Editing SSC/Stability/InstabilityOnsetOverview (section)

====Related Thoughts====

=====D&eacute;j&agrave; Vu=====
It should not be a surprise, but is nevertheless comforting to see, that our general expression for the polytropic displacement function, 

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<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>
</div>
morphs into the analytically defined, polytropic displacement functions that had previously been identified, namely, for n = 3 and n = 5 configurations.  The first of these transformations is trivial to demonstrate because, when n = 3, the leading coefficient of the radially dependent portion of the expression for <math>~x_P</math> is zero.  Hence,
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<math>~x_P\biggr|_{n=3} \rightarrow \biggl[\frac{3(n-1)}{2n}\biggl]_{n=3} =  1 \, .</math>
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In the second case, drawing on the definition of <math>~\theta(\xi)</math> for n = 5 polytropes, as given [[SSC/Structure/Polytropes#Primary_E-Type_Solution_2|in an accompanying chapter]], we have,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~x_P\biggr|_{n=5}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{6}{5}\biggl[1 + \frac{1}{2}\biggl( \frac{1}{\xi \theta^{5}}\biggr) \frac{d\theta}{d\xi}\biggr]_{n=5} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{6}{5} - \frac{3}{5\xi} \biggl( 1 + \frac{\xi^2}{3} \biggr)^{5/2} \frac{\xi}{3} \biggl( 1 + \frac{\xi^2}{3} \biggr)^{-3/2} 
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{6}{5} - \frac{1}{5} \biggl( 1 + \frac{\xi^2}{3} \biggr)
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>~
1 - \frac{\xi^2}{15} \, .
</math>
  </td>
</tr>
</table>
</div>

=====Multiple Harmonics=====
In addition to the green circular marker, two small open circular markers have been placed along the dashed segment of the isothermal displacement function that is displayed in the left panel of Figure 4 &#8212; see also the top panel of Figure 2.  As has been [[#Harmonics|explained above]], in connection with the center and bottom panels of Figure 2, these additional markers have been placed at the next two locations along the undulating displacement function where <math>~d\ln x_Y/d\ln\xi = -3</math>.  They also correspond to the next two positions along the isothermal equilibrium sequence where an extremum in the mass occurs.  

In the left panel of Figure 4, the undulating displacement function that provides a solution to the LAWE for pressure-truncated, n = 6 polytropic configurations has been similarly marked at the two additional locations where <math>~d\ln x_P/d\ln\xi = -3</math>.  It is clear that these two additional markers correspond to the next two positions along the companion n = 6 equilibrium sequence (Figure 3) where an extremum in the mass occurs.  By analogy with the isothermal case, then, we can immediately identify the segments of the n = 6 displacement function that represent the eigenfunctions of the marginally unstable, first harmonic and second harmonic modes of radial oscillation.  Presumably, similar undulations occur in, and precisely this same type of structural information about multiple harmonic-mode eigenfunctions can be extracted from, our analytically specified polytropic displacement function, <math>~x_P</math>, for configurations having any value of the polytropic index within the range, <math>~5 < n < \infty</math>.

=====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]], 
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<math>~\Theta_H = \frac{\sin\xi}{\xi} \, .</math>
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Hence,
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<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>

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

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  <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>.