User:Tohline/SSC/Stability/Polytropes
Radial Oscillations of Polytropic Spheres
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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
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<math>~r_0</math> |
<math>~=</math> |
<math>~a_n \xi \, ,</math> |
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<math>~\rho_0</math> |
<math>~=</math> |
<math>~\rho_c \theta^{n} \, ,</math> |
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<math>~P_0</math> |
<math>~=</math> |
<math>~K\rho_0^{(n+1)/n} = K\rho_c^{(n+1)/n} \theta^{n+1} \, ,</math> |
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<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,
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<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:
- P. LeDoux & Th. Walraven (1958, Handbuch der Physik, 51, 353) —
- R. F. Christy (1966, Annual Reviews of Astronomy & Astrophysics, 4, 353) — Pulsation Theory
- M. Hurley, P. H. Roberts, & K. Wright (1966, ApJ, 143, 535) — The Oscillations of Gas Spheres
- J. P. Cox (1974, Reports on Progress in Physics, 37, 563) — Pulsating Stars
Tables
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Quantitative Information Regarding Eigenvectors of Oscillating Polytropes <math>~(\Gamma_1 = 5/3)</math> |
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<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 <math>~\sigma_0^2 R^3/(GM)</math> |
<math>\frac{(n+1) *\mathrm{Cox74}}{3 *\mathrm{HRW66}} \cdot \frac{\bar\rho}{\rho_c}</math> |
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<math>~0</math> |
<math>~1</math> |
<math>~1/3</math> |
<math>~1</math> |
<math>~1</math> |
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<math>~1</math> |
<math>~3.30</math> |
<math>~0.38331</math> |
<math>~1.892</math> |
<math>~0.997</math> |
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<math>~1.5</math> |
<math>~5.99</math> |
<math>~0.37640</math> |
<math>~2.712</math> |
<math>~1.002</math> |
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<math>~2</math> |
<math>~11.4</math> |
<math>~0.35087</math> |
<math>~4.00</math> |
<math>~1.000</math> |
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<math>~3</math> |
<math>~54.2</math> |
<math>~0.22774</math> |
<math>~9.261</math> |
<math>~1.000</math> |
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<math>~3.5</math> |
<math>~153</math> |
<math>~0.12404</math> |
<math>~12.69</math> |
<math>~1.003</math> |
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<math>~4.0</math> |
<math>~632</math> |
<math>~0.04056</math> |
<math>~15.38</math> |
<math>~1.000</math> |
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