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.

Introducing the dimensionless frequency-squared, ${\sigma }_c^2\equiv 3{\omega }^2/(2\pi G{\rho }_c)$, we can rewrite this LAWE as, $0$ $=$ $\frac{d^{2}x}{d{r}_0^2}+[4-(\frac{g_0}{r_0}\cdot \frac{{\rho }_0{r}_0^2}{P_0}\left)\right]\frac{1}{r_0}\cdot \frac{dx}{d{r}_0}+\left(\frac{{\rho }_0{r}_0^2}{P_0}\right)[\frac{2\pi G{\rho }_c{\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\frac{g_0}{r_0}\right]\frac{x}{r_0^2},$ where, as a reminder, $g_0\equiv GM(r_0)/r_0^2$. Now, for our $(n_c,n_e)=(5,1)$ bipolytrope, we have found it useful to adopt the following four dimensionless variables: ${\rho }^{*}$ $\equiv$ $\frac{{\rho }_0}{{\rho }_c}$ ;     $r^{*}$ $\equiv$ $\frac{r_0}{[K_c^{1/2}/(G^{1/2}{\rho }_c^{2/5})]}$ $P^{*}$ $\equiv$ $\frac{P_0}{K_c{\rho }_c^{6/5}}$ ;     $M_r^{*}$ $\equiv$ $\frac{M_r}{[K_c^{3/2}/(G^{3/2}{\rho }_c^{1/5})]}$ This means that, $\frac{g_0}{r_0}=\frac{GM(r_0)}{r_0^3}$ $=$ $G{M}_r^{*}\left[K_c^{3/2}{G}^{-3/2}{\rho }_c^{-1/5}\right]r_{*}^{-3}\left[K_c^{-3/2}{G}^{3/2}{\rho }_c^{6/5}\right]=\left[G{\rho }_c\right]M_r^{*}{r}_{*}^{-3};$ $\frac{{\rho }_0{r}_0^2}{P_0}$ $=$ ${\rho }^{*}{\rho }_c(r^{*}{)}^2[K_c{G}^{-1}{\rho }_c^{-4/5}](P^{*}{)}^{-1}\left[K_c^{-1}{\rho }_c^{-6/5}\right]=\left[G^{-1}{\rho }_c^{-1}\right]{\rho }^{*}(r^{*}{)}^2(P^{*}{)}^{-1};$ $\frac{g_0}{r_0}\cdot \frac{{\rho }_0{r}_0^2}{P_0}$ $=$ $\left[G{\rho }_c\right]M_r^{*}{r}_{*}^{-3}\cdot \left[G^{-1}{\rho }_c^{-1}\right]{\rho }^{*}(r^{*}{)}^2(P^{*}{)}^{-1}=\frac{M_r^{*}{\rho }^{*}}{P^{*}{r}^{*}}.$ Making these substitutions, the LAWE can be rewritten as, $0$ $=$ $\frac{d^{2}x}{d{r}_0^2}+[4-\frac{M_r^{*}{\rho }^{*}}{P^{*}{r}^{*}}]\frac{1}{r_0}\cdot \frac{dx}{d{r}_0}+\frac{1}{G{\rho }_c}\left[\frac{{\rho }^{*}(r^{*}{)}^2}{P^{*}}\right][\frac{2\pi G{\rho }_c{\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\frac{G{\rho }_c{M}_r^{*}}{(r^{*}{)}^3}\right]\frac{x}{r_0^2};$ then, multiplying through by $[K{G}^{-1}{\rho }_c^{-4/5}]$ allows us to everywhere switch from $(r_0{)}^2$ to $(r^{*}{)}^2$, namely, $0$ $=$ $\frac{d^{2}x}{d(r^{*}{)}^2}+[4-\frac{M_r^{*}{\rho }^{*}}{P^{*}{r}^{*}}]\frac{1}{r^{*}}\cdot \frac{dx}{d(r^{*})}+\left[\frac{{\rho }^{*}(r^{*}{)}^2}{P^{*}}\right][\frac{2\pi {\sigma }_c^2}{3{\gamma }_{\mathrm{g}}}-(3-\frac{4}{{\gamma }_{\mathrm{g}}}\left)\frac{M_r^{*}}{(r^{*}{)}^3}\right]\frac{x}{(r^{*}{)}^2}.$

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,

$0$

$=$

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

In shorthand, we can rewrite this equation in the form,

$0$

$=$

$x^{″}+\frac{{\mathscr{H}}}{r^{*}}{x}^{\prime }+{𝒦}x,$

where,

$x^{\prime }$

$=$

$\frac{dx}{d{r}^{*}}$

and

$x^{″}$

$=$

$\frac{d^{2}x}{d(r^{*}{)}^2};$

and,

${𝒦}\to \left(\frac{{\sigma }_c^2}{{\gamma }_{\mathrm{g}}}\right){{𝒦}}_1-{\alpha }_{\mathrm{g}}{{𝒦}}_2;$

and,

${\mathscr{H}}$

$\equiv$

$\{4-(\frac{{\rho }^{*}}{P^{*}}\left)\frac{M_r^{*}}{(r^{*})}\right\}$

,

${{𝒦}}_1$

$\equiv$

$\frac{2\pi }{3}\left(\frac{{\rho }^{*}}{P^{*}}\right)$

and

${{𝒦}}_2$

$\equiv$

$\left(\frac{{\rho }^{*}}{P^{*}}\right)\frac{M_r^{*}}{(r^{*}{)}^3}.$

Drawing from our "Table 2" profiles,

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