11institutetext: Institute of Physics, Laboratory of Astrophysics, École
Polytechnique Fédérale de Lausanne (EPFL), 1290 Versoix, Switzerland 11email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
# A 0.9% calibration of the Galactic Cepheid luminosity scale
based on Gaia DR3 data of open clusters and Cepheids ††thanks: Tables 1 \- 5,
7, 11 and 17 are available in electronic form at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-
bin/qcat?J/A+A/

Mauricio Cruz Reyes  ID 11 Richard I. Anderson ID 11

(Received 19 August 2022; accepted 27 January 2023)

We have conducted a search for open clusters in the vicinity of classical
Galactic Cepheids based on high-quality astrometry from the third data release
(DR3) of the ESA mission Gaia to improve the calibration of the Leavitt law
(LL). Our approach requires no prior knowledge of existing clusters, allowing
us to both detect new host clusters and cross-check previously reported
associations. Our Gold sample consists of 34 Cepheids residing in 28 open
clusters, including 27 fundamental mode and seven overtone Cepheids. Three new
bona fide cluster Cepheids are reported (V0378 Cen, ST Tau, and GH Lup) and
the host cluster identifications for three others (VW Cru, IQ Nor, and SX Vel)
are corrected. The fraction of Cepheids occurring in open clusters within
$2\,$kpc of the Sun is $f_{\mathrm{CC,2kpc}}=0.088^{+0.029}_{-0.019}$.
Nonvariable cluster members allow us to determine cluster parallaxes to $\sim
7\,\mu$as in the range $12.5<G<17$ mag, where recent studies found that
parallax corrections by Lindegren et al. (2021) (L21) are accurate and require
no further offset corrections. By comparing Cepheids in MW clusters to
Cepheids in the LMC, we confirm these independent results and the adequacy of
the L21 corrections for the cluster members in this range. By combining
cluster and field Cepheids, we calibrate the LL for several individual
photometric passbands, together with reddening-free Wesenheit magnitudes based
on Gaia and HST photometry, while solving for the residual offset applicable
to Cepheid parallaxes, $\Delta\varpi_{\mathrm{Cep}}$. The most direct
comparison of our results with the SH0ES distance ladder yields excellent
($0.3\sigma$) agreement for both the absolute magnitude of a $10$ d solar
metallicity Cepheid in the near-IR HST Wesenheit magnitudes,
$M_{H,1}^{W}=-5.914\pm 0.017$ mag, and the residual parallax offset,
$\Delta\varpi_{\mathrm{Cep}}=-13\pm 5\,\mu$as. Despite the use of a common set
of photometry, this is an important cross-check of the recent Hubble constant
measurement by Riess et al. (2022a) based on independently determined cluster
membership and average parallaxes. Using the larger sample of $26$ Gold
cluster Cepheids and $225$ MW Cepheids with recent Gaia DR3 astrometry and
photometry, we determine $M_{G,1}^{W}=-6.051\pm 0.020$ mag in the optical Gaia
Wesenheit magnitude at the sample average iron abundance of
($\langle\mathrm{[Fe/H]}\rangle=0.069$) and
$\Delta\varpi_{\mathrm{Cep}}=-22\pm 3\,\mu$as. Correcting to solar metallicity
yields $M_{G,1}^{W}=-6.004\pm 0.019$ mag and
$\Delta\varpi_{\mathrm{Cep}}=-19\pm 3\,\mu$as. These results mark the
currently most accurate absolute calibrations of the Cepheid luminosity scale
based purely on observations of Milky Way Cepheids, and it is also the most
precise determination of the residual Cepheid parallax offset at a
significance of $6-7\sigma$.

###### Key Words.:

Stars:distances – Stars: variables: Cepheids – open clusters and associations:
general – distance scale

##  1 Introduction

The absolute calibration of the classical Cepheid luminosity scale is
fundamental for a distance estimation in the nearby Universe and the accurate
measurement of Hubble’s constant, $H_{0}$. The third data release (DR3) of the
ESA mission Gaia has provided astrometry of unprecedented quantity and quality
(The Gaia collaboration et al., 2016, 2021) for approximately 1.5 billion
stars in the magnitude range $3<G<21$, including 14992 classical Cepheid stars
(Eyer et al., 2022; Ripepi et al., 2022b) with an average parallax uncertainty
of $70\,\mu$as. Because the parallax is generally considered the gold standard
of geometric distance measurements, the Gaia parallaxes are of crucial
importance for the absolute calibration of Leavitt’s law Leavitt & Pickering
(1912, henceforth: LL), also known as the period-luminosity relation, and they
are of great interest for all further applications of Cepheids as distance
tracers. In particular, Gaia parallaxes are required to clarify the
implications of the current $5\sigma$ discrepancy between the value of $H_{0}$
measured using a distance ladder composed of classical Cepheids and type Ia
supernovae (e.g., Riess et al., 2022b) and the value of $H_{0}$ inferred from
observations of the ESA mission Planck of the cosmic microwave background
assuming a flat $\Lambda$CDM Universe (Planck Collaboration et al., 2020).

However, Gaia-based LL calibrations based on Cepheid parallaxes must currently
simultaneously solve for a residual parallax offset due to systematics of the
Gaia data processing in addition to the LL intercept and slope (e.g., Riess et
al., 2021). Because this simultaneous parallax offset determination reduces
the precision to which Gaia can calibrate the LL, strategies for mitigating
this problem are needed.

Lindegren et al. (2021, henceforth: L21) derived corrections to the zeropoint
offset of about $10-30\,\mu$arcsec, whose exact value depends nontrivially on
the magnitude of the observed source, its position in the sky, and its color.
Several studies (not necessarily based on Cepheids) have investigated residual
(compared to Lindegren’s correction) zeropoint offsets, generally finding good
agreement at magnitudes ($G\gtrsim 13$ mag) at which L21 is well calibrated
(e.g., Huang et al., 2021; Riess et al., 2021; El-Badry et al., 2021), whereas
an offset remains at the brighter end, where the L21 calibration was based on
fewer sources. The origins of these residual offsets are complex and not yet
fully understood, although it is likely that they originate from differences
between the Cepheid and quasar samples, with Cepheids being systematically
brighter, of redder intrinsic color, and photometrically and chromatically
variable. Moreover, the Milky Way Cepheids that were used to calibrate the LL
fall within a magnitude range ($G\lesssim 13$ mag) that requires special
observational and data-processing steps to avoid saturation (including the
gating mechanism to avoid saturation and changing from 2D to 1D image
processing for the astrometric model, cf. L21).

An interesting possibility for avoiding difficulties related to this zeropoint
systematic could be the use of parallax information derived from stars that
are observationally as similar as possible to the objects used to determine
the Gaia systematics. Because Cepheids are relatively young stars ($<300$
Myr), they are occasionally found in open star clusters (cf. Anderson et al.,
2013, and references therein), whose brightest main-sequence members will tend
to be bluer than Cepheids, and several magnitudes fainter. At the same time,
open clusters contain many stars, so that an average cluster parallax will
benefit from a $\sqrt{N}$ improvement in precision, eventually limited by the
angular covariance of the Gaia parallaxes (Lindegren et al., 2021; Apellániz
et al., 2021; Vasiliev & Baumgardt, 2021; Zinn, 2021).

The currently most common approach to identifying cluster Cepheids is to
consider cluster input lists from studies based on Gaia astrometry (Cantat-
Gaudin & Anders, 2020; Hunt & Reffert, 2021; Castro-Ginard et al., 2022; Zhou
& Chen, 2021; He et al., 2022) and to then compare the astrometric parameters
of Cepheids with the average cluster parameters (Anderson et al., 2013;
Breuval et al., 2020; Zhou & Chen, 2021; Medina et al., 2021). However, there
is no guarantee that all Cepheid-hosting clusters have been detected so far,
and the selection function of clusters is not well known. It is also rather
common for Cepheids to reside in the coronae of their host clusters, that is,
farther from the center than the typical cluster core radius of $\sim 4\,$pc
(e.g., Anderson et al., 2013). This is to some extent expected from the
clustered star formation process that causes the majority of birth clusters to
disperse into the field over timescales of tens of million years (Dinnbier et
al., 2022). Tidal deformations further cause cluster shapes to deviate from
circular over hundreds of million years, thus breaking the symmetry of the
appearance and complicating the detection of cluster members against a highly
contaminated background (Boffin et al., 2022). Additionally, it is quite
common for multiple clusters to exist relatively close to each other on the
sky (Turner, 1998) because of the high density of clusters in spiral arms and
the superposition on sky of multiple spiral arms. Substantial and spatially
variable extinction can further complicate the issue. To most reliably
determine the most complete sample of cluster Cepheids detectable with Gaia
DR3 data, we therefore adopted the approach of searching for clusters in the
vicinity of Cepheids, rather than the other way around.

A major improvement of the extragalactic distance ladder built by the _SH0ES_
project (Riess et al., 2022b) has been the photometric homogeneity of Cepheid
observations carried out exclusively in the Hubble Space Telescope (HST)
photometric system. With the release of time-series observations in Gaia DR3,
there is now an additional data set of very high quality, well-resolved
multichromatic observations based on a well-characterized and homogeneous
photometric system that includes observations of Milky Way and Local Group
Cepheids, reaching Cepheids as far as M31 and M33 (Evans et al., 2022), albeit
with increased uncertainties due to higher instrumental noise and higher
crowding. The goal of this paper is to leverage these unprecedented data sets
to achieve the most accurate absolute calibration of the MW LL in well-
characterized filters, notably including the reddening-free near-IR HST
Wesenheit function used by the SH0ES team to measure the Hubble constant
(Riess et al., 2022b), while simultaneously solving for the residual parallax
offset of Cepheids.

This article is organized as follows. Section 2 describes our method for
detecting and estimating the parameters of clusters in the physical vicinity
of MW Cepheids based on Gaia data and the estimation of membership
probabilities for the Cepheids. Section 3 separates the sample of cluster
Cepheid candidates into Gold, Silver, and Bronze samples. Section 4 presents
the simultaneous calibration of the Cepheid LL in multiple photometric bands
and an LMC-based cross-check of the L21 corrections applied to cluster member
stars. Section 5 presents an additional discussion, and Sect. 6 lists our
conclusions. Additional tables and figures are provided in the online
appendix.

##  2 Method

The starting point of our analysis was the list of positions of 3352 Milky Way
classical Cepheids that were classified by the OGLE collaboration based on a
large combination of all-sky time-series survey data (Pietrukowicz et al.,
2021), which we extended by 230 additional classical Cepheids that were
reported by Gaia DR3 in June 2022 (Ripepi et al., 2022b; Eyer et al., 2022).
While there can be disagreements over Cepheid classifications, especially for
overtone Cepheids with sinusoidal light curves, we note that the list by
Pietrukowicz et al. (2021) was used to validate the Gaia DR3 sample, and that
the samples of MW Cepheids overlap by $\sim 85\%$ . Extragalactic Cepheids and
Cepheids that are too distant for identifying clusters were removed from the
Gaia DR3 sample by the quality cuts explained in Sec. 2.1 and by requiring
Cepheids to be brighter than $G<16$ mag. For each Cepheid we considered, we
retrieved all stars within a radius of one degree from the Gaia
archive111https://gea.esac.esa.int/archive/ and then searched for host
clusters as explained in the following and as illustrated schematically in
Fig. 1.

Figure 1: Schematic overview of the pipeline designed to detect cluster
Cepheids.

###  2.1 Cluster detection

Because clusters are gravitationally bound systems, cluster members share
similar positions (RA, DEC), proper motions ($\mu_{\alpha^{*}},\mu_{\delta}$),
parallaxes ($\varpi$), and radial velocities. Thus, stars belonging to a
common cluster can be separated from fore- or background stars as
overdensities in the multidimensional space spanned by the available
membership constraints.

Gaia DR3 provides information for all six of these parameters, although radial
velocity information is only available for a rather limited number of stars
owing to the faintness of most member stars. Hence, our analysis employs only
positions, proper motions, and parallaxes for the cluster identification.
Where available, radial velocity information was used to assess membership
probabilities of Cepheids (cf. Sec. 2.3).

We detected clusters using the publicly available code called hierarchical
density-based spatial clustering of applications with noise (McInnes et al.,
2017, HDBSCAN). As is common practice, we included only stars in our analyis
whose parallax signal-to-noise ratio $\varpi/\sigma_{\varpi}\geq 5$, whose
renormalised unit weight error (RUWE) is smaller than $1.4$ to exclude sources
with poor astrometry (Fabricius et al., 2021, e.g., due to the presence of
companions), and that are brighter than $G=18\,$mag (parameter phot_g_mean_mag
in table gaiadr3.gaia_source), where the Gaia astrometry is most precise. In
practice, this magnitude cut represents no serious limitation for our work and
allows us to clearly recover the main sequences of Cepheid-hosting clusters
that are several magnitudes fainter than their Cepheid members. Gaia
parallaxes of all stars considered for membership were corrected for
systematics using the recipe provided by L21.

The code HDBSCAN uses the $n-$dimensional distance between objects to identify
overdense regions. As Cepheids and open clusters are located in the Galactic
plane, we used Galactic coordinates ($l,b$) for the positional constraints
rather than RA and DEC. The ability of HDBSCAN to detect arbitrarily shaped
clusters was particularly useful for our purposes because the physical shape
of clusters in various stages of dispersal was not known a priori. The only
fixed input parameter required by HDBSCAN is the number of stars $p$ that are
expected to qualify an overdensity as a cluster. Deviations of the number of
cluster stars $s$ from $p$ will cause overdensities with $s<p$ to remain
undetected by HDBSCAN and may sometimes result in a single cluster being split
into multiple parts if $s>p$. To ensure that our analysis was not sensitive to
these undesirable side effects, we repeated our analysis using ten different
values for $p$ ranging from 10 to 100 in increments 10 and found a consistent
number of cluster members in each case. The mean (median) number of member
stars reported per cluster is 230 (152) (cf. Sect. 3).

Following Castro-Ginard et al. (2018) and Hunt & Reffert (2021), we rescaled
each of the Gaia astrometric parameters to variables with zero mean and unit
standard deviation by subtracting the mean from each parameter and rescaling
parameters such that the $25-75\%$ percentile has unit variance. This
procedure ensures equal weighting among the five dimensions and improves
robustness against outliers.

Inspection of the parallax distributions returned by HDBSCAN revealed outliers
in parallax. To retain only likely cluster members, we determined the mode of
the parallax distribution returned by HDBSCAN and retained all cluster members
whose parallaxes agreed to within $3$ standard deviations of a Gaussian fit to
the parallax distribution centered on the mode.

At distances beyond $2$ kpc, cluster identification becomes increasingly
limited due to the current parallax and proper motion uncertainties of Gaia.
Because our goal of calibrating the Galactic LL requires utmost accuracy and
precision, we prioritized greater purity (lower contamination) at the
potential cost of completeness. We thus visually inspected all identified
cluster candidates to ensure that cluster stars were overdense in each of the
membership constraints considered and that the resulting color-magnitude
diagrams indicated a coeval population being detected, as evidenced by a
clearly visible main sequence. Additionally, we discarded clusters in which a
majority of main-sequence stars exceeded the brightness of their candidate
Cepheid members.

For each cluster, HDBSCAN provided a list of likely cluster members together
with membership probabilities. By design, all identified clusters were within
the projected vicinity of Cepheids. However, these same Cepheids were not
necessarily selected as cluster members by HDBSCAN, requiring a separate
membership analysis for Cepheids in the detected clusters (cf. Sec. 2.3).

###  2.2 Cluster parameters

For each cluster that passed the first visual screening, we computed the
center position in RA and DEC. We additionally computed averages and
dispersions in both proper motion directions, parallax, and radial velocity,
where available.

####  2.2.1 Cluster parallaxes

Figure 2: Difference between individual and cluster average parallax for all
member stars considered. Left: Comparison of the parallax difference as a
function of the G magnitude, where the number of stars per bin is color-coded
according to the color bar on the right. Right: Same as the left plot, but as
a function of the color $Bp-Rp$. The vertical dotted lines in both panels
illustrate the magnitude and color range we used to estimate the cluster
parallaxes.

For a given source, the Gaia parallax systematics are well known to depend on
its sky position as well as its magnitude and color (L21). Magnitude and color
trends are likely related to the sophisticated on-board processing of Gaia,
which was implemented to avoid saturation across the extreme dynamic range of
the survey (limit $21.7$ mag). Using the cluster members returned by HDBSCAN,
we investigated whether an optimal magnitude and color range could be
identified to obtain the most reliable and precise average cluster parallax.
We calculated the deviation from the cluster average,
$\Delta\varpi=\langle\varpi\rangle-\varpi_{i}$, for all member stars of all
host clusters. We combined all values of $\Delta\varpi$ into a single set,
which we divided into bins of $0.2$ mag in $G$ band. For each bin, we
estimated the weighted mean and the weighted error of $\Delta\varpi$. Figure 2
illustrates this result and shows a noticeable decrease in the variance of
$\Delta\varpi$ for $12.5<G<17$ and $0.23<Bp-Rp<2.75$. The observed variance is
dominated by the uncertainty in parallaxes for fainter stars and by lower
statistics for bright stars ($G\lesssim 9$ mag). Systematic trends at $G<12.5$
mag can be partially due to the gating mechanism of Gaia or to differences in
photometric processing 222According to Fig. 1 in L21, no gating is applied to
stars fainter than $12.5$ mag. However, the WC0b and WC1 calibration models of
the astrometric field overlap in the range $12.5<G<13$ mag, which implies a
transition from 2D images to binned 1D images, respectively.. We note that the
exact magnitude range is not critical for the estimation of the cluster
parallaxes. For example, restricting the magnitude range further to $13.5-17$
mag changes the mean cluster parallax by less than $2\,\mu$as, while
increasing the uncertainty in average parallax for clusters with $\lesssim
100$ members (e.g., CWNU 175 or vdBergh 1) by approximately $1\,\mu$as (cf.
Sect. 3). Because of these clear and consistent trends and to avoid
sensitivity to gating-related issues, we adopted the range of $12.5<G<17$ mag
as the optimal range for determining high-fidelity average cluster parallaxes
and their uncertainties. We further restricted the color range of member stars
to $0.23<Bp-Rp<2.75$ to avoid the color range for which Fig. 2 shows
increasing deviations from zero residuals, accompanied by increasing
uncertainties due to low statistics.

