# A study of the Czernik 2 and NGC 7654 open clusters using CCD UBV
photometric and Gaia EDR3 data

B. Akbulut 11affiliationmark:  S. Ak 22affiliationmark:  T. Yontan
22affiliationmark:  S. Bilir 22affiliationmark:  T. Ak22affiliationmark:  T.
Banks33affiliationmark:  4 4affiliationmark:  E. Kaan Ulgen11affiliationmark:
E. Paunzen 55affiliationmark:

###### Abstract

We analysed the open clusters Czernik 2 and NGC 7654 using CCD UBV photometric
and Gaia Early Data Release 3 (EDR3) photometric and astrometric data.
Structural parameters of the two clusters were derived, including the physical
sizes of Czernik 2 being $r=5^{\prime}$ and NGC 7654 as $8^{\prime}$. We
calculated membership probabilities of stars based on their proper motion
components as released in the Gaia EDR3. To identify member stars of the
clusters, we used these membership probabilities taking into account location
and the impact of binarity on main-sequence stars. We used membership
probabilities higher than $P=0.5$ to identify 28 member stars for Czernik 2
and 369 for NGC 7654. The mean proper motion components
($\mu_{\alpha}\cos\delta$, $\mu_{\delta}$) of Czernik 2 were derived as
($-4.03\pm 0.04$,$-0.99\pm 0.05$) mas yr-1 and for NGC 7654 as ($-1.89\pm
0.03,-1.20\pm 0.03$) mas yr-1. We estimated colour-excesses and metallicities
separately using $(U-B)\times(B-V)$ two-colour diagrams to derive
homogeneously determined parameters. The derived $E(B-V)$ colour excess is
$0.46\pm 0.02$ mag for Czernik 2 and $0.57\pm 0.04$ mag for NGC 7654. [Fe/H]
metallicities were obtained for the first time for both clusters, $-0.08\pm
0.02$ dex for Czernik 2 and $-0.05\pm 0.01$ dex for NGC 7654. Keeping the
reddening and metallicity as constant quantities, we fitted PARSEC models
using $V\times(B-V)$ and $V\times(U-B)$ colour-magnitude diagrams, resulting
in estimated distance moduli and ages of the two clusters. We obtained the
distance modulus for Czernik 2 as $12.80\pm 0.07$ mag and for NGC 7654 as
$13.20\pm 0.16$ mag, which coincide with ages of $1.2\pm 0.2$ Gyr and $120\pm
20$ Myr, respectively. The distances to the clusters were calculated using the
Gaia EDR3 trigonometric parallaxes and compared with the literature. We found
good agreement between the distances obtained in this study and the
literature. Present day mass function slopes for both clusters are comparable
with the value of Salpeter (1955), being $X=-1.37\pm 0.24$ for Czernik 2 and
$X=-1.39\pm 0.19$ for NGC 7654.

††slugcomment: Not to appear in Nonlearned J., 45.00footnotetext: Istanbul
University, Institute of Graduate Studies in Science, Programme of Astronomy
and Space Sciences, 34116, Beyazıt, Istanbul, Turkey
Corresponding Author<EMAIL_ADDRESS>
00footnotetext: Istanbul University, Faculty of Science, Department of
Astronomy and Space Sciences, 34119 University, Istanbul, Turkey
00footnotetext: Istanbul University, Faculty of Science, Department of
Astronomy and Space Sciences, 34119 University, Istanbul, Turkey
00footnotetext: Istanbul University, Faculty of Science, Department of
Astronomy and Space Sciences, 34119 University, Istanbul, Turkey
00footnotetext: Istanbul University, Faculty of Science, Department of
Astronomy and Space Sciences, 34119 University, Istanbul, Turkey
00footnotetext: Nielsen, Data Science, 200 W Jackson Blvd #17, Chicago, IL
60606, USA
00footnotetext: Physics & Astronomy, Harper College, 1200 W Algonquin Rd,
Palatine, IL 60067, USA
00footnotetext: Istanbul University, Institute of Graduate Studies in Science,
Programme of Astronomy and Space Sciences, 34116, Beyazıt, Istanbul, Turkey
00footnotetext: Department of Theoretical Physics and Astrophysics, Masaryk
University, Kotlár̆ská 2, 611 37 Brno, Czech Republic

Keywords Galaxy: open cluster and associations: individual: Czernik 2 and NGC
7654, stars: Hertzsprung Russell (HR) diagram

##  1 Introduction

Table 1 : Basic parameters for Czernik 2 and NGC 7654 derived by this study
and compiled from the literature: columns present cluster names, colour
excesses $E(B-V$), distance moduli $\mu_{V}$, distances $d$, iron abundances
[Fe/H], age $t$, trigonometric distances from Gaia $\langle d_{Gaia}\rangle$
and proper motion components $\langle\mu_{\alpha}\cos\delta\rangle$,
$\langle\mu_{\delta}\rangle$.

