Collections of stars have many similarities to a plasma of electrons and ions. These similarities arise from the fact that in both cases the interaction between the individual particles (stars, or ions and electrons) is a radial, 1/r² force. The principal difference is that the force between stars is always attractive, so there is no analog of Debye shielding. One consequence of this difference is that a plasma can be spatially homogeneous and static, when one averages over lengthscales large compared to the interparticle separation; but a collection of stars cannot be—the stars congregate into clusters that are held together by the stars’ mutual gravity.

A globular star cluster is an example. A typical globular cluster is a nearly spherical swarm of stars with the following parameters: cluster radius ≡ R = 10 light-years; total number of stars in the cluster ≡ N = 10⁶; and mass of a typical star ≡ m = 0.4 solar masses = 8 × 10³² g. Each star moves on an orbit of the average, “smeared out” gravitational field of the entire cluster. Since that smeared-out gravitational field is independent of time, each star conserves its total energy (kinetic plus gravitational) as it moves. Actually, the total energy is only approximately conserved. Just as in a plasma, gravitational “Coulomb collisions” of the star with other stars produce changes in the star’s energy.

(a) What is the mean time t_E for a typical star in a globular cluster to change its energy substantially? Express your answer, accurate to within a factor of ∼3, in terms of N, R, and m. Evaluate it numerically and compare it with the age of the universe.

(b) The cluster evolves substantially on the timescale t_E. What type of evolution would you expect to occur? What type of stellar energy distribution would you expect to result from this evolution?