Several studies have shown the existence of nonzero residual parallax offsets
for stars brighter than $G<13$ mag after the L21 corrections were applied
(e.g., Huang et al., 2021; Zinn, 2021; El-Badry et al., 2021; Riess et al.,
2021, 2022a). However, analyses of open and globular clusters, as well as of
the LMC, have shown the L21 procedure to accurately correct parallax
systematics to within $\sim 1\,\mu$as (Flynn et al., 2022; Maíz Apellániz,
2022) in the optimal magnitude and color range established above. As a result,
a significant nonzero residual parallax offset is expected for (bright)
Cepheids, while no residual parallax offset is expected for cluster members
after the L21 corrections are applied.

The final cluster parallaxes were computed as the weighted mean of the
retained cluster members. The total parallax uncertainty sums the statistical
uncertainty determined as the error on the weighted mean in quadrature and the
systematic contribution due to angular covariance determined by Apellániz et
al. (2021). Because our initial search radius around Cepheids is $1$ deg, the
full diameters of all clusters is significantly smaller than $2$ deg. This
allowed us to consider the estimation of angular covariance based on the LMC
alone as given by $V_{\varpi,\mathrm{LMC}}$ in their Eq. 10 (cf. also Sect.
2.2 in Ripepi et al., 2022c), neglecting wide-angle contributions estimated
using quasars. This is analogous to the approach taken by Zinn (2021) in
conjunction with the angular covariance estimates based on the Kepler field.
In practice, this reduces the error floor for average cluster parallaxes from
$\lesssim 10$ to $\lesssim 7\,\mu$as. Because the mean separation of our
Cepheid clusters is very large, covariance among clusters is negligible.

####  2.2.2 Maximum angular separations

We calculated the projected distance of the Cepheid from cluster center
assuming that both objects are at the distance of the cluster. Candidate
associations with separations greater than $25$ pc were discarded in favor of
sample purity and to ensure that cluster average parallaxes can be used as
accurate proxies for Cepheid parallaxes. Hypothetical Cepheids residing in
extended tidal tails (Jerabkova et al., 2021) would thus be excluded from our
analysis. We refer to Cepheids as coronal cluster members if their projected
separation from cluster center exceeds $8$ pc but does not exceed $25$ pc.

####  2.2.3 Proper motions

We computed bulk cluster proper motions as the mean of all clusters members as
well as proper motion dispersions using cluster members in the color and
magnitude range used for parallaxes. We used proper motions to reject cluster
candidates as spurious asterisms if the resulting velocity dispersion exceeded
reasonable values for gravitationally bound systems following Cantat-Gaudin &
Anders (2020) and Hunt & Reffert (2021). Specifically, up to $\varpi=0.67$
mas, we rejected associations whose projected velocity dispersion exceeds
$5\times\sqrt{2}\,\mathrm{[mas/yr]}$ ($5$ $\mathrm{km\,s^{-1}}$), whereas a
maximum difference of $1\,\text{mas yr}^{-1}$ was allowed for clusters with a
smaller parallax to reflect the increased uncertainties, in particular, of the
fainter main-sequence cluster members. Thus, we required333We recall that
$\mu\mathrm{[mas/yr]}\approx v\mathrm{[km/s]}\cdot\varpi\mathrm{[mas]}/4.74$,
so that $1\,\mathrm{[mas/yr]}\approx 5\mathrm{[km/s]}\times\sqrt{2}$ at
$\varpi=0.67$ mas (1.5 kpc).

$\displaystyle\sqrt{\sigma_{\mu_{\alpha}^{*}}^{2}+\sigma_{\mu_{\delta}}^{2}}\leq\left\\{\begin{matrix}1\quad\text{mas
yr}^{-1}\quad\text{if}\quad\varpi\leq 0.67\quad\text{mas}\\\
5\sqrt{2}\frac{\varpi}{4.74}\quad\text{mas
yr}^{-1}\quad\text{if}\quad\varpi>0.67\quad\text{mas}.\end{matrix}\right.$ (1)

In practice, however, all retained clusters exhibit a significantly lower
velocity dispersion, with a mean value of $2.8$ $\mathrm{km\,s^{-1}}$(cf. Fig.
A). Inspection revealed that the proper motion dispersion estimated using only
spatially densely concentrated cluster members returned by our clustering
analysis underestimated the intrinsic velocity dispersion of true cluster
members observed at large angular separations, which require a statistically
greater velocity dispersion to reach their large separations from cluster
centers. To avoid unrealistically low membership probabilities for coronal
cluster Cepheids (cf. Sec. 2.3), we therefore adopted twice the standard
deviation determined based on the member stars recovered by HDBSCAN as the
more conservative estimate of true cluster proper motion dispersion when
assessing Cepheid membership in clusters.

####  2.2.4 Radial velocity

Cluster radial velocities (RV) are computed using Gaia DR3 mean radial
velocities (Katz et al., 2022, parameter radial_velocity from table
gaia_source). For each cluster with available DR3 RVs, Table 1 lists the
number of (non-Cepheid) cluster member stars, their median RV, standard error
on the cluster median RV, and the Cepheid paramaters.

We did not consider cluster RVs based on few stars ($\lesssim 3$) sufficiently
reliable for further analysis. Thus, we did not consider RV as a membership
constraint for the candidate host clusters of WX Pup, CV Mon, IQ Nor, and SX
Vel.

###  2.3 Cepheid membership determination

We computed cluster membership probabilities for Cepheids whose proper motions
and parallaxes separately agreed to within approximately $3\sigma$ of their
potential host cluster parameters. This subsection presents our method, and
the resulting probabilities are presented in Sec. 3. A tolerance of up to
$0.5\sigma$ was permitted in this initial screening. In this context, $\sigma$
refers to combined (square-summed) dispersions or uncertainties, depending on
the parameter, of clusters and Cepheids as follows. For proper motions, the
cluster dispersion as described in Sec. 2.2.3 was combined with the Cepheid
uncertainties reported by Gaia. For parallaxes, $\sigma$ contains the squared
sum of uncertainties of the weighted cluster average (no significant internal
dispersion expected), the individual Cepheid parallax uncertainty, and an
additional $15\,\mu$as uncertainty to reflect the magnitude dependence of the
residual parallax offset after applying the L21 corrections.

We computed Cepheid membership probabilities using the likelihood formalism
developed in Anderson et al. (2013) and the membership constraints $\varpi$,
$\mu_{\alpha}^{*}$, $\mu_{\delta}$, and RVs. Strictly speaking, this approach
performs a hypothesis test under the null hypothesis of the Cepheid cluster
membership and can only reject this null hypothesis, not prove it. As in
Anderson et al. (2013), we computed the Bayesian likelihood $P(B|A)=1-p(c)$,
where $p(c)$ is the $p-$value of the $\chi^{2}_{N_{\mathrm{dof}}}$ distributed
quantity

$c=\mathbf{x^{T}\Sigma^{-1}x},$ (2)

where the vector $x$ contains the differences between Cepheid and cluster
parameters, that is,

$x=(\varpi_{\mathrm{Cl}}-\varpi_{\mathrm{Cep}},v_{r,\mathrm{Cl}}-v_{\gamma,\mathrm{Cep}},...)\
,$ (3)

and $\Sigma$ is the diagonal covariance matrix containing the squared values
of $\sigma$ for the various membership constraints, as explained above. Our
threshold for rejecting the membership hypothesis was $P(B|A)<0.0027$, which
corresponds to a $3\sigma$ rejection criterion. Stars with a higher
probability are considered bona fide cluster Cepheids provided the host
cluster detection is sufficiently robust.

Radial velocities were included in this calculation if cluster average RVs
($v_{r,\mathrm{Cl}}$) could be estimated using at least three member stars and
if Cepheid systemic radial velocities, $v_{\gamma}$, could be determined using
a Fourier series fit to time-series data from either the velocities of
Cepheids project (cf. Anderson et al. in prep., VELOCE I) or the literature
(e.g., Anderson et al., 2016a). In addition to cluster average values, Table 1
lists RV data for Cepheids, including $v_{\gamma}$, its uncertainty,
references to data used, and the difference between cluster median and Cepheid
$v_{\gamma}$, the total uncertainty (summed in quadrature), and the difference
between cluster and Cepheid in units of the total uncertainty. The only
Cepheid for which RV information significantly contradicts membership is XZ
Car, which is part of our Silver sample (cf. Sect. 3.2). All other stars are
found to agree to within $1.35\sigma$ with their host cluster median
velocities. Further information about Cepheid RVs and Gaia DR3 radial
velocities of Cepheids will be provided as part of the VELOCE project
(Anderson et al. in prep.).

Table 1: Radial velocity information for clusters and Cepheids

Cluster | $N_{\mathrm{RV}}$ | median RV | $\sigma_{\mathrm{RV,Cl}}$ | Cepheid | $v_{\gamma}$ | $\sigma_{v_{\gamma}}$ | Refs | $\Delta_{\mathrm{Cl-Cep}}$ | $\sigma_{\mathrm{tot}}$ | $N_{\sigma}$
---|---|---|---|---|---|---|---|---|---|---
|  | ($\mathrm{km\,s^{-1}}$) | ($\mathrm{km\,s^{-1}}$) |  | ($\mathrm{km\,s^{-1}}$) | ($\mathrm{km\,s^{-1}}$) |  | ($\mathrm{km\,s^{-1}}$) | ($\mathrm{km\,s^{-1}}$) |
UBC 129 | 9 | -14.84 | 6.18 | X Vul | -14.54 | 0.14 | a;b;c | -0.30 | 6.18 | 0.05
NGC 103 | 7 | -77.88 | 9.89 | NO Cas | -79.13 | 0.17 | V | 1.25 | 9.89 | 0.13
UBC 130 | 3 | -0.20 | 2.08 | SV Vul | -0.68 | 2.80 | V | 0.49 | 3.49 | 0.14
NGC 6067 | 93 | -39.04 | 3.10 | QZ Nor | -38.60 | 0.13 | V | -0.44 | 3.10 | 0.14
UBC 290 | 19 | -23.50 | 5.17 | X Cru | -24.26 | 0.15 | d | 0.76 | 5.17 | 0.15
NGC 6067 | 93 | -39.04 | 3.10 | V0340 Nor | -39.63 | 0.09 | V | 0.59 | 3.10 | 0.19
Berkeley 58 | 3 | -71.36 | 28.59 | CG Cas | -77.52 | 0.56 | e;c;V | 6.16 | 28.59 | 0.22
UBC 533 | 9 | -17.96 | 7.95 | GH Lup | -16.11 | 0.25 | f | -1.85 | 7.95 | 0.23
UBC 375 | 6 | -13.13 | 9.86 | V0438 Cyg | -10.06 | 2.05 | e | -3.07 | 10.07 | 0.30
FSR 0951 | 16 | 44.02 | 3.05 | RS Ori | 42.94 | 0.04 | V | 1.08 | 3.05 | 0.35
Cl X Pup | 4 | 74.70 | 10.07 | X Pup | 71.05 | 0.38 | V | 3.65 | 10.08 | 0.36
NGC 6649 | 21 | -1.44 | 24.70 | V0367 Sct | -10.60 | 1.26 | e;g;b;c | 9.16 | 24.73 | 0.37
Lynga 6 | 14 | -46.98 | 24.41 | TW Nor | -56.60 | 0.29 | g;b | 9.62 | 24.41 | 0.39
NGC 129 | 5 | -52.12 | 6.51 | DL Cas | -46.53 | 0.32 | V | -5.59 | 6.52 | 0.86
NGC 6664 | 12 | 9.33 | 8.54 | Y Sct | 14.03 | 0.15 | V | -4.70 | 8.54 | 0.55
IC 4725 | 184 | 2.03 | 1.71 | U Sgr | 3.16 | 0.09 | V | -1.13 | 1.71 | 0.66
CWNU 175 | 4 | 6.19 | 11.53 | VW Cru | -2.82 | 5.30 | h | 9.01 | 12.69 | 0.71
NGC 7790 | 6 | -68.61 | 11.94 | CF Cas | -77.76 | 0.15 | e;h;b;c | 9.15 | 11.94 | 0.77
UBC 106 | 28 | 41.88 | 2.84 | CM Sct | 39.63 | 0.24 | e;g;h | 2.25 | 2.85 | 0.79
NGC 6087 | 69 | 2.91 | 2.67 | S Nor | 6.09 | 2.12 | V | -3.18 | 3.41 | 0.93
Cl ST Tau | 15 | 9.01 | 6.79 | ST Tau | 1.73 | 0.26 | a;b;c;i;V | 7.28 | 6.80 | 1.07
Cl V0378 Cen | 7 | -25.57 | 8.50 | V0378 Cen | -16.24 | 0.04 | V | -9.33 | 8.50 | 1.10
Ruprecht 79 | 11 | 34.55 | 6.28 | CS Vel | 27.10 | 0.19 | g;b | 7.45 | 6.29 | 1.19
NGC 5662 | 89 | -26.60 | 3.59 | V Cen | -21.74 | 0.08 | V | -4.86 | 3.59 | 1.35
UBC 231 | 3 | 75.64 | 7.54 | WX Pup | 52.98 | 0.03 | V | 22.66 | 7.54 | (3.01)
Ruprecht 93 | 13 | -27.77 | 2.73 | XZ Car | 5.72 | 0.42 | V | -33.49 | 2.76 | 12.12
vdBergh 1 | 1 | 9.35 | 0.00 | CV Mon | 19.44 | 0.11 | a;g;c;j | -10.09 | 0.11 | (91.71)
Cl IQ Nor | 1 | 6.07 | 0.00 | IQ Nor | -24.51 | 0.13 | V | 30.58 | 0.13 | (244.66)
Cl SX Vel | 1 | -22.52 | 0.00 | SX Vel | 29.79 | 0.05 | V | -52.31 | 0.05 | (1046.18)

444RV differences between clusters and Cepheids are considered significant
only if a sufficient number (here: $\gtrsim 3$) of cluster stars was available
to determine an accurate median for the cluster. The last column shows
apparently highly discrepant values in parentheses if they are based on an
insufficient number of stars. References listed in Column ‘Refs’ a: Barnes et
al. (1988), b: Bersier et al. (1994), c: Gorynya et al. (1992), d: Bersier
(2002), e: Metzger et al. (1991), f: Coulson & Caldwell (1985), g: Metzger et
al. (1992), h: Pont et al. (1994), i: Imbert (1999), j: Storm et al. (2004),
V: VELOCE (Anderson et al. in prep.)

In contrast to Anderson et al. (2013), we did not explicitly use the angular
separation as an external multiplicative prior because individual cluster
members were already separated from the background by our clustering analysis.
However, our use of a maximum allowed projected separation of $25$ pc could be
seen as a flat prior with $P(A)=1$ for absolute projected separations smaller
than this cutoff value. Ages and chemical compositions were not considered in
the calculation of the likelihood.

##  3 Cluster Cepheids

We grouped our sample of cluster Cepheids into Gold, Silver, and Bronze
samples according to the following criteria. The Gold sample contains cluster
Cepheids whose host cluster detections were robust and whose membership
likelihoods exceeded the threshold for rejecting the membership hypothesis
(cf. Sec. 2.3). This sample is best suited for LL calibration. The Silver
sample contains cases where the host cluster detection is solid, whereas the
likelihood computation quantitatively rejects cluster membership due to a
difference slightly larger than $3\sigma$ in individual constraints. This
sample is of particular interest for the further study to refine possible
membership, for instance, taking uncertainties related to stellar multiplicity
into account. The Bronze sample is composed of two cases for which the host
cluster detection is not as clean as in the Gold sample.

Tables 2-5 list the Cepheids and their host clusters for the Gold, Silver, and
Bronze samples, along with their main astrometric information. Representative
examples of each set are shown in Figs. 3 and 4. We applied an additional
uncertainty of $15\,\mu$as when we computed the significance of the
disagreement in parallax (cf. Sec. 2.3). Table 17 provides a list of the Gaia
EDR3 source ids for all the cluster members and their L21 corrected
parallaxes.

