Eliminating Error in Chemical Abundances of Extragalactic H ii Regions

# Eliminating Error in the Chemical Abundance Scale for Extragalactic H  II Regions

Á.R. López-Sánchez M. A. Dopita, L. J. Kewley, H. J. Zahid, D. C. Nicholls & J. Scharwächter
Australian Astronomical Observatory, PO BOX 296, Epping, NSW 1710, Australia
Department of Physics and Astronomy, Macquarie University, NSW 2109, Australia
Research School of Astronomy and Astrophysics, Australian National University, Cotter Rd. Weston, ACT 2611, Australia
Astronomy Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822
E-mail: alopez@aao.gov.au
Received date: 26 Jan 2012; accepted date: 22 Mar 2012
###### Abstract

In an attempt to remove the systematic errors which have plagued the calibration of the H ii region abundance sequence, we have theoretically modeled the extragalactic H ii region sequence. We then used the theoretical spectra so generated in a double blind experiment to recover the chemical abundances using both the classical electron temperature + ionization correction factor technique, and the technique which depends on the use of strong emission lines (SELs) in the nebular spectrum to estimate the abundance of oxygen. We find a number of systematic trends, and we provide correction formulae which should remove systematic errors in the electron temperature + ionization correction factor technique. We also provide a critical evaluation of the various semi-empirical SEL techniques. Finally, we offer a scheme which should help to eliminate systematic errors in the SEL-derived chemical abundance scale for extragalactic H ii regions.

###### keywords:
galaxies: abundances, ISM: abundances, H ii regions, methods: data analysis, techniques: spectroscopic

## 1 Introduction

To understand how galaxies in the early universe evolved into those that we see locally requires an understanding of both chemical and star formation history of galaxies over cosmic time. Indeed, observational data compiling hundreds of thousands galaxies at different redshifts are suggesting that there actually is a deep physical connection between metallicities, masses and star-formation rate (e.g. Tremonti et al., 2004; Brinchmann et al., 2004; Kewley & Ellison, 2008; Lara-López et al., 2010a, b), although these results are mainly based on trends of the chemical abundances and not on their absolute metallicities.

Theoretical simulations can now predict the evolution of chemical abundances in galaxies (e.g. Mollá & Díaz, 2005; Davé & Oppenheimer, 2007). However, it is difficult to put these predictions to the test because the absolute calibration of the chemical abundance scale from H ii regions is currently discrepant by more than a factor of two. See, for example the work of Kewley & Ellison (2008) where abundances were calculated for 45,000 galaxies from the Sloan Digital Sky Survey (SDSS) (York et al., 2000; Abazajian et al.., 2003) using 10 different calibrations given in the literature. Although all methods show some correlation with abundance, there is typically a scatter ranging over a factor of 2-3 in the absolute oxygen abundance derived from the various techniques. The same result was found by López-Sánchez & Esteban (2010), who also considered in their analysis the metallicities derived using a direct determination of the electron temperature of the ionized gas. Without a proper estimation of the absolute chemical abundances in galaxies, the comparison between observational data and theoretical models is not entirely satisfactory.

Thanks to its strong emission lines, the easiest element to measure in H ii region emission spectra is oxygen. This is fortunate, since O is an -process element made directly in short-lived massive stars. It comprises about 50% of all the heavy elements by mass throughout the universe, and is therefore representative of all the heavy elements. By contrast, Fe is made in lower mass stars, and is not released promptly into the interstellar medium of galaxies. The O/H abundance in H ii regions –typically expressed in terms of 12+log(O/H), where O/H is the ratio of the number of oxygen to hydrogen atomic density . This has been determined in three semi-empirical ways, as well as through direct theoretical photoionization modeling:

1. By direct measurement of the electron temperature, from line ratios such as the [O iii] 4363/5007 (see Osterbrock & Ferland, 2006), followed by an analysis of the various ionization fractions in the zones of the H ii region which produce optical emission lines. We dub this the  method. This has been very extensively applied by observers over the past 30 years (e.g. Peimbert & Costero, 1969; Stasińska, 1978; Aller, 1984; Díaz et al., 1987; Izotov, Thuan, & Lipovetski, 1994; Esteban & Peimbert, 1995; Vílchez & Esteban, 1996; Stasińska, 2005; Bresolin et al., 2005; Izotov et al, 2006; Esteban et al., 2004; López-Sánchez & Esteban, 2009).

2. By direct measurement of recombination lines of ions of heavy elements such as O ii and, in some cases, O i and comparing these with the recombination lines of hydrogen (e.g. Esteban et al., 1998, 2004, 2009; García-Rojas et al., 2004; García-Rojas & Esteban, 2007; López-Sánchez et al., 2007; Peimbert et al., 2007). We refer to this technique as the RL method, although it is generally not useful in the determination of the cosmic abundance scale, since the recombination lines of heavy elements are generally too weak ( times fainter than H) to measure.

3. By measurement of the strong emission lines in the H ii region spectrum calibrated using

1. photoionization modeling (e.g. Pagel et al., 1979; McGaugh, 1991; Kewley & Dopita, 2002; Kobulnicky & Kewley, 2004; Dopita et al., 2006a; Dors & Copetti, 2006), or

2. H ii regions for which the oxygen abundance is known from the  method (e.g. Pilyugin, 2001a, b; Denicoló, Terlevich & Terlevich, 2002; Pettini & Pagel, 2004; Pilyugin & Thuan, 2005; Pérez-Montero & Díaz, 2005; Pilyugin et al., 2010; Pilyugin & Mattsson, 2011).

We will call this the Strong Emission Line (SEL) method, and it is nowadays extensively used to determine metallicities in large galaxy surveys.

Each of these methods has its particular problems, and range of application. More importantly, where all methods can be used, the derived O/H abundances often show systematic disagreements between the various methods. These may amount to factors of two or more. In this paper, we seek to resolve some of the causes of these discrepancies by re-deriving the chemical abundances of model H ii regions using the method and the standard SEL techniques. We use a double-blind approach, so that the analysis of the photoionizaton models was conducted without any a priori knowledge of the chemical abundances and physical parameters used in the photoionization models.

We will now briefly indicate how these alarming abundance discrepancies between the various methods may arise.

### 1.1 The Te and RL methods

At normal nebular temperatures ( K), the auroral [O iii4363 line is only a few percent as strong as the nebular 5007 line, so that high quality spectra are required to give an adequate signal-to-noise ratio to measure the fainter line. When the S/N ratio is inadequate, the strength of the fainter line tends to be overestimated, leading to an underestimate in the derived abundance. This is also true for other commonly-used temperature sensitive line ratios.