Cluster | $E(B-V)$ | $\mu_{V}$ | $d$ | [Fe/H] | $t$ | $\langle d_{\rm Gaia}\rangle$ | $\langle\mu_{\alpha}\cos\delta\rangle$ | $\langle\mu_{\delta}\rangle$ | Ref
---|---|---|---|---|---|---|---|---|---
| (mag) | (mag) | (pc) | (dex) | (Myr) | (pc) | (mas yr-1) | (mas yr-1) |
Czernik 2 | 0.74$\pm$0.10 | 13.54$\pm$0.10 | 1775$\pm$80 | — | 100 | — | — | — | (1)
| — | — | — | — | — | 1907$\pm$377 | -4.06$\pm$0.01 | -0.90$\pm$0.01 | (2)
| — | — | — | — | 1590$\pm$95 | 2016$\pm$155 | -4.05$\pm$0.02 | -0.96$\pm$0.02 | (3)
| 0.45 | 12.80 | 1899 | — | 370 | 2020$\pm$202 | -4.06$\pm$0.01 | -0.90$\pm$0.01 | (4)
| 0.46$\pm$0.02 | 12.80$\pm$0.07 | 1883$\pm$63 | -0.08$\pm$0.02 | 1200$\pm$200 | 1919$\pm$189 | -4.03$\pm$0.04 | -0.99$\pm$0.05 | (5)
NGC 7654 | — | — | — | — | — | 1600$\pm$263 | -1.94$\pm$0.01 | -1.13$\pm$0.01 | (2)
| — | — | — | — | 380$\pm$23 | 1681$\pm$165 | -1.93$\pm$0.01 | -1.13$\pm$0.01 | (3)
| 0.60 | 12.94 | 1653 | — | 155 | 1678$\pm$130 | -1.94$\pm$0.01 | -1.13$\pm$0.01 | (4)
| 0.57$\pm$0.04 | 13.20$\pm$0.16 | 1935$\pm$146 | -0.05$\pm$0.01 | 120$\pm$20 | 1640$\pm$82 | -1.89$\pm$0.03 | -1.20$\pm$0.03 | (5)
| 0.64 | 12.75 | 1421 | — | 58 | — | — | — | (6)
| 0.66 | 13.04$\pm$0.20 | 1580$\pm$150 | — | — | — | — | — | (7)
| 0.70 | 13.98 | 2300 | — | 10 | — | — | — | (8)
| 0.57 | — | — | — | 96 | — | — | — | (9)
| 0.57 | 12.60 | 1470 | — | 35 | — | — | — | (10)
| 0.58$\pm$0.02 | 12.53$\pm$0.29 | 1400$\pm$200 | — | 158 | — | — | — | (11)
| 0.62$\pm$0.05 | 12.82$\pm$0.20 | 1510$\pm$145 | — | 100 | — | — | — | (12)
| 0.57 | 12.50$\pm$0.10 | 1380$\pm$70 | — | 160 | — | — | — | (13)
| 0.58$\pm$0.03 | 12.60$\pm$0.10 | 1400$\pm$200 | — | 60$\pm$10 | — | — | — | (14)
| 0.73$\pm$0.15 | 13.11$\pm$1.15 | 1480$\pm$470 | — | 10 | — | — | — | (15)

(1) Tadross (2009), (2) Cantat-Gaudin et al. (2018), (3) Liu & Pang (2019),
(4) Cantat-Gaudin et al. (2020), (5) This study, (6) Lundby (1946), (7)
Schmidt (1977), (8) Pfau (1980), (9) Kaltcheva (1990), (10) Battinelli,
Brandimarti, & Capuzzo–Dolcetta (1994), (11) Viskum (1997), (12) Choi et al.
(1999), (13) Pandey et al. (2001), (14) Bonatto & Bica (2006), (15)
Maciejewski & Niedzielski (2007)

