# Gradients of chemical abundances in the Milky Way from H ii regions:
distances derived from Gaia EDR3 parallaxes and temperature inhomogeneities

J. E. Méndez-Delgado1,2 , A. Amayo3, K. Z. Arellano-Córdova4, C. Esteban1,2 J.
García-Rojas1,2, L. Carigi3 and G. Delgado-Inglada3

1Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Spain
2Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna,
Spain
3Instituto de Astronomía, Universidad Nacional Autónoma de México, Ap. 70-264,
04510 CDMX, México
4Department of Astronomy, The University of Texas at Austin, 2515 Speedway,
Stop C1400, Austin, TX 78712, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a reassessment of the radial abundance gradients of He, C, N, O,
Ne, S, Cl, and Ar in the Milky Way using the deep optical spectra of 42 H ii
regions presented in Arellano-Córdova et al. (2020, 2021) and Méndez-Delgado
et al. (2020) exploring the impact of: (1) new distance determinations based
on Gaia EDR3 parallaxes and (2) the use of Peimbert’s temperature fluctuations
paradigm ($t^{2}>0$) for deriving ionic abundances. We find that distances
based on Gaia EDR3 data are more consistent with kinematic ones based on
Galactic rotation curves calibrated with radio parallaxes, which give less
dispersion and uncertainties than those calibrated with spectrophotometric
stellar distances. The distances based on the Gaia parallaxes –DR2 or EDR3–
eliminate the internal flattening observed in previous determinations of the
Galactic gradients at smaller distances than $\sim 7$ kpc. Abundances and
gradients determined assuming $t^{2}>0$ —not only for O but also for the rest
of elements— are not affected by the abundance discrepancy problem and give
elemental abundances much consistent with the solar ones for most elements. We
find that our radial abundance gradient of He is consistent with the most
accurate estimates of the primordial He abundance. We do not find evidence of
azimuthal variations in the chemical abundances of our sample. Moreover, the
small dispersion in the O gradient —indicator of metallicity in photoionized
regions— indicate that the gas of the H ii regions is well mixed in the
sampled areas of the Galaxy.

###### keywords:

Stars: distances–ISM: abundances–Galaxy: abundances–Galaxy: disc–Galaxy:
Evolution–ISM: H ii regions.

††pubyear: 2021††pagerange: Gradients of chemical abundances in the Milky Way
from H ii regions: distances derived from Gaia EDR3 parallaxes and temperature
inhomogeneities–Gradients of chemical abundances in the Milky Way from H ii
regions: distances derived from Gaia EDR3 parallaxes and temperature
inhomogeneities

##  1 Introduction

Chemical evolution models of galaxies require precise observational
constraints as the determinations of abundance gradients. In order to obtain
good determinations of these gradients, it is necessary to have accurate
radial distances and precise determinations of the chemical abundances. In the
Milky Way, the determination of the Galactocentric distances is particularly
problematic, since both the Sun and the observed nebulae are placed within the
Galactic disk. Although it is common for Galactic abundance gradient studies
to focus mainly on the determination of chemical abundances, in this paper we
will emphasize the effects of using the more precise distance determinations.

Distances to Galactic H ii regions can be determined using different
techniques. Some of the most common methodologies are based on the study of
radial velocities of the nebulae or photometry of their associated stars (see
e. g. Caplan et al., 2000; Russeil, 2003; Quireza et al., 2006; Russeil et
al., 2007; Balser et al., 2011; Moisés et al., 2011; Foster & Brunt, 2015;
Anderson et al., 2015; Wenger et al., 2018, and references therein). However,
their results are strongly dependent on the rotation and extinction models of
the Galaxy, which are still matter of debate. This problem can be avoided by
using direct parallax-based distance determinations of their stellar
populations (e. g. Menten et al., 2007; Kraus et al., 2007; Hirota et al.,
2007; Bartkiewicz et al., 2008; Reid et al., 2009; Brunthaler et al., 2009;
Zhang et al., 2009; Xu et al., 2009, and references therein). However, until a
few years ago, there were not enough reliable parallax measurements. Since the
availability of the optical parallaxes from the Second Data Release of Gaia
(hereinafter Gaia DR2, Gaia Collaboration et al., 2018) there are billions of
reliable optical stellar parallaxes, from which distance can be determined
directly. Moreover, in the recent Gaia Early Data Release 3 (hereinafter EDR3,
Gaia Collaboration et al., 2021), these parallaxes have substantially improved
their precision.

Regarding the determination of chemical abundances of H ii regions, there is a
significant source of uncertainty due to the Abundance Discrepancy (AD)
problem, which is the systematic difference between the derived abundances
based on the intensity of collisionally excited lines (CELs) and recombination
lines (RLs), where the RLs give always higher abundances. The origin of this
problem is still not clear. However, García-Rojas & Esteban (2007) showed that
the Abundance Discrepancy Factor (ADF)111Measured as the ratio between the
abundances derived with RLs and CELs. is rather constant for O2+ in Galactic H
ii regions –presenting a different behavior from what is observed in planetary
nebulae–, being consistent with the predictions of the temperature
fluctuations paradigm proposed by Peimbert (1967).

In this work, we derive the Galactic radial abundance gradients of He, C, N,
O, Ne, S, Cl, and Ar based on the 42 Galactic H ii regions studied by Méndez-
Delgado et al. (2020) and Arellano-Córdova et al. (2020, 2021). We analyze the
distances based on the Gaia EDR3 parallaxes and the probabilistic approach of
Bailer-Jones et al. (2021) and derive the chemical composition assuming the
temperature fluctuations ($t^{2}>0$) paradigm (Peimbert, 1967).

In Sec. 2 we describe the sample of spectra adopted from the literature. In
Sec. 3, we analyze the distances to the Galactic H ii regions of our sample
using the Gaia EDR3 parallaxes, as well as the kinematic distance
determinations. In Sec. 4, we analyze the determination of chemical abundances
in the Galactic H ii regions of our sample under the paradigm of temperature
fluctuations – $t^{2}>0$ – as well as the resulting chemical abundance
gradients. In Sec. 5 we discuss our results and summarize them in Sec. 6. In
the Appendix we include some data tables and figures.

##  2 Observational sample

The spectra of the 42 Galactic H ii regions used in this paper are of high and
intermediate spectral resolution, taken with different telescopes with
diameters ranging from 4 m to 10 m. The objects which spectra are published in
Esteban et al. (2004); Esteban et al. (2013) and García-Rojas et al. (2004,
2005, 2006, 2007)222See Table A4 to know in which reference the spectra of
each object were published. were observed with the UT2 (Kueyen) of the Very
Large Telescope (VLT) using the Ultraviolet–Visual Echelle Spectrograph
(UVES), covering the spectral range from 3570 to 10,400 Å and with $R\sim$
8000\. The spectra of the H ii regions observed by Esteban et al. (2017);
Esteban & García-Rojas (2018) and Arellano-Córdova et al. (2021) were taken
with the 10.4 m Gran Telescopio Canarias (GTC) using OSIRIS (Optical System
for Imaging and low-Intermediate-Resolution Integrated Spectroscopy)
spectrograph in longslit mode, covering from 3600 to 7750 Å with $R\sim$ 1000
or 2500. The spectra of three H ii regions were observed by García-Rojas et
al. (2014) or Fernández-Martín et al. (2017) with the Intermediate dispersion
Spectrograph and Imaging System (ISIS) attached to the 4.2 m William Herschel
Telescope (WHT), the spectral coverage was from 3600 to 9200 Å or 3200 to
10,000 Å with $R\sim$ 1000\. In addition, we have included 17 Galactic H ii
regions and ring nebulae around massive stars for the distance analysis but
not in the abundance one. This is because we do not have the emission line
ratios of their spectra available or, in the case of ring nebulae, their He,
O, and N abundances may be polluted by stellar ejecta.

##  3 Revising the distances of a sample of Galactic H ii regions with Gaia
EDR3

We conduct an extensive literature search on ionizing sources of our sample of
Galactic H ii regions in order to determine their distances. The ionizing
stars are expected to be O, early B or Wolf-Rayet-type stars, so they should
be among the most luminous stars contained in the boundaries or immediate
surroundings of the H ii regions. In addition, their parallax should be more
precise than those of the rest of their associated stellar population.
However, some of these stars may be located in dusty environments or belong to
multiple systems that may be not well spatially resolved. In these cases, we
look for other stars or young stellar objects (YSOs) that belong to the H ii
region, even if they are not their ionizing sources.

After finding the stars contained in the nebulae, we locate them in the Gaia
DR2 and EDR3 catalogs using their coordinates, to finally obtain their
parallax-based distances (Bailer-Jones et al., 2018, 2021). Besides the
parallaxes, the Gaia EDR3 data contain color and magnitude measurements,
allowing to estimate photometric distances. Taking advantage of this, Bailer-
Jones et al. (2021), have estimated the distance to the stars using two
methods, obtaining the so-called geometric and photogeometric distances.
Geometric distances are estimated from the parallaxes, considering the non-
linearity of the transformation and the asymmetry of the probability
distribution through a prior parameter that varies smoothly as a function of
Galactic longitude and latitude, independently of assumptions of stellar
physics or interstellar extinction. Geometric distances were also calculated
with the Gaia DR2 parallaxes in Bailer-Jones et al. (2018). The photogeometric
distances, meanwhile, use the same distance prior, in addition to a color-
magnitude one based on a model of the Galactic interstellar extinction.

In Table A1 we present the identified ionizing/associated stars in each
nebula, as well as their spectral type, the Gaia DR2 ID, heliocentric
geometric distances derived from their parallaxes in the Gaia DR2 and EDR3
data (Bailer-Jones et al., 2018, 2021) and the references to the
identification of ionizing/associated stars (I/A) and spectral type (SpT). For
each distance determination, we specify the uncertainty associated with the
parallax. In general, these uncertainties decrease considerably between the
Gaia DR2 and Gaia EDR3 data except in two cases: the stars MFJ SH 2-212 2 and
HD 253327, associated with the H ii regions Sh 2-212 and Sh 2-257,
respectively. In both stars, the EDR3 parallax is noticeably worse than the
DR2 one and give very different distances. Moffat et al. (1979) cataloged 14
possible ionizing stars in Sh 2-212. However, Deharveng et al. (2008) showed
that several of these sources are evolved stars that may not belong to the
same star cluster. Moffat et al. (1979) and Deharveng et al. (2008) agree that
the main ionizing star of the region is MFJ SH 2-212 2, which has the most
uncertain parallax. However, the confirmed membership of MFJ SH 2-212 5, MFJ
SH 2-212 7 and MFJ SH 2-212 11 to the same stellar group as MFJ SH 2-212 2
allow us to estimate the distance to Sh 2-212 with a high confidence. The odd
value of the parallax of MFJ SH 2-212 2 does not seem to be related to the
presence of dust in the center of Sh 2-212 since most of its optical
extinction is of interstellar origin (Deharveng et al., 2008). It is more
likely that the discrepant parallax is due to MFJ SH 2-212 2 being a binary or
multiple system. This is suggested by its renormalised unit weight error
(RUWE) value of 8.48, which is larger than the expected threshold of 1.4 for
single stars (Fabricius et al., 2021).

The Gaia parallax of HD 253327 (in Sh 2-257) is also very different when
comparing DR2 and EDR3 data. Ojha et al. (2011) showed that Sh 2-255 and Sh
2-257 are associated with the same gas complex and therefore they are at the
same distance. The lower uncertainty of the EDR3 parallax of the ionizing star
of Sh 2-255, LS 19, indicates that its distance is the most suitable one for
HD 253327. In this case, the RUWE value of 1.209 does not point to HD 253327
being a binary or multiple system (although it does not rule it out).
Additionally, the similar optical extinction coefficients of Sh 2-255 and Sh
2-257 (see Table 7 from Shaver et al. (1983a)) seems to rule out dust
extinction as the cause of the problem. There are several ionizing stars for
some nebulae included in Table A1. If the distance obtained for each ionizing
source is consistent, we take the more precise individual parallax as the
representative one for the nebula. In the cases of NGC 3603, M 16, M 17 and M
42, we adopt the distances derived by Drew et al. (2019) and Binder & Povich
(2018) which are based on the analysis of the Gaia DR2 parallaxes of multiple
stars associated with the photoionized regions. The use of several stars
diminishes the statistical errors, reaching a higher precision in the distance
estimates.

A large fraction of the sample of stars analyzed in this work has Gaia EDR3
parallaxes with uncertainties smaller than 10 per cent
($e_{\text{plx}}/\text{plx}<0.10$), enabling us to make a reliable estimate of
the distance directly from inverting the parallax (Bailer-Jones et al., 2021).
In Gaia DR2, quasars observations indicate a global zeropoint bias in the
parallax measurements of about $-$0.03 mas, with variations dependent on
magnitude, color and position of the stars (Lindegren et al., 2018). For Gaia
EDR3, Lindegren et al. (2021b, a) provide a tentative expression for a better
correction of the zeropoint bias including its non-trivial dependence on the
ecliptic latitude, magnitude and color. For our sub-sample with the best
parallaxes (with uncertainties smaller than 10 per cent), this zeropoint
varies from $-$0.055 mas to $-$0.011 mas, although the median is $-$0.030 mas
with $1\sigma$ deviation of 0.010 mas. This indicates that the deviation from
a constant value is rather small, minimizing the dependence of the distance
with the zeropoint correction and the variables on which it depends. From the
Gaia EDR3 data we can have up to 3 distance estimates for the same star which
will be analyzed in Sec. 3.1, along with some relevant kinematic distance
estimates.

###  3.1 Galactocentric Distances

We adopt a solar Galactocentric distance of $R_{0}=8.2\pm 0.1$ kpc to estimate
the Galactocentric distances of the nebulae from the heliocentric parallax-
based ones. That value of $R_{0}$ is the best estimate to the overall
distribution of 26 recent values with tracers in the Galactic centre, bulge,
disc and halo (Bland-Hawthorn & Gerhard, 2016), and it is very close to the
geometric distance to the Galactic centre black hole estimated by Gravity
Collaboration et al. (2019) of $R_{0}=8.178\pm 0.013\text{\it(stat)}\pm
0.022\text{\it(sys)}$ kpc, where stat and sys stands for the contribution of
statistical and systematic uncertainties, respectively. This same value was
adopted by Méndez-Delgado et al. (2020), Arellano-Córdova et al. (2020, 2021),
so the comparison with results of those papers is straightforward.

In Table A3 we compare our parallax-based estimates of Galactocentric
distances with the kinematic determinations of Russeil (2003); Russeil et al.
(2007) and Wenger et al. (2019) in the nebulae in common. Although there is a
large number of articles devoted to the estimation of kinematic distances, the
ones of Russeil (2003) and Russeil et al. (2007) are representative of the
usual method based on the Brand & Blitz (1993) rotation curve, calibrated
through the observation of around 400 Galactic H ii regions with
spectrophotometric distances to their associated stars (Brand & Wouterloot,
1988), and the IAU-defined solar motion parameters333$R_{0}=8.5$ kpc,
$\theta_{0}=220\text{ km s}^{-1}$, $U_{\odot}=10.27\text{ km s}^{-1}$,
$V_{\odot}=15.32\text{ km s}^{-1}$, $W_{\odot}=7.74\text{ km s}^{-1}$ (Kerr &
Lynden-Bell, 1986). On the other hand, the kinematic determinations of Wenger
et al. (2019) are based on a Monte Carlo technique adapted to the more recent
rotation curve of Reid et al. (2014), which was calibrated with around 100
parallax-distances from masers observed at radio wavelengths, with updated
solar parameters444$R_{0}=8.34\pm 0.16$ kpc, $\theta_{0}=240\pm 8\text{ km
s}^{-1}$, $U_{\odot}=10.5\pm 1.7\text{ km s}^{-1}$, $V_{\odot}=14.4\pm
6.8\text{ km s}^{-1}$, $W_{\odot}=8.9\pm 0.9\text{ km s}^{-1}$ (although the
sample has been expanded to $\approx 200$ regions in Reid et al., 2019). The
results of this method generally present smaller uncertainties than the direct
use of the rotation curve of Reid et al. (2014) (Wenger et al., 2018).

In Fig. A1 we show the comparison between the Gaia EDR3 geometric
Galactocentric distances and other estimates obtained from Gaia Parallaxes. In
the top panel, the consistency with the geometric Gaia DR2 distance is
remarkable, besides the reduction of the uncertainties in the EDR3 values. The
middle panel shows that the photogeometric values are practically the same
than the geometric ones for the whole sample, with the exception of Sh 2-127,
where the difference is appreciable, although both estimates are consistent
within the error bars. In the more distant regions, the uncertainties in the
photogeometric distances are slightly smaller due to the use of more
information about the stars. However, all the objects we analyze in this paper
are located at almost zero Galactic latitudes, which makes the correction for
extinction less accurate (Bailer-Jones et al., 2021). Therefore, there is
little reward on adopting the photogeometric distances in some particular
cases, while it seems more reasonable to use the geometric EDR3 ones in the
whole sample as the representative parallax-based distances. The bottom panel
shows the consistency between the geometric EDR3 Galactocentric distances and
those simply calculated as the inverse of the parallax for the sub-sample with
parallax uncertainties smaller than 10 per cent.

Fig. A2 shows a comparison between the geometric Gaia EDR3 Galactocentric
distances and the kinematic estimations by Russeil (2003), Russeil et al.
(2007) and Wenger et al. (2019). Besides the adoption of different solar
parameters, the main difference between both kinematic estimates is in the
Galactic rotation curve used. As we can see in Fig. A2, at Galactocentric
distances $<13$ kpc, the kinematic determinations from Russeil (2003) and
Russeil et al. (2007) show a moderate good agreement with our estimates based
on Gaia EDR3 parallaxes, although there is a slight overestimation of the
kinematic distances due to the larger solar $R_{0}$ adopted. More important is
the underestimation of the kinematic distances in a group of regions with
$R_{G}$ between 5 and 7 kpc, which includes Sh 2-29, Sh 2-32, M 8 and M 20.
Finally, at distances greater than $\approx 13$ kpc, the kinematic distances
from Russeil (2003) and Russeil et al. (2007), seem to be significantly
overestimated for most objects. The bottom panel of Fig. A2 shows that the
kinematic distances from Wenger et al. (2019) are in fairly good agreement
with our determinations. Although there is debate about which rotation curve
of the Milky Way is the most appropriate (see Russeil et al., 2017), in our
case, the one proposed by Reid et al. (2014) gives more consistent results.
However, it should be kept in mind that our sample of Galactic H ii regions
and ring nebulae is quite limited and only covers two quadrants of the
Galactic disc.

Considering observational evidences, we define the following criteria to adopt
the representative distances of our sample of Galactic nebulae: (i) we adopt
the EDR3 geometric distance when the uncertainties of the parallax are smaller
than 20 per cent; (ii) when (i) is not satisfied, we adopt the kinematic
distances of Wenger et al. (2019) if their uncertainties are smaller than 20
per cent; (iii) when criteria (i) and (ii) cannot be satisfied (the only case
is Sh 2-48), we adopt the EDR3 geometric distance, however, it must be used
with caution since its uncertainty is relatively high. Exceptions to the
aforementioned procedure are Sh 2-209 and Sh 2-270, and both cases will be
discussed in Sec. 3.2. Table 1 presents the adopted heliocentric and
Galactocentric distances for our sample of Galactic nebula while Fig. 1 shows
their Galactic distribution.

Figure 1: Galactic distribution of the H ii regions and ring nebulae of our
sample. The adopted solar Galactocentric distance is $R_{0}=8.2$ kpc.  Table
1: Adopted heliocentric and Galactocentric distance estimates for the H ii
regions and ring nebulae of our sample.

