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\begin{document}


\begin{center}
{\large{\bf Astro 201 -- Radiative Processes -- Problem Set 3}}

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

\end{center}

\vspace{0.1in}
\ni
Readings: Rybicki \& Lightman 8.1, Mihalas pages 81--84 (his pages, not the Course Reader's), Mihalas section 4-5. Also however much of Rybicki \& Lightman pages 90--102 that you wish.

\vspace{0.1in}
\ni
{\large{\bf Problem 1.}} Eddington Limit

\ni Rybicki and Lightman problem 1.4

\vspace{0.1in}
\ni
{\large{\bf Problem 2.}} Pulsar Dispersion Measure

\ni A radio astronomer notes that pulses from a certain pulsar
observed at a radio frequency of $\nu = 2$ GHz arrive slightly
ahead of the pulse train observed at $\nu = 1$ GHz. The lead time is 1 s.

\ni (a) Use the dispersion relation derived in class for a cold, ionized
plasma, and the information above,
to derive the column density of electrons
along the line of sight to this pulsar. Express in standard
pulsar-community units of $\cm^{-3}\, \pc$.

\ni (b) For an assumed density of electrons in the interstellar medium
of $0.03 \cm^{-3}$, calculate the distance to this pulsar.
Does this seem reasonable?

\ni (c) For such an assumed
electron density, is the radio astronomer safely observing above the plasma cut-off
frequency?

\ni (d) Calculate the optical depth to Thomson scattering along
the line-of-sight to this pulsar.


\vspace{0.1in}
\ni
{\large{\bf OPTIONAL Problem 3.}} Hyperfine $^3$He$^+$

\ni NOTE: While this problem is optional, it concerns
material covered in lecture, and all material covered in lecture will
be tested on the mid-term and final exams.

\ni Observations of the hyperfine transition in $^3$He$^+$ are used
to probe the $^3$He/H abundance in the galaxy. This abundance
reflects the primordial yield from big bang nucleosynthesis
and galactic chemical evolution.

\ni (a) Estimate, using the scaling relations presented in class
and whatever facts you remember,
the wavelength of the ground-state hyperfine transition
in $^3$He$^+$. Compare to the true answer of 3.46 cm.

\ni (b) Estimate the Einstein A coefficient (transition probability)
of this line. Compare to the true answer of $1.95 \times 10^{-12} \, \s^{-1}$.

\vspace{0.1in}
\ni
{\large{\bf OPTIONAL Problem 4.}} Shades of the Sun

\ni (a) When the sun sits 5 degrees above the horizon
and is about to set, it looks red. Give a semi-quantitative
explanation why. Neglect dust.

\ni (b) During a total solar eclipse, the sun's corona
appears white to the naked eye. The corona is composed
of plasma. State why the corona looks white (one sentence
will suffice).

\vspace{0.1in}
\ni
{\large{\bf Problem 5.}} Back-of-the-envelope CO 1-0

\ni Most astronomers know that the Einstein A coefficient
for the Lyman alpha ($n=2$ to $n=1$) transition in atomic hydrogen
is of order $10^9 \s^{-1}$ (actually, $5 \times 10^8 \s^{-1}$)
 We estimated
this result in class to order-of-magnitude by considering an
(accelerating) electron on a spring that displaces a Bohr radius,
$a_0$, and has natural angular frequency $\omega$. We took
the inverse lifetime of the excited state as
$A \sim P/\hbar \omega$, where $P$ is the power radiated by an
accelerating charge.

\ni The ubiquitous carbon monoxide molecule, $^{12}$CO, is used
by astronomers to trace the presence and temperature
of molecular gas in everything
from galaxies to circumstellar disks. We would rather try
to detect H$_2$, but sadly H$_2$ has no permanent electric dipole moment
because of its symmetry. Carbon monoxide does have a permanent
dipole moment.

\ni (a) Estimate the wavelength of the lowest energy, rotational
transition in CO ($J=1$ to $J=0$, where $J$ is the
rotational quantum number). Do this by considering
a barbell spinning about its axis of greatest moment of inertia
and recognizing that angular momentum comes quantized in units
of $\hbar$.

\ni Compare your estimate to the true answer of 2.6 mm.

\ni (b) Use the scalings of our semi-classical spring model to estimate
the Einstein A coefficient of this transition, i.e., the inverse
lifetime of the excited $J=1$ state.

\ni Bring to bear one fact that is difficult
to guess from first principles: the dipole moment of the CO
molecule is 0.1 Debyes (1 Debye = 10$^{-18}$ cgs. Note that $ea_0 = 2.5$
Debyes, where $e$ is the electron charge), not $\sim$1 Debye,
as one might have guessed naively. The smaller-than-usual dipole
moment of CO is a consequence of the strong double bond connecting
C to O. Most other molecules---e.g., H$_2$O,
CS, SiS, SiO, HCN, OCS, HC$_3$N---have
dipole moments that are all of order 1 Debye.

\ni Compare your estimate to the true answer of $6 \times 10^{-8} \s^{-1}$.

\ni (c) Estimate $\sigma$, the cross-section of the transition
at line center, in $\cm^{2}$. Assume the line to be thermally broadened
at a typical molecular cloud temperature of $20 \K$.

\ni (d) If the number abundance of CO molecules
to H$_2$ molecules is $n_{\rm CO}/n_{{\rm H}_2} \sim 10^{-4}$ (i.e.,
close to the solar abundance ratio of $n_{\rm C}/n_{\rm H} \sim 10^{-4}$),
estimate the column of hydrogen molecules required to
produce optical depth unity at the center of the CO line.
Is this column likely to be exceeded in a molecular cloud?
Take a typical molecular cloud H$_2$ density of $10^{4} \cm^{-3}$
and a cloud dimension of 100 pc.\footnote{In Spitzer's book on the
interstellar medium,
molecular cloud densities of molecular hydrogen are said to range from
10$^3$ to 10$^6$ $\cm^{-3}$. The density can be orders of magnitude
higher than even $10^6 \cm^{-3}$ as you approach star forming
regions. Molecular clouds are clumpy structures.}

\ni (e) Based on your answer to (d), would you conclude
that this transition is a good way to measure the mass
of molecular gas in a galaxy? What about the temperature?

\end{document}

