A&A 474, 807-814 (2007)
DOI: 10.1051/0004-6361:20077884

Checking the reliability of equivalent width R₂₃ for estimating the metallicities of galaxies

Y. C. Liang^1,2 - F. Hammer² - S. Y. Yin^1,3,4

1 - National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, PR China
2 - GEPI, Observatoire de Paris-Meudon, 92195 Meudon, France
3 - Department of Physics, Hebei Normal University, Shijiazhuang 050016, PR China
4 - Department of Physics, Harbin University, Haerbin 150086, PR China

Received 15 May 2007 / Accepted 8 August 2007

Abstract
Aims. We verify whether the O/H abundances of galaxies can be derived from the equivalent width (${\it EW}$) R₂₃ instead of the extinction-corrected flux R₂₃, and eventually propose a method of improving the reliability of this empirical method, which is often used for the non-flux calibrated spectra of galaxies.
Methods. We select 37 173 star-forming galaxies from the Second Data Release of the Sloan Digital Sky Survey (SDSS-DR2), which offers a wide range of properties to test the EW method.
Results. The EW-R₂₃ method brings with it a significant bias: for the bulk of SDSS galaxies, it may affect the determination of $\log$ (O/H) by factors ranging from -0.2 to 0.1 dex and for some galaxies by factors ranging from -0.5 to 0.2 dex. We characterize this discrepancy (or bias) by $\alpha = ({I_{\rm [O~II]}/I_{\rm H\beta}}$)/( ${EW_{\rm [O~II]}/EW_{\rm H\beta}})$, which is virtually independent of dust extinction, while tightly correlating with D_n(4000), although at a lower significance, with (g-r) colors.
Conclusions. The EW-R₂₃ method cannot be used as a proxy for the extinction-corrected flux R₂₃ method. From analytical third-order polynomial fits of $\alpha$ versus (g-r) colors, we have been able to correct the EW-R₂₃ method. With this additional and easy correction, the EW-R₂₃ method provides O/H abundance values similar to those derived from the extinction-corrected flux R₂₃ method with an accuracy of $\pm$0.1 dex for >92% of the SDSS galaxies.

Key words: galaxies: abundances - galaxies: evolution - galaxies: ISM - galaxies: photometry - galaxies: spiral - galaxies: starburst

1 Introduction

Chemical abundance is a fundamental parameter for tracing the history of star formation and evolution of galaxies. Oxygen is the most commonly used metallicity indicator in the interstellar medium (ISM) by virtue of its high relative abundance and strong emission lines in the optical part of the spectrum (e.g., [O  II]$\lambda$3727 and [O  III] $\lambda\lambda$4959, 5007). However, the "direct'' method of estimating oxygen abundances from electron temperature ($T_{\rm e}$) is generally only available for metal-poor galaxies (12+ $\log~{\rm (O/H)}<8.5$), where the [O  III]$\lambda$4363 emission line can possibly be detected, which is needed for measuring $T_{\rm e}$ by its ratio to [O  III] $\lambda\lambda$4959, 5007 (Pagel et al. 1992; Skillman & Kennicutt 1993). For metal-rich galaxies, the most commonly used are empirical strong-line methods, such as R₂₃:

$$R_{23} = \frac{I([{\rm O II}]\lambda3727) + I([{\rm O III}]\lambda\lambda4959,5007)}{I(\rm H\beta)},$$ (1)

i.e. the flux ratios of [O  II] and [O  III] to H$\beta$(Pagel et al. 1979; McGaugh 1991; Zaritsky et al. 1994; Kobulnicky et al. 1999; Tremonti et al. 2004, and the references therein).

However, flux calibrations are frequently problematic in the current generation of wide-field galaxy surveys of multiobject spectrography, because of unfavorable observing conditions or instrumental effects such as a variation in system response over the field of view, nevertheless, one still expects to derive metallicity properties of the star-forming galaxies detected in the large data sets from surveys. Then, equivalent widths (EWs) are being used to replace the fluxes of their R₂₃ values for estimating metallicities of the galaxies, i.e., from the EW R₂₃:

$$EW~R_{23} = \frac{EW([{\rm O~II}]\lambda3727) + EW([{\rm O~III}]\lambda\lambda4959,5007)}{EW(\rm H\beta)}\cdot$$ (2)

This replacement was first checked by Kobulnicky & Phillips (2003, KP03 hereafter) on the basis of a sample of 243 nearby galaxies. Consequently, this method has been used by some researchers to estimate metallicities of nearby and even intermediate-zgalaxies (e.g. Kobulnicky et al. 2003; Kobulnicky & Kewley 2004; Lamareille et al. 2005a,b; Mouhcine et al. 2006a,b, etc.).

However, it is known that there is a continuum ( $F_{C\lambda}$) scale factor between the flux ( $F_{\lambda}$) and EW ( $W_{\lambda}$) values of the emission line:

$$W_\lambda = \frac {F_{\lambda}} {F_{C\lambda}}\cdot$$ (3)

This will naturally make us ask whether the continua underlying [O  II] and H$\beta$ (and [O  III]) are very similar or not. If not, $F_{C\lambda}$ could be far from a unique constant, and then the metallicities derived from EW R₂₃ could have large discrepancies from those derived from flux R₂₃.

The SDSS provides a complete dataset and measurements up to several ten-thousand galaxies, making it a very good database for studying this question. In this paper, we selected 37 173 star-forming galaxies from the SDSS-DR2 to further check the reliability of using EW R₂₃ replacing the extinction-corrected flux R₂₃to estimate the metallicities of galaxies. Moreover, this is a large homogeneous database observed by one single highly efficient facility, which will minimize the effect of using various instruments. Some other characteristic parameters provided by the SDSS can also help for understanding the question further, e.g., D_n(4000), g and r photometric magnitudes, etc.