Nebula |  $d$ (kpc) |  $R_{G}$ (kpc) | Method | Notes
---|---|---|---|---
Sh 2-29 | $1.15^{+0.04}_{-0.03}$ | $7.06\pm 0.14$ | Plx |
Sh 2-32 | $1.46\pm 0.05$ | $6.75\pm 0.15$ | Plx |
Sh 2-47 | $1.54\pm 0.05$ | $6.73^{+0.15}_{-0.14}$ | Plx |
Sh 2-48 | $3.59^{+0.70}_{-0.61}$ | $4.88^{+0.72}_{-0.66}$ | Plx | 1
Sh 2-53 | $3.38\pm 0.18$ | $5.10^{+0.24}_{-0.26}$ | Plx |
Sh 2-54 | $1.91\pm 0.05$ | $6.42\pm 0.15$ | Plx |
Sh 2-57 | $2.10^{+0.06}_{-0.08}$ | $6.32\pm 0.17$ | Plx |
Sh 2-61 | $2.39^{+0.07}_{-0.08}$ | $6.15\pm 0.16$ | Plx |
Sh 2-82 | $0.78\pm 0.06$ | $7.76\pm 0.13$ | Plx |
Sh 2-83 | $16.12\pm 1.23$ | $13.24\pm 1.13$ | Kin | 2
Sh 2-88 | $2.06^{+0.08}_{-0.07}$ | $7.44^{+0.12}_{-0.11}$ | Plx |
Sh 2-90 | $3.36^{+0.39}_{-0.28}$ | $7.33^{+0.12}_{-0.10}$ | Plx |
Sh 2-93 | $3.64^{+0.08}_{-0.07}$ | $7.66^{+0.19}_{-0.21}$ | Kin | 2
Sh 2-98 | $11.24^{+2.59}_{-2.52}$ | $11.21^{+2.00}_{-1.79}$ | Plx |
Sh 2-100 | $11.40^{+1.14}_{-1.01}$ | $11.61\pm 0.87$ | Plx |
Sh 2-127 | $10.13^{+1.53}_{-1.22}$ | $13.76^{+1.31}_{-0.87}$ | Kin | 2
Sh 2-128 | $7.27^{+1.10}_{-0.82}$ | $11.78^{+0.78}_{-0.64}$ | Kin | 2
Sh 2-132 | $4.53^{+0.29}_{-0.27}$ | $10.21^{+0.28}_{-0.26}$ | Plx |
Sh 2-152 | $4.54^{+0.86}_{-0.74}$ | $10.61^{+0.71}_{-0.43}$ | Kin | 2
Sh 2-156 | $2.56^{+0.22}_{-0.19}$ | $9.40^{+0.21}_{-0.22}$ | Plx |
Sh 2-158 | $2.84^{+0.14}_{-0.16}$ | $9.61\pm 0.20$ | Plx |
Sh 2-175 | $2.02\pm 0.05$ | $9.38\pm 0.13$ | Plx |
Sh 2-203 | $2.39^{+0.08}_{-0.09}$ | $10.22\pm 0.17$ | Plx |
Sh 2-206 | $2.96^{+0.17}_{-0.15}$ | $10.88^{+0.25}_{-0.26}$ | Plx |
Sh 2-207 | $3.59^{+0.23}_{-0.26}$ | $11.48\pm 0.34$ | Plx |
Sh 2-208 | $4.02^{+0.27}_{-0.25}$ | $11.87^{+0.37}_{-0.34}$ | Plx |
Sh 2-209 | $9.33\pm 0.70$ | $17.00\pm 0.70$ | Avg | 5
Sh 2-212 | $6.65^{+1.36}_{-1.26}$ | $14.76\pm 1.30$ | Kin | 2
Sh 2-219 | $4.16^{+0.32}_{-0.28}$ | $12.17^{+0.39}_{-0.41}$ | Plx |
Sh 2-228 | $2.56\pm 0.09$ | $10.72\pm 0.19$ | Plx |
Sh 2-235 | $1.66\pm 0.07$ | $9.85^{+0.16}_{-0.17}$ | Plx |
Sh 2-237 | $2.07\pm 0.06$ | $10.26^{+0.16}_{-0.17}$ | Plx |
Sh 2-255 | $1.96^{+0.12}_{-0.09}$ | $10.13\pm 0.21$ | Plx |
Sh 2-257 | $1.96^{+0.12}_{-0.09}$ | $10.13\pm 0.22$ | Plx |
Sh 2-266 | $4.60^{+0.53}_{-0.51}$ | $12.69^{+0.62}_{-0.64}$ | Plx |
Sh 2-270 | $8.08\pm 1.29$ | $16.10\pm 1.40$ | Avg | 6
Sh 2-271 | $3.25^{+0.11}_{-0.12}$ | $11.34\pm 0.22$ | Plx |
Sh 2-283 | $5.37^{+0.58}_{-0.41}$ | $13.12^{+0.62}_{-0.65}$ | Plx |
Sh 2-285 | $4.38^{+0.42}_{-0.31}$ | $12.07^{+0.51}_{-0.47}$ | Plx |
Sh 2-288 | $5.06^{+0.49}_{-0.47}$ | $12.54^{+0.54}_{-0.52}$ | Plx |
Sh 2-297 | $1.08^{+0.04}_{-0.05}$ | $8.98^{+0.14}_{-0.13}$ | Plx |
Sh 2-298 | $4.12^{+0.54}_{-0.40}$ | $11.38^{+0.58}_{-0.51}$ | Plx |
Sh 2-301 | $3.21\pm 0.14$ | $10.50\pm 0.20$ | Plx |
Sh 2-308 | $1.51^{+0.10}_{-0.08}$ | $9.14^{+0.16}_{-0.17}$ | Plx |
Sh 2-311 | $5.28^{+0.40}_{-0.35}$ | $11.60^{+0.37}_{-0.39}$ | Plx |
IC 5146 | $0.76\pm 0.01$ | $8.29\pm 0.10$ | Plx |
RCW 52 | $2.33\pm 0.07$ | $7.83\pm 0.10$ | Plx |
RCW 58 | $2.70^{+0.13}_{-0.11}$ | $7.60\pm 0.10$ | Plx |
G2.4+1.4 | $2.65\pm 0.15$ | $5.57^{+0.25}_{-0.26}$ | Plx |
NGC 2579 | $5.19^{+0.48}_{-0.35}$ | $10.82^{+0.41}_{-0.44}$ | Plx |
NGC 3576 | $2.48^{+0.11}_{-0.10}$ | $7.66\pm 0.10$ | Plx |
NGC 3603 | $7.02\pm 0.10$ | $8.61\pm 0.11$ | Plx | 3
NGC 6888 | $1.67\pm 0.04$ | $7.95\pm 0.10$ | Plx |
NGC 7635 | $2.83^{+0.13}_{-0.10}$ | $9.64^{+0.17}_{-0.18}$ | Plx |
M 8 | $1.22\pm 0.04$ | $6.99\pm 0.14$ | Plx |
M 16 | $1.71\pm 0.18$ | $6.58^{+0.26}_{-0.28}$ | Plx | 4
M 17 | $1.82\pm 0.16$ | $6.46\pm 0.25$ | Plx | 4
M 20 | $1.42^{+0.09}_{-0.08}$ | $6.79\pm 0.18$ | Plx |
M 42 | $0.41\pm 0.01$ | $8.54\pm 0.11$ | Plx | 4

1: Uncertainties in the parallax larger than 20 per cent.

2: Kinematic determinations taken from Wenger et al. (2019).

3, 4: Geometric determinations based on the Gaia DR2 parallaxes taken from
Drew et al. (2019) and Binder & Povich (2018), respectively.

5, 6: Average of several spectrophotometric and/or kinematic determinations
obtained by Esteban et al. (2017) and Esteban & García-Rojas (2018),
respectively.

Figure 2: Comparison between the updated Galactocentric distances adopted in
this work with those used by Méndez-Delgado et al. (2020); Arellano-Córdova et
al. (2020, 2021).

In Fig. 2 we show the comparison between the values adopted by Méndez-Delgado
et al. (2020) and Arellano-Córdova et al. (2020, 2021) and our updated values
given in Table 1. In general, there is a good agreement between both sets of
distances. In Méndez-Delgado et al. (2020) and Arellano-Córdova et al. (2020,
2021), the ionizing/associated stars of some of the nebulae have very
uncertain Gaia DR2 parallaxes. In those cases, the adopted distance was the
average of different kinematic and spectrophotometric estimates. However, most
of the kinematic distances used in the calculation of the aforementioned
average were based on the rotation curve of Brand & Blitz (1993), which, as
shown in Fig. A2, seems to overestimate the distance of the objects located at
Galactocentric distances $\gtrsim 13$ kpc, which explains the larger previous
distance determinations of several nebula located between $\approx 13$ and
$\approx 14$ kpc shown in Fig. 2.

###  3.2 On the distances of Sh 2-209 and Sh 2-270

From our sample of Galactic H ii regions, only Sh 2-209 and Sh 2-270 show
differences between parallax-based and kinematic/spectrophotometric distances
that can not be explained by methodological issues. The Galactocentric
distances adopted in Esteban et al. (2017) and Esteban & García-Rojas (2018)
are $R_{G}=17.00\pm 0.70\text{ kpc}$ and $R_{G}=16.10\pm 1.40\text{ kpc}$ for
Sh 2-209 and Sh 2-270, respectively, but the parallax-based distances derived
from Gaia EDR3 are only about 65 per cent of those values (see Table A3). Sh
2-209 and Sh 2-270 were precisely among the objects which stars have very
uncertain Gaia DR2 parallaxes and for which Esteban et al. (2017) and Esteban
& García-Rojas (2018) adopted the average value of several spectrophotometric
and/or kinematic distances, as commented at the end of the previous section.
If we assume that the Gaia EDR3 parallax-based distances are the true ones for
these regions, Sh 2-209 would be located only at 0.36 kpc from Sh 2-206 in the
Galactic plane (both nebulae are less than one degree apart). There are
several determinations of the Galactic rotation velocity of Sh 2-209 in the
local standard of rest (Brand & Blitz, 1993; Caplan et al., 2000; Russeil,
2003; Balser et al., 2011), all being consistent with the most precise one by
Balser et al. (2011) of $V_{\text{lsr}}=-51.18\pm 0.09\text{ km s}^{-1}$. The
same occurs with Sh 2-206, where Balser et al. (2011) determined
$V_{\text{lsr}}=-26.61\pm 0.05\text{ km s}^{-1}$. These numbers imply a ratio
of 1.92 between the Galactic rotation velocity of Sh 2-209 with respect to
that of Sh 2-206, which is entirely inconsistent for two neighbor objects
located at very similar Galactocentric distance.

In the case of Sh 2-270, if we assume the Gaia EDR3 parallax-based distance as
the true one, this region would be only 0.22 kpc apart from Sh 2-255 and Sh
2-257. Balser et al. (2011) estimated a $V_{\text{lsr}}=-5.05\pm 0.09\text{ km
s}^{-1}$ for Sh 2-257 (which is located at the same distance as Sh 2-255),
while Russeil (2003) reported a $V_{\text{lsr}}=25.6\text{ km s}^{-1}$ for Sh
2-270. However, the three objects are in the general direction of the Galactic
anticentre and such velocity difference is not conclusive.

Foster & Brunt (2015) performed a spectrophotometrical analysis of the
ionizing/associated stars of Sh 2-209 and Sh 2-270 included in Table A1. From
their distance modulus, they found heliocentric distances of $d=10.58\pm
0.57\text{ km s}^{-1}$ and $d=9.27\pm 1.85\text{ km s}^{-1}$ for Sh 2-209 and
Sh 2-270, respectively, which are consistent with the distances adopted by
Esteban et al. (2017) and Esteban & García-Rojas (2018). This rejects the
misidentification of the ionizing/associated stars as the cause of the
discrepancy in the distances of these objects and points out to possible
errors in the reported Gaia EDR3 parallaxes. Although these errors are rather
rare, there are cases – as MFJ SH 2-212 2 and HD 253327, commented before –
where the Gaia DR2 and EDR3 parallaxes can differ beyond the formal
uncertainty bars.

As a result of this discussion, we adopt the distances reported by Esteban et
al. (2017) and Esteban & García-Rojas (2018) for Sh 2-209 and Sh 2-270, both
having convenient conservative error bars. Improved optical or radio
parallaxes for the stars associated with these nebulae would help to solve the
distance inconsistencies.

##  4 Effects of temperature fluctuations on the chemical abundances

Peimbert (1967) introduced the concept and the formalism of the paradigm of
temperature fluctuations in gaseous nebulae. In his eqs. (9) and (12), he
defines the average temperature ($T_{0}$) and the root mean square temperature
fluctuation parameter ($t^{2}$), that accounts for the level of
inhomogeneities or fluctuations in the spatial distribution of the electron
temperature within the observed volume. According to this paradigm, when we
estimate the electron temperature from an auroral-nebular ratio of CELs – as
[O iii] I($\lambda 4363$)/I($\lambda 5007$) –, the resulting value will be
conditioned to $t^{2}$. The derived temperature will favor the hottest areas
within the line of sight and consequently, ionic abundance determinations
based on CELs will be underestimated.

On the other hand, the intensity ratios of RLs have a very slight dependence
on temperature and the impact of temperature fluctuations is practically
negligible. Because of this, it is possible that temperature inhomogeneities
could cause or contribute to the AD problem, making abundances based on CELs
systematically underestimated, while those of RLs are not.

As mentioned in Sec. 1, García-Rojas & Esteban (2007) conclude that for M 8, M
16, M 17, M 20, M 42, NGC 3576, NGC 3603 and Sh 2-311, the ADF(O2+) is rather
constant around a factor of 2 and consistent with the temperature fluctuations
paradigm. However, the ADF of other ions, such as O+, does not have to be the
same as ADF(O2+). In fact, if there is a global value of $t^{2}>0$ in a given
nebula, this will affect differently to each ion, depending on the excitation
energies of the upper levels of their observed CELs as shown in eq. (11) from
Peimbert et al. (2004). Therefore, considering a constant ADF on all ions is
different from considering a global constant $t^{2}$.

There are several methods to estimate $T_{0}$ and $t^{2}$. It is possible to
calculate them by comparing different temperature diagnostics of the gas that
belongs to the same volume. It is also common to determine them from the ADF,
although this procedure assumes that temperature fluctuations are the cause of
the AD problem. From our sample, M 8, M 16, M 17, M 20, M 42, NGC 2579, NGC
3576, NGC 3603 and Sh 2-311 have deep high spectral-resolution spectra
obtained with the Very Large Telescope (VLT), with reliable estimates of
$t^{2}$. Fig. 3 shows that the radial distribution of $t^{2}$ in the
aforementioned sample of H ii regions along the Galactic disc seems to be
rather flat, at least at $R_{G}$ values between 6.45 and 11.6 kpc. Therefore,
it seems reasonable to adopt a weighted average of $t^{2}=0.038\pm 0.004$ as a
representative-$t^{2}$ value for the Galactic H ii regions. We must say that
although its uncertainty of $t^{2}$ is the smallest of the whole sample, the
Orion Nebula has been eliminated in the calculation of the weighted average.
This is because the Orion Nebula shows the lowest value, by far, of $t^{2}$.
Considering it to calculate the weighted mean we obtain $t^{2}=0.029\pm
0.008$. A value that does not seem to represent properly the rest of the H ii
regions included in Fig. 3. In any case, if included, both determinations of
the weighted average of $t^{2}$ are consistent within the $1\sigma$ error
bars. The conclusions of the paper are not altered nor do they depend on the
inclusion or exclusion of this point.

###  4.1 Ionic and total abundances

We use eq. (15) from Peimbert (1967) as well as eq. (8), eq. (9) and eq. (11)
from Peimbert et al. (2004) to calculate the impact of assuming
$t^{2}=0.038\pm 0.004$ in the ionic abundances. In this process, we also use
the excitation energies of the atomic levels associated with each
corresponding emission line, together with the physical conditions and the
ionic abundances determined by Arellano-Córdova et al. (2020, 2021) assuming
$t^{2}=0$. In these previous works, the electron density ($n_{\rm e}$) was
obtained using the [O ii] $\lambda$3729/$\lambda$3726, [S ii]
$\lambda$6717/$\lambda$6731 and/or [Cl iii] $\lambda$5518/$\lambda$5538 line
intensity ratios (although the [Cl iii] diagnostic was discarded if it led to
large uncertainties). For cases where the derived $n_{\rm e}$ was too low
($n_{\rm e}<100$ cm-3), a value of $100$ cm-3 was adopted. On the other hand,
the electronic temperatures ($T_{\rm e}$) were derived using a low- and high-
ionization zones scheme, using the [N ii]
($\lambda$6548+$\lambda$6584)/$\lambda$5755 ($T_{\rm e}$([N ii])) and the [O
iii] ($\lambda$4959+$\lambda$5007)/$\lambda$4363 ($T_{\rm e}$([O iii])) line
intensity ratios, respectively. The temperature relation proposed by Esteban
et al. (2009) was used to derive either $T_{\rm e}$([N ii]) or $T_{\rm e}$([O
iii]) when [O ii] $\lambda$4363 or [N ii] $\lambda$5755 was not detected. The
resulting ionic abundances derived for $t^{2}=0.038\pm 0.004$ are shown in
Tables A4 and A5. Table A4 also shows the references of the spectroscopic data
as well as the values of $n_{\rm e}$ and $T_{\rm e}$ computed by Arellano-
Córdova et al. (2020) and Arellano-Córdova et al. (2021).

Figure 3: $t^{2}$ values determined for the Galactic H ii regions M 8, M 16, M
17, M 20, M 42, NGC 2579, NGC 3576, NGC 3603 and Sh 2-311 from the literature
(see text for references) with respect to their adopted Galactocentric
distance. The horizontal line represents the weighted average of these values
and its uncertainty.

To obtain the total abundances of C, N, Ne, Cl, S and Ar, it is necessary to
use ionization correction factors (ICFs) that take into account the
contribution of the unseen ions to the total abundances of these elements. In
this work, we adopt the ICFs from Amayo et al. (2021). These ICFs have been
determined from a carefully selected large set of realistic photoionization
models for extragalactic H ii regions, covering a wide range of ionization
degrees. Moreover, they allow us to include the uncertainties associated to
each ICF as a function of the ionization degree, which leads to a more robust
determination of the chemical abundances. When comparing their selected sample
of $\sim$1800 models (from which the ICFs where derived) to our sample of
Galactic H ii regions, we find good consistency with the vast majority of our
objects, supporting the validity of using these ICFs in this work even though
they were proposed for a different kind of objects. In some objects, it is
possible to obtain the total abundances of N, S and Cl by considering only
their ionic abundances (without using an ICF). We find good agreement between
the derived total abundances of N, S and Cl and the predictions of the ICFs.
The differences are smaller than 0.08 dex, a value smaller than the mean
uncertainties obtained for the total abundances of these elements (of
$\sim$0.12 dex).

In order to propagate the uncertainties in the total abundances, we generate a
log-normal distribution of 400 Monte Carlo experiments with a $1\sigma$
defined by the lower and upper uncertainties of the ionic abundances
(following eq. 13 from Amayo et al. 2021) and choose a random ICF from a
second log-normal distribution, defined in the same way by the lower and upper
uncertainties of each ICF. We then compute the total abundances and report the
nominal values and the 16 and 84 percentiles as their lower and upper
uncertainties, respectively.

In the estimates of the total abundance of Ar, we only consider the regions
where both [Ar iii] and [Ar iv] emission lines are observed, as Amayo et al.
(2021) recommends. This criterion reduces the number of objects where the
total abundance of Ar is derived to 10 H ii regions.

