# Internal Pressure of a Star

dats13

## Homework Statement

Assume a star is in hydrostatic equilibrium and that the density of the star is follows
$$\rho \propto \frac{1}{r^{a}}$$

where $$\r$$ is the distance from the centre of the star and $$\r$$ is a constant.

Derive an relation for the pressure of the star as a function of $$\r$$.

## Homework Equations

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

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

## The Attempt at a Solution

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

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

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

$$\rho \propto \frac{1}{r^{a}}$$

If I assume
$$\rho(r) = \rho_c(1-(r/R)^{a})$$

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

Once I get the suitable function for $$\rho$$, 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 $$\rho$$.

Any suggestions would be greatly appreciated.