SSC/Stability/BiPolytropes/RedGiantToPN/Pt2

Main Sequence to Red Giant to Planetary Nebula (Part 2)

Part I:  Background & Objective

Part II:

Part III:

Part IV:

Foundation

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

Adiabatic Wave (or Radial Pulsation) Equation

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

whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. Assuming that the underlying equilibrium structure is that of a bipolytrope having $(n_c,n_e)=(5,1)$, it makes sense to adopt the normalizations used when defining the equilibrium structure, namely,

${\rho }^{*}$	$\equiv$	$\frac{{\rho }_0}{{\rho }_c}$	;	$r^{*}$	$\equiv$	$\frac{r_0}{[K_c^{1/2}/(G^{1/2}{\rho }_c^{2/5})]}$
$P^{*}$	$\equiv$	$\frac{P_0}{K_c{\rho }_c^{6/5}}$	;	$M_r^{*}$	$\equiv$	$\frac{M_r}{[K_c^{3/2}/(G^{3/2}{\rho }_c^{1/5})]}$

We note as well that,

$g_0$	$=$	$\frac{GM(r_0)}{r_0^2}$
	$=$	$G\left[M_r^{*}{\rho }_c^{-1/5}\right(\frac{K_c}{G}{)}^{3/2}\left]\right[r^{*}{\rho }_c^{-2/5}(\frac{K_c}{G}{)}^{1/2}{]}^{-2}$
	$=$	$\frac{G{M}_r^{*}}{(r^{*}{)}^2}\left[{\rho }_c^{3/5}\right(\frac{K_c}{G}{)}^{1/2}].$

Hence, multiplying the LAWE through by $(K_c/G){\rho }_c^{-4/5}$ gives,

$0$

$=$

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

$0$	$=$	$\frac{d^{2}x}{dr{*}^2}+[\frac{4}{r^{*}}-{\rho }_c^{-2/5}(\frac{K_c}{G}{)}^{1/2}\left(\frac{g_0{\rho }_0}{P_0}\right)]\frac{dx}{dr*}+{\rho }_c^{-4/5}(\frac{K_c}{G}\left)\right(\frac{{\rho }_0}{{\gamma }_{\mathrm{g}}{P}_0}\left)\right[{\omega }^2+(4-3{\gamma }_{\mathrm{g}})\frac{g_0}{r_0}]x$
	$=$	$\frac{d^{2}x}{dr{*}^2}+\{\frac{4}{r^{*}}-{\rho }_c^{-2/5}(\frac{K_c}{G}{)}^{1/2}\frac{G{M}_r^{*}}{(r^{*}{)}^2}\left[{\rho }_c^{3/5}\right(\frac{K_c}{G}{)}^{1/2}\left]\right[\frac{{\rho }_c{\rho }^{*}}{P^{*}{K}_c{\rho }_c^{6/5}}\left]\right\}\frac{dx}{dr*}+{\rho }_c^{-4/5}\left(\frac{K_c}{G}\right)\left[\frac{{\rho }_c{\rho }^{*}}{{\gamma }_{\mathrm{g}}{P}^{*}{K}_c{\rho }_c^{6/5}}\right]\{{\omega }^2+(4-3{\gamma }_{\mathrm{g}})\frac{G{M}_r^{*}}{(r^{*}{)}^2}[{\rho }_c^{3/5}\left(\frac{K_c}{G}{)}^{1/2}\right]\frac{{\rho }_c^{2/5}}{r^{*}}\left(\frac{G}{K_c}{)}^{1/2}\right\}x$
	$=$	$\frac{d^{2}x}{dr{*}^2}+\{\frac{4}{r^{*}}-\frac{M_r^{*}}{(r^{*}{)}^2}[\frac{{\rho }^{*}}{P^{*}}\left]\right\}\frac{dx}{dr*}+\left(\frac{1}{G{\rho }_c}\right)\left[\frac{{\rho }^{*}}{{\gamma }_{\mathrm{g}}{P}^{*}}\right]\{{\omega }^2+(4-3{\gamma }_{\mathrm{g}})\frac{G{\rho }_c{M}_r^{*}}{(r^{*}{)}^3}\}x$
	$=$	$\frac{d^{2}x}{dr{*}^2}+\{\frac{4}{r^{*}}-(\frac{{\rho }^{*}}{P^{*}}\left)\frac{M_r^{*}}{(r^{*}{)}^2}\right\}\frac{dx}{dr*}+\left(\frac{{\rho }^{*}}{P^{*}}\right)\{\frac{{\omega }^2}{{\gamma }_{\mathrm{g}}G{\rho }_c}+(\frac{4}{{\gamma }_{\mathrm{g}}}-3\left)\frac{M_r^{*}}{(r^{*}{)}^3}\right\}x$
	$=$	$\frac{d^{2}x}{dr{*}^2}+\{4-(\frac{{\rho }^{*}}{P^{*}}\left)\frac{M_r^{*}}{(r^{*})}\right\}\frac{1}{r^{*}}\frac{dx}{dr*}+\left(\frac{{\rho }^{*}}{P^{*}}\right)\{\frac{2\pi {\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-\frac{{\alpha }_{\mathrm{g}}{M}_r^{*}}{(r^{*}{)}^3}\}x.$

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