How do you weigh a galaxy? The same way you weigh the Sun: watch something orbit it. In a spiral galaxy, stars and gas clouds circle the center, and Doppler shifts give their speeds at every radius. A plot of orbital speed against radius is a rotation curve. In Dark Matter you saw what it revealed; here you will do the calculation, using the shell theorem.
💡 Why it works
For a roughly spherical distribution of mass, a body orbiting at radius $r$ feels only the mass $M(r)$ inside its orbit, as if it were at the center. A circular orbit needs $v^2/r = GM(r)/r^2$, so $$v(r) = \sqrt{\frac{GM(r)}{r}}, \qquad M(r) = \frac{v^2r}{G}.$$ If nearly all of the mass sat in the bright center, $M(r)$ would stop growing beyond it and orbital speeds would fall as $1/\sqrt{r}$, like the planets. Instead, from the 1970s Vera Rubin and Kent Ford and later radio astronomers found that the curves of spiral galaxies stay flat far beyond the visible disk.
🔑 Key idea
A flat rotation curve ($v$ constant) means $M(r) = v^2r/G$ keeps growing in proportion to $r$, out where there are few stars and little gas. Most of a galaxy's mass is invisible: dark matter, in a halo with density falling roughly as $1/r^2$.
Orbital speed against distance from the center. The dashed curve is what the visible stars and gas alone would produce; the measured curve stays flat.
Worked examples
✏️ Worked example: Weighing the Milky Way inside the Sun's orbit
The Sun orbits the galactic center at about 230 km/s, 8.2 kpc out. How much mass lies inside the Sun's orbit, and how long is one orbit? ($1\text{ kpc} = 3.09 \times 10^{19}$ m.)
$\rho \approx 9 \times 10^{-22}$ kg/m³, or about $0.014\,M_\odot$ per cubic parsec.
Answer: A thin haze: roughly half a proton's mass per cubic centimeter, but spread through an enormous volume.
⚠️ Common mistake
Missing why a flat curve is a surprise. With visible matter alone, the mass would be nearly all inside the bright inner disk, and stars and gas farther out should orbit more slowly, $v \propto 1/\sqrt{r}$, as the outer planets do. A flat curve means the enclosed mass keeps growing with radius even where we see almost nothing. Check with numbers: $M(r) = v^2r/G$ doubles from 10 kpc to 20 kpc when $v$ stays the same.
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 2 XP.