Blog Post 7, Worksheet 3.1, Problem 4

4. We actually observe a flat rotation curve in our own Milky Way. This means v(r) is nearly constant for a large range of distances. 

(a) Let's call this constant rotational velocity \( V_c\). 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 \(V_c\), r, and G?

From Problem 3, we know that relationship between velocity, mass enclosed, and distance from the center is expressed by the equation: \[v(r) = (\frac{GM_{\text{enc}}}{r})^{1/2}\] Now, we set this equation for velocity equation to a constant rotational velocity \(V_c\): \[v_C = (\frac{GM_{\text{enc}}}{r})^{1/2}\] From here, we solve for \(M_{\text{enc}}\), as we are searching for the M(<r) as a function of r: \[V_C = (\frac{GM_{\text{enc}}}{r})^{1/2}\] \[ \frac{GM_{\text{enc}}}{r} = V_c^2\] \[M(<r) = \frac{V_c^2r}{G}\] Thus, M(<r) as a function of r in therms of \(V_c\), r, and G is: \[M_{\text{enc}} = \frac{V_c^2r}{G}\]
(b) How does this compare with the picture of the galaxy you drew last week with most of the mass appearing in the bulge?

Unlike the picture we drew last week, this equation shows that the mass enclosed in the galaxy increases proportionally to the distance from the center of the galaxy - that is, when the distance increases, the mass increases. In the picture from last week, this did not appear to be true, as most of the mass appeared to be located in the center of the galaxy.

(c) If the Milky Way rotation curve is observed to be float (\(V_c \approx\) 250 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 that the total mass of stars in the Milky Way, a number you have been given in your first assignment and should commit to memory. 

Using the given values for \(V_c\) (240 km/s) and r (100 kpc), we can plug these values into the equation for \(M_\text{enc}\): \[M_{\text{enc}} = \frac{V_c^2r}{G}\] \[M_{\text{enc}} = \frac{(240 \text{ km/s})^2(100 kpc)}{G}\] \[M_{\text{enc}} = \frac{((240 \text{ km/s})(10^5 \text{ cm/km}))^2(100 \text{ kpc} (\frac{3 * 10^23 \text{ cm}}{100 \text{ kpc}})))}{7 * 10^{-8} \text{ cm}^3 \text{ s}^{-1} \text{ g}^{-1}}\] \[M_\text{ enc} = 2.6 * 10^{45} \text{ g  } (\frac{1 M_\odot}{2 * 10^{33} \text{ g}})\] \[M_\text{ enc} = 1.3 * 10^{12} M_\odot\]
If we compare this value to the stellar mass of the galaxy, \(10^{10} M_\odot\), we see that the stellar mass makes up only around 1% of the matter that should be in our galaxy: \[\frac{10^{10} M_\odot}{1.3 * 10^{12} M_\odot} * 100 \%≈ 1\% \]