© The Authors 2023

1. Introduction

Galera-Rosillo et al. (2022, hereinafter GR22) presented an observational study of a sample of the brightest planetary nebulae (PNe) of M31. The deep PN spectra allowed the authors to derive precise chemical abundances for the nine nebulae studied and estimate the masses of their central stars. These data constrain the progenitor masses of the tip of the PN luminosity function in the disc of M31 to a narrow range around 1.5 M_⊙, and raise some questions regarding the inconsistency with the expected and computed N/O abundance ratios for five of those PNe.

The GR22 results were called into doubt by Ueta & Otsuka (2022, hereinafter UO22), who claimed that they were plagued by a systematic underestimation of the reddening coefficient c(Hβ). This argument was mainly based on the fact that GR22 only considered the Hα and Hβ lines to derive c(Hβ), and that no iterative process was applied to tune the theoretical ratios to the specific plasma conditions (electron density and temperature) of each nebula. By applying such an iterative process, and determining c(Hβ) using additional Balmer lines (exactly which ones is not specified in their article), UO22 claim significantly different results from GR22. This discrepancy is discussed in this Letter.

2. Reddening from ground-based optical observations: General considerations

In the optical range, the extinction coefficient c(Hβ) is usually computed by comparing the intensities of Balmer and/or (less frequently) Paschen lines relative to Hβ with their case B values (Storey & Hummer 1995). Some caution has to be taken when using these lines. In particular, lines blended or affected by telluric emission (Paschen lines) should be excluded, as well as those with relatively large principal quantum numbers (n > 7) that depart from their expected case B values (see Mesa-Delgado et al. 2009; Domínguez-Guzmán et al. 2022).

Furthermore, observational experience over decades has shown that the relative flux calibration of Balmer lines with n > 4 can be affected by random and systematic errors that are difficult to quantify, and are generally related to the flux calibration process in the blue region of the spectrum rather than to photon noise. This results in significant uncertainties in the determination of the reddening coefficient, especially when considering the smaller difference in wavelength between these high-order Balmer lines. This is the reason why, when the real errors cannot be safely estimated, it is a common practice to give a prevailing weight to the observed Hα/Hβ line ratio (e.g., Johnson et al. 2006). Most importantly, whichever procedure is used to determine c(Hβ), it has to be ensured that the procedure provides physical results in terms of the de-reddened Balmer decrement.

The iterative determination of c(Hβ), T_e, and n_e is a common practice in the interstellar medium community. It is, for example, used by Sánchez et al. (2007). Although this iterative determination as recommended by Ueta & Otsuka (2021), in what they call proper plasma analysis practice (PPAP), is indeed a good practice, experience has also shown that this iterative process would only change the determined abundances by a few percent compared to the use of the T_e = 10 000 K and n_e = 1000 cm⁻³ classical hypothesis to adopt the intrinsic Hα/Hβ line ratio. This is due to the rather stable value for Hα/Hβ against these parameters. This point is further developed in the next section.

In extreme cases, when metal-rich clumps at very low temperatures are present in the gas, which boosts the HI line emissivities, as often found in PNe with high abundance discrepancy factors (see, e.g., García-Rojas et al. 2022; Gómez-Llanos & Morisset 2020), the interpretation of the Hα/Hβ line ratio can become a complex task. But in these cases, the Hα/Hβ line ratio will be higher than the standard value of 2.86 (not lower, as found by UO22), and the problem of determining the exact contribution of hot and cold gas to the H I emissivity becomes a complicated issue, which is beyond the scope of this article.

3. Revision of the reddening determination of the M31 PNe observed by GR22

While we cannot reproduce in detail the analysis of UO22, who performed a Monte Carlo run of 1500 simulations, we can consider the issues they raised by performing some simple tests.

First, we considered the effects of skipping the iteration of the reddening determination on the determination of the plasma conditions (T_e and n_e). Indeed, Ueta & Otsuka (2021) and UO22 advocate for the use of the PPAP, based on a coherent and simultaneous determination of the reddening correction and physical properties (namely T_e and n_e), before determining any ionic and elemental abundances. Their main point is to avoid the errors produced by computing c(Hβ) by comparing the observed H I line ratios with theoretical ones determined using a priori values for T_e and n_e. The protocol they prescribe is described in Ueta & Otsuka (2021, their Fig. 1; it is an iterative process that includes the determination of c(Hβ), T_e (from, e.g., the [N II] line ratio), and n_e (from, e.g., the [S II] line ratio). They claim that if such a practice is not followed, the resulting abundances may be strongly affected. While this is a correct scheme and a good practice, we show in the following the effects of applying or not the PPAP. We performed all the following computations using the PYNEB v1.1.16 Python package (Luridiana et al. 2015).

