How Is Gravitational Pressure Calculated in a Contracting Gaseous Sphere?

[Himanshu]

Homework Statement

Consider a cloud of gaseous hydrogen contracting under gravity to form a star. The cloud is assumed to be spherical of volume $$\Omega$$ and mass $$M$$ distributed uniformly.

Show that the gravitational presure of the cloud is

P($$\Omega$$)=-(1/5)(4pi/3)^(1/3)GM^2/$$\Omega$$^2

Homework Equations

Gravitational Potential Energy of a sphere is given by -(3/5)GM^2/R.

This means that a Gravitational force is acting on the particles of the sphere to keep the sphere intact. This force is given by

F=-grad(P.E.)
F=-(3/5)GM^2/R^2
This force is acting on the entire surfacr of the sphere whose area is A= 4pi R^2
Therefore gravitational pressure is F/A

P($$\Omega$$)=-[(3/5)GM^2/R^2]/4pi R^2

on getting rid of R I get the required result.

The Attempt at a Solution

Where's the problem. Well this was a delebrate derivation. First of all I don't clearly understand what does gravitational pressure mean(ie. is it calculated form F/A or there are other ways of calculating it). Second, the area that I have considered is the surface area of the sphere. Well the Gravitational force is acting in the inside of the sphere also. Moreover gravitational pressure should be a function of r(the radial distance from the core) instead of R as common sense tells that gravitational pressure should be maximum at the core and minimum at the surface of the sphere.

Please help me remove my doubts.

There further parts of the question which I wish to ask later..

Thanks.

 

[farmerwa]

I can resolve the second problem. When you calculate the potential energy of the sphere, the final integral that you do does not have to be performed to the surface of the sphere. You can, instead, integrate to an arbitrary position, r, inside the sphere. This will, in turn, give the energy due to the interaction of all of the particles inside the sphere with all of the particles in the gravitational object. You have now derived the potential energy as a function of radius from the center of the star.

Now, when you take the gradient of the potential energy, you are in fact finding the force on a spherical shell at distance r from the origin.

If you do not like this derivation, consider a spherical shell of thickness dr inside the star. Get the force on the interior of the shell and on the exterior of the shell, subtract the two, and you should be left with the net force on the shell. From here, divide by the area of the shell, and you should be left with the pressure. I have not done the derivation myself, but it is a standard method.