###  3.1 Gold sample

Cluster parameters |  | Cepheid parameters
---|---|---
Cluster | $\alpha(^{\circ})$ | $\delta(^{\circ})$ | N |  $\varpi$ ($\mu$as) |  $\mu^{*}_{\alpha}$ (mas/yr) |  $\mu_{\delta}$ (mas/yr) |  | Cepheid |  $\varpi$ ($\mu$as) |  $\mu^{*}_{\alpha}$ (mas/yr) |  $\mu_{\delta}$ (mas/yr) | Sep (pc) | M.P
Czernik 41 | $297.746$ | $25.279$ | 112 | $407\pm 8$ | $-2.949\pm 0.102$ | $-6.180\pm 0.104$ |  | ATO J297$\dagger^{(*)}$ | $340\pm 18$ | $-3.025\pm 0.012$ | $-6.294\pm 0.017$ | 0.82 | 0.05
NGC 7790 | $359.619$ | $61.206$ | 149 | 322 $\pm$ 7 | -3.229 $\pm$ 0.115 | -1.729 $\pm$ 0.085 |  | CE Cas A | 332 $\pm$ 15 | -3.298 $\pm$ 0.015 | -1.873 $\pm$ 0.017 | 2.3 | 0.80
NGC 7790 | $359.619$ | $61.206$ | 149 | 322 $\pm$ 7 | -3.229 $\pm$ 0.115 | -1.729 $\pm$ 0.085 |  | CE Cas B | 333 $\pm$ 15 | -3.301 $\pm$ 0.014 | -1.809 $\pm$ 0.016 | 2.3 | 0.90
NGC 7790 | $359.619$ | $61.206$ | 149 | 322 $\pm$ 7 | -3.229$\pm$ 0.115 | -1.729 $\pm$ 0.085 |  | CF Cas | 316 $\pm$ 12 | -3.240 $\pm$ 0.012 | -1.766 $\pm$ 0.012 | 1.5 | 0.95
Berkeley 58 | $0.076$ | $60.947$ | 183 | 336 $\pm$ 7 | -3.430 $\pm$ 0.237 | -1.791 $\pm$ 0.130 |  | CG Cas | 296 $\pm$ 14 | -3.241 $\pm$ 0.013 | -1.673 $\pm$ 0.015 | 4.8 | 0.43
UBC 106 | $280.469$ | $-5.411$ | 495 | 443 $\pm$ 7 | -1.048 $\pm$ 0.108 | -1.365 $\pm$ 0.134 |  | CM Sct | 444 $\pm$ 16 | -1.064 $\pm$ 0.015 | -1.414 $\pm$ 0.014 | 6.7 | 0.96
Ruprecht 79 | $145.253$ | $-53.850$ | 152 | 281 $\pm$ 7 | -4.615 $\pm$ 0.183 | 3.086 $\pm$ 0.200 |  | CS Vel | 272 $\pm$ 12 | -4.567 $\pm$ 0.014 | 3.131 $\pm$ 0.014 | 2.7 | 0.81
vdBergh 1 | $99.283$ | $3.074$ | 60 | 585 $\pm$ 10 | 0.411 $\pm$ 0.204 | -0.704 $\pm$ 0.185 |  | CV Mon | 601 $\pm$ 15 | 0.349 $\pm$ 0.016 | -0.666 $\pm$ 0.014 | 0.5 | 0.92
NGC 129 | $7.606$ | $60.200$ | 297 | 556 $\pm$ 7 | -2.594 $\pm$ 0.120 | -1.169 $\pm$ 0.104 |  | DL Cas | 580 $\pm$ 27 | -2.706 $\pm$ 0.025 | -1.189 $\pm$ 0.027 | 1.9 | 0.82
NGC 6664 | $279.115$ | $-8.190$ | 361 | 504 $\pm$ 7 | -0.065 $\pm$ 0.176 | -2.514 $\pm$ 0.351 |  | EV Sct(∗) | 526 $\pm$ 18 | -0.209 $\pm$ 0.018 | -2.546 $\pm$ 0.015 | 1.9 | 0.79
UBC 533 | $230.311$ | $-53.176$ | $68$ | $878\pm 9$ | $-1.766\pm 0.192$ | $-1.381\pm 0.207$ |  | GH Lup | $864\pm 21$ | $-1.337\pm 0.021$ | $-2.202\pm 0.020$ | 12.7 | 0.25
Cl IQ Nor | $228.444$ | $-54.590$ | 48 | 544 $\pm$ 9 | -0.999 $\pm$ 0.433 | -1.790 $\pm$ 0.173 |  | IQ Nor | 535 $\pm$ 18 | -0.897 $\pm$ 0.015 | -1.821 $\pm$ 0.020 | 7.4 | 0.99
NGC 103 | $6.391$ | $61.315$ | 243 | 317 $\pm$ 7 | -2.695 $\pm$ 0.383 | -1.031 $\pm$ 0.149 |  | NO Cas(∗) | 298 $\pm$ 13 | -2.828 $\pm$ 0.012 | -1.208 $\pm$ 0.012 | 11.3 | 0.87
NGC 6067 | $243.304$ | $-54.229$ | 1085 | 513 $\pm$ 7 | -1.948 $\pm$ 0.149 | -2.595 $\pm$ 0.186 |  | QZ Nor(∗) | 484 $\pm$ 20 | -1.896 $\pm$ 0.023 | -3.848 $\pm$ 0.019 | 11 | 0.01
FSR 0951 | $95.572$ | $14.6358$ | 176 | 610 $\pm$ 7 | 0.214 $\pm$ 0.102 | 0.032 $\pm$ 0.142 |  | RS Ori | 589 $\pm$ 30 | 0.196 $\pm$ 0.036 | 0.005 $\pm$ 0.028 | 1.5 | 0.97
NGC 6087 | $244.742$ | $-57.912$ | 196 | $1073\pm 7$ | -1.603 $\pm$ 0.272 | -2.424 $\pm$ 0.247 |  | S Nor | 1099 $\pm$ 22 | -1.608 $\pm$ 0.025 | -2.136 $\pm$ 0.020 | 0.3 | 0.72
Cl ST Tau | $85.740$ | $13.719$ | 79 | 953 $\pm$ 8 | 0.691 $\pm$ 0.248 | -3.636 $\pm$ 0.288 |  | ST Tau | 916 $\pm$ 34 | 0.188 $\pm$ 0.035 | -2.318 $\pm$ 0.022 | 10.0 | 0.08
UBC 130 | $298.043$ | $27.443$ | 42 | 425 $\pm$ 9 | -2.103 $\pm$ 0.071 | -5.872 $\pm$ 0.102 |  | SV Vul | 402 $\pm$ 21 | -2.158 $\pm$ 0.016 | -5.962 $\pm$ 0.021 | 6.5 | 0.90
Cl SX Vel | $131.268$ | $-46.095$ | 77 | 497 $\pm$ 7 | -5.057 $\pm$ 0.160 | 4.976 $\pm$ 0.120 |  | SX Vel | 501 $\pm$ 19 | -4.345 $\pm$ 0.019 | 4.921 $\pm$ 0.022 | 9.6 | 0.17
Lyngå 6 | $241.209$ | $-51.952$ | 173 | 421 $\pm$ 8 | -1.915 $\pm$ 0.187 | -2.762 $\pm$ 0.183 |  | TW Nor | 360 $\pm$ 20 | -1.891 $\pm$ 0.021 | -2.806 $\pm$ 0.017 | 0.6 | 0.25
IC 4725 | $277.958$ | $-19.126$ | 492 | 1554 $\pm$ 6 | -1.685 $\pm$ 0.239 | -6.159 $\pm$ 0.319 |  | U Sgr | 1605 $\pm$ 22 | -1.795 $\pm$ 0.025 | -6.127 $\pm$ 0.017 | 0.1 | 0.42
NGC 5662 | $218.731$ | $-56.665$ | 241 | 1337 $\pm$ 6 | -6.461 $\pm$ 0.157 | -7.189 $\pm$ 0.180 |  | V Cen | 1409 $\pm$ 22 | -6.697 $\pm$ 0.016 | -7.068 $\pm$ 0.018 | 17.2 | 0.05
FSR 0384 | $342.921$ | $56.106$ | 63 | 520 $\pm$ 8 | -3.446 $\pm$ 0.198 | -1.752 $\pm$ 0.096 |  | V Lac | 496 $\pm$ 16 | -3.237 $\pm$ 0.016 | -1.439 $\pm$ 0.017 | 16.9 | 0.27
NGC 6067 | $243.304$ | $-54.229$ | 1085 | 513 $\pm$ 7 | -1.948 $\pm$ 0.149 | -2.595 $\pm$ 0.186 |  | V0340 Nor | 491 $\pm$ 25 | -2.066 $\pm$ 0.027 | -2.634 $\pm$ 0.021 | 0.5 | 0.94
NGC 6649 | $278.359$ | $-10.399$ | 425 | 514 $\pm$ 7 | 0.025 $\pm$ 0.132 | -0.121 $\pm$ 0.156 |  | V0367 Sct | 473 $\pm$ 20 | 0.082 $\pm$ 0.021 | -0.273 $\pm$ 0.019 | 1.7 | 0.60
Cl V0378 Cen | $199.642$ | $-62.593$ | 107 | 518 $\pm$ 8 | -4.325 $\pm$ 0.319 | -1.744 $\pm$ 0.194 |  | V0378 Cen(∗) | 524 $\pm$ 19 | -5.656 $\pm$ 0.014 | -2.282 $\pm$ 0.019 | 7.7 | 0.11
NGC 129 | $7.608$ | $60.199$ | 311 | 557 $\pm$ 7 | -2.591 $\pm$ 0.123 | -1.169 $\pm$ 0.113 |  | V0379 Cas(∗) | 524 $\pm$ 14 | -2.696 $\pm$ 0.012 | -1.313 $\pm$ 0.015 | 25.0 | 0.42
UBC 375 | $304.617$ | $40.062$ | 160 | 562 $\pm$ 7 | -3.073 $\pm$ 0.461 | -5.219 $\pm$ 0.284 |  | V0438 Cyg | 530 $\pm$ 16 | -3.324 $\pm$ 0.017 | -4.559 $\pm$ 0.019 | 2.9 | 0.48
CWNU 175 | $188.395$ | $-63.506$ | 37 | 732 $\pm$ 9 | -3.987 $\pm$ 0.277 | -1.187 $\pm$ 0.165 |  | VW Cru | 738 $\pm$ 16 | -3.903 $\pm$ 0.015 | -1.134 $\pm$ 0.015 | 1.1 | 0.96
UBC 231 | $115.571$ | $-25.265$ | 68 | $345\pm 8$ | $-2.177\pm 0.240$ | $2.297\pm 0.340$ |  | WX Pup | 387 $\pm$ 15 | -2.163 $\pm$ 0.010 | 2.559 $\pm$ 0.014 | 22.2 | 0.01
UBC 290 | $191.807$ | $-59.373$ | 253 | 639 $\pm$ 6 | -5.918 $\pm$ 0.279 | -0.273 $\pm$ 0.198 |  | X Cru | 654 $\pm$ 19 | -5.926 $\pm$ 0.015 | -0.173 $\pm$ 0.017 | 7.6 | 0.98
FSR 0384 | $342.921$ | $56.106$ | 63 | 520 $\pm$ 8 | -3.446 $\pm$ 0.198 | -1.752 $\pm$ 0.096 |  | X Lac(∗) | 520 $\pm$ 18 | -3.296 $\pm$ 0.017 | -1.442 $\pm$ 0.017 | 17.2 | 0.43
UBC 129 | $299.131$ | $26.464$ | 131 | 880 $\pm$ 7 | -1.031 $\pm$ 0.138 | -4.363 $\pm$ 0.189 |  | X Vul | 864 $\pm$ 22 | -1.352 $\pm$ 0.016 | -4.247 $\pm$ 0.020 | 4.9 | 0.77
NGC 6664 | $279.115$ | $-8.190$ | 361 | 504 $\pm$ 7 | -0.065 $\pm$ 0.176 | -2.514 $\pm$ 0.351 |  | Y Sct | 558 $\pm$ 20 | -0.737 $\pm$ 0.025 | -2.878 $\pm$ 0.019 | 16.23 | 0.08
Table 2: Gold sample of cluster Cepheids. Left: Host cluster parameters.
Right: Cepheid parameters. The average cluster parallaxes were estimated using
stars in the range $12.5<G<17$ as explained in Sect. 3. The uncertainty
includes the contribution from angular covariance. (∗) denotes first overtone
pulsators. The second last column states the projected separation of the
Cepheid from cluster center in pc. The last column states the membership
probability if HDBSCAN considers the Cepheid a member and ”-” if not. ATO
J297$\dagger$ abbreviates the full identifier of ATO J297.7863+25.3136.

The Gold sample consists of 34 Cepheids residing in 28 distinct Galactic open
clusters. Out of the 34 Cepheids, 27 Cepheids pulsate in the fundamental mode,
and 7 Cepheids pulsate in the first overtone. We identify ST Tau, V0378 Cen,
and GH Lup as bona fide cluster Cepheids for the first time.

We cross-matched all 28 Gold sample host clusters with cluster catalogs from
the literature (Anderson et al., 2013; Usenko et al., 2019; Cantat-Gaudin et
al., 2018, 2020; He et al., 2022; Hunt & Reffert, 2021; Medina et al., 2021).
We found cluster parameters in agreement to within $1\sigma$ of the previously
reported parameters in the literature for 24 of them. However, we found
disagreements greater than $2\sigma$ among at least one of the astrometric
parameters for the host clusters of SX Vel, IQ Nor, and VW Cru.

Last but not least, we identified four entirely new clusters that host one
Cepheid each. We denoted them by the prefix Cl followed by the Cepheid name.
Additional information for a subset of Gold sample cluster Cepheids is
provided below.

##### SX Vel

is found to be a member of a newly detected host cluster (Cl SX Vel,
$d=2012\pm 29$ pc) at a projected separation of $9.6\,$pc. The presence of
multiple clusters in close proximity somewhat complicates this membership
analysis. Anderson et al. (2013) investigated multiple possible host clusters,
including Bochum 7, NGC 2660, FSR 1441, SAI 94, and Ruprecht 70, and we here
add NGC 2659. Membership in Bochum 7 ($5754$ pc; cf. Kharchenko et al., 2005)
and SAI 94 ($3515\pm 60$ pc; cf. Elsanhoury & Amin, 2019) is readily excluded
based on distance, while proper motion differences exclude membership in NGC
2660, and FSR 1441 (Cantat-Gaudin et al., 2018). However, NGC 2659 and
Ruprecht 70 require some discussion because the computed likelihoods for
cluster membership are consistent with the hypothesis of membership for both
and because the likelihood obtained for NGC 2659 is even higher ($0.65$) than
for Cl SX Vel ($0.17$). However, closer inspection revealed that the higher
likelihood for NGC 2659 is driven by weaker proper motion constraints (twice
larger dispersion). The parallaxes of both clusters agree to within
$1.1\sigma$ ($497\pm 7\,\mu$as vs $508\pm 7\,mu$as). Additionally, the
observed separation of $43$ pc is inconsistent with our maximum allowed
separation of $25\,$pc. Similarly, for Ruprecht 70, the separation of $34$ pc
rejects this association, although the likelihood alone ($0.004$) would not
reject membership according to our criteria.

##### IQ Nor

is associated with a cluster at a distance of $1839\pm 32$ pc. Previously,
Anderson et al. (2013) investigated the membership of IQ Nor in the following
clusters: NGC 5582, NGC 5925, or Loden 2115. Membership in NGC 5822 is
excluded based on parallax ($12\sigma$ difference Cantat-Gaudin et al., 2018),
whereas proper motions exclude membership in NGC 5925 (Cantat-Gaudin et al.,
2018) and Loden 2115 (Kharchenko et al., 2013).

##### VW Cru

resides in a cluster reported independently as CWNU 175 while this article was
in preparation (He et al., 2022). Although Anderson et al. (2013) previously
investigated possible membership in Loden 624 (Kharchenko et al., 2013), we
note that CWNU 175 is a different physical object separated by $1.9$ deg from
Loden 624.

##### WX Pup

is a coronal member of the cluster UBC 231 (see also Zhou & Chen, 2021) and a
good example of how the Gaia systematics can limit the ability to detect host
clusters because the cluster and Cepheid parallaxes differed by $3.6\sigma$
prior to applying L21 parallax corrections. After applying L21 corrections,
this differences reduces to $1.8\sigma$. While the membership likelihood of WX
Pupis a relatively low $1\%$ and the projected separation of 22.2 pc is close
to our cutoff, its membership in UBC 231 is not rejected according to the
criteria we specified. We searched for other cases where the chronological
order of the L21 corrections would affect the conclusion concerning
membership, but found none.

##### ATO J297.7863+25.3136

was discovered recently (Heinze et al., 2018) and identified as a member of
Cluster 41 by Medina et al. (2021). We here confirm this association at a
distance of $2456\pm 49$ pc. However, this cluster is located in a highly
reddened region of the sky, limiting its usefulness for LL calibration (cf.
Fig. 5).

##### SV Vul

is especially valuable for LL calibration due to its long period because the
majority of Cepheids in distant supernova-host galaxies have periods
$\log{P}>1.2$ (e.g., Riess et al., 2018). We find a very high likelihood of
$90\%$ for this cluster Cepheid combination at a distance of $2354\pm 49$ pc,
and we note the small $\sim 6.5$ pc separation from cluster center. Thus, our
analysis confirms previous statements of the SV Vul cluster membership
reported by Negueruela et al. (2020) and Medina et al. (2021). Moreover,
inspection of several membership constraints as well as the residuals from our
LL calibration does not corroborate the possibility that the parallaxes of SV
Vul are unreliable, reported by Owens et al. (2022) (cf. Fig. 6 and Sect.
4.3). We therefore find no reason to discard this valuable star from LL
calibration.

###  3.2 Silver sample

The Silver sample contains three Cepheids with likelihoods that are formally
inconsistent with membership in well-defined clusters according to our
criteria. However, disagreements among the individual membership constraints
are sufficiently small to warrant additional discussion and inspection.