This paper is organized as follows. The sample selection criteria are described in Sect. 2. In Sect. 3, we check how big the difference is between the underlying continua of [O  II] and H$\beta$, which is quantified as a parameter $\alpha$ (by 1/$\alpha$). The emission-line quantities of the sample galaxies are analyzed in Sect. 4, as well as the discrepancies between their EW R₂₃ and flux R₂₃, and the discrepancies between the derived $\log$ (O/H) abundances from them. In Sect. 5, we try to find the factors that mostly affected the $\alpha$ parameter, then to find the relations between them and $\alpha$, hence to modify the EW R₂₃ method, which includes the stellar population indicators D_n(4000) and colors. In Sect. 6, we discuss the boundary of the upper branch of 12+$\log$ (O/H) abundances from the $\log R_{23}$ calibration, then summarize and conclude this paper in Sect. 7.

  
2 Sample selection

The data analyzed in this study were drawn from the SDSS-DR2 (Abazajian et al. 2004). These galaxies are part of the SDSS "main'' galaxy sample used for large-scale structure studies (Strauss et al. 2002). We selected a sample of star-forming galaxies with metallicity estimates from the SDSS-DR2 database. The selection criteria are similar to those of Tremonti et al. (2004) and Liang et al. (2006). We summarize the criteria for selection as follows, and mark the selected number of the sample galaxies in parenthesis at the end of each step:
(i) SDSS-DR2 (261 054), 14.5< r <17.77 mag (Petrosian magnitude, 193 890 left);
(ii) 12+$\log$ (O/H) $_{\rm SDSS} > 0$ (50 385 left) (SDSS refers to the metallicity values provided by the MPA/JHU group, which were obtained on the basis of the 2001 Charlot & Longhetti model and Bayesian technique, see Tremonti et al. 2004);
(iii) 0.04<z<0.25 (40 693 left) (this allows the spectral coverage from [O II]$\lambda$3727 to [S II]$\lambda$6731 for the sample galaxies);
(iv) the emission lines [O II], H$\beta$, [O III], H$\alpha$, and [N II] are detected and their fluxes and EWs are reasonable; fluxes of emission lines H$\beta$, H$\alpha$, and [N II]6584 are detected at levels higher than 5$\sigma$ (39 029 left);
(v) $8.5<12+\log$ (O/H) _R23 <9.3 (38 932 left) (we adopt the formula of Tremonti et al. 2004 to convert R₂₃ to the oxygen abundance 12+$\log$ (O/H)_R23, which is only suitable for metal-rich galaxies; therefore, we only select the galaxies with $12+\log~{\rm (O/H)}>8.5$ here; 9.3 is almost the upper limit of the abundances in the samples);
(vi) the discrepancy between $\log$ (O/H)_R23 and $\log$ (O/H) $_{\rm SDSS}$ is less than 0.1 dex (37 173 left finally), which removes some scattered data points from the R₂₃ calibration, as shown by Fig. 1.

	Figure 1: Comparison between the oxygen abundances obtained by the MPA/JHU group (marked as "SDSS'') and those derived from R23 calibration of Tremonti et al. (2004) (marked as "R23''). The dashed lines mark the 0.1 dex discrepancies from the equal values (solid line).
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Finally, we obtained a sample of 37 173 star-forming galaxies from the SDSS-DR2 as our sample in this study.

The criteria (i)-(iv) are almost the same as what was adopted in Tremonti et al. (2004), except the lower limit for redshift z was increased from 0.005 to 0.04 by following Kewley et al. (2002) to minimize the aperture effects of the SDSS. In Sect. 6, we discuss criterion (v) in particular, i.e. 12+$\log$ (O/H) $\sim 8.5$ as the lower boundary of the upper branch of oxygen abundances from $\log~R_{23}$ calibration.

  
3 Quantifying the difference between EW R₂₃ and flux R₂₃: determining the $\alpha$ parameter

The MPA/JHU collaboration has put the measurements of emission-lines and some physical parameters for a large sample of SDSS galaxies on the MPA SDSS website (Kauffmann et al. 2003; Brinchmann et al. 2004; Tremonti et al. 2004, etc.). These measurement values were obtained from the stellar-feature subtracted spectra with the spectral population synthesis code of Bruzual & Charlot (2003).

Fluxes of the emission lines should be corrected for dust extinction, so we estimate the attenuation of the sample galaxies using the Balmer line ratio H$\alpha$/H$\beta$ and assuming case B recombination, with a density of 100 cm^-3, a temperature of 10⁴ K, and the intrinsic H$\alpha$/H$\beta$ ratio of 2.86 (Osterbrock 1989), following the relation of

$$\left(\frac{ I_{\rm H\alpha}}{ I_{\rm H \beta}}\right)_{\rm o... ...eta0}}\right)_{\rm intr}10^{-c {\rm (f(H\alpha)- f(H\beta))}}.$$ (4)

Using the average interstellar extinction law given by Osterbrock (1989), we obtain f(H $\alpha)- f$(H $\beta) = - 0.37$. For the data points with c < 0, c = 0 is assumed since their intrinsic H$\alpha$/H$\beta$ may be lower than 2.86 if their electron temperature is high (Osterbrock 1989).

