Question

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6.8 The Doppler formula for the observed frequency f from a source moving with velocity along the line of sight of an observer is where fo is the frequency in the rest frame of the source. (a) What is the distribution in frequency of a partícular spectral line radiated from a gas at temperature T? (b) Find the breadth of the line, de fined as the variance (f -fo)2. (c) Atomic hydrogen and atornic oxygen are both present in a hot gas. How muclı broader is the hydrogen line compared to the oxygen line, of rouglıly the same frequency?

Answer 1

Given f = f_0 (1 + v_x/c). By using maxwellian probability distribution, The velocities in v_x to v_x + d v_x region given by probability function p(v_x). p(theta_x) = (M/2 pi k T)^1/2 exp (-M/2 k T v^2_x) dv_x corresponding frequency distribution G(f) is given by in df region is. G(f) df = (M/2 pi k T)^1/2 integral^-infinity_infinity exp (-M/2 kT v^2_1) dv_x exp (-M/2 kT) c^2/f^2_0 (f - f_0)^2) c/f_0 df. we know that integral^infinity_-infinity (exp (-(m/2 k T)^1/2 v^2_x) dv_x = (2 pi k T/M)^1/2 therefore G(f) = c/f_0 (M/2 pi kT)^1/2 exp ((-M/2 KT) (c^2/2 kT) (c^2/2 kT) (c^2 f^2_0) (f - f^2_0)) \r\n therefore width given by Delta f - f - f_0 = 2f_0 squareroot 2(n_2) kT/mc^2 = 7.16 times 10^-7 f_0 squareroot T/M_M Delta^bar^2 f = 51.26 times 10^-14 f^2_0 (T/M_M) Delta f_M = 7.16 times 10^-7 f_0 squareroot T/M_NH Delta f_0 = 7.16 times 10^-7 f_0 squareroot T/N_No therefore Delta f_M