2.1. M 31

The well known M 31 is the most massive spiral galaxy in the Local Group. It is at a distance of 785 kpc from the Milky Way (^{McConnachie et al. 2005}).

Chemical gradients calculated from the abundances of H II regions have been studied by several authors, mostly based on abundances derived by using strong-line methods because due to the high metallicity in this galaxy, the auroral lines indicative of electron temperature, such as [OIII]λ4363 and [NII]λ5755, are faint and difficult to detect. The largest sample analyzed in this way is the one by ^{Sanders et al. (2012)}, which included 192 H II regions. They reported an O abundance gradient of about −0.0195 ± 0.0055 dex kpc⁻¹, but found that the slope depended on the choice of the strong-line method used.

^{Zurita & Bresolin (2012)} obtained abundances based on the direct-method for 31 H II regions con-centrated at two galactocentric distances of 3.9 and 16.1 kpc. These authors discussed the O/H abun-dance gradient finding a robust negative slope of ΔO/H / ΔR = −0.023 dex kpc⁻¹, based on both the direct-method and the strong-line method. This value is similar to the values reported by other authors, based on strong-line methods (^{Zaritsky, Kennicutt, & Huchra 1994}; ^{Sanders et al. 2012}).

Interestingly, ^{Zurita & Bresolin (2012)} found that abundances determined with the direct-method for their H II regions are lower, by about 0.3 dex, than the values determined with the strong-line methods and the values derived for supergiant stars, at any galactocentric distance. The authors attribute this discrepancy to a bias in their sample, which would not be representative of the mean H II region population (because only high-temperature, and thus low-metallicity regions could be detected in their search), and to the probable depletion of O in dust grains. The abundance data for H II regions used in this work, are taken from ^{Zurita & Bresolin (2012)}.

Data for PNe were collected from the articles by ^{Kwitter et al. (2012)}; ^{Sanders et al. (2012)}; ^{Balick et al. (2013)}; and ^{Fang et al. (2013}, ²⁰¹⁵⁾. The use of data from different authors can introduce undesirable inhomogeneities. Therefore, in order to analyze chemical abundances on a more homogeneous system, physical conditions and ionic abundances of PNe were recalculated by us from the extinction-corrected line intensities published by the cited authors. The IRAF 2.16 five level nebular modelling package “stsdas.analysis.nebular”, with the tasks temden and ionic , was employed to determine the physical conditions and ionic abundances. For the full sample the [O III] temperature was derived and occasionally the [N II] temperature was available, but all the ionic abundances were calculated with the [O III] temperature (one-temperature zone model). Total abundances were derived from the ionic abundances by using the ionization correction factors (ICFs) by ^{Delgado-Inglada, Morisset, & Stasińska (2014)} which correct for the ions not seen. The results, although not too different from the ones published by the cited authors, are now in a homogeneous system.

The PN distribution in this galaxy extends up to R/R₂₅ ≈ 5 (R ≥ 100 kpc) while data for the H II regions cover only up to R/R₂₅ ≈ 1 (R ≈ 20 kpc).

The galactocentric distances used in the diagrams for the gradients were collected from the same references used for the abundance ratios.

2.2. M 33

M 33 is the third most-massive spiral galaxy in the Local Group. Its Hubble type is very late, SA(s)cd, and its mass is much lower than the masses of the Milky Way and M 31, by factors of 16 and 30 respectively. It is located at 849 kpc from the Milky Way and appears almost face-on. The central metallicity in M 33 as given by H II regions, 12 + log O/H = 8.5, is lower than that of the bigger spirals, and it is even lower than that of NGC 300 (see Table 2).

Gradients from PN abundances were given by ^{Magrini, Stanghellini, & Villaver (2009)} and ^{Bresolin et al. (2010)} who found that the slopes were equal to the ones of H II regions, or flatter.

For this galaxy, PN data were collected from ^{Bresolin et al. (2010)} and ^{Magrini et al. (2010)}, and H II region data from ^{Magrini et al. (2010)}. The distributions of H II regions and PNe extend up to R/R₂₅ ≃ 1 (R ≈ 9 kpc). As said in the Introduction, in the sample of PNe we excluded those objects showing [OIII]λ5007/Hβ intensity ratios smaller than 3 and, consistently, for the H II regions we excluded the objects showing ratios larger than 3. This is to avoid contamination by compact H II regions in the PN sample, and to avoid contamination by supernova remnants, W-R nebulae and other highly ex-cited objects in the H II region sample. Our diagrams then contain only genuine PNe and H II regions.

