NEW IMPROVED CALIBRATION RELATIONS FOR THE DETERMINATION OF ELECTRON TEMPERATURES AND OXYGEN AND NITROGEN ABUNDANCES IN H ii REGIONS

We wish to assemble here a sample of calibrating H ii regions with metallicities spanning a large range. Spectrophotometric measurements of moderate- and high-metallicity (12 + log(O/H) ≳ 8.0) H ii regions, with accurate measured electron temperatures t₃ = t(O iii) or/and t₂ = t(N ii), include those of H ii regions in disks of nearby spiral galaxies—NGC 300 (Bresolin et al. 2009), NGC 1365 (Bresolin et al. 2005), NGC 5194 = M 51 (Bresolin et al. 2004; Garnett et al. 2004), NGC 5236 = M 83 (Bresolin et al. 2005), NGC 5457 = M 101 (Esteban et al. 2009; Garnett & Kennicutt 1994; Izotov et al. 2007; Kennicutt et al. 2003; Kinkel & Rosa 1994; Luridiana et al. 2002; McCall et al. 1985; Rayo et al. 1982; Sedwick & Aller 1981; Torres-Peimbert et al. 1989; van Zee et al. 1998)—and in irregular galaxies—NGC 6822 (Hernández-Martínez et al. 2009; Hidalgo-Gámez et al. 2001; Miller 1996; Peimbert et al. 2005) and the Small Magellanic Cloud (SMC; Dufour et al. 1982; Testor 2001; Testor et al. 2003).

To select accurate measurements of H ii regions in spiral galaxies, we require that the deviation of the oxygen abundance of the H ii region from the general linear relation that describes the radial oxygen abundance gradient in the disk to be less than ∼0.1 dex. In the case of the spiral galaxy NGC 1365, the number of H ii regions with a measured electron temperature is too small to establish with certainty the radial distribution of oxygen abundances in the disk (Bresolin et al. 2005). However, the oxygen abundance of H ii region 15, based on the measurement of the electron temperature t(N ii), is confirmed independently by the one based on the measured electron temperature t(S iii), as well as by the abundances of H ii regions 5 and 14, at nearly the same galactocentric distance (Bresolin et al. 2005). Therefore, H ii region 15 in the disk of NGC 1365 has been included in our sample of calibrating H ii regions. As for the dwarf irregular galaxies NGC 6822 and the SMC, since the oxygen abundance is relatively constant over their extent (e.g., Croxall et al. 2009), we have required the deviations of oxygen abundances from the mean value to be less than ∼0.1 dex. Moreover, if temperature estimates from both the [O iii]λ4363 and [N ii]λ5755 auroral lines are available for the same H ii region, they are considered to be independent measurements. Using the above selection criteria, we have selected 72 measurements of moderate- and high-metallicity (12 + log(O/H) ≳ 8.0) H ii regions as calibrating data points. As for low-metallicity (12 + log(O/H) ≲ 8.0) calibrating H ii regions with accurate electron temperature t₃, the data (46 data points) have been taken from Izotov et al. (2007). Thus, our calibrating sample contains a total of 118 data points.

To ensure that we have a relatively homogeneous data set, we have recalculated electron temperatures and oxygen and nitrogen abundances for all the calibrating H ii regions, within the framework of the standard H ii region model with two distinct temperature zones within the nebula. The equations used to derive electron temperatures from line fluxes are given in the next section.

To convert the values of the line fluxes to the electron temperatures t_3,O and t_2,N and to the ion abundances O⁺⁺/H⁺, O⁺/H⁺, and N⁺/H⁺, we have used the five-level atom model for the O⁺⁺, O⁺, and N⁺ ions, updated by recent atomic data. The Einstein coefficients for the spontaneous transitions A_jk for the five low-lying levels for all ions above have been taken from Froese Fisher & Tachiev (2004). The energy level data are from Edlén (1985) for O⁺⁺, from Wenåker (1990) for O⁺, and from Galavís et al. (1997) for N⁺. The effective cross sections (or effective collision strengths) for electron impact Ω_jk as a function of temperature are from Aggarwal & Keenan (1999) for O⁺⁺, from Pradhan et al. (2006) for O⁺, and from Hudson & Bell (2005) for N⁺. To derive the effective cross section for a given electron temperature, we have fitted a second-order polynomial to these data.

