Problem 3.5 Use Equation 3.20 to show that, if the density rho in a spherical galaxy is constant, then a star following a circular orbit moves so that its angular speed Ω(r)=V(r)/r is constant. Show that a star moving on a radial orbit, i.e., in a straight line through the center, would oscillate harmonically in radius with period
P=√3π/Gp ~ 3tff, where tff=√1/Gp.
The free-fall time tff is roughly the time that a gas cloud of density rho would take to collapse under its own gravity, if it is not held up by pressure. Show that, if you bored a hole through the center of the Earth to the other side, and dropped an egg down it, then (ignoring air resistance, outflows of molten lava, etc.) you could return about an hour and a half later to retrieve the egg as it returned to its starting point.