condition at the surface is, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dh}{dx}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{h}{x} \biggr[  3 - \frac{4}{\gamma}  + \cancelto{0}{\frac{x s^2}{\gamma q}} \biggr]</math>
  </td>
  <td align="left" colspan="2">
&nbsp;
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{n-3}{n+1} \biggl(\frac{h}{x} \biggr) \, .</math>
  </td>
  <td align="left">
&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~x = x_0 \, .</math>
  </td>
</tr>
</table>
</div>
-->
===Behavior Approaching Center===
<span id="CentralValue">It would also be good to know</span> what the limiting value of this displacement function is at the center of the polytropic configuration.  Drawing on some of our already-developed [[Appendix/Ramblings/PowerSeriesExpressions#Approximate_Power-Series_Expressions|power-series expressions]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^' }{\xi \theta^{n}} \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \biggl(\frac{n-3}{n-1}\biggr)\frac{1}{\xi} 
\biggl[1- \frac{\xi^2}{6} + \frac{n\xi^4}{120} + \cdots \biggr]^{-n}
\frac{d}{d\xi}\biggl[1- \frac{\xi^2}{6} + \frac{n\xi^4}{120} + \cdots \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~1 + \biggl(\frac{n-3}{n-1}\biggr) 
\biggl[1 + \frac{n\xi^2}{6} \biggr]
\biggl[- \frac{1}{3} + \frac{n\xi^2}{30} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~1 - \frac{1}{3}\cdot \biggl(\frac{n-3}{n-1}\biggr) 
\biggl[1 + \frac{n\xi^2}{15}  \biggr]
</math>
  </td>
</tr>
</table>
</div>

So, let's rescale the polytropic displacement function such that the central value is unity for all values of the polytropic index.  We have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_P</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~A_0 \biggl[ 1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^' }{\xi \theta^{n}} \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A_0</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ 1 - \frac{1}{3}\cdot \biggl(\frac{n-3}{n-1}\biggr) \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[  \frac{3(n-1)- (n-3)}{3(n-1)} \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3(n-1)}{2n} \, .
</math>
  </td>
</tr>
</table>
</div>
Note that this even provides the correct normalization factor for the isothermal case <math>~(n=\infty)</math>.

{{ SGFworkInProgress }}

===What if Eigenfrequency not Zero===
Here we should keep in mind that the [[SSC/Perturbations#Ensure_Finite_Amplitude_Fluctuations|''conventional'' surface boundary condition]] is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x}{d\ln\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\gamma_\mathrm{g}} \biggl( 4 - 3\gamma_\mathrm{g} + \frac{\omega^2 R^3}{GM_\mathrm{tot}} \biggr)</math>
  </td>
</tr>
</table>
</div>

Recalling that the ratio of the mean to central density is given by the [[SSCpt1/Virial/FormFactors#PTtable|structural form factor]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\bar\rho}{\rho_c} = f_\mathrm{M}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( - \frac{3\theta^'}{\xi} \biggr) \, ,</math>
  </td>
</tr>
</table>
</div>
we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\omega^2 R^3}{GM_\mathrm{tot}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\omega^2 R^3}{G}\biggl(\frac{3}{4\pi \bar\rho R^3}  \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3\omega^2 }{4\pi G}\biggl[\rho_c\biggl( - \frac{3\theta^'}{\xi} \biggr)  \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sigma_c^2 \biggl( - \frac{\xi}{6\theta^'} \biggr) \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c} \, .</math>
</div>
Hence, the conventional surface boundary can be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~-~\frac{d\ln x}{d\ln\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\alpha + \frac{\sigma_c^2}{6 \gamma_\mathrm{g}} \biggl( \frac{\xi}{\theta^'} \biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[
\biggl( \frac{\sigma_c^2}{6 \gamma_\mathrm{g}}\biggr) \frac{\xi^2}{\theta} -\alpha Q \biggr] \frac{\theta}{\xi \theta^'}
</math>
  </td>
</tr>
</table>
</div>



<div align="center">
{{ Math/EQ_RadialPulsation02 }}
</div>

Now, if we return to the [[#ForceSigmaZero|above derivation]] but, this time, do ''not'' set <math>~\sigma_c^2 = 0</math>, the polytropic LAWE gives,

