Finite Rank Approximation Based Method for Solving the Rte in Stellar Atmospheres and Application to an Inverse Problem

The Finite Rank Approximation (FRA) based method is well known in operator approximation theory but it is also useful for suggesting numerical methods for solving integral equations. In this document we describe two FRA methods for the numerical resolution of the integral formulation of the 1D Radiative Transfer Equation (RTE) posed in a static slab; we browse some advantages of them (especially the possibility to control the error) and we give some reduction of computation technics (iterative refinement schemes) which can be used with these methods. Numerical results obtained for a realistic Sun atmosphere model are given. In the last section we give an example of an inverse problem associated to the RTE: we show the iterative recovering of the albedo from the measurement of the outgoing specific intensity at a surface of the considered domain. 1 The transfer equation in stellar atmospheres We consider a simplified 1D steady-state transfer equation posed in a static slab, stratified in plane-parallel homogeneous layers (Mihalas 1970): for all τ ∈]0, τ⋆[ and all μ ∈ [−1, 1], μ ∂I ∂τ (τ, μ) = I(τ, μ)− ̟(τ) 2 ∫ 1 −1 I(τ, μ) dμ − S⋆(τ), (1.1) The first author acknowledges FONDECYT 1030808 and ECOS CO4E08 grants. The second author acknowledges the support of ECOS-Sud postdoctoral grants: www.ecos.univ-paris5.fr. The authors are grateful to Bernard Rutily (Centre de Recherche Astronomique de Lyon, France) for fruitful discussion on this topic. 1 Departamento de Ingenieŕıa Matemática Universidad de Chile, Blanco Encalada 2120, Santiago de Chile, Chile; e-mail: [email protected] 2 Centro de Modelamiento Matemático UMI 2807 CNRS, Universidad de Chile, Blanco Encalada 2120, Santiago de Chile, Chile 3 Centro de Modelamiento Matemático UMI 2807 CNRS, Universidad de Chile, Blanco Encalada 2120, Santiago de Chile, Chile; e-mail: [email protected] c © EDP Sciences 2006 DOI: (will be inserted later) 2 Title : will be set by the publisher where I, ̟ and S⋆ denote the specific intensity, the albedo and the primary creation rate respectively. The position variable τ is the optical depth at a given frequency, whose maximum value τ⋆ is the optical thickness of the atmosphere. The cosine of the angle of incidence (zenith angle) is denoted by μ. We suppose moreover that we know the specific intensity of the incoming radiation on both boundary planes, i.e. { I(0, μ) = I 0 (μ) −1 ≤ μ < 0, I(τ⋆, μ) = I + ⋆ (μ) 0 < μ ≤ 1, (1.2) where I 0 and I + ⋆ are given functions (see Figure 8a section 5). We define the source function by S(τ) = S⋆(τ) + ̟(τ) 2 ∫ 1 −1 I(τ, μ) dμ, τ ∈ [0, τ⋆]. (1.3) By injecting (1.3) in (1.1) we get an expression of I in terms of S and the boundary values I 0 and I + ⋆ : for all τ ∈ [0, τ⋆], I(τ, μ) =    I 0 (μ) exp [ τ μ ] − 1 μ ∫ τ 0 S(s) exp [ τ − s μ ] ds, μ < 0, S(τ), μ = 0, I ⋆ (μ) exp [ − τ⋆ − τ μ ] + 1 μ ∫ τ⋆ τ S(s) exp [ − s− τ μ ] ds, μ > 0. (1.4) The source function satisfies the following weakly integral equation S(τ) = S0(τ) + ̟(τ) 2 ∫ τ⋆ 0 E1(|τ − σ|)S(σ) dσ, τ ∈ [0, τ⋆], (1.5) where the free term S0 is given for all τ ∈ [0, τ⋆] by S0(τ) = S⋆(τ) + ̟(τ) 2 (∫ 0 −1 I 0 (μ) exp(τ/μ) dμ