**4.4. Radiative Transfer**

The photoionization calculations described above are relatively straight forward, provided the internally produced radiation can freely escape the cloud. This is the case in most galactic nebulae, and possibly also in AGN NLR clouds, but not in the BLR, where the optical depth to line and continuum radiation is significant. In this case the calculations must be modified, and the following pages describe some commonly used ways to do so.

**4.4.1 Continuum transfer.** The internally
produced recombination (bound-free)
radiation can have a large effect on the degree of ionization by
interacting with
the gas far from its point of creation. Sophisticated, iterative methods
have
been developed to account for the propagation of this radiation in the
cloud.
They depend on the gas distribution and geometry and cannot be applied, in
a simple way, to all configurations.

Approximate methods have been developed too, to shorten the calculations and reduce the number of iterations. The "on-the-spot" approximation is based on the assumption that the diffuse radiation is absorbed very close to its point of creation. This is normally a good assumption for the ground level recombination of helium and hydrogen, in nebulae that are very optically thick to the Lyman ionizing radiation. In this case the mean free path of the ground level recombination radiation is very short, and the assumption of local absorption works very well. The approximation is easy to apply since all that is required is to omit recombination to the ground level from the total recombination coefficient in the ionization (3) and thermal equilibrium (10) equations. The method cannot be used for treating recombination to excited levels, where the mean free path of the bound-free radiation is of the order of the cloud size or larger. It also fails near the boundaries of the cloud, where the optical depth to the surface is short even for the ground level recombination radiation.

There are ways to improve this simplified treatment. In the "modified
on-the-spot" approximation, a correction factor is applied to each of the
recombination coefficients, depending on the optical depth to the
surface at the relevant
frequency. The application is particularly simple in a slab geometry,
where the
radiation can be divided into inward going and outward going beams, and only
optical depths to two surfaces need to be computed. For example, the
recombination
coefficient for hydrogen in equation (3), at a point inside the cloud
where the Lyman continuum optical depth to the inner (illuminated) side
is _{in}
and to the outer side is
_{out}, can be
written in the following way:

(25) |

where _{1} is
the recombination coefficient to *n* = 1 and
_{B} is the sum of
all recombination coefficients to levels with *n* > 1. The
factor *a*_{1}, which is
of the order of 2, takes into account the oblique escape of the ground level
recombination photons and the frequency dependence of the optical depth. A
modification of
_{2},
_{3} etc. can be
included in a similar way. Here the frequency
dependence of the optical depth must be calculated with great care, which
means that *a*_{1} is a strong function of the location in
the cloud.

In a second approximation, named "outward only", the locally produced diffuse radiation is added to the incident flux and carried into the cloud in one, or more directions. Its obvious limitations is near the illuminated surface, where no diffuse radiation is allowed to escape. The process puts much of the heat deep in the cloud, causing an unrealistic temperature structure.

The free-free optical depth is never very large and the above approximate
transfer methods are not adequate in this case. In many cases the
optical depth
is so small that no correction term is required. In other cases the
free-free
optical depth, (ff) must be
calculated at all frequencies and included in the
free-free heating integral (11). The free-free cooling rate is then
modified using
the exp(-h_{cut} /
*kT*_{e}) factor mentioned in
4.2.2. This local treatment
is only a
first order approximation to the rather complex full treatment of the
free-free radiation transfer.

**4.4.2 Line transfer.** Standard radiative
transfer techniques require a numerical
solution of the radiation field everywhere in the gas. Each individual
line profile
is divided into several frequency bins, and the redistribution in frequency,
following an absorption-emission process, is taken into account at all
points.
This is successfully applied in stellar atmosphere calculations, where
conditions
are close to LTE. Under such conditions, the local temperature and the level
populations are not very sensitive to the emitted line flux and good
solutions
are obtained even when a small number of transitions are considered.

This is not the case in gaseous nebulae, where conditions are far from LTE, and a complete solution of the statistical equilibrium equations is required in order to calculate the temperature. Realistic photoionization calculations for AGN clouds involved the computation of several hundred emission lines, the large majority of which are optically thick. Neglecting some lines in the energy balance calculations, for the sake of treating the transfer of others in a more complete way, may result in a poor estimate of the kinetic temperature and wrong line ratios. Combining the two types of treatments, by solving the full radiative transfer in all lines, is beyond the capability of the most sophisticated computer codes available. We are thus faced with the choice of treating the radiative transfer in detail, at the expense of the atomic physics, or vice versa.

The alternative, so far preferred in most advanced calculations, is to treat
the atomic physics in the most accurate way and use a simplified method for
the line transfer. The method is known as *the escape probability
method* and is
demonstrated here for the simple case of a two level atom.

