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Astron. Astrophys. 331, 347-360 (1998)

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2. Numerical scheme

The hydrodynamical equations coupled with the energy and ionization equations are solved with a finite-difference method, originally described by Tenorio-Tagle, Bodenheimer & Noriega-Crespo (1986). The method is an extension of a previous one (Tenorio-Tagle 1976) that accounted only for the ionization of hydrogen, and the radiative transport was realized through the "grey" approximation at the hydrogen threshold frequency ([FORMULA] eV). The new method accounts for the time-dependent ionization of hydrogen and helium and radiative cooling by collisionally excited lines, obtained from the ionization structure of carbon, nitrogen, oxygen and neon, calculated under a steady-state approximation, at every time step. Other processes such as bremsstrahlung, collisional ionization and a chain of charge-exchange reactions have been considered. The ionizing radiation field can have different spectral forms (from a black body to a power-law, etc.), and the method allows for its evolution as a function of time. For the diffuse radiation field, able to ionize both hydrogen and helium, one of two different approximations has been made: either the on-the-spot approximation (OTS) was assumed for both hydrogen and helium, or the diffuse field was transported under the assumption of the outward only approximation (see Williams 1967). The description of both approximations is given in appendix A, and a comparison of the results obtained using both possibilities is given in the Sect. 3.

The hydrodynamic equations are written in a finite-difference Lagrangian formulation (see Tenorio-Tagle et al. 1986) following the scheme described by Richtmyer & Morton (1967). A non-linear artificial viscosity term has been used in order to smooth shocks over several grid points.

The ionization equations of H0, He0 and [FORMULA] are solved simultaneously with the energy equation by a Newton-Raphson iteration. The equation of state and the energy equation are,

[EQUATION]

Here temperature (T) was adopted as the hydrodynamic variable instead of the frequently used internal energy ([FORMULA]). The factor [FORMULA], which accounts for the gas-particle degrees of freedom was set to [FORMULA], corresponding to a monoatomic gas. The summations in Eq. (2) on h and c indicate different heating and cooling mechanisms.

The ionization equations of H0, He0 and [FORMULA] take into account radiative recombination and all processes that can cause ionization and lead to the time-dependent equations

[EQUATION]

The summation on f indicates the contribution from the stellar radiation field (s) and the diffuse radiation field (d). To resume, and to offer a better explanation of the whole set of variables, we list below the symbols used and give a brief explanation of their meanings:

  • k = Boltzmann constant,
  • p = pressure,
  • [FORMULA] and [FORMULA] = total mass and particle number density, respectively,
  • [FORMULA] = mean molecular weight,
  • [FORMULA] = atomic masses of the various elements in grams,
  • [FORMULA] = relative abundances,
  • [FORMULA] = density of free electrons,
  • T = temperature,
  • Q = artificial viscosity,
  • [FORMULA] (term that accounts for the total number of particles: atoms, ions and electrons),
  • [FORMULA] = ionization degree of H0, He0, [FORMULA] and [FORMULA] respectively, with [FORMULA],
  • [FORMULA] = photoionization rates,
  • C = collisional ionization rates,
  • [FORMULA] = recombination rates,
  • [FORMULA] = heating rates, and
  • [FORMULA] = cooling rates.

2.1. Ionizing flux

The ionizing source is located at the centre of the computational grid ([FORMULA]) and we assume that its dimensions are negligible compared with other scales involved in the problem. The photoionization rate of an i ion to an [FORMULA] ion can been written as

[EQUATION]

and the corresponding heating rates in the Eq. (2) are

[EQUATION]

where [FORMULA] is the ionization threshold frequency, [FORMULA] is the photoionization cross-section from the ground level and [FORMULA] is the mean intensity of the radiation. This radiative field is composed of two parts: [FORMULA], the field [FORMULA], produced by the central source and the diffuse field [FORMULA], produced through recombinations. If we assume that the central source is a star with radius [FORMULA] and an emergent flux [FORMULA], the radiative transport equation is

[EQUATION]

The optical depth [FORMULA] is measured from the central source position and is dominated by the most abundant elements in the region, hydrogen and helium. The contribution from heavier elements is negligible due to their low abundance. All photoionization terms [FORMULA] and the corresponding heating terms [FORMULA], i.e. all the integral terms in the ionization and energy equations, can be determined as a function of one single parameter if the photoionization cross-sections of the H0, He0 and [FORMULA] have the same frequency dependence. The assumption made here for three frequency intervals is shown in Fig. 1.

