The abundance discrepancy in ionized nebulae: which are the correct abundances?

To explain differences between the values obtained from several temperature diagnostics in ionized nebulae, Peimbert (1967) proposed the existence of temperature inhomogeneities within the ionized gas. Since the CEL-emissivities have an exponential dependency on temperature (Osterbrock & Ferland, 2006), in the case of internal variations, their emissions will be substantially enhanced in the hotter volumes. On the other hand, RLs have emissivities depending on $\sim T_{\rm e}^{-1}$, so their emission is favored in cooler regions, although with substantially smaller bias. Mathematically, this implies that the temperature inferred from the CEL intensity ratios (such as [O iii] 4363/5007) will be systematically higher than the average gas temperature within the volume where the CEL-emitter ion coexists (in the case of [O iii] 4363/5007, the ion would be O²⁺). In such case, the temperature structure of each ionization volume will be characterized by two parameters: $T_{\rm 0}$, which is the average gas temperature, and $t^{2}$, which quantifies how inhomogeneous the temperature is. The mathematical formalism is presented in equations (1) and (2).

In these equations, in a specific volume of the nebula, $dV$, $T_{\rm e}$, $n_{\rm e}$ are the electron temperature and density and $n(X^{i+})$ is the ion’s particle density.

Under this paradigm, when the estimated temperature from CEL intensity ratios (e.g., $T_{\rm e}$([O iii] 4363/5007)) is adopted to calculate chemical abundances relative to H using CEL/RL intensity ratios (e.g., [O iii] 5007/H$\beta$), the overestimation of temperature will result in an underestimation of ionic abundances. Conversely, if chemical abundances relative to H are calculated using RL/RL intensity ratios (e.g., O ii V1 4649+/H$\beta$), both lines will cancel their linear dependence on temperature, and the derived ionic abundances will be correct.

Despite the simplicity and elegance of this proposal, its adoption as a generalized explanation for the observed ADF has not been universal, and various arguments against the existence of temperature variations have been put forward. Although classical photoionization models predict the existence of temperature inhomogeneities, these are too small ($t^{2}\sim 0.004$) (Kingdon & Ferland, 1995; Stasińska & Szczerba, 2001; Ercolano et al., 2007; Ferland et al., 2017) to account for the observed ADF in H ii regions ($t^{2}\sim 0.04$) or PNe ($t^{2}\sim 0.06$) (Peimbert et al., 2017). An exception might be nebula models with very high metallicity, where temperature gradients are more pronounced (Stasińska, 2005). Temperature inhomogeneities in the gas tend to balance out over relatively short timescales (Ferland et al., 2016), so their existence requires an additional physical mechanism to create and maintain them. Additionally, some PNe (Liu et al., 2000; Corradi et al., 2015; García-Rojas et al., 2022) exhibit ADF values that are too large and appear irreconcilable with the $t^{2}$ formalism.

However, observational evidence seems to favour temperature variations as the cause of the ADF in H ii regions with precise measurements of O ii RLs. For instance, García-Rojas & Esteban (2007) found that the ADF of various ions in several Galactic H ii regions depends on the excitation energy of the atomic levels giving rise to CELs, which is one of the predictions of the formalism by Peimbert (1967). Recently, Méndez-Delgado et al. (2023b) analyzed 32 spectra from extragalactic H ii regions, 20 from Galactic H ii regions, and 8 from Galactic ring nebulae with simultaneous detections of O ii RLs, $T_{\rm e}$([O iii] 4363/5007) and $T_{\rm e}$([N ii] 5755/6584). As shown in Fig1, these authors demonstrated that $t^{2}(\text{O}^{2+})$, measured by comparing O ii RLs and [O iii] CELs (Peimbert & Peimbert, 2013), is linearly correlated with the temperature difference $T_{\rm e}$([O iii] 4363/5007) $-$ $T_{\rm e}$([N ii] 5755/6584). This shows that $t^{2}(\text{O}^{2+})$ is indeed related to the gas temperature structure. Moreover, it shows that temperature variations are concentrated in regions of higher ionization degree.

The fact that temperature variations are concentrated in the zones of higher ionization, closer to the ionizing stars of the gas, points to stellar feedback as a possible cause of temperature variations. Phenomena such as shocks, winds, or even variations in the ionizing spectrum of stars can cause significant inhomogeneities in the temperature of the surrounding gas (Peimbert et al., 1991; Perez, 1997; Ercolano et al., 2007). These are not entirely understood physical phenomena, and their incorporation into photoionization models is not widespread. Méndez-Delgado et al. (2023b) then analyzed 8 spectra of ring nebulae, bubbles of ionized gas created by the ejection of material from young massive stars. Five of these analyzed spectra are from NGC 6888, a nebula associated with a Wolf–Rayet WN6 star, and three others from NGC 7635 associated with an O6.5(n)fp star (Esteban et al., 2016). The authors found that the observed trend between $t^{2}(\text{O}^{2+})$ and $T_{\rm e}$([O iii] 4363/5007) $-$ $T_{\rm e}$([N ii] 5755/6584) in the ring nebulae is very similar to what is observed in H ii regions. Interestingly, the spectra of NGC 6888 exhibit the most extreme temperature variations ($t^{2}(\text{O}^{2+})$ $>$ 0.08), while the values for NGC 7635 are lower ($t^{2}(\text{O}^{2+})$ $<$ 0.06).

