**1.2. Heating and cooling**

During the photoionization process, the absorption of a photon
creates a free electron which
rapidly shares its energy with the other electrons present in the gas
by elastic collisions, and thus heats the gas. The energy gains are
usually dominated by photoionization of hydrogen atoms, although
photoionization of helium contributes significantly. Intuition might
suggest that *T*_{e}
will decrease away from the ionizing source, since
the ionizing radiation field decreases because of geometrical dilution
and absorption in the intervening layers. This is actually not the
case.
The total energy gains per unit volume and unit time at a distance *r*
from the ionizing source are schematically given by:

(1.13) |

where

(1.14) |

If ionization equilibrium is achieved in each point of the nebula, one has (in the "on-the-spot case")

(1.15) |

Therefore, *G* can be written

(1.16) |

where

(1.17) |

Thus < *E* > can be seen as the average energy gained per
photoionization, and is roughly independent of *r*.
It can be shown (see e.g.
Osterbrock 1989),
that when the ionization source is a blackbody of temperature
*T*_{}, one
has < *E* >
(3/2)*kT*_{}.
Therefore:

(1.18) |

meaning that the energy gains are roughly proportional to the temperature of the ionizing stars.

Thermal losses in nebulae occur through recombination, free-free
radiation
and emission of collisionally excited lines. The dominant process
is usually due to collisional excitation of ions from heavy
elements (with O giving the largest contribution, followed by C, N, Ne and
S). Indeed, these ions have low-lying energy levels which can easily be
reached at nebular temperatures. The excitation potentials of hydrogen
lines are much higher, so that collisional excitation of H^{0}
can become important only at high electron temperatures.

For the transition *l* of ion *j* of an element
*X*^{i}, in a simple two-level
approach and when each excitation is followed by a radiative
deexcitation, the cooling rate can be schematically written as

(1.19) |

where _{ijl} is
the collision strength,
_{ijl} is the
statistical weight of the upper level, and
_{ijl} is the
excitation energy.

If the density is sufficiently high, some collisional deexcitation may occur and cooling is reduced. In the two-level approach one has:

(1.20) |

So, in a first approximation, one can write that the electron temperature is determined by

(1.21) |

where *G* is given by Eq. (1.18) and
*L*_{coll}^{ijl} by Eq. (1.20).

The following properties of the electron temperature are a consequence of the above equations:

- *T*_{e} is expected to be usually rather uniform in nebulae,
its variations are mostly determined by the mean energy of the
absorbed stellar photons, and by the populations of the main cooling
ions. It is only at high metallicities (over solar) that large
*T*_{e} gradients
are expected: then cooling in the O^{++} zone is dominated by
collisional excitation of fine structure lines in the ground level of
O^{++}, while the absence of fine structure lines in the ground
level of O^{+} forces the temperature to rise in the outer zones
(Stasinska 1980a,
Garnett 1992).

- For a given
*T*_{} ,
*T*_{e} is generally lower at higher
metallicity.

- For a given metallicity, *T*_{e} is generally lower for
lower *T*_{} .

- For a given
*T*_{} and
given metallicity, *T*_{e} increases with density in
regions where *n* is larger than a critical density for
collisional deexcitation of the most important cooling lines (around
5 × 10^{2} - 10^{3} cm^{-3}).