A few questions about using power laws

[sfbsoccer25]

Have a homework question about power law density distributions that I could use a little help on...

Given a power-law distribution, ρ(R) \propto R^{-\propto}, show that a flat rotation curve can be obtained if \propto = 2 and that solid body rotation is obtained if \propto = 0.

Also, I'm really not sure what this next question is asking for... Any help?

Suppose the rotation curve of the Milky Way is flat out to 2R_{0}. What mass does that imply out to that distance? If all the luminosity of the Milky Way is contained inside 2R_{0} what is the mass-to-light ratio of the Milky Way in solar units? What is the significance of this value?

 

[Mordred]

There seems to be something wrong with the wording of the question. In regards to the last part. I cannot decide if they are looking for the visible mass as opposed to the total mass or the luminosity of the visible mass.
Is this the same wording or your variation if the wording?

 

[sfbsoccer25]

This is the exact wording from the assignment. I believe we are looking for the total mass out to 2R. Then the M/L ratio of the MW if all the luminosity is also in this 2R.

 

[cepheid]

You can show (using a gravitational equivalent of Gauss' law or something like that) that the gravitational acceleration on a test particle at radius R depends only on the total mass enclosed M_enc by that radius. The mass enclosed has a ρ*R³ dependence, of course. By equating centripetal acceleration to gravitational accleration, you get that the speed v, of a particle at radius R should be $v = \sqrt{GM_{enc} / R}$

Of course, if M_enc depends on ρR³, then M_enc / R depends on ρR². If ρ goes like R^-2, then M_enc/R goes like R⁰. In other words, it has no R dependence. It is constant.

For a solid spinning body, how should v depend on R? How does v actually depend on R for alpha = 0?