In the left panel of Fig. 1 we plot the variation in Hα/Hβ for a huge range of T_e − n_e values, and the right panel shows a close-up of the range of T_e and n_e that corresponds to the values obtained for the PNe in M31. We see that the Hα/Hβ line ratio does not change by more than 8% when T_e ranges from 8500 to 14 000 K and n_e ranges from 320 to 32 000 cm⁻³ (see the discussion of the effect of this practice in Sect. 4).

Fig. 1. Hα/Hβ line ratio dependence on the electron temperature and density. The right panel is a close-up covering the ranges for these parameters obtained for the PNe of M31. The underlying data used to draw the contours are the raw data from Storey & Hummer (1995).

In Table 1, for each PN observed by GR22 we list the values of c(Hβ) obtained from the observed Hα/Hβ line ratio using different sets of (T_e, n_e) to compute the theoretical line ratio. Case 1 corresponds to adopting T_e = 10 000 K and n_e = 1000 cm⁻³; case 2 uses the specific plasma conditions for each nebula as derived from the data in GR22; and case 3 uses the values from UO22. The results for cases 2 and 3 were obtained with the PPAP prescription, and they show only very small differences compared to the simplification of assuming T_e = 10 000 K and n_e = 1000 cm⁻³, as done by GR22.

From Fig. 1 it is also evident that the extremely low values of the de-reddened Hα/Hβ ratio obtained by UO22, as low as 2.16 and 2.11 for PNe M2538 and M2860, respectively, would correspond to huge densities (above 10¹⁰ cm⁻³), incompatible with the values they obtained from the [S II] line ratio, or to very high temperatures that are well outside the range spanned by ionized nebulae. The unrealistic UO22 results are also illustrated in Fig. 2, where the theoretical Hα/Hβ, Hγ/Hβ, and Hδ/Hβ line ratios for four combinations of n_e and T_e are compared, as are the de-reddened ratios for the M31 nebulae in GR22 and UO22. The four colored dots in the figure (the leftmost symbols) are the values obtained from PYNEB when T_e is set to 10 000 K and 15 000 K and n_e is set to 2500 and 25 000 cm⁻³, covering the values of the PNe under study. As already mentioned, the H I/Hβ line ratio in this range of nebular conditions is nearly invariant on the logarithmic scale of the figure. The next points are the de-reddened H I/Hβ line ratios reported by GR22 (diamonds) and UO22 (stars) for the M31 PNe. Two groups of PNe can be defined: on one side, M1687, M50, M1596, and M1074, where the corrected line ratios are very close in both papers to the theoretical values; on the other side, for M2538, M1675, M2068, M2471, and M2860, the GR22 line ratios are very close to the theoretical values, while the line ratios reported by UO22 are far from them. The differences between the UO22 line ratios and the theoretical values cannot be explained by differences in the adopted values of T_e and/or n_e, as they are much higher than expected from any combination of reasonable values for these parameters. The patterns in UO22, with de-reddened Hα/Hβ values lower than the range of realistic values of the plasma’s parameters and Hγ/Hβ and Hδ/Hβ higher than expected, point to spectra with a blue excess artificially produced by an oversized reddening correction. Such an incorrect correction translates to an incorrect determination of the N/O abundance ratios determined by UO22, as we next show.

Fig. 2. Hα/Hβ, Hγ/Hβ, and Hδ/Hβ line ratios in logarithmic scale, from top to bottom. The first four values of each row (filled circles) correspond to the theoretical values obtained using (log Te, log ne) = (4.0, 3.4), (4.0, 4.4), (4.2, 3.4), and (4.2, 4.4), respectively. The horizontal lines correspond to the value obtained for (4.0, 3.4). The following nine values correspond to the de-reddened line ratios obtained by GR22 (diamonds) and UO22 (stars) for the nine PNe considered in both papers.

