## Clusters of galaxies and the virial theorem

The only objects with substantially greater masses than galaxies, other than the universe itself, are collections of galaxies, known as clusters of galaxies. The nearby (~55 MLY) Virgo cluster contains about 2500 galaxies. The galaxies move about under the influence of all the other galaxies in the cluster; think of a swarm of bees. The galaxy motions can be used to estimate the mass of the cluster.

For this purpose, astronomers make use of the virial theorem which applies to a gravitationally bound system of discrete objects ("particles") in stable equilibrium. In our case, galaxies are the particles. The theorem states that twice the total kinetic energy of all the particles plus the sum of the potential energies equals zero, where the summations are over the individual particles and rtj is the separation distance between mt and mj. Note that V = 0 when the particles are widely dispersed, rtj ^ to, the usual convention.

The virial theorem 2K + V = 0 can only be satisfied if the potential energy is negative and twice the value of K, since K is always positive. Particles in free fall from a great distance toward the barycenter would have K + V = 0 from energy conservation and would not satisfy the virial theorem. In contrast, particles swarming around the barycenter in a stable (unchanging configuration) do obey the

virial theorem. In this case, the total energy E = K + V must be less than zero, an indication that such a system is gravitationally bound.

The orbit of a satellite around the earth, or of a planet around the sun, is a simple example of the virial theorem. From Newton's second law, one readily determines that the kinetic energy of the satellite is one-half the negative of the potential energy.

Measurements of the particle separations rtj and speeds vt, in particular the averages of vt2 and r,j-1, allow one to find a mass for the entire cluster. The mass determined is that required to provide the gravity necessary to keep particles of speed vt in a cluster with galaxy separations r,j. At a given typical speed, a more compact cluster (small r,j) indicates more gravity is present. The mass so determined is called the virial mass.

Typically, one finds that the virial mass of a cluster is far greater than the visible mass. In other words, it is greater than the total mass of the visible galaxies, even including estimates of their dark matter. The virial mass in clusters is typically 10-50 times greater than the visible mass. Again there is an apparent prevalence of dark matter, but this time on an even larger scale.

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