lecture notes9

Thus most of our interest in the appearance of lyman

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Unformatted text preview: rrect” treatment requires quantum electrodynamics, but the answer can be obtained by treating the wave function of the excited state as a damped harmonic oscillator with damping ~ exp( ­ALyαt/2). The response of a resonant system with this damping leads to a smearing out of the energy in the oscillator, ∝ [Δω2 + (ALyα/2)2]−1. Putting in the normalization suggests: ALyα / 4 π 2 , δ (Δν = ν − ν Lyα ) → φ nat (Δν ) = Δν 2 + ( ALyα / 4 π ) 2 which is correct for an atom at rest. Doppler broadening: The atoms are not actually at rest, but moving with a € Maxwellian velocity distribution. Each component of v is thus a Gaussian with variance kT/mH. This implies a distribution of absorption frequencies: 1 −Δν 2 δ (Δν = ν − ν Lyα ) → φ th (Δν ) = 1 / 2 exp , 2 π Δν D Δν D where € 2 kT Δν D = ν Lyα . mHc 2 Turbulent broadening: If the gas is turbulent then the velocity of the eddies can add to the Doppler broadening. We will neglect it here, which is appropriate for € hydrogen if the turbulence is subsonic (recall that the sound speed in an ideal gas is comparable to the speed of the atoms that make it up). The true broadening ϕ(Δν) is the convolution of both of these, which is called a Voigt distribution. The natural broadening width is ALyα/4π = 50 MHz, whereas the Doppler broadening is 106T41/2 GHz. These are both small compared to νLyα = 2465 THz. Under ordinary circumstances, the Doppler broadening is larger and dominates. The exception is that the natural broadening has power ­law tails to large |Δν|, and so the occasional absorption far from line center is dominated by natural broadening. 7 In the case where we are dominated by the Doppler broadening, t...
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