4. Effects on the abundance determinations

We performed a simple test to check the N⁺/H⁺ abundance ratios computed by GR22, which is a main concern of UO22. This abundance ratio was recalculated using Hα as the reference H I ion, instead of Hβ as commonly done. The advantages are that the dependence on the adopted extinction and of the relative flux calibration is removed, given the closeness in wavelength of Hα (λ6562) and the nebular [N II]λ6548+84 doublet, and that the measurement of Hα has a higher S/N than Hβ. For simplicity, only the [N II]λ6584 line was used. The results are shown in Fig. 3. For each nebula, we adopted the extinction-corrected fluxes reported by either UO22 and GR22, their reported T_e([N II]) for the emissivity of the [N II]λ6584 line and the H I lines, and an average value of n_e given by the [S II] and [Ar IV] lines, which has little or no influence on the results given the very small density dependence of the abundances on the considered T_e range.

Fig. 3. Values of 12 + log(N+/H+) obtained using the data from GR22 (diamonds) and UO22 (stars) and using Hβ (larger symbols) and Hα (smaller symbols outlined in black). The error bars reported by UO22 are also shown.

For each PN, we plotted the values of 12 + log(N⁺/H⁺) obtained using the data from GR22 (diamonds) and UO22 (stars), using Hβ (larger symbols) and Hα (smaller black-outlined symbols). The error bars reported by UO22 are also shown. For all PNe, the newly calculated N⁺/H⁺ ratio from [N II] lines normalized by Hα (using the GR22 data) are within 2% of those obtained when normalizing by Hβ (black vs. colored diamonds). This result is somewhat obvious, as the de-reddened Hα/Hβ ratio in GR22 is consistent with the theoretical value for the typical physical conditions of these ionized nebulae. However, the differences using the UO22 fluxes are much greater and, in the case of five PNe, on the order of their adopted uncertainties, albeit systematically lower. We consider this as further evidence that UO22 overestimated the extinction correction.

A final test was made by computing the intensities of the Hα, Hβ, [N II]λ5755, 6584 Å, [S II]λ6716, 6731 Å, and [O III]λ4363, 5007 Å lines using a set of (T_e, n_e) for given ionic abundances. These emission lines were reddened using an arbitrary c(Hβ) = 0.3 and then de-reddened using the c(Hβ) derived from the Hα/Hβ line ratio and adopting T_e = 10 000 K and n_e = 1000 cm⁻³. The T_e and n_e values were then computed from the corresponding diagnostic line ratios, and the N⁺/H⁺ and O⁺⁺/H⁺ ionic abundances were recovered and compared with their original values. This very simple procedure allowed us to estimate the error on abundances caused by not taking the correct values for T_e and n_e coherently into account in the determination of c(Hβ). Table 2 reports the errors obtained on N⁺/H⁺ and O⁺⁺/H⁺ by applying this procedure. We first see that even taking extreme values for T_e and n_e leads to errors lower than 10% in all cases, and on the order of a few percent for 8000 K < T_e < 12 000 K. We also see that the main effect comes from using a wrong value for T_e. These results confirm that it is sensible to adopt a PPAP procedure, but that the errors implied by not using it are far smaller than claimed by Ueta & Otsuka (2021). In this respect, it is hard to understand the results reported by Ueta & Otsuka (2021), who claim to find errors as large as 50% for lower differences between the real value of (T_e, n_e) and the standard ones used to compute the reddening correction.

It can be noticed in Table 2 that the error on O⁺⁺/H⁺ for T_e = 10 000 K and n_e = 1000 cm⁻³ (bold values) is 0.1, instead of zero as expected since the correct (T_e, n_e) combination is used to recover c(Hβ) and the un-reddened line intensities. This difference is due to the simplified method used in PYNEB to look for the crossing point between the T_e and the n_e diagnostic line ratios. This gives an idea of the rather good, though not absolute, precision of the determination of ionic abundances using PYNEB.