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

<tr>
  <td align="right">
LAWE
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{d\xi^2} + \biggl[ 4 + (n+1) \biggl(\frac{\xi \theta^'}{\theta} \biggr) \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi}  
+ \alpha (n+1) \biggl(\frac{\xi \theta^'}{\theta} \biggr)  \frac{x}{\xi^2} + \biggl( \frac{\sigma_c^2}{6}\biggr) \frac{n x}{\theta}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{n-3}{n-1}\biggr) \biggl\{ \frac{4}{\xi^2} + \frac{2n(\theta^')}{\xi \theta} 
+ \frac{12\theta^' }{\xi^3 \theta^{n}}+ \frac{8n(\theta^')^2}{\xi^2 \theta^{n+1}}  
+ (n+1) \frac{n(\theta^')^3 }{\xi \theta^{n+2}} \biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl(\frac{n-3}{n-1}\biggr) \biggl[ 4 + (n+1) \biggl(\frac{\xi \theta^'}{\theta} \biggr) \biggr] \biggl[ \frac{1}{\xi^2} 
+ \frac{n(\theta^')^2 }{\xi^2 \theta^{n+1}} + \frac{3\theta^' }{\xi^3 \theta^{n}} \biggr]
+ \alpha (n+1) \biggl(\frac{\theta^'}{\xi \theta} \biggr) \biggl[ 1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^' }{\xi \theta^{n}} \biggr] 
+ \biggl( \frac{\sigma_c^2}{6}\biggr) \frac{n}{\theta}\biggl[ 1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^' }{\xi \theta^{n}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\alpha (n+1) \biggl(\frac{\theta^'}{\xi \theta} \biggr) + 
\biggl(\frac{n-3}{n-1}\biggr) \biggl\{ \frac{4}{\xi^2} + \frac{2n(\theta^')}{\xi \theta} 
+ \frac{12\theta^' }{\xi^3 \theta^{n}}+ \frac{8n(\theta^')^2}{\xi^2 \theta^{n+1}}  
+ (n+1) \frac{n(\theta^')^3 }{\xi \theta^{n+2}}  
- \frac{4}{\xi^2} 
- \frac{4n(\theta^')^2 }{\xi^2 \theta^{n+1}} - \frac{12\theta^' }{\xi^3 \theta^{n}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-  (n+1) \biggl(\frac{ \theta^'}{\xi\theta} \biggr) 
-  (n+1)  \biggl[  \frac{n(\theta^')^3 }{\xi \theta^{n+2}}  \biggr]
- (n+1) \biggl[  \frac{3 (\theta^')^2 }{\xi^2 \theta^{n+1}} \biggr]
+ \alpha (n+1) \biggl[ \frac{(\theta^')^2 }{\xi^2 \theta^{n+1}} \biggr] \biggr\}
+ \biggl( \frac{\sigma_c^2}{6}\biggr) \frac{n}{\theta}\biggl[ 1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^' }{\xi \theta^{n}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\alpha (n+1) \biggl(\frac{\theta^'}{\xi \theta} \biggr) + 
\biggl(\frac{n-3}{n-1}\biggr) \biggl\{ (n-1) \biggl(\frac{ \theta^'}{\xi\theta} \biggr) 
+ \frac{(\theta^')^2}{\xi^2 \theta^{n+1}}  \biggl[4n - 3(n+1)  + \alpha (n+1) \biggr]\biggr\}
+ \biggl( \frac{\sigma_c^2}{6}\biggr) \frac{n}{\theta}\biggl[ 1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^' }{\xi \theta^{n}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \alpha (n+1) +  (n-3) \biggr] \biggl[ \biggl(\frac{ \theta^'}{\xi\theta} \biggr) 
+ \biggl( \frac{n-3}{n-1}\biggr) \frac{(\theta^')^2}{\xi^2 \theta^{n+1}}  \biggr] 
+ \biggl( \frac{\sigma_c^2}{6}\biggr) \frac{n}{\theta}\biggl[ 1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^' }{\xi \theta^{n}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \alpha (n+1) +  (n-3) \biggr]  \biggl(\frac{ \theta^'}{\xi\theta} \biggr)\biggl[ 1
+ \biggl( \frac{n-3}{n-1}\biggr) \frac{\theta^'}{\xi \theta^{n}}  \biggr] 
+ \biggl( \frac{\sigma_c^2}{6}\biggr) \frac{n}{\theta}\biggl[ 1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^' }{\xi \theta^{n}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1 + \biggl( \frac{n-3}{n-1}\biggr) \frac{\theta^'}{\xi \theta^{n}}  \biggr] \biggl\{[ \alpha (n+1) +  (n-3) ]  \biggl(\frac{ \theta^'}{\xi\theta} \biggr)
+ \biggl( \frac{\sigma_c^2}{6}\biggr) \frac{n}{\theta}\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{x}{\xi^2} \biggl\{[ \alpha (n+1) +  (n-3) ]  \biggl(\frac{ \xi \theta^'}{\theta} \biggr)
+ \biggl( \frac{\sigma_c^2}{6}\biggr) \frac{n\xi^2 }{\theta}\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{x}{\xi^2} \biggl\{ \biggl( \frac{\sigma_c^2}{6}\biggr) \frac{n\xi^2 }{\theta} -  [ \alpha (n+1) +  (n-3) ]  Q
\biggr\}
</math>
  </td>
</tr>
</table>
</div>

