I n t r o d u c t o r y   n o t e s  


1. Light scattering characteristics used in radiative transfer calculations

In order to solve the radiative transfer problem, the following optical properties of dust grains must be defined beforehand:
  • the extinction cross-sections Cext for calculations of the optical distance inside a dusty object;
  • particle albedo and the scattering matrix for description of the process of light scattering when the polarization is considered and the phase function when it is not considered (in some cases only the asymmetry parameter g is used);
  • the absorption cross-sections Cabs which are used for calculations of the dust temperature and emitted radiation.

2. A brief survey of the main approaches on calculation of radiative transfer in dusty media

Approach Applicability range Speed Accuracy Source and dust configurations Polarized radiation
Single scattering approximation (tau<<1) tau < 0.1-0.5 (depends on albedo, g) very fast (depends on albedo, g or scatterer shape?) may be not accurate any 3D configuration yes
Double scattering approximation tau<1 (depends on albebo, g?) rather fast not very accurate any 3D configuration yes
Lambda-iteration mathod (generalization of 1,2,...-scattering approximations) taumax ~ albedo, g? quick (if number scattering is small) depends on the numbers of itarations and scatterings? any 3D configuration yes
Numerical methods developed for a given geometry any tau not fast not very exact? usually 1D configurations no
Grid-point methods ? very slow ? 3D configurations yes
Monte Carlo method tau about 1-100? not fast (depends on tau and configuration) not very accurate any 3D configuration yes



3. Light scattering codes used in astrophysical applications

The table below contains some characteristics of the radiative transfer programs created during the last 25 years and their applications to interpretation of the data observed for different dusty cosmic objects. As usual, many modern radiative transfer codes are modifications of earlier versions created at the same institute or university. We do not intend to search for the origin and roots of codes and just note that the author of the original code usually appears as the co-author of further publications.

The major part of the papers mentioned in the following table are based on two methods: a) iterative scheme to solve the moment equations of radiative transfer equation (methods of moments, MoM) which was originally formulated by Hummer and Rybicki [1971] for spherical geometry with a central point source (1D geometry) and b) Monte Carlo (MC) simulation. In some cases, for simplification of calculations the phase function is taken in the approximate form suggested by Henyey and Greenstein [1941] (HG function). The standard applications include: (CS) shells and envelopes around early (pre-main-sequence; PMS) and late-type stars and young stellar objects (YSO), reflection nebulae (RN), clouds and globules, diffuse galactic light (DGL) and in recent years galaxies and active galactic nuclei (AGN). A short description of recent progress in continuum radiative transfer modelling is also given in the review of Henning [2000].

Note that the old and still troublesome problem is verification of numerical codes. It was seriously analyzed by Ivezic et al. [1997] who compared three different radiative transfer codes and obtained the benchmark results for temperature and emerging spectra in the case of spherical geometry. In principle astronomers can use results and experience gained in other fields of science. For example, 3D radiative transfer codes are widely used in the terrestrial atmosphere applications as described, for example, by Cahalan [2000].

Author(s) Method Particles Geometry Output Applications

Leung [1976]		 MoM, quasi-diffusion approximation homogeneous and coated spheres 1D (sphere) intensity, SED dust clouds with central source, CS shells

Witt [1977]		 MC spheres, HG phase function plane-parallel layer intensity RN

White [1979]		 Doubling method MRN mixture homogeneous layer (optically thick but geometrically thin) intensity, polarization RN

Daniel [1980]		 MC single size spheres homogeneous sphere polarization cool stars

Yorke [1980]		 MoM single size spheres 1D (sphere), isotropic scattering SED cocoon stars

Rowan-Robinson [1980]		 Ray tracing method single size homogeneous and coated spheres 1D (inhomogeneous sphere), isotropic scattering SED, intensity profiles hot-centred interstellar clouds, M giants and supergiants

Lefevre et al. [1982]		 MC graphite and silicate spheres inhomogeneous sphere, anisotropic scattering SED late-type stars

Lefevre et al. [1983]		 MC graphite and silicate spheres homogeneous ellipsoid, anisotropic scattering SED,images young and late-type stars

Warren-Smith [1983]		 MC spheres of different sizes plane layers surface brightness, polarization RN
Spagna and Leung  [1983] Newton-Raphson iterative scheme spheres (up to 5 constituents) 1D (homogeneous sphere) SED CS shells, clouds