The galactocentric distances employed to build the gradient diagrams were calculated by us, according to the procedure described by ^{Cioni (2009)}.

2.3. NGC 300

The almost face-on spiral NGC 300 is the least massive galaxy of the sample studied here and the only one outside the Local Group. Also, it has the latest Hubble type, SA(s)d. It is similar to M 33 in several aspects. Several authors have studied the H II regions. ^{Bresolin et al. (2009)} were the first to determine the oxygen abundance gradient based on direct-method abundance determinations. They found ΔO/ΔR = −0.077 dex kpc⁻¹ with a central abundance of 12+log O/H ≃ 8.57.

^{Stasińska et al. (2013)} analyzed the chemical gradients provided by H II region and PN abundances finding that gradients of PNe appear flatter than those of H II regions.

In this work we have re-analyzed the PN data presented by ^{Stasińska et al. (2013)}, to compare the case of NGC 300 with the other galaxies of the sample. The abundances presented by ^{Stasińska et al. (2013)} were derived by adopting the ICFs proposed by ^{Kingsburgh & Barlow (1994)}, and in this work we have used the more recent ICFs presented by ^{Delgado-Inglada, Morisset, & Stasińska (2014)}. With these new ICFs we found that our O/H values were equal to the ^{Stasínska et al. (2013)} ones, with differences of less than 0.03 dex, our Ne/H values show differences, on average, of less than 0.05 dex, while our Ar/H values show differences of 0.07 dex on average. Data for H II regions are from ^{Bresolin et al. (2009)}.

In this galaxy PNe and H II regions have been found up to R/R₂₅ ≃ 1 (R ≈ 5 kpc). Galactocentric distances were collected from the same authors as the abundances.

2.4. Milky Way

Metallicity gradients have been widely studied in the MW by means of H II regions, PNe, Cepheid stars, stellar clusters and other objects; see e.g., ^{Deharveng et al. (2000)}; ^{Maciel, Costa, & Ushida (2003)}; ^{Henry et al. (2010)}; ^{Stanghellini & Haywood (2018}, ²⁰¹⁰⁾; ^{Esteban et al. (2017)}; the compilation by ^{Mollá et al. (2019)}, etc. The reported results for PNe have been contradictory. It has been claimed that the PN gradients coincide with those of H II regions or that the PN gradients are shallower, that gradients have flattened (or steepened) with time, or that the gradient changes slope at certain distance from the galactic center. One of the main problem in these determinations is the large uncertainties in the distances to PNe.

Determining the distance to galactic PNe is a difficult task. The trigonometric parallax method is available only for a handful of nearby objects. Even the parallaxes measured by GAIA are limited to distances of a few kpc around the Solar System, and in GAIA Data Release 2 less than a hundred PNe have confidently measured parallaxes (^{Kimeswenger & Barría 2018}). Therefore, the distances for a large sample of PNe are based on model-dependent statistical methods which, on occasion, lead to different results. At present, the most used distance scales are those proposed by ^{Stanghellini & Haywood (2010)}, hereafter S10, and by ^{Frew, Parker, Bojiĕcié (2016)}, hereafter F16. The latter authors established a robust optical statistical distance indicator, the Hα surface brightness vs. radius (S_Hα-r) indicator, where the intrinsic radius is calculated by using the angular size, the integrated Hα flux, and the reddening to the PNe. This radius, combined with the angular size, yields directly the distance. On the other hand, S10 determine statistical distances based on apparent diameters and 5 Ghz fluxes of PNe. A comparison of both distance scales are presented below.

