Editing Appendix/Ramblings/PatrickMotl (section)

==October 11==
Here's one reasonable way to consider plotting the eigenfunction at various points in time from the nonlinear simulations of oscillating/collapsing, pressure-truncated n = 5 polytropes.  (It assumes, even though Patrick's simulation is conducted on a cylindrical-coordinate mesh, that he is able to extract a radially dependent <math>~M_r</math> function from the density data.)

Drawing from an [[Appendix/Ramblings/NonlinarOscillation#Exploration|accompanying discussion]], we recognize that, in the unperturbed equilibrium state,

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

<tr>
  <td align="right">
<math>~m_\xi \equiv \frac{M_r}{M_\mathrm{tot}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{\xi}{\tilde\xi}\biggr)^3 \biggl(1 + \frac{\xi^2}{3}\biggr)^{-3/2} 
\biggl(1 + \frac{\tilde\xi^2}{3}\biggr)^{3/2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r_\xi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi \biggl\{ 
\biggl[ \frac{4\pi}{2^5\cdot 3}\biggr]^{1/2} \tilde\xi^{-6}
\biggl( 1+\frac{\tilde\xi^2}{3} \biggr)^{3}\biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>

where the "tilde" indicates the value of <math>~\xi</math> at the surface of the pressure-truncated configuration.  Usually this parametric relationship between mass and radius is used to generate a plot showing how the mass interior to <math>~r</math> varies with the Lagrangian coordinate, <math>~r</math>, in the equilibrium state.  But it can just as well be used to show how <math>~r</math> varies with the Lagrangian coordinate, <math>~m_\xi</math>, in the equilibrium state.  Note that these two expressions can be combined to give <math>~r_\xi</math> in the unperturbed configuration &#8212; hereafter referred to as <math>~r_0</math> &#8212; directly in terms of the fractional mass, <math>~m_\xi</math>; namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
r_0 (m_\xi)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tilde{r}_\mathrm{edge}
\biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C} - 3 m_\xi^{2/3}}\biggr]^{1/2} \, ,
</math>
  </td>
</tr>
</table>
</div>
<span id="DefineTildeC">where,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tilde{C}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{3^2}{\tilde\xi^2}\biggl( 1 + \frac{\tilde\xi^2}{3} \biggr) \, ,
</math>&nbsp; &nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>~\tilde{r}_\mathrm{edge}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\pi}{2^3\cdot 3}\biggr]^{1/2} {\tilde\xi}^{-6} \biggl(1+\frac{\tilde\xi^2}{3}\biggr)^3
\, .
</math>
  </td>
</tr>
</table>
</div>

<font color="red">'''TEST:'''</font> &nbsp; After Patrick introduces a model of given <math>~\tilde\xi</math> into the hydrocode, he should integrate over the density distribution to generate a ''numerically determined'' plot of <math>~r</math> versus <math>~m_\xi \equiv M_r/M_\mathrm{tot}</math>, then compare the plot to this analytically specified function for <math>~r_0(m_\xi)</math>.  The degree to which the plots match at time, <math>~t = 0</math>, will give a measure of the uncertainty in the ''numerically determined'' eigenfunction at later times in the evolution.

At any later time in an evolution, Patrick can again integrate over the density distribution to generate new, numerically determined values of <math>~r(m_\xi)</math>.  From this new data, he can generate a plot of the fractional displacement,

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

<tr>
  <td align="right">
<math>~x(m_\xi)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{ [r_0(m_\xi) - r(m_\xi) ]}{r_0(m_\xi)} \, .</math>
  </td>
</tr>
</table>

Ideally, if oscillations of the model are due entirely to a single radial mode &hellip;
<ul>
  <li>The ''shape'' of this fractional displacement function should not change with time; but</li>
  <li>The overall ''amplitude'' is expected to vary with time at a frequency that is predicted by linear theory.
</ul>
[[#Fig6|Figure 6, above]], quantifies the eigenfrequency and the shape of the displacement function for fifteen different models; for the marginally unstable model, the eigenfrequency is zero and the eigenfunction (displacement function) is exactly a parabola.

<font color="red">'''NOTE:'''</font>&nbsp; Because the model evolutions are being performed on an Eulerian mesh while the numerically determined quantity, <math>~m_\xi</math>, is a Lagrangian tracer, the array of <math>~m_\xi</math> grid points at which <math>~r</math> is measured will vary with time.  This could pose a problem because both of the variables on the right-hand-side of the <math>~x(m_\xi)</math> expression need to be evaluated on the same Lagrangian-tracer grid.  Fortunately, we have an analytic expression for <math>~r_0(m_\xi)</math>; so, each time a new Lagrangian-tracer grid arises during the numerical evaluation of <math>~r(m_\xi)</math>, this analytic expression for <math>~r_0(m_\xi)</math> should be called upon to give values of <math>~r_0</math> on the revised Lagrangian-tracer grid.
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