## 2. Numerical schemeThe 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 ( 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 H Here temperature ( The ionization equations of H The summation on *k*= Boltzmann constant,*p*= pressure,- and = total mass and particle number density, respectively,
- = mean molecular weight,
- = atomic masses of the various elements in grams,
- = relative abundances,
- = density of free electrons,
*T*= temperature,*Q*= artificial viscosity,- (term that accounts for the total number of particles: atoms, ions and electrons),
- = ionization degree of H
^{0}, He^{0}, and respectively, with , - = photoionization rates,
*C*= collisional ionization rates,- = recombination rates,
- = heating rates, and
- = cooling rates.
## 2.1. Ionizing fluxThe ionizing source is located at the centre of the computational
grid () and we assume that its dimensions are
negligible compared with other scales involved in the problem. The
photoionization rate of an and the corresponding heating rates in the Eq. (2) are where is the ionization threshold frequency, is the photoionization cross-section from the ground level and is the mean intensity of the radiation. This radiative field is composed of two parts: , the field , produced by the central source and the diffuse field , produced through recombinations. If we assume that the central source is a star with radius and an emergent flux , the radiative transport equation is The optical depth 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 and the corresponding heating terms
, 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 H
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 processesSeveral 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.
where is the photoionization rate from the
ground level of the ion ( where 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 where are the collisional excitation and
de-excitation rates and are the radiative decay
rates respectively for the transition Moreover
© European Southern Observatory (ESO) 1998 Online publication: February 4, 1998 |