**4.2. Thermal Equilibrium**

The equilibrium temperature of the gas is the result of heating, by
absorption
of the central source radiation, and cooling via several atomic
processes. The
temperature referred to is the kinetic temperature of the charged particles,
or the *electron temperature*, *T*_{e}, which is well
defined for all particles with a
Maxwellian distribution of kinetic energies. It should not be confused
with the
*radiation temperature* (or color temperature),
*T*_{rad}, characterizing the
radiation field or the *excitation temperature*,
*T*_{ex}, describing the populations of the
atomic levels. Conditions in gaseous nebulae are far from thermodynamic
equilibrium and *T*_{e} is usually different from both
*T*_{rad} and *T*_{ex}. However, in some
AGN clouds the density and optical depth are large enough so that for some
ions *T*_{ez} = *T*_{ex}, i.e. the level
populations are given by the Boltzmann excitation equation.

The most important heating-cooling processes in AGNs clouds are:

**4.2.1. Bound-free heating-cooling.** Consider
the absorption of a photon with
energy *h* by the ion
*X*^{i} in a level whose threshold ionization frequency is
_{0}.
The initial kinetic energy of the freed electron is
(*h* -
*h*_{0}), and
this energy is
quickly spread, by elastic collisions, among the charged particles.
^{(2)} The heating
rate per unit volume due to this ionization is

(9) |

Summation over all levels of all elements gives the total bound-free heating. A correction due to stimulated recombination, as in equation (7), must be included in some cases.

The average energy of a recombining electron is close to
*kT*_{e} and the total
energy loss due to spontaneous radiative recombination is obtained by
summing over expressions of the form

(10) |

for all ions and all levels. Here
_{T} is an
*energy averaged recombination
coefficient* which is somewhat different from the coefficient
used in equation
(3).

**4.2.2. Free-free heating-cooling.** The
free-free heating rate per unit volume, due
to the ion *X*^{+Z}, is

(11) |

where

(12) |

In this equation *g*_{ff}(, *T*_{e}) is the thermal average of the Gaunt
factor and allowance is made for stimulated emission.

Free-free absorption is a significant heating source for the gas in cases of
intense low frequency radiation and large columns of ionized gas. Most low
density nebulae are optically thin to free-free absorption, but some BLR
clouds,
with densities exceeding 10^{10} *cm*^{-3} and
large column densities, may become
opaque to this radiation, especially at low frequencies. For example, a
unit free-free
optical depth at a wavelength of 30 *µ*m, for
*N*_{e} = 10^{10} *cm*^{-3} and
*T*_{e} = 10^{4}*K*,
is obtained for a column density of
10^{22}
*cm*^{-2}.

Free-free cooling is the result of bremsstrahlung events converting some kinetic energy into radiation via electron-ion Coulomb collisions. The rate, per unit volume, is given to a good approximation by

(13) |

The large abundance of hydrogen and helium ensures that their contribution to free-free cooling is the most important one.

A modification of the heating-cooling rate is required in cases of
significant
free-free optical depth, since some of the radiation is re-absorbed and
heating by the diffuse free-free radiation was not included. An approximate
way to introduce the correction is to multiply the cooling rate by the
factor exp(-*h*
_{cut} /
*kT*_{e}), where
_{cut} is the depth
dependent frequency, where the gas
becomes optically thin to free-free absorption.

**4.2.3. Collisional excitation and de-excitation
heating-cooling.** Inelastic
collisions of free electrons with ions, followed by a radiative decay,
convert kinetic
energy into excitation energy and contribute to the cooling of the
gas. Collisional
de-excitation returns energy to the electron gas and is thus a heating
process. It is convenient to discuss the net cooling, which is the
cooling minus heating, per unit volume and time.

Consider the two level system, *i* and *j* (*j* >
*i*), with statistical weights *g*_{i}
and *g*_{j} respectively. The levels are coupled by an
optically thin line, of energy
*E*_{ij} and a radiative transition rate
*A*_{ji}. The collisional excitation rate between the levels is

(14) |

and the collisional de-excitation rate

(15) |

where _{ij}
is the effective, temperature averaged, collision strength. In the
absence of other populating mechanisms, the relative population of the
levels is

(16) |

A useful concept is the so called "critical density", which is obtained
for each
transition by solving for the electron density for which
*C*_{ji} = *A*_{ji}. Collisional
de-excitation can be neglected for densities much smaller than this critical
density, while the Boltzmann excitation equation can replace (16) for
densities much above it.

The net cooling is the energy emitted by the atoms per unit volume and time

(17) |

In the limit of low density this is reduced to

(18) |

Thus, in this limit, the net cooling is proportional to
*N*_{e}^{2}. In the high density limit
*N*_{j}
*N*_{i}
and the net cooling is proportional to the gas density. Since the
net cooling is basically the line emission, this is also the density
dependence of the emergent line flux.

The formalism used here is easily generalized to a multi-level system and to the case where other atomic processes contribute to the level populations.

The steady state electron temperature is obtained by solving the simple energy conservation equation

(19) |

This requires the full solution of the statistical equilibrium equations at all points in the cloud. The ionization and thermal solutions are of course coupled and iterative methods must be used to solve them, simultaneously.

^{2} see however the note below about
secondary electrons.
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