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

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5. Results

The numerical code used for the calculations treats the H2 chemistry as a time-dependent problem and has been described by Hollenbach & Natta (1995). At [FORMULA], the hydrogen in the shell is all molecular. As the temperature of the central star increases, the gas begins to dissociate and, at somewhat later times, to ionize. For our PNe modelling, the 1995 code is modified to include the formation of H2 by H-, the destruction of H2 by X-ray ionization and by C+, O and O+, and the time-dependent change of density and column density caused by the expansion of the shell and the advance (or recession) of the ionization front caused by the time dependence of the incident ionizing flux and the changing density.

We have run a grid of models, varying the shell parameters (e.g., [FORMULA], [FORMULA], [FORMULA]) and the central core mass [FORMULA]. In this section, we will discuss the properties and evolution of the shell by referring to a standard case (see Table 1; He, C and O abundances are from Perinotto 1991, other metal abundances are solar), which provides a useful template for the understanding of the physical processes. The template model has a core with mass [FORMULA]=0.6 [FORMULA], which is typical for PNe in general, but which may be lower than the stellar masses associated with H2-emitting PNe. As discussed in Sect. 1, the latter group is observed at low galactic latitudes and are thought to originate from more massive progenitors than average. The model has a value of [FORMULA] cm-3, which ensures that a portion of the shell remains neutral for all relevant times, yet which results in an electron density [FORMULA] cm-3 at [FORMULA] cm that does not greatly exceed observational limits. We note that if [FORMULA], [FORMULA] refers to the high density HII gas directly adjacent to (and in thermal pressure equilibrium with) the neutral PDR gas, whereas considerable HII may be at lower densities. The main purpose of the template model is to allow us to discuss the various physical processes and their relative importance in different phases of the evolution. As we will see, the same phases occur for higher [FORMULA], but the duration of some is much shorter.


[TABLE]

Table 1. Parameters of the Template Model


We will first (Sects. 5.1 to 5.5) discuss the structure of the shell as a function of time and the H2 emission which originates in the irradiated neutral shell. For simplicity, we will define this emission as PDR emission. The H2 emission produced in the shock between the expanding shell and the precursor red giant wind will be discussed in Sect. 5.9. The shock emission does not depend on the shell structure, as long as there is sufficient column in the shell to shield the preshock gas and maintain the H2 abundance in the red giant wind. The shock emission depends only on the shell speed and on the preshock density and speed of the red-giant precursor wind (cf. Eqs. 17 and 18).

Fig. 4-8 summarize the properties of the PN shell in the template model. Fig. 4 shows the mass of the ionized, atomic and molecular gas and the density of the atomic gas as function of time. The structure of the neutral shell at selected times is shown in Fig. 5, which plots temperature, electron abundance and atomic and molecular hydrogen abundance as functions of the depth in the shell. The slight discontinuities at [FORMULA] cm-2 are due to our simplified chemistry, which abruptly introduces CO at that column. Fig. 6 and 7 show for the same times the rates of gas heating and cooling and the dominant processes of formation and destruction of H2. Fig. 8 plots the intensity of the PDR H2 1-0S(1) line and the ratio of 2-1S(1)/1-0S(1) as a function of time.

[FIGURE] Fig. 4. The dependence of the density on time in the template model is shown by the solid curve. The fractional shell mass in ionized (dashed line), atomic (dotted line) and molecular (dot-dashed line) hydrogen are also plotted as a function of time. Note that this shell remains mostly neutral; the HII region contains only a small fraction of the total mass, 1% at most.

[FIGURE] Fig. 5. Temperature (solid line) and fractional abundance of H I (dotted line), H2 (dot-dashed line) and electrons (long-dashed line) are shown as function of the column density of hydrogen nuclei in the neutral shell at different times for the template model. The star illuminates the shell from the left. At each time, column density zero marks the edge between the ionized and neutral regions of the shell. The effect of soft X-rays on the electron abundances is clearly seen at columns [FORMULA] cm-2 for [FORMULA] yrs. There is considerable H2 at [FORMULA] K at [FORMULA] cm-2.

