A precision determination of the Effect of Metallicity on Cepheid absolute magnitudes in VIJHK bands from Magellanic Cloud Cepheids

TL;DR

This study precisely measures how metallicity influences Cepheid absolute magnitudes across multiple bands, finding effects smaller than 2%, which supports their use as reliable distance indicators in cosmology.

Contribution

It provides the first high-accuracy comparison of Cepheid P-L relations between LMC and SMC, demonstrating negligible metallicity effects across optical and infrared bands.

Findings

Metallicity effects are smaller than 2% in all bands.

Results are consistent with zero metallicity effect within 3% uncertainty.

Supports the use of Cepheids as standard candles across different metallicities.

Abstract

Using high-quality observed period-luminosity relations in both Magellanic Clouds in VIJHKs bands and optical and near-infrared Wesenheit indices we determine the effect of metallicity on Cepheid P-L relations by comparing the relative distance between LMC and SMC as determined from the Cepheids to the distance difference between the Clouds which has been derived with very high accuracy from late-type eclipsing binary systems. Within an uncertainty of 3% which is dominated by the uncertainty on the mean metallicity difference between the Cepheid populations in LMC and SMC we find metallicity effects smaller than 2% in all bands and in the Wesenheit indices, consistent with a zero metallicity effect. This result is valid for the metallicity range from -0.35 dex to -0.75 dex corresponding to the mean [Fe/H] values for classical Cepheids in LMC and SMC, respectively. Yet most Cepheids in galaxies beyond the Local Group and located in the less crowded outer regions of these galaxies do fall into this metallicity regime, making our result important for applications to determine the distances to spiral galaxies well beyond the Local Group. Our result supports previous findings which indicated a very small metallicity effect on the near-infrared absolute magnitudes of classical Cepheids, and resolves the dispute about the size and sign of the metallicity effect in the optical spectral range. It also resolves one of the most pressing problems in the quest towards a measurement of the Hubble constant with an accuracy of 1% from the Cepheid-supernova Ia method.

Tables

Table 1. Table 1: Free linear fits to the observed P-L relations in the LMC and SMC.

Filter	$a$	$b$	$\sigma$	$N$
LMC
$V$	$-2.779\pm 0.021$	$17.543\pm 0.014$	$0.228$	$1794$
$I$	$-2.977\pm 0.015$	$16.892\pm 0.010$	$0.154$	$1769$
$J$	$-3.118\pm 0.011$	$16.431\pm 0.008$	$0.120$	$1740$
$H$	$-3.224\pm 0.009$	$16.152\pm 0.006$	$0.097$	$1743$
$Ks$	$-3.247\pm 0.009$	$16.095\pm 0.006$	$0.089$	$1747$
$W_I$	$-3.332\pm 0.008$	$15.904\pm 0.006$	$0.083$	$1778$
$W_{JKs}$	$-3.334\pm 0.008$	$15.857\pm 0.005$	$0.084$	$1743$
SMC
$V$	$-2.644\pm 0.036$	$17.792\pm 0.026$	$0.282$	$955$
$I$	$-2.947\pm 0.027$	$17.264\pm 0.020$	$0.223$	$963$
$J$	$-3.087\pm 0.023$	$16.858\pm 0.017$	$0.185$	$907$
$H$	$-3.184\pm 0.021$	$16.577\pm 0.016$	$0.166$	$906$
$Ks$	$-3.206\pm 0.021$	$16.530\pm 0.015$	$0.160$	$901$
$W_I$	$-3.330\pm 0.019$	$16.385\pm 0.014$	$0.146$	$946$
$W_{JKs}$	$-3.311\pm 0.020$	$16.322\pm 0.014$	$0.150$	$896$

Table 2. Table 2: Zero points of SMC Cepheid P-L relations with slopes adopted from the LMC

Filter	$b_{SMC}$	$\sigma$	$N$	$\Delta \left(m-M\right)$
$V$	$17.883\pm 0.009$	$0.282$	$954$	$0.340\pm 0.017$
$I$	$17.285\pm 0.007$	$0.222$	$962$	$0.393\pm 0.012$
$J$	$16.880\pm 0.006$	$0.186$	$908$	$0.449\pm 0.010$
$H$	$16.604\pm 0.006$	$0.166$	$906$	$0.452\pm 0.009$
$Ks$	$16.558\pm 0.005$	$0.161$	$901$	$0.463\pm 0.008$
$W_{VI}$	$16.386\pm 0.005$	$0.145$	$946$	$0.482\pm 0.008$
$W_{JKs}$	$16.338\pm 0.005$	$0.150$	$896$	$0.481\pm 0.008$

