Astro 201 – Radiative Processes – Problem Set 12 Due in class.

Readings: RL 8.1, 8.2; RL 1.7, 1.8; Reader pages 203–212 (photocopied from Chamberlain & Hunten’s textbook, “Theory of Planetary Atmospheres”) Problem 1. Faraday Rotation Consider the propagation of light through a magnetized plasma. The magnetic field is uniform B⃗_{0} = B_{0}ẑ. Light travels parallel to ẑ. An electron in the plasma feels a force from the electromagnetic wave, and a force from the externally imposed B-field. Its equation of motion reads

m˙v⃗ = −eE⃗ − \frac{e}{c} v⃗ × B⃗_{0}  (1)

where the electric field E⃗ can be decomposed into right-circularly-polarized (RCP) and left-circularly-polarized (LCP) waves:

E⃗ = E_{0}(x̂ ∓ iŷ)e^{i(k_{∓}z−ωt)}  (2)

where it is understood that the real part should be taken. The upper sign (-) corresponds to RCP waves, while the lower sign (+) corresponds to LCP waves. In the equation of motion above, we have neglected the Lorentz force from the wave’s B-field, since it is small (by v/c) compared to the force from the wave’s E-field.

(a) Prove that the solution of the equation of motion reads

v⃗ = \frac{−ie}{m(ω ± ω_{cyc})}E⃗  (3)

where ω_{cyc} = eB_{0}/mc. This is the hardest part of the derivation for the dispersion relation of RCP and LCP waves. But it is fairly straightforward.

(b) Rybicki & Lightman Problem 8.3 OPTIONAL Problem 2. Combined Scattering and Absorption Rybicki & Lightman Problem 1.10 Problem 3. Greenhouse Warming

Taken from Chamberlain & Hunten Problem 1.3

Assume that solar radiation is absorbed only at the Earth’s surface where the albedo is 40%. The re-radiated energy is absorbed mainly by water vapor, which we approximate as a gray absorber with a density scale height of 2 km and total optical depth τ = 2. Plot the temperature distribution with height for radiative equilibrium. What is the temperature discontinuity at the ground? What is the gradient in the air temperature near the ground, in K/km?