"Direct" Gas-phase Metallicity in Local Analogs of High-redshift Galaxies: Empirical Metallicity Calibrations for High-redshift Star-forming Galaxies

We select local reference galaxies located in the ±0.05 dex region of the local star-forming sequence defined by Equation (3) in Kewley et al. (2013a) on the BPT diagram as follows (red small data points in Figure 1(a)):

$$\begin{eqnarray}&&\mathrm{log}([{\rm{O}}\,{\rm{III}}]/{\rm{H}}\beta )\gt \displaystyle \frac{0.61}{\mathrm{log}([{\rm{N}}\,{\rm{II}}]/{\rm{H}}\alpha )-0.08}+1.05,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\mathrm{log}([{\rm{O}}\,{\rm{III}}]/{\rm{H}}\beta )\lt \displaystyle \frac{0.61}{\mathrm{log}([{\rm{N}}\,{\rm{II}}]/{\rm{H}}\alpha )-0.08}+1.15,\end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray}&&\mathrm{log}([{\rm{N}}\,{\rm{II}}]/{\rm{H}}\alpha )\lt -0.5.\end{eqnarray} \tag{ 3 }$$

The following [S ii]$\lambda \lambda 6717,6731$ and [O i]λ6300 diagnostics are applied to further remove galaxies with AGN/shock contamination to the emission-line flux (Kewley et al. 2006)

$$\begin{eqnarray}&&\mathrm{log}([{\rm{O}}\,{\rm{III}}]/{\rm{H}}\beta )\lt \displaystyle \frac{0.72}{\mathrm{log}([{\rm{S}}\,{\rm{II}}]/{\rm{H}}\alpha )-0.32}+1.30,\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\mathrm{log}([{\rm{O}}\,{\rm{III}}]/{\rm{H}}\beta )\lt \displaystyle \frac{0.73}{\mathrm{log}([{\rm{O}}\,{\rm{I}}]/{\rm{H}}\alpha )-0.59}+1.33.\end{eqnarray} \tag{ 5 }$$

A total of 22428 unique SDSS galaxies are selected. In general, these galaxies represent low-redshift star-forming galaxies, which are located on the local BPT star-forming sequence.

The sample of local analogs of high-redshift galaxies has been selected from the ±0.04 dex region of the z ∼ 2.3 star-forming sequence defined by Equation (9) in Steidel et al. (2014) on the BPT diagram (blue small data points in Figure 1(a)):

$$\begin{eqnarray}&&\mathrm{log}([{\rm{O}}\,{\rm{III}}]/{\rm{H}}\beta )\gt \displaystyle \frac{0.67}{\mathrm{log}([{\rm{N}}\,{\rm{II}}]/{\rm{H}}\alpha ))-0.33}+1.09,\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\mathrm{log}([{\rm{O}}\,{\rm{III}}]/{\rm{H}}\beta )\lt \displaystyle \frac{0.67}{\mathrm{log}([{\rm{N}}\,{\rm{II}}]/{\rm{H}}\alpha ))-0.33}+1.17,\end{eqnarray} \tag{ 7 }$$

and

$$\begin{eqnarray}&&\mathrm{log}([{\rm{N}}\,{\rm{II}}]/{\rm{H}}\alpha )\lt -0.5.\end{eqnarray} \tag{ 8 }$$

All the galaxies selected by Equations of (6)–(8) are classified as galaxies by the Kewley et al. (2001) criterion (Figure 1(a)), and all the galaxies with log([N ii]λ6584/Hα$)\lt -0.75$ are classified as galaxies by the Kauffmann et al. (2003a) criterion. We further apply the [S ii] and [O i] BPT diagnostic to reduce the AGN/shock contaminants. Studies have shown that z ∼ 2 star-forming galaxies exhibit a higher [O iii]/Hβ ratio for a given [S ii]/Hα ratio (e.g., Strom et al. 2017b). Thus, we shift the Kewley et al. (2006) criteria on the [O iii]/Hβ ratio by +0.05 dex for high-redshift galaxies.

$$\begin{eqnarray}&&\mathrm{log}([{\rm{O}}\,{\rm{III}}]/{\rm{H}}\beta )\lt \displaystyle \frac{0.72}{\mathrm{log}([{\rm{S}}\,{\rm{II}}]/{\rm{H}}\alpha )-0.32}+1.35,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\mathrm{log}([{\rm{O}}\,{\rm{III}}]/{\rm{H}}\beta )\lt \displaystyle \frac{0.73}{\mathrm{log}([{\rm{O}}\,{\rm{I}}]/{\rm{H}}\alpha )-0.59}+1.38.\end{eqnarray} \tag{ 10 }$$

A total of 443 galaxies are selected based on the above selection criteria. Equations (9) and 10 remove about 20%, and these two criteria do not affect the distribution of the local analogs on the optical diagnostics diagrams. We cross-correlate the positions of the analogs of high-redshift galaxies with the ROSAT ALL-Sky Survey Faint Source Catalog (Voges et al. 2000) and find that none of the analogs are detected at X-ray wavelengths. We further check the local analogs with high-redshift galaxies with high full width half maximum of the Balmer emission lines to remove potential AGNs hosting low-mass supermassive black holes (${10}^{6}\,{M}_{\odot }$ with $\mathrm{FWHM}\,\gt 600$ km s⁻¹, Greene & Ho 2004). We find only one galaxy with $\mathrm{FWHM}\gt 400$ km s⁻¹ among our local analogs, and the galaxy is removed from our further analysis. We also visually inspect the image of the local analogs and their fiber positions to remove spurious galaxies being targeted due to poor photometric deblending. The properties of these local analogs can be summarized as follows. The median stellar mass is $\mathrm{log}({M}_{* }/{M}_{\odot })={8.8}_{-0.02}^{+0.06}$. The median SFR and sSFR are ${3.9}_{-0.2}^{+0.7}$ M_☉ yr⁻¹ and ${10.0}_{-0.5}^{+1.0}$ Gyr⁻¹, respectively. The sSFR of the local analogs is comparable to that in z ∼ 2 star-forming galaxies with similar stellar mass (e.g., Rodighiero et al. 2011). Furthermore, these analogs closely resemble the ISM conditions in high-redshift galaxies, including high ionization parameters ($\mathrm{log}q\simeq 7.9$ cm⁻¹) and high electron densities (${n}_{e}\simeq 120$ cm⁻³). The median ionization parameters and electron densities in these analogs are also comparable to those in the z ∼ 2–3 galaxies (e.g., Nakajima & Ouchi 2014; Sanders et al. 2016b). We refer readers to Bian et al. (2016, 2017) for more details.

