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.

 

[Bernie G]

I think the best density profile (for a gas star and probably a neutron star) and the one worth solving for is ρ(r)=ρc(1−(r/R)a) with a=2. Using this I calculated ρc = 2.5M/V where M/V = average star density. So ρ(r)=2.5M(1−(r/R)2)/V. I think this should be used in the pressure integral. I'm interested in accurately calculating the core pressure for neutron stars and theoretical compact gas stars. I want to show that neutron star core pressure is significantly less than ρ(c)2. Anybody there?

 

[Bernie G]

Rewrite using different format: I think the best density profile (for a gas star and probably a neutron star) and the one worth solving for is ρ = ρc(1−(r/R)^a) where ρc = core density and a = 2. Using this I calculated ρc = 2.5M/V where M/V = average star density. So ρ = 2.5M[1−(r^2/R^2)]/V. I think this should be used in the pressure integral. I'm interested in accurately calculating the core pressure for neutron stars and theoretical compact gas stars. Probably neutron star core pressure at collapse is less than relativistic pressure of (ρc^2)/3. Anybody there?

 

[berkeman]

The OP has graduated. Thread closed.