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\begin{document}


\begin{center}
{\large{\bf Astro 201 -- Radiative Processes -- Problem Set 2}}

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

\end{center}

\vspace{0.1in}
\ni
Readings: Mihalas section 2-1.

\vspace{0.1in}
\ni
{\large{\bf Problem 1.}} Flat Disk SED

\ni We return to the perfectly flat, blackbody disk encircling
a blackbody star of problem set 1.

\ni (a) Explain why $\nu F_{\nu}$ is a measure
of the ``flux radiated by an object per logarithmic
interval in frequency.'' Here $F_{\nu}$
is the flux density [units of energy per time per area
per frequency], and $\nu$ is the frequency of radiation.
(Some refer more loosely to $\nu F_{\nu}$ as the
flux radiated per octave in frequency. An octave, in either
the acoustic or electromagnetic spectrum,
represents a factor of 2 in frequency.)

\ni Can you understand why $\nu F_{\nu}$, and not $F_{\nu}$,
is a quantity
of interest to those who wish to understand the
overall energetics of an object? It is called
the ``broadband spectral energy distribution,''
or ``broadband SED,'' or ``SED'' for short.

\ni Hint: Most macroscopic objects in the universe are {\it broadband emitters};
that is, they radiate continuum radiation at all wavelengths.
Plotting their spectrum would give a curve that varies smoothly
with frequency. For any object, you can ask whether it is putting out
its energy predominantly in
X-rays, gamma rays, radio waves, infrared waves, etc (is it
an ``X-ray object?'' ``a gamma-ray object?'' ``an infrared object?'')
Now ask yourself why a person asking such questions should
plot $\nu F_{\nu}$ and not $F_{\nu}$.

\ni (b) Is $\nu F_{\nu}$ equal to $\lambda F_{\lambda}$,
where $\lambda$ is the wavelength of the radiation,
and $F_{\lambda}$ is per wavelength rather than
per frequency? Show why or why not.

\ni (c) Write down an expression for $\nu F_{\nu}$
for the blackbody disk. That is, write down the
formula for the spectrum of the disk as measured
by an observer for whom the disk is a point source.

\ni Recall that the disk has a temperature $T(r)$
at every stellocentric radius, $r$.
The inner radius of the disk is $r_i$ and
the outer radius is $r_o$. The disk is a distance $D$
away from the observer, and is inclined by an angle
$i$ ($i = 0$ corresponds to a face-on disk).
Your expression should take the form of an integral.

\ni (d) OPTIONAL: {\it Sketch} (no heroics necessary, but get the
orders of magnitude right and label the axes correctly)
$\nu F_{\nu}$ vs. $\nu$. If the spectrum
exhibits power-law behavior, give the slope of
the power law (i.e., $d\ln (\nu F_{\nu})/d\ln \nu$).
Overlay on your sketch the SED of the central
stellar blackbody. Log-log space is best.

\vspace{0.1in}
\ni
{\large{\bf Problem 2.}} Practice with $j_{\nu}$, $\alpha_{\nu}$, $S_{\nu}$, $B_{\nu}$, $I_{\nu}$

\ni (a) A plane-parallel
slab of uniformly dense
gas is known to be in LTE (local thermodynamic equilibrium)
at a uniform temperature T. Its thickness normal to its surface
is $s$. Its absorption coefficient
is $\alpha_{\nu,\rm{gas}}$. Write down the specific intensity, $I_{\nu}$,
viewed normal to the slab, in terms of the variables given.

\ni (b) The same slab is now filled uniformly with non-emissive dust having
absorption coefficient $\alpha_{\nu,\rm{dust}}$. The dust is non-emissive,
so its emissivity
$j_{\nu,\rm{dust}} = 0$. Write down $I_{\nu}$ viewed normal to the slab,
in terms of all variables given so far.

\ni (c) The slab of gas and dust is further mixed with a third
component: an emissive, non-absorptive uniform medium having emissivity
$j_{\nu,\rm{med}}$ and absorption coefficient $\alpha_{\nu, \rm{med}} = 0$.
Write down $I_{\nu}$ viewed normal to the slab, in terms of all variables
given.\footnote{A physical realization of this problem might be
an HII region surrounding an ionizing O star. The material in LTE
would be the fully ionized plasma, emitting thermal bremsstrahlung radiation.
The dust would be dust. The emissive, non-absorptive medium would
be the same ionized plasma emitting recombination (line) radiation.
For the assumptions stated in the problem to be valid, we would have
to evaluate $\nu$ at, say, an optical recombination line like
H${\alpha}$.}

\vspace{0.1in}
\ni
{\large{\bf OPTIONAL Problem 3.}} Photon Pinball in Clouds

\ni {\bf NOTE:} While this problem is optional, those students who have
done it report back that they enjoyed it and found that the problem contained
useful insights.
Nevertheless, it covers material that is not traditionally part
of a radiation course, and because of past criticism regarding
excessive homework, I am leaving it optional.