In some objects, it is possible to derive the total abundances of N, S and Cl
without an ICF by simply adding the ionic abundances of the available ions
because either the ionization degree or the depth of the spectra allow to
observe emission lines of all the ions of each element that should be present
in the nebula. Here, we follow the same criteria used by Arellano-Córdova et
al. (2020) to identify these objects, although in some cases the number of H
ii regions changes with $t^{2}>0$ because it can produce a slight change in
the ionization degree. The resulting total abundances are shown in Table A6.

For Sh 2-83, Sh 2-100, Sh 2-128, Sh 2-209, M 17, M 42, NGC 2579, NGC 3576 and
NGC 3603 Méndez-Delgado et al. (2020) estimated a small value of the radiation
softness parameter ($\text{log(}\eta)<0.9$, see Vilchez & Pagel, 1988, for a
definition of $\eta$). In these cases, the amount of neutral helium within the
ionized gas is negligible (Pagel et al., 1992). Therefore, we only consider
these regions to our analysis of the total helium abundances, assuming
$\text{He}/\text{H}\approx\text{He}^{+}/\text{H}^{+}$.

###  4.2 The radial abundance gradients

Using the abundances obtained and the updated Galactocentric distances, we
compute the radial abundance gradients assuming $t^{2}=0.0$ and $t^{2}>0.0$.
We have strict control of the uncertainties in both quantities: distances and
total abundances (which include uncertainties from the ICFs). The fit
parameters (slope and intercept) of the revised radial gradients are given in
Table 2.

The radial gradients for He, C, N, O, Ne, S, Cl and, Ar are shown in Fig. 4.
For the abundances determined from CELs, the black dashed lines indicate the
gradients obtained assuming $t^{2}>0$ while the green ones indicate those with
$t^{2}=0.0$. Although the abundances of C2+ have been determined from RLs (and
therefore they are independent on $t^{2}$), the total abundances of C depend
on the ICF(C+), which varies slightly with $t^{2}$. Following Arellano-Córdova
et al. (2020, 2021), we discard several objects to compute the radial
abundance gradients of N and Ne, because of their odd abundances and high
uncertainties. In the case of Cl, we discarded the abundances obtained for Sh
2-219, Sh 2-237 and Sh 2-271, because of their very high and uncertain Cl/H
ratios. In fact, these objects are of very low ionization degree, implying
that their associated ICFs are very high and uncertain. The red circles and
the purple diamonds in the aforementioned figure represent the present-day
photospheric solar abundances given by Lodders (2019) and Asplund et al.
(2021), respectively.

The comparison between the chemical composition of the Sun, which was formed
4.57 Gyr ago (Asplund et al., 2021), and that of the H ii regions, which have
been formed much more recently, may not be straightforward. Nuclear
processing, radioactive decay and diffusion can change the derived chemical
composition of the Sun (Asplund et al., 2021). Also, most of the proto-solar
abundances are expected to be lower than abundances of young objects, due to
the chemical enrichment of the ISM from the Sun’s birth until the present-day.
According to Asplund et al. (2021), the difference between the proto-solar and
photospheric abundances of helium is 0.070 dex, and of heavier elements is
0.064 dex, although the latter may be even smaller, of 0.03 dex, depending on
the mixing scheme considered. On the other hand, Carigi et al. 2019 estimate
an O enrichment of $\sim$0.1 dex for the last 4.6 Gyr in our galaxy at
$R_{G}=8$ kpc, based on chemical evolution models. Since both the impact of
the solar diffusion as well as the chemical enrichment during the last $\sim$5
Gyr are similar or even smaller than the typical uncertainties in the
abundance determinations in ionized nebulae and stellar abundances, we will
use the present-day photospheric solar abundances in our comparisons, keeping
in mind its possible associated uncertainty.

When comparing with the solar abundances, it should be considered the possible
radial migration of the Sun since its formation. Recent dynamical and
chemodynamical models for the Milky Way disk have estimated a low radial
migration for the Sun (e.g., Klačka et al., 2012; Martínez-Barbosa et al.,
2015; Martinez-Medina et al., 2017). Klačka et al. found a radial-migration
average of $0.23\pm 0.68$ kpc, Martínez-Barbosa et al. claimed that the Sun
has unlikely migrated, and Martinez-Medina et al. estimated an average of
radial migration of $0.32\pm 1.40$ kpc. Hence, we consider the Sun is
representative of the solar neighborhood and consequently the comparison of
the chemical gradients (at $R_{G}\sim 8$ kpc) with the solar composition is
appropriate, if the age difference between H ii regions and the Sun is
neglected, as we mention above.

Figure 4: Radial distribution of He, C, N, O, Ne, S, Cl, and Ar abundances for the sample of Galactic H ii regions assuming $t^{2}>0$ as a function of their revised distances. The black dashed line represents the linear fit using chemical abundances based on CELs assuming $t^{2}>0$ or RLs, while the green dashed line represents the linear fit using CELs assuming $t^{2}=0$. The red circle and purple diamond indicate the solar values recommended by Lodders (2019) and Asplund et al. (2021), respectively. Blue circles correspond to chemical abundances derived from RLs. Table 2: Galactic radial abundance gradients determined from the spectra of H ii regions assuming $t^{2}=0.0$ and $t^{2}>0$. | $t^{2}=0.0$ | $t^{2}=0.038\pm 0.004$
---|---|---
12+log(X/H) | Slope (dex kpc-1) | intercept (dex) | Slope (dex kpc-1) | intercept (dex)
He | $-0.0072^{+0.0037}_{-0.0035}$ | $11.02^{+0.03}_{-0.04}$ | $-0.0072^{+0.0037}_{-0.0035}$ | $11.02^{+0.03}_{-0.04}$
C | $-0.077\pm 0.019$ | $9.05\pm 0.16$ | $-0.077\pm 0.017$ | $9.12\pm 0.15$
N | $-0.063\pm 0.009$ | $8.20\pm 0.09$ | $-0.078\pm 0.013$ | $8.60\pm 0.14$
O | $-0.044\pm 0.009$ | $8.86\pm 0.09$ | $-0.059\pm 0.012$ | $9.22\pm 0.12$
Ne | $-0.035\pm 0.018$ | $8.10\pm 0.16$ | $-0.048\pm 0.019$ | $8.48\pm 0.16$
S | $-0.035\pm 0.011$ | $7.20\pm 0.10$ | $-0.049\pm 0.012$ | $7.50\pm 0.12$
Cl | $-0.021\pm 0.010$ | $5.23\pm 0.09$ | $-0.027\pm 0.010$ | $5.44\pm 0.09$
Ar | $-0.029\pm 0.008$ | $6.50\pm 0.07$ | $-0.042\pm 0.011$ | $6.79\pm 0.09$

##  5 Discussion

The use of distances based on Gaia DR2 parallaxes by Méndez-Delgado et al.
(2020); Arellano-Córdova et al. (2020, 2021) and our revision based on Gaia
EDR3 have been an important step forward to improve the quality of the
abundance gradients determinations. Comparing the results obtained assuming
kinematic or parallax-based distances have raised interesting issues. In the
case of Esteban & García-Rojas (2018), the underestimation of the
Galactocentric distances of Sh 2-61, Sh 2-90, M 8, M 16, M 17, M 20 and NGC
3576, produced an apparent flattening at $R_{G}<8\text{ kpc}$ in the abundance
gradients of several elements. If true, as Esteban & García-Rojas (2018)
discuss, this would imply a somewhat different chemical evolution of the
central parts of the Galaxy, invoking the possible action of the Galactic bar.
However, this seems to be a spurious result produced by the underestimation of
small $R_{\rm G}$ values due to the use of rotation model-dependent distance
estimates. It is important to note that the latest calibrations made to the
Galactic rotation curves by Reid et al. (2014, 2019), produce consistent
results with the parallax-based distances from Gaia EDR3 for our sample of H
ii regions.

###  5.1 The reliability of the derived gradient of helium

Méndez-Delgado et al. (2020) found a negative slope in the radial abundance
gradient of He determined from H ii regions of the Milky Way. Although there
was marginal evidence of a negative helium gradient in some previous works
(see Hawley, 1978; Peimbert et al., 1978; Talent & Dufour, 1979; Shaver et
al., 1983a; Deharveng et al., 2000; Fernández-Martín et al., 2017), the
uncertainties of those determinations made them compatible with a flat
distribution. There are several factors that make it difficult to determine
the abundance gradient for this element: (i) unlike the heavier elements, the
abundance of primordial He is relatively high, so its relative increase due to
stellar nucleosynthesis is rather small, (ii) the known metastability of the
23S level of the He atom in its triplet configuration makes it very sensitive
to self-absorptions and collisions, which generates deviations in the
populations of the triplet levels expected by pure recombination, and (iii)
estimating the He0/H+ abundance is a difficult issue, and its contribution is
expected to be important for the range of ionization degree of most Galactic H
ii regions. Considering these factors, we restrict ourselves to 9 highly-
ionized H ii regions where
$\text{He}^{+}/\text{H}^{+}\approx\text{He}/\text{H}$ due to the hardness of
their ionizing sources, covering a range of Galactocentric distances between
6.45 and 17 kpc.

Figure 5: Estimation of the fraction of primordial baryonic mass in helium
$Y_{p}$ using the abundances of He and O in the sample of regions used in the
determination of the helium gradient in the Milky Way. Top panel: Considering
$t^{2}=0$. Bottom panel: Considering $t^{2}=0.038\pm 0.004$.

If the thus determined radial gradient of He is accurate, the relations
between $Y$ (He abundance by mass) and O/H must be used to extrapolate the
primordial helium value. The most recent determinations of the primordial
helium abundance are consistent with a fraction of primordial baryonic mass in
helium $Y_{p}$ ranging between 0.243 to 0.247 (Aver et al., 2015; Peimbert et
al., 2016; Fernández et al., 2019; Hsyu et al., 2020; Planck Collaboration et
al., 2020; Kurichin et al., 2021; Valerdi et al., 2021). In Fig. 5 we show the
determinations of $Y_{p}$ considering the sample of regions adopted for the
determination of the Galactic radial abundance gradient of He. For the
normalization, we have considered that oxygen per unit of mass represents
$55\pm 10$ per cent of the metallicity $Z$ (Peimbert et al., 2007). The value
of $Y_{p}$ considering $t^{2}=0$ is $0.229\pm 0.014$ while the one obtained
considering $t^{2}>0$ is $0.236\pm 0.012$. Since both values are formally
consistent with the most precise determinations of $Y_{p}$ within the
uncertainty bars, it gives further support to the reliability of our
determination of the Galactic radial abundance gradient of He.

###  5.2 The radial gradients of heavy elements under the $t^{2}$ paradigm

The resulting gradients of C, N, O, Ne, and S considering $t^{2}=0.0$ are in
excellent agreement with those reported by Arellano-Córdova et al. (2020,
2021) and therefore, the results and the discussion regarding these elements
remain unchanged. However, we find some differences in the gradients of Cl and
Ar. In this work we only consider those regions with estimates of both Ar+2
and Ar+3 since the ICFs constrained only with the Ar+2 abundance are rather
uncertain (Amayo et al., 2021). Due to this methodological difference, our Ar
gradient with $t^{2}=0.0$ is slightly flatter in comparison to what is found
by Arellano-Córdova et al. (2020), although both are consistent within the
uncertainties. In the case of Cl, Arellano-Córdova et al. (2020) used the ICF
scheme derived by Esteban et al. (2015), obtaining a steeper gradient than our
results, although they are also consistent within the uncertainties.

As mentioned in Sec 4, it is usual to estimate the value of $t^{2}$ from the
measured ADF(O2+). However, there are fewer studies that also calculate the
ADF for other ions such as C2+, O+, N2+ or Ne2+ –getting that the ADF can be
different in each ion– and essentially none considering other different ions.
Therefore, the consistency between the solar abundances of O and the
abundances of nearby H ii regions with $t^{2}>0$ is not surprising, since in
these regions the RLs provide abundances closer to the solar ones. However,
this does not necessarily imply that this consistency must be replicated in
the abundances of other elements such as N, S, Cl or Ar, since by
construction, the determination of $t^{2}$ does not consider the ADF of any
ion of these elements. Therefore, the discussion about the gradients of
elements different than O and their comparison with the solar abundances is
pertinent and it is not a circular argument.

The definition of $t^{2}$ is not incompatible with the temperature
stratification of the nebulae – which explains that generally $T_{\rm e}$([N
ii]) is higher than $T_{\rm e}$([O iii]) –; it just quantifies the level of
temperature inhomogeneities within the emission volume (out of the expected
stratification). Assuming a global value of $t^{2}>0$ in each H ii region
affects the determination of the abundances of low and high ionization ions in
different proportions since, generally, the temperatures associated with each
volume are different in addition to the different dependence of $t^{2}$ with
the excitation energy of the emission lines (see eq.(11) in Peimbert et al.,
2004). This means that the parameters of the linear fits to the gradients –
slope and intercept – for $t^{2}>0$ are susceptible to change with respect to
the case of assuming $t^{2}=0$.

Fig. 4 shows the Galactic radial abundance gradients derived in this work by
considering $t^{2}=0.038\pm 0.004$ and $t^{2}=0$. In the second panel of the
right column, we show the O gradient. The blue circles represent the total O
abundances derived from RLs for M 8, M 16, M 17, M 20, M 42, NGC 3576, NGC
3603, Sh 2-311 and NGC 2579 (Esteban et al., 2005; Esteban et al., 2013). The
slope with $t^{2}>0$ is slightly steeper with respect to the $t^{2}=0$ case.
The resulting gradient is consistent with the recent O gradient of Luck
(2018), based on an updated sample of $\sim$400 classical Cepheids in the
Galactic disc. If the paradigm of temperature fluctuations is valid, then the
agreement between the O gradient with $t^{2}>0$ and the solar abundances
implies that the depletion of O into dust grains in the solar neighborhood
regions is smaller than the uncertainty bars, in consistency with the results
of Mesa-Delgado et al. (2009) and Peimbert & Peimbert (2010) who estimated a
depletion of O into dust of up to 0.11 dex.

The radial distribution of the Ar/O abundance ratio, shown in Fig. 6, is
almost flat, both in the case with $t^{2}=0$ and $t^{2}>0$ (with slopes of
$-0.011\pm 0.010$ and $-0.014\pm 0.010$ dex kpc-1, respectively). Since both
elements are produced by $\alpha$-particle captures in massive stars, their
abundance ratio should be constant, although there may be a small fraction of
Ar produced in Type-Ia supernovae, according to the Galactic chemical
evolution models of Kobayashi et al. 2020, which would add an extra scattering
factor in the Ar/O distribution, as discussed in Amayo et al. (2021). The
dispersion of the order of 0.1 dex in the distribution of Ar/O can serve as an
upper limit to the possible depletion of O into dust grains, since there are
several factors that contribute to the observed dispersion, such as
uncertainties in the ionic abundances of both O and Ar as well as of the ICF
of Ar and the possible production of Ar in supernovae.

Figure 6: Ar/O radial gradient with the same layout than Fig. 4

The solar N/H given by Lodders (2019) and Asplund et al. (2021) is between the
values predicted by the radial gradient of N considering $t^{2}=0$ and
$t^{2}>0.0$. The slope obtained with $t^{2}>0$, shown in Table 2, is slightly
steeper when compared to some values in the literature (e.g., Shaver et al.,
1983b; Afflerbach et al., 1996; Esteban & García-Rojas, 2018; Arellano-Córdova
et al., 2021). It has better consistency with the slope value of $-0.085\pm
0.020$ dex kpc-1 determined by Carigi et al. (2005). Additionally, we
determine the gradient of log(N/O) when $t^{2}>0$, since this ratio has
important implications for the N production mechanisms. Taking into account
temperature inhomogeneities, we obtain a radial gradient of $\log$(N/O) with a
slope of $-0.018\pm 0.015$ dex kpc-1. This means that $t^{2}>0$ do not alter
the N/O gradient, which is consistent with that found by Arellano-Córdova et
al. (2020, 2021) considering $t^{2}=0$. When comparing N/O to the O/H, we do
not observe a clear plateau as metallicity decreases, and all objects seems to
follow a slightly positive trend instead. The distribution of N/O versus O/H,
shown in Fig. 7 has a positive slope of $0.12\pm 0.20$ when $t^{2}>0$, a value
that is maintained even when the most metallic and uncertain points are
removed. This supports the contribution of some secondary N in the Galactic
disc and is expected due to the metallicity range of our sample, since
previous works in our Galaxy and other star forming galaxies suggest the
primary N production mechanism dominates only at $12+\log$(O/H) $\lesssim 8.0$
(van Zee et al., 1998; Henry et al., 2000; Vincenzo et al., 2016; Esteban et
al., 2020). However, a flat distribution cannot be ruled out considering the
uncertainty of the slope of our N/O versus O/H distribution.

Figure 7: N/O as a function of $12+\log$(O/H) with the same layout than Fig.
4.

The radial gradients of Ne, S and Ar under the paradigm of $t^{2}>0$ are
consistent with the solar values given by Lodders (2019) and Asplund et al.
(2021), while those obtained with $t^{2}=0$ fail to reproduce both sets of
solar abundances. The slopes obtained for these gradients, taking into account
temperature inhomogeneities, are steeper than those derived by Arellano-
Córdova et al. (2020) with $t^{2}=0$. These gradients are also consistent with
the O one within the uncertainties, as expected because they are
$\alpha$-elements and are produced by the same population of massive stars
(Kunth & Sargent, 1983). In the case of $t^{2}>0$, we find that the S
abundances of Sh2 257, Sh2 271 and IC 5146 can be calculated without an ICF
due to their low degree of ionization (following the criterion of Arellano-
Córdova et al. (2020) of O++/(O++O++) $\leq 0.03$).

In the case of Cl, we do not use an ICF for M 20, M 17, M 42, NGC 2579, NGC
3576, and NGC 3603 when $t^{2}>0$, either because they have low degree of
ionization (O++/(O++O++) $\leq 0.26$ (Arellano-Córdova et al., 2020) or
because they show [Cl ii], [Cl iii] and [Cl iv] emission lines in their
spectra, allowing to compute the Cl abundance directly by summing up ionic
abundances. In Fig. 4 it can be seen that the Cl gradient with $t^{2}>0$
obtained with the rest of objects is consistent with the solar values from
Lodders (2019) and Asplund et al. (2021), while the homogeneous temperature
scenario fails to reproduce these values (a behaviour also found by Arellano-
Córdova et al., 2020). Nevertheless, it also must be considered that the solar
values of Cl/H are somewhat uncertain, since they can only be obtained from
sunspot spectra because the solar spectrum does not include atomic features of
Cl. This implies that its determination relies additionally on the molecular
data used and on the atmosphere model to be suitable for a sunspot with
unknown temperature, which makes their uncertainty probably larger than quoted
(Asplund et al., 2021).

As discussed by Arellano-Córdova et al. (2020), we also expect a lockstep
evolution of Cl and the rest of $\alpha-$elements (O among them), because they
are produced in the same nucleosynthesis processes (being Cl an intermediate
product of these). We find that the Cl gradient slope is consistent with those
of all the $\alpha-$elements, except for O, both when $t^{2}=0$ and when
$t^{2}>0$. Both in the case $t^{2}=0$ and $t^{2}>0$, the slopes of the
gradients of O and Cl differ by 50%. This difference do not rely on the ICF
used as it remains even when the ICF from Esteban et al. (2015) is used.
Instead, this inconsistency seems to be originated in a statistical difference
between the sample of regions used to derive the O and Cl gradients. The
number of objects used to obtain the gradient of O is larger than that of Cl,
sampling a wider range of Galactocentric distances. If we constrain our sample
to those regions in the range of Galactocentric distances between $6.0\text{
kpc}<R_{G}<15.0\text{ kpc}$ – which covers the well-sampled area in the Cl
gradient – we obtain slopes of the O gradient of $-0.038\pm 0.010$ dex kpc-1
and $-0.051\pm 0.010$ dex kpc-1 for $t^{2}=0$ and $t^{2}>0$, respectively.
These values are in better agreement with the slopes of the Cl gradients for
each case, as expected under the mentioned scenario.