##### AP Vel

was previously reported as a member of the cluster Ruprecht 65 (Chen et al.,
2015) located at a distance of $2085\pm 32$ pc. The low membership probability
is dominated by the $3.3\sigma$ parallax difference. We do note, however, that
the proper motion parameters of AP Vel ($\mu_{\alpha}^{*}$, $\mu_{\delta}$)
are within $2.3$ and $1.7\sigma$ of the cluster averages, and that the Cepheid
is located rather close to (0.21 deg) from cluster center.

##### X Pup

is located at a rather large separation of $\sim 24.3$ pc from the center of
its possible newly identified host cluster. The low likelihood is driven by
proper motion differences between Cepheid and cluster, which are significant
at the level of $\sim 3.1$ and $3.3\sigma$ for $\mu_{\alpha}^{*}$ and
$\mu_{\delta}$, respectively. However, we note that the total velocity
dispersion of Cl X Pup is merely $3.3\,$$\mathrm{km\,s^{-1}}$, which may
indicate that an underestimated proper motion dispersion was used to calculate
the membership. Additionally, the comparatively large separation from the
cluster (cf. Sec. 2.2.3) as well as orbital motion tentatively reported by
Anderson et al. (2016a) may contribute to deviations in proper motion. We note
the good agreement in parallax ($1.4\sigma$) and radial velocity, where the
Cepheid barycentric velocity is $71.02\pm 0.16$ km/s (Anderson et al. in
prep), which is fully consistent with the median cluster radial velocity based
on four stars reported in Gaia DR3 ($74\pm 10$ km/s) (cf. Table 1). We
therefore consider the cluster membership of X Pup to be potentially
underestimated due to an underestimated cluster proper motion dispersion.
Further study is required to ascertain its membership before X Pup is included
in the Gold sample.

##### XZ Car

is situated at a projected separation of 15 pc from its potential newly
identified host cluster Ruprecht 93. Although the parallax of XZ Car fully
agrees with that of the cluster, we find a low membership probability due to
differences in the kinematic membership constraints, notably radial
velocities, which differ by $33\,$$\mathrm{km\,s^{-1}}$ between the pulsation-
averaged Cepheid RV and the median RV of the 13 cluster members with DR3
radial velocities (cf. Table 1). Although XZ Car is a long-term spectroscopic
binary and exhibits a trend of its pulsation-averaged velocity $v_{\gamma}$
that exceeds $5$$\mathrm{km\,s^{-1}}$ over a baseline of $\sim 40$ yr
(Anderson et al., 2016a, Shetye et al. in prep.), we caution that orbital
motion is unlikely to explain the large RV difference. Additionally,
$\mu_{\alpha}^{*}$ and $\mu_{\delta}$ differ by $2.9\sigma$ and $2.6\sigma$.
We note that evidence of orbital motion has also been found using Gaia proper
motion anomalies (Kervella et al., 2019), however, indicating that proper
motion may also provide incorrect membership indications for XZ Car. It would
be intriguing (but beyond scope for this article) to investigate the nature of
the orbit and the companion required to explain these differences. However, XZ
Car does not appear to be gravitationally bound to Ruprecht 93. Further
membership analysis using the full Gaia temporal baseline might clarify this
high-interest association.

Cluster parameters |  | Cepheid parameters |  |  |  |
---|---|---|---|---|---|---
Cluster | $\alpha(^{\circ})$ | $\delta(^{\circ})$ | N |  $\varpi$ ($\mu$as) |  $\mu^{*}_{\alpha}$ (mas/yr) |  $\mu_{\delta}$ (mas/yr) |  | Cepheid |  $\varpi$ ($\mu$as) |  $\mu^{*}_{\alpha}$ (mas/yr) |  $\mu_{\delta}$ (mas/yr) | Sep (pc) | M.P
Ruprecht 65 | $129.818$ | $-44.062$ | 133 | $478\pm 7$ | $-4.795\pm 0.270$ | $4.424\pm 0.360$ |  | AP Vel(∗) | 545 $\pm$ 12 | -6.030 $\pm$ 0.013 | 5.647 $\pm$ 0.013 | 8.5 | 0
Ruprecht 93 | $166.029$ | $-61.371$ | 206 | 482 $\pm$ 7 | -6.437 $\pm$ 0.143 | 3.130 $\pm$ 0.097 |  | XZ Car | 473 $\pm$ 18 | -7.277 $\pm$ 0.019 | 2.622 $\pm$ 0.017 | 15.3 | 0
Cl X Pup | $113.353$ | $-20.481$ | 125 | 363 $\pm$ 6 | -2.166 $\pm$ 0.149 | 3.125 $\pm$ 0.205 |  | X Pup | 397 $\pm$ 20 | -1.236 $\pm$ 0.016 | 1.786 $\pm$ 0.018 | 24.4 | 0

Table 3: Silver sample of cluster Cepheids.

Cluster parameters |  | Cepheid parameters |  |  |
---|---|---|---|---|---
Cluster | $\alpha(^{\circ})$ | $\delta(^{\circ})$ | N |  $\varpi$ ($\mu$as) |  $\mu^{*}_{\alpha}$ (mas/yr) |  $\mu_{\delta}$ (mas/yr) |  | Cepheid |  $\varpi$ ($\mu$as) |  $\mu^{*}_{\alpha}$ (mas/yr) |  $\mu_{\delta}$ (mas/yr) | Sep (pc) | M.P
Asterism BB Cen | $178.358$ | $-62.608$ | 284 | 316 $\pm$ 6 | -6.196 $\pm$ 0.378 | 0.781 $\pm$ 0.173 |  | BB Cen | 330 $\pm$ 11 | -6.43 $\pm$ 0.011 | 0.978 $\pm$ 0.013 | 15.0 | 0.83
Asterism V620 Pup | $119.732$ | $-29.490$ | 118 | 293 $\pm$ 7 | -2.346 $\pm$ 0.232 | 2.895 $\pm$ 0.350 |  | V620 Pup | 268 $\pm$ 14 | -2.282 $\pm$ 0.012 | 3.668 $\pm$ 0.013 | 17.3 | 0.45

Table 4: Bronze sample of cluster Cepheids.

Cepheid | Cluster | Reason of rejection | Reference
---|---|---|---
BB Sgr | Collinder 394 |  $7\sigma$ difference in parallax | Gieren et al. (1997); Usenko et al. (2019)
OGLE-GD-CEP-1175 | NGC 6193 | Position in the color magnitude diagram | Medina et al. (2021)
WISE J124231.0-625132 | NGC 4609 | Position in the color magnitude diagram | Medina et al. (2021)
SX Car | ASCC 61 | No host cluster detected at this location | Anderson et al. (2013); Chen et al. (2015)
S Mus | ASCC 69 | No host cluster detected at this location | Anderson et al. (2013); Chen et al. (2015)

Table 5: Cepheids considered as possible cluster members in the literature
that were not found to be bona fide cluster Cepheids here.

###  3.3 Bronze sample

Clusters reported as part of the Gold and Silver samples can be clearly
distinguished from field stars in position and proper motion. However, these
distinctions were less clear in the case of possible host clusters
(tentatively labeled asterisms) reported here as part of the Bronze sample.
Additionally, the Gaia CMDs exhibit two main sequences, suggesting likely
fore- or background contamination, perhaps by spiral arms being crossed (cf.
Figure 4). Unfortunately, the cluster membership probabilities provided by
HDBSCAN do not allow us to filter out contaminants. However, there appear to
be a clear overdensities in parallax space for stars in the vicinity of both
BB Cen and V0620 Pup, and we note that the computed likelihoods for the
Cepheid are high and fully consistent with cluster membership, assuming the
cluster is real.

Figure 3: Position in the sky, position in the proper motion space, and color
magnitude diagram for different cluster Cepheids. Background stars are shown
in gray, and the cluster membership probability is color-coded. Light colors
indicate high probability. Cepheids are shown as labeled using large filled
red circles. Cepheids detected as cluster members by HDBSCAN also feature an
overplotted symbol to illustrate membership probability. Figure 4: Graphical
representation of membership constraints for specific examples of cluster
Cepheids in the Silver (XZ Car) and Bronze (BB Cen, V620 Pup) samples, as well
as two rejected associations. Figure 5: Cluster Czernik 41 and the Cepheid ATO
J297.7863+25.3136. The cluster is a clear overdensity on the sky and in the
proper motion space. However, the CMD does not exhibit a clean main sequence
with member stars at atypically red colors, indicating high extinction, which
is also reflected by varying levels of background stars amid the gray points
on the left.

###  3.4 Rejected associations

Our analysis refuted the cluster membership of several Cepheids previously
considered as cluster members in the literature, and these cases are listed in
Table 5. Specifically, OGLE-GD-CEP-1175 and WISE J124231.0-625132 were
previously reported as members of NGC 6193 and NGC 4609 (Medina et al., 2021),
respectively. They are both too faint to be among the most luminous evolved
stars in the recovered host clusters, however (cf. Fig. 4). BB Sgr was
previously considered to be a member of Collinder 394 (Gieren et al., 1997;
Usenko et al., 2019). However, Medina et al. (2021) found that neither
parallax nor proper motion match for this combination. Notably, the parallax
differs by $7\sigma$, which reliably excludes membership.

Data from the All-Sky Compiled Catalogue (Kharchenko, 2001) led Anderson et
al. (2013) and Chen et al. (2015) to consider the Cepheids SX Car and S Mus
potential cluster members of the clusters ASCC 61 and 69, respectively. Evans
et al. (2014) reported the detection of a population of X-ray sources near the
position of S Mus and interpreted this population as evidence for young stars
in the cluster ASCC 69. At the reported cluster distances of $1700$ and $1000$
pc, Gaia data should allow the detection of even relatively small clusters
near the positions of the two Cepheids. However, because no clusters are
detected, we conclude that S Mus and SX Car are not bona fide cluster
Cepheids.

##  4 LL and Gaia zeropoint offset calibration

In this section, we calibrate period luminosity relations for MW Cepheids that
pulsate in the fundamental mode while simultaneously investigating residual
parallax offsets that are applicable after applying the L21 corrections.
Section 4.1 describes the observational data for MW Cepheids, Sect. 4.2
contains a cross-check of the expected zero residual offset applicable to
cluster parallaxes using the LMC, and Sect. 4.3 describes the calibration of
the MW LL using combined cluster and field Cepheids.

###  4.1 Milky Way Cepheids

Table 6: Astrometric and photometric constraints applied to the MW Cepheid
sample.

Astrometric constraints | Photometric constraints
---|---
RUWE $<1.4$ | $0.8<Bp-Rp<2.75$
$\varpi/\sigma_{\varpi}\geq 8.5$ | num_clean_epochs_g $>15$
astrometric_chi2_al $<3000$ | num_clean_epochs_bp $>15$
astrometric_excess_noise $<0.25$ | num_clean_epochs_rp $>15$
astrometric_excess_noise_sig $<70$ | ipd_frac_multi_peak$<7$
| int_average_g$>6$
666The constraints relate to parameters given in Gaia DR3 data tables
gaia_source and vari_cepheid. Astrometric constraints555Descriptions available
here:
https://gea.esac.esa.int/archive/documentation/GDR3/Gaia_archive/chap_datamodel/sec_dm_main_source_catalogue/ssec_dm_gaia_source.html
are applied to all Cepheids used in this work and reproduce the sample of
Cepheids used by Riess et al. (2021). astrometric_chi2_al quantifies the
goodness of fit in the along-scan direction without taking into account
astrometric_excess_noise. Positive values of astrometric_excess_noise indicate
that the source may not be astrometrically well behaved, and this excess noise
may be relevant if astrometric_excess_noise_sig $>2$. Since currently, a
detailed guidance for how to use these parameters is lacking, we adopted very
conservative cuts to remove the clearest outliers. Photometric constraints are
applied only to the sample of Cepheids for which Gaia photometry is used. In
particular, the parameter ipd_frac_multi_peak specifies the percentage of
multiply peaked Gaia windows that were accepted by the image parameter
determination. We adopted a constraint in this parameter to avoid blending of
the Cepheid photometry with nearby sources, which particularly applies to the
$Bp$ and $Rp$ spectrophotometry. An overview of these samples is given in
Table 7.

Table 7: MW Cepheid samples used to calibrate the Galactic Cepheid LL in
various passbands

Cepheid | Gaia DR3 source id | $P$ | $\varpi$ | $\varpi_{\rm corr}$ | $W_{H}$ | $W_{G}$ | $Bp$ | $V$ | $G$ | $Rp$ | $F160W$
---|---|---|---|---|---|---|---|---|---|---|---
|  | (d) | (mas) | (mas) |  |  |  |  |  |  |
AA Gem | 3430067092837622272 | 11.297 | 0.274 | 0.311 | $W_{H}$ |  |  |  |  |  | $F160W$
AA Mon | 3102535635624415872 | 3.937 | 0.313 | 0.316 |  |  | $Bp$ | $V$ | $G$ | $Rp$ |
AB Cam | 473239154746762112 | 5.788 | 0.212 | 0.241 |  |  | $Bp$ | $V$ | $G$ | $Rp$ |
… | … | … | … | … | … | … | … | … | … | … | …
777The complete version of this table is available at the CDS. $\varpi$ is the
Cepheid parallax as obtained from Gaia DR3, and $\varpi_{\rm corr}$ lists the
parallax corrected for the L21 offset.

We compiled samples of fundamental-mode MW Cepheids based on the astrometric
and photometric quality criteria tabulated in Table 6. The astrometric
constraints were compiled such as to reproduce the sample of 68 low-reddening
MW Cepheids observed by the SH0ES team using HST (Riess et al., 2018, 2021).
However, a larger sample of Cepheids is considered in other photometric bands
and using Gaia photometry. Hence, we added cuts based on astrometric goodness-
of-fit parameters to remove Cepheids whose astrometry was very likely flawed,
such as RX Cam, the only Cepheid for which an orbital parallax solution is
available in Gaia DR3. The photometric criteria we adopted include a magnitude
cut to avoid saturated stars, a color cut to limit exposure to reddening, a
cut on the number of available photometric epochs based on which the mean
magnitudes were computed, and the parameter ipd_frac_multi_peak $<7,$ which
was adopted to limit exposure to blended sources. We further adopted a cut on
period for the Gaia sample $P>3.9$d to avoid exposure to misclassified
overtone Cepheids. The most stringent cut in practice is that we require
individual iron abundance measurements based on high-resolution spectroscopy
for all sample stars (Genovali et al., 2014, 2015). The final sample of
fundamental-mode classical Cepheids for $W_{G}$ contains $225$ stars and is
listed in Table 7.

We compiled ground-based photometry in the Johnson $V$ and Cousins $I$ bands
from Groenewegen (2018) and Breuval et al. (2020, 2021). This dataset has been
homogenized by Groenewegen (2018) and was studied extensively. It mainly
includes $V-$ and $I-$band data reported by Mel’nik et al. (2015), which are
based on observations by L. Berdnikov (e.g., Berdnikov, 2008). Reddening
values, $\mathrm{E(B-V)}$ for Galactic Cepheids are taken from Fernie et al.
(1995) and scaled by a factor of $0.94$ following Groenewegen (2018). We also
computed reddening-free Wesenheit magnitudes (Madore, 1982) using $V$ and
$I-$band data, $W_{VI}$ (cf. below).

We collected Gaia DR3 photometry in Gaia $G$ band, as well as integrated $Bp$
and $Rp$ spectrophotometry (Ripepi et al., 2022b; Riello et al., 2021).
Specifically, we used intensity-averaged magnitudes from Gaia CU7 Specific
objects studies (parameters int_average_g, int_average_g_error and analogous
for $Bp$ and $Rp$ from table gaiadr3.vari_cepheid) published as part of the
Gaia DR3 variability analysis for Cepheids (Ripepi et al., 2022b; Eyer et al.,
2022). We also computed reddening-free Wesenheit magnitudes, $W_{G}$, based on
$G$, $Bp$, and $Rp$ as stated below.

Finally, we collected HST WFC3-IR F160W photometry for MW Cepheids from Riess
et al. (2019) and Ripepi et al. (2022c), as well as their reported reddening-
free NIR Wesenheit magnitudes $W_{H}$. Benefits of this homogeneous HST
dataset include the excellent calibration of the HST photometric system,
homogeneity with respect to extragalactic Cepheids, high spatial resolution,
and the lack of time- and location-specific calibration issues typical of
ground-based NIR photometry. We also experimented with ground-based near-IR
photometry available from a range of literature references following Breuval
et al. (2021), notably combining ground-based $J,H,K_{s}$ photometry from
Laney & Stobie (1992), Monson & Pierce (2011), and Genovali et al. (2014).
However, the homogenization of these data sets is not as straightforward due
to different photometric systems in use (e.g., improvements in detector
technology), the calibration of atmospheric absorption in the NIR, and the
standardization of NIR passbands. After some tests, and notably in comparison
with the HST F160W photometry available from Riess et al. (2019), we discarded
ground-based NIR photometry as not sufficiently accurate for the purposes of
our study.

HST WFC3-IR observations are subject to count-rate nonlinearity (CRNL) at the
level of $0.0075\pm 0.006$ mag/dex (Riess et al., 2019). We took these CRNL
corrections into acount when we compared Cepheid samples spanning a
significant dynamic range, that is, when we compared MW Cepheids to
extragalactic Cepheids, such as those in the LMC, or when we compared them to
Cepheids in supernova-host galaxies (SN-hosts). CRNL corrections to offset
differences among MW Cepheids alone are at the level of $1-2$ mmag and were
therefore neglected.

