The Milky Way Cepheid Leavitt law based on Gaia DR2 parallaxes of companion stars and host open clusters populations

Classical Cepheids provide the foundation for the empirical extragalactic distance ladder. Milky Way Cepheids are the only stars of this class accessible to trigonometric parallax measurements. However, the parallaxes of Cepheids from the second Gaia data release (GDR2) are affected by systematics due to the absence of chromaticity correction and occasionally by saturation. As a proxy for the parallaxes of 36 Galactic Cepheids, we adopt either the GDR2 parallaxes of their spatially resolved companions or the GDR2 parallax of their host open cluster. This novel approach allows us to bypass the systematics on the GDR2 Cepheids parallaxes that is induced by saturation and variability. We adopt a GDR2 parallax offset of 0.046 mas with an uncertainty of 0.015 mas that covers most of the recent estimates. We present new Galactic calibrations of the Leavitt law in the V, J, H, K_S and Wesenheit W_H bands. We compare our results with previous measurements from the Hubble Space Telescope (HST) and compute a revised value for the Hubble constant anchored to Milky Way Cepheids. From an initial Hubble constant of 76.18 +/- 2.37 km/s/Mpc based on parallax measurements without Gaia, we derive a revised value by adopting companion and average cluster parallaxes in place of direct Cepheid parallaxes and we find H0 = 73.07 +/- 1.75 (stat. + syst.) +/- 1.88 (ZP) km/s/Mpc when all Cepheids are considered, and H0 = 73.51 +/- 1.76 (stat. + syst.) +/- 1.91 (ZP) km/s/Mpc for fundamental mode pulsators only.


Introduction
Classical Cepheids (CCs) have historically a major importance among variable stars because of the simple correlation between their pulsation period and their intrinsic luminosity, also called the Leavitt law or the period-luminosity (hereafter PL) relation (Leavitt 1908;Leavitt & Pickering 1912). However, after more than a century of active research, the absolute calibration of the Leavitt law is still unsatisfactory due to the lack of precise and direct distance measurements for a sizeable sample these stars. A careful calibration of this relation and especially of its zero-point is fundamental, as it is used to anchor extragalactic distances and to derive the expansion rate of the Universe, i.e. the Hubble constant H 0 . In fact, the determination of H 0 from the Cosmic Microwave Background (CMB) based on the standard Λ-Cold-Dark-Matter (ΛCMD) model (Planck Collaboration et al. 2018) is currently found to be in ∼ 5σ tension with the empirical or direct distance ladder measurements . This tension may have important implications in cosmology, and may even point towards new physics beyond ΛCMD (Verde et al. 2019).
The Leavitt law calibration requires the independent and accurate distance measurement for a sample of CCs. Unfortunately, Gaia's second data release (hereafter GDR2) contains a number of systematic effects that may reduce the precision of the parallaxes of CCs (Gaia Collaboration et al. 2018). First, CCs are bright stars, so a small number with G < 6 mag are affected by saturation, making their parallaxes unreliable. In addition, CC colors cycle through many variations during the parallax cycle: the effective temperature of a Cepheid changes on average by 1000 K over a full pulsation cycle, which means ∼0.5 mag in op-Article number, page 1 of 14 arXiv:2006.08763v1 [astro-ph.SR] 15 Jun 2020 A&A proofs: manuscript no. Breuval_AA_Letter tical bands, so this may add additional noise to their astrometry due to the chromaticity of the PSF. Future Gaia data releases are expected to include chromaticity corrections for variable stars and incorporate a better model of the PSF to deal with saturation. While recent analyses of Gaia DR2 parallaxes for CCs with G>6 mag do not appear to suffer excess noise (Groenewegen 2018;Riess et al. 2018b;Gaia Collaboration et al. 2017;Clementini et al. 2019), it is important to pursue alternative approaches to extract parallaxes from Gaia DR2 for CCs that are insensitive to these systematics.
Even in the absence of systematic errors, the use of open cluster parallaxes for CCs they host can provide enhanced precision over the use of a single CC parallax. Because open cluster parallaxes are based on many stars, the increased precision from averaging as well as the ability to reject outliers for stars in astrometric binaries is extremely valuable.
In the present letter, we seek to calibrate the Milky Way (MW) Cepheid Leavitt law using stars that are not affected by these issues and to benefit from the gain in precision afforded by cluster average parallaxes. In Sect. 2, we introduce our sample of stars and their associated parallaxes. In Sect. 3.1, we derive calibrations of the Leavitt law in various bands. Then, in Sect. 3.2, we compare our GDR2 parallaxes with the corresponding expected parallaxes from Hubble Space Telescope (HST) measurements and in Sect. 3.3 we derive a value for the Hubble constant anchored to Milky Way Cepheids.