In this work we used the PN chemical abundances reported by ^{Henry, Kwitter, & Balick (2004)}, ^{Milingo et al. (2010)}), and ^{Henry et al. (2010)}. These authors belong to the same group; therefore, all physical conditions were calculated with the same method, and total chemical abundances were obtained using the ICFs described in ^{Kwitter & Henry (2001)}. The Galactic sample consists of 156 PNe covering galactocentric distances in the range 0.21 kpc ≤ R ≤ 22.73 kpc (0.02 ≤ R/R₂₅ ≤ 1.97). More than 40 of these PNe lie at distances larger than the solar galactocentric distance and are crucial for the gradient determination.

Oxygen abundances for H II regions were taken from ^{Esteban et al. (2017)}, while neon and argon abundances were collected from ^{García-Rojas et al. (2004)}), ^{García-Rojas et al. (2005)}, ^{García-Rojas et al. (2006)}, ^{García-Rojas et al. (2007)}, ^{Esteban et al. (2004)}, ^{Esteban et al. (2013)}, and ^{Fernández-Martín et al. (2017)}. As said above, the use of data processed by different authors can introduce some in-homonogeneities and uncertainties in the Ne and Ar gradients of H II regions, that should be considered carefully.

In Figure 6 we present the comparison of the metallicity gradients of oxygen, argon and neon for the Galactic H II regions and the PN sample. This figure will be discussed in detail in § 3.4, Here we want to show that for the case of PNe we are using the distances by F16 and S10 for a comparison. As seen in this figure the oxygen, neon, and argon abundances show a very large dispersion at all galactocentric distances, but the linear regressions for PNe are similar for both distance indicators. Therefore in the following F16 distances will be used as an independent way to compare with other works.

3.1. M 31

In Figure 1 the radial gradients for O/H, Ne/H and Ar/H, for PN and H II region abundances are plotted vs. R/R₂₅. The gradients vs. R (kpc) are presented in Table 2.

Fig. 1 Radial metallicity gradients of O (top), Ne (center) and Ar (bottom) for M 31, are presented. Black circles represent PNe and red squares, H II regions. Black dashed lines correspond to the linear fit for PN data and dashed-dotted red lines correspond to the linear fits for H II regions. The color figure can be viewed online. 

A linear fit to the gradients is included in each case. The abundances of elements in PNe present a large dispersion at any given galactocentric distance, but in particular in the central zone. It is worth to notice that there are some PNe in the central region with very low O/H abundance, which do not have Ne/H or Ar/H abundance determinations. H II regions also show a large dispersion in the elemental abundances at any galactocentric distance (^{Zurita & Bresolin 2012}; ^{Sanders et al. 2012}).

For the case of O in H II regions, we obtain:

or equivalently

The O gradient found is equal, within the uncertainties, to the one derived by ^{Zurita & Bresolin (2012)}. For Ne, we find

and for Ar, we find

In all the cases, the errors correspond to 1 sigma.

PN abundances seem unrelated to the galactocentric distance; the PN gradients are really flat, showing slopes of −0.001 ± 0.001 dex kpc⁻¹ for O, −0.002 ± 0.001 dex kpc⁻¹ for Ne, and −0.002 ± 0.001 dex kpc⁻¹ for Ar, which considering the errors are consistent with 0. This indicates that at large galactocentric distances PNe present on average the same O/H abundances as the central zones; the same is true for Ar and Ne.

The O/H value at the intercept for H II regions, 12+log O/H = 8.76±0.10, seems slightly larger than the value for PNe, 12+log O/H = 8.46±0.03, while the Ne/H and Ar/H central values are similar for H II regions and PNe, within uncertainties. However, due to the negative gradients for H II regions, at large distances (R ≤ R₂₅,) the O/H, Ne/H and Ar/H abundances in PNe are always larger than the abundances in H II regions.

The larger value of O at the intercept for H II regions seems to be an artifact due to the limited sample of ^{Zurita & Bresolin (2012)}, which moreover presents large uncertainties. In addition, there are a large number of PNe with low O abundances in the central zone, with no Ne and Ar measurements, that could be contributing to the low O/H central value for PNe. In this zone a very large dispersion is observed.

In the central zone the value at R = 0 of log Ne/O=−0.77 for H II regions is about solar and excludes the possibility indicated by ^{Zurita & Bresolin (2012)} of a large O depletion in dust grains. On the other hand, Ne in H II regions is similar to Ne in PNe in the central zone. Relative to Ar, log Ar/O (H II) = −2.38 and log Ar/O (PNe) = −2.24 which correspond well with the solar or Orion values.