Table 3. Table 3: Metallicity effect on Cepheid P-L relations in VIJHK, W V ​ I subscript 𝑊 𝑉 𝐼 W_{VI} and W J ​ K subscript 𝑊 𝐽 𝐾 W_{JK} bands

Filter	$R_{\lambda }$	$\Delta \left(m-M\right)$	$\Delta {\left(m-M\right)}_0^{cep}$	$\Delta {\left(m-M\right)}_0^{cep}-\Delta {\left(m-M\right)}_0^{ecl}$	$\gamma$
$V$	$3.134$	$0.340\pm 0.017$	$0.481\pm 0.017$	$0.009\pm 0.031$	$-0.022\pm 0.076$
$I$	$1.894$	$0.393\pm 0.012$	$0.478\pm 0.012$	$0.006\pm 0.029$	$-0.015\pm 0.071$
$J$	$0.892$	$0.449\pm 0.010$	$0.489\pm 0.010$	$0.017\pm 0.028$	$-0.042\pm 0.069$
$H$	$0.553$	$0.452\pm 0.009$	$0.477\pm 0.009$	$0.005\pm 0.028$	$-0.012\pm 0.069$
$Ks$	$0.363$	$0.463\pm 0.008$	$0.479\pm 0.009$	$0.007\pm 0.028$	$-0.017\pm 0.069$
$W_{VI}$	-	-	$0.482\pm 0.008$	$0.010\pm 0.027$	$-0.025\pm 0.067$
$W_{JKs}$	-	-	$0.481\pm 0.008$	$0.009\pm 0.027$	$-0.022\pm 0.067$

Table 4. Table 4: Dependence of the metallicity effect on chosen sample cutoffs

Periods in LMC, SMC	$\Delta {\left(m-M\right)}_0$	$\Delta E\left(B-V\right)$	${\gamma }_V$	${\gamma }_I$	${\gamma }_J$	${\gamma }_H$	${\gamma }_{Ks}$	${\gamma }_{W_{VI}}$	${\gamma }_{W_{JKs}}$
$\left(0:100\right)$,$\left(2.5:100\right)$	$0.480\pm 0.004$	$-0.045\pm 0.003$	$-0.022$	$-0.015$	$-0.042$	$-0.012$	$-0.017$	$-0.025$	$-0.022$
$\left(2.5:100\right)$,$\left(2.5:100\right)$	$0.480\pm 0.003$	$-0.045\pm 0.003$	$-0.015$	$-0.012$	$-0.039$	$-0.012$	$-0.017$	$-0.022$	$-0.025$
both $>$ 10d	$0.472\pm 0.005$	$-0.037\pm 0.005$	$-0.029$	$0.030$	$-0.022$	$0.007$	$-0.010$	$-0.010$	$-0.017$
free fits mags at 10d	$0.478\pm 0.010$	$-0.033\pm 0.008$	$-0.037$	$0.017$	$-0.037$	$0.022$	$-0.039$	$-0.027$	$-0.039$

Equations

$$(m-M{)}_0^{SMC}-(m-M{)}_0^{LMC}=\Delta (m-M{)}_0=\Delta (m-M{)}_{\lambda }-\Delta E(B-V)\ast R_{\lambda }$$

$$\Delta (m-M{)}_0=0.481\pm 0.004\mathrm{mag},$$

$$\Delta E(B-V)=-0.045\pm 0.003\mathrm{mag}.$$

$$\Delta (m-M{)}_0^{ecl}=0.472\pm 0.026\mathrm{mag}$$

$$\gamma =\frac{\Delta (m-M{)}_0^{cep}-\Delta (m-M{)}_0^{ecl}}{\Delta [Fe/H]}$$

Full text

Our result supports previous findings which indicated a very small metallicity effect on the near-infrared absolute magnitudes of classical Cepheids, and resolves the dispute about the size and sign of the metallicity effect in the optical spectral range. It also resolves one of the most pressing problems in the quest towards a measurement of the Hubble constant with an accuracy of $1\%$ from the Cepheid-supernova Ia method.

