Editing Appendix/Ramblings/PatrickMotl (section)

===Additional Information on Frequencies===

In our [[SSC/Perturbations#The_Eigenvalue_Problem|accompanying derivation of the LAWE]], we define the perturbations in pressure ( p ), density ( d ), and Lagrangian radial position ( x ) such that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_0(m) + P_1(m,t) = P_0(m) \biggl[1 + p(m) e^{i\omega t}  \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0(m) + \rho_1(m,t) = \rho_0(m) \biggl[1 + d(m) e^{i\omega t}  \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_0(m) + r_1(m,t) = r_0(m) \biggl[1 + x(m) e^{i\omega t}  \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Let's focus on the time-dependent behavior of the central density.  Whenever <math>~\omega^2</math> is negative, we appreciate that <math>~\omega</math> will be imaginary so the exponent of the exponential will be real.  In this case, we expect the central density to grow exponentially (cloud collapse); its behavior will be described by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho(t)}{\rho_0} \biggr|_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 + d_c e^{\sqrt{\omega^2}t} \, .
</math>
  </td>
</tr>
</table>
(Note:  In what follows, we will assume for simplicity that the radial eigenfunction for the perturbed density, <math>~d(m)</math>, is normalized such that it is unity at the center (''i.e.,'' at m = 0); that is, we will set <math>~d_c = 1</math>.)

Now, the frequency, <math>~\omega</math>, has units of inverse time.  Also, in the [[SSC/Stability/n5PolytropeLAWE#General_Form_of_the_LAWE_for_Spherical_Polytropes|chapter where we specifically study pressure-truncated n = 5 ploytropes]], we define the dimensionless frequency, 
<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>

Realizing that, in Patrick's plots, 
<div align="center">
<math>~T_0 \equiv \biggl[\frac{3\pi}{32 G \bar\rho}\biggr]^{1 / 2} \, ,</math>
</div>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bar\rho</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{eq}^3} \, ,</math>
  </td>
</tr>
</table>

<span id="convert">we see that the exponent in the exponential may be rewritten as,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sqrt{\omega^2} t</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sqrt{\sigma_c^2} \biggl(\frac{2\pi G \rho_c}{3}\biggr)^{1 / 2}  \biggr(\frac{t}{T_0}\biggr) \biggl[\frac{3\pi}{32 G \bar\rho}\biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sqrt{\sigma_c^2} \biggr(\frac{t}{T_0}\biggr) \frac{\pi}{4}\biggl[\frac{\rho_c}{\bar\rho}\biggr]^{1 / 2}
</math>
  </td>
</tr>
</table>

Now, from our [[SSCpt1/Virial/FormFactors#Mean-to-Central_Density|accompanying discussion of the integration form factors for truncated n = 5 polytopes]], we know that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{f}_M \biggr|_{n=5}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\bar\rho}{\rho_c} \biggr|_{n=5} = 
\biggl[ 1 + \frac{{\tilde\xi}^2}{3}  \biggr]^{-3 / 2} \, .
</math>
  </td>
</tr>
</table>


<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="6">'''Selected Models'''</td>
</tr>
<tr>
  <td align="center"><math>~\tilde\xi</math></td>
  <td align="center"><math>~\frac{M_\mathrm{tot}}{M_\mathrm{SWS}}</math></td>
  <td align="center"><math>~\frac{R_\mathrm{eq}}{R_\mathrm{SWS}}</math></td>
  <td align="center"><math>~\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>~\sigma_c^2</math></td>
  <td align="center"><math>~\sqrt{\omega^2} t</math></td>
</tr>
<tr>
  <td align="center">2.75</td>
  <td align="center">-</td>
  <td align="center">-</td>
  <td align="center">6.6065</td>
  <td align="center">+0.061925</td>
  <td align="center">0.50235  <math>~\biggl(\frac{t}{T_0} \biggr)</math></td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">1.77408</td>
  <td align="center">0.47309</td>
  <td align="center">8</td>
  <td align="center">0.0</td>
  <td align="center">0.0  <math>~\biggl(\frac{t}{T_0} \biggr)</math></td>
</tr>
<tr>
  <td align="center">3.25</td>
  <td align="center">-</td>
  <td align="center">-</td>
  <td align="center">9.6123</td>
  <td align="center">- 0.039629</td>
  <td align="center">0.48474 <math>~\biggl(\frac{t}{T_0} \biggr)</math></td>
</tr>
<tr>
  <td align="center">3.5</td>
  <td align="center">-</td>
  <td align="center">-</td>
  <td align="center">11.4610</td>
  <td align="center">- 0.065282</td>
  <td align="center">0.67936 <math>~\biggl(\frac{t}{T_0} \biggr)</math></td>
</tr>
</table>

As we have detailed in a [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|separate discussion]], for pressure-truncated n = 5 polytropes,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
\frac{M_\mathrm{tot}}{M_\mathrm{SWS}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggr( \frac{3^{2} \cdot 5^3 }{4\pi } \biggr)^{1/2} \frac{\tilde\xi^{3}}{(3 + \tilde\xi^2)^{2}}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\frac{R_\mathrm{eq}}{R_\mathrm{SWS}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggr( \frac{3^{2} \cdot 5}{2^2 \pi } \biggr)^{1/2}  \frac{\tilde\xi}{(3 + \tilde\xi^2)} \, .
</math>
  </td>
</tr>
</table>
</div>