On the estimation of the total kinetic energy of clusters of stars or galaxies

Abstract

The kinetic energy of some model clusters was estimated by simulated observation of the line-of-sight velocities of the members, and this estimate was then compared with the actual kinetic energy which was known for each model. The model clusters were all self-consistent and had ellipsoidal velocity distributions of the type first discussed by Eddington some sixty years ago and give a reasonable run of density with distance from the centre of the cluster. Estimates of the total kinetic energy varied with the part (as seen in the plane of the sky) of the cluster sampled, since velocity distributions of this sort have a preponderance of velocities along a radius of the cluster at great distances from the centre. Overestimates of the total kinetic energy—even by as much as a factor of 2—were found to be rare; and, in practice, a factor of 1.5 would be a fair summary of the more extreme results obtained from samples near the apparent centre. These results were found to be fairly insensitive to the degree of anisotropy in the velocity distribution. Since the mass of a cluster as determined from the virial theorem is directly proportional to the estimated kinetic energy, these same factors represent the systematic overestimation of the total mass of the cluster due to a lack of isotropy in the cluster's velocity field.