Galaxy Rotation Curve problem -

[tetris11]

Hi folks,

I'm supposed to derive the func. form of the rotation curve for the outer parts of our galaxy, in the absence of dark matter.

Im assuming that I treat this as a linear curve, since in reality, dark matter flattens out the curve, when it should continue following the linear(?)

What I've done:

1) v= rw v = Vel, r = distance from centre w = angular freq. (d(theta)/dt)
2) v²/r = GM(r)/r²

What do I do now? If I subs. 1 into 2, I get stuck with w and M(r).

How am I supposed to treat M(r): (M(r) = [(4/3)*Pi*r³]*p)
Do I just introduce p (density) into the eqn. and assume p is constant?

 

[NateD]

Thanks for your help!Yes, you can assume that the density is constant. Then you can solve for w in terms of the density, and then substitute it back into the equation. Since the density is constant, the angular frequency w should be proportional to the radius r. So the rotation curve should have the form v=kr, where k is a constant.