Sample of parallaxes
We consider two sets of parallaxes: one based on Cepheid companions and one based on average cluster parallaxes. The benefits of these samples are flux and color constancy (companions and clusters) and averaging over a large sample (clusters).

Parallaxes of Cepheids resolved companions
Recently, Kervella et al. (2019) presented a sample of 28 Galactic Cepheids that are members of gravitationally bound and spatially resolved stellar systems. In these systems, Cepheids companions are photometrically stable stars and their GDR2 parallaxes are therefore not affected by such a strong chromatic effect as Cepheids. As the CCs and their companions share the same parallax (their relative distance is negligible compared to the distance to Gaia), the GDR2 parallaxes of the companions provide a natural proxy for those of the CCs. The companions parallaxes are precise at 15% in average.
The angular separation between the CCs and their companions is in most cases larger than 10 arcsec, which is large enough to prevent flux contamination, given the brightness of the CCs. At 10" separation for stars hundreds to thousands of parsec distant, there is no expected effect of orbital motion on parallax or proper motion measurements: the parallaxes of the CCs and companions are not sensitive to the binarity of these wide systems.
The GDR2 astrometry is generally of poor quality for very bright stars (G < 6 mag), due to calibration issues and saturation (Riess et al. 2018b;Drimmel et al. 2019;Lindegren 2019). This occurs independently of the chromaticity issue raised previously, whether the star is variable or not. While several Cepheids of our sample are close to this limit, with an average G magnitude of 8 mag, their companions are in average 7 mag fainter than their parent Cepheids. The companions are therefore not as affected as CCs by the saturation issue and they are far off from the sen-sitivity limit. They consequently belong to the best dynamical range for Gaia.
We perform a selection based on GDR2 quality criteria and pulsation modes (see Appendix A.2 and C) that results in a sample of 22 GDR2 parallaxes of Cepheids resolved companions, listed in Table A.1.

Parallaxes of Cepheids in Open Clusters
Open clusters (OCs) contain a significant number of stars located at the same distance and are numerous in the Milky Way. Therefore, identifying Cepheids in OCs allows us to estimate their distances, with a gain in precision by taking the average over a population compared to individual parallax measurements. We performed a cross-match between the Ripepi et al. (2019) reclassification of GDR2 Cepheids and the Cantat-Gaudin et al. (2018) catalog of Milky Way Open Clusters. This catalog provides parallaxes for 1229 OCs that are precise at ∼1%, based on the GDR2 parallaxes of their member stars. Our comparison is based on 5 membership constraints: the separation θ, parallax , proper motion µ * α and µ δ , and age. More details on the data and the cross-match are given in Appendix A.2. This selection and a crossing with the literature resulted in a final sample of 14 cluster Cepheids, whose data are provided in Table A

