Addition of grav potentials / Orbital velocity

[ValarDohaeris]

The Attempt at a Solution

I think I may of been too general with the volume in finding $$M_0$$
I'm assuming $$r_0 = r_M$$ although I'm not sure why different notation is used thougj

http://imageshack.us/photo/my-images/3/gravscan.jpg/

 

[gneill]

The Attempt at a Solution

I think I may of been too general with the volume in finding $$M_0$$
I'm assuming $$r_0 = r_M$$ although I'm not sure why different notation is used thougj

http://imageshack.us/photo/my-images/3/gravscan.jpg/

Yup. You need to integrate to find the mass due to the dark matter since its density is not constant with r.

 

[ValarDohaeris]

I integrated betwee r₀ and 0 and got.

M(r) = 4∏ρ₀r³₀

v² = GM/r
Not sure where to go from here to get the second term.

 

[gneill]

I integrated betwee r₀ and 0 and got.

M(r) = 4∏ρ₀r³₀

That doesn't look quite right. What happened to the constant r_o from the density function?

v² = GM/r
Not sure where to go from here to get the second term.

You should have two mass terms to add together. The first one is the trivial M for the central stars, while the second is due to the mass of the dark matter.

 

[ValarDohaeris]

Ah cheers got it now, misread the question was thinking M was the total mass of sphere rather than just the stars contributions.. $$M(r)_{dm} = 4πρ_0r^2_0r$$