Cepheids — distance scale — galaxies: distances and redshifts — Magellanic Clouds

1 Introduction

Classical Cepheids are prime distance indicators and have been widely used to calibrate the first rungs of the extragalactic distance scale using the famous period-luminosity (PL) relation (more recently also called the ”Leavitt Law”). In the ongoing quest to achieve a determination of the Hubble constant with an accuracy of 1%, Cepheid variables have been particularly useful to calibrate the peak luminosities of Type Ia supernovae which seem to offer the most promising route to achieve such an accurate measurement of $H_{0}$ (Riess et al., 2016). The two main limitations to further reduce the systematic uncertainty on $H_{0}$ as determined from the Cepheid-SN Ia method currently come from the uncertainty on the distance of the adopted fiducial galaxy to which the Cepheid PL relations observed in SN host galaxies are tied, and from the ill-known effect metallicity might have on Cepheid absolute magnitudes, particularly in near- and mid-infrared bands which are used in modern Cepheid distance work to reduce the effect of interstellar extinction on the results (e.g. Gieren et al., 2005a; Freedman et al., 2012).

While the situation with respect to the absolute distance of a suitable Cepheid fiducial galaxy has considerably improved with the 2.2% determination of the distance of the Large Magellanic Cloud (LMC) by Pietrzyński et al. (2013) from a sample of late-type eclipsing binaries, and the potential to push the accuracy of the LMC distance to even 1% with the same method by improving the surface brightness-color relation which serves to calculate the angular diameters of the binary component stars (e.g. Graczyk et al., 2017), the situation is much more confused with respect to the knowledge of how exactly Cepheid absolute magnitudes depend on metallicity. A number of empirical tests have been made over the years (Freedman & Madore, 1990; Kennicutt et al., 1998; Macri et al., 2006; Romaniello et al., 2008; Scowcroft et al., 2009) which mostly favored the view that in optical bands metal-rich Cepheids are intrinsically brighter than their metal-poor counterparts of the same pulsation period, but this result has been challenged by the work of Romaniello et al. (2008) who found an effect in the opposite sense. Freedman & Madore (2011) suggested that the metallicity effect vanishes in the near-infrared (HK) domain while being significant at optical and mid-infrared wavelengths, but with opposite signs. Storm et al. (2011), using the Infrared Surface Brightness variant of the Baade-Wesselink method (Fouque & Gieren, 1997; Gieren et al., 2005b) to determine the distances to Milky Way, LMC and SMC Cepheid samples, and comparing the resulting absolute PL relations of Milky Way and Magellanic Cloud Cepheids found a small effect in all optical and near-infrared bands, but the uncertainty of this determination was in the order of $\pm 0.1$ mag/dex which needs to be beaten down significantly in order to achieve a 1% measurement of $H_{0}$. Theoretical work has been done, among others, by Bono et al. (2010) but the uncertainties on the adopted input physics for the models are currently too large to allow to clearly support either of the empirical results. Clearly, a truly accurate empirical test is long overdue.

As stated above, a breakthrough in the century-long effort to accurately measure the distances to the Magellanic Clouds has been made by using the very rare well-detached eclipsing binary systems being composed of two similar red giant stars for this purpose. Pietrzyński et al. (2013) have shown that the distance of one such system can be determined with an accuracy close to 2% if the data for the analysis (optical light curves, orbital radial velocity curve, and the out-of-eclipse K-band magnitude of the system) are of very high precision. From eight systems, Pietrzynski et al. determined the LMC distance with an accuracy of 2.2%; the dominant source of systematic error in this process is the adopted surface brightness-color relation which our group is currently improving by interferometric angular diameter measurements of a large number of nearby red clump giants. The distance determined in this way is practically independent of metallicity and the assumed reddening. Similarly, our group has succeeded to determine the distance to the central part of the Small Magellanic Cloud (SMC) with an accuracy of 3% using five late-type systems resembling those in the LMC used by Pietrzyński et al. (2013), and using exactly the same prescripts in the analysis of these systems, and the distance calculations (Graczyk et al., 2014). As a consequence, the relative distance between LMC and SMC is an extremely well-determined number from this work since the systematic errors cancel out, and the statistical uncertainties are in the order of 1% or less.

