Rotation Curves: Blog 6, Worksheet 3.1, Problem 4

This week we went further into our exploration of the Milky Way, but something didn't quite add up. The mass of observable matter in the galaxy, does not actually match the mass of the galaxy. This is the basic idea that lead to the astronomical mystery of dark matter. For this weeks three blog posts, I will type up questions 4-6 on worksheet 3.1, because they flow well together and build evidence that the mass of the galaxy is not quite what we think. The first step of this is measuring the rotation curve of the galaxy:

We actually observe a flat rotation curve in our own Milky Way. (You will show this with a radio telescope in your second lab!) This means v(r) is nearly constant for a large range of distances. 

(a) Lets call this constant rotational velocity Vc. If the mass distribution of the Milky Way is spherically symmetric, what must be the M(< r) as a function of r in this case, in terms of Vc, r, and G? 


In question number three we found a formula for the Keplerian rotation curve, v(r) for a mass m if the central mass is concentrated in a single point at the center (with mass Menc). The formula we found for that problem ended up being \[ v^2 = \frac{G M_{enc} }{r} \] Rearranging this formula for this specific question leads us to: \[M(< r) = \frac{ V_c^2 r }{G} \]

(b) How does this compare with the picture of the galaxy you drew last week with most of the mass appearing to be in bulge? 

This approximation assumes the mass of the galaxy is constant as you go out from the center. We know that is not true because there is a quite heavy bulge in the middle of the milky way.

Source

(c) If the Milky Way rotation curve is observed to be flat (Vc < 240 km/s) out to 100 kpc, what is the total mass enclosed within 100 kpc? How does this compare with the mass in stars? Recall the total mass of stars in the Milky Way, a number you have been given in your first assignment and should commit to memory.

In our first assignment we learned that the mass of the Milky Way is \(10^{10} M_{\odot} \). Lets see how our calculations compare: \[ M_{MW} = 10^{10} M_{\odot} = 10^{10} \times 2 \times 10^{33} = 2 \times 10^{43} \: grams \] \[M(< 100 \: kpc) =  \frac{ V_c^2 r }{G} = \frac{ (240)^2 (2 \times 10^{43}) }{6.67 \times 10^{-8}} = 8.6 \times 10^{54} \: grams \]
That is a huge discrepancy between the two masses. Something suspicious is afoot...