Editing SSC/Stability/Polytropes (section)

===Adiabatic (Polytropic) Wave Equation===
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>

<!--
<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave Equation'''</font><br />

<math>
\frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} + \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr]  x = 0 \, ,
</math>
</div>
-->

whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations.  If the initial, unperturbed equilibrium configuration is a [[SSC/Structure/Polytropes#Polytropic_Spheres|polytropic sphere]] whose internal structure is defined by the function, <math>~\theta(\xi)</math>, then 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a_n \xi \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_c \theta^{n} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~K\rho_0^{(n+1)/n} = K\rho_c^{(n+1)/n} \theta^{n+1} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~g_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{GM(r_0)}{r_0^2} = \frac{G}{r_0^2} \biggl[ 4\pi a_n^3 \rho_c \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr) \biggr]
\, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{(n+1)K}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>
Hence, after multiplying through by <math>~a_n^2</math>, the above adiabatic wave equation can be rewritten in the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4}{\xi} - \frac{g_0}{a_n}\biggl(\frac{a_n^2 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{d\xi} + \biggl(\frac{a_n^2\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{a_n\xi} \biggr]  x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
In addition, given that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{g_0}{a_n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G \rho_c \biggl(-\frac{d \theta}{d\xi} \biggr) \, ,</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\frac{a_n^2 \rho_0}{P_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(n+1)}{(4\pi G\rho_c)\theta} = \frac{a_n^2 \rho_c}{P_c} \cdot \frac{\theta_c}{\theta}\, ,</math>
  </td>
</tr>
</table>
</div>
we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} + 
\biggl[\omega^2 \biggl(\frac{a_n^2 \rho_c }{\gamma_g P_c} \biggr) \frac{\theta_c}{\theta} - 
\biggl(3-\frac{4}{\gamma_g}\biggr)  \cdot \frac{(n+1)V(x)}{\xi^2} \biggr]  x </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>0 \, ,</math>
  </td>
</tr>
</table>
</div>
where we have adopted the function notation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~V(\xi)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{\xi}{\theta} \frac{d \theta}{d\xi} \, .</math>
  </td>
</tr>
</table>
</div>

As can be seen in the following set of retyped expressions, this is the form of the ''polytropic'' wave equation published by {{ MF85bfull }}, at the beginning of their discussion. 

<div align="center">
<table border="2" cellpadding="10" width="80%">
<tr>
  <td align="center">
''Polytropic'' Wave Equation extracted<sup>&dagger;</sup> from<br />
{{ MF85bfigure }}<br />
&copy; Astronomical Society of Australia
  </td>
<tr>
  <td>
[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline:  There appears to be a sign error in the numerator of the second term of the polytropic wave equation as published by Murphy &amp; Fiedler and as retyped here; there also appears to be an error in the definition of the coefficient, &alpha;*, as given in the text of their paper.]]<!-- [[File:MurphyFiedler1985b.png|500px|center|Murphy &amp; Fiedler (1985b)]] -->
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right">
<math>
\frac{d^2\eta}{d\zeta^2} 
+ \biggl( \frac{4+(n + 1)V}{\zeta}\biggr) \frac{d\eta}{d\zeta}
+ \biggl( \frac{\omega_k^2 \theta_c}{\theta} - \frac{\alpha^*(n+1)V}{\zeta^2}\biggr)\eta
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>0 \, ,</math>
  </td>
</tr>
</table>
<div align="center">where: &nbsp; &nbsp;<math>\omega_k^2 = \frac{\sigma_k^2 \alpha_n^2 \rho_c}{\gamma P_c} \, .</math></div>
  </td>
</tr>
<tr><td align="left"><sup>&dagger;</sup>As displayed here, the layout of the equations has been modified from the original publication.</td></tr>
</table>
</div>


It is also the same as the radial pulsation equation for polytropic configurations that appears as equation (56) in {{ HRW66full }}; hereafter, {{ HRW66hereafter }}.  The relevant set of equations from {{ HRW66hereafter }} is retyped here.

<div align="center" id="HRW66excerpt">
<table border="2" cellpadding="10" width="80%">
<tr>
  <td align="center">
Radial Pulsation Equation as Presented<sup>&dagger;</sup> by<br />
{{ HRW66figure }}<br />
&copy; American Astronomical Society
  </td>
<tr>
  <td align="left">
<!-- [[File:HRW66_PolytropicWaveEquation.png|600px|center|Hurley, Roberts &amp; Wright (1966)]] -->
The radial pulsation equation is

<table border="0" align="center" cellpadding="8" width="100%">
<tr>
  <td align="right">
<math>\frac{d^2X}{dx^2}</math>
  </td>
  <td align="center" width="4%"><math>=</math></td>
  <td align="left">
<math>
-~\frac{1}{x} \frac{dX}{dx}\biggl[4 + (n+1)x\frac{\theta '}{\theta} \biggr]
- \frac{\theta ' X}{\gamma \theta x}\biggl[ (n+1)(3\gamma - 4) - \frac{x s^2}{\theta '} \biggr]\, ,
</math>
  </td>
  <td align="center" width="5%">(56)</td>
</tr>
</table>

with end-point conditions

<table border="0" align="center" width="100%">
<tr>
  <td align="center">
<math>X ' = 0 \, ,</math>&nbsp; &nbsp; at &nbsp; &nbsp; <math>x = 0 \, ,</math>
  </td>
  <td align="center" width="5%">(57)</td>
</tr>
</table>