[FIGURE] Fig. 6. The dominant heating and cooling rates are shown as function of the column density in the neutral shell at various times for the template model. t=2200 yr : the solid line is the rate of net grain photolectric heating, the dot-long-dashed curve the heating due to FUV pumping of H2 followed by collisional de-excitation, the long-dashed line the cooling rate for H2 rotational and vibrational line emission, the dot-short-dashed line (labelled H2d) the rate of cooling due to dissociation of H2, the short-dashed line the rate of OH+H2O cooling, the dotted line the rate of CO cooling. At this time, two other cooling processes (cooling by emission in the OI 6300Å, SII 6700Å and FeII 1.26µm and FeII 1.64µm lines and cooling due to collisions with grains) contribute to the total cooling at small column density. We have omitted them to avoid confusion. t=4000 yr : the heating is due to FUV pumping of H2 (dot-long-dashed), photoelectric effect (solid) and X-ray heating (dotted), the cooling to emission in the H2 vibrational and rotational lines (long-dashed), O I 63µm (short-dashed) and optical and near-IR lines (dot-short-dashed). t=7000 yr : the heating is due to X-rays (dotted) and H2 FUV pumping followed by collisional deexcitation (dot-long-dashed), the cooling to emission of Ly[FORMULA] (solid), O I 63µm (short-dashed), optical and near-IR lines (dot-short-dashed), and H2 (long dashed).

[FIGURE] Fig. 7. H2 formation and destruction rates are shown as function of the column density in the neutral shell at various times for the template model. At [FORMULA] and [FORMULA] yr, the solid curves indicate the rate of destruction of H2 by collisions with other particles (H, H2 and electrons), the short-dashed curves the rate of photodissociation by FUV photons, the dot-long-dashed lines the sum of the rates of destruction due to reaction of H2 with C+ and O, the long-dashed line the rate of formation of H2 on grain surface. The rates of destruction by reaction with C+ and O have a very similar dependence on N, and we find convenient to plot them together. At [FORMULA] yr, three additional processes are important: destruction by reaction with O+ (short-dashed), by ionization caused by non-thermal electrons from soft X-rays (dot-short-dashed) and formation of H2 by reaction of H- with H (dotted). The formation rates are per H atom while the destruction rates are per H2 molecule.

[FIGURE] Fig. 8. PDR H2 1-0 S(1) intensity as a function of time in the template model. The intensity is the integrated emissivity in the PDR slab divided by 4[FORMULA]. The different components of the line are shown separately: the fluorescent emission due to the FUV stellar photons as a solid line, the thermal emission as a dashed line. The thin dotted vertical line marks the position of [FORMULA], i.e., the time when [FORMULA]=30000 K. The inset plots the ratio of the two H2 lines 2-1S(1)/1-0 S(1) as a function of time. The secondary plateau in thermal H2 emission at [FORMULA]5000-8000 yr is caused by the heating due to soft X-rays.

5.1. FUV-dominated phases: thermal and fluorescent

At t=0 the shell is completely molecular. As time proceeds, the fraction [FORMULA] begins to increase on the shell side facing the central star, as [FORMULA], or equivalently [FORMULA], increases (Fig. 4). The hydrogen chemistry is dominated by formation of H2 on grain surfaces and by destruction of H2 by FUV photons and by reaction with C+ and O. The maximum of [FORMULA] for the template star occurs at about 2200 yr. At this time, the shell is mostly atomic to a column of about [FORMULA] cm-2 and there is a large column of warm H2 (T[FORMULA] 1000 K), heated by the grain photoelectric heating mechanism and FUV pump heating of H2 (Fig. 5 and 6). The average temperature of the H2 2 µm-emitting layers is [FORMULA] K and the ratio of the 2-1S(1) to 1-0S(1) line is of the order of 0.15. The PN is very bright in the H2 lines, with an intensity in the 1-0S(1) of the order of 10-3 erg cm-2 s-1 sr-1 between [FORMULA] 500 and 3000 yr. This thermal FUV-dominated phase lasts until about [FORMULA] yr.

For [FORMULA] yrs, [FORMULA], n, and [FORMULA] all decrease. The dissociation front moves to lower column density ([FORMULA] cm-3 at t=4000 yr) and the molecular gas is cooler and less dense. The H2 emission decreases and fluorescent emission becomes relatively more important and dominates the spectrum for [FORMULA] 5000-7000 yr. The H2 ratio 2-1S(1)/1-0S(1) increases and may reach [FORMULA]0.5, characteristic of fluorescent emission. In this fluorescent FUV-dominated phase at [FORMULA] yr, the template PN has a typical surface brightness in the 1-0S(1) line of [FORMULA] erg cm-2 s-1 sr-1 .

5.2. X-ray dominated phase

In the later stages of the evolution ([FORMULA] yrs in the template model) 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 increased importance of the soft X-ray relative to FUV photons affects not only the line emission but the hydrogen chemistry as well. In a relatively cool shell, H2 forms on grain surfaces and is destroyed by FUV photodissociation. When X-rays are present, the chemistry becomes much more complex. A number of other processes, such as destruction of H2 by collisions in [FORMULA] K gas, by reaction with C+, O and O+ and, in some cases, by direct dissociation from the non-thermal electrons produced by the X-rays, as well as formation of H2 by reaction with H-, become important. Cooling by collisionally excited metal optical lines and Ly[FORMULA] dominates at small column densities where [FORMULA] K.

