Ionized Gaseous Nebulae Abundance Determination from the Direct Method

The T_e of [O iii] can be calculated from the emission-line ratio,

$$\begin{eqnarray}&&{R}_{O3}=\displaystyle \frac{I(4959)+I(5007)}{I(4363)},\end{eqnarray} \tag{ 7 }$$

given that, according to Osterbrock & Ferland (2006), temperature can be obtained from the ratio of collisional transitions that have a similar energy but occupy different levels. In this calculation, it is not required to measure the two strong [O iii] lines as there is a theoretical ratio between them (I(5007) = 3· I(4959)). The fitting between the ratio and the electronic temperature was obtained using the program pyneb assuming a five-level atom and the following non-linear fitting for n_e = 100 cm⁻³:

$$\begin{eqnarray}&&t([{\rm{O}}\,{\rm{III}}])=0.7840-0.0001357\cdot {R}_{O3}+\displaystyle \frac{48.44}{{R}_{O3}}\end{eqnarray} \tag{ 8 }$$

in units of 10⁴ K, valid in the range t = 0.7–2.5 and using collisional strengths from Aggarwal & Keenan (1999). This fitting gives precisions better than 1% for 1.0 < t([O iii]) < 2.5 and better than 3% for 0.7 < t([O iii]) < 1.0. It was calculated for a density of 100 cm⁻³, but considering a density of 1000 cm⁻³ reduces the temperature only in a 0.1%. This relation with the resulting fitting can be seen in left panel of Figure 4.

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For [O ii] the quotient for the electron temperature is calculated from

$$\begin{eqnarray}&&{R}_{O2}=\displaystyle \frac{I(3726)+I(3729)}{I(7319\,\mathring{\rm A} )+I(7330\,\mathring{\rm A} )}.\end{eqnarray} \tag{ 9 }$$

One has to be careful in this case, because the [O ii] auroral lines might be contaminated by recombination emission. Such emission, however, can be quantified and corrected, according to Liu et al. (2001), as such contribution can be fitted (for 0.5 ≤ t ≤ 1.0) by the function

$$\begin{eqnarray}&&\displaystyle \frac{{I}_{R}(7319+7330)}{I(H\beta )}=9.36\cdot {t}^{0.44}\cdot \displaystyle \frac{{O}^{2+}}{{H}^{+}}.\end{eqnarray} \tag{ 10 }$$

Moreover, the ratio of the [O ii] lines is strongly dependent on the electron density. Ideally, one should know the [O ii] density from the ratio I(3726 Å)/I(3729 Å) but very frequently we lack resolution to separate the doublet in which case one has to resort to the [S iii] density, also representing the low-excitation zone. The fitting obtained is

$$\begin{eqnarray}&&t([{\rm{O}}\,{\rm{II}}])={a}_{0}(n)+{a}_{1}(n)\cdot {R}_{O2}+\displaystyle \frac{{a}_{2}(n)}{{R}_{O2}},\end{eqnarray} \tag{ 11 }$$

which also gives t in units of 10⁴ K and where the coefficients are, respectively,

$$\begin{eqnarray*}{a}_{0}(n) & = & 0.2526-0.000357\cdot n-\displaystyle \frac{0.43}{n},\\ {a}_{1}(n) & = & 0.00136+\displaystyle \frac{0.00481}{n},\end{eqnarray*}$$

$$\begin{eqnarray}&&{a}_{2}(n)=35.624-0.0172\cdot n-\displaystyle \frac{25.12}{n},\end{eqnarray} \tag{ 12 }$$

being n the electron density in units of cm⁻³ and using the collisional coefficients from Pradhan et al. (2006) and Tayal (2007). The fittings were made in the range t = 0.8–2.5 in units of 10⁴ K with an uncertainty better than 2%. The resulting fittings for different electron densities can be seen in right panel of Figure 4.

Often, the auroral [O ii] lines are not observed with good S/N or they are outside our observed spectral range. In that case, it is practical to use some relation based on photoionization models in order to infer t[O ii] from t[O iii]. For instance, the relation

$$\begin{eqnarray}&&t([{\rm{O}}\,{\rm{II}}])=\displaystyle \frac{2}{t{([{\rm{O}}{\rm{III}}])}^{-1}+0.8},\end{eqnarray} \tag{ 13 }$$

which is based on Stasińska (1990) models is frequently accepted. However, such an expression neglects the dependence of t[O ii] on the density, consistent with the dispersion for the objects for which both temperatures have been derived from observations as can be seen in left upper panel of Figure 5. In the same panel are shown fittings to grids of models, presented in Pérez-Montero & Díaz (2003), with different electron densities. The density-dependent calibration obtained in this case and given in Hägele et al. (2006) is

$$\begin{eqnarray}&&t([{\rm{O}}\,{\rm{II}}])=\displaystyle \frac{1.2+0.002\cdot n+\tfrac{4.2}{n}}{t{([{\rm{O}}{\rm{III}}])}^{-1}+0.08+0.003\cdot n+\tfrac{2.5}{n}}.\end{eqnarray} \tag{ 14 }$$

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The [S ii] line ratio is commonly used to determine electron density for the low-excitation zone. It is generally assumed that the nebula has constant density, although there is growing evidence for the existence of a density profile instead. Fortunately, the high-excitation species have diagnostic ratios that are almost no density sensitive.

