How Is the Total Potential Energy of a Plummer Mass Distribution Calculated?

[leonne]

Homework Statement

Now calculate the total potential energy of the Plummer mass distribution. Hint:
The answer should depend only on G, M, and a

Homework Equations

M(r)=Mr³/(r²+a²)^3/2
p(r)=3Ma²/4pie(r²+a²)^5/2

The Attempt at a Solution

looking at my notes i find a formula to find total potential energy
U=-4pieG integral from 0 to R M(r)P(r)r dr
What i am confused about is that it says hint answer only depends on G M a, but what about the R from the integral? it also said in the problem that hint integral from o to infinity x⁴dx/(x²+1)⁴ = pie/32

so looking at the hint integral, instead of doing integral from 0 to R would I just do it from 0 to infinity? That than would get rid of the R and just have answer of just G M a.
Is this right?
thanks

 

[Feldoh]

The total potential energy is from R=0 to R=inf. You need to essentially sum up the potential energy at each distance.

 

[leonne]

o ok thxs

 

[drdoubleu89]

sorry I am a little bit confused so, after setting the integral from 0 to infinty, your answer should come out to be

-3M^2Ga(pie/32)?

 

[leonne]

i don't know i didn't do the integral yet ill let you know later when I do it.

 

[leonne]

yep that is the answer but its a^2 not a ... wow that was easy