Editing SSC/Stability/InstabilityOnsetOverview (section)

====Analyses of Radial Oscillations====
In one [[SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres|accompanying discussion]], we have reviewed and replicated many aspects of the computational analysis presented by [http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild (1941)] of radial modes of oscillation in isolated, n = 3 polytropic spheres.  In [[SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_Spheres|another chapter]] we have presented results from our own analysis of radial oscillations in ''pressure-truncated'', n = 5 polytropic spheres.  

From Schwarzschild's work, we have come to appreciate that, if the adiabatic exponent is assigned the value, <math>~\gamma_g = (n+1)/n = 4/3</math> &#8212; in which case the parameter, <math>~\alpha = 0</math> &#8212; the following analytically defined eigenvector provides an 
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3"><font color="maroon"><b>Exact Solution to the (n = 3) Polytropic LAWE</b></font></td>
</tr>
<tr>
  <td align="right">
<math>~\sigma_c^2 = 0</math>
  </td>
  <td align="center">
&nbsp; and &nbsp;
  </td>
  <td align="left">
<math>~x = 1 </math>
  </td>
</tr>
</table>
</div>
that satisfies both the desired central boundary condition and the conventional surface boundary condition.  This is easy to verify.  Because the displacement function is unity and, therefore, independent of the radial coordinate, <math>~\xi</math>, its first and second derivatives &#8212; and, hence, the first two terms on the RHS of the polytropic LAWE &#8212; are zero.  The third term on the RHS of the polytropic LAWE is also zero because both <math>~\sigma_c^2</math> and <math>~\alpha</math> are zero.  Hence the sum of these three terms is zero, so the LAWE is satisfied.  And, because <math>~n =3</math> and <math>~\sigma_c^2=0</math>, we expect from the conventional surface boundary condition that <math>~d\ln x/d\ln \xi = 0</math> at <math>~\xi = \xi_\mathrm{surf}</math>, which of course it is because the the first derivative of the displacement function is zero at all radial locations, including the surface.

<span id="n6Analytic">In the course of our study</span> of the properties of radial oscillations in pressure-truncated, n = 5 polytropes, [[SSC/Stability/n5PolytropeLAWE#Search_for_Analytic_Solutions_to_the_LAWE|we discovered]] that, if the adiabatic exponent is assigned the value, <math>~\gamma_g = (n+1)/n = 6/5</math> &#8212; in which case the parameter, <math>~\alpha = - \tfrac{1}{3}</math> &#8212; the following analytically defined eigenvector provides an 
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3"><font color="maroon"><b>Exact Solution to the (n = 5) Polytropic LAWE</b></font></td>
</tr>
<tr>
  <td align="right">
<math>~\sigma_c^2 = 0</math>
  </td>
  <td align="center">
&nbsp; and &nbsp;
  </td>
  <td align="left">
<math>~x =  1 - \frac{\xi^2}{15} </math>
  </td>
</tr>
</table>
</div>
<span id="NeutralMode">which satisfies</span> both of the desired boundary conditions if the configuration is truncated at the radial coordinate, <math>~\tilde\xi = 3</math>.  From our [[#SpecificN5Reference|above reference]] to the n = 5 equilibrium sequence, we know that <math>~\tilde\xi = 3</math> identifies the configuration that sits at the maximum-mass turning point.  In this case, then, we recognize that our analytically defined eigenvector describes the properties of the (marginally unstable) fundamental mode of radial oscillation specifically for the configuration at the maximum-mass turning point &#8212; which, recall, is also the configuration at the <math>~P_e</math>-max turning point of the P-V sequence.  Our ability to identify this analytic solution to the polytropic LAWE was significantly aided by the fact that the structural properties of the underlying equilibrium configuration are analytically specifiable.

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 19 March 2017:  As far as we have been able to determine, it has not previously been recognized that this eigenvector provides a precise solution to the Polytropic LAWE.]]Building on our recognition of these two analytically specifiable eigenvectors for polytropes, and Yabushita's extraordinary discovery of the set of analytically specifiable eigenvectors that are associated with pressure-truncated isothermal spheres, [[SSC/Stability/Isothermal#Try_to_Generalize|we have made the following additional, and more broadly applicable discovery]]:  &nbsp;For any value of the polytropic index in the range, <math>~3 \le n < \infty</math>, if the adiabatic exponent is assigned the value, <math>~\gamma_g = (n+1)/n</math> &#8212; in which case the parameter, <math>~\alpha = (3-n)/(n+1)</math> &#8212; the following eigenvector specification provides an,

<div align="center" id="ExactPolytropicSolution">
<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>

Furthermore, for all values of <math>~n</math> in this specified range, this displacement function will satisfy both of the desired boundary conditions if the associated polytropic configuration is truncated at the radial coordinate, <math>~\tilde\xi</math>, that satisfies the condition,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl[ \frac{1}{\theta^{n+1}} \biggl(\frac{d\theta}{d\xi}\biggr)^2\biggr]_{\tilde\xi}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{n-3} \, .</math>
  </td>
</tr>
</table>
</div>

As has been pointed out, [[#MaximumMass|above, in the context of our discussion of equilibrium sequences]], in all cases <math>~(3 \le n < \infty)</math>, this is a condition that also is associated with the "maximum mass" (and <math>~P_e</math>-max) turning point along the corresponding equilibrium sequence &#8212; see the green markers in Figure 3.  We conclude, therefore, that for all polytropic indexes greater than or equal to three (including isothermal structures), the configuration that is marginally [dynamically] unstable <math>~(\sigma_c^2 = 0)</math> is precisely associated with the maximum-mass and <math>~P_e</math>-max turning point along the relevant equilibrium sequence!