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 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?

 

[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?

 

[pmacias]

Power laws are a commonly used mathematical model in various scientific fields, including astronomy and physics. In the context of density distributions, a power law can be used to describe the relationship between the density of a certain object and its size or distance.

To address your first question, a power law distribution with \propto = 2 means that the density decreases as the radius (R) increases, following the relationship ρ(R) \propto R^{-2}. This type of distribution is often seen in systems with flat rotation curves, such as spiral galaxies. This is because the mass distribution in these galaxies is dominated by dark matter, which has a constant density and thus produces a flat rotation curve. Therefore, a power law with \propto = 2 can be used to explain the flat rotation curve observed in these galaxies.

On the other hand, a power law with \propto = 0 represents a solid body rotation, where the density remains constant throughout the object. This type of distribution is often seen in systems with a high concentration of mass, such as a planet or a star. In this case, the rotation curve would increase with distance, reaching a maximum at the edge of the object.

Moving on to your second question, let's consider the rotation curve of the Milky Way. If it is flat out to 2R_{0}, this means that the mass distribution in the Milky Way is dominated by dark matter out to that distance. Using the power law model, we can estimate the mass contained within this distance by integrating ρ(R) over the range 0 to 2R_{0}. This would give us the total mass of the Milky Way within 2R_{0}. Similarly, if we know the luminosity of the Milky Way within 2R_{0}, we can calculate the mass-to-light ratio by dividing the mass by the luminosity.

The significance of this value lies in its ability to provide insight into the composition of the Milky Way. A high mass-to-light ratio would suggest a larger amount of dark matter, while a lower ratio would indicate a higher proportion of luminous matter. This can help us understand the overall structure and evolution of our galaxy and other galaxies in the universe.

In summary, power laws are a useful tool for describing density distributions and can provide valuable information about the mass and structure of objects in our universe. I hope this helps with your homework question. If you need further clarification or assistance, please don't hesitate to ask