Rogers and Martin  [1986]		 Half-range MoM spheres 1D (sphere) with power density distribution SED CS shells (IRC +10 216)

Chini et al. [1986]		 MoM MRN mixture 1D (sphere) with power density distribution SED CS shells

Wolfire and Cassinelli [1986]		 Modified MoM MRN mixture 1D (sphere) with power density distribution SED protostars

Spagna and Leung [1987]		 MoM, quasi-diffusion approximation homogeneous and coated spheres 2D (disks) intensity, SED disk dust clouds, CS disks, disk galaxies

Bastien and Menard [1988]		 MC spheres 3D, arbitrary geometry, inhomogeneous density distribution images, polarization maps YSO, CS shells

Egan et al. [1988]		 Updated and comprehensive versions of codes of Leung [1976] and Spagna and Leung [1983] for one-dimensional geometries (sphere, plane-parallel, cylindrical)        

Efstathiou and Rowan-Robinson [1990]		 Ray tracing method single and multi-component mixtures of spheres 2D axisymmetric inhomogeneous configurations (disks, ellipsoids, tori) SED late type stars, AGN

Hoefflich [1991]		 MC Thomson scattering axisymmetric photospheres polarization SN 1987A

Collison and Fix [1991]		 Iterative scheme single size silicate spheres 2D axisymmetric inhomogeneous shells, isotropic scattering SED, images CS shells

Whitney and Hartmann [1992]		 MC spheres, HG phase function 3D (disks) images, polarization maps PMS objects

Pier and Krolik [1992]		 Multi-dimensional Newton-Raphson technique MRN mixture 2D (homogeneous torus) SED AGN, Seyfert galaxies

Bosma [1993a], [1993b]		 New iterative scheme in MoM anisotropic scattering, randomly oriented particles 1D (sphere) intensity and polarization  

Fischer [1993]		 MC MRN mixture 3D, arbitrary geometry, inhomogeneous density distribution images, polarization maps protostellar sources

Groenewegen [1993]		 Iterative scheme spheres 1D (sphere) with power density distribution, isotropic scattering, determination of inner radius as dust condensation boundary SED shells around AGB stars

Voshchinnikov and Karjukin [1994]		 MC, method of symmetrized trajectories spheres (Rayleigh, MRN mixture) 2D (inhomogeneous spheroid) intensity, polarization CS shells around young stars

Code and Whitney [1995]		 MC electrons, spherical particles 3D (illuminated spherical blobs) intensity, polarization supergiants, RCB stars, DGL, RN

Sonnhalter et al. [1995]		 Frequency dependent flux-limited diffusion approximation mixture of carbon, silicate and silicate-ice spheres 2D axially-symmetric dusty disks with different density distribution intensities, images at different wavelengths  

Lopez et al. [1996]		 MC carbon spheres inhomogeneous axisymmetric shell, anisotropic scattering SED, images AGB stars, Red Rectangle

Steinacker and Henning [1996]1		 Direct solution to discretized radiative transfer equation spheres 3D arbitrary configuration SED, images CS shells, YSO
Men'shchikov and Henning [1997] MoM spheres of different sizes and materials 2D axially-symmetric CS disks with arbitrary density distribution SED, images CS shells, YSO

Ivezic and Elitzur [1997]		 Numerical integration spheres (6 types of materials and 2 size distributions) 1D (sphere and plane-parallel slab) SED CS shells, clouds, YSO

Varosi and Dwek [1999]		 Analytical approximation and MC spheres spherically-symmetric two-phase clumpy medium fluxes star-forming regions, starburst galaxies

Wolf and Henning [2000]		 MC including calculations of dust temperature spheres of different sizes and materials 3D, arbitrary number, shape and geometrical configuration of illuminating sources and dust density distribution SED, images, polarization maps CS shells, YSO, AGN

Gordon et al. [2001] and Misselt et al. [2001]		 MC mixture of carbonaceous and silicate grains, polycyclic aromatic hydrocarbons (PAHs) 3D, arbitrary distribution of stars and dust SED, images, polarization maps RN, clusters of stars, galaxies

Wolf et al. [2002] 1		 Dissemination of MC code of Wolf and Henning [2000] on non-spherical particles        
Notes:
   1. First results only.
   2. All notations are explained in the text above the table.


