Appendix/Ramblings/51BiPolytropeStability/CleanMethodicalExaminatioin

Radial Oscillations of Polytropic Spheres

Adiabatic (Polytropic) Wave Equation

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

Adiabatic Wave (or Radial Pulsation) Equation

$\frac{d^{2}x}{d{r}_0^2}+[\frac{4}{r_0}-(\frac{g_0{\rho }_0}{P_0}\left)\right]\frac{dx}{d{r}_0}+\left(\frac{{\rho }_0}{{\gamma }_{\mathrm{g}}{P}_0}\right)[{\omega }^2+(4-3{\gamma }_{\mathrm{g}})\frac{g_0}{r_0}]x=0$

BiPolytrope with n_c = 5 and n_e = 1

Copied from here … Here we construct a bipolytrope in which the core has an $n_c=5$ polytropic index and the envelope has an $n_e=1$ polytropic index. This system is particularly interesting because the entire structure can be described by closed-form, analytic expressions. In deriving the properties of this model, we will follow the general solution steps for constructing a bipolytrope that we have outlined elsewhere.

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