# A 1.3% distance to M33 from HST Cepheid photometry

Louise Breuval Department of Physics and Astronomy, Johns Hopkins University,
Baltimore, MD 21218, USA<EMAIL_ADDRESS>Adam G. Riess Department of Physics
and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA  Space
Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
Lucas M. Macri Department of Physics & Astronomy, Texas A&M University,
College Station, TX 77843, USA  Siyang Li Department of Physics and Astronomy,
Johns Hopkins University, Baltimore, MD 21218, USA  Wenlong Yuan Department of
Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA
Stefano Casertano  Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218, USA  Tarini Konchady Department of Physics & Astronomy,
Texas A&M University, College Station, TX 77843, USA  Boris Trahin Institut
d’Astrophysique Spatiale, Université Paris-Saclay, CNRS, Batiment 121, F-91405
Orsay Cedex, France  Meredith J. Durbin Department of Astronomy, University of
Washington, Box 351580, U.W., Seattle, WA 98195-1580, USA  Benjamin F.
Williams Department of Astronomy, University of Washington, Box 351580, U.W.,
Seattle, WA 98195-1580, USA

###### Abstract

We present a low-dispersion period-luminosity relation (PL) based on 154
Cepheids in Messier 33 (M33) with Hubble Space Telescope (HST) photometry from
the PHATTER survey. Using high-quality ground-based light curves, we recover
Cepheid phases and amplitudes for multi-epoch HST data and we perform template
fitting to derive intensity-averaged mean magnitudes. HST observations in the
SH0ES near-infrared Wesenheit system significantly reduce the effect of
crowding relative to ground-based data, as seen in the final PL scatter of
$\sigma=0.11\,\rm mag$.

We adopt the absolute calibration of the PL based on HST observations in the
Large Magellanic Cloud (LMC) and a distance derived using late-type detached
eclipsing binaries to obtain a distance modulus for M33 of $\mu=24.622\pm
0.030\,\rm mag$ ($d=840\pm 11\,\rm kpc$), a best-to-date precision of 1.3%. We
find very good agreement with past Cepheid-based measurements. Several TRGB
estimates bracket our result while dissagreeing with each other.

Finally, we show that the flux contribution from star clusters hosting
Cepheids in M33 does not impact the distance measurement and we find only
$\sim$ 3.7% of the sample is located in (or nearby) young clusters. M33 offers
one of the best sites for the cross-calibration of many primary distance
indicators. Thus, a precise independent geometric determination of its
distance would provide a valuable new anchor to measure the Hubble constant.

longtable

##  1 Introduction

Cepheid variables are the best-calibrated primary distance indicators and are
commonly used to form the first rung of the empirical distance ladder (e.g.,
Riess et al., 2022). Their Period-Luminosity (PL) relation, also known as the
“Leavitt Law” (Leavitt & Pickering, 1912), is calibrated geometrically in the
Milky Way (MW) from Gaia DR3 parallaxes (Riess et al., 2021), in the Large
Magellanic Cloud (LMC) from detached eclipsing binaries (Riess et al., 2019a;
Pietrzyński et al., 2019), and in NGC 4258 with water masers (Reid et al.,
2019). Cepheid distances are used to calibrate the second rung of the distance
ladder, type Ia supernovae (SNe Ia), which allows us to measure the distance
to further galaxies in the Hubble flow and to derive the value of the Hubble
constant, $H_{0}$.

Figure 1: Map of M33: the template (i.e. ground-based) sample is shown in red
while the HST (i.e. PHATTER) sample is shown in blue. Dark blue and light blue
markers are Cepheids from the gold and silver sample respectively. Empty blue
markers are excluded Cepheids (see §3.2). Note the HST sample is a subset of
the template sample.

Messier 33 (hereafter M33) is a nearby type Sc II-III spiral galaxy and the
third largest member of the Local Group. As early as 1926, Edwin Hubble used
this galaxy as one of the spiral nebulae to learn about the structure of the
Universe and observed 35 Cepheid variables to measure its distance (Hubble,
1926). Since then, it has been extensively studied, and is still a crucial
object for the distance scale (Freedman et al., 1991; Lee et al., 2022): M33
has intermediate inclination ($i=57\pm 4^{\circ}$, Kourkchi et al.,
2020)111http://edd.ifa.hawaii.edu, Table “CF4 Initial Candidates”, which
limits the effects of reddening and of geometry that can produce additional
scatter in the PL relation. Additionally, M33 is known for its steep
metallicity gradient, which was measured using red giant branch (RGB) stars
(Tiede et al., 2004), planetary nebulae (Magrini et al., 2009), and H II
regions (Bresolin, 2011; Toribio San Cipriano et al., 2016; Rogers et al.,
2022).

Cepheids are numerous in M33, and large samples have been obtained by various
programs (Macri et al., 2001; Hartman et al., 2006; Pellerin & Macri, 2011).
Recently, the PHATTER collaboration (PI: J. Dalcanton) published a detailed
catalog222https://archive.stsci.edu/hlsp/phatter of UV to NIR photometry for
22 million stars in the central disk of M33 (Williams et al., 2021) using the
Hubble Space Telescope (HST). Although they are not time-series observations,
serendipitous overlaps between the PHATTER fields of view in a given filter
provide multiple data points randomly spread across the phase of M33 Cepheids.
Out of 250 Cepheids in the HST sample (defined in §2.2), 225 variables have
more than one epoch in $F475W$ and $F814W$, and 66 objects have more than one
epoch in $F160W$ (due to smaller overlaps of the WFC3/IR fields). Knowledge of
the date and time of observation for each HST exposure, combined with periods
previously measured from other surveys, enable the correction of these random-
phase observations to mean magnitude. Finally, past studies (e.g. Macri et
al., 2001; Riess et al., 2012; Wagner-Kaiser et al., 2015; Kodric et al.,
2018) have revealed the advantages of space-based observations such as HST in
limiting crowding effects and their impact for the PL dispersion, as well as
providing homogeneous photometry including in the near-infrared. In this paper
we aim to take advantage of the recently published high-quality PHATTER
catalog in order to provide a new PL calibration for M33 Cepheids in HST
filters and to improve the M33 distance measurement.

The outline of this paper is the following. In §2 we present the samples of
M33 Cepheids used in this study. In §3 we describe the construction of
template light curves from ground-based data and the procedure to recover mean
magnitudes from random-epoch photometry. In §4 we calibrate the Cepheid PL
relation and determine the M33 distance modulus. Lastly, in §5 we investigate
the effects of Cepheids located in star clusters, we estimate their occurrence
rate in M33 and compare it with that of other Local Group galaxies.

##  2 Photometric data

In order to recover intensity-averaged mean magnitudes from random-phase HST
data (hereafter the HST sample), we use templates obtained by compiling a
large number of well-sampled ground-based light curves of M33 Cepheids
(hereafter the template sample). Both samples are described below.

###  2.1 The template sample

Figure 2: Distribution of phase uncertainties for Cepheids of the template
sample at the epoch of HST observations, estimated from the period uncertainty
and the interval between the midpoint of the HST and of the ground
observations. The dashed horizontal line represents our threshold for the gold
sample (see below): we only keep Cepheids with $\sigma(\phi)<0.05$.

We used a sample of 609 previously-known Cepheids (Macri et al., 2001;
Pellerin & Macri, 2011) with homogeneous gri light curves obtained by Konchady
et al. (in prep.) using archival CFHT/MegaCam observations (proposal ID
04BF26, PI Beaulieu; proposal ID 04BH98, PI Hodapp). They are represented in
red in Fig. 1 and constitute the template sample.

The majority of the original CFHT observations (associated with proposal ID
04BF26) are extensively described in Hartman et al. (2006); they span roughly
one-and-a-half years (2003 August to 2005 January) and were obtained on 27
separate nights. We supplemented these with an additional four nights of $i$
observations obtained in 2004 August and September (associated with proposal
ID 04HB98). Konchady et al. (in prep.) performed an independent analysis of
these images, carrying out time-series PSF photometry that was calibrated
against Pan-STARRS DR1 (Chambers et al., 2016). The periods and phases of the
Cepheids were redetermined by simultaneously fitting the CFHT gri photometry
and the WIYN BVI photometry of Pellerin & Macri (2011) using the Yoachim et
al. (2009) templates. We solved for a common period and phase across the six
bands, and independent mean magnitudes and light curve amplitudes in each
band.

Cepheid light curves of the template sample have two purposes: they are used
to build templates thanks to their complete phase coverage (see §3.1) and to
recover the amplitudes and phases of the HST light curves (see §3.2). For this
reason their periods must be known precisely. From the period uncertainty, we
estimate the uncertainty in the phase-shift between the mid-date of the ground
observations (MJD = 52170) and of PHATTER observations (MJD = 57989), and we
flag Cepheids for which this uncertainty $\sigma(\phi)$ is larger than 0.05
(or $\log\sigma(\phi)>-1.3$, dashed horizontal line in Fig. 2). They
constitute the ”silver” sample (see Sect. 3.2). Additionally we only keep
Cepheids which have optimal ground-based light curves (Table 3 of Pellerin &
Macri, 2011). This leaves a total of 420 Cepheids, for which we perform a
visual inspection of each light curve’s quality. Cepheids have an average of
45, 31 and 44 data points per light curve in $g$, $r$, and $i$, respectively.

We note that our ground-based sample is minimally affected by blending given
the relatively high image quality of the CFHT and WIYN observations and the
rejection of outliers by Pellerin & Macri (2011).

###  2.2 The HST sample

The PHATTER survey (Williams et al., 2021) contains photometric measurements
for 22 million stars in M33 with 6 UV to NIR filters (Advanced Camera for
Surveys and Wide Field Camera 3) on the Hubble Space Telescope (HST). The
survey focuses on the inner disk of the galaxy and covers $\sim 300\sq\arcmin$
(equivalent to a de-projected area of $\sim 38\,\rm kpc^{2}$), extending up to
$\sim 14\arcmin$ from the center (equivalent to a distance of $\sim 3.5\,\rm
kpc$). The observations were taken between 2017 February 21 and 2018 February
25. The catalog reaches 26 to 28 mag in $V$ depending on crowding. It is the
largest and most complete catalog to date for stellar populations in M33. We
identified 250 Cepheids from the template sample in the PHATTER catalog: they
are represented in blue in Fig. 1 and are hereafter referred to as the HST
sample.

Figure 3: Wavelength coverage of HST filters used in this analysis (top
panel), CFHT filters adopted to build optical template light curves (middle
panel) and 2MASS filters used for NIR templates (Inno et al., 2015, bottom
panel). We adopt $g$, $i$ and $H$ templates to fit light curves in $F475W$,
$F814W$ and $F160W$, respectively.

We matched the Cepheid coordinates to the full-frame PHATTER catalogs using an
initial search radius of 0.1 arcsec, and found that all had matches within
$<0.5$ mas with expected magnitudes ($19<F475W<23$). We then used the pixel
coordinates from these catalogs to retrieve the exposure-level photometry from
the original DOLPHOT .phot outputs. These files contain columns with
photometry from each individual input frame in addition to the combined
measurements.

The PHATTER survey does not provide time-series observations, which might make
it in principle poorly suited to study variable stars such as Cepheids.
However, in the optical $F475W$ and $F814W$ filters, successive PHATTER
pointings show a significant overlap, and therefore up to 4 epochs can be
available for a given Cepheid. In the NIR, the pointings have smaller
overlaps, which gives one to two epochs per Cepheid. Each epoch can be
decomposed into 4 or 5 separate dithers/exposures and the phase-coverage of
each epoch is random. We note that the first exposure of each $F475W$ and
$F814W$ visit sequence is significantly shallower than the rest, as these are
short exposures targeting the brightest stars (M. Durbin, 2023, private
communication). They were therefore excluded as they are not useful for
Cepheids. The date and time of a given HST observation provide the relative
phase of the corresponding measurement. Then, mean magnitudes can be recovered
from sparse data by applying a template-fitting procedure (§3).

