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Astron. Astrophys. 337, 517-538 (1998)

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7. Summary and conclusions

We present in this paper the results of theoretical models of the emission expected from neutral shells, tori or clumps surrounding the hot central stars of PNe. We include the effects of shocks, FUV (6 eV [FORMULA] eV), and soft X rays (50 eV [FORMULA] KeV) on the predominantly neutral gas and follow the time dependent chemistry for H2 (Hollenbach & Natta 1995), solving for the chemical and temperature structure and the emergent spectrum of the evolving shell. We consider a large interval of values of the mass of the central core (from 0.6 to 0.836 [FORMULA]) and of the shell properties, namely its density and filling factor. The calculations give as function of time the physical and chemical properties of the shell (temperature, fractional abundances of HII, HI, H2 and electrons), as well as the intensities of a number of lines, which can be compared to the observations and potentially used to determine the physical parameters of the ejection process.

The star warms quickly to [FORMULA][FORMULA]30,000 K, initiating a rapid rise in the luminosity of FUV and Ly-c photons. It is important to realize that only shells with high density (more precisely, high values of the ratio [FORMULA]) can trap the ionization front and remain in part neutral. The value of the minimum density required to trap the ionization front depends on the stellar parameters and on the evolution of the shell density with time and can be inferred from Fig. 3. As time proceeds, the fractional mass of ionized gas first increases, then decreases, and finally increases again as the star evolves along the white dwarf branch. For [FORMULA]=0.836 [FORMULA], most of the observable PN lifetime occurs in this last phase, where soft X rays are important.

If the density is sufficiently high, a three-layered shell is produced with an inner HII region, a central HI region, and an outer H2 region. In this case, we can identify three phases in the PN evolution. i) The early evolution of PN (around the peak of [FORMULA]) is dominated by FUV photons. The H2 is FUV pumped to vibrationally-excited levels, but the densities are sufficiently high that this energy is collisionally de-excited into heat. The shell has a large column of warm H2 and is very bright in all lines. The vibrationally excited v=1-0 and 2-1 H2 lines at 2µm are dominated by thermal emission from collisionally excited levels. ii) As [FORMULA] decreases with time, the molecular gas becomes cooler, the density drops, and the line intensities decrease rapidly. Finally the FUV pumping of H2 is no longer appreciably modified by collisions, and the v=1-0 H2 lines are mainly produced by the fluorescent cascade. This is the fluorescent phase. iii) At even later times, the star heats to [FORMULA][FORMULA]100,000 K and soft X-rays heat and ionize the neutral gas well above the values determined by the FUV stellar radiation. This results in an enhanced emission of all those lines that are temperature sensitive, such as the thermal component of the H2 2 µm lines as well as optical (e.g., OI 6300Å and near-IR metal (e.g., FeII 1.26µm and FeII 1.64µm) lines. The H2 emission in the vibrationally excited lines is dominated by the thermal emission from the hot molecular gas.

The exact properties and duration of these three stages depend on the mass of the central core, which, in turn, determines the time dependence of the radiation field to which the shell is exposed. In particular, a higher mass core results in a faster evolution of the physical properties and emission spectrum of the surrounding shell. The fluorescent phase is suppressed in high-mass cores, but does appear in neutral shells around low-mass central stars.

Models with high-mass cores are particularly interesting. They are the only case where we predict intense line emission in old nebulae (unless shocks are exceptionally strong). Since high-mass cores reach high [FORMULA], soft X-rays are very important and dominate most of the evolution of the shell. In these models, the decrease with time of the intensity of the H2 lines (both in the near and mid-infrared) is much less rapid than in PNe with low-mass cores. Time-dependent H2 chemistry is important in these objects, enhancing the intensity of the H2 lines. We believe that old (large) nebulae with strong H2 emission likely belong to this group.

Somewhat disappointingly, the shell density does not seem to affect in any simple way the line intensities, as long as the shell remains at least partially neutral. We have not been able to identify, among our results, any line ratio that, alone, could constrain the density of the neutral gas. The H2 v=0 lines in the mid-infrared and their ratio to the 1-0S(1) can provide important clues to the density. CII 158µm and OI 63µm are also density sensitive in shells around high mass cores. In order to find the density it is clear that several lines will have to be fitted simultaneously, together with other information, such as the size (age) of the nebula and the effective temperature of the central star. Measurement of molecular transitions (e.g., CS and HCO+)in the millimeter are useful probes of the density of the cool molecular gas deep inside the PDR.

Bachiller et al. (1997) have presented evidence that the molecular gas in PNe is highly clumped and that the density of the clumps ([FORMULA] cm-3) does not appear to change with the age of the nebula. Although this is difficult to understand from a theoretical point of view, we comment here on how this would change our model results, which assume [FORMULA] or [FORMULA]. At these high densities, time dependent effects tend to be unimportant, so that one can refer to a model run which produces the desired density at the observed distance from the star. We note, however, that we do not expect our models to be valid in the case of the tiny photoevaporating cometary globules observed in the Helix (Huggins et al. 1992; Meaburn et al. 1992; O'Dell & Handron 1996; Young et al. 1997; Cox et al. 1998). In this situation, photoevaporation into the diffuse HII gas leads to much more rapid advection of H2 into the PDR, and to significantly higher H2 intensities, than our models would predict. Models such those of Bertoldi & McKee (1990), Störzer & Hollenbach (1997), Johnstone et al. (1998) should be applied in this case.

The importance of the H2 2µm emission in the shocked layer of gas formed by the impact of the expanding shell on the precursor red-giant wind has also been examined. The emission in these lines does not depend on the physical and chemical structure of the shell, but only on the two parameters [FORMULA], [FORMULA], which characterize the red giant wind, and on the shock velocity [FORMULA]. Compared to the PDR emission, the shock contribution is never dominant for typical red-giant wind parameters ([FORMULA] km s-1, [FORMULA] [FORMULA] yr-1, [FORMULA]=1). However, with high ratios of [FORMULA] and somewhat dependent on the mass of the central star, the shock contribution can become significant.

The amount of atomic hydrogen predicted by our models is modest and rather hot. This is consistent with the scarcity of detections of the 21 cm line, either in emission or in absorption (see Schneider et al. 1987). Most of the shell mass in objects which emit in the vibrationally excited lines of H2 is likely to be molecular. A considerable mass in old shells may be in gas which is H2 but where the carbon and oxygen are in C+ and O, and not in CO. We suggest that in old PNe, molecular masses inferred from CO observations may be underestimated.

The results presented in this paper are suitable for further developments. A careful treatment of the carbon and oxygen chemistry is required to provide, for example, OH, H2O, C, CO, CO+, and HCO+ abundances and line intensities, and modifications in the H2 and [CII] line intensities. (Latter et al, in preparation). The predicted emission in the 2µm band of both ionized and molecular hydrogen can be compared to the existing observations to gain insight into the properties of the central stars and, possibly, of the neutral gas itself. Finally, ISO is now providing us with a wealth of data on middle and far infrared lines, which will allow a better understanding of the shell properties.

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© European Southern Observatory (ESO) 1998

Online publication: August 17, 1998