SSC/FreeEnergy/PolytropesEmbedded/Pt3A

Background

Index to original, very long chapter

Free-Energy of Bipolytropes

In this case, the Gibbs-like free energy is given by the sum of four separate energies,

$\mathfrak{𝔊}$

$=$

$[W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}+{\mathfrak{𝔖}}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+[W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}+{\mathfrak{𝔖}}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}{]}_{\mathrm{e}\mathrm{n}\mathrm{v}}.$

In addition to specifying (generally) separate polytropic indexes for the core, $n_c$, and envelope, $n_e$, and an envelope-to-core mean molecular weight ratio, ${\mu }_e/{\mu }_c$, we will assume that the system is fully defined via specification of the following five physical parameters:

• Total mass, $M_{\mathrm{t}\mathrm{o}\mathrm{t}}$;
• Total radius, $R$;
• Interface radius, $R_i$, and associated dimensionless interface marker, $q\equiv R_i/R$;
• Core mass, $M_c$, and associated dimensionless mass fraction, $\nu \equiv M_c/M_{\mathrm{t}\mathrm{o}\mathrm{t}}$;
• Polytropic constant in the core, $K_c$.

In general, the warped free-energy surface drapes across a five-dimensional parameter "plane" such that,

$\mathfrak{𝔊}$

$=$

$\mathfrak{𝔊}(R,K_c,M_{\mathrm{t}\mathrm{o}\mathrm{t}},q,\nu ).$

See Also