Editing SSC/Stability/InstabilityOnsetOverview (section)

=====Configurations Having an Index Less Than Three=====

Up to this point, we have focused our stability analysis on pressure-truncated equilibrium sequences for which the polytropic index, <math>~n \ge 3</math>, because these sequences exhibit turning points associated with physically interesting mass/pressure limits.  As it turns out, if we assume that <math>~\sigma_c^2</math> is zero, the generalized <math>~x_P(\xi)</math> displacement function defined above also provides a solution to the polytropic LAWE when <math>~n < 3</math>.  This can be demonstrated explicitly when <math>~n=1</math> because the equilibrium structural function, <math>~\Theta_H(\xi)</math>, is expressible analytically; [[SSC/Structure/Polytropes#Primary_E-Type_Solution_2|specifically]], 
<div align="center">
<math>~\Theta_H = \frac{\sin\xi}{\xi} \, .</math>
</div>

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

<tr>
  <td align="right">
<math>~x_P\biggr|_{n=1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-3 \biggl[ \biggl( \frac{1}{\xi \theta}\biggr) \frac{d\theta}{d\xi}\biggr]_{n=1} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{\xi^2}\biggl[ 1- \xi \cot\xi  \biggr]  = 1 + \frac{\xi^2}{15} + \frac{2\xi^4}{315} + \frac{\xi^6}{1575} + \cdots \, .
</math>
  </td>
</tr>
</table>
</div>

In an [[SSC/Stability/n1PolytropeLAWE#Succinct_Demonstration|accompanying discussion]] <font color="red"><b>[<== Very Useful Link]</b></font> we show that this displacement function precisely satisfies the n = 1, polytropic LAWE when <math>~\sigma_c^2 = 0</math>.
 
Does this mean that at least one configuration along the equilibrium sequence of pressure-truncated, n = 1 polytropes &#8212; see [[SSC/Structure/PolytropesEmbedded#Additional.2C_Numerically_Constructed_Polytropic_Configurations|the right panel of Figure 3 in an accompanying discussion]] &#8212; is marginally [dynamically] unstable? The answer is, "No," because, when it is evaluated at the surface <math>~(\tilde\xi)</math> of the truncated configuration, the logarithmic derivative of the displacement function ,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x_P}{d\ln\xi} \biggr|_{n=1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1- \tilde\xi \cot\tilde\xi  )^{-1}  
( \tilde\xi \cot\tilde\xi  + \tilde\xi^2 + \tilde\xi^2 \cot^2\tilde\xi -2)\, ,
</math>
  </td>
</tr>
</table>
</div>

is positive along the entire equilibrium sequence <math>~(0 < \tilde\xi < \xi_\mathrm{surf} = \pi)</math>.  Hence, the desired surface boundary condition, <math>~d\ln x/d\ln\xi = - 3</math>, is not satisfied at any location along the sequence.  As a consequence, this displacement function cannot serve as a physically satisfactory, radial-mode eigenfunction.  

Presumably the same logic &#8212; and ultimate consequence &#8212; applies to all other equilibrium, pressure-truncated polytropic sequences that have <math>~n < 3</math>.