Dark Matter in the Coma Cluster

http://gregg.physics.ucdavis.edu/gregg/coma/coma.html

The Coma Cluster represents the nearest developing cluster of galaxies to earth, at a distance of approximately 100 Mpc. Although it was originally considered to be in virial equilibrium, the Coma Cluster is actually highly dynamic. Astronomers have identified more than 1,000 galaxies within the Coma Cluster! (http://gregg.physics.ucdavis.edu/gregg/coma/coma.html)

From Zwicky's publication, we are given several bits of information about the cluster. This is outlined below:

• Differences in speed of 1500 to 2000 km/sec
• \(R_{cluster} = 1 \, million \, light \, years = 10^{24} \, cm \)
• Contains 800 individual Nebulae
• Each nebulae has a mass of \(M_{nebulae} = 10^9 \, solar \, masses\)

Although Zwicky calculated out the average velocities of the stars, in his paper, we are going to reproduce his results using our expert method!

First, we'll sketch out the system: 

In our sketch, each of the nebulae is represented by a star in motion. To get started, we'll begin by assuming the system is in a state of equilibrium, and by applying the virial theorem. \[ K = - \frac{1}{2}U\] Now let's expand the expressions for both K and U: \[K = \frac{1}{2}M \bar{v}^2 \] \[U = - \frac{GM^2}{R}\] Inserting these into the virial theorem equation, we get: \[ \frac{1}{2}M \bar{v}^2 = \frac{1}{2} \frac{GM^2}{R} \] Solving for the average velocity, we get: \[ \bar{v} = \sqrt{\frac{GM}{R}} \] Now, we can apply the values that Zwicky has provided for the Coma Cluster to calculate this average velocity! \[ \bar{v} = \sqrt{\frac{(6.67 \times 10^{-8}) \times (800 \times 10^9) \times (2 \times 10^{33}) }{10^{24}}} \approx 3 \times 10^6 \, \frac{cm}{sec} \approx 30 \, \frac{km}{sec} \]

On the surface, this calculation is a bit concerning. According to Zwicky, observations have revealed differences in speed between 1500 to 2000 km/sec. Even if we adjusted our assumption that this system is in equilibrium, the change would only change by a relatively small factor. Our calculation, however, is off my several orders of magnitude.

Quite remarkably, this disparity is where Zwicky made an amazing discovery. All of our calculations are based solely on visible mass, but clearly, there is something causing the velocities to be much greater than expected. Zwicky credits this difference to some additional mass that is not visible. This appropriately named matter has been dubbed "Dark Matter."

Another hypothesis tested by Zwicky from these calculations is how the apparent velocities of the Coma Cluster would be attributed to the Einstein redshift. This is given by the relative change in wavelength shown below (considering just visible mass): \[ \frac{ \Delta \lambda } { \lambda} \sim - \frac{e_p}{c^2} \sim 3.5 \times 10^{-8} \] This translates to a speed of ~ 10 m/s. Again, for the immense different in velocity to be accounted for, there would need to be an immense amount of dark matter for the system to hold!