4. Bibliography

Bastien, P. and Menard, F. (1988) Astrophys. J., 326, 334

Bosma, P.B. (1993a) Astron. Astrophys., 276, 303

Bosma, P.B. (1993b) Astron. Astrophys., 279, 572

Cahalan, R.F. (2000) In IRS 2000: Current Problems in Atmospheric Radiation, Abstracts, p. 61

Chini, R., Krügel, E. and Kreysa, E. (1986) Astron. Astrophys., 167, 315

Code, A.D. and Whitney, B.A. (1995) Astrophys. J., 441, 440

Collison, A.J. and Fix, J.D. (1991) Astrophys. J., 368, 545

Daniel, J.-Y. (1980) Astron. Astrophys., 87, 204

Efstathiou, A. and Rowan-Robinson, M. (1990) Monthly Notices RAS, 245, 275

Egan, M.P., Spagna, G.F. and Leung, C.M. (1988) Computer Physics Comm., 48, 271

Fischer, O. (1993) PhD Thesis, Friedrich-Schiller-Universität, Jena

Gordon, K.D., Misselt, K.A., Witt, A.N. and Clayton, G.C. (2001)

Groenewegen, M.A.T. (1993) PhD Thesis, University of Amsterdam

Henning, Th. (2000) In The Formation of Binary Stars, ed. by B. Mathieu and H. Zinnecker, IAU Symp., 200, in press

Henyey, L.G. and Greenstein, J.K. (1941) Astrophys. J., 93, 70

Hoefflich, P. (1991) Astron. Astrophys., 246, 481

Hummer, D.G. and Rybicki, G.B. (1971) Monthly Notices RAS, 152, 1

Ivezic, Z. and Elitzur, M. (1997) Monthly Notices RAS, 287, 799

Ivezic, Z., Groenewegen, M.A.T., Men'shchikov, A.B. and Szczerba, R. (1997) Monthly Notices RAS, 291, 121

Lefevre, J., Bergeat, J. and Daniel, J.-Y. (1982) Astron. Astrophys., 114, 346

Lefevre, J., Bergeat, J. and Daniel, J.-Y. (1983) Astron. Astrophys., 121, 51

Leung, C.M. (1976) J. Quant. Spectrosc. Rad. Transfer, 16, 559

Lopez, B., Mékarnia, D. and Lefèvre, J. (1996) Astron. Astrophys., 296, 752

Men'shchikov, A.B. and Henning, Th. (1997) Astron. Astrophys., 318, 879

Misselt, K.A., Gordon, K.D., Clayton, G.C. and Wolff, M.J. (2001) Astrophys. J., 551, 277

Pier, E.A. and Krolik, J.H. (1992) Astrophys. J., 401, 99

Rogers, C. and Martin, P.G. (1986) Astrophys. J., 311, 800

Rowan-Robinson, M. (1980) Astrophys. J. Suppl., 44, 403

Sonnhalter, C., Preibisch, Th. and York, H.W. (1995) Astron. Astrophys., 299, 545

Spagna, G.F. and Leung, C.M. (1983) Computer Physics Comm., 28, 337

Spagna, G.F. and Leung, C.M. (1987) J. Quant. Spectrosc. Rad. Transfer, 37, 565

Steinacker, J. and Henning, Th. (1996) In The Role of Dust in the Formation of Stars, ed. by H.U. Käufl and R. Siebenmorgen, p. 355

Varosi, F. and Dwek, E. (1999) Astrophys. J., 523, 265

Voshchinnikov, N.V. and Karjukin, V.V. (1994) Astron. Astrophys., 288, 883

Warren-Smith, R.F. (1983) Monthly Notices RAS, 205, 337

White, R.L. (1979) Astrophys. J., 230, 116

Whitney, B.A. and Hartmann, L. (1992) Astrophys. J., 395, 529

Witt, A.N. (1977) Astrophys. J. Suppl., 35, 1

Wolf, S. and Henning, Th. (2000) Computer Physics Comm., 132, 166

Wolf, S., Voshchinnikov, N.V. and Henning, Th. (2002) Astron. Astrophys., 385, 365

Wolfire, M.G. and Cassinelli, J.P. (1986) Astrophys. J., 310, 207

Yorke, H.W. (1980) Astron. Astrophys., 86, 286
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Last modified: 05/04/03, V.I.