Evidence for dark matter from galaxy motions

Evidence for the existence of dark matter in galaxy clusters dates from the 1930s. The Swiss astronomer Fritz Zwicky used the velocity dispersion of galaxies in clusters as determined from Doppler shifts to estimate their dynamical mass. Suppose all galaxies have the same mass $m$, then the kinetic energy of the system of $N$ galaxies is

$$K={1\over 2}\sum m\,v^2={1\over 2}\,M\,\sigma^2\,,$$ (9.1)

where $M=N\,m$ is the total mass of the cluster galaxies, and the velocity dispersion $\sigma^2$ is defined as

$$\sigma^2 = {\sum m\,v^2\over M}\,.$$ (9.2)

The potential energy in the system is of order

$$U=-{G\,M^2\over R}\,,$$ (9.3)

where $R$ is a measure of the size of the system. When the system is in virial equilibrium, $2K=\vert U\vert$ hence

$$M={R\,\sigma^2\over G}\,.$$ (9.4)

Note that the measured rms velocity is just the line-of-sight one, say $\sigma_z^2=\langle(v_z-\langle v_z\rangle)^2\rangle$, but if the velocities are isotropic, then $\sigma^2=3\sigma_z^2$. So from that, and from an estimate of $R$, one can estimate the dynamical mass $M$.

Zwicky estimated the masses of the galaxies, and compared the result with the dynamical mass. Not only was the dynamical mass much larger, the velocities of the cluster galaxies was so high that, given the amount of mass inferred to be in the galaxies, the cluster was not even a gravitationally bound system. Left on its own, it would cease to be a high density region of galaxies on a short time scale because the galaxies would fly apart very rapidly. So either clusters were just chance superpositions of galaxies, or there was much more mass in them than was inferred from just the galaxies. Zwicky concluded that most of the mass in clusters must be invisible.