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## Homework Statement

Assume a star is in hydrostatic equilibrium and that the density of the star is follows

[tex]\rho \propto \frac{1}{r^{a}}[/tex]

where [tex]\r[/tex] is the distance from the centre of the star and [tex]\r[/tex] is a constant.

Derive an relation for the pressure of the star as a function of [tex]\r[/tex].

## Homework Equations

Hydrostatic Equilibrium:

[tex]

\frac{dP}{dr} = - \rho(r) \frac{G m(r)}{r^2} (1)

[/tex]

Mass Equation:

[tex]

\frac{\mathrm{d} M}{\mathrm{d} r}=\rho (r)4\pi r^{2} dr (2)

[/tex]

## The Attempt at a Solution

I am assuming that the following boundary conditions must be satisfied.

[tex]

\rho(0) = \rho_c \; \; \; \rho(R) = 0

[/tex]

[tex]

m(0) = 0 \; \; \; m(R) = M_{tot}

[/tex]

The problem that I run into is satisfying these equations with the assumption

[tex]\rho \propto \frac{1}{r^{a}}[/tex]

If I assume

[tex] \rho(r) = \rho_c(1-(r/R)^{a}) [/tex]

then the boundary conditions are satisfied, but the power law assumption is not.

Once I get the suitable function for [tex]\rho[/tex], then I can solve Eq. (2) for the Mass as a function of r and then (1) to get a function for the pressure.

But I am unsure of how to satisfy both the power law and boundary conditions for [tex]\rho[/tex].

Any suggestions would be greatly appreciated.