4 Photoionization analysis: New limits on the oxygen abundance

4.1 The modelling procedure

The observational data do not permit a direct estimate of the oxygen abundance, since nothing is known of the electron temperature or of the presence of oxygen ions more charged than O⁺⁺. It is therefore necessary to rely upon photoionization models.

We have constructed sequences of photoionization models in which the oxygen abundance varies over several orders of magnitude. The models are constrained by the available observations, which consist of the observed line intensities, the equivalent width of HH$\beta$, the total flux in HH$\beta$, the size of the nebula, and the radial distribution of H$\alpha$ shown in Fig. 3. As already noted by Tovmassian et al. (2001), the shape of the stellar continuum only implies that the star is hotter than 50 000 K. The models are computed with the photoionization code PHOTO, using the atomic data listed in Stasinska & Leitherer (1996). The central star is assumed to radiate as a blackbody of temperature $T_{\star }$. The hydrogen density at a radius r is taken to be n = $n_{\rm c}$ exp -(r/h)², where $n_{\rm c}$ is a free parameter and $h = 2.8 d / 2.06 \times 10^{5}$, where d, the distance to the nebula (in the same units as h and r), is also a free parameter. The ionizing radiation field is treated in the outward-only approximation. The computations start close to the star and are stopped when the equivalent width in HH$\beta$, W(HH$\beta$), becomes equal to the observed value, taken to be 70 Å.

   

Table 5: Some global properties of the sequences of models investigated.

$T_{\star }$	$Q({\rm {H^{0}}})$	$L({\rm H\beta})$	d	z	$R_{{\rm out}}$	$M_{{\rm neb}}$	$n_{\rm c}$
[K]	[erg s-1]	[erg cm-2 s-1]	[kpc]	[kpc]	[cm]	[$M_{\odot}$]	[cm-3]
100 000	6 $\times$ 1047	0.48	25.0	20.7	1.8 $\times$ 1018	0.78	125
100 000	1 $\times$ 1047	0.080	10.2	8.5	6.3 $\times$ 1017	0.070	190
125 000	6 $\times$ 1047	0.29	19.2	15.9	1.6 $\times$ 1018	0.45	148
125 000	1 $\times$ 1047	0.048	7.8	6.5	6.1 $\times$ 1017	0.044	223
150 000	6 $\times$ 1047	0.20	16.0	13.3	1.3 $\times$ 1018	0.29	163
150 000	1 $\times$ 1047	0.032	6.5	5.4	4.8 $\times$ 1017	0.027	245

With such a representation of the nebula, it is easy to show that, for each assumed stellar temperature $T_{\star }$ and total luminosity $L_{\star}$, the total observed flux in HH$\beta$ implies a certain distance to the planetary nebula. The observed angular size of the image then fixes the outer radius of the nebula. By trial and error, we determine the value of $n_{\rm c}$ for which the radius $R_{{\rm out}}$ corresponding to W(HH$\beta$) = 70 Å is close to $5.6 d / 2.06 \times 10^{5}$. Table 5 gives some global properties of selected sequences of photoionization models. This includes the distance, d, and the height above the Galactic plane, z, implied by these models. Note that $R_{{\rm out}}$and the nebular mass, $M_{{\rm neb}}$, are decreasing functions of the electron temperature. The values given in the table correspond to the low metallicity end of the model sequences.

The chemical abundances in each sequence of models are parametrized by O/H, with the abundances of the heavy elements with respect to oxygen following the recipe of McGaugh (1991). Note that the abundance ratios in PNG 135.9+55.9, as in any PN, may be significantly different from those assumed in McGaugh's recipe, especially for carbon and nitrogen. For helium, we assume an abundance of 0.08 with respect to hydrogen in all models. This is close to the value estimated from the observed HeII 4686/HH$\beta$ ratio and using the upper limit for He I $\lambda$5876 given by the CFHT spectrum (see Sect. 5). In any event, our estimate of the chemical composition of PNG 135.9+55.9 is independent of the relative abundances adopted in the model sequences since our estimate uses the observed line intensities and relies on the ionization and temperature structure of the nebula, which depend essentially on hydrogen and helium in the domain of interest.

