Derivation of OH sum rules without Rayleigh--Jeans approximation: a powerful tool for measuring the variation of fundamental physical constants

In this paper, we show that the optical depths of any OH Λ-doublet quartet obey one exact linear relation in every velocity channel. No approximation is needed. The only exception is the squaredΠ 1/2 J = 1/2 level, which has three transitions only, because its F = 0 0 line is strictly forbidden. The relation is τ S 1(v)/μ S 1 squared+τ S 2(v)/μ S 2 squared=τ M 1(v)/(aμ M 1 squared)+τ M 2(v)/(bμ M 2 squared), where a=2J squared+J-1, b=2J squared+3J, and the μ i squared are reduced transition dipole moments derived from the catalogued Einstein A coefficients. It holds equally for gas in local thermodynamic equilibrium (LTE), for non-LTE gas, for masers and for conjugate gas. We suggest that it is a powerful tool for measuring the variation of fundamental physical constants.