Fritz Zwicky: Dark Matter in the Coma Cluster

Source: http://upload.wikimedia.org/wikipedia/commons/1/1d/Hubble_close-up_on_the_Coma_Cluster.jpg

In 1933, Fritz Zwicky made an observation in the Coma Cluster that defied anything known in astronomy at the time. The Coma Cluster is a cluster of over 1000 observed galaxies. Looking through Zwicky's paper, we see that Zwicky observed the individual galaxies moving away from each other at very fast speeds. However, the gravitational effects of the total mass of the cluster could not account for these large velocities. Therefore, Zwicky concluded that there must be other matter besides the "luminous matter" that was present in the Coma Cluster, which we now refer to as dark matter.

In order to see how Zwicky came to this conclusion, let's walk through the steps that he did. However, we will replace some of his older terms with modern terminology to get a better understanding of how the existence of dark matter was first detected.

First, Zwicky claims that the Coma Cluster "has mechanically reached a stationary state", which implies that the Coma Cluster is in equilibrium, and can therefore be modeled with the Virial Theorem. Zwicky models the Virial Theorem as follows:
\begin{align}
\overline{\varepsilon_k} = -\frac{1}{2} \overline{\varepsilon_p}
\end{align}
Which can be rewritten using modern terminology as:
\begin{align}
K = -\frac{1}{2} U
\end{align}
where $K$ is the total kinetic energy of the system, and $U$ is the total potential energy of the system.

Zwicky then tells us some of the details about the assumptions he made about the Coma Cluster which we can also use to help with the calculations. For example, the Coma Cluster can be assume to have a uniform density, has a radius $R$ of 1 million light years ($1 \times 10^{24}$ cm), and is comprised of "800 individual nebulae each of a mass corresponding to $10^9$ solar masses". Using this information, Zwicky calculated the total mass $M$ of the system (note that a solar mass is $2 \times 10^{33}$ g):

\begin{align}
M = \text{number of nebulae} \times 10^9 \text{solar masses} \times \text{solar mass} = 800 \times 10^9 \times (2 \times 10^{33}) = 1.6 \times 10^{45} \text{ g}
\end{align}

So now that we know the mass of the Coma Cluster, we can figure out how fast the cluster is going. Zwicky did this by finding solving for the equation $\overline{\varepsilon_p} = \frac{\Omega}{M}$ for the average potential energy of the system, and $\overline{\varepsilon_k} = \frac{v^2}{R}$ for the average kinetic energy of the system, where $\Omega$ is the total potential energy of the system, $M$ is the total mass of the system, $v$ is the velocity of the system, and $R$ is the radius of the Coma Cluster. Zwicky used these variables to solve for the velocity $v$ of the system. We can solve for the velocity using the Virial Theorem as follows. It is important to note that in the Virial Theorem, the $U = \Omega = -\frac{3}{5}\frac{GM^2}{R}$, which are both variables representing the total gravitational potential energy of the Coma Cluster:

\begin{align}
K &= -\frac{1}{2} U\\
\frac{1}{2} M v^2 &= -\frac{1}{2} \left( -\frac{3}{5} \frac{GM^2}{R} \right)\\
v^2 &= \frac{3}{5}\frac{GM}{R}\\
v &= \sqrt{\frac{3}{5}\frac{GM}{R}}\\
v &= \sqrt{\frac{3}{5}\frac{(6.67 \times 10^{-8} \frac{\text{cm}^3}{g \cdot s})(1.6 \times 10^{45} \text{ g})}{1 \times 10^{24} \text{ cm}}}\\
v &= 8 \times 10^6 \frac{m}{s}
\end{align}

Interestingly enough, Zwicky shows that based on the observations of the doppler shift, the galaxies were moving away at an average speed of 1000 $\frac{km}{s}$. In order for this to happen, the Coma Cluster must be at least 400 times denser than the calculated mass! This suggests that perhaps the mass calculated for the Coma Cluster was not all of the observed mass. Zwicky argued that perhaps there was other matter present in the Coma Cluster, dark matter, which was present alongside the "luminous matter", and that the mass of the dark matter was not taken into account.

However, the other possibility was that Zwicky assumed to the contrary that the Coma Cluster was at stationary equilibrium, and if that was not the case, the Virial Theorem would not hold. Therefore, if the Coma Cluster was not at stationary equilibrium, Zwicky argued that "the entire available potential energy appears as kinetic energy", as represented:

\begin{align}
K = -U
\end{align}

Even under this constraint, the calculations are only affected by a factor of 2, and the results don't change much from the initial calculation. This suggests that even if the Coma Cluster was not in stationary equilibrium, and that the total kinetic energy of the system was equal to the total potential energy of the system, the observed speeds of the galaxies was still too fast for the cluster to stay together. At the observed speeds, the Coma Cluster would fly apart, unless there was gravitational influence from matter that we did not account for. Again, this suggested that dark matter might exist in the Coma Cluster.

Another proposition that Zwicky made was that, assuming that the mass of the Coma Cluster was actually the mass calculated before, $M$, without any other "dark matter" in the system, and that the observed velocities of an average 1000 km/hr were real, then the Coma Cluster would break down into 800 individual galaxies flying apart from each other.