The importance of the X-ray emission of the central star in the later stages of the PN evolution can be better appreciated by comparing the thermal component of the H2 intensity to the results of models with the same parameters where the X-ray flux of the central star is artificially set to zero (Fig. 9, top panel). The line intensity is the same in the two models at early times, when the shell physics is dominated by the FUV photons, but diverges at later times ([FORMULA]5000 yr for [FORMULA]=0.6 [FORMULA]), following the decrease of [FORMULA]. It should be emphasized that the intrinsic soft X-ray luminosity of the central star is quite uncertain; therefore the observation of X-ray heated H2 emission at late times may probe the soft X-ray spectrum from a given core.

[FIGURE] Fig. 9. Thermal component of the PDR H2 1-0S(1) intensity as a function of time for models with (solid curves) and without (dashed curves) X-rays. The top panel shows the results for [FORMULA]=0.6 [FORMULA] the bottom panel for [FORMULA]=0.836 [FORMULA]. All other parameters as in the template model. The X-rays are particularly important at late times ([FORMULA] yr) in the high mass central star case. The position of [FORMULA] for [FORMULA] [FORMULA] is shown by the thin dotted vertical line in the upper panel. For [FORMULA]=0.836[FORMULA], [FORMULA] coincides with the beginning of the curves.

Note that in the template model the H2 emission in the vibrationally excited lines is dominated by fluorescent emission for [FORMULA] yr, and the effects of the X-rays on the total H2 line intensity is negligible. Fig. 9 shows only the thermal component of the line emission. In terms of the H2 2 µm emission, it is more important to include X-rays in the PDR description of PNe with high mass cores (see Sect. 5.3) than with low mass cores.

5.3. High mass central stars

The shell evolution depends strongly on the mass of the central star, which, in turn, determines the time dependence of the radiation field to which the shell is exposed. A higher mass core results in a faster evolution of the physical properties and emission spectrum of the surrounding shell. The FUV thermal phase and the X-ray thermal phase still exist (the fluorescent phase never dominates the 1-0S(1) emission for high mass cores), but occur on much shorter timescales. Moreover, since higher mass cores reach higher [FORMULA], soft X-rays are more important and dominate most of the evolution of the shell.

Fig. 10 compares the H2 predicted emission for models with increasing core mass. The other parameters are as in the template model (Table 1). All the curves begin at the time [FORMULA] when [FORMULA]=30,000 K. (this corresponds roughly to the 1-0S(1) peak in the thermal FUV-dominated phase). The curves initially decrease rapidly with time as [FORMULA], n, and [FORMULA] decline (see Fig. 2). This sharp decline stops as soon as the stars reach the white dwarf cooling tracks.

[FIGURE] Fig. 10. PDR H2 1-0S(1) intensity as a function of time for models with template model parameters but different core masses: [FORMULA]=0.6 [FORMULA] (dot-short-dashed curve), [FORMULA]=0.64 [FORMULA] (dot-long-dashed curve), [FORMULA]=0.696 [FORMULA] (dashed curve), and [FORMULA]=0.836 [FORMULA] (solid curve). The inset on the top right shows the ratio of the two H2 lines 2-1S(1)/1-0S(1) as a function of time for the same models. The curve for [FORMULA]=0.836 [FORMULA] is not shown here but in Fig. 11c to avoid confusion. All the curves start at [FORMULA].

The different components of the H2 1-0S(1) line for [FORMULA]=0.836 [FORMULA] are shown in Fig. 11 panel a). We use a logarithmic time scale, to show the early evolution of this model in detail. The X-ray peak at [FORMULA] yr corresponds to the peak in [FORMULA] (see Fig. 2), which is also traced by the peak in the contribution from direct excitation of H2 by the nonthermal electrons produced by the X-rays. Fig. 9, bottom panel compares the H2 intensity to the results of a model where the X-ray flux is set to zero at all times. For this extremely high mass core the FUV-dominated phase lasts only few hundreds years, and the X-ray radiation heats the gas and maintains the H2 excitation at later times. Without X-rays the H2 intensity falls below [FORMULA] erg cm-2 s-1 sr-1 after [FORMULA]1300 yr, while, when the X-rays are included, the intensity of this line is still [FORMULA] erg cm-2 s-1 sr-1 at [FORMULA] yr.

