Editing SSC/Stability/n5PolytropeLAWE (section)

=Radial Oscillations of n = 5 Polytropic Spheres=

==Background==

===General Form of the LAWE for Spherical Polytropes===
In an [[SSC/Perturbations#2ndOrderODE|accompanying discussion]], we derived the so-called,

<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
</div>

whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. Because this widely used form of the radial pulsation equation is not dimensionless but, rather, has units of inverse length-squared, we have found it useful to also recast it in the following [[SSC/Perturbations#Dimensionless_Expression|dimensionless form]]:
<div align="center">
<math>
\frac{d^2x}{d\chi_0^2} + \biggl[\frac{4}{\chi_0} - \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0} + \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl(\frac{1}{\gamma_\mathrm{g}} \biggr)\biggl[\tau_\mathrm{SSC}^2 \omega^2 + (4 - 3\gamma_\mathrm{g})\biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{1}{\chi_0} \biggr]  x = 0 ,
</math><br />
</div>
where,
<div align="center">
<math>~g_\mathrm{SSC} \equiv \frac{P_c}{R\rho_c} \, ,</math> &nbsp; &nbsp; &nbsp; and  &nbsp; &nbsp; &nbsp; <math>~\tau_\mathrm{SSC} \equiv \biggl[\frac{R^2 \rho_c}{P_c}\biggr]^{1/2} \, .</math>
</div>

In a [[SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres|separate discussion]], we showed that specifically for isolated, ''polytropic'' configurations, this linear adiabatic wave equation (LAWE) can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} + 
\biggl[\frac{\omega^2}{\gamma_g \theta} \biggl(\frac{n+1 }{4\pi G \rho_c} \biggr) - 
\biggl(3-\frac{4}{\gamma_g}\biggr)  \cdot \frac{(n+1)V(x)}{\xi^2} \biggr]  x </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4}{\xi} - \frac{(n+1)}{\theta} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] \frac{dx}{d\xi} + 
\frac{(n+1)}{\theta}\biggl[\frac{\sigma_c^2}{6\gamma_g } - 
\frac{\alpha}{\xi} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr]  x \, ,</math>
  </td>
</tr>
</table>
</div>
where we have adopted the dimensionless frequency notation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma_c^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{3\omega^2}{2\pi G \rho_c}  \, .</math>
  </td>
</tr>
</table>
</div>

===Specifically for n=5 Configurations===

Here we focus on an analysis of the ''specific'' case of isolated, <math>~n=5</math> polytropic configurations, whose unperturbed equilibrium structure can be prescribed in terms of analytic functions.  Our hope &#8212; as yet unfulfilled &#8212; is that we can discover an analytically prescribed eigenvector solution to the governing LAWE.  

From our discussion of the [[SSC/Structure/Polytropes#Primary_E-Type_Solution_2|equilibrium structure of isolated, <math>~n=5</math> polytropes]], we know that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( 1 + \frac{\xi^2}{3} \biggr)^{-1/2} =3^{1/2} ( 3 + \xi^2 )^{-1/2}\, .</math>
  </td>
</tr>
</table>
</div>
Hence, we know as well that,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

The LAWE therefore becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4}{\xi} - \frac{(n+1)}{\theta} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] \frac{dx}{d\xi} + 
\frac{(n+1)}{\theta}\biggl[\frac{\sigma_c^2}{6\gamma_g } - 
\frac{\alpha}{\xi} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr]  x </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4}{\xi} - \frac{6}{\theta} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] \frac{dx}{d\xi} + 
\biggl[\frac{\sigma_c^2}{\gamma_g } \cdot \frac{1}{\theta} - 
\frac{6\alpha}{\xi} \cdot \frac{1}{\theta} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr]  x</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[4 - \frac{6\xi^2}{(3+\xi^2)} \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} + 
\biggl[\frac{\sigma_c^2}{3^{1/2} \gamma_g } \cdot (3+\xi^2)^{1/2} - 
\frac{6\alpha}{(3+\xi^2)}\biggr)\biggr]  x \, .</math>
  </td>
</tr>
</table>
</div>

