3. Assume a spherical cloud of gas, in the form of molecular hydrogen. The cloud has mass M, radius R, uniform density po and temperature T. (i) Show that the potential energy of the cloud is 3 GM² 5 R (ii) The cloud is stable if the virial theorem holds. Show that the cloud will contract if 3MkBT 3 GM² MH₂ < 5 R where you may use the fact that the kinetic energy of a particle of tem- perature T is (3/2)kBT. Show that this corresponds to a lower limit to the mass of the cloud: 3/2 1/2 5kBT 3 Mj GMH2 Απρο Here, MJ is called the Jeans mass, and it represents the smallest mass of the cloud which can initiate star formation. Note that it depends only on the cloud's temperature and density. (iii) Calculate the Jeans mass for a dense cloud of T = 10K, and number density of molecules nH₂ = 10¹0 m³.