In the low density regime (n_e ≲ 100 cm⁻³), the following simple expressions provide approximations to the numerical results with accuracy better than 1%. The electron temperatures are related to the measured line fluxes in the following way:

$$\begin{equation} t_{\rm 3,O} = \frac{1.467}{\log Q_{\rm 3,O} -0.876 - 0.193\log t_{\rm 3,O} + 0.033\,t_{\rm 3,O} } \end{equation} \tag{ 1 }$$

and

$$\begin{equation} t_{\rm 2,N} = \frac{1.118}{\log Q_{\rm 2,N} -0.891 - 0.177\log t_{\rm 2,N} + 0.030\,t_{\rm 2,N} } \end{equation} \tag{ 2 }$$

or

$$\begin{equation} t_{\rm 2,O} = \frac{0.871}{\log Q_{\rm 2,O} -0.871 - 0.134\log t_{\rm 2,O} + 0.011\,t_{\rm 2,O} }, \end{equation} \tag{ 3 }$$

where Q_3,O = [O iii](λ4959+λ5007)/[O iii]λ4363 is the ratio of nebular to auroral oxygen line intensities in the t_3,O zone, Q_2,N = [N ii](λ6548 + λ6584)/[N ii]λ5755 is the ratio of nebular to auroral nitrogen line intensities, and Q_2,O = [O ii](λ3727 + λ3729)/[O ii](λ7320 + λ7330) is the ratio of nebular to auroral oxygen line intensities in the t_2,O zone.

The equations relating ion abundances to measured line fluxes are:

$$\begin{equation} 12+\log \frac{\rm N^+}{\rm H^+} = \log N_{2} + 6.263 + \frac{0.893}{t_2} - 0.603\,\log t_2 - 0.003\,t_2, \end{equation} \tag{ 4 }$$

$$\begin{equation} 12+\log \frac{\rm O^+}{\rm H^+} = \log R_{2} + 5.929 + \frac{1.617}{t_2} - 0.568\,\log t_2 - 0.008\,t_2, \end{equation} \tag{ 5 }$$

and

$$\begin{equation} 12+\log \frac{\rm O^{++}}{\rm H^+} = \log R_{3} + 6.251 + \frac{1.204}{t_3} - 0.613\,\log t_3 - 0.015\,t_3. \end{equation} \tag{ 6 }$$

The total oxygen abundance is determined from

$$\begin{equation} \frac{\rm O}{\rm H} = \frac{\rm O^{++}}{\rm H^+} + \frac{\rm O^{+}}{\rm H^+}. \end{equation} \tag{ 7 }$$

The total nitrogen abundance is determined from

$$\begin{equation} \log \frac{\rm N}{\rm H} = \log \frac{\rm O}{\rm H} + \log \frac{\rm N}{\rm O}, \end{equation} \tag{ 8 }$$

assuming

$$\begin{equation} \frac{\rm N^+}{\rm O^+} = \frac{\rm N}{\rm O}. \end{equation} \tag{ 9 }$$

The N⁺/O⁺ ion abundance ratio is derived from

$$\begin{equation} \log \frac{\rm N^+}{\rm O^+} = \log \frac{N_2}{R_2} + 0.334 - \frac{0.724}{t_2} - 0.035 \log t_2 + 0.005 t_2 \end{equation} \tag{ 10 }$$

which is obtained by combining Equation (4) with Equation (5).

The relation between the electron temperatures t₂ within the O⁺ zone and t₃ within the O⁺⁺ zone is

$$\begin{equation} t_2 = 0.672\,t_3 + 0.314. \end{equation} \tag{ 11 }$$

This relation derived by Pilyugin et al. (2009) is very similar to the commonly used ones of Campbell et al. (1986) and Garnett (1992).

Here, we examine how the fluxes and flux ratios of various emission lines in the calibrating H ii regions vary as a function of electron temperature t₂ (Figure 1) and of oxygen (Figure 2) and nitrogen (Figure 3) abundances. Inspection of those variations shows that none of the considered emission line fluxes and flux ratios displays a monotonic behavior with electron temperature and oxygen and nitrogen abundances over the whole temperature (or metallicity) range. Instead, the curves show bends. Close examination suggests that there are three distinct regimes, depending on the temperature of the H ii region, i.e., on whether it belongs to the class of cool, warm, or hot H ii regions. We have therefore decided to construct distinct calibrations for the determination of temperature and abundances for each of these classes of H ii regions.

