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

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3. Test calculations

Many tests have been performed in order to check the reliability of the code. The hydrodynamic scheme, coupled with simple ionization structures, had already been implemented to solve many astrophysical problems (see Sect. 2). However, the calculation of the ionization structure with several elements, several ionization stages and a frequency dependent radiative transport, require further checks in order to warrant its accuracy. The most feasible comparison is with widely used steady-state photoionization codes, such as CLOUDY (Ferland 1990). CLOUDY includes a large number of physical processes and has a high accuracy on photoionization calculations. To compare both codes, the same initial conditions were used, i.e. a constant-density medium with [FORMULA], a star with a black-body spectrum of 40000 K and a radius of [FORMULA] cm. The selected heavy element abundances with respect to hydrogen, were those CLOUDY uses by default for H II region calculations: helium, 0.0095; carbon, [FORMULA] ; nitrogen, [FORMULA] ; oxygen, [FORMULA] ; and neon, [FORMULA].

Fig. 2 shows the comparison with CLOUDY. The various curves correspond to the ionization degree of hydrogen (labelled X), helium (labelled Y and Z; to indicate the distributions of [FORMULA] and He [FORMULA], respectively), temperature (T) and the ionization structure of nitrogen (N). Three different models (identified with the acronym HIS, which stands for h ydrodynamics and i onization s tructure, and a number) were calculated in order to compare with CLOUDY. The different models show results obtained with different approximations to the treatment of the diffuse radiation field (see Appendix A for a detailed description of the approximations). HIS-1 applied the OTS approximation only to hydrogen, in HIS-2 the OTS approximation was used for both hydrogen and helium, and in HIS-3 the diffuse radiation was treated in the outward approximation. All our models were calculated time dependently from [FORMULA], until the leading IF reached the Strömgren radius and a shock began to develop (t [FORMULA] yr). Note that there are slight differences in the position of the IF between the various models. This is due to the slightly different times at which the calculations were stopped. Note also that for HIS-3, the hydrogen (X) IF is located [FORMULA] pc behind the other IFs. This is due to the greater pressure at the IF, caused by the larger temperature established through photoionization, which leads to an earlier development of the outer shock in this case. At [FORMULA] yr the IF reached in HIS-3 the Strömgren radius ([FORMULA] pc) but its structure was modified by the emergent shock, and hence the different conditions inhibit a clear comparison with CLOUDY.

[FIGURE] Fig. 2. Comparison with CLOUDY. From top to bottom the panels show: (X) degree of ionization of [FORMULA], (YZ) degrees of ionization of [FORMULA] and He [FORMULA], (T) the temperature distribution and (N) the ionization structure of nitrogen.

The [FORMULA] and [FORMULA] ionization fronts (X and Y in Fig. 2) in our models present a steeper profile than in CLOUDY. This is due to the fact that recombinations there have not had enough time to balance the ionization, and the IF width is close to its theoretical value ([FORMULA] cm); the photon mean free path. However, the IFs of He [FORMULA] soon reaches equilibrium and resembles the output from CLOUDY (Z in Fig. 2).

There are slight differences between [FORMULA] pc and [FORMULA] pc, at low-level values of z (shown on a logarithmic scale), between the different runs. Note that a full comparison with CLOUDY in this zone was not possible as CLOUDY sets z to 0 whenever its value falls below [FORMULA]. The increase in He [FORMULA] for HIS-2 is due to ionization of [FORMULA] through the high-energy tail photons produced by recombinations to the ground level of H0 and He0. The enhancement increase in HIS-3, of nearly three orders of magnitude, results from photons produced by recombinations to the ground level of [FORMULA], occurring at the inner He [FORMULA] zone and whose ionizing photons under the OTS approximation (HIS-2) are absorbed locally. By way of contrast, in the outward approximation (HIS-3) these photons penetrate deeper into the region to be absorbed in the outer parts, increasing the value of z up to [FORMULA] at the IF (see curve Z in Fig. 2 and to note that the z -scale is logarithmic).

The temperature distributions of the HIS and CLOUDY models (Fig. 2) show a gradual increase towards the edge due to the hardening of the radiation. However, the most conspicuous difference is the existence of a temperature peak at the IF with a temperature of 15000 K in our models which does not appear in CLOUDY. This is an important characteristic of the time-dependent calculations and is caused by the cooling length that newly ionized gas needs to approach the equilibrium temperature of the H II region. There is, however, a substantial discrepancy between CLOUDY and HIS of almost 3000 K in the inner regions dropping to 1000 K at the IF. These differences, we believe arise from the inclusion in CLOUDY of further cooling agents, such as Lyman [FORMULA], and perhaps by the more accurate treatment than our two-level atom approximation for the collisionally excited lines. Both will be the subject of further improvements in our code.

The various approximations to the treatment of the diffuse radiation field also led to some temperature differences between our models (see Fig. 2). In HIS-1 the temperature falls at the centre down to 4000 K due to the cooling through the [FORMULA] line of the O [FORMULA]. However, for HIS-2 the temperature at the centre rises to 15000 K. This increment is caused by the photons produced by the recombinations to excited levels of [FORMULA] ([FORMULA] eV) which are entirely absorbed in this zone. In HIS-1 these photons were not taken into account. Rubin (1968) showed that the effect of the OTS approximation on the central part of the region was exactly the opposite. The OTS approximation overestimated the radiation field in the central part and the fraction of neutrals was lowered and hence the photoheating. As a consequence the temperature falls to 2000 K at the centre. Rubin, however, did not account for the ionization of [FORMULA] to He [FORMULA], nor its recombinations, which are what cause the heating. The outward approximation in HIS-3 avoids this problem by allowing the transport of these photons farther into the nebula. Thus the central peak vanishes and the equilibrium temperature is increased in the whole region through heating by diffuse photons. Finally, Fig. 2 also compares the ionization structure of nitrogen. Note that except for slight differences at the N IV -N III interface the structure for HIS-1, HIS-3 and CLOUDY are identical. HIS-2 presents slight deviations at the N III -N II interface because the optical depth is lowered by the He0 depletion produced by the diffuse photons. This lower optical depth allows a slightly larger zone of N III and [FORMULA] (Y in Fig. 2).

The comparison of our time dependent models with CLOUDY is more than satisfactory. All the variables calculated with HIS-1 coincide well with the output from CLOUDY. Inclusion of the diffuse radiation due to helium leads to slight discrepancies which will require further analysis. CLOUDY gives a very different treatment of the diffuse radiation through local escape probabilities but without considering further ionizations. This could explain the similarity between CLOUDY and our HIS-1 model where the diffuse radiation arises only through hydrogen recombination.

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Online publication: February 4, 1998