Figure 1(a) shows the selection of the analogs of high-redshift galaxies and normal SDSS galaxies in the reference sample in the [O iii]λ5007/Hβ versus [N ii]λ6584/Hα BPT diagram. In addition, we compare the location of our local analogs and normal SDSS galaxies in other optical diagnostic diagrams in Figures 1(b)–(e). We find that our analogs share the same regions with z ∼ 2 star-forming galaxies in all these diagnostic diagrams (Steidel et al. 2014; Shapley et al. 2015; Sanders et al. 2016b). Compared to normal SDSS galaxies, our analogs do not show offsets in the [O iii]λ5007/Hβ versus [S ii]λλ6717,6731/Hα (S2-) BPT diagram (Figure 1(b)) and O32 versus R23 diagram (Figure 1(d)), but do show shifts in the O32 versus N2 and O32 versus O3N2 diagrams (Figures 1(e) and (f)). We also find no significant offset between our analogs and local SDSS galaxies in the [O iii]λ5007/Hβ versus [O i]λ6300/Hα (O1-) BPT diagram (Figure 1(c)). The O1-BPT diagram has not been studied in z ∼ 2 galaxies, and our result can be tested in the future studies of z ∼ 2 galaxies.

We cannot detect the weak [O iii]λ4363 emission line in most of our individual spectra. To increase the signal-to-noise ratio, we combine the spectra and generate stacked spectra for both our local analogs and selected SDSS galaxies. We adopt the procedure that has been used in Andrews & Martini (2013) and Brown et al. (2016), except we carry out dust extinction correction before stacking the spectra. The procedure is as follows:

• 1.  
  We obtain the reduced one-dimensional SDSS spectra from the SDSS DR9 data release. The spectra were taken using the multi-object, fiber-fed spectrographs (Smee et al. 2013) mounted on the 2.5 m telescope of SDSS (Gunn et al. 2006). The spectra have been reduced by the SDSS spectro2d pipeline (Stoughton et al. 2002).
• 2.  
  Each of the SDSS galaxy spectra is corrected for dust extinction using the Balmer decrement (Hα/Hβ) assuming Case B recombination and the Cardelli et al. (1989) dust extinction law.
• 3.  
  Each SDSS spectrum is shifted to the rest frame based on the redshift from the MPA/JHU catalog.
• 4.  
  Each of the SDSS spectra is resampled onto a wavelength grid in the 3700–7300 Å range with ${\rm{\Delta }}\lambda =1$Å while maintaining flux conservation.
• 5.  
  We normalize the spectra with the mean flux density in the 4400–4450 Å wavelength range.
• 6.  
  We divide the local analogs and the selected SDSS galaxies separately into 0.25 dex bins in [N ii]λ6584/Hα from log([N ii]λ6584/Hα) = x to $x+0.25$, where x = [−2.25, −2.00, −1.75, −1.50, −1.25, −1.00, −0.75]. We use ${\rm{N}}{2}_{x}^{x+0.25}$ to represent the log([N ii]λ6584/Hα) between x and $x+0.25$ for the remainder of the paper.
• 7.  
  We stack the spectra falling into each of the [N ii]λ6584/Hα bins using the mean flux density at each wavelength.

Following the above procedure, we generate a total of 14 stacked spectra: 7 for the local analogs of high-redshift galaxies and 7 for the SDSS reference sample.

The underlying stellar absorption features in the stacked spectra affect the emission-line flux measurements. In particular, the Hγ stellar absorption feature is close to the [O iii]λ4363 emission line. Therefore, to measure the line fluxes accurately, it is important to subtract the stellar continuum from the stacked spectra. We fit the stellar continuum of each stacked spectrum with a set of stellar synthesis models using the STARLIGHT stellar population synthesis code (Cid Fernandes et al. 2005). We mask out the emission-line region before fitting the spectra. The Bruzual & Charlot (2003) stellar synthesis models with a Chabrier initial mass function (Chabrier 2003) are adopted to fit the stacked spectra. These stellar synthesis models cover a broad range of metallicity ($0.05\mbox{--}2.5{Z}_{\odot }$) and stellar age ($1\,\mathrm{Myr}\mbox{--}13$ Gyr). The upper panel of Figure 2 shows an example of the observed stacked spectrum (${F}_{\mathrm{obs}}$) and the model spectrum (${F}_{\mathrm{model}}$) from the best-fit stellar synthesis model at the wavelength close to the Hγ and [O iii $]\lambda 4363$ lines. The model spectrum follows the continuum of the observed spectrum and captures most of the spectral absorption features, suggesting our stellar synthesis model fitting process is robust.

