Editing SSC/Stability/NeutralMode (section)

=Radial Pulsation Neutral Mode=

==Background==
The integro-differential version of the statement of hydrostatic balance is

<div align="center">
{{Math/EQ_SShydrostaticBalance01}}
</div>

[[SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations|From our separate discussion]], we have found that,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3"><font color="maroon"><b>Exact Solution to the <math>~(3 \le n < \infty)</math> Polytropic LAWE</b></font></td>
</tr>
<tr>
  <td align="right">
<math>~\sigma_c^2 = 0</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~x_P \equiv \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \, .</math>
  </td>
</tr>
</table>
</div>

Let's rewrite the significant functional term in this expressions in terms of basic variables.  That is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{1}{\xi \theta^n}\biggr)\frac{d\theta}{d\xi}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\biggl(\frac{a_n \rho_c}{r_0 \rho_0}\biggr)\frac{g_0}{4\pi G \rho_c a_n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{M(r_0)}{4\pi r_0^3 \rho_0 }
\, .
</math>
  </td>
</tr>
</table>

==Trial Eigenfunction &amp; Its Derivatives==
Let's adopt the following ''trial'' solution:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>x_t</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
a -\frac{bM_r}{4\pi r_0^3 \rho_0 } = a - \frac{bg_0}{4\pi G r_0 \rho_0 }
\, .
</math>
  </td>
</tr>
</table>
Then we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>-\biggl(\frac{1}{b}\biggr)\frac{dx_t}{dr_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d}{dr_0} \biggl[\frac{M_r}{4\pi r_0^3 \rho_0 }\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{1}{4\pi r_0^3 \rho_0 }\biggr]\frac{dM_r}{dr_0} 
-
\biggl[\frac{M_r}{4\pi r_0^3 \rho_0^2 }\biggr]
\frac{d\rho_0 }{dr_0} 
- 
\biggl[\frac{3M_r}{4\pi r_0^4 \rho_0 }\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{r_0 } 
- 
\biggl[\frac{3M_r}{4\pi r_0^4 \rho_0 }\biggr]
-
\biggl[\frac{M_r}{4\pi r_0^3 \rho_0^2 }\biggr]
\frac{d\rho_0 }{dr_0} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>-\biggl(\frac{1}{b}\biggr)\frac{d^2 x_t}{dr_0^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d}{dr_0}\biggl\{
\frac{1}{r_0 } 
- 
\biggl[\frac{3M_r}{4\pi r_0^4 \rho_0 }\biggr]
-
\biggl[\frac{M_r}{4\pi r_0^3 \rho_0^2 }\biggr]
\frac{d\rho_0 }{dr_0} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{1}{r_0^2 } 
-
\biggl[\frac{3}{4\pi r_0^4 \rho_0 }\biggr]\frac{dM_r}{dr_0}
+
\biggl[\frac{3M_r}{4\pi r_0^4 \rho_0^2 }\biggr]\frac{d\rho_0}{dr_0}
+4
\biggl[\frac{3M_r}{4\pi r_0^5 \rho_0 }\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
-
\biggl[\frac{M_r}{4\pi r_0^3 \rho_0^2 }\biggr]\frac{d^2\rho_0 }{dr_0^2} 
-
\biggl[\frac{1}{4\pi r_0^3 \rho_0^2 }\biggr]\frac{d\rho_0 }{dr_0} \cdot \frac{dM_r}{dr_0}
+
\biggl[\frac{3M_r}{4\pi r_0^4 \rho_0^2 }\biggr]\frac{d\rho_0 }{dr_0}
+
\biggl[\frac{2M_r}{4\pi r_0^3 \rho_0^3 }\biggr]\biggl(\frac{d\rho_0 }{dr_0}\biggr)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{4}{r_0^2 } 
+
\frac{3M_r}{4\pi r_0^5 \rho_0 }\biggl[4
+
\frac{d\ln \rho_0}{d \ln r_0}\biggr]
+
\frac{1}{r_0^2 }\biggl[ \frac{3M_r}{4\pi r_0^3 \rho_0 }
-
1 \biggr] \frac{d \ln \rho_0 }{d\ln r_0} 
-
\biggl[\frac{M_r}{4\pi r_0^3 \rho_0^2 }\biggr]\frac{d^2\rho_0 }{dr_0^2} 
+
\biggl[\frac{2M_r}{4\pi r_0^5 \rho_0 }\biggr]\biggl(\frac{d\ln \rho_0 }{d \ln r_0}\biggr)^2 \, .
</math>
  </td>
</tr>
</table>
Given that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Delta \equiv \frac{M_r}{4\pi r_0^3\rho_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{4\pi G} \biggl( \frac{g_0}{r_0\rho_0}\biggr) 
=
- \biggl[\frac{P_0}{4\pi G r_0^2 \rho_0^2} \cdot \frac{d\ln P_0}{d \ln r_0} \biggr]

