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. Assume that the atmosphere is composed of 80% N2 and 20% O2 by number and that these atoms are well-mixed throughout the atmosphere. Take the cross section to Rayleigh scattering of a single N2 molecule to be σN, and the cross section to Rayleigh scattering of a single O2 molecule to be σO. The total pressure of the air at ground level is P0. Take the atmosphere to be in hydrostatic equilibrium at constant temperature T, and constant gravity. The mass of the Earth is M⊕ and the radius of the Earth is R⊕. (a) What is the mean molecular weight μ of the atmosphere? (b) Derive a symbolic expression for the optical depth τ 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. 3 −27 2 −27 2 cm and σO ≈ 2×10 cm . Looking up values for the (c) Assume σN ≈ 10 other variables in your answer for (b), estimate the numerical value of τ. An answer good to within a factor of 2 will get full credit. In the absence of clouds and hazes, is it possible for astronauts on the space station to see straight down to the ground at visible wavelengths?
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.
Assume that the atmosphere is composed of 80% N2 and 20% O2 by number and that these atoms are well-mixed throughout the atmosphere. Take the cross section to Rayleigh scattering of a single N2 molecule to be σN, and the cross section to Rayleigh scattering of a single O2 molecule to be σO. The total pressure of the air at ground level is P0. Take the atmosphere to be in hydrostatic equilibrium at constant temperature T, and constant gravity. The mass of the Earth is M⊕ and the radius of the Earth is R⊕.
(a) What is the mean molecular weight μ of the atmosphere?
(b) Derive a symbolic expression for the optical depth τ 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.
3
−27 2 −27 2
cm and σO ≈ 2×10 cm . Looking up values for the
(c) Assume σN ≈ 10
other variables in your answer for (b), estimate the numerical value of τ. An answer good to within a factor of 2 will get full credit. In the absence of clouds and hazes, is it possible for
astronauts on the space station to see straight down to the ground at visible wavelengths?
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