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In the cool high-metallicity H ii regions, the R₂ and R₃ line fluxes strongly decrease with decreasing electron temperature, or equivalently, with increasing oxygen and nitrogen abundances. On the contrary, the N₂ line fluxes remain approximately constant with temperature, forming a sort of plateau. The behavior difference between the R₂ and N₂ line fluxes have a natural explanation as the interplay of two factors. On the one hand, the line-emissivity j(R₂) is temperature sensitive, i.e., the emissivity decreases with decreasing electron temperature, causing a decrease of the R₂ line flux for lower electron temperatures. On the other hand, the oxygen abundance increases on average with decreasing electron temperature, resulting in an increase of the R₂ line flux with decreasing electron temperature. The combination of these two factors results in a decrease of the R₂ line flux with decreasing electron temperature. Since at 12 + log(O/H) ≳ 8.3, secondary nitrogen becomes dominant and the nitrogen abundance increases at a faster rate than the oxygen abundance (Henry et al. 2000), the change of nitrogen abundances with decreasing electron temperature is larger than that of oxygen abundances and, as a consequence, compensates the decrease of the line-emissivity j(N₂). Thus, the N₂ line fluxes remain approximately constant in cool H ii regions.

Inspection of Figure 4 shows that cool H ii regions can be selected with the criteria logN₂ ≳ −0.1. They are shown in Figures 1, 2, 3, and 4 by the gray (light green in the color online version) filled circles. The adopted boundary between the cool H ii regions and the warm H ii regions is shown by the dotted line in Figure 4. It should be emphasized that the R₂, R₃ line fluxes and the N₂/R₂, S₂/R₂ flux ratios in cool H ii regions are very sensitive to electron temperature and oxygen and nitrogen abundances. One can thus expect these parameters to act as reliable temperature and metallicity indexes.

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Close examination of the behavior of emission line fluxes and flux ratios with electron temperature and oxygen and nitrogen abundances suggests that there is a second bend in the curves at t₂ ∼ 1.2 or at 12 + log(O/H) ∼ 8.0, although this bend is less prominent than the previous one. This bend suggests that we should consider separately warm H ii regions with 12 + log(O/H) ≳ 8.0 and hot H ii regions with 12 + log(O/H) ≲ 8.0. The warm H ii regions are shown in Figures 1, 2, 3, and 4 by the dark (black in the color version) plus signs, while the hot ones are shown by the gray (light blue in the online color version) filled triangles. Examination of Figure 4 shows that the value of log(N₂/S₂) = −0.25 can be taken as indicating roughly the boundary between warm and hot H ii regions, although this criterion is not very precise. Another way for distinguishing between warm and hot H ii regions in disks of spiral galaxies will be discussed below. It should be noted that the strong line fluxes and flux ratios of warm and hot H ii regions are less sensitive to electron temperature and oxygen and nitrogen abundances than those of cool H ii regions.

To summarize, we propose to divide H ii regions into three classes: cool, warm, and hot ones. The adopted classification criteria and parameters of these three H ii region classes are summarized in Table 1. Distinct calibration relations for the determination of temperatures and abundances are derived below for each of these classes.

Here, we search for a calibration that makes use of both N₂/R₂ and S₂/R₂ ratios as temperature and metallicity indexes. This calibration will be referred to as the ONS calibration.

3.1.1. Calibration Relations for Temperature

We adopt the following simple expression to relate the electron temperature to the above surrogate indicators:

$$\begin{equation} t = \frac{1}{a_0 + a_1\,P + a_2\,\log R_3 + a_3\,\log (N_2/R_2) + a_4\,\log (S_2/R_2)}. \end{equation} \tag{ 12 }$$

This expression is analogous to the equation for determining electron temperature in the classic T_e method, see Equations (1) and (2). The numerical values of the coefficients in Equation (12) can be derived from the condition that the scatter of the residuals σ_t defined as

$$\begin{equation} \sigma _t = \sqrt{\frac{1}{n}\sum \limits _{j=1}^{j=n} \left(\frac{t_{j}^{\rm CAL}}{t_{j}^{\rm OBS}}-1\right)^2} \end{equation} \tag{ 13 }$$

is minimal. Here, t^CAL_j is the electron temperature t₂ calculated from Equation (12) and t^OBS_j is the measured t₂. With this requirement, we have derived the following relations for the determination of the electron temperature:

$$\begin{eqnarray} t_{\rm ONS} & = & 1/[1.111 + 0.505\,P - 0.446\log R_3\nonumber\\\ms && +\, 0.081\log (N_2/R_2) - 0.008\log (S_2/R_2)]\nonumber \\\ms && {\rm if}\; \log N_2 > -0.1\nonumber \\\ms t_{\rm ONS} & = & 1/[1.325 - 0.007\,P - 0.229\log R_3\nonumber \\\ms &&+\, 0.362\,\log (N_2/R_2) - 0.173\log (S_2/R_2)] \\\ms && {\rm if}\; \log N_2 < -0.1, \; \log (N_2/S_2) > -0.25\nonumber \\\ms t_{\rm ONS} & = & 1/[1.318 - 0.649\,P + 0.283\log R_3\nonumber\\\ms &&+\, 0.115\,\log (N_2/R_2) + 0.151\log (S_2/R_2)]\nonumber \\\ms && {\rm if}\; \log N_2 < -0.1, \;\log (N_2/S_2) < -0.25.\nonumber \end{eqnarray} \tag{ 14 }$$