Electron densities are necessary for the derivation of chemical abundances of ions of the type np², such as O⁺. These densities were derived using the following emission-line ratio:

$$\begin{eqnarray}&&{R}_{S2}=\displaystyle \frac{I(6716)}{I(6731)}\end{eqnarray} \tag{ 15 }$$

The following expression is proposed to derive the electron density

$$\begin{eqnarray}&&{{\rm{n}}}_{e}([{\rm{S}}\,{\rm{II}}])={10}^{3}\cdot \displaystyle \frac{{R}_{S2}\cdot {a}_{0}(t)+{a}_{1}(t)}{{R}_{S2}\cdot {b}_{0}(t)+{b}_{1}(t)}\end{eqnarray} \tag{ 16 }$$

with n_e in units of cm⁻³ and t in units of 10⁴ K. Using the appropriate fittings and pyneb with collision strengths from Tayal & Zatsarinny (2010) gives these polynomial fittings to the coefficients

$$\begin{eqnarray*}&&{a}_{0}(t)=16.054-7.79/t-11.32\cdot t,\end{eqnarray*}$$

$$\begin{eqnarray*}&&{a}_{1}(t)=-22.66+11.08/t+16.02\cdot t,\end{eqnarray*}$$

$$\begin{eqnarray*}&&{b}_{0}(t)=-21.61+11.89/t+14.59\cdot t.\end{eqnarray*}$$

$$\begin{eqnarray}&&{b}_{1}(t)=9.17-5.09/t-6.18\cdot t.\end{eqnarray} \tag{ 17 }$$

This expression fits the density calculated by pyneb better than a 1% for temperatures in the range $0.6\lt {t}_{e}\lt 2.2$ and densities in the range $10\lt {n}_{e}\lt 1000$. Here, t is generally t[O iii], although an iterative process could be used to calculate it with t[S iii] given that this temperature, like t[O ii], a type np³ ion, is density dependent. The ratio used in this case is

$$\begin{eqnarray}&&{R}_{S2}^{\prime }=\displaystyle \frac{I(6717)+I(6731)}{I(4068)+I(4076)}.\end{eqnarray} \tag{ 18 }$$

For the [S ii] auroral lines it is enough to measure one of them, as they are related by a fixed theoretical ratio, I(4068 Å) ≈3 · I (4076 Å). We can calculate in this way the [S ii] temperature:

$$\begin{eqnarray}&&t([{\rm{S}}\,{\rm{II}}])={a}_{0}(n)+{a}_{1}(n)\cdot {R}_{S2}^{\prime }+\displaystyle \frac{{a}_{2}(n)}{{R}_{S2}^{\prime }}+\displaystyle \frac{{a}_{3}(n)}{{R}_{S2}^{\prime 2}},\end{eqnarray} \tag{ 19 }$$

where

$$\begin{eqnarray*}&&{a}_{0}(n)=0.99+\displaystyle \frac{34.79}{n}+\displaystyle \frac{321.82}{{n}^{2}},\end{eqnarray*}$$

$$\begin{eqnarray*}&&{a}_{1}(n)=-0.0087+\displaystyle \frac{0.628}{n}+\displaystyle \frac{5.744}{{n}^{2}},\end{eqnarray*}$$

$$\begin{eqnarray*}&&{a}_{2}(n)=-7.123+\displaystyle \frac{926.5}{n}-\displaystyle \frac{94.78}{{n}^{2}},\end{eqnarray*}$$

$$\begin{eqnarray}&&{a}_{3}(n)=102.82+\displaystyle \frac{768.852}{n}-\displaystyle \frac{5113}{{n}^{2}}\end{eqnarray} \tag{ 20 }$$

also fitted with the coefficients by Tayal & Zatsarinny (2010) in the range t = 0.8 to 2.5 in units of 10⁴ K and n_e up to 500 cm⁻³ with a precision better than 2%.

When the [S ii] auroral lines are not available it is usually assumed that t([S ii]) ≈ t[O ii]. There is evidence, however, suggesting a somewhat lower value. From the models used in Pérez-Montero (2003), it is obtained a lineal fitting:

$$\begin{eqnarray}&&t([{\rm{S}}\,{\rm{II}}])=0.71\cdot t([{\rm{O}}\,{\rm{II}}])+0.12\end{eqnarray} \tag{ 21 }$$

for a 100 cm⁻³ number density. For lower densities, this expression seems to be valid. In any case, for the few objects for which we have a simultaneous measurement of both temperatures, the dispersion is quite large as can be seen in lower left panel of Figure 5 for H ii galaxies.

Direct measurements of the [S iii] temperature become possible with the availability of the collisional lines in the near-IR:

$$\begin{eqnarray}&&{R}_{S3}=\displaystyle \frac{I(9069)+I(9532)}{I(6312)}.\end{eqnarray} \tag{ 22 }$$

This expression can be simplified in case of lacking one of the near-IR lines, knowing that I(9532 Å) ≈ 2.44 · I(9069 Å). With this ratio, it is possible the corresponding temperature with the following fitting in the range t = 0.6–2.5 using the collision strengths from Hudson et al. (2012),

$$\begin{eqnarray}&&t([{\rm{S}}\,{\rm{III}}])=0.5147+0.0003187\cdot {R}_{S3}+\displaystyle \frac{23.64041}{{R}_{S3}},\end{eqnarray} \tag{ 23 }$$

with a precision better than 1% in the range 0.6 < t([S iii]) < 1.5 and better than 3% up to values t([S iii]) = 2.5. These values enhance in less than a 3% when the considered density goes from 100 to 1000 cm⁻³.