In Figure 2 we show the abundance gradients for PNe separated by ^{Peimbert types. Peimbert (1978)} called Type I those PNe with a N/O abundance ratio larger than 0.5 and a He/H abundance ratio larger than 0.14. This group includes the PNe with central stars with initial masses larger than about 3.0 M_⊙ and therefore they are the youngest objects among PNe. They have enriched their nitrogen abundances by nucleosynthesis processes, such as CNO and hot-bottom-burning (HBB) and the newly formed N has been transported to the surface in different dredge-up events. In the Galaxy, Type I PNe belong to the thin disk and their ages are about 1 Gyr. Non-Type I PNe include the Peimbert Types II and III, which are classified based on their radial velocity, being smaller or larger than 60 km s⁻¹, and are located in the thick disk. They correspond to older objects with initial masses smaller than 2 M_⊙, where no large N-enrichment is present. Their ages are between 2 to 8 Gyr.

Fig. 2 Radial metallicity gradients for different Peimbert Type PNe in M 31. The data for Type I PNe are shown as red triangles, the data for non-Type I PNe as blue crosses. The color figure can be viewed online. 

Thus our Figure 2 represents an effort to determine the behavior of the abundance gradient over time. It is evident that although the gradients are very flat and the uncertainties are large, Type I PNe show slightly steeper gradients for the three elements: O, Ne and Ar. Additionally, Type I PNe seem to be O-poorer than non-Type I, possibly showing the effect of CNO and HBB processes which operates in these massive stars. This phenomenon will be discussed in more detail in § 4.1.

It should be mentioned here that the value of the N/O abundance ratio, used to define a Peimbert Type I PN is slightly dependent on the metallicity. The N/O ratio defined by Peimbert (1978) applies to the metallicities of the Milky Way and M 31, and should be slightly lower for M 33 and NGC 300, but the difference is not important for this work and the results are not much affected.

We will discuss the results for M 31 in sections ahead, together with the results for the other galaxies.

3.2. M 33

Metallicity gradients for H II regions and PNe in M 33 are presented in Figures 3 and 4. In the latter one, PNe are separated into Peimbert Type I and non-Type I objects. Differently to what happens in M 31, in this galaxy O, Ne and Ar abundances of PNe and H II regions are similar, within uncertainties, at any galactocentric distance.

Fig. 3 Radial metallicity gradients in M 33. Symbols are as in Figure 1. The color figure can be viewed online. 

Fig. 4 Radial metallicity gradients for PNe of different Peimbert Type in M 33. In red the data for Type I PNe are shown and in blue, the data for non-Type I PNe. The color figure can be viewed online. 

For PNe the straight-line fit of the gradients gives 12+log O/H = (8.34±0.07) −(0.038±0.016) R (kpc) and for H II regions, 12+log O/H=(8.48±0.03) − (0.047 ± 0.008) R (kpc). That is, the slope for PNe seems shallower than the slope for H II regions, by about 30% for O and by larger factors for Ne and Ar, but the uncertainties are large making these results inconclusive. However, we consider them indicative of shallower PN gradients.

In Figure 4, the gradients of PNe with different Peimbert Types are shown. There are only a few Type I PNe in the sample (about 20%) and the un-certainties are very large. However, the central values can be considered to be equal, within errors, for both PN types. Despite the uncertainties, slightly steeper gradients are apparent for Type I PNe, while the gradients for non-Type I PNe are flatter. The slopes shown by Type I PNe are more similar to those of H II region. The huge uncertainties in these results make them inconclusive, but only indicative. We consider them reliable because M 33 shows results similar to M 31 and NGC 300.

3.3. NGC 300

PN data in this galaxy were analyzed by ^{Stasińska et al. (2013)}, where O, Ne, S and Ar abundance gradients were presented. These authors found that the formal O/H, NeH and Ar/H abundance slopes for PNe are shallower than those of H II regions and attributed this to a steepening of the metallicity gradients during the last Gyr. The O/H central value of PNe is smaller by 0.15 dex than the central value of H II regions. Ne/H and Ar/H, on the other hand, present the same central abundances for PNe and H II regions, and almost flat gradients, although affected by large dispersion at any galactocentric distance. According to ^{Stasińska et al. (2013)}, due to the difference in the O/H value at R = 0, O abundances in PNe could be affected by nucleosynthesis of their central stars.

