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\begin{document}


\begin{center}
{\large{\bf Astro 201 -- Radiative Processes -- Problem Set 6}}

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

\end{center}

\vspace{0.1in}
\ni
Readings: Rybicki \& Lightman 9.5, 10.6

% Use Saha to derive H- abundance and opacity (see Fluids solution set #3)!

\vspace{0.1in}
\ni
{\large{\bf Problem 1.}} Saha and Redshift of Recombination

\ni
Rybicki \& Lightman problem 9.4, parts (a), (b), and (c)

\ni (d) Calculate the redshift, $z_{rec}$, at which recombination
occurred in the early universe. Use the temperature-redshift
relation

\begin{equation}
T_{\rm photon} = T_0 (1+z)
\end{equation}

\ni and the relation between baryonic number density and redshift

\begin{equation}
n = n_0 (1+z)^3 \, .
\end{equation}

\ni Here $T_0 = 2.73 \K$ is the temperature of the cosmic
microwave photon gas today, and $n_0 = 10^{-7} \cm^{-3}$
is the number density of baryons (read: hydrogen atoms)
today, grossly averaged over the entire universe.
You may define the epoch of recombination to be when
0.5 of the protons have recombined to form neutral hydrogen.

\ni (Redshift is a cosmologist's measure of time. A redshift $z = 0$
corresponds to today. As one goes back in time, the redshift increases.
Photons that have travelled from an epoch corresponding to a redshift
$z$ have their original wavelengths, $\lambda$, stretched
to a longer wavelength, $\lambda'$, by the expansion of the universe.
By definition, $\lambda' / \lambda = 1+z$.)

\vspace{0.1in}
\ni
{\large{\bf Problem 2.}} The Orion Nebula

\ni (a) Sketch the flux density, $F_{\nu}$, for the Orion Nebula
due to its free-free emission from a wavelength
of $\lambda = 100\mum$ to $\lambda = 100\cm$.
Express $F_{\nu}$ in Janskys ($10^{-23}$ is cgs units).
Plot the spectrum on a log-log plot, and indicate
power-law indices where appropriate.

\ni Take the density of electrons to be $n_e = 2\times 10^3 \cm^{-3}$,
the electron and proton temperatures to be $T = T_e = T_p = 8\times 10^3 \K$,
the dimension of the ionized cloud to be $R = 1 \pc$,
and the distance to the Orion Nebula to be $d = 500 \pc$.
Take whatever geometry (cube, sphere) for the cloud is most convenient.

\ni {\it Please do not forget free-free self-absorption.}

\ni OPTIONAL (b) Overlay on your plot $F_{\nu}$ from pure electron-electron
scatterings. Here aim only for order-of-magnitude accuracy.
Remember that electron-electron collisions involve
time-varying quadrupole moments, not time-varying
dipole moments, and recall the quadrupole vs. dipole
scalings from our discussion of Einstein A's.

\ni 
For this part, consider the emissivity ($j_{\nu}$) purely
from electron-electron scatterings. Drop the emissivity
from electron-proton scatterings. However, for the absorptivity
($\alpha_{\nu}$), consider the contributions from both
electron-proton and electron-electron scatterings.
If you think one absorption process is more important  
than the other, justify why you think that is the case 
and proceed by including just the one.

\vspace{0.1in}
\ni
{\large{\bf Problem 3.}} Blowing Stromgren Bubbles

\ni Consider a lone O star emitting $\eta$ Lyman limit
photons per second. It sits inside hydrogen gas of infinite extent
and of number density $n$. The star ionizes
an HII region---a.k.a. a ``Stromgren sphere,'' after Bengt
Stromgren, who understood that such spheres have sharp
boundaries---in which hydrogen is nearly completely ionized.

\ni {\it Every} Lyman limit photon goes towards ionizing a neutral
hydrogen atom. That is, every photon emitted by the star
goes towards maintaining the Stromgren bubble. Put yet another
way, no Lyman limit photon emitted by the star travels
past the radius of the Stromgren sphere.

\ni The rate at which Lyman limit photons
are emitted by the central star equals the rate
of radiative recombinations in the ionized gas.
The sphere is {\it nearly} completely ionized.\footnote{The sphere
cannot be 100\% ionized because then there would be no neutrals
to absorb the Lyman limit photons that are continuously
streaming out of the star.} These
facts of photo-ionization equilibrium
determine the approximate radius of the Stromgren sphere.

