AN IMPLICIT INTEGRAL METHOD TO SOLVE SELECTED RADIATIVE-TRANSFER PROBLEMS .1. NON-LTE LINE FORMATION

In this paper we present a new method to solve those radiative transfe r problems where the scattering term in the source function, i.e., the frequency-integrated mean intensity J(phi), is independent of both fr equencies and directions. This particular form of the source function, together with an implicit description of the evolution of the specifi c intensities I-(tau(L); mu, nu) and I+(tau(L + 1); mu, nu) incoming t o an individual layer (tau(L), tau(L + 1)) from the neighboring ones, allows one to solve implicitly the radiative transfer equation layer b y layer. Consequently, J(phi) can be expressed as an explicit function of the (as yet unknown) specific intensities I+(tau(L + 1); mu, nu), without any need to solve numerically a system of equations or to inve rt matrices. In this way, the global problem is reduced to a series of one-layer two-point boundary problems. The resulting algorithm is the representation of the actual physical process. This, together with th e fact that it does not require a matricial formalism, brings about se lf-evident advantages in terms of reliability and numerical accuracy, to say nothing of the conspicuous saving of both computational time an d memory storage. As an application, the instance of the spectral line formation in a two-level atomic model is considered here. This import ant paradigm case not only is at the basis of many actual problems of line transfer but also is often the key to their solution. Moreover, i t offers a template to check the numerical accuracy of any solution of the radiative transfer equation. We have left to a second paper in th is series the application of our method to the problem of the temperat ure correction within an LTE stellar atmosphere. This certainly will b e of use for modeling purposes.