Calibration of the Leavitt law
In this section, we combine the 22 Cepheids companions with the 14 open cluster Cepheids. Their parameters are listed in Table 1. The Leavitt law calibration is obtained by performing a fit of the Astrometry Based Luminosity function (ABL) on our sample, for which we applied a Monte Carlo simulation. More details on the method are provided in Appendix D.
We found 5 Cepheids to be present in both samples. For these 5 stars, the companion parallax and the cluster parallax agree within 1σ, except for U Sgr that is at 1.4σ. In order to avoid any correlation between our two sets of parallaxes, we recomputed for these 5 stars the Cantat-Gaudin et al. (2018) cluster parallaxes as the median of all stars parallaxes after excluding the companion. We found our new cluster parallaxes to differ by 0.5 µas at most from the original values, so we adopted these new parallax values and considered the two sources of measurement to be independent and non-correlated. For these 5 Cepheids, both parallax measurements (cluster and companion) are considered independently in the linear fit.
GDR2 parallaxes are subject to a zero-point offset, whose value was studied extensively but is still debated. The different estimates are discussed in Appendix E. In the following, we adopt ZP GDR2 = −0.046 mas (Riess et al. 2018b), which is close to the median of all values (see Table E.1).
In order to correct for the extinction, different formulations are available (Savage & Mathis 1979;Cardelli et al. 1989). We adopt the Fitzpatrick (1999) formulation with R V = 3.3, that yields R J = 0.86, R H = 0.55, and R K = 0.37. This will allow a direct comparison of our calibration with that of Riess et al. (2016), based on HST Fine Guidance Sensor (FGS) and HST Wide Field Camera 3 (WFC3) measurements.
The PL coefficients obtained in different bands for ZP GDR2 = −0.046 mas are listed in Table 2. For different ZP GDR2 values, the coefficients are provided in Table E.2 in the Appendix. The Leavitt law calibration in the K S band is displayed in Fig. 1   the difference between the input parallax and the parallax given by the best fit. This calibration gives a reduced χ 2 of 0.36 and a dispersion of σ = 0.14 mag.
An equivalent calibration, based this time on direct Cepheid parallaxes, is presented in Fig. 2. When the CC parallaxes are adopted, we obtain χ 2 r = 0.86 and a dispersion of σ = 0.19 mag. The dispersion of the PL relation based on Cepheid parallaxes ( Fig. 2) does not appear to be systematic, but rather results in a larger spread not accounted in the uncertainties. The PL coefficients derived from GDR2 parallaxes of Cepheids are provided in Table F.1.

Comparison between GDR2 and HST parallaxes
In this section, we compare our sample of GDR2 parallaxes with a set of parallaxes available before the Gaia era. Riess et al. (2016), hereafter R16, combined 10 Cepheid parallaxes from HST/FGS (Benedict et al. 2007) with three Hipparcos measurements and 2 Cepheids with parallaxes measured by spatial scanning with the HST/WFC3 (Riess et al. 2014;Casertano et al. 2016), and obtain: 3) ensures the consistency of this comparison. To account for the width of the instability strip (σ = 0.07 mag) and for the photometric transformations from ground to HST system (σ = 0.06 mag), we set the apparent magnitudes uncertainties to 0.09 mag. Fig.  3 shows the comparison between the GDR2 parallaxes of our sample corrected by a 0.046 mas offset and the predicted parallaxes from R16. The GDR2 parallaxes appear to be slightly underestimated compared with the predicted values, especially for Cepheids with large parallax values.
The prototype δ Cep is particularly interesting for this study: it hosts a resolved companion with a GDR2 parallax and it is also present in the sample of HST/FGS parallaxes by Benedict et al. (2007). The GDR2 parallax of its companion is 3.393 ± 0.049 mas while its HST/FGS parallax is 3.66 ± 0.15 mas. These two measurements differ by 1.7σ (7% in relative terms), which  (Fernie et al. 1995), to which we applied a multiplicative factor of 0.94. Mean apparent magnitudes in V, J, H, K S bands are from the catalog compiled by Groenewegen (2018): V band magnitudes are originally from Mel'nik et al. (2015) and NIR magnitudes are given in the 2MASS system with the original references provided in the last column. Apparent magnitudes in the W H system are also provided, their uncertainties include the photometric transformation errors.  Notes. ( ) Cepheid pulsating in the first-overtone mode. In that case, the period was converted following the approach described in Appendix C. ( * ) The parallaxes presented in this agrees with the general trend observed in Fig. 3. We note that δ Cep has no valid parallax in GDR2, so its companion parallax is the only possible alternative to HST/FGS measurements.

