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h3s - Homework 3 Solutions 1 Calculate the size of a...

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Homework # 3 Solutions 1. Calculate the size of a Stromgren sphere in a gas of density 10 3 cm - 3 sur- rounding an O3 star. This is the volume of gas nearly completely ionized by the star. Assume the recombination rate of protons and electrons to form neutral hydrogen is dn e dt = - 2 × 10 - 13 n e n p T 10 4 K - 3 / 4 cm - 3 s - 1 . A Stromgren sphere represents a balance between the rate of production of ionizaing photons and recombinations. The ionization rate doesn’t matter once equilibrium is established since it is generally faster than recombination. The flux of ionizing photons is F ν = Z ν 1 L * ( ν ) 4 πr 2 = R 2 * r 2 Z ν 1 2 ν 2 hc 2 1 e hν/kT - 1 The spectral distribution peaks near p 2 kT . When ν 1 ν p the lower limit of the integral can be taken to be 0 and most of the star’s photons ionize, but when ν 1 < ν p only a fraction of the star’s photons can ionize. For these limits Z a x 2 dx e x - 1 ( a 2 + 2 a + 2 ) e - a a > 0 . 9 2 ζ (3) - a 2 / 2 2 . 4 - a 2 / 2 a < 0 . 9 For an O3 star, T = 44 , 500 K, a = 1 /kT = 3 . 6. For a G3 star, T = 5780 K, a = 27 . 3. Thus, the fraction of photons available is 25% for the O3 star, but
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