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On the nitrogen abundance of FLIERs: the outer knots of the planetary nebula NGC 7009

We have constructed a 3D photoionisation model of a planetary nebula (PN) similar in structure to NGC 7009 with its outer pair of knots (also known as FLIERs --fast, low-ionization emission regions). The work is motivated by the fact that the strong [N II]6583A line emission from FLIERs in many planetary nebulae has been attributed to a significant local overabundance of nitrogen. We explore the possibility that the apparent enhanced nitrogen abundance previously reported in the FLIERs may be due to ionization effects. Our model is indeed able to reproduce the main spectroscopic and imaging characteristics of NGC 7009's bright inner rim and its outer pairs of knots, assuming homogeneous elemental abundances throughout the nebula, for nitrogen as well as all the other elements included in the model. Because of the fact that the (N+/N)/(O+/O) ratio predicted by our models are 0.60 for the rim and 0.72 for the knots, so clearly in disagreement with the N+/N=O+/O assumption of the ionization correction factors method (icf), the icfs will be underestimated by the empirical scheme, in both components, rim and knots, but more so in the knots. This effect is partly responsible for the apparent inhomogeneous N abundance empirically derived. The differences in the above ratio in these two components of the nebula may be due to a number of effects including charge exchange --as pointed out previously by other authors-- and the difference in the ionization potentials of the relevant species --which makes this ratio extremely sensitive to the shape of the local radiation field. Because of the latter, a realistic density distribution is essential to the modelling of a non-spherical object, if useful information is to be extracted from spatially resolved observations, as in the case of NGC 7009.


