The complexity of the physical and chemical processes which occur at different times in the neutral shells of PNe is evident in the description of the results of our model calculations given in Sect. 5. Although some important characteristics are displayed by the models, it is clear that many different spectral lines need to be reproduced simultaneously, in order to derive the parameters of the shell and the nature of the central star. In a forthcoming paper (Vicini et al. 1998), we show how the PDR models can account for the observed properties of the neutral gas in the nebula NGC 2346. Another obvious test case is NGC 7027 (Natta & Hollenbach, in preparation), the well-known, young ( 600 yr; Masson 1989) PN, whose intense emission in the 1-0S(1) line seems consistent with the predictions of PDR models for high-mass cores.
When comparing the model results to PNe observations, several caveats should be kept in mind. In order to follow the time-dependent PDR physics and chemistry evolution, we have used a number of approximations that are described and justified in Sect. 3. In particular, we have used a greatly simplified scheme for the chemistry of carbon and oxygen. The sense of the error is that we may have somewhat overestimated the C+ abundance at and beyond the H2 dissociation front when the temperature there is K so that the C+ + H2 reaction proceeds rapidly. Thus, we have underestimated the H2 abundance and may have overestimated the temperature in this region. We note that we have included the effect of the production of the coolants OH and H2O by the reaction of H2 with O in these high temperature regions, so that we have not greatly overestimated the temperature. We have tested for the maximum error in our predicted intensities due to the simplified chemistry by running models without the C++H2 reaction. In the latter models the H2 2 µm intensities rise by factors as much as 10 at the thermal peak in the template model, and for late times, years in the high mass core case with cm-3. The H2 mid-infrared lines decrease by an order of magnitude at years for cm-3 and . The FeII 1.26 µm line decreases by an order of magnitude at years for cm-3 and . For all other regions of parameter space and for all other species, the observable line intensities change by much smaller factors. Clearly, the models presented in this paper somewhat underestimate the H2 2µm intensities, but the true intensities must lie closer to these models than the opposite (and unphysical) extreme of no C+ + H2 reaction. Therefore, we estimate that the inclusion of carbon/oxygen chemistry will not change the results presented here by more than a factor of about 2. Clearly the next step is to include the full carbon and oxygen chemistry into the coupled radiative transfer, hydrodynamics, chemistry and thermal balance code we have developed. Such a step is essential if model [CI], CO, or other molecular intensities are desired. However, this would require a major streamlining of the current code, or much faster computers, and we defer this task to future work.
The simplification in the geometry and evolution of the shell, as well as the uncertainty in the evolutionary tracks of the central stars and, in particular, the tracks of the X-ray fluxes, introduce an inevitable crudeness to our models. The soft X-ray luminosity of the central star is of great importance in the late stages of the evolution of the neutral shell, and it is not well constrained either by theory or observations. The results of the late phases of the PNe evolution are therefore only indicative. They do show, however, that, under reasonable assumptions, soft X-rays dominate the physics and chemistry of the neutral shells around old PNe. For high-mass cores, all the observable life of PNe occurs in a X-ray dominated phase.
Some of the parameters which define the shell structure and evolution have been kept constant in our grid of models. In some cases, they turned out to be quite irrelevant. This is true, for example, of the mass of gas in the shell, that we have fixed to be =0.3 . The results do not depend on , as long as . In other cases (i.e., , , ), most of our results can be simply scaled using the relations discussed in Sects. 3 and 4. The adopted abundances of O and C are typical of carbon-rich PNe. Some sparse models we have computed for oxygen-rich objects indicate that for low-mass cores the H2 line intensity does not vary significantly with the chemical composition, while metal line intensities are, of course, proportional to the metal abundance.
In our calculations, we have assumed normal ISM dust properties. This was due in part to the need to reduce the number of free parameters, but also to the consideration that most dust is in the neutral shell and has been exposed neither to the harsh environment of the HII region nor to high-velocity shocks. It should therefore retain the properties of the red giant wind dust. The choice of the dust properties affects our results in various ways. Different dust properties can change the FUV attenuation and the rate of H2 formation on grains, affecting the H2 abundance. More importantly, grain photoelectric heating depends on the assumed grain properties (inter alia , on the fraction of very small grains and PAHs). If these vary, the neutral gas temperature (and hence the H2 line intensity) will also vary. Note however that photoelectric effects dominate the heating only in the very early stages of the PN evolution (cf. Fig. 6).
We have neglected the effect of dust absorption on the lines. This may be important in the very early stages of the PN evolution, when the dust column density through the shell is large. The exact amount of extinction to the observer depends not only on the total column density through the shell but also on the filling factor and the viewing angle. In the later stages of the evolution dust extinction through the shell becomes negligible, as the column density in the shell decreases. The maximum extinction occurs when the neutral gas is in a torus seen edge-on and is . Assuming standard dust-to-gas ratios, and taking into account that the 2µm H2 lines form very close to the dissociation front, we estimate that for , =0.1 the 2µm optical depth drops below unity at yr, depending on the expansion velocity at early times, i.e., on the core mass. The effect of dust extinction is more important for optical lines such as OI 6300Å (A6300/A), which can be entirely suppressed in young, compact PNe. The intensities given in Fig. 17 should, therefore, be considered as upper limits to the true values.
Our calculations indicate that a significant amount of molecular (H2) gas exists in which oxygen is mostly atomic and carbon is mostly C+, rather than CO. Fig. 20 compares the true mass in H2 to the mass of molecular gas in the region of the shell where CO exists. This last quantity ( in Fig. 20) decreases as the nebula expands and the column density decreases, while the true H2 mass remains almost constant. In the approximation we used for the C and O chemistry, neither of them is sensitive to the model parameters. The crude treatment of C and O chemistry in our models does not allow us a quantitative discussion of the molecular masses derived for many PNe from CO millimeter observations (see Huggins et al. 1996 for the most recent set of data). We only point out that CO inferred molecular masses are likely to be lower limits to the true values. This effect needs to be taken into account when modeling the precise (observed) inverse correlation between the ratio of the molecular to ionized gas mass and R. It may also explain those PNe detected in the H2 1-0S(1) but not in CO (Huggins et al. 1996). Clearly, better models are needed to clarify this point.
Fig. 20 also shows the mass of atomic hydrogen as a function of time for four different models with different . In all our template ( cm-3) models we find that the fraction of the shell mass in H I is at any time quite small, 10%. The problems of determining the amount of H I in PNe by observing the 21 cm line, either in absorption or in emission, have been discussed by Schneider et al. (1987). Our values of the H I mass ( 0.03 ) are in agreement with the upper limits they derive from emission measurements of 22 PNe. Schneider et al. have also searched for 21 cm absorption in 10 PNe, with only one detection. We have computed the model-predicted 21 cm optical depth from the expression km s-1, where km s-1 is the velocity difference across the shell and the spin temperature. We find that, since much of the atomic hydrogen is quite warm, is considerably lower than one might naively estimate. We find that in high mass cores , so that no significant 21 cm absorption should be expected. However, in low mass cores can be as large as 0.5-1, which may explain the single detection. CII 158µm or OI 63µm do trace the HI gas, but, as noted above, they also trace part of the H2 gas.
© European Southern Observatory (ESO) 1998
Online publication: August 17, 1998