[FIGURE] Fig. 11. a shows the different components of the PDR H2 1-0S(1) intensity as a function of time for [FORMULA]=0.836 [FORMULA]: thermal emission (dashed line), fluorescent emission (solid line), direct excitation by the nonthermal electrons produced by X-rays (dot-dashed line). b shows the same for the 2-1S(1) line. c plots the ratio of the two H2 lines 2-1S(1)/1-0S(1). Note the logarithmic time scale. The other model parameters are as in the template model.

The intensity of the 2-1S(1) line is shown in Fig. 11, panel b), and the ratio of the two lines in Fig. 11, panel c). The 2-1S(1) line has a large fluorescent contribution at [FORMULA] yr. The relatively high 2-1/1-0 ratio at earlier time is due to the high temperature of the H2-emitting gas. The small thermal peak in 1-0S(1) at [FORMULA] years, with the corresponding dip in the 2-1/1-0 ratio, may not be real but may result from the H2 dissociation front, the dominant 2 µm emitting region, and the (artificial) C+/CO transition all occurring at the same column at this time. More realistic C+/CO chemistry will likely remove this glitch. We have tested and found it not to be a product of (space or time) numerical grid size, but a product of the combination of time dependent H2 chemistry with the C+ reaction on H2 at the C+/CO boundary.

5.4. Time-dependent H2 chemistry

The importance of properly taking into account the time dependence of H2 chemistry can be assessed by looking at the results shown in Fig. 12, where we plot the ratio of the H2 1-0S(1) intensity to the predictions of equilibrium models with the same parameters. Time-dependent models predict an H2 intensity higher than equilibrium models in those phases of the evolution where the mass of HII increases with time (see Fig. 3). At each time step, a layer of mostly atomic gas becomes ionized, and a new layer of molecular matter is exposed to the UV or X-ray heating radiation. In other words, there is rapid advection of molecular material into the PDR. This results in a larger amount of hot molecular gas than equilibrium calculations predict, and, as a consequence, in a larger intensity of the H2 2µm lines.

[FIGURE] Fig. 12. Ratio of the PDR H2 1-0S(1) intensity to the prediction of steady-state models as a function of time. The upper panel plots the results for [FORMULA]=0.6 [FORMULA] and three different values of [FORMULA], 105 (solid curve), 106 (short- dashed curve) and 107 (dot-long-dashed curve) cm-3. Note that the shell with [FORMULA] cm-3 becomes completely ionized at [FORMULA] yr. The thin dotted vertical line marks the position of [FORMULA]. The bottom panel shows the same ratio for [FORMULA]=0.836 [FORMULA], same densities. All other parameters as in the template model.

The non-equilibrium effect is not strong for [FORMULA] [FORMULA]. The exception is a brief phase in models with [FORMULA] cm-3, just before the shell becomes completely ionized. The effect is, however, of great importance for those PNe with high mass cores. In these objects, most of the lifetime of the PN is spent in a phase where the mass of ionized gas increases, as the star evolves very slowly along the white dwarf branch and the shell continues to expand. At each timestep, a new layer of molecular gas is exposed to the stellar radiation and heated by the strong X-ray flux from the very hot central star to high temperatures. The effect is stronger at lower density. For [FORMULA]=0.836 [FORMULA] and [FORMULA] cm-3, the enhancement of the H2 1-0S(1) intensity is [FORMULA] for [FORMULA] 2000 yr.

5.5. Dependence of H2 2µm emission on shell parameters

Our calculations show that the main parameter that controls the H2 emission is [FORMULA], the mass of the core, which determines the evolution of the FUV, EUV and X-ray luminosity of the central star. As long as the shell is not completely ionized, the H2 line intensities show little dependence on the shell parameters, density, filling factor [FORMULA] and [FORMULA], over most of the observable life of PNe. We note that the observed intensity would depend on [FORMULA] if, for example, the clumps from a partial shell did not fill the beam. The parameter [FORMULA] only enters our calculation in determining the column through the partial shell, for fixed [FORMULA].

Fig. 13 shows the H2 1-0S(1) intensity as a function of time for two core masses and three different densities. The variation of n with time is the same in each set of models ([FORMULA]=2), so that at any given time we can compare shells whose density varies by a factor 102. In the [FORMULA] [FORMULA] case (top panel) the dependence on n is quite small at all times. Note again that the [FORMULA] cm-3 case terminates at [FORMULA] yr, where the shell is completely ionized. At [FORMULA] yr, when thermal emission dominates, the density dependence is negligible largely because the FUV pumping of H2 dominates the heating and H2 rovibrational lines dominate the cooling. This can be qualitatively understood by comparing the density sensitivity of the heating and cooling rates. The heating rate per unit volume [FORMULA] by FUV pumping of H2 is proportional to [FORMULA], where [FORMULA] is the local FUV dissociation rate (see Eqs. 12 and 13) which depends on [FORMULA]. The cooling [FORMULA] by H2 for [FORMULA] cm-3 is proportional to [FORMULA], since the H2 rovibrational levels are in LTE at these densities. Equating the heating to cooling cancels the density dependence, so that in this case the temperature is the same at the same [FORMULA], regardless of n. Therefore, the H2 intensities, which depend only on [FORMULA] and T for LTE conditions, are not sensitive to n.

