Let's define the intensity of an emission line $I(v',J'\rightarrow v'',J'')=N(v',J')A(v',J'\rightarrow v'',J'')$ as the number of photons emitted per unit time per unit volume due to the $(v',J')\rightarrow (v'',J'')$ line. Also, let's define the intensity of an emission band $I(v'\rightarrow v'')=N(v')A(v'\rightarrow v'')$ as the number of photons emitted per unit time per unit volume due to all transitions with $v'\rightarrow v''$. I would assume that one should have $$ I(v'\rightarrow v'')=\sum_{J'}\sum_{J''}I(v',J'\rightarrow v'',J'')$$ I would like to know how I can carry out the summation in terms of e.g., the rotational partition function and the $A(v',J'\rightarrow v'',J'')$ transition probability expression [e.g., Morton and Noreau, The Astrophysical Journal Supplement Series, 95, 1, 301-343, 1994] to prove this equality if it indeed is valid.
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