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\begin{document}

\begin{center}
{\large{\bf Astro 201 -- Radiative Processes -- Problem Set 8}}

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

\end{center}

\vspace{0.1in}
\ni
Readings: Rybicki \& Lightman Chapter 5

\vspace{0.1in}
\ni
{\large{\bf Problem 1.}} Seeing the Forest Through the Clouds

\ni Optical spectra of high redshift quasars often exhibit a
dense thicket of absorption lines (see examples from
Wolfe et al.~1993; ApJ, 404, 480). Many (but not all) of these
absorption features are thought to be from the same transition:
Lyman $\alpha$ ($n=1$ to $n=2$) in hydrogen. The absorption lines
are located at different wavelengths because they arise
from different hydrogen clouds located at different redshifts between
us and the quasar (for the latter, read: convenient, bright, continuum
light source). This so-called ``Lyman $\alpha$ forest'' of absorption
lines is diagnostic of the degree of clumpiness in gas at high
redshift; i.e., it is diagnostic of structure formation.

\ni In this problem, we relate the observed width of each
line to the column density of hydrogen in the
corresponding ``Lyman $\alpha$ cloud.''
Most of this problem can be done in ignorance of cosmology.

\ni Take the Ly$\alpha$ line profile presented by a single cloud
to be both Doppler broadened and naturally broadened.
As described by Rybicki \& Lightman page 291, the convolved
line profile is described by the Voigt function.

\ni (a) If the ``$a$''-parameter in the Voigt function is much
less than one, the Voigt function can be described in two parts.
Define $u \equiv (\nu-\nu_0)/\Delta \nu_D$, where I am using the notation
of Rybicki \& Lightman. If $u \ll 1 + a^{-2}$, then

\begin{equation}
\phi (u \ll 1 + a^{-2}) \approx \frac{1}{\sqrt{\pi} \Delta \nu_D} e^{-u^2} \, ,
\end{equation}

\ni while if $u \gg 1 + a^{-2}$, then

\begin{equation}
\phi (u \gg 1+ a^{-2}) \approx \frac{a}{\pi \Delta \nu_D u^2} \, .
\end{equation}

\ni In other words, the core of the line, near
line center, is dominated by the Doppler Gaussian profile, while
the wings of the line, far from line center, are dominated by
the Lorentzian profile. You can verify that these expressions
are nothing more than (10.68) and (10.73) taken in the appropriate limits.

\ni Calculate ``$a$'' for the Ly$\alpha$ line and verify that
it is much less than one. Use a Doppler width of
$\Delta \nu_D = (10 \km/\s) \nu_0 / c$.

\ni (b) The {\it equivalent width} of a spectral absorption line is,
by definition,

\begin{equation}
EW \equiv \int_{0}^{\infty} \frac{F_{\lambda,0} - F_{\lambda}}{F_{\lambda,0}} d\lambda
\end{equation}

\ni where $F_{\lambda}$ is the actual flux density, and $F_{\lambda,0}$ is the
flux density of the spectrum {\it if the absorption line were not present} (i.e., the ``continuum'' flux density.)

\ni Measurements of the equivalent widths of spectral lines can be used
to infer column densities of intervening material.

\ni First suppose that the Ly$\alpha$ cloud is optically thin in
the line, so that the line profile looks Gaussian (we cannot
probe the Lorentzian wings of the line). Derive
a symbolic expression between $EW$ and the column density, $N_{HI}$, of
neutral hydrogen in the cloud, using whatever fundamental
constants you need.

\ni Hint: The frequency width of the line is still narrow compared
to the line frequency (the line is not relativistically broadened).
Approximate integrals in light of this fact.

\ni (c) Now suppose that the Ly$\alpha$ cloud is so optically
thick in the line that we can make out the naturally broadened
Lorentzian wings of the line. That is, the core of the line
is completely black, but far from line center, the flux
rises back up according to the quasi-power-law Lorentzian. Derive
a symbolic expression between $EW$ and $N_{HI}$ in this regime.

\ni Hint: Using the Lorentzian function in the core is
a permitted mathematical convenience because the core is completely blackened.