A more insidious problem was first pointed out by Peimbert & Costero (1969). This still has not been resolved to everyone’s satisfaction. Normal H ii regions are not homogeneous but contain temperature gradients, dense inclusions in which collisional de-excitation of cooling lines leads to higher temperatures, as well as colliding supersonic flows in which both density and electron temperatures may be raised as a result of shocks. In all such regions of enhanced temperature, the [O iii4363/5007 emission line flux ratio is raised by a factor which depends only on the temperature. This would not be a problem except for the fact that the line emissivity is also raised in such regions by a factor proportional to the square of the electron density, . Thus, the temperature estimate provided by the forbidden line ratio is dominated by the line ratio characterizing these over-dense inclusions, rather than providing a measure of the electron temperature of the nebula as a whole. The electron temperature is therefore systematically overestimated. Hence, the presence of temperature gradients or fluctuations in the ionized gas leads to the abundances based on the  method being systematically underestimated (Peimbert, 1967; Peimbert et al., 2007; Stasińska, 2002, 2005).

Indeed, detailed studies that compared heavy elements abundances derived using both the  method and the RL method in Galactic (Esteban et al., 2004; García-Rojas et al., 2005, 2006; García-Rojas & Esteban, 2007; Mesa-Delgado et al., 2009b; Mesa-Delgado & Esteban, 2010; Mesa-Delgado et al., 2011) and extragalactic (Tsamis et al., 2003; Peimbert, 2003; López-Sánchez et al., 2007; Esteban et al., 2009) H ii regions find a very good agreement between both results when temperature fluctuations are considered. However, their origin and existence is still controversial because they are not well reproduced by standard photoionization models (Kingdon & Ferland, 1995; Rodríguez & García-Rojas, 2010), and hence additional mechanisms are proposed to explain the presence of temperature fluctuations in the ionized gas (see reviews by Esteban, 2002; Peimbert & Peimbert, 2006).

### 1.2 The SEL method

The strong emission line (SEL) method was first proposed by Pagel et al. (1979). It relies upon the ratio of the sum of the strong forbidden oxygen emission lines with respect to H, the so-called ratio; =  (  [O ii]  3726,3729 + [O iii]  4959,5007 )  /  (H). Since then many other such ratios have been proposed. The most widely used of these are the calibrations of McGaugh (1991) and Kewley & Dopita (2002), based on detailed photoionization modelling, and the empirical relations provided by Pilyugin (2001a, b); Pilyugin & Thuan (2005) and Pilyugin, Vílchez & Thuan (2010). Both kinds of calibrations strive to improve the accuracy by making use of the [O iii]/[O ii] ratio as ionization parameter, which accounts for the large scatter found in the versus oxygen abundance calibration, which is larger than observational errors (Kobulnicky et al., 1999)

Hence, the SEL method is ideally suited to global abundance determinations in distant galaxies, since only the strong lines are visible against the stellar continuum. The abundance scale for ensembles of H ii regions in galaxies has recently been calibrated by Dopita et al. (2006a). The basic problem with all ratios of optical forbidden lines to hydrogen recombination lines is that they are two-valued as a function of chemical abundance. This abundance ambiguity can only be raised by the simultaneous use of several such ratios. An additional problem is that not all SEL method give similar oxygen abundances: the SEL methods based on calibrations using photoionization models generally tend to overpredict (but not always, see Pérez-Montero et al., 2010; Dors et al., 2011) the observed oxygen abundances derived using the  method by 0.2–0.4 dex (Peimbert et al., 2007; Bresolin et al., 2009; López-Sánchez & Esteban, 2010; Moustakas et al., 2010). As we should expect, that is not the case of the SEL techniques that are based on calibrations with objects for which the oxygen abundances are well known from the  method. Generally speaking, the SEL method generally returns abundances which are higher than the method, but in better agreement with those derived from recombination lines of heavy elements (López-Sánchez & Esteban, 2010).

### 1.3 A double-blind test

In this paper, we seek to eliminate some of the systematic errors in abundance determinations by applying the  method and the SEL method to theoretically-generated H ii region spectra, rather than using observed spectra. This has the advantage of eliminating observational errors but, more importantly, it also provides a direct test of whether the various methods can recover the chemical abundances and physical conditions which were fed into the theoretical models. In order to ensure objectivity, López-Sánchez (who applied the and the SEL method) was provided a randomized selection of theoretical H ii region models generated by Dopita. The data provided consisted of a set of line fluxes (relative to H) as delivered by the model. The models were identified only by letters, and no information was given about the input physical parameters which were input to generate the models. This is the classical double-blind technique often applied in medicine, but rarely in astrophysics.

In Section 2, we provide details of the techniques used to generate the theoretical spectra, in Section 3 we apply the to the theoretical spectra and examine how well the input parameters have been recovered. Where possible, we provide correction formulae to apply. Section 4 presents the results provided by the most common SEL techniques. In this section, we also compare the oxygen abundances provided by the SEL methods with the derived  and RL abundances in high-quality Galactic and extragalactic H ii spectra. Finally, Section 5 gives the conclusions and provides our recommended procedure needed to accurately derive the oxygen abundance in H ii regions.

## 2 The Theoretical Models

We have generated a sequence of photoionization models using the Mappings iiis code, an updated version of the code originally described in Sutherland & Dopita (1993). These models are very similar to those produced by Dopita et al. (2000), and subsequently used by Kewley & Dopita (2002), but the chemical abundances have been specifically modified so as to provide a better fit to the Sloan Digital Sky Survey (SDSS) observed sequence of extragalactic H ii regions (Kewley et al., 2006).

Our grid of photoionization models uses as stellar input the fluxes computed by the Starburst 99 code (Leitherer et al., 1999) appropriate for continuous star formation. The input spectrum and the assumed stellar IMF input into the Starburst 99 code is the same as used by Dopita et al. (2000). The later Starburst 99 models (Vázquez & Leitherer, 2005), which incorporate a fully self-consistent mass-loss formulation were not used as these generate a stellar radiation field that is somewhat too soft relative to these earlier models (see Dopita et al., 2006a, for a detailed discussion of the issue). To summarise, the most likely cause of this discrepancy is probably that the stellar wind is clumpy as a result of radiation pressure instabilities, so a multi-dimensional model atmosphere would need to be constructed. Effects of stellar rotation can also modify the emergent ionizing spectrum (Levesque et al., 2010). For consistency, we have interpolated the stellar fluxes to match the chemical abundances actually used in the model, allowing for the abundance scale shift caused by the re-calibration of the solar metallicity (Grevesse et al., 2010).

The extragalactic H ii region sequence is clearly defined in the observational plane by the standard diagnostic line ratio plots of Baldwin, Phillips & Terlevich (1981) and Veilleux & Osterbrock (1987). These plot the ratios [N ii] /H, [S ii] /H and [O i] /H against the [O iii] /H emission line ratio. Kewley et al. (2006) gave these diagnostics for some 45,000 galaxies drawn from the SDSS. Our photoionisation models closely reproduce the observational sequence of extragalactic H ii regions, bearing in mind that the ionization parameter tends to become higher as the abundance decreases, thanks to the decreasing importance of the ram pressure of the stellar winds (Dopita et al., 2006a).

