Louise Breuval

Pierre Kervella

Piotr Wielgórski

Wolfgang Gieren

Dariusz Graczyk

Boris Trahin

Grzegorz Pietrzyński

Frédéric Arenou

Behnam Javanmardi

Bartlomiej Zgirski

The influence of metallicity on the Leavitt Law from geometrical distances of Milky Way and Magellanic Clouds Cepheids

Keywords: parallaxes -stars, distances -stars, variables, Cepheids -distance scale -metallicity

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The Cepheid Period-Luminosity (PL) relation, discovered by Henrietta Leavitt (Leavitt & Pickering 1912) about a century ago, is an essential tool for measuring astronomical distances since it represents the first rung of the extragalactic distance ladder. This law is used to measure distances to type Ia supernovae (SNe Ia) host galaxies, and thus plays a key role in the determination of the Hubble constant (H 0 ). This parameter currently exhibits a tension of at least ∼ 4σ between its measurement in the early Universe by Planck Collaboration et al.

louise.breuval@obspm.fr (2020) assuming a Λ-CDM cosmology and the local estimate based on Cepheid distances (Riess et al. 2021). The precise calibration of the PL relation is therefore of paramount importance to reach a 1% determination of the Hubble constant.

While the slope (α) and intercept (β) of the Leavitt law are generally consistent between various studies, the value and even the sign of the metallicity term (γ, defined as M = α log P + β + γ[Fe/H]) are still debated and constitute 0.5% of the error budget of H 0 (Riess et al. 2016). Some empirical studies report a metallicity dependence consistent with γ ∼ 0 mag/dex: Udalski et al. (2001) concludes with a null-effect from the study of a metal-poor galaxy in optical bands, Storm et al. (2011a) finds a null effect in all bands except in W V I and Wielgórski et al. (2017) derive a gamma value consistent with zero in optical and NIR bands. Still, a large majority of the analysis investigating the metallicity effect derived a negative sign, with values ranging between -0.2 and -0.5 mag/dex (Freedman & Madore 1990;Macri et al. 2006;Saha et al. 2006;Gieren et al. 2018;Groenewegen 2018). This trend would indicate that metal-rich Cepheids are brighter than metal-poor ones. However, the study by Romaniello et al. (2008) yielded a metallicity effect of the opposite sign, confirming the theoretical predictions (Caputo et al. 2000;Bono et al. 2008;Fiorentino et al. 2013).

In this paper, we aim at determining the effect of metallicity on the PL relation by combining samples of Cepheids in the Milky Way (MW) and in the Magellanic Clouds (MCs), taking advantage of the large range of metallicity covered by the Cepheids in these 3 galaxies (from +0.08 dex to -0.75 dex). Most of the Cepheids located in distant galaxies hosting SNIa have metallicities within this range, therefore our results are directly applicable to extragalactic studies of the distance scale (e.g. Javanmardi et al. 2021).

Recently, Pietrzyński et al. (2019) and Graczyk et al. (2020) measured the most precise distances to date for the Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC) respectively, based on enhanced samples of late-type detached eclipsing binaries (DEBs). These distances allow us to obtain a precise calibration of the PL relation in the LMC and SMC. For Milky Way (MW) Cepheids, we use the early third Gaia Data Release (EDR3) which recently provided parallaxes of unprecedented precision for hundreds of galactic Cepheids.

In Sect. 2, we present our samples of Cepheids in the three galaxies and in Sect. 3 we provide the distances we adopted for each sample. Then in Sect. 4 we estimate the metallicity effect by fitting the Period-Luminosity-Metallicity (PLZ) relation in the three galaxies. We discuss the results in Sect. 5.

SAMPLES OF CEPHEIDS

MW Cepheids

We gather a sample of Milky Way Cepheids for which well-covered light curves are available. In the NIR J, H and K bands, we combine the catalogs by Welch et al. (1984), Laney & Stobie (1992), Barnes et al. (1997) and Monson & Pierce (2011). The data from these four studies are found to be in close agreement, with residuals of 0.013, 0.010 and 0.002 mag in J, H and K respectively (Monson & Pierce 2011). We adopt these values as photometric zero-point uncertainties for the NIR photometry. Additional NIR data were also found in Feast et al. (2008), we consider that including this source of data does not impact the homogeneity and the dispersion of the data since it only affects four stars of the sample. In the optical V and I bands, we use the catalog from Berdnikov (2008) that provides photometry in the Johnson-Cousins system for a large number of Cepheids. Since it is a compilation of data from various catalogs by the same author, we adopt a photometric zero-point uncertainty of 0.010 mag.

For each star and in each filter, we phase the data at the date of maximum luminosity and we obtain intensity-averaged mean apparent magnitudes by performing light curve fitting using Fourier series. Depending on the properties of the different light curves (such as the presence of bumps, steep variations, or to prevent the introduction of unphysical oscillations when the data are too dispersed or not dense enough), we adapt the number of Fourier modes, and thus of free parameters, in order to obtain a satisfactory representation of the light curve. A Fourier decomposition of order three is generally sufficient for an usual Cepheid light curve such as δ Cep, and is up to order six for a more complex star such as RS Pup. We derived the statistical uncertainties on the mean magnitudes from the scatter of each light curve. In some few cases, a very large number of data points are available (> 300) and result in unrealistic small errors: in these cases we adopt a minimum error of 0.006 mag.

For long period Cepheids, large phase shifts may degrade the quality of the fit, the photometry being spread over four decades. Therefore, period changes were taken into account for the phasing of long period stars such as SV Vul, GY Sge or RS Pup (Kervella et al. 2017). We adopted a polynomial model of up to degree five for the pulsation period.

We carefully analyse the light curves: we exclude Cepheids for which less than 8 data-points are available (MW Cepheids have on average 35 data points in NIR and 160 in optical) and Cepheids that have poor quality photometry or insufficient phase coverage. Finally, we convert all the NIR data in the 2MASS system using the transformations from Monson & Pierce (2011). The systematics related to these transformations are negligible. Examples of a well covered light curve and of a poor-quality light curve are provided in Fig. 8 and 9 in Appendix.

We select Cepheids pulsating in the fundamental mode according to the reclassification by Ripepi et al. (2019). For stars that were not available in this catalog, we adopted by order of priority the pulsation modes from Groenewegen (2018), from the Variable Star indeX (VSX, Watson et al. 2006) and from Luck (2018). We adopt reddening values from Fernie et al. (1995) with a 0.94 scaling factor as suggested by Groenewegen (2018), and from Acharova et al. (2012) if not available in the latter. We adopt an uncertainty of 0.05 if it is not provided.