INTRODUCTION
It is well known that planetary nebulae (PNe) possess a number of small-scale structures that, at variance with their  Empirically derived parameters, for the different structures of the NGC 7009, for both, the Eastern and Western sides of the nebula, with "R", "K" and "NEB" standing for rim, outer knots and the whole nebula (see Tables 1 and 3 of Paper i).
For over ten years, the strong [N ii] line emission from LISs has been attributed to a significant local overabundance of nitrogen (Balick et al. 1993(Balick et al. , 1994(Balick et al. , 1998; and references therein). Balick et al. (1993) interpreted the Nenrichment in FLIERs as evidence of their origins in recent high-velocity ejections of the PN central star. The above work and successively Balick et al. (1994), which included the derivation of the ionic and elemental abundances of LISs, pointed out "an apparent enhancement of nitrogen relative to hydrogen by factors of 2-5" in FLIERs. However, since then, further work casted doubts over the latter statement (Hajian et al. 1997;Alexander & Balick 1997;Gonçalves et al. 2003;Perinotto et al. 2004a).
Abundances in PNe can be derived from the analysis of collisionally excited line (CEL) or optical recombination line (ORL) spectra using empirical methods or tailored photoionisation models (e.g. Stasińska 2004) or a combination of the two. For some elements (e.g. He) and ions (e.g. N ++ if only the optical spectrum is available) the use of ORL is mandatory, whilst for others the use of CELs may be unavoidable. But, when both CEL and ORL determinations are available, the former have been commonly preferred because they are generally stronger and easier to detect than ORLs. Moreover, there is a well known discrepancy between CEL-and ORL-abundances, as well as electron temperature determinations (Liu et al. 1995). Empirical abundance analysis rely on ionization correction factors (icfs) to account for the unseen ions (e.g. Kingsburgh & Barlow, 1994). Results obtained with the icf method can be somewhat uncertain in some cases, particularly when they are applied to spatially resolved long-slit spectra (Alexander & Balick 1997), as has been the case for the work of Balick et al. (1994); Hajian et al. (1997) and Gonçalves et al. (2003, hereafter Paper i).
In fact, the abundances derived in Paper i and shown here in Table 1, from optical long-slit spectra of NGC 7009, using the icf scheme of Kingsburgh & Barlow (1994), showed only a marginal evidence for overabundance of N/H in the outer knots of the nebula (the ansae), reinforcing the doubts over previous results (Balick et al. 1994) where the N/H enhancement of a factor of 2-5 in the ansae were reported.
One of the major shortcomings of empiricallydetermined chemical abundances lies in the fact that a number of assumptions on the ionization structure of the gas need to be made in order to obtain the icfs. A preferred alternative would be the construction of a tailored photoinisation model for a given object, aiming to fit the emission line spectrum and, in the case of spatially resolved objects, projected maps in a number of emission lines. NGC 7009, the "Saturn Nebula", is a PN comprising a bright elliptical rim. Its small-scale structures include a pair of jets and two pairs of low-ionization knots. On a larger scale, it is known that NGC 7009 possess a tenuous halo with a diameter of more than 4 arcmin (Moreno et al. 1998), whose inner regions display a system of concentric rings ) like those observed in NGC 6543 and other few PNe (Balick et al 2001). Highexcitation lines dominate the inner regions along the minor axis, while emission from low-ionization species is enhanced at the extremities of the major axis. The ionization structure is further enriched by the fact that the lowexcitation regions present strong variations in excitation level and clumpiness. NGC 7009 was classified as an oxygenrich PN (Hyung & Aller 1995a), with an O/C ratio exceeding 1, and anomalous N, O, and C abundances (Baker 1983;Balick et al. 1994;Hyung & Aller 1995a). Its central star is an H-rich O-type star, with effective temperature of 82 000 K (Méndez et al. 1992;Kingsburgh & Barlow 1992). The kinematics of NGC 7009 was studied first by Reay & Atherton (1985) and Balick et al. (1987), who showed that the ansae are expanding near the plane of the sky at highly supersonic velocities. The derived inclination of the inner (caps) and outer (ansae) knots, with respect to the line of sight, are i ∼ = 51 • and i ∼ = 84 • , respectively (Reay & Atherton 1985). More recently, Fernández et al. (2004) have measured the proper motion and kinematics of the ansae in NGC 7009, assuming that they are equal and opposite from the central star, obtaining Vexp = 114 ± 32 km s −1 , for the distance of ∼ 0.86 ± 0.34 kpc.
In this paper we will present a simple 3D photoionisation model, aiming at reproducing the observed geometry and spectroscopic "peculiarities" of a PN like NGC 7009, exploring the possibility that the enhanced [N ii] emission observed in the outer knots may be due to ionization effects. We will use the 3D photoinisation code mocassin (Ercolano et al. 2003a) together with the long-slit data presented in Paper i and the HST images available for this nebula. Our model and the observational data are described in Section 2. The results for our main model are presented and discussed in Section 3, while in Section 4 we discuss an alternative model, aimed at highlighting the relevance of the geometry and density distribution in such a kind of modelling. The work is briefly summarised in Section 5, where our final conclusions are also stated.

Observational data
The observations used to constrain the photoionisation model were described in detail in Paper i. We refer to Figure 1 of Paper i, the HST [O iii] and [N ii] images of NGC 7009, in which the nebular features are outlined. Here we will employ the same nomenclature introduced in Paper i. See also the left panel of Figure 2, in Section 3.5.

The 3D photoionisation code: mocassin
The nebula was modeled using the 3D photoinisation code, mocassin, of Ercolano et al. (2003a). The code employs a Monte Carlo technique to the solution of the radiative transfer of the stellar and diffuse field, allowing a completely geometry-independent treatment of the problem without the need of imposing symmetries or approximations for the transfer of the diffuse component.
The reliability of the code was demonstrated via a set of benchmarks described by Péquignot et al. (2001) and Ercolano et al. (2003a). A number of axy-symmetric planetary nebulae have already been modeled using mocassin, examples include NGC 3918 (Ercolano et al. 2003b), NGC 1501 (Ercolano et al. 2004) and the H-deficient knots of Abell 30 (Ercolano et al. 2003c).

