Analytical properties of the radiance in atmospheric radiative transfer theory

•Proof of existence of the radiance within the scope of the theory of atmospheric radiative transfer.•Proof of relations between the photon number and photon energy density function and the radiance.•Strictly mathematical derivation of the analytical properties of these state density functions.

It is demonstrated mathematically strictly that state density functions, as the radiance (specific intensity), exist to describe certain state properties of transported photons on microscopic and the state of the radiation field on macroscopic scale, which have independent physical meanings. Analytical properties as boundedness, continuity, differentiability and integrability of these functions to describe the photon transport are discussed. It is shown that the density functions may be derived based on the assumption of photons as real particles of non-zero and finite size, independently of usual electrodynamics, and certain historically postulated functional relationships between them were proved, that is, these functions can be derived mathematically strictly and consistently within the framework of the theory of the phenomenological radiative transfer if one takes the theory seriously by really assuming photons as particles. In this sense these functions may be treated as fundamental physical quantities within the scope of this theory, if one considers the possibility of the existence of photons.