For MW Cepheids, we search for individual metallicities in Genovali et al. (2015). This catalog provides mean abundances based on high resolution spectra for 75 Cepheids. For stars that are not available in this catalog, we adopt the values from Genovali et al. (2014): they provide homogeneous Cepheid metallicities from their group and compiled from the literature, rescaled to their solar abundance. The individual metallicities are represented in Fig. 1 by colored points, they range from -0.33 dex to +0.55 dex. The gradient of metallicity in the MW is particularly visible, with metalrich Cepheids located closer to the galactic center than metal-poor ones. These individual metallicities have a weighted mean value of 0.083 ± 0.019 dex with a scatter of 0.14 dex. In the following, we adopt this weighted mean value for all MW Cepheids for consistency and homogenity with the LMC and SMC samples that only have a mean metallicity, but also because the current precision of the individual metallicities is not sufficient for a thorough calibration of the metallicity effect.

The Cepheids of our MW sample are represented in Fig. 1 and their main parameters are listed in Table 4 and Table 5 in Appendix.

LMC Cepheids

We build a sample of LMC Cepheids by combining the OGLE-IV photometry in V and I bands (Soszyński et al. 2015) with the multi-epoch observations from the LMC Near-Infrared Synoptic Survey by Macri et al. (2015) taken with the CPAPIR camera on the 1.5m CTIO telescope. We update their NIR mean magnitudes to bring them into better agreement with the 2MASS system using the following relations (L. Macri, priv. comm.). These were derived by matching ∼ 34 000 stars in common between their Table A1 and the 2MASS Point Source Catalog [START_REF] Cutri | 2MASS All Sky Catalog of point sources[END_REF], with 12 < H < 13.5, K > 11.5 and -0.5 < J -K < 1.4 mag:

J 2MASS = J M15 -0.0167 + 0.0205 (J M15 -K M15 -0.4) +0.0101 (J M15 -K M15 -0.4) 2 H 2MASS = H M15 + 0.0116 -0.0054 (J M15 -K M15 -0.4) -0.0189 (J M15 -K M15 -0.4) 2 K 2MASS = K M15 + 0.0162 + 0.0227 (J M15 -K M15 -0.4) -0.0595 (J M15 -K M15 -0.4) 2
We adopt a photometric zero-point uncertainty of 0.02 mag in all bands. Since some Cepheids exhibit large brightness variations during a pulsation cycle, we consider that single-epoch photometry is not precise enough to derive reliable mean magnitudes, therefore we discarded the mean magnitudes derived by Inno et al. (2016) from template fitting on 2MASS single-point data and IRSF measurements.

We perform a quality check on this initial sample: we reject stars with magnitude uncertainties larger than 1% and with less than 5 data points (LMC Cepheids have on average 43 data points in NIR and 147 in optical), and we only consider fundamental mode Cepheids. We reject Cepheids located outside a radius of 3 degrees around the LMC center in order to avoid outliers such as stars that do not belong to the LMC or that are strongly affected by its geometrical effects (see Sect. LMC Cepheids, we adopt the mean metallicity used by Gieren et al. (2018), that compiles several estimates from various studies: [Fe/H] LMC = -0.34 ± 0.06 dex.

The uncertainties take into account the homogenization of the different measurements.

SMC Cepheids

We assemble a sample of SMC Cepheids by taking the mean magnitudes from the VISTA survey of the Magellanic Clouds (VMC) (Ripepi et al. 2016) crossmatched with OGLE IV photometry by Soszyński et al. (2015). Unfortunately, we do not have H-band photometry for SMC Cepheids because we rejected data from single epoch photometry and template fitting. Results in the H band are therefore derived from the combination of MW and LMC Cepheids only. Magnitudes in the VISTA system were converted into the 2MASS system using the equations from Ripepi et al. (2016):

J = J VMC + 0.070 (J VMC -K VMC ) K = K VMC -0.011 (J VMC -K VMC )
We perform an additional correction (L. Macri, priv. comm.) derived by matching ∼ 7000 stars in common between the VMC DR4 and the 2MASS Point Source Catalog, with J > 12.25, K > 11.5 and -0.5 < J -K < 1.4 mag:

J 2MASS = J -0.0087 -0.0010 (J -K -0.4) K 2MASS = K + 0.0011 -0.0087 (J -K -0.4)
We adopt a photometric zero-point uncertainty of 0.02 mag for all bands. As we did for the LMC sample, we also reject SMC Cepheids with magnitude uncertainties larger than 1%, with less than 5 data points (SMC Cepheids have on average 17 data points in NIR and 46 in optical) and we only keep Cepheids pulsating in the fundamental mode. As for the LMC sample, we adopt reddening values from the Górski et al. (2020) reddening map.

While the LMC has a rather simple geometry, the SMC is very elongated along the line of sight: we select Cepheids located in a region of 0.6 deg around the SMC center, which covers an area of 1.3 kpc width. Since the SMC distance is derived from detached eclipsing binaries (DEBs), this selection ensures that the Cepheids are located in the same region as these DEBs. The final SMC sample has 284 stars in the V band and 295 stars in K S , it is listed in Table 7 in Appendix and provided as supplementary material. A map of our final sample of SMC Cepheids is represented in Fig. 3.

For SMC Cepheids, we adopt the mean metallicity used by Gieren et al. (2018), that compiles several estimates from various studies: [Fe/H] SMC = -0.75 ± 0.05 dex. Similar to the LMC value, the uncertainty takes into account the homogenization of the different measurements.

DISTANCES

In order to calibrate the Leavitt law, one needs to derive the absolute magnitude of each Cepheid from its apparent luminosity and from its distance.

Distances to Milky Way Cepheids

Recently, the early third Gaia Data Release provided new parallaxes for Milky Way Cepheids (Gaia Collaboration 2020). We perform a first quality check of Gaia EDR3 parallaxes based on the Renormalised Unit Weight Error (RUWE) provided in the catalog. This parameter reflects the quality of the parallax of a star compared to other stars of the same color and brightness. Its value is expected to be close to 1 for wellbehaved sources (Lindegren et al. 2020a). In particular, the RUWE is sensitive to the photocentric motion of unresolved objects, therefore it can be used to detect possible astrometric binaries. We discard the Cepheids of our sample that have a RUWE > 1.4: this selection corresponds to approximately 13 % of our MW sample and removes the stars that are possibly affected by saturation or contamination by a bright neighbour companion. In particular, all the outliers noticed by eye on the PL relation are affected by a large RUWE, therefore the quality check based on this parameter appears to be relevant for our purpose. Riess et al. (2021) use a different indicator: they identify stars with a goodness of fit (GOF) larger than 12.5 as having a compromised parallax. We find the GOF and the RUWE selections to have a very similar effect on our sample: adopting this GOF criteria for the quality check instead of the RUWE leads to rejecting exactly the same stars, except T Mon, V0496 Aql and VW Pup, that have a RUWE of 1.72,1.56 and 1.41,and a GOF of 12.11,10.90 and 11.36 respectively. The RUWE criterion seems slightly more selective than the GOF limit adopted by Riess et al. (2021). Adopting a threshold of RUWE < 1.4 corresponds to a limit GOF of 10.