Then, the extinction-corrected R₂₃ parameter and its relation with the EW R₂₃ are:

 

$${{I_{\rm [O~II]} + I_{\rm [O~III]}} \over {I_{\rm H\beta}}}$$                                R₂₃ =

$${ {F_{\rm [O~II]}} \over {F_{H\beta}} }10^{c({f(\rm [O~II])}-f(\r... ..._{\rm [O~III]}} \over {F_{\rm H\beta}} } 10^{c{(f(\rm [O~III])}-f(\rm H\beta))}$$ =

$${ {W_{\rm [O~II]}F_{\rm C,[O~II]}} \over {W_{\rm H\beta}F_{\rm C,H\beta}} }10^{c{(f(\rm [O~II])}-f(\rm H\beta))}$$ =

$$+ { {W_{\rm [O~III]}F_{\rm C,[O~III]}} \over {W_{\rm H\beta}} F_{\rm C,H\beta}} 10^{c{(f(\rm [O~III])}-f(\rm H\beta))}$$

$${{ \alpha {W_{\rm [O~II]}} + {W_{\rm [O~III]}}} \over {W_{\rm H\beta}}},$$ = (5)

where $\alpha = ({F_{\rm C,[O~II]}}/F_{\rm C,H\beta}) 10^{c{(f(\rm [O~II])}-f(H\beta))}$, and $({F_{\rm C,[O~III]}}/F_{\rm C,{\rm H\beta}}) 10^{c{(f(\rm [O~III])}-f(H\beta))}$ is about equal to 1 since [O  III] $\lambda\lambda$4959,5007 and H$\beta$ are very close in wavelength. The expression $I_{\rm [O~II],[O~III],{\rm H\beta}}$ refers to the dereddened, calibrated flux values of the corresponding lines; $F_{\rm [O~II],[O~III],{\rm H\beta}}$ refers to the observed flux values; and $W_{\rm [O~II],[O~III],{\rm H\beta}}$ represents the ${\it EW}$ values of the related emission lines. The parameter c is the extinction coefficient (also see KP03).

The $\alpha$ values of these SDSS galaxies can be calculated directly from $\alpha = ({F_{\rm C,[O~II]}}/F_{\rm C,H\beta}) 10^{c{(f(\rm [O~II])}-f({\rm H\beta}))}$, where $F_{\rm C,[O~II]}$ and $F_{\rm C,H\beta}$ can be estimated from their ratios of fluxes to ${\it EW}$ values, and c is the extinction coefficient. The derived $\alpha$ parameters of these sample galaxies show a median value of 0.85 and an average value of 0.86 in a range from 0.1 to 2.6. KP03 estimate the typical value of the $\alpha$ parameters of their 243 sample galaxies to be $\alpha = 0.84 \pm 0.3$. However, KP03 still adopted $\alpha=1$ when they used the ${\it EW}$ R₂₃ to estimate the oxygen abundances of galaxies.

4 Analysis of emission-line quantities

Several relations are analyzed in this section, including those of the emission-line quantities, e.g. ${\it EW}$(H$\beta$), with the $\alpha$ parameter, with the discrepancies of ${\it EW}$ R₂₃ and flux R₂₃, and with the discrepancies of the $\log$ (O/H) abundances derived from these two R₂₃ values.

4.1 Relations between the line strengths and $\alpha$ parameters

Figure 2: a) Emission-line strengths ${\it EW}$(H$\beta$) of the sample galaxies as a function of the ratios of equivalent width ratios to dereddened emission-line fluxes ratios for [O  II]/H$\beta$. b) ${\it EW}$(H$\beta$) as a function of the discrepancy between the quantity of ${\it EW}$ R23 and the flux R23 (extinction-corrected). c) ${\it EW}$(H$\beta$) as a function of the discrepancy between the two metallicity estimates from ${\it EW}$ R23 and flux R23.

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Figure 2a shows the relations between the emission-line strengths ${\it EW}$(H$\beta$) and the line-ratio ${\it EW}_{\rm [O~II]} \over {\it EW}_{\rm H\beta}$/ $I_{\rm {[O~II]}}\over I_{\rm H\beta}$ (=1/$\alpha$) of the sample galaxies. The solid line marks the one-to-one correspondence. It seems that there is some correlation between the $\alpha$ parameter and the emission-line strengths: from the galaxies with stronger emission lines, the ratios of $\log\big({{{{EW}_{\rm [O~II]}} \over EW_{\rm H\beta}}/ {{\rm I_{[O~II]}}\over {\rm I_{H\beta}}}}\big)$ (=$\log$ (1/$\alpha$)) change from -0.2 (mostly) to 0.5 monotonicly, though the scatters are large and show some exceptional points with $\log$ (1/$\alpha$)< -0.2. This range is similar to what KP03 found for their sample galaxies. The line strengths of [O  II] and [O  III], i.e., ${\it EW}_{\rm [O~II]}$ and ${\it EW}_{\rm [O~III]}$, also show similar trends to ${\it EW}$(H$\beta$) in these kinds of relations, but with somewhat larger scatter. The ratio $\log\big({{{{\it EW}_{\rm [O~II]}} \over {{\it EW}_{H\beta}}}/{{I_{\rm {[O~II]}}}\over {I_{\rm H\beta}}}}\big)$ (=$\log$ (1/$\alpha$)) of all the sample galaxies show a median value of 0.069 dex (with a scatter of 0.060 dex) and a mean value of 0.077 dex (with a scatter of 0.104 dex).