Input parameters
A thorough investigation of the vast parameter space was carried out in this work. This involved experimenting with various gas density distributions, central star parameters and nebular elemental abundances. The model input parameters that best fitted all observational constraints are summarised in Table 2.3, and discussed in more detail in the following subsections.
A common problem when studying galactic PNe is the large uncertainties associated with the distance estimates, that propagate to the determination of the nebula geometry and central star parameters. We adopted a distance of 0.86 kpc for our models of NGC 7009, as computed by Fernández et al. (2004) from the weighted average of 14 values (Acker et al. 1992) determined with statistical methods; the value was quoted with an error of ± 0.34 kpc.

Stellar parameters
After having experimented with various stellar atmosphere models to describe the ionising continuum, we reverted to using a blackbody of T ef f = 80,000 K and log L * = 3.50, as this resulted in the best fit of the nebular emission line spectrum. Méndez et al. (1992) determined the T ef f and log g for the central star of NGC 7009 using non-LTE model atmosphere analysis of the stellar H and He absorption line profiles, finding T ef f = 82 000 K and log L * = 3.97, in solar units. Méndez et al. (1992) assumed a distance of 2.1 kpc in their analysis which therefore resulted in their value for the stellar luminosity being higher than that inferred from our modelling.

Elemental Abundances
The nebular elemental abundances used for the photoionisation model are listed in Table 2, where they are given by number with respect to H. Although the mocassin code can handle chemical inhomogeneities, these were not included in our models as they proved to be not necessary to reproduce the CEL spectra of the R and K regions of NGC 7009. The values shown result from an iterative process, where the initial guesses at the elemental abundances of He, N, O, Ne and S, taken from Paper i (see Table 1), and those of C and Ar, from Pottasch (2000), were successively modified to fit the spectroscopic observations.

Density distribution
The simplest possible density distribution model was constructed in order to demonstrate that the spectroscopic peculiarities often found in LISs can be the product of simple (and well-known) photoionisation effects. The case of NGC 7009 is taken as an example, but the emphasis is not in the construction of a detailed model for this object in particular. With this in mind we described the nebula by an ellipsoidal rim with a H number density, NH, peaking to 9000 cm −3 in the short axis direction exponentially decreasing to a minimum value of 4000 cm −3 in the long axis direction. The short and long axes of the inner and outer ellipsoids measure 3.84×10 16 cm and 9.99×10 16 cm, and 7.06×10 16 cm and 1.84×10 17 cm, respectively, at the distance assumed for NGC 7009. The rim is surrounded by a spherical shell of less opaque, homogeneous density gas, with NH = 1600 cm −3 . The diameter of the sphere is equal to the long axis of the outer ellipsoid defining the rim. Cylindrical jets, 1.75×10 16 cm in diameter, connect the rim to a pair of disk-shaped knots aligned at a distance of 3.49×10 17 cm from the central star, along the long axis of the ellipsoid. The cylindrical jets widen into cone-shapes at the knot ends in order to simulate the effect of material accumulating at the knots, as suggested by the HST images (particularly for knot K4, as seen in the right panels of Figure 2). The diameter of the base of the cones equals that of the disk-shaped knots. The centres of the 3.49×10 16 cm diameter circular disks representing the knots are aligned with the centres of the cylindrical jets (hence they are seen almost edge on). The width of the disks is assumed to be 3.88×10 16 cm in our model, although only a fraction of these is ionised, as is clear from the right panels of Figure 2. The H number density in the jets and knots is taken to be homogeneous and equal to 1250 cm −3 and 1500 cm −3 , respectively, consistently with the values derived in Paper i.
Results from the ellipsoidal rim and the spherical outer shell are combined into a single R-component to enable us to carry out a direct comparison with the slit spectra from Paper i, as we show in Table 1.
The jets (J-component) are included in our simulation as the radiation field has to be transferred through them before reaching the outer ansae (K-component). However, given that the emission detected from this region is very faint and that such structures may not be in equilibrium, we take our results for the J-component as very uncertain and omit them from any further discussion.
Finally, neither the inner caps nor the tenuous halo were included in our model. The former because they do not lie on the same axis as the K-component (Reay & Atherton 1985), and are therefore not expected to have a major influence on the ionization structure of the outer knots. And the latter because it is very faint, and therefore it is not expected to contribute significantly to the integrated emission line spectrum. Figure 1 shows model profiles along the major axis, in agreement with our assumed density, as Ne/NH ratio is correlated to the level of ionisation of the gas (see Section 3.4).