[FIGURE] Fig. 1. The solid lines are the photoionization coefficients for H0, He0 and [FORMULA] according to the empiric formula of Seaton (1958). The dotted line are the approximations used in each of the three selected frequency intervals. In the box the adopted frequency dependence for each interval is given.

In this way all the integral terms of photoionization can be calculated prior to their use in the time-dependent calculations and stored, ready to use, in tables as functions of a single parameter; the optical depth. This approximation saves an enormous amount of computational time.

2.2. Physical processes

Several microphysical processes are considered in the numerical scheme. Here we discuss the different approximations, our choice of numerical coefficients and their range of validity. As a general rule we have looked for simple analytic formulae to obtain the fastest computational speed.

Recombination. We have used the expressions given by Seaton (1959) in the hydrogenic approximation to the radiative recombination coefficients [FORMULA] and those averaged in energy [FORMULA] for [FORMULA], [FORMULA] and [FORMULA]. For the recombination coefficients of heavier elements, [FORMULA] (where m indicates the element and k indicates the ion), from the ground level of the ion (m, [FORMULA]) to all levels of the ion (m,k), a good empirical fit is given by Aldrovandi & Pequinot (1973), which includes both the radiative and the dielectronic contribution.

Collisional ionization. For the collisional ionization coefficients [FORMULA] the expression given by Canto & Daltabuit (1974) have been used taking the numerical values of Franco (1981).

Charge-exchange reactions. For the charge-exchange reactions of heavy elements with hydrogen and helium we have used the rate coefficients tabulated as a function of temperature by Butler, Heil & Dalgarno (1980) and Butler & Dalgarno (1980).

Cooling by collisionally excited lines. The most important cooling source in H II regions is emission through the collisionally excited lines of heavy elements. In order to obtain this cooling, the ionization structure of carbon, nitrogen, oxygen and neon is calculated. It requires that in each position in the nebula the ionization number be equal to the recombination number to all levels. For two successive ionization states, k and [FORMULA], of every element we can calculate the relation

[EQUATION]

where [FORMULA] is the photoionization rate from the ground level of the ion (m,k) with a threshold frequency [FORMULA]. For the photoionization cross-sections [FORMULA], we have used a linear interpolation of the tables given by Reilman & Manson (1979). The set of equations (9) for each element together with

[EQUATION]

where [FORMULA] is equal to the selected abundance, here assumed constant across the nebula, allows the determination of the ionization structure of the element. Once the ionization structure has been calculated, the cooling through forbidden-line radiation is obtained by solving the two-level approximation for the population levels of each ion. Thus for level i we have

[EQUATION]

where [FORMULA] are the collisional excitation and de-excitation rates and [FORMULA] are the radiative decay rates respectively for the transition l between the levels [FORMULA] and [FORMULA]. The values tabulated by Mendoza (1983) for [FORMULA] and [FORMULA] were used.

Moreover bremsstrahlung, or free-free emission produced by the deceleration of electrons in the Coulomb field from the atoms of hydrogen and helium has been included.

Cooling in the neutral gas. When the fractional ionization due to photoionization is low (x [FORMULA]), other cooling sources operate in the neutral gas: excitation by impact of hydrogen neutral atoms on the fine-structure lines of C [FORMULA], O0, Si [FORMULA] and Fe [FORMULA]. As the code does not account for the ionization of some of these elements, the interstellar cooling law from Dalgarno & Mc Cray (1972) has been used in this range of ionization degree.

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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998
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