In our long-term Araucaria Project, we have demonstrated how the combination of optical and near-infrared photometry of Cepheids in combination with a reddening law can be used to very accurately measure the distance and mean reddening of a target galaxy with respect to the LMC (e.g. Gieren et al., 2005a; Pietrzyński et al., 2006; Soszyński et al., 2006). In this paper, we are going to employ the very extensive optical and near-infrared photometric datasets of Cepheids in both Clouds to construct PL relations (see section 2) and measure the relative distance between SMC and LMC from the observed PL relations in optical (V,I) and near-infrared (J,H,Ks) photometric passbands. We can then compare the distance difference between the Clouds measured from the Cepheids with the value obtained from the late-type eclipsing binaries, and interpret the offset from the binary-based distance difference as due to the effect of metallicity on the Cepheid absolute magnitudes in the corresponding band, assuming an average metallicity difference between the classical Cepheid populations in LMC and SMC of 0.406 dex, the SMC Cepheids being more metal-poor by this amount (see section 5 of this paper). Since the Cepheid PL relations in LMC and SMC are extremely well determined from large samples of stars with very accurate photometry, the magnitude offsets, for a given band, between LMC and SMC can be measured very accurately, so we can finally expect a measurement of the size, and sign of the metallicity effect in the VIJHKs bands which is approaching the 1% accuracy needed in the context of a precision measurement of $H_{0}$.

2 Cepheid samples in the LMC and SMC

In our work we used data from two surveys covering major parts of the Magellanic Clouds. The optical data come from the Optical Gravitational Lensing Experiment (OGLE, e.g. Udalski et al., 2015). In the course of OGLE project, the Magellanic Clouds have been observed since 1996 using the 1.3 meter telescope at the Las Campanas Observatory (Chile). In particular during the third phase of this project (OGLE-III), V and I band light curves of about 1700 classical Cepheids pulsating in the fundamental mode (FU) were obtained in the LMC (Soszyński et al., 2008), and about 2000 FU Cepheids were observed in the SMC (Soszyński et al., 2010). Multi-epoch photometry collected over 8-13 years was carefully transformed onto the system of Landolt (Soszyński et al., 2008). From the OGLE-III catalog we adopted magnitudes, coordinates and periods of the Magellanic Cloud Cepheids. All stars brighter than about 12.5 mag are saturated on the OGLE images. This saturation limit corresponds to Cepheids with periods of 30 days. Therefore we complemented the LMC Cepheid sample with 28 bright FU Cepheids observed in the OGLE-III shallow survey (Ulaczyk et al., 2013). This complementary survey was tailored for bright stars. All the data were collected with the same instrumental system and were tied to the OGLE III photometric system. As a result we have an extremely homogenous sample of optical light curves for all Cepheids in both Magellanic Clouds.

The near-Infrared data in the J, H and Ks bands come from observations collected with the 1.4 meter IRSF telescope located at the South African Astronomical Observatory Sutherland site (Kato et al., 2007). This catalog contains single epoch observations. From these data we calculated mean magnitudes following the procedure described in Soszyński et al. (2005). We estimated mean magnitudes in all bands with a precision close to 0.03 mag. In order to transform the mean magnitudes of the Cepheids to the 2MASS system we applied the formulae given in Kato et al. (2007).

Due to the observed nonlinearity of the Cepheid period-luminosity relation in the SMC at the very short-period end (Bauer et al., 1999) we rejected Cepheids pulsating with periods shorter than 2.5 days in this galaxy. In this way we ended up with samples of nearly 1800 LMC and about 900 SMC fundamental mode Cepheids, each Cepheid in these samples having accurate mean magnitudes in 5 filters (VIJHKs). Moreover, we calculated Wesenheit indices for combinations of V, I and J, Ks filters. The Wesenheit index for a set of two filters (X and Y) is given by the following formula: $W_{XY}=m_{Y}-\frac{A_{Y}}{E(X-Y)}(m_{X}-m_{Y})$. Such combinations of magnitudes for V, I and J, Ks filters are widely used for distance determination since they do not depend on reddening (if assumed reddening law is correct, e.g. Madore, 1985). We adopted extinction coefficients for our filters using the O’Donnell (1994) and Cardelli et al. (1989) formulas, which lead to $\frac{A_{I}}{E(V-I)}=1.55$ and $\frac{A_{Ks}}{E(J-Ks)}=0.69$.