One method to check if a Cepheid is an astrometric binary is to look for proper motion anomalies between the observations by Hipparcos and Gaia. Using the approach described in Kervella et al. (2019), we find 20 Cepheids with a high proper motion anomaly signal. However, none of them were identified by their high RUWE or GOF and they do not appear as outliers, therefore we do not exclude them.

Cepheids are variable stars and therefore their brightness and colour can change significantly during a pulsation cycle. This effect was not taken into account in the processing of Gaia DR2 astrometry and resulted in additional systematics, noise and dispersion for variable stars parallaxes (Breuval et al. 2020). The correction for this chromatic effect on Cepheid parallaxes is still absent from Gaia EDR3 (Lindegren et al. 2020a). However, the number of observations obtained for each star increased consequently between Gaia DR2 (∼22 months) and Gaia EDR3 (∼34 months). We assume in this paper that the noise induced by this effect is negligible for Gaia EDR3 parallaxes.

For each Cepheid we correct for the parallax zeropoint (ZP) by using the Python tool1 described in Lindegren et al. (2020a). This ZP correction takes into account the ecliptic latitude, magnitude and colour of each star. Our MW Cepheids cover a range of magnitudes from G = 3 to G = 12 mag. For our sample of MW Cepheids, we find the ZP to vary between -4 and -54 µas with a median value of -27 µas (σ = 10 µas), which is very similar to the median parallax offset derived by Riess et al. (2021). Following Lindegren et al. (2020a) who recommend to include an uncertainty of a few micro arcseconds in the ZP, we adopt a systematic error of 5 µas on this quantity. Considering our sample of Cepheids, this error is equivalent to an average systematic uncertainty of 0.020 mag in distance modulus. In Sect. 5.1, we discuss the influence of adopting this individual ZP correction compared with the uniform ZP of -17 µas derived from quasars.

We find 13 Cepheids to fall in the range between G = 10.8 and G = 11.2 mag, where a transition of window classes occurs (Fig. 1 in Lindegren et al. 2020b). In this particular range, the value of the parallax zero-point can possibly be affected so we quadratically add 10 µas to the parallax uncertainty.

Finally, we increase all Gaia EDR3 parallax uncertainties by 10%, following Riess et al. (2021) to account for potential excess uncertainty. This correction has significantly reduced since Gaia DR2, where it was recommended to increase parallax uncertainties by 30%.

Distances to LMC Cepheids

Recently, Pietrzyński et al. (2019) estimated the distance to the LMC with a 1% precision based on detached-eclipsing binaries (DEBs): d LMC = 49.59 ± 0.09 (stat.) ± 0.54 (syst.) kpc. This method for measuring distances is independent from Cepheids and relies on surface-brightness relations, established by precise interferometric measurements. We use this value as initial distance to our Cepheids and we add a corrective term depending on the position of each Cepheid in the LMC, assuming the disc geometry derived by OGLE from Cepheids by Jacyszyn-Dobrzeniecka et al. (2016). First we compute the cartesian coordinates (x i , y i , z i ) of each Cepheid from their equatorial coordinates (α i , δ i ) using the transformations:

         x i = -d LMC cos δ i sin(α i -α LMC ) y i = d LMC sin δ i cos δ LMC -cos δ i sin δ LMC cos(α i -α LMC ) z i = c 1 x i + c 2 y i + d LMC
where (α LMC , δ LMC ) = (80.05, -69.30) deg are the coordinates of the LMC center and the coefficients (c 1 , c 2 ) = (0.395 ± 0.014, -0.215 ± 0.013) are from Jacyszyn-Dobrzeniecka et al. (2016). The corrected distance of each LMC Cepheid is:

d i = x 2 i + y 2 i + z 2 i
The distances of each LMC Cepheid derived with this correction are located in a range of ± 1.5 kpc around the mean LMC distance from Pietrzyński et al. (2019). They are represented by the colors in Fig. 2.

Distances to SMC Cepheids

The distance to the SMC was recently measured by Graczyk et al. (2020) with a precision of 1.5 % using the same method as used in Pietrzyński et al. (2019) for the LMC: from a sample of 15 DEBs, a distance of d SMC = 62.44 ± 0.47 (stat.) ± 0.81 (syst.) kpc is derived. However, the SMC has a large extension along the line of sight (Subramanian & Subramaniam 2012;Jacyszyn-Dobrzeniecka et al. 2016;Ripepi et al. 2017), which makes the distance to its core region particularly difficult to measure, contrary to the LMC that has a rather simple geometry. In this section, we take into account the SMC elongated shape in order to derive corrected distances to each of its Cepheids. For each SMC Cepheid of coordinates (α i , δ i ), we compute the cartesian coordinates (x i , y i ) such that:

     x i = -d SMC cos δ i sin(α i -α SMC ) y i = d SMC sin δ i cos δ SMC -cos δ i sin δ SMC cos(α i -α SMC )
where (α SMC , δ SMC ) = (12.54, -73.11) deg (Ripepi et al. 2017). Then we used the equations corresponding to the blue lines in Fig. 4 of Graczyk et al. (2020):

d i (x) = (3.086 ± 0.066) x i + d SMC d i (y) = (-3.248 ± 0.118) y i + d SMC
We adopt the mean value of d i (x) and d i (y) as the final distance of each Cepheid. The elongated shape of the SMC is highlighted by the dispersion of the derived distances between +5 kpc and -6 kpc around the mean value d SMC , which represents almost 10% of the mean value. The distances of our sample of SMC Cepheids are represented on the map in Fig. 3. A discussion about the elongated shape of the SMC and its impact on our results is provided in Sect. 5.2.

THE METALLICITY EFFECT FROM MILKY WAY AND MAGELLANIC CLOUD CEPHEIDS

In this section, we aim at estimating the metallicity term γ of the Leavitt law. In Sect. 4.1, we start by fitting the α and β coefficients of the PL relation in each of the three galaxies, without considering the metallicity term. In Sect. 4.2, we include the metallicity for each galaxy and derive the third term of the PLZ relation by combining the three galaxies.

The Period-Luminosity relation

We adopt the Cepheid samples described in Sect. 2. In a first place, we correct apparent magnitudes for the extinction by adopting the reddening law from Cardelli et al. (1989) andO'Donnell (1994) assuming R V = 3.135 which yields A λ = R λ E(B -V ) with R I = 1.894, R J = 0.892, R H = 0.553 and R K S = 0.363. We also derive optical and NIR Wesenheit indices (Madore 1982) as defined by W V I = I -1.526 (V -I) and W JK = K S -0.686 (J -K S ). Wesenheit magnitudes are particularly convenient for calibrating the PL relation since they are independent of reddening.

