Editing SSC/Stability/Isothermal (section)

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

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  <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,
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<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,
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<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,
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<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>),
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<math>\mathcal{L}(\mathfrak{M}) = \mathcal{L}\biggl(\xi^2 \frac{d\psi}{d\xi } \biggr)</math>
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  <td align="center">
<math>=</math>
  </td>
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<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>

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  <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>

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  <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>

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&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
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<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>

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&nbsp;
  </td>
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<math>=</math>
  </td>
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<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,
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<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,
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<math>\mathcal{L}(\xi^3 e^{-\psi})</math>
  </td>
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<math>=</math>
  </td>
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<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>

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&nbsp;
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<math>=</math>
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<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>
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</tr>

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&nbsp;
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<math>=</math>
  </td>
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<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>
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</tr>

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&nbsp;
  </td>
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<math>=</math>
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<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>
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</tr>

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&nbsp;
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<math>=</math>
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<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>

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&nbsp;
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<math>=</math>
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<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,

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<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)]] -->
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<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">

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  <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.