lecture notes9

Putting in the normalization suggests aly 4 2

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Unformatted text preview: isions begin to dominate – usually of order 100 – the departure coefficient increases with n. In some cases, this can lead to a population inversion and maser activity. 3. Additional Processes The spectrum of an H II region consists largely of the recombination processes described above, collisionally excited cooling lines, and (in the radio) the free ­free continuum. There are, however, a few additional features of the spectrum that should be pointed out. A. LYMAN α EMISSION We need to understand the transport of Lyman ­α radiation if we are to understand the emission line strength. The key factor here is the scattering cross section, σ(ν). This can be determined by the principle of detailed balance. We note that in thermal equilibrium at temperature T, the 2p:1s population ratio is 3 exp( ­ELyα/kT). Therefore, the upward transition rate (in s−1) in a blackbody spectrum must be 3 exp( ­ELyα/kT) ALyα for ELyα>>kT.1 This means that: ȹ − hν ȹ ȹ −hν ȹ 8πν 2 ∫ c 2 σ (ν ) expȹ kT ȹdν = 3ALyα expȹ kTLyα ȹ . ȹ Ⱥ ȹ Ⱥ This implies that the cross section is: € 1 This restriction is necessary so that we can ignore stimulated emission of Lyman ­α. 6 σ (ν ) = 3λ2 α ALyα Ly δ (ν − ν Lyα ) . 8π This formula is not directly useful as written because for most applications we need to resolve the width of the Dirac δ ­function. This has 3 major sources: € Natural broadening: The exited atom H(2p) has a finite lifetime of ~ 1.6 ns. Therefore its energy must be ill ­defined in accordance with the uncertainty principle and the δ ­function correspondingly smeared out. A “co...
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