As discussed in Garnett (1992), t[S iii] is in between the temperatures of [O iii] and of [O ii] and allows us to calculate the ${{\rm{S}}}^{2+}$ abundance from just the 6312 Å line in high-metallicity objects. In case that the [[S iii]] cannot be measured, it is possible to derive t([[S iii]]) from t([O iii]). and vice versa The empirical fitting given by Hägele et al. (2006) is

$$\begin{eqnarray}&&t([{\rm{S}}\,{\rm{III}}])=(1.19\pm 0.08)\cdot t([{\rm{O}}\,{\rm{III}}])-(0.32\pm 0.10),\end{eqnarray} \tag{ 24 }$$

which can be seen in left lower panel of Figure 5.

The T_e of [N ii] can be calculated using the ratio

$$\begin{eqnarray}&&{R}_{N2}=\displaystyle \frac{I(6548)+I(6583)}{I(5755)}\end{eqnarray} \tag{ 25 }$$

which, with the corresponding fitting leads to the expression

$$\begin{eqnarray}&&t([{\rm{N}}\,{\rm{II}}])=0.6153-0.0001529\cdot {R}_{N2}+\displaystyle \frac{35.3641}{{R}_{N2}},\end{eqnarray} \tag{ 26 }$$

also in units of 10⁴ K in the range t = 0.6–2.2 using collision strengths from Tayal (2011). This fit gives a precision better than 1% in the range 0.7 < t([[N ii]]) < 2.2 ,and better than 3% in the range 0.6 < t([[N ii]]) < 0.7. It was calculated for a density of 100 cm⁻³, but for a density of 1000 cm⁻³, the temperature is reduced in less than a 1%.

The nebular lines of [N ii] are very close to Hα, so they sometimes appear blended to this line; therefore, it is not possible to measure both of them. In this case, it is often assumed the following theoretical relation between them, I(6583) ≈ 2.9 · I(6548). Besides, the auroral line of [N ii] is affected by recombination emission, which can be corrected using the next expression proposed by Liu et al. (2001):

$$\begin{eqnarray}&&\displaystyle \frac{{I}_{R}(5755)}{I(H\beta )}=3.19\cdot {t}^{0.30}\cdot \displaystyle \frac{{N}^{2+}}{{H}^{+}}\end{eqnarray} \tag{ 27 }$$

in the range between 5000 and 20,000 K.

Unfortunately, the auroral line has very low signal-to-noise ratio (S/N), so it is usually considered the approximation t([N ii]) ≈ t([O ii]) as valid. This relation is confirmed by photoionization models, but is quite sensitive to density and the inner ionization structure of the nebula and it is possible to reach values closer to t([S iii]) in some cases; therefore, in all cases it cannot be taken without any sort of uncertainty.

Another possibility is to calculate t([N ii]) directly from t([O iii]) using the expression derived using photoionization models by Pérez-Montero & Contini (2009), as can be seen in lower right panel of Figure 5:

$$\begin{eqnarray}&&t([{\rm{N}}\,{\rm{II}}])=\displaystyle \frac{1.85}{t{([{\rm{O}}{\rm{III}}])}^{-1}+0.72}\end{eqnarray} \tag{ 28 }$$

The Balmer temperature is an alternative to other temperatures calculated using collisionally excited lines. It depends on the value of the Balmer jump (BJ) in emission (i.e., the continuum nebular emission at a bluer wavelength than the Balmer series). To measure this value, it is necessary to fit the continuum in both sides of the discontinuity (${\lambda }_{B}$ = 3646 Å). An example of the measurement of the BJ in an the optical spectrum of an H ii galaxy can be seen in Figure 6.

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The contribution of the underlying stellar population affects, between other factors, to the emission of hydrogen lines near from the BJ. The increment in the number of lines at shorter wavelengths produces blends between them that trend to reduce the level of the continuum at a redder wavelength of the discontinuity, so it is necessary to take into the account all this in the final uncertainty. Once measured, the BJ and the Balmer temperature (T(Bac)) is measured from the quotient of the flux of the jump and the emission of the line H11 by means of the expression given in Liu et al. (2001):

$$\begin{eqnarray}T({Bac}) & = & 368\times (1+0.259{y}^{+}+3.409{y}^{++})\\ & & \times {\left(\displaystyle \frac{{BJ}}{H11}\right)}^{-3/2}\,K,\end{eqnarray} \tag{ 29 }$$

where ${y}^{+}$ and ${y}^{++}$ are the ionic abundances of helium once and twice ionized, respectively, and BJ is in ergs cm⁻² s⁻¹ Å⁻¹.