The quantity t_ONS denotes here the electron temperature t₂ derived from the calibration relation.

3.1.2. Calibration Relations for Oxygen and Nitrogen Abundances

As discussed above, the combination of these parameters can also serve as an indicator of the oxygen abundance in cool H ii regions. The following simple expression

$$\begin{eqnarray} 12+\log {\rm (O/H)} &=& a_0 + a_1P + a_2\log R_3\nonumber\\ &&+ a_3\log (N_2/R_2) + a_4\log (S_2/R_2) \end{eqnarray} \tag{ 15 }$$

is adopted to relate the oxygen abundance to the strong line fluxes. Again, the numerical values of the coefficients in Equation (15) can be derived from the requirement that the scatter of the residuals σ_O/H defined as

$$\begin{equation} \sigma _{\rm O/H} = \sqrt{\frac{1}{n}\sum \limits _{j=1}^{j=n} \big(\log {\rm (O/H)}_{j}^{\rm CAL} - \log {\rm (O/H)}_{j}^{\rm OBS}\big)^2} \end{equation} \tag{ 16 }$$

is minimal. Here, (O/H)^CAL_j is the oxygen abundance calculated from Equation (15) and (O/H)^OBS_j is the measured T_e-based oxygen abundance.

The relations

$$\begin{eqnarray} &&12+\log {\rm (O/H)}_{\rm ONS} = 8.277 + 0.657\,P - 0.399\log R_3\nonumber\\ &&\qquad-\, 0.061\log (N_2/R_2) + 0.005\log (S_2/R_2)\nonumber \\ &&\qquad {\rm if}\; \log N_2 > -0.1\nonumber \\ &&12+\log {\rm (O/H)}_{\rm ONS} = 8.816 - 0.733\,P + 0.454\log R_3\nonumber\\ &&\qquad +\, 0.710\log (N_2/R_2) - 0.337\log (S_2/R_2) \nonumber\\ &&\qquad {\rm if}\; \log N_2 < -0.1, \;\log (N_2/S_2) > -0.25 \nonumber\\ &&12+\log {\rm (O/H)}_{\rm ONS} = 8.774 - 1.855\,P + 1.517\log R_3\nonumber\\ &&\qquad +\; 0.304\log (N_2/R_2) + 0.328\log (S_2/R_2)\nonumber \\ &&\qquad {\rm if}\; \log N_2 < -0.1, \;\log (N_2/S_2) < -0.25. \end{eqnarray} \tag{ 17 }$$

have been obtained. A few points show large (>0.2 dex) deviations. These points have been excluded in deriving the final calibration relations.

In the same way, calibration relations for nitrogen abundance determination in H ii regions have been constructed:

$$\begin{eqnarray} && 12 +\log {\rm (N/H)}_{\rm ONS} = 7.811 + 0.290\,P - 0.081\log R_3\nonumber\\ &&\qquad + 0.877\,\log (N_2/R_2) + 0.002\log (S_2/R_2)\nonumber \\ &&\qquad {\rm if}\; \log N_2 > -0.1 \nonumber\\ &&12 +\log {\rm (N/H)}_{\rm ONS} = 8.241 - 0.781\,P + 0.612\log R_3\nonumber\\ &&\qquad + 1.455\log (N_2/R_2) - 0.209\log (S_2/R_2) \nonumber\\ &&\qquad {\rm if}\; \log N_2 < -0.1, \;\log (N_2/S_2) > -0.25\nonumber \\ && 12 +\log {\rm (N/H)}_{\rm ONS} = 8.080 - 1.476\,P + 1.349\log R_3\nonumber\\ && \qquad + 1.259\log (N_2/R_2) + 0.004\log (S_2/R_2)\nonumber \\ && \qquad {\rm if}\; \log N_2 < -0.1, \;\log (N_2/S_2) < -0.25. \end{eqnarray} \tag{ 18 }$$

Again, a few points with large (>0.2 dex) deviations have been excluded in deriving the final calibration relations.

