User:Tohline/SSC/Stability/BiPolytrope0 0

Radial Oscillations of a Zero-Zero Bipolytrope

This is a chapter that summarizes an accompanying, detailed derivation.

Two Separate Eigenvectors

Core

<math>~\alpha_c \equiv 3-\frac{4}{\gamma_c}</math>
<math>~g = \frac{1}{1+2q^3}</math>

Mode	Core Eigenvector	<math>~\frac{3\omega_\mathrm{core}^2}{2\pi \gamma_c G \rho_c} = 2\alpha_c + 2j(2j+5)</math>
<math>~j=0 </math>	<math>~x_\mathrm{core} = a_0 </math>	<math>~6-8/\gamma_c</math>
<math>~j=1 </math>	<math>~x_\mathrm{core} = a_0 \biggl[ 1 - \frac{7}{5}\biggr(\frac{\xi^2}{g^2}\biggr) \biggr]</math>	<math>~20-8/\gamma_c</math>
<math>~j=2 </math>	<math>~x_\mathrm{core} = a_0 \biggl[ 1 - \frac{18}{5}\biggr(\frac{\xi^2}{g^2}\biggr) + \frac{99}{35}\biggr(\frac{\xi^2}{g^2}\biggr)^2 \biggr]</math>	<math>~42-8/\gamma_c</math>

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