There are other collisional excited lines for other ions in the optical spectrum, but their corresponding temperatures or densities cannot be directly measured using the appropriate emission line ratios in the optical or their auroral lines are too weak to be detected. This is the case for [Ar iv], for instance, for which we can derive the density in the inner nebula, or the temperatures of the rest of ions for which the corresponding auroral line are not available in the observed spectral range, or there is not enough S/N to detect their recombination lines. In this case, it usually takes some assumptions about the gas structure, and the position of each species is associated to another ion whose temperature is known. This is the case of the next ions, assuming the three-zone temperature sketch proposed by Garnett (1992):

$$\begin{eqnarray}t([\mathrm{Ne}\,{\rm{III}}]) & \approx & t([\mathrm{Fe}\,{\rm{III}}])\approx t(\mathrm{He}\,{\rm{II}})\approx t([\mathrm{Ar}\,{\rm{IV}}])\\ & \approx & t([{\rm{O}}\,{\rm{III}}])\approx {t}_{h}\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&t([\mathrm{Ar}\,{\rm{III}}])\approx t([{\rm{S}}\,{\rm{III}}])\approx {t}_{m}\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&t([{\rm{S}}\,{\rm{II}}])\leqslant t([{\rm{N}}\,{\rm{II}}])\approx t([{\rm{O}}\,{\rm{II}}])\approx {t}_{l}.\end{eqnarray} \tag{ 32 }$$

Helium lines, as equal as hydrogen ones in the visible spectrum, have a recombination nature and they are generally strong and numerous, although many of them are usually blended with other lines. Besides, they are affected by absorption of underlying stellar population by fluorescence or by collisional contribution.

The total He abundance can be calculated in the optical spectrum using lines of He i and He ii. Following the same equations as those described in Olive & Skillman (2004), once subtracted, the stellar absorption and reddening correction has been carried out and the abundance of He⁺ can be derived for each He i RL as

$$\begin{eqnarray}&&{y}^{+}(\lambda )=\displaystyle \frac{{\rm{I}}(\lambda )}{{\rm{I}}(H\beta )}\displaystyle \frac{{F}_{\lambda }(n,t)}{{f}_{\lambda }(n,t,\tau )},\end{eqnarray} \tag{ 34 }$$

where ${F}_{\lambda }(n,t)$ is the theoretical emissivity scaled to Hβ and ${f}_{\lambda }$ is the optical depth function. It generally uses the strongest lines, including He i $\lambda \lambda$ 4471, 5876, 6678, and 7065 Å, and then  ${y}^{+}$ is calculated using a weighted mean; however, a precise determination of He abundance requires the use of as many lines as possible (see Peimbert et al. 2002).

Using pyneb with the atomic data from Porter et al. (2013), the theoretical emissivities related to Hβ can be calculated:

$$\begin{eqnarray*}&&{F}_{4471}=(2.0301+1.5\cdot {10}^{-5}\cdot n)\cdot {t}^{0.1463-0.0005\cdot n},\end{eqnarray*}$$

$$\begin{eqnarray*}&&{F}_{5876}=(0.745-5.1\cdot {10}^{-5}\cdot n)\cdot {t}^{-0.226-0.0011\cdot n},\end{eqnarray*}$$

$$\begin{eqnarray*}&&{F}_{6678}=(2.612-0.000146\cdot n)\cdot {t}^{0.2355-0.155\cdot n},\end{eqnarray*}$$

$$\begin{eqnarray}&&{F}_{7065}=(4.329-0.0024\cdot n)\cdot {t}^{-0.368-0.0017\cdot n}\end{eqnarray} \tag{ 35 }$$

with t in units of 10⁴ K and n in units of cm⁻³ and calculated for temperatures between 8000 and 25,000 K and densities between 10 and 1000 cm⁻³. The precision of the fittings is better than 1% in all cases. Normally, it takes the temperature of [O iii], as it is the most precise. The optical depth functions can be taken from Olive & Skillman (2004), but these are only important for high-extinction objects, and when it is required, a very precise measurement of this abundance (e.g., for the search for the primordial helium abundance).

The abundance of He²⁺ can be calculated using the He ii 4686 Å line and a high-excitation temperature by means of the recombination coefficients by Storey & Hummer (1995) using the following expression:

$$\begin{eqnarray}&&{y}^{2+}=\displaystyle \frac{{\rm{I}}(4686)}{{\rm{I}}(H\beta )}\cdot 0.0416\cdot {t}^{-0.146}\end{eqnarray} \tag{ 36 }$$

calculated in the same range as He i emissivities.

Then, the total He abundance can be derived the addition of the abundances of the two ions

$$\begin{eqnarray}&&y={y}^{+}+{y}^{2+},\end{eqnarray} \tag{ 37 }$$

where y is the relative abundance of helium (He/H), ${y}^{+}$ is the abundance of He⁺, and ${y}^{2+}$ is the abundance of He²⁺.

The chemical abundance of O⁺ can be derived with the relative intensity of [O ii] 3726, 3729 Å emission lines to Hβ, and the corresponding temperature using the following expression obtained from fittings to pyneb using the default collision strengths from Pradhan et al. (2006) and Tayal (2007):

$$\begin{eqnarray*}12+\mathrm{log}\left(\displaystyle \frac{{O}^{+}}{{H}^{+}}\right) & = & \mathrm{log}\left(\displaystyle \frac{I(3726)+I(3729)}{I(H\beta )}\right)\\ & & +5.887+\displaystyle \frac{1.641}{{t}_{l}}-0.543\cdot \mathrm{log}({t}_{l})\\ & & +0.000114\cdot {n}_{e},\end{eqnarray*} \tag{ 38 }$$

with a precision better than 0.01 dex in the temperature range 0.7 < t(O⁺) < 2.5 and density of 100 cm⁻³. For a density of 1000 cm⁻³, the precision is better than 0.02 dex.