We account for the width of the instability strip by adding quadratically to the photometry uncertainties the intrinsic scatter in each band: this quantity is obtained by subtracting quadratically the measurement errors (photometric inhomogeneities, differential extinction, geometrical effects, phase corrections, etc) from the scatter of the PL relation: we adopt a width of the instability strip of 0.07 mag in NIR bands (J, H, K S and W JK ) from Persson et al. (2004), 0.15 mag in V and 0.09 mag in I from Macri et al. (2006) and finally 0.08 mag in W V I from Madore et al. (2017). We derive the absolute magnitude M λ of each Cepheid from their distance d (in kpc) and dereddened apparent magnitude

m λ : M λ = m λ -5 log d -10 (1)
In the Milky Way, the distance is obtained at the first order by taking the inverse of the parallax. In order to avoid biases due to this inversion, we adopt the approach introduced by Feast & Catchpole (1997) and [START_REF] Arenou | Harmonizing Cosmic Distance Scales in a Post-HIPPARCOS Era[END_REF], consisting in fitting the Astrometric Based Luminosity (ABL) function instead of absolute magnitudes: ABL = π (mas) 10 0.2m λ -2 = 10 M λ /5

(2)

where: We adopt a pivot period of log P 0 = 0.7 which represents the median period of our Cepheid sample. This approach ensures minimum correlations between the fitted coefficients. We perform a 3σ clipping procedure on the PL relation to remove possible outliers.

M λ = α λ (log P -log P 0 ) + β λ (3)
A non-linearity in the SMC PL relation was highlighted at the short-periods end (log P < 0.4) (EROS Collaboration et al. 1999).

For LMC and SMC Cepheids, Chown et al. (2021) detect a break in the PL relation at log P = 0.29 and also at very long periods (log P = 1.72). Cepheids beyond these limits are found to deviate from the global PL fit and can affect both the slope and the zero-point. Additionally, the shortperiod edge of the PL relation is potentially affected by first-overtone contamination. In the following, we exclude all Cepheids with periods shorter than 2.5 days (log P = 0.4) and longer than 52 days (log P = 1.72). Finally, we include the systematics on the LMC and SMC distance moduli (respectively 0.026 mag and 0.032 mag) and the photometric zero-points provided in Sect. 2 on the intercept error. We use the curve fit function from the scipy Python library in a Monte Carlo algorithm to derive the PL coefficients and the 16th and 84th percentiles of the distribution to derive the uncertainties. The PL relations derived for each galaxy are provided in Table 1, where both the slope and intercept are fitted.

In each band, the intercept increases with decreasing metallicity, i.e. it becomes less negative from the MW to the LMC and in turn to the SMC. In the NIR, the intercept changes by ∼ 0.18 mag between the MW and the SMC, possibly indicating a strong dependence with metallicity. We note that our K S band calibration in the MW is in good agreement with the result by Breuval et al. (2020) based on Gaia DR2 parallaxes. The fit of the PL relation in the K S band performed in each of the three galaxies is represented in Fig. 4.

The Period-Luminosity-Metallicity relation

In this section, we now calibrate the dependence of the PL intercept β with metallicity. First, we fit the PL relation of the form M = α(log P -0.7) + β in each of the three galaxies separately with a common slope fixed to the LMC value. As in previous section, the systematics due to the LMC and SMC distance and to the photometric zero-point are included in quadrature to the intercept random error. The intercept β contains the metallicity term such that:

β = γ [Fe/H] + δ (4)
In Fig. 5 are represented the intercepts of the PL relations in the MW, LMC and SMC as a function of metallicity. We fit Eq. 4 with a Monte Carlo algorithm to derive the γ and δ coefficients, and we adopt the 16th and 84th percentiles of the distribution to derive the random errors. A histogram representing the distribution of the γ values obtained with the Monte Carlo algorithm is represented in Fig. 6. The results of the fit are listed in Table 2. In the NIR, we report a strong metallicity effect of -0.208 ± 0.051 mag/dex in J, -0.152 ± 0.092 mag/dex in H and -0.221 ± 0.050 mag/dex in K S . The NIR Wesenheit index W JK shows a similar dependence with -0.214 ± 0.057 mag/dex. These results agree by 1 σ with Gieren et al. (2018), who used the Infrared Surface Brightness Technique (Fouqué & Gieren 1997;Storm et al. 2011b) to derive the distances to the Cepheids in their MW, LMC and SMC samples, an approach different and independent from the one used in the present study. In the NIR Wesenheit index W H , Riess et al. ( 2019) find an effect of -0.170 ± 0.060 mag/dex, which is also close to our results in the NIR. In optical bands, we derive a weaker effect than in the NIR with -0.048±0.051 mag/dex in V and of -0.138±0.051 mag/dex in I. These values also agree at 1 σ with Gieren et al. (2018), and the value in V is also consistent at 1 σ with the differential study of LMC and SMC PL relations by Wielgórski et al. (2017). On average, our results are located between the values by Wielgórski et al. (2017), consistent with a null metallicity effect, and the work by Gieren et al. (2018) that derive a strong negative effect. In the H band, we derive a metallicity effect weaker than in other NIR bands, likely because it is derived from the MW and LMC samples only (due to the lack of H-band photometry for SMC Cepheids). We conclude with the general trend being that the sensitivity to metallicity increases in absolute sense and becomes more negative from optical to NIR wavelengths. This trend is particularly visible in Fig. 7.

We note that the PL slope was fixed to the LMC value because this sample contains significantly more stars than the two other ones. However, if the slope is fixed to the value found in the Milky Way or in the SMC, the intercepts agree by 0.2% in NIR and by 1.4% in optical. Similarly, the γ values agree at 0.2 σ and 0.8 σ in NIR and optical respectively.

DISCUSSION

The metallicity term of the PL relation can be sensitive to many different effects. In this section we study the stability γ after varying some parameters.

Influence of Gaia EDR3 parallax zero-point

In Sect. 3.1, we corrected each Gaia EDR3 parallax for their individual zero-point by using the Python tool described in Lindegren et al. (2020a). However, in Lindegren et al. (2020b), a uniform parallax zero-point of -17 µas is derived from quasars. The results of the PLZ fit obtained after adopting this uniform zero-point are provided in the second part of Table 3 in appendix. They are consistent at the 1σ level with the values derived using the individual zero-point, although it gives a slightly more negative metallicity effect in each band. For example, in K S we obtain γ = -0.271 ± 0.051 mag/dex compared with γ = -0.221 ± 0.050 mag/dex with individual zero-points. This effect can be explained by the individual zero-points being on average more negative than -17 µas for our sample of MW Cepheids.

We also investigate whether the individual zero-point correction by Lindegren et al. (2020a) is adapted to the most distant Cepheids: we remove from our sample the Cepheids with a parallax smaller than 0.

