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(la) Consider a spherically symmetrical galaxy with mass Mg and radius Rg in the crude approximation that the density p is constant and equal to its average value. Show that the gravitational potential energy of such a galaxy can be approximated [8 marks] 3 GM U 5 RG (1b) Using the virial theorem and treating the galaxy as a collection of identical stars with a characteristic velocity dispersion o, show that the Jeans mass (the critical mass for gravita- tional collapse) can be approximated (12 marks] 3/2 1/2 M) = (3627)"() o 3G Απρ (1c) Show that the corresponding critical radius for gravitational collapse (Jeans length) can be written (4 marks] 502 1/2 R = 47GP (10) The characteristic velocity dispersions, masses, and radii of the smallest dwarf spheroidal galaxies are o ~ 10 km/s, MG ~10 M, and RG200 pc. Find the average density, Jeans mass and Jeans length for these galaxies, based on the approach in this problem. Comment on your results. [6 marks)