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1 Radiative transfer codes
In order to solve the radiative transfer problem, the following optical properties of dust grains must be defined
beforehand:
ffl extinction cross­sections (see Eq. (??)) for calculations of the optical distance inside a dusty object;
ffl particle albedo (Eq. (??)) and the scattering matrix (Eq. (??)) for description of the process of light
scattering if the polarization is considered or phase function (Eq. (??)) if not (in some cases only the
asymmetry parameter g (Eq. (??)) is used);
ffl absorption cross­sections which are used for calculations of dust temperature and emitted radiation.
Table 1 contains some characteristics of radiative transfer programs created during the last 25 years and
their applications to the interpretation of observed data on 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
used to appear as the co­author of further publications. The major part of the papers mentioned in Table 1
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 F (Eq. (??)) is taken in an approximate form suggested
by Henyey and Greenstein (1941; HG function, see discussion in Part II, Sect. 6). The standard applications
include: circumstellar (CS) shells and envelopes around early (pre­main­sequence; PMS) and late­type stars
and young stellar objects (YSO), reflection nebulae (RN), interstellar clouds and globules, diffuse galactic light
(DGL) and in recent years galaxies and active galactic nuclei (AGN).
Table 1: Papers presenting new approaches and schemes in radiative
transfer in dusty media.
Author(s) Method; particles; geometry; output; applications
Leung (1976) MoM, quasi­diffusion approximation; homogeneous and coated spheres; 1D
(sphere); intensity, SED; dust clouds with central source, CS shells
Witt (1977) MC; spheres, HG phase function; plane­parallel layer; intensity; RN
White (1979) Doubling method; MRN mixture; homogeneous layer (optically thick but geomet­
rically 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
Rowan­Robinson (1980) Ray tracing method; single size homogeneous and coated spheres; 1D (inhomogeneous sphe­
re), isotropic scattering; SED, intensity profiles; hot­centred interstellar clouds; M
giants and supergiants
Lef`evre et al. (1982) MC; graphite and silicate spheres; inhomogeneous sphere, anisotropic scattering;
SED; late­type stars
Lef`evre et al. (1983) MC; graphite and silicate spheres; homogeneous ellipsoid, anisotropic scattering;
SED, images; young and late­type stars
Warren­Smith (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 (homoge­
neous sphere); SED; CS shells, interstellar clouds
Rogers and Martin (1986) Half­range 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, quasi­diffusion approximation; homogeneous and coated spheres; 2D (disks);
intensity, SED; disk dust clouds; CS disks; disk galaxies
Bastien and M'enard (1988) MC; spheres; 3D, arbitrary geometry, inhomogeneous density distribution; images,
polarization maps; YSO, CS shells
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Table 1: (Continued.)
Author(s) Method; particles; geometry; output; applications
Egan et al. (1988) Updated and comprehensive versions of codes of Leung (1976) and Spagna and
Leung (1983) for one­dimensional geometries (sphere, plane­parallel, cylindrical)
Efstathiou and Rowan­Robinson
(1990)
Ray tracing method; single and multi­compo­ nent mixtures of spheres; 2D axisym­
metric inhomogeneous configurations (disks, ellipsoids, tori); SED; late type stars,
AGN
H¨offlich (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) Multi­dimensional 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, po­
larization; supergiants, RCB stars, DGL, RN
Sonnhalter et al. (1995) Frequency dependent flux­limited diffusion approximation; mixture of carbon, sili­
cate and silicate­ice spheres; 2D axially­symmetric 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) \Lambda 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 axially­symmetric CS disks with
arbitrary density distribution; SED, images; CS shells, YSO
Ivezi'c and Elitzur (1997) Numerical integration; spheres (6 types of materials and 2 size distributions); 1D
(sphere and plane­parallel slab); SED; CS shells, interstellar clouds, YSO
V'arosi and Dwek (1999) Analytical approximation and MC; spheres; spherically­symmetric two­phase
clumpy medium; fluxes; star­forming regions, starburst galaxies
Wolf and Henning (2000) MC including calculations of dust temperature; spheres of different sizes and ma­
terials; 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) \Lambda Dissemination of MC code of Wolf and Henning (2000) on non­spherical particles
\Lambda First results only
A short description of recent progress in continuum radiative transfer modelling is given in the review of
Henning (2000).
The old and still troublesome problem is verification of numerical codes. It was seriously analyzed by Ivezi'c
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.
Note that 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
2

Cahalan (2000).
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Cahalan, R.F. (2000) In IRS 2000: Current Problems in Atmospheric Radiation, Abstracts, p. 61
Chini, R., Kr¨ugel, 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
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