We used the following definitions for Wesenheit magnitudes $W_{VI}$ (Breuval
et al., 2022), $W_{H}$ (Riess et al., 2016), and $W_{G}$ (Ripepi et al.,
2019):

$W_{VI}=I-1.239\cdot(V-I)\ ,$ (4) $W_{H}=F160W-0.386\cdot(F555W-F814W)\
,\mathrm{and}$ (5) $W_{G}=G-1.921\cdot(Bp-Rp)\ .$ (6)

Extinction corrections were applied using reddening coefficients calculated
for a Fitzpatrick (1999) reddening law with $R_{V}=3.3$ and a spectral energy
distribution representative of a $10$ d Cepheid near the center of the
instability strip (cf. Anderson, 2022) as given by a Castelli & Kurucz (2003)
model atmosphere with $T_{\rm eff}=5400K$, [Fe/H]$=0.0$, $\log{g}=1.5$.
Specifically, this yields $R_{V_{\text{Johnson}}}=3.553$,
$R_{I_{\text{Cousins}}}=2.095$, $R_{F160W}=0.674$ and $R_{Bp}=3.701$,
$R_{G}=2.991$, $R_{Rp}=2.196$, where the subscript F160W refers to the HST
WFC3-IR system. All filter profiles were downloaded from the Spanish VO filter
profile service888http://svo2.cab.inta-csic.es/theory/fps/. These values were
used in conjunction with color excess values defined for Johnson-Cousins
$E(B-V)$ to estimate extinction in the respective bands. We also compiled
individual iron abundances from the literature ensuring a common solar iron
abundance (cf. Sec. 4.2).

###  4.2 Confirming the adequacy of L21 parallax corrections for cluster
parallaxes using the LMC

Lindegren et al. (2021) provided a recipe for correcting systematic parallax
errors related to source magnitude, color, and sky-position (ecliptic
latitude) based on millions of quasars and LMC stars as well as 7000 bright
physical stellar pairs. However, previous articles have presented evidence
that residual parallax offsets need to be applied even after the L21
corrections are applied. For example, Riess et al. (2021) determined an
additional constant parallax offset of $14\pm 6$ $\mu$as based on 75 Galactic
Cepheids in the magnitude range $6<G<12$, and these residual offsets are now
well documented using different methods and stellar types (e.g., Zinn et al.,
2019; Zinn, 2021; Khan et al., 2019; Schönrich et al., 2019; Stassun & Torres,
2021; Ren et al., 2021; Wang et al., 2022; Flynn et al., 2022). Hence, an
accurate LL calibration based on Cepheid parallaxes requires solving for the
residual offset applicable to the sample of stars used in the calibration.

However, recent work based on open and globular clusters as well as the
Magellanic Clouds has shown that the L21 recipe accurately corrects parallax
systematics of stars fainter than $G>13$ mag (Flynn et al., 2022; Maíz
Apellániz, 2022). As a result, average cluster parallaxes based on
L21-corrected member stars in this magnitude range are particularly useful for
LL calibration because no further offsets need to be determined, that is,
$\Delta\varpi_{\mathrm{Cl}}=0$. Average cluster parallaxes can therefore
inform the residual parallax offset applicable to Cepheid parallaxes,
$\Delta\varpi_{\mathrm{Cep}}$. This is done in Sect. 4.3. However, prior to
adopting $\Delta\varpi_{\mathrm{Cl}}=0$, we decided to verify the validity of
this approach using observations of Cepheids in the LMC, whose distance
$\mu_{\text{DEBs}}$ is known with an accuracy of $1.3\%$ from detached
eclipsing binary stars (Pietrzyński et al., 2019).

We compiled Johnson-Cousins $V-$ and $I-$band photometry of LMC Cepheids from
the OGLE-III catalog of variable stars (Soszyński et al., 2017) and selected
fundamental-mode OGLE-III Cepheids within the matching period range of cluster
Cepheids ($3.9-45$ d) and cross-matched their positions (maximum search radius
$2^{\prime\prime}$) with Gaia DR3 positions to obtain Gaia $G-$band, $Bp$, and
$Rp$ photometry from the SOS Cepheid list (Ripepi et al., 2022b,
gaiadr3.vari_cepheid). The accuracy of the cross-match was verified by
considering the agreement between periods reported by OGLE and Gaia. We
adopted the OGLE-III Cepheid sample instead of the Gaia DR3 list of Cepheids
in the LMC direction because a) geometric corrections (cf. below) are well
described for this sky region (Pietrzyński et al., 2019), and b) the
classification of Cepheids in OGLE-III benefits from longer time series and
long-standing human experience in classification. Since OGLE-III covers the
main part of the LMC disk, and thus the bulk of the Cepheid population,
including outer regions from Gaia is not expected to add a significantly
greater number of Cepheids outweighing the downsides related to the geometric
correction. We used reddening maps based on red clump stars (Skowron et al.,
2021) to correct for extinction using the values of $R_{\lambda}$ mentioned in
Sect. 4.1 and the conversion $E(V-I)=0.686\cdot E(B-V)$ derived analogously.

For our NIR analysis, we used HST WFC3 observations of 70 LMC Cepheids (Riess
et al., 2019) because they can be directly compared to the HST observations of
MW cluster Cepheids (Riess et al., 2022a) after the appropriate CRNL
corrections are applied. Because the NIR Wesenheit magnitudes, $W_{H}^{HST}$,
reported by Riess et al. (2019) already include a CRNL correction applicable
for the comparison with Cepheids in the SN-host sample, we recomputed $W_{H}$
using Eq. 5 and their original HST observations in the individual passbands
$F555W$, $F814W,$ and $F160W$. We then applied appropriate CRNL corrections
(average of $0.010$ mag) to account for the flux difference of $0.9-1.8\,$dex
between MW cluster and LMC Cepheids.

We applied geometric corrections to apparent magnitudes following Jacyszyn-
Dobrzeniecka et al. (2016), effectively treating all LMC Cepheids as though
they were observed at the same distance, determined to an accuracy of $1.3\%$
using detached eclipsing binary systems (Pietrzyński et al., 2019). As a
result of this correction, the effect of the LMC intrinsic depth on the
observed scatter in the LL is minimized. This is necessary due to the large
sky region covered by OGLE-III ($1.7$ kpc) and to ensure that the distance
estimate to the LMC reflects the distance to the barycenter of the detached
eclipsing binaries. Moreover, the correction decreases the observed scatter in
the LL, resulting in a slight ($\sim 0.004\,$mag) improvement in the
uncertainties for the LL intercept $\beta$.

For LMC Cepheids, we fit linear LLs of the form $m=\alpha(\log P-\log
P_{0})+\beta^{{}^{\prime}}$ using a least-squares fitting procedure and a
$2.7\sigma$ outlier rejection (applying Chauvenet’s criterion for the HST LMC
Cepheid sample); $m$ denotes apparent magnitudes corrected to the LMC
barycenter. Depending on the photometric data set, the samples used in the fit
contained between $68$ and $712$ LMC Cepheids. The results for a range of
individual photometric bands and Wesenheit magnitudes are listed in Table 8,
including the number of available Cepheids, and the assumed intrinsic width of
the LL.

Table 8: LLs of the form $m=\alpha(\log{P-10})+\beta$ fit to LMC Cepheids

Band | LMC | WIS
---|---|---
| $\beta$ | $\alpha$ | $n_{\text{LMC}}$ |
| (mag) | (mag/$\log{P}$) |  | (mag)
$Bp$ | $14.329\pm 0.016$ | $-2.717\pm 0.055$ | 546 | 0.23
$V$ | $14.187\pm 0.014$ | $-2.669\pm 0.049$ | 701 | 0.22
$G$ | $14.066\pm 0.013$ | $-2.857\pm 0.046$ | 546 | 0.19
$Rp$ | $13.611\pm 0.011$ | $-2.962\pm 0.037$ | 546 | 0.16
$I$ | $13.580\pm 0.011$ | $-2.872\pm 0.038$ | 712 | 0.14
$F160W^{a}$ | $12.923\pm 0.014$ | $-3.229\pm 0.056$ | 68 | 0.09
$W_{VI}$ | $12.870\pm 0.008$ | $-3.181\pm 0.029$ | 684 | 0.08
$W_{G}$ | $12.669\pm 0.005$ | $-3.347\pm 0.020$ | 543 | 0.10
$W_{H}^{a}$ | $12.643\pm 0.011$ | $-3.290\pm 0.044$ | 68 | 0.06
999 $\beta$ is expressed here in apparent magnitudes. The last column
indicates the intrinsic width of the LL due to the finite width of the
instability strip (WIS), adopted from Breuval et al. (2022). Superscript a
indicates that no HST F160W-IR CRNL corrections were applied to observations
of LMC Cepheids for this comparison. Magnitudes in the NIR Wesenheit function
were recomputed using Eq. 5 based on the observations reported by Riess et al.
(2019).

To determine the validity of the expected $\Delta\varpi_{\mathrm{Cl}}=0$, we
computed the absolute magnitude of LMC Cepheids by applying the distance
modulus measured obtained using detached eclipsing binaries (Pietrzyński et
al., 2019, $\mu_{\mathrm{DEB}}=18.477\pm 0.004\,(\mathrm{stat})\pm
0.026\,(\mathrm{syst})$ mag),

M $\displaystyle=\alpha(\log{P-10\
\mathrm{[d]}})+\beta^{{}^{\prime}}-\mu_{\mathrm{DEB}}$ (7)
$\displaystyle=\alpha(\log{P-10\
\mathrm{[d]}})+\delta^{{}^{\prime}}-\mu_{\mathrm{DEB}}+\gamma\Delta_{\mathrm{[Fe/H]}}\
.$ (8)

These absolute magnitudes of LMC metallicity Cepheids were then compared to MW
Cepheids using the astrometry-based luminosity (Arenou & Luri, 1999, ABL),
which avoids the issue of inverting parallaxes to obtain distances,

ABL
$\displaystyle=10^{\frac{M}{5}}=(\varpi+\Delta\varpi_{\mathrm{Cl}})10^{\frac{m-10}{5}}\
.$ (9)

Superscript ${}^{{}^{\prime}}$ in Eq. 8 implies that $\beta$ and $\delta$ are
given in apparent magnitudes after applying geometric corrections to the LMC
Cepheids. $\beta$ denotes the LL intercept at the average sample metallicity,
$\delta=\beta-\gamma\mathrm{[Fe/H]}$ is the LL intercept corrected to solar
metallicity, and $\Delta_{\mathrm{[Fe/H]}}$ is the difference in iron
abundance between the MW and LMC Cepheid samples. Table 9 lists the results of
this comparison for six individual photometric bands and three Wesenheit
formulations.

The metallicity difference between LMC and MW Cepheid requires careful
consideration. For the LMC, we adopted a common mean iron abundance,
$\mathrm{[Fe/H]_{LMC}}=-0.409\pm 0.003$, based on the recently remeasured
average iron abundances of LMC Cepheids that has been shown to be consistent
with a single value (Romaniello et al., 2022, dispersion $0.076$ dex). For MW
cluster Cepheids, we adopted individual iron abundances as described above and
compiled in Table 11. Although several improvements in the determination of
$\gamma$ have been recently presented (Gieren et al., 2018; Breuval et al.,
2021, 2022; Ripepi et al., 2022a), we here preferred to use $\gamma$ as a free
parameter, while first fixing $\Delta\varpi_{\mathrm{Cl}}=0$ and then
performing the same comparison while fitting for $\gamma$ and
$\Delta\varpi_{\mathrm{Cl}}$ simultaneously. The individual MW Cepheid
abundances are compiled in Table 11.

Table 9: Metallicity term $\gamma$ and zeropoint offset $\Delta\varpi$
obtained by comparing Gold sample cluster Cepheids to the LMC LL using
Equation (9).

Band |  $\gamma$ free and $\Delta\varpi=0$ |  $\gamma$ and $\Delta\varpi$ free |  |
---|---|---|---|---
| $\gamma$ | $\Delta\varpi$ | $\gamma$ | $\Delta\varpi$ | $n_{\text{MW}}$ |
| (mag/dex) | ($\mu$as) | (mag/dex) | ($\mu$as) |  |
$Bp$ | $-0.041\pm 0.098$ | $0$ | $-0.191\pm 0.255$ | $-18\pm 27$ | 23 |
$V$ | $0.015\pm 0.090$ | $0$ | $0.029\pm 0.216$ | $2\pm 24$ | 22 |
$G$ | $-0.300\pm 0.102$ | $0$ | $-0.447\pm 0.265$ | $-18\pm 28$ | 23 |
$Rp$ | $-0.168\pm 0.075$ | $0$ | $-0.187\pm 0.183$ | $-2\pm 21$ | 23 |
$I$ | $-0.083\pm 0.102$ | $0$ | $-0.150\pm 0.217$ | $-8\pm 23$ | 15 |
$F160W^{a}$ | $-0.259\pm 0.062$ | $0$ | $-0.216\pm 0.142$ | $5\pm 15$ | 15 |
$W_{VI}$ | $-0.035\pm 0.098$ | $0$ | $-0.146\pm 0.206$ | $-14\pm 22$ | 15 |
$W_{G}$ | $-0.394\pm 0.058$ | $0$ | $-0.448\pm 0.141$ | $-6\pm 15$ | 26 |
$W_{H}^{a}$ | $-0.287\pm 0.059$ | $0$ | $-0.297\pm 0.134$ | $-1\pm 14$ | 15 |
101010Wesenheit magnitudes of cluster Cepheids, $W_{H}$, were computed using
Eq. 5 using the photometric data for the individual passbands presented by
Riess et al. (2022a). Cluster parallaxes were bias corrected using the L21
approach. Superscript a indicates that HST WFC-IR CRNL corrections have been
applied to account for the $0.9$ to $1.8$ dex difference in flux among MW
cluster Cepheids and LMC Cepheids (mean correction $0.010$ mag). The weighted
mean and associated uncertainty for $\Delta\varpi_{\mathrm{Cl}}$ is $-4\pm
6\,\mu$as.

Our results for $\gamma$ listed in Table 9 show that metal-rich Cepheids are
typically brighter than metal poor Cepheids in each of the photometric bands
as well as the three Wesenheit formulations. This echoes recent results by
Breuval et al. (2022), albeit at lower precision because the metallicity range
we considered is limited. Additionally, as noted by Breuval et al., our
results are consistent with predictions of $\gamma$ derived from Geneva
stellar evolution models (Anderson et al., 2016b). We further confirm the
particularly strong metallicity dependence in Gaia $G$ band and the Gaia
Wesenheit function $W_{G}$ reported by Breuval et al. (2022) and Ripepi et al.
(2022a), while neither $Bp$ nor $Rp$ exhibit such a steep trend with
metallicity.

Concerning $\Delta\varpi_{\mathrm{Cl}}$, we find residual offsets consistent
with $0$ to within $1\sigma$ in all nine cases, and a weighted mean value of
$\Delta\varpi=-4\pm 6\,\mu$as. Additionally, we note that the values of
$\gamma$ obtained when fixing $\Delta\varpi_{\mathrm{Cl}}=0$ are consistent to
within their uncertainties with $\gamma$ values obtained when both parameters
are free, as well as with recent literature results. In summary, our
comparison involving the LMC thus strongly supports that the average cluster
parallaxes determined above exhibit no evidence of residual parallax offsets
beyond the L21 corrections.

Table 10: Information used for determining the Galactic LL using cluster
Cepheids.

