Editing Apps/ImamuraHadleyCollaboration (section)

===Radial Modes in Homogeneous Spheres===

Before attempting to analyze natural modes of oscillation in a polytropic torus, it is useful to review what is known about radial oscillations in the geometrically simpler, uniform-density sphere. As we have reviewed in [[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|a separate chapter]], [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S Sterne (1937)] was the first to recognize that the set of eigenvectors that describe radial modes of oscillation in a homogeneous, self-gravitating sphere can be determined analytically.  In the present context, it is advantageous for us to pull Sterne's solution from a discussion of the same problem presented by [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland (1964)].  As we have summarized in [[SSC/PerspectiveReconciliation#Eulerian_Reformulation|a separate chapter]], the relevant eigenvalue problem is defined by the following one-dimensional, 2<sup>nd</sup>-order ODE:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
( 1-\chi_0^2 ) \frac{\partial^2 \xi}{\partial \chi_0^2}
+ \frac{2}{\chi_0}\biggl[ 1 - 2\chi_0^2 \biggr] \frac{\partial \xi}{\partial \chi_0}
+ \biggl(4 - \frac{2}{\chi_0^2} \biggr) \xi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \mathfrak{F} \xi \, ,
</math>
  </td>
</tr>
</table>
</div>
where, it is understood that the expression for the (spatially and temporally varying) radial location of each spherical shell is,
<div align="center">
<math>~r(r_0,t) = r_0 + \delta r(r_0)e^{i\sigma t} \, ,</math>
</div>
and in the present context we are adopting the variable notation, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi_0</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{r_0}{R} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\xi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\delta r}{R} \, ,</math>
  </td>
</tr>
</table>
</div>
and <math>~R</math> is the initial (unperturbed) radius of the sphere.  Here, the eigenvalue is related to the physical properties of the homogeneous sphere via the relation,
<div align="center">
<math>~\mathfrak{F}  \equiv \frac{2}{\gamma_\mathrm{g}} \biggl[\biggl(\frac{3\sigma^2}{4\pi G\rho_0}\biggr) + (4 - 3\gamma_\mathrm{g}) \biggr]  \, .</math>
</div>


Drawing from [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland's (1964)] presentation &#8212; see his p. 29, and [[SSC/PerspectiveReconciliation#Eulerian_Reformulation|our related discussion]] &#8212; the following table details the eigenvectors (radial eigenfunction and associated eigenfrequency) for the three lowest radial modes (m = 2, 4, 6) that satisfy this wave equation; the figure displayed in the right-most column has been extracted directly from p. 29 of Rosseland and shows the behavior of the lowest five radial modes (m = 2, 4, 6, 8, 10, as labeled) over the interval <math>~0 \le \chi_0 \le 1</math>.

<div align="center">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="4">
<font size="+1"><b>Table 2</b></font><p></p>
Rosseland's (1964) Eigenfunctions for Homogeneous Sphere<p></p>
Figure in the right-most column extracted from p. 29 of [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland (1964)]<p></p>
"''The Pulsation Theory of Variable Stars''" (New York:  Dover Publications, Inc.)
  </th>
</tr>
<tr>
  <td align="center">Mode</td>
  <td align="center" colspan="1">Eigenfunction</td>
  <td align="center">Square of Eigenfrequency:<p></p><math>~3\sigma^2/(4\pi G\rho)</math></td>
  <td align="center" rowspan="5">[[File:RosselandModesFigure3.png|350px|center|Rosseland (1937)]]</td>
</tr>
<tr>
  <td align="center"><math>~m</math></td>
  <td align="center">As Published</td>
  <td align="center"><math>~\frac{m}{2}(m+1)\gamma - 4</math></td>
</tr>
<tr>
  <td align="center"><math>~2</math></td>
  <td align="left"><math>~\xi = -2\chi_0</math></td>
  <td align="center"><math>~3\gamma - 4</math></td>
</tr>
<tr>
  <td align="center"><math>~4</math></td>
  <td align="left"><math>~\xi = -\frac{20}{3}\chi_0 + \frac{28}{3}\chi_0^3</math></td>
  <td align="center"><math>~10\gamma-4</math></td>
</tr>
<tr>
  <td align="center"><math>~6</math></td>
  <td align="left"><math>~\xi = -14\chi_0 + \frac{252}{5} \chi_0^3 - \frac{198}{5} \chi_0^5</math></td>
  <td align="center"><math>~21\gamma-4</math></td>
</tr>
</table>
</div>

We should point out that, except for the lowest (m = 2) mode, each of the radial eigenfunctions crosses zero at least once at some location(s) that resides between the center <math>~(\chi_0=0)</math>  of and the surface <math>~(\chi_0 = 1)</math> of the sphere.  More specifically, for each mode, <math>~m</math>, the number of such radial "nodes" is <math>~(m-2)/2</math>.  The locations of these nodes is apparent from even a casual inspection of the figure presented in the right-most column of Table 2.

When these radial modes of oscillation are discussed in the astrophysics literature, the conditions that give rise to a dynamical instability are often emphasized.  Specifically, each mode becomes unstable when <math>\sigma^2</math> becomes negative, which translates into a value of <math>~\gamma < \gamma_\mathrm{crit}(m)</math> &#8212; see our [[SSC/Stability/UniformDensity#Properties_of_Eigenfunction_Solutions|related discussion of the properties of eigenfunction solutions]] in the context of [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S Sterne's (1937)] analysis of this stability problem.  Here we want to emphasize that all of natural modes of oscillation ''exist'' even when the configuration is dynamically stable.