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\begin{document}

\begin{center}
{\large{\bf Astro 201 -- Radiative Processes -- Problem Set 12}}

{\large{\bf Due in class.}}

\end{center}

\vspace{0.1in}
\ni
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'')

\vspace{0.1in}
\ni
{\large{\bf Problem 1.}} Faraday Rotation

\ni Consider the propagation of light through a magnetized
plasma. The magnetic field is uniform $\vec{B_0} = B_0 \hat{z}$.
Light travels parallel to $\hat{z}$.

\ni 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

\begin{equation}
m\dot{\vec{v}} = - e \vec{E} - \frac{e}{c} \, \vec{v} \times \vec{B_0}
\end{equation}

\ni where the electric field $\vec{E}$ can be decomposed
into right-circularly-polarized (RCP) and left-circularly-polarized (LCP)
waves:

\begin{equation}
\vec{E} = E_0 (\hat{x} \mp i\hat{y}) e^{i(k_{\mp}z - \omega t)}
\end{equation}

\ni 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.

\ni 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.

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

\begin{equation}
\vec{v} = \frac{-ie}{m(\omega\pm\omega_{cyc})} \vec{E}
\end{equation}

\ni where $\omega_{cyc} = eB_0/mc$. This is the hardest part of
the derivation for the dispersion relation of RCP and LCP
waves. But it is fairly straightforward.

\ni (b) Rybicki \& Lightman Problem 8.3

\vspace{0.1in}
\ni
{\large{\bf OPTIONAL Problem 2.}} Combined Scattering and Absorption

\ni Rybicki \& Lightman Problem 1.10

\vspace{0.1in}
\ni
{\large{\bf Problem 3.}} Greenhouse Warming

\ni Taken from Chamberlain \& Hunten Problem 1.3

\ni 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 $\tau = 2$. 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?

\end{document}


