Find the total gravitational potential energy stored in a sphere with a 1/r2 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 1041 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:
Then integrated to obtain an equation of the total mass of material already assembled:
After this, i plugged these equations into dU(r)=-((GM(r)dM)/r) and integrated again for an equation for gravitational potential energy:
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 ((GM2)/R), as i can't simply plug in ρ=((M)/(4/3)πr3) like i did for the sphere of constant density...
Any help would be greatly appreciated - this has been irritating me for a while now!