Zoom In Zoom Out Reset image size

We measure the following emission-line fluxes in the stellar continuum subtracted stacked spectra: [O ii]λλ3726,3729, Hγ, [O iii]λ4363, Hβ, Hα, [N ii]λλ6548,6584, and [S ii]λλ6717,6731. We use the IDL MPFIT package to fit the emission lines by assuming that the emission-line profile follows the Gaussian distribution. A single Gaussian function is used to fit the emission lines that are well separated ($\gt 30$ Å) from other emission lines, including the Hβ, [O iii]λ4959, [O iii]λ5007. Two Gaussians are fit simultaneously to those pairs of lines closer than 30 Å, including the Hγ and [O iii]λ4363 pair, the [O ii]λλ3726,3729 doublet, and the [S ii]λλ6717,6731 doublet. As for the Hα and [N ii]λλ6548,6584 complex, three Gaussians are used to fit simultaneously. In the fitting process, the central wavelength is fixed to the vacuum wavelength of the emission lines. We adopt the standard deviation of the continuum region close to ($\lt 100$ Å) the emission-line-fitting regions as the 1σ noise of the spectra. We carry out Monte Carlo simulations to estimate the uncertainties of line fluxes. A thousand simulated spectra are generated. The flux densities of these simulated spectra at each wavelength follow a Gaussian distribution whose center is at the actual flux density, and the standard derivation is consistent with 1σ error at that wavelength. The bottom panel of Figure 2 shows the model-subtracted spectrum (${F}_{\mathrm{obs}}-{F}_{\mathrm{model}}$) and best Gaussian fit for the Hγ and [O iii]λ4363 line, which demonstrates that our line-fitting is robust. Tables 1 and 2 summarize the emission-line fluxes in each stacked spectrum.

Table 1.  The Relative Line Flux from the Stacked Spectra of the SDSS Reference Galaxies

line	${\rm{N}}{2}_{-0.75}^{-0.50}$	${\rm{N}}{2}_{-1.00}^{-0.75}$	${\rm{N}}{2}_{-1.25}^{-1.00}$	${\rm{N}}{2}_{-1.50}^{-1.25}$	${\rm{N}}{2}_{-1.75}^{-1.50}$	${\rm{N}}{2}_{-2.00}^{-1.75}$	${\rm{N}}{2}_{-2.25}^{-2.00}$
[O ii]$\lambda 3727$	3.301 ± 0.038	3.513 ± 0.023	3.088 ± 0.014	2.523 ± 0.011	1.750 ± 0.009	1.071 ± 0.007	0.566 ± 0.011
[Ne iii]$\lambda 3869$	0.125 ± 0.008	0.198 ± 0.007	0.287 ± 0.007	0.368 ± 0.009	0.427 ± 0.008	0.451 ± 0.007	0.512 ± 0.010
[O iii]$\lambda 4363$	0.006 ± 0.004	0.015 ± 0.002	0.027 ± 0.001	0.045 ± 0.001	0.077 ± 0.001	0.116 ± 0.001	0.156 ± 0.002
Hβ	1.000 ± 0.073	1.000 ± 0.073	1.000 ± 0.080	1.000 ± 0.102	1.000 ± 0.235	1.000 ± 0.383	1.000 ± 0.865
[O iii]$\lambda 4959$	0.290 ± 0.003	0.679 ± 0.003	1.069 ± 0.003	1.407 ± 0.003	1.672 ± 0.006	1.824 ± 0.008	2.198 ± 0.011
[O iii]$\lambda 5007$	0.878 ± 0.007	2.038 ± 0.009	3.199 ± 0.009	4.190 ± 0.010	5.022 ± 0.017	5.453 ± 0.022	6.660 ± 0.033
[O i]$\lambda 6100$	0.107 ± 0.002	0.095 ± 0.001	0.076 ± 0.000	0.060 ± 0.000	0.042 ± 0.001	0.026 ± 0.001	0.014 ± 0.001
[N ii]$\lambda 6548$	0.233 ± 0.002	0.130 ± 0.001	0.073 ± 0.000	0.043 ± 0.001	0.023 ± 0.001	0.014 ± 0.001	0.010 ± 0.001
Hα	2.860 ± 0.016	2.860 ± 0.009	2.860 ± 0.006	2.860 ± 0.005	2.860 ± 0.007	2.860 ± 0.009	2.860 ± 0.010
[N ii]$\lambda 6584$	0.720 ± 0.004	0.402 ± 0.001	0.228 ± 0.001	0.134 ± 0.001	0.074 ± 0.001	0.044 ± 0.001	0.026 ± 0.001
[S ii]$\lambda 6717$	0.529 ± 0.001	0.421 ± 0.000	0.311 ± 0.000	0.233 ± 0.000	0.155 ± 0.001	0.094 ± 0.001	0.051 ± 0.001
[S ii]$\lambda 6731$	0.372 ± 0.001	0.299 ± 0.000	0.222 ± 0.000	0.166 ± 0.000	0.111 ± 0.001	0.069 ± 0.001	0.040 ± 0.001
used	n	n	y	y	y	y	n
Te Metallicity	⋯	⋯	8.26 ± 0.02	8.18 ± 0.009	8.00 ± 0.008	7.80 ± 0.006	⋯

Download table as:  ASCIITypeset image

Table 2.  The Relative Line Flux in the Spectra of the Local Analogs of High-redshift Galaxies

