You will need: the gravitational energy Egrav theorem Egrav = -3 == PdV, the relation P : = -GM/R, the virial Pgas + Prad (where Pgas is assumed to be ideal, not degenerate), and the internal energy densities Ugas Pgas and urad == = 3 Prad In the absence of any nuclear energy production, dE/dt = -L, and the star will contract. If the star was much larger at the start of contraction (t = 0), then its initial energy can be ignored, so E = == SL dt. (a) Show that if L ty with y < 1, then the time to contract from infinite radius to radius R is where tH = t(R) = (1 y)tH |E|/L is the Kelvin-Helmholtz time scale. (b) Show that, if R t, then z = 1 y.