4.2 Discrepancies between EW R₂₃ and R₂₃

When directly using the ${\it EW}$ R₂₃ to replace the flux R₂₃, namely, $\alpha=1$ is adopted for ${\it EW}$ R₂₃, the derived metallicities could have some discrepancies. Figure 2b shows the discrepancies between the ${\it EW}$ R₂₃ and the extinction-corrected flux R₂₃ as a function of line strengths ${\it EW}$(H$\beta$). The general trend is that, from galaxies with stronger line strengths, the differences between $\log$ (${\it EW}$ R₂₃) and $\log(R_{23})$ increase from -0.15 (mostly) to 0.5 dex monotonicly. KP03 also find a similar trend for their sample galaxies. Our much larger sample shows this systematic discrepancy more clearly. The median value of these discrepancies is about 0.061 dex (with a scatter of 0.050 dex), and the mean value of them is about 0.069 dex (with a scatter of 0.086 dex).

4.3 Discrepancies between (O/H) _EW R23 and (O/H) _R23

One of the most important results of this work is given in Fig. 2c, which shows the difference between $\log$ (O/H) $_{{\it EW}~R_{23}}$ and $\log$ (O/H)_R23as functions of the emission-line strengths ${\it EW}$(H$\beta$). The general trend is that, from the galaxies with stronger line strengths, the differences in the two $\log$ (O/H) estimates change from -0.5 to 0.2 dex, with most of them ranging from -0.2 to 0.1 dex. KP03 do not present such a direct comparison for the O/H abundances.

The large sample of SDSS star-forming galaxies that we use generally show that the ${\it EW}$ R₂₃ will underestimate the oxygen abundances of the sample galaxies by a factor of 0.041 (the median offset, with a scatter of 0.036 dex) or 0.054 dex (the mean offset, with a scatter of 0.078 dex). Here the $\log$ (O/H) abundances were obtained from the R₂₃ calibration of Tremonti et al. (2004). We also adopted some other R₂₃ calibration formulas, i.e. Kobulnicky et al. (1999, the analysis formulas for the models of McGaugh 1991), Zaritsky et al. (1994), and Kobulnicky & Kewley (2004, the average of McGaugh 1991 and Kewley & Dopita 2002), to check these discrepancies, the results of which are quite similar. Moustakas & Kennicutt (2006) find similar discrepancies to ours by comparing the abundances derived from the extinction-corrected flux- and ${\it EW}$-R₂₃ of 12 nearby spiral galaxies. They find that the integrated abundances determined from the emission-lines systematically underestimated the characteristic abundances. The mean offset is - $0.06\pm0.09$ dex using the McGaugh (1991) calibration, or - $0.11\pm0.13$ dex using the Pilyugin & Thuran (2005) calibration, and the corresponding median offsets are -0.04 and -0.09 dex, respectively.

These results show that the oxygen abundances derived from the ${\it EW}$ R₂₃ and the extinction-corrected flux-R₂₃ are not seriously different for these star-forming galaxies, generally less than 0.1 dex. However, the global discrepancy of them does show a wide distribution, from -0.5 to 0.2 dex, which means that this replacement of ${\it EW}$ R₂₃ to flux R₂₃ for metallicity estimates could cause different effects on the individual galaxies, and should be considered carefully.

5 Modifying the EW R₂₃-method

It is interesting to further check the main factors that affect the discrepancy between the $\log$ (O/H) abundances derived from ${\it EW}$ R₂₃ and flux R₂₃. We may then find some ways to modify the ${\it EW}$ R₂₃ method in order to obtain almost identical oxygen abundances to the flux R₂₃ on the basis of this large set of SDSS galaxies.

5.1 Factors that affect the $\alpha$ parameter

In Sect. 3, we point out that the $\alpha$ parameter is the ratio of the intrinsic continua underlying [O  II]$\lambda$3727 and H$\beta$ and that it is related to the observed continua and the line extinction c (see Eq. (5)). In this section, we discuss the dominant effect on the $\alpha$ parameter.

Equation (5) shows $\alpha$ is a function of the dust extinction and stellar populations of the galaxies, so then we have

$$\alpha {{F_{\rm C,[O~II]}} \over {F_{\rm C,H\beta}}} 10^{c{(f(\rm [O~II])}-f({\rm H}\beta))}$$ =

$${{F^0_{\rm C,[O~II]}} \over {F^0_{\rm C,H\beta}}} 10^{(c-c^*){(f(\rm [O~II])}-f({\rm H}\beta))},$$ = (6)

where $F^0_{\rm C,[O~II]}$ and $F^0_{\rm C,H\beta}$ are the dereddened continua underlying [O  II] and H$\beta$, c the dust attenuation on emission-line, and c^* characterizes the dust attenuation on the continuum. Here we assume c^* and c follow the same reddening law (also see KP03).

As a rough estimate, we assume that the sample galaxies follow the same extinction law as the starburst galaxies studied by Calzetti et al. (1994), who found that the difference between the optical depths of the continua underlying the Balmer lines is about one-half of the difference between the optical depths of the Balmer emission lines (their Eq. (26)). Thus, we assume $c^*\approx 0.5c$, and then obtain the following equation:

$$\alpha = {{F^0_{\rm C,[O~II]}} \over {F^0_{\rm C,H\beta}}} 10^{0.5c{(f(\rm [O~II])}-f({\rm H}\beta))},$$ (7)

where ${{F^0_{\rm C,[O~II]}}/{F^0_{\rm C,H\beta}}}$ characterizes the stellar populations of the galaxies, and c-term characterizes the dust extinction.

It is easy to check the relation between $\alpha$ and the dust extinction c since we have obtained both of them for the individual galaxies. However, we cannot obtain the intrinsic values of ${{F^0_{\rm C,[O~II]}}/{F^0_{\rm C,H\beta}}}$ directly. Fortunately, the MPA/JHU group and the SDSS provide D_n(4000) parameters and several photometric magnitudes for this large sample of galaxies, which can characterize the stellar populations of the galaxies. We check the relations between $\alpha$ and dust extinction c, D_n(4000), g-r colors for the sample galaxies in the next three sections.