Implications on the distance scale
The Following the method presented in Section 4 in Riess et al. (2018a), we translate our previous parallax comparison into a comparison in terms of the Hubble constant. We want to examine the impact of changing the MW anchor alone on the H 0 measurement that depends on three anchors. Hence, we look at the H 0 value from R16 that pertains only to the MW. We use the relation H 0, GDR2 = α H 0, R16 where α = GDR2 / R16 and H 0, R16 is the value anchored to Milky Way Cepheids only and is equal to 76.18 ± 2.37 km s −1 Mpc −1 .
For each star of the sample, we derive the corresponding α value and we adopt a Monte Carlo approach to estimate the final average α value of the sample. We performed this calculation on different samples and listed the resulting H 0 values in Table 3. The uncertainties on H 0 include the final error on the R16 estimate excluding the anchors (1.8%), the error on the estimation of α and finally the uncertainties on the photometric relations to convert ground-based magnitudes into HST magnitudes (1.5%).
Changing the GDR2 parallax offset by 0.015 mas results in a change of 2.6% in the Hubble constant, therefore, we adopted a confidence interval of 0.015 mas around the -0.046 mas zeropoint and added a 2.6% uncertainty to account for this effect. We obtain a final value of 73.51 ± 2.60 km s −1 Mpc −1 for fundamental modes only and of 73.07 ± 2.58 km s −1 Mpc −1 for all stars included. Both values are very consistent with the LMC and NGC 4258 anchor results derived by Riess et al. (2019a), and also very close to the result by Reid et al. (2019). The last value agrees at the 1.1 σ level with that of Freedman et al. (2020) and at the 2.2 level with the Planck Collaboration et al. (2018) measurement.
We note that the CCs used to calibrate the PL relation and H 0 have lower mean periods than most extragalactic Cepheids found by HST. Though there is no evidence of a break in the PL relation at log P = 1 for the W H magnitude system , it remains important to add longer period Cepheids to the parallax calibration to maintain low systematics.

Conclusions
We presented an original calibration of the Milky Way Leavitt law based on GDR2 parallaxes of resolved Cepheid companions and on GDR2 parallaxes of open clusters hosting Cepheids. Companion and cluster members are not subject to large amplitude photometric and color variability, which reduces the potential for systematic parallax uncertainties. Additionally, average cluster parallaxes established for solid cluster member stars allows to improve the precision by a factor √ N where N is the number of member stars and is generally larger than 100. The comparison of our calibration with a previous work based on HST Cepheid parallaxes indicates a systematic offset between both measurements. By replacing the trigonometic parallaxes used in R16 by companion and cluster average parallaxes, we render the Milky Way, the LMC, and NGC4258 Leavitt Laws more consistent with one another: we find a MW estimate of 73.51 ± 2.60 km s −1 Mpc −1 for fundamental modes only and of H 0 = 73.07 ± 2.58 km s −1 Mpc −1 for all stars included.
The inclusion of the variability of CCs is not expected in the astrometric processing of the third Gaia data release. However, the effects of the systematics due to the absence of chromaticity correction on Cepheids parallaxes should be reduced in the next releases thanks to the larger number of measurements. The future developments will help to pursue the community goal to measure H 0 with utmost precision and accuracy.
Acknowledgements. We thank D. Graczyk, S. Borgniet and L. Inno for their comments that led to improvements of the present study. We are grateful to T. J. Calderwood from AAVSO for the photometry of Polaris. 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 ( For a given Cepheid, when more than one companion was found by Kervella et al. (2019), we selected the companion with the smallest uncertainty on its parallax.
Various quality indicators are introduced in the second release of Gaia data, such as the re-normalised unit weight error (RUWE, noted in the following). It is particularly pertinent to use because it evaluates the quality of the parallax of a star compared to other stars of the same type. This parameter is defined by  as: where UWE = χ 2 /(N − 5) is the unit weight error and u 0 is an empirical normalisation factor which is not directly available in the Gaia release but which can be computed from the lookup The star CE Cas B is a particular case because its companion, CE Cas A, is also a Cepheid. The two components of CE Cas are present in the GDR2, but with statistically different parallaxes ( A = 0.317 ± 0.031 mas; B = 0.262 ± 0.030 mas). We exclude both stars from our sample as a precaution.
The star α UMi is extremely bright, with K ≈ 0.5 mag. Therefore, measuring accurate photometry for this star is particularly challenging. It has no valid parallax in GDR2 and appears saturated in most catalogs (Skrutskie et al. 2006). The only accurate average magnitudes based on several pulsation cycles were found in the AAVSO database that provides J = 0.93 ± 0.01 mag and H = 0.67 ± 0.01 mag in the UKIRT system. Additionally, the uncertain pulsation mode as well as the age difference between the Cepheid and its companion raise questions concerning the properties of α UMi and whether it should be included in PL-relation fits (Anderson 2018;Bond et al. 2018;Groenewegen 2018). We decided to exclude this star from our sample.
The Cepheid RS Pup has been studied in details by Kervella et al. (2014) who estimated its parallax to 0.524 ± 0.022 mas, using polarimetric HST images of the light echoes propagating in its circumstellar nebula (see also Kervella et al. (2017)). We note that this independent estimate is in very good agreement with the GDR2 parallax of RS Pup companion (0.503 ± 0.045 mas), but differs by 0.060 mas from the GDR2 parallax of the Cepheid itself (0.584 ± 0.026 mas).
A comparison between direct GDR2 Cepheid parallaxes and GDR2 companion parallaxes is displayed in Fig. A.1.

