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\begin{document}

\begin{center}
{\large{\bf Astro 201 -- Radiative Processes -- Problem Set 11}}

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

\end{center}

\vspace{0.1in}
\ni
Readings: Article by Kellerman; Rybicki \& Lightman 7.1, 7.2, 7.4,
as much as you like of 7.6, 7.7; however much of Blandford's lecture notes
but watch for errors.

\vspace{0.1in}
\ni
{\large{\bf Problem 1.}} Compton Catastrophe

\ni We derived in class that for a synchrotron-self-Compton (SSC) emitting
source that is static in bulk, the ratio of the luminosity due to
first-generation inverse Compton scattering of synchrotron
photons to the luminosity due to synchrotron processes is

\begin{equation}
f \equiv \frac{L_{IC,1}}{L_{sync}} = C \nu_{m} T_{bm}^5
\end{equation}

\ni where $\nu_m$ is the frequency at which the synchrotron
spectrum peaks, and $T_{bm}$ is the brightness temperature
of the synchrotron radiation at that frequency.

\ni (a) Derive an estimate for the constant $C$ in terms of
fundamental constants. Assume whatever geometry you
like for the source. I found I had to use the (usual) constants
$e$, $k$, $c$, and $m_e$.
% [Try not to let the ``k'' throw you;
%there's no temperature in the strict thermodynamic
%(i.e., Maxwellian) sense, but that doesn't
%stop the astronomers from making up a ``brightness'' temperature.]

\ni You might find the heuristic derivation of the optically
thick portion of synchrotron spectra helpful (indirectly).

\ni (b) If $\nu_m \sim 1 \,\rm{GHz}$ (as it typically is
for compact radio sources), what is the maximum $T_{bm}$
above which $f > 1$? Express in Kelvin.

\ni (c) Given $f$, what is $L_{IC,2} / L_{IC,1}$? That is,
what is the ratio of second-generation to first-generation scattered
power? Explain your answer.

\ni (d) Consider the situation after $N$ scatterings, where
$N \rightarrow \infty$. What is the critical $f$ for which the total
luminosity of the source becomes unbounded?

\vspace{0.1in}
\ni
{\large{\bf Problem 2.}} Compton Saturation

\ni Rybicki \& Lightman problem 7.1b (only b. We answered
the other parts in lecture.)

\vspace{0.1in}
\ni
{\large{\bf Problem 3.}} Typical vs.~Average (or Lies, Damn Lies, and Statistics)

\ni Consider a non-relativistic electron moving at speed $v \ll c$.
A single photon of original energy $\epsilon$ Compton scatters off the
electron and suffers an energy shift of $\Delta \epsilon$.
Work in the same Thomson limit that we have been working in
throughout lecture.

\ni (a) Estimate a typical value for the fractional
change in photon energy, $\Delta \epsilon / \epsilon$.
Express your answer symbolically. By ``typical,'' we mean
any old photon that comes in from any old direction.

\ni Be sure to specify the possible signs of the change.

\ni (b) Consider this same electron flying through
a sea of photons all of energy $\epsilon$. Many photons
get inverse Compton scattered to and fro.

\ni Averaged over all the photons that got scattered,
what is the mean fractional change in photon energy,
$\langle \Delta \epsilon / \epsilon \rangle$?
You may use our expression for the inverse Compton
power scattered by a single electron if you wish.
Neglect the change in the electron's velocity
as it is bombarded by photons.

\ni Be sure the specify the possible signs of the change.

\ni (c) Explain physically why parts (a) and (b)
give different answers. (And then read Blandford
page 175, if you wish. It is better to think
about this problem and achieve your own understanding
than to head straight for Blandford right away).


\end{document}