5.2 The relation between $\alpha$ and dust extinction

The [O  II]$\lambda$3727 is bluer in wavelength and is affected more strongly by dust extinction than H$\beta$, thus the $\alpha$ parameter may be correlated with dust extinction A_V. Figure 3 shows the $\alpha$ parameter as a function of dust extinction A_V(= cR_V/1.47, Seaton 1979, R_V=3.1). The derived A_V is from 0 to 2.7, with the median and average values of 0.87 and 0.89, respectively. It shows that there is no clear correlation between $\alpha$ and A_V for these SDSS sample galaxies, and the linear least-square fit is $\alpha=0.056A_V+0.821$ with a very slight slope. This means that the differences in the continua underlying [O  II]3727 and H$\beta$ are not affected much by dust extinction. The reasons may be that the related two lines are not very far away at wavelength and that the dust extinction coefficients of these SDSS star-forming galaxies are not so large.

	Figure 3: Relation of $\alpha$ parameter and dust extinction AV for the sample galaxies.
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5.3 The corrected EW R₂₃ method using D_n(4000) index

The break occurring at 4000 Å is the strongest discontinuity in the optical spectrum of a galaxy, and it arises because of the accumulation of a large number of spectral lines in a narrow wavelength region. The main contribution to the opacity comes from ionized metals. In hot stars, the elements are multiply ionized and the opacity decreases, so the 4000 Å break will be small for young stellar populations and large for old, metal-rich galaxies (Kauffmann et al. 2003). Kauffmann et al. (2003) adopt the narrow definition from Balogh et al. (1999) and denote this index as D_n(4000) for the SDSS galaxies. It is the ratio of the average flux density $F_\nu$ in the bands 3850-3950 and 4000-4100 $\AA$, 100 Å narrower than the definition by Bruzual (1983). The parameter D_n(4000) is one of the main ones used by Kauffmann et al. (2003) to trace the stellar formation history of the SDSS galaxies, which shows a monotonic increase after the instantaneous burst of star formation.

Figure 4: a) Relation of the $\alpha$ and Dn(4000) parameters for the sample galaxies. The big squares are the 16 median-value points in the bins of 0.05 in Dn(4000) from 1.0 to 1.8. The solid line represents the third-order polynomial fit for these median-value points, and it is given as Eq. (8). b) Comparison between the metallicities derived from the $\alpha$-modified ${\it EW}$ R23 by using Dn(4000) following Eq. (8) (marked as the subscript "modified''). The diff refers to the difference between the oxygen abundances from the modified ${\it EW}$ R23 and flux R23. c) The direct comparison between the two oxygen abundances derived from the modified ${\it EW}$ R23 and flux R23 (extinction-corrected). In b) and c), the solid lines are the equal-value lines, and the dashed lines show the 0.1 dex discrepancy.

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We plot the relations of $\alpha$ against D_n(4000) for the sample galaxies in Fig. 4a, which clearly shows a correlation. A third-order polynomial fit for this relation is obtained by fitting the 16 median-value points in bins of 0.05 in D_n(4000) from 1 to 1.8, and given as:

 

$$\alpha {\rm 10.88} - {\rm 18.31}~ x + {\rm 11.18}~ x^2 - {\rm 2.34} ~x^3,$$ (8)

with a standard error of 0.164, where x=D_n(4000). This relation of $\alpha$ vs. D_n(4000) could be used to correct the ${\it EW}$ R₂₃ and then to obtain the consistent metallicities of galaxies with the flux R₂₃. We propose using R₂₃(${\it EW}$) $_{\rm corrected}$= $\alpha$$\times$${\it EW}$([O  II])/${\it EW}$(H$\beta$) + ${\it EW}$([O  III])/${\it EW}$(H$\beta$) to then estimate the metallicities of galaxies.

Figure 4b shows the consistency of the metallicity estimates from the corrected ${\it EW}$ R₂₃ and flux R₂₃ after we apply the correction of D_n(4000) for $\alpha$. Figure 4c shows more direct comparisons for the two O/H estimates with such a correction. Both of them show that the two derived abundances are very consistent. The median and mean discrepancies between $\log$ (O/H) $_{{\it EW}~R_{23}}$ and $\log$ (O/H)_R23 now decrease to about -0.005 dex (with a scatter of 0.024 dex), and -0.010 dex (with a scatter of 0.054 dex), respectively. Then the corrected ${\it EW}$ R₂₃ method provides $\log$ (O/H) abundances similar to those of the extinction-corrected flux R₂₃ method within an accuracy of $\pm$0.1 dex for >94% of the galaxies.

Unfortunately this correction is difficult to handle with non-calibrated spectra, simply because reliable measurements for D_n(4000) require flux-calibrated spectroscopy. Since D_n(4000) amplitude depends on the stellar population, age, and metallicity, it also correlates with colors. In the following we aim at generating a correction usable by a large community, using a non-calibrated spectrum and one color as input (e.g. g-r for the SDSS data).

5.4 The corrected EW R₂₃ method using g-r color

Colors can provide important information for the stellar populations of the galaxies and can be obtained directly from the photometric observations. SDSS has made the u, g, r, i, z band magnitudes of the galaxies available publicly. Since the u magnitude has large uncertainty (20 per cent; Kauffmann et al. 2003) and the images in u- and z-band are relatively shallow, whereas i-band image may suffer from the "red halo'' effect (Michard 2002; Wu et al. 2005), we use g-r color here to study such corrections for $\alpha$. The g and r magnitudes can be converted to other band magnitudes, e.g. B, V, following some conversions, for example, Smith et al. (2002) and Jordi et al. (2006).

