A generalized $\sqrt{\epsilon}$-law

Štěpán, Jiří; Bommier, V.

doi:10.1051/0004-6361:20066507

Cited by 6 publications

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“…This generalized √ -law concerns the surface value of only one quantity, which moreover combines intensity and polarization. This inconvenience can be overcome if some polarization is artificially introduced into the primary source of photons (Frisch 1999;Štěpàn & Bommier 2007). Siewert et al (1981) were able to construct an exact solution for a matrix H(µ), which, loosely speaking, replaces the functions H l (µ) and H r (µ) and provides the emergent radiation For µ → ∞, H l (µ) goes to infinity as √ 5 µ and H r (µ) to √ 2.…”

Section: Introductionmentioning

confidence: 99%

Nonconservative Rayleigh scattering

Frisch

2019

A&A

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Context. The continuous spectrum of stellar and planetary atmospheres can be linearly polarized by Rayleigh or Thomson scattering. The polarization rate depends on the ratio κc/(κc + σc), κc and σc being the absorption coefficients due to photo-ionizations and scattering processes, respectively. The scattering process is conservative if κc = 0, and in this case the center-to-limb variation of the polarization rate follows Chandrasekhar’s law. Deviations from this law appear if the scattering is nonconservative, that is, if photons have a probability ϵ = κc/(κc + σc) of being destroyed at each scattering. Aims. Nonconservative Rayleigh scattering is addressed here with a perturbation point of view, using ϵ, assumed to be a constant, as an expansion parameter. The goal is to obtain a perturbation expansion of the polarized radiation field that can be used to measure of the effects of a nonzero ϵ on the polarization rate of the emergent radiation and to check the accuracy of numerical codes. Methods. The expansion method is an application to Rayleigh scattering of a general perturbation approach developed for scalar monochromatic transport equations. The introduction of a space variable, rescaled by a factor √ϵ, transforms the radiative transfer equation into a new equation from which one can extract simpler equations to describe the field in the interior of the medium and in boundary layers. Results. The perturbation method is applied to a plane-parallel slab with no incident radiation and an unpolarized primary source of photons. The interior and boundary layer fields are expanded in powers of √ϵ. The expansion of the interior radiation field shows that it is unpolarized at leading order, with an intensity i0(τ̃) satisfying a diffusion equation, and that the polarization appears at order ϵ. The emergent radiation is calculated up to and including order ϵ. The leading term yields the polarization rate of the Chandrasekhar’s law. The following one, of order √ϵ, accurately predicts the decrease of the polarization rate for values of ϵ up to 10−3 and shows that it varies roughly as (1 − μ) for any unpolarized primary source. Methods for testing the accuracy of numerical schemes are proposed. The perturbation method is also applied to a slab with an incident radiation field and a polarized primary source of photons.

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Section: Introductionmentioning

confidence: 99%