In the previous section, we have derived the ONS calibration, using both N₂/R₂ and S₂/R₂ ratios as temperature and metallicity indexes. The question arises then whether a calibration, where only one ratio is used as a temperature and metallicity index, can provide reliable estimates of oxygen and nitrogen abundances in H ii regions as well. We have considered the ON calibration based on the N₂/R₂ ratio and the OS calibration based on the S₂/R₂ ratio. We found that the accuracies of the oxygen and nitrogen abundances and of the electron temperatures obtained with the ON calibration are comparable with those of the same quantities determined with the ONS calibration, for all classes of H ii regions. On the other hand, the OS calibration provides reliable estimates for some characteristics of H ii regions, such as oxygen abundances in cool H ii regions, but somewhat less reliable estimates of other characteristics, such as nitrogen abundances in warm H ii regions. Thus, we will give only the ON calibration relations below and will not discuss further the OS calibration.

We have derived the following relations for the determinations of the electron temperature t_ON = t₂, of the oxygen abundance 12 + log(O/H)_ON, and of the nitrogen abundance 12 + log(N/H)_ON:

$$\begin{eqnarray} &&\hspace{-1.5pc} 12 +\log {\rm (O/H)}_{{\rm ON}} = 8.606 - 0.105\,\log R_3 - 0.410\log R_2\nonumber\\ &&\hspace{-1.5pc}\quad -\; 0.150\log (N_2/R_2) \;{\rm if}\; \log N_2 > -0.1 \nonumber\\ &&\hspace{-1.5pc}12 +\log {\rm (O/H)}_{{\rm ON}} = 8.642 + 0.077\log R_3 + 0.411\log R_2 \nonumber\\ &&\hspace{-1.5pc}\quad+\; 0.601\log (N_2/R_2) \; {\rm if}\; \log N_2 < -0.1,\nonumber \\ &&\hspace{-1.5pc}\quad\log (N_2/S_2) > -0.25\nonumber \\ &&\hspace{-1.5pc}12 +\log {\rm (O/H)}_{{\rm ON}} = 8.013 + 0.905\log R_3 + 0.602\log R_2\nonumber \\ &&\hspace{-1.5pc}\quad +\; 0.751\,\log (N_2/R_2)\; {\rm if}\; \log N_2 < -0.1,\nonumber\\ &&\hspace{-1.5pc}\quad\log (N_2/S_2) < -0.25. \end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray} &&\hspace{-1.5pc} 12+\log {\rm (N/H)}_{{\rm ON}} = 7.955 + 0.048\log R_3 - 0.171\log N_2 \nonumber\\ &&\hspace{-1.5pc}\quad+ 1.015\,\log (N_2/R_2)\; {\rm if}\; \log N_2 > -0.1\nonumber \\ &&\hspace{-1.5pc}12+\log {\rm (N/H)}_{{\rm ON}} = 7.928 + 0.291\log R_3 + 0.454\log N_2 \nonumber\\ &&\hspace{-1.5pc}\quad + 0.953\log (N_2/R_2)\; {\rm if}\; \log N_2 < -0.1,\nonumber \\ &&\hspace{-1.5pc}\quad \log (N_2/S_2) > -0.25\nonumber \\ &&\hspace{-1.5pc}12+\log {\rm (N/H)}_{{\rm ON}} = 7.505 + 0.839\log R_3 + 0.492\,\log N_2 \nonumber\\ &&\hspace{-1.5pc}\quad + 0.970\log (N_2/R_2)\; {\rm if}\; \log N_2 < -0.1,\nonumber\\ &&\hspace{-1.5pc}\quad\log (N_2/S_2) < -0.25. \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} t_{{\rm ON}} & = & 1/[1.373 - 0.217\log R_3 - 0.325\log R_2\nonumber\\ && + 0.006\,\log (N_2/R_2)]\; {\rm if}\; \log N_2 > -0.1 \nonumber\\ t_{{\rm ON}} & = & 1/[1.437 - 0.254\log R_3 + 0.013\log R_2 \nonumber\\ && + 0.302\,\log (N_2/R_2)] {\rm if} \; \log N_2 < -0.1, \nonumber\\ &&\log (N_2/S_2) > -0.25\nonumber \\ t_{{\rm ON}} & = & 1/[0.999 + 0.110\,\log R_3 + 0.169\,\log R_2\nonumber\\ && + 0.278\,\log (N_2/R_2)]\; {\rm if}\; \log N_2 < -0.1,\nonumber\\ && \log (N_2/S_2) < -0.25. \end{eqnarray} \tag{ 21 }$$

Again, a few points with large deviations were excluded in the final derivation of the calibration relations.