\ni The temperature inside the sphere is about 10000 K.
(A class on the interstellar medium can show you why.)

%\ni (a) Radiative recombinations and radiative
%bound-bound decays in the Stromgren sphere produce
%H$\alpha$ photons ($n=3$ to $n=2$; $\lambda = 6563$ Angstroms).
%Calculate the rate
%at which H$\alpha$ photons are emitted by the sphere,
%and compare to $\eta$. HII regions are famous for their
%prodigious output of H$\alpha$ photons.

%\ni In full, this is a nasty problem because it requires
%that we look at all possible routes to all possible states
%(in Osterbrock's language, we need to compute the ``cascade matrix.'')
%Make the drastic simplification that we need consider
%only recombinations from the free state to the $n=3$ state.
%After an electron lands in the $n=3$ state, assume further
%that it has only three
%choices: either it can transition to $n=2$, it can transition
%to $n=1$, or it can be re-ionized. 
% Eugene: this problem is too complicated; requires bound-free cross-section
% from n=3
%\ni Tables of radiative recombination coefficients from
%Osterbrock, and formulae from Rybicki \& Lightman 10.5 for
%oscillator strengths of bound-bound transitions in hydrogen
%should prove useful.

%\ni Attention to detail will be rewarded.
%Please work as much as you can symbolically and leave
%numerical evaluations as final last steps.

\ni OPTIONAL (a) Derive a symbolic expression for the radius
of the Stromgren sphere using the above variables and whatever
variables were introduced in lecture.

\ni (b) What is the timescale, $t_{rec}$, over which a free proton
radiatively recombines in the sphere? That is, how long
would a free proton have to wait before undergoing
a radiative recombination? Give both
a symbolic expression, and a numerical evaluation
for $n = 1 \cm^{-3}$.

\ni OPTIONAL (c) If the star were initially ``off,'' and the gas
surrounding it initially neutral, what is the timescale
for the Stromgren sphere to develop after the star
were turned ``on''? That is, how long does the
star take to blow an ionized bubble?
Think simply and to order-of-magnitude;
you should get the same answer as (b).

\vspace{0.1in}
\ni
{\large{\bf Problem 4.}} Time to Relax in the Stromgren Sphere

\ni It is often assumed that velocity distributions
of particles are Maxwellian. The validity of this assumption
rests on the ability of particles to collide elastically
with one another and share their kinetic energy. For a Maxwellian
to be appropriate, the timescale for a collision must be
short compared to other timescales of interest.
This problem tests these assumptions for the case
of the Stromgren sphere of nearly completely ionized
hydrogen of problem 3.

\ni (a) Establishing the electron (kinetic) temperature: what
is the timescale, $t_{e}$, for free electrons in the Stromgren
sphere to collide with one another?
Consider collisions occurring at relative velocities
typical of those in an electron gas at temperature $T_e$.
Work only to order-of-magnitude and express your
answer in terms of $n$, $T_e$, and other fundamental constants.

\ni I think it is fair to say that this problem is done most
easily in cgs units.

\ni (b) Establishing the proton (kinetic) temperature:
repeat (a), but for protons, and consider collisions at
relative velocities typical of those in a proton
gas of temperature $T_p$. Call the proton relaxation time $t_p$.

\ni (c) Establishing a common (kinetic) temperature: suppose
that initially, $T_e > T_p$. What is the timescale
over which electrons and protons equilibrate to a common kinetic
temperature? This is {\it not} merely the timescale for a proton to collide
with an electron. You must consider also the amount of energy exchanged
between an electron and proton during each encounter.
Estimate, to order-of-magnitude, the time it takes a cold proton
to acquire the same kinetic energy as a hot electron.
Call this time $t_{ep}$. Again, express your answer
symbolically.

\ni Hint: you might find it helpful to switch the charge
on the electron and consider head-on collisions between
the positive electron and positive proton.

\ni (d) Numerically
evaluate $t_e / t_{rec}$, $t_p / t_{rec}$, and $t_{ep}/t_{rec}$,
for $T_e \sim T_p \sim 10^4 \K$ (but $T_e \neq T_p$ so that
$t_{ep}\neq 0$). Is assuming a Maxwellian
distribution of velocities at a common temperature for both
electrons and ions a good approximation in Stromgren spheres?

\end{document}


