SSC/Stability/BiPolytropes/RedGiantToPN

Red Giant to Planetary Nebula

Following the Lead of Yabushita75

Here in the context of $(n_c,n_e)=(5,1)$ bipolytropes, we want to construct mass-versus-central density plots like the one displayed for truncated isothermal spheres in Figure 1 of an accompanying discussion, and as displayed for a $(n_c,n_e)=(\mathrm{\infty },3/2)$ bipolytrope in Figure 1 (p. 445) of 📚 S. Yabushita (1975, MNRAS, Vol. 172, pp. 441 - 453).

In our accompanying chapter that presents example models of $(n_c,n_e)=(5,1)$ bipolytropes, we have adopted the following normalizations:

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

Also, from the relevant interface conditions, we find,

$\left(\frac{K_e}{K_c}\right)$

$=$

${\rho }_0^{-4/5}(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4}.$

And, inverting this last expression gives,

${\rho }_0^{4/5}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4}\right(\frac{K_e}{K_c}{)}^{-1}$
$\Rightarrow {\rho }_0$	$=$	$\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4}(\frac{K_e}{K_c}{)}^{-1}{]}^{5/4}.$

for our normalizations we can replace ${\rho }_0$ with $K_e/K_c$ everywhere, via the relati

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