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Astron. Astrophys. 332, 1044-1054 (1998)

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2. Modeling planetary nebulae and abundances determinations

To investigate the issue of abundance determinations in planetaries we proceed in two steps: in the first one hydrodynamical models are calculated with assigned input chemical abundances which serve as substitutes of real planetaries. This first step contains two items:

  • [i)] radiation gasdynamical modeling along theoretical stellar evolutionary tracks in the HR diagram in the framework of the interacting-winds theory;
  • ii) generation of "equilibrium" models from selected models of the hydrodynamical sequences, which are equivalent to stationary photoionization models.

The second step contains the determination of the chemical abundances with the "constant [FORMULA], [FORMULA] " method and their comparison with the input values. It will then provide us with information on the errors inherent in this method.

2.1. The physical model

It is an established fact that a PN forms out of the envelope the central star looses during its evolution on the asymptotic giant branch (AGB) due to slow but massive stellar winds. Unfortunately, to date neither observations nor theoretical models can provide a final picture of the mass loss history on the AGB for given stellar parameters. On the theoretical side there seems to be some consensus that pulsationally induced shock waves in connection with dust formation are the driving force of the winds. The subsequent radiative acceleration of the grains and their momentum coupling to the gas can account for the observed mass loss rates and velocities. Despite the substantial progress that has been made in the self-consistent modeling of such winds during the last few years (see e.g. Fleischer et al. 1995; Höfner et al. 1995 and the references therein) one is far from incorporating such models into stellar evolutionary calculations. The latter is, however, needed to derive the resulting density distribution of the circumstellar shell at the end of the AGB phase from first principles. This density structure is the crucial initial quantity needed for the computation of realistic radiation gasdynamical models of PNe. It has, in general, still to be prescribed assuming a stationary outflow and different mass loss phases on the AGB (see also Sect. 2.3).

According to the interacting stellar-winds model (Kwok et al. 1978) this slow wind is hit by the tenuous but fast, radiation-driven wind emitted by the central star during its evolution towards white dwarf dimensions. The terminal velocity of this fast outflow increases with the shrinking stellar size and therefore the character of the flow pattern that results in the interaction with the slow wind changes with time.

The properties of the stellar wind in the early phases of post-AGB evolution are rather uncertain. The mass loss rate should, however, decrease in this period from the values characteristic of dust-driven AGB winds ([FORMULA]) to the ones of hot radiation driven winds which in the case of planetary nebulae nuclei are of the order of [FORMULA]. The velocity on the other hand is expected to increase from about 10 km/sec to more than 1000 km/sec. As long as its value is small (of the order of 100 km/sec) a nearly isothermal shock forms in the interaction region; i.e. the thermalized mechanical energy of the fast wind is effectively radiated away in the dense cooling region behind the shock front. This is the so called "momentum conserving" phase of the wind interaction.

However, the fast wind's increasing velocity will quickly (i.e. within a few hundred years) lead to a situation where radiative cooling in the expanding and diluting gas can no longer compete with the increasing mechanical energy input. When this happens, a zone of hot, low-density gas forms behind the shock. This high-temperature region - also called the "hot bubble" - is separated by a contact discontinuity from the slow wind and sweeps up the actual dense nebular shell. The latter is bounded on its outer rim by a second shock which separates the swept-up gas from the rest of the slow wind material (the "halo" in the following). This "energy conserving" form of wind interaction then prevails for the largest part of a typical PN's life.

Already Schmidt-Voigt & Köppen (1987) - and later Marten & Schönberner (1991) and Mellema (1994) - have shown that this picture of PN evolution is incomplete. Using stellar models with masses of about [FORMULA] evolving along recent evolutionary tracks these PNe models made clear that for densities typically found in the circumstellar shells of AGB stars the onset of hydrogen photoionization leads to strong dynamical effects in the early evolutionary phases: a D-type ionization front forms in the slow wind which drives a shock into the neutral gas ahead of it long before an outer shock resulting from wind interaction can form. The effects of photoionization must therefore be included in hydrodynamical computations as well as a great number of cooling processes. Finally, Marten (1994, 1995) has shown that the evolutionary time scale of the central star may become comparable to the heating/cooling and ionization/recombination time scales of the gas, so that even a fully time-dependent treatment of the ionization and energy balance may become necessary.

