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# In the branching probability we understand that

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Unformatted text preview: . 2 B. LEVEL NETWORKS To study the quantitative implications of the above process, we introduce the level ­ resolved recombination coefficient αnl, which is the contribution to α coming from recombinations to the nl level. We further introduce the cascade matrix C(nl,n’l’), which is the probability that an atom in the nl level will decay via a path that includes n’l’. We may also define the branching probability, Anl ,n ' l ' P ( nl, n ' l' ) ≡ , ∑ Anl,n'' l'' n '' l '' which is the probability that an atom in the nl level will decay directly to n’l’. In the branching probability, we understand that channels blocked by radiative transfer effects € (the Lyman lines) are excluded from the denominator. The rate of production of atoms in the nl level (in cm−3 s−1) is then given by ∑ C(n' ' l' ', nl)α n'' l'' ne n p . n '' l '' The rate of emission of photons in the nln’l’ line is then this rate times the branching probability, and the line emissivity (in erg cm−3 s−1) is obtained by multiplying by hν: € 4 πj nl →n ' l ' = hν nl →n ' l ' P ( nl, n ' l' ) ∑ C ( n ' ' l' ', nl)α n '' l '' n e n p . n '' l '' We know how to compute all of the coefficients here except for the cascade matrix. The latter is most easily obtained via induction. It is trivial to see that: € Ⱥ 0 n ' > n C ( nl, n ' l' ) = Ⱥ . Ⱥ δll ' n ' = n For n’<n, we can find the cascade matrix for each n’l’ by considering the probability to reach n’l’ from any higher level: € n C ( nl, n ' l' ) = n ' ' −1 ∑ ∑ C(nl, n' ' l' ' )P (n' ' l' ', n' l' ) . n ' ' = n ' +1 l ' ' = 0 By this method, we can calculate the hydrogen line ratios in the low ­density limit. These in principle depend on temperature, but the dependence is extremely weak, since all of the € αnl’s decrease slowly with tem...
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## This document was uploaded on 03/08/2014 for the course AY 102 at Caltech.

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