Influence of the SMC sample

As mentionned in Sect. 2.3 and 3.3, the distance to the core region of the SMC is particularly difficult to measure. From their sample of DEBs, Graczyk et al. (2020) derive an uncertainty of about 2% for the distance to the SMC core region. These DEB systems are unevenly distributed in the central region of the galaxy and their individual distances show a large dispersion around the mean value, ranging from 57 kpc to 67 kpc (see their Fig. 3), which corresponds to ∼ 16% of the SMC distance. In order to avoid including Cepheids located too far away from the SMC center, we restricted our sample to a region of radius 0.6 degree around the SMC center. With a smaller radius, the contribution of the SMC sample in the PLZ fit becomes smaller than the MW contribution, therefore we consider that the number of retained SMC Cepheids is insufficient. On the other hand, if we assume a radius larger than 0.6 deg around the SMC center, the number of outlier stars increases and the distance of some Cepheids may not correspond to the distance of the SMC core region. In order to test the validity of our hypothesis, we perform the same PLZ fit with a radius of 0.5 and 0.7 deg around the SMC center and report the coefficients in Table 3.

After extending the SMC sample to a radius of 0.7 degree around the galaxy centrer, we find γ values in very good agreement (better than 1σ) with the values derived in the initial conditions. When the radius is reduced to 0.5 degree, the metallicity effect still agrees at 1σ with the initial conditions in all bands. Considering a smaller region around the SMC center results in a slightly stronger (i.e. more negative) metallicity effect. These results highlight the sensitivity of the metallicity effect with respect to the adopted SMC sample, and in particular to the spatial distribution of the Cepheids considered. Moreover, it emphasizes the necessity to correct each Cepheid distance according to their position in the SMC plane, as we did in Sect. 3.3.

We consider the variation of γ within a region of 0.5 • < R < 0.7 • around the SMC center as an additional source of systematic uncertainties: this source of error is at the level of 0.02 mag/dex in optical bands and of 0.01 mag/dex in NIR (see Table 3). We adopt the same additional source of uncertainty for the intercept δ, although the latter coefficient is particularly stable when the radius around the SMC center is changed. These systematics are included in the results presented in Table 2.

CONCLUSIONS

We build large samples of Cepheids in the Milky Way and in the Magellanic Clouds and make use of the most recent and precise distances available to estimate the metallicity effect on the Cepheid PL relation. In the K S band we derive an effect of γ = -0.221±0.051 mag/dex, in agreement with the value found by Gieren et al. (2018) but more precise. In the V band we derive a weaker effect of γ = -0.048 ± 0.055 mag/dex, which is consistent with both Wielgórski et al. (2017) and Gieren et al. (2018) within the error bars. We conclude with a nonzero dependence of Cepheid magnitude with metallicity and we confirm its negative sign: metal-rich Cepheids are brighter than metal-poor ones.

The improved precision reached in this work was made possible thanks to the high quality of Gaia EDR3 parallaxes and to the new distances of the two Magellanic Clouds obtained by the Araucaria Project. Combining Milky Way and Magellanic Cloud Cepheids also allows to reach a better precision than previous studies based on Magellanic Clouds only, by the larger range of metallicities they cover. A refined analysis of each light curve ensures the use of accurate mean magnitudes. However, the elongated shape of the SMC in the line of sight remains a source of systematic uncertainty in our study, despite continuous efforts to improve our knowledge of its structure. In this study, we assumed a linear dependence of the PL relation with metallicity, but it might as well be non-linear (Gieren et al. 2018). Additional high resolution spectroscopic metallicity mea-surements of both Milky Way and Magellanic Cloud Cepheids should be carried out in the future to even better constrain the metallicity effect, particularly in the NIR, in our effort to further reduce the systematic uncertainty on the determination of the Hubble constant from the Cepheid-SN Ia method. Note-( * ) Initial conditions corresponds to Gaia EDR3 parallaxes corrected for individual zero-point and SMC Cepheids limited to a radius of 0.6 degree around the SMC center. 2014); (G14b): metallicity from the literature (Genovali et al. 2013;Lemasle et al. 2007;Luck et al. 2011;Luck & Lambert 2011;Pedicelli et al. 2010;Romaniello et al. 2008;Sziládi et al. 2007;Yong et al. 2006)

APPENDIX

Figure 1 .

 1 Figure 1. Galactic maps projected on the sky (left) and on the galactic plane (right) showing the distribution of the MW Cepheid sample. The color scale represents the metallicity [Fe/H] and the red cross is the position of the Solar System.

Figure 2 .

 2 Figure 2. Map of the LMC Cepheids considered in our study. Yellow stars are the eclipsing binaries from Pietrzyński et al. (2019) and the red cross is the center of the LMC. The dashed circle represents a radius of 3 degrees around the LMC center.

Figure 3 .

 3 Figure 3. Map of the SMC Cepheids considered in our study. Yellow stars are the eclipsing binaries from Graczyk et al. (2020) and the red cross is the center of the SMC. The dashed circle represents a radius of 0.6 degree around the SMC center.

  Note-(*)The number of stars is given after the sigma clipping procedure and the period cuts. (a) Mean [Fe/H] = +0.083 ± 0.019 dex (b) Mean [Fe/H] = -0.34 ± 0.06 dex (c) Mean [Fe/H] = -0.75 ± 0.05 dex

Figure 4 .

 4 Figure 4. Fit of the PL relation in KS for MW, LMC and SMC Cepheids. The lower panel shows the residual between the Cepheid absolute magnitudes and the corresponding PL fit for each of the three galaxies. The LMC and SMC relations were offset by +1 and +2 mag, respectively, for visualization purposes.

Figure 5 .

 5 Figure 5. Intercept of the PL relation represented as a function of metallicity in J, H, K, V , I, and Wesenheit bands.

Figure 6 .

 6 Figure 6. Histogram of the γ values obtained in the KS band by the Monte Carlo algorithm iterated 10 000 times.

Figure 7 .

 7 Figure7. Metallicity effect as a function of wavelength, compared with values from the literature. The error bars include the systematics discussed in Sect. 5.2. For visualization purposes, the X axis was slightly shifted for our values so that the error bars do not overlap, but they correspond to the same wavelength as the literature values.

  value p.t.p.: 0.42 mag) Mean mag from flux: mK = 5.458 +/-0.006 mag['Laney+ 1992']

Figure 8 .

 8 Figure8. Example of a well covered light curve in the K band for the Galactic Cepheid KN Cen. The solid blue line represents the best fit of the light curve, the dashed green line is the mean magnitude derived from the best fit and the green region is the uncertainty on the intensity-averaged mean magnitude.

Figure 9 .

 9 Figure9. Example of a poor-quality light curve in the K band for the Galactic Cepheid BK Aur. The solid blue line represents the best fit of the light curve, the dashed green line is the mean magnitude derived from the best fit and the green region is the uncertainty on the intensity-averaged mean magnitude.

Table 1 .

 1 Results of the PL fit of the form M = α(log P -0.7) + β in the Milky Way, the Large Magellanic Cloud and the Small Magellanic Cloud.

