Rigid body rotation near galactic center

[clandarkfire]

Homework Statement

Equate the expression for centripetal acceleration with the gravitational acceleration to show that the central parts of the galactic rotation curve are consistent with rigid body rotation.

The attempt at a solution
Say a star near the galactic center has mass m and the rest of the mass inside its orbit has mass M. Then:
$$F=ma=\frac{GMm}{r^2}$$
$$a=\frac{v^2}{r}$$
$$\frac{GM}{r^2}=\frac{v^2}{r}\Rightarrow{v^2} \propto{r}$$
But what I want to show is that v is proportional to r, not that v^2 is proportional to r. What have I done wrong?

 

[tiny-tim]

hi clandarkfire!

M is proportional to … ?

 

[clandarkfire]

hi clandarkfire!

M is proportional to … ?

I'm actually not sure. If it's proportional to r (seems reasonable), I get $$G \propto {v^2}$$, which seems pretty nonsensical.
If it's r^2 or r^3, I still don't think I get $$r \propto v$$, which is what I'm looking for.

 

[haruspex]

I'm actually not sure. If it's proportional to r (seems reasonable), I get $$G \propto {v^2}$$, which seems pretty nonsensical.
If it's r^2 or r^3, I still don't think I get $$r \propto v$$, which is what I'm looking for.

Try r^3 again, and if it still fails please post your working.