The solar metallicity for our photoionisation models was defined by the solar oxygen abundance given by Grevesse et al. (2010), 12+log(O/H)=8.690.05. However, the abundance of helium and carbon were assumed to vary according the formula given by Dopita et al. (2006a). For nitrogen, we use a form similar to that given in Dopita et al. (2006b):

 12+log(N/H)=7.6+log[Z/Z⊙+(Z/Z⊙)2]. (1)

Here the zero point of the nitrogen abundance has been adjusted to prove a better fit in the corresponding Veilleux & Osterbrock (1987) diagnostic diagram. Because nitrogen is an important coolant at high metallicity, this has the incidental effect of improving the fit of the models to the observed extragalactic H ii region abundance sequence in the other two diagnostic diagrams. For all models we include dust, and dust physics according to the description in Dopita et al. (2006a), where the depletion factors of the various elements from the gaseous phase are also given.

Our models form a grid in chemical abundance or metallicity, defined by the quantity 12+log(O/H), and the ionization parameter, , the ratio of the number of photons passing per unit area and time to the number density of hydrogen atoms. The parameter is related to the dimensionless ionization parameter by , where is the speed of light.

Our photoionization models are spherical, with a target ionization parameter for the inner surface. This is unlike the Kewley & Dopita (2002) models, which were plane parallel. Our models are also isobaric with cm K, where is the Boltzmann’s Constant, so that the density varies throughout the model, but is typically cm. Unlike the Kewley & Dopita (2002) models, our models allow for photoelectric heating of the gas by dust grains, and include the effects of radiation pressure, which is becoming appreciable at the higher values of the ionization parameter.

### 2.1 The Double-Blind Test Models

From the theoretical model H ii region grid we have taken, for a fixed metallicity, every second model in , and applied to these the standard observational techniques for deriving abundances. In order that no bias was incurred in the analysis, the spectra were randomly selected from the input grid of models, and the raw spectra labelled simply A,B,C….Z, AA..AG were passed by Dopita to López-Sánchez for analysis. This analysis returned physical conditions, ionization parameters, and chemical abundances given by each of the semi-empirical techniques commonly used by observers, which included both the  and the SEL methods. We can then compare the results of this analysis to the known inputs into the original models to discover where there exist systematical errors, and to evaluate the magnitude of the random errors that can be generated by these techniques. The advantage of this approach is that it totally eliminates errors generated by the observations, and provides spectra of essentially infinite signal to noise for analysis.

In an ideal world, the parameters returned by the abundance analysis would be the same as those input into the photoionization models. However, in practice the different abundance techniques have different implicit assumptions about both the atomic physics relating to particular ionic species, and more generally, about the physics of the H ii region itself. Therefore, we should not expect the results to agree exactly.

The physical and abundance parameters of the input models are given in Table 1. Note that we give three values of the ionization parameter, the nominal ionization parameter input into the model, , the ionization parameter at the inner surface of the ionized gas, , and mean ionization parameter in the model, . These are related to each other, but not in any obvious way. For example, is calculated on the basis of an assumption that the temperature in the inner zone of the H ii region is 10,000 K. Because of our isobaric assumption, is lower than for the high-abundance models, and higher than for the low-abundance models. The ratio of to is always greater than two due to the spherical divergence of the radiation field, and can reach values as high as ten where radiation pressure gradients start to play a role in determining the gas pressure.

As stated above, the input stellar spectrum, the physical conditions and the chemical abundance set were chosen so as to replicate, as far as possible, the observed sequence for extragalactic H ii regions (Dopita et al., 2000; Kewley et al., 2006). In Figure 1 we show the two main diagnostic diagrams from Veilleux & Osterbrock (1987) and the diagrams which Dopita et al. (2000) used to separate the chemical abundance from the ionization parameter at the high-abundance end of the scale.

These models display quite a tight relationship between the oxygen abundance, 12+log(O/H), and the electron temperature, , given by the high-ionization zone from the [O iii] (4959+5007)/4363 ratio, and in the low-ionization (H + He) zone emitting the [O ii] (3726+3729), [N ii6584 and [S ii] (6717+6731) lines. This is shown in Fig 2. Note that there is larger scatter in the plot for the [O iii] zone, because the temperature in this zone is more sensitive to the input ionization parameter. On the left hand plot, the left-hand upper points correspond to high ionization parameter, while the reverse is true for the points in the lower right-hand side of the diagram. There is much less variation in the local ionization parameter in the low-ionization zone of the model H ii regions, so that the scatter is less for the right-hand panel of Fig. 2.

For each zone of the model H ii region we have generated empirical fits to the oxygen abundance – electron temperature relationship. For [O iii] we find

 Te[O\textsciii]=25200−1400x−2000|x−7.0|1.7, (2)

while for the low-ionization zone we have

 Te(low)=9500+400x−2000|x−6.9|2.35, (3)

where in both cases. These curves emphasize once again that the electron temperature is a key diagnostic for the chemical abundance in H ii regions.

## 3 Application of the Te method

### 3.1 Physical conditions

We have taken care to analyze the model spectra in exactly the same way as an observer would do when presented with the spectrum of real H ii regions. If there are no systematic effects in the analysis technique, this exercise should yield the same electron temperatures and chemical abundances as was input into the models. One potential source of difference is the fact that the atomic data used in the nebular models is not necessarily the same as are used in the analysis of the spectra, although in many cases they are. The atomic data used for our models are detailed in Sutherland & Dopita (1993) (see their Section 3.5.2), which are similar to the atomic data included in IRAF task temden in the NEBULAR package by Shaw & Dufour (1995) (see their Table 1). The output of the various nebular modeling codes have been compared against each other for a number of standard models on more than one occasion through the so-called Lexington Benchmarks code-off exercise (Ferland, 1995). Any egregious errors made by any one of them have subsequently been eliminated from the appropriate codes. We can therefore be fairly certain that other photoionization codes such as Cloudy (e.g. Ferland et al., 1994) will produce very similar output to the Mappings IIIs code used here.

The model spectra were created without considering any extinction or reddening, and not stellar absorption in the H i Balmer lines, to decrease the uncertainties when analyzing the data. Thus, for the analysis of the model spectra we have used a reddening coefficient, (H), and a equivalent width of the stellar absorption underlying the H i Balmer lines, , equal to 0, following López-Sánchez & Esteban (2009).

We derived the electron temperature, , and density, , of the ionized gas of the models using several emission line ratios. The values obtained for each region are compiled in Table 7. All determinations were computed with the IRAF task temden in the NEBULAR package (Shaw & Dufour, 1995). We used the updated atomic dataset for the O, S, and S ions as input to the NEBULAR routine. The references are indicated in Table 4 of García-Rojas et al. (2005).

In accord with Fig. 2, and with equations 2 and 3, we assumed a two-zone approximation for the H ii region to describe the temperature structure of the nebulae. In the high-ionization H + He zone, we used both the [O iii] temperature, [O iii], and the mean of [O iii], [S iii] and [Ar iii], , as the representative temperature for high ionization potential ions. In the low-ionization zone containing H + He we adopted the mean of [O ii], [S ii] and [N ii], , for the low ionization potential ion temperatures.