However, Méndez-Delgado et al. (2023b) tested this same hypothesis in Galactic PNe, and these do not exhibit a similar correlation as in H ii regions, as shown in Fig. 2. This indicates that the cause of the ADF in H ii regions and PNe is different. Although obvious, it is important to note that even if temperature variations alone are not able to explain the observed ADF in PNe, this does not rule out their presence and impact on the determination of chemical abundances in these objects. The ADF in PNe may be caused by various linked physical phenomena. For example, the observational evidence presented by Richer et al. (2022) for NGC 6153 shows that while most of the ADF in this region may be due to chemical inhomogeneities, temperature variations are not negligible, finding values of $t^{2}(\text{O}^{2+})\approx 0.03$, which is typically found in H ii regions.

An alternative explanation to the ADF is the existence of chemical inhomogeneities. If in a gas with a typical nebular temperature of $\sim 10000$K there are gas components of different metallicity, they will induce temperature variations (Zhang et al., 2007). However, if the metallicity of these inclusions is extremely high compared to their hydrogen abundance, their temperature can drop to a few thousand kelvin or even lower. Under such circumstances, the flux of O ii RLs may surpass the flux of the optical [O iii] CELs (Liu, 2003), and a different mathematical formalism than that proposed by Peimbert (1967) must be applied (note the $dV$ dependence on Eq. (2)), as the emission from RLs and CELs will essentially come from different volumes (Stasińska, 2002).

The existence of these high-metallicity clumps has been particularly important to explain cases of extreme ADF (Liu et al., 2000; Yuan et al., 2011; Corradi et al., 2015; Gómez-Llanos & Morisset, 2020; García-Rojas et al., 2022) that have been associated with the existence of central binary stellar systems (Corradi et al., 2015; Jones et al., 2016; Wesson et al., 2018). Narrow-band imaging and Integral Field Unit (IFU) observations of Galactic PNe have been particularly crucial (García-Rojas et al., 2016; Ali & Dopita, 2019; García-Rojas et al., 2022) as they have revealed morphological differences in the distribution of heavy-element RLs and their optical CEL counterparts. In addition, there are indications of kinematical differences between [O iii] optical CELs and O ii RLs (Richer et al., 2013, 2022; Peña et al., 2017), which strengthens this scenario.

However, although this scenario provides a plausible explanation for the extreme ADF in some PNe, it does not entirely answer the initial question of which of the abundances, based on CELs or RLs, is correct. One of the most interesting proposals to assess how important the error in chemical abundance determinations is under this scenario is the ”Abundance Contrast Factor (ACF)” formalism (see Gómez-Llanos & Morisset, 2020, and the proceeding by Morisset et al. in this series). The ACF measures the proportion of flux emitted by the cold component relative to the total emission. If the cold component is extremely hydrogen-poor, heavy-element RLs will have the most significant errors. However, H i RLs can also have a significant bias by increasing their emission in the cold component, affecting the inferred abundances from heavy-element CEL to H i RL intensity ratios.

Even in cases where the high-metallicity clumps are so hydrogen-poor that the impact on H i RLs is negligible, abundances observationally determined with CELs may not be reliable. Besides potential existence of real temperature inhomogeneities in the “hot” gas component emitting CELs, the existence of extremely cold clumps emitting a large amount of heavy-element RLs could bias the $T_{\rm e}$ values inferred from auroral to nebular CEL ratios by contaminating the emission of auroral lines (e.g., [O iii] $\lambda$4363, [N ii] $\lambda$5755) (Rubin, 1986; Gómez-Llanos et al., 2020). In order to “decontaminate” the temperatures determined with CEL ratios, it is necessary to know precisely the temperature and ionic abundances of the high-metallicity clump that is emitting the “RL-contamination”. This requires determining the physical conditions using heavy-element RLs such as O ii $\lambda$4089/$\lambda$4649 to determine $T_{\rm e}$ or O ii $\lambda$4661/$\lambda$4649 to determine $n_{\rm e}$ (Wesson et al., 2005; Peimbert & Peimbert, 2013; Storey et al., 2017). This involves measuring heavy-element RLs that are comparatively fainter than those traditionally used to determine chemical abundances. A correction like the one proposed by Liu et al. (2000) in their equations (1) and (2) using physical conditions and ionic abundances determined in the “hot” gas component is conceptually incorrect, and it has not been demonstrated to be “less incorrect” than not making any correction.

In addition to the difficulty of explaining the creation and survival of high-metallicity clumps (Henney & Stasińska, 2010), extrapolating the existence of chemical inhomogeneities to explain the ADF in all PNe, not just extreme cases, could imply the existence of a wide variety of ACF values. If the high-metallicity component is not so hydrogen-poor, the emission of both CELs and RLs can come from both volumes. Such a scenario would imply that both estimates of CELs and RLs using the standard “direct method” are essentially incorrect. On the other hand, there are still some interesting questions about the observational evidence of the high-metallicity components. For example, García-Rojas et al. (2022) has shown that the emission of O ii RLs has a different morphology than its collisionally excited counterparts, as shown in Fig. 3. This has been argued as evidence of the presence of a cold gas component with a temperature of $\sim 800$ K. However, excellent MUSE maps show that highly ionized CELs such as [Cl iv] or [Ar IV] have a similar distribution to O ii RLs. This does not seem to be a contrast problem since weaker lines than O ii $\lambda$4649, such as [Cl iii] $\lambda$5518, have a distribution similar to that of [O iii] CELs. Since both Cl and Ar are third-row elements in the periodic table, this also does not seem to be a recombination contribution to these CELs (Barlow et al., 2003). How could a clump with a temperature of $\sim 800$ K could emit these highly ionized CELs?