line	${{\rm{N}}}_{-0.75}^{-0.50}$	${{\rm{N}}}_{-1.00}^{-0.75}$	${{\rm{N}}}_{-1.25}^{-1.00}$	${{\rm{N}}}_{-1.50}^{-1.25}$	${{\rm{N}}}_{-1.75}^{-1.50}$	${{\rm{N}}}_{-2.00}^{-1.75}$	${{\rm{N}}}_{-2.25}^{-2.00}$
[O ii]$\lambda 3727$	2.757 ± 0.030	2.404 ± 0.023	2.071 ± 0.011	1.881 ± 0.008	1.264 ± 0.005	0.819 ± 0.012	0.532 ± 0.007
[Ne iii]$\lambda 3869$	0.245 ± 0.010	0.313 ± 0.008	0.400 ± 0.007	0.464 ± 0.008	0.514 ± 0.007	0.514 ± 0.007	0.537 ± 0.010
[O iii]$\lambda 4363$	0.017 ± 0.003	0.024 ± 0.001	0.043 ± 0.001	0.063 ± 0.001	0.095 ± 0.001	0.132 ± 0.001	0.167 ± 0.002
Hβ	1.000 ± 0.007	1.000 ± 0.007	1.000 ± 0.004	1.000 ± 0.003	1.000 ± 0.003	1.000 ± 0.005	1.000 ± 0.007
[O iii]$\lambda 4959$	0.899 ± 0.007	1.244 ± 0.009	1.563 ± 0.006	1.797 ± 0.005	2.059 ± 0.006	2.251 ± 0.011	2.348 ± 0.016
[O iii]$\lambda 5007$	2.761 ± 0.019	3.761 ± 0.025	4.713 ± 0.018	5.427 ± 0.015	6.196 ± 0.016	6.705 ± 0.031	6.912 ± 0.047
[O i]$\lambda 6100$	0.082 ± 0.001	0.068 ± 0.002	0.051 ± 0.001	0.045 ± 0.001	0.030 ± 0.000	0.024 ± 0.001	0.014 ± 0.001
[N ii]$\lambda 6548$	0.208 ± 0.002	0.109 ± 0.001	0.063 ± 0.001	0.034 ± 0.001	0.022 ± 0.001	0.014 ± 0.001	0.010 ± 0.001
Hα	2.860 ± 0.015	2.860 ± 0.014	2.860 ± 0.008	2.860 ± 0.006	2.860 ± 0.006	2.860 ± 0.010	2.860 ± 0.014
[N ii]$\lambda 6584$	0.626 ± 0.004	0.354 ± 0.002	0.201 ± 0.001	0.118 ± 0.001	0.066 ± 0.001	0.045 ± 0.001	0.025 ± 0.001
[S ii]$\lambda 6717$	0.269 ± 0.002	0.220 ± 0.002	0.174 ± 0.001	0.153 ± 0.001	0.104 ± 0.001	0.076 ± 0.001	0.050 ± 0.001
[S ii]$\lambda 6731$	0.223 ± 0.002	0.172 ± 0.002	0.132 ± 0.001	0.116 ± 0.001	0.080 ± 0.001	0.057 ± 0.001	0.037 ± 0.001
used	n	y	y	y	y	y	y
Te Metallicity	⋯	8.37 ± 0.02	8.25 ± 0.01	8.16 ± 0.009	8.03 ± 0.006	7.91 ± 0.006	7.81 ± 0.008

Download table as:  ASCIITypeset image

Figure 1(a) shows the location of the stacked spectrum on the [O iii]λ5007/Hβ versus the [N ii]λ6584/Hα BPT diagram. We find that all the stacked spectra closely trace the local star-forming sequence and high-redshift star-forming sequence on the BPT diagram except the stacked spectrum of normal SDSS galaxies in the ${\rm{N}}{2}_{-2.00}^{-2.25}$ bin. However, only two galaxies in the SDSS reference galaxy sample fall into the ${\rm{N}}{2}_{-2.00}^{-2.25}$ bin, and both of them are located above the local star-forming sequence, biasing the stack spectrum toward the high-redshift star-forming sequence. Therefore, this data point is discarded for further analysis.

To achieve reliable direct oxygen abundance measurements, we only use the stacked spectra with S/Ns of [O iii]λ4363 greater than ten (${\rm{S}}/{\rm{N}}\gt 10$) for further analysis.

We adopt the Izotov et al. (2006) recipe to derive the direct T_e oxygen abundance. This method builds up a relation between the [O iii]λλ4959,5007/[O iii]λ4363 ratio and the electron temperature in the ${{\rm{O}}}^{++}$ zone (T_e(O iii)). This relation also weakly depends on the electron density, which can be estimated from the [S ii]λλ6717,6731 doublet ratio. We do not have directly measured electron temperatures in the O⁺ zone (T_e(O ii)). We adopt a relation between T_e(O iii) and T_e(O ii) from Garnett (1992) and Campbell et al. (1986) as follow: T_e(O ii) = 0.7T_e(O iii)+3000K. We estimate the ${{\rm{O}}}^{++}$ abundance using the T_e(O iii) temperature and [O iii]λλ4959,5007/Hβ ratio and O⁺ abundance using the T_e(O ii) temperature, [O ii]λλ3726,3729/Hβ and electron density. We compute the oxygen abundance by summing the ${{\rm{O}}}^{++}$ abundance and the ${{\rm{O}}}^{+}$ abundance. We refer the reader to Section 3.1 in Izotov et al. (2006) for more details.

We study the ISM conditions, including the ionization parameter and electron density in the SDSS reference galaxies and local analogs of high-redshift galaxies. We estimate the ionization parameter in stacked spectrum in each N2 bin by adopting the recipe that first introduced in Kobulnicky & Kewley (2004; see also Kewley & Ellison 2008). This method is based on the Kewley & Dopita (2002) photoionization models. The ionization parameter is calculated by primarily using the O32. For a fixed O32, the oxygen abundance has a secondary effect on the ionization parameter measurement. So we further correct for the ionization parameter based on the oxygen abundance from the R23 and N2O2. We refer readers to Section A2.3 in Kewley & Ellison (2008) for more details on the ionization parameter measurements. We estimate the electron density in each of the stacked spectra based on the [S ii]λ6717/[S ii]λ6731 ratio by adopting the nebular.temden routine in IRAF(Shaw & Dufour 1995). Tables 3 and 4 summarize the results of the ionization parameters and electron densities in the SDSS reference galaxies and local analogs. We also list the median stellar mass⁴ and SFR for each N2 bin in Tables 3 and 4. The stellar masses of the local analogs and SDSS reference galaxies are consistent with each other in each of the N2 bins. The local analogs have higher ionization parameters (∼0.3 dex), higher SFRs, and higher electron densities (a factor of ∼5) than SDSS reference galaxies in all N2 bins except the ${{\rm{N}}}_{-2.00}^{-2.25}$ bin,⁵ suggesting that the ISM conditions change significantly between the SDSS reference galaxies and local analogs.

