utesfan100

2012-Aug-30, 05:53 AM

The mass distribution of the Milky Way is given in http://arxiv.org/pdf/astro-ph/9612059v2.pdf.

As a simplification, consider that 1/4 of the mass of the Milky Way is in a spherically symmetric central region with a radius of 10,000 LY, and 3/4 of the mass is in a uniform disk-like region with a radius of 40,000 LY. A factor or 17/16 by making the disk occupy the entire region is not significant to this order of magnitude estimate of force.

For R<<10,000 most of the mass is due to the central mass. Only M=\frac{1}{4}\left(\frac{r}{10,000}\right)^3 impacts the dynamic. This gives us \frac{GM}{r^2}=\frac{G r}{4\times10^{12}}=\frac{v^2}{r}, using the equation of circular motion. This gives us that velocity is proportional to radius within the spherically symmetric mass, which is observed in galactic rotation curves.

For R>10,000, the internal mass is M=\frac{1}{4}+\frac{3}{4}\left(\frac{r}{40,000}\ri ght)^2.

This gives us \frac{G}{4r^2}+\frac{3G}{64\times10^8}=v^2/r. Finally we get the velocity distribution v^2=\frac{G}{4r}+\frac{3Gr}{64\times10^8}

Taking the derivative relative to r gives us 2V\frac{dV}{dr}=\frac{-G}{4r^2}+\frac{3G}{64\times10^8}, which is 0 at roughly 20,000 LY. Thus for the region 20,000<r<40,000 we should expect the orbital velocity to be increasing as we move further out. This produces a profile very similar to that shown at wikipedia.

http://en.wikipedia.org/wiki/Milky_Way#Galactic_rotation

How to galactic orbital velocities differ from this classical model?

As a simplification, consider that 1/4 of the mass of the Milky Way is in a spherically symmetric central region with a radius of 10,000 LY, and 3/4 of the mass is in a uniform disk-like region with a radius of 40,000 LY. A factor or 17/16 by making the disk occupy the entire region is not significant to this order of magnitude estimate of force.

For R<<10,000 most of the mass is due to the central mass. Only M=\frac{1}{4}\left(\frac{r}{10,000}\right)^3 impacts the dynamic. This gives us \frac{GM}{r^2}=\frac{G r}{4\times10^{12}}=\frac{v^2}{r}, using the equation of circular motion. This gives us that velocity is proportional to radius within the spherically symmetric mass, which is observed in galactic rotation curves.

For R>10,000, the internal mass is M=\frac{1}{4}+\frac{3}{4}\left(\frac{r}{40,000}\ri ght)^2.

This gives us \frac{G}{4r^2}+\frac{3G}{64\times10^8}=v^2/r. Finally we get the velocity distribution v^2=\frac{G}{4r}+\frac{3Gr}{64\times10^8}

Taking the derivative relative to r gives us 2V\frac{dV}{dr}=\frac{-G}{4r^2}+\frac{3G}{64\times10^8}, which is 0 at roughly 20,000 LY. Thus for the region 20,000<r<40,000 we should expect the orbital velocity to be increasing as we move further out. This produces a profile very similar to that shown at wikipedia.

http://en.wikipedia.org/wiki/Milky_Way#Galactic_rotation

How to galactic orbital velocities differ from this classical model?