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Astro 210 Homework 8 Solutions

# Astro 210 Homework 8 Solutions - Astronomy 210 Spring 2011...

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Astronomy 210 Spring 2011 Homework Set #8: Solutions 1. The random walk of light inside the Sun. (a) [5 points] At each step, each particle has an equal chance of going to the left or to the right. This symmetry means that the entire collection (“ensemble”) of particles, and the average motion of a single particle, has not net displacement left or right. Thus We must have ( D N ) = 0 for all N . To infer this in a more mathematical way, we can use mathematical induction. The particle starts (at “step N = 0”) at the origin: D 0 = 0. A particle at step N has distance D N , with some (as-yet unkown) average ( D N ) . The next step has a 50% chance of being at distance D N +1 = D N + , and a 50% chance of being at distance D N +1 = D N . The average of these is ( D N +1 ) = ( D N ± ) = 1 2 ( ( D N ) + ) + 1 2 ( ( D N ) − ) = ( D N ) (1) And thus we infer that ( D N +1 ) = ( D N ) = ( D N 1 ) = . . . = ( D 0 ) = D 0 = 0 (2) (b) [5 points] The first step has equal probabilities of D 1 = ± , and thus both possibilties have D 2 1 = 2 . Taking the average of these two equal values just gives the same value: ( D 2 1 ) = 2 .

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Astro 210 Homework 8 Solutions - Astronomy 210 Spring 2011...

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