Name | Period | $\varpi$ | $\sigma_{\text{stat}}$ | $\sigma_{\text{cov}}$ | $\sigma_{\text{total}}$ | $W_{G}$ | $W_{H}^{a}$ | [Fe/H] | Reference | E(B-V)
---|---|---|---|---|---|---|---|---|---|---
| (d) | ($\mu$as) | ($\mu$as) | ($\mu$as) | ($\mu$as) | (mag) | (mag) |  |  | (mag)
CE Cas A | 5.141 | $322.5$ | 3.1 | 6.7 | 7.4 | $7.628\pm 0.073$ |  |  |  |
CE Cas B | 4.479 | $322.5$ | 3.1 | 6.7 | 7.4 | $7.759\pm 0.081$ |  |  |  |
CF Cas | 4.875 | $322.5$ | 3.1 | 6.7 | 7.4 | $7.642\pm 0.010$ | $7.641\pm 0.030$ | $0.02\pm 0.06$ | LL11 | 0.556 $\pm$ 0.021
CG Cas | 4.366 | $335.7$ | 3.0 | 6.7 | 7.3 | $7.687\pm 0.030$ | $7.652\pm 0.034$ | $0.09\pm 0.06$ | LL11 | 0.667 $\pm$ 0.009
CM Sct | 3.917 | $442.9$ | 2.2 | 6.2 | 6.6 | $7.046\pm 0.014$ |  | $0.15\pm 0.06$ | LL11 | 0.775 $\pm$ 0.045
CS Vel | 5.905 | $281.2$ | 2.8 | 6.7 | 7.3 | $7.676\pm 0.006$ | $7.669\pm 0.029$ | $0.12\pm 0.06$ | G14 | $0.716\pm 0.027$
CV Mon | 5.379 | $585.2$ | 6.9 | 6.6 | 9.6 | $6.198\pm 0.006$ | $6.205\pm 0.029$ | $0.09\pm 0.09$ | G14 | 0.705 $\pm$ 0.018
DL Cas | 8.001 | $556.6$ | 2.2 | 6.5 | 6.9 | $5.597\pm 0.006$ | $5.645\pm 0.021$ | $-0.01\pm 0.08$ | L11 | $0.487\pm 0.005$
EV Sct(∗) | 3.090 | $504.0$ | 2.8 | 6.4 | 7.0 | $6.541\pm 0.010$ |  | $0.09\pm 0.07$ | G15 |
GH Lup | 9.276 | $877.8$ | 6.1 | 6.4 | 8.8 | $4.448\pm 0.008$ |  | $0.13\pm 0.06$ | G14 | $0.328\pm 0.011$
IQ Nor | 8.220 | $543.6$ | 7.2 | 6.1 | 9.4 | $5.711\pm 0.016$ |  | $0.22\pm 0.07$ | G15 | $0.676\pm 0.044$
NO Cas(∗) | 2.582 | $317.0$ | 2.7 | 6.4 | 7.0 | $8.113\pm 0.018$ |  |  |  |
QZ Nor(∗) | 3.786 | $513.4$ | 1.4 | 6.5 | 6.6 | $6.379\pm 0.006$ |  | $0.21\pm 0.06$ | G15 | $0.289\pm 0.020$
RS Ori | 7.567 | $609.6$ | 3.5 | 6.3 | 7.2 | $5.571\pm 0.036$ | $5.603\pm 0.030$ | $0.11\pm 0.09$ | G15 | $0.332\pm 0.010$
S Nor | 9.754 | $1073.2$ | 3.8 | 6.1 | 7.2 | $3.932\pm 0.014$ | $3.940\pm 0.014$ | $0.02\pm 0.09$ | R08 | $0.182\pm 0.008$
ST Tau | 4.034 | $953.0$ | 5.6 | 6.0 | 8.3 | $5.608\pm 0.019$ |  | $-0.14\pm 0.15$ | L07 | $0.328\pm 0.006$
SV Vul | 44.993 | $424.8$ | 5.9 | 6.6 | 8.9 | $3.393\pm 0.014$ | $3.589\pm 0.035$ | $0.05\pm 0.08$ | L11 | $0.474\pm 0.024$
SX Vel | 9.550 | $497.0$ | 4.0 | 6.0 | 7.2 | $5.746\pm 0.019$ |  | $-0.18\pm 0.07$ | L07 | $0.237\pm 0.014$
TW Nor | 10.786 | $421.2$ | 3.7 | 6.6 | 7.6 | $5.653\pm 0.022$ | $5.855\pm 0.030$ | $0.27\pm 0.10$ | G15 | $1.190\pm 0.023$
U Sgr | 6.745 | $1554.1$ | 2.7 | 5.4 | 6.0 |  | $3.565\pm 0.028$ | $0.08\pm 0.08$ | L11 | $0.408\pm 0.007$
V Cen | 5.494 | $1336.5$ | 2.2 | 6.1 | 6.5 | $4.293\pm 0.014$ | $4.258\pm 0.028$ | $0.04\pm 0.09$ | R08 | 0.265 $\pm$ 0.016
V Lac | 4.982 | $519.7$ | 4.7 | 6.6 | 8.1 | $6.304\pm 0.013$ |  | $0.06\pm 0.06$ | LL11 | $0.293\pm 0.034$
V0340 Nor | 11.289 | $513.2$ | 1.4 | 6.5 | 6.6 | $5.293\pm 0.005$ | $5.313\pm 0.029$ | $0.07\pm 0.07$ | G15 | $0.312\pm 0.050$
V0367 Sct | 6.295 | $513.6$ | 2.7 | 6.7 | 7.2 | $5.865\pm 0.033$ | $6.125\pm 0.054$ | $0.05\pm 0.08$ | G15 |
V0378 Cen(∗) | 6.459 | $517.8$ | 3.7 | 6.5 | 7.5 | $5.468\pm 0.002$ |  | $0.08\pm 0.06$ | LL11 | $0.374\pm 0.049$
V0379 Cas(∗) | 4.305 | $556.6$ | 2.2 | 6.5 | 6.9 | $5.876\pm 0.005$ |  | $0.06\pm 0.08$ | L11 |
V0438 Cyg | 11.211 | $561.9$ | 3.6 | 6.4 | 7.3 | $5.142\pm 0.021$ |  | $0.33\pm 0.06$ | LL11 |
VW Cru | 5.265 | $731.8$ | 6.1 | 6.3 | 8.7 | $5.601\pm 0.003$ |  | $0.19\pm 0.06$ | LL11 | 0.640 $\pm$ 0.046
WX Pup | 8.933 | $344.7$ | 4.6 | 6.5 | 8.0 | $6.323\pm 0.004$ | $6.326\pm 0.016$ | $-0.15\pm 0.15$ | G14 | $0.306\pm 0.018$
X Cru | 6.220 | $639.2$ | 2.5 | 6.0 | 6.5 | $5.677\pm 0.008$ |  | $0.15\pm 0.06$ | LL11 | $0.294\pm 0.019$
X Lac(∗) | 5.443 | $519.7$ | 4.7 | 6.6 | 8.1 | $5.766\pm 0.004$ |  | $0.08\pm 0.06$ | LL11 |
X Pup | 25.972 | $363.4$ | 3.7 | 5.1 | 6.4 | $4.089\pm 0.345$ | $5.023\pm 0.012$ | $0.02\pm 0.08$ | G15 | $0.396\pm 0.015$
X Vul | 6.320 | $879.5$ | 3.2 | 6.0 | 6.8 | $4.727\pm 0.007$ | $4.860\pm 0.031$ | $0.07\pm 0.08$ | L11 | $0.775\pm 0.021$
XZ Car | 16.652 | $482.1$ | 2.5 | 6.6 | 7.1 | $5.269\pm 0.033$ | $5.169\pm 0.010$ | $0.19\pm 0.06$ | LL11 | $0.372\pm 0.026$
Y Sct | 10.341 | $503.4$ | 2.8 | 6.4 | 7.0 | $5.121\pm 0.171$ |  | $0.23\pm 0.06$ | LL11 | $0.792\pm 0.021$

111111Cluster average parallaxes include the corrections as described by L21.
Iron abundances were rescaled by Genovali et al. (2015) to the common solar
abundance $A({\rm Fe})=7.50$ (Grevesse & Sauval, 1998). Color excess values
E(B-V) are taken from Fernie et al. (1995) and scaled by a factor 0.94 (cf.
Groenewegen, 2018). The symbol (∗) denotes Cepheids pulsating in the first
overtone mode. a: Observations reported in the HST system ($W_{H}$) are
computed using Eq. 5 and the individual passband data from Riess et al.
(2022a), that is, they do not contain the CRNL correction needed for
comparison with the SN-host Cepheid sample.
References: (LL11): Luck & Lambert (2011), (G14): Genovali et al. (2014),
(L11): Luck et al. (2011), (G15): Genovali et al. (2015), (R08): Romaniello et
al. (2008), (L07): Lemasle et al. (2007).

Figure 6: ABL for fundamental-mode Cepheids in the Gold sample using different
photometric systems. Open circles indicate the two Cepheids of the Silver
sample that are not part of the LL fits. The ABL values and the residuals were
shifted by constant offsets as indicated in the legend to facilitate visual
inspection. Cluster parallaxes were determined after applying the L21 parallax
corrections.

###  4.3 Galactic LL and residual parallax offset for Cepheids

We calibrated the Milky Way LL and the residual parallax offset applicable to
MW Cepheid parallaxes, $\Delta\varpi_{\rm Cep}$, using our Gold sample of
cluster Cepheids. We note that the following exclusively considers MW Cepheid
information and is thus independent of the LMC, which was merely used as a
cross-check in Sect. 4.2.

We fit the MW LL while simultaneously determining the residual parallax offset
for Cepheids, $\Delta\varpi_{\rm Cep}$, using

$\displaystyle\text{ABL}=10^{M/5}=\begin{cases}\hfil\varpi
10^{\frac{m-10}{5}}&\text{for clusters},\\\
\hfil(\varpi+\Delta\varpi_{\text{Cep}})10^{\frac{m-10}{5}}&\text{for
Cepheids},\end{cases}$ (10)

with
$M=\alpha\cdot(\log{P}-1)+\beta=\alpha\cdot(\log{P}-1)+\delta+\gamma\cdot\mathrm{[Fe/H]}$.
Both LL slope and zeropoint were used as free parameters, and
$\Delta\varpi_{\rm Cl}=0$ as explained above.

Figure 7: ABL for $W_{H}$ based on HST photometry for the joint sample of Gold
cluster Cepheids (N = 15) and the Cepheids in the R21a sample (N = 67). Black
error bars are derived using Gaia EDR3 parallaxes of Cepheids, and colored
error bars are based on cluster parallaxes. Specific cases are colored
individually to help identify Cepheids with cluster parallaxes discussed in
the text. U Sgr, S Nor, and SV Vul appear twice in the plot because we use the
Cepheid and cluster parallaxes to estimate its ABL. The Cepheids in the Silver
sample XZ Car and X Pup are not included in the fit. In the plot, the
zeropoint offset of the Cepheids has been already applied.

We first performed this fit at the sample average iron abundance and then
repeated the fit assuming a fixed value of $\gamma$ from the literature,
specifically, $\gamma_{W_{H}}=-0.217\pm 0.046$ (Riess et al., 2022b) and
$\gamma_{W_{G}}=-0.384\pm 0.051$ (Breuval et al., 2022). We used individual
Cepheid iron abundances, not the sample average, to determine the zeropoint at
solar metallicity, $\delta$. Using fixed literature slopes for $\gamma$ has
the significant benefit of $\gamma$ being informed by a wider range of
metallicities, while both the range of [Fe/H] in the MW sample and the
correction to the solar value are small. Although we propagated the errors,
this metallicity correction has virtually no effect on the final results due
to the only slightly supersolar metallicity of MW Cepheids. Following common
practice (e.g., Kodric et al., 2018; Riess et al., 2022b), we applied a
$2.7\sigma$ outlier rejection. This step removed $24$ of $249$ Cepheids for
the Gaia-only sample, the vast majority of which are $>3\sigma$ outliers. The
ABL fit results are illustrated in Figures 7 and 8.

Figure 8: ABL in the Gaia Wesenheit magnitude $W_{G}$ (cf. Figure 7 for sample
details).

At the average sample metallicity, we find for the NIR Wesenheit LL ($W_{H}$)

$\displaystyle\beta=-5.953\pm 0.020,\quad\alpha=-3.406\pm 0.052,$ (11)
$\displaystyle\Delta\varpi_{\text{Cep}}=-18\pm
5\,\mu\text{as},\quad\chi^{2}=1.1,$ (12) $\displaystyle\text{[Fe/H]}=0.086\ ,$
(13)

which at solar metallicity becomes

$\displaystyle\delta=-5.930\pm 0.020,\quad\alpha=-3.383\pm 0.052,$ (14)
$\displaystyle\Delta\varpi_{\text{Cep}}=-17\pm
5\,\mu\text{as},\quad\gamma^{\text{fixed}}_{W_{H}}=-0.217\pm 0.046,$ (15)
$\displaystyle\chi^{2}=1.1,\quad\text{N}^{\text{MW}}_{\text{Cl}}=15,\quad\text{N}^{\text{MW}}_{\text{Cep}}=67.$
(16)

We note that CRNL corrections ($\sim 0.05$ mag) were applied to the apparent
WFC3/IR F160W and NIR Wesenheit magnitudes to facilitate the comparison with
Cepheids in supernova-host galaxies and simplify the comparison with the SH0ES
distance ladder.

Both results establish a nonzero residual parallax offset for MW Cepheid
parallaxes at $\gtrsim 3\sigma$ significance, and this result is fully
consistent with the $-14\pm 6\,\mu$as offset determined by Riess et al.
(2021). This provides additional evidence that clusters and Cepheids require
different residual parallax offsets.

To directly compare our results to the value of $M_{H,1}^{W}$ determined as
part of the SH0ES distance ladder (Riess et al., 2022b, a), we fixed the LL
slope to the SH0ES baseline value and obtained

$\displaystyle\delta=-5.914\pm 0.017,\quad\alpha_{\text{fixed}}=-3.299\pm
0.015,$ (17) $\displaystyle\Delta\varpi_{\text{Cep}}=-13\pm
5\,\mu\text{as},\quad\gamma^{\text{fixed}}_{W_{H}}=-0.217\pm 0.046,$ (18)
$\displaystyle\chi^{2}=1.2.$ (19)

Our result for $\delta$ agrees to within $0.3\sigma$ with the value of
$M_{H,1}^{W}$ determined by the SH0ES team via the two-parameter Gold sample
fit in Table 5 of Riess et al. (2022a), where $M_{H,1}^{W}=-5.907\pm 0.018$
mag. Nevertheless, our approach to determine $\delta$ using the NIR Wesenheit
function $W_{H}$ (Eq. 14) differs from their approach in three important
elements. First, we used a combined fit of Cepheid and cluster parallaxes to
obtain an absolute calibration based exclusively on Gaia astrometry. Second,
our clustering analysis in Sect. 2.1 was conducted entirely independently of
Riess et al. (2022a). Third, the samples of cluster member stars differ
between our study and Riess et al. (2022a), resulting in an average difference
of $\sim 5\,\mu$as among cluster parallaxes. We therefore consider our result
an important cross-check based on mostly independent astrometric information.

For the corresponding Gaia Wesenheit function ($W_{G}$) at sample average
metallicity, we obtain

$\displaystyle\beta=-6.051\pm 0.020,\quad\alpha=-3.303\pm 0.049,$ (20)
$\displaystyle\Delta\varpi_{\text{Cep}}=-22\pm
3\,\mu\text{as},\quad\chi^{2}=1.2,$ (21) $\displaystyle\text{[Fe/H]}=0.069\ ,$
(22)

and, after correcting to solar metallicity using the individual Cepheid iron
abundances,

$\displaystyle\delta=-6.004\pm 0.019,\quad\alpha=-3.242\pm 0.047,$ (23)
$\displaystyle\Delta\varpi_{\text{Cep}}=-19\pm
3\,\mu\text{as},\quad\gamma^{\text{fixed}}_{W_{G}}=-0.384\pm 0.051,$ (24)
$\displaystyle\chi^{2}=1.1,\quad\text{N}^{\text{MW}}_{\text{Cl}}=26,\quad\text{N}^{\text{MW}}_{\text{Cep}}=225.$
(25)

We thus find $1\sigma$ agreement for $\Delta\varpi_{\rm Cep}$ regardless of
whether HST or Gaia photometry is used, and using different, albeit not
independent, sets of Cepheids and cluster parallaxes. In particular, we note
the improved precision on $\Delta\varpi_{\rm Cep}$ determined using Gaia
photometry, for which we obtain a $6\sigma$ detection that is consistent with
the value determined using the independent HST photometry. We further note
that metallicity corrections do not challenge the accuracy of our
determination of $\Delta\varpi_{\rm Cep}$. To illustrate our results in a more
conventional LL form, we plot the absolute Wesenheit magnitudes as a function
of $\log{P}$ in Figure 9.

Figure 9: LL in the $H$ and $G$ Wesenheit bands. Given the high precision of
the Cepheid parallaxes, their individual distances were calculated as
$1/\varpi$. The plots are shown for illustration purposes, and they were not
used to fit the data.  Table 11: Combined fits to cluster Cepheids and (field)
MW Cepheids.

Filter | $\alpha$ | $\beta$ | $\Delta\varpi_{\rm Cep}$ | $\langle\mathrm{[Fe/H]}\rangle$ | NCep | Ncl
---|---|---|---|---|---|---
| (mag/$\log{P}$) | (mag) | ($\mu$as) |  |  |
$W_{H}^{a}$ | $-3.412\pm 0.053$ | $-6.003\pm 0.020$ | $-19\pm 5$ | 0.086 | 67 | 15
$W_{G}$ | $-3.303\pm 0.049$ | $-6.051\pm 0.020$ | $-22\pm 3$ | 0.069 | 225 | 26
$Bp$ | $-2.513\pm 0.080$ | $-4.225\pm 0.036$ | $-21\pm 6$ | 0.069 | 243 | 23
$V$ | $-2.553\pm 0.071$ | $-4.377\pm 0.033$ | $-22\pm 5$ | 0.069 | 246 | 22
$G$ | $-2.751\pm 0.077$ | $-4.612\pm 0.035$ | $-22\pm 5$ | 0.068 | 238 | 23
$Rp$ | $-2.804\pm 0.058$ | $-4.984\pm 0.025$ | $-19\pm 4$ | 0.068 | 234 | 23
$F160W^{a}$ | $-3.353\pm 0.060$ | $-5.729\pm 0.023$ | $-23\pm 6$ | 0.088 | 67 | 15
$W_{H}^{b}$ | $-3.406\pm 0.052$ | $-5.953\pm 0.020$ | $-18\pm 5$ | 0.086 | 67 | 15
$F160W^{b}$ | $-3.346\pm 0.060$ | $-5.679\pm 0.023$ | $-22\pm 6$ | 0.088 | 67 | 15
121212a Does not include the CRNL correction. b Includes the CRNL correction
(mean $0.010$ mag) to facilitate comparison with extragalactic Cepheid samples
(Riess et al., 2022a).

