2.1.2. Strong line or statistical methods
When the electron temperature cannot be determined, for example because the observations do not cover the appropriate spectral range or because temperature sensitive lines such as [O III] 4363 cannot be observed, one has to go for statistical methods or "strong line methods". These methods have first been introduced by Pagel et al. (1979) to derive metallicities in giant extragalactic H II regions. They have since then being reconsidered and recalibrated by many authors, among which Skillman (1989), McGaugh (1991, 1994), Pilyugin (2000, 2001).
Pagel et al. (1979) proposed to use the 4 strongest lines of O and H : H, H, [O II] 3727 and [O III] 5007. From Sect. 1, the main parameters governing the relative intensities of the emission lines in a nebula are : <T>, the mean effective temperature of the ionization source, the gas density distribution (parametrized by U in the case of homogeneous spheres), and the metallicity, represented by O/H. Luckily oxygen is at the same time the main coolant in nebulae, and the element whose abundance is most straightforwardly related to the chemical evolution of galaxies. The spectra must be corrected for reddening, which is done by comparing the observed H / H ratio with the case B recombination value at a typical Te and assuming a reddening law (see Sect. 3.3). Therefore two independent line ratios, [O II] 3727 / H and [O III] 5007 / H, remain to determine three quantities. Statistical methods rely on the assumption that <T> (and possibly U) are closely linked to the metallicity, and that it is the metallicity which drives the observed line ratios. Basing on available photoionization model grids, Pagel et al. showed that ([O II] 3727 + [O III] 5007) / H, later called O23, could be used as an indicator of O/H at metallicities above half-solar. Skillman (1989) later argued that this ratio could also be used in the low metallicity regime, in cases when the observations did not have sufficient signal-to-noise to measure the [O III] 4363 line intensity. McGaugh (1994) improved the method and proposed to use both [O III] 5007 /[O II] 3727 and O23 to determine simultaneously O/H and U (his method should perhaps be called the O23 + method). For the reasons explained above, the same value of ([O II] 3727 + [O III] 5007) / H can correspond to either a high or a low value of the metallicity. A useful discriminator is [N II] 6584 / [O II] 3727, since it is an empirical fact that [N II] 6584 / [O II] 3727 increases with O/H (McGaugh 1994).
The expected accuracy of statistical methods is typically 0.2 - 0.3 dex, the method being particularly insensitive in the turnover region at O/H around 3 × 10-4.
On the low metallicity side, the method can easily be calibrated with data on metal-poor extragalactic H II regions where the [O III] 4363 line can be measured. Recently, Pilyugin (2000) has done this using the large set of excellent quality observations of blue compact galaxies by Izotov and coworkers (actually, the strong line method proposed by Pilyugin differs somewhat from the O23 method, but it relies on similar principles). He showed that the method works extremely well at low metallicities (with an accuracy of about 0.04 dex). This is a priori surprising, since giant H II regions are powered by clusters of stars that were formed almost coevally. The most massive stars die gradually, inducing a softening of the ionizing radiation field on timescales of several Myr, which should affect the O23 ratio, as shown by McGaugh (1991) or Stasinska (1998). As discussed by Stasinska et al. (2001), data on H II regions in blue compact dwarf galaxies are probably biased towards the most recent starbursts, and the dispersion in <T> is not as large as could be expected a priori. Another possibility, advocated by Bresolin et al. (1999) in their study of giant H II regions in spiral galaxies is that some mechanism must disrupt the H II regions after a few Myr. Of course, the O23 method is expected of much lower accuracy when applied to H II regions ionized by only a few stars, since in that case the ionizing radiation field varies strongly from object to object.
On the high metallicity side (O/H larger than about 5 × 10-4), the situation is much more complex. In this regime, there is so far no direct determination of O/H to allow a calibration of the O23 method since the [O III] 4363 line is too weak to be measured (at least with 4 m class telescopes). The calibrations rely purely on models but it is not known how well these models represent real H II regions. Besides, at these abundances, the [O II] 3727 and [O III] 5007 line intensities are extremely sensitive to any change in the nebular properties (Oey & Kennicutt 1993, Henry 1993, Shields & Kennicutt 1995). Note that the calibration proposed by Pilyugin (2001) of his related X23 method in the high metallicity regime actually refers to O/H ratios that are lower than 5 × 10-4.
Other methods have been proposed as substitutes to the O23 method. The S23 method, proposed by Vílchez & Esteban (1996) and Díaz & Pérez-Montero (2000) relies on the same principles as the O23 method, but uses ([S II] 6716, 6731 + [S III] 9069, 9532) / H (S23) instead of ([O II] 3727 + [O III] 5007) / H. One advantage over the O23 method is that the relevant line ratios are less affected by reddening. Besides, the excitation levels of the [S II] 6716, 6731 and [S III] 9532 lines are lower than those of the [O II] 3727 and [O III] 5007 lines, so that S23 increases with metallicity in a wider range of metallicities than O23 (the turnover region for S23 is expected at O/H around 10-3). Unfortunately, [S III] 9532 is more difficult to observe than [O III] 5007. Oey & Shields (2000) argue that the S23 method is more sensitive to U than claimed by Diaz & Perez-Montero (2000). This would require futher checks, but in any case, the S23 method could be refined into an S23 + method in the same way as the O23 was refined into the O23 + method.
Stevenson et al. (1993) proposed to use [Ar III] 7136] / [S III] 9532 as an indicator of the electron temperature in metal-rich H II regions, and therefore of their metallicity. This method relies on the idea that the Ar/S ratio is not expected to vary significantly from object to object, and that the Ar++ and S++ zones should be coextensive. However, photoionization models show that, because of the strong temperature gradients expected at high metallicity, this method could lead to spurious results.
Alloin et al. (1979) proposed to use [O III] 5007 / [N II] 6584 as a statistical metallicity indicator. While this line ratio depends on an additional parameter, namely N/O, the accuracy of this method turns out to be similar to that of statistical methods mentioned above. More recently, Storchi-Bergman et al. (1994), van Zee et al. (1998) and Denicolo et al. (2001) advocated for the use of the [N II] 6584 / H ratio (N2) as metallicity indicator. Similarly to [N II] 6584 / [O III] 5007, this ratio shows to be correlated with O/H over the entire range of observed metallicities in giant H II regions. The reason why, contrary to the O23 ratio, it increases with O/H even at high metallicity is due to a conjunction of [N II] 6584 / H being less dependent on Te than O23, N/O being observed to increase with O/H in giant H II regions (at high metallicity at least) and U tending to decrease with metallicity. The advantage of this ratio is that it is independent of reddening and of flux calibration, and is only weakly affected by underlying stellar absorption in the case of observations encompassing old stellar populations. This makes it extremely valuable for ranking metallicities of galaxies up to redshifts about 2.5.
As mentioned above, statistical methods for abundance determinations assume that the nebulae under study form a one parameter family. This is why they work reasonably well in giant H II regions. They are not expected to work in planetary nebulae, where the effective temperatures range between 20000 K and 200000 K. Still, it has been shown empirically that there is an upper envelope in the [O III] 5007 / H vs. O/H relation (Richer 1993), probably corresponding to PNe with the hottest central stars. The existence of such an envelope can be used to obtain lower limits of O/H in PNe located in distant galaxies.