##  3 Template fitting

In this section we describe the construction of template light curves from
ground-based data (§3.1) and the procedure to recover mean magnitudes from
PHATTER photometry in HST filters (§3.2). The HST and template samples were
observed in different filters. The HST $F475W$ filter is very similar to the
$g$ one from CFHT/MegaCam, and $F814W$ corresponds to the $i$ filter (see Fig.
3). Finally, the template sample does not cover the NIR up to the $F160W$
filter, therefore we use the 2MASS $H$-band templates by Inno et al. (2015),
based on a large sample of LMC Cepheid light curves. We adopt $g$-band and
$i$-band templates to derive HST mean magnitudes in $F475W$ and $F814W$
respectively, and $H$-band templates to derive $F160W$ mean magnitudes.

###  3.1 Building template light curves

We use the well-sampled light curves from the template sample to build
template light curves in the $g$, $r$ and $i$ filters of CFHT/MegaCam. These
ground-based light curves are ideal to build templates and to recover the mean
magnitudes from HST random phase observations: they are representative of
Cepheids from the HST sample as they belong to the same host galaxy and have a
very similar period distribution (Fig. 4). Other templates from the literature
(e.g. Yoachim et al., 2009, from LMC Cepheids) could have been used instead of
creating new ones. However, adopting templates built from a population similar
to the HST sample avoids possible differences in light curve shapes for
Cepheids from different galaxies (possibly due to metallicity effects,
Antonello et al., 2000).

In order to account for changes in light curve shape as a function of period
(Hertzsprung, 1926), we split the sample into four different period bins. They
are described in Table 1. The number of bins was determined by the size and by
the distribution of our calibrating sample: having a larger sample would have
allowed us to use more bins.

Figure 4: Period distribution of Cepheids from the template sample and from the HST sample. Table 1: Number of Cepheids in each period bin for the template sample. Bin | $\log P$ | $N_{g}$ | $N_{r}$ | $N_{i}$
---|---|---|---|---
1 | $0.3-0.9$ | 148 | 136 | 143
2 | $0.9-1.2$ | 91 | 84 | 81
3 | $1.2-1.5$ | 46 | 47 | 44
4 | $1.5-2.0$ | 20 | 19 | 16

While the reference phase of a Cepheid is often defined by the epoch of
maximum brightness, this quantity can be biased by the presence of a bump in
the light curve that varies in shape and phase as a function of period along
the Hertzsprung (1926) progression. This bump coincides with maximum light for
Cepheids with periods around 10 days. To overcome this issue, Inno et al.
(2015) adopted another feature to determine the phase of a Cepheid light
curve: the mean magnitude along the rising branch (MRB). As mean magnitudes
are known with great precision for our template sample, this approach is more
reliable than using the maximum to set the phase (see more details in Inno et
al., 2015) and we adopt it in our analysis:

$\phi_{obs}=\rm mod\left(\frac{JD_{obs}-JD_{MRB}}{P}\right)$ (1)

For a filter $\lambda$, we normalize the magnitude values $m_{i}$ by deriving
the quantity:

$T_{\lambda}=\frac{m_{i}-\langle m_{i}\rangle}{A_{\lambda}}$ (2)

where $\langle m_{i}\rangle$ is the mean magnitude and $A_{\lambda}$ is the
amplitude. Finally, we merge all phased and normalized light curves into a
single template for each period bin. The final templates and compiled light
curves are shown in Fig. 5 in the $g$, $r$ and $i$ filters and for each of the
four period bins. We follow Inno et al. (2015) and fit the merged light curves
with a seventh order Fourier series of the form:

$F_{7}(\phi)=A_{0}+\sum_{i=1}^{7}A_{i}\,cos(2\pi i\phi+\Phi_{i})$ (3)

The resulting coefficients are listed in Table 2.

Figure 5: Merged light curves of M33 Cepheids (amplitude-scaled) used for the templates in the $g$, $r$ and $i$ band (1st, 2nd and 3rd lines respectively). The four columns correspond to the four period bins listed in Table 1. The point in the bottom left corner shows the typical error bar, multiplied by a factor of 3 for better visibility. Table 2: Fourier parameters for template light curves obtained with the calibrating sample of Cepheids. Bin | $A_{0}$ | $A_{1}$ | $A_{2}$ | $A_{3}$ | $A_{4}$ | $A_{5}$ | $A_{6}$ | $A_{7}$ | $\Phi_{1}$ | $\Phi_{2}$ | $\Phi_{3}$ | $\Phi_{4}$ | $\Phi_{5}$ | $\Phi_{6}$ | $\Phi_{7}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$g$
1 | 0.004 | 0.431 | 0.161 | 0.062 | 0.022 | 0.007 | 0.004 | -0.005 | 1.665 | 1.502 | 1.132 | 0.918 | 0.043 | 0.036 | 1.649
2 | 0.007 | 0.450 | -0.073 | 0.019 | 0.020 | 0.008 | 0.005 | -0.004 | 1.615 | -1.554 | 0.747 | -0.567 | -1.423 | -13.477 | 1.027
3 | 0.004 | 0.424 | 0.134 | 0.074 | 0.052 | 0.032 | 0.025 | 0.014 | 1.802 | 1.615 | 1.113 | 0.924 | 0.714 | 0.511 | 0.289
4 | -0.001 | 0.415 | 0.169 | 0.082 | -0.040 | -0.023 | 0.014 | 0.008 | 1.656 | 1.587 | 1.555 | -1.682 | -1.854 | 0.464 | 0.749
$r$
1 | 0.003 | 0.429 | -0.162 | 0.064 | 0.023 | -0.007 | 0.005 | -0.005 | 1.642 | -7.820 | 1.409 | 1.078 | 3.979 | -0.036 | -4.029
2 | 0.005 | 0.451 | -0.076 | 0.019 | 0.022 | 0.009 | 0.007 | 0.008 | 1.616 | -1.273 | 1.251 | 0.046 | -0.583 | -0.073 | -0.397
3 | 0.002 | 0.431 | -0.135 | 0.081 | 0.057 | 0.039 | 0.024 | 0.013 | 1.694 | -1.447 | 1.322 | 1.181 | 1.090 | 1.003 | 0.952
4 | -0.001 | 0.422 | -0.173 | -0.082 | -0.037 | 0.022 | -0.017 | -0.009 | 1.477 | 4.767 | -7.740 | -7.727 | 1.217 | -1.784 | -1.636
$i$
1 | 0.005 | 0.429 | -0.156 | -0.065 | 0.019 | 0.008 | 0.008 | -0.003 | 1.595 | -1.378 | -1.505 | 1.399 | 1.025 | 0.346 | -2.680
2 | 0.007 | 0.462 | -0.080 | 0.024 | 0.022 | 0.005 | -0.005 | 0.002 | 1.599 | -1.124 | 1.579 | 0.527 | -4.821 | 16.604 | 14.585
3 | 0.004 | 0.448 | -0.131 | 0.082 | 0.058 | 0.039 | 0.029 | 0.012 | 1.601 | -1.400 | 1.422 | 1.424 | 1.365 | 1.382 | -11.207
4 | -0.002 | 0.440 | -0.171 | -0.082 | -0.037 | 0.023 | -0.015 | 0.010 | 1.382 | -1.433 | -1.285 | -1.166 | 1.450 | -1.479 | 1.365

###  3.2 Template fitting procedure

Before performing the fit, we set the first-guess $V$-band ($F475W$) amplitude
$A_{V}$ to that of the ground-based $g$-band light curve. Then, we fix the
amplitude ratios to $A_{I}=0.58\,A_{V}$ from Yoachim et al. (2009) and to:

$A_{H}=\left\\{\begin{array}[]{ll}0.34\,A_{V}&\mbox{if $P\leq 20$ d}\\\
0.40\,A_{V}&\mbox{if $P>20$ d}\end{array}\right.$ (4)

from Inno et al. (2015). The first-guess phase in $F475W$ is set to the phase
in $g$. By comparing CFHT $g$ and $i$ light curves, we derive a small phase
lag between $V$ and $I$ of: $\phi_{I}=\phi_{V}+0.027$, and we adopt the
$H$-band phase lag from Inno et al. (2015) for $F160W$:
$\phi_{H}=\phi_{V}+0.080-0.002\,\log P.$ We note that Soszyński et al. (2005)
derived a different phase lag of about 0.3 between $H$ and $V$, but that is
largely due to the choice of a different reference phase (maximum brightness).
In §4.5 we discuss the sensitivity of our results to the phase lag.

Figure 6: Example of light curves obtained from the template fitting
procedure. Their quality is representative of that of the entire sample. The
shaded area represents the mean magnitude error in each filter. For a given
star, the three filters are shown with the same scale in magnitude.

We fit the templates to the HST measurements in the three filters
simultaneously by performing a grid-search on $A_{V}$ and $\phi_{V}$, where
$\phi_{V}$ has a narrow, informative prior from the template sample. The
amplitude ratios are fixed throughout the procedure and we retain as final
parameters the solution that minimizes the quantity:

$Z=\chi^{2}_{\rm tot}+Q(A_{V})$ (5)

where $\,\chi^{2}_{\rm tot}=\chi^{2}_{H}+\chi^{2}_{V}+\chi^{2}_{I}\,$ and each
$\chi^{2}$ is defined as:

$\chi^{2}=\sum_{i}\frac{(O_{i}-C_{i})^{2}}{\sigma_{i}^{2}}$ (6)

with $O_{i}$ the data points, $C_{i}$ the fit values and $\sigma_{i}$ the
error of each data point. The quantity $Q(A_{V})$ is a penalty function that
prevents the fitted HST amplitudes to diverge too far from the expected values
(i.e. the $g$-band amplitude of each Cepheid). It is defined as:

$Q(A_{V})=\frac{(A_{V,\,\rm fitted}-A_{V,\,\rm ground})^{2}}{\sigma_{A}^{2}}$
(7)

The dispersion in the difference in amplitudes is set to
$\sigma_{A}=0.030\,\rm mag$ from the ground based sample. Finally, the errors
on each apparent magnitude are estimated from a $\chi^{2}$ distribution
assuming $\chi^{2}<\chi^{2}_{\rm min}+1$ (Press et al., 1992). Figure 6 shows
a few examples of light curves obtained from the template fitting procedure.

In the following we consider two different subsamples. The gold sample
includes Cepheids for which the phase is known with good confidence from
ground-based light curves: these Cepheids have at least a valid $g$-band light
curve or a valid $i$-band light curve (or both ideally). In the case where
only one light curve is available among $g$ and $i$, amplitudes and phases in
the missing band can be easily recovered from the relations adopted above.
Cepheids of the gold sample must also have a phase uncertainty of
$\sigma_{\phi}<0.05$ to allow for a precise rephasing of HST observations. As
their phase and amplitude are assumed to be known from the ground, the grid-
search is performed on a limited range of parameters: across
[$A_{V}-0.4;A_{V}+0.4$] in amplitude and [$\phi_{V}-0.05;\,\phi_{V}+0.05$] in
phase. The silver sample includes Cepheids with no $g$ and no $i$ light curves
or with a larger phase uncertainty $\sigma_{\phi}>0.05$. For these stars, the
phasing is considered unknown and we perform the grid-search over [0; 1] in
phase. The expected amplitude of these silver sample Cepheids is also unknown:
as the mean V-band peak-to-peak amplitude of our sample is 0.8 mag, we perform
the grid search within [0.3, 1.3] in amplitude, which corresponds to $0.8\pm
0.5$ mag. After a visual inspection of each light curve, we find 9 Cepheids
from the gold sample which appear to have an incorrect phasing (i.e. reaching
the boundaries of the grid-search in $\phi_{V}$): for these stars we allow the
search to cover [0; 1] in phase (but generally the final phase stays within
$\pm 0.1$ from the first guess) and we find much lower $\chi^{2}$ and better
fit quality. These nine stars are moved to the silver sample.