Our analysis of these data, calculated with the ICFs by ^{Delgado-Inglada, Morisset, & Stasińska (2014)}, shows similar results, presented in Figure 5. O, Ne and Ar metallicity gradients for PNe in NGC 300 are flatter than the values for H II regions. Their ΔX/ΔR values are about half the values found for H II regions, in very good agreement with the results of ^{Stasińska et al. (2013)}. In the central zone, H II regions show 12+log O/H = 8.57±0.03, log Ne/O = −0.86±0.008 and log Ar/O= −2.20±0.07. These abundance ratios are similar to the solar or Orion values, while PNe show values 12+log O/H= 8.37±0.03, log Ne/O =−0.73±0.06, and log Ar/O = −2.06±0.05, also similar to the solar and Orion values but with an apparent O decrease by 0.2 dex, relative to the O of H II regions, as already indicated by ^{Stasińska et al. (2013)}. This will be discussed in §4. The central values of Ne/H and Ar/H are similar for PNe and H II regions.

Fig. 5 Radial metallicity gradients in NGC 300. Symbols are as in Figure 1. The color figure can be viewed online. 

In this case, an analysis discriminating between the Peimbert Type PNe is not possible, because there are very few Type I PNe in the sample.

3.4. The Milky Way

Metallicity gradients have been widely studied in the MW by means of many kind of objects. See references in §2.4.

^{Henry et al. (2010)}, using distances by ^{Cahn, Kaler, & Stanghellini (1992)} derived an O gradient for PNe of −0.058 ± 0.006 dex kpc⁻¹, which changes to −0.042±0.004 dex kpc⁻¹ if the distances by Stanghellini, Shaw, & Villaver (2015) are used. ^{Henry et al. (2010)} suggested that the gradient steepens beyond a galactocentric distance of 10 kpc. In a recent work, ^{Stanghellini & Haywood (2018)}, using distances given by S10, reported that out to R/R₂₅ ≃ 2.4 (R ≈ 28 kpc) the radial gradient of oxygen for PNe is shallow, with a slope of ≈ −0.02 dex kpc⁻¹ and a central abundance of 12+log(O/H)≃ 8.68. These authors suggest that the gradient changes with R in the sense that the significant slope is limited to R between 10 and 13.5 kpc, and outside this range the gradient is almost flat.

We analyzed the gradients derived from the data for PNe and H II regions mentioned in §2.4. PN abundances were calculated by us in a homogeneous way. The results are presented in Figures 6 and 7. In the latter figure the data have been binned in distance taking bins of 1 kpc, for clarity. The distances by F16 are used for PNe. Clearly, PN gradients are flatter that those of II regions. ΔX/ΔR is about twice larger for H II regions (see Figure 7 and Table 2). The binning of data with distances introduced an artifact in Figure 7 in the sense that the O and Ne abundances for PNe seem larger than the values for H II regions at R = 0. But this does not occur in Figure 6, where the original data, not binned, were used. In these graphs it is observed that the O and Ne values at the central zones coincide for PNe and H II regions, within uncertainties, while the Ar/H abundance is lower in PNe by 0.7 dex. However, due to the shallower gradients for PNe, at galactocentric distances larger than a few kpc the average abundances of PNe are larger than those of H II regions, similarly to what happens in M 31 and NGC 300. This can explain in part the results by ^{Rodríguez & Delgado-Inglada (2011)}, who found that in the solar vicinity, PNe appear richer than H II regions.

Fig. 6 Radial metallicity gradients of O, Ne and Ar, in the Milky Way. Open blue circles and black triangles are PNe with distances from ^{Frew et al. (2016)} and ^{Stanghellini & Haywood (2010)} respectively. The dotted-dashed line is the linear fit for F16 distances and the solid line that for Stanghellini & Haywood distances. The green dashed line is the linear fit for H II regions by ^{Esteban et al. (2016)}. The color figure can be viewed online. 