The line ratios used to measure the temperatures are:

[O iii] (4959+5007)/4363,

[S iii] (9069+9530)/6311,

[Ar iii] (7135+7751)/5192,

[N ii] (6548+6583)/5755,

[O ii] (3726+3729)/(7319+7330), and

[S ii] (6717+6731)/(4069+4076).

We note that some analyses of high-quality, deep spectrophotometric data of bright H ii galaxies (e.g. Hägele et al., 2008, 2011; Pérez-Montero et al., 2010) consider the electron temperature derived for a particular ion only to obtain the ionic abundance of that particular ion. However, the two-zone approximation for the H ii region is most widely used in extragalactic analyses.

Since, in the analysis of the spectra, no information about the pressures or density adopted in the models was given, we used the standard density-sensitive line ratios [O ii3726/3729, [N i5198/5200, [Cl iii5517/5537 and [S ii6717/6731 to derive . In practice, all models have a common pressure cmK, all density-sensitive ion line ratios should be at their low-density limit. Table 7 compiles the value derived in all cases, and the average value adopted for each model. Once was obtained we used it to derive using the [O iii], [S iii], [Ar iii], [O ii], [S ii], and [N ii] line ratios, and we iterated until convergence was attained.

### 3.2 Results for Te

The temperatures derived from the [O iii] (4959+5007)/4363 ratio, and the mean temperature derived for the high-ionization species [O iii], [S iii] and [Ar iii], are compared with the [O iii] emission-line weighted temperature given by the model in Fig 3. The agreement for [O iii] is very good, while the mean of the high-ionization species tends to underestimate the true temperature.

We empirically derive the following third-order correction formula to transform the measured [O iii] line temperatures into the model [O iii] line temperatures:

 tc4=t4(1−0.05(t4−1)2), (4)

where  K is the temperature measured from the spectra, and is the temperature corrected to that delivered by the corresponding model in units of  K. The corrections to [O iii] are almost always smaller than the intrinsic uncertainty in the derived temperature – typically of order 500–1000 K.

For the mean of the high excitation species [O iii] , [S iii] and [Ar iii] the corresponding correction formula is:

 tc4=t4(1.12+0.02t24)−0.087, (5)

with as the average value of the electron temperature derived using the [O iii], [S iii] and [Ar iii] ratio.

For the low-ionization zones of the model H ii regions we use the mean of the [O ii] , [S ii] and [N ii] temperatures given by the models, as was also done in the fitting exercise. The result of the comparison is shown in Fig 4. As we see, we find a clear linear correlation between model and fit, although the slope is not exactly unity. The empirical fit is given by the formula:

 tc4=t4(1.28−0.03t24)−0.15, (6)

where and are the temperature measured from the spectrum and the corrected to that delivered by the corresponding model in units of  K, respectively. The correlation coefficients of the fits in Eqn. 4, 5 and 6 are all in excess of 0.98.

These differences between the temperatures delivered by the model are the result of the temperature structure within the ionized region. The temperatures delivered by the Mappings IIIs code are weighted by the emissivity of, for example, the [O iii5007 line for [O iii]. The [O iii4363 line is more strongly affected by temperature stratification as pointed out by Peimbert & Costero (1969), so that the temperature delivered by measurement of the [O iii] (4959+5007)/4363 ratio is different to the strong-line weighted mean temperature delivered by the code.

For observers who are only able to measure the [O iii] temperature in the high-ionization zone (H + He), and need a correction formula to estimate the temperature in the low-ionization (H + He) zone, the models provide a convenient relationship that fits to an accuracy of  K for temperatures above 10,000 K:

 Te(low)=Te[O \textsciii]+450−70exp[(Te[O \textsciii]/5,000)1.22]. (7)

Note that the temperature sensitivity to the abundance is strong, and that in high-abundance H ii regions, the low ionization species deliver a lower electron temperature. Note however, that the form and shape of Eq. 7 is rather different from the one proposed by Garnett (1992), . However, both the temperature offset and the slope are similar in the range where most of the high-quality nebula spectra have been obtained.

### 3.3 Ionic abundances

The IRAF package NEBULAR (Shaw & Dufour, 1995) has been used to analyze the intensities of the forbidden lines using the temperatures and densities derived for the high- and low-ionization zones to derive ionic abundances of O, O, N, S, S, Ne, Ar, Ar, Cl, and Cl. The ionic abundances so derived are listed in the Appendix in Table 8.

### 3.4 Total abundances

In all cases we adopted O/H = O/H + O/H to determine the total oxygen abundance. Although a very weak nebular He ii 4686 line is produced in several models, the relative contribution of He to the total amount of helium is negligible, implying that O has also a very low abundance in the nebula, thus we did not consider its contribution to the total O/H ratio. For the rest of the elements, we have to adopt a set of ionization correction factors (ICFs) to correct for the unseen ionization stages. The ICFs adopted here are basically the same used by López-Sánchez & Esteban (2009).

To derive the nitrogen abundance we assumed the standard ICF by Peimbert & Costero (1969): N/O = N/O, which is the typical assumption considered in the analysis of the ionized gas in extragalactic H ii regions (e.g. López-Sánchez & Esteban, 2010).

We measured two ionization stages of sulphur, S and S, in all spectra. However, a significant contribution of S is expected. We adopted the ICF given by Stasińska (1978), which is based on photoionization models of H ii regions and is expressed as a function of the O/O ratio.

In the case of neon we also applied the classical ICF proposed by Peimbert & Costero (1969), that assumes that the ionization structure of Ne is similar to that of O. This is a good approximation for high ionization objects, where a small fraction of Ne is expected.

For argon we have determinations of the Ar and Ar abundances. However, some contribution of Ar is expected. Hence, the total argon abundance was calculated by considering the ICFs proposed by Izotov et al. (1994).

We measured lines Cl and Cl. As can be seen in Table 8, the dominant ionization stage is Cl and the contribution of Cl to the total abundance is rather small. To take into account the Cl fraction we have adopted the relation by Peimbert & Torres-Peimbert (1977).

The results we obtain depend on whether we use all of the high-ionization species to determine in the high-ionization zone, (high), and all the low-ionization species to compute in the low-ionization zone, (low), or whether we use only the temperature derived from [O iii] line ratio, [O iii]. The total abundances we obtain by these two approaches are shown in Tables 2 and 3, and are shown in graphical form in Fig 5 and 6, respectively.

When all of the high ionization species are used to derive (high), the oxygen abundance is systematically overestimated by  dex and the other species seen principally in their high-excitation stages, Ne and Ar, are overestimated by somewhat larger amounts. For 12+log(O/H)=9.39, the  method fails because IRAF task ionic does not allow electron temperatures lower than 5,000 K for the low ionization ions. If we use only the temperature derived from the [O iii] lines, [O iii] using the [O iii] (4959+5007)/4363 ratio in the high-ionization zone, and the temperature derived from the [O ii] lines, [O ii] using the [O ii] (3726+3729)/(7319+7330) ratio in the low-ionization zone, the systematic error is much less, specially for the total oxygen abundance, although it is still slightly overpredicted.