%\ni {\bf NOTE:} Those who have already done this problem are not
%permitted to repeat their experience. They must proceed to problem 4.
%You know who you are, and I know who you are.

\noindent
This problem shows how the usual $e^{-\tau}$ attenuation
factor for flux passing through an absorbing medium can be incorrect
for flux passing through a purely scattering medium.
(And thank
goodness, or else it would be really dark on cloudy days.)
Clouds can be viewed as a purely scattering medium.

\noindent
Idealize the cloud as a uniform, 1-dimensional slab comprising particles that
can only scatter light. A uniform flux of photons
irradiates the top of the cloud deck (that is, just above the cloud
deck, there exists a uniformly bright sheet).
A photon passing through
the cloud gets bounced like a pinball from cloud droplet
to cloud droplet, preserving its frequency and never
getting absorbed by any droplet. A few photons
are lucky enough to make it through the cloud, while most
get pinballed back out the way they came. We will calculate the fraction
that make it through.

\noindent
Take the cloud to have a droplet density [droplets per cubic volume] $\eta$,
the droplet radius to be $R$, and the vertical thickness of the
cloud to be $z_{max}$. Measure vertical distance through the cloud
by $z$, where the top of the cloud is located at $z=0$
and the base of the cloud is located at $z=z_{max}$.

\noindent
(a) Write down the optical depth of the cloud, $\tau$.

\ni
(b) {\it Incident} photons from the sun strike the top of the cloud.
The photons have a number flux, $F_i$ [number per time per area].
What is the {\it number density of incident photons} at the top
of the cloud? Call this photon number density $n_i$.
These incident photons have NOT been scattered yet by any
droplet. Remember that flux is a number density multiplied by a speed.

\noindent
(c) These photons random walk through the scattering droplets.
Random walks are described by the diffusion equation,

\begin{equation}
\frac{\partial n}{\partial t} = D \frac{\partial^2 n}{\partial z^2}
\end{equation}

\noindent where $D$ is the diffusion coefficient and $n$ is the photon number
density.

\noindent Express $D$ in terms of symbols defined above and
whatever fundamental constants you deem appropriate.
Hint: dimensional analysis may prove useful.

\noindent (d) In steady-state, $\partial n/ \partial t = 0$ (the number
density of photons everywhere in the cloud does not change
with time). Write down the
solution to the diffusion equation for $n(z)$. You should have two,
as yet unknown, constants of integration.

\noindent
(e) To solve for the two constants of integration, you need two
boundary conditions. The first condition is that $n_t = n(z=z_{max})$.
%Here $n_r$ is the number density of photons at the top of the cloud
%that are being reflected back out into space.
Here $n_t$ is the number density of photons at the base of the cloud.
These photons comprise the {\it transmitted} flux.

\noindent
The second condition is that the (net, number) flux, $F$, of photons at $z=0$
equals the incident flux, $F_i$ (directed down into the
cloud) MINUS the outgoing, {\it reflected} flux, $F_r$ (directed up, away
from the cloud into space). Recall Fick's law, which is just
another way of writing the diffusion equation, that
the (net) flux $F = - D \partial n / \partial z$.
So we have $F(z=0) = F_i - F_r$.

\noindent Use the above, and the fact that the incident flux, $F_i$,
must equal the reflected flux, $F_r$, PLUS the transmitted flux, $F_t$,
to calculate $T$, the ratio of the transmitted flux
to the incident flux, in terms of $\tau$. Are you glad that
clouds scatter but do not absorb light?

\vspace{0.1in}
\ni
{\large{\bf Problem 4.}} Galaxies

\ni In an idealized model of a galaxy, stars are uniformly
distributed within a cylinder of radius $R$ and height $H$.
The number density of stars is $n$ [in units of stars per volume].
Model each star
as a spherical blackbody of temperature $T_{\ast}$ and radius $R_{\ast}$.
The density of stars is so low that of all possible lines of sight
running through the galaxy, the fraction that intersect a star
is $\ll 1$.

\ni An observer resolves the galaxy in a face-on viewing geometry, but
does not resolve the individual stars making up the galaxy. In other words,
the observer detects the galaxy as a circular disk that is very nearly
uniformly bright from center to edge.

\ni Write down the specific intensity the observer measures. Neglect terms
of order $H/d$, where $d$ is the distance to the galaxy.

%\vspace{0.1in}
%\ni
%{\large{\bf Problem 4.}} Planet Hydrogen

%\ni {\bf NOTE:} This problem is required only for people who
%are not permitted to do problem 3. But it isn't hard, and I would
%encourage everyone to try it.

%\ni (a) If every gas molecule in the
%Earth's atmosphere were replaced by molecular hydrogen,
%what color would the sky be?

%\ni (b) Explain why the atmosphere is/is not
%in local thermodynamic equilibrium.

%\ni (c) Estimate the optical depth of the (real) Earth's atmosphere
%to blue light, and to red light, when looking at zenith.

\end{document}