Out of the 250 initial PHATTER Cepheids, we only keep the 220 which have
optimal ground-based light curves (Table 3 from Pellerin & Macri, 2011). We
rejected 29 stars with only one epoch per filter or with multiple but very
close epochs ($\Delta\phi<0.01$), which did not allow the fit to converge
successfully. We also excluded 26 stars for which the fit was not satisfactory
and 11 stars which yielded a fitted $V$-band amplitude different by more than
0.5 mag from the expected amplitude from ground-based light curves. This
leaves a total of 154 Cepheids which constitute the ”gold+silver” sample. The
final intensity-averaged mean magnitudes obtained for our sample of Cepheids
in $F160W$, $F475W$ and $F814W$ are listed in Table LABEL:table:all_data
(Appendix A).

Figure 7: Period-luminosity relation in $m_{H}^{W}$ for M33 Cepheids. The dark
solid line is the PL fit of the gold + silver sample assuming a free slope,
and the red dashed line shows the same fit when the slope is fixed to
$-3.26\,\rm mag/dex$ (Riess et al., 2019a).

##  4 Period-Luminosity relation and distance to M33

###  4.1 Photometric transformations to WFC3

In order to derive the distance to M33, we will compare its PL relation with
that established in the LMC, which has the most precise and Cepheid-
independent distance measurement (Pietrzyński et al., 2019). The LMC PL
relation (Riess et al., 2019a) is calibrated in the SH0ES photometric system
based on HST/WFC3 filters ($F160W$, $F555W$ and $F814W$). On the other hand,
the mean magnitudes obtained in the present work from PHATTER photometry were
measured with the WFC3/IR camera for the $F160W$ filter and with ACS/WFC for
the optical $F475W$ and $F814W$ filters. We transform the PHATTER color
$(F475W-F814W)_{\rm ACS}$ into the SH0ES color $(F555W-F814W)_{\rm WFC3}$
using synthetic populations based on PARSEC isochrones generated by the CMD
tool333http://stev.oapd.inaf.it/cgi-bin/cmd developed by Bressan et al. (2012)
(version v3.7) for HST/WFC3 and HST/ACS bandpasses. We consider a population
of Cepheid-like stars with ages of $1-500\,\rm Myr$, $\log g<2$, masses of
$3-7M_{\odot}$ and temperatures of $4800-6500\,\rm K$. We derive the following
transformation with a scatter of 0.003 mag:

$(F555W-F814W)_{\rm WFC3}=0.065\\\ +0.658\,(F475W-F814W)_{\rm ACS}$ (8)

The mean Cepheid color of the sample is $(F475W-F814W)_{\rm ACS}=1.41\,\rm
mag$ (sample standard deviation = 0.29 mag), for which Eq. 8 gives
$(F555W-F814W)_{\rm WFC3}=0.99\,\rm mag$ (sample standard deviation = 0.19
mag). In the following we adopt the near-infrared HST/WFC3 Wesenheit index
defined in Riess et al. (2022) assuming the reddening law from Fitzpatrick
(1999) with reddening paramneter $R_{V}=3.3$:

$m_{H}^{W}=F160W-0.386\,(F555W-F814W)_{\rm WFC3}$ (9)

We also derive optical Wesenheit magnitudes defined in Riess et al. (2019a)
as:

$m_{I}^{W}=F814W-1.3\,(F555W-F814W)_{\rm WFC3}$ (10)

Table 3: Calibration of the PL relation in M33 ($m=\alpha\log P+\beta$) and
resulting distance modulus. The second column gives the fitted PL slope
$\alpha$. The third and fourth columns give the PL intercept $\beta$ when the
slope is a free parameter and when the slope is fixed to the LMC value (Riess
et al., 2019a) respectively.
Band | $\alpha$ | $\beta_{\,\rm free}$ | $\beta_{\,\rm fixed}$ | $\sigma$ | $\chi^{2}_{\rm dof}$ | Nstars | Sample |  $\mu_{\rm M33}$ (mag)
---|---|---|---|---|---|---|---|---
$m_{H}^{W}$ | $-3.207\pm 0.039$ | $21.993\pm 0.041$ | $22.046\pm 0.010$ | 0.110 | 0.96 | 99 | Gold | 24.619 $\pm$ 0.030
$m_{H}^{W}$ | $-3.193\pm 0.032$ | $21.980\pm 0.034$ | $22.048\pm 0.008$ | 0.113 | 0.72 | 154 | Gold + Silver | 24.622 $\pm$ 0.030
$m_{I}^{W}$ | $-3.167\pm 0.046$ | $21.909\pm 0.051$ | $22.065\pm 0.014$ | 0.141 | 1.32 | 99 | Gold | 24.617 $\pm$ 0.032
$m_{I}^{W}$ | $-3.179\pm 0.037$ | $21.933\pm 0.041$ | $22.072\pm 0.010$ | 0.130 | 1.01 | 154 | Gold + Silver | 24.624 $\pm$ 0.030

###  4.2 Count-rate Nonlinearity Correction

The WFC3-IR instrument, which is used in the SH0ES distance ladder to measure
nearby bright Cepheids as well as distant stars in supernovae host galaxies,
is affected by count-rate nonlinearity (CRNL, or reciprocity failure). This
effect dims faint sources relative to bright ones due to a decreased photon
collection efficiency. Its most recent calibration gives a correction of
$0.0077\,\rm mag/dex$ (Riess et al., 2019b). In order to derive the distance
to M33, $m_{H}^{W}$ magnitudes in the LMC and in M33 must be corrected for the
CRNL consistently. The PL intercept of $15.898\,\rm mag$ calibrated in the LMC
by Riess et al. (2019a) does not include the CRNL term (despite its note to
the contrary, see footnote to table 5 of Yuan et al., 2020). Finally, M33
Cepheids are fainter than LMC Cepheids by about 2 dex (Li et al., 2021), so we
add $0.015\pm 0.005$ mag to the LMC intercept from Riess et al. (2019a) to
account for this difference (which is equivalent to subtracting 0.015 mag to
our $m_{H}^{W}$ apparent magnitudes in M33).

###  4.3 Geometric correction

We take into account the position of each Cepheid relative to the center of
M33 ($\alpha=23.4625^{\circ}$, $\delta=30.6602^{\circ}$ from van der Marel et
al., 2019) by applying a geometric correction. Our HST sample is located very
near the center of M33 and the galaxy has a moderate inclination, which limits
the effects of projection and of reddening. We adopt an inclination angle of
$i=57\pm 4^{\circ}$ and a position angle of $\rm PA=22.5^{\circ}$ (both from
Kourkchi et al., 2020), obtaining a mean correction of 0.0007 mag with a
dispersion of 0.003 mag, with values ranging between -0.005 mag and +0.008
mag. A positive geometric correction corresponds to a Cepheid farther than the
center of M33.

###  4.4 Period-luminosity relation in M33

In this section, we adopt the apparent Wesenheit $m_{H}^{W}$ mean magnitudes
obtained from template fitting for our sample of M33 Cepheids. We include an
additional $0.07\,\rm mag$ in quadrature to all magnitude errors to account
for the finite width of the instability strip (Riess et al., 2019a). The PL
relation is then calibrated for the two subsamples defined in §3 (gold and
silver samples).

For the gold sample we obtain a slope of $-3.207\pm 0.039\,\rm mag/dex$ in
$m_{H}^{W}$, which agrees well with that derived by Riess et al. (2019a) in
the LMC. The PL scatter is $0.110\,\rm mag$ for a total of 99 stars. Including
the silver sample yields a slightly shallower slope of $-3.193\pm 0.032\,\rm
mag/dex$, still in good agreement with Riess et al. (2019a), and slightly
raises the scatter to $0.113\,\rm mag$. In the optical Wesenheit index
$m_{I}^{W}$, we adopt the slope of $-3.31\,\rm mag/dex$ as well as the LMC PL
intercept of 15.935 from Riess et al. (2019a) and we obtain a PL dispersion of
0.13 mag for the gold + silver sample, and 0.14 for the gold sample. We note
that Li et al. (2021) obtained a PL scatter of $0.13\,\rm mag$ in M31 for
their gold sample with 42 Cepheids, which shows the great precision of our PL
calibration. Our PL coefficients are listed in Table 3 and the PL relation is
shown in Fig. 7.

###  4.5 Distance to M33

To obtain the distance modulus for M33 ($\mu_{\rm M33}$), we compare the
intercept of our $m_{H}^{W}$ PL relation in M33 with that of the LMC obtained
by Riess et al. (2019a), $m_{H}^{W}=15.898-3.26\log P$. We add the CRNL term
of 0.015 mag to the LMC intercept to account for the difference in brightness
between LMC and M33 Cepheids (see §4.2). We fix our PL slope to $-3.26$ for
consistency with the LMC and we derive:

$(\mu_{\rm M33}-\mu_{\rm LMC})=(\beta_{\rm M33}-\beta_{\rm LMC})+\Delta m$
(11)

where $\mu_{\rm LMC}=18.477\pm 0.026\,\rm mag$ is the most direct and precise
geometric distance to the LMC available (Pietrzyński et al., 2019). The term
$\Delta m$ is the correction for the difference in metallicity between M33 and
LMC Cepheids:

$\Delta m=-\gamma\,(\rm[O/H]_{M33}-[O/H]_{LMC})$ (12)

Romaniello et al. (2022) gives $\rm[O/H]_{LMC}=-0.32\pm 0.01\,dex$ from a
sample of 89 Cepheids. In M33 we use the metallicity gradient by Bresolin
(2011) which gives:

$12+\rm\log(O/H)=8.50_{\pm 0.02}-0.045_{\pm 0.006}\,R_{kpc}$ (13)

relative to 8.69 for solar (Asplund et al., 2009). For our HST sample, [O/H]
metallicities range from -0.20 dex to -0.36 dex. We adopt the mean metallicity
of $\rm[O/H]_{M33}=-0.27\pm 0.03\,dex$. Using the metallicity correction of
$\gamma=-0.217\pm 0.046\,\rm mag/dex$ from Riess et al. (2022), we obtain a
correction of $\Delta m=0.011\pm 0.007\,\rm mag$. Using the metallicity
correction from Breuval et al. (2022) returns $\Delta m=0.014\,\rm mag$ which
is very similar, but we adopt the former as it is more suited for measurements
in the Wesenheit $m_{H}^{W}$ index. From Eq. 11 we obtain a final distance
modulus of $24.622\pm 0.030$ mag based on the gold and silver samples combined
in the NIR Wesenheit index $m_{H}^{W}$. Using only the pure gold sample
results in a very similar value (see Table 3). Finally, the optical Wesenheit
index yields a very consistent distance of $24.617\pm 0.032$ and $24.624\pm
0.030$ mag from the gold sample only and gold + silver samples combined,
respectively. We retain as final distance modulus the one from the gold +
silver sample in the $m_{H}^{W}$ filter, $24.622\pm 0.030\,\rm mag$, as it is
based on the most precise PL intercept. The full error budget is detailed in
Table 4.

In §3.2, we adopted the phase lag from Inno et al. (2015) between $H$ and $V$
light curves ($\sim 0.08$, with a scatter of 0.03 in the relation), derived by
using the mean magnitude along the rising branch as a reference for the phase.
We made this choice for consistency, as we also use the NIR templates by Inno
et al. (2015). On the other hand, Soszyński et al. (2005) found a larger phase
lag of $\sim 0.3$ (with a larger scatter of about $0.1$ mag) by using the
phase at maximum brightness as a reference. Inno et al. (2015) show that these
two methods to phase the data are very different; using the Soszyński et al.
(2005) phase lag with the Inno et al. (2015) template is formally
inconsistent, but it leads to a difference of only 0.006 mag in the distance
modulus.