RESULTS
The best fit to the observed spectra was obtained assuming the parameters given in Table 2, as discussed in Section 2.3. Table 3 shows the predicted and observed intensities of some important collisionally excited lines, in which values are given relative to that of Hβ(=100) for each nebular component (R and K) and integrated over the nebula (NEB).
The model Hβ fluxes are given in the first row of the table. The dereddened line intensities quoted in the Obs column of Table 3 were obtained from the spectroscopic data presented in Table 1 of Paper i, by using a logarithmic extinction constant c Hβ =0.16 (Gonçalves et al. 2003) and the reddening law of Cardelli et al. (1989). For each line inten-sity, in each component, the upper row (of the Obs column) shows the values for the North-East (R1 and K1) regions of NGC 7009, while those for the South-Western R2 and K4 components are given in the lower row (see Figure 1 and Table 1 of Paper i).
The line intensities predicted by our model were convolved with a long-slit profile (assumed to be rectangular) aligned along the long axis of the PN. This is a necessary correction for an extended object with a complex geometry, such as that of NGC 7009, if any meaningful conclusions regarding the ionisation and temperature structures are to be gained from the comparison of the model with the observations. The dimensions of 1.5 ′′ vs. 4 ′ at a distance of 0.86 kpc were assumed in order to be consistent with Paper i. For each nebular component listed in Table 3 we give the slit and no-slit line intensities in adjacent columns.
The absolute value for the observed Hβ flux of NGC 7009 of 3.197 × 10 −10 erg cm −2 s −1 , quoted in Table 3, was obtained from the VLA radio recombination line flux (Garay et al. 1989). This flux should be compared to the nebula-integrated value (NEB) predicted by our model for the no-slit case. A very good agreement (better than 2%) is found.

Narrow slit effects
From the comparison of the model slit and no-slit columns it appears that some emission lines are more affected than others. This is very easy to understand if one considers the electron temperature distribution and the physical extension of the various regions where each of the relevant ionic species are most abundant. He iiλ4686, for example, is enhanced by ∼31% in the slit results for the R component and ∼29% overall. The opposite behaviour is shown by the He i lines for which the slit results show a ∼5% depletion for He iλ5876 and He iλ6678 in both the R component and NEB component. Similar beaviours are observed for other species as well, particularly we note the enhancement of intensities in the slit column for [N ii], [S ii] and [S iii] lines. We also note that different lines from the same ionic species appear to be enhanced/depleted by different amounts, like [S ii]λ4069, λ4076, enhanced by ∼10-12% and λ6717,λ6731 enhanced by ∼15-18%. This is due to the different sensitivities of the various transitions to changes in the electron temperatures.