Table 3.  The Properties of the SDSS Reference Galaxies

Bin	#a	SFRb	Stellar Massc	$\mathrm{log}q$ d	nee	Te Metallicity
		${M}_{\odot }$ yr−1	$\mathrm{log}({M}_{* }/{M}_{\odot })$	cm s−1	cm−3	$12+\mathrm{log}({\rm{O}}/{\rm{H}})$
${\rm{N}}{2}_{-0.75}^{-0.50}$	15927	${1.1}_{-0.7}^{+1.7}$	${9.6}_{-0.3}^{+0.3}$	7.39 ± 0.02	${5}_{-2}^{+9}$	⋯
${\rm{N}}{2}_{-1.00}^{-0.75}$	4491	${0.9}_{-0.6}^{+2.2}$	${9.2}_{-0.3}^{+0.3}$	7.57 ± 0.03	${16}_{-14}^{+18}$	⋯
${\rm{N}}{2}_{-1.25}^{-1.00}$	1595	${0.7}_{-0.5}^{+1.9}$	${8.9}_{-0.5}^{+0.4}$	7.72 ± 0.04	${22}_{-19}^{+25}$	8.26 ± 0.02
${\rm{N}}{2}_{-1.50}^{-1.25}$	333	${0.5}_{-0.3}^{+1.2}$	${8.3}_{-0.5}^{+0.5}$	7.74 ± 0.03	${21}_{-17}^{+25}$	8.18 ± 0.009
${\rm{N}}{2}_{-1.75}^{-1.50}$	63	${0.5}_{-0.3}^{+1.6}$	${7.8}_{-0.4}^{+0.5}$	7.92 ± 0.07	${34}_{-25}^{+42}$	8.00 ± 0.008
${\rm{N}}{2}_{-2.00}^{-1.75}$	17	${0.5}_{-0.2}^{+0.9}$	${7.7}_{-0.5}^{+0.2}$	8.10 ± 0.11	${51}_{-31}^{+72}$	7.80 ± 0.006
${\rm{N}}{2}_{-2.25}^{-2.00}$	2	${0.6}_{-0.2}^{+1.5}$	${7.3}_{-0.4}^{+0.4}$	8.44 ± 0.15	${167}_{-132}^{+204}$	⋯

Notes.

^aNumber of galaxies for a given N2 bin. ^bThe uncertainty in the SFR represents the 16th and 84th percentiles of the SFR distribution in a given N2 bin. ^cThe uncertainty in the stellar mass represents the 16th and 84th percentiles of the stellar mass distribution in a given N2 bin. ^dIonization parameter derived from the stacked spectrum. ^eElectron density derived from the stacked spectrum.

Download table as:  ASCIITypeset image

Table 4.  The Properties of the Local Analogs of High-redshift Galaxies

Bin	#a	SFRb	Stellar Massc	$\mathrm{log}q$ d	nee
		${M}_{\odot }$ yr−1	$\mathrm{log}({M}_{* }/{M}_{\odot })$	cm s−1	cm−3	$12+\mathrm{log}({\rm{O}}/{\rm{H}})$
${\rm{N}}{2}_{-0.75}^{-0.50}$	45	${3.4}_{-2.9}^{+12.3}$	${9.6}_{-0.7}^{+0.3}$	7.77 ± 0.09	${255}_{-235}^{+277}$	−
${\rm{N}}{2}_{-1.00}^{-0.75}$	69	${13.5}_{-11.2}^{+20.3}$	${9.2}_{-0.3}^{+0.3}$	7.88 ± 0.18	${159}_{-141}^{+177}$	8.37 ± 0.02
${\rm{N}}{2}_{-1.25}^{-1.00}$	110	${8.0}_{-6.5}^{+20.3}$	${8.9}_{-0.6}^{+0.4}$	7.97 ± 0.16	${110}_{-102}^{+119}$	8.25 ± 0.01
${\rm{N}}{2}_{-1.50}^{-1.25}$	99	${2.6}_{-2.0}^{+6.6}$	${8.4}_{-0.5}^{+0.4}$	7.95 ± 0.08	${111}_{-98}^{+124}$	8.16 ± 0.009
${\rm{N}}{2}_{-1.75}^{-1.50}$	44	${1.2}_{-0.8}^{+1.7}$	${8.0}_{-0.3}^{+0.3}$	8.14 ± 0.10	${132}_{-109}^{+156}$	8.03 ± 0.006
${\rm{N}}{2}_{-2.00}^{-1.75}$	11	${1.8}_{-1.7}^{+2.1}$	${7.8}_{-0.4}^{+0.2}$	8.33 ± 0.14	${95}_{-56}^{+137}$	7.91 ± 0.006
${\rm{N}}{2}_{-2.25}^{-2.00}$	2	${0.4}_{-0.2}^{+1.7}$	${7.3}_{-0.1}^{+0.4}$	8.50 ± 0.14	${97}_{-54}^{+144}$	7.81 ± 0.008

Notes.

^aNumber of galaxies for a given N2 bin. ^bThe uncertainty in the SFR represents the 16th and 84th percentiles of the SFR distribution in a given N2 bin. ^cThe uncertainty in the stellar mass represents the 16th and 84th percentiles of the stellar mass distribution in a given N2 bin. ^dIonization parameter derived from the stacked spectrum. ^eElectron density derived from the stacked spectrum.

