Annu. Rev. Astron. Astrophys. 2013. 51:63-104
Copyright © 2013 by Annual Reviews. All rights reserved |

**2.1. The Radiative Transfer Equation**

The stationary radiation field is described by the specific intensity
*I*(* x*,

(1) |

The left-hand side of this equation represents the change in intensity
over an infinitesimal distance along the path determined by the position
* x* and the propagation direction

An alternative form of the RTE (Equation 1) uses explicitly the distance
*s* along the path defined by a position * x* and
propagation direction

(2) |

If we assume for now that the source term *j* does not depend on
the intensity *I*, we can readily solve the differential equation

(3) |

with the optical depth between two positions defined as

(4) |

Equation 3 has a simple physical interpretation: It shows that the
intensity at any position *s* along a path results from the
emission at all anterior points *s'* along the path, reduced
by a factor of e^{?τ(s, s', λ)} to
account for the extinction from the intervening matter. We stress that
Equation 3 is only a formal solution of the RTE but not a very useful
one. Indeed, the emissivity generally does not only depend on position,
direction, and wavelength, but also on the specific intensity
itself. The formal solution (Equation 3) is then no more than an
integral equation of the RTE itself; this is particularly the case for
dust RT. The remainder of Section 2 gradually expands the dust RTE by
including the various relevant physical processes.

**2.2. Primary Emission and Absorption**

Primary emission and absorption in a dusty medium are two important and
obvious processes to take into account. Primary emission accounts for
the radiative energy added to the radiation field– this is often
stellar emission, but it can also include, for example, radiation from
an AGN, emission line radiation from ionized gas, or Bremsstrahlung. In
general form, it can be characterized by a function
*j*_{*}(* x*,

When we take only primary emission and absorption by dust into account, the RTE (Equation 1) becomes

(5) |

This equation is a simple first-order differential equation, which can be solved by simply integrating along the line of sight, as is done for the formal solution (Equation 3). For general 3D geometries, this integration is done numerically.

The complexity of the RTE increases substantially when we take
scattering into account. Scattering, as with absorption, removes
radiation from a beam and hence accounts for a second sink term in the
RTE, the efficiency of which is quantified by the scattering coefficient
κ_{sca}. Rather than converting the radiation to internal
energy, it emits the same radiation in a different direction. Scattering
therefore does not only imply a second sink term, but also a second
source term. The scattering phase function Φ(* n*,

(6) |

Adding to the RTE the sink and source terms due to scattering, we obtain

(7) |

where the extinction coefficient κ_{ext} =
κ_{abs} + κ_{sca}. Unlike the simple
differential equation (Equation 5), Equation 7 is an
integro-differential equation in which the radiation fields at all
different positions and in all different directions are coupled.

In many RT calculations, scattering is important and adds significant complexity, given that dust scattering is anisotropic. This is especially true for UV to NIR wavelengths: Observations indicate that the dust scattering albedo at these wavelengths is at least 50% and that scattering off dust grains is strongly anisotropic (Mattila 1970, Calzetti et al. 1995, Burgh, McCandliss & Feldman 2002, Gordon 2004). [http://www.stsci.edu/~kgordon/Dust/Scat_Param/scat_data.html contains the up-to-date versions of the original plots of albedo and scattering phase function asymmetry versus wavelength from Gordon 2004.] Using an isotropic phase function or other approximations such as an isotropic two-stream approximation or a (effective) forward scattering (see e.g., Code 1973, Natta & Panagia 1984, Calzetti, Kinney & Storchi-Bergmann 1994) might not always be physically justifiable. Several authors demonstrated that improper treatment of anisotropic scattering leads to significant errors (e.g. Bruzual, Magris & Calvet 1988, Witt, Thronson & Capuano 1992, Baes & Dejonghe 2001). Even for radiation at mid-infrared (MIR) wavelengths or longer, where the scattering off common ISM grains is low, an application calculating the heating of the grains will need to properly handle the scattering (Nielbock et al. 2012).

The Henyey-Greenstein (HG) phase function (Henyey & Greenstein 1941) is the most widely used parametrization of the dust phase function and provides a good single-parameter approximation. Dust grain models predict small deviations from an HG phase function, and other parametrizations or numerical phase functions can be used for increased accuracy (Kattawar 1975, Hong 1985, Draine 2003b).