	Band	α	β	σ	N ( * )
		MW (a)
	V	-2.443 ± 0.031 -3.296 ± 0.024 0.25 178
	I	-2.780 ± 0.028 -3.981 ± 0.024 0.23 150
	WV I	-3.289 ± 0.026 -5.030 ± 0.025 0.21 149
	J	-3.050 ± 0.029 -4.498 ± 0.026 0.18	97
	H	-3.160 ± 0.028 -4.762 ± 0.024 0.17	97
	KS	-3.207 ± 0.028 -4.848 ± 0.022 0.17	97
	WJK -3.317 ± 0.028 -5.086 ± 0.026 0.17	97
		LMC (b)
	V	-2.704 ± 0.007 -3.284 ± 0.033 0.23 1446
	I	-2.916 ± 0.005 -3.910 ± 0.033 0.15 1460
	WV I	-3.281 ± 0.008 -4.877 ± 0.038 0.08 1432
	J	-3.127 ± 0.005 -4.385 ± 0.033 0.12 805
	H	-3.160 ± 0.005 -4.696 ± 0.033 0.11 808
	KS	-3.217 ± 0.005 -4.737 ± 0.033 0.10 807
	WJK -3.272 ± 0.008 -4.974 ± 0.039 0.10 806
		SMC (c)
				17 294
	H	-	-	-	-
	KS	-3.163 ± 0.002 -4.670 ± 0.038 0.15 295
	WJK -3.326 ± 0.002 -4.916 ± 0.043 0.14 295

V -2.594 ± 0.012 -3.196 ± 0.038 0.28 284 I -2.871 ± 0.008 -3.841 ± 0.038 0.22 297 WV I -3.334 ± 0.014 -4.834 ± 0.043 0.12 283 J -2.956 ± 0.004 -4.317 ± 0.038 0.

Table 2 .

 2 Final results of the PLZ fit of the form M = α(log P -0.7) + δ + γ [Fe/H] and associated uncertainties.

	Band	α	σ	δ	σ	γ	σ
	V	-2.704 0.007 -3.252 0.020 -0.048 0.055
	I	-2.916 0.005 -3.948 0.020 -0.138 0.053
	WV I	-3.281 0.008 -5.005 0.022 -0.251 0.057
	J	-3.127 0.005 -4.463 0.022 -0.208 0.052
	H	-3.160 0.005 -4.748 0.020 -0.152 0.092
	KS	-3.217 0.004 -4.826 0.019 -0.221 0.051
	WJK -3.273 0.008 -5.075 0.022 -0.214 0.057
	Note-The uncertainties include the systematics discussed
	in Sect. 5.2.

Table 3 .

 3 Results of the PLZ linear fit of the form M = α(log P -0.7) + δ + γ [Fe/H] for Milky Way, Large Magellanic Cloud and Small Magellanic Cloud Cepheids fitted together for different conditions. The slope values are the same as in Table2.

	Band	δ	γ	N	Comments
	V	-3.252 ± 0.020 -0.048 ± 0.051 1908 Initial conditions ( * )
	I	-3.948 ± 0.020 -0.138 ± 0.051 1907 Initial conditions
	W V I	-5.005 ± 0.022 -0.251 ± 0.057 1864 Initial conditions
	J	-4.463 ± 0.022 -0.208 ± 0.051 1196 Initial conditions
	H	-4.748 ± 0.020 -0.152 ± 0.092	905	Initial conditions
	K S	-4.826 ± 0.019 -0.221 ± 0.050 1199 Initial conditions
	W JK -5.075 ± 0.022 -0.214 ± 0.057 1198 Initial conditions
	V	-3.274 ± 0.020 -0.084 ± 0.051 1908 Gaia EDR3 parallax ZP = -17µas
	I	-3.966 ± 0.020 -0.165 ± 0.050 1907 Gaia EDR3 parallax ZP = -17µas
	W V I	-5.020 ± 0.022 -0.275 ± 0.058 1864 Gaia EDR3 parallax ZP = -17µas
	J	-4.495 ± 0.022 -0.258 ± 0.052 1196 Gaia EDR3 parallax ZP = -17µas
	H	-4.778 ± 0.020 -0.241 ± 0.099	905	Gaia EDR3 parallax ZP = -17µas
	K S	-4.857 ± 0.019 -0.271 ± 0.051 1199 Gaia EDR3 parallax ZP = -17µas
	W JK -5.106 ± 0.022 -0.263 ± 0.058 1198 Gaia EDR3 parallax ZP = -17µas
	V	-3.252 ± 0.020 -0.036 ± 0.052 1952 R=0.7 • around SMC center
	I	-3.948 ± 0.020 -0.130 ± 0.052 1951 R=0.7 • around SMC center
	W V I	-5.005 ± 0.022 -0.252 ± 0.058 1908 R=0.7 • around SMC center
	J	-4.464 ± 0.022 -0.206 ± 0.052 1241 R=0.7 • around SMC center
	K S	-4.826 ± 0.019 -0.218 ± 0.051 1244 R=0.7 • around SMC center
	W JK -5.075 ± 0.022 -0.208 ± 0.058 1242 R=0.7 • around SMC center
	V	-3.250 ± 0.020 -0.077 ± 0.051 1845 R=0.5 • around SMC center
	I	-3.947 ± 0.020 -0.160 ± 0.053 1842 R=0.5 • around SMC center
	W V I	-5.005 ± 0.022 -0.251 ± 0.059 1800 R=0.5 • around SMC center
	J	-4.462 ± 0.022 -0.225 ± 0.054 1132 R=0.5 • around SMC center
	K S	-4.825 ± 0.019 -0.234 ± 0.050 1132 R=0.5 • around SMC center
	W JK -5.075 ± 0.022 -0.222 ± 0.057 1132 R=0.5 • around SMC center

Table 4 .

 4 Sample of Milky Way Cepheids and main parameters. Parallaxes from Gaia EDR3 include zero point correction. Stars with a RUWE parameter larger than 1.4 were marked with a star and excluded from the sample. References: (F95): reddening from Fernie et al. (1995) multiplied by 0.94; (A12): reddening from Acharova et al. (2012); (G14): metallicity from Genovali et al. (

Table 5 .

 5 rescaled toGenovali et al. (2014) solar abundance; (G15): metallicity fromGenovali et al. (2015). Optical and NIR mean apparent magnitudes for the sample of Milky Way Cepheids. The magnitudes do not include the reddening correction. The uncertainties are only the random errors and do not include photometric zero point errors. References: (B97):Barnes et al. (1997); (F08): Feast et al. (2008); (L92): Laney & Stobie (1992); (M11): Monson & Pierce (2011); (W84):Welch et al. (1984). All magnitudes in V and I are fromBerdnikov (2008).

