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,

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

Gratifyingly, we find as expected that,

$[\frac{d{P}^{*}}{d{r}^{*}}{]}_{\mathrm{e}\mathrm{n}\mathrm{v},i}$

$=$

$\frac{{\mu }_e}{{\mu }_c}[\frac{d{P}^{*}}{d{r}^{*}}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e},i}.$

Discrete LAWE for Bipolytrope

Summary Set of Nonlinear Governing Relations

In summary, the following three one-dimensional ODEs define the physical relationship between the three dependent variables $\rho$, $P$, and $r$, each of which should be expressible as a function of the two independent (Lagrangian) variables, $t$ and $M_r$:

Equation of Continuity
$\frac{d\rho }{dt}=-4\pi {\rho }^2{r}^2\frac{d}{d{M}_r}\left(\frac{dr}{dt}\right)-\frac{2\rho }{r}\left(\frac{dr}{dt}\right)$
,

Euler + Poisson Equations
$\frac{d^{2}r}{d{t}^2}=-4\pi r^2\frac{dP}{d{M}_r}-\frac{G{M}_r}{r^2}$

Adiabatic Form of the
First Law of Thermodynamics
$\rho \frac{dP}{dt}-{\gamma }_{\mathrm{g}}P\frac{d\rho }{dt}=0.$

March from the Center, Outward

We know the analytic structure of the equilibrium configuration. Let's choose a Lagrangian grid that is labeled by $(r_0{)}_j$ and the corresponding enclosed mass, $m_j(r_0)$, where the center of the spherical bipolytrope is labeled by $j=0$ while each subsequent grid "line" is labeled $j+\frac{1}{2}$. We will identify the mean density of each mass shell by the expression,

${\overline{\rho }}_{j-1/2}$

$=$

$[\frac{\mathrm{\Delta }m}{\mathrm{V}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}}{]}_{j-1/2}=[m_j-m_{j-1}\left]\right[\frac{4\pi }{3}(r_j^3-r_{j-1}^3){]}^{-1}$

See Also