[FIGURE] Fig. 13. PDR H2 1-0S(1) intensity in PNe with different densities. The upper panel plots the results for [FORMULA]=0.6 [FORMULA] and three different values of [FORMULA], 105 (solid curve), 106 (short- dashed curve) and 107 (dot-long-dashed curve) cm-3. The thin dotted vertical line marks the position of [FORMULA]. The bottom panel is for [FORMULA]=0.836 [FORMULA], same densities. All other parameters as in the template model. The bottom panel plots also the prediction of steady-state models with the same densities ([FORMULA] cm-3, dotted curve; [FORMULA] cm-3, long-dashed curve; [FORMULA] cm-3, dot-short-dashed curve).

The H2 intensity is more sensitive to n when H2 does not dominate both heating and cooling. For example, in the [FORMULA] [FORMULA] case, the H2 intensity increases with n at [FORMULA] yr, when the OI 63µm line dominates the cooling (see Fig. 6, 13). For [FORMULA] cm-3, the fine structure levels of OI are in LTE and the cooling rate is given [FORMULA]. Therefore, for a given column [FORMULA] the temperature is an increasing function of [FORMULA], the [FORMULA] abundance at that point. Since [FORMULA] increases with n, the H2 1-0S(1) line is stronger at higher n in this case.

At [FORMULA] yr, fluorescent emission dominates, and the density dependence is also very small. Higher density means greater FUV pumping of vibrational levels because H2 has higher abundances and therefore competes more effectively with dust for FUV photons. However, the increased pumping is nearly exactly offset by the increase in the collisional de-excitation of the vibrationally-excited levels at higher density.

In high-mass core models (Fig. 13, bottom panel) the dependence of the H2 line intensities on n is also weak, with a variation in the 1-0S(1) line of less than a factor of 3 in the time interval [FORMULA] yr and a factor of 4 at [FORMULA] yr. Fig. 13, bottom panel, plots also the predictions of models with the same parameters, but steady-state H2 chemistry. One can see that the dependence on density is small also in the steady-state models, but that in these models the line intensity decreases with the density. The moderate sensitivity to variations of n is due to the fact that X-rays contribute a major fraction of the heating in the dissociation front, while the cooling is dominated by O I 63µm and the OH+H2O lines. This effect alone tends to increase the H2 intensity with density. However, the enhancement of the H2 lines due to time-dependent chemistry is larger at lower density (see also Fig. 12) and this reverses the behaviour of the density dependence in time-dependent models.

All the models discussed in this section have a time density dependence [FORMULA] for [FORMULA], as expected if the shell evolution is driven by pressure from the central star. If the shell is expanding freely ([FORMULA]=3), the density decreases faster and the shell becomes completely ionized for a larger range of parameters (see Fig. 3). If this does not happen, and a significant column of molecular gas exists, then the H2 intensity is not very sensitive to the exact value of n, and the two cases ([FORMULA]=2 and [FORMULA]=3) do not differ significantly.

The lack of dependence of the PDR H2 2µm line intensity on the shell density n over a large interval of time (as long as the shell is dense enough to stay mainly neutral) is one of the main results of our calculations and has important consequences for the interpretation of the observations. We show below that other observations may help constrain the density.

5.6. Mid-infrared H2 lines

In addition to the two H2 lines at 2µm 1-0S(1) and 2-1S(1), we have computed the intensity of 5 lines from v=0 [namely, the J=6-4 line at 8.0µm (S(4)), the J=5-3 at 9.7µm (S(3)), the J=4-2 at 12µm (S(2)), the J=3-1 at 17µm (S(1)) and the J=2-0 at 28µm (S(0))], assuming LTE population of the J levels and an ortho-to-para ratio of 3. Fig. 14 shows the results for [FORMULA] [FORMULA], Fig. 15 for [FORMULA]=0.836[FORMULA]. These lines have excitation temperatures between 3474 and 512 K, and are excited in molecular gas cooler ([FORMULA] 200-500 K) than that emitting the 2 µm lines (the excitation temperature of the 1-0S(1) line is 6880 K).

