Blog #7, Worksheet #3.1, Problem 4, Flat Rotational Curve

4. 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 3 we solved for \(v(r)\) in terms of M, r and g: \[v(r) = (\frac{GM}{r})^{\frac{1}{2}}\]All we then need to do is reorder the equation and to solve for M. \[M_{enc} = \frac{r{V_c}^2}{G}\]

(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? 

In our drawing of the galaxy, a majority of the mass of the galaxy was centered in the bulge of the galaxy, within a small portion of the entire radius of the Milky Way. However, in the flat rotation model that we just found, the mass of the galaxy is directly proportional to the distance away from the center of the galaxy, (given that Vc and G are constants). We know this is not true because a majority of the mass is concentrated at the center of the galaxy like we saw in the picture. To explain how the constant velocity is attained, we need some way to make up for this large mass that is seemingly existent as you move further and further from the center of the galaxy. This is the main proposal for why Dark Matter must exist.

(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.

We found that \[M_{enc} = \frac{r{V_c}^2}{G}\] Plugging in the values we found as \[V_c = 240 \frac{km}{s} \qquad r = 100 kpc = 3 x 10^{18} km \qquad G = 6.67 x 10^11 \frac{m^3}{kg \cdot s^2} \]

Thus \[M_{enc} \thickapprox 10^{40} kg \]However the observed mass of the galaxy is \[10^{42} kg\]We can only see about 1% of the mass of the galaxy. The other 99% of the mass is hypothesized as being made up of "Dark Matter."