It is interesting to compare our gradients with $t^{2}>0$ with those gradients
derived from CELs in the FIR. These lines, due to their origin, have a much
lower dependence on temperature than optical CELs. Therefore, if chemical
abundances based on optical CELs are underestimated in the case of $t^{2}=0$,
the abundances derived with FIR should be less affected, being more similar to
our predictions with $t^{2}>0$. Rudolph et al. (2006) determined the gradients
of N, O and S using FIR CELs. For the O gradient, our slope is steeper and our
intercept is higher in both cases: $t^{2}=0$ and $t^{2}>0$. This may be due to
the fact that the determination of the total O abundance in the work of
Rudolph et al. (2006) is not direct, requiring an ICF to correct for the lack
of [O II] lines in their FIR spectra. Errors in the ICF are especially
critical for the innermost regions of the Galaxy where O+ is usually the
dominant ion. In the case of N, although our determinations and those of
Rudolph et al. (2006) require the use of ICFs to estimate the contribution of
N2+ and N+, respectively, both kinds of results are in agreement when the
abundance derived with $t^{2}>0$ are considered in the optical data. In the
case of S, both ions S+ and S3+ may have relevant contributions to the total
abundance. The slope of the radial gradient of S from Rudolph et al. (2006) is
consistent with ours in the case of $t^{2}>0$, this is not the case when
comparing the intercept.

In the case of the N/O ratio, the slopes of both gradients are consistent
within the uncertainties. Moreover the N/O vs O/H distribution have a similar
qualitative behavior: the observed trend in the optical with $t^{2}=0$ is
practically flat (In our case log(N/O) = -0.77 $\pm$ 0.08, Arellano-Córdova et
al., 2021), while that observed in both FIR and optical CELs with $t^{2}>0$
has a positive slope. Both the FIR and the optical data with $t^{2}>0$ may
support the contribution of some secondary N in the Galactic disc. However,
the uncertainties are large enough to prevent us from being conclusive. The
comparison between the optical data results assuming $t^{2}=0$ and the
gradients obtained for H ii regions by other groups or other objects has been
extensively discussed in Arellano-Córdova et al. (2020, 2021).

In our sample of Galactic H ii regions, we found no evidence of azimuthal
variations in the chemical abundances and, essentially, their distribution
depends only on the Galactocentric distance. However, our sample is
concentrated in two Galactic quadrants, as shown in Fig. 1, whose
characteristics may be different from the H ii regions located in the rest of
the Galaxy (Wenger et al., 2019).

Recently, De Cia et al. (2021) have claimed that there are variations of up to
1 dex in the metallicity of the neutral gas of the solar neighborhood. We
definitively do not observe such variations in H ii regions. The O abundances
– the proxy of metallicity in ionized nebulae– present a rather uniform linear
distribution with the Galactocentric radius, independently of considering
$t^{2}=0$ or $t^{2}>0$. In fact, the radial gradient of O is already
perceptible even if we limit ourselves to heliocentric distances less than 3
kpc – as the data of De Cia et al. (2021) – given the quality of our spectra
and the detail of our analysis. As mentioned before and contrary to what
happens with Fe and Ni atoms, the impact of O depletion onto dust grains is
limited to a factor of the order of 0.1 dex at maximum, an order of magnitude
below the variations acclaimed by De Cia et al. (2021). This implies that any
hypothetical pristine gas inclusions in the solar neighborhood must be well
mixed in present-day Galactic H ii regions.

##  6 Conclusions

In this work, we determine the radial abundances gradients of He, C, N, O, Ne,
S, Cl and Ar in the Milky Way assuming the temperature fluctuations paradigm,
$t^{2}>0$, and the most recent parallax-based distances using Gaia EDR3 data.
The analysis of the physical conditions and chemical abundances of this work
is complementary to the previous ones by Méndez-Delgado et al. (2020) and
Arellano-Córdova et al. (2020, 2021), where a homogeneous temperature
structure, $t^{2}=0$, is assumed.

We calculate the parallax-based distances to the Galactic H ii regions through
the identification of their ionizing sources and using the Bailer-Jones et al.
(2021) statistical approach. The comparison of the Gaia EDR3 parallax-based
distances with kinematic ones shows that the Galactic rotation curve of Reid
et al. (2014, 2019), calibrated with maser parallaxes obtained at radio
wavelengths, provide more consistent results than the classic rotation curve
of Brand & Blitz (1993) for the regions of our sample.

To apply the Peimbert’s $t^{2}$ formalism (Peimbert, 1967), we use the
weighted mean of the $t^{2}$ values obtained for M 8, M 16, M 17, M 20, NGC
3576, NGC 3603, Sh 2-311 and NGC 2579 assuming that the ADF(O++) is produced
by temperature fluctuations (García-Rojas & Esteban, 2007; Esteban et al.,
2013). That sample of H ii regions covers $R_{G}$ values between 6.5 and 11.6
kpc. We find that $t^{2}$ shows a rather constant distribution with $R_{G}$,
which suggests that a $t^{2}=0.038\pm 0.004$ may be representative for our
sample. We apply the Peimbert (1967) formalism using the representative
$t^{2}$ value and the ionic abundances determined by Arellano-Córdova et al.
(2020, 2021), obtained under the assumption of $t^{2}=0$. To estimate the
total abundances of C, N, Ne, S, Cl, and Ar, we use the ICFs from Amayo et al.
(2021), including the uncertainties of the ICFs in our calculations.

We find that total abundances determined from CELs of all elements at a given
$R_{G}$ increase by up to 0.3 dex when $t^{2}>0$ is considered, becoming more
consistent with the solar values recommended by Lodders (2019) and Asplund et
al. (2021). The radial distributions of the $\alpha$-elements are consistent
with a common slope –which is expected due to their common nucleosynthesis–
when using $t^{2}>0$. The $t^{2}$ formalism does not seem to introduce
inconsistencies in the distributions of the chemical abundances studied.
Therefore it can be invoked as a likely explanation of the ADF problem in
Galactic H ii regions. However, the demonstration of the physical processes
that may generate these temperature fluctuations is a matter of debate and
there is still no definitive solution to this problem.

## Acknowledgements

We are very grateful to Dr. Laura Magrini for her useful comments to the first
version of the paper. JEM-D is very grateful to Žofia Chrobáková for her
suggestions and discussions about this work. JEM-D appreciates the friendly
communication of Trey Wenger and the exchange of views regarding kinematic
distances. JEM-D thanks the support of the Instituto de Astrofísica de
Canarias under the Astrophysicist Resident Program and acknowledges support
from the Mexican CONACyT (grant CVU 602402). JEM-D, CE and JG-R acknowledge
support from (i) the Agencia Estatal de Investigación del Ministerio de
Ciencia e Innovación (AEI-MCINN) under grant Espectroscopía de campo integral
de regiones H 2 locales. Modelos para el estudio de regiones H 2
extragalácticas with reference 10.13039/501100011033 and (ii) the grant
P/308614 financed by funds transferred from the Spanish Ministry of Science,
Innovation and Universities (MCIU), charged to the General State Budgets and
from the General Budgets of the Autonomous Community of the Canary Islands by
the MCIU. AM-A thanks Grazyna Stasińska for her suggestions and several
conversations and CONACyT for her PhD. scholarship (CVU No. 825508). AM-A,
GD-I, and LC thank support from PAPIIT (DGAPA-UNAM) grant no. IN$-$103820\. LC
acknowledges support from the grants IA-100420, IG-100622, and IN-100519
(DGAPA-PAPIIT,UNAM) and funding from the CONACYT grant CF19-39578. JG-R
acknowledges support from the Severo Ochoa excellence program
CEX2019-000920-S. This work has made use of data from the European Space
Agency (ESA) mission Gaia (https://www.cosmos.esa.int/Gaia), processed by the
Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/Gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

## Data availability

This research is based on public data available in cited the references. Our
results are entirely present in the tables of this work.

## References

  * Afflerbach et al. (1996) Afflerbach A., Churchwell E., Acord J. M., Hofner P., Kurtz S., Depree C. G., 1996, ApJS, 106, 423
  * Amayo et al. (2021) Amayo A., Delgado-Inglada G., Stasińska G., 2021, MNRAS, 505, 2361
  * Anderson et al. (2015) Anderson L. D., Armentrout W. P., Johnstone B. M., Bania T. M., Balser D. S., Wenger T. V., Cunningham V., 2015, ApJS, 221, 26
  * Arellano-Córdova et al. (2020) Arellano-Córdova K. Z., Esteban C., García-Rojas J., Méndez-Delgado J. E., 2020, MNRAS, 496, 1051
  * Arellano-Córdova et al. (2021) Arellano-Córdova K. Z., Esteban C., García-Rojas J., Méndez-Delgado J. E., 2021, MNRAS, 502, 225
  * Asplund et al. (2021) Asplund M., Amarsi A. M., Grevesse N., 2021, arXiv e-prints, p. arXiv:2105.01661
  * Avedisova & Kondratenko (1984) Avedisova V. S., Kondratenko G. I., 1984, Nauchnye Informatsii, 56, 59
  * Aver et al. (2015) Aver E., Olive K. A., Skillman E. D., 2015, J. Cosmology Astropart. Phys., 2015, 011
  * Bailer-Jones et al. (2018) Bailer-Jones C. A. L., Rybizki J., Fouesneau M., Mantelet G., Andrae R., 2018, AJ, 156, 58
  * Bailer-Jones et al. (2021) Bailer-Jones C. A. L., Rybizki J., Fouesneau M., Demleitner M., Andrae R., 2021, AJ, 161, 147
  * Balser et al. (2011) Balser D. S., Rood R. T., Bania T. M., Anderson L. D., 2011, ApJ, 738, 27
  * Bartkiewicz et al. (2008) Bartkiewicz A., Brunthaler A., Szymczak M., van Langevelde H. J., Reid M. J., 2008, A&A, 490, 787
  * Binder & Povich (2018) Binder B. A., Povich M. S., 2018, ApJ, 864, 136
  * Bland-Hawthorn & Gerhard (2016) Bland-Hawthorn J., Gerhard O., 2016, ARA&A, 54, 529
  * Bohigas & Tapia (2003) Bohigas J., Tapia M., 2003, AJ, 126, 1861
  * Brand & Blitz (1993) Brand J., Blitz L., 1993, A&A, 275, 67
  * Brand & Wouterloot (1988) Brand J., Wouterloot J. G. A., 1988, A&AS, 75, 117
  * Brunthaler et al. (2009) Brunthaler A., Reid M. J., Menten K. M., Zheng X. W., Moscadelli L., Xu Y., 2009, ApJ, 693, 424
  * Camargo et al. (2011) Camargo D., Bonatto C., Bica E., 2011, MNRAS, 416, 1522
  * Caplan et al. (2000) Caplan J., Deharveng L., Peña M., Costero R., Blondel C., 2000, MNRAS, 311, 317
  * Carigi et al. (2005) Carigi L., Peimbert M., Esteban C., García-Rojas J., 2005, ApJ, 623, 213
  * Carigi et al. (2019) Carigi L., Peimbert M., Peimbert A., 2019, ApJ, 873, 107
  * Chini & Wink (1984) Chini R., Wink J. E., 1984, A&A, 139, L5
  * Chu (1982) Chu Y. H., 1982, ApJ, 254, 578
  * Chu et al. (1982) Chu Y. H., Gull T. R., Treffers R. R., Kwitter K. B., Troland T. H., 1982, ApJ, 254, 562
  * Churchwell & Walmsley (1973) Churchwell E., Walmsley C. M., 1973, A&A, 23, 117
  * Copetti et al. (2007) Copetti M. V. F., Oliveira V. A., Riffel R., Castañeda H. O., Sanmartim D., 2007, A&A, 472, 847
  * Damiani et al. (2017) Damiani F., et al., 2017, A&A, 604, A135
  * De Becker et al. (2005) De Becker M., Rauw G., Blomme R., Pittard J. M., Stevens I. R., Runacres M. C., 2005, A&A, 437, 1029
  * De Cia et al. (2021) De Cia A., Jenkins E. B., Fox A. J., Ledoux C., Ramburth-Hurt T., Konstantopoulou C., Petitjean P., Krogager J.-K., 2021, Nature, 597, 206
  * Deharveng et al. (2000) Deharveng L., Peña M., Caplan J., Costero R., 2000, MNRAS, 311, 329
  * Deharveng et al. (2008) Deharveng L., Lefloch B., Kurtz S., Nadeau D., Pomarès M., Caplan J., Zavagno A., 2008, A&A, 482, 585
  * Drew et al. (2019) Drew J. E., Monguió M., Wright N. J., 2019, MNRAS, 486, 1034
  * Esteban & Fernandez (1998) Esteban C., Fernandez M., 1998, MNRAS, 298, 185
  * Esteban & García-Rojas (2018) Esteban C., García-Rojas J., 2018, MNRAS, 478, 2315
  * Esteban et al. (1989) Esteban C., Vilchez J. M., Manchado A., Edmunds M. G., 1989, Ap&SS, 157, 3
  * Esteban et al. (2004) Esteban C., Peimbert M., García-Rojas J., Ruiz M. T., Peimbert A., Rodríguez M., 2004, MNRAS, 355, 229
  * Esteban et al. (2005) Esteban C., García-Rojas J., Peimbert M., Peimbert A., Ruiz M. T., Rodríguez M., Carigi L., 2005, ApJ, 618, L95
  * Esteban et al. (2009) Esteban C., Bresolin F., Peimbert M., García-Rojas J., Peimbert A., Mesa-Delgado A., 2009, ApJ, 700, 654
  * Esteban et al. (2013) Esteban C., Carigi L., Copetti M. V. F., García-Rojas J., Mesa-Delgado A., Castañeda H. O., Péquignot D., 2013, MNRAS, 433, 382
  * Esteban et al. (2015) Esteban C., García-Rojas J., Pérez-Mesa V., 2015, MNRAS, 452, 1553
  * Esteban et al. (2017) Esteban C., Fang X., García-Rojas J., Toribio San Cipriano L., 2017, MNRAS, 471, 987
  * Esteban et al. (2020) Esteban C., Bresolin F., García-Rojas J., Toribio San Cipriano L., 2020, MNRAS, 491, 2137
  * Fabricius et al. (2021) Fabricius C., et al., 2021, A&A, 649, A5
  * Fernández-Martín et al. (2017) Fernández-Martín A., Pérez-Montero E., Vílchez J. M., Mampaso A., 2017, A&A, 597, A84
  * Fernández et al. (2019) Fernández V., Terlevich E., Díaz A. I., Terlevich R., 2019, MNRAS, 487, 3221
  * Forbes (1989) Forbes D., 1989, A&AS, 77, 439
  * Foster & Brunt (2015) Foster T., Brunt C. M., 2015, AJ, 150, 147
  * Gaia Collaboration et al. (2018) Gaia Collaboration et al., 2018, A&A, 616, A1
  * Gaia Collaboration et al. (2021) Gaia Collaboration et al., 2021, A&A, 649, A1
  * García-Rojas & Esteban (2007) García-Rojas J., Esteban C., 2007, ApJ, 670, 457
  * García-Rojas et al. (2004) García-Rojas J., Esteban C., Peimbert M., Rodríguez M., Ruiz M. T., Peimbert A., 2004, ApJS, 153, 501
  * García-Rojas et al. (2005) García-Rojas J., Esteban C., Peimbert A., Peimbert M., Rodríguez M., Ruiz M. T., 2005, MNRAS, 362, 301
  * García-Rojas et al. (2006) García-Rojas J., Esteban C., Peimbert M., Costado M. T., Rodríguez M., Peimbert A., Ruiz M. T., 2006, MNRAS, 368, 253
  * García-Rojas et al. (2007) García-Rojas J., Esteban C., Peimbert A., Rodríguez M., Peimbert M., Ruiz M. T., 2007, RMxAA, 43, 3
  * García-Rojas et al. (2014) García-Rojas J., Simón-Díaz S., Esteban C., 2014, A&A, 571, A93
  * Georgelin et al. (1973) Georgelin Y. M., Georgelin Y. P., Roux S., 1973, A&A, 25, 337
  * Gravity Collaboration et al. (2019) Gravity Collaboration et al., 2019, A&A, 625, L10
  * Hamann et al. (2006) Hamann W. R., Gräfener G., Liermann A., 2006, A&A, 457, 1015
  * Harten et al. (1978) Harten R. H., Felli M., Tofani G., 1978, A&A, 70, 205
  * Hawley (1978) Hawley S. A., 1978, ApJ, 224, 417
  * Henry et al. (2000) Henry R. B. C., Edmunds M. G., Köppen J., 2000, ApJ, 541, 660
  * Hernández et al. (2004) Hernández J., Calvet N., Briceño C., Hartmann L., Berlind P., 2004, AJ, 127, 1682
  * Hirota et al. (2007) Hirota T., et al., 2007, PASJ, 59, 897
  * Hsyu et al. (2020) Hsyu T., Cooke R. J., Prochaska J. X., Bolte M., 2020, ApJ, 896, 77
  * Hunter & Massey (1990) Hunter D. A., Massey P., 1990, AJ, 99, 846
  * Ji et al. (2012) Ji W. G., Zhou J. J., Esimbek J., Wu Y. F., Wu G., Tang X. D., 2012, A&A, 544, A39
  * Kerr & Lynden-Bell (1986) Kerr F. J., Lynden-Bell D., 1986, MNRAS, 221, 1023
  * Kirsanova et al. (2008) Kirsanova M. S., Sobolev A. M., Thomasson M., Wiebe D. S., Johansson L. E. B., Seleznev A. F., 2008, MNRAS, 388, 729
  * Klačka et al. (2012) Klačka J., Nagy R., Jurči M., 2012, MNRAS, 427, 358
  * Kobayashi et al. (2020) Kobayashi C., Karakas A. I., Lugaro M., 2020, ApJ, 900, 179
  * Kraus et al. (2007) Kraus S., et al., 2007, A&A, 466, 649
  * Kunth & Sargent (1983) Kunth D., Sargent W. L. W., 1983, ApJ, 273, 81
  * Kurichin et al. (2021) Kurichin O. A., Kislitsyn P. A., Klimenko V. V., Balashev S. A., Ivanchik A. V., 2021, MNRAS, 502, 3045
  * Lindegren et al. (2018) Lindegren L., et al., 2018, A&A, 616, A2
  * Lindegren et al. (2021a) Lindegren L., et al., 2021a, A&A, 649, A2
  * Lindegren et al. (2021b) Lindegren L., et al., 2021b, A&A, 649, A4
  * Lodders (2019) Lodders K., 2019, arXiv e-prints, p. arXiv:1912.00844
  * Luck (2018) Luck R. E., 2018, AJ, 156, 171
  * Lynds & Oneil (1983) Lynds B. T., Oneil E. J. J., 1983, ApJ, 265, 803
  * Lynds & Oneil (1986) Lynds B. T., Oneil E. J. J., 1986, ApJ, 306, 532
  * Maíz Apellániz et al. (2016) Maíz Apellániz J., et al., 2016, ApJS, 224, 4
  * Mallick et al. (2012) Mallick K. K., Ojha D. K., Samal M. R., Pand ey A. K., Bhatt B. C., Ghosh S. K., Dewangan L. K., Tamura M., 2012, ApJ, 759, 48
  * Martínez-Barbosa et al. (2015) Martínez-Barbosa C. A., Brown A. G. A., Portegies Zwart S., 2015, MNRAS, 446, 823
  * Martinez-Medina et al. (2017) Martinez-Medina L. A., Pichardo B., Peimbert A., Carigi L., 2017, MNRAS, 468, 3615
  * Mehner et al. (2016) Mehner A., et al., 2016, A&A, 585, A81
  * Méndez-Delgado et al. (2020) Méndez-Delgado J. E., Esteban C., García-Rojas J., Arellano-Córdova K. Z., Valerdi M., 2020, MNRAS, 496, 2726
  * Menten et al. (2007) Menten K. M., Reid M. J., Forbrich J., Brunthaler A., 2007, A&A, 474, 515
  * Mesa-Delgado et al. (2009) Mesa-Delgado A., Esteban C., García-Rojas J., Luridiana V., Bautista M., Rodríguez M., López-Martín L., Peimbert M., 2009, MNRAS, 395, 855
  * Moffat et al. (1979) Moffat A. F. J., Fitzgerald M. P., Jackson P. D., 1979, A&AS, 38, 197
  * Moisés et al. (2011) Moisés A. P., Damineli A., Figuerêdo E., Blum R. D., Conti P. S., Barbosa C. L., 2011, MNRAS, 411, 705
  * Molina Lera et al. (2018) Molina Lera J. A., Baume G., Gamen R., 2018, MNRAS, 480, 2386
  * Nazé (2009) Nazé Y., 2009, A&A, 506, 1055
  * Neckel & Staude (1984) Neckel T., Staude H. J., 1984, A&A, 131, 200
  * Ojha et al. (2011) Ojha D. K., et al., 2011, ApJ, 738, 156
  * Pagel et al. (1992) Pagel B. E. J., Simonson E. A., Terlevich R. J., Edmunds M. G., 1992, MNRAS, 255, 325
  * Pandey et al. (2013) Pandey A. K., et al., 2013, ApJ, 764, 172
  * Paron et al. (2013) Paron S., Weidmann W., Ortega M. E., Albacete Colombo J. F., Pichel A., 2013, MNRAS, 433, 1619
  * Peimbert (1967) Peimbert M., 1967, ApJ, 150, 825
  * Peimbert & Peimbert (2010) Peimbert A., Peimbert M., 2010, ApJ, 724, 791
  * Peimbert et al. (1978) Peimbert M., Torres-Peimbert S., Rayo J. F., 1978, ApJ, 220, 516
  * Peimbert et al. (2004) Peimbert M., Peimbert A., Ruiz M. T., Esteban C., 2004, ApJS, 150, 431
  * Peimbert et al. (2007) Peimbert M., Luridiana V., Peimbert A., 2007, ApJ, 666, 636
  * Peimbert et al. (2016) Peimbert A., Peimbert M., Luridiana V., 2016, Rev. Mex. Astron. Astrofis., 52, 419
  * Persi et al. (1987) Persi P., Ferrari-Toniolo M., Shivanandan K., Spinoglio L., 1987, A&AS, 70, 437
  * Planck Collaboration et al. (2020) Planck Collaboration et al., 2020, A&A, 641, A6
  * Povich et al. (2017) Povich M. S., Busk H. A., Feigelson E. D., Townsley L. K., Kuhn M. A., 2017, ApJ, 838, 61
  * Quireza et al. (2006) Quireza C., Rood R. T., Bania T. M., Balser D. S., Maciel W. J., 2006, ApJ, 653, 1226
  * Reed (2003) Reed B. C., 2003, AJ, 125, 2531
  * Reid et al. (2009) Reid M. J., Menten K. M., Zheng X. W., Brunthaler A., Xu Y., 2009, ApJ, 705, 1548
  * Reid et al. (2014) Reid M. J., et al., 2014, ApJ, 783, 130
  * Reid et al. (2019) Reid M. J., et al., 2019, ApJ, 885, 131
  * Rho et al. (2004) Rho J., Ramírez S. V., Corcoran M. F., Hamaguchi K., Lefloch B., 2004, ApJ, 607, 904
  * Rho et al. (2006) Rho J., Reach W. T., Lefloch B., Fazio G. G., 2006, ApJ, 643, 965
  * Rolleston et al. (1994) Rolleston W. R. J., Dufton P. L., Fitzsimmons A., 1994, A&A, 284, 72
  * Roman-Lopes & Roman-Lopes (2019) Roman-Lopes A., Roman-Lopes G. F., 2019, MNRAS, 484, 5578
  * Rudolph et al. (1996) Rudolph A. L., Brand J., de Geus E. J., Wouterloot J. G. A., 1996, ApJ, 458, 653
  * Rudolph et al. (2006) Rudolph A. L., Fich M., Bell G. R., Norsen T., Simpson J. P., Haas M. R., Erickson E. F., 2006, ApJS, 162, 346
  * Russeil (2003) Russeil D., 2003, A&A, 397, 133
  * Russeil et al. (2007) Russeil D., Adami C., Georgelin Y. M., 2007, A&A, 470, 161
  * Russeil et al. (2017) Russeil D., Zavagno A., Mège P., Poulin Y., Molinari S., Cambresy L., 2017, A&A, 601, L5
  * Samal et al. (2010) Samal M. R., et al., 2010, ApJ, 714, 1015
  * Samal et al. (2014) Samal M. R., et al., 2014, A&A, 566, A122
  * Shaver et al. (1983a) Shaver P. A., McGee R. X., Newton L. M., Danks A. C., Pottasch S. R., 1983a, MNRAS, 204, 53
  * Shaver et al. (1983b) Shaver P. A., McGee R. X., Newton L. M., Danks A. C., Pottasch S. R., 1983b, MNRAS, 204, 53
  * Skiff (2014) Skiff B. A., 2014, VizieR Online Data Catalog, p. B/mk
  * Smith et al. (1996) Smith L. F., Shara M. M., Moffat A. F. J., 1996, MNRAS, 281, 163
  * Sota et al. (2014) Sota A., Maíz Apellániz J., Morrell N. I., Barbá R. H., Walborn N. R., Gamen R. C., Arias J. I., Alfaro E. J., 2014, ApJS, 211, 10
  * Talent & Dufour (1979) Talent D. L., Dufour R. J., 1979, ApJ, 233, 888
  * Townsley et al. (2011) Townsley L. K., Broos P. S., Chu Y.-H., Gruendl R. A., Oey M. S., Pittard J. M., 2011, ApJS, 194, 16
  * Valerdi et al. (2021) Valerdi M., Peimbert A., Peimbert M., 2021, MNRAS, 505, 3624
  * Vilchez & Pagel (1988) Vilchez J. M., Pagel B. E. J., 1988, MNRAS, 231, 257
  * Vincenzo et al. (2016) Vincenzo F., Belfiore F., Maiolino R., Matteucci F., Ventura P., 2016, MNRAS, 458, 3466
  * Wenger et al. (2018) Wenger T. V., Balser D. S., Anderson L. D., Bania T. M., 2018, ApJ, 856, 52
  * Wenger et al. (2019) Wenger T. V., Balser D. S., Anderson L. D., Bania T. M., 2019, ApJ, 887, 114
  * Xu et al. (2009) Xu Y., Reid M. J., Menten K. M., Brunthaler A., Zheng X. W., Moscadelli L., 2009, ApJ, 693, 413
  * Zhang et al. (2009) Zhang B., Zheng X. W., Reid M. J., Menten K. M., Xu Y., Moscadelli L., Brunthaler A., 2009, ApJ, 693, 419
  * van Zee et al. (1998) van Zee L., Salzer J. J., Haynes M. P., O’Donoghue A. A., Balonek T. J., 1998, AJ, 116, 2805

