hw3 - Physics 160: Stellar Astrophysics Homework #3...

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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 9.1-9.4, 10.1-10.6 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 Rayleigh-Jeans 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 G-type star the Sun, and the 33,000 K O-type star λ Orionis. Assume that the atmospheres of both stars are pure H and have a density of 10-6 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 main-sequence 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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