Alternatively, Kniazev et al. (2003) suggest the use of the 7319,7330 Å doublet in those objects observed with a setup that does not cover the 3727 [O ii] line:

$$\begin{eqnarray*}12+\mathrm{log}\left(\displaystyle \frac{{{\rm{O}}}^{+}}{{{\rm{H}}}^{+}}\right) & = & \mathrm{log}\left[\displaystyle \frac{I(7320+7330)}{I({\rm{H}}\beta )}\right]\\ & & +7.21+\displaystyle \frac{2.511}{{t}_{l}}-0.422\cdot \mathrm{log}{t}_{l}\\ & & +{10}^{-3.40}\,{n}_{{\rm{e}}}(1-{10}^{-3.44}\,\cdot \,{n}_{{\rm{e}}}).\end{eqnarray*} \tag{ 39 }$$

Regarding ${{\rm{O}}}^{2+}$, its chemical abundance was derived using the relative intensity of [O iii] 4959, 5007 Å emission lines to Hβ and the corresponding temperature using the following expression obtained from fittings to pyneb:

$$\begin{eqnarray}12+\mathrm{log}\left(\displaystyle \frac{{O}^{2+}}{{H}^{+}}\right) & = & \mathrm{log}\left(\displaystyle \frac{I(4959)+I(5007)}{I(H\beta )}\right)+6.1868\\ & & +\displaystyle \frac{1.2491}{{t}_{h}}-0.5816\cdot \mathrm{log}({t}_{h})\end{eqnarray} \tag{ 40 }$$

with a precision better than 0.01 dex in the temperature range 0.7 < t(O²⁺) < 2.5. A change in the density from 10 to 1000 cm⁻³ implies a decrease of less than 0.01 dex in the derived abundance.

The total oxygen abundance can be approximated by

$$\begin{eqnarray}&&\displaystyle \frac{O}{H}=\displaystyle \frac{{{\rm{O}}}^{+}+{{\rm{O}}}^{2+}}{{{\rm{H}}}^{+}}\end{eqnarray} \tag{ 41 }$$

given that due to the charge exchange reaction, the relative fractions of neutral oxygen and hydrogen are similar:

$$\begin{eqnarray}&&\displaystyle \frac{{O}^{0}}{O}=\displaystyle \frac{{H}^{0}}{H}\end{eqnarray} \tag{ 42 }$$

However, in some high-excitation spectra where the He ii 4586 Å is seen, it can be considered that part of the O is under the form of ${{\rm{O}}}^{3+}$. In that case, it can be assumed that

$$\begin{eqnarray}&&\mathrm{ICF}({O}^{+}+{O}^{2+})=1+\displaystyle \frac{{y}^{2+}}{{y}^{+}}.\end{eqnarray} \tag{ 43 }$$

The abundances are obtained from the 6717 and 6731 Å lines for S⁺, and by the 9069 and 9532 Å lines for ${{\rm{S}}}^{2+}$, although for the latter, the 6312 Å line can also be used using the following expressions:

$$\begin{eqnarray}12+\mathrm{log}\left(\displaystyle \frac{{S}^{+}}{{H}^{+}}\right) & = & \mathrm{log}\left(\displaystyle \frac{I(6717)+I(6731)}{I(H\beta )}\right)\\ & & +5.463+\displaystyle \frac{0.941}{{t}_{l}}-0.37\cdot \mathrm{log}{t}_{l})\end{eqnarray} \tag{ 44 }$$

and

$$\begin{eqnarray}12+\mathrm{log}\left(\displaystyle \frac{{S}^{2+}}{{H}^{+}}\right) & = & \mathrm{log}\left[\displaystyle \frac{I(9069)+I(9532)}{I(H\beta )}\right]\\ & & +5.983+\displaystyle \frac{0.661}{{t}_{m}}-0.527\mathrm{log}({t}_{m}).\end{eqnarray} \tag{ 45 }$$

In case the near-IR [S iii] lines cannot be measured but the auroral line at 6312 Å is available with good S/N, it is possible to derive ${{\rm{S}}}^{2+}$ abundances from the expression:

$$\begin{eqnarray}12+\mathrm{log}\left(\displaystyle \frac{{S}^{2+}}{{H}^{+}}\right) & = & \mathrm{log}\left(\displaystyle \frac{I(6312)}{I(H\beta )}\right)\\ & & +6.695+\displaystyle \frac{1.664}{{t}_{m}}-0.513\cdot \mathrm{log}({t}_{m}).\end{eqnarray} \tag{ 46 }$$

These were derived using the same collisional coefficients as used for the derivation of expressions for temperature. The fittings have a precision better than 0.01 dex in the temperature range 0.7 < t.

