Abstract
The determination of the average path-length of photons emerging from a finite planeparallel atmosphere with molecular scattering is discussed. We examine the effects of polarisation on the average path-length of the emergent radiation by comparing the results with those obtained for the atmosphere where the scattering obeys the scalar Rayleigh function. Only the axial radiation field is considered for both cases.
To solve this problem we have used the integro-differential equations of Chandrasekhar for the diffuse scattering and transmission functions (or matrices). By differentiation of these equations with respect to the albedo of single scattering we obtain new equations the solution of which gives us the derivatives of the intensities of the emergent radiation at the boundaries.
As in the case of scalar transfer the principles of invariance by Chandrasekhar may be used to find an adding scheme to obtain both the scattering and transmission matrices and their derivatives with respect to the albedo of single scattering. These derivatives are crucial in determining the average path length.
The numerical experiments have shown that the impact of the polarisation on the average pathlength of the emergent radiation is the largest in the atmospheres with optical thickness less than, or equal to, three, reaching ≈ 6.9% in the reflected radiation.
Similar content being viewed by others
References
Bellman, R.E., Kagiwada, H.H., Kalaba, R.E., and Prestrud, M.C.: 1964,Invariant Imbedding and Time-Dependent Processes, Elsevier Publ. Co., Inc., N.Y., 256.
Burniston, E.E., and Siewert, C.E.: 1970,J. Math. Phys. 11, 3416.
Chandrasekhar, S.: 1960,Radiative Transfer, Dover Publ., Inc., N.Y., 393.
Domke, H.: 1972,Transfer of Polarized Radiation, Thesis, 195.
Ivanov, V.V., Gutshabash, S.D.: 1974,Izv. Acad. Sci. USSR, Atmos. Oceanic Phys. 10, 851, (In Russian.)
Penndorf, R.: 1957,J. Opt. Soc. Amer. 56, 394.
Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T.: 1986,Numerical Recipes, Cambridge Univ. Press, Cambridge, 818.
van de Hulst, H.C.: 1963,A New Look at Multiple Scattering, Unnumbered mimeographed report, NASA Inst. for Space Sci., N.Y.
van de Hulst, H.C.: 1980,Multiple Light Scattering, Academic Press, N.Y., 739.
Viik, T.: 1991,Astrophys. Space Sci. 178, 133.
Viik, T.: 1993a,Astrophys. Space Sci. 204, 97.
Viik, T.: 1994a,Astrophys. Space Sci. 222, 47.
Viik, T.: 1994b, admitted byAstrophys. Space Sci.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Viik, T. Average photon path-length of radiation emerging from finite atmospheres. Astrophys Space Sci 232, 49–64 (1995). https://doi.org/10.1007/BF00627543
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00627543