The filled gray (light blue in the online version) circles in Figure 5 show the comparison between ONS-calibration-determined (upper panel) and ON-calibration-determined (lower panel) electron temperatures with measured ones in the calibrating H ii regions. Three hot H ii regions (HS 0837+4717, HS 1028+3843, and UM 420), represented by plus signs, show large deviations from the equality solid line. These outliers show significantly enhanced nitrogen line intensities and enhanced N/O ratios. López-Sánchez & Esteban (2010) have found that Wolf-Rayet features are detected in all galaxies with a high N/O ratio. They suggest that it is the ejecta of Wolf-Rayet stars that locally enrich the interstellar medium of these galaxies in nitrogen. López-Sánchez et al. (2007) have quantitatively analyzed the effect of local enrichment in the blue compact dwarf galaxy NGC 5253. They found that the enrichment pattern agrees with that expected for local pollution by the ejecta of Wolf-Rayet stars. Pustilnik et al. (2004) have found that the number of Wolf-Rayet stars in the region of the current starburst in HS 0837+4717 is about 1000. It is not clear however whether Wolf-Rayet ejecta can enhance nitrogen by as much as a factor of 2–3, as is observed. More data are needed to know with more certainty what makes these three galaxies to be outliers. It should be noted that galaxies with significantly enhanced nitrogen abundances are very rare (Pustilnik et al. 2004; López-Sánchez & Esteban 2010). Because of their nitrogen enhancement, the three galaxies HS 0837+4717, HS 1028+3843, and UM 420 have been misclassified as warm H ii regions according to our criteria, while they should have been classified as hot H ii regions. The filled dark squares in Figure 5 show electron temperatures calculated using the calibration for hot H ii regions. It can be seen that the deviations become considerably smaller.

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Inspection of Figure 5 shows that the scatter in the t_ONS–t_OBS and t_ON–t_OBS diagrams is smaller for cool H ii regions than that for hot ones. This is not surprising since it has been noted above that our temperature indexes are more sensitive to the electron temperature in cool H ii regions than in warm and hot ones (Figure 1). The rms deviations of electron temperatures for the 115 calibrating H ii regions, excluding the three peculiar objects, are equal $\sigma _{t_{\rm ONS}}$ = $\sigma _{t_{{\rm ON}}}$ = 0.054.

We next compare the abundances determined from the calibrations with T_e-based abundances. In the upper left panel of Figure 6, we plot the oxygen abundances (O/H)_ONS obtained from Equation (17) against the T_e-based oxygen abundances. Each calibrating H ii region is represented by a gray (light blue in the online version) filled circle. Equality is shown by the solid line. As before, the plus signs show oxygen abundances in the three peculiar misclassified hot H ii regions determined with the calibration for warm H ii regions, while the filled dark squares show abundances in the same objects determined with the calibration for hot H ii regions. Again, the deviations from the equality line are considerably smaller for the latter. The rms deviation of oxygen abundances for the 115 calibrating H ii regions, excluding the three peculiar objects, is $\sigma _{{\rm (O/H)}_{\rm ONS}}$ = 0.074.

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In the upper right panel of Figure 6, we plot the oxygen abundances (O/H)$_{t_{\rm ONS}}$ obtained by the T_e method equations (Section 2.2) and with electron temperatures t₂ estimated from Equation (14), against the T_e-based oxygen abundances. The general behavior of the (O/H)$_{t_{\rm ONS}}$ abundances is similar to that of the (O/H)_ONS abundances. The rms deviation of (O/H)$_{t_{\rm ONS}}$ abundances for the 115 calibrating H ii regions (excluding again the three peculiar objects) is $\sigma _{{\rm (O/H)}_{t_{\rm ONS}}}$ = 0.075. It may be surprising that the scatter of the (O/H)$_{t_{\rm ONS}}$ abundances is not appreciably smaller in the cool H ii regions than in the warm and hot H ii regions since, as discussed before, the scatter in the t_ONS–t_OBS diagram is significantly smaller for cool H ii regions as compared with warm and hot H ii regions. Examination of the equations of the T_e method for the determination of oxygen ionic abundances (see Equations (5) and (6)) gives a natural explanation for this apparent inconsistency. Since the oxygen ionic abundance varies in inverse proportion to the electron temperature, the smaller errors in the electron temperatures for H ii regions with t ∼ 0.6 and the twice as larger errors in the electron temperatures for H ii regions with t ∼ 1.2 result in similar errors in the oxygen abundance. Therefore, the smaller scatter in the t_ONS–t_OBS diagram for cool H ii regions does not result in a smaller scatter in the (O/H)$_{t_{\rm ONS}}$–(O/H)$_{t_{\rm OBS}}$ diagram for the same cool H ii regions.