We further applied the same approach for Johnson $V-$band, Gaia $G$, $Bp$, and
$Rp$, and HST $F160W$ photometry. The results are listed in Table 12. In
particular, we note that the value of $\Delta\varpi_{\rm Cep}$ is consistent
within less than $1\sigma$ for all nine rows in Table 12. Figure 10
illustrates the results for individual photometric passbands together with
linear fits of the LL parameters as a function of the inverse of the effective
central wavelength $\lambda$ of each filter. The average iron abundances of
the samples differ by $<0.02$ dex, and we thus expect a difference of $\sim$
0.02 dex $\cdot$ 0.2 mag/dex = 0.004 mag at most between the values of $\beta$
evaluated at the lower and upper metallicity of our sample. This difference is
well contained within the uncertainties. Fitting the wavelength dependence of
$\alpha$ and $\beta$ as a function of inverse wavelength, we determine the
following dependence of LL slope and zeropoint on central wavelength:

$\displaystyle\alpha$ $\displaystyle=(-3.769\pm 0.083)+(0.683\pm
0.059)/\lambda$ (26) $\displaystyle\beta$ $\displaystyle=(-6.526\pm
0.056)+(1.208\pm 0.041)/\lambda\ .$ (27)

Figure 10: Linear fit of the LL parameters as a function of the inverse of the
effective wavelength in different photometric filters.

##  5 Discussion

###  5.1 Using Silver sample Cepheids for LL calibration

Our criteria placed two long-period Cepheids with uncertain cluster
membership, X Pup and XZ Car, in the Silver sample, which we conservatively
did not use for LL calibration. As explained in Sec. 3.2, both stars featured
low membership likelihoods due primarily to mismatching kinematic information.
However, closer inspection suggested that X Pup is possibly a true cluster
Cepheid that can be used for LL calibration (cf. Sec. 3.2). We determine the
impact of including these stars in our analysis below.

Including X Pup and XZ Car in the cross-check of involving the LMC (Sect. 4.2)
would not significantly affect the results. For $W_{G}$ we find
$\Delta\varpi_{\rm Cl}=-8\pm 17\,\mu$as, $\gamma=-0.418\pm 0.150$ mag/dex, and
for $W_{H}$, we obtain $\Delta\varpi_{\rm Cl}=-7\pm 16\,\mu$as,
$\gamma=-0.205\pm 0.148$ mag/dex. All these values agree to within much less
than one standard deviation with those obtained using only the Gold sample of
Cepheids.

Including XZ Car in the combined LL fit in Sec. 4.3 has no impact because it
is a $3.5\sigma$ LL outlier that would be rejected by the $\sigma-$clipping
procedure. Including X Pup in the fit does not significantly affect the LL
calibration ($\alpha=-3.313\pm 0.049$, $\beta=-6.051\pm 0.020$,
$\Delta\varpi_{\rm Cep}=-21\pm 3$ all agree to much better than $1\sigma$ with
results in Eq. 20) and marginally increases the reduced $\chi^{2}$ by $0.008$.
Furthermore, X Pup has not been identified as an LL outlier by Riess et al.
(2022a) in the NIR Wesenheit formulation.

###  5.2 Fraction of Cepheids in clusters within $2$ kpc

The fraction of Cepheids residing in clusters is of interest for understanding
clustered star formation (Dinnbier et al., 2022) and the extragalactic
distance scale (Anderson & Riess, 2018), among other things. Using our Gold
sample of cluster Cepheids and data from the recent Gaia DR3, we updated
previous estimates of this fraction,
$f_{\mathrm{CC,2kpc}}=N_{\mathrm{Cl,2kpc}}/N_{\mathrm{Cep,2kpc}}$. Assuming
that all Cepheid-hosting clusters within $2$ kpc could be identified by our
method, we have $N_{\mathrm{Cl,2kpc}}=22$ (Gold sample), which includes $11$
coronal members separated by projected distances of $8-25$ pc from their host
cluster centers.

We estimated the total number of Cepheids within $2$ kpc,
$N_{\mathrm{Cep,2kpc}}$ using the photometric parallaxes obtained with our
$W_{G}$ LL calibration for all stars classified as DCEP in Gaia DR3 table
gaiadr3.vari_cepheid. This yields $180^{+32}_{-38}$ fundamental mode Cepheids
as well as $70^{+9}_{-16}$ first-overtone or multimode Cepheids, where
overtone periods were fundamentalized using the period ratios determined by
Kovtyukh et al. (2016, assuming a mean metallicity
$\langle\mathrm{[Fe/H]}\rangle=0.032$). For multimode Cepheids, either the
fundamental or first-overtone period was used to compute the distance. We also
sought to estimate $N_{\mathrm{Cep,2kpc}}$ using distances provided by the
parameter distance_gspphot in Gaia DR3 table gaiadr3.gaia_source as well as
Gaia parallaxes (including the residual offset determined in Eq. 24). However,
this reduced the size of Cepheid samples by approximately $20\%$ due to
limited data availability. We therefore considered the estimation based on
photometric distances our baseline result due to greater completeness. The
results are tabulated in Table 13, where asymmetric uncertainties reflect the
range of stars defined by the $1\sigma$ distance or parallax uncertainties.

We thus estimate $f_{\mathrm{CC,2kpc}}=0.088^{+0.029}_{-0.019}$, where the
uncertainties provided denote the full range of possibilities. We further find
a slightly higher fraction of fundamental mode Cepheids in clusters, with
$f_{\mathrm{CC,2kpc,FM}}=0.089^{+0.030}_{-0.018}$ and
$f_{\mathrm{CC,2kpc,FO}}=0.081^{+0.026}_{-0.023}$, assuming the OGLE
classification of cluster Cepheids (Pietrukowicz et al., 2021). If the
pulsation modes assigned in Gaia DR3 (Ripepi et al., 2022b) were used instead,
the difference would be slightly larger, with
$f_{\mathrm{CC,2kpc,FM}}=0.100^{+0.033}_{-0.020}$ and
$f_{\mathrm{CC,2kpc,FO}}=0.065^{+0.021}_{-0.022}$. This difference could be
explained by the dependence of $f_{\rm CC}$ on age due to clusters dissolving
into the field over time combined with the tendency of overtone Cepheids to
originate from older lower-mass stars than fundamental-mode Cepheids, which
can be rather young.

We note that a few bright Cepheids, such as Polaris and the cluster Cepheid U
Sgr, are not included in the vari_cepheid table. However, their absence does
not change the overall result. Our new estimate supersedes our previous
slightly lower estimate of $f_{CC,2kpc}=15/217=6.9\%$ reported in Dinnbier et
al. (2022) due to improvements in our membership determination and the input
data from Gaia DR3.

Figure 11 illustrates the fraction of Cepheids residing in clusters within 2
kpc of the Sun as a function of age. Cepheid ages were computed using period-
age relations for fundamental and first-overtone Cepheids (Anderson et al.
(2016b)). We confirmed that ages based on periods of overtone Cepheids matched
ages computed using period-age relations for fundamental-mode Cepheids after
fundamentalizing the pulsation periods of first-overtone Cepheids using period
ratios of Milky Way double-mode Cepheids (Kovtyukh et al., 2016). Figure 11
thus illustrates the dispersal of Cepheid host clusters over time, an effect
previously reported by Anderson & Riess (2018) and also seen in dynamical
NBODY simulations (Dinnbier et al., 2022). We caution that young ages are
rather poorly sampled within $2$ kpc of the Sun due to the low volumetric rate
of long-period Cepheids. At ages above 132 Myr, no cluster Cepheids are found
within $2\,$kpc of the Sun.

Figure 11: Clustered Cepheid fraction as a function of Cepheid age estimated
using the period-age relations for solar metallicity (Anderson et al., 2016b).
The size of the error bars illustrates the full range of possible fractions.
Different numbers of bins were used to illustrate the dependence on binning.
Young long-period Cepheids are rare within 2 kpc of the Sun, increasing the
scatter at ages below $80$ Myr. No Cepheids older than $132$ Myr are found in
clusters within 2 kpc of the Sun. Table 12: Number of Cepheids within 2 kpc of
the Sun.

Sample | Method | Fundamental mode | Overtone | Multi mode | Total
---|---|---|---|---|---
Field | LL $W_{G}$ | $180^{+32}_{-38}$ | $62^{+9}_{-15}$ | $8^{+0}_{-1}$ | $250^{+41}_{-54}$
Field | distance_gspphot | $139^{+12}_{-13}$ | $55^{+7}_{-9}$ | $7^{+0}_{-1}$ | $201^{+19}_{-23}$
Field | Parallax | $130^{+10}_{-13}$ | $61^{+6}_{-13}$ | $7^{+1}_{-0}$ | $198^{+17}_{-26}$
Clusters | LL $W_{G}$, OGLE classifiers | $16^{+1}_{-1}$ | $5^{+0}_{-1}$ | $1^{+0}_{-0}$ | $22^{+1}_{-2}$
Clusters | LL $W_{G}$, Gaia classifiers | $18^{+1}_{-1}$ | $4^{+0}_{-1}$ | $0^{+0}_{-0}$ | $22^{+1}_{-2}$

131313The upper and lower indexes are an estimate of the maximum and minimum
number of Cepheids, they are not standard errors, and for this reason, they
are not are not added in quadrature.

###  5.3 Expected improvements

Astrometric uncertainties tend to increase with distance, complicating the
identification of distant open clusters. For Gaia EDR3, the number of false
cluster detections at distances greater than $3$ kpc increases rapidly, so
that significant work is required to ascertain the veracity of the recovered
cluster candidates. However, upcoming Gaia data releases will improve the
ability to correctly identify clusters at large distances, which can be
expected to result in much improved cluster Cepheid samples with Gaia DR4 and
beyond. Whereas Gaia EDR3 was based on 34 months of observations, the DR4
astrometric solution of Gaia will be based on approximately 66 months of
observations, and the Gaia Collaboration expects improvements in proper motion
proportional to $t^{-3/2}$ and in parallax proportional to $t^{-1/2}$. Hence,
DR4 proper motion uncertainties may be about $0.35$ times their DR3
uncertainties, whereas DR4 parallax uncertainties could be approximately
$0.70$ times those reported in DR3. As Eq. 1 illustrates (cf. also footnote
3), the ability of detecting clusters against the background depends on
distance and proper motion uncertainties. However, it is unlikely that the
full gain in proper motion precision will directly map to a greater volume
limit for detecting clusters because parallax errors improve less rapidly. To
obtain a rough estimate of future improvements, we therefore considered a mean
improvement by a factor of approximately 2 (counting parallax and both proper
motion directions separately), which would double the distance within which
cluster Cepheids can be detected. Based on their location in the Galactic
plane, the number of clusters increases proportional to $d^{2}$, resulting in
a potential quadrupling of cluster-hosting Cepheids with DR4, and thus, in a
potential improvement of a factor of $2$ for the LL calibration. Since most
long-period Cepheids are located at distances beyond $2\,$kpc, this will be
particularly useful to increase the number of these high-priority targets.

Calibrating the cosmic distance ladder to within $1\%$ requires parallaxes of
Cepheids measured to an accuracy of $\sim 5\,\mu$as (Riess et al., 2021). At
present, cluster Cepheids appear to be the most viable route to this goal.
However, the angular covariance of the (E)DR3 parallaxes currently still sets
an error floor of $\sim 7\,\mu$as and is therefore in urgent need of further
improvement. It is very noteworthy that cluster members apparently do not
require residual parallax offset corrections, since solving for this offset
has thus far limited the power of Gaia parallaxes for measuring $H_{0}$ (e.g.,
Riess et al., 2018, 2021). Additionally, new HST observations of cluster
Cepheids will be crucial to avoid uncertainties related to photometric
transformations from the ground to the HST system. In summary, identifying new
cluster Cepheids and measuring their photometry using HST will provide the
most accurate basis for calibrating the distance ladder for a $1\%$ $H_{0}$
measurement. We are optimistic that future Gaia data releases will continue to
improve the error floor set by angular covariance and that other mitigation
strategies can be identified to leverage the power of Gaia for the
extragalactic distance scale and cosmology.

##  6 Conclusions

We carried out a systematic search for MW cluster Cepheids using Gaia EDR3 and
DR3 data. The improved proper motion precision of EDR3 over DR2 allowed us to
obtain a more detailed and accurate view of cluster membership for previously
discussed cluster Cepheids. Since our method requires no advance knowledge of
clusters being present in the vicinity of Cepheids, we a) determined cluster
astrometry without the need for prior literature search on the host clusters,
and b) avoided confusion of cluster identification in case of complex sky
areas featuring multiple clusters. We thus established a Gold sample of 34
Cepheids residing in 28 distinct MW open clusters. They include the three new
bona fide cluster Cepheids ST Tau, V0378 Cen, and GH Lup. Additionally, we
corrected the host cluster identification for three Cepheids previously
discussed in the literature, namely SX Vel, IQ Nor, and VW Cru. We find SV Vul
to be a bona fide cluster Cepheid that falls squarely on the Galactic LL. We
find three Silver sample cluster Cepheid candidates of interest, of which X
Pup is a likely cluster Cepheid, whereas the XZ Car cluster membership is
tentatively excluded by kinematic constraints and the AP Vel parallax narrowly
contradicts membership in Ruprecht 65. Additional combinations of possible
interest are included in a Bronze sample.

Using photometric distances of Cepheids in the Gold sample and the
concatenated list of Cepheids from Pietrukowicz et al. (2021) and Ripepi et
al. (2022b), we estimate the fraction of clustered Cepheids within $2$ kpc to
be in the range of
$f_{\mathrm{CC,2kpc}}=N_{\mathrm{Cl,2kpc}}/N_{\mathrm{Cep,2kpc}}=0.088^{+0.029}_{-0.019}$.
We find a slightly larger fraction for Cepheids pulsating in the fundamental
mode compared to the first overtone, which may be related to the dependence of
$f_{\rm CC}$ on age and cluster dispersal timescales.

Cluster parallaxes are superior for LL calibration compared to individual
Cepheid parallaxes because cluster member stars combine several benefits,
including a) greater statistical precision, b) better systematics in a fainter
magnitude range that does not require special processing, c) the absence of
high-amplitude variability, and d) greater consistency in brightness and color
with LMC stars and quasars used to determine the EDR3 parallax systematics
(L21). The uncertainty of average cluster parallaxes is currently dominated by
angular covariance, which limits average parallax uncertainties to $\gtrsim
7\,\mu$as, although the statistical uncertainty can be as low as $1.4\,\mu$as.

We identified the magnitude and color ranges of $12.5<G<17$ mag and $0.23<Bp-
Rp<2.75$ as a sweet spot for determining average cluster parallaxes. Previous
studies (e.g,. Flynn et al., 2022; Maíz Apellániz, 2022) found that parallaxes
of cluster member stars in this magnitude range are adequately corrected by
the L21 recipes, and we cross-checked this result using Cepheids in the LMC,
taking the metallicity difference between MW and LMC Cepheids into account.
Using the LL metallicity slope $\gamma$ yielded negative values in six
individual photometric passbands and three reddening-free Wesenheit
magnitudes, confirming recent results by Breuval et al. (2022). Allowing for a
nonzero offset for cluster parallaxes yields a weighted average of
$\Delta\varpi_{\rm Cl}=-4\pm 6\,\mu$as, with each individual offset consistent
with $0$ to within $1\sigma$. Hence, we confirm that cluster parallaxes
determined using member stars in this magnitude and color range require no
further correction of residual parallax offsets beyond the L21 corrections. We
stress that the LMC was used only for comparison and does not otherwise enter
the results of this study.

Setting $\Delta\varpi_{\rm cl}=0$, we calibrated the Galactic Cepheid LL in
the several passbands and reddening-free Wesenheit magnitudes while
simultaneously solving for a residual parallax offset of Gaia parallaxes of
Cepheids, $\Delta\varpi_{\rm Cep}$. In particular, we calibrated the absolute
luminosity scale of $10$ d fundamental-mode Cepheids at solar metallicity to a
precision of $0.94\%$ using NIR HST Wesenheit magnitudes and to a precision of
$0.87\%$ using optical Gaia Wesenheit magnitudes. The LL slope and metallicity
effect from the SH0ES analysis provide the most direct comparison of our
results of relevance for the Hubble constant and reveals excellent
($0.3\sigma$) agreement with the recent results by Riess et al. (2022a). Using
NIR HST and optical Gaia Wesenheit magnitudes, we obtained $\Delta\varpi_{\rm
Cep}=-17\pm 5$ and $-19\pm 3\,\mu$as, respectively. This $7\sigma$ measurement
of the residual parallax offset for Cepheids is the most precise to date and
provides strong independent confirmation of the Cepheid parallax offset of
$-14\pm 6\,\mu$as measured by the SH0ES team.

Cluster Cepheids can play a crucial role for the measurement of $H_{0}$ by
providing an accurate absolute trigonometric scale based on Gaia astrometry
without the need to solve for further offsets while determining the Hubble
constant. Future developments, such as improved proper motion membership
constraints for cluster detection through the longer astrometric baselines of
Gaia in future data releases, improved corrections of the Gaia parallax
systematics and angular covariance, and high-quality photometry of MW Cepheids
in and out of clusters will particularly improve the base calibration of the
distance scale toward a $1\%$ Hubble constant measurement.

