Question

Consider a uniform density sphere of mass M and radius R in hydrostatic equilibrium with zero surface pressure. Derive expressions for the pressure P(r) and the gravitational potential phi(r) in terms of r, M, R, G and constants.

Answer 1

Let the uniform density be .

Consider a thin shell of thickness dr at a distance r from the centre of the star. Mass of the shell dm will be,

Then the Gravitational force on the shell will be due to the interior mass of the star, Therfore we can write the gravitational force as.

The mass of the interior portion of the star can be written as,

Substituting we get,

This inward force is balanced by the outward pressure of the gas. The outward pressure will be

P(r) - P(r+dr) (Since P(r+dr) < P(r), The surface pressure is 0) (Here the sign followed is inward force is +ve, as in the case for gravitational force and outward force would be -ve)

Therfore we can write,

Now, we have to integrate both sides from R to distance r (since pressure at surface is 0), so as to find P(r)

Substituting value of density we can write,

Here, R is the radius of the star.

Gravitational potential is defined as work done per unit mass of the particle to bring a particle of mass m from infinity to the point at a distance of r, it can be evaluated by,

The gravitational potential for distance r>R is given by,