=References=

Here we have drawn primarily from the following three sources:
* &sect;5.3.8 (p. 372) of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt's (2004)] treatise on ''Polytropes: &nbsp; Applications in Astrophysics and Related Fields''
* [http://adsabs.harvard.edu/abs/1968MNRAS.140..109Y S. Yabushita (1968, MNRAS, 140, 109)] &#8212; ''Jeans's Type Gravitational Instability of Finite Isothermal Gas Spheres''
* [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T L. G. Taff &amp; H. M. Horn (1974, MNRAS, 168, 427-432)] &#8212; ''Radial Pulsations of Finite Isothermal Gas Spheres''

See also:
* [http://adsabs.harvard.edu/abs/1975MNRAS.171…85Y S. Yabushita (1975, MNRAS, 171, 85)]  &#8212; ''Jeans's Type Gravitational Instability of Finite Isothermal Gas Spheres &#8212; II''
* [http://adsabs.harvard.edu/abs/1992Ap%26SS.193..173Y S. Yabushita (1992, Astrophys. &amp; Space Sciences, 193, 173 - 183)] &#8212; ''Similarity between the structure and stability of isothermal and polytropic gas spheres''
* [http://adsabs.harvard.edu/abs/1987A%26A...171..225C J. P. Chieze (1987, A&amp;A, 171, 225 - 232)] &#8212; ''The fragmentation of molecular clouds. I - The mass-radius-velocity dispersion relations''
* [[SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres|Radial Oscillations of Polytropic Spheres]]
* [http://adsabs.harvard.edu/abs/2001A%26A…375.1091L M. Lombardi &amp; G. Bertin (2001, AA, 375, 1091-1099)], ''Boyle's law and gravitational instability''
* [http://adsabs.harvard.edu/abs/2003RPPh…66.1651L R. B. Larson (2003, Reports on Progress in Physics, 66, 1651-1697)], ''The physics of star formation''
* From [http://adsabs.harvard.edu/abs/2013ApJ...773..131H G. P. Horedt (2013, ApJ, 773,131)]: "At present it is not clear whether the gravitational instability triggering the contraction of pre stellar cores is due to Jeans or Bonnor-Ebert instability (Padoan &amp; Nordlund [http://adsabs.harvard.edu/abs/2004ApJ...617..559P 2004], [http://adsabs.harvard.edu/abs/2011ApJ...741L..22P 2011]; Hennebelle &amp; Chabrier [http://adsabs.harvard.edu/abs/2009ApJ…702.1428H 2009]).
** [http://adsabs.harvard.edu/abs/2011ApJ...741L..22P P. Padoan &amp; A. Nordlund (2011, ApJL, 741, L22)], ''The Observable Prestellar Phase of the Initial Mass Function.'' In the introduction, they state: &nbsp; <font color="darkgreen">Once a core mass grows beyond its Bonnor-Ebert mass, the core rapidly collapses within about a free-fall time &hellip;</font>
** "Elmegreen (1999), for example, has proposed that the minimum stellar mass is of the order of this Bonnor-Ebert mass."
* [http://www.am.ub.edu/~robert/Documents/bonnor-ebert.pdf Discussion of Bonnor-Ebert configurations]
* [https://arxiv.org/abs/1108.5275 On the Stability of non-isothermal Bonnor-Ebert spheres]
* [http://astro1.physics.utoledo.edu/~megeath/ph6820/lecture6_ph6820.pdf Power-point presentation focused on Bonnor-Ebert spheres]
* [http://www3.mpifr-bonn.mpg.de/staff/schilke/StarFormation2006/Collapse.pdf Another power-point presentation]
* [https://astrobear.pas.rochester.edu/trac/wiki/u/BonnorEbert Good, complimentary wiki discussion of BE spheres]; includes dynamical collapse simulation.
* [http://adsabs.harvard.edu/abs/1993ApJ...416..303F P. N. Foster and R. A. Chevalier (1993, ApJ, 416, 303)], ''Gravitational Collapse of an Isothermal Sphere''
* [https://ui.adsabs.harvard.edu/abs/2013RMxAA..49..127R/abstract A. C. Raga, J. C. Rodr&iacute;guez-Mam&iacute;rez, A. Rodr&iacute;guez-Gonz&aacute;lez, V. Lora, and A. Esquivel (2013, Revista Mexicana de Astronom&iacute;a y Astrof&iacute;sica, Vol. 49, pp. 127 - 135)], ''Analytic and Numerical Calculations of the Radial Stability of the Isothermal Sphere''

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