Figure 5: a) Relation of the $\alpha$ and g-r colors for the sample galaxies. The big squares refer to the 10 median-value points in the bins of 0.1 in g-r from 0.1 to 1.1. The solid line represents the third-order polynomial fit for these median-value points, and it is given by Eq. (9). b) Comparison between the metallicities derived from the $\alpha$-modified ${\it EW}$ R23 by using g-r colors following Eq. (9) (marked as the subscript "modified''). The " diff'' is as a function of g-r colors. c) The direct comparison between the two oxygen abundances from the $\alpha$-modified ${\it EW}$ R23 and the flux R23 (extinction-corrected). The solid lines are the equal-value lines, and the dashed lines show the 0.1 dex discrepancy from the equal-value lines in b) and c)

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Figure 5a shows the correlation between the g-r colors (in Petrosian magnitudes) and $\alpha$ for the sample galaxies. The basic trend shows that the redder galaxies have relatively lower $\alpha$ values, corresponding to the larger differences between the underlying continua of [O  II] and H$\beta$. The third-order polynomial fits for these correlations were obtained and given as

 

$$\alpha = {\rm 1.20} - {\rm 1.11}~ x + {\rm 1.15}~ x^2 - {\rm0.53} ~x^3,$$ (9)

with a standard error of 0.186, where x refers to g-r color. Then we propose using R₂₃(${\it EW}$) $_{\rm corrected}= \alpha\times {\it EW}$([O  II])/${\it EW}$(H$\beta$) + ${\it EW}$([O  III])/${\it EW}$(H$\beta$) to estimate the metallicities of galaxies.

Figure 5b shows the consistency of the metallicity estimates from the corrected ${\it EW}$ R₂₃ and flux R₂₃ after we apply the correction from g-r color for $\alpha$. Figure 5c shows the comparison between the two O/H estimates with this correction more directly. They show that the median and mean discrepancies between log(O/H) $_{{\it EW}~R_{23}}$ and $\log$ (O/H)_R23 now decrease to about -0.004 dex (a scatter of 0.030 dex) and -0.012 dex (a scatter of 0.062 dex), respectively. Then the two oxygen abundance estimates are almost identical to each other. Namely, the corrected ${\it EW}$ R₂₃ method provides $\log$ (O/H) abundances similar to those of the extinction-corrected flux R₂₃ method within an accuracy of $\pm$0.1 dex for >92% of the galaxies.

5.5 Correction for the EW R₂₃ method by a constant  $\alpha =0.85$

The calculated median $\alpha$ value of this large sample of SDSS local star-forming galaxies is about 0.85 (the average value is $\sim$0.86). We may also use this constant $\alpha$-factor to modify the ${\it EW}$ R₂₃. However, this constant correction is only useful for estimating the global metallicity distribution of a large dataset, for example, from the database of a survey. As for the individual galaxy, this may reversely enlarge the uncertainty for some of them, for example, the object with about ${\it EW}$(H $\beta) \sim10$ Å (see Fig. 2a). Indeed, the correction on ${\it EW}$ R₂₃ for the individual galaxies correlates tightly with their stellar populations and star formation histories.

  
6 Discussions about the boundary of the upper branch of 12+log (O/H)

In this work, we adopt 12+$\log$  ${\rm (O/H)}\sim8.5$ as the low boundary of the upper branch of oxygen abundances from the $\log~R_{23}$ calibration. The main reason is that the R₂₃ calibration used (taken from Tremonti et al. 2004) is valid above this metallicity. In Sect. 6.1, we present more observational data with O/H abundances derived from electron temperature $T_{\rm e}$, especially from the recent SDSS, and some photoionization models to further identify the reason we adopt 12+$\log$ (O/H) $\sim$ 8.5 as the boundary of the upper branch of oxygen abundances to compare the ${\it EW}$ R₂₃ and flux R₂₃ methods. However, this boundary value is a bit higher than used by some other researchers, e.g., Pilyugin (2000, 2001a,b) and Pilyugin & Thuan (2005), who use 12+$\log$ (O/H)$\sim$8.2 on the basis of a sample of H  II regions having their $\log$ (O/H) abundances estimated from $T_{\rm e}$. Therefore, we extend the boundary to compare the ${\it EW}$ R₂₃ and flux R₂₃ methods and check whether our relations are valid or not in the region of 12+$\log$ (O/H) $_{T_{\rm e}}\sim 8.2$-8.5, which will be presented in Sect. 6.2.

6.1 The observational data with (O/H) $_{T_{\rm e}}$ and the photoionization models

In Fig. 6a we plot the (O/H) $_{T_{\rm e}}$-R₂₃ relations for the sample galaxies having $T_{\rm e}$-based O/H abundances, which are taken from Kniazev et al. (2004, 624 samples), Izotov et al. (2006, 409 samples), and Yin et al. (2007, 695 samples). Figure 6b presents the predictions of photoionization models for the relations of 12+$\log$ (O/H) vs. $\log~R_{23}$ taken from Kewley & Dopita (2002) and Kobulnicky et al. (1999, K99), which was obtained by analyzing those of McGaugh (1991).