Table A1: Sample of Galactic H ii regions and ring nebulae for which distances have been analyzed in this work. The table contains the name of the ionizing/associated stars of each nebula, spectral type, Gaia DR2 ID and the geometric distances determined from their parallax in the heliocentric reference system ($d$). All objects have parallax accuracy better than 20 per cent unless otherwise indicated. Nebula | Ion./Assoc. Star | Spectral Type | Gaia DR2 ID |  $d$ DR2 |  $d$ EDR3 | Ref. | Ref.
---|---|---|---|---|---|---|---
|  |  |  | (kpc) | (kpc) | I/A | SpT
Sh 2-29 | HD 165921 | O7V+B0V | 4066278846098710016 | $1.07^{+0.07}_{-0.06}$ | $1.15^{+0.04}_{-0.03}$ | 1 | 32
Sh 2-32 | ALS 17181 | B1V | 4066292830512043520 | $1.59^{+0.15}_{-0.13}$ | $1.46\pm 0.05$ | 1 | 33
Sh 2-47 | BD-15 4913 | B0.5III | 4098093077571330816 | $1.66^{+0.14}_{-0.12}$ | $1.54\pm 0.05$ | 1 | 1
Sh 2-48 | BD-14 5014 | O7.5V | 4098272263591120128 |  $4.35^{+2.09}_{-1.22}$ b |  $3.59^{+0.70}_{-0.61}$ a | 1 | 34
Sh 2-53 | TIC 413308470 | O6V-O5V | 4152566800618412928 |  $4.36^{+1.70}_{-1.08}$ b | $3.38\pm 0.18$ | 2 | 35
TIC 333769778 | O8V-9V | 4152566628819783424 |  $3.36^{+1.64}_{-0.91}$ b |  $4.27^{+3.24}_{-1.20}$ a | 2 | 2
Sh 2-54 | HD 167971 | O8Ia-O4/5 | 4153657378750644864 |  $1.98^{+0.64}_{-0.39}$ a | $1.34^{+0.15}_{-0.12}$ | 44 | 32
WR 113 | WC8+O8/9 | 4153716198275554176 | $1.82^{+0.19}_{-0.16}$ | $1.91\pm 0.05$ | 1 | 36
Sh 2-57 | BD-08 4623 | B0.5III | 4156562838196105600 | $2.10^{+0.24}_{-0.20}$ | $2.10^{+0.06}_{-0.08}$ | 1 | 1
Sh 2-61 | EM* AS 310 | B1 | 4256460509022075008 | $2.39^{+0.30}_{-0.24}$ | $2.39\pm 0.08$ | 3 | 37
Sh 2-82 | HD 231616 | B0.5V | 4323127850373105536 | $0.85^{+0.10}_{-0.08}$ | $0.78\pm 0.06$ | 1, 4 | 1, 4
Sh 2-88 | HD 338916 | O7.5V | 2020947043840246144 | $1.93^{+0.16}_{-0.14}$ | $2.06^{+0.08}_{-0.07}$ | 1 | 34
HD 338926 | O8.5Ib | 2020923679216984064 | $2.08^{+0.19}_{-0.16}$ | $1.99\pm 0.06$ | 1 | 1
LS II +25 09 | B0.5V | 2020883959358967552 |  $4.38^{+2.98}_{-1.89}$ b | N/A${}^{\text{(I)}}$ | 1 | 1
Sh 2-90 | TIC 287521091 | O8–O9MS | 2027904753426401152 |  $6.47^{+2.74}_{-1.77}$ b | $3.36^{+0.39}_{-0.28}$ | 5 | 5
Sh 2-93 | [F89b] S93 1 | B2V | 2027545659769078016 |  $3.30^{+2.65}_{-1.61}$ b |  $3.54^{+1.85}_{-1.35}$ b | 4 | 38
Sh 2-98 | WR 130 | WN8 | 2030934212864924032 |  $10.12^{+3.69}_{-2.71}$ b |  $11.24^{+2.59}_{-2.52}$ b | 1 | 43
Sh 2-100 | s D | O5 | 2058236705827295872 |  $9.46^{+2.39}_{-1.70}$ b | $11.40^{+1.14}_{-1.01}$ | 6 | 6
Sh 2-127 | WB 85 B | O8.5 | 2176011164377605888 |  $9.45^{+2.59}_{-1.89}$ b |  $12.29^{+5.19}_{-2.38}$ b | 7 | 7
Sh 2-128 | ALS 19702 | O7 | 2177724310884924288 |  $8.62^{+2.24}_{-1.60}$ b |  $7.85^{+1.61}_{-1.11}$ a | 8 | 8
Sh 2-132 | WR 153 | WN6o+O6I | 2006219978921880192 | $4.14^{+0.57}_{-0.45}$ | $4.53^{+0.29}_{-0.27}$ | 9 | 39
Sh 2-152 | Sh2-152 4 | O8.5V | 2013435042930935936 |  $4.96^{+2.14}_{-1.48}$ b |  $4.45^{+1.78}_{-1.34}$ b | 10 | 10
Sh 2-156 | Anon | O7 | 2013668306897124736 | $2.57^{+0.30}_{-0.25}$ | $2.56^{+0.22}_{-0.19}$ | 11 | 11
Sh 2-158 | TYC 4279-1463-1 | O3.5V+O9.5V | 2014960718396556160 | $2.67^{+0.27}_{-0.22}$ | $2.84^{+0.14}_{-0.16}$ | 51 | 34
Sh 2-175 | LS I +64 26 | B1.5V | 527233525086096768 | $2.12^{+0.14}_{-0.12}$ | $2.02\pm 0.05$ | 1 | 10
Sh 2-203 | LS I +55 47 | O9.5IV | 446947804498934272 | $2.55^{+0.25}_{-0.21}$ | $2.39^{+0.08}_{-0.09}$ | 48 | 49
Sh 2-206 | BD+50 886 | O4V | 250748269579173248${}^{\text{(II)}}$ |  $3.14^{+0.82}_{-0.56}$ a | $2.96^{+0.17}_{-0.15}$ | 1 | 34
Sh 2-207 | MFJ SH 2-207 1 | O9.5V | 275291258896477312 | $3.53^{+0.52}_{-0.40}$ | $3.59^{+0.23}_{-0.26}$ | 13 | 50
Sh 2-208 | MFJ SH 2-208 3 | B0V | 275284696186500096 |  $4.98^{+0.95}_{-0.70}$ a | $4.02^{+0.27}_{-0.25}$ | 13 | 13
Sh 2-209 | ALS 18696 | B1III | 271701112917796096 | $2.60^{+0.26}_{-0.22}$ | $2.61\pm 0.14$ | 12 | 12
ALS 18697 | O9III | 271701009838634752 | $2.55^{+0.59}_{-0.41}$ | $3.12^{+0.61}_{-0.49}$ | 12 | 12
[CW84] S209-3 | BI | 271701112917794176 |  $2.41^{+0.82}_{-0.51}$ a |  $3.18^{+0.86}_{-0.62}$ a | 12 | 12
Sh 2-212 | MFJ SH 2-212 2 | O7 | 260166892342134528 |  $0.78^{+0.53}_{-0.23}$ a |  $7.95^{+4.94}_{-2.71}$ b | 13, 14 | 32
MFJ SH 2-212 5 | ? | 260166681885774208 |  $5.39^{+1.62}_{-1.09}$ b | $4.30^{+0.65}_{-0.51}$ | 13, 14 | -
MFJ SH 2-212 7 | ? | 260167270299248384 |  $5.10^{+1.06}_{-0.77}$ a | $3.94^{+0.36}_{-0.35}$ | 13, 14 | -
MFJ SH 2-212 11 | ? | 260167339018718208 |  $5.98^{+2.17}_{-1.46}$ b |  $4.90^{+1.22}_{-0.96}$ b | 13, 14 | -
Sh 2-219 | LS V +47 22 | B0.5V | 254926272030205440 | $3.84^{+0.66}_{-0.50}$ | $4.16^{+0.32}_{-0.28}$ | 1 | 1
Sh 2-228 | ALS 19710 | O8V | 187250236123135360 | $3.23^{+0.52}_{-0.40}$ | $2.56\pm 0.09$ | 12 | 12
Sh 2-235 | BD+35 1201 | O9.5V | 3455729793008409088 | $1.65^{+0.12}_{-0.11}$ | $1.66\pm 0.07$ | 15, 16 | 40
Sh 2-237 | LS V +34 46 | B2V | 182584041919665024 | $2.16^{+0.35}_{-0.27}$ | $2.07\pm 0.06$ | 1, 17 | 17
Sh 2-255 | LS 19 | O9.5V | 3373362495052179456 | $2.38^{+0.43}_{-0.32}$ | $1.96^{+0.12}_{-0.09}$ | 18 | 18
Sh 2-257 | HD 253327 | B0.5V | 3373408159144488320 | $0.29^{+0.07}_{-0.05}$ |  $2.55^{+1.99}_{-1.03}$ b | 18 | 18
Sh 2-266 | MWC 137 | sgB[e] | 3344973478481675264 |  $5.52^{+1.73}_{-1.17}$ b | $4.60^{+0.53}_{-0.51}$ | 19 | 41
Sh 2-270 | [NS84] 8 | B0.5V | 3344023436011171200 |  $3.84^{+2.47}_{-1.54}$ b |  $1.79^{+1.27}_{-0.42}$ b | 20 | 20
Sh 2-271 | ALS 18672 | O9V | 3331937432404833792 | $3.64^{+0.58}_{-0.44}$ | $3.25^{+0.11}_{-0.12}$ | 13, 21 | 13
Sh 2-283 | MFJ SH 2-283 8 | B0V | 3119828097371908992 |  $6.79^{+1.75}_{-1.24}$ b | $5.37^{+0.58}_{-0.41}$ | 13 | 13
Sh 2-285 | [L85] S285 1 | B0V | 3112496794360485248 | $3.82^{+0.75}_{-0.55}$ | $4.38^{+0.42}_{-0.31}$ | 13, 22 | 22
Sh 2-288 | RAFGL 5223 | B0V | 3107524257323390208 |  $5.26^{+1.53}_{-1.06}$ b | $5.06^{+0.49}_{-0.47}$ | 3 | 42
Sh 2-297 | HD 53623 | B0V | 3045713939855362944 | $1.06^{+0.08}_{-0.07}$ | $1.08^{+0.04}_{-0.05}$ | 23 | 23
Sh 2-298 | WR 7 | WN4b | 3032940844556081408 |  $4.32^{+0.90}_{-0.66}$ a | $4.12^{+0.54}_{-0.40}$ | 24 | 39
Sh 2-301 | MFJ SH 2-301 1 | O6.5V | 2934257408213035264${}^{\text{(III)}}$ |  $4.45^{+1.04}_{-0.73}$ a | $3.21\pm 0.14$ | 13 | 34
Sh 2-308 | WR 6 | WN4b | 2922367976673391232 | $2.27^{+0.30}_{-0.24}$ | $1.51^{+0.10}_{-0.08}$ | 25 | 39
Sh 2-311 | HD 64315 | O5.5V+O7V | 5602025904044961536 |  $7.51^{+2.56}_{-1.82}$ b |  $6.62^{+2.38}_{-1.72}$ b | 1 | 32
HD 64568 | O3V | 5602033390154015744 |  $5.39^{+1.26}_{-0.90}$ a | $4.26^{+0.25}_{-0.34}$ | 1 | 32
LSS 830 | O7V | 5602027755165929344 | $4.72^{+0.87}_{-0.65}$ | $5.28^{+0.40}_{-0.35}$ | 1 | 34
Table A2: Continued.

