Homework 3 Solutions

Points depend on the dimensionality of your solution

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Unformatted text preview: (z ) gH Substituting this into the previous equation, assuming H is constant (because T is constant), and integrating from z = 0 to z = 3H yields: ln (I (0)/I (z )) = κλ P (0) −3 (e − 1) = −29.1κλ g Where I’ve used P(0) = 300 Pa (note how the combination of g/κλ has the same units as pressure – this is important!). For our two wavelengths: Optical (κ0.5µm = 10 ­4 m2/kg): I(0)/I(z) = 0.997, or a decline of only 0.003 mag – our atmosphere is pretty transparent at visible wavelengths! Near ­infrared (κ1µm = 0.1 m2/kg): I(0)/I(z) = 0.054, or a decline of 3.2 mag. This is actually too much, and is because our assumption that water vapor opacity looks like liquid water opacity is wrong at these wavelengths. Water vapor has strong peaks in opacities like those sene above due to rotovibration features (stretching and bending of molecular bonds), but in between these featuers opacity is quite low. (4) Random walk [20 pts] 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 program that performs a series of random walks. Each walk should be 1000 steps of unit length, with the direction being random at each step. You are to compute 1000 such walks, and plot the distribution of final distances from the origin, as well as compute th...
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