The gray (light blue in the online version) filled circles in the lower left panel of Figure 6 show the nitrogen abundances (N/H)_ONS obtained from Equation (18) against the T_e-based nitrogen abundances for all the calibrating H ii regions. As before, the solid line shows equality, the plus signs the nitrogen abundances in the three peculiar misclassified hot H ii regions determined with the calibration for warm H ii regions, and the filled dark squares the abundances in these peculiar objects determined with the calibration for hot H ii regions. The rms deviation of nitrogen abundances for the 115 calibrating H ii regions is $\sigma _{{\rm (N/H)}_{\rm ONS}}$ = 0.047. Finally, in the lower right panel we plot the nitrogen abundances (N/H)$_{t_{\rm ONS}}$ against the T_e-based nitrogen abundances. The general behavior of the (N/H)$_{t_{\rm ONS}}$ abundances is similar to that for (N/H)_ONS abundances. The rms deviation of (N/H)$_{t_{\rm ONS}}$ abundances for the 115 calibrating H ii regions is $\sigma _{{\rm (N/H)}_{t_{\rm ONS}}}$ = 0.049. It should be emphasized that the calibrations for nitrogen provide more accurate abundances than those for oxygen.

Figure 7 shows the comparison of observed and ON-calibration-determined abundances. Comparison of Figures 6 and 7 shows that the general behavior of the ONS-calibration-determined and the ON-calibration-determined abundances are similar. Direct comparison in Figure 8 of ONS-calibration-determined and ON-calibration-determined abundances and temperatures shows that they are very close to each other.

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In conclusion, the small scatter in Figures 5–7 constitutes evidence that our criteria for selecting calibrating H ii regions with accurate measurements are sound, and that our proposed temperature and metallicity indexes are reliable.

Analysis of the O/H–N/O diagram provides another means for checking the reliability of our calibrations. Moreover, this diagram provides a way to evaluate the credibility of abundances determinations in H ii regions where abundances cannot be derived by direct methods. The O/H–N/O diagram is shown in Figure 9. The dark filled circles show the T_e-based oxygen and nitrogen abundances in the calibrating H ii regions. The gray (light blue in the online version) filled circles in the upper panel show (O/H)_ONS versus (N/H)_ONS abundances for the 118 calibrating H ii regions and 411 H ii regions in nearby spiral and irregular galaxies taken from van Zee et al. (1998), van Zee & Haynes (2006), Bresolin et al. (1999), and Bresolin et al. (2005). The plus signs represent again the three peculiar misclassified hot H ii regions with abundances determined with the calibration for warm H ii regions, and the open squares represent the abundances of those objects determined with the calibration for hot H ii regions. We see that the H ii regions with abundances derived with our calibrations and those with T_e-based abundances lie in the same general region in the O/H–N/O diagram. In the lower panel, we plot the (O/H)_ON and (N/H)_ON abundances for the same H ii regions. The general trends in the lower panel are the same as those in the upper panel. The overlap of the dark and gray points is a testimony to the reliability of our calibrations.

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The radial distribution of oxygen abundances in the disk of the spiral galaxy M 101, determined with the T_e method, provides us with one more possibility for checking the credibility of our calibrations. It has long been known that the disks of spiral galaxies show radial abundance gradients, with the oxygen abundance being higher in the central part of the disk and decreasing outward. There exists a reliable way to distinguish between hot and warm H ii regions in the disks of spiral galaxies (Pilyugin et al. 2004): move outward from the central part of the disk until the oxygen abundance decreases to 12 + log(O/H) = 8.0, then define the galactocentric radius where that happens to be R*_G. Then, the abundances in H ii regions beyond R*_G are determined with the calibration for hot H ii regions, and those in H ii regions closer than R*_G with the calibration for warm H ii regions. In the case of M 101, R*_G is equal to ∼ 0.9R₂₅, where R₂₅ is the isophotal radius at the surface brightness of B = 25 mag arcsec⁻². The sources of the spectrophotometric measurements of H ii regions in the disk of M 101 are given in Section 2.1.

