Astronomy 722: Radiative Transfer and Gas
Dynamics
Thornton 326, San Francisco State University
c
2016
Andisheh Mahdavi
Spring 2016, TuTh 5:10PM
Homework 1 Due 5:10PM 2/11
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%) Find the peak frequency νmax of the Sun’s specific intensity when it is approximated by a blackbody in
frequency space, Bν (ν). Convert νmax to a wavelength, λmax = c/νmax . Repeat the exercise in wavelength
space, i.e., find the maximum of Bλ (λ), and covert it back to a frequency. Comment on the results.
2. (40%) An spherical distribution of optically thin plasma has radius R and constant emission coefficient jν .
Find the observed flux of the plasma Fν everywhere in space. Your answers should be in terms of jν , R, and
the distance of the observer D.
3. (40%) The habitable zone around a star is defined as the zone within which water on the surface of a rocky
body can be liquid.
(a) (15%) Estimate the inner and outer radius of the “habitable zone” around a main-sequence star of
temperature T? . To do this you will need to make the following simplifying assumptions:
• The radii of main-sequence stars scale roughly as the square of their temperatures, and they emit
blackbody radiation.
• The planet in the habitable zone rotates quickly, has no atmosphere, and reflects a constant fraction f
of all the light incident on it, absorbing the rest and being heated by it. The heating allows the planet
to achieve thermal equilibrium such that its surface reradiates blackbody emission of temperature Tp .
(b) (10%) Compare the total outgoing flux at frequencies less than ν = 3kTp /h on the surface of the planet
to the total flux of incoming radiation from the star at the same wavelength. Find the value of f at which
the two are equal.
(c) (10%) Compare the size of the habitable zone around the Sun as computed above to the Earth’s location
and comment on any differences. Look up any quantities as needed. I’ll start you off with the mean
albedo of the Earth, f = 0.35.
(d) (15%) Now imagine that the planet has a constant density atmosphere that has opacity equal to
κ0 ν < ν0
κν =
0
ν > ν0
where κ0 is a constant. Assume that all the energy absorbed goes into heating the planet. Calculate the
temperature of the planet as a function of time. Introduce heretofore unmentioned quantities as needed,
explaining why you are introducing each of them.
(e) (10%) Apply the above to the Earth-Sun system. Look up quantities as needed. A good approximation
is κ0 = 10−4 cm2 /g and ν0 = c/(10µm)for the atmosphere. Comment on the results.
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