Fritz Zwicky - Dark Matter in the Coma Cluster

This week we will be walking through Fritz Zwicky's 1933 paper "Dark Matter in the Coma Cluster."

Coma Cluster

Source

The Coma Cluster, which is 321 million light years away from Earth, is home to over 1000 identified galaxies (1). In the middle shines two supergiant elliptical galaxies, on of which (NGC 4839) is pictured above. Together with the Leo Cluster, it forms the Coma Supercluster.

In 1933, Zwicky examined the Coma Cluster. Based on the assumption that the cluster has reached a stationary state, he uses the property $K= - \frac{1}{2}U$ from the virial theorem and applies it to the cluster. Yet before he can use this formula, important information must be set up. The cluster has a radius of $10^{24}$ cm with about 800 nebulae, each with an average mass of $10^9$ solar masses. Because the sun's mass is about $2 \times 10^{33}$ g he can find the mass of the cluster:

$$M  \approx M_{\odot} \times 10^9 \times Number_{nebulae}$$ $$M  \approx 2 \times 10^{33} \times 10^9 \times 800 \approx 1.6 \times 10^{45} g$$

Next, he sets up a formula we found this week and relates it to the virial theorem to find veloctity: $$U = -\frac{3GM^2}{5R}$$ $$K= \frac{1}{2}Mv^2= - \frac{1}{2} U =  \frac{3GM^2}{10R}$$ Solving for v gives: $$v = \sqrt{ \frac{3GM}{5R}} = \sqrt{ \frac{3 \times 6.7 \times 10^{-8} \times 1.6 \times 10^{45}}{5 \times 10^{24}}} = 8.0 \times 10^6 \frac{cm}{s}$$

However, there is something off about this calculated velocity, it is very different from the actual velocity. The density of the Coma Cluster would have to be at least 400 times greater than the density calculated based on the matter we can see. This must mean there is something we cannot see adding density to the cluster, which Zwicky names dark matter.

Yet before Zwicky concludes the presence of dark matter, he checks for any sources of error. First, he examines the fact that the Coma Cluster might not be in equilibrium, meaning that instead of the viriral theorem, he would have to use $$U= -K$$ but that only changes the answer by a factor of 2, not 400.

Next, he examines the possibility that the observed velocities are real, meaning the cluster if flying apart. However, there is not enough proof for this as all other data suggests nebulae do not reach those speeds.

Finally, Zwicky corrects for redshift to see if that could be a source of error. However, this only corresponds to a speed of 10 m/sec and is not large enough to account for our answer.

In conclusion, Zwicky discovered the existence of dark matter, one of the most exciting subjects in astronomy today!