1. The problem statement, all variables and given/known data "Imagine a galaxy with mass m at a distance r away from the center of a sphere, within which a total mass M reside. As viewed by an observer in the center, the galaxy appears to be receding according to the Hubble's law, v = H0r. To heuristically derive the critical density m0, we associate a kinetic energy to the galaxy's hubble ow, symbolically,1/2 mv^2, and balance this against its gravitational energy. If the density in the sphere is equal to the critical density, the total binding energy (kinetic plus gravitational) is zero. Show that this yields the following expression for the critical density." 2. Relevant equations What am I doing wrong in my process to find the critical density formula based on the given information? 3. The attempt at a solution My logic is: 1/2(M-m)v^2 = GMm/r So using v = H0r and the volume of a sphere, I plug density(volume of sphere) into M-m, and H0r into v. after doing this, my solution looks like this. density = GMm/2∏r^6H0^2 I can't figure out the next step.