We attempted to independently constrain the value of the metallicity effect of
the PL relation, $\gamma$. However, the very narrow range of abundances
spanned by these Cepheids yields uncertainties $\sigma(\gamma)$ of $0.24-0.42$
mag/dex which do not improve upon previous measurements (Breuval et al., 2021,
2022; Riess et al., 2022).

Table 4: Error budget for the distance to M33. Error | Value | Source
---|---|---
LMC DEBs | 1.20 % | Pietrzyński et al. (2019)
LMC PLR mean | 0.41 % | Riess et al. (2019a)
M33 PLR mean | 0.38 % | Measured here
Metallicity correction | 0.33 % | Riess et al. (2022)
CRNL across 2 dex | 0.23 % | Riess et al. (2019b)
Total |  1.38 % |
Table 5: M33 distance modulus from the literature. The last column gives the
LMC distance modulus adopted to obtain the distances given in the first
column.
$\mu_{\rm M33}$ | Reference | Method | $\mu_{\rm LMC}$
---|---|---|---
$24.64\pm 0.09$ | Freedman et al. (1991) | Cepheids | 18.50
$24.56\pm 0.10$ | Freedman et al. (2001) | Cepheids | 18.50
$24.65\pm 0.12$ | Macri (2001) | Cepheids | 18.50
$24.50\pm 0.06$ | McConnachie et al. (2004) | TRGB | -
$24.67\pm 0.08$ | Sarajedini et al. (2006) | RR Lyrae | 18.50
$24.71\pm 0.04$ | Rizzi et al. (2007) | TRGB | -
$24.53\pm 0.11$ | Scowcroft et al. (2009) | Cepheids | 18.40
$24.84\pm 0.10$ | U et al. (2009) | TRGB | 18.50
$24.76\pm 0.05$ | Pellerin & Macri (2011) | Cepheids | 18.50
$24.57\pm 0.05$ | Conn et al. (2012) | TRGB | -
$24.62\pm 0.07$ | Gieren et al. (2013) | Cepheids | 18.50
$24.62\pm 0.06$ | Bhardwaj et al. (2016) | Cepheids | 18.47
$24.80\pm 0.06$ | Yuan et al. (2018) | Miras | 18.493
$24.57\pm 0.06$ | Zgirski et al. (2021) | JAGB | 18.477
$24.67\pm 0.05$ | Lee et al. (2022) | JAGB | -
$24.72\pm 0.07$ | Lee et al. (2022) | TRGB | -
$24.71\pm 0.04$ | Lee et al. (2022) | Cepheids | -
$24.67\pm 0.06$ | Ou et al. (2023) | Miras | 18.49
24.622 $\pm$ 0.030 | Present work | Cepheids | 18.477

###  4.6 Comparison with the literature

####  4.6.1 Cepheids, RR Lyrae, Miras

Figure 8 shows our final distance modulus for M33 and compares it with other
values from the literature based on various indicators (listed in Table 5, all
corrected to a common LMC distance modulus of 18.477 mag (Pietrzyński et al.,
2019). Our distance agrees very well with other estimates based on Cepheids,
especially with Freedman et al. (1991), Macri (2001), Scowcroft et al. (2009)
and Bhardwaj et al. (2016). In particular, Cepheids appear to provide the most
consistent distance measurements among all other distance indicators. The
error of the Pellerin & Macri (2011) distance was revised to 0.05 mag to
include the systematic uncertainties from the photometric comparison with
Massey et al. (2006) in their section 3. The Cepheid distance to M33 by Lee et
al. (2022) is larger than our value by 0.10 mag ($1.7\sigma$). This difference
matches the size and direction of the metallicity dependence of Cepheids,
$\sim$ -0.2 mag/dex (Breuval et al., 2022), where metal poor Cepheids are
fainter. (It is the wrong direction for crowding which, if uncorrected, would
make ground-based observations of Cepheids appear too bright). The Lee et al.
(2022) sample is in the outer regions of M33, around 5 kpc from the center,
which corresponds to a metallicity of about [O/H] $\sim-0.4\,\rm dex$,
somewhat metal poor. They derive their distance to M33 relatively to the
absolute PL calibration by Monson et al. (2012), based on Milky Way Cepheids
which are metal rich with [O/H] $\sim 0.1\,\rm dex$. This 0.5 dex difference
in metallicity produces an expected difference of $\sim$ 0.10 mag, bringing it
into agreement with the study here. An alternative to correcting for
metallicity is to use a reference with a similar metallicity as M33, i.e., the
metal poor Cepheids in the LMC, $\sim$ -0.3 dex, and the geometric DEB
distance as a reference for the Lee et al. (2022) sample which we find yields
24.65 mag in good agreement with our result. We also find good agreement with
the RR Lyrae distance by Sarajedini et al. (2006). Finally, the two Mira-based
distances (Yuan et al., 2018; Ou et al., 2023) differ by 0.13 mag, which can
be attributed to the use of different data sets and methodologies, differences
in periods and possible calibration systematics between ground and space-based
data.

Figure 8: M33 distance modulus from the present work compared with values from
the literature. All values shown here are rescaled to the recent LMC distance
modulus of 18.477 from Pietrzyński et al. (2019) (see their original adopted
LMC distance in Table 5).

####  4.6.2 Tip of the Red Giant Branch

Distance moduli based on TRGB show a larger dispersion than Cepheid-based
measurements with differences as large as 0.34 mag between different studies.
For example, McConnachie et al. (2004) derive a distance modulus of $24.50\pm
0.06$ mag based on an annulus region between $0.5-0.8^{\circ}$ in the outer
disk of M33. They investigate the impact of crowding and conclude that this
effect is negligible in regions farther than 0.5∘ from the center of M33. They
also rule out the possibility of contamination from AGB stars. More recently,
Lee et al. (2022) selected a TRGB field in the southern part of M33 at a
distance of about $0.25-0.45^{\circ}$ from the galactic center, and therefore
no less likely to be affected by blending than the McConnachie et al. (2004)
sample, yet obtained a much higher distance modulus of $24.72\pm 0.06$ mag. We
speculate some of this difference could be attributed in part to a difference
in sample color, as the redder color cut from Lee et al. (2022) extends
further into the region where the TRGB is fainter and may benefit from a color
correction (see Jang & Lee, 2017). We note that although the mean color of the
sample from Lee et al. (2022) is located at the boundary where Jang & Lee
(2017) states a color correction is not necessary, Jang & Lee (2017) do not
calibrate their color relation based on the mean color of their entire sample.
It is also possible the different TRGB-based measurements are due to
population differences as recently seen in Hoyt (2023), Anderson et al. (2023)
and Wu et al. (2022) which all identify significant, intrinsic variations in
the TRGB brightness with location, sub-RGB-population or the apparent ratio of
RGB to AGB stars (that may relate to age or metallicity). Durbin et al. (2020)
also identify additional systematics associated with the TRGB by comparing
different calibration approaches. The $\chi^{2}$ of the TRGB measurements
around their weighted mean value is 3.67, compared to 1.42 for Cepheid-based
distances (see Table 6). Finally, both measurements based on the JAGB method
(Zgirski et al., 2021; Lee et al., 2022) agree to $1\sigma$ with our value.

Table 6: Statistics of the M33 distances from the literature based on various
distance indicators. Column (2) is the weighted mean distance modulus for each
indicator. Columns (3) and (4) respectively give the $\chi^{2}$ of the
measurements from the literature around the weighted mean distance modulus and
around the distance modulus of the present paper. Column (5) gives the number
of estimates from the literature considered for each distance indicator.
Indicator | Mean $\mu_{\rm M33}$ | $\chi^{2}_{mean,\nu}$ | $\chi^{2}_{B23,\nu}$ | N
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
Cepheids | 24.644 | 1.08 | 1.39 | 9
TRGB | 24.639 | 3.67 | 2.96 | 5
JAGB | 24.625 | 1.65 | 0.84 | 2
Miras | 24.721 | 2.24 | 3.84 | 2

####  4.6.3 Eclipsing binaries

A distance to M33 based on detached eclipsing binaries (DEBs) was published by
Bonanos et al. (2006). However, unlike the well-established LMC and SMC
distances by Pietrzyński et al. (2019) and Graczyk et al. (2020) respectively,
which are based on late-type DEBs (a purely empirical method calibrated
geometrically through red giant interferometry), the M33 distance by Bonanos
et al. (2006) relies on early-type DEBs which depend on surface flux
calculated from non-local thermodynamic equilibrium models and is strongly
affected by model uncertainties. Therefore we will not compare our result with
this measurement as we limit our comparisons to empirical measures.

In the future, the ability to measure many primary distance indicators in the
same host offers the best chance to identify and rectify differences between
distance indcators. M33 offers one of the best such opportunities.

##  5 Photometric bias from Cluster Cepheids

###  5.1 Motivation

Contamination from crowded backgrounds such as star clusters can bias
photometric measurements of Cepheids in nearby galaxies. Photometric
measurements of extragalactic Cepheids are usually corrected for crowding
effects by injecting artificial stars in the vicinity of Cepheids and by
remeasuring their contribution (Riess et al., 2009). However, this test may
not properly reproduce the impact of stars physically associated with
Cepheids, which might be unresolved and whose light properties might differ
from those of the background field stars. Anderson & Riess (2018) found that
blending due to cluster Cepheids was responsible for a 0.23% overestimate of
$H_{0}$, although cluster Cepheids are a relatively rare phenomenon. They
concluded that chance superposition of Cepheids with clusters was not a limit
for a 1% measurement of the Hubble constant. In this section, we estimate the
blending contribution from cluster Cepheids in M33: we measure the occurrence
rate of Cepheids in clusters in M33 and we derive the typical flux
contribution of the clusters in order to determine by how much they affect
Cepheid photometry and our M33 distance modulus.

###  5.2 Crossmatch of Cepheid and cluster catalogs

First, we estimate the number of M33 Cepheids located in or near clusters.
Anderson & Riess (2018) reported a fraction of 2.4% cluster Cepheids in the
M31 galaxy, lower than in the Milky Way, LMC and SMC (with 8.5%, 11% and 6%
respectively). In order to obtain the fraction of cluster Cepheids in M33, we
crossmatch an initial sample of 609 fundamental mode Cepheids from Pellerin &
Macri (2011) with the catalog of 2137 star clusters in M33 from Johnson et al.
(2022). We adopt a separation of $\theta_{\rm sep}<1.2\,r_{\rm ap}$ as
membership criteria (Senchyna et al., 2015), with $r_{ap}$ the mean cluster
radius provided in Johnson et al. (2022). From this crossmatch we find a total
of 10 cluster Cepheids, listed in Table 7.

Table 7: Cluster Cepheids found by crossmatching Cepheids from Pellerin & Macri (2011) and star clusters from Johnson et al. (2022) with $\theta_{\rm sep}<1.2\,r_{\rm ap}$, where $r_{\rm ap}$ is the average cluster radius. Cepheid | Cluster |  $r_{\rm ap}$ (″ ) |  $\theta_{\rm sep}$ (″ )
---|---|---|---
01334331+3043559 | J22-241 | 1.17 | 0.85
01340959+3036215 | J22-477 | 1.55 | 1.46
01340060+3050079 | J22-521 | 1.83 | 1.77
01332060+3034584 | J22-665 | 1.49 | 0.12
01342512+3034381 | J22-722 | 1.41 | 0.27
01335311+3048343 | J22-836 | 1.43 | 0.49
01335809+3045568 | J22-900 | 1.31 | 1.51
01332768+3034238 | J22-1464 | 1.32 | 0.26
01334212+3032109 | J22-1492 | 1.60 | 1.74
01342784+3041012 | J22-1762 | 1.35 | 1.04
Figure 9: Average cluster contribution (curve of growth) for the 10 confirmed
M33 cluster Cepheids (Table 7) in the $m_{H}^{W}$, $F160W$, $F475W$, $F814W$
filters.