Open star clusters are valuable tools to understand the structure and
evolution of the Galaxy. The stars which make up open clusters are formed by
the collapse of a molecular cloud under similar physical conditions and
subsequently loosely bound to each other by weak mutual gravitational forces.
For these reasons the distances, metallicities, and ages of cluster member
stars are similar to each other, although their masses differ from star to
star. These basic astrophysical properties of open star clusters are important
in understanding star formation and evolution. One of the important findings
obtained from the study of open cluster stars having similar ages and chemical
structures is that they reflect the physical properties of the cloud they
formed from. Cluster stars are a useful ‘tool’ to explore star formation and
evolution, stellar interactions, stellar nucleosynthesis, and the dynamical
evolution of the cluster to which they are gravitationally bound.

Open clusters are young stellar groups that consist of around a hundred to
several thousand stars. The member stars are gravitationally bound to each
other and share the same origin (as explained above). Such clusters represent
an important component of the spiral arms of the Milky Way. Also known as
Galactic clusters, these objects are rich in gas and dust understood to be the
remaining matter from the initial period of star formation. Morphologies vary
from cluster to cluster. Open clusters may appear as sparse star distributions
and irregularly mixed with field stars, or as crowded, dense, and relatively
spherical forms containing several thousand stars. The gravitational potential
effects of the Galactic disc can make it difficult to determine the structural
properties and internal dynamics of these objects, as well as influencing the
life time of the cluster as an observable entity. The level of stellar density
affects the determination of structural parameters for clusters: a small open
cluster can be dispersed by internal interactions such as the evolution of
stars, ejection and evaporation; whereas a large open cluster can be dispersed
by external interactions such as giant molecular clouds, the galactic tidal
field, and shock waves in the spiral arms (Carraro, 2006; Andersen &
Nordstrom, 2000). Since the differences in cluster size also affects their
dynamic evolution it is necessary to examine the stellar densities of
different radius intervals from the cluster centers to understand the
evolution process and determine sensitive parameters.

Photometry, astrometry, and spectroscopy are all methods applied to the
analysis of open clusters. An analysis depending on only one of these methods
can lead to results conflicting with an analysis based on another technique; a
better approach is to apply as many of these methods together (hereafter
called homogenisation) to a given cluster to derive more self-consistent
results, such as in this study. Improving our understanding of star formation,
stellar evolution, and the structure of the Galaxy requires determining
complete spatial, structural, kinematic and astrophysical parameters of many
known open clusters. Therefore data of open clusters have been compiled in
catalogues by working groups and continuously updated databases created.
Examples of databases and catalogues containing such photometric,
spectroscopic and astrometric data of open clusters include
WEBDA111https://webda.physics.muni.cz/ (Mermilliod, 1995),
DAML02222https://wilton.unifei.edu.br/ocdb/ (Dias et al., 2002),
SAI333http://ocl.sai.msu.ru/(Koposov et al., 2008),
MWSC444https://heasarc.gsfc.nasa.gov/W3Browse/all/mwsc.html (Kharchenko et
al., 2012) and UPK555https://sites.google.com/ushs.hs.kr/upk(Sim et al.,
2019).

In photometric analyses of open clusters, fundamental astrophysical parameters
such as colour excesses, metallicity, distance, and age of these objects are
determined by using colour-magnitude diagrams (CMDs) and two-colour diagrams
(TCDs). The relevant parameters are estimated through the comparison of
observational data with stellar models and theoretical isochrones on CMDs and
TCDs. Differences in the morphologies of open clusters affect the
distributions, densities, and placements of the main sequence, turn off, and
giant stars on CMDs and TCDs, which in turn can lead to interpretative issues.
Therefore to get reliable results one should take into account only cluster
member stars which have been accurately selected, ensure the quality and
homogeneity of data used, and employ uniform analysis methods across the
clusters being studied.