Fig. 7 Radial metallicity gradients. The black circles correspond to PNe with F16 distances in bins of 1 kpc. Solid lines are the fits for PNe and dashed lines those for H II regions. The top panel shows the radial gradient for O, the center panel the gradient for Ne, and the bottom the gradient for Ar. The color figure can be viewed online. 

The gradients derived for PNe of different Peimbert types are shown in Figure 8. In this case, gradients of Type I and non-Type I PNe are equal within uncertainties. A possible change in the slope, at R≈ 14 kpc, is appreciated, which will be discussed in the next section.

Fig. 8 The PNe of the MW with distances binned as in Figure 7. Red diamonds represent Type I PNe and blue stars non-Type I PNe. Solid lines show the linear fits for Type I SNe, the dotted lines those for non-Type I SNe. The color figure can be viewed online. 

We used the known GAIA distances of PNe to analyze the gradient for PNe. Using the same set of abundances as for the MW, we searched for those PNe that have calculated parallaxes in GAIA Data Release 2 (GAIA DR2). We did not take into account those objects with negative parallax and those with errors in parallax (dp/p) larger than 0.4. We found 22 PNe meeting these requirements. No gradient was found in this interval for the three analyzed elements, because GAIA galactocentric distances are limited from 6 to 10 kpc, and this interval is too short to show any gradient, due to the large dispersion in abundances.

4.1. M 31

Our work extends the PN sample, including objects from 2 kpc up to a distance larger than 100 kpc (0.2 - 5 R₂₅). In this interval the abundance gradients for PNe are flat, consistent with a slope of zero. The average abundances of PNe are the same at all galactocentric distances, showing a very large dispersion.

On the other hand the gradients for H II regions are always negative, with values −0.030 ± 0.010 dex kpc⁻¹ for O, −0.036 ± 0.016 dex kpc⁻¹ for Ne and −0.021 ± 0.013 dex kpc⁻¹ for Ar. These slopes are much shallower than the slopes in the other three galaxies. The very flat gradients found in M 31 could indicate, according to ^{Sánchez et al. (2014)}, that M 31 has been perturbed by interactions or merging. It is clear that these phenomena have had an important role in the formation and growth of M 31, a galaxy that shows numerous stellar sub-structures in its outskirts (^{McConnachie et al. 2009}, and references therein).

At the central position PNe appear to have an average O/H abundance slightly lower than the average of H II regions, but similar Ne/H and Ar/H abundances. This could be due to the presence of several PNe with very low O abundance (and not known Ne and Ar) in the central region, as it was explained in § 3.1

Similarly to our work, ^{Sanders et al. (2012)} and ^{Magrini et al. (2016)} reported that the O/H gradient from PN abundances is flat in M 31. ^{Sanders et al. (2012)} in particular, from the analysis of O abundances of 52 PNe derived with the direct-method and located at galactocentric distances from 5 to 25 kpc (0.2 to 1.2 R₂₅), found no relation between PN abundances and their galactocentric distances. The same PN sample was re-analyzed by ^{Magrini et al. (2016)}, who reported that radial migration plays an important role in PNe of M 31, which possibly explains the extreme flatness of PNe gradients, since PNe may have migrated far from their place of origin.

When the sample is divided into Peimbert Type I PNe (young objects) and non-Type I PNe (older objects), we found that the young objects show a steeper gradient, although still very flat (Figure 2). Due to their youth (ages lower than about 1 Gyr), Type I PNe have had less time to migrate from their birth places. Thus they might be showing the gradient at about 1 Gyr ago, but considering that these gradients are very flat (compared to those of H II regions) migration could have been an important role for these young PNe too.

It is important to mention that the results found for M 31 corroborate models of galactic chemical evolution, which besides including the star formation rate, gas in fall rate accross the disk, inflows and outflows of gas, stellar evolution and yields, among other processes, also include radial redistribution of stars (stellar migration). Models by ^{Ruiz-Lara et al. (2017)} predict that radial redistribution and accretion increase the metallicity dispersion, and flat-ten the age and metallicity profiles of galaxies; the greater the efficiency of the redistribution, the larger the flattening effect and, as a consequence, a steeper metallicity gradient should be expected at the birth of the objects. Type I PNe being closer to their birth places, they show steeper gradients.