Only a self-consistent radiation gasdynamical description of all the processes mentioned can lead to a reliable modeling of PN evolution. Numerically, this is a difficult task since it results in an ionization rate network for a large number of ions which has to be solved together with the hydrodynamics. For a self-consistent computation of the cooling caused by collisonally excited lines - which yields the main contribution to the loss of thermal energy - this system has to be supplemented by the equations of statistical equilibrium for a large number of transitions. We have used a newly developed radiation gasdynamical code to compute our PNe models which will be outlined in the next section.

2.2. The radiation gasdynamical code

Since a detailed description of the code and a discussion of the results obtained with it so far is given elsewhere (Marten & Szczerba 1997; Schönberner et al. 1997; Schönberner et al., in prep.), we only mention here some basic aspects. Our code solves the Eulerian equations of hydrodynamics supplemented by the continuity equations for the single particle densities in spherical symmetry

[EQUATION]

with state- and flux-vectors, [FORMULA] and [FORMULA], respectively given by

[EQUATION]

and the geometrical source terms

[EQUATION]

All quantities have their usual meanings with [FORMULA] denoting the total advected energy density (made up of a thermal and a kinetic part), [FORMULA] the particle density for the ion of species [FORMULA] and ionization state l, and [FORMULA] the electron density. The energy stored in the ionization state of the gas does not need to be advected explicitely since this is already accounted for by the separate advection of the single ion densities. In order to account for the strong shocks and to minimize numerical diffusion across the contact discontinuity a high-resolution, second-order Godunov-type advection scheme (LeVeque 1997) including a Riemann solver is used for the integration of Eq. (1). The equation of state is that of an ideal gas.

A fully time-dependent ionization rate network given by the system of ordinary differential equations

[EQUATION]

along with the particle conservation of species [FORMULA]

[EQUATION]

and the condition of charge neutrality

[EQUATION]

is solved implicitly in step with the hydrodynamics to account for the source terms of the thermal energy and particle densities originating from heating/cooling and ionization/recombination processes. Here [FORMULA] is the net rate for ion [FORMULA]. It includes contributions due to photo- and collisional ionizations, radiative and dielectronic recombinations as well as charge-exchange reactions. See Marten & Szczerba (1997) for a detailed description of all terms including the total energy gain, [FORMULA], the total energy loss of the gas, [FORMULA], and the last term on the rhs. of Eq. (5) which describes cooling by collisional ionizations (with [FORMULA] denoting the ionization potential of ion [FORMULA]). Currently, up to 76 ions of the elements H, He, C, N, O, Ne, S, Cl and Ar, up to the ionization stage XII, are accounted for. Ionizations are always assumed to take place from the ground state of the respective ions. Radiative cooling is computed self-consistently and includes thermal bremsstrahlung, contributions due to recombinations, collisionally excited transitions of [FORMULA] and up to approximately 300 collisionally excited lines of heavy ions.

The local mean intensity of the ionizing radiation field, [FORMULA], which enters into Eqs. (4) and (5) via the net rate [FORMULA] consists of two terms: the geometrically diluted and exponentially weakened radiation field of the star, and a second term which represents the diffuse radiation field [FORMULA]:

[EQUATION]

Only contributions due to the opacity caused by the photoionizations were considered in the optical depth [FORMULA]. Radiative transfer is solved for photon energies between 4 and 500 eV on a frequency grid containing up to 344 grid points in case the full network of 76 ions is used. The diffuse field [FORMULA] is treated in the "on the spot" approximation (Osterbrock 1989).

As already implied by writing the fully coupled radiation gasdynamical equations in the form (1), (4) and (5) an operator splitting technique is used to advance the solution of the full system numerically in time by alternating between the integration of the advection step Eq. (1) and the solution of the radiative sources Eqs. (4) and (5) along the radial grid. This, apart from simplifying code design, development and testing, has the advantage that it easily allows us to compute the relaxation of a full radiation hydrodynamical model to thermal and ionization equilibrium. The procedure employed for this purpose will be described in Sect. 2.4.