The European Space Agency’s (ESA) Gaia Early Data Release 3 (hereafter Gaia
EDR3, Gaia collaboration et al., 2021) supplies updated equatorial ($\alpha$,
$\delta$) and Galactic ($l$, $b$) coordinates, trigonometric parallaxes and
proper motions ($\varpi$, $\mu_{\alpha}\cos\delta$, $\mu_{\delta}$), and
photometric magnitudes ($G$, $G_{\rm BP}$ and $G_{\rm RP}$) of nearly 1.8
billion objects to high accuracies, based on 34 months of observational data.
The astrometric data ($\alpha$, $\delta$, $\varpi$, $\mu_{\alpha}\cos\delta$,
$\mu_{\delta}$) have a limiting magnitude of $G\sim 21$ mag. At $G\leq 20$ mag
the uncertainty in the $G$-band photometry ranges across 0.2-6 mmag. For the
sources brighter than $G\leq 15$ mag, the median error of trigonometric
parallaxes and proper motion components are up to 0.03 mas and 0.03 mas yr-1,
respectively, while these limits reach 1.3 mas and 1.4 mas yr-1 up to $G\sim
21$ mag, respectively (Gaia collaboration et al., 2021). Such accurate data
allow us to obtain membership probabilities of stars, mean proper motion
components and distance values for the open clusters with high precision.

In this study we determined the structural, astrophysical and astrometric
parameters of the Czernik 2 and NGC 7654 open clusters. We used Gaia EDR3
astrometric and photometric data together with CCD UBV photometric
observations, utilising independent methods, presenting detailed UBV and Gaia
results for both clusters. The literature summaries of the clusters are as
follows:

###  1.1 Czernik 2

Czernik 2 ($\alpha=00^{\rm h}43^{\rm m}49^{\rm s}$, $\delta=+60^{\rm
o}11^{\rm{}^{\prime}}49^{\rm{}^{\prime\prime}}$, $l=121^{\rm o}.98$,
$b=-2^{\rm o}.66$) was classified as Trumpler III1r (Ruprecht, 1966) and
presented in detail in the literature for the first time by Czernik (1966) as
an open cluster with an $r=10$ arcmin angular diameter. 128 stars were
reported for the cluster in the study. Although this cluster consists of stars
with apparent magnitudes to be considered faint, it is classified by Trumpler
(1930) as easily definable due to its relative greater stellar density
compared to the surrounding field stars. The first CCD photometric
observations were carried out by Phelps & Janes (1994). From an analysis of
the CMD, they stated that Czernik 2 did not clearly show characteristics of
being an open cluster. Maciejewski & Niedzielski (2008) used CCD BV and 2MASS
JHKS observational data to make a photometric and astrometric analysis of the
cluster, examining whether Czernik 2 was an open cluster. Analysing together
CMDs and vector-point diagrams (VPDs) constructed for different distances,
Maciejewski & Niedzielski (2008) could not define a central concentration in
the cluster region nor could they even separate the main sequence of cluster
from field stars on the CMDs. For these reasons they claimed that Czernik 2 is
not an open cluster. Tadross (2009) examined 60 poorly studied open clusters,
including Czernik 2, using JHKS near-IR 2MASS photometry (Skrutskie et al.,
2006). They obtained the diameter, reddening, distance and age of Czernik 2 as
$r=5.8$ arcmin, $E(B-V)=0.74\pm 0.10$ mag, $d=1775\pm 80$ pc and $t=100$ Myr
respectively. Cantat-Gaudin et al. (2018) made use of Gaia DR2 data alone to
obtain a list of members, mean distance and proper motion components of 1,229
open clusters including new discovered 60 open clusters. They calculated the
distance of Czernik 2 from Gaia DR2 trigonometric parallaxes as $d=1907\pm
377$ pc, and mean proper motion components as $\mu_{\alpha}\cos\delta=-4.06\pm
0.01$ and $\mu_{\delta}=-0.90\pm 0.01$ mas yr-1. Liu & Pang (2019) applied the
Star cluster Hunting Pipeline (SHiP) to identify star clusters in Gaia DR2
data, covering 2,443 open clusters. As a result of their analysis, they
determined the distance of the Czernik 2 cluster as $d=2016\pm 155$ pc, its
age as $t=1590\pm 95$ Myr, and the metal abundance as
$\rm\log(Z/Z_{\odot})=-1$. Moreover, they calculated mean proper motion
components of Czernik 2 as $\mu_{\alpha}\cos\delta=-4.05\pm 0.02$ and
$\mu_{\delta}=-0.96\pm 0.02$ mas yr-1. Cantat-Gaudin et al. (2020) put to use
the homogeneous Gaia DR2 photometry within the upper magnitude $G\leq 18$ mag
to derive the main parameters (distance, age, and interstellar extinction) of
1,867 clusters identified with Gaia DR2 astrometry. Determinations of the main
cluster parameters were done via isochrone fitting, which resulted in the
values of distance, age, and extinction of the Czernik 2 being estimated as
$d=1899$ pc, $t=370$ Myr, and $A_{\rm V}=1.41$ mag (which corresponds to
$E(B-V)=0.45$ mag), respectively. Cantat-Gaudin et al. (2020) also estimated a
mean distance $d=2020\pm 202$ pc to the cluster through use of the
trigonometric parallaxes of cluster member stars.