[FIGURE] Fig. 14. Intensity as function of time of the 5 mid-infrared H2 lines v=0-0 S(0) at 28µm, S(1) at 17µm, S(2) at 12 µm, S(3) at 9.7µm, S(4) at 8µm for [FORMULA] [FORMULA]. For each line, we show the results for two densities, [FORMULA] (dashed curves) and [FORMULA] (dot-dashed curves) cm-3. All the curves start at [FORMULA]. The shell with [FORMULA] cm-3 becomes completely ionized at [FORMULA] yr and is not plotted here. The right bottom panel shows the ratio of the 1-0S(1) line at 2.12 µm to the 17 µm line for the same values of [FORMULA].

[FIGURE] Fig. 15. Same as Fig. 14 for [FORMULA]=0.836[FORMULA]. The solid curves are for [FORMULA] cm-3.

Like the 2 µm lines, the intensity of the mid-infrared lines decreases with time, as the shells move further from the central star and become cooler. The decrease is much more rapid in models with low-mass cores. The exception to this trend occurs between 4000-5000 years in the [FORMULA] [FORMULA], [FORMULA] cm-3 case. The increase in the pure rotational H2 emission with time during this interval is caused by the C+ + H2 reaction, whose rate coefficient is proportional to exp(-4640/T). During this time interval the H2 emission arises from a warm ([FORMULA] K), largely atomic region at columns [FORMULA] cm-2. This region is heated by the FUV pumping of H2 and cooled by species other than H2. The H2 destruction is dominated by the reaction with C+. As t increases from 4000 to 5000 years, the shell expands outwards, the star evolves, and the FUV flux from the central star decreases. This causes a drop in the temperature at these columns from [FORMULA] to [FORMULA] K. Because of the temperature sensitivity of the C+ reaction, the drop in temperature results in roughly a factor of 30 increase in the H2 abundance (and column) in this region. Although the temperature drop alone would reduce the H2 intensities, the increase in H2 column is dominant so that the intensities increase with time.

The line intensities also decrease with time for the high mass cores, but less rapidly, especially for the lower density shells. At [FORMULA] yr, lines brighter than [FORMULA] erg cm-2 s-1 sr-1 are only produced around high-mass cores with neutral densities characterized by [FORMULA] cm-3. The line intensities are stronger for lower densities because the heating has a weaker density dependence ([FORMULA]) than the cooling, which has strong contributions from OH and H2O. As a result, dense gas is cooler and less emissive in these lines.

We have inspected the ratio of the various mid-infrared lines and of the 1-0S(1) to the 17µm line (this last is shown in the bottom-right panel of both figures), looking for possible density indicators. For high-mass cores, the line ratios do not show any strong density-dependence; the variation of any line ratio in the density interval we considered is a factor of three at most. The situation is somewhat more favorable for low-mass cores, where at [FORMULA]5000 yr the ratio of the 1-0S(1) line, which is mostly due to fluorescence, over the 17µm line is very large ([FORMULA] for [FORMULA] cm-3), and increases as the density decreases.

5.7. Metal lines

Our model calculations produce, as a byproduct, the intensity of the metal lines which are important coolants of the atomic and molecular gas. Among them, we show for various models the far-infrared lines CII 158µm, OI 63µm, and SiII 35µm (Fig. 16). Fig. 17 shows the intensity of the red line OI 6300Å and the FeII 1.26µm line. The OI 63µm and, to a lesser extent, the CII 158µm lines are often optically thick, especially at early times. Therefore, much of their flux emerges from the (hotter) side of the shell facing the central star. The actual observed intensity depends on the geometry of the shell, [FORMULA] and on the viewing angle. We have computed the intensity in the direction of the star and in the direction opposite to it, and plot in Fig. 16 the intensity from the bright side of the PDR, i.e., in the direction of the star. At [FORMULA] yr, the OI 63µm line can be more than ten times weaker when seen from the dark side of the shell, because of the self-absorption by cooler atomic oxygen deeper in the shell.

[FIGURE] Fig. 16. Intensity of the infrared metal lines as function of time. The three top panels plot the results for [FORMULA]=0.836 [FORMULA] and three different densities ([FORMULA] cm-3, solid line; [FORMULA] cm-3, dashed line; [FORMULA] cm-3, dot-long-dashed line). The left panel shows the intensity of the CII 158µm line; the central panel the ratio of the OI 63µm to the CII 158µm line; the right panel the ratio of the SiII 35µm to the CII 158µm line. The three bottom panels show the same quantities for models with [FORMULA]=0.6 [FORMULA]. All other parameters as in the template model.