The computational domain usually has to cover three orders of magnitude in radius (from about [FORMULA] up to [FORMULA]) in order to be able to follow the evolution from the instant when the star leaves the AGB up to the late stages when it evolves along the white dwarf cooling track. Since a rather high resolution is necessary in the nebula proper for an accurate treatment of the radiative transfer and the time dependent ionization problem (see Marten & Szczerba 1997), we used a constant distance ratio mesh with 1000 grid points for the calculations described below. The size of a mesh cell increases from [FORMULA] at the inner to about five times that value at the outer grid boundary since high resolution is not necessary in the outer parts of the halo. But still, the large number of mesh points resulted in long computing times. In the late evolutionary stages the increasing velocity of the fast wind, which constitutes our inner boundary condition, even worsens the situation since it leads to very small time steps due to the Courant-Friedrichs-Levy stability condition. Our problem poses extreme difficulties for an explicit hydrodynamics scheme on a fixed Eulerian grid. Clearly, adaptive mesh refinement will become a necessity in future computations, especially for large scale parameter studies and the extension of the calculations to two spatial dimensions. Alternatively, an implicit adaptive grid method may turn out to be of benefit for the 1D-case. This would require major code changes and introduce large complexity both in code design and use, though. To our knowledge no implicit adaptive grid method has been coupled to such large ionization networks, yet.

2.3. Radiation gasdynamical models

Two different radiation gasdynamical PNe model sequences (Schönberner et al. 1997; Schönberner et al., in prep.) computed along the [FORMULA] stellar evolutionary track of Blöcker (1995b) were used for our investigations.

The first sequence (hereafter termed sequence A) has been computed with initial conditions basically taken from the three-wind-model of Marten & Schönberner (1991), the only difference being the assumption of a constant wind velocity of 15 km/sec for both the AGB wind and the superwind phase (see also Marten 1994). All nine elements and the full network of 76 ions were included into these computations.

In an attempt to estimate the effects of a more realistic initial density and velocity distribution on our results a second PN model sequence (B in the following) was taken from Schönberner et al. (1997). As described therein it is the result of a first attempt to follow the hydrodynamical evolution of a circumstellar shell from the AGB towards a planetary nebula. A combination of (parameterized) mass loss rates as used in recent stellar evolutionary calculations (Blöcker 1995a) with time-dependent hydrodynamical modeling of the resulting gas/dust-shell and subsequent computation of the PN evolution was used for this purpose. The elements included in these computations were H, He, C, N, O and Ne and with each ionization stage considered. In either case the slow wind is assumed to be initially neutral. The spectral photon flux distribution was determined from the stellar effective temperature under the assumption of a black body spectrum. The velocity and mass loss rate of the fast central star wind were computed from the analytical approximation to the radiation driven wind theory as given by Kudritzki et al. (1989), with the line-force parameters suggested by Pauldrach et al. (1988). In the lower effective temperature range (below about 25 000 K) where the theory of radiation driven winds is not applicable we used the interpolation described in Marten & Schönberner (1991). The dynamical impact of the stellar wind is rather unimportant in this transition region, making a more sophisticated description not really necessary. Finally, we wish to emphasize that the post-AGB mass loss rates used in our models are the same as those in the calculations of Blöcker (1995b), thus avoiding inconsistencies between stellar and nebular evolution.

The first models of each sequence, i.e. models A0 and B0 (cf. Tables 2 and 3) represent nebulae that are optically thick in the Lyman continuum of hydrogen while the remaining models are optically thin. The elemental abundances given in Table 1 were used for all computations. We did not consider any temporal changes in the chemical composition of the gas expelled from the star.