Appendix A.2: Cluster Cepheids sample
Following Anderson et al. (2013), we start the search for potential cluster members by looking at the proximity in the sky: we selected all Cepheids located in a region of 10r 50 around each cluster (where r 50 is the radius containing half of the members) and we find a total of 2647 couples. For these couples, we compared the parallaxes, the proper motions and the age of both component. Since GDR2 parallaxes of Cepheids may be affected by systematics due to the absence of chromaticity correction, we account for this effect by including 20% error in quadrature. The proper motions for Cepheids and open clusters are taken from Ripepi et al. (2019) and Cantat-Gaudin et al. (2018) respectively.
The age of open clusters is provided by Kharchenko et al. (2013), and the age of Cepheid is derived using period-age relations from Anderson et al. (2016).
We also searched in the literature for additional combinations and examined if they satisfy our membership constraints. Indeed, some Cepheids are not present in the Ripepi et al. (2019) reclassification, so they could not be found by means of our crossmatch. Anderson et al. (2013) previously presented many of our couples, and provide 3 additional combinations that verify our membership criteria: TW Nor, CV Mon and V0367 Sct respectively in Lyngå 6, vdBergh 1 and NGC 6649.
We find a total of 14 Cepheids being candidate members of open clusters. They are listed in Table A.2, where filled circles stands for the agreement of a parameter at 1σ or less. In this table is also provided the separation in arcmin between a Cepheid and the center of its host cluster. The field charts of each open cluster Cepheid is displayed in Fig. A.2, A.3 and A.4.
The Cepheid QZ Nor is a particular case: located at 18 arcmin of NGC 6067, it is a peripherical member of this cluster. The 9σ difference in µ δ could be explained by the fact that the Cepheid is leaving the cluster. This membership was identified by Anderson et al. (2013) as bona fide. Moreover, QZ Nor is also present in the sample of companions found by Kervella et al. (2019): the stable star Gaia DR2 5932565899990412672 is located at 16" (30 kau) from the Cepheid. Its GDR2 parallax of 0.452 ± 0.132 mas agrees particularly well with the 0.443 ± 0.002 mas parallax of NGC 6067 from Cantat-Gaudin et al. (2018). Therefore, we decide not to exclude this couple.
The cross-match also resulted in potential members that only have 2MASS single epoch photometry available. Since average The NIR magnitudes from Genovali et al. (2014) are derived by template fitting and provided in the 2MASS system. For the remaining stars, the mean magnitude is computed as the median of the available data in Welch et al. (1984), Schechter et al. (1992) and 2MASS (Skrutskie et al. 2006). For RS Nor, the averaged NIR magnitudes were derived by fitting the photometric light curves using the SPIPS algorithm (Mérand et al. 2015). In the V band, all mean magnitudes are provided in the standard Johnson system and taken from Mel'nik et al. (2015). An uncertainty of 0.03 mag on those magnitudes is adopted.
From apparent magnitudes, we build the reddening-free Wesenheit magnitudes m W H (Madore 1982) which are a combination of HST bands defined by Riess et al. (2018a) as: where R = 0.386 is derived from the Fitzpatrick (1999) formulation with R V = 3.3. The correspondance between HS T filters F160W, F555W and F814W and ground based magnitudes in J, H and V is given by: These HST relations have a scatter of 0.06 mag and are computed with the average apparent J, H and V magnitudes from Table 1. We note that the transformation from ground-based magnitudes into the HST system requires to account for the Count-Rate Non-Linearity (CRNL) effect (Riess et al. 2018b). This bias affects the infrared detectors on WFC3 and has the consequence  of decreasing the magnitude of faint stars as extragalactic CCs, compared to bright stars as Milky Way CCs. This correction is performed by adding 0.026 mag to HST F160W apparent magnitudes (Riess et al. 2019b).
We also account for the width of the instability strip in the photometry errors. In the V band, Macri et al. (2006) find a dispersion of of 0.23 mag: an intrinsic width is 0.22 mag and is obtained after subtracting the estimated measurement errors. For J, H and K bands, Madore et al. (2017) finds a scatter of 0.12 mag, which leaves an intrinsic width of 0.11 mag in NIR bands. Finally, Riess et al. (2019a) find a dispersion of 0.075 mag in the W H band, yielding an intrinsic width of 0.07 mag for the instability strip.
In order to compute absolute magnitudes, we need to correct the apparent magnitudes from the interstellar absorption. We take the E(B − V) values from the DDO database (Fernie et al. 1995), which is a compilation of various E(B − V) from the literature determined in the same system. Following Groenewegen (2018), we apply a multiplicative factor of 0.94 to these reddening values. For BP Cir and DK Vel, different pulsation modes were found: they are both classified as FO Cepheids by GDR2 and other studies (Zabolotskikh et al. 2004;Ripepi et al. 2019), while they are listed as fundamentals by Luck (2018). The two stars are also consistent with fundamental pulsators in the PL plane. Given the disagreement between the different references about the pulsation mode of BP Cir and DK Vel, we decide to exclude them from the sample.
In order to establish accurate PL and PW relations without excluding the first overtones, we converted their observed periods P FO into the fundamental mode equivalent period P F using the equation by Kovtyukh et al. (2016): Field and cluster Cepheids have a similar distribution in the Galactic plane, so they have similar metallicity distributions and both of them can be assumed close to solar (Romaniello et al. 2008). The first overtones of the sample have periods P FO comprised between 3 and 4 days. In this range of periods, we can approximate the previous equation by the linear relation: The conversion of first overtones into fundamentals is listed in Table C.2. The positions of these Cepheids in the PL plane after the transformation are consistent with the distribution of fundamental pulsators.
Even though converting first overtones into fundamentals may introduce a small uncertainty on periods, we decide to include them in the sample for the calibration of the Leavitt law. Indeed, the periods obtained after conversion with the relations from Feast & Catchpole (1997) and Kovtyukh et al. (2016) only differ by 0.006 days. Gallenne et al. (2018) find a difference of less than 1% between an empirical conversion law and a theoretical one. Including the five first overtones of the sample with their modified periods instead of rejecting them induces only a very small change on the intercept of the PL relation and allows to improve the precision of the fit.