Although there are small systematic errors in this procedure, all are correctable. It is clear that the  method returns reliable abundances. Provided that the [O iii] (4959+5007)/4363 ratio is used to determine the in the high-ionization zone, the scatter in the derived abundances of O, N, S and Cl is typically  dex, provided that the O abundance is below 12+log(O/H) . For high abundances, the  method returns systematically low abundances, where it can be applied. This is in agreement with the work of Stasińska (2002) and Stasińska (2005), who predicted that temperature gradients in these high-metallicity H ii regions can cause the abundances to be underestimated by as much as  dex.

Both for Ne and Ar, the systemic error and the scatter are distressingly large. We do not think that this is a consequence of using slightly different atomic data for models and line analysis using IRAF. However the cause is likely to be different in the two cases. In the case of Ne, the error is probably caused by the fact that only Ne is observable, and the [Ne iii3869, 3968 doublet has sufficiently high excitation energy that the line emissivity is strongly biased towards the hottest part of the H ii region. In the case of Ar, charge-exchange processes are very important in determining the ionic balance, and these strongly affect the ionization balance and therefore the ionization correction factors (Dopita et al., 1997). Furthermore, the assumptions of the ICFs used to derived the total abundances of Ne and Ar from their ionic abundances may be not valid in some cases (e.g. Pérez-Montero et al., 2007).

Given that Fig 2 shows that the electron temperature and the abundance are closely correlated, it is interesting to see just how well the oxygen abundance can be recovered from the measured [O iii] temperature alone. To do this we used our empirical relation (Eq. 4) to correct the measured temperature and then applied the relationship between [O iii] and 12+log(O/H) given by Eq. 2 to obtain the oxygen abundance. The result is shown in Fig. 7. For abundances 12+log(O/H) the abundances are recovered to an accuracy of dex. At lower abundances, significant scatter is introduced by variations in the ionization parameter , and it becomes essential to determine this parameter independently.

## 4 Application of the SEL Method

The strong emission line (SEL) technique to derive nebular abundances is typically used when the spectra are of too low signal-to-noise ratio for the temperature sensitive emission lines to be detectable. This typically encompasses the majority of global determinations of metallicity in external galaxies, and all galaxies with redshifts large enough to be of cosmological interest. An extensive review of 10 metallicity calibrations, including theoretical and empirical methods, has been presented by Kewley & Ellison (2008), using data gathered from the SDSS. López-Sánchez & Esteban (2010) also review the most common empirical calibrations and compare their results with those derived using the  method. The majority of the empirical calibrations rely on ratios between bright emission lines to estimate the oxygen abundance. The most commonly used of these are known by their shorthand contractions; , , , , , and . Their definitions are:

 R3=I([\textscOiii])λ4959+I([\textscOiii])λ5007Hβ, (8)
 R2=I([\textscOii])λλ3726,3729Hβ, (9)
 R23=R3+R2, (10)
 P=R3R23, (11)
 y=logR3R2=log1P−1−1, (12)
 N2=I([\textscNii])λ6584Hα, (13)
 O3N2=I([\textscOiii])λ5007I([\textscNii])λ6584, (14)
 N2O2=I([\textscNii])λ6584I([\textscOii])λλ3726,3729, (15)
 S23=I([\textscSii])λλ6717,6731+I([\textscSiii])λλ9076,9532Hβ, (16)
 S23O23=S23R23. (17)

Simple line ratios such as (Jensen et al., 1976; Pagel et al., 1979), (Vílchez & Esteban, 1996; Christensen et al., 1997; Díaz & Pérez-Montero, 2000) –or indeed any forbidden line to recombination line ratio– suffer from being two-valued, and hence a separate calibration for the low and high metallicity regimes is usually needed. In the case of the index, the calibrations are given for 12+log(O/H)8.1 (low metallicity) and 12+log(O/H)8.4 (high metallicity). That means that a very large fraction of the star-forming regions lie in the ill-defined turning zone around 12+log(O/H)8.20–8.30, where regions with the same value have oxygen abundances that differ by almost an order of magnitude (see, for example, Figure A.1 in López-Sánchez & Esteban, 2010). The reason of this behavior is that the intensity of the oxygen (or any other heavy element) lines do not monotonically increase with metallicity. At low abundance, the lines of the heavy element being considered are weak relative to a hydrogen recombination line due to their low abundance. However, at high abundance they are weak due to the low electron temperature, which suppresses collisional excitations into the excited state responsible for the forbidden line. The turn-around in the line ratio can be extended to higher abundance by choosing lines with lower excitation potentials which make the ratio less temperature-sensitive. Hence the utility of over . The use of multiple line ratios also helps to remove the abundance ambiguities.

Ratios such as (Alloin et al., 1979; Pettini & Pagel, 2004) were introduced in an attempt to sidestep the abundance ambiguity entirely. This ratio is much more monotonic in abundance because of the strong secondary component of N enrichment alluded to above; see Eq. 1. This helps at high abundance, but the ratio once again becomes ambiguous when 12 + log(O/H) . A ratio such as (Dopita et al., 2000; Kewley & Dopita, 2002) also provides a useful calibration for metallicity at high abundance. Its advantage is that both N and O co-exist in the same zone of the nebula, the [O iii] lines become weaker at high metallicity due to the low electron temperature, while the [N ii] lines become stronger due to the higher relative abundance of this element. However, its disadvantage is that the nitrogen and oxygen emission lines are widely separated in wavelength, making flux calibration and reddening corrections more critical.

The fundamental weakness in the use of all these strong-line ratios is that the overall spectrum of an H ii region depends not only upon the chemical abundances, but also upon the ionization parameter or . That this is so is abundantly obvious from Fig. 1. The importance of this was recognized by Baldwin et al. (1981), who were amongst the first to use line ratio diagnostic diagrams. However, the first empirical calibration involving an ionization parameter was presented by McGaugh (1991), who developed models of H ii regions using the photoionization code Cloudy. McGaugh (1991) introduced the parameter to derive (together with the ratio) the oxygen abundance using only the bright oxygen lines. The analytical expressions for these models were given by Kobulnicky et al. (1999).

It is clear from Fig. 1c, and as emphasized by Dopita et al. (2000) and Kewley & Dopita (2002), that the [O iii]/[O ii] ratio is probably the best one to use to determine the ionization parameter, particularly if the [N ii]/[O ii] ratio is also measured. To a lesser extent can also be used. Kewley & Dopita (2002) provided a procedure to allow a simultaneous solution of both and 12 + log(O/H), which involves the and parameters, although they also give a parametrization of the oxygen abundance and using other parameters such as [S iii]/[S ii] (which is also a good estimator of the ionization parameter), , , and . Later, Kobulnicky & Kewley (2004) provided a parametrization of the Kewley & Dopita (2002) method using the and parameters with a form similar to that given by McGaugh (1991) calibration. Following this method, and the oxygen abundance are computed iteratively using only the bright oxygen lines.