###  1.2 NGC 7654

NGC 7654, also known as M52, ($\alpha=23^{\rm h}24^{\rm m}47^{\rm s}$,
$\delta=+61^{\rm o}35^{\rm{}^{\prime}}36^{\rm{}^{\prime\prime}}$, $l=112^{\rm
o}.82$, $b=+0^{\rm o}.43$) is an open cluster classified as Trumpler I2r with
an angular diameter of 6′ and a distinct central stellar density. The cluster
has been studied by several researchers who used UBV, uvby$\beta$ and 2MASS
photometries (Lundby, 1946; Pesch, 1960; Schmidt, 1977; Pfau, 1980; Danford &
Thomas, 1981; Kaltcheva, 1990; Battinelli, Brandimarti, & Capuzzo–Dolcetta,
1994; Bonatto & Bica, 2006). These studies closely agree on the cluster’s
astrophysical parameters. The ranges of colour excess and age of NGC 7654 were
determined by these different authors as being within $0.57\leq E(B-V)\leq
0.70$ mag and $10\leq t\leq 96$ Myr respectively. The mean distance of the
cluster is given as $d=1450$ pc (Lundby, 1946; Pesch, 1960; Schmidt, 1977;
Choi et al., 1999). Pandey et al. (2001) used CCD UBVRIc photometry to examine
17,860 stars that are located in the cluster vicinity up to a limit of $V=20$
mag. They showed that the colour excess in the direction of cluster field
ranges over $0.46\leq E(B-V)\leq 0.80$ mag and calculated the mean distance of
NGC 7654 as $d=1380\pm 70$ pc. Using the isochrones of Bertelli et al. (1994),
Pandey et al. (2001) calculated the age of cluster as $t=160$ Myr and
determined the mass function slope as $X=-1.40\pm 0.07$ for the main-sequence
stars in the mass range $0.8<M/M_{\odot}<4.5$. NGC 7654 was also studied using
the Gaia DR2 data (Gaia collaboration et al., 2018): Cantat-Gaudin et al.
(2018) gave the distance of NGC 7654 as $d=1600\pm 263$ pc, and mean proper
motion components as $\mu_{\alpha}\cos\delta=-1.94\pm 0.01$ and
$\mu_{\delta}=-1.13\pm 0.01$ mas yr-1; Liu & Pang (2019) estimated distance,
age and metal abundance of NGC 7654 as $d=1681\pm 165$ pc, $t=380\pm 23$ Myr
and $\rm\log(Z/Z_{\odot})=-0.75$, respectively. In addition to this, they
measured the mean proper motion components of the cluster as
$\mu_{\alpha}\cos\delta=-1.93\pm 0.01$ and $\mu_{\delta}=-1.13\pm 0.01$ mas
yr-1. Cantat-Gaudin et al. (2020) derived a distance estimate based on
isochrone fitting as $d=1653$ pc, and from the trigonometric parallaxes of
member stars as $d=1678\pm 130$ pc. The age and interstellar extinction of the
cluster are given in the study as $t=155$ Myr and $A_{\rm V}=1.85$ mag (which
corresponds to $E(B-V)=0.60$ mag) respectively (see Table 1).

