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\begin{document}


\begin{center}
{\large{\bf Astro 201 -- Radiative Processes -- Problem Set 4}}

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

\end{center}

\vspace{0.1in}
\ni
Readings: However much of Rybicki \& Lightman pages 77--90 that you wish; the goal is to feel comfortable with the Larmor power formula, Figure 3.2, near and far zones, and multipole expansions; you are free, of course, to achieve this goal by reading any text you like.

\vspace{0.1in}
\ni
{\large{\bf Problem 1.}} Hyperfine Emission from Neutral Hydrogen

\ni This problem is an exercise in learning more astronomy jargon
and in practicing some of the formalism in Rybicki \& Lightman
Chapter 1.

\ni Neutral hydrogen in the electronic ground state can be
in one of two hyperfine states. Denote the number density of
atoms in the ground hyperfine level (singlet state)
as $n_0$, and the number density of atoms in the excited
hyperfine level (triplet state) as $n_1$. DEFINE the {\it excitation
temperature}, $T_{ex}$, of the transition as

\begin{equation}
\frac{n_1}{n_0} = \frac{g_1}{g_0} e^{-h\nu/kT_{ex}} \, .
\end{equation}

\ni Here $h\nu = hc/\lambda$ is the mean energy difference between the levels,
and $g_0 = 1$ and $g_1 = 3$ are the statistical weights of the levels.
The excitation temperature is merely another way of expressing
the ratio of ground state to excited state populations.
By definition, if a gas is in local thermodynamic equilibrium (LTE)
at some gas kinetic temperature $T$, then $T_{ex} = T$;
the level populations are distributed in Boltzmann fashion at
the local temperature $T$.
Some people refer to the excitation temperature for the $\lambda = 21 \cm$
transition as the {\it spin temperature}. But use
of the term ``excitation temperature'' is general to any line transition;
it is simply a measure of how excited an atom is.

\ni (a) Define $T_{\ast} = h\nu/k$ and compute its value.

\ni It is likely that $T_{ex} \gg T_{\ast}$. For the remainder
of this problem, work in the $T_{\ast}/T_{ex} \ll 1$ limit.

\ni (b) Write down the absorption coefficient, $\alpha_{\nu}$
(units of per length), for this transition. Express
your answer in terms of $\phi (\nu)$ (the line profile function),
$A_{21}$ (Einstein A coefficient), $\lambda$, whatever
densities you need, and $T_{\ast}/T_{ex}$. {\it Do not forget
the correction for stimulated emission.}

\ni (c) Write down the volume emissivity, $j_{\nu}$
(units of $\erg \s^{-1} \cm^{-3} \Hz^{-1} \sr^{-1}$),
for this transition. Use whatever quantities defined above
that you need.

\ni (d) Write down the source function, $S_{\nu}$,
for this transition.

\ni (e) Write down the specific intensity, $I_{\nu}$, of a cloud
of HI that is optically thin along the
line-of-sight (l-o-s). Take the l-o-s dimension
of the cloud to be $L$, and give the answer only
to leading order in $\tau \ll 1$, where $\tau$
is the optical depth at an arbitrary wavelength.

\ni Does your answer depend on $T_{ex}$?

\ni If someone gives you a spectrum of the 21
cm line that appears in emission
and tells you that the line was emitted from an optically thin cloud,
what physical quantities can you infer from the spectrum?

\ni (f) Write down the optical depth of the cloud.
Does your answer depend on $T_{ex}$?

\ni (g) How large would $L$ have to be for the cloud
to be marginally optically thick? Use a
gas density of $n = 1 \cm^{-3}$, a gas temperature of
$T = 100\K$, and an excitation temperature $T_{ex} = T$.
Assume the line is only thermally broadened.

%\ni (h) For such an $L$, what is the optical depth
%at $\lambda = 21 \cm$ due to {\it dust}? Use
%the dust model you constructed in problem 2.
%Would you expect radio astronomers to worry about dust?

\vspace{0.1in}
\ni
{\large{\bf Problem 2.}} Multiple Multipoles

\ni (a) Write down the electric field a distance $r$
away from a monopole of charge $q$.

\ni (b) Someone moves another monopole of charge $-q$ next
to the original monopole. The two charges are separated
by a distance $b$. Derive, to order-of-magnitude,
the factor by which this dipole electric field is reduced
from the monopole field.

\ni (c) Someone moves two more charges, $-q$ and $q$,
into position to form a square. The edge of the square
has length $b$. Going around the square, the charges
are $-q$, $q$, $-q$, and $q$. Derive, to order-of-magnitude,
the factor by which this quadrupole electric field
is reduced from the dipole field.

\ni (d) Draw a picture of a pure electric octopole, that is,
a charge distribution for which the electric field
decreases as $1/r^5$ (and no less gradually). For bonus
points, draw a picture of an electric hexadecapole
(electric field dies no less gradually than $1/r^6$).

\ni (e) Now imagine the dipole and quadrupole configurations
in (b) and (c) rotating about their centers-of-charge
with frequency $\nu$. We have a rotating barbell
and a rotating square, respectively. (Think CO and H$_2$).

\ni To order-of-magnitude, what is the maximum distance from each
object inside of which the electric fields are nearly perfectly
in phase with the rotation?

\ni This is the boundary of the {\it near zone}, inside
of which the electric field geometry rotates with frequency
$\nu$ as if it were a rigid body.

\ni (f) Electromagnetic waves are emitted
at the boundary of the near zone, into the far (radiation) zone.

\ni Write down (one line of argument suffices)
the factor by which the power
carried by waves emitted by the rotating
quadrupole is smaller than the power
carried by waves emitted by the rotating
dipole.

\vspace{0.1in}
\ni
{\large{\bf Problem 3.}} Rotating Magnetic Dipole

\ni Rybicki \& Lightman Problem 3.1.

\ni Hint: The Larmor power formula gives the power radiated by an accelerating electric dipole moment. This problem is similar except that it concerns an accelerating magnetic dipole moment.

\end{document}


