You can weigh anything with an orbit. Apply that to the stars and gas circling the center of the Milky Way and other galaxies, and the answer is startling: galaxies have far more mass than all their stars and gas.
Why it works
For a star on a circular orbit of radius $r$, gravity from the mass $M$ inside the orbit supplies the centripetal force: $\frac{GM}{r^2} = \frac{v^2}{r}$, so $$M = \frac{v^2 r}{G}.$$ Most of a galaxy's light comes from its center. If the light showed where the mass is, then far out, beyond most of the stars, $M$ would stop growing, and speeds would fall off as $v \propto 1/\sqrt{r}$, like planets around the Sun. Measured rotation curves, pioneered by Vera Rubin and Kent Ford in the 1970s, show instead that speeds stay about the same, far beyond the visible disk. A flat curve ($v$ constant) means $M \propto r$: the enclosed mass keeps growing with distance, in a huge halo of matter we cannot see: dark matter.
Key idea
Stars and gas far out in galaxies orbit too fast for the visible mass. Using $M = v^2r/G$, galaxies hold several times more mass than we see. This dark matter makes up about $85\%$ of all matter; what it is remains unknown.
Worked examples
Worked example: The mass inside the Sun's orbit
The Sun orbits $26{,}000$ light-years ($2.46 \times 10^{20}$ m) from the galactic center at $220$ km/s. How much mass lies inside its orbit?
- $M = \frac{v^2 r}{G} = \frac{(2.2 \times 10^{5})^2 \times 2.46 \times 10^{20}}{6.674 \times 10^{-11}}$.
- $M = 1.8 \times 10^{41}$ kg, which is $\frac{1.8 \times 10^{41}}{1.99 \times 10^{30}} \approx 9 \times 10^{10}$ solar masses.
Answer: About 90 billion solar masses.
Worked example: What the visible mass predicts
Stars $30{,}000$ light-years out orbit at $220$ km/s. If nearly all the mass were inside that distance, how fast would stars at $120{,}000$ light-years move?
- With the same mass inside, $v \propto \frac{1}{\sqrt{r}}$: four times as far means half as fast.
- Predicted: $110$ km/s. Measured speeds that far out stay near $200$ km/s, so the mass keeps growing: dark matter.
Answer: About $110$ km/s, if there were no dark matter.
Common mistake
A reasonable first guess is that dark matter is just ordinary dust, gas, dim stars and planets we have not counted. Those have been searched for: dust glows in infrared, gas in radio and X-rays, and dim stars and black holes would bend the light of stars behind them, which surveys looked for. Above all, the amounts of light elements made in the Big Bang and the pattern in the cosmic microwave background both cap ordinary matter at about one-sixth of all matter. Check: other evidence agrees, from the speeds of galaxies in clusters to gravitational lensing, which maps mass by how it bends light.
What dark matter is made of is an open question. The leading idea is a kind of particle that feels gravity but does not interact with light; experiments are searching for it, so far without a confirmed detection.