Theory of Line Emission and Absorption

Consider a gas which is exposed to radiation at the resonant frequency v. The molecules undergo transitions between the lower and the upper energy levels. In the case of statistical equilibium the number of molecules per unit time undergoing transitions from the lower state to the upper state (= absorption) equals the number of molecules making the reverse transition (caused by spontaneous and stimulated emission) 

Furthermore the Einstein coefficient Aui may be given in terms of the B coefficients by use of Planck s radiation law 

The absorption coefficient a v the fractional change in energy density per unit length, is times the net absorption per cm3 per secondbl r00 h 

In the case of local thermodynamical equilibrium (LTE) at a temperature T, the distribution of the molecules over different states is given by the Boltzmann distribution  

However, if there is no thermodynamic equilibrium, as is common for interstellar cases, it is convenient to define between two energy levels an excitation temperature Tex which is determined by fitting the observed molecular distribution to the Boltzmann formula, Eq. (13). 

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