Consider a two level atom with an energy separation between the levels of
*E*_{12} and a normalized line profile
_{}, which is assumed to be
identical for both absorption and emission. Let
*I*_{} be the
radiation intensity and *J* the intensity
averaged over angles and frequencies

(28) |

Consider only radiative processes; spontaneous emission, with a rate of
*n*_{2}*A*_{21},
absorption, with a rate of
*n*_{1}*B*_{12}*J* and induced emission,
with a rate of *n*_{2}*B*_{21}*J*.
The rate equation for the level population is:

(29) |

For isotropic line emission the emission coefficient is:

(30) |

and the absorption coefficient is:

(31) |

where stimulated emission is counted as negative absorption. The line source
function, *S*_{},
is therefore

(32) |

where we have made use of the fact that *A*_{21} =
(2*h* ^{3} /
*c*^{2})*B*_{21} and
*g*_{1}*B*_{12} =
*g*_{2}*B*_{21}.

Let _{21}
be the probability of a line photon to escape the cloud and (1 -
_{21})
the probability to be trapped. In the escape probability method we assume
that

(33) |

which, by using the definition of *S* (32) and substituting into
the rate equation (29), simplifies to

(34) |

In the same way the emergent line flux, per unit volume, is:

(35) |

This is the essence of the escape probability method. It shows that the
equations are similar to the optically thin case except that an
*effective Einstein coefficient*,
_{21}
*A*_{21}, replaces *A*_{21}. The method
ensures local energy conservation
and the local temperature is well determined. The scheme is easily
generalized
to a many level atom, by replacing *A*_{ji} for each
transition, by
_{ji}
*A*_{ji}.

The escape probability approximation gives a correct solution where all scatterings are local and there is little diffusion in space (i.e. the photon is scattered many times close to its point of creation and then escapes the cloud without any further interaction). It is also formally correct for the uniform case, where the temperature and degree of ionization are the same throughout the clouds. In such cases is a "mean escape probability" which is a function of the total cloud optical depth.

Most realistic nebulae are not uniform throughout. Moreover, the line scattering process cannot be entirely local and some diffusion in space must occur. Thus the escape probability describing the trapping of the radiation, in the statistical equilibrium equation (34), is not necessarily the same function needed for calculating the emergent flux (35). Despite this, the advantage of this technique, especially the ability to treat hundreds of optically thick transitions simultaneously, is so great that it is currently being used in many photoionization calculations. The emphasis so far has been on getting reliable estimates of for different line profiles and cloud geometries.

The scattering of resonance line photons is a well studied problem and
various excellent calculations are available to estimate it under a
variety of
conditions. The number of scattering depends on the geometry, the optical
depth and the line profile (or more accurately, the "redistribution
function"). A
general result of such calculations is that the scattering of line
photons is mostly
local (i.e. little diffusion in space) if the re-emitted photon is in
the core of the
line, within 3 Doppler widths of the line center. Such photons escape
the cloud
by diffusion into the line wings where scattering is coherent and a
small number
of scattering carry the photon a large distance in space. It means that
the "local
scattering" assumption, used in the escape probability approximation, is
quite adequate for all resonance lines whose optical depth does not exceed
about 10^{4} (the optical depth corresponding to 3 Doppler widths).

The main result of the numerical transfer calculations mentioned above is
expressed as the number of scatterings before escape,
*Q*(). This is related to
via

(36) |

It is usually found that the number of scatterings is roughly linear
with the *line center* optical depth, calculated from

(37) |

where *f*_{12} is the oscillator strength and
*v*_{Doppler} is the line Doppler width.
^{(3)} Thus

(38) |

where *k*() is a weak
function of and is of the
order of 2-5.

The total path length traveled by the photon before escape is also
proportional
to , with a different
dependence factor, *k'*().
Numerical calculations
show that *k'*()
*k*(), i.e. the time it
takes optically thick line photons to
escape the cloud is several times longer than the time it takes the
optically
thin photons. This is important for dynamical reasons, since the trapped
line radiation increases the internal pressure in the cloud
(see chapter 9). The
implication is that the radiation pressure in optically thick lines is
enhanced by a factor of ~ 5, almost regardless of the optical depths.

The method most commonly applied in modeling the broad line clouds is
the "local escape probability", whereby the escape probability at each
point is
a function of the optical depth at that location. Thus, in a slab model,
at a
point in the cloud where the line center optical depth to one surface is
and to the other surface is
(_{tot} -
), the local escape
probability is:

(39) |

Obviously, _{tot} is
not known a-priori and two or more iterations are required
for a complete convergence of the calculations.

The following expressions for are similar to what is used in most current calculations:

- For
*L*,*HeI*10830 and most resonance lines a good approximation is(40) - For all other hydrogen lines, and non-resonance transitions
(41) where

*a*is the damping constant for the line. The notable difference from the resonance line case is the dependence on^{-l/2}at large (~ 5000) optical depths. This different functional form is a question of some debate and is of great significance for lines like*H*. - Stark broadening of the upper levels of some transitions, changes the
escape probability at high densities. Some calculations of the modified
are
available for hydrogen. They must be incorporated in the calculations for
*N*_{e}>> 10^{10}*cm*^{-3}.

