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Unformatted text preview: Physics 160: Stellar Astrophysics Homework #3 Due Tuesday October 18th at 5pm in SERF 340 Reading: Carroll & Ostlie sections 9.19.4, 10.110.6
Exercises [80 pts + 20 pts BONUS]:
(1) [10 pts] Show that Planck’s relation 1
2hc2
BT (λ) = 5 hc/λkT
λe
−1
reduces to the Wein relation for small wavelengths (λ << hc/kT): BT (λ) = A −B/λT
e
λ5 and the RayleighJeans relation at large wavelengths (λ >> hc/kT):: BT (λ) = C T
λ4 and solve for the constants A, B and C
(2) [35 pts] Consider two stars: the 5,800 K Gtype star the Sun, and the
33,000 K Otype star λ Orionis. Assume that the atmospheres of both stars
are pure H and have a density of 106 kg/m3.
(a) [10 pts] For both stars, what is the ratio of neutral H atoms in the
first and second excited states (n = 2 and 3)?
(b) [15 pts] For both stars, what fraction of H atoms are ionized? Note
that the electron density used in the Saha equation is equal to the
density of ionized H atoms.
(c) [10 pts] H I Balmer α absorption comes from the n = 3→2
transition; H I Lyman α absorption comes from the n = 2→1 transition.
What is the relative strength of these two lines in the Sun and λ
Orionis? Does this depend at all on the degree of H ionization?
(3) [10 pts] The center of the Sun has a temperature of roughly 1.5x107 K
and density of 1.6x105 kg/m3. What is the ionization fraction of H at that
temperature? Is this consistent with our usual assumption that the Sun’s core is pure plasma?
(4) [10 pts] A 0.5 M mainsequence star and a 1.2 M giant star can have
the same spectral type (M0) but not necessarily the same atmospheric
temperature. Using an estimate of the mean density of the two stars as a
proxy for the atmospheric pressures (see Appendix G for radii), argue why
this would be the case based on the Saha equation.
(5) [15 pts] The opacity of water vapor in Earth’s atmosphere is about 0.05
m2/kg at visible wavelengths. Assume standard temperature and pressure
(STP): T = 0ºC and P = 100 kPa.
(a) [5 pts] Calculate the mass densities (in kg/m3) of pure water vapor
(100% H2O) and dry air (78% N2, 21% O2) in our atmosphere. Why do
clouds float?
(b) [5 pts] How thick does a cloud of pure water vapor have to be to
block out the Sun? Assume you have to decrease the apparent
magnitude of the Sun to V = 6 for it to be invisible. Does this answer
seem reasonable? What might we be missing?
(c) [5 pts] Estimate the opacity of dry air (hint: how far can you see on
a clear day?). Roughly how thick would our atmosphere have to be for
us to not see any stars at night?
(6) [20 pts BONUS] Random walk experiment: As discussed in section 9.3,
the path a light ray takes to escape an optically thick medium can be
described as a random walk due to absorption and scattering, with steps
equal to the mean free path, L. In this question, you are going to write a
simple program that computes a 1D random walk.
(a) [10 pts] Write a program that calculates the position of a particle
after a series of 100 random steps. A simple algorithm is to iteratively
add to a position variable (initially 0), at each step randomly choosing a
step of 1 or +1. Repeat these steps 100 times and save the final
distance. Provide a printout of the program you used to perform this
calculation.
(b) [10 pts] Perform this calculation 1000 times, and plot the histogram
of these distances. What does this distribution look like? What is the
mean, median and standard deviation of your distance values? How are
these numbers related to the number of steps taken? ...
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 Fall '08
 Norman,M
 Physics, Work

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