Pilyugin and his co workers have attempted an empirical calibration of the and parameters using the bright [O iii4959,5007 and [O ii3726,3729 nebular lines. To get this calibration, they used a combination of photoionization models and empirical data from observed H ii regions where the auroral [O iii4363 line was available. Indeed, the so-called Pilyugin parameter, , was introduced by Pilyugin (2000), after confirming that the parameter has a systematic error depending on the hardness of the ionizing radiation. Hence, the excitation parameter also considers the ionization degree of the H ii region. His first empirical calibrations involving both the and the parameter were presented in Pilyugin (2001a) and Pilyugin (2001b) for high- and low-metallicity H ii regions. Subsequently, Pilyugin & Thuan (2005) and Pilyugin, Vílchez & Thuan (2010) tried to improved such calibrations including more spectroscopic measurements of H ii regions in spiral and irregular galaxies derived using the -method. Finally, Pilyugin & Mattsson (2011) provided another empirical calibration considering only the , and parameters (the so-called NS calibration), which was derived to estimate oxygen abundances in galaxies whose spectrum is lacking of the [O ii] 3726,3729 emission lines.

### 4.1 Comparison of the SEL Techniques

We have applied all of these commonly-used SEL techniques to our model spectra to derive oxygen abundances, again via a double-blind procedure. The different SEL techniques fall into various classes, depending on which combination of strong line ratios are used. Table 9 lists the values of all these parameters derived for each model. This table also includes the value derived for the parameter obtained from the optimal calibration provided by Kewley & Dopita (2002) and Kobulnicky & Kewley (2004). Table 4 compiles the oxygen abundances derived for each model, indicating which branch (high or low metallicity) the model spectra would fall into when using the parameter. We considered the values of the oxygen abundances derived from the  method to choose the expressions of the lower or the upper branch provided by the empirical calibrations. For models F, U, W and AF, for which we obtained oxygen abundances of 12+log(O/H)8.4 using the  method, the results provided by both the low- and high- metallicity branches are tabulated, and both sets of data are plotted in Figs. 8 to 11. Note that the calibrations that invoke the , , or parameters give the same result for the low and high metallicity branches. The Kewley & Dopita (2002) calibration using the parameter is only valid for objects in the high-metallicity branch.

To quantify the goodness of these SEL techniques, we have performed a linear fit between the oxygen abundances derived from these empirical methods and the oxygen abundances assumed by the models. In all cases, for models with 12+log(O/H)=8.39 we have assumed the average value between the high and low metallicity branches. We note that not all models have been considered in this fit, but only those within the validity range of each SEL calibration. Table 5 compiles the results for these linear fits, including the correlation coefficient, the dispersion of the data and the average value of the difference between the abundance given by the SEL method and that assumed by the models (offset). First row of Table 5 lists the results of a linear fit between the oxygen abundances derived following the  method and those imposed by the models. This fit does not consider the models with 12+log(O/H)=9.39 (see Fig. 6).

#### 4.1.1 N2 and O3n2 methods

Let us first consider the techniques which rely upon the parameter; that of Denicoló, Terlevich & Terlevich (2002) and those drawn from the Pettini & Pagel (2004) which involve a linear interpolation of the parameter, a cubic fit to this same parameter, and their fit using the parameter. These are shown in Fig. 8. All these calibrations have similar scatter  dex. However, they are well-behaved (if rather curved) in their abundance sequence. This type of curvature was also observed by Kewley & Ellison (2008), if we can interpret the mass sequence of galaxies as equivalent to an abundance sequence.

Yin et al (2007) indicated that the and the indices are only useful for calibrating metallicities of galaxies with 12+log(O/H)8.5, while Pérez-Montero & Contini (2009) established that empirical calibrations using the parameter are not valid for objects with 12+log(O/H)8.0. Following Fig. 8, we suggest that the method should not be applied for 12+log(O/H) 8.7-9.0, while the calibration provided by Pettini & Pagel (2004) is only valid for 12+log(O/H) 8.7. Importantly, in all cases the result has a high uncertainty, 0.25 dex, which is a consequence of lacking of any parameter which considers the ionization degree of the gas. Furthermore, in the case of galaxies showing an overabundance of nitrogen (e.g. Pustilnik et al., 2004; López-Sánchez et al., 2007; López-Sánchez & Esteban, 2010; Monreal-Ibero et al., 2010), the application of any or calibration will provide misleading oxygen abundances.

#### 4.1.2 S23 and S23o23 methods

Figure 9 compares the oxygen abundances assumed by our models with the results provided using the Pérez-Montero & Díaz (2005) calibrations, that consider the and parameters. In the case of using the index, the calibration seems to be valid up to 12+log(O/H)9.0, having a dispersion of 0.24 dex. However, this method cannot be applied for 12+log(O/H)9.0. On the other hand, the calibration works quite well at high metallicities, 12+log(O/H)8.5. But it is not valid for 12+log(O/H)8.2 as the result basically does not depend on metallicity, having a constant value which depends on , and a high dispersion, 0.6 dex.

Much of the scatter in the Denicoló, Terlevich & Terlevich (2002), Pettini & Pagel (2004) and Pérez-Montero & Díaz (2005) calibrations is due to a failure to include the effects of the ionization parameter on the line ratios. That is specially important in the analysis of 2D spectroscopic data, as the oxygen abundance maps derived using these calibrations may directly reflect the ionization structure of the H ii region and not actual changes in metallicity (López-Sánchez et al., 2011). It is clear that the scatter could be reduced to 0.1 dex if the ionization parameter is taken into account, since the vertical separation of the points on the figures is almost entirely the result of the different ionization parameters used in the models. However, this would not solve the curvature issue.

#### 4.1.3 Pilyugin method

We next consider the Pilyugin et al. empirical calibrations of and (Pilyugin, 2001a, b; Pilyugin & Thuan, 2005; Pilyugin et al., 2010; Pilyugin & Mattsson, 2011). As we discussed before, is an empirical parameter that traces the ionization degree of the H ii region. The results we obtain are shown in Fig. 10. The main characteristic of these SEL calibrations is that there is a systematic offset in the sense that the Pilyugin method tends to underestimate the abundance by up to 0.25 dex (Pilyugin, 2001a, b), 0.4 dex (Pilyugin et al., 2010) and 0.5 dex (Pilyugin & Thuan, 2005; Pilyugin & Mattsson, 2011). It is also clear that the scatter for the Pilyugin (2001a, b) and Pilyugin & Thuan (2005) empirical calibrations is typically 0.3 dex. In both calibrations, the scatter becomes considerably larger in the 12+log(O/H)8.0–8.4 range, as a result of the ambiguity in determining the appropriate branch on which the is located. We note that for this intermediate-metallicity regime it is common to give an average value between the oxygen abundances derived for the high and low metallicity branches if using the Pilyugin (2001a, b) and Pilyugin & Thuan (2005) method (e.g. Moustakas et al., 2010; López-Sánchez & Esteban, 2010).