The ICF for sulfur takes into account the ${{\rm{S}}}^{3+}$ abundance, which cannot be determined in the optical range. A good approximation is given by Stasińska (1978):

$$\begin{eqnarray}&&\mathrm{ICF}({S}^{+}+{S}^{2+})={\left[1-{\left(\displaystyle \frac{{O}^{2+}}{{O}^{+}+{O}^{2+}}\right)}^{\alpha }\right]}^{-1/\alpha }.\end{eqnarray} \tag{ 47 }$$

Although it is customary to write this expression as a function of the O⁺/(O⁺ + O²⁺) ionic fraction, we reformulated it in terms of O²⁺/(O⁺ + O²⁺) because the errors that are associated to ${{\rm{O}}}^{2+}$ are considerably smaller than for ${{\rm{O}}}^{+}$, and for a sample of objects with observed [S iv] line at 10.5 μm, it was derived $\alpha \approx 3.27$ (Dors et al. 2016).

We can calculate N⁺ abundance from the 6548 and 6583 Å lines. The expression to calculate ${N}^{+}/{H}^{+}$ abundance using these lines and assuming a low-excitation temperature is

$$\begin{eqnarray}12+\mathrm{log}\left(\displaystyle \frac{{N}^{+}}{{H}^{+}}\right) & = & \mathrm{log}\left(\displaystyle \frac{I(6548)+I(6583)}{I(H\beta )}\right)\\ & & +6.291+\displaystyle \frac{0.90221}{{t}_{l}}-0.5511\cdot \mathrm{log}({t}_{l})\end{eqnarray} \tag{ 48 }$$

with a precision better than 0.01 dex in the temperature range 0.6 < t < 2.2. It decreases less than 0.01 dex when the considered density goes from 100 to 1000 cm⁻³.

One can calculate quite precisely the total abundance of Nitrogen assuming that

$$\begin{eqnarray}&&\displaystyle \frac{{N}^{+}}{N}=\displaystyle \frac{{O}^{+}}{O},\end{eqnarray} \tag{ 49 }$$

owing to the similarity of ionization potentials of O⁺ and ${N}^{+}$, which leads to the corresponding ICF:

$$\begin{eqnarray}&&\mathrm{ICF}({N}^{+})=\displaystyle \frac{O}{{O}^{+}}.\end{eqnarray} \tag{ 50 }$$

The ratio N⁺/O⁺ can be derived directly from the expression

$$\begin{eqnarray}\mathrm{log}\left(\displaystyle \frac{{N}^{+}}{{O}^{+}}\right) & = & \mathrm{log}\left(\displaystyle \frac{I(6583)}{I(3726)+I(3729)}\right)\\ & & +0.493-0.025\cdot {t}_{l}\displaystyle \frac{0.687}{{t}_{l}}\\ & & +0.1621\cdot \mathrm{log}({t}_{l}).\end{eqnarray} \tag{ 51 }$$

Ne has [Ne iii] prominent emission lines in the blue part of the spectrum and can be a good tracer of metallicity, as it is not depleted into dust grains. Because one of the emission lines (3968 Å) usually appears blended with and H i recombination line (H7), the Ne²⁺ ionic abundance can be derived using the following expression and the electron temperature of the high-excitation zone (usually t([O iii])):

$$\begin{eqnarray}12+\mathrm{log}\left(\displaystyle \frac{{{Ne}}^{2+}}{{H}^{+}}\right) & = & \mathrm{log}\left(\displaystyle \frac{I(3968)+I(6583)}{I(H\beta )}\right)\\ & & +6.947-\displaystyle \frac{1.614}{{t}_{h}}-0.4291\cdot \mathrm{log}({t}_{h}),\end{eqnarray} \tag{ 52 }$$

with a precision better than 2% in the range of temperature from 0.6 to 2.2 in units of 10⁴ K. The collisional coefficients are from McLaughlin et al. (2000).

The ionization correction factor for neon can be calculated according to the expression given by Pérez-Montero et al. (2007) based on photoionization models:

$$\begin{eqnarray}&&\mathrm{ICF}({{Ne}}^{2+})=0.753+0.142\,x+\displaystyle \frac{0.171}{x},\end{eqnarray} \tag{ 53 }$$

where x = O²⁺/(O⁺ + O²⁺). This expression deviates from the classical approximation Ne/O ≈ Ne²⁺/O²⁺ used to derive total neon abundances.

For argon, we use the [Ar iii] 7137 Å line. It is possible to measure the lines of [Ar iv] at 4713 and 4740 Å. as well. Nevertheless, the first of them usually appears blended with another line of He ii at 4711 Å that is difficult to correct, so it is better to use the second and brighter to calculate the ionic abundance of Ar³⁺,

$$\begin{eqnarray}12+\mathrm{log}\left(\displaystyle \frac{{{Ar}}^{2+}}{{H}^{+}}\right) & = & \mathrm{log}\left(\displaystyle \frac{I(7135)}{I(H\beta )}\right)\\ & & +6.100+\displaystyle \frac{0.86}{{t}_{m}}-0.404\cdot \mathrm{log}({t}_{m})\end{eqnarray} \tag{ 54 }$$

$$\begin{eqnarray}12+\mathrm{log}\left(\displaystyle \frac{{{Ar}}^{3+}}{{H}^{+}}\right) & = & \mathrm{log}\left(\displaystyle \frac{I(4740)}{I(H\beta )}\right)\\ & & +6.306+\displaystyle \frac{1.232}{{t}_{h}}-0.703\cdot \mathrm{log}({t}_{h}),\end{eqnarray} \tag{ 55 }$$

both with a precision better than 2% in the fitting in the same temperature range than in Ne using the collisional coefficients from Galavis et al. (1995) for Ar²⁺ and Ramsbottom & Bell (1997) for Ar⁺³.

