Radiative transfer in the stochastic interstellar medium

A Boltzmann-like mathematical model is developed for photon transfer in an interstellar cloud, containing one or more clumps, whose centers are stochasticly distributed. The outline of the method is given considering the simplest approach: time-independent transport through a purely absorbing medium composed of two randomly mixed fluids. The emphasis of the work is on the statistical description of the two immiscible fluids. The density distribution of the mixture is described by means of a random field function which maps the structure of the medium. As a consequence, each realization of the statistics corresponds to a possible configuration of a “real” interstellar cloud, as it is possible to infer from the observations. An equation for the expected value of the photon intensity is derived using the method of smoothing. This equation contains an infinite formal Neumann series which includes multiple applications of the inverse transport operator. The reliability of the truncated series is discussed and the explicit expression of the generic term of the expansion and of the general multipoint autocorrelation functions of the cross-section is given. Finally, it is shown that, up to an arbitrary order, the radiative transfer equation is actually a Volterra equation. Possible extension to include time-dependence as well as scattering process are also discussed.