User:Tohline/SSC/Stability/BiPolytrope0 0
Radial Oscillations of a Zero-Zero Bipolytrope
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Groundwork
In an accompanying discussion, we derived the so-called,
Adiabatic Wave (or Radial Pulsation) Equation
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<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. According to our accompanying derivation, if the initial, unperturbed equilibrium configuration is an <math>~(n_c, n_e) = (0,0)</math> bipolytrope, then we know that the relevant functional profiles are as follows for the core and envelope, separately.
Core
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<math>~r_0</math> |
<math>~=</math> |
<math>~\biggl( \frac{P_c}{G\rho_c^2}\biggr)^{1 / 2} \chi \, ,</math> |
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<math>~\rho_0</math> |
<math>~=</math> |
<math>~\rho_c \, ,</math> |
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<math>~\frac{P_0}{P_c}</math> |
<math>~=</math> |
<math>~1 - \frac{2\pi}{3} \chi^2 \, ,</math> |
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<math>~M_r</math> |
<math>~=</math> |
<math>~\frac{4\pi}{3} \biggl( \frac{P_c^3}{G^3 \rho_c^4} \biggr)^{1 / 2}\chi^3 \, .</math> |
Envelope
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