Nebula | Ion./Assoc. Star | Spectral Type | Gaia DR2 ID |  $d$ DR2 |  $d$ EDR3 | Ref. | Ref.
---|---|---|---|---|---|---|---
|  |  |  | (kpc) | (kpc) | I/A | SpT
IC 5146 | BD+46 3474 | B0V | 1974546106933956608 | $0.72\pm 0.02$ | $0.76\pm 0.01$ | 1 | 1
RCW 52 | LS 1887 | O8V | 5350529730352416384 | $2.45^{+0.20}_{-0.17}$ | $2.33\pm 0.07$ | 1 | 42
RCW 58 | WR 40 | WN8h | 5240040631514998272 | $3.90^{+0.54}_{-0.43}$ | $2.70^{+0.13}_{-0.11}$ | 26 | 43
G2.4+1.4 | WR 102 | WR | 4067059872964459648 | $2.67^{+0.22}_{-0.19}$ | $2.65\pm 0.15$ | - | 42
NGC 2579 | VdBH 13a B | O5V | 5542914544020602880 | $5.85^{+0.73}_{-0.59}$ | $5.19^{+0.48}_{-0.35}$ | 27 | 27
VdBH 13b A | O6.5V | 5542914715819276160${}^{\text{(IV)}}$ |  $2.99^{+2.04}_{-1.11}$ b |  $2.75^{+1.33}_{-0.88}$ a | 27 | 27
NGC 3576 | HD 97484 | O8+O8V | 5337239658437912960 | $2.76^{+0.39}_{-0.30}$ | $2.48^{+0.11}_{-0.10}$ | 28, 45 | 45
NGC 3603 | - | - | - | $7.02\pm 0.10$ | - | 29 | -
NGC 6888 | WR 136 | WN6 | 2061690233159124352 | $1.95^{+0.14}_{-0.12}$ | $1.67\pm 0.04$ | - | 43
NGC 7635 | BD+60 2522 | O6.5 | 2014149897293278848 | $2.50^{+0.20}_{-0.17}$ | $2.83^{+0.13}_{-0.10}$ | - | 32
M 8 | HD 165052 | O7V+O7.5V | 4066064956700837248 | $1.24^{+0.08}_{-0.07}$ | $1.22\pm 0.04$ | 30 | 30
M 16 | - | - | - | $1.71\pm{0.18}$ | - | 31 | -
M 17 | - | - | - | $1.82\pm{0.16}$ | - | 31 | -
M 20 | HD 164492 | O7V+B6V+A2Ia+aBe? | 4069268658691324672 | $1.53^{+0.26}_{-0.19}$ | $1.42^{+0.09}_{-0.08}$ | 31 | 46, 47
M 42 | - | - | - | $0.41\pm{0.01}$ | - | 31 | -

(1)Avedisova & Kondratenko (1984), (2)Paron et al. (2013), (3)Hunter & Massey
(1990), (4)Forbes (1989), (5)Samal et al. (2014), (6)Samal et al. (2010),
(7)Rudolph et al. (1996), (8)Bohigas & Tapia (2003), (9)Harten et al. (1978),
(10)Russeil et al. (2007), (11)Lynds & Oneil (1983), (12)Chini & Wink (1984),
(13)Moffat et al. (1979), (14)Deharveng et al. (2008), (15)Kirsanova et al.
(2008), (16)Camargo et al. (2011), (17)Pandey et al. (2013), (18)Ojha et al.
(2011), (19)Esteban & Fernandez (1998), (20)Neckel & Staude (1984), (21)Persi
et al. (1987), (22)Rolleston et al. (1994), (23)Mallick et al. (2012),
(24)Esteban et al. (1989), (25)Chu et al. (1982), (26)Chu (1982), (27)Copetti
et al. (2007), (28)Townsley et al. (2011), (29)Drew et al. (2019), (30)Damiani
et al. (2017), (31)Binder & Povich (2018), (32)Sota et al. (2014), (33)Nazé
(2009), (34)Maíz Apellániz et al. (2016), (35)Ji et al. (2012), (36)Skiff
(2014), (37)Hernández et al. (2004), (38)Hunter & Massey (1990), (39)Smith et
al. (1996), (40)Georgelin et al. (1973), (41)Mehner et al. (2016), (42)Reed
(2003), (43)Hamann et al. (2006), (44)De Becker et al. (2005), (45)Povich et
al. (2017), (46)Rho et al. (2004), (47)Rho et al. (2006), (48)Churchwell &
Walmsley (1973), (49)Roman-Lopes & Roman-Lopes (2019), (50)Molina Lera et al.
(2018), (51)Lynds & Oneil (1986),

${}^{\text{(I)}}$ The parallax could not be estimated in EDR3.

${}^{\text{(II)}}$ In the EDR3 the new ID is 250748269580520320

${}^{\text{(III)}}$ In the EDR3 the new ID is 2934257408221360384

${}^{\text{(IV)}}$ In the EDR3 the new ID is 5542914715819272960

a: $0.2\leq e_{\text{plx}}/\text{plx}<0.3$

b: $e_{\text{plx}}/\text{plx}\geq 0.3$ or negative.

Table A3: Galactocentric distance estimates for the sample of Galactic H ii
regions and ring nebulae.

Nebula |  $R_{G}$ (kpc)
---|---
| Geometric | Photogeometric | 1/Plx | Kinematic
| DR2 | EDR3 | EDR3 | EDR3 | W19 | R03-R07
Sh 2-29 | $7.13\pm 0.17$ | $7.06\pm 0.14$ | $7.07\pm 0.13$ | $7.07\pm 0.13$ | - | $5.81\pm 0.89$
Sh 2-32 | $6.62\pm 0.25$ | $6.75\pm 0.15$ | $6.76\pm 0.16$ | $6.75\pm 0.15$ | - | $5.69\pm 1.00$
Sh 2-47 | $6.62\pm 0.24$ | $6.73\pm 0.15$ | $6.71\pm 0.14$ | $6.72\pm 0.14$ | - | $6.59\pm 0.47$
Sh 2-48 | $4.28\pm 1.80$ | $4.86\pm 0.69$ | - | - | - | $4.86\pm 0.27$
Sh 2-53 | $4.35^{+1.51}_{-1.17}$ | $5.10\pm 0.25$ | $5.07\pm 0.28$ | $5.10\pm 0.27$ | $4.68\pm 0.28$ | $5.11\pm 0.25$
Sh 2-54 | $6.52\pm 0.26$ | $6.42\pm 0.15$ | $6.43\pm 0.16$ | $6.43\pm 0.15$ | $6.08\pm 0.27$ | $6.16\pm 0.37$
Sh 2-57 | $6.33\pm 0.30$ | $6.32\pm 0.17$ | $6.33\pm 0.17$ | $6.33\pm 0.16$ | - | $6.52\pm 0.26$
Sh 2-61 | $6.16\pm 0.34$ | $6.15\pm 0.16$ | $6.16\pm 0.16$ | $6.16\pm 0.15$ | - | $5.97\pm 0.24$
Sh 2-82 | $7.73\pm 0.16$ | $7.76\pm 0.13$ | $7.75\pm 0.14$ | $7.77\pm 0.13$ | - | $7.57\pm 0.23$
Sh 2-83 | - | - | - | - | $13.24\pm 1.13$ | -
Sh 2-88 | $7.47\pm 0.14$ | $7.44\pm 0.12$ | $7.44\pm 0.11$ | $7.44\pm 0.11$ | $7.52\pm 0.19$ | $10.94^{+12.83}_{-3.29}$
Sh 2-90 | $7.92\pm 1.34$ | $7.33\pm 0.12$ | $7.33\pm 0.12$ | - | $7.78\pm 0.25$ | $11.38^{+13.36}_{-3.52}$
Sh 2-93 | $7.56^{+0.54}_{-0.26}$ | $7.47\pm 0.39$ | $7.58^{+0.70}_{-0.28}$ | - | $7.66\pm 0.19$ | $8.48^{+4.44}_{-0.76}$
Sh 2-98 | $10.33^{+2.81}_{-2.15}$ | $11.10\pm 1.97$ | $11.49^{+1.72}_{-1.47}$ | - | $11.53\pm 0.88$ | $8.53\pm 0.26$
Sh 2-100 | $10.13^{+1.74}_{-1.31}$ | $11.57\pm 0.88$ | $11.08\pm 1.27$ | - | $9.37\pm 0.41$ | $9.83\pm 0.29$
Sh 2-127 | $13.08\pm 2.15$ | $15.59\pm 4.40$ | $14.25\pm 2.32$ | - | $13.76\pm 1.31$ | $15.06\pm 0.39$
Sh 2-128 | $12.64\pm 1.87$ | $12.13\pm 1.24$ | $11.52\pm 0.99$ | - | $11.78\pm 0.78$ | $13.02\pm 0.46$
Sh 2-132 | $9.98\pm 0.44$ | $10.21\pm 0.27$ | $10.23\pm 0.26$ | $10.18\pm 0.30$ | - | $10.98\pm 0.33$
Sh 2-152 | $10.77^{+1.75}_{-1.35}$ | $10.50\pm 1.30$ | $10.15\pm 1.10$ | - | $10.61^{+0.71}_{-0.43}$ | $9.55\pm 0.11$
Sh 2-156 | $9.40\pm 0.26$ | $9.39\pm 0.22$ | $9.38\pm 0.22$ | $9.38\pm 0.24$ | $10.75\pm 0.58$ | $9.86\pm 0.48$
Sh 2-158 | $9.52\pm 0.25$ | $9.62\pm 0.19$ | $9.62\pm 0.21$ | $9.60\pm 0.19$ | $9.62\pm 0.09$ | $9.57\pm 0.09$
Sh 2-175 | $9.45\pm 0.19$ | $9.38\pm 0.13$ | $9.38\pm 0.13$ | $9.38\pm 0.14$ | - | $9.09\pm 0.14$
Sh 2-203 | $10.35\pm 0.32$ | $10.22\pm 0.18$ | $10.21\pm 0.17$ | $10.22\pm 0.17$ | - | $9.91\pm 0.28$
Sh 2-206 | $11.09\pm 0.78$ | $10.88\pm 0.25$ | $10.88\pm 0.26$ | $10.86\pm 0.26$ | $10.77\pm 0.59$ | $11.36\pm 0.81$
Sh 2-207 | $11.44\pm 0.56$ | $11.50\pm 0.33$ | $11.44\pm 0.34$ | $11.41\pm 0.34$ | - | $12.39\pm 0.91$
Sh 2-208 | $12.74\pm 0.99$ | $11.88\pm 0.34$ | $11.71\pm 0.46$ | $11.91\pm 0.45$ | - | $12.36\pm 0.94$
Sh 2-209 | $10.55\pm 0.34$ | $10.57\pm 0.23$ | $10.50\pm 0.23$ | $10.53\pm 0.24$ | $14.75^{+1.56}_{-1.10}$ | $16.43\pm 1.92$
Sh 2-212 | $13.84\pm 2.20$ | $12.81\pm 1.25$ | $12.02\pm 1.02$ | - | $14.76\pm 1.30$ | $14.27^{+1.38}_{-1.55}$
Sh 2-219 | $11.86\pm 0.74$ | $12.18\pm 0.42$ | $12.16\pm 0.40$ | $12.12\pm 0.36$ | - | $12.97\pm 0.81$
Sh 2-228 | $11.42\pm 0.57$ | $10.73\pm 0.18$ | $10.71\pm 0.20$ | $10.71\pm 0.19$ | - | $10.83^{+1.72}_{-1.37}$
Sh 2-235 | $9.84\pm 0.23$ | $9.86\pm 0.17$ | $9.85\pm 0.15$ | $9.85\pm 0.17$ | - | $103.95^{+103.77}_{-64.79}$
Sh 2-237 | $10.34\pm 0.43$ | $10.26\pm 0.16$ | $10.22\pm 0.18$ | $10.25\pm 0.17$ | - | $11.85^{+3.46}_{-2.32}$
Sh 2-255 | $10.53\pm 0.50$ | $10.12\pm 0.22$ | $10.14\pm 0.20$ | $10.11\pm 0.20$ | $9.89\pm 0.08$ | $10.61\pm 0.38$
Sh 2-257 | $10.53^{+0.50}_{-0.52}$ | $10.13\pm 0.22$ | $10.13\pm 0.20$ | $10.10\pm 0.20$ | $9.89\pm 0.08$ | $10.61\pm 0.36$
Sh 2-266 | $13.69\pm 1.79$ | $12.67\pm 0.65$ | $12.87\pm 0.77$ | - | - | $16.17\pm 3.14$
Sh 2-270 | $12.09^{+2.51}_{-2.15}$ | $10.00\pm 1.25$ | $10.39\pm 1.32$ | - | - | $15.18\pm 2.21$
Sh 2-271 | $11.72\pm 0.64$ | $11.34\pm 0.22$ | $11.37\pm 0.23$ | $11.33\pm 0.23$ | - | $11.89\pm 1.49$
Sh 2-283 | $14.39\pm 1.89$ | $13.08\pm 0.66$ | $13.20\pm 0.63$ | - | - | $15.98\pm 0.29$
Sh 2-285 | $11.52\pm 0.80$ | $12.09^{+0.46}_{-0.50}$ | $11.99\pm 0.50$ | $12.10\pm 0.44$ | - | $15.84\pm 0.27$
Sh 2-288 | $12.71\pm 1.50$ | $12.54^{+0.50}_{-0.51}$ | $12.50\pm 0.85$ | - | $12.99\pm 0.86$ | $14.15\pm 1.08$
Sh 2-297 | $8.97\pm 0.16$ | $8.99\pm 0.14$ | $8.99\pm 0.13$ | $8.99\pm 0.14$ | $8.56\pm 0.28$ | $9.24\pm 0.35$
Sh 2-298 | $11.59\pm 0.83$ | $11.42\pm 0.53$ | $11.38\pm 0.58$ | - | $11.73\pm 0.72$ | $13.06\pm 0.67$
Sh 2-301 | $11.48\pm 0.91$ | $10.51\pm 0.21$ | $10.55\pm 0.20$ | $10.52\pm 0.21$ | $11.60\pm 0.73$ | $11.40\pm 0.47$
Sh 2-308 | $9.67\pm 0.32$ | $9.14\pm 0.17$ | $9.10\pm 0.19$ | $9.13\pm 0.16$ | - | $9.37\pm 0.28$
Sh 2-311 | $11.13\pm 0.84$ | $11.60\pm 0.39$ | $11.60\pm 0.44$ | $11.55\pm 0.45$ | $10.84\pm 0.73$ | $11.51\pm 0.40$
IC 5146 | $8.29\pm 0.10$ | $8.29\pm 0.10$ | $8.29\pm 0.10$ | $8.29\pm 0.10$ | - | -
RCW 52 | $7.83\pm 0.10$ | $7.83\pm 0.10$ | $7.83\pm 0.10$ | $7.83\pm 0.10$ | - | $8.36^{+1.34}_{-0.22}$
RCW 58 | $7.64\pm 0.16$ | $7.60\pm 0.10$ | $7.60\pm 0.10$ | $7.60\pm 0.10$ | - | $8.53\pm 0.24$
G2.4+1.4 | $5.53\pm 0.32$ | $5.55\pm 0.27$ | $5.60\pm 0.26$ | $5.60\pm 0.23$ | - | -
NGC 2579 | $11.26\pm 0.61$ | $10.80\pm 0.40$ | $10.74\pm 0.39$ | $10.74\pm 0.39$ | - | -
NGC 3576 | $7.65\pm 0.11$ | $7.66\pm 0.10$ | $7.66\pm 0.10$ | $7.66\pm 0.10$ | - | -
NGC 3603 | $8.61\pm 0.11$ | - | - | - | - | $9.21\pm 0.28$
NGC 6888 | $7.94\pm 0.10$ | $7.95\pm 0.10$ | $7.95\pm 0.10$ | $7.95\pm 0.10$ | - | -
NGC 7635 | $9.42\pm 0.22$ | $9.64\pm 0.18$ | $9.61\pm 0.17$ | $9.62\pm 0.17$ | $10.28^{+0.66}_{-0.44}$ | -
M 8 | $6.97\pm 0.18$ | $6.98\pm 0.14$ | $7.00\pm 0.14$ | $7.00\pm 0.14$ | - | $5.74\pm 0.98$
M 16 | $6.58\pm 0.27$ | - | - | - | - | $6.16\pm 0.38$
M 20 | $6.69\pm 0.35$ | $6.79\pm 0.19$ | $6.78\pm 0.22$ | $6.83\pm 0.19$ | - | $5.08\pm 0.70$
M 17 | $6.45\pm 0.25$ | - | - | - | $6.49\pm 0.09$ | $6.21\pm 0.45$
M 42 | $8.54\pm 0.11$ | - | - | - | $8.70\pm 0.01$ | $9.43\pm 0.56$

W19: Wenger et al. (2019).

R03-R07: Russeil (2003), Russeil et al. (2007).

Figure A1: Comparison of the Gaia EDR3 geometric Galactocentric distance with
other determinations based on Gaia parallaxes. Top panel: Gaia DR2 geometric
distance. Middle panel: Gaia EDR3 photogeometric distance. Bottom panel:
Distance obtained directly as the inverse of Gaia EDR3 parallax in a sub-
sample with uncertainties in the parallax smaller than 10 per cent.