Download table as:  ASCIITypeset image

Here, we investigate the N2 and O3N2 empirical metallicity calibrations in the local analogs and the SDSS reference galaxies using the direct oxygen abundance and line ratios measured in the stacked spectrum. Figure 3 shows the relation between the direct T_e oxygen abundance and the N2 and O3N2 indices in the local analogs of high-redshift galaxies (blue squares) and the local normal star-forming galaxies (red squares). The relation between the direct oxygen abundance and the N2 and O3N2 indices is fit with a linear equation for both the local analogs of high-redshift galaxies and SDSS reference galaxies. The results for the least-squares fit for our local analogs are as follows (blue solid lines in Figure 3):

$$\begin{eqnarray}&&12+\mathrm{log}({\rm{O}}/{\rm{H}})=8.82+0.49\times {\rm{N}}2,\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&12+\mathrm{log}({\rm{O}}/{\rm{H}})=8.97-0.39\times {\rm{O}}3{\rm{N}}2.\end{eqnarray} \tag{ 12 }$$

The results for the SDSS reference galaxies are as follows (red solid lines in Figure 3):

$$\begin{eqnarray}&&12+\mathrm{log}({\rm{O}}/{\rm{H}})=8.98+0.63\times {\rm{N}}2,\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&12+\mathrm{log}({\rm{O}}/{\rm{H}})=9.05-0.47\times {\rm{O}}3{\rm{N}}2,\end{eqnarray} \tag{ 14 }$$

where $7.8\lt 12+\mathrm{log}({\rm{O}}/{\rm{H}})\lt 8.4.$

Zoom In Zoom Out Reset image size

In Figure 3, we compare our N2 and O3N2 calibrations in the local analogs, the SDSS reference galaxies, and the local H ii regions. For the local H ii region, we use the metallicity calibrations from Steidel et al. (2014) rather than those from PP04. Steidel et al. (2014) repeated the fits to N2 and O3N2 metallicity calibrations in PP04, but only focused on the H ii regions with direct T_e metallicity and in the low-metallicity range ($-1.7\lt {\rm{N}}2\lt -0.3$ and $-0.4\lt {\rm{O}}3{\rm{N}}2\lt 2.1$). Thus, the metallicity calibrations from Steidel et al. (2014) provide better comparisons with the relations in this work. The new N2 and O3N2 calibrations based on the local analogs are not consistent with those found in both the SDSS reference galaxies and the local H ii region PP04. The discrepancies between the SDSS reference galaxies and the local H ii regions are mainly caused by the contribution of diffused ionized gas emission in the SDSS reference galaxies (Sanders et al. 2017) and slightly different recipes for deriving the direct T_e metallicity. The contribution of the diffused ionized gas emission in the local analogs is negligible due to their high Hα surface density (Zhang et al. 2017). The direct T_e-based metallicity in the local analogs is systematically higher than that in the SDSS reference galaxies for a given N2 or O3N2 in the metallicity range of $7.8\lt 12+\mathrm{log}({\rm{O}}/{\rm{H}})\lt 8.4$. This discrepancy is mainly due to the changes of ISM conditions between our analogs and normal SDSS star-forming galaxies. The high ionization parameter and electron density increase the [N ii]λ6584 and [O iii]λ5007 line fluxes for a given metallicity, which causes the lower N2 and higher O3N2 values in the analogs (Dopita et al. 2016; see Section 5.3 for a further discussion).

The Hα and [N ii]λ6584 emission lines move out of the K-band atmospheric transmission window for galaxies at $z\gt 2.65$. Therefore, metallicity diagnostics based on the emission lines at shorter wavelengths are essential for galaxies at $z\gt 2.65$. We study the empirical metallicity calibrations for the R23, O32 and [O iii]λ5007/Hβ, and [Ne iii]λ3869/[O ii] ratios. The [O ii], [O iii], Hβ, [Ne iii] lines can be accessed in the H- and K-bands for galaxies at z ∼ 3.5, and the [Ne iii]/[O ii] ratio can be measured in galaxies with redshift as high as z = 5.0. These lines provide a powerful tool to measure the metallicity in galaxies at high redshift by assuming the ISM conditions did not change dramatically from z = 2 to z = 5⁶

Figure 4 shows the relation between the T_e metallicity and the four metallicity diagnostics mentioned previously. We find that the our local analogs and normal SDSS galaxies do not follow the same relation in all four relations. For a given metallicity, the R23, O32, [O iii]λ5007/Hβ, and [Ne iii]/[O ii] ratios in the local analogs are higher that those in the normal SDSS galaxies. We fit the relations between the metallicity and R23, O32, [O iii]λ5007/Hβ, and [Ne iii]/[O ii] ratios for the local analogs of high-redshift galaxies. These relations are suitable for the metallicity range of $7.8\lt 12+\mathrm{log}({\rm{O}}/{\rm{H}})\lt 8.4$.

Zoom In Zoom Out Reset image size

For the R23–Z relation, our data only cover the R23 upper branch and the transition zone of the R23 upper and lower branches. We fit the upper branch of the R23–Z relation using a third-order polynomial:

$$\begin{eqnarray}&&y=138.0430-54.8284x+7.2954{x}^{2}-0.32293{x}^{3},\end{eqnarray} \tag{ 15 }$$

where $y=R23$ and $x=12+\mathrm{log}({\rm{O}}/{\rm{H}})$. It is worth noting that the metallicity is not very sensitive to the R23 values in the metallicity range of $7.8\lt 12+\mathrm{log}({\rm{O}}/{\rm{H}})\lt 8.4$, because most of the data points are located in the transition zone of the R23 upper and lower branches.

