7. The disc of a galaxy can be modelled as a static infinite slab of incompressible (p = constant) self-gravitating fluid in the region |z| R. Solve Poisson's equation AV el = 4nGp to find the gravitational field inside and outside such a disc. Show that the gravitational force infinitely far away from the disc i1s a non-zero constant, and give a brief physical explanation for why this 1s the case. A star is released from rest at z = R. Show that the star undergoes simple harmonic motion in the galactic disc. If the mean Milky Way density is p = 107 kg/m? and R = 10' m how long does it take a star to cross the Milky Way disk and what 1s its speed at the galactic mid-plane (z = 0)