3 Relative distance between the SMC and LMC from classical Cepheids

3.1 Cepheid P-L relations in the Magellanic Clouds

Applying the method of least squares and $3\sigma$ clipping we fitted straight lines $m=a\log P+b$ to obtain the coefficients of the observed period-luminosity (P-L) relations for the LMC Cepheids. Results are presented in Table 1. Data and fitted relations are also shown in Figure 1. To check the agreement of the slopes of P-L relations in LMC and SMC we also obtained free fits to the SMC data, with results also given in Table 1 and fits plotted in Figure 2. As can be appreciated the agreement between the slope measured in the LMC and SMC is very good, close to the combined $1\sigma$ in all bands except V, where the discrepancy is close to the combined $3\sigma$ uncertainty on the slopes. Since the slopes of the LMC P-L relations are determined with higher accuracy than those of the corresponding SMC relations, we adopted them for the SMC. Forcing the LMC slopes to the SMC P-L relations leads to the zero points given in Table 2 which we used to measure the observed SMC distance moduli in the different photometric bands. As previously, we used $3\sigma$ clipping to remove outstanding points. The last column of Table 2 gives the relative distance moduli between the LMC and SMC obtained in this process for the different bands. In the case of the Wesenheit indices, these values are already the true, reddening-free relative distance moduli, while in the case of the other bands the values are reddened, e.g. not corrected for extinction. Our results are generally in very good agreement with the results obtained by other authors based on similar data for Cepheids in the Magellanic Clouds (e.g. Persson et al., 2004; Ripepi et al., 2012; Inno et al., 2013; Macri et al., 2015; Ngeow et al., 2015; Ripepi et al., 2016). Observed small differences are probably due to different samples of Cepheids used by different authors.

3.2 Relative distance and reddening from multiband Cepheid P-L relations

Based on the observed relative distance moduli obtained in the previous section and assuming the extinction law of O’Donnell (1994) and Cardelli et al. (1989) we will now measure the true relative LMC-SMC distance modulus, and the relative reddening following our previous work in the Araucaria Project (e.g. Gieren et al., 2009; Pietrzyński et al., 2006). The relative true distance modulus can be written as:

[TABLE]

where $\lambda$ is a given filter, $\Delta(m-M)_{\lambda}$ are the corresponding values from the last column of Table 2, $R_{\lambda}$ are the ratios of total to selective absorption calculated from the O’Donnell (1994) and Cardelli et al. (1989) reddening law, and $\Delta E(B-V)=E(B-V)_{SMC}-E(B-V)_{LMC}$ is the relative mean reddening affecting the Cepheid samples in the LMC and SMC.

Fitting a straight line to this relation with the data given in columns 2 and 3 of Table 3, we obtain the following results:

[TABLE]

[TABLE]

The fit is shown in Fig. 3. Our result shows that the Cepheids in the SMC are on average by 0.045 mag less reddened than the LMC Cepheids. In order to check on this result we used the reddening maps of the LMC and SMC obtained by Haschke et al. (2011) and calculated the reddening for all studied Cepheids. From these maps which are based on red clump stars we find that our sample of Cepheids in the SMC is on average 0.036 mag less reddened than the sample in the LMC, in excellent agreement with our result from the multiband Cepheid P-L relations in the present study.

4 Relative distance between SMC and LMC from late-type eclipsing binaries

Recently our group has measured precise and accurate geometrical distances to both Magellanic Clouds using late-type, fully detached eclipsing binary systems consisting of pairs of red giants having typically 3-4 solar masses, very similar to the masses of short-period classical Cepheids. From 8 such systems, Pietrzyński et al. (2013) obtained an LMC distance modulus of 18.493 $\pm$ 0.008 (stat) $\pm$ 0.048 (sys) mag. In Pietrzyński et al. (2013) it was demonstrated that the effect of the geometrical depth of the LMC on this result is negligible.

From the analysis of 5 systems in the SMC resembling those in the LMC we measured a distance modulus of 18.965 $\pm$ 0.025 (stat) $\pm$ 0.048 (sys) for the SMC barycenter (Graczyk et al., 2014). As discussed in that paper, this value of the SMC distance should be only marginally affected by the more complicated, as compared to the LMC, geometrical structure of the SMC.