For the O32–Z relation, we fit the data points using a linear equation:

$$\begin{eqnarray}&&12+\mathrm{log}({\rm{O}}/{\rm{H}})=8.54-0.59\times {\rm{O}}32.\end{eqnarray} \tag{ 16 }$$

For the [O iii]λ5007/Hβ–Z relation, we fit the data points using a third-order polynomial:

$$\begin{eqnarray}&&y=43.9836-21.6211x+3.4277{x}^{2}-0.1747{x}^{3},\end{eqnarray} \tag{ 17 }$$

where $y\,=$ [O iii]λ5007/Hβ and $x=12+\mathrm{log}({\rm{O}}/{\rm{H}})$.

For the [Ne iii]/[O ii]–Z relation, we fit the data points using a linear equation:

$$\begin{eqnarray}&&12+\mathrm{log}({\rm{O}}/{\rm{H}})=7.80-0.63\times \mathrm{log}([\mathrm{Ne}\,{\rm{III}}]/[{\rm{O}}\,{\rm{II}}]).\end{eqnarray} \tag{ 18 }$$

The PP04 metallicity diagnostics cause a discrepancy between the N2 and O3N2 metallicities in z ∼ 2 star-forming galaxies. The O3N2-based metallicity is systematically smaller than the N2-based metallicity by $0.10-0.15$ dex (e.g., Hainline et al. 2009; Bian et al. 2010; Newman et al. 2014; Steidel et al. 2014; Sanders et al. 2015). This discrepancy is primarily due to the offset between the local and z ∼ 2 star-forming galaxies in the [O iii]λ5007/Hβ and [N ii]λ6584/Hα BPT diagram. Various physical mechanisms have been proposed to interpret this offset, including changes of ionization parameters, electron densities, radiation fields, and nitrogen-to-oxygen ratio over cosmic time. These proposed physical mechanisms are crucial input parameters for photoionization models and play an important role for the metallicity estimations. Our stacked spectra also shed light on what is the major physical mechanism(s) to drive the offset between high-redshift and low-redshift galaxies on the BPT diagram (F. Bian 2018, in preparation).

To test our new calibrations in high-redshift galaxies, we apply the BKD18 N2 and O3N2 empirical metallicity calibrations (Equations (11) and 12) to two samples of z ∼ 2 star-forming galaxies. One sample is selected based on their UV emission properties, UV-selected star-forming galaxies from Steidel et al. (2014), and the other is selected based on their stellar mass, mass-selected star-forming galaxies from Sanders et al. (2015). The local analog selection criteria in this work are based on the location of the UV-selected galaxies on the BPT diagram. Therefore, the ISM conditions in our local analogs are more representative to those in the UV-selected galaxies, and the BKD18 metallicity calibration is more suitable for this type of galaxy. Figure 5 shows the comparison between the N2- and O3N2-based metallicities in these UV-selected galaxies (blue filled circles) using our new metallicity calibrations. In the metallicity range of $8.1\lt \mathrm{log}({\rm{O}}/{\rm{H}})\lt 8.4$ (shaded region in Figure 5), the mean difference between the BKD18 N2- and O3N2-metallicity is -0.03 dex. For a comparison, the mean difference between the PP04 N2- and O3N2-metallicity is 0.13 dex. The BKD18 calibrations successfully solve the discrepancy between N2- and O3N2-based metallicity when applying local PP04 calibrations to high-redshift galaxies (e.g., Steidel et al. 2014). The UV-selected star-forming galaxies may only represent 50%–70% of the whole population of high-redshift star-forming galaxies (Reddy et al. 2005). Therefore, we also apply our metallicity calibrations to a sample of mass-selected galaxies (Sanders et al. 2015), which is a more representative sample of z ∼ 2 star-forming galaxies. Shapley et al. (2015) found that the [O iii]/Hβ ratio in mass-selected star-forming galaxies is not as high as UV-selected star-forming galaxies at z ∼ 2, i.e., the BPT locus of mass-selected galaxies is lower than the BPT locus of UV-selected galaxies at z ∼ 2. Our local analogs, therefore, may not fully resemble the ISM conditions of mass-selected galaxies, which could introduce systematical uncertainty when applying the new calibrations to mass-selected galaxies. In Figure 5, we plot the new N2- and O3N2-based metallicity based on the four stacked spectra of the mass-selected galaxies from Sanders et al. (2016b). In the metallicity range of $8.1\lt \mathrm{log}({\rm{O}}/{\rm{H}})\lt 8.4$, the mean difference between the BKD18 N2- and O3N2-metallicity is -0.02 dex for mass-selected star-forming galaxies. For comparison, the mean difference between the PP04 N2- and O3N2-metallicity is 0.08 dex. Though the z ∼ 2 mass-selected galaxies is slight offset from the z ∼ 2 UV-selected galaxies on the BPT diagram (e.g., Figure 1(a) in Bian et al. 2017), the BKD18 calibrations are also suitable for z ∼ 2 mass-selected galaxies.

Zoom In Zoom Out Reset image size

It is worth noting that the N2 metallicity becomes systematically larger than the O3N2 metallicity in the BKD18 metallicity calibration, when $12+\mathrm{log}({\rm{O}}/{\rm{H}})\gt 8.4$. The discrepancy increases when metallicity becomes larger. At $12+\mathrm{log}({\rm{O}}/{\rm{H}})=8.6$, the O3N2 metallicity is about 0.1 dex higher than N2 metallicity. This discrepancy suggests that the BKD18 calibration is not suitable for high-metallicity ($12+\mathrm{log}({\rm{O}}/{\rm{H}}\gt 8.4$) cases. We do not recommend using the BKD18 calibration out of the metallicity range of $7.8\,\lt 12+\mathrm{log}({\rm{O}}/{\rm{H}})\lt 8.4$. It is difficult to use our local analogs to study the empirical calibration at $12+\mathrm{log}({\rm{O}}/{\rm{H}})\gt 8.4$. The direct T_e method becomes insensitive (or saturated) to the high-metallicity ($12+\mathrm{log}({\rm{O}}/{\rm{H}})\gt 8.5$) galaxies, because [O iii]λ4363 is only emitted in the hottest nebulae, which biases toward the low-metallicity region. Therefore, the direct T_e method underestimates the oxygen abundance in such galaxies.

