Difference between revisions of "User:Tohline/SSC/Stability/Polytropes"

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whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations.
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 [[User:Tohline/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>
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where,
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<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_n}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>


==Overview==
==Overview==

Revision as of 22:43, 2 April 2015

Radial Oscillations of Polytropic Spheres

Whitworth's (1981) Isothermal Free-Energy Surface
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Groundwork

In an accompanying discussion, we derived the so-called,

Adiabatic Wave Equation

<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>

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 polytropic sphere whose internal structure is defined by the function, <math>~\theta(\xi)</math>, then

<math>~r_0</math>

<math>~=</math>

<math>~a_n \xi \, ,</math>

<math>~\rho_0</math>

<math>~=</math>

<math>~\rho_c \theta^{n} \, ,</math>

<math>~P_0</math>

<math>~=</math>

<math>~K\rho_0^{(n+1)/n} = K\rho_c^{(n+1)/n} \theta^{n+1} \, ,</math>

<math>~g_0</math>

<math>~=</math>

<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>

where,

<math>~a_n</math>

<math>~=</math>

<math>~\biggl[\frac{(n+1)K_n}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr]^{1/2} \, .</math>

Overview

The eigenvector associated with radial oscillations in isolated polytropes has been determined numerically and the results have been presented in a variety of key publications:

Tables

Quantitative Information Regarding Eigenvectors of Oscillating Polytropes

<math>~(\Gamma_1 = 5/3)</math>

<math>~n</math>

<math>~\frac{\rho_c}{\bar\rho}</math>

Excerpts from Table 1 of

Hurley, Roberts, & Wright (1966)

<math>~s^2 (n+1)/(4\pi G\rho_c)</math>

Excerpts from Table 3 of

J. P. Cox (1974)

<math>~\sigma_0^2 R^3/(GM)</math>

<math>\frac{(n+1) *\mathrm{Cox74}}{3 *\mathrm{HRW66}} \cdot \frac{\bar\rho}{\rho_c}</math>

<math>~0</math>

<math>~1</math>

<math>~1/3</math>

<math>~1</math>

<math>~1</math>

<math>~1</math>

<math>~3.30</math>

<math>~0.38331</math>

<math>~1.892</math>

<math>~0.997</math>

<math>~1.5</math>

<math>~5.99</math>

<math>~0.37640</math>

<math>~2.712</math>

<math>~1.002</math>

<math>~2</math>

<math>~11.4</math>

<math>~0.35087</math>

<math>~4.00</math>

<math>~1.000</math>

<math>~3</math>

<math>~54.2</math>

<math>~0.22774</math>

<math>~9.261</math>

<math>~1.000</math>

<math>~3.5</math>

<math>~153</math>

<math>~0.12404</math>

<math>~12.69</math>

<math>~1.003</math>

<math>~4.0</math>

<math>~632</math>

<math>~0.04056</math>

<math>~15.38</math>

<math>~1.000</math>

Related Discussions

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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