###  5.3 Creation of stamp images

We identified these 10 cluster Cepheids in the PHATTER mosaics and we produced
stamp images (cutouts centered on each Cepheid) in each filter. The HST stamp
images of the 10 crossmatched cluster Cepheids are shown in Fig. 11 (Appendix
B), where the clusters are easy to identify by eye. In particular, UV filters
($F275W$ and $F336W$) are well suited for detecting hot blue cluster stars,
while background red giant stars with luminosity similar to that of the
Cepheid may contribute more in the infrared. We note that some stamps are
blank because the Cepheid is located outside the limit of the PHATTER fields
in NIR and UV.

###  5.4 Visual inspection of stamp images

In order to make sure that the cluster count is complete, we inspected each
Cepheid stamp image for the presence of any additional undetected clusters. We
report an additional 13 suspect cluster Cepheids, listed in Table 8 and shown
in Fig. 13 (Appendix B). Three of them (01343182+3043050, 01343169+3043002,
and 01340910+3036296) are listed in the Johnson et al. (2022) catalog but at a
distance greater than $1.2r_{\rm ap}$ from the Cepheid (on average at $\sim
2r_{\rm ap}$), therefore they were not found by the crossmatch procedure. Two
of them are also listed in the Sarajedini & Mancone (2007) catalog at about 1″
and 4.5″ from the Cepheid (this catalog does not provide the cluster radii).

###  5.5 Flux contribution from the clusters

We followed the approach used in Anderson & Riess (2018) (see their §3.2.1) to
separate three contributions: the flux from the Cepheid, the flux of the
cluster, and the background contribution. The average cumulative light
contribution from clusters $\Delta m$ (or curve of growth) is obtained using a
series of apertures of increasing radius, starting from the Cepheid in the
center ($r=1$ pixel) to a radius of about 2″ . We note that $\Delta m$ can be
negative (if a light contribution from the cluster is detected) or positive
(if the cluster flux is low or if its location is statistically sparser than
the nearby environment of the Cepheid). At the distance to M33 found in §4.5,
a separation of 1″ corresponds to 4.1 pc along the major axis and 3.7 pc along
the minor axis as projected on the plane of the disk. Anderson & Riess (2018)
find that the cluster light contribution in M31 flattens off at a separation
of about 3.8 pc, which corresponds to approximately twice the average cluster
radius.

Fig. 9 shows the average light contribution from the 10 confirmed M33 clusters
(§5.2, Table 7). Similarly to Anderson & Riess (2018), the contribution in the
optical is significant, with $\Delta m$ around $-0.5$ mag in $F475W$ at a
separation of 4 pc, and becomes lower towards the near infrared with $\Delta
m=-0.20$ mag and $-0.17$ mag in $F814W$ and $m_{H}^{W}$ respectively. These
values are about a factor of 2 smaller than those found by Anderson & Riess
(2018) in M31 at the same separation.

Table 8: Cluster Cepheids found by inspecting visually the image cutouts from
PHATTER mosaics centered on each Cepheid. The second column indicates if the
Cepheid is found nearby a cluster listed in J22 or in S07, and ”$-$” indicates
that no known cluster was found around this Cepheid in the literature.
Cepheid | Cluster |  $r_{\rm ap}$ (″) |  $\theta_{\rm sep}$ (″ )
---|---|---|---
01333896+3034140 | $-$ | $-$ | $-$
01343182+3043050 | J22-40 | 1.60 | 3.73
01333015+3038039 | $-$ | $-$ | $-$
01333438+3035307 | $-$ | $-$ | $-$
01333433+3034270 | $-$ | $-$ | $-$
01343169+3043002 | J22-40 | 1.60 | 2.96
01340910+3036296 | J22-27, S07-325 | 2.23 | 4.71
01341217+3036362 | $-$ | $-$ | $-$
01340474+3049181 | S07-292 | $-$ | 1.12
01333348+3033210 | $-$ | $-$ | $-$
01334821+3038001 | $-$ | $-$ | $-$
01342988+3047541 | $-$ | $-$ | $-$
01340084+3049551 | $-$ | $-$ | $-$
Figure 10: Average cluster contribution (curve of growth) for the 13
additional cluster Cepheids found by visual inspection (Table 8) in the
$m_{H}^{W}$, $F160W$, $F475W$, $F814W$ filters.

The contribution of the 13 additional cluster Cepheids obtained by visual
inspection of the stamp images (§5.4, Table 8) is represented in Fig. 10. In
all filters, it is lower than the contribution from the 10 confirmed clusters
(Fig. 9) with only $\Delta m=-0.02$ mag at 4 pc in $m_{H}^{W}$. This suggests
that the 13 additional possible cluster Cepheids do not contribute to the
contamination of the Cepheid flux.

Following Anderson & Riess (2018), we estimate the average photometric bias
produced by the 10 confirmed cluster Cepheids from Table 7 by multiplying
their occurence rate (1.6 %) by their flux contribution in $m_{H}^{W}$, which
gives a bias of 0.003 mag. This shows that cluster Cepheids do not impact the
distance measurement of M33. If we conservatively assume that all 13
additional Cepheids from Table 8 are associated with clusters, we obtain a
maximum occurence rate of 3.7%, still lower than the fraction of cluster
Cepheids in the Milky Way, LMC and SMC, and a correspondingly lower bias per
cluster Cepheid (since the additional 13 do not produce a significant
difference).

##  6 Summary

We take advantage of the recently published high-quality PHATTER photometric
survey of the M33 galaxy and we construct the Cepheid PL relation in the SH0ES
near-infrared Wesenheit system ($m_{H}^{W}$). We use well-sampled ground-based
light curves for the same Cepheid sample to recover the phases and amplitudes
and we correct the random-epoch PHATTER measurements to mean magnitude. We
also present new optical template light curves based on the same population of
M33 Cepheids. These can be directly applied to fit sparsely-sampled light
curves.

We improve the uncertainty in the Cepheid distance to M33 to the 1.3% level
and we present the tightest PL relation to date in this galaxy, with a scatter
of only 0.11 mag. In particular, the use of HST photometry allows to
significantly reduce the effect of crowding over past studies based on ground-
based observations. This new Cepheid distance provides groundwork for
including M33 as an anchor galaxy in the empirical distance scale (Riess et
al., 2022), with a similar role as the Milky Way, the LMC and NGC 4258. In
order to consider M33 as an independent anchor of the distance scale, a
precise geometric calibration of its distance is required, using for example a
large sample of late-type detached eclipsing binaries. Future facilities such
as the ELTs and the Roman Space Telescope could enable significant
improvements in that matter.

We discuss differences between past measurements of the distance to M33,
especially based on the TRGB method, and we identify factors that explain
these discrepancies such as the possible effect of blending and the choice of
color cut when defining the red-giant branch in the color-magnitude diagram.
Finally, we investigate the bias from cluster Cepheids and estimate that at
most 3.7% of M33 Cepheids are in these systems, resulting in a negligible
contamination of 0.003 mag to our distance measurement. Our result, compared
with other distance measurements from the literature, highlights the
unprecedented reliability and precision of Cepheids as standard candles.

## Acknowledgements

We thank Abigail Lee for discussions about the TRGB and Cepheid distances to
M33. L.B. is deeply grateful to Arshia Jacob for her constant support and
kindness during the preparation of this paper. This research has made use of
Astropy, a community-developed core Python package for Astronomy (Astropy
Collaboration et al., 2018), as well as of the SVO Filter Profile
Service444http://svo2.cab.inta- csic.es/theory/fps/. Some of the data
presented in this paper were obtained from the Mikulski Archive for Space
Telescopes (MAST) at the STScI.

## References

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##  Appendix A Output parameters for HST M33 Cepheids