[FIGURE] Fig. 17. Intensity of the OI 6300Å and FeII 1.26µm lines as function of time for different models. The two top panels plot the results for [FORMULA]=0.836 [FORMULA] and three densities ([FORMULA] cm-3, solid line; [FORMULA] cm-3, dashed line; [FORMULA] cm-3, dot-long-dashed line). The two bottom panels for [FORMULA]=0.6 [FORMULA] and [FORMULA] cm-3. All other parameters as in the template model.

As for the majority of the lines we have considered, the intensity of the far-infrared lines decreases with time, more slowly in PNe with high-mass cores, so that old (large) objects with very bright emission are very likely powered by high-mass central stars. These lines, however, should be detectable by ISO in a large number of cases. In all our models the CII 158µm line flux (assuming a distance of 1 Kpc and [FORMULA]=0.1) is [FORMULA] erg cm-2 s-1 at all times, well within ISO capability (see Liu et al. 1996).

Is it possible to use the ratio of different metal lines to determine model parameters, density in particular? Inspection of Fig. 16 shows one clear trend, namely that relative young ([FORMULA]yr) PNe with high-mass cores and high PDR density are the only objects for which OI 63µm/CII 158µm [FORMULA]. It is somewhat counter-intuitive that denser PDR gas has smaller [OI]/[CII] ratios, considering that [OI] has a higher critical density than [CII]. However, in these models the denser gas is cooler, which suppresses the [OI]/[CII] ratio because of the higher excitation energy of [OI]. Unfortunately, the [OI]/[CII] ratio may not be a very useful density diagnostic as it should be kept in mind that the values in Fig. 16 are maximum values, for PDRs seen from their bright side. Different orientations with respect to the observer may result in much lower values of the ratio. We have checked the other extreme case, i.e., PDR seen from their dark side, and found that at [FORMULA] yr other combinations of model parameters result into OI 63µm/CII 158µm[FORMULA]. However, observed intense CII 158µm emission and low ([FORMULA]) ratios OI 63µm/CII 158µm in mid-size PNe are likely to indicate high-density, high-core mass objects.

The ratio SiII 35µm/CII 158µm is shown in Fig. 16, right panels. We have assumed that elemental Si has solar abundance, that 1/2 of it is locked into grains, and that all gaseous Si is singly ionized in the PDR. The SiII 35µm line is quite strong, unless considerably more depletion of Si into grains occurs. Measurement of this line will therefore constrain the amount of Si incorporated into grains in the ejecta of PNe. Some SiII 35µm emission originates from the HII gas, so care must taken in interpreting observations.

The OI 6300Å line is very intense. As a function of time it shows a double-peak structure, which is clearly visible in models with [FORMULA] [FORMULA]. The first peak occurs in the UV-heated atomic layer (cf. Fig. 5). The intensity shown in Fig. 17 is the intensity emerging from the side of the shell facing the star. Depending on viewing angle, the observer may see the OI 6300Å through much of the shell, and dust may significantly attenuate these intensities (see Sect. 6). The second peak occurs at later times, in the X-ray heated neutral gas (cf. the behaviour of the H2 1-0S(1) intensity in Fig. 8). At these times, the shell has low column and dust attenuation is negligible. For high-mass cores, X-rays dominate the heating at all relevant times. The O I line is intense and it may be best detectable at [FORMULA], when, as in the case of lower-mass cores, dust extinction becomes negligible.

Fig. 17 shows also the intensity of the FeII 1.26µm line (the FeII 1.64µm line intensity is in all cases 0.7 times weaker), assuming no depletion of Fe into grains. Both depletion and dust attenuation are important in the early stages of the evolution. The emission at [FORMULA] 5000 yr is very weak in PNe with high-mass cores, and of the order of few[FORMULA] erg cm-2 s-1 sr-1 in low-mass cores. A significant fraction of FeII emission could arise from the HII region.

5.8. Br[FORMULA]

The intensity of Br[FORMULA] depends only on the number of ionizing photons [FORMULA] and on the radius of the shell R (see Eq. 7), not on shell parameters such as density and filling factor. Fig. 18, top panel plots as function of time the Br[FORMULA] intensity for the four values of core mass we have considered. Note that, at times [FORMULA] yr, PNe with low-mass cores have the highest Br[FORMULA] intensity. At later times, when the PN core reaches the white dwarf cooling track, all the curves coincide.