Appendix D: Methods
In order to calibrate the PL relations as well as Period-Wesenheit (PW) relations, we used the approach introduced by Feast &   Catchpole (1997) and Arenou & Luri (1999) and we computed the Astrometric Based Luminosity (ABL), defined as: where M λ is the absolute magnitude, m λ is the dereddened apparent magnitude and is the parallax in milliarcseconds. Calibrating the Leavitt Law following this approach is equivalent to determine the coefficients a and b in the equation: We performed a weighted fit of the ABL function by using the curve_fit function from the python Scipy library. The robustness of the fit and of the uncertainties is ensured by a Monte Carlo approach, applied with 10 000 iterations. The distributions of the slope and zero-point of our K S Leavitt law obtained by this technique are displayed on the histograms in Fig. D.1.
We used the formalism detailed in Gallenne et al. (2017), i.e. we adopted the following linear parametrization: where a λ and b λ are respectively the slope and the zero-point of the PL relation. Such a parametrization removes the correlation between a λ and b λ and minimizes their respective uncertainties. The optimum value of log P 0 depends on the dataset (see Gallenne et al. (2017)  where log P i are the periods of the stars, and e i are the uncertainties on their parallax; denotes the averaging operator. We find our sample centered around log P 0 = 0.84. values were derived by Riess et al. (2018b) and Groenewegen (2018), who estimate −0.046 mas and −0.049 mas respectively. The recent determinations of ZP GDR2 are listed in Table E.1. Table E.2 gives the PL coefficients obtained with different parallax offsets. Changing the zero-point from −0.029 mas to −0.070 mas results in a change of 0.7% in slope and 2.8% (or 0.153 mag) in intercept for the K S band PL relation. Both slope and intercept increase with this offset variation.