### 5. COMMENTS ON THE VELOCITY DISPERSION IN THE COMA CLUSTER OF NEBULAE

As we have seen in sect. 3, there exist in the Coma cluster apparent differences in velocity of at least 1500 to 2000 km/s. In relation with this enormous velocity dispersion one can make the following considerations.

1. If one assumes that the Coma system has reached a mechanically stationary state, it follows from the Virial theorem

 (4)

where k and p denote the mean kinetic and potential energies, e.g. per unit mass in the system. For the purpose of estimation, we assume that matter is distributed uniformly in the cluster. The cluster has a radius R of approximately one million light years (equal to 1024 cm) and contains 800 individual nebulae each of a mass of 109 solar masses. The total mass M of the system is therefore

 (5)

From this we have for the total potential energy Ω:

 (6)

where Γ = gravitational constant

or

 (7)

and furthermore

 (8)

In order to obtain, as observed, a medium-sized Doppler effect of 1000 km/s or more, the average density in the Coma system would have to be at least 400 times greater than that derived on the basis of observations of luminous matter [This would be in approximate accordance with the opinion of Einstein and de Sitter as discussed in Sect. 4.]. If this should be verified, it would lead to the surprising result that dark matter exists in much greater density than luminous matter.

2. One may also assume that the Coma system is not in stationary equilibrium, but that the entire available potential energy appears as kinetic energy. We would then have

 (9)

One may thus save only a factor of 2 compared to the assumption 1, and the need for an enormous density of dark matter remains.

3. Let the average density in the Coma cluster be determined purely by luminous matter (M as mentioned above). Then the large speeds cannot be explained on the basis of considerations of type 1 or 2 above. If the observed speeds are real anyway, the Coma system should fly apart in the course of time. The end result of this expansion would be 800 single nebulae (field nebulae), which would have proper speeds, as shown in 2., of the order of the original ones (1000 to 2000 km/s). In analogy, one would have to expect that single nebulae with such large proper speeds can also be observed in the present evolutionary state of the Universe. This conclusion hardly matches the experimental facts, as the spread of proper speeds of individually occurring nebulae does not exceed 200 km/s.

4. One may also attempt to consider the speeds as apparent ones, interpreting them as caused by Einstein's redshift. Assuming the above mass M one would have for the relative change of the wavelength

 (10)

which is equivalent to a speed of only 10 m/s. Thus, in order to arrive to an explanation of the large velocity dispersion, one would have to permit a much greater density of dark matter than under assupmtions 1 or 2.

These considerations indicate that the large velocity dispersion in the Coma system (and other dense clusters of nebulae) holds an unsolved problem.