**4.4.3 Line and continuum fluorescence.**
Wavelength coincidences between
emission lines ("line fluorescence") can be an important source of
radiative
excitation. The best known examples are the *HeII - OIII* Bowen
Fluorescence, at a wavelength of about 304Å, and the
OI-L
fluorescence at 1025Å. The first
involves the excitation of OIII lines by the absorption of HeII
*L* photons. It is
important in both the NLR and the BLR clouds, as indicated by the observed
OIII Bowen lines, at wavelengths around 3000Å. The second is
illustrated in
Fig. 6 and results in extra excitation of the
OI^{3}*D*^{0} level by the hydrogen
L
line. It is very important in the BLR, where the scattering of the
*H* photons
increases the
*L*
radiation intensity in the part of the cloud where the
*H*
optical depth is large. Observable lines that are enhanced by this
process are *OI*1302
and *OI*8446
(see diagram).

Wavelength coincidences among FeII lines can be important too. There
are *several hundred* such coincidences and some may be more
important than
others. Another interesting possibility is a wavelength coincidence
between *L*,
which has a broad profile due to its large optical depth, and several
FeII lines. Other possibilities that have been mentioned involved
*MgII*2798,
*NV*124O and more.

Accurate treatment of line fluorescence requires a complete transfer calculation. There is also a local, less accurate solution, based on the escape probability method, that is simple to use and easy to incorporate into the statistical equilibrium equations. It involves the assumption of rectangular line profiles, (or more accurately a constant source function across the line profile), and gives quite good results. Its main disadvantage is the local treatment and the poor approximation at the line wings, where the source function is not constant.

Line fluorescence in AGNs has been a source of some confusion. Such
processes are efficient in removing line photons from one transition,
and pumping
them into another, at frequencies where the radiation field is most
intense, i.e.
close enough to the line center for the source function to be
constant. Further
out into the line wings the source function is smaller and the pumping
efficiency
reduced. A separation of only a few Doppler widths between lines, can
result in
almost a zero fluorescence efficiency. This is the case even in very
large optical depth lines, such as
*L*, where the line
profile is many Doppler widths wide.

Line photons can be destroyed by continuum absorption processes. This is
sometimes called "continuum fluorescence" and is particularly important for
optically thick lines, where the effective absorption optical depth is
increased
by the increased path length of the photons (see the
*k'*() factor mentioned
earlier). Important examples are the ionization of hydrogen *n* = 1
by resonance
lines with <
912Å, the ionization of hydrogen from the *n* = 2 and *n*
= 3 levels
by *L*,
*MgII*2798,
*H* and *FeII*
lines, and the ionization of neutral helium
from the 2^{1}*S* and 2^{3}*P*
levels. Absorption by dust grains, that are mixed in with
the gas, and by *H*^{-}, are other examples.

The escape probability method provides a simple local treatment for this
situation. Consider again the two level atom, a line absorption cross
sections of
_{l} and a
continuum absorption cross section, at the line frequency,
_{c}. Define

(42) |

and

(43) |

The escape probability formalism suggests that in the presence of a continuum opacity source, an effective escape probability

(44) |

is to replace
_{21}
in the statistical equilibrium equation (34). In this case, the
emergent line flux, per unit volume, is

(45) |

and the number of continuum absorptions (e.g. photoionizations) is

(46) |

This treatment is local and does not take into account the absorption of
line
photons away from their point of creation. A possible way to improve it,
in cases
of large continuum optical depth, is to multiply Eqn. (45) by
exp(-_{c}), where
_{c} is a typical
continuum optical depth, e.g. toward the inner surface of the
cloud. The extra amount of continuum absorption should then be added to the
expression in (46). This is not the only way to treat the continuum
absorption
process, and other, rather different methods, have also been suggested.

Absorption of external continuum radiation by spectral lines can be
computed with the same formalism. Consider a point in the cloud where
the optical depth to the illuminated surface, in a certain line, is
_{in}. The
probability of a
photon emitted towards the continuum source to escape is
(_{in}), which is also
the probability of the external radiation to reach that point in the
cloud. The
local *J* (33) is thus increased by an amount corresponding to the
unattenuated external flux multiplied by
(_{in}). In the case of a
central point source with luminosity
*L*_{}, the
increase in *J* is

(47) |

and the rate equation, omitting collisional and ionization processes, takes the form

(48) |

where
_{21}
is the two-directional escape probability of equation (33). The process
is important in cases of large ionization parameter. It can become
significant
in the partly neutral zone, where continuum absorption in spectral lines is
immediately followed by collisional ionization from an excited state.

^{3} A component of microturbulence has been
suggested to increase the line width and to
reduce the optical depth. These are not considered here. We also do not
consider velocity
gradient (e.g. expansion) inside the clouds, that require a different
escape probability function.
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