SSC/Structure/BiPolytropes/AnalyzeStepFunction

More Careful Examination of Step Function Behavior

The ideas that are captured in this chapter have arisen as an extension of our accompanying "renormalization" of the Analytic51 bipolytrope.

Discontinuous Density Distribution

Expectations

From among the set of governing relations that apply to spherically symmetric configurations, we focus, first, on the combined,

Euler + Poisson Equations

$\frac{d{v}_r}{dt}=-\frac{1}{\rho }\frac{dP}{dr}-\frac{G{M}_r}{r^2}$

At the interface between the core and envelope of an equilibrium bipolytrope, both the core and the envelope must satisfy the relation,

$\frac{1}{\rho }\frac{dP}{dr}$

$=$

$-\frac{G{M}_r}{r^2}.$

Now, the quantity on the right-hand side of this expression must be the same at the interface, when viewed either from the perspective of the core or from the perspective of the envelope. Therefore, at the interface, the equilibrium configuration must obey the relation,

$[\frac{1}{\rho }\frac{dP}{dr}{]}_{\mathrm{e}\mathrm{n}\mathrm{v},i}$

$=$

$[\frac{1}{\rho }\frac{dP}{dr}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e},i}.$

Now, if we set ${\rho }_e=({\mu }_e/{\mu }_c){\rho }_c$ at the interface, then,

$[\frac{dP}{dr}{]}_{\mathrm{e}\mathrm{n}\mathrm{v},i}$

$=$

$\frac{{\mu }_e}{{\mu }_c}[\frac{dP}{dr}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e},i}.$

Check Behavior

In step 4 of our accompanying analysis, we find that from the perspective of the core,

$P^{*}$

$=$

$(1+\frac{1}{3}{\xi }^2{)}^{-3},$

and,

$r^{*}$

$=$

$(\frac{3}{2\pi }{)}^{1/2}\xi .$

Hence, at the interface,

$\frac{d{P}^{*}}{d{r}^{*}}{|}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}=\{\frac{d\xi }{d{r}^{*}}\cdot \frac{d{P}^{*}}{d\xi }{\}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	$=$	$-2\left(\frac{2\pi }{3}{)}^{1/2}\right[(1+\frac{1}{3}{\xi }^2{)}^{-4}\xi {]}_i$
	$=$	$-\left(\frac{4\pi }{3}\right){\theta }_i^8{r}_i^{*}.$

While, in step 8 of that analysis, we find from the perspective of the envelope,

$P^{*}$

$=$

${\theta }_i^6{\varphi }^2,$

and,

$r^{*}$

$=$

$(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta .$

Hence, at the interface,

$\frac{d{P}^{*}}{d{r}^{*}}{|}_{\mathrm{e}\mathrm{n}\mathrm{v}}=\{\frac{d\eta }{d{r}^{*}}\cdot \frac{d{P}^{*}}{d\eta }{\}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	$=$	$\left\{\right(\frac{{\mu }_e}{{\mu }_c}){\theta }_i^2(2\pi {)}^{1/2}\left\}2{\theta }_i^6\right[\varphi \cdot \frac{d\varphi }{d\eta }{]}_i$
	$=$	$2(2\pi {)}^{1/2}{\theta }_i^8\{\left(\frac{{\mu }_e}{{\mu }_c}\right)\left\}\right[3^{1/2}{\theta }_i^{-3}\left(\frac{d\theta }{d\xi }{)}_i\right]$
	$=$	$-\frac{2}{3}(6\pi {)}^{1/2}{\theta }_i^8(\frac{{\mu }_e}{{\mu }_c}){\xi }_i$
	$=$	$-2\left(\frac{2\pi }{3}{)}^{1/2}{\theta }_i^8\right(\frac{{\mu }_e}{{\mu }_c}\left)\right[\left(\frac{2\pi }{3}{)}^{1/2}{r}_i^{*}\right]$
	$=$	$-\left(\frac{4\pi }{3}\right){\theta }_i^8\left(\frac{{\mu }_e}{{\mu }_c}\right)r_i^{*}.$

See Also