Comparison of the emission lines spectrum
We show in the last three columns of Table 3 a comparison of our model with the observations integrated over the whole slit. Whilst a satisfactory agreement is obtained for many emission lines, some discrepancies do remain, including the case of the [N ii] and the [S ii] lines. These discrepancies can be readily understood by noticing the different emission lines intensities measured for the North-East and the South-West sides of the nebula (Table 1 of Paper i). In our models we have assumed the nebula to be symmetric about the x-, y-and z-axis and cannot therefore reproduce such asymmetries, which will be reflected into the integrated nebular spectrum for the species that are affected the most. This is not a great concern for us, given that a detailed model The Model spectra are given with and without considering the narrow slit effect. Upper rows give the observed intensities of the North-East (R1 and K1) part of the nebula, while those for the South-West (R2 and K4) zone are given in the lower rows. See Figure 1 of Paper i for the sizes of R1, K1, R2 and K4. "0.00" as the model predicted Heiiλ4686 emission and as the observed [Cliii]λλ5517,5737 from K, former means that model does not produce any such emission and latter means that such lines were not detected in the spectra of the knots. *The absolute value of the observed Hβ flux for the whole nebula was obtained from Garay et al. (1989) specific to NGC 7009 is not being sought, our goal being the construction of a model able to explain the apparent enhancement of some low-ionisation species in LISs of PNe such as NGC 7009 in terms of photoionisation effects only, without the need of assuming enhancements in the total elemental abundances of those regions. For this reason, in the modelling we aimed at reproducing the spectra of the in-dividual regions (R and K) with values falling between or being close to one of the observational data for North-East or the South-West regions.
A good agreement is shown in the rim and shell (namely R, as justified in Section 2.3.2) with nearly all predicted lines falling at intermediate values between the two sides of the nebula (upper and lower rows of the Obs column), or being within 10-30% of one of the two values. The only notable exception is the [S ii]λλ4068.6, 4076.4 auroral doublet, with both component being depleted by factors of 4 to 10. The same behaviour is not observed in the knots. We also note that mocassin predicts slightly enhanced values for the [S ii]λλ6717,6731 nebular doublet in the R-component. Since the auroral lines have much higher critical densities than the nebular lines, it is possible that they are being produced in a region that is not accounted for in our model. It would be useful to check the behaviour of the [O ii]λ7320 doublet, also auroral transitions, but unfortunately these fall outside the spectral range explored by Paper i. The agreement between model and dereddened intensities of the K-component is even better than that of the Rcomponent. All model predictions fall in between the data observed at the two sides of the nebula, or are within 10-30% of one of the two values. Our model does not predict any He iiλ4686 emission from the knots. We argue that the, however quite low, emission reported at the locations of K1 and K4 could be attributed to intervening material in the PN halo not included in our model.
In the following we will discuss in more detail the predicted mean temperatures and ionic abundances derived from our modelling.

Mean temperatures
Mean temperatures weighted by the ionic abundances are given in Table 4. Expressions for the volume integrated fractional ionic abundances (discussed in Section 3.4) as well as those for the mean temperatures weighted by ionic abundances were defined in Ercolano et al. (2003b), their eqs.
(2) and (3). The values listed in Table 4 show that the temperature structure of the nebula is well reproduced by our model.  Table 1) for the two sides of the nebula, namely 10,100 K, 10,000 K, and 10,100K for rim, knots and NEB, respectively. Similarly, the values of Te[N ii] of the three components agree with the empirical values to better than 5%, while for Te[S ii] the agreement is to within 33% and 11% for the two knots, due to our model underestimating of the [S ii] auroral lines (see Section 3.2).
We confirm that the electron temperature distribution is fairly homogeneous across the various regions of the nebula, see for instance, Rubin et al. (2002) and Sabbadin et al. (2004). The mean Te for the whole nebula, from the neutral (I) to the highly ionized ions (VII), for all species, are respectively 10,200±200K, 10,050±150K, 10,050±850K, 11,400±1050K, 12,600±700K, 13,100±200K and 12,650±1,300K.  For each element the first row is for the R, the second is for K, and the last one is for NEB.