As in the case of neon, the total abundance of argon can be calculated using the ionization correction factors (ICF(Ar²⁺) and the ICF(Ar²⁺+Ar³⁺)) given by Pérez-Montero et al. (2007). The first one can be used only when we cannot derive a value for Ar³⁺. The expressions for these ICFs are

$$\begin{eqnarray}&&\mathrm{ICF}({{Ar}}^{2+})=0.596+0.967(1-x)+\displaystyle \frac{0.077}{(1-x)}\end{eqnarray} \tag{ 56 }$$

$$\begin{eqnarray}&&\mathrm{ICF}({{Ar}}^{2+}+{{Ar}}^{3+})=0.928+0.364(1-x)+\displaystyle \frac{0.006}{(1-x)}\end{eqnarray} \tag{ 57 }$$

where x = O²⁺/(O⁺+O²⁺).

The abundance of Fe is usually derived from the absorption stellar features, but there are some high-excitation [Fe iii] lines that can be used to derive Fe²⁺ abundances in the ionized gas-phase. For instance, from 4658 Å from pyneb it is obtained:

$$\begin{eqnarray}12+\mathrm{log}\left(\displaystyle \frac{{{Fe}}^{2+}}{{H}^{+}}\right) & = & \mathrm{log}\left(\displaystyle \frac{I(4658)}{I(H\beta )}\right)\\ & & +6.288+\displaystyle \frac{1.408}{{t}_{h}}-0.203\cdot \mathrm{log}({t}_{h})\end{eqnarray} \tag{ 58 }$$

using the collisional coefficients of Fe²⁺ from Zhang (1996) in the range t = 0.8–2.5. The ICF to derive the total Fe abundance was proposed by Rodríguez & Rubin (2004):

$$\begin{eqnarray}&&\mathrm{ICF}({{Fe}}^{2+})={\left(\displaystyle \frac{{O}^{+}}{{O}^{2+}}\right)}^{0.09}\left[1+\displaystyle \frac{{O}^{2+}}{{O}^{+}}\right].\end{eqnarray} \tag{ 59 }$$

Sometimes it is possible to derive an electron temperature and its corresponding ionic abundance. Nevertheless, the total abundance cannot be calculated, because the lines corresponding to other excitation lines cannot be measured. This is the case of O/H when the [O ii] 3727 Å lines are out of the spectral range or their corresponding auroral lines at 7319 and 7330 Å are too faint.

In this case, the total oxygen abundance can be estimated directly from the [O iii] electron temperature as proposed by Amorín et al. (2015), which can be seen in left panel of Figure 7:

$$\begin{eqnarray}&&12+\mathrm{log}(O/H)=9.22(\pm 0.03)-0.89(\pm 0.02)\cdot t([{\rm{O}}\,{\rm{III}}]),\end{eqnarray} \tag{ 60 }$$

with a dispersion of 0.15 dex. This expression can also be used to derive the temperature from a strong-line derivation of O/H. The electron temperature is useful, for example, to quantify the extinction from the Balmer decrement.

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The most widely used parameter based on oxygen lines that can be related to metallicity is R23 given by Pagel et al. (1979):

$$\begin{eqnarray}&&R23=\displaystyle \frac{I(3727)+I(4959,5007)}{I(H\beta )}.\end{eqnarray} \tag{ 61 }$$

However, there are some known limitations to give a calibration of this parameter with O/H. The main limitation is that the relation is double-valued: R23 grows with O/H for low-Z and decreases for high-Z, as the main source of cooling when the temperature is lower are hydrogen recombination lines. In the so-called turnover region (around 12+log(O/H) ≈ 8.0–8.3, as can be seen in right panel of Figure 7), the dispersion is very high and it is very difficult to provide an accurate determination.

For this tutorial and for the sake of consistency, I provide fittings to the sample of objects with a direct determination of O/H in Pérez-Montero & Contini (2009), following the same functional expressions given by Kobulnicky et al. (1999) based on R23 and on the excitation as a function of the ratio of [O ii] and [O iii]. The fittings for both branches at an average excitation in each branch are shown in Figure 7.

In the upper branch (12+log(O/H) > 8.25),

$$\begin{eqnarray}&&12+{\rm{log}}(O/H)\\ &&\,=8.656-0.411\cdot x-0.586\cdot {x}^{2}+0.469\cdot {x}^{3}\\ &&\,-y\cdot (-0.224+0.309\cdot x+0.741\cdot {x}^{2}+0.722\cdot {x}^{3}),\end{eqnarray} \tag{ 62 }$$

where x = log(R23) and y = log([O ii]3727/[O iii]4959+5007). The dispersion of this fitting is taken as the standard deviation of the residuals to the abundances from the direct method is 0.17 dex.

For the lower branch (12 + log(O/H) < 8.0),

$$\begin{eqnarray}&&12+{log}(O/H)\\ &&\,=7.247-1.196\cdot x+4.078\cdot {x}^{2}-2.194\cdot {x}^{3}\\ &&\,-y\cdot (1.076-3.735\cdot x+5.671\cdot {x}^{2}-2.947\cdot {x}^{3}),\end{eqnarray} \tag{ 63 }$$

with a dispersion of 0.14 dex.