Peimbert Type I PNe present slightly lower O abundances than non-Type I’s, while the Ne and Ar abundances are similar. This can be due to nucleosynthesis, because CNO and HBB processes are expected to occur in these more massive central stars, modifying their initial O abundance. Such an O decrease is predicted by recent sophisticated evolutionary stellar models for low-intermediate mass stars of the MW, computed at different metallicities, by Ventura et al. (2017). Using the sequence of models with z ≈ 0.014 (solar metallicity) it is found that stars with masses larger than 3 M_⊙ decrease their O/H abundance by up to 0.2 dex at the end of their evolution, while stars of lower masses do not modify their original O/H. This is the effect we are finding in the comparison of Type I and non-Type I PNe in M 31.

4.2. M 33

For M 33, a late Hubble-type galaxy, the results are presented in Figures 3 and 4. In this case O, Ne and Ar in PNe show central values similar to those of H II regions. Therefore there was no important enrichment of the ISM since the time of formation of these PNe. Similarly, the O abundance gradients of PNe and H II regions are equal, within uncertainties, but the Ne and Ar abundance slopes in PNe seem significantly flatter, although the large dispersion and large uncertainties make these results doubtful.

Due to the similarity of metallicity gradients and the large dispersion, ^{Magrini, Stanghellini, & Villaver (2009)} claimed that gradients are equal for PNe and H II regions. In a recent paper, Magrini et al. (2016) analyzed the possible effects of radial migration in M 33 and concluded that it is not im-portant. Also ^{Bresolin et al. (2010)} declared that PNe and H II regions have gradients equal within uncertainties, but an analysis of their Table 8 shows that the slopes of PNe are systematically flatter than those of H II regions, even when the uncertainties are large.

It is interesting to notice that PNe in M 33 do not show O reduction compared to H II regions, as occurs in NGC 300 (see next section), despite the similar low metallicity of both galaxies. It seems that the initial masses of the central stars in this galaxy are not as large as in NGC 300. Thus, they are not affected by nucleosynthesis in the same way as seen in NGC 300. According to ^{Ventura et al. (2016)} models, the initial masses should have been not much larger than 2 M_⊙.

Although the uncertainties are huge due to the small number of objects, Type I PNe in M 33 seem to present slopes steeper than non-Type I PNe, and more similar to the ones of H II regions. Again these results are very uncertain, but we consider them indicative of a behavior similar to that of M 31.

4.3. NGC 300

For NGC 300, the latest Hubble-type galaxy, with a metallicity similar to M 33, the abundance gradients of PNe are about twice smaller than the gradients of H II regions (see Figure 5). The gradients presented by H II regions are the largest (ΔO / R = −0.077 ± 0.008 dex kpc⁻¹) of the whole galaxy sample.

The average O/H central abundance of PNe is lower than the value of H II regions by 0.2 dex, while Ne and Ar have the same central values. Such an O decrease, which does not occur for Ne and Ar, could be the result of stellar nucleosynthesis and dregde-up events. Stellar evolution models performed by ^{Ventura et al. (2016)} for PNe in the SMC (which has a metallicity similar to that of NGC 300) predict a de-crease of the initial O occurring in stars with masses larger than 3 M_⊙, due to HBB; simultaneously, a large N-enrichment occurs. Such an N-enrichment is observed for the PNe of NGC 300 (^{Stasińska et al. 2013}). Therefore, we conclude that the central stars of PNe analyzed in this galaxy had, in general, large initial masses; they are younger than 1-2 Gyr. This is certainly due to a bias in the sample because only the brightest objects were observed at the distance of NGC 300 (^{Peña et al. 2012}).

The central values of Ne/H and Ar/H are similar for PNe and H II regions. Once again, this indicates that PNe are young objects, which had initial abundances similar to the present ISM.

An interesting fact is that at R/R₂₅ larger than 0.6, the Ar/H abundances of PNe are definitely larger than those of H II regions, independently of the large dispersion. This is also found for Ne, but it is less marked. Since Ar it is not expected to have been modified by stellar nucleosynthesis of the central stars, this is indicating that radial migration should have been important for PNe (despite their youth), and that PNe have changed their initial galactocentric distances, being churned in the galaxy, although not at the level of migration found in M 31.

