Astronomy 722: Radiative Transfer and Gas
Dynamics
TH 326, San Francisco State University
c
2016
Andisheh Mahdavi
Spring 2016 Tu Th 5:10PM
Homework 2 Due 5:10PM 2/18
While I may have consulted with other students in the class regarding this homework, the solutions presented here
are my own work. I understand that to get full credit, I have to show all the steps necessary to arrive at the answer,
and unless it is obvious, explain my reasoning using diagrams and/or complete sentences.
Name
Signature:
1. (20%) Consider a plot of the flux spectrum Fν on the y axis vs. frequency ν on the x axis. Show that the
area under that curve between any two frequencies ν1 and ν2 is the same as the corresponding area under the
curve which plots (ν × Fν × ln 10) vs. log10 ν. Explain why an astronomer might think this result is useful or
interesting. Plot the blackbody spectrum which is most similar to the Sun’s using both methods—you should
plot Fν as observed above Earth’s atmosphere. Use cgs units for the x and y axes.
2. (20%) Try working through problems 1.8 and 1.9 on your own, without consulting the back of the book. Then
check the answer in the back of the book and correct as needed. Provide explanations in your own words.
3. (60%) Imagine a spherically symmetric cloud of dust of mass M and radius R. The cloud has a spherical star
at the center. The star is much, much smaller than the cloud of dust. Within the frequency range accessible
to you, the dust has no emission, but does absorb the star’s light with unknown frequency-dependent opacity
κν .
The cloud of dust has a power law density distribution which I’ve inferred through separate (molecular gas)
observations:
ρ(r) ∝ r−2
where r is the 3D distance from the center of the cloud.
The star has radius R? and gives off blackbody emission of temperature T . I already know R? and T (from
separate observations that don’t factor into this problem).
I sit with a telescope a distance D from the star and observe its spectrum.
(a) (25%) On my telescope, I observe this object as a point (i.e. infinitesimally small) source. I find it has a
smooth spectrum with no absorption lines, no local minima, and with just one peak (global maximum) at
frequency ν = kT /h. Find a frequency-dependent opacity law κν that would cause you to observe this
spectrum. There are many opacity laws possible.
(b) (10%) What generic features are shared by all opacity laws that produce this type of absorption?
(c) (15%) What is the frequency-dependent optical depth from the edge of the cloud to the surface of the
star as viewed by me? At which frequencies does the cloud become optically thin (τν < 1)? Can you
think of a clever way to use this to constrain some combination of M and R?
(d) (10%) What is the total flux (integrated over all frequencies) that arrives at my location?
N.B. Numerical integration will be required for this problem.
1