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline:  As is shown in the subsection on "Boundary Conditions," below, it appears as though the term on the right-hand-side of {{ HRW66hereafter }}'s equation (58) is incorrect, as published and as retyped here; it should be preceded with a negative sign.]]and

<table border="0" align="center" cellpadding="8" width="80%">
<tr>
  <td align="right">
<math>(n+1)\frac{dX}{dx}</math>
  </td>
  <td align="center" width="4%"><math>=</math></td>
  <td align="left">
<math>\frac{X}{\gamma x} \biggl[ (n+1)(3\gamma - 4) + \frac{x s^2}{q} \biggr] \, ,</math> &nbsp; &nbsp; at  &nbsp; &nbsp; <math>x = x_0 \, .</math>
  </td>
  <td align="center" width="5%">(58)</td>
</tr>
</table>

  </td>
</tr>
<tr><td align="left">
<sup>&dagger;</sup>Set of equations and accompanying text displayed here, as a single digital image, exactly as they appear in the original publication.
</td></tr>
</table>
</div>

In order to make clearer the correspondence between our derived expression and the one published by {{ HRW66hereafter }}, we will rewrite the {{ HRW66hereafter }} radial pulsation equation:  (1) Gathering all terms on the same side of the equation; (2) making the substitution, 
<div align="center">
<math>\theta^' \rightarrow -\frac{\theta V}{x} \, ;</math>
</div>
and (3) reattaching a "prime" to the quantity, <math>~s</math>, to emphasize that it is a ''dimensionless'' frequency.

<div align="center">
<table border="1" align="center" width="80%" cellpadding="5">
<tr><td align="left">
<font color="maroon">'''ASIDE:'''</font>  In their equation (46), {{ HRW66hereafter }} convert the eigenfrequency, <math>~s</math> &#8212; which has units of inverse time &#8212; to a dimensionless eigenfrequency, <math>~s^'</math>, via the relation,
<div align="center">
<math>~s = \biggl( \frac{4\pi G \rho_c}{1+n} \biggr)^{1/2} s^' ~~~~~~~\cdots\cdots~~~~~~~(46)</math>
</div>
Then, immediately following equation (46), they state that they will "omit the prime on <math>~s</math> henceforward."   As a result, the ''dimensionless'' eigenfrequency that appears in their equations (56) and (58) is unprimed.  This is unfortunate as it somewhat muddies our efforts, here, to demonstrate the correspondence between the {{ HRW66hereafter }} ''polytropic'' radial pulsation equation and ours.  In our subsequent manipulation of equation (56) from {{ HRW66hereafter }} we reattach a prime to the quantity, <math>s</math>, to emphasize that it is a ''dimensionless'' frequency.  But this prime on <math>~s</math> should not be confused with the prime on <math>\theta</math> ({{ HRW66hereafter }} equation 56) or with the prime on <math>~X</math> ({{ HRW66hereafter }} equation 57), both of which denote differentiation with respect to the radial coordinate.
</td></tr>
</table>
</div>

With these modifications, the {{ HRW66hereafter }} radial pulsation equation 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^2 X}{dx^2} + \biggl[\frac{4 - (n+1)V }{x}\biggr]\frac{dX}{dx}
- \frac{V}{\gamma x^2}\biggl[\frac{x^2 (s^')^2}{\theta V} + (3\gamma -4)(n+1) \biggr]X
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{d^2 X}{dx^2} + \biggl[\frac{4 - (n+1)V }{x}\biggr]\frac{dX}{dx}
+ \biggl[-\frac{(s^')^2 }{\gamma \theta } - \biggl(3 -\frac{4}{\gamma}\biggr)\frac{(n+1)V}{x^2} \biggr]X \, .
</math>
  </td>
</tr>
</table>
</div>

<span id="HRW66frequency">The correspondence with our derived expression is complete, assuming that,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(s^')^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\omega^2 \biggl(\frac{a_n^2 \rho_c }{P_c} \biggr) \theta_c</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\omega^2 \biggl[\frac{n+1 }{4\pi G \rho_c} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

As has been explained in the above "<font color="maroon">'''ASIDE'''</font>," this is exactly the factor that {{ HRW66hereafter }} use to normalize their eigenfrequency, <math>~s</math>, and make it dimensionless <math>(s^')</math>.  It is clear, as well, that {{ HRW66hereafter }} have adopted a sign convention for the square of their eigenfrequency that is the opposite of the sign convention that we have adopted for <math>~\omega^2</math>.  That is, it is clear that,
<div align="center">
<math>~s^2 ~~\leftrightarrow~~ - \omega^2 \, .</math>
</div>