Question

In the context of the virial theorem applied to a galaxy cluster, if the velocity dispersion of the member galaxies is measured to be 1000 km/s, what does this value reveal about the cluster&#x27;s total gravitational mass?

Accepted Answer

The virial theorem states that for a stable, gravitationally bound system like a galaxy cluster, the total kinetic energy is equal to half the magnitude of the total gravitational potential energy. This relationship allows astronomers to calculate the total mass of the cluster using the observed velocity dispersion, which is the statistical spread of the velocities of the individual galaxies moving within the cluster. A velocity dispersion of 1000 km/s indicates a very high kinetic energy because kinetic energy is proportional to the square of the velocity. According to the virial theorem, because the cluster is gravitationally bound, this high kinetic energy must be balanced by an equally large gravitational potential energy. Since potential energy is determined by the total mass and the spatial distribution of the galaxies, a high velocity dispersion reveals that the cluster must have a massive amount of gravitational pull to prevent the galaxies from flying away. When calculating this mass, astronomers often find that the total mass derived from the 1000 km/s dispersion is significantly higher than the mass visible in the form of stars and gas. This discrepancy is the primary evidence for the existence of dark matter, an invisible form of matter that provides the additional gravity required to keep the fast-moving galaxies contained within the cluster.

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In the context of the virial theorem applied to a galaxy cluster, if the velocity dispersion of the member galaxies is measured to be 1000 km/s, what does this value reveal about the cluster's total gravitational mass?