[TABLE]

Table 1. Elemental abundances used for the computation of our hydrodynamical models given on a logarithmic scale


2.4. Equilibrium models

The full radiation hydrodynamical models described in the previous sections differ from static photoionization models used in classical methods of abundance determinations in several respects:

  • [a)] they have a density structure consistent with the evolutionary state of the central star whereas in static photoionization models usually some form of simple density structure is assumed;
  • b) they include additional physical processes: viz. heating behind shocks and expansion cooling which lead to deviations from the equilibrium temperature;
  • c) apart from dynamical effects, deviations from thermal and ionization equilibrium can also result if the evolutionary time scale of the central star becomes comparable to the heating/cooling and ionization/recombination time scales of the gas, so that the time-derivatives in Eqs. (4) and (5) become important (see Marten 1995).

To estimate the effects of items b) and c) on the line fluxes, corresponding equilibrium models were computed for a number of our dynamical models along the central star's track. Using a given radiation hydrodynamical model as initial condition the luminosity and effective temperature of the star were fixed and the advection step Eq. (1) "switched off", i.e. the velocity, v, was set to zero everywhere on the grid. Then, only Eqs. (4) and (5) describing the radiative sources were solved until the time-derivatives vanished in the spatial regions of interest, i.e. the temperature and ionization structures are relaxed to their equilibrium values.

This procedure was carried out for the actual high-density nebula, where essentially all of the line radiation is formed, and a part of the surrounding low-density halo (see Figs. 2 and 4). The computational grid of the dynamical models extends farther in and outwards than shown in Figs. 2 and 4 but the method becomes numerically very expensive in regions of low densities, where the corresponding time scales for relaxation become large. To avoid long computing times one could of course compare the dynamical models directly to models obtained from a static photoionization code using the density structures of the dynamical models as input to the former. However, we refrained from such a procedure in order to avoid systematic errors which would almost inevitably result by the use of two distinct codes due to differences in atomic data, physical approximations and numerical methods.


[FIGURE] Fig. 1. The dynamical model belonging to instant A0 ([FORMULA], [FORMULA], [FORMULA]). The three upper panels show the hydrodynamical structure over the largest part of the computational grid while the remaining ones give a detailed picture of the actual nebula itself. The total ion density as well as the electron density (dashed curve) are depicted in the density plots. The following line styles are used for the ionization fractions: dotted: neutrals, dashed: singly ionized ions, dashed-dotted: doubly ionized ions, dashed-three-dotted: triply ionized ions. Note the hot bubble in the temperature plot of the third upper panel and the shock caused by the D-type ionization front at a radius of about [FORMULA].

[FIGURE] Fig. 2a and b. Temperature structure of two optically thin models from sequence A. The actual nebula is bounded by the contact discontinuity at the inner rim, where the temperature jumps to the values encountered in the hot bubble, and a nearly isothermal shock at the outer rim (local temperature maximum at [FORMULA] and [FORMULA] for models A1 and A4 respectively). Beyond the latter extends the halo. Note that deviations from the corresponding equilibrium models become the more important the lower the density is, i.e. they are most apparent in the halo and in the model with larger age.

[FIGURE] Fig. 3. Complete radial structure of our dynamical model belonging to instant B0 ([FORMULA], [FORMULA], [FORMULA]). Note the very high density behind the outer shock accompanying the D-Front compared to model A0. Otherwise, qualitatively the same structures as in the latter case are visible. The density dip near a radius of [FORMULA] is a result of the temporarily decreased mass loss rate which followed the last thermal pulse on the AGB.

Figs. 2 and 4 give an impression of the temperature structures obtained. The deviations turned out to be quite small especially in model sequence B where the initial density distribution resulted in a nebula which is quite dense even when the central star has nearly reached its maximum temperature after 5462 years of evolution. Larger deviations are to be expected, however, for later evolutionary phases, at least for sequence A where the densities are lower (see also Marten 1995). In our new calculations which do not yet cover stages as late as the ones in Marten (1995), the line fluxes between equilibrium and dynamical models differed by only a few percent. Therefore, the use of the equilibrium models for the determination of the chemical abundances resulted in values comparable to those gained with the dynamical models (see Sect. 3.3).