\, ,
</math>
  </td>
</tr>
</table>
these expression can be rewritten as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>-\biggl(\frac{r_0^2}{b}\biggr)\frac{dx_t}{dr_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>r_0 \biggl\{
1 - 3\Delta - \Delta \cdot \frac{d\ln \rho_0 }{d\ln r_0} 
\biggr\} \, ,
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>-\biggl(\frac{r_0^2}{b}\biggr)\frac{d^2 x_t}{dr_0^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 4 + 3\Delta \biggl[4 + \frac{d\ln \rho_0}{d \ln r_0}\biggr]
+ \biggl[ 3\Delta - 1 \biggr] \frac{d \ln \rho_0 }{d\ln r_0} 
- \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} 
+ 2\Delta \biggl(\frac{d\ln \rho_0 }{d \ln r_0}\biggr)^2 
\, .
</math>
  </td>
</tr>
</table>

==Plug Trial Eigenfunction Into LAWE==

&nbsp;<br />
<table border="1" width="60%" align="center" cellpadding="8"><tr><td align="center">
<div align="center">'''LAWE'''</div>
{{Math/EQ_RadialPulsation01}}
</td></tr></table>


Plugging our ''trial'' radial displacement function, <math>x_t</math>, into the LAWE gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
LAWE
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\biggl(\frac{r_0^2}{b}\biggr)\frac{d^2 x_t}{dr_0^2}
-\biggl(\frac{r_0}{b}\biggr)\biggl[ 4 + \frac{d\ln P_0}{d\ln r_0}\biggr]\frac{dx_t}{dr_0}
- \biggl(\frac{r_0^2}{b}\biggr) \biggl( \frac{\rho_0}{\gamma_g P_0}\biggr)
\biggl[ (4-3\gamma_g)\frac{g_0}{r_0} + \sigma_c^2\biggl(\frac{2\pi G\rho_c}{3}\biggr)\biggr]x_t
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\biggl(\frac{r_0^2}{b}\biggr)\frac{d^2 x_t}{dr_0^2}
-\biggl(\frac{r_0}{b}\biggr)\biggl[ 4 + \frac{d\ln P_0}{d\ln r_0}\biggr]\frac{dx_t}{dr_0}
+ \biggl(\frac{1}{b}\biggr) \frac{1}{\gamma_g} \biggl( \frac{d\ln P_0}{d \ln r_0}\biggr)
(4-3\gamma_g)x_t
- \biggl(\frac{r_0^2}{b}\biggr) \biggl( \frac{\rho_0}{\gamma_g P_0}\biggr)
\biggl[ \sigma_c^2\biggl(\frac{2\pi G\rho_c}{3}\biggr)\biggr]x_t
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 4 + 3\Delta \biggl[4 + \frac{d\ln \rho_0}{d \ln r_0}\biggr]
+ \biggl[ 3\Delta - 1 \biggr] \frac{d \ln \rho_0 }{d\ln r_0} 
- \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} 
+ 2\Delta \biggl(\frac{d\ln \rho_0 }{d \ln r_0}\biggr)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+\biggl[ 4 + \frac{d\ln P_0}{d\ln r_0}\biggr]\biggl\{
1 - 3\Delta - \Delta \cdot \frac{d\ln \rho_0 }{d\ln r_0} 
\biggr\}
+ \biggl(\frac{1}{b}\biggr) \frac{1}{\gamma_g} \biggl( \frac{d\ln P_0}{d \ln r_0}\biggr)
(4-3\gamma_g)(a - b\Delta)
- \biggl(\frac{1}{b}\biggr) \biggl( \frac{\rho_0r_0^2}{\gamma_g P_0}\biggr)
\biggl[ \sigma_c^2\biggl(\frac{2\pi G\rho_c}{3}\biggr)\biggr](a - b\Delta)
\, .
</math>
  </td>
</tr>
</table>
Now, if we set <math>\sigma_c^2 = 0</math> and <math> d\ln P_0/d\ln r_0 = \gamma_g(d\ln \rho_0/d\ln r_0)</math>, this expression becomes,

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

<tr>
  <td align="right">
LAWE
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 4 + 3\Delta \biggl[4 + \frac{d\ln \rho_0}{d \ln r_0}\biggr]
+ \biggl[ 3\Delta - 1 \biggr] \frac{d \ln \rho_0 }{d\ln r_0} 
- \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} 
+ 2\Delta \biggl(\frac{d\ln \rho_0 }{d \ln r_0}\biggr)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+\biggl[ 4 + \frac{d\ln P_0}{d\ln r_0}\biggr]\biggl\{
1 - 3\Delta - \Delta \cdot \frac{d\ln \rho_0 }{d\ln r_0} 
\biggr\}
+ \biggl(\frac{1}{b}\biggr) \frac{1}{\gamma_g} \biggl( \frac{d\ln P_0}{d \ln r_0}\biggr)
(4-3\gamma_g)(a - b\Delta)
- \biggl(\frac{1}{b}\biggr) \biggl( \frac{\rho_0r_0^2}{\gamma_g P_0}\biggr)
\biggl[ \sigma_c^2\biggl(\frac{2\pi G\rho_c}{3}\biggr)\biggr](a - b\Delta)
\, .
</math>
  </td>
</tr>
</table>
Notice that the key components of this last term may be rewritten as,