We recall that the distances of these systems were measured using the simple equation: $d(pc)=1.337\times 10^{-5}r(km)/\varphi(mas)$. The linear diameter (r) comes from the analysis of the system while the angular diameter ($\varphi$) is derived from a surface brightness-color relation (e.g. $m_{0}=S-5log(\varphi)$, where S is the surface brightness in a given band, and $m_{0}$ is the unreddened magnitude of a given star in this band). The surface brightness - color relation (SBCR) using the $(V-K)$ color is very well established (to about 2 %) from interferometric observations (e.g. Di Benedetto, 2005; Kervella et al., 2004). This method depends only very weakly on reddening and the adopted reddening law. It is also practically independent of metallicity (Pietrzyński et al., 2013; Thompson et al., 2001). A change of 1 dex would produce a change of Sv of about 0.006 mag, which corresponds to a 0.3 % change in the distance determination. Since our samples of eclipsing systems in both Clouds differ in metallicity by 0.4 dex (see section 5) this could introduce an error at the level of 0.1% only, therefore we will neglect it.

The systematic uncertainty on our distance measurement of LMC and SMC from late-type eclipsing binaries comes from the calibration of the SBCR, and the accuracy of the zero points in our photometry. Since the same photometric system and the same SBCR was used to calculate the distances to the LMC and SMC, the relative distance is dominated only by statistical errors and it equals to:

[TABLE]

We emphasize that the studied eclipsing binaries are composed of stars which are in the same evolutionary phase as Cepheids (e.g. helium-burning giants), which should imply that their spatial distributions in the Clouds are very similar to those of the classical Cepheids. Therefore we do not expect any significant influence of the LMC and SMC depth effects on our results.

5 Metallicities of the young stellar populations in the Magellanic Clouds

We have surveyed the literature for metallicity determinations of young stars in the Magellanic Clouds focussing on studies, which employ exactly the same technique in their analyses of both Clouds. In this way we intend to obtain abundance differences between the two Clouds which are as little as possible influenced by any systematic effects caused by the analysis method and the atmospheric structures of stars of different type. Hill et al. (1995) and Hill (1999) obtained a [Fe/H] difference between LMC and SMC of -0.485 $\pm$ 0.12 dex from F and K supergiants. From classical Cepheids Luck et al. (1998) determined a difference of -0.38 $\pm$ 0.13 dex, while Romaniello et al. (2008) found a value of -0.42 $\pm$ 0.15 dex. All three studies were carried out in LTE. From Davies et al. (2015) we obtained a logarithmic metallicity difference [Z] of -0.17 $\pm$ 0.12 dex from their NLTE study of red supergiant stars, which included the elements Fe, Ti, Si, and Mg (note that we have excluded two outliers in the SMC and one in the LMC from their sample).

We have also included two investigations of hotter stars. From Trundle et al. (2007) we infer a difference in metallicity [Z] of -0.36 $\pm$ 0.04 dex including the elements O, Mg, Si, and Fe. Note that the former three elements were treated in NLTE whereas the analysis of Fe is based on LTE. Urbaneja et al.. (2017) and Schiller (2010) in their study of A supergiant stars used the same set of NLTE radiative transfer calculations for Fe and Mg in the LMC and SMC, respectively, to obtain a metallicity difference of -0.385 $\pm$ 0.16 dex.

The mean metallicity difference between SMC and LMC obtained from these six studies is -0.367 $\pm$ 0.106 dex. With respect to this mean value the Davies et al. difference is a 2-$\sigma$ outlier. Excluding the Davies et al. value we obtain a mean difference of $-$0.406 with a standard deviation of $\sigma$ = 0.048 dex. In the following, we will use this value.

6 Effect of metallicity on Cepheid absolute magnitudes in the optical and near-infrared

We adopt the relative distance moduli obtained for the VIJHKs, and the two Wesenheit bands (last column of Table 2) and correct them for the relative reddening (0.045 mag) measured from the analysis in section 3.2. This yields the values quoted in column 4 of Table 3. Assuming that the differences of the values of the unreddened relative distances obtained from the Cepheids on the one hand, and eclipsing binaries on the other hand are due to the $-$0.406 dex difference in mean metal content of the Cepheids in LMC and SMC, we calculate the metallicity effect $\gamma$ in each filter in the following way:

[TABLE]

The values of $\gamma$ for the different filters and their uncertainties are given in the last column of Table 3.