Fractional ionic abundances
Results for the fractional ionic abundances are shown in Table 5. Hydrogen and helium are fully at least singly-ionized in R, K and NEB, and significant fractions of the heavy elements are in higher ionization stages in the rim as well as in the nebula as a whole (NEB). We also note the lower ionization of the knots. An important issue that should be noted here is the N/N + ratio being higher than the O/O + ratio by a factor of 1.39 in the knots, 1.66 rim and 1.61 in the total nebula. This result is at variance with the N/N + =O/O + (Kingsburgh & Barlow 1994;Perinotto et al. 2004b) generally assumed by the icf method, with the consequent errors on empirically derived total elemental abundances, like those in Paper i. We note from Table 5, that only a small fraction ( 0.6%, 14% and 0.7% for R, K and NEB, respectively) of the total nitrogen in the nebula is in the form of N 0 and N + . As only lines from these ions were observed (see Table 3 of Paper i), the nitrogen abundance determination is particularly uncertain. Sulfur abundances also suffer from similar problems.
Combining the values in Table 5 with those for the total elemental abundances used by our model (Table 2), fractional ionic abundances relative to H are readily available. The total abundance that we have used for all regions modeled returns ionic abundances, for R and K respectively, of He + = 0.091 and 0.111; N + = 1.17×10 −6 and 2.72×10 −6 ; O +2 = 3.88×10 −4 and 3.64×10 −4 ; Ne +2 = 9.72×10 −5 and 9.89×10 −5 ; and finally S +2 = 3.09×10 −6 and 6.79×10 −6 . These -which are the most important ions of the optical range for determining the total abundances-are compara-ble to the ionic abundances derived from the observations in Paper i, with an agreement better than 90% for all of the above ions, except the S +2 /H of R, whose discrepancy reaches 70%.
In Figure 1 we show the ionic abundance profiles of the model, as well as the electron density and temperature profiles along the axis which includes the knots. The topleft panel in this Figure, reflects the Ne (continuous line) and Te (dashed line) variations through the long axis of the nebula. It shows that Ne peaks at the innermost region of the R-component, having a mean value through the component of 4,100 cm −3 . The J-component as well as the K-component have mean Ne of 1385 cm −3 and 1640 cm −3 , respectively, while somewhat smoothed profiles are present at the edges. The electron density profile is a result of the ionization structure combined with the density distribution assumed in Section 2.3.3. As for the Te profiles, although varying somewhat within the components, have mean values that are in close agreement with the values measured in Paper i, see numbers quoted in the previous subsection. This will be discussed further in Section 4. Figure 1 also highlights the strong dependence of the ionisation level on the geometry and density distribution of the gas. It is therefore clear that an apparent overabundance of N + in the knots can be produced by providing the correct gas opacity to screen this region from the direct stellar photons. We should add at this point that an even larger N + abundance can be obtained by further enhancing the gas density at the rim-jet interface, without significantly changing the [S ii] density ratio in the R-component. After the inner cavity the inner wall of the ellipsoidal shell is encountered, this has N H = 4,000 cm −3 and Ne ∼ 5,000 cm −3 . The value of Ne decreases through the first shaded area that represents the ellipsoidal shell, as the level of ionisation of the gas decreases. After the first shaded area we find the jet (N H =1,250 cm −3 , Ne ∼ 1,500 cm −3 ). At the end of the jet we find the knot, where and the N H = 1,500 cm −3 , while Ne reaches about 1,700 cm −3 .

Visualization of the model results
NGC 7009 has been observed with the HST WFPC2 with filters centred in the [O iii]λ5007 and [N ii]λ6583 emission lines (see Section 2.1). These archive images, already published in Paper i, are compared to the model predicted emission maps in Figure 2. The maps were produced at an inclination of 84 • with respect to the line of sight, as indicated by the kinematics of the PN polar axis (Reay & Atherton 1985), for [O iii] (right top panel) and [N ii] (lower right panel) emission lines. First of all we call the readers attention to the fact that the polar bubble, which appear in the HST [O iii] image, at fainter intensity levels, as an extension of the shell, and also the inner pair of knots, K2 and K3 were not considered in our modelling. Thus, excluding the polar bubble and the inner knots, the images and our maps are at least qualitatively in good agreement (note that a photometric comparison is beyond the scope of this paper).
From the maps we note the higher excitation of the equatorial axis of the PN as compared to the polar one, because of the higher density of the rim, and following the pattern known from previous studies, as stated in the Introduction. One also clearly sees that the [N ii] map is more extended than the [O iii] one, as expected from a nebula excited by a central star. Finally, one notes that the knots are fainter (as compared to the inner regions) in [O iii] and brighter in [N ii], due to the enhanced recombination in the knots. All the above suggest that our model can satisfactorily reproduce the main features in the images of NGC 7009, that could in principle be extended to other PNe with a pair of outer low-ionization knots (see, for more examples, Gonçalves et al. 2001).

