Agalaxy cluster contains n = 1000 galaxies within its half-mass radius of rh = 1.5 mpc. to the zeroth order, these galaxies are moving in random directions not unlike ideal gas particles. the three-dimensional mean square velocity of the cluster is σ = ^1/2 = 1000 km s^−1 . (a) what is the mean number density of galaxies within the half-mass radius? (b) suppose that the typical cross-sectional area of a galaxy is a = 10^−3 mpc^2 . how far will a galaxy in the cluster travel, on average, before it collides with another galaxy? (c) what is the average time between collisions for a galaxy in the cluster? how does this time compare with the hubble time?