Figure 5 presents the results of our models as a function of $12 + \log ({\rm O}/{\rm H})$. Each row of panels corresponds to a different central star temperature: $T_{\star }$ =100 000 K, $T_{\star }$ =125 000 K, and $T_{\star }$ =150 000 K, as indicated in the first panel. Each column of panels displays a different line ratio. In each panel, two series of models are represented with different symbols. Circles correspond to models with central stars having a total number of ionizing photons, $Q({\rm {H^{0}}})$, equal to $10^{47}~{\rm photons~ s}^{-1}$, while squares correspond to models with a total number of ionizing photons of $6\times 10^{47}~{\rm photons~ s}^{-1}$, values that roughly bracket the luminosities of the central stars of planetary nebulae as computed by post-AGB evolutionary models (Blöcker 1995). The horizontal lines show the observed values: thick lines for measured values or limits, thin lines indicating the uncertainties in measured values, and upward- and downward-pointing arrows denoting lower and upper limits, respectively. For the observational data, we adopted the CFHT observations, since they provide the most accurate measurements and the most stringent limits. For the H$\alpha$/HH$\beta$ ratio, however, we plot the value derived from the SPM1 observations. As noted earlier, our observations give inconsistent results for the H$\alpha$/HH$\beta$ ratio. Values of H$\alpha$/HH$\beta$ around 3.1 are easily accounted for by our models, but lower values are extremely difficult to explain. We will return to this issue in Sect. 4.3.

Figure 5: This figure presents our grid of photoionization models. Each row of panels corresponds to a particular central star temperature, indicated in the first panel. Each column of panels plots a different line intensity ratio. In each panel, two sequences of models are included: circles denote models with ionizing luminosities of $10^{47}~{\rm photons}~{\rm s}^{-1}$, squares denote models with ionizing luminosities of $6\times 10^{47}~{\rm photons}~{\rm s}^{-1}$. The thick horizontal lines indicate the observed values or their limits from the CFHT spectrum. In the case of limits, upward- and downward-pointing arrows denote whether they are lower or upper limits, respectively. Thin horizontal lines denote the uncertainty range of measured values.

Figure 6: Here, we compare two models with $T_{\star }$ =150 000 K and $Q({\rm {H^{0}}})$  $=10^{47}~{\rm photon}~{\rm s}^{-1}$, but with the circles denoting a filling factor of 1 (the model from Fig. 5) and squares denoting a filling factor of 0.1. As this experiment demonstrates, the maximum oxygen abundance allowed by the models decreases as the filling factor decreases.

4.2 Discussion of the models

The new observations allow us to eliminate the models with $T_{\star }$  $\geq 125~000$ K, since these produce a [Ne  V]/[Ne  III] ratio much larger than is observed. Our models show that $T_{\star }$ should be around 100 000 K. $T_{\star }$ cannot be much lower than this value since He  I $\lambda$5876 would then be observed. Note that the observed lower limit to [O  III]/[O  II] provides no useful constraint upon the ionization structure (or central star temperature) of our models.

For the models with $T_{\star }$ = 100 000 K shown in Fig. 5, the oxygen abundances compatible with the observed [O  III]/HH$\beta$ ratio are $5.8 < 12 + \log {\rm O}/{\rm H} < 6.3$ dex. Allowing for a reasonable uncertainty in $T_{\star }$ implies $12 + \log {\rm O}/{\rm H} \leq 6.5$ dex. The upper limit to [Ne  V]/[Ne  III] eliminates models excited by a star with a temperature significantly above 100 000 K which would otherwise permit $12 + \log {\rm O}/{\rm H}$ of the order of 7-8 dex.

We have constructed other series of models to test the robustness of these conclusions. For example, we have calculated models in which the nebular radius, total nebular flux, and HH$\beta$ equivalent width were varied, but the conclusions remain unchanged.

