USAAAO — 2021 First Round

A stable open cluster of about N = 1000 Sun-like stars has an angular diameter of θ = 60 arcminutes and distance d = 500 pc. Assuming the cluster can be approximated by a sphere of uniform density, estimate the average velocities of stars in the cluster. The gravitational potential energy of a sphere of uniform density and radius r is U_sphere = −(3/5)GM_sphere²/r. [Angular diameter updated from the original exam’s 30 arcminutes so that the keyed answer follows from the stated numbers.]