In cases where the [O iii] are not observed owing the studied spectral range, Pérez-Montero et al. (2007) propose to use the empirical relation with [Ne iii] at 3968 Å (=[O iii] 5007/15.37) and also used a hydrogen recombination line at a blue wavelength, such as Hγ or Hδ.

The simplest strong-line ratio used to derive O/H using [N ii] lines is the N2 parameter, defined by Storchi-Bergmann et al. (1994):

$$\begin{eqnarray}&&N2=\mathrm{log}\left(\displaystyle \frac{I(6583)}{I({\rm{H}}\alpha )}\right).\end{eqnarray} \tag{ 64 }$$

Notice that, contrary to R23, the log is in the definition of the parameter. This parameter has the advantage of being totally independent on reddening correction or flux calibration uncertainties. The linear fitting proposed by Pérez-Montero & Contini (2009) gives

$$\begin{eqnarray}&&12+\mathrm{log}(O/H)=9.07+0.79\cdot N2,\end{eqnarray} \tag{ 65 }$$

valid for all the range of metallicity with a dispersion of 0.34 dex. This relation can be seen in Figure 8. This is usually used to decide to the branch of the R23 parameter, but the dispersion is very high owing to a large dependence on ionization parameter and also because of the dispersion in the O/H-N/O relation.

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The dependence on log U can be reduced through the definition of the O3N2 parameter (Alloin et al. 1979):

$$\begin{eqnarray}&&O3N2=\mathrm{log}\left(\displaystyle \frac{I(5007)}{I(H\beta )}\cdot \displaystyle \frac{I(H\alpha )}{I(6583)}\right).\end{eqnarray} \tag{ 66 }$$

The linear fitting proposed by Pérez-Montero & Contini (2009) leads to

$$\begin{eqnarray}&&12+\mathrm{log}(O/H)=8.74-0.31\cdot O3N2\end{eqnarray} \tag{ 67 }$$

with a dispersion of 0.32 dex, but for O3N2 > 2.0 as the O3N2 parameter is constant for lower metallicities, as can be seen in Figure 8.

Among other strong emission line ratios the parameter N2O2,

$$\begin{eqnarray}&&N2O2=\mathrm{log}\left(\displaystyle \frac{I(6583)}{I(3727)}\right),\end{eqnarray} \tag{ 68 }$$

can be used to derive N/O, as can be seen in Figure 8. The linear fitting proposed by Pérez-Montero & Contini (2009) gives

$$\begin{eqnarray}&&\mathrm{log}(N/O)=0.93\cdot N2O2-0.20\end{eqnarray} \tag{ 69 }$$

with a dispersion of 0.24 dex for all N/O values. A similar parameter for a lower wavelength baseline is the N2S2 parameter,

$$\begin{eqnarray}&&N2S2=\mathrm{log}\left(\displaystyle \frac{I(6583)}{I(6717,6731)}\right),\end{eqnarray} \tag{ 70 }$$

whose linear fitting gives

$$\begin{eqnarray}&&\mathrm{log}(N/O)=1.26\cdot N2S2-0.86\end{eqnarray} \tag{ 71 }$$

with a dispersion of 0.31 dex according to Pérez-Montero & Contini (2009).

Strong-line methods based on sulfur emission lines provide accurate oxygen abundances when the red and near-IR spectral optical range is observed and measured. This is based on the fact that both O and S are primary elements, hence the S/O ratio is not expected to vary. Following an analogous definition as for oxygen lines, Vilchez & Esteban (1996) defined the S23 parameter based on [S ii] and [S iii] lines as

$$\begin{eqnarray}&&S23=\displaystyle \frac{I(6717,6731)+I(9099,9532)}{I(H\beta )}\end{eqnarray} \tag{ 72 }$$

The polynomial fitting proposed by Pérez-Montero & Díaz (2005) to derive total oxygen abundance is

$$\begin{eqnarray}&&12+\mathrm{log}(O/H)=8.15+0.85\cdot x+0.58\cdot {x}^{2},\end{eqnarray} \tag{ 73 }$$

where x = log(S23). This can be used up to solar metallicity (i.e., 12+log(O/H) = 8.69) with a dispersion of 0.20 dex as the standard deviation of the residuals to the O/H derived from the direct method. This relation can be seen in left panel of Figure 9. Similarly, Pérez-Montero et al. (2006) propose a fitting to derive total sulfur abundances:

$$\begin{eqnarray}&&12+\mathrm{log}(S/H)=6.622+1.860\cdot x+0.382\cdot {x}^{2},\end{eqnarray} \tag{ 74 }$$

with a dispersion of 0.185 dex in the range of −1.0 ≤ log(S23) ≤ 0.5.

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For high metallicities, Pérez-Montero & Díaz (2005) propose a polynomial fitting to a combination of both S12 and R23, as can be seen in right panel of Figure 9. This fitting gives

$$\begin{eqnarray}&&12+\mathrm{log}(O/H)=9.09+1.03\cdot x-0.22\cdot {x}^{2},\end{eqnarray} \tag{ 75 }$$

where x = log(S23/R23) and the fitting to the objects has a dispersion of 0.27 dex.