We have also considered models in which the gas is distributed in small clumps with the same global density law as above, but with an overall filling factor of 0.1. Although the H$\alpha$ image is smooth at our resolution, we cannot a priori exclude the presence of small scale clumps or filaments. Clumpy models that fit the observational constraints will result in a lower global ionization level than smoothly-distributed models. For the purposes of illustration, Fig. 6 compares the smooth model of Fig. 5 with $T_{\star }$ = 150 000 K and $Q({\rm {H^{0}}})$  $= 10^{47}~{\rm ph~s}^{-1}$ (circles) with a clumpy model whose filling factor is 0.1 (squares). Because the ionization is lower in the clumpy model, the ${\rm O}^{+++}/{\rm O}^{++}$ ratio is lowered with respect to the smooth model, while the [O  III] $\lambda$5007/HH$\beta$ ratio is raised. Consequently, the oxygen abundance compatible with the observed [O  III] $\lambda$5007/HH$\beta$ ratio is smaller than in the smooth case. Note that the model presented here gives $12 + \log {\rm O}/{\rm H} \simeq 6.5$ dex, but still violates the [Ne  V] $\lambda$3426/[Ne  III] $\lambda$3869 and H$\alpha$/HH$\beta$ constraints. The total nebular mass of a model of given total HH$\beta$ luminosity and given radius is roughly proportional to the square of the filling factor. Therefore, clumpy models that satisfy the observational constraints will have lower nebular masses than the corresponding models listed in Table 5. Finally, if the density in the clumps were extremely high, models with high metallicities, even as high as solar, could account for the weak intensities of the forbidden lines, because these lines would be quenched by collisional de-excitation. For this to occur, densities exceeding $10^{6}~{\rm cm}^{-3}$ are required for both [O  III] $\lambda$5007 and [Ne  III] $\lambda$3869. Given that the HH$\beta$ flux and equivalent width as well as the size of the nebula are known, such high densities would imply that the filling factor would have to be of the order of 10^-8 or less, which is highly unrealistic.

Our computations were made assuming that the star radiates as a blackbody. However, a more realistic stellar atmosphere would give a different spectral energy distribution for the ionizing photons, particularly at the largest energies. One expects that extended, metal-poor atmospheres could have larger fluxes at energies above 100 eV. This would increase the [Ne  V]/[Ne  III] ratio and consequently strenghten our conclusion that $12 + \log {\rm O}/{\rm H} \le 6.5$ dex. On the other hand, absorption by metals could depress the number of photons able to produce [Ne  V] (Rauch 2002), in which case a star with $T_{\star }$  $\sim 150~000$ K could become acceptable. In this case, however, the excitation of the nebula would be lower than that predicted by blackbody models with $T_{\star }$ = 150 000 K and the line ratios would resemble those produced by the blackbody model with $T_{\star }$ =100 000 K, again implying $12 + \log {\rm O}/{\rm H} \leq 6.5$ dex. In any case, the amount of metals in the atmosphere is not expected to be large, unless the atmosphere contains dredged-up carbon. A definitive answer to this problem can only come from a direct measurement of lines from more highly charged ions.

Note that the distance we obtain for our object (see Table 5) indicates that it is located in the Galactic halo, in agreement with its radial velocity (Tovmassian et al. 2001). Its derived nebular mass is compatible with the range of nebular masses derived for Magellanic Cloud PNe (Barlow 1987).

To conclude this section, we emphasize that our new observational data allow us to confirm that PNG 135.9+55.9 is an extremely oxygen-poor planetary nebula, with an oxygen abundance less than 1/50 of the solar value. Our models favour a value of $12 + \log {\rm O}/{\rm H}$ between 5.8 and 6.5 dex, compared with the solar value of $12 + \log {\rm O}/{\rm H} = 8.83$ dex (Grevesse & Sauval 1998). Our modelling experiments indicate that this conclusion is independent of plausible changes in the properties assumed for the central star or nebular envelope.

4.3 The H $\mathsf{\alpha}$/H $\mathsf{\beta}$ problem

Our preferred models with $T_{\star }$ = 100 000 K and low metallicities are compatible with the H$\alpha$/HH$\beta$ ratio derived from the SPM1 data, as well as with the highest values found in individual spectra from the other SPM observations (Table 2, Fig. 5). However, as discussed earlier, we find apparently significant variations of the H$\alpha$/HH$\beta$ ratio between the different observing runs and even among individual spectra obtained during individual runs. The majority of the individual spectra indicate that H$\alpha$/HH$\beta$ is below 3, as do the observations reported by Tovmassian et al. (2001).