Figure A2: Comparison of the Gaia EDR3 geometric Galactocentric distances with
kinematic distance determinations. Top panel: Determinations from Russeil
(2003) and Russeil et al. (2007) based on the rotation curve of Brand & Blitz
(1993). Botton panel: Determinations from Wenger et al. (2019) based on the
rotation curve of Reid et al. (2014) using a Monte Carlo approach.  Table A4:
Physical conditions, Ionic abundances derived assuming $t^{2}=0.038\pm 0.004$
in units of 12+log(Xi+/H+) and references of spectroscopic data

Region | Reference |  $n_{\rm e}$[O ii] |  $n_{\rm e}$[S ii] |  $n_{\rm e}$[Cl iii] |  $T_{\rm e}$[N ii] |  $T_{\rm e}$[O iii] |  He+ |  C2+ |  N+ |  O+ |  O2+ |  O2+ |  O2+/O
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  (cm-3) |  (cm-3) |  (cm-3) | (K) | (K) | (RLs) | (RLs) |  |  |  | (RLs) | (CELs)
Sh 2-29 | 1 | - | $100\pm 40$ | $700_{-600}^{3000}$ | $7500\pm 200$ | $6300\pm 200^{a}$ | $10.52\pm 0.01$ | - | $8.05^{+0.06}_{-0.05}$ | $8.83\pm 0.07$ | $7.98^{+0.15}_{-0.13}$ | - | 0.12
Sh 2-32 | 1 | - | $100\pm 50$ | - | $7100\pm 200$ | $5800\pm 300^{a}$ | - | - | $8.02^{+0.07}_{-0.08}$ | $8.78\pm 0.11$ | $7.52^{+0.25}_{-0.17}$ | - | 0.05
Sh 2-47 | 1 | - | $100\pm 70$ | - | $7000\pm 500$ |  $5600_{-700}^{+600}$ a | $9.92\pm 0.03$ | - | $8.06^{+0.18}_{-0.19}$ | $8.93^{+0.34}_{-0.32}$ | $8.53^{+0.76}_{-0.57}$ | - | 0.29
Sh 2-48 | 1 | - | $100\pm 60$ | - | $6800\pm 500$ | $5300\pm 700^{a}$ | $10.87\pm 0.07$ | - | $7.99^{+0.20}_{-0.19}$ | $8.77^{+0.37}_{-0.34}$ | $8.99^{+0.82}_{-0.50}$ | - | 0.62
Sh 2-53 | 1 | - | $300\pm 100$ | - | $7000\pm 400$ | $5600\pm 600^{a}$ | $10.59\pm 0.04$ | - | $8.29^{+0.16}_{-0.15}$ | $8.98\pm 0.28$ | $8.52^{+0.50}_{-0.42}$ | - | 0.26
Sh 2-54 | 1 | - | $200\pm 50$ | $200_{-610}^{900}$ | $7500\pm 200$ | $8800_{-1000}^{+800}$ | $10.95\pm 0.03$ | $8.49\pm 0.11$ | $7.76^{+0.06}_{-0.07}$ | $8.55^{+0.13}_{-0.12}$ | $7.91^{+0.36}_{-0.35}$ | - | 0.19
Sh 2-61 | 2 | - | $1100\pm 200$ | - | $7500\pm 400$ | $6300\pm 500^{a}$ | - | - | $8.09^{+0.10}_{-0.09}$ | $8.64^{+0.14}_{-0.15}$ | $6.34^{+0.35}_{-0.25}$ | - | 0.01
Sh 2-82 | 1 | - | $100\pm 70$ | - | $7000\pm 900$ | $5600\pm 1200^{a}$ | $10.14\pm 0.02$ | - | $8.06^{+0.29}_{-0.30}$ | $8.89^{+0.44}_{-0.49}$ | - | - | $\sim 0.0$
Sh 2-83 | 3 | - | $300\pm 100$ | - | $11900\pm 600$ | $10400\pm 500$ | $10.899\pm 0.007$ | - | $6.40\pm 0.08$ | $7.24^{+0.13}_{-0.14}$ | $8.43\pm 0.09$ | - | 0.94
Sh 2-88 | 1 | - | $900\pm 100$ | $<100$ | $8300\pm 200$ | $7300\pm 300^{a}$ | $10.67\pm 0.04$ | - | $7.83\pm 0.05$ | $8.64\pm 0.09$ | $8.03^{+0.14}_{-0.12}$ | - | 0.20
Sh 2-90 | 2 | - | $200\pm 100$ | - | $8200\pm 200$ | $7200\pm 300^{a}$ | $10.92\pm 0.01$ | - | $7.92^{+0.06}_{-0.05}$ | $8.53\pm 0.08$ | $8.37^{+0.15}_{-0.12}$ | - | 0.41
Sh 2-93 | 1 | - | $500\pm 50$ | - | $7900\pm 300$ | $7000\pm 400^{a}$ | $10.47\pm 0.02$ | - | $8.04\pm 0.08$ | $8.61^{+0.14}_{-0.15}$ | $7.68^{+0.23}_{-0.22}$ | - | 0.11
Sh 2-100 | 3 | - | $400\pm 200$ | - | $8600\pm 300$ | $8200\pm 100$ | $10.984\pm 0.015$ | $8.36\pm 0.08$ | $6.93^{+0.07}_{-0.08}$ | $7.90^{+0.11}_{-0.10}$ | $8.70^{+0.06}_{-0.05}$ | $8.52\pm 0.06$ | 0.87
Sh 2-127 | 3 | - | $600\pm 100$ | - | $9800\pm 200$ | $9500\pm 200^{a}$ | $10.799\pm 0.008$ | - | $7.41\pm 0.03$ | $8.33\pm 0.05$ | $7.61\pm 0.06$ | - | 0.16
Sh 2-128 | 3 | - | $500\pm 100$ | - | $10600\pm 200$ | $9900\pm 300$ | $10.928\pm 0.015$ | - | $6.91^{+0.05}_{-0.04}$ | $7.94\pm 0.06$ | $8.18^{+0.08}_{-0.07}$ | - | 0.63
Sh 2-132 | 4 | - | $300\pm 100$ | - | $9300\pm 500$ | $8800\pm 600^{a}$ | $10.85\pm 0.06$ | - | $7.60\pm 0.09$ | $8.50\pm 0.15$ | $7.38\pm 0.20$ | - | 0.07
Sh 2-152 | 2 | - | $700\pm 100$ | - | $8200\pm 100$ | $7300\pm 100^{a}$ | $10.802\pm 0.007$ | $7.97\pm 0.14$ | $7.74\pm 0.03$ | $8.63^{+0.04}_{-0.03}$ | $8.11\pm 0.07$ | - | 0.23
Sh 2-156 | 4 | - | $900\pm 100$ | - | $9400\pm 400$ | $9000\pm 400$ | $10.90\pm 0.05$ | - | $7.41\pm 0.07$ | $8.37\pm 0.13$ | $7.93^{+0.10}_{-0.11}$ | - | 0.26
Sh 2-175 | 2 | - | $<100$ | - | $7000\pm 1300$ | $5600\pm 400^{a}$ | - | - | $7.98^{+0.18}_{-0.12}$ | $8.77^{+0.23}_{-0.15}$ | - | - | $\sim 0.0$
Sh 2-209 | 3 | - | $300\pm 300$ | - | $10600\pm 800$ | $10700_{-1200}^{+1000}$ | $10.893\pm 0.046$ | - | $6.84\pm 0.14$ | $7.74\pm 0.32$ | $8.05^{+0.24}_{-0.23}$ | - | 0.67
Sh 2-212 | 3 | - | $<100$ | - | $8300\pm 700$ | $11100\pm 1000$ | $11.019\pm 0.019$ | - | $6.78^{+0.17}_{-0.16}$ | $8.35\pm 0.31$ | $7.98^{+0.20}_{-0.18}$ | - | 0.30
Sh 2-219 | 2 | - | $<100$ | - | $8600\pm 300$ | $7800\pm 500^{a}$ | $10.35\pm 0.04$ | - | $7.60^{+0.07}_{-0.06}$ | $8.45^{+0.11}_{-0.12}$ | $6.38^{+0.22}_{-0.19}$ | - | 0.01
Sh 2-235 | 2 | - | $<100$ | - | $8100\pm 200$ | $7100\pm 200^{a}$ | $10.59\pm 0.01$ | - | $7.73^{+0.05}_{-0.04}$ | $8.60^{+0.08}_{-0.07}$ | $7.31^{+0.12}_{-0.11}$ | - | 0.05
Sh 2-237 | 2 | - | $400\pm 100$ | - | $8700\pm 300$ | $8000\pm 400^{a}$ | $9.67\pm 0.07$ | - | $7.69\pm 0.05$ | $8.50\pm 0.09$ | $6.71\pm 0.13$ | - | 0.02
Sh 2-257 | 2 | - | $100\pm 100$ | - | $7900\pm 200$ | $6900\pm 300^{a}$ | $10.24\pm 0.02$ | - | $7.78\pm 0.07$ | $8.60\pm 0.10$ | $7.00^{+0.17}_{-0.16}$ | - | 0.02
Sh 2-266 | 2 | - | $300\pm 200$ | - | $8300\pm 400$ | $7400\pm 600^{a}$ | $10.32\pm 0.07$ | - | $7.69^{+0.11}_{-0.12}$ | $8.36^{+0.18}_{-0.16}$ | $6.44^{+0.27}_{-0.26}$ | - | 0.01
Sh 2-270 | 2 | - | $400\pm 100$ | - | $9300\pm 1000$ | $8800\pm 1300^{a}$ | - | - | $7.38^{+0.20}_{-0.18}$ | $8.24^{+0.28}_{-0.30}$ | - | - | $\sim 0.0$
Sh 2-271 | 2 | - | $<100$ | - | $8700\pm 200$ | $7900\pm 300^{a}$ | $10.22\pm 0.03$ | - | $7.65\pm 0.06$ | $8.44\pm 0.08$ | $6.16\pm 0.12$ | - | 0.01
Sh 2-285 | 2 | - | $<100$ | - | $8500\pm 300$ | $7700\pm 40^{a}$ | - | - | $7.59\pm 0.07$ | $8.38\pm 0.11$ | - | - | $\sim 0.0$
Sh 2-288 | 3 | - | $400\pm 300$ | - | $9400\pm 300$ | $9200\pm 500$ | $10.792\pm 0.010$ | - | $7.34\pm 0.07$ | $8.35\pm 0.10$ | $7.99^{+0.15}_{-0.13}$ | - | 0.30
Sh 2-297 | 2 | - | $<100$ | - | $7800\pm 200$ | $6700\pm 300^{a}$ | $10.17\pm 0.04$ | - | $7.89^{+0.06}_{-0.05}$ | $8.67\pm 0.08$ | $7.17^{+0.16}_{-0.14}$ | - | 0.03
Sh 2-298 | 2 | - | $<100$ | - | $11700\pm 500$ | $11700\pm 200$ | $10.971\pm 0.017$ | - | $7.41\pm 0.07$ | $8.24\pm 0.11$ | $8.26^{+0.04}_{-0.03}$ | - | 0.51
Sh 2-311 | 5 | $300\pm 100$ | $300\pm 100$ | - | $9300\pm 200$ | $8900\pm 100$ | $10.914\pm 0.006$ | $8.01\pm 0.05$ | $7.44^{+0.04}_{-0.05}$ | $8.42\pm 0.06$ | $8.08\pm 0.04$ | $8.04\pm 0.04$ | 0.32
IC 5146 | 6 | $<100$ | $<100$ | - | $7100\pm 100$ | $5700\pm 200^{a}$ | - | - | $8.03^{+0.06}_{-0.05}$ | $8.80^{+0.11}_{-0.10}$ | - | - | $\sim 0.0$
NGC 2579 | 7 | $1200\pm 300$ | $800\pm 200$ | - | $8600\pm 300^{a}$ | $9300\pm 200$ | $10.934\pm 0.026$ | $8.18\pm 0.05$ | $6.99\pm 0.06$ | $8.13^{+0.10}_{-0.11}$ | $8.44\pm 0.05$ | $8.46\pm 0.03$ | 0.67
NGC 3576 | 8 | $1700\pm 200$ | $1100\pm 300$ | - | $8800\pm 200$ | $8400\pm 100$ | $10.941\pm 0.015$ | $8.44\pm 0.02$ | $7.21^{+0.06}_{-0.05}$ | $8.21\pm 0.06$ | $8.65\pm 0.04$ | $8.62\pm 0.05$ | 0.74
NGC 3603 | 9 | $2700_{-500}^{+800}$ | $3100_{-700}^{+1300}$ | - | $11200\pm 500$ | $9000\pm 100$ | $10.996\pm 0.027$ | $8.49\pm 0.07$ | $6.57^{+0.07}_{-0.08}$ | $7.44\pm 0.14$ | $8.67\pm 0.05$ | $8.69\pm 0.05$ | 0.94
M 8 | 10 | $1600\pm 600$ | $1300\pm 200$ | - | $8400\pm 100$ | $8000\pm 100$ | $10.838\pm 0.009$ | $8.31\pm 0.02$ | $7.70\pm 0.03$ | $8.52\pm 0.06$ | $8.20^{+0.05}_{-0.04}$ | $8.23\pm 0.03$ | 0.32
M 16 | 9 | $1200\pm 200$ | $1100\pm 200$ | - | $8300\pm 100$ | $7600\pm 200$ | $10.892\pm 0.009$ | $8.39\pm 0.04$ | $7.89^{+0.04}_{-0.05}$ | $8.64^{+0.07}_{-0.06}$ | $8.26^{+0.10}_{-0.07}$ | $8.30\pm 0.04$ | 0.30
M 17 | 10 | $500\pm 100$ | $400\pm 100$ | - | $8900\pm 200$ | $8000\pm 100$ | $10.967\pm 0.013$ | $8.73\pm 0.03$ | $6.98^{+0.06}_{-0.05}$ | $7.95^{+0.08}_{-0.07}$ | $8.77^{+0.06}_{-0.05}$ | $8.68\pm 0.02$ | 0.87
M 20 | 9 | $300\pm 100$ | $300\pm 100$ | - | $8300\pm 200$ | $7800\pm 300$ | $10.854\pm 0.009$ | $8.19\pm 0.05$ | $7.76^{+0.05}_{-0.04}$ | $8.66^{+0.06}_{-0.05}$ | $8.06^{+0.11}_{-0.10}$ | $8.00\pm 0.22$ | 0.20
M 42 | 11 | $6900_{-1500}^{+2400}$ | $4900_{-1800}^{+2900}$ | - | $10200\pm 200$ | $8300\pm 100$ | $10.946\pm 0.014$ | $8.34\pm 0.02$ | $7.00\pm 0.06$ | $7.85\pm 0.12$ | $8.73\pm 0.04$ | $8.57\pm 0.01$ | 0.88

(1) Arellano-Córdova et al. (2021), (2) Esteban & García-Rojas (2018), (3)
Esteban et al. (2017), (4) Fernández-Martín et al. (2017), (5) García-Rojas et
al. (2005), (6) García-Rojas et al. (2014), (7) Esteban et al. (2013), (8)
García-Rojas et al. (2004), (9) García-Rojas et al. (2006), (10) García-Rojas
et al. (2007), (11) Esteban et al. (2004).

a $T_{\rm e}$ estimated using the temperature relation from Esteban et al.
(2009).

