Addition of grav potentials / Orbital velocity

AI Thread Summary
The discussion focuses on calculating gravitational potentials and orbital velocity by integrating mass contributions from both central stars and dark matter. Participants clarify the need to account for variable density in dark matter when determining mass, leading to the equation M(r) = 4πρ₀r₀³. There is confusion regarding notation, specifically the relationship between r₀ and rₘ. It is emphasized that two mass terms must be considered: one for the stars and another for dark matter. The conversation concludes with a correction on the interpretation of total mass versus just the stars' contributions.
ValarDohaeris
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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/
 

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ValarDohaeris said:

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.
 
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.
 
ValarDohaeris said:
I integrated betwee r0 and 0 and got.

M(r) = 4∏ρ0r30
That doesn't look quite right. What happened to the constant ro 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.
 
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
 
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