Physics 521, Stars
Stanimir Metchev, Nov 19, 2011
Homework Assignment 5
Due in class on Tuesday, Dec 1, 2011
Material covered: Lectures 17–21; HKT §2, §4 and Gray §5, §7.
1. (20 points) Exercise 2.22 of HKT.
2. (10 points) Exercise 4.1 of HKT.
3. (10 points) Exercise 1 from §5 of Gray: Experiment with some numerical
manipulation of spectra to help understand their nature as a distribution (histogram) of
photons. For example, make a graph of Planck’s radiation law
2h" 3
1
S" = 2 h" kT
.
c e
#1
What units does Iν = Sν have? As with any histogram, one has some choice of the binning
interval. What binning interval have you chosen? How would your graph change if the
binning interval were reduced by a factor of 10? Now use the relation Iν dν = Iλ dλ to
!
convert Iν to Iλ. Make a graph of Iλ vs. λ. Compare this new graph to the Iν graph. Do
they peak at the same wavelength? If not, why not? On the Iλ graph, plot Iν as a function
of λ. Why are these two plots not the same?
4. (20 points) Line radiation is emitted from an optically thin, thermal source, so that
the spectrum of the emitted radiation is proportional to the emission function:
h"
j" = 0 n 2 A21$ (" ),
4#
where ν0 is the central line frequency, n2 is the number density of atoms with electrons in
level 2, A21 is the Einstein A coefficient for the transition, and φ(ν) is the line profile
function. Assuming that the only broadening mechanisms are Doppler and natural
!
broadening, show that the observed half-width of the line is independent of the
temperature T for T << Tc and increases as the square root of T for T >> Tc, where Tc is
some critical temperature. For the Lyman-α (1s→2p) line of hydrogen estimate Tc in
terms of fundamental constants, and give its numerical value.
Hint: Express the natural width γ for the Lyman-α transition in terms of its Einstein
A coefficient, and use the following expression for A21:
8" 2e 2# 02 g1
A21 =
f12 ,
me c 3 g2
where e is the electron charge, me is the electron mass, g1 and g2 are the statistical weights
of the 1s and 2p states of hydrogen (evaluate these), and the product of g1 and the
absorption oscillator strength f12 is g1f12 = 214/39. Make sure to also express ν0 of Lyman! constants, and to then evaluate it numerically.
α in terms of fundamental