Astronomy: Speed at the edge of the galaxy

1. Apr 7, 2015

razidan

1. The problem statement, all variables and given/known data
The number of stars in the galaxy is N=10^12 and the radius of the galaxy is Rgalaxy = 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)

2. Relevant equations

3. 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.

2. Apr 7, 2015

3. Apr 7, 2015

razidan

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.

4. Apr 7, 2015

phyzguy

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

5. Apr 7, 2015

razidan

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