Because of collisional excitation of the hydrogen lines, H$\alpha$/HH$\beta$ cannot have the recombination value, but is expected to be larger. Collisional excitation is unavoidable at electron temperatures above $\sim$15 000 K as soon as there is a small fraction of residual neutral hydrogen. The amount by which H$\alpha$/HH$\beta$ exceeds the recombination value (2.75 at 20 000 K, 2.70 at 30 000 K, using the case B coefficients of Storey & Hummer 1995) depends upon both the electron temperature, which is higher for higher values of $T_{\star }$, and on the amount of neutral hydrogen. For example, the models with $T_{\star }$ = 125 000 K predict a value of H$\alpha$/HH$\beta$ around 3.4 at low metallicities. With our observational constraints, there is not much room for a drastic reduction of the proportion of neutral hydrogen in our models. Indeed, for a given $T_{\star }$ and chemical composition, the proportion of H⁰ at each point in the nebula is completely determined by (and roughly proportional to) $n_{\rm e}/(L_{V}/R^{2})$, where $n_{\rm e}$ is the local electron density, L_V is the stellar luminosity in the V band and R is the distance of this point to the star. L_V/R² is a distance-independent quantity that relates the stellar flux in the V band to the angular distance of this point to the star; $n_{\rm e}$ is determined by the density distribution law obtained from our H$\alpha$ images and the value of $n_{\rm c}$ is imposed by the observational constraints on W(HH$\beta$) and the total angular radius, as explained in Sect. 4.1. Our models of PNG 135.9+55.9 with $T_{\star }$ = 100 000 K indicate that the H$\alpha$/HH$\beta$ ratio should not be lower than 3 if the nebula is metal-poor. Values of $T_{\star }$ significantly below 100 000 K could be consistent with some of the observed values of H$\alpha$/HH$\beta$ but they are excluded by the failure to observe He  I $\lambda$5876 in this object.

Does this mean that the object is not as oxygen-poor as inferred above? There are several arguments against a higher abundance. In our $T_{\star }$ = 100 000 K models, H$\alpha$/HH$\beta$ < 3 implies $12 + \log {\rm O}/{\rm H} > 7.5$ dex, but this is clearly incompatible with the observed value of [O  III] $\lambda$5007/HH$\beta$, as seen in Fig. 5. At the other extreme, for our $T_{\star }$ = 150 000 K models, H$\alpha$/HH$\beta$ < 3 implies $12 + \log {\rm O}/{\rm H}> 8$ dex, which, though marginally compatible with the observed [O  III] $\lambda$5007/HH$\beta$, predicts a value for the [Ne  V] $\lambda$3426/[Ne  III] $\lambda$3869 ratio far larger than observed.

Nor is it likely that any possible variability of the H$\alpha$/HH$\beta$ ratio affects our oxygen abundances. Supposing that this variation is real and refers entirely to the nebular radiation, one would then expect the [O  III] $\lambda$5007/HH$\beta$ ratio to vary as a consequence of variable ionization or temperature conditions, but this is not seen. All of the line intensities apart from H$\alpha$ remain remarkably constant, including the [O  III] $\lambda$5007 line. Therefore, we believe that the H$\alpha$/HH$\beta$ problem does not affect our conclusions as regards the oxygen abundance in PNG 135.9+55.9.

However, this H$\alpha$/HH$\beta$ problem is extremely puzzling. One does not expect this ratio to vary in nebular conditions for an extended object. One possibility could be that PNG 135.9+55.9 harbours a compact disk. Such a suggestion has been made for other planetary nebulae based upon either morphological, spectroscopic, or variability arguments (He 2-25: Corradi 1995; IC 4997: Miranda & Torrelles 2000; Lee & Hyung 2000; and M 2-9: Livio & Soker 2001). Accretion disks in close, interacting binary systems have the particularity of both being variable (Warner 1995) and having H$\alpha$/HH$\beta$ ratios much smaller than 3, sometimes attaining values below unity (e.g., Williams 1995). If PNG 135.9+55.9 contained such an accretion disk and if this disk contributed to the emission of the hydrogen and helium lines in the central part of the nebula, this could explain both the variability of the H$\alpha$/HH$\beta$ ratio in PNG 135.9+55.9 and the values of 2.7 or lower observed in some of our spectra. However, an accretion disk would be an unresolved point source in our images, and our deconvolution experiments found no significant contribution to the H$\alpha$ emission from the central source. Likewise, the lack of Dopper-broadened emission lines (Tovmassian et al. 2001) also argues against an accretion disk as the origin of a significant fraction of the line emission (Warner 1995).