Abstract
The paper treats the quadratic and bilinear integrals of the transfer equation for the homogeneous plane-parallel atmospheres. The method for derivation of these integrals differs from that developed previously by the author with use of Ambartsumian’s invariance principle. The variational formalism is used to show that they are a direct consequence of the form-invariance of the proper Lagrangian with respect to the depth translation transformation. The concept of a group of the transfer problems reducible to the source-free problem is defined. These problems admit the closed-form quadratic and bilinear integrals and are coupled with each other by means of non-linear relations. It is shown that the quadratic and bilinear equations are obtainable for atmospheres with the energy sources distributed according to polynomial law of arbitrarily high degree. The finite atmosphere is considered as well. We derive a number of new relations that link the solutions of the transfer problems in the finite and semi-infinite atmospheres. The four-point bilinear relations for the Green’s function we obtain represent the most general and informative manifestation of the invariance properties of the transfer problems discussed.
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