Or, 
<div align="center" id="n5LAWE">
<table border="1" cellpadding="8" align="center">
<tr><th align="center">LAWE for <math>~n=5</math> Polytropes</th></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left"> 
<math>~(3+\xi^2) \frac{d^2x}{d\xi^2} + \biggl[12 - 2\xi^2 \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} + 
\biggl[\frac{\sigma_c^2}{3^{1/2} \gamma_g } \cdot (3+\xi^2)^{3/2} - 
6\alpha \biggr]  x </math>
  </td>
</tr>
</table>
</td></tr></table>
</div>

==Numerical Integration of LAWE==

By numerically integrating the above LAWE using the algorithm outlined in a [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward|separate chapter]], we have examined the properties of the displacement function that describes radial modes of oscillation in pressure-truncated, n = 5, polytropic configurations. Our brief description, here, of these modes parallels the more detailed description of radial oscillation modes in truncated isothermal spheres that has been [[SSC/Stability/Isothermal#Previously_Published_Eigenvalues_and_Eigenfunctions|presented in a separate chapter]]. 

The animation sequence that appears in the right panel of Composite Display 1 shows how our numerically derived displacement function, <math>~x(\xi)</math>, varies with radius &#8212; from the center of the n=5 polytropic sphere, out to <math>~\xi = 10</math> &#8212; for sixteen different values of the square of the eigenfrequency, <math>~\sigma_c^2</math>, as denoted at the top of each animation frame. The segment of the <math>~x(\xi)</math> curve that has been drawn in blue identifies the ''eigenfunction'' that corresponds to the specified value of the eigenfrequency.  In each frame, the radial location at which the blue segment terminates simultaneously identifies: (a) the radius at which the logarithmic derivative of the displacement function, <math>~d\ln x/d\ln\xi </math>, is negative three; and (b) the radius, <math>~\tilde\xi</math>, at which the n = 5 polytropic configuration has been truncated.  As displayed here, in every frame, the <math>~x(\xi)</math> function has been normalized such that the displacement amplitude is unity at the truncated configuration's surface.  

The left panel of Composite Display 1 is also animated and has been provided in support of the animation on the right.  Specifically, the number written at the top of each left-panel frame quantitatively identifies the radial location, <math>~\tilde\xi</math>, of the surface of the relevant truncated polytropic configuration; and, on each frame, "&times;" marks the location of that truncated configuration on the mass-radius equilibrium sequence. 

<div align="center" id="n5TruncatedMovie">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="2">Composite Display 1: &nbsp; Numerically Generated Fundamental-Mode Eigenvectors</th>
</tr>
<tr>
  <td align="center">
[[File:N5Truncated2.gif|600px|n5 Truncated movie]]
  </td>
  <td align="center">
Excel File:<br />
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/LinearPerturbation/n5Eigenvectors/n5TruncatedSphere.xlsx --- worksheet = OursPt1]]
Movie File:<br />
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/n5movie/ --- worksheet = n5Truncated2.gif]]
  </td>
</tr>
</table>
</div>

Each full loop through the left-panel animation sequence can be viewed as evolution along the equilibrium sequence from <math>~\tilde\xi = 0.75</math> to  <math>~\tilde\xi = 5</math>, then back again.  During this evolution, the "&times;" marker moves through both turning points along the sequence:  the maximum radius configuration &#8212; at <math>~\tilde\xi= \sqrt{3}</math> &#8212; and the maximum mass configuration  &#8212; at <math>~\tilde\xi= 3</math>.  Notice that <math>~\sigma_c^2</math> is positive for all models having <math>~\tilde\xi < 3</math> while it is negative for all models having  <math>~\tilde\xi > 3</math>.  Hence, models having <math>~\tilde\xi > 3</math> are dynamically unstable and, as best we have been able to determine via these numerical integrations, the transition from stable to unstable models &#8212; that is, the ''marginally'' unstable model &#8212; occurs at <math>~\tilde\xi = 3</math>.  (Via an ''analytic'' analysis, we prove, below, that this association is precise.)  For emphasis, the "&times;" marker (left panel) and the numerical value recorded for <math>~\sigma_c^2</math> (right panel) have been colored red for models that are not stable.