	Star	Period	πEDR3	RUWE	E(B -V )	Ref.	[Fe/H]	Ref.
		(day)	(mas)		(mag)		(dex)
	AA Gem	11.302 0.311 ± 0.018	1.25	0.345 ± 0.036 F95 -0.08 ± 0.05 G15
	AC Mon	8.014 0.383 ± 0.019	1.38	0.507 ± 0.033 F95 -0.03 ± 0.06 G14b
	AD Gem	3.788 0.370 ± 0.020	0.97	0.206 ± 0.048 F95 -0.14 ± 0.06 G15
	AD Pup	13.596 0.254 ± 0.017	1.36	0.363 ± 0.020 F95 -0.20 ± 0.15 G14b
	AE Vel	7.134 0.369 ± 0.012	0.97	0.691 ± 0.055 F95	0.14 ± 0.06	G14b
	AG Cru	3.837 0.758 ± 0.020	1.02	0.242 ± 0.020 F95	0.08 ± 0.06	G14b
	AP Pup	5.084 0.924 ± 0.020	1.05	0.250 ± 0.034 F95 -0.16 ± 0.15 G14b
	AP Sgr	5.058 1.217 ± 0.024	0.88	0.184 ± 0.015 F95	0.10 ± 0.08	G14b
	AQ Car	9.769 0.361 ± 0.016	1.07	0.168 ± 0.013 F95 -0.30 ± 0.15 G14b
	AQ Pup	30.149 0.294 ± 0.023	1.18	0.531 ± 0.017 F95	0.06 ± 0.05	G15
	AS Per	4.973 0.650 ± 0.016	1.08	0.684 ± 0.041 F95	0.14 ± 0.06	G14b
	AT Pup	6.665 0.604 ± 0.016	1.04	0.166 ± 0.011 F95 -0.22 ± 0.15 G14b
	AV Sgr	15.415 0.404 ± 0.025	0.84	1.238 ± 0.027 F95	0.35 ± 0.17	G15
	AW Per	6.464 1.093 ± 0.029	1.16	0.479 ± 0.016 F95	0.04 ± 0.06	G14b
	AY Cas	2.872 0.414 ± 0.019	1.07	0.760 ± 0.049 F95	0.02 ± 0.06	G14b
	AY Cen	5.310 0.574 ± 0.014	0.95	0.357 ± 0.066 F95	0.08 ± 0.06	G14b
	AY Sgr	6.570 0.546 ± 0.019	0.85	0.840 ± 0.009 F95	0.11 ± 0.06	G15
	BB Her	7.508 0.280 ± 0.015	1.06	0.392 ± 0.039 A12	0.26 ± 0.06	G14b
	BB Sgr	6.637 1.188 ± 0.024	0.82	0.285 ± 0.011 F95	0.08 ± 0.08	G14b
	BE Mon	2.706 0.504 ± 0.017	1.17	0.549 ± 0.036 F95	0.05 ± 0.09	G15
	BF Oph	4.068 1.189 ± 0.024	0.84	0.261 ± 0.016 F95	0.14 ± 0.06	G14b
	BG Vel	6.924 1.045 ± 0.017	0.99	0.434 ± 0.011 F95 -0.10 ± 0.15 G14b
	BK Aur	8.002 0.426 ± 0.015	1.01	0.393 ± 0.026 F95 -0.07 ± 0.15 G14b
	BM Per	22.952 0.334 ± 0.022	0.99	0.919 ± 0.059 F95	0.23 ± 0.06	G14b
	BM Pup	7.199 0.302 ± 0.013	1.18	0.575 ± 0.058 F95 -0.07 ± 0.08 G15
	BN Pup	13.673 0.301 ± 0.015	1.25	0.422 ± 0.017 F95	0.03 ± 0.05	G15
	BP Cas	6.273 0.442 ± 0.013	1.02	0.864 ± 0.014 F95	0.09 ± 0.06	G14b
	BZ Cyg	10.142 0.500 ± 0.014	1.14	0.832 ± 0.018 F95	0.19 ± 0.08	G14b
	CD Cas	7.801 0.412 ± 0.014	1.06	0.745 ± 0.012 F95	0.13 ± 0.06	G14b
	CD Cyg	17.074 0.394 ± 0.016	1.01	0.512 ± 0.021 F95	0.15 ± 0.06	G14b
	CE Pup	49.326 0.114 ± 0.014	0.82	0.740 ± 0.074 A12 -0.04 ± 0.09 G14
	CF Cas	4.875 0.316 ± 0.012	1.04	0.556 ± 0.021 F95	0.02 ± 0.06	G14b
	CG Cas	4.366 0.296 ± 0.017	1.03	0.667 ± 0.009 F95	0.09 ± 0.06	G14b
	CK Sct	7.415 0.490 ± 0.020	0.97	0.816 ± 0.024 F95	0.21 ± 0.06	G14b
	CN Car	4.933 0.342 ± 0.014	0.96	0.438 ± 0.049 F95	0.21 ± 0.06	G14b
	CP Cep	17.859 0.279 ± 0.021	1.01	0.681 ± 0.045 F95 -0.01 ± 0.08 G14b
	CR Cep	6.233 0.699 ± 0.013	1.06	0.704 ± 0.009 F95 -0.06 ± 0.08 G14b
	CR Ser	5.301 0.578 ± 0.020	1.19	0.974 ± 0.017 F95	0.12 ± 0.08	G15
	CS Mon	6.732 0.324 ± 0.014	1.04	0.528 ± 0.032 F95 -0.08 ± 0.06 G14b
	CS Ori	3.889 0.257 ± 0.022	1.33	0.373 ± 0.030 F95 -0.25 ± 0.06 G15
	CS Vel	5.905 0.272 ± 0.016	0.91	0.716 ± 0.027 F95	0.12 ± 0.06	G14b

Table 5

 5

	(continued)

Table 5

 5

	(continued)

Table 6 .

 6 Sample of Large Magellanic Cloud Cepheids and their main parameters. The Cepheid names in the first column are of the form OGLE-LMC-CEP-XXXX. The uncertainties on V and I band mean magnitudes are 0.02 mag and the uncertainty on E(B -V ) values is 0.017 mag. The distances listed in column (5) are corrected for their position in the LMC by the equations provided in Sect. 3.2. Apparent magnitudes in this table are not corrected for the reddening. The full table is available as supplementary material.

