Astro 160: The Physics of Stars

We know thatwhereN 7 (t) = N 5(t) e λ5t − 1N 8 (t) e λ8t − 1 · N 6(t)N 5 (t) e λ5t − 1N 8 (t) e λ8t − 1 = constant• Given that the current ratio of naturraly occurring U 235 to U 238 is 0.0071, evaluate the gradient ofthe straight line for rock samples of age (a) 1 billion years, (b) 3 billion years and (c) 5 billion years.We know that the gradient of the straight line is just the constant in front of N 6 (t) so we just have to plugin numbers(a). t = 1 billion years.We know thatλ 5 ∼ 9.90 × 10 −10 yr −1 λ 8 ∼ 1.5 × 10 −10 yr −1given these and the fact that we know the ratio between U 235 and U 238 we can find the gradient, for 1bilion years we get0.0071 · eλ 5t − 1e λ 8t − 1 = 0.0715For 3 billion years we get0.0071 · eλ 5t − 1e λ 8t − 1 = .231and finally for 5 billion years we get0.0071 · eλ 5t − 1e λ 8t − 1 = .891Problem # 4 Radiative AtmospheresIn this problem we will solve for the structure of the outer part of a star assuming that energy istransported solely by radiative diffusion (which is not the case in the sun, but is the case in stars moremassive than the sun). The star has a mass M and a luminosity L. Assume that the luminosity and massare approximately constant at the large radii of interest, that gas pressure dominates, and that the opacity isdue to electron scattering. Do not assume that the atmosphere is thin (i.e even though M r ≈ constant = M,because rchanges, the gravitional acceleartion is not constant).Write down the equations for hydrostatic equlibrium and energy transport by radiative diffusion. Usethese to calculate dP rad /dP, the change in radiatio pressure with pressure in the atmosphere. What doesthis result imply for how the ratio of gas pressure to radiation pressure changes as a function of thedistance in the atmosphere? Show that your result for dP rad /dP implies that ρ ∝ T 3 and P ∝ ρ 4/3 forradiative atmospheres (in the language that we will use in the next week, this means that the radiative partof the star is an n=3 polytrope).since we know what the radiation pressure is we can find what the change is with respect to rP rad = 1 3 aT 4dP raddr= 1 3 a d dr (T 4 ) = 4 3 T 4 ∇T6