Statistical tests have revealed slight nonlinearities near 10 days in the JHK P-L relations in the LMC (e.g. Bhardwaj et al., 2016). In order to test the stability of the derived $\gamma$ values with regard to the chosen samples of LMC and SMC Cepheids, we repeated the analysis a) introducing the same period cutoff at 2.5 days for the LMC Cepheid sample as done for the SMC sample; b) retaining in both Clouds only Cepheids with periods longer then 10 days; and c) comparing the magnitudes derived from the free fits to the full Cepheid samples at a specific intermediate period of 10 days. In Table 4, we give the results for the relative true distance between SMC and LMC for these cases, as well as the reddening differences, and the resulting metallicity effects together with their uncertainties. As can be appreciated from the numbers in Table 4, in case a) the changes are totally negligible; in case b) the derived metallicity effect is smaller than 1% for all bands except V and I where it is about 1.5%; and in case c) the metallicity effect is smaller than 2% in all bands. Most importantly, in all these cases the metallicity effect is consistent with a zero effect in all bands, within the very small uncertainties. We conclude that the details of the selection of the Cepheid samples in LMC and SMC do not have any significant effect on the values for the metallicity effect we obtain in the considered photometric bandpasses.

7 Discussion

As can be seen from the values in Table 3, our study reveals a very small metallicity effect, less than 2% in all bands investigated in this study. In particular, the effect is less than 1% in the optical I and near-infrared H and Ks bands, as well as in the two Wesenheit indices. Within the small uncertainty of 3% of the $\gamma$ values in Table 3, the metallicity effect derived in this paper is clearly consistent with a null effect in all optical and near-infrared bands; the same holds for the Wesenheit indices. The same conclusion was reached earlier by Majaess et al. (2011) for the optical VI Wesenheit index. Obviously, this conclusion of a vanishing metallicity effect is restricted to the metallicity range of $-$0.35 dex $>$ [Fe/H] $>$ $-$0.75 dex comprised by the classical Cepheid populations in LMC and SMC.

The sign of the small metallicity effect we find in all bands is in agreement with the study of Romaniello et al. (2008), and at odds with all the studies which derived the metallicity effect by comparing inner- and outer-field Cepheid samples in spiral galaxies and assuming a value of the metallicity gradient in the disc of these galaxies (usually determined from H II region oxygen abundances), a method which was pioneered by Freedman & Madore (1990) by a comparison of three fields at different galactocentric distances in the Andromeda galaxy. The most likely cause that these studies consistently found the more metal-rich Cepheids located in the inner galaxy regions to be significantly brighter than their more metal-poor counterparts in the outer part of the galaxy is the effect of blending which affects the crowded inner regions more severely than the outer regions of a spiral galaxy, even at HST resolution as for example in the study of M 101 Cepheids by Kennicutt et al. (1998).

We note that using gas-phase oxygen abundances in place of the stellar metallicities we presented in Sect. 5 does not change our conclusions. Taking the compilation of O/H abundances for HII regions in the LMC and the SMC by Bresolin (2011), who consistently derived them from the calculation of the gas electron temperature, we obtain a difference of -0.33 $\pm$ 0.13 dex between the SMC and the LMC. This difference yields a $\gamma$ value that is fully compatible with the one based on stellar metallicities alone.

Regarding the near-infrared spectral region, our current result is in line with the results of Freedman & Madore (2011), and of Storm et al. (2011) who also found with different methods a vanishing metallicity effect in the near-infrared regime, particularly in the H and K bands. The results of Storm et al. (2011) obtained from an application of the Near-Infrared Surface Brightness Technique to large samples of Milky Way and LMC, and a small sample of SMC classical Cepheids are confirmed and strenghtened by the extension of this work to a much larger sample of SMC Cepheids (Gieren et al., 2017). The present determination however is more accurate, by a factor of at least 2, than these previous measurements. Regarding the optical regime, the cited previous studies did find significant metallicity effects, particularly in the V band, whereas our current study indicates that the effect in V and I is at most 2% and consistent with a null effect in these bands, too.