The upper panel of Figure 10 shows the radial distribution of oxygen abundances in the disk as determined by the T_e method. The linear least-square best fit to those data is shown by the solid line in all three panels of Figure 10. The middle panel shows the radial distribution of (O/H)_ONS abundances, and the lower panel that of (O/H)_ON abundances. Here, we have used the criteria listed in Table 1 to assign each H ii region to one of three classes. Inspection of Figure 10 shows that all three abundance radial distributions are in good agreement. This is a testimony to the reliability of our calibrations and of our H ii region classification criteria.

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We now compare our calibrations with some previous calibrations. For the sake of definiteness, we will discuss the ONS calibration. López-Sánchez & Esteban (2010) have recently compared the abundances obtained by the T_e-method with those derived from previous calibrations. Those authors found the parametric or P calibration (the most recent version being that of Pilyugin & Thuan 2005) to yield the most accurate abundances. We have thus limited ourselves to comparing the oxygen abundances derived from the present calibrations to those obtained from the parametric calibration and from the often-used calibrations of Tremonti et al. (2004):

$$\begin{equation} 12+\log {\rm (O/H)}_{Tr} = 9.185 - 0.313\,x - 0.264\,x^2 - 0.321\,x^3 \end{equation} \tag{ 22 }$$

and of Kobulnicky & Kewley (2004):

$$\begin{eqnarray} 12+\log {\rm (O/H)}_{{\rm KK}} & = & 9.11 - 0.218\,x - 0.0587\,x^2\nonumber\\ &-& 0.330\,x^3 - 0.199\,x^4 \nonumber\\ & -& y\,(0.00235- 0.01105\,x -0.051\,x^2 \nonumber \\ & -& 0.04085\,x^3 - 0.003585\,x^4), \end{eqnarray} \tag{ 23 }$$

where x = logR₂₃ and y = log(R₃/R₂).

In each panel of Figure 11, we show by dark (black in the color version) circles the oxygen abundances determined with the ONS calibration against the T_e-based abundances in the calibrating H ii regions. For comparison, we show with gray (light blue in the color version) circles the (O/H)_P abundances estimated with the parametric calibration of Pilyugin & Thuan (2005; upper panel), the (O/H)_KK abundances obtained with the calibration of Kobulnicky & Kewley (2004; middle panel), and the (O/H)_Tr abundances determined with the calibration of Tremonti et al. (2004; lower panel) against those obtained by the T_e method. The solid line shows equality.

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The upper panel shows that there is a general good agreement between $({\rm O/H})_{T_e}$ and (O/H)_P abundances. This was also found by López-Sánchez & Esteban (2010). However, the current calibration has several advantages over the P calibration. First, it provides more accurate oxygen abundances, especially in the high-metallicity range. Second, the current calibration is valid over the whole range of metallicity while the parametric calibration is not valid for H ii regions in the transition zone, with 8.0 ≲ 12 + log(O/H) ≲ 8.3. Third, for a correct application of the P calibration, one needs to know a priori on which of the two branches the H ii region lies. In the case of the current calibration, this difficulty arises only for hot low-metallicity H ii regions with enhanced nitrogen abundances. However this difficulty can usually be overcome. Indeed, the auroral line [O iii]λ4363 is often detected in these objects, i.e., a direct measurement of the electron temperature is usually possible. Furthermore, in some types of investigations (for example, the study of radial abundance gradients in the disks of spiral galaxies), there usually exist additional data that help to bypass the difficulty. Finally, this does not happen often as nitrogen-enhanced low-metallicity galaxies are very rare (Pustilnik et al. 2004; López-Sánchez & Esteban 2010).

The middle and lower panels of Figure 11 show that the (O/H)_KK and the (O/H)_Tr abundances are shifted toward higher values relative to the T_e-based abundances. We have also computed (O/H)_KK and (O/H)_Tr abundances for the H ii regions in M 101 and have found that a similar shift occurs in Figure 10. This shift is not surprising since Tremonti et al. (2004) have noted themselves that their calibration may systematically overestimate oxygen abundances by as much as a factor of two.

Thus, we have shown that the present calibrations yield several advantages over previous ones. They give reliable and improved estimates of electron temperatures and oxygen and nitrogen abundances in H ii regions. To refine the values of the coefficients in the derived calibrations, as well as the dividing boundaries between cool, warm, and hot H ii regions, we need to have a larger sample of high-precision spectrophotometric measurements of H ii regions with fluxes of the auroral oxygen [O iii]λ4363 and/or nitrogen [N ii]λ5755 lines.