A. Belleni-Morante

An Inverse Problem for Photon Transport in Interstellar Clouds

Abstract We study an inverse problem for UV-photons in an interstellar cloud that occupies a bounded and convex region V⊂ R 3. We assume that the photon number density is known (e.g., experimentally) at some , where is a location “far” from V and the unit vector is such that the straight line, passing through and parallel to , crosses the interior of V. We show that the knowledge of is sufficient to determine uniquely the total cross section σ(x), x∈ V, within a suitable family .

RADIATIVE TRANSFER IN THE STOCHASTIC INTERSTELLAR MEDIUM

Abstract 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 discu...

Time dependent photon transport in a three dimensional interstellar cloud with stochastic clumps

We consider time dependent photon transport in a three dimensional interstellar cloud which occupies a three dimensional regionV. One or more clumps of given shapes are present withinV and their positions are determined by a suitable set of stochastic variables. Iff is the photon number density in the cloud or in the clumps, then our mathematical model leads to two coupled initial value problems for the average photon density over the stochastic variables 〈f〉 and forf* =f -〈f〉. By using the theory of semigroups, we prove existence and uniqueness of a strongly continuous solution and examine the small fluctuation approximation of such a solution.

A kinetic model for dust coagulation

Abstract A Boltzmann-like model is developed for particle transport in presence of coagulation, in which evolution equations for the number densities of small and large particles are derived. Unlike the standard Boltzmann equation, number densities have a dependence on the particle mass.

Mathematical methods for photon transport in random media

Abstract Radiative transfer through random media is studied into an abstract Banach space setting. The attention is focused on systems whose stochastic behaviour can be described by means of a finite number of random parameters, such as a cloud with stochastic clumps. By using two different projection methods, sequences of evolution equations, which approximate the exact equation for the expectation, are deduced.

Some remarks on densely defined streaming operators

We show that the streaming operator in particle transport problems and in population dynamics is densely defined, even if the boundary conditions are not the standard nonreentry boundary conditions.

GLOBAL SOLUTION FOR A PROBLEM OF NEUTRON TRANSPORT WITH TEMPERATURE FEEDBACK

Publisher Summary This chapter describes the global solution for a problem of neutron transport with temperature feedback. The behavior of a nuclear reactor with temperature-dependent feedback was studied by using the diffusion approximation of neutron transport and by assuming that feedback was acting only through the multiplication factor. In another study, the same problem was briefly examined under the assumption that the macroscopic cross-sections were linearly dependent on temperature. The nonlinear neutron transport problem under consideration was first transformed into a nonlinear abstract problem; the theory of evolution equations in Banach spaces was then used to prove existence and uniqueness of a solution, defined on a suitably small time interval [0, t]. This chapter discusses the same problem referring for simplicity to a slab of thickness 2L with cross sections depending on temperature in a sufficient smooth way. In this way, by introducing two auxiliary linear neutron transport problems, it can be proved that the nonlinear problem under consideration admits a unique solution belonging to a suitable Banach space and defined on any arbitrarily fixed finite time interval [0, t].

Two inverse problems in photon transport theory: evaluation of a time-dependent source and of a time-dependent cross section

In photon transport theory, two types of inverse problems are considered: (a) identification of some physical or geometrical quantity (such as a cross section, or a photon source, or the shape of the surface that bounds the host medium), evaluating its dependence on spatial and/or angle variables, under the assumption that photon transport is time independent and starting, for instance, from the knowledge of the exiting photon flux; (b) identification of some physical or geometrical quantity that characterizes the host medium, evaluating its dependence on spatial and/or angle variables and on time, under the assumption that photon transport is time dependent and starting, for instance, from the knowledge of the time behaviour of the exiting photon flux.

A numerical approach for inverse problems in photon transport inside an interstellar cloud

In the paper we formulate and solve two inverse problems interesting the physical structure of interstellar clouds. In the problems the unknowns are the dimension of the cloud and its total cross-section of absorption. We solve numerically an integro-differential equation of photon transport by using, as additional data, measured values of the photon density outside the cloud. Numerical results are provided by an iterative scheme proposed in the paper.