The main assumption underlying our study is the universality of the slope of the Cepheid P-L relation in all bands we have considered, at least in the range from $-$0.35 to $-$0.75 dex corresponding to the Cepheid metallicities in LMC and SMC. While there have been a few studies challenging this assumption (e.g. Sandage & Tammann, 2008), most empirical evidence, particularly from studies of Cepheids in nearby Local Group galaxies with excellent datasets as in our own Araucaria Project have confirmed that the observed slope of the P-L relation is, within small uncertainties, identical for the Cepheid populations in a number of galaxies having a range of metallicities of their young stellar populations, from LMC abundances down to about -1.0 dex in the dwarf irregular galaxies WLM and IC 1613 (Bresolin et al., 2006, 2007). This seems to be particularly true for the near-infrared JHK bands. For example, Pietrzyński et al. (2006), in their distance study of the very metal-poor Local Group galaxy IC 1613 found excellent agreement between the observed slopes of the P-L relations in the J and K bands and those of the corresponding LMC Cepheid relations of Persson et al. (2004). Claims that P-L relation slopes in different optical bands might vary with metallicity (e.g. Bono et al., 2010) were almost exclusively based on Cepheid data from external galaxies well beyond the Local Group where crowding and blending of the Cepheids is likely to be a problem which affects the Cepheid photometry at an unknown level. Such blending will affect the Cepheid photometry in a differential way, affecting the short-period fainter Cepheid more strongly than the brighter long-period Cepheids, which could introduce spurious changes in the slopes of the observed P-L relations. It is certainly much safer to rely on Local Group Cepheid data and observed P-L relations, where the best Cepheid photometric data are reasonably unaffected by the very complicated blending and crowding problem.

8 Conclusions

We have used the very high-quality optical photometric data of the OGLE Project in the V and I bands together with near-infrared JHKs photometry obtained on the IRSF telescope in South Africa to establish very precise period-luminosity relations in both Magellanic Clouds. These data have been used to determine the effect of metallicity on Cepheid absolute magnitudes in these bands, and on the optical and near-infrared Wesenheit indices. We did this by comparing the relative distance between LMC and SMC as determined from the Cepheids in the different photometric bands to the distance difference between the Clouds which has been measured very accurately with late-type eclipsing binaries in both Clouds. The comparison yields a metallicity effect smaller than 2% in all bands and in the Wesenheit indices, which within the 3% uncertainty of the determination means that there is no significant metallicity effect in neither of the bands.

We selected for our study Cepheids in the central parts of the Magellanic Clouds where observed depth effects are small. Since the components of our binary systems (helium burning giants) are expected to have the same spatial distribution as the Cepheids and we have been able to use sizeable samples of eclipsing binaries in both Clouds, our results should not be affected in any significant way by the geometrical extension of both Magellanic Clouds (Pietrzyński et al., 2013; Graczyk et al., 2014). Reported differences between P-L relations based on different samples of Cepheids (e.g. Persson et al., 2004; Ripepi et al., 2012; Inno et al., 2013; Macri et al., 2015; Ngeow et al., 2015; Ripepi et al., 2016) are very small and confirm that our results are not significantly affected by the depth effect.

While the present result is restricted to the metallicity range comprised by the Cepheids in both Magellanic Clouds, it is this the metallicity range into which the Cepheids observed in distant spiral galaxies at several Mpc typically fall. While in massive spiral galaxies Cepheid variables very close to the centers may have abundances even above solar, it is the Cepheids at relatively large galactocentric distances in these galaxies which are typically used for determining the distances of their host galaxies because they are better resolved and less affected by crowding in the images. Therefore the metallicity range from LMC to SMC abundances seems to be the most relevant range for Cepheid distance work to spiral galaxies located at distances of about 5-20 Mpc, which makes the present results for the metallicity effect on Cepheids relevant and important for the determination of Cepheid distances to SN Ia host galaxies in the near-infrared (H-band) regime using HST or, in the near future, the JWST space telescope.

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 program (grant agreement No 695099). WG, MG and DG gratefully acknowledge financial support for this work from the Millenium Institute of Astrophysics (MAS) of the Iniciativa Cientifica Milenio del Ministerio de Economia, Fomento y Turismo de Chile, project IC120009. We (WG, GP and DG) also very gratefully acknowledge financial support for this work from the BASAL Centro de Astrofisica y Tecnologias Afines (CATA) PFB-06/2007. We also acknowledge support from the Polish National Science Center grant MAESTRO DEC-2012/06/A/ST9/00269. Based on observations made with ESO telescopes under programme 098.D-0263(A,B), 097.D-0400(A), 097.D-0150(A), 097.D-0151(A) and CNTAC programme CN2016B-38, CN2016A-22, CN2015B-2, CN2015A-18. Last not least, we are grateful to the OGLE and IRSF team members for providing data of outstanding quality which made this investigation possible.