Editing SSC/Structure/OtherAnalyticModels (section)

====Overview====
{{ Prasad49 }} performed a semi-analytic analysis of the radial oscillations and stability of structures having a parabolic density distribution.  Let's examine his tabulated results to see if they help us understand more fully whether or not our analysis is on the right track.  For example, from his Table I, we see that <math>~\mathfrak{F} = 0</math> when <math>~\alpha = 0</math>, where, according to his equation (3),

<div align="center">
<math>~\mathfrak{F} \equiv \sigma^2 - 5\alpha \, .</math>
</div>

This means that, also, <math>~\sigma^2 = 0</math>.  Now, from our derived [[#Second_Constraint|second constraint]], we deduce that,

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

<tr>
  <td align="right">
<math>~\mathfrak{F} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5[s_{nm} -n(1  - 2\lambda)] \, .</math>
  </td>
</tr>
</table>
</div>
Hence, since <math>~\mathfrak{F} = 0</math>, we conclude that,

<div align="center">
<math>~s_{nm} = n(1  - 2\lambda) \, .</math>
</div>

Also, since by definition <math>~s_{nm} = n + m</math>, we conclude that,

<div align="center">
<math>~\frac{m}{n} = - 2\lambda \, .</math>
</div>

Next, given that <math>~\alpha = 0</math>, we conclude from our derived [[#First_Constraint|first constraint]], that
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~s_{nm} = \frac{3^2}{2^2}\biggl[ -1 \pm 1\biggr] </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow</math>&nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~s_{nm}^{+} =0 </math>&nbsp; &nbsp; and &nbsp; &nbsp; <math>~s_{nm}^{-} = -\frac{9}{2} \, .</math>
  </td>
</tr>
</table>
</div>

=====The Minus Root=====

Combining these two results for the "minus" solution, we furthermore conclude that, for this ''specific'' mode, the relationship between the two exponents and <math>~\lambda</math> are,


<div align="center">
<math>~n^- = - \frac{9}{2(1-2\lambda)}</math> &nbsp;  &nbsp;  &nbsp; and  &nbsp;  &nbsp;  &nbsp;  <math>~m^- = (s_{nm} - n^-) = \frac{9\lambda}{(1-2\lambda)} \, .</math>
</div>

=====The Plus Root=====
Next, let's examine the "plus" solution.  Because <math>~s_{nm}^{+} =0 </math>, this solution implies that,


<div align="center">
<math>~m^+ = -n^+</math> &nbsp; &nbsp; &nbsp;<math>~\Rightarrow</math>&nbsp; &nbsp; &nbsp;<math>~\frac{m}{n} = -1</math>.
</div>

In this case, then, we deduce that,


<div align="center">
<math>~\lambda = -\frac{1}{2}\biggl(\frac{m}{n}\biggr) = +\frac{1}{2}</math>.
</div>

So, even though these first two constraints have not revealed the ''value'' of either of the exponents, <math>~n</math> and <math>~m</math>, we see that the resulting trial eigenfunction must be,
<div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center">
<math>~\mathcal{G}_\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a_0^n(1 + \lambda x^2)^n \cdot (2 - x^2)^m </math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a_0^n\biggl[\frac{(1 + \tfrac{1}{2} x^2)}{(2 - x^2)}\biggr]^n </math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{a_0}{2}\biggr)^n\biggl[\frac{(2 + x^2)}{(2 - x^2)}\biggr]^n \, .</math>
  </td>
</tr>
</table>

Interesting!