4.4. The Milky Way

The results for the MW are presented in Figures 6 and 7. The latter figure shows PNe data in distance bins of 1 kpc. The PN sample covers a galactocentric distance interval 0.21 kpc ≤ R ≤ 22.73 kpc (0.02 ≤ R/R₂₅ ≤ 1.97). The gradient obtained for in PNe is −0.024 dex kpc⁻¹, similar to the ones derived for Ne and Ar. As occurs in M 31, and NGC 300, PN gradients are flatter, by about a factor of 2, than the gradients of H II regions.

The gradients for Type I and non-Type I PNe are shown in Figure 8. In this figure the slopes of both kinds of nebulae seem indistinguishable. Our result is different from that reported by ^{Stanghellini & Haywood (2010)} and ^{Stanghellini & Haywood (2018)} for their young and old PN samples, using the distances by S10. They claim that young PNe definitely present a steeper gradient of −0.027 dex kpc⁻¹ versus −0.015 dex kpc⁻¹ for old PNe. We are not sure if our different result is due to the different PN samples (a much larger sample from the literature, not homogenized, was used by Stanghellini & Haywood) or to the different distance scale used. In any case we concur with them in that PN gradients are flatter than those of H II regions. Therefore, it appears than PNe have moved from their birth places due to radial migration and are showing flatter gradients. Alternatively, it is possible that the abundance gradients were flatter several Gyr ago.

Although a linear fit to these gradients produces acceptable results, it should be noticed that at distances larger than about 14 kpc, the observed abundances (in particular O/H and Ar/H) decrease to a value below the linear fit and continue flat outwards, showing a break. Unfortunately, in our sample there are few PNe there and this result is not conclusive. A break at a distance R ≈13.5 kpc was reported by ^{Stanghellini & Haywood (2018)} for the same sample of outer PNe. Following ^{Halle et al. (2015)} they attributed this behavior to the effect of the galactic bar, whose outer Lindblad resonance would be located at about this distance according to N-body simulations. It is crucial to observe a larger number of PNe in the outskirts of the Galaxy to verify this behavior.

It is interesting to compare the gradients of H II regions (−0.040 dex kpc⁻¹) and PNe (−0.024 dex kpc⁻¹) with other gradients provided by well measured indicators of different ages. ^{Mollá et al. (2019)} prepared a compilation of gradients for objects of different ages, computed by various authors, in order to compare them with results from their chemical evolution models. From this compilation we selected the gradients calculated by Mollá et al. for Cepheid stars, (which are young objects with ages of about 0.1 Gyr and show an O/H gradient of −0.049 dex kpc⁻¹) and open clusters (OC) which span ages from 2 to more than 8 Gyr. The O/H gradient for OC younger than 2 Gyr is −0.030 dex kpc⁻¹, and for ages between 2 and more than 8 Gyr, it is −0.027 dex kpc⁻¹. It is clear that the gradients of H II regions and Cepheid stars are equal within uncertainties, indicating that in the Galaxy the chemical gradients have not changed significantly at least in the last 0.1 Gyr, while gradients of PNe (Type I’s and non-Type I’s) are equal to those of OC older than 2 Gyr. Therefore for objects of 2-8 Gyr the gradient is flatter (−0.027 dex kpc⁻¹) than those for Cepheids and H II regions (−0.049 and −0.040 dex kpc⁻¹). PNe and OC certainly could have been affected by migration, but not in as an extreme way as in M 31. Alternatively, PN and OC gradients could correspond to the true gradients present several Gyr ago.

^{Mollá et al. (2019)} discussed the possible effects of migration in the MW. By analyzing models by different authors, their conclusion was that radial migration seems not to be important for stars younger than 4 Gyr. Only for objects older than 8 Gyr, radial migration may be important. Models by Mollá et al. of the time evolution of chemical gradients in the Milky Way (without considering migration, bar or spiral arms) predict a very smooth evolution of the radial gradient within the optical disk. Some models show a steepening of the gradient, from −0.02 to −0.04 dex kpc⁻¹ over a time of 10 Gyr. Therefore the gradients shown by PNe and OC could be the gradients at the time of formation of these objects, not affected by migration.