2 Black Hole Sun [30 points] (a) [12 points] Suppose the temperature of all of the gas in the Sun were to cool instantly to zero degrees K. Estimate how long it would take for the Sun to collapse to shrink its radius by a factor of 2. Express your answer both symbolically in terms of M , R , and fundamental constants and numerically, in seconds. An answer good to a factor of 3 suffices. Call this time the "free-fall time." (b) [12 points] Return to the normal hot Sun. Pressure disturbances in the Sun travel at the speed of sound, as given approximately by "sound = ,KT/(umn).' Using the virial theorem, estimate the "sound-crossing time": the time for a sound wave to travel a solar radius. Express your approximate answer both symbolically in terms of M , R , and fundamental constants-and numerically, in seconds. As with (a), your answer should only be approximate. (c) [6 points] Compare (a) to (b) and comment. If your answer in (a) were much shorter than your answer in (b), what would happen? 3 Optical Depth of Earth's Atmosphere [36 points] The sky is blue because sunlight at visible wavelengths scatters off nitrogen and oxygen molecules in the Earth's atmosphere. The scattering is called Rayleigh scattering. This problem computes the optical depth of the Earth's atmosphere to Rayleigh scattering. Take the atmosphere to be composed of 80% Na and 20% O, by number. Assume the nitrogen and the oxygen are well-mixed throughout the atmosphere in these proportions. Take the cross section to Rayleigh scattering of a single Na molecule to be oy; and the cross section to Rayleigh scattering of a single O, molecule to be do. The total pressure of the air at ground level is P. Take the atmosphere to be in hydrostatic equilibrium at constant temperature T', and constant gravity. The mass of the Earth is Me and the radius of the Earth is Re. (a) [8 points] What is the mean molecular weight @ of the atmosphere? (b) [20 points] Derive a symbolic expression for the optical depth + of the Earth's atmosphere to Rayleigh scattering. Measure the optical depth in the vertical direction: start from the ground and go straight up to infinity. Express your answer in terms of the variables given above and fundamental constants. This answer should be exact. (c) [8 points] Assume ON - 10 27 em and go - 2 x 10 27 em. Looking up values for the other 'Note how the speed of sound happens to be comparable to the typical speeds with which individual particles move