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Astron. Astrophys. 347, 617-629 (1999)

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2. Basic equations

An N-level-atom with continuum denoted by [FORMULA] is considered. We assume that the characteristic time-scales of the atomic processes (excitation and ionisation) are short compared to those for changes in the ambient medium (both gas parameters and radiation field). In this case, the level populations [FORMULA] ([FORMULA]) can be calculated by means of the statistical equations

[EQUATION]

together with the auxiliary condition [FORMULA]. The rate coefficients [FORMULA] describe the various physical processes taken into account, which are collisional (de-)excitation, line absorption, spontaneous and stimulated line emission, collisional ionisation and three-body-recombination, photo-ionisation and direct photo-recombination:

[EQUATION]

These formulations include the Milne-type relations between the respective forward and reverse rates (see Woitke et al. 1996). h, c and k are Planck's constant, the speed of light and Boltzmann's constant. T is the kinetic temperature and [FORMULA] the electron density. [FORMULA] is the energy difference between the upper (u) and lower ([FORMULA]) state with degeneracies [FORMULA] and [FORMULA]. [FORMULA] is the frequency and [FORMULA] the line center frequency. [FORMULA] is the Einstein coefficient for spontaneous emission. [FORMULA] and [FORMULA] are the rate coefficients for collisional de-excitation and collisional ionisation, respectively. [FORMULA] is the photo-ionisation cross section, [FORMULA] the Saha function and [FORMULA] the threshold frequency of level [FORMULA]. [FORMULA] is the ionisation potential of level [FORMULA].

In order to approximately account for optical depth effects in the lines, an escape probability method is used. The mean escape probabilities [FORMULA] are calculated by applying Sobolev theory according to the mean velocity gradient [FORMULA] (see Woitke et al. 1996 for further details). The radiation field is thereby formally split into continuum plus lines and the local continuum (background) mean intensity is expressed in terms of the dimensionless quantity [FORMULA]. The gas is assumed to be optically thin in the continuum, i. e. the background mean intensity [FORMULA] is used for the calculation of the bound-free radiative rates (Eqs. 4 and 5).

The radiative net heating rate is calculated after having solved the statistical equations. The net heating rate is defined as the total net gain of photon energy per time and volume. It can be split into the rates caused by the radiative bound-bound and bound-free transitions:

[EQUATION]

In order to study the role of the spectral lines of different strength, we define the following classes of spectral lines by

[EQUATION]

The heating/cooling rates of these classes are obtained by calculating Eq. (6) separately for all lines of the respective transition type

[EQUATION]

In statistical equilibrium, the energy contained in the atom in the form of electronic excitation and ionisation potential energy is constant. The atom only transmits energy from the radiative to the thermal kinetic pool of energy or vice versa, and the net heating rate is equal to the total net gain of thermal kinetic energy per time and volume (see Fig. 1). This rate can be split into the rates caused by (de-)exciting collisions and by the rates due to bound-free transitions

[EQUATION]

[EQUATION]

Eq. (14) can be used to check the quality of the solution of the statistical equations.

[FIGURE] Fig. 1. Energy fluxes between the pools of thermal kinetic energy ("translation"), ionisation potential and electronic excitation energies, and radiation energy. Positive fluxes ("heating") are directed to the left.

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© European Southern Observatory (ESO) 1999

Online publication: June 30, 1999
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