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. Multiplying this LAWE through by $(K_c/G){\rho }_c^{-4/5}$ and recognizing that,

$g_0$

$=$

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

we have,

$0$

$=$

$\frac{d^{2}x}{dr{*}^2}+\{\frac{4}{r^{*}}-(\frac{{\rho }^{*}}{P^{*}}\left)\frac{M_r^{*}}{(r^{*}{)}^2}\right\}\frac{dx}{dr*}+\left(\frac{{\rho }^{*}}{P^{*}}\right)\{\frac{{\omega }^2}{{\gamma }_{\mathrm{g}}G{\rho }_c}+(\frac{4}{{\gamma }_{\mathrm{g}}}-3\left)\frac{M_r^{*}}{(r^{*}{)}^3}\right\}x$

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,

Related Discussions