###### Acknowledgements.

We thank the anonymous referee for comments that allowed to improve the
quality of the manuscript. We are thankful for useful discussions with Adam
Riess and Stefano Casertano as well as their comments on an earlier version of
the manuscript. We further acknowledge useful discussions in the framework of
the ISSI International Team project #490 meeting in Bern in July 2022.
MC & RIA acknowledge support from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (Grant
Agreement No. 947660). RIA further acknowledges support through a Swiss
National Science Foundation Eccellenza Professorial Fellowship (award
PCEFP2_194638). This work has made use of data from the European Space Agency
(ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia
Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research has made use
of NASA’s Astrophysics Data System; the SIMBAD database and the VizieR catalog
access tool141414http://cdsweb.u-strasbg.fr/ provided by CDS, Strasbourg;
Astropy151515http://www.astropy.org, a community-developed core Python package
for Astronomy (Robitaille et al., 2013; Price-Whelan et al., 2018);
TOPCAT161616http://www.star.bristol.ac.uk/~mbt/topcat/ (Taylor, 2005).

##  Appendix A Detected host clusters

Figure 12 shows the estimated dispersion in proper motion for the detected
clusters in this study. Table 17 provides the Gaia EDR3 source ids for all
cluster members, as well as the parallax corrections that were used to compute
the residual offset for MW Cepheids in Sect. 4.3.

Figure 12: Proper motion dispersion of the detected clusters in the Gold,
Silver, and Bronze samples.  Table 13: Cluster members of the Gold, Silver,
and Bronze samples.

Cluster | Gaia EDR3 source id | $\varpi_{corr}$ | $corr$
---|---|---|---
|  | (mas) | (mas)
Czernik 41 | 2026716726778925568 | 0.3398 | -0.0148
Czernik 41 | 2026716692420403328 | 0.4367 | -0.0408
Czernik 41 | 2026716731097672704 | 0.3948 | -0.0470
… | … | … | …
171717The complete version of this table is available at the CDS. $\varpi_{\rm
corr}$ is the parallax corrected applying the L21 offset, and $\rm corr$ is
the corresponding value of the correction.

## References

  * Anderson et al. (2016a) Anderson, R., Casertano, S., Riess, A., et al. 2016a, The Astrophysical Journal Supplement Series, 226, 18
  * Anderson (2022) Anderson, R. I. 2022, A&A, 658, A148
  * Anderson et al. (2013) Anderson, R. I., Eyer, L., & Mowlavi, N. 2013, Monthly Notices of the Royal Astronomical Society, 434, 2238
  * Anderson & Riess (2018) Anderson, R. I. & Riess, A. G. 2018, The Astrophysical Journal, 861, 36
  * Anderson et al. (2016b) Anderson, R. I., Saio, H., Ekström, S., Georgy, C., & Meynet, G. 2016b, Astronomy & Astrophysics, 591, A8
  * Apellániz et al. (2021) Apellániz, J. M., González, M. P., & Barbá, R. 2021, Astronomy & Astrophysics, 649, A13
  * Arenou & Luri (1999) Arenou, F. & Luri, X. 1999, in Astronomical Society of the Pacific Conference Series, Vol. 167, Harmonizing Cosmic Distance Scales in a Post-HIPPARCOS Era, ed. D. Egret & A. Heck, 13–32
  * Barnes et al. (1988) Barnes, III, T. G., Moffett, T. J., & Slovak, M. H. 1988, ApJS, 66, 43
  * Berdnikov (2008) Berdnikov, L. 2008, VizieR Online Data Catalog, II
  * Bersier (2002) Bersier, D. 2002, ApJS, 140, 465
  * Bersier et al. (1994) Bersier, D., Burki, G., Mayor, M., & Duquennoy, A. 1994, A&AS, 108, 25
  * Boffin et al. (2022) Boffin, H. M. J., Jerabkova, T., Beccari, G., & Wang, L. 2022, MNRAS, 514, 3579
  * Breuval et al. (2020) Breuval, L., Kervella, P., Anderson, R. I., et al. 2020, Astronomy & Astrophysics, 643, A115
  * Breuval et al. (2021) Breuval, L., Kervella, P., Wielgórski, P., et al. 2021, The Astrophysical Journal, 913, 38
  * Breuval et al. (2022) Breuval, L., Riess, A. G., Kervella, P., Anderson, R. I., & Romaniello, M. 2022, ApJ, 939, 89
  * Cantat-Gaudin & Anders (2020) Cantat-Gaudin, T. & Anders, F. 2020, Astronomy & Astrophysics, 633, A99
  * Cantat-Gaudin et al. (2020) Cantat-Gaudin, T., Anders, F., Castro-Ginard, A., et al. 2020, A&A, 640, A1
  * Cantat-Gaudin et al. (2018) Cantat-Gaudin, T., Jordi, C., Vallenari, A., et al. 2018, Astronomy & Astrophysics, 618, A93
  * Castelli & Kurucz (2003) Castelli, F. & Kurucz, R. L. 2003, in Modelling of Stellar Atmospheres, ed. N. Piskunov, W. W. Weiss, & D. F. Gray, Vol. 210, A20
  * Castro-Ginard et al. (2022) Castro-Ginard, A., Jordi, C., Luri, X., et al. 2022, A&A, 661, A118
  * Castro-Ginard et al. (2018) Castro-Ginard, A., Jordi, C., Luri, X., et al. 2018, Astronomy & Astrophysics, 618, A59
  * Chen et al. (2015) Chen, X., de Grijs, R., & Deng, L. 2015, Monthly Notices of the Royal Astronomical Society, 446, 1268
  * Coulson & Caldwell (1985) Coulson, I. M. & Caldwell, J. A. R. 1985, South African Astronomical Observatory Circular, 9, 5
  * Dinnbier et al. (2022) Dinnbier, F., Anderson, R. I., & Kroupa, P. 2022, A&A, 659, A169
  * El-Badry et al. (2021) El-Badry, K., Rix, H.-W., & Heintz, T. M. 2021, Monthly Notices of the Royal Astronomical Society
  * Elsanhoury & Amin (2019) Elsanhoury, W. & Amin, M. Y. 2019, Serbian Astronomical Journal, 45
  * Evans et al. (2022) Evans, D. W., Eyer, L., Busso, G., et al. 2022, arXiv e-prints, arXiv:2206.05591
  * Evans et al. (2014) Evans, N. R., Pillitteri, I., Wolk, S., et al. 2014, ApJ, 785, L25
  * Eyer et al. (2022) Eyer, L., Audard, M., Holl, B., et al. 2022, arXiv e-prints, arXiv:2206.06416
  * Fabricius et al. (2021) Fabricius, C., Luri, X., Arenou, F., et al. 2021, Astronomy & Astrophysics, 649, A5
  * Fernie et al. (1995) Fernie, J., Beattie, B., Evans, N., & Seager, S. 1995, IBVS No, 4148
  * Fitzpatrick (1999) Fitzpatrick, E. L. 1999, Publications of the Astronomical Society of the Pacific, 111, 63
  * Flynn et al. (2022) Flynn, C., Sekhri, R., Venville, T., et al. 2022, MNRAS, 509, 4276
  * Genovali et al. (2014) Genovali, K., Lemasle, B., Bono, G., et al. 2014, Astronomy & Astrophysics, 566, A37
  * Genovali et al. (2015) Genovali, K., Lemasle, B., Da Silva, R., et al. 2015, Astronomy & Astrophysics, 580, A17
  * Gieren et al. (2018) Gieren, W., Storm, J., Konorski, P., et al. 2018, Astronomy & Astrophysics, 620, A99
  * Gieren et al. (1997) Gieren, W. P., Fouqué, P., & Gómez, M. 1997, The Astrophysical Journal, 488, 74
  * Gorynya et al. (1992) Gorynya, N. A., Irsmambetova, T. R., Rastorgouev, A. S., & Samus, N. N. 1992, Soviet Astronomy Letters, 18, 316
  * Grevesse & Sauval (1998) Grevesse, N. & Sauval, A. J. 1998, Space Sci. Rev., 85, 161
  * Groenewegen (2018) Groenewegen, M. 2018, Astronomy & Astrophysics, 619, A8
  * He et al. (2022) He, Z., Li, C., Zhong, J., et al. 2022, ApJS, 260, 8
  * Heinze et al. (2018) Heinze, A., Tonry, J. L., Denneau, L., et al. 2018, The Astronomical Journal, 156, 241
  * Huang et al. (2021) Huang, Y., Yuan, H., Beers, T. C., & Zhang, H. 2021, The Astrophysical Journal Letters, 910, L5
  * Hunt & Reffert (2021) Hunt, E. L. & Reffert, S. 2021, Astronomy & Astrophysics, 646, A104
  * Imbert (1999) Imbert, M. 1999, A&AS, 140, 79
  * Jacyszyn-Dobrzeniecka et al. (2016) Jacyszyn-Dobrzeniecka, A. M., Skowron, D. M., Mróz, P., et al. 2016, Acta Astron., 66, 149
  * Jerabkova et al. (2021) Jerabkova, T., Boffin, H. M. J., Beccari, G., et al. 2021, A&A, 647, A137
  * Katz et al. (2022) Katz, D., Sartoretti, P., Guerrier, A., et al. 2022, arXiv e-prints, arXiv:2206.05902
  * Kervella et al. (2019) Kervella, P., Gallenne, A., Evans, N. R., et al. 2019, Astronomy & Astrophysics, 623, A117
  * Khan et al. (2019) Khan, S., Miglio, A., Mosser, B., et al. 2019, Astronomy & Astrophysics, 628, A35
  * Kharchenko (2001) Kharchenko, N. 2001, Kinematika i Fizika Nebesnykh Tel, 17, 409
  * Kharchenko et al. (2005) Kharchenko, N., Piskunov, A., Röser, S., Schilbach, E., & Scholz, R.-D. 2005, Astronomy & Astrophysics, 438, 1163
  * Kharchenko et al. (2013) Kharchenko, N., Piskunov, A., Schilbach, E., Röser, S., & Scholz, R.-D. 2013, Astronomy & Astrophysics, 558, A53
  * Kodric et al. (2018) Kodric, M., Riffeser, A., Hopp, U., et al. 2018, AJ, 156, 130
  * Kovtyukh et al. (2016) Kovtyukh, V., Lemasle, B., Chekhonadskikh, F., et al. 2016, Monthly Notices of the Royal Astronomical Society, 460, 2077
  * Laney & Stobie (1992) Laney, C. & Stobie, R. 1992, Astronomy and Astrophysics Supplement Series, 93, 93
  * Leavitt & Pickering (1912) Leavitt, H. S. & Pickering, E. C. 1912, Harvard College Observatory Circular, 173, 1
  * Lemasle et al. (2007) Lemasle, B., François, P., Bono, G., et al. 2007, Astronomy & Astrophysics, 467, 283
  * Lindegren et al. (2021) Lindegren, L., Bastian, U., Biermann, M., et al. 2021, Astronomy & Astrophysics, 649, A4
  * Luck et al. (2011) Luck, R., Andrievsky, S., Kovtyukh, V., Gieren, W., & Graczyk, D. 2011, The Astronomical Journal, 142, 51
  * Luck & Lambert (2011) Luck, R. E. & Lambert, D. L. 2011, The Astronomical Journal, 142, 136
  * Madore (1982) Madore, B. F. 1982, ApJ, 253, 575
  * Maíz Apellániz (2022) Maíz Apellániz, J. 2022, A&A, 657, A130
  * McInnes et al. (2017) McInnes, L., Healy, J., & Astels, S. 2017, The Journal of Open Source Software, 2
  * Medina et al. (2021) Medina, G. E., Lemasle, B., & Grebel, E. K. 2021, Monthly Notices of the Royal Astronomical Society, 505, 1342
  * Mel’nik et al. (2015) Mel’nik, A. M., Rautiainen, P., Berdnikov, L. N., Dambis, A. K., & Rastorguev, A. S. 2015, Astronomische Nachrichten, 336, 70
  * Metzger et al. (1991) Metzger, M. R., Caldwell, J. A. R., McCarthy, J. K., & Schechter, P. J. 1991, ApJS, 76, 803
  * Metzger et al. (1992) Metzger, M. R., Caldwell, J. A. R., & Schechter, P. L. 1992, AJ, 103, 529
  * Monson & Pierce (2011) Monson, A. & Pierce, M. 2011, VizieR Online Data Catalog, J
  * Negueruela et al. (2020) Negueruela, I., Dorda, R., & Marco, A. 2020, Monthly Notices of the Royal Astronomical Society, 494, 3028
  * Owens et al. (2022) Owens, K. A., Freedman, W. L., Madore, B. F., & Lee, A. J. 2022, The Astrophysical Journal, 927, 8
  * Pietrukowicz et al. (2021) Pietrukowicz, P., Soszyński, I., & Udalski, A. 2021, Acta Astron., 71, 205
  * Pietrzyński et al. (2019) Pietrzyński, G., Graczyk, D., Gallenne, A., et al. 2019, Nature, 567, 200
  * Planck Collaboration et al. (2020) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6
  * Pont et al. (1994) Pont, F., Burki, G., & Mayor, M. 1994, A&AS, 105, 165
  * Price-Whelan et al. (2018) Price-Whelan, A. M., Sipőcz, B., Günther, H., et al. 2018, The Astronomical Journal, 156, 123
  * Ren et al. (2021) Ren, F., Chen, X., Zhang, H., et al. 2021, ApJ, 911, L20
  * Riello et al. (2021) Riello, M., De Angeli, F., Evans, D. W., et al. 2021, A&A, 649, A3
  * Riess et al. (2022a) Riess, A. G., Breuval, L., Yuan, W., et al. 2022a, ApJ, 938, 36
  * Riess et al. (2021) Riess, A. G., Casertano, S., Yuan, W., et al. 2021, The Astrophysical Journal Letters, 908, L6
  * Riess et al. (2018) Riess, A. G., Casertano, S., Yuan, W., et al. 2018, The Astrophysical Journal, 855, 136
  * Riess et al. (2018) Riess, A. G., Casertano, S., Yuan, W., et al. 2018, ApJ, 861, 126
  * Riess et al. (2019) Riess, A. G., Casertano, S., Yuan, W., Macri, L. M., & Scolnic, D. 2019, The Astrophysical Journal, 876, 85
  * Riess et al. (2016) Riess, A. G., Macri, L. M., Hoffmann, S. L., et al. 2016, ApJ, 826, 56
  * Riess et al. (2019) Riess, A. G., Narayan, G., & Calamida, A. 2019, Calibration of the WFC3-IR Count-rate Nonlinearity, Sub-percent Accuracy for a Factor of a Million in Flux, Instrument Science Report WFC3 2019-1, 13 pages
  * Riess et al. (2022b) Riess, A. G., Yuan, W., Macri, L. M., et al. 2022b, ApJ, 934, L7
  * Ripepi et al. (2022a) Ripepi, V., Catanzaro, G., Clementini, G., et al. 2022a, A&A, 659, A167
  * Ripepi et al. (2022b) Ripepi, V., Clementini, G., Molinaro, R., et al. 2022b, arXiv e-prints, arXiv:2206.06212
  * Ripepi et al. (2022c) Ripepi, V., Clementini, G., Molinaro, R., et al. 2022c, arXiv e-prints, arXiv:2206.06212
  * Ripepi et al. (2019) Ripepi, V., Molinaro, R., Musella, I., et al. 2019, A&A, 625, A14
  * Robitaille et al. (2013) Robitaille, T. P., Tollerud, E. J., Greenfield, P., et al. 2013, Astronomy & Astrophysics, 558, A33
  * Romaniello et al. (2008) Romaniello, M., Primas, F., Mottini, M., et al. 2008, Astronomy & Astrophysics, 488, 731
  * Romaniello et al. (2022) Romaniello, M., Riess, A., Mancino, S., et al. 2022, Astronomy and Astrophysics, 658, A29
  * Schönrich et al. (2019) Schönrich, R., McMillan, P., & Eyer, L. 2019, MNRAS, 487, 3568
  * Skowron et al. (2021) Skowron, D. M., Skowron, J., Udalski, A., et al. 2021, ApJS, 252, 23
  * Soszyński et al. (2017) Soszyński, I., Udalski, A., Szymański, M. K., et al. 2017, Acta Astron., 67, 103
  * Stassun & Torres (2021) Stassun, K. G. & Torres, G. 2021, ApJ, 907, L33
  * Storm et al. (2004) Storm, J., Carney, B. W., Gieren, W. P., et al. 2004, A&A, 415, 531
  * Taylor (2005) Taylor, M. B. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed. P. Shopbell, M. Britton, & R. Ebert, 29
  * The Gaia collaboration et al. (2021) The Gaia collaboration, Brown, A., Vallenari, A., et al. 2021, Astronomy & Astrophysics, 650, C3
  * The Gaia collaboration et al. (2016) The Gaia collaboration, Prusti, T., De Bruijne, J., et al. 2016, Astronomy & astrophysics, 595, A1
  * Turner (1998) Turner, D. G. 1998, The Astronomical Journal, 116, 274
  * Usenko et al. (2019) Usenko, I., Kniazev, A. Y., Katkov, I. Y., et al. 2019, Odessa Astronomical Publications
  * Vasiliev & Baumgardt (2021) Vasiliev, E. & Baumgardt, H. 2021, Monthly Notices of the Royal Astronomical Society
  * Wang et al. (2022) Wang, C., Yuan, H., & Huang, Y. 2022, AJ, 163, 149
  * Zhou & Chen (2021) Zhou, X. & Chen, X. 2021, Monthly Notices of the Royal Astronomical Society, 504, 4768
  * Zinn (2021) Zinn, J. C. 2021, AJ, 161, 214
  * Zinn (2021) Zinn, J. C. 2021, The Astronomical Journal, 161, 214
  * Zinn et al. (2019) Zinn, J. C., Pinsonneault, M. H., Huber, D., & Stello, D. 2019, The Astrophysical Journal, 878, 136