Fig. 1 : Inverse coloured (negative) $21^{\prime}\times 21^{\prime}$ $V$-band images of the two clusters: the total exposure time is 1200 sec for Czernik 2 (left panel) and 800 sec for NGC 7510 (right panel). In these printed images, East and North correspond to the left and up directions, respectively. Table 2 : Observation log: columns denote name of clusters, observation date, filters, exposure times (in seconds), and the number of exposures ($N$). Dates are day-month-year. |  | Filter/Exposure Time (s) $\times N$
---|---|---
Cluster | Obs. Date | $U$ | $B$ | $V$
Czernik 2 | 06.10.2018 | 90$\times$5, 900$\times$3 | 8$\times$5, 90$\times$5, 900$\times$2 | 5$\times$5, 60$\times$5, 600$\times$2
NGC 7654 | 07.10.2018 | 30$\times$1, 50$\times$5, 600$\times$3, 1200$\times$4 | 3$\times$5, 30$\times$5, 400$\times$4 | 1.5$\times$5, 15$\times$5, 200$\times$4

##  2 Observations

Photometric observations of Czernik 2 and NGC 7654 were made using Bessell UBV
filters with the 1-m T100 telescope located at TÜBİTAK National Observatory
(TUG) in Antalya, Turkey. The T100 is a Ritchey Chretien telescope, which was
equipped for this project with a Spectral Instruments 1100 Cryo-cooler cooled
CCD camera placed in the focal plane. The CCD camera is a 4K$\times$4K pixel
Fairchild model 486 BI (back-illuminated and with improved ultraviolet
sensitivity). The physical scale of the pixels making up the CCD chip is
15$\times$15 microns. The pixel scale of the telescope-camera system is
$0^{{}^{\prime\prime}}.31$ pixel-1, leading to a $21^{\prime}\times
21^{\prime}$ field of view on the sky. The CCD gain is 0.57 electrons ADU-1
with a readout noise of 4.11 electrons. The “bias” level of CCD camera is
around 500 ADU, the well capacity is about 142,900 electrons, the dark current
is 0.0002 e- per pixel per second, and the CCD can be cooled down to
$-100^{\rm o}$ C reducing the dark current to the reported level. The
telescope-camera system is advantageous in that it has a wide field of view
which allows observation of many open star clusters in a single exposure and
has high quantum efficiency at both long and short wavelengths. During the
observations we used different exposure times in UBV filters so as to obtain
images of bright and faint stars in the regions of clusters with high signal
to noise ($S/N$) ratio and to avoid saturation of CCD pixels. See Fig. 1 for
example $V$-band images with inverted colours of the two clusters. See Table 2
for the observation log. Photometric calibrations were based on Landolt (2009)
standard stars in a total of 14 areas whose observation details are given in
Table 3. Frames were stacked. We utilised IRAF’s standard CCD calibration
processes, and applied PyRAF and astrometry.net routines for astrometric
corrections to all cluster images. We computed instrumental magnitudes of
standard stars via aperture photometry using the relevant IRAF packages. Then
we applied multiple linear regression to these magnitudes to obtain
photometric extinction and transformation coefficients for the two nights of
observation (see Table 4). Instrumental magnitudes of the stars in cluster
direction were measured with PSF Extractor (PSFEx) routines (Bertin & Arnouts,
1996). Aperture corrections were then applied to these magnitudes before
finally transforming instrumental magnitudes to standard brightnesses in the
Johnson & Morgan (1953) photometric system via the transformation equations
given by Janes & Hoq (2011).

Table 3 : Information on the observations of standard stars from selected Landolt (2009) fields. The columns are the observation date, star field name as from Landolt, the number of standard stars ($N_{\rm st}$) observed in a given field, the number of pointings to each field ($N_{\rm obs}$, i.e., observations), and the airmass range the fields (on a given night) were observed over ($X$). Dates are day-month-year. Date | Star Field | $N_{\rm st}$ | $N_{\rm obs}$ | $X$
---|---|---|---|---
| SA92 | 6 | 2 |
| SA93 | 4 | 1 |
| SA95 | 9 | 1 |
06.10.2018 | SA96 | 2 | 1 | 1.230 - 2.017
| SA98 | 19 | 1 |
| SA100 | 5 | 1 |
| SA114 | 5 | 1 |
| SA93 | 4 | 1 |
| SA94 | 2 | 2 |
| SA96 | 2 | 1 |
| SA97 | 2 | 2 |
| SA98 | 19 | 1 |
07.10.2018 | SA99 | 3 | 1 | 1.242 - 1.872
| SA109 | 2 | 1 |
| SA110 | 4 | 2 |
| SA112 | 6 | 2 |
| SA113 | 15 | 1 |
| SA114 | 5 | 1 |
Table 4 : Transformation and extinction coefficients derived for the two observation nights: $k$ and $k^{\prime}$ are the primary and secondary extinction coefficients, respectively. $\alpha$ and $C$ are the transformation coefficients. Dates are day-month-year. Filter/Colour index | Obs. Date | $k$ | $k^{\prime}$ | $\alpha$ | $C$
---|---|---|---|---|---
$U$ | 06.10.2018 | 0.5795$\pm$0.0624 | -0.0747$\pm$0.0793 | — | —
$B$ |  | 0.2620$\pm$0.0467 | -0.0303$\pm$0.0567 | 0.9775$\pm$0.0842 | 1.3908$\pm$0.0714
$V$ |  | 0.1482$\pm$0.0189 | — | — | —
$U-B$ |  | — | — | 0.9881$\pm$0.1196 | 3.5383$\pm$0.0968
$B-V$ |  | — | — | 0.1009$\pm$0.0106 | 1.4750$\pm$0.0312
$U$ | 07.10.2018 | 0.2154$\pm$0.0588 | +0.1071$\pm$0.0766 | — | —
$B$ |  | 0.1651$\pm$0.0458 | +0.0616$\pm$0.0520 | 0.8214$\pm$0.0821 | 1.5979$\pm$0.0713
$V$ |  | 0.1212$\pm$0.0164 | — | — | —
$U-B$ |  | — | — | 0.7120$\pm$0.1161 | 4.1107$\pm$0.0883
$B-V$ |  | — | — | 0.0773$\pm$0.0069 | 1.5709$\pm$0.0253