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

<tr>
  <td align="right">
<math>
\biggl( \frac{\rho_0r_0^2}{\gamma_g P_0}\biggr)
\biggl[ \sigma_c^2\biggl(\frac{2\pi G\rho_c}{3}\biggr)\biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi G \rho_0^2r_0^2}{P_0}\biggr)
\biggl[ \frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr)\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-
\biggl( \frac{1}{\Delta}\biggr) \frac{d\ln P_0}{d\ln r_0}\biggl[ \frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr)\biggr] 
\, .
</math>
  </td>
</tr>
</table>
So, for our ''trial'' eigenfunction, we have &hellip;

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

<tr>
  <td align="right">
LAWE
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 2\Delta - 1 \biggr] \frac{d \ln \rho_0 }{d\ln r_0} 
- \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} 
+ 2\Delta \biggl(\frac{d\ln \rho_0 }{d \ln r_0}\biggr)^2
+ \frac{d\ln P_0}{d\ln r_0} \cdot \biggl\{\biggl[
1 - 3\Delta - \Delta \cdot \frac{d\ln \rho_0 }{d\ln r_0} 
\biggr]
+ \frac{(4-3\gamma_g)}{\gamma_g} 
\biggl[ \frac{a}{b} - \Delta \biggr]
+ 
\frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr) 
\biggl[ \frac{a}{b\Delta} - 1\biggr]
\biggr\}
\, .
</math>
  </td>
</tr>
</table>

==Consider Polytropic Structures==

Referring back to, for example, [[SSC/Stability/Polytropes#Groundwork|a separate review of polytropic structures]], we recognize that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Delta = \frac{1}{4\pi G} \biggl( \frac{g_0}{r_0\rho_0}\biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\xi^3} \biggl[ \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr) \biggr]\theta^{-n}
=
\frac{1}{\xi} \biggl(- \frac{d\theta}{d\xi}\biggr) \theta^{-n}
=
-\frac{\theta^'}{\xi \theta^n} 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{d\ln \rho_0}{d\ln r_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
n\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{d\ln P_0}{d\ln r_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(n+1) \, .
</math>
  </td>
</tr>
</table>
Also,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl(\frac{r_0^2}{\rho_0}\biggr) \frac{d^2\rho_0}{dr_0^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\xi^2}{\rho_c \theta^n}\biggr) \frac{d}{d\xi}\biggl[n\rho_c \theta^{n-1} \theta^'\biggr]
</math>
  </td>
</tr>

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
n(n-1)(\xi^{n+1} \theta^{n-1} \Delta)^2 + n\xi^2 \theta^{n-1}\, .
</math>
  </td>
</tr>
</table>

Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
LAWE
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
n( 2\Delta - 1 )  
- \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} 
+ 2n^2 \Delta 
+ (n+1) \biggl[
1 - 3\Delta - n\Delta  
\biggr]
+ (n+1) \biggl\{\frac{(4-3\gamma_g)}{\gamma_g} 
\biggl[ \frac{a}{b} - \Delta \biggr]
+ 
\frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr) 
\biggl[ \frac{a}{b\Delta} - 1\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2n\Delta - n   
+ 2n^2 \Delta 
+ n - 3n\Delta - n^2\Delta 
+ 1 - 3\Delta - n\Delta 
- \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} 
+ (n+1) \biggl\{\frac{(4-3\gamma_g)}{\gamma_g} 
\biggl[ \frac{a}{b} - \Delta \biggr]
+ 
\frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr) 
\biggl[ \frac{a}{b\Delta} - 1\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 + n^2 \Delta 
- (2n+3)\Delta   
- \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} 
+ (n+1) \biggl[ \frac{4}{\gamma_g} -3\biggr]
\biggl[ \frac{a}{b} - \Delta \biggr]
+ 
(n+1) \biggl\{\frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr) 
\biggl[ \frac{a}{b\Delta} - 1\biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>
If, <math>\gamma_g = (n+1)/n</math>, we can further simplify and obtain,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
LAWE
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 + n^2 \Delta 
- (2n+3)\Delta   
- \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} 
+ \biggl[ n-3\biggr]
\biggl[ \frac{a}{b} - \Delta \biggr]
+ 
(n+1) \biggl\{\frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr) 
\biggl[ \frac{a}{b\Delta} - 1\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 + (n-3)\frac{a}{b} + n^2 \Delta 
+ (3-n)\Delta - (2n+3)\Delta   
- \Delta \biggl( \frac{r_0^2}{\rho_0}\biggr)\frac{d^2\rho_0 }{dr_0^2} 
+ 
(n+1) \biggl\{\frac{\sigma_c^2}{6\gamma_g} \biggl(\frac{\rho_c}{\rho_0}\biggr) 
\biggl[ \frac{a}{b\Delta} - 1\biggr]
\biggr\}
</math>
  </td>
</tr>
</table>