Table 9: Full sample of HST M33 Cepheids used in this analysis and their mean magnitudes obtained from template fitting. The values given here do not include subtraction of 0.015 mag to correct CRNL for 2 dex between LMC and M33 Cepheids. They do not include addition of 0.069 mag to $m_{H}^{W}$ errors for intrinsic scatter. The sample names in the last column are: (G) = gold and (S) = silver. Cepheids marked with (*) are listed as cluster Cepheids in Sect. 5.  ID | RA | DEC | $\log P$ | $F160W$ | $F475W$ | $F814W$ | $m_{H}^{W}$ | sample
---|---|---|---|---|---|---|---|---
01332610+3033200 | 23.3586 | 30.5555 | 1.3009 | $18.261_{\pm 0.091}$ | $20.678_{\pm 0.079}$ | $19.171_{\pm 0.024}$ | $17.853_{\pm 0.093}$ | G
01332735+3035515 | 23.3638 | 30.5976 | 1.0496 | $19.154_{\pm 0.071}$ | $21.093_{\pm 0.084}$ | $19.791_{\pm 0.046}$ | $18.798_{\pm 0.075}$ | G
01332768+3034238 (∗) | 23.3652 | 30.5732 | 1.1715 | $18.402_{\pm 0.053}$ | $20.330_{\pm 0.035}$ | $19.103_{\pm 0.035}$ | $18.065_{\pm 0.054}$ | G
01332803+3032133 | 23.3667 | 30.5370 | 1.1975 | $18.575_{\pm 0.079}$ | $21.810_{\pm 0.113}$ | $19.868_{\pm 0.037}$ | $18.057_{\pm 0.085}$ | G
01332878+3034403 | 23.3698 | 30.5778 | 0.9494 | $19.179_{\pm 0.034}$ | $20.922_{\pm 0.038}$ | $19.891_{\pm 0.037}$ | $18.892_{\pm 0.037}$ | G
01332922+3031360 | 23.3716 | 30.5267 | 0.8148 | $19.920_{\pm 0.045}$ | $21.726_{\pm 0.062}$ | $20.463_{\pm 0.051}$ | $19.574_{\pm 0.049}$ | G
01332923+3037449 | 23.3717 | 30.6291 | 1.6632 | $16.942_{\pm 0.165}$ | $20.028_{\pm 0.050}$ | $18.235_{\pm 0.014}$ | $16.462_{\pm 0.166}$ | G
01332946+3035574 | 23.3726 | 30.5993 | 1.4868 | $17.655_{\pm 0.116}$ | $20.634_{\pm 0.125}$ | $18.886_{\pm 0.156}$ | $17.186_{\pm 0.127}$ | G
01332984+3034529 | 23.3742 | 30.5813 | 0.9080 | $19.398_{\pm 0.044}$ | $21.553_{\pm 0.087}$ | $20.243_{\pm 0.008}$ | $19.040_{\pm 0.049}$ | G
01333012+3036381 | 23.3754 | 30.6106 | 1.2546 | $18.325_{\pm 0.144}$ | $20.271_{\pm 0.025}$ | $18.926_{\pm 0.011}$ | $17.958_{\pm 0.144}$ | G
01333039+3035555 | 23.3765 | 30.5987 | 1.3438 | $17.940_{\pm 0.132}$ | $20.124_{\pm 0.015}$ | $18.725_{\pm 0.020}$ | $17.560_{\pm 0.132}$ | G
01333073+3034495 | 23.3779 | 30.5804 | 0.7492 | $19.809_{\pm 0.102}$ | $21.667_{\pm 0.013}$ | $20.567_{\pm 0.043}$ | $19.505_{\pm 0.103}$ | G
01333080+3031113 | 23.3782 | 30.5198 | 1.2438 | $18.372_{\pm 0.134}$ | $20.982_{\pm 0.174}$ | $19.409_{\pm 0.111}$ | $17.947_{\pm 0.144}$ | G
01333165+3039314 | 23.3817 | 30.6587 | 1.1194 | $18.743_{\pm 0.172}$ | $20.180_{\pm 0.026}$ | $19.316_{\pm 0.173}$ | $18.498_{\pm 0.178}$ | S
01333216+3039457 | 23.3839 | 30.6627 | 1.1751 | $18.712_{\pm 0.081}$ | $21.129_{\pm 0.223}$ | $19.564_{\pm 0.160}$ | $18.289_{\pm 0.107}$ | G
01333241+3031437 | 23.3849 | 30.5288 | 0.9012 | $19.506_{\pm 0.171}$ | $21.925_{\pm 0.108}$ | $20.460_{\pm 0.015}$ | $19.109_{\pm 0.173}$ | S
01333242+3034094 | 23.3849 | 30.5692 | 0.8484 | $19.545_{\pm 0.076}$ | $20.997_{\pm 0.127}$ | $20.087_{\pm 0.086}$ | $19.289_{\pm 0.085}$ | G
01333291+3035490 | 23.3870 | 30.5969 | 1.4807 | $17.557_{\pm 0.138}$ | $20.698_{\pm 0.181}$ | $18.914_{\pm 0.156}$ | $17.079_{\pm 0.151}$ | G
01333308+3042308 | 23.3877 | 30.7085 | 0.7790 | $19.828_{\pm 0.198}$ | $21.925_{\pm 0.115}$ | $20.678_{\pm 0.235}$ | $19.486_{\pm 0.209}$ | S
01333327+3037476 | 23.3885 | 30.6298 | 1.3551 | $18.266_{\pm 0.174}$ | $21.417_{\pm 0.394}$ | $19.512_{\pm 0.194}$ | $17.757_{\pm 0.207}$ | S
01333348+3033210 (∗) | 23.3894 | 30.5558 | 0.9546 | $19.300_{\pm 0.200}$ | $21.277_{\pm 0.013}$ | $20.067_{\pm 0.057}$ | $18.968_{\pm 0.201}$ | S
01333424+3035009 | 23.3925 | 30.5835 | 1.1956 | $18.629_{\pm 0.082}$ | $21.051_{\pm 0.312}$ | $19.664_{\pm 0.171}$ | $18.252_{\pm 0.122}$ | G
01333438+3035307 (∗) | 23.3931 | 30.5918 | 1.1995 | $18.459_{\pm 0.143}$ | $20.760_{\pm 0.263}$ | $19.367_{\pm 0.109}$ | $18.080_{\pm 0.160}$ | S
01333548+3044180 | 23.3977 | 30.7383 | 0.9782 | $19.173_{\pm 0.035}$ | $21.503_{\pm 0.167}$ | $20.039_{\pm 0.106}$ | $18.776_{\pm 0.061}$ | G
01333552+3033307 | 23.3978 | 30.5585 | 1.0616 | $18.869_{\pm 0.085}$ | $21.022_{\pm 0.011}$ | $19.710_{\pm 0.010}$ | $18.511_{\pm 0.085}$ | G
01333557+3036496 | 23.3980 | 30.6138 | 1.0288 | $19.166_{\pm 0.052}$ | $21.463_{\pm 0.211}$ | $20.072_{\pm 0.104}$ | $18.788_{\pm 0.079}$ | G
01333628+3037313 | 23.4010 | 30.6253 | 1.0713 | $19.110_{\pm 0.035}$ | $21.900_{\pm 0.037}$ | $20.144_{\pm 0.029}$ | $18.639_{\pm 0.037}$ | G
01333649+3030536 | 23.4020 | 30.5148 | 0.8802 | $19.243_{\pm 0.081}$ | $20.923_{\pm 0.227}$ | $20.045_{\pm 0.140}$ | $18.995_{\pm 0.106}$ | G
01333680+3034348 | 23.4031 | 30.5763 | 0.9571 | $19.254_{\pm 0.041}$ | $20.811_{\pm 0.166}$ | $19.752_{\pm 0.052}$ | $18.960_{\pm 0.060}$ | G
01333747+3031388 | 23.4059 | 30.5274 | 1.7605 | $16.988_{\pm 0.084}$ | $20.102_{\pm 0.136}$ | $18.180_{\pm 0.073}$ | $16.475_{\pm 0.093}$ | G
01333754+3033054 | 23.4062 | 30.5515 | 0.9531 | $19.297_{\pm 0.027}$ | $21.627_{\pm 0.070}$ | $20.234_{\pm 0.023}$ | $18.918_{\pm 0.033}$ | S
01333791+3033550 | 23.4078 | 30.5652 | 0.9521 | $19.418_{\pm 0.200}$ | $21.500_{\pm 0.075}$ | $20.205_{\pm 0.040}$ | $19.064_{\pm 0.201}$ | S
01333880+3037515 | 23.4115 | 30.6309 | 1.1389 | $18.981_{\pm 0.069}$ | $21.503_{\pm 0.025}$ | $19.875_{\pm 0.030}$ | $18.542_{\pm 0.070}$ | G
01334058+3045421 | 23.4189 | 30.7617 | 0.8682 | $19.584_{\pm 0.069}$ | $21.989_{\pm 0.211}$ | $20.444_{\pm 0.138}$ | $19.166_{\pm 0.094}$ | G
01334167+3043115 | 23.4234 | 30.7198 | 0.9064 | $19.402_{\pm 0.209}$ | $21.789_{\pm 0.027}$ | $20.397_{\pm 0.040}$ | $19.023_{\pm 0.209}$ | S
01334331+3043559 (∗) | 23.4304 | 30.7322 | 1.3045 | $18.196_{\pm 0.123}$ | $20.940_{\pm 0.070}$ | $19.246_{\pm 0.092}$ | $17.741_{\pm 0.126}$ | G
01334390+3032452 | 23.4328 | 30.5458 | 1.8731 | $16.197_{\pm 0.091}$ | $19.803_{\pm 0.055}$ | $17.745_{\pm 0.031}$ | $15.649_{\pm 0.092}$ | G
01334456+3043132 | 23.4355 | 30.7203 | 1.3510 | $18.135_{\pm 0.076}$ | $20.891_{\pm 0.068}$ | $19.202_{\pm 0.031}$ | $17.681_{\pm 0.078}$ | G
01334582+3044207 | 23.4408 | 30.7390 | 1.1178 | $18.983_{\pm 0.094}$ | $21.239_{\pm 0.090}$ | $19.893_{\pm 0.013}$ | $18.616_{\pm 0.097}$ | G
01334596+3030303 | 23.4414 | 30.5084 | 0.5382 | $20.445_{\pm 0.087}$ | $22.468_{\pm 0.246}$ | $21.529_{\pm 0.081}$ | $20.181_{\pm 0.109}$ | G
01334654+3046449 | 23.4438 | 30.7791 | 1.2743 | $18.341_{\pm 0.104}$ | $20.247_{\pm 0.053}$ | $18.945_{\pm 0.026}$ | $17.985_{\pm 0.105}$ | G
01334681+3043335 | 23.4449 | 30.7260 | 0.7779 | $19.771_{\pm 0.064}$ | $21.715_{\pm 0.136}$ | $20.578_{\pm 0.112}$ | $19.457_{\pm 0.078}$ | G
01334691+3044112 | 23.4453 | 30.7365 | 0.7498 | $19.852_{\pm 0.132}$ | $22.002_{\pm 0.008}$ | $20.699_{\pm 0.019}$ | $19.496_{\pm 0.132}$ | S
01334720+3035365 | 23.4465 | 30.5934 | 1.4232 | $17.767_{\pm 0.110}$ | $20.319_{\pm 0.154}$ | $18.804_{\pm 0.008}$ | $17.357_{\pm 0.117}$ | G
01334821+3038001 (∗) | 23.4507 | 30.6333 | 1.1780 | $18.557_{\pm 0.115}$ | $20.428_{\pm 0.020}$ | $19.166_{\pm 0.025}$ | $18.211_{\pm 0.115}$ | G
01334879+3049143 | 23.4531 | 30.8206 | 0.7486 | $19.980_{\pm 0.047}$ | $21.885_{\pm 0.071}$ | $20.668_{\pm 0.043}$ | $19.646_{\pm 0.052}$ | G
01334886+3034159 | 23.4534 | 30.5711 | 0.9011 | $19.351_{\pm 0.041}$ | $21.181_{\pm 0.191}$ | $20.118_{\pm 0.071}$ | $19.056_{\pm 0.066}$ | G
01334929+3032182 | 23.4552 | 30.5383 | 0.9912 | $19.302_{\pm 0.175}$ | $21.790_{\pm 0.062}$ | $20.264_{\pm 0.033}$ | $18.889_{\pm 0.176}$ | S
01334955+3047437 | 23.4563 | 30.7955 | 0.7788 | $19.902_{\pm 0.046}$ | $21.931_{\pm 0.078}$ | $20.721_{\pm 0.054}$ | $19.570_{\pm 0.052}$ | G
01334960+3044008 | 23.4565 | 30.7336 | 0.7978 | $19.742_{\pm 0.038}$ | $21.926_{\pm 0.113}$ | $20.697_{\pm 0.015}$ | $19.405_{\pm 0.048}$ | G
