\documentstyle[11pt]{article}
\input answermac.tex
\input units.tex

\addtolength{\oddsidemargin}{-.4in}
\addtolength{\evensidemargin}{-.4in}
\addtolength{\topmargin}{0in}
\addtolength{\textwidth}{0.8in}
\addtolength{\textheight}{1.00in}
\setlength{\parskip}{0.15in}
% This last command gives the possibility of variable line spacing.
%\renewcommand{\baselinestretch}{1.5}

\def\ni{\noindent}
\def\J{{\rm\,J}}
\def\W{{\rm\,W}}
\def\s{{\rm\,s}}
\def\erg{{\rm\,erg}}
\def\cm{{\rm\,cm}}
\def\m{{\rm\,m}}
\def\km{{\rm\,km}}
\def\kg{{\rm\,kg}}
\def\mm{{\rm\,mm}}
\def\gm{{\rm\,g}}
\def\g{{\rm\,g}}
\def\d{\rm\,d}
\def\h{\rm\,h}
\def\mum{\,\mu{\rm m}}
\def\K{{\rm\,K}}
\def\yr{{\rm\,yr}}
\def\Hz{{\rm\,Hz}}
\def\days{{\rm\,days}}
\def\cals{{\rm\,cals}}
\def\mole{{\rm\,mole}}
\def\gal{{\rm\,gal}}
\def\calo{\,{\rm calorie}}
\def\dyne{\,{\rm dyne}}
\def\s{\,{\rm s}}
\def\at{\, {\rm atmosphere}}
\def\J{\,{\rm joule}}
\def\sr{\,{\rm sr}}
\def\pomega{\tilde{\omega}}

\begin{document}

\begin{center}
{\large{\bf Astro 201 -- Radiative Processes -- Problem Set 9}}

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

\end{center}

\vspace{0.1in}
\ni
Readings: Rybicki \& Lightman 4.1, 4.8, 4.9, 6.1--6.3, 6.6, skim 6.4--6.5,
skim 6.8; copy of Shu's chapter on the ``simple version''
of synchrotron theory; however much of the classic and readable paper
by Scheuer and Williams

\ni NOTE FOR ALL PROBLEMS: In this class, we are not going to stress
over the $\sin \alpha$ term in any expression involving
synchrotron emission, nor are we going to worry about
other factors of order unity, like gamma ($\Gamma$) functions.
If you see a $\sin \alpha$ or gamma function in your
travels, just set it equal to 1.

\vspace{0.1in}
\ni
{\large{\bf Problem 1.}} Synchrotron Losses

\ni (a) Obtain an analytic expression for the energy
of a single relativistic electron as a function of time, $E(t)$,
taking into account its energy loss by synchrotron radiation.
Your expression should contain only the variables $E(0)$ (the
initial energy of the electron), $B$ (the magnetic field,
here held fixed with time, following the rest of the world,
though one should worry in general about the field changing with time
just as the electron energy spectrum changes with time),
time $t$, and fundamental constants. Assume $\sin \alpha = 1$
(the electron pitch angle is 90 degrees) for simplicity.

\ni For (synchrotron)
problems of interest to us, the electron always remains relativistic.
It merely evolves from a large $\gamma \gg 1$ to a smaller $\gamma \gg 1$.

\ni (b) How can you reconcile the loss of energy of the electron
with the bald statement of Rybicki \& Lightman on page
168 that ``$\gamma$ is constant''?

\ni OPTIONAL (c) We have made arguments in class that power-law
distributions of electrons in astrophysical sources
are maintained against synchrotron losses by continuous
energization by central engines (a.k.a.~{\it injection}). The
injection (input) spectrum of electrons is modified
by synchrotron losses to produce a steady-state (output) distribution.

\ni Call $\eta (E,t) = dN/dE$ the differential
energy spectrum of electrons as discussed repeatedly in class.
Continuity of electrons in energy space reads

\begin{equation}
\frac{\partial \eta (E,t)}{\partial t} + \frac{\partial}{\partial E} [\dot{E}\eta (E,t)] = I
\end{equation}

\ni where $I$ is the rate of injection of electrons with some
input distribution and $\dot{E}$ is the rate of energy loss
of a single electron by synchrotron radiation. This equation should not mystify
you; it merely describes how the number of electrons
in a given energy bin changes with time, taking into account
a flux divergence (the second term on the left-hand-side)
and a source term (the right-hand-side).

\ni We have assumed in class a {\it steady-state}
distribution of electron energies for which $\eta \propto E^p$.
Given $p$, how must $I$ scale with $E$?
Give only the scaling and forget
about the numerical coefficients.

\ni As with most scaling problems, you don't have to solve
anything in detail. Ruthlessly work to order of magnitude.

\ni OPTIONAL (d) Electrons having a given energy must wait a characteristic time
before synchrotron losses become important. Before
this time elapses for all such electrons, how does $\eta$
scale with $E$?

\ni OPTIONAL (e) The {\it spectral} index ($\alpha = d\ln F_{\nu}/d\ln \nu$)
of radiation in a fixed frequency range
from a radio jet flattens with
increasing distance from the central galaxy.
That is, $\alpha = -0.5$ at the remote edge of the jet (the ``hot spot''),
and $\alpha = -1$ closer in. Given your understanding
in (c) and (d), where are the ``freshest'' electrons located, i.e.,
those newly injected into the energy spectrum?
Are they at the end of the jet, or are they closer in?
In other words, where is the principal site of particle acceleration?

\ni OPTIONAL (f) Sketch several profiles of $\eta$ vs.~$E$ at various times,
assuming $I$ is constant in time.

\vspace{0.1in}
\ni
{\large{\bf Problem 2.}} Great Balls of Relativistic Fire

\ni Rybicki \& Lightman Problem 4.1 (Don't worry
if you can't reproduce the answer to within a factor of 2;
the answer is meant to be a rule of thumb.)

\vspace{0.1in}
\ni
{\large{\bf Problem 3.}} Superluminal Motion, or Photons Chasing Photons

\ni Rybicki \& Lightman Problem 4.7

\ni (c) Plot $v_{app}/c$ vs.~$\theta$ for $\gamma = 10^2$. Does the
viewing angle $\theta$ need to be especially small for superluminal
motion to be perceived?

\ni For culture: see the beautiful illustration of superluminal
motion in the optical M87 jet by Biretta in the accompanying .jpg
on the class website.

\end{document}
