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## Homework Statement

Find the total gravitational potential energy stored in a sphere with a 1/r

^{2}density distribution if the total mass is 6.7 solar mass and the radius is 1.3 solar radius. Express you answer in units of 10

^{41}Joules.

## The Attempt at a Solution

To derive the equation, i attempted to follow the same procedure in deriving the gravitational potential energy of a constant sphere, but with also considering the altering value of density throughout.

I started off by figuring out an equation for the mass in a thin shell of mass dM and width dr at radius r:

dM=4πr

^{2}ρ

_{0}dr

Then integrated to obtain an equation of the total mass of material already assembled:

M(r)=(4/3)πr

^{3}ρ

_{0}

After this, i plugged these equations into dU(r)=-((GM(r)dM)/r) and integrated again for an equation for gravitational potential energy:

U=G(16/15)π

^{2}ρ

_{0}

^{2}R

^{5}

The problem is that i can't for the life of me right now figure out how to re-arange this into an equation of the form ((GM

^{2})/R), as i can't simply plug in ρ=((M)/(4/3)πr

^{3}) like i did for the sphere of constant density...

Any help would be greatly appreciated - this has been irritating me for a while now!

Thanks.