Editing SSC/Stability/Isothermal (section)

====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,
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<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\}
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&nbsp;
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<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\}
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<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\} \, .
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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:

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Equation extracted from p. 427 of<br />{{ TVH74figure }} 
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<!-- [[File:TaffAndVanHornEq1.png|500px|center|Taff &amp; Van Horn (1974)]] -->
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\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 \, .
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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,
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<font color="maroon"><b>Isothermal LAWE</b></font><br />
{{ Math/EQ_RadialPulsation03 }}
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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.

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<font size="+1"><b>Table 2:</b></font>&nbsp; Mapping between our notation and that employed by {{ TVH74 }}
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  <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>
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  <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>
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