Astronomy: Speed at the edge of the galaxy

[razidan]

Homework Statement

The number of stars in the galaxy is N=10^12 and the radius of the galaxy is R_galaxy = 20 kPc
Let m be the average mass of a star in the galaxy.
what is the velocity of a star at the edge of the galaxy (relative to the center of the galaxy)

Homework Equations

The Attempt at a Solution

I had to make 2 assumptions here:
1) uniform distribution of stars, $σ=\frac{Nm}{\pi R^2}$ .
2) the star in question is at a distance of R+$\epsilon$ from the center, and then take the limit of $\epsilon$ goes to 1 light year (or any other distance that is way smaller then R).

In order to solve the question I need to find the force, F, on the star and solve $F=\frac{mv^2}{R}$ for v.

In order to find the force, one has to solve the integral $F=Gmσ \int\limits_0^R\int_0^{2\pi}\frac{rdrd\theta}{(R+\epsilon)^2+r^2-2rRCos(\Theta)}$.
And that is where I'm stuck.
Ideas?

Thanks,
Raz.

 

[phyzguy]

Try looking up the Shell Theorem.

 

[razidan]

Try looking up the Shell Theorem.

Hmm... Haven't thought of that. But what if the distribution is exponential ($e$)? will the shell theorem still hold?
My intuition says that different distributions will result in different speeds.

 

[phyzguy]

The shell theorem always holds as long as the distribution is spherically symmetric.

 

[razidan]

The shell theorem always holds as long as the distribution is spherically symmetric.

Thanks. that simplifies things...
any ideas as to why Mathematica won't solve that integral, though?