Astrophysics and Space Science

, 362:151 | Cite as

The generalized Milne problem in gas-dusty atmosphere

  • N. A. Silant’ev
  • G. A. Alekseeva
  • V. V. Novikov
Original Article


We consider the generalized Milne problem in non-conservative plane-parallel optically thick atmosphere consisting of two components—the free electrons and small dust particles. Recall, that the traditional Milne problem describes the propagation of radiation through the conservative (without absorption) optically thick atmosphere when the source of thermal radiation is located far below the surface. In such case, the flux of propagating light is the same at every distance in an atmosphere. In the generalized Milne problem, the flux changes inside the atmosphere. The solutions of both the Milne problems give the angular distribution and polarization degree of emerging radiation. The considered problem depends on two dimensionless parameters \(W\) and (\(a+b\)), which depend on three parameters: \(\eta \)—the ratio of optical depth due to free electrons to optical depth due to small dust grains; the absorption factor \(\varepsilon \) of dust grains and two coefficients—\(\overline{b}_{1}\) and \(\overline{b}_{2}\), describing the averaged anisotropic dust grains. These coefficients obey the relation \(\overline{b}_{1}+3\overline{b}_{2}=1\). The goal of the paper is to study the dependence of the radiation angular distribution and degree of polarization of emerging light on these parameters. Here we consider only continuum radiation.


Radiative transfer Scattering Polarization 



This research was supported by the Program of Presidium of Russian Academy of Sciences N 17, by the Program of the Department of Physical Sciences of Russian Academy of Sciences N 2 and the president program “the leading scientific schools” N 7241.

The authors are very grateful to Dr. H. Frisch for a number of useful remarks, especially for new factorization (9).


  1. Abhyankar, K.D., Fymat, A.L.: Astrophys. J. Suppl. Ser. 23, 35 (1971) ADSCrossRefGoogle Scholar
  2. Chandrasekhar, S.: Radiative Transfer. Dover, New York (1960) zbMATHGoogle Scholar
  3. Dolginov, A.Z., Gnedin, Yu.N., Silant’ev, N.A.: Propagation and Polarization of Radiation in Cosmic Media. Gordon & Breach, Amsterdam (1995) Google Scholar
  4. Frisch, H.: Private communication (2017) Google Scholar
  5. Frölich, H.: Theory of Dielectrics. Clarendon, Oxford (1958) Google Scholar
  6. Greenberg, J.M.: Interstellar grains. In: Stars and Stellar Systems. Vol. VII. University of Chicago Press, Chicago (1968), Chap. 6 Google Scholar
  7. Lenoble, J.: J. Quant. Spectrosc. Radiat. Transf. 10, 533 (1970) ADSCrossRefGoogle Scholar
  8. Schnatz, T.W., Siewert, C.E.: Mon. Not. R. Astron. Soc. 152, 491 (1971) ADSCrossRefGoogle Scholar
  9. Silant’ev, N.A.: Sov. Astron. 24, 341 (1980) ADSGoogle Scholar
  10. Silant’ev, N.A.: Astron. Rep. 51, 67 (2007) ADSCrossRefGoogle Scholar
  11. Silant’ev, N.A., Alekseeva, G.A., Novikov, V.V.: Astrophys. Space Sci. 357, 53 (2015) ADSCrossRefGoogle Scholar
  12. Sobolev, V.V.: Course in theoretical astrophysics. NASA Technical Translation F-531, Washington (1969) Google Scholar
  13. van de Hulst, H.G.: Multiple Light Scattering. Academic Press, New York (1980) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  • N. A. Silant’ev
    • 1
  • G. A. Alekseeva
    • 1
  • V. V. Novikov
    • 1
  1. 1.Central Astronomical Observatory at Pulkovo of Russian Academy of SciencesSaint-PetersburgRussia

Personalised recommendations