# A few questions about using power laws

1. May 15, 2013

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

2. May 15, 2013

### 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?

3. May 17, 2013

### 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.

4. May 17, 2013

### cepheid

Staff Emeritus
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 Menc 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 Menc depends on ρR3, then Menc / R depends on ρR2. If ρ goes like R-2, then Menc/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?