Table A5: Ionic abundances derived assuming $t^{2}=0.038\pm 0.004$ in units of 12+log(Xi+/H+). Nebula | Ne2+ | S+ | S2+ | Cl+ | Cl2+ | Cl3+ | Ar2+ | Ar3+
---|---|---|---|---|---|---|---|---
Sh 2-29 | - | $6.96^{+0.06}_{-0.05}$ | $7.06\pm 0.20$ | - | $4.90^{+0.13}_{-0.12}$ | - | $6.09\pm 0.10$ | -
Sh 2-32 | - | $6.96\pm 0.08$ | - | - | - | - | $5.37\pm 0.17$ | -
Sh 2-47 | - | $6.99^{+0.18}_{-0.17}$ | - | - | - | - | $5.42^{+0.28}_{-0.25}$ | -
Sh 2-48 | - | $6.73^{+0.18}_{-0.16}$ | - | - | - | - | $6.59\pm 0.36$ | -
Sh 2-53 | - | $6.79^{+0.20}_{-0.17}$ | - | - | - | - | $6.28^{+0.19}_{-0.20}$ | -
Sh 2-54 | - | $6.36\pm 0.08$ | $6.97^{+0.30}_{-0.26}$ | - | $5.26^{+0.10}_{-0.09}$ | - | $6.23^{+0.16}_{-0.13}$ | -
Sh 2-61 | - | $6.98\pm 0.10$ | - | - | - | - | - | -
Sh 2-82 | - | $6.94^{+0.35}_{-0.29}$ | - | - | - | - | $5.83^{+0.51}_{-0.47}$ | -
Sh 2-83 | $7.90^{+0.13}_{-0.12}$ | $5.28\pm 0.07$ | $6.39^{+0.10}_{-0.12}$ | - | $4.68\pm 0.09$ | - | $5.81\pm 0.06$ | -
Sh 2-88 | - | $6.39^{+0.06}_{-0.05}$ | $7.19^{+0.16}_{-0.15}$ | - | $5.04^{+0.12}_{-0.14}$ | - | $6.24^{+0.09}_{-0.08}$ | -
Sh 2-90 | - | $6.50^{+0.05}_{-0.06}$ | - | - | $5.29^{+0.09}_{-0.10}$ | - | $6.49^{+0.11}_{-0.09}$ | -
Sh 2-93 | - | $6.71^{+0.09}_{-0.08}$ | $7.07^{+0.28}_{-0.26}$ | - | - | - | $6.14\pm 0.15$ | -
Sh 2-100 | $8.15^{+0.07}_{-0.06}$ | $5.68\pm 0.07$ | $7.09^{+0.13}_{-0.11}$ | - | $5.15^{+0.08}_{-0.07}$ | - | $6.48\pm 0.06$ | $5.09\pm 0.11$
Sh 2-127 | - | $6.03^{+0.04}_{-0.03}$ | $6.87^{+0.07}_{-0.06}$ | - | $4.86\pm 0.05$ | - | $5.82\pm 0.05$ | -
Sh 2-128 | $7.35^{+0.12}_{-0.11}$ | $5.72\pm 0.03$ | $6.60^{+0.07}_{-0.06}$ | - | $4.82\pm 0.05$ | - | $5.97\pm 0.03$ | -
Sh 2-132 | - | $6.51^{+0.10}_{-0.09}$ | $6.73\pm 0.05$ | - | - | - | $6.24^{+0.11}_{-0.10}$ | -
Sh 2-152 | - | $6.32\pm 0.03$ | $7.21^{+0.06}_{-0.05}$ | - | $5.17\pm 0.04$ | - | $6.34\pm 0.04$ | -
Sh 2-156 | $6.74^{+0.14}_{-0.13}$ | $6.00^{+0.09}_{-0.08}$ | $6.98\pm 0.06$ | - | $4.99\pm 0.09$ | - | $6.27\pm 0.04$ | -
Sh 2-175 | - | $6.94^{+0.16}_{-0.10}$ | - | - | - | - | - | -
Sh 2-209 | - | $5.61^{+0.11}_{-0.12}$ | $6.44^{+0.31}_{-0.29}$ | - | - | - | - | -
Sh 2-212 | $7.06^{+0.24}_{-0.22}$ | $5.32\pm 0.17$ | $6.75\pm 0.29$ | - | $5.06\pm 0.19$ | - | $6.16^{+0.19}_{-0.18}$ | -
Sh 2-219 | - | $6.46\pm 0.06$ | - | - | $4.96^{+0.29}_{-0.30}$ | - | $5.42\pm 0.08$ | -
Sh 2-235 | - | $7.09\pm 0.08$ | $6.94^{+0.09}_{-0.08}$ | - | $4.93\pm 0.07$ | - | $6.11^{+0.06}_{-0.05}$ | -
Sh 2-237 | - | $6.57^{+0.07}_{-0.06}$ | - | - | $5.01\pm 0.22$ | - | $5.29^{+0.12}_{-0.11}$ | -
Sh 2-257 | - | $7.19^{+0.18}_{-0.17}$ | $7.00^{+0.19}_{-0.20}$ | - | $4.83^{+0.12}_{-0.11}$ | - | $5.61^{+0.15}_{-0.14}$ | -
Sh 2-266 | - | $6.77^{+0.10}_{-0.09}$ | - | - | - | - | - | -
Sh 2-270 | - | $6.42^{+0.18}_{-0.17}$ | - | - | - | - | - | -
Sh 2-271 | - | $6.84\pm 0.09$ | $6.68\pm 0.09$ | - | $4.87\pm 0.10$ | - | $5.25^{+0.09}_{-0.11}$ | -
Sh 2-285 | - | $5.64^{+0.07}_{-0.08}$ | - | - | - | - | - | -
Sh 2-288 | - | $6.06\pm 0.07$ | $6.81^{+0.11}_{-0.12}$ | - | $4.87\pm 0.07$ | - | $5.96^{+0.07}_{-0.06}$ | -
Sh 2-297 | - | $6.65^{+0.06}_{-0.05}$ | - | - | $4.91\pm 0.11$ | - | $5.71^{+0.09}_{-0.10}$ | -
Sh 2-298 | $8.01^{+0.05}_{-0.04}$ | $6.56\pm 0.06$ | $6.61\pm 0.06$ | - | $4.93^{+0.10}_{-0.09}$ | - | $6.05\pm 0.03$ | $4.48^{+0.08}_{-0.10}$
Sh 2-311 | $7.33\pm 0.05$ | $6.34\pm 0.04$ | $6.81\pm 0.04$ | $4.41\pm 0.04$ | $4.95^{+0.05}_{-0.04}$ | - | $6.24\pm 0.04$ | -
IC 5146 | - | $7.00\pm 0.06$ | $6.52^{+0.07}_{-0.06}$ | - | - | - | - | -
NGC 2579 | $7.58\pm 0.06$ | $5.66\pm 0.07$ | $6.80^{+0.05}_{-0.06}$ | $3.67\pm 0.06$ | $5.05\pm 0.07$ | $2.97^{+0.07}_{-0.06}$ | $6.34\pm 0.04$ | $4.19^{+0.14}_{-0.12}$
NGC 3576 | $8.09\pm 0.04$ | $5.94\pm 0.05$ | $7.00^{+0.06}_{-0.05}$ | $3.93\pm 0.05$ | $5.04\pm 0.04$ | $3.45^{+0.06}_{-0.05}$ | $6.51^{+0.04}_{-0.03}$ | $4.71\pm 0.06$
NGC 3603 | $8.15\pm 0.06$ | $5.16^{+0.09}_{-0.08}$ | $6.81\pm 0.08$ | $3.26\pm 0.07$ | $4.86\pm 0.08$ | $4.07\pm 0.05$ | $6.36\pm 0.05$ | $5.06\pm 0.10$
M 8 | $7.47\pm 0.06$ | $6.22^{+0.05}_{-0.04}$ | $7.04\pm 0.04$ | $4.36\pm 0.03$ | $5.14^{+0.04}_{-0.03}$ | - | $6.41\pm 0.03$ | $4.24^{+0.12}_{-0.11}$
M 16 | $7.61^{+0.11}_{-0.10}$ | $6.60^{+0.04}_{-0.05}$ | $6.96\pm 0.05$ | $4.57^{+0.06}_{-0.05}$ | $5.13\pm 0.04$ | - | $6.45^{+0.04}_{-0.03}$ | $4.49^{+0.23}_{-0.21}$
M 17 | $8.19^{+0.06}_{-0.05}$ | $5.67\pm 0.06$ | $6.99^{+0.04}_{-0.05}$ | $3.73^{+0.10}_{-0.11}$ | $5.05\pm 0.06$ | $3.41^{+0.20}_{-0.22}$ | $6.50\pm 0.03$ | $4.57^{+0.16}_{-0.14}$
M 20 | $7.13\pm 0.11$ | $6.47^{+0.04}_{-0.03}$ | $6.96\pm 0.04$ | $4.60^{+0.04}_{-0.05}$ | $5.12\pm 0.05$ | - | $6.46\pm 0.05$ | $4.59^{+0.27}_{-0.24}$
M 42 | $8.17\pm 0.04$ | $5.57^{+0.18}_{-0.16}$ | $6.79\pm 0.06$ | $3.61\pm 0.08$ | $4.92^{+0.06}_{-0.05}$ | $3.90^{+0.07}_{-0.06}$ | $6.41^{+0.05}_{-0.04}$ | $4.95^{+0.07}_{-0.06}$
Table A6: Total abundances of C, N, O, Ne, S, Cl, and Ar derived assuming $t^{2}=0.0$ and $t^{2}=0.038\pm 0.004$ in units of 12+log(X/H). Region | C/H | N/H | O/H | Ne/H | S/H | Cl/H | Ar/H
---|---|---|---|---|---|---|---
| $t^{2}=0.0$ | $t^{2}>0.0$ | $t^{2}=0.0$ | $t^{2}>0.0$ | $t^{2}=0.0$ | $t^{2}>0.0$ | $t^{2}=0.0$ | $t^{2}>0.0$ | $t^{2}=0.0$ | $t^{2}>0.0$ | $t^{2}=0.0$ | $t^{2}>0.0$ | $t^{2}=0.0$ | $t^{2}>0.0$
Sh 2-29 | — | — | $7.87_{-0.05}^{+0.06}$ | $8.11_{-0.05}^{+0.07}$ | $8.65\pm 0.07$ | $8.89_{-0.06}^{+0.07}$ | — | — | $6.99_{-0.08}^{+0.09}$ | $7.29_{-0.09}^{+0.15}$ | $5.07_{-0.13}^{+0.15}$ | $5.17\pm 0.11$ | — | —
Sh 2-32 | — | — | $7.80_{-0.07}^{+0.08}$ | $8.03_{-0.07}^{+0.09}$ | $8.54\pm 0.11$ | $8.79\pm 0.11$ | — | — | — | — | — | — | — | —
Sh 2-47 | — | — | $7.92_{-0.09}^{+0.30}$ | $8.30_{-0.21}^{+0.59}$ | $8.75_{-0.13}^{+0.32}$ | $9.08_{-0.19}^{+0.54}$ | — | — | — | — | — | — | — | —
Sh 2-48 | — | — | $8.02_{-0.21}^{+0.34}$ | $8.47_{-0.27}^{+0.86}$ | $8.70_{-0.14}^{+0.38}$ | $9.17_{-0.22}^{+0.68}$ | — | — | — | — | — | — | — | —
Sh 2-53 | — | — | $8.11_{-0.14}^{+0.26}$ | $8.46_{-0.18}^{+0.35}$ | $8.78_{-0.20}^{+0.25}$ | $9.10_{-0.17}^{+0.37}$ | — | — | — | — | — | — | — | —
Sh 2-54 | — | — | $7.66_{-0.08}^{+0.16}$ | $7.87_{-0.10}^{+0.15}$ | $8.42_{-0.08}^{+0.13}$ | $8.65_{-0.10}^{+0.13}$ | — | — | $6.80\pm 0.18$ | $7.05\pm 0.20$ | $5.29_{-0.13}^{+0.09}$ | $5.49\pm 0.12$ | — | —
Sh 2-61 | — | — | $7.89\pm 0.12$ | $8.09\pm 0.12$ | $8.41\pm 0.17$ | $8.63\pm 0.15$ | — | — | — | — | — | — | — | —
Sh 2-82 | — | — | $7.81_{-0.18}^{+0.32}$ | $8.05\pm 0.31$ | $8.60_{-0.31}^{+0.49}$ | $8.88\pm 0.45$ | — | — | — | — | — | — | — | —
Sh 2-83 | — | — | $7.53_{-0.16}^{+0.15}$ | $7.68_{-0.14}^{+0.17}$ | $8.27_{-0.08}^{+0.09}$ | $8.45_{-0.10}^{+0.09}$ | $7.76\pm 0.13$ | $7.95\pm 0.11$ | $6.44\pm 0.11$ | $6.59\pm 0.12$ | $4.72_{-0.09}^{+0.10}$ | $4.81\pm 0.10$ | — | —
Sh 2-88 | — | — | $7.73_{-0.05}^{+0.06}$ | $7.95_{-0.06}^{+0.09}$ | $8.53_{-0.07}^{+0.08}$ | $8.74_{-0.06}^{+0.09}$ | — | — | $6.98_{-0.11}^{+0.09}$ | $7.24\pm 0.12$ | $5.14\pm 0.13$ | $5.27\pm 0.13$ | — | —
Sh 2-90 | — | — | $7.98\pm 0.07$ | $8.24_{-0.08}^{+0.09}$ | $8.51_{-0.06}^{+0.07}$ | $8.76_{-0.06}^{+0.08}$ | — | — | — | — | $5.27\pm 0.10$ | $5.41_{-0.09}^{+0.12}$ | — | —
Sh 2-93 | — | — | $7.88_{-0.07}^{+0.10}$ | $8.09\pm 0.10$ | $8.44\pm 0.13$ | $8.65\pm 0.12$ | — | — | $6.92_{-0.17}^{+0.13}$ | $7.19_{-0.18}^{+0.25}$ | — | — | — | —
Sh 2-100 | — | — | $7.63\pm 0.12$ | $7.90\pm 0.12$ | $8.49_{-0.03}^{+0.04}$ | $8.77_{-0.04}^{+0.06}$ | $7.94\pm 0.05$ | $8.24_{-0.06}^{+0.08}$ | $6.94\pm 0.10$ | $7.20\pm 0.11$ | $5.06_{-0.06}^{+0.08}$ | $5.24_{-0.07}^{+0.08}$ | $6.31\pm 0.05$ | $6.50_{-0.06}^{+0.05}$
Sh 2-127 | — | — | $7.36\pm 0.03$ | $7.50\pm 0.04$ | $8.26\pm 0.04$ | $8.40_{-0.04}^{+0.05}$ | — | — | $6.74_{-0.05}^{+0.06}$ | $6.92\pm 0.05$ | $5.00\pm 0.05$ | $5.10\pm 0.05$ | — | —
Sh 2-128 | — | — | $7.26\pm 0.06$ | $7.42_{-0.07}^{+0.08}$ | $8.21\pm 0.05$ | $8.38\pm 0.05$ | $7.44\pm 0.11$ | $7.61\pm 0.11$ | $6.52_{-0.05}^{+0.06}$ | $6.67\pm 0.06$ | $4.80_{-0.05}^{+0.04}$ | $4.90_{-0.04}^{+0.05}$ | — | —
Sh 2-132 | — | — | $7.49_{-0.07}^{+0.10}$ | $7.63_{-0.08}^{+0.15}$ | $8.37_{-0.15}^{+0.14}$ | $8.54_{-0.17}^{+0.14}$ | — | — | $6.75\pm 0.051$ | $6.90\pm 0.06$ | — | — | — | —
Sh 2-152 | — | — | $7.66\pm 0.03$ | $7.90_{-0.04}^{+0.05}$ | $8.53\pm 0.03$ | $8.75_{-0.02}^{+0.04}$ | — | — | $6.98_{-0.03}^{+0.04}$ | $7.26\pm 0.05$ | $5.24\pm 0.03$ | $5.37\pm 0.04$ | — | —
Sh 2-156 | — | — | $7.43\pm 0.10$ | $7.60_{-0.09}^{+0.11}$ | $8.34\pm 0.10$ | $8.51\pm 0.11$ | $7.14\pm 0.14$ | $7.38_{-0.17}^{+0.14}$ | $6.88\pm 0.05$ | $7.02\pm 0.05$ | $5.06\pm 0.10$ | $5.17_{-0.09}^{+0.11}$ | — | —
Sh 2-175 | — | — | $7.74_{-0.09}^{+0.11}$ | $7.99_{-0.11}^{+0.17}$ | $8.51\pm 0.15$ | $8.76\pm 0.16$ | — | — | — | — | — | — | — | —
Sh 2-209 | — | — | $7.23_{-0.22}^{+0.30}$ | $7.39_{-0.22}^{+0.38}$ | $8.08_{-0.12}^{+0.26}$ | $8.21\pm 0.16$ | — | — | — | $6.53_{-0.19}^{+0.32}$ | — | — | — | —
Sh 2-212 | — | — | $6.85\pm 0.22$ | $7.00_{-0.22}^{+0.26}$ | $8.34_{-0.13}^{+0.26}$ | $8.51_{-0.20}^{+0.28}$ | $7.50\pm 0.28$ | $7.70\pm 0.32$ | $6.57\pm 0.25$ | $6.74\pm 0.31$ | $5.05_{-0.19}^{+0.26}$ | $5.23\pm 0.22$ | — | —
Sh 2-219 | — | — | $7.45\pm 0.06$ | $7.60\pm 0.07$ | $8.28\pm 0.12$ | $8.44_{-0.09}^{+0.14}$ | — | — | — | — | $5.96_{-0.38}^{+0.23}$ | $5.98_{-0.33}^{+0.40}$ | — | —
Sh 2-235 | — | — | $7.56\pm 0.04$ | $7.73_{-0.05}^{+0.04}$ | $8.42\pm 0.07$ | $8.61_{-0.06}^{+0.08}$ | — | — | $7.11\pm 0.06$ | $7.27_{-0.06}^{+0.07}$ | $5.31\pm 0.11$ | $5.36_{-0.08}^{+0.10}$ | — | —
Sh 2-237 | — | — | $7.54\pm 0.05$ | $7.69_{-0.06}^{+0.05}$ | $8.34_{-0.09}^{+0.08}$ | $8.51\pm 0.09$ | — | — | — | — | $5.74\pm 0.25$ | $5.79_{-0.26}^{+0.22}$ | — | —
Sh 2-257 | — | — | $7.60\pm 0.06$ | $7.78\pm 0.07$ | $8.41\pm 0.10$ | $8.61\pm 0.10$ | — | — | $7.18\pm 0.11$ | $7.40\pm 0.11$ | $5.44\pm 0.17$ | $5.45\pm 0.17$ | — | —
Sh 2-266 | — | — | $7.52\pm 0.11$ | $7.69_{-0.11}^{+0.09}$ | $8.19\pm 0.16$ | $8.36\pm 0.18$ | — | — | — | — | — | — | — | —
Sh 2-270 | — | — | $7.26\pm 0.19$ | $7.40\pm 0.17$ | $8.09\pm 0.26$ | $8.23\pm 0.31$ | — | — | — | — | — | — | — | —
Sh 2-271 | — | — | $7.50\pm 0.05$ | $7.65\pm 0.05$ | $8.27\pm 0.09$ | $8.44_{-0.08}^{+0.09}$ | — | — | $6.89_{-0.05}^{+0.08}$ | $7.08_{-0.06}^{+0.07}$ | $6.04\pm 0.18$ | $6.05\pm 0.18$ | — | —
Sh 2-285 | — | — | $7.44_{-0.08}^{+0.06}$ | $7.59_{-0.06}^{+0.07}$ | $8.21\pm 0.11$ | $8.38\pm 0.10$ | — | — | — | — | — | — | — | —
Sh 2-288 | — | — | $7.40_{-0.09}^{+0.10}$ | $7.58_{-0.11}^{+0.09}$ | $8.34\pm 0.08$ | $8.51_{-0.06}^{+0.10}$ | — | — | $6.68_{-0.09}^{+0.08}$ | $6.87\pm 0.10$ | $4.92\pm 0.07$ | $5.03\pm 0.09$ | — | —
Sh 2-297 | — | — | $7.71\pm 0.05$ | $7.90_{-0.05}^{+0.06}$ | $8.48_{-0.07}^{+0.09}$ | $8.68_{-0.07}^{+0.08}$ | — | — | — | — | $5.45_{-0.15}^{+0.14}$ | $5.47\pm 0.15$ | — | —
Sh 2-298 | — | — | $7.68\pm 0.08$ | $7.80_{-0.09}^{+0.08}$ | $8.43_{-0.05}^{+0.06}$ | $8.55_{-0.04}^{+0.06}$ | $8.27_{-0.07}^{+0.08}$ | $8.40_{-0.08}^{+0.09}$ | $6.80\pm 0.04$ | $6.89_{-0.04}^{+0.03}$ | $4.94_{-0.08}^{+0.10}$ | $5.02_{-0.09}^{+0.10}$ | $6.00\pm 0.03$ | $6.09_{-0.03}^{+0.04}$
Sh 2-311 | $8.21\pm 0.10$ | $8.26\pm 0.09$ | $7.50\pm 0.05$ | $7.68\pm 0.06$ | $8.41_{-0.04}^{+0.05}$ | $8.58_{-0.04}^{+0.05}$ | $7.70\pm 0.05$ | $7.93_{-0.07}^{+0.05}$ | $6.78_{-0.02}^{+0.03}$ | $6.93\pm 0.03$ | $4.93\pm 0.03$ | $5.11\pm 0.05$ | — | —
IC 5146 | — | — | $7.81\pm 0.05$ | $8.04_{-0.05}^{+0.06}$ | $8.57\pm 0.09$ | $8.81_{-0.09}^{+0.10}$ | — | — | $6.89\pm 0.04$ | $7.13_{-0.04}^{+0.05}$ | — | — | — | —
NGC 2579 | $8.18_{-0.06}^{+0.05}$ | $8.20_{-0.06}^{+0.05}$ | $7.36\pm 0.09$ | $7.55\pm 0.10$ | $8.41_{-0.05}^{+0.06}$ | $8.61_{-0.04}^{+0.06}$ | $7.60_{-0.08}^{+0.09}$ | $7.79_{-0.07}^{+0.10}$ | $6.71\pm 0.05$ | $6.85\pm 0.05$ | $4.94\pm 0.07$ | $5.07_{-0.07}^{+0.06}$ | $6.21\pm 0.03$ | $6.36_{-0.03}^{+0.04}$
NGC 3576 | $8.48\pm 0.03$ | $8.50_{-0.03}^{+0.04}$ | $7.62\pm 0.06$ | $7.86_{-0.08}^{+0.09}$ | $8.54\pm 0.02$ | $8.79_{-0.03}^{+0.04}$ | $8.01\pm 0.03$ | $8.26_{-0.04}^{+0.06}$ | $6.90_{-0.04}^{+0.05}$ | $7.07_{-0.05}^{+0.06}$ | $4.92_{-0.04}^{+0.03}$ | $5.08_{-0.04}^{+0.05}$ | $6.35\pm 0.03$ | $6.53\pm 0.04$
NGC 3603 | $8.53_{-0.07}^{+0.08}$ | $8.54\pm 0.07$ | $7.69\pm 0.15$ | $7.91\pm 0.14$ | $8.46_{-0.03}^{+0.04}$ | $8.70_{-0.04}^{+0.05}$ | $7.93_{-0.05}^{+0.06}$ | $8.18_{-0.06}^{+0.07}$ | $6.85_{-0.09}^{+0.08}$ | $7.01_{-0.09}^{+0.08}$ | $4.82\pm 0.07$ | $4.93_{-0.06}^{+0.07}$ | $6.24_{-0.04}^{+0.06}$ | $6.37\pm 0.05$
M 8 | $8.47\pm 0.06$ | $8.53\pm 0.06$ | $7.73\pm 0.04$ | $7.95\pm 0.05$ | $8.48\pm 0.04$ | $8.69_{-0.04}^{+0.05}$ | $7.79_{-0.04}^{+0.05}$ | $8.07_{-0.07}^{+0.06}$ | $6.92\pm 0.03$ | $7.10_{-0.03}^{+0.04}$ | $5.04\pm 0.02$ | $5.30\pm 0.05$ | $6.28\pm 0.02$ | $6.48\pm 0.03$
M 16 | $8.53_{-0.10}^{+0.09}$ | $8.62_{-0.10}^{+0.08}$ | $7.89_{-0.05}^{+0.06}$ | $8.12\pm 0.07$ | $8.57_{-0.04}^{+0.05}$ | $8.79_{-0.05}^{+0.06}$ | $7.91_{-0.09}^{+0.11}$ | $8.23\pm 0.13$ | $6.93\pm 0.03$ | $7.11_{-0.03}^{+0.04}$ | $5.06\pm 0.04$ | $5.29\pm 0.05$ | $6.30\pm 0.02$ | $6.52\pm 0.03$
M 17 | $8.79_{-0.03}^{+0.04}$ | $8.80_{-0.04}^{+0.03}$ | $7.67_{-0.07}^{+0.09}$ | $7.95_{-0.09}^{+0.10}$ | $8.54\pm 0.04$ | $8.83\pm 0.05$ | $7.96\pm 0.04$ | $8.27_{-0.05}^{+0.06}$ | $6.90_{-0.04}^{+0.05}$ | $7.10\pm 0.05$ | $4.93_{-0.05}^{+0.06}$ | $5.08_{-0.05}^{+0.06}$ | $6.32\pm 0.02$ | $6.51\pm 0.03$
M 20 | $8.44\pm 0.20$ | $8.52\pm 0.19$ | $7.69\pm 0.05$ | $7.90_{-0.05}^{+0.06}$ | $8.55_{-0.05}^{+0.04}$ | $8.76\pm 0.05$ | $7.50\pm 0.11$ | $7.82\pm 0.10$ | $6.88_{-0.02}^{+0.03}$ | $7.07_{-0.03}^{+0.04}$ | $5.06_{-0.03}^{+0.04}$ | $5.23_{-0.03}^{+0.04}$ | $6.31\pm 0.04$ | $6.53\pm 0.05$
M 42 | $8.46_{-0.05}^{+0.04}$ | $8.47_{-0.05}^{+0.03}$ | $7.77\pm 0.13$ | $8.02\pm 0.14$ | $8.52\pm 0.02$ | $8.78_{-0.03}^{+0.05}$ | $7.96\pm 0.03$ | $8.25_{-0.04}^{+0.06}$ | $6.74_{-0.06}^{+0.07}$ | $6.92_{-0.06}^{+0.08}$ | $4.86_{-0.04}^{+0.05}$ | $4.98\pm 0.05$ | $6.27\pm 0.04$ | $6.43\pm 0.04$