	Cepheid	P	α	δ	d	V	I	J	H	K S	E(B -V )
		(days) (J2000) (J2000)	(kpc)	(mag) (mag)	(mag)	(mag)	(mag)	(mag)
	0107	8.739	72.214	-69.356 50.63 ± 0.56 14.761 13.947 13.332 ± 0.018 13.027 ± 0.015 12.941 ± 0.014	0.182
	0174	15.863	72.719	-69.316 50.55 ± 0.56 14.739 13.666 12.876 ± 0.022 12.454 ± 0.020 12.312 ± 0.019	0.174
	0328	34.460	73.599	-70.902 50.68 ± 0.56 13.124 12.088 11.460 ± 0.031 11.111 ± 0.027 11.001 ± 0.024	0.133
	0467	22.718	74.301	-67.383 50.06 ± 0.55 13.704 12.775 12.113 ± 0.033 11.767 ± 0.031 11.668 ± 0.029	0.112
	0473	2.634	74.331	-68.821 50.25 ± 0.56 16.335 15.590 15.093 ± 0.082 14.856 ± 0.116 14.641 ± 0.114	0.141
	0478	2.764	74.355	-69.567 50.36 ± 0.56 16.160 15.471 14.961 ± 0.057 14.666 ± 0.073 14.624 ± 0.076	0.150
	0480	4.035	74.364	-69.355 50.33 ± 0.56 16.865 15.779 15.107 ± 0.049 14.513 ± 0.052 14.515 ± 0.080	0.161
	0482	7.466	74.370	-69.227 50.31 ± 0.56 15.655 14.661 13.977 ± 0.081 13.439 ± 0.031 13.315 ± 0.030	0.151
	0487	3.109	74.422	-69.406 50.33 ± 0.56 16.221 15.469 15.002 ± 0.124 14.614 ± 0.041 14.488 ± 0.090	0.165
	0488	3.647	74.422	-68.800 50.24 ± 0.56 16.535 15.608 14.921 ± 0.037 14.517 ± 0.045 14.451 ± 0.066	0.140
	0494	2.727	74.441	-69.062 50.27 ± 0.56 16.973 16.013 15.441 ± 0.087 14.953 ± 0.100 14.658 ± 0.091	0.134
	0498	3.630	74.455	-68.720 50.22 ± 0.56 15.914 15.154 14.617 ± 0.071 14.295 ± 0.059 14.286 ± 0.061	0.139
	0514	3.504	74.554	-69.203 50.28 ± 0.56 16.276 15.458 14.879 ± 0.063 14.458 ± 0.055 14.381 ± 0.068	0.152
	0518	3.249	74.577	-69.367 50.30 ± 0.56 16.550 15.589 14.988 ± 0.044 14.565 ± 0.044 14.543 ± 0.046	0.174
	0529	2.856	74.637	-69.041 50.24 ± 0.56 16.603 15.777 15.168 ± 0.087 14.746 ± 0.084 14.698 ± 0.087	0.134
	0539	3.455	74.672	-68.865 50.21 ± 0.55 16.064 15.320 14.758 ± 0.056 14.373 ± 0.069 14.299 ± 0.054	0.131
	0540	3.750	74.673	-69.526 50.31 ± 0.56 15.841 15.069 14.574 ± 0.061 14.198 ± 0.058 14.188 ± 0.056	0.160
	...	...	...	...	...	...	...	...	...	...	...

Table 7 .

 7 Sample of Small Magellanic Cloud Cepheids and their main parameters. The Cepheid names in the first column are of the form OGLE-SMC-CEP-XXXX. The uncertainties on V and I band mean magnitudes are 0.02 mag and the uncertainty on E(B -V ) values is 0.015 mag. The distances listed in column (5) are corrected for their position in the SMC by the equations provided in Sect. 3.3. Apparent magnitudes in this table are not corrected for the reddening. The full table is available as supplementary material.

	Cepheid	P	α	δ	d	V	I	J	K S	E(B -V )
		(days) (J2000) (J2000)	(kpc)	(mag) (mag)	(mag)	(mag)	(mag)
	0443	4.037	10.525	-73.061 63.36 ± 0.95 16.443 15.671 15.139 ± 0.018 14.742 ± 0.006	0.096
	0489	3.242	10.734	-73.160 63.42 ± 0.95 16.493 15.822 15.096 ± 0.006 14.492 ± 0.010	0.101
	0494	4.758	10.742	-73.335 63.72 ± 0.96 15.884 15.025 14.447 ± 0.006 14.030 ± 0.004	0.103
	0495	6.312	10.743	-73.092 63.30 ± 0.95 16.038 15.217 14.604 ± 0.012 14.164 ± 0.004	0.097
	0499	6.229	10.751	-73.304 63.66 ± 0.96 15.986 15.258 14.740 ± 0.004 14.374 ± 0.006	0.108
	0514	2.542	10.774	-73.082 63.27 ± 0.95 16.842 16.145 15.687 ± 0.010 15.306 ± 0.010	0.097
	0518	15.773	10.802	-73.326 63.67 ± 0.96 15.184 14.173 13.449 ± 0.002 12.948 ± 0.004	0.108
	0524	10.527	10.828	-73.339 63.68 ± 0.96 15.383 14.536 13.934 ± 0.008 13.492 ± 0.006	0.105
	0533	3.909	10.859	-73.254 63.52 ± 0.95 16.122 15.390 14.895 ± 0.006 14.489 ± 0.010	0.105
	0551	3.262	10.905	-73.129 63.28 ± 0.95 16.473 15.807 15.302 ± 0.006 14.954 ± 0.016	0.101
	0570	10.883	10.947	-73.241 63.45 ± 0.95 15.213 14.354 13.738 ± 0.010 13.294 ± 0.004	0.112
	0571	4.897	10.948	-73.335 63.62 ± 0.95 15.872 15.178 14.662 ± 0.006 14.297 ± 0.008	0.104
	0576	14.426	10.963	-73.333 63.61 ± 0.95 15.110 14.122 13.470 ± 0.016 12.923 ± 0.018	0.104
	0584	4.654	10.989	-73.192 63.35 ± 0.95 16.058 15.296 14.777 ± 0.006 14.418 ± 0.006	0.111
	0596	3.072	11.013	-73.277 63.48 ± 0.95 17.170 16.282 15.673 ± 0.006 15.184 ± 0.012	0.120
	...	...	...	...	...	...	...	...	...	...

https://www.cosmos.esa.int/web/gaia/edr3-code

ACKNOWLEDGMENTS

We thank the anonymous referee for the very constructive comments that helped us to improve the manuscript.

We are grateful to Lucas Macri for providing the photometric transformations. The research leading to these results has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 695099 (project CepBin). This work has made use of data from the European Space Agency (ESA) mission Gaia (http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, http://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. The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-15-CE31-0012-01 (project UnlockCepheids). W.G. and G. P. gratefully acknowledge financial support for this work from the BASAL Centro de Astrofisica y Tecnologias Afines (CATA) AFB-170002. W.G. and D. G. acknowledge financial support from the Millenium Institute of Astrophysics (MAS) of the Iniciativa Cientifica Milenio del Ministerio de Economia, Fomento y Turismo de Chile, project IC120009. This research made use of Astropy7, a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2018). Support from the Polish National Science Centre grants MAESTRO UMO-2017/26/A/ST9/00446 and from the IdPII 2015 0002 64 and DIR/WK/2018/12 grants of the Polish Ministry of Science and Higher Education is also acknowledged. We used the SIMBAD and VIZIER databases and catalog access tool at the CDS, Strasbourg (France), and NASA's Astrophysics Data System Bibliographic Services.