ALTERNATIVE MODEL
As noted in previous sections, the ionisation structure of the model is strongly dependent on the input 3D density distribution. Because of that, here we also present the results obtained with a slightly different geometry, which we call the alternative model. The only difference in terms of input density distribution between our initial model and the alternative one is that in the latter the jets start 3.3% further away from the central star than in the previous model. Since the H number density in the jet is four times lower than the value in the rim at the position of the polar axis, our change results into an increase in the line optical depth along the path connecting the central star to the outer pair of knots. The intensity of the ionizing stellar field reaching the knots is therefore decreased in our alternative model, hence resulting in a lower ionization level. Other model parameters were also adjusted in order to optimise our fit of the spectrum and these are summarised in Table 6. More importantly, with this alternative model we aimed at reproducing the more extreme [N ii] emission of the Eastern knot of the nebula (Knot1 in Paper i and in Figure 2).
As in the case of our original model discussed in Section 3, we found a good agreement of the alternative model with the dereddened fluxes of Paper i, for the rim, the knots as well as for the whole nebula (NEB). Again, most of the  ii] of 10,200 K and 9,950 K, 10,600 K and 10,600 K, and 10,200 K and 10,100 K, for rim, knots and NEB, respectively. And the Ne are automatically matched with the empirical values, since they are constraining the model as input parameters.
The corresponding ionisation structure is shown in Table 8, from which we obtain the N/N + and O/O + ratios of the different zones in the nebula, as being: 1.64 (Rcomponent), 1.33 (K-component) and 1.56 (NEB), that, as above, are in contradiction with the ratios adopted in the empirical icf scheme. We chose to include Table 8 in the paper, although qualitatively similar to Table 5, to allow for model icfs to be derived at a later time should one wish to do so.
Finally, the projected emission maps obtained from this model are not shown here as they are qualitatively identical to those of our original model (right panels of Figure 2).

CONCLUSIONS
This work focused on the study of the apparent N overabundance in the outer knots of NGC 7009 with respect to the main nebular rim and shell. Alexander & Balick (1997) and Gruenwald & Viegas (1998) showed that long-slit data may give spurious overabundances of N and other elements in the outer regions of model PNe. These authors have identified the different charge-exchange reaction rates of N and O as the main responsible for the effect in the low-ionization regions of the nebulae, therefore affecting the (N + /N)/(O + /O) ratio, that as discussed in Section 3.4, is commonly used to obtain the icf for nitrogen. However, as shown by Mampaso (2004) charge-exchange reactions can, in this case, account at most by 20% of the nitrogen overabundance of the knots of NGC 7009.
In this paper we have presented a model that was able to reproduce the main spectroscopic characteristics of the various spatial regions of NGC 7009 without the need of assuming an inhomogeneous set of abundances. We investigated the importance of taking into account the effects of a narrow slit, and our results in Table 3   As in Table 2, with the top/bottom rows given intensities of the Eastern/Western size of the nebula.
rim, knots and the whole nebula, respectively, all at variance with the icf assumption of unity for this ratio. The icfs will therefore be underestimated by the empirical scheme, both in the case of the R and K components, but more so in the former (by a factor of 1.21). Therefore this effect may partly be responsible for the apparent inhomogeneous N abundance derived from observations. The differences in the (N + /N)/(O + /O) in these two components may be due to a number of effects, including charge exchange, as stated above, and the difference in the ionization potentials of the relevant species, which makes the (N + /N)/(O + /O) ratio extremely sensitive to the shape of the local radiation field. For this reason, a realistic density distribution is essential to the modelling of a non-spherical object, if useful information is to be extracted from spatially resolved observations. The density distribution of the gas modifies the shape of the local radiation field via the gas opacities and (as demonstrated in Section 3.4 and Section 4) matching the emission line spectrum of a given nebular region relies, among other things, on the careful construction of a model capable of providing the correct optical depths in various directions.
Our main conclusions discussed here may also be extended to other PNe exhibiting FLIERs, such as NGC 6543