Abstract
For scalar scattering problems, we showed in Part I that many exact results for semi-infinite media can be derived with the so-called resolvent method, based on convolution-type integral equations satisfied by the Green and the resolvent functions. For a polarized radiation field, these functions become matrices or vectors. We show in this Chapter that they also satisfy convolution integral equations, which we use to construct polarized \(\sqrt {\epsilon }\)-laws and nonlinear integral equations for H-matrices. We treat the Rayleigh scattering, the resonance polarization and the Hanle effect.
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References
Abhyankar, K.D., Fymat, A.L.: Imperfect Rayleigh scattering in a semi-infinite atmosphere. Astron. Astrophys. 4, 101–110 (1970)
Abhyankar, K.D., Fymat, A.L.: Tables of auxiliary functions for the nonconservative Rayleigh phase matrix in semi-infinite atmospheres. Astrophys. J. Suppl. Ser. 195, 35–101 (1971)
Ambartsumian, V.A.: Light scattering by planetary atmospheres. Astron. Zhurnal 19, 30–41 (1942)
Bommier, V., Štěpán, J.: A generalized \(\sqrt {\epsilon }\)-law. The role of unphysical source terms in resonance line polarization transfer and its importance as an additional test of NLTE radiative transfer. Astron. Astrophys. 468, 797–801 (2007)
Bond, G.R., Siewert, C.E.: On the nonconservative equation of transfer for a combination of Rayleigh and isotropic scattering. Atrophys. J. 164, 97–110 (1971)
de Rooij, W.A., Bosma, P.B., van Hooff, J.P.C.: 1989, A simple method for calculating the H-matrix for molecular scattering. Astron. Astrophys. 226, 347–356 (1989)
Frisch, H.: Resonance polarization and the Hanle effect; the integral equation formulation and some applications. In: Nagendra, K.N., Stenflo, J.O. (eds.) Solar Polarization, 2nd Solar Polarization Workshop, Kluwer, Dordrecht, pp. 97–113 (1999)
Frisch, U., Frisch, H.: Non-LTE transfer. \(\sqrt {\epsilon }\) revisited. Mon. Not. R. Astron. Soc. 173, 167–182 (1975)
Frisch, U., Frisch, H.: Universality of escape from a half-space for symmetrical random walks. In: Lévy Flights and Related Topics in Physics, eds. M. Shlesinger, G. Zaslavsky, U. Frisch, Lectures Notes in Physics, vol. 450, pp. 262–268. Springer Verlag, Berlin (1995)
Fymat, A.L.: Theory of radiative transfer in atmospheres exhibiting polarized resonance fluorescence, Ph.D. thesis, Los Angeles, UCLA
Ivanov, V.V.: Nonmagnetic polarization of the Doppler core of strong Fraunhofer lines. Sov. Astron. 34, pp. 621–625 (1990); translation from Astron. Zhurnal 67, pp. 1233–1242 (1990)
Ivanov, V.V.: Generalized Rayleigh scattering I. Basic theory. Astron. Astrophys. 303, 609–620 (1995)
Ivanov, V.V.: Generalized Rayleigh scattering III. Theory of I-matrices. Astron. Astrophys. 307, 319–331 (1996)
Ivanov, V.V., Kasaurov, A.M., Loskutov, V.M., Viik, T.: Generalized Rayleigh scattering II. Matrix source functions. Astron. Astrophys. 303, 621–634 (1995)
Ivanov, V.V., Kasaurov, A.M., Loskutov, V.M.: Generalized Rayleigh scattering IV. Emergent radiation. Astron. Astrophys. 307, 332–346 (1996)
Ivanov, V.V, Grachev, S.I., Loskutov, V.M.: Polarized line formation by resonance scattering I. Basic formalism. Astron. Astrophys. 318, 315–326 (1997)
Kriese, J.T., Siewert, C.E.: An expedient method for calculating H-matrices. Astrophys. J. 164, 389–391 (1971)
Landi Degl’Innocenti, E., Bommier, V.: Resonance line polarization for arbitrary magnetic fields in optically thick media III. A generalization of the \(\sqrt {\epsilon }\)-law. Astron. Astrophys. 284, 865–873 (1994)
Lenoble, J.: Importance de la polarisation dans le rayonnement diffusé par une atmosphère planétaire. J. Quant. Spectrosc. Radiat. Transf. 10, 533–556 (1970)
Nagendra, K.N., Sampoorna, M.: Numerical methods in polarized line formation theory. In: Solar Polarization 5, S. Berdyugina, K.N. Nagendra, R. Ramelli (eds), ASP Conference Series 405, 261–273 (2009)
Pahor, S. : Albedo and Milne ’s problem for thermal neutrons. Nuclear Sci. Eng. 31, 110–116 (1968)
Siewert, C.E., Burniston, E.E.: On existence and uniqueness theorems concerning the H-Matrix of radiative transfer. Astrophys. J. 174, 629–641 (1972)
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Frisch, H. (2022). The \(\sqrt {\epsilon }\)-Law, the Nonlinear H-Equation, and Matrix Singular Integral Equations. In: Radiative Transfer . Springer, Cham. https://doi.org/10.1007/978-3-030-95247-1_15
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DOI: https://doi.org/10.1007/978-3-030-95247-1_15
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