# Astronomy: Speed at the edge of the galaxy

## Homework Statement

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)

## 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.

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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