**2.4. Radiative Transfer in Dust Mixtures**

In any real dust medium, there is a range of different types of dust
grains present, with various chemical compositions, sizes, shapes, and
number densities. Each grain type *i* is characterized by its own
absorption coefficient κ_{abs,i}(λ),
scattering coefficient κ_{sca,i}(λ), and
scattering phase function Φ_{i}(* n*,

(8) |

Clearly, Equation 8 is formally identical to Equation 7, if we define the absorption coefficient, scattering coefficient, extinction coefficient, and scattering phase function of a dust mixture as, respectively,

(9a) (9b) (9c) |

and

(9d) |

In terms of primary emission, absorption, and scattering, RT in dust mixtures is completely identical to RT in a dust medium with a single average dust grain, and no approximations are needed (Martin 1978, Wolf 2003a).

In addition to primary emission, absorption, and scattering, a fourth
physical process in dust RT, the thermal emission by the dust itself,
must be taken into account. Dust grains that absorb radiation re-emit
the acquired radiative energy at wavelengths longward of ~ 1
*µ*m. To account for this astrophysical process, we need to
incorporate the third source term *j*_{d}(* x*,
λ) in our RTE:

(10) |

The dust emissivity term could be perceived as merely an extra source term similar to the primary stellar emissivity term. Its exact form is strongly dependent on which physical emission processes are important, and often it depends, in a complicated and nonlinear way, on the intensity of the radiation field itself.

A common assumption is that the dust grains are in thermal equilibrium
with the local interstellar radiation field. In this case, the
emissivity of the population of grains of type *i* can be written
as a modified blackbody emission characterized by the equilibrium
temperature
*T*_{i}(* x*). Summing over all
populations, we obtain

(11) |

where *B*(*T*, λ) is the Planck function. The
equilibrium temperature of each type of grain is determined by the
balance equation, i.e., the condition that the total amount of energy
absorbed equals the total amount of emitted energy:

(12) |

where *J*(* x*, λ) represents the mean intensity
of the radiation field,

(13) |

The equilibrium temperature of the dust grains depends explicitly on their size and chemical composition. At the same location, dust grains of different sizes and/or chemical compositions will obtain different equilibrium temperatures. So far, we have easily combined the absorption and scattering due to different kinds/sizes of dust grains in the RTE without any approximations. The same cannot be done for the thermal re-emission term. One could compute the average temperature for the different grains on the basis of Equation 2.4 and obtain a mean temperature. This results in reducing the complexity of the dust mixture at a given position to a single mean grain that will reach a single equilibrium temperature. Although this could be useful, sufficient, or even necessary for certain applications, this is a physically incorrect simplification of the RT problem (e.g. Wolf 2003a).

Although the assumption of thermal equilibrium is useful in some applications, it breaks down in others. Particularly important is the case where the dust medium contains very small dust grains [including polycyclic aromatic hydrocarbons (PAHs)]. Large dust grains reach thermal equilibrium and emit as modified blackbodies with an equilibrium temperature. However, small dust grains have small heat capacities, and the absorption of even a single UV/optical photon can substantially heat the grain. These small grains will not reach the equilibrium temperature but will instead undergo temperature fluctuations that lead to grain emission at temperatures well in excess of the equilibrium temperature. The emission from small grains is necessary to explain the observed MIR emission of many objects (see, e.g., Sellgren 1984, Boulanger & Perault 1988, Helou et al. 2000, Smith et al. 2007, Draine et al. 2007). When including such transiently heated dust grains, the dust emissivity changes from Equation 14 to

(14) |

Here, *P*_{i}(*T*, * x*) is the
temperature distribution for dust grains of
type

Finally, thermal emission is not the only emission process of dust
grains: Additional nonthermal processes are extended red emission
(Witt
& Boroson 1990,
Smith
& Witt 2002)
and blue luminescence
(Vijh, Witt & Gordon 2004).
These nonthermal processes can account for a
substantial fraction of the surface brightness of interstellar clouds at
optical wavelengths (see, e.g.,
Gordon, Witt & Friedmann 1998,
Witt
et al. 2008).
Both processes can be included as an additional term in the dust source
term *j*_{d}(* x*, λ) in Equation 10.

**2.6. Radiative Transfer of Polarized Radiation**

The specific intensity *I*(* x*,

The most common description of polarized radiation uses the Stokes
vector * S* = (

The RT formalism we describe above can be extended to include polarized radiation. Instead of a single RTE, we then obtain a vector RTE, or equivalently, a set of four coupled scalar equations:

(15) |

The first complication is the scattering source term, where a 4×4
scattering (or Mueller) matrix
(* n*,

(16) |

A second complication is the thermal emission term: The full Stokes vector is used for thermal emission from aligned grains.