Figure 6: a) The $T_{\rm e}$-based abundances and $\log~R_{23}$ of the galaxies and H  II regions taken from Yin et al. (2007, Yin07), Kniazev et al. (2004, Kniazev04), and Izotov et al. (2006, Izotov06). Associated with the calibration from SDSS-DR2 observations given by Tremonti et al. (2004; the thick solid line, marked by "T04''). b) The predictions of photoionization models for the relations of 12+$\log$ (O/H) vs. $\log(R_{23}$): the two dashed lines marked with y values are taken from Kobulnicky et al. (1999, K99), which are the analytical formulas for McGaugh (1991); the seven thin lines down to 12+$\log$ (O/H)$\sim$7.6 are taken from Kewley & Dopita (2002) with different ionization parameter q values. The thick solid line is the same as a). Both a) and b) show that it is very difficult to derive the reliable O/H abundance from the R23 parameter for the cases of 12+$\log$ (O/H)<8.5. (See the online color versions of the plots for the discrepancies of the data and models.)

Open with DEXTER

Figures 6a and 6b show that both the large sample of observational data and photoionization models confirm that it is very difficult to derive a reliable O/H abundance from the R₂₃ parameter for the region of 12+$\log$  ${\rm (O/H)}<8.5$ because of the large scatter of the data points there, the weak dependence of O/H on R₂₃ (down to 12+$\log$ (O/H) < 7.9), and the strong effects of ionization parameters. Yin et al. (2007) has used their Fig. 3b to show the large discrepancy, up to 0.4 dex, of the two sets of log(O/H) estimates derived from the upper branch and lower branch of R₂₃ calibrations for the samples within 7.9<12+log ${\rm (O/H)}<8.4$. Stasinska (2002) discusses the weak dependence of (O/H) on R₂₃ in the transition region from nebular physics.

Considering the discussions above, and also to avoid that some galaxies included may lie in the lower branch of the R₂₃-(O/H) relation, we select the galaxies having 12+$\log$  ${\rm (O/H)}>8.5$ to check the ${\it EW} R_{23}$ method in this study. However, it could be useful to check this and the validity of our relations in an extended upper branch range, 12+$\log$  ${\rm (O/H)}\sim8.2$-8.5, as used in some studies.

6.2 Checking the extended upper-branch range of 12 + log (O/H) $\sim$ 8.2-8.5

We select a subsample from Fig. 6a to check the situation in the extended range of upper branch, $12+\log$ (O/H)$\sim8.2$-8.5. To be consistent with other parts of this work, we select this subsample based on the SDSS-DR2 catalog. Finally, 37 independent objects with metallicities of $8.2<12+\log$ (O/H) $_{T_{\rm e}}<8.5$ are selected by cross-correlating the DR2 catalog with the lists of Kniazev et al. (2004, from DR1), Izotov et al. (2006, from DR3), and Yin et al. (2007, from DR4). Then we use the ${\it EW}$ and flux measurements of the related emission lines provided by the MPA/JHU group to estimate their ${\it EW}$ R₂₃ and flux R₂₃, hence, the resulted abundances.

These 37 galaxies show a discrepancy between $\log({\it EW} R_{23}$) and $\log$ (R₂₃) (as Fig. 2b) within a range of -0.2 to 0, which may result in overestimated $\log$ (O/H) abundances of about 0-0.2 dex by the ${\it EW} R_{23}$ as shown in Fig. 2c. Their D_n(4000) values are around 1.0, with g-r colors around 0.0 (ranging from -0.2 to 0.3), which confirms that they are low-metallicity objects that will distribute in the left hand parts in Figs. 4a and 5a. Their $\alpha$ values are within 1-2, with the average value about 1.5. If we apply the $\alpha$ corrections for their ${\it EW}$ R₂₃, the correction factors will be about $\alpha \sim 1.4$ and 1.2 by extrapolating Eqs. (8) and (9), respectively.

If we do not consider the discrepancy among the different photoionization parameters and the weak dependence of O/H on R₂₃ in this metallicity region (shown as Fig. 6b) and try to extrapolate the R₂₃ calibration of Tremonti et al. (2004) down to these objects with 8.2<12+$\log$ (O/H) $_{T_{\rm e}}<8.5$, then the open squares in Fig. 7a show that the R₂₃ method will often overestimate the abundances of the galaxies and even underestimate the abundances of some galaxies. The stars in Fig. 7a show that the ${\it EW}$ R₂₃ method provides a higher log(O/H) abundance than the flux R₂₃, generally about 0.2 dex, which is consistent with the discussions above and with Fig. 2c. Figure 7a also shows that the correction by $\alpha$(g-r) will not improve the consistency for these objects much, and the correction by $\alpha$(D_n(4000)) gives more consistent O/H abundances for the objects with 12+$\log$ (O/H) _R23>8.2. We changed to the R₂₃ calibration of Zaritsky et al. (1994) (which is an average of three previous calibrations) to obtain these comparisons again, and find the results are very similar to Fig. 7a.

Figure 7: Comparisons between the $\log$ (O/H) abundances obtained from flux R23 and those from $T_{\rm e}$ or from the corrected or uncorrected ${\it EW}$ R23 method: a) using the R23 calibration of Tremonti et al. (2004, T04); b) using the R23 calibration of Pilyugin (2000, P00). The different symbols represent those what obtained from the different methods marked by the labels in the top left corner.