In the case of the Pilyugin et al. (2010) and Pilyugin & Mattsson (2011) methods (that also considers the and indices), three calibrations are provided for low, intermediate and high metallicity. In these cases, the scatter is slightly lower, 0.14 dex and 0.25 dex, respectively, than the scatter found using the previous calibrations. The offset is also large, 0.4-0.5 dex, when comparing with that provided by the models.

The very interesting point here is that Pilyugin method is giving the best results when it is applied for real objects for which the oxygen abundance has been computed using the  method (e.g. Bresolin et al., 2009; López-Sánchez & Esteban, 2010; Moustakas et al., 2010). However, this method is clearly failing when predicting the metallicities of the theoretical models using the bright emission lines. We will further discuss this issue in Sect. 4.2.

#### 4.1.4 Kewley and Dopita method

Although Pilyugin method considers a parameter which is related to ionization degree of the gas, the scatter provided by this calibration is still relatively high. However, the scatter is much reduced when using a SEL method which is based on photoionization models, as those provided by the McGaugh (1991), Kewley & Dopita (2002) and Kobulnicky & Kewley (2004) methods. All these models explicitly consider an ionization parameter, . Fig. 11 shows their results. As we see, in all these cases the agreement is excellent. We remind the reader that we are plotting both the high and low metallicity results for the models with 12+log(O/H)=8.39.

Left panel of Fig. 11 shows the case of the McGaugh (1991) calibration using the Kobulnicky et al. (1999) parametrization. This calibration systematically underestimates the abundances given by the models by 0.15 dex, but their dispersion is only 0.07 dex. In the case of using the Kewley & Dopita (2002) or Kobulnicky & Kewley (2004) methods considering the and parameter, the results are much better. Their method allows to iteratively determine the ionization parameter and the oxygen abundance. Both, the techniques by Kewley & Dopita and by Kobulnicky & Kewley, eliminate the scatter in the SEL technique to a very large extent. For the Kewley & Dopita (2002) technique it is 0.1 dex, while for the Kobulnicky & Kewley (2004) technique it is 0.07 dex excepting the ambiguous region around 12+log(O/H)=8.4.

Far right panel of Fig. 11 shows a fit using the calibration provided by Kewley & Dopita (2002) for the high-metallicity branch, 12+log(O/H)8.35, and relied on the and the technique they use for obtaining the ionization parameter for the lower metallicities. This assumption provides very good results, as this method has a very low offset, 0.05 dex, and dispersion, 0.09 dex.

The slight offset in the abundances derived following the Kewley & Dopita (2002) and Kobulnicky & Kewley (2004) methods is probably due to the fact that they used a different grid of models. The models we use in this paper are more closely related to the earlier Kewley & Dopita (2002) grid, except in respect of the much more sophisticated treatment of dust physics employed here.

However, the recent analysis of the ionized gas within a sample of 40 strong star-forming galaxies performed by López-Sánchez & Esteban (2010) shows that the oxygen abundances provided by photoionization models (McGaugh, 1991; Kewley & Dopita, 2002; Kobulnicky & Kewley, 2004) are, systematically, 0.2–0.3 dex higher than the oxygen abundances provided by the  method. This observational result has been also found by Yin et al (2007), Bresolin et al. (2009), Moustakas et al. (2010) and López-Sánchez et al. (2011); in some objects the differences reach up to 0.6 dex. Interesting, however, the photoionization models developed by Dors et al. (2011) provided O/H values close to those obtained using the  method. Detailed tailor-made photoionization models of individual galaxies (e.g. Pérez-Montero et al., 2010) also provided results similar to those derived from the observed electron temperatures of the ionized gas.

### 4.2 Comparison of the SEL techniques with real data

Hence, we have further investigated the validity of the SEL techniques using real data. For this, we aim to use the deepest, highest quality observational data of Galactic and extragalactic H ii regions nowadays available. Particularly, we also want to compare the results provided by the  and the most common SEL techniques (the Pilyugin and the Kewley & Dopita methods) with the oxygen abundances derived using oxygen recombination lines (RL). As it is well known (e.g. Peimbert, 2003; Esteban et al., 2002, 2004, 2009; García-Rojas et al., 2004, 2005; García-Rojas & Esteban, 2007; Peimbert et al., 2007), in all cases the oxygen abundances determined using RL are 0.2-0.3 dex higher than the oxygen abundances derived using the  method, which is based on collisionally excited lines (CEL). The effect does not depend on metallicity, electron density, electron and Balmer temperatures or ionization degree (García-Rojas & Esteban, 2007; Mesa-Delgado et al., 2009a) and it may be attributed to the presence of temperature fluctuations in H ii regions.

Moreover, recent analyses (Peimbert, Peimbert & Ruiz, 2005; Przybilla, Nieva & Butler, 2008; Simón-Díaz & Stasińska, 2011) have found that the oxygen nebular abundance based on RL agrees better with the abundances of the stars associated with the nebulae than the one derived from the  method, suggesting that RL abundances are more reliable than the abundances derived from CEL.

Table 6 lists all the 22 Galactic and extragalactic H ii regions found in the literature for which oxygen abundances are available using both the RL and the  methods. Indeed, the average offset between  and RL abundances is 0.22 dex. Table 6 also compiles the derived , , and indexes derived from the reddening-corrected data. We must emphasize that, in almost all the cases, the error in the [O ii3726,3729 lines is less than 5%. Using these observational parameters, we then applied the Pilyugin et al. method and the Kewley & Dopita method to determine the oxygen abundances using these SEL techniques. Table 6 compiles the average value derived for the Pilyugin (2001a, b); Pilyugin & Thuan (2005); Pilyugin et al. (2010) techniques (P) and the average value obtained for the Kewley & Dopita (2002) (using the , and parameters) and the Kobulnicky & Kewley (2004) method (KD).

As we see in Table 6, we find again that the results obtained using the P technique agree quite well with the oxygen abundance determined using the  method (offset of 0.03 dex), but the results provided by the KD method are systematically 0.30 dex larger. That is exactly the opposite behavior we discussed in the previous section. This result is graphically shown in Fig. 12, which compares the  abundance (-axis) with the oxygen abundances derived from RL (black squares), the P method (blue diamonds) and the KD technique (red triangles).

Interestingly, the results provided using the KD technique have a better agreement with the values determined from RL. These are systematically 0.08 dex lower than the oxygen abundances given by the KD method, but both agree within the errors (the dispersion of the RL data is 0.06 dex, while the dispersion of the KD data is 0.07 dex). This observational result, which was previously noted by Peimbert et al. (2007) and López-Sánchez & Esteban (2010), is actually suggesting that the real metallicities of the H ii regions may be not those given by the standard  method based on line ratios of collisionally excited lines (CELs), but the values derived using recombination lines (RL). If this is correct, then the KD method is the only SEL technique that is reproducing those metallicities.