Finally, it is also worth mentioning here that, while formally correct, caution has to be taken when interpreting the N/O determinations by GR22 in terms of stellar nucleosynthesis. When dealing with total N abundances and N/O ratios, GR22 adopted an ionization correction factor ICF(N) N/O = N⁺/O⁺ (their Sect. 3.3), and therefore the N/O values in Figs. 5 and 6 from GR22 should be strictly interpreted as dealing with N⁺/O⁺. Their subsequent discussion, which compares their N/O determinations with previous N/O determinations also based on that ICF (as referred to by GR22 in Sect. 4.3), is justified by their deeper and better data. However, the comparison with theoretical N/O values from models could be quite uncertain, in both GR22 and in previous literature determinations, if the adopted ICF(N) N/O = N⁺/O⁺ is proven to be wrong. And this might be the case in PNe with high ω = O⁺⁺/(O⁺ + O⁺⁺) > 0.9 ratios, where both O⁺ and N⁺ are residual ions (see Sect. 4.4 and Fig. 6 in Delgado-Inglada et al. 2014). For high ionization PNe, it is absolutely mandatory to use more ion ratios (e.g., Ar IV/Ar III) to determine a correct ICF, as already discussed in GR22 and shown in Sabin et al. (2022) and García-Rojas et al. (2022).

5. Summary and conclusions

We tested the results of GR22 on nine bright PNe of M31, which had been questioned by UO22. Our conclusions are as follows:

• Given the mild variations in the brightest H I Balmer lines under the plasma conditions in the range of the considered PNe, skipping the iterative process indicated by UO22 to compute the nebular extinction only causes errors of 0.01–0.02 in the values of the reddening correction c(Hβ). This is within the uncertainties quoted by GR22.

• The de-reddened line ratios for the main H I Balmer lines computed by GR22 are very close to the theoretical values for all nebulae.

• Adopting the larger reddening from UO22 leads to unrealistic de-reddened line ratios (or nebular conditions) for several of the target PNe.

• The reddening correction applied by GR22 provides consistent results for the N⁺/H⁺ abundance ratio when it is estimated using (extinction-independent) Hα instead of Hβ as the reference H I ion. Contrarily, the results of UO22 determined using the two different H I lines are inconsistent, beyond the estimated uncertainties.

• Errors on N⁺/H⁺ and O⁺⁺/H⁺ generated by skipping the iterative procedure are on the order of a few percent for the range of temperatures displayed by the target M31 PNe (8000 K < T_e < 12 000 K), much less than what Ueta & Otsuka (2021) obtained.

This analysis therefore confirms the results obtained by GR22 and their conclusions. All the figures and tables presented in this Letter were obtained using the Jupyter notebook, which can be found on GitHub¹.

Acknowledgments

We are very grateful to Sebastián Sánchez, who refereed this Letter. We want to thank support and advice from Grazyna Stasińska and Bruce Balick. C.M. acknowledges the support from UNAM/DGAPA/PAPIIT grant IN101220. J.G.-R., R.C. and A.M. acknowledge support under grant P/308614 financed by funds transferred from the Spanish Ministry of Science, Innovation and Universities, charged to the General State Budgets and with funds transferred from the General Budgets of the Autonomous Community of the Canary Islands by the MCIU. J.G.-R. also acknowledges support from an Advanced Fellowship under the Severo Ochoa excellence program CEX2019-000920-S and financial support from the Canarian Agency for Research, Innovation and Information Society (ACIISI), of the Canary Islands Government, and the European Regional Development Fund (ERDF), under grant with reference ProID2021010074.

References

All Tables

All Figures

	Fig. 1. Hα/Hβ line ratio dependence on the electron temperature and density. The right panel is a close-up covering the ranges for these parameters obtained for the PNe of M31. The underlying data used to draw the contours are the raw data from Storey & Hummer (1995).
In the text

Fig. 2. Hα/Hβ, Hγ/Hβ, and Hδ/Hβ line ratios in logarithmic scale, from top to bottom. The first four values of each row (filled circles) correspond to the theoretical values obtained using (log Te, log ne) = (4.0, 3.4), (4.0, 4.4), (4.2, 3.4), and (4.2, 4.4), respectively. The horizontal lines correspond to the value obtained for (4.0, 3.4). The following nine values correspond to the de-reddened line ratios obtained by GR22 (diamonds) and UO22 (stars) for the nine PNe considered in both papers.

In the text

	Fig. 3. Values of 12 + log(N+/H+) obtained using the data from GR22 (diamonds) and UO22 (stars) and using Hβ (larger symbols) and Hα (smaller symbols outlined in black). The error bars reported by UO22 are also shown.
In the text