The BKD18 metallicity calibrations provide a useful tool to study the mass–metallicity relation in high-redshift galaxies, particularly at the low-metallicity end. We measure metallicities in a sample of z ∼ 2 star-forming galaxies from Steidel et al. (2014) using N2 and O3N2 indicators. Figure 6 shows the comparison between the mass–metallicity relation based on the BKD18 calibrations (blue squares) and that based on the PP04 calibrations (black circles). In the metallicity range of $8.0\lt Z\lt 8.4$, we find that in both the N2 and O3N2-based metallicities the BKD18 calibrations are systematically higher than those based on the PP04 calibrations by 0.05 to 0.1 dex.

Recently, a few other studies have recalibrated the N2 and O3N2 metallicity for high-redshift galaxies (e.g., Brown et al. 2016; Cowie et al. 2016). Most of these studies used galaxies that are selected by matching their global properties to high-redshift galaxies, such as sSFRs (Brown et al. 2016) or Hβ luminosities (Cowie et al. 2016). However, studies have shown that low-redshift galaxies selected by matching the global properties of high-redshift galaxies do not have the same ISM conditions as their high-redshift counterparts (e.g., Shirazi et al. 2014; Bian et al. 2016). Therefore, simply matching global properties may miss crucial information on the change of the ISM conditions, which play an important role in regulating the relation between metallicity and emission-line ratios. In Figure 7, we compare the N2- and O3N2-based metallicities in a sample of mass-selected star-forming galaxies at z ∼ 2 from Sanders et al. (2015) using the BKD18 calibrations and the Brown et al. (2016) calibration. We adopt the measurements of the sSFR in the Table 1 of Sanders et al. (2015). The mean difference between the Brown et al. (2016) calibration N2 and O3N2 metallicities is 0.05 dex. We find the BKD18 calibration results in the better consistency between the N2- and O3N2 metallicities compared with the Brown et al. (2016) calibration in the metallicity range of 8.1–8.4.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We study the evolution of the metallicity calibrations across cosmic time using the MAPPINGs photoionization models (Sutherland & Dopita 1993; Dopita et al. 2013). We use the MAPPINGS IV code to generate the relation between the diagnostic line ratios and the oxygen abundance for two cases. In the first case, the input ionization parameter is $\mathrm{log}q=7.5$ and the ISM pressure is $\mathrm{log}(P/k)=5.2$, which corresponds to an electron density of ${n}_{e}\approx 10$ cm⁻³. These parameters are comparable to those in normal SDSS galaxies (e.g., Dopita et al. 2006; Nakajima & Ouchi 2014; Bian et al. 2016). In the second case, the input ionization parameter is $\mathrm{log}q=8.0$ and the ISM pressure is $\mathrm{log}(P/k)=6.2$, which corresponds to an electron density of ${n}_{e}\approx 100$ cm⁻³. These parameters are consistent with those in z ∼ 2 galaxies and our local analogs (e.g., Nakajima & Ouchi 2014; Bian et al. 2016; Kaasinen et al. 2017). Figure 8 shows the relations between the different diagnostic line ratios and the oxygen abundance derived from the photoionization models. The metallicity calibrations between the high-redshift and low-redshift conditions derived by the photoionization models (solid lines in Figure 8) follow the same trend as those between the local analogs and the SDSS reference galaxies (dashed lines in Figure 8). This suggests that the different metallicity calibration relations between the high- and low-redshift galaxies are due to the changes of the ISM conditions across cosmic time.

Zoom In Zoom Out Reset image size

Although the T_e metallicity and theoretical metallicity show the same relative trend for the dependence of ISM conditions for a given strong line ratio, the absolute metallicity derived from the direct T_e method is significantly smaller than that from photoionization models by about 0.5 dex (see also Kewley & Ellison 2008; Kewley 2018). This discrepancy is mainly due to (1) the inhomogeneous temperature distribution in H ii regions. The temperature fluctuation and/or gradients in H ii regions bias the electron temperature measurements to the high temperature (low-metallicity) regions, thus the direct T_e method underestimates the oxygen abundance. Furthermore, (2) κ-distributions of electron energy in H ii regions are also a factor. Nicholls et al. (2012) suggested that the electron energy in H ii regions follows the κ-distributions rather than the Maxwell–Boltzmann distribution, and the electron temperatures derived based on the Maxwell–Boltzmann distribution are significantly higher than those from κ-distributions (Nicholls et al. 2013). Finally, (3) the relation between t3 (electron temperature in ${{\rm{O}}}^{++}$ zone) and t2 (electron temperature in O⁺ zone) also plays a role. In this study, the T_e(O ii)-T_e(O iii) relation from Campbell et al. (1986) is adopted for a better comparison with previous studies. However, it is different from the T_e(O ii)-T_e(O iii) relation in the MAPPINGS photoionization models (T_e(O ii) = 0.685T_e(O iii) + 2100K Dopita et al. 2013)). The metallicities derived based on the T_e(O ii)–T_e(O iii) in the MAPPINGS photoionization models are 0.1–0.2 dex higher than the metallicities in this study. Meanwhile, the uncertainties of the input parameters in the photoionization models, particularly the shape of ionizing radiation field (e.g., Kewley et al. 2013a; Maier et al. 2014) and abundance scale (e.g., Steidel et al. 2014; Nicholls et al. 2017), can significantly affect the metallicity estimations. These parameters have not been well studied, particularly at high-redshift. Our local analogs of high-redshift galaxies provide great opportunities to study what constrains these key input parameters in photoionization models (F. Bian et al. 2018, in preparation).