As we have seen along this paper, the oxygen abundances of our model spectra are well reproduced using the  method and SEL techniques based on photoionization models (McGaugh, 1991; Kewley & Dopita, 2002; Kobulnicky & Kewley, 2004) but 0.3–0.5 dex higher than those estimated using the Pilyugin method. However, the situation is inverted in other works based on the comparison of observations with the results of the SEL methods (e.g. Peimbert et al., 2007; Yin et al, 2007; Bresolin et al., 2009; López-Sánchez & Esteban, 2010; Moustakas et al., 2010; López-Sánchez et al., 2011; Rosales-Ortega et al., 2011), i.e., the oxygen abundance in real nebulae is better reproduced by those SEL methods which rely in an empirical calibration derived using H ii regions where a direct estimation of the electron temperature exists.

A possible explanation of this puzzling situation is that current photoionization models are not properly reproducing the behavior of collisionally excited lines (CELs) in real nebulae. As we said before, several authors have invoked the presence of temperature fluctuations in the nebulae (Peimbert, 1967) in order to address the differences between the abundances determined from CELs and RLs. Due to the strong dependence on  of the intensity of CELs, such temperature fluctuations cause the metallicities derived from CELs to be underestimated. Although our models consider global temperature gradients due to the hardening of the ionizing radiation field along the nebula, they did not reproduce the values of the temperature fluctuations parameter () that are necessary to make the O/H ratios determined from RLs and CELs to converge (e.g. García-Rojas & Esteban, 2007). These hypothetical fluctuations must be of small spatial scale and may be produced by, for example, density inhomogeneities or localized deposition of mechanical energy that heat the gas.

Summarizing, assuming the validity of the scenario outlined above, the oxygen abundances determined from RLs and using SEL techniques based on photoionization models (KD method) would be closer to the real ones, while values determined from CELs and empirical calibrations (P method) would provide abundances about 0.2–0.3 dex lower. This should have a profound impact in current galaxy metallicity estimations that have been obtained using CELs or any empirical calibration–such as those provided by Pilyugin et al. or Pettini & Pagel (2004) – that should be revised upwards.

Finally, we want to note that the small offset of 0.08 dex between the RL and the KD may be a consequence of the depletion of the oxygen atoms in dust grains. Our models are taking into account this effect, but observations suggest that oxygen depletion in dust grains may be decreasing the actual gas-phase metallicity (i.e., that derived from RL) of H ii regions by 0.08-0.12 dex (Mesa-Delgado et al., 2009b; Peimbert & Peimbert, 2010; Simón-Díaz & Stasińska, 2011).

## 5 Conclusions

In this paper, we have subjected the commonly-used techniques for deriving chemical abundances to a critical “double-blind” analysis. We find that the originally-used  method is capable of delivering reliable abundances for O, N, S, and Cl. However, the abundances inferred for the noble gases Ne and Ar are considerably less secure, although probably for different physical reasons.

The electron temperature itself is a reasonably good indicator of the oxygen abundance, as the one is the dominant controller of the other. Error arises because there is also a weaker dependence of electron temperature on the ionization parameter. For the [O iii] lines, this renders an abundance obtained from the electron temperature alone unreliable below 12+log(O/H) .

The temperature derived from the temperature-sensitive collisionally excited line ratios is always somewhat different from emission-weighted temperature defined for the principal emission lines – in the case of [O iii]–. This is due to the variability of electron temperature through the zone containing the ion of interest due to hardening of the radiation field and spherical divergence of the radiation field (which changes the photoelectric heating rate), and change in ionization state (which influences the cooling rate of the plasma). In this sense, the models naturally produce electron temperature gradients inside H ii regions, such as those discussed in Stasińska (1978), Garnett (1992) and Stasińska (2005).

Although our models can reproduce these “global” temperature fluctuations, they cannot simulate the small-scale temperature fluctuations, such as those first discussed by Peimbert (1967) and Peimbert & Costero (1969), produced by turbulent or photo-evaporative flows, shocks, or radiation shielding by dense inclusions or globules of un-ionized material. Thus we can be reasonably sure that the true temperature fluctuations in real H ii regions should be greater than the models predict. The effect of this would be to raise the temperature as measured by the [O iii] (4959+5007)/4363, which would then feed into a systematic underestimate in the oxygen abundance as delivered by the  method. Hence, accepting the existence of small-scale temperature fluctuations within the ionized gas and assuming that the RLs give the true metallicities, those oxygen abundances determined from CELs and empirical calibrations (P method) would provide abundances which are 0.2–0.3 dex lower than the real ones. That is because of the strong dependence of the intensity of CELs on : the existence of such small-scale temperature fluctuations will cause an underestimation of the oxygen abundances using CELs lines or strong-line methods which are based on the bright nebular lines. However, oxygen abundances derived using SEL techniques based on photoionization models (such as the KD method, for which the  is fixed a priori) would be closer to the real values.

We have demonstrated that those SEL techniques which do not explicitly solve for the ionization parameter are quite unreliable, resulting in large scatter, systemic error, and abundance-dependent errors. To obtain the ionization parameter, we need a measurement of the [O iii5007/[O ii3726,3729 ratio. For abundances 12+log(O/H) the oxygen abundance could then be estimated from the [O iii] 5007/H line ratio. This is verified in Fig 13, which provides a new diagnostic plot for this low abundance regime.

At high abundance, the most sensitive diagnostic plot is that of [O iii5007/[O ii] 3726, 3729 vs. [N ii6584/[O ii3726,3729, shown in panel c of Fig. 1. The main error here is the degree to which the N/O abundance has been accurately calibrated against the O abundance. We believe that the relationship given in Eq. 1 is probably good, since the model grids reproduce the Veilleux & Osterbrock (1987) diagnostic plots shown in Fig. 1, panels (a) and (b).

Figures 1, panel (c) and 13 thus provide a SEL technique that is applicable to the determination of both metallicity and ionization parameter over the full abundance range, with the caveat that the sensitivity of this technique is rather poor in the range 12 + log(O/H) . In this range, common to all SEL techniques, rather accurately-measured line ratios are required.

## Acknowledgements

We thank our anonymous referee for his/her helpful comments. We thank Maritza Lara-López, Mercedes Mollá, Enrique Pérez-Montero, José M. Vílchez, César Esteban, Stuart Ryder, Guillermo Hägele, and Fabián Rosales-Ortega, for very fruitful discussions and comments about this study. Dopita acknowledges support the Australian Research Council (ARC) for support under Discovery project DP0984657. This research has made extensive use of the SAO/NASA Astrophysics Data System Bibliographic Services (ADS).

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## Appendix A Electron temperatures, electron densities and ionic abundances for the model H ii regions.

Here we compile the derived parameters for the model H ii regions. The first table, Table 7, gives the derived densities and temperatures in the various zones of the H ii region. Table 8 gives the derived ionic abundances. Finally, Table 9 gives the line ratio values used to compute the oxygen abundance using the empirical calibrations as well as the ionization parameters derived for the models using the Kewley & Dopita (2002) and Kobulnicky & Kewley (2004) techniques.