01334983+3037587 | 23.4575 | 30.6329 | 1.1105 | $18.690_{\pm 0.110}$ | $20.738_{\pm 0.020}$ | $19.470_{\pm 0.020}$ | $18.343_{\pm 0.110}$ | G
01335051+3047537 | 23.4603 | 30.7982 | 1.4233 | $17.912_{\pm 0.104}$ | $20.897_{\pm 0.116}$ | $19.077_{\pm 0.079}$ | $17.425_{\pm 0.110}$ | G
01335067+3034459 | 23.4609 | 30.5794 | 0.7803 | $19.842_{\pm 0.207}$ | $22.022_{\pm 0.340}$ | $20.689_{\pm 0.218}$ | $19.478_{\pm 0.231}$ | S
01335067+3047146 | 23.4610 | 30.7873 | 0.9875 | $19.088_{\pm 0.046}$ | $21.349_{\pm 0.240}$ | $20.036_{\pm 0.090}$ | $18.729_{\pm 0.080}$ | G
01335075+3035444 | 23.4613 | 30.5956 | 1.3676 | $18.191_{\pm 0.120}$ | $20.776_{\pm 0.020}$ | $19.121_{\pm 0.020}$ | $17.746_{\pm 0.120}$ | G
01335090+3033361 | 23.4619 | 30.5600 | 1.5745 | $17.638_{\pm 0.060}$ | $21.241_{\pm 0.201}$ | $19.103_{\pm 0.094}$ | $17.070_{\pm 0.082}$ | G
01335094+3031174 | 23.4621 | 30.5214 | 0.6782 | $19.998_{\pm 0.040}$ | $21.830_{\pm 0.055}$ | $20.726_{\pm 0.026}$ | $19.693_{\pm 0.043}$ | G
01335104+3043598 | 23.4626 | 30.7332 | 1.0915 | $18.935_{\pm 0.074}$ | $21.353_{\pm 0.137}$ | $19.833_{\pm 0.102}$ | $18.524_{\pm 0.086}$ | G
01335172+3050370 | 23.4654 | 30.8436 | 0.9564 | $19.405_{\pm 0.200}$ | $21.505_{\pm 0.483}$ | $20.209_{\pm 0.226}$ | $19.051_{\pm 0.242}$ | S
01335182+3033109 | 23.4657 | 30.5529 | 0.8132 | $19.753_{\pm 0.018}$ | $22.289_{\pm 0.009}$ | $20.764_{\pm 0.043}$ | $19.341_{\pm 0.021}$ | G
01335198+3048485 | 23.4664 | 30.8135 | 1.1251 | $18.920_{\pm 0.172}$ | $20.610_{\pm 0.215}$ | $19.459_{\pm 0.142}$ | $18.603_{\pm 0.184}$ | S
01335232+3046026 | 23.4678 | 30.7674 | 0.7752 | $19.957_{\pm 0.294}$ | $22.292_{\pm 0.024}$ | $20.793_{\pm 0.240}$ | $19.551_{\pm 0.300}$ | S
01335247+3038442 | 23.4685 | 30.6456 | 1.5544 | $17.443_{\pm 0.139}$ | $19.865_{\pm 0.080}$ | $18.448_{\pm 0.020}$ | $17.058_{\pm 0.141}$ | G
01335311+3048343 (∗) | 23.4711 | 30.8095 | 0.9821 | $19.156_{\pm 0.174}$ | $20.813_{\pm 0.401}$ | $19.761_{\pm 0.138}$ | $18.864_{\pm 0.205}$ | S
01335345+3033085 | 23.4726 | 30.5523 | 1.0606 | $19.106_{\pm 0.060}$ | $21.593_{\pm 0.071}$ | $20.079_{\pm 0.078}$ | $18.696_{\pm 0.066}$ | G
01335428+3041107 | 23.4760 | 30.6863 | 1.4476 | $17.642_{\pm 0.165}$ | $20.302_{\pm 0.123}$ | $18.699_{\pm 0.064}$ | $17.210_{\pm 0.169}$ | G
01335478+3041061 | 23.4781 | 30.6850 | 1.5309 | $17.420_{\pm 0.105}$ | $20.191_{\pm 0.013}$ | $18.479_{\pm 0.006}$ | $16.960_{\pm 0.105}$ | G
01335479+3045181 | 23.4782 | 30.7550 | 0.7621 | $19.692_{\pm 0.198}$ | $21.582_{\pm 0.059}$ | $20.474_{\pm 0.033}$ | $19.385_{\pm 0.199}$ | S
01335482+3045317 | 23.4783 | 30.7588 | 1.5529 | $17.543_{\pm 0.099}$ | $19.167_{\pm 0.279}$ | $17.781_{\pm 0.018}$ | $17.166_{\pm 0.122}$ | G
01335502+3035372 | 23.4791 | 30.5936 | 1.0056 | $19.054_{\pm 0.068}$ | $21.186_{\pm 0.008}$ | $19.907_{\pm 0.022}$ | $18.704_{\pm 0.068}$ | G
01335514+3048309 | 23.4796 | 30.8085 | 0.9493 | $19.528_{\pm 0.200}$ | $21.798_{\pm 0.516}$ | $20.328_{\pm 0.282}$ | $19.130_{\pm 0.250}$ | S
01335523+3043429 | 23.4800 | 30.7285 | 1.4209 | $18.011_{\pm 0.232}$ | $20.786_{\pm 0.454}$ | $19.056_{\pm 0.117}$ | $17.547_{\pm 0.261}$ | S
01335612+3039029 | 23.4837 | 30.6508 | 1.0180 | $19.106_{\pm 0.056}$ | $21.335_{\pm 0.018}$ | $19.998_{\pm 0.018}$ | $18.741_{\pm 0.056}$ | G
01335615+3043424 | 23.4838 | 30.7284 | 1.0779 | $18.951_{\pm 0.058}$ | $21.309_{\pm 0.037}$ | $19.795_{\pm 0.077}$ | $18.541_{\pm 0.062}$ | G
01335645+3046435 | 23.4851 | 30.7788 | 0.8630 | $19.646_{\pm 0.075}$ | $21.755_{\pm 0.020}$ | $20.370_{\pm 0.024}$ | $19.269_{\pm 0.075}$ | G
01335646+3044420 | 23.4851 | 30.7449 | 0.7388 | $19.847_{\pm 0.103}$ | $21.739_{\pm 0.117}$ | $20.702_{\pm 0.105}$ | $19.559_{\pm 0.110}$ | G
01335737+3041133 | 23.4889 | 30.6870 | 1.0648 | $18.885_{\pm 0.172}$ | $21.487_{\pm 0.054}$ | $19.907_{\pm 0.046}$ | $18.459_{\pm 0.173}$ | S
01335760+3048341 | 23.4898 | 30.8094 | 0.6849 | $20.248_{\pm 0.054}$ | $22.664_{\pm 0.203}$ | $21.193_{\pm 0.107}$ | $19.849_{\pm 0.079}$ | G
01335761+3038053 | 23.4899 | 30.6348 | 1.0915 | $18.852_{\pm 0.172}$ | $21.290_{\pm 0.128}$ | $19.790_{\pm 0.230}$ | $18.446_{\pm 0.185}$ | S
01335809+3045568 (∗) | 23.4919 | 30.7658 | 1.4986 | $17.818_{\pm 0.228}$ | $20.706_{\pm 0.204}$ | $18.935_{\pm 0.118}$ | $17.343_{\pm 0.236}$ | S
01335852+3043596 | 23.4937 | 30.7332 | 1.0470 | $18.905_{\pm 0.052}$ | $21.599_{\pm 0.182}$ | $20.026_{\pm 0.023}$ | $18.480_{\pm 0.070}$ | G
01335870+3033166 | 23.4945 | 30.5546 | 0.8863 | $19.371_{\pm 0.043}$ | $21.328_{\pm 0.058}$ | $20.094_{\pm 0.059}$ | $19.032_{\pm 0.048}$ | G
01335886+3037198 | 23.4951 | 30.6221 | 1.3903 | $18.027_{\pm 0.179}$ | $20.603_{\pm 0.114}$ | $19.116_{\pm 0.070}$ | $17.624_{\pm 0.182}$ | G
01335947+3032266 | 23.4977 | 30.5407 | 1.6990 | $17.186_{\pm 0.067}$ | $20.559_{\pm 0.035}$ | $18.604_{\pm 0.024}$ | $16.664_{\pm 0.068}$ | G
01335989+3037600 | 23.4994 | 30.6333 | 0.7779 | $19.916_{\pm 0.070}$ | $21.263_{\pm 0.121}$ | $20.381_{\pm 0.076}$ | $19.667_{\pm 0.079}$ | G
01340058+3036306 | 23.5023 | 30.6085 | 0.8024 | $19.643_{\pm 0.195}$ | $21.814_{\pm 0.119}$ | $20.535_{\pm 0.063}$ | $19.293_{\pm 0.198}$ | S
01340102+3043097 | 23.5041 | 30.7193 | 1.4843 | $17.804_{\pm 0.108}$ | $20.225_{\pm 0.351}$ | $18.811_{\pm 0.135}$ | $17.420_{\pm 0.144}$ | G
01340120+3048131 | 23.5048 | 30.8036 | 0.6645 | $20.350_{\pm 0.221}$ | $22.574_{\pm 0.358}$ | $21.258_{\pm 0.141}$ | $19.991_{\pm 0.242}$ | S
01340123+3031135 | 23.5050 | 30.5204 | 1.3965 | $17.746_{\pm 0.083}$ | $19.537_{\pm 0.253}$ | $18.430_{\pm 0.143}$ | $17.440_{\pm 0.111}$ | G
01340137+3040264 | 23.5056 | 30.6739 | 1.1737 | $18.586_{\pm 0.174}$ | $21.979_{\pm 0.165}$ | $20.129_{\pm 0.241}$ | $18.091_{\pm 0.189}$ | S
01340166+3031030 | 23.5067 | 30.5174 | 1.1982 | $18.359_{\pm 0.116}$ | $21.288_{\pm 0.331}$ | $19.544_{\pm 0.140}$ | $17.891_{\pm 0.148}$ | G
01340167+3049102 | 23.5068 | 30.8195 | 0.6306 | $20.286_{\pm 0.035}$ | $21.956_{\pm 0.064}$ | $20.970_{\pm 0.050}$ | $20.010_{\pm 0.041}$ | G
01340178+3039229 | 23.5073 | 30.6563 | 1.3358 | $18.084_{\pm 0.112}$ | $20.439_{\pm 0.133}$ | $18.936_{\pm 0.144}$ | $17.677_{\pm 0.123}$ | G
01340187+3041483 | 23.5077 | 30.6967 | 0.8015 | $19.859_{\pm 0.055}$ | $21.689_{\pm 0.209}$ | $20.572_{\pm 0.090}$ | $19.550_{\pm 0.080}$ | G
01340219+3037418 | 23.5090 | 30.6282 | 0.8642 | $19.655_{\pm 0.038}$ | $21.206_{\pm 0.156}$ | $20.198_{\pm 0.084}$ | $19.374_{\pm 0.059}$ | G
01340223+3042423 | 23.5092 | 30.7117 | 0.8943 | $19.331_{\pm 0.089}$ | $21.406_{\pm 0.012}$ | $20.082_{\pm 0.008}$ | $18.970_{\pm 0.089}$ | G
01340259+3036282 | 23.5107 | 30.6078 | 1.1346 | $18.887_{\pm 0.174}$ | $21.318_{\pm 0.085}$ | $19.763_{\pm 0.037}$ | $18.467_{\pm 0.176}$ | S
01340296+3047273 | 23.5122 | 30.7909 | 0.8503 | $19.550_{\pm 0.079}$ | $21.515_{\pm 0.064}$ | $20.337_{\pm 0.035}$ | $19.226_{\pm 0.081}$ | G
01340367+3045279 | 23.5152 | 30.7577 | 0.7919 | $19.820_{\pm 0.090}$ | $21.371_{\pm 0.102}$ | $20.329_{\pm 0.038}$ | $19.530_{\pm 0.094}$ | G
01340399+3036158 | 23.5165 | 30.6044 | 1.2114 | $18.389_{\pm 0.104}$ | $20.650_{\pm 0.206}$ | $19.251_{\pm 0.026}$ | $18.009_{\pm 0.117}$ | G
01340426+3041150 | 23.5176 | 30.6875 | 0.8822 | $19.441_{\pm 0.048}$ | $21.748_{\pm 0.177}$ | $20.336_{\pm 0.124}$ | $19.057_{\pm 0.073}$ | G
01340501+3035576 | 23.5208 | 30.5993 | 0.7702 | $19.777_{\pm 0.049}$ | $21.702_{\pm 0.053}$ | $20.524_{\pm 0.048}$ | $19.453_{\pm 0.052}$ | G
01340516+3038511 | 23.5214 | 30.6475 | 1.8421 | $16.855_{\pm 0.118}$ | $19.970_{\pm 0.634}$ | $18.018_{\pm 0.358}$ | $16.334_{\pm 0.219}$ | S
