The survey is devoted to applications of nonlinear integral equations to linear convolution equations, their discrete analogues, and also the connection of these equations with problems of radiative transfer, in particular, with the Ambartsumyan equations.

Convolution equations and nonlinear functional equations

The diurnal variation of the vertical structure of Titan's thermosphere is calculated through simultaneous solution of the equations of heat transfer and hydrostatic equilibrium. The temperature and density profiles are found above the mesopause. The dynamical response of the thermosphere to heating is for the most part neglected. Nevertheless, we are able to draw some interesting qualitative and quantitative conclusions regarding the vertical structure. Heating of the upper thermosphere occurs primarily through absorption of solar Lyman α radiation by methane, with an additional amount of heating (≲20%) due to low-energy magnetospheric electron precipitation. The heat is conducted downward to the mesopause, where it is removed by IR cooling due principally to acetylene. The mesopause is found to occur where the density is 2.2×10^(12) cm^(−3) (736 km), and has a temperature of ∼110 K. The exospheric temperature is unlikely to exceed 225 K in the course of a Titan day. The diurnally averaged exospheric temperature is in the range 187–197 K depending on the amount of magnetospheric electron heating that is included in the model. The amplitude of the diurnal variation is found to be ≲28 K. We find that the vertical extent of the hydrogen cloud is too large to be explained in terms of simple thermal escape of hydrogen from a ∼225-K exosphere, and we conclude that other processes must be important for populating or heating the neutral torus.

The thermosphere of Titan

Comparisons with Experiments

Collision-Induced Absorption (CIA)

Radiative Cross Sections

The paper is devoted to the consideration of the problems of non-coherent scattering in a spectral line for an arbitrary redistribution law. The representation of the redistribution function r(x′,x) in the form r(χ′,χ) = ∑ k = 1 ∞ A k α k (χ′)α k (χ) is used to study the various mechanisms of line broadening. The form of representation plays an important role in this investigation. New methods for solving radiative transfer integro—differential equations, as well as the corresponding integral equations are suggested. The solution of integral equations involves factorization of matrix integral operators. The generalized classical problems of absorption-line formation in stellar atmospheres are formulated and solved (Shuster's and Eddington's models). The results of numerical calculations, carried out to the fourth order, are given for Doppler broadening.

N.B. Yengibarian

Papers

Non-coherent scattering