[FIGURE] Fig. 4a and b. Left: Total ion and electron density (solid and dashed lines respectively) of our dynamical model B2. Note that compared to Fig. 3 the density has decreased appreciably and the spike caused by the D-Front at the outer rim has disappeared. The increased pressure of the hot bubble has meanwhile swept up a new shell at the inner rim. Right: Temperature structure of the same model compared to the corresponding equilibrium model. Deviations from the equilibrium temperature structure are negligible due to the relatively high densities of [FORMULA] in the main nebula.

2.5. Determination of the "observed" chemical abundances

As said in the Introduction, we restrict ourselves to determinations made with method ii). Among the various choices of the "constant [FORMULA], [FORMULA] " method, we adopt the one made in the accurate many-objects study by Kingsburgh & Barlow (1994, KB in the following). This recent work is homogenous with respect to:

  • [a)] the set of recent atomic data, and
  • b) the adopted scheme of the ionization correction factors (updated relative to previous schemes).

We recall here the temperatures and densities used by us to derive the abundances of the various ions. The procedure follows the one used by KB except for the electron density which we have everywhere derived from the [S II ] doublet while KB used [O II ] or [S II ]. It is evident that the procedure is rather more elaborate than the simple "constant [FORMULA], [FORMULA] " statement appears to indicate. The temperature from the 5755/6584 [N II ] line ratio for neutral and singly ionized species was used and that from the 4363/5007 [O III ] line ratio for all other ions with the following specifications. A temperature one third of the way between [FORMULA] (N II) and [FORMULA] (O III) was adopted for [FORMULA] and [FORMULA]. [FORMULA] (O III) + 1000 K was chosen for all triply ionized species, except for [FORMULA], where [FORMULA] (O III) + 650 K was used instead. [FORMULA] (O III) + 2270 K was taken for all 4-times ionized species. The neutral species were not used to derive the total elemental abundance. The reason is that fractions of neutrals, e.g. for oxygen or nitrogen, were assumed to be the same as that of neutral hydrogen, in which case final abundance ratios with respect to hydrogen would not be much affected, at least if the neutrals are of minor importance relative to the sum of the ionized species. Note that the density from low ionization species was used also for high ionization species. This is because the density from the [Cl III ] and [Ar IV ] doublets, although sometimes observed, is of much lower accuracy than that from the [O II ] and [S II ] stronger doublets. On the other hand a very precise density is not required when deriving abundances relative to hydrogen.

We have used the same lines used by KB (listed in their Table 8), except for the space UV lines, which we have omitted. These lines have not been observed in the vast majority of relatively well observed PNe (see e.g. the compilation by Perinotto 1991). When observed (by IUE), they cannot always be safely put on the same basis as the optical lines, because of the different entrance apertures of the IUE spectrographs relative to the ground optical instruments and because of lack of lines in common between the two sets of instruments. We have instead added the 9069, 9532 [S III ] doublet lines which can be observed from the ground (with the same instrument used to acquire optical data). We have thus preferred in his work to limit our analysis in order for it to be adequate for objects well observed by ground-based instruments. Both the hydrodynamical models already described and the procedure by KB just explained make use of very recent temperature dependent collision strengths of heavy ions. However, to avoid any effect of residual differences in the atomic data used, we have computed the line fluxes of the hydrodynamical models from their temperature, density and ionization structure using precisely the same atomic data subsequently employed in the calculations of the elemental abundances. These atomic data are those used of KB. A comparison with the line fluxes calculated directly with the hydrodynamical code did in fact show agreement in all cases to within a few per cent. From the line fluxes we get the "observed" elemental abundance presented in Tables 2 and 3 for sequences A and B, respectively.


[TABLE]

Table 2. Derived elemental abundances for models of sequence A. The calculated final abundances for helium are 0.062 and 0.080 for models A0dyn and A0equil, respectively (see text). The input abundances are given in the second column



[TABLE]

Table 3. Derived elemental abundances for models of sequence B. The calculated final abundances for helium are 0.060 and 0.035 for models B0dyn and B0equil, respectively (see text). The input abundances are given in the second column


We consider the abundances of He, N, O, Ne, Ar and S relative to Hydrogen. Carbon is omitted because it is not implemented in the present version of our code.

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Online publication: March 30, 1998
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