\ni (d) Clouds in regime (b) are referred to as members
of the ``Ly$\alpha$ forest.'' Clouds in regime (c) are referred
to as ``damped Ly$\alpha$ systems''---the ``damped'' refers
to the fact that we are sensitive to the ``naturally damped''
Lorentzian wings of the line.

\ni In Wolfe et al.'s paper, we can distinguish Ly$\alpha$
forest clouds having observed $EW_{obs} \sim 0.2 \AA$. Numerically
estimate $N_{HI}$ for such clouds, assuming they are located
at redshift $z = 3$.
%% EUGENE: Use 0.2 \AA for EW_obs instead so that you are safely
%% in the optically thin regime.

\ni Remember that
spectral line profiles get stretched with the expansion
of the universe, so $EW_{obs} = (1+z) EW_{int}$, where
$EW_{int}$ is the intrinsic equivalent width of the line
(the $EW$ you would measure if you were positioned right
behind the cloud). All of your expressions for (a)--(c)
pertain to $EW_{int}$.

\ni (e) In the same quasar spectrum, we can
also distinguish damped Ly$\alpha$ systems having
$EW_{obs} \sim 100 \AA$. Numerically estimate
$N_{HI}$ for such clouds, and compare your answer
to the column in our Galaxy, $N_{HI,Gal} \sim 10^{21} \cm^{-2}$.
The fact that they are comparable argues that
damped Ly$\alpha$ systems are full-fledged galaxies
that happen to lie between us and the quasar.

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

\ni Why do galaxies have the sizes and masses that they do?

\ni Here is a rule of thumb 
governing the sizes of objects that
can collapse under the weight of their
own self-gravity: objects can only collapse
if their {\it cooling time} is shorter
than their {\it gravitational collapse time.}
The cooling time of gas is the time it takes to lose
order unity (say, 0.5) of its thermal energy.
The collapse time is the time it takes to
shrink its radius by order unity.
(This rule of thumb is
subject to details regarding the exact cooling mechanisms; i.e.,
the effective adiabatic index of collapsing material.
The adiabatic index doesn't have to
be exactly one (isothermal) for collapse to proceed.)

\ni (a) Present an order-of-magnitude derivation,
using only bremsstrahlung radiation to cool off
gas, of the {\it maximum} radius $R$ of an object that can form
out of a {\it virialized} lump of hydrogen plasma.
(Recall that if a gas is virialized, its kinetic
temperature is uniquely related to the depth of its gravitational
potential well. Gas becomes virialized as it
collapses under its own self-gravity and shocks;
the kinetic energy of collapse is converted efficiently
into heat in these shocks.)

\ni Express in kpc.

\ni (b) For an object having
such a maximum radius $R$, present an order-of-magnitude derivation
of its {\it minimum} mass $M$ by considering
how hot the gas must be to be a fully ionized plasma.

\ni Express in solar masses.

\vspace{0.1in}
\ni
{\large{\bf OPTIONAL Problem 3.}} Sobolev Escape

\ni Imagine a homogeneous sphere of uniform
density that emits and absorbs photons of a certain
frequency. Every volume element of the sphere produces
a certain number of photons per second,
and these photons are unleashed from each
volume element isotropically. The
material is purely absorbing; neglect scattering.

\ni The radial optical depth of the sphere
to these photons is $\tau$ (integrated
over the radius of the sphere).

\ni (a) What fraction of the photons that
are unleashed per second (throughout
the entire sphere) actually escape
the sphere per second (unabsorbed)? Express
in terms of $\tau$. You have calculated
what is called the ``escape probability
for a uniform absorbing sphere,''
useful for more technical analyses of radiative transfer.

%\ni Remember that the probability that
%a single photon can traverse an optical
%depth of $\tau'$ in a certain direction
%without getting absorbed is $e^{-\tau'}$.

\ni (b) Explain why your answer in (a) makes sense
in the limits that $\tau \ll 1$ and $\tau \gg 1$.

\ni You may find it helpful to solve (a) to order-to-magnitude
in these limits before trying to attack (a) in full generality.
This problem can, with a modicum of effort, be solved
exactly.

%\ni (c) OPTIONAL BONUS QUESTION: What fraction
%of the photons that are unleashed per second in a {\it shell}
%that extends from an inner radius of $R' > 0$ to the
%outer edge of the sphere at radius $R$ escape?

\end{document}