01340593+3038192 | 23.5246 | 30.6387 | 1.0305 | $19.197_{\pm 0.172}$ | $21.720_{\pm 0.317}$ | $20.192_{\pm 0.179}$ | $18.784_{\pm 0.195}$ | S
01340593+3039285 | 23.5246 | 30.6579 | 1.1639 | $18.491_{\pm 0.174}$ | $20.841_{\pm 0.468}$ | $19.413_{\pm 0.299}$ | $18.103_{\pm 0.224}$ | S
01340660+3038167 | 23.5273 | 30.6379 | 0.9338 | $19.195_{\pm 0.200}$ | $21.178_{\pm 0.190}$ | $19.912_{\pm 0.070}$ | $18.848_{\pm 0.207}$ | S
01340737+3030483 | 23.5305 | 30.5134 | 0.7710 | $19.957_{\pm 0.062}$ | $22.091_{\pm 0.109}$ | $20.796_{\pm 0.038}$ | $19.603_{\pm 0.069}$ | G
01340794+3038312 | 23.5330 | 30.6420 | 0.7710 | $19.805_{\pm 0.198}$ | $21.712_{\pm 0.196}$ | $20.526_{\pm 0.218}$ | $19.479_{\pm 0.212}$ | S
01340817+3039318 | 23.5339 | 30.6588 | 1.1246 | $18.857_{\pm 0.172}$ | $21.385_{\pm 0.080}$ | $19.963_{\pm 0.070}$ | $18.471_{\pm 0.175}$ | S
01340873+3045431 | 23.5363 | 30.7619 | 1.1311 | $18.535_{\pm 0.113}$ | $20.827_{\pm 0.135}$ | $19.432_{\pm 0.111}$ | $18.156_{\pm 0.121}$ | G
01340883+3039462 | 23.5367 | 30.6628 | 1.0050 | $19.132_{\pm 0.042}$ | $21.307_{\pm 0.007}$ | $19.944_{\pm 0.008}$ | $18.761_{\pm 0.042}$ | G
01340910+3036296 (∗) | 23.5377 | 30.6082 | 1.1645 | $18.426_{\pm 0.174}$ | $20.352_{\pm 0.361}$ | $19.193_{\pm 0.202}$ | $18.107_{\pm 0.203}$ | S
01340936+3029565 | 23.5389 | 30.4990 | 0.9619 | $19.283_{\pm 0.026}$ | $21.328_{\pm 0.053}$ | $20.057_{\pm 0.033}$ | $18.935_{\pm 0.030}$ | G
01340959+3036215 (∗) | 23.5398 | 30.6059 | 1.0206 | $18.932_{\pm 0.175}$ | $21.144_{\pm 0.206}$ | $19.881_{\pm 0.108}$ | $18.586_{\pm 0.185}$ | S
01341017+3044502 | 23.5423 | 30.7472 | 1.0185 | $19.253_{\pm 0.053}$ | $21.528_{\pm 0.143}$ | $20.131_{\pm 0.070}$ | $18.873_{\pm 0.067}$ | G
01341027+3034077 | 23.5426 | 30.5688 | 1.2866 | $18.334_{\pm 0.012}$ | $21.147_{\pm 0.079}$ | $19.530_{\pm 0.005}$ | $17.898_{\pm 0.023}$ | G
01341121+3047547 | 23.5466 | 30.7985 | 1.0938 | $18.753_{\pm 0.173}$ | $20.646_{\pm 0.018}$ | $19.484_{\pm 0.013}$ | $18.433_{\pm 0.173}$ | S
01341125+3041558 | 23.5467 | 30.6988 | 1.4251 | $17.682_{\pm 0.098}$ | $20.116_{\pm 0.051}$ | $18.659_{\pm 0.021}$ | $17.287_{\pm 0.099}$ | G
01341199+3035191 | 23.5498 | 30.5886 | 1.1258 | $18.879_{\pm 0.173}$ | $21.201_{\pm 0.114}$ | $19.759_{\pm 0.275}$ | $18.488_{\pm 0.189}$ | S
01341217+3036362 (∗) | 23.5505 | 30.6100 | 1.2147 | $18.569_{\pm 0.112}$ | $21.245_{\pm 0.114}$ | $19.628_{\pm 0.061}$ | $18.133_{\pm 0.117}$ | G
01341217+3046411 | 23.5506 | 30.7781 | 1.5546 | $17.423_{\pm 0.092}$ | $19.816_{\pm 0.088}$ | $18.420_{\pm 0.052}$ | $17.043_{\pm 0.096}$ | G
01341226+3045507 | 23.5509 | 30.7641 | 1.0014 | $19.118_{\pm 0.175}$ | $21.574_{\pm 0.443}$ | $20.049_{\pm 0.282}$ | $18.706_{\pm 0.220}$ | S
01341241+3047206 | 23.5516 | 30.7890 | 0.6306 | $20.278_{\pm 0.193}$ | $22.276_{\pm 0.083}$ | $21.049_{\pm 0.077}$ | $19.941_{\pm 0.195}$ | S
01341259+3041262 | 23.5523 | 30.6906 | 0.9201 | $19.430_{\pm 0.026}$ | $21.757_{\pm 0.020}$ | $20.309_{\pm 0.035}$ | $19.037_{\pm 0.028}$ | G
01341330+3043069 | 23.5552 | 30.7185 | 0.9316 | $19.396_{\pm 0.200}$ | $21.851_{\pm 0.064}$ | $20.358_{\pm 0.019}$ | $18.992_{\pm 0.201}$ | S
01341343+3043340 | 23.5558 | 30.7261 | 0.9610 | $19.237_{\pm 0.178}$ | $21.420_{\pm 0.085}$ | $20.042_{\pm 0.012}$ | $18.862_{\pm 0.179}$ | S
01341356+3030274 | 23.5564 | 30.5076 | 0.8637 | $19.575_{\pm 0.035}$ | $21.689_{\pm 0.040}$ | $20.395_{\pm 0.015}$ | $19.221_{\pm 0.037}$ | G
01341383+3044184 | 23.5575 | 30.7384 | 1.0259 | $19.302_{\pm 0.172}$ | $21.796_{\pm 0.004}$ | $20.255_{\pm 0.015}$ | $18.886_{\pm 0.172}$ | S
01341387+3043240 | 23.5576 | 30.7233 | 0.9006 | $19.461_{\pm 0.023}$ | $21.780_{\pm 0.059}$ | $20.371_{\pm 0.040}$ | $19.078_{\pm 0.029}$ | G
01341389+3032122 | 23.5577 | 30.5366 | 0.9586 | $19.283_{\pm 0.200}$ | $21.671_{\pm 0.395}$ | $20.240_{\pm 0.297}$ | $18.894_{\pm 0.236}$ | S
01341396+3048375 | 23.5580 | 30.8104 | 0.7530 | $19.938_{\pm 0.066}$ | $21.839_{\pm 0.073}$ | $20.739_{\pm 0.080}$ | $19.634_{\pm 0.072}$ | G
01341425+3037138 | 23.5592 | 30.6205 | 0.9766 | $19.220_{\pm 0.030}$ | $21.670_{\pm 0.142}$ | $20.195_{\pm 0.086}$ | $18.820_{\pm 0.052}$ | G
01341471+3046097 | 23.5612 | 30.7694 | 1.0706 | $19.099_{\pm 0.172}$ | $22.208_{\pm 0.413}$ | $20.344_{\pm 0.227}$ | $18.600_{\pm 0.210}$ | S
01341554+3044116 | 23.5646 | 30.7365 | 0.9086 | $19.562_{\pm 0.038}$ | $21.784_{\pm 0.008}$ | $20.333_{\pm 0.005}$ | $19.168_{\pm 0.038}$ | G
01341590+3046436 | 23.5661 | 30.7788 | 0.9322 | $19.414_{\pm 0.034}$ | $21.830_{\pm 0.103}$ | $20.383_{\pm 0.012}$ | $19.021_{\pm 0.043}$ | G
01341889+3044412 | 23.5786 | 30.7448 | 1.3053 | $18.464_{\pm 0.128}$ | $20.761_{\pm 0.082}$ | $19.349_{\pm 0.012}$ | $18.080_{\pm 0.130}$ | G
01341986+3043020 | 23.5826 | 30.7172 | 0.9163 | $19.538_{\pm 0.200}$ | $21.899_{\pm 0.069}$ | $20.437_{\pm 0.300}$ | $19.142_{\pm 0.215}$ | S
01342038+3041010 | 23.5848 | 30.6836 | 1.2032 | $18.613_{\pm 0.137}$ | $20.968_{\pm 0.043}$ | $19.449_{\pm 0.050}$ | $18.202_{\pm 0.138}$ | G
01342055+3042440 | 23.5855 | 30.7122 | 0.7557 | $19.795_{\pm 0.085}$ | $21.595_{\pm 0.163}$ | $20.508_{\pm 0.051}$ | $19.494_{\pm 0.095}$ | G
01342102+3044152 | 23.5874 | 30.7375 | 0.9993 | $19.168_{\pm 0.175}$ | $20.997_{\pm 0.472}$ | $19.778_{\pm 0.172}$ | $18.833_{\pm 0.217}$ | S
01342122+3045369 | 23.5883 | 30.7602 | 1.2511 | $18.207_{\pm 0.086}$ | $20.391_{\pm 0.159}$ | $19.101_{\pm 0.081}$ | $17.854_{\pm 0.097}$ | G
01342142+3046180 | 23.5891 | 30.7716 | 0.6772 | $20.039_{\pm 0.058}$ | $22.129_{\pm 0.016}$ | $20.885_{\pm 0.007}$ | $19.698_{\pm 0.058}$ | G
01342241+3044080 | 23.5933 | 30.7355 | 0.8842 | $19.570_{\pm 0.145}$ | $21.678_{\pm 0.029}$ | $20.345_{\pm 0.016}$ | $19.206_{\pm 0.145}$ | S
01342254+3049055 | 23.5938 | 30.8182 | 1.2970 | $18.214_{\pm 0.185}$ | $20.823_{\pm 0.619}$ | $19.201_{\pm 0.038}$ | $17.777_{\pm 0.243}$ | S
01342419+3047389 | 23.6007 | 30.7941 | 0.8859 | $19.420_{\pm 0.111}$ | $21.220_{\pm 0.102}$ | $20.113_{\pm 0.052}$ | $19.114_{\pm 0.115}$ | G
01342472+3044311 | 23.6028 | 30.7420 | 0.7713 | $19.964_{\pm 0.198}$ | $21.907_{\pm 0.019}$ | $20.703_{\pm 0.002}$ | $19.633_{\pm 0.198}$ | S
01342913+3043388 | 23.6213 | 30.7274 | 1.1693 | $18.850_{\pm 0.174}$ | $19.794_{\pm 0.562}$ | $18.908_{\pm 0.197}$ | $18.600_{\pm 0.231}$ | S
01343020+3044567 | 23.6257 | 30.7491 | 0.6031 | $20.372_{\pm 0.029}$ | $22.500_{\pm 0.106}$ | $21.224_{\pm 0.034}$ | $20.023_{\pm 0.041}$ | G
01342988+3047541 (∗) | 23.6244 | 30.7983 | 1.0902 | $19.015_{\pm 0.172}$ | $21.594_{\pm 0.318}$ | $19.990_{\pm 0.151}$ | $18.583_{\pm 0.194}$ | S
01335230+3045008 | 23.4678 | 30.7502 | 0.9642 | $19.338_{\pm 0.200}$ | $21.840_{\pm 0.293}$ | $20.273_{\pm 0.265}$ | $18.915_{\pm 0.224}$ | S
01334228+3037474 | 23.4260 | 30.6298 | 1.1931 | $18.734_{\pm 0.185}$ | $20.824_{\pm 0.025}$ | $19.411_{\pm 0.020}$ | $18.350_{\pm 0.185}$ | S
01341344+3033177 | 23.5558 | 30.5549 | 1.0603 | $18.898_{\pm 0.173}$ | $21.533_{\pm 0.028}$ | $20.041_{\pm 0.298}$ | $18.494_{\pm 0.189}$ | S
01333576+3033007 | 23.3988 | 30.5501 | 0.7801 | $19.765_{\pm 0.199}$ | $21.964_{\pm 0.352}$ | $20.697_{\pm 0.167}$ | $19.418_{\pm 0.222}$ | S
01333825+3042510 | 23.4092 | 30.7141 | 0.9057 | $19.378_{\pm 0.200}$ | $21.799_{\pm 0.069}$ | $20.407_{\pm 0.005}$ | $18.999_{\pm 0.201}$ | S

##  Appendix B Postage stamps of cluster Cepheids

Figure 11: Postage stamps in HST filters for the 10 Cluster Cepheids detected
by the crossmatch procedure (§5).

Figure 12: Continuation of Fig. 11.

Figure 13: Postage stamps in HST filters for the 13 additional cluster
Cepheids found by visual inspection. The position of suspected clusters is
shown in blue.

Figure 14: Continuation of Fig. 13.

Figure 15: Continuation of Fig. 13.