Open with DEXTER

We also used the R₂₃ calibration of Pilyugin (2000) (their Eq. (5)) to do these comparisons, and present the results in Fig. 7b. It shows that mostly the flux R₂₃ method provides consistent $\log$ (O/H) abundances with the $T_{\rm e}$ method, except for some data points in the left hand part of the data with underestimated $\log$ (O/H)_R23 and some reversed ones in the right hand section. It also shows that the corrected ${\it EW}$ R₂₃ method by $\alpha$(D_n(4000)) provides more consistent abundances with the flux R₂₃ for the objects with 12+$\log$ (O/H) _R23>8.1 than the uncorrected ${\it EW}$ R₂₃ or corrected by $\alpha$(g-r), which are similar to Fig. 7a. These differences among the ${\it EW}$ R₂₃ abundances presented in Fig. 7 are not difficult to understand since the average $\alpha$ value of these low-metallicity objects is about 1.5, which is higher than the correction relations provided, $\sim$1.4 or 1.2.

In short, in the lower metallicity range of 8.2 < 12+$\log$ (O/H) $_{T_{\rm e}}$ < 8.5, we would not recommend using our relations to correct the ${\it EW}$ R₂₃ method for the oxygen abundance calibrations, since there are several situations affecting the results, such as the big scatter of the data, the strong effects of ionization parameters, and the weak dependence of O/H on R₂₃ in the range.

  
7 Summary and conclusion

The goal of this paper is to check the reliability of using ${\it EW}$ R₂₃ (= ${\it EW}({\rm [O~II]})+{\it EW}({\rm [O~III])}\over {{\it EW}({\rm H\beta)}}$) to replace the extinction-corrected flux R₂₃(= ${{I({\rm [O~II]})+I({\rm [O~III]})}\over {I({\rm H\beta})}}$) to estimate the metallicities of star-forming galaxies on the basis of a large sample (37 173) of galaxies with 12+$\log$ (O/H)_R23>8.5, selected from the SDSS-DR2. This replacement is often adopted when there are some problems dealing with proper flux calibrations for the spectral observations. This large sample can provide some obvious statistical trends.

The results show that the logarithm values of ${\it EW}$ R₂₃ and extinction-corrected flux-R₂₃have a discrepancy from -0.4 to 0.5 dex, with a median value of about +0.061 dex and a mean value about +0.069 dex. Thus, the discrepancies between the log(O/H) abundances obtained from ${\it EW}$ R₂₃ and those from the flux-R₂₃ range from -0.5 to 0.2 dex and have a median value of about -0.041 dex, a mean value of about -0.054 dex. These discrepancies are caused by the different continua ( $F_{C\lambda}$) underlying the emission lines [O  II] and H$\beta$ ([O  III], as well). These differences are characterized by the $\alpha$ parameter as ( $F_{C,{\rm H\beta}}$)/( $F_{C,[{\rm O~II}]}$), which changes from 0.1 to 2.6, and by a median value of 0.85 and a mean value of 0.86.

Then we discuss the factor that affects this discrepancy mostly. Our large sample of data shows that the $\alpha$ parameter is almost independent of the dust extinction inside the galaxies and depends closely on stellar populations of the galaxies, which can be quantified by the D_n(4000) parameters and colors of the galaxies. Third-order polynomial fits have been obtained for the observed relations of $\alpha$ versus D_n(4000) and $\alpha$ versus g-r colors for the sample galaxies, which can be used to modify the ${\it EW}$ R₂₃ method. After applying this correction by D_n(4000), the median and mean discrepancies between log(O/H) $_{{\it EW}~R_{23}}$ and $\log$ (O/H)_R23 decrease to about -0.005 dex and -0.010 dex, respectively. After applying this correction by g-r colors, the median and mean discrepancies between $\log{\rm (O/H)}_{EW~R_{23}}$ and $\log{\rm (O/H)}_{R_{23}}$ decrease to about -0.004 dex and -0.012 dex, respectively. The two derived sets of $\log{\rm (O/H)}$ abundances are almost identical now.

In summary, when there are problems with flux calibrations of the spectra, the ${\it EW}$ R₂₃ could be used roughly to replace the extinction-corrected flux R₂₃ to estimate the metallicities of star-forming galaxies. The discrepancy caused by this replacement can be from -0.2 to 0.1 dex generally. This factor is consistent with those found by KP03 and Moustakas & Kennicutt (2006). However, this discrepancy could be large for the different individual galaxies, from -0.5 to 0.2, if the underlying continua of [O  II] and H$\beta$ ([O  III]) are quite different. The D_n(4000) parameters and colors of the galaxies are very useful for correcting the ${\it EW}$ R₂₃ method, which will then greatly decrease the discrepancies and result in an almost identical oxygen abundance to the flux-R₂₃.

Nevertheless, the modified ${\it EW}$ R₂₃still suffers from the drawback of the "double-value'' of R₂₃for the O/H estimates, and some other strong-line ratios are also useful for estimating the metallicities of galaxies then, such as [N  II]/H$\alpha$, ([O  III]/H$\beta$)/([N  II]/H$\alpha$), [N  II]/[S  II] and [O  III]/H$\beta$, etc. (Kewley & Dopita 2002; Pettini & Pagel 2004; Liang et al. 2006; Yin et al. 2007).

Acknowledgements

We thank our referee for the valuable comments and suggestions, which helped to improve this work. We thank Rob Kennicutt, Lisa Kewley, Hong Wu, Licai Deng, Bo Zhang, and Hector Flores for valuable discussions related to this study, and thank Ruixiang Chang, Shiying Shen, Caina Hao, Jing Wang, and Chen Cao for helpful discussions about the SDSS database. We thank James Wicker for his help in improving the English expression in the text. This work was supported by the Natural Science Foundation of China (NSFC) Foundation under No.10403006, 10433010, 10673002, 10573022, 10333060, and 10521001; and the National Basic Research Program of China (973 Program) No. 2007CB815404.

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