[FIGURE] Fig. 18. Top panel: Br[FORMULA] intensity as function of time for different core masses: [FORMULA]=0.6 [FORMULA] (dot-short-dashed curve), [FORMULA]=0.64 [FORMULA] (dot-long-dashed curve), [FORMULA]=0.696 [FORMULA] (dashed curve), and [FORMULA]=0.836 [FORMULA] (solid curve). Bottom panel: ratio of Br[FORMULA] to H2 1-0S(1) intensity for the same core masses. In these models, [FORMULA] cm-2, [FORMULA], other parameters as in the template model.

The bottom panel of Fig. 18 shows the ratio of Br[FORMULA] over H2 1-0S(1) intensity for the same four values of [FORMULA]. This ratio depends only weakly on the density, since both the Br[FORMULA] and the H2 1-0S(1) are density insensitive. The curves in Fig. 18 are for [FORMULA] cm-3. This figure illustrates an important characteristic of the models. Namely, we find that Br[FORMULA] is more intense than the H2 1-0S(1) only in low-mass core PNe in the time interval [FORMULA]1000-8000 yr. At later times, and in high-mass core PNe, Br[FORMULA] is generally less intense than the molecular line, and significantly so for [FORMULA]=0.836 [FORMULA]. The Br[FORMULA] luminosity can be increased relative to the H2 luminosity by increasing [FORMULA] relative to [FORMULA].

5.9. Shock H2 2µm emission

The H2 intensity produced in the shock between the expanding shell and the precursor red giant wind does not depend on the physical and chemical structure of the shell (as long as the FUV does not penetrate the shell and dissociate the preshock H2 in the red giant wind), but only on the shell velocity [FORMULA]. The H2 emission also depends on the preshock density and the velocity [FORMULA] of the red giant wind, or equivalently [FORMULA] and [FORMULA]. The shock velocity is then given by [FORMULA], which together with the preshock density determines the H2 line intensities.

Fig. 19, upper panel shows the intensity of the H2 1-0S(1) line, computed according to Eq. 17, for [FORMULA] [FORMULA] yr-1, [FORMULA]=1 (i.e, assuming that the red giant wind is isotropic), [FORMULA]=8 km s-1. The different curves correspond to different [FORMULA] (see figure caption). The slight dependence on [FORMULA] is caused by the assumed dependence of [FORMULA] on [FORMULA].

[FIGURE] Fig. 19. Top panel: H2 1-0S(1) intensity produced in the shock between the shell and the red giant precursor wind for four values of [FORMULA], [FORMULA]=0.6 [FORMULA] (dot-short-dashed curves), [FORMULA]=0.64 [FORMULA] (dot-long-dashed curves), [FORMULA]=0.696 [FORMULA] (dashed curves), and [FORMULA]=0.836 [FORMULA] (solid curves). The results depend only on the parameters [FORMULA] [FORMULA] yr-1, [FORMULA]=8 km s-1, [FORMULA]. The slight difference between different [FORMULA] is due to the assumed dependence of [FORMULA] on [FORMULA] at early times. Note that the curves for [FORMULA]=0.696 and 0.836 [FORMULA] coincide. The bottom panel plots the ratio of the shock vs. PDR intensity for the same four values of [FORMULA], other parameters as in the template model.

The shock emission is compared to the PDR emission in the lower panel of Fig. 19, which plots the ratio of the shock/PDR line intensity for models with different [FORMULA] and shell parameters as in the template model. In these models, the shock contribution is never larger than the PDR emission for [FORMULA]. We ignore the comparison at [FORMULA], before the central star emits significant FUV. Obviously, at these very early times, shock emission could dominate if the shell has achieved supersonic speeds relative to the red giant wind. For low mass cores, the ratio of shock to PDR intensity is at first [FORMULA] (for [FORMULA] yr for [FORMULA] [FORMULA], [FORMULA] yr for [FORMULA] [FORMULA]), and of order unity later on when the H2 2µm intensity is low. For high mass cores, the shock contribution is larger than 10% over most of the lifetime of the PNe.

This result on the shock versus PDR dominance does not depend on the shell parameters, which, as we have seen, do not affect either the shock or PDR emission, but is very sensitive to the red giant wind properties. The shock emission is directly proportional to the ratio [FORMULA]. If [FORMULA] [FORMULA] yr-1, i.e., a factor 10 larger than in Fig. 19, the H2 emission would be dominated by the shock component over most of the observable life of PNe with high-mass cores; if a factor 10 smaller, the shock contribution to the H2 emission would never be detectable. Our adopted value 10-5 [FORMULA] yr-1 is typical of red giant winds (Loup et al. 1993). We cannot, however, exclude the possibility of larger values, possibly because [FORMULA] or because high-mass cores may have higher red giant mass loss rates.

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© European Southern Observatory (ESO) 1998

Online publication: August 17, 1998
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