##  3 Data Analysis

###  3.1 Photometric Data

The photometric catalogues of Czernik 2 and NGC 7654 list all identified stars
located in the cluster areas, providing information such as positions
($\alpha$, $\delta$), apparent $V$ magnitude, $U-B$ and $B-V$ colours indices,
proper motion components ($\mu_{\alpha}\cos\delta$, $\mu_{\delta}$),
trigonometric parallaxes ($\varpi$) from Gaia EDR3, and membership
probabilities ($P$). Photometric errors for Johnson ($V$, $U-B$, $B-V$) and
Gaia EDR3 ($G$, $G_{\rm BP}-G_{\rm RP}$) magnitudes and colour indices are
taken as internal errors. Table 5 lists the mean photometric errors in
consecutive $V$-magnitude intervals. At $V=21$ mag the mean internal errors of
photometric measurements for Czernik 2 reach 0.04 for the $V$-band magnitude
and 0.07 mag for the $B-V$ colour index. For stars brighter than $V=20$ mag
the errors reach 0.06 mag for the $U-B$ colour index. Similarly, for NGC 7654
these mean internal errors are about 0.04, 0.07, and 0.07 mag for $V$
magnitude, $U-B$, and $B-V$ colour indices, respectively (see Table 5). The
mean photometric errors are smaller than 0.003 and 0.04 mag in $G$ -band and
$G_{\rm BP}-G_{\rm RP}$ colour index, respectively, for the stars brighter
than $V=20$ mag for Czernik 2. Similarly, these errors reach up to 0.003 mag
in $G$-band and 0.20 mag in the $G_{\rm BP}-G_{\rm RP}$ colour index for the
stars within $V=20$ mag in NGC 7654.

Table 5 : The mean internal photometric errors ($\sigma_{\rm V}$, $\sigma_{\rm U-B}$, $\sigma_{\rm B-V}$, $\sigma_{\rm G}$, $\sigma_{G_{\rm BP}-G_{\rm RP}}$) and number of measured stars ($N$) in the corresponding $V$ magnitude interval for each cluster. Czernik 2 | NGC 7654
---|---
$V$ | $N$ | $\sigma_{\rm V}$ | $\sigma_{\rm U-B}$ | $\sigma_{\rm B-V}$ | $\sigma_{\rm G}$ | $\sigma_{G_{\rm BP}-G_{\rm RP}}$ | $N$ | $\sigma_{\rm V}$ | $\sigma_{\rm U-B}$ | $\sigma_{\rm B-V}$ | $\sigma_{\rm G}$ | $\sigma_{G_{\rm BP}-G_{\rm RP}}$
( 7, 12] | 14 | 0.001 | 0.002 | 0.001 | 0.004 | 0.006 | 20 | 0.001 | 0.001 | 0.001 | 0.003 | 0.005
(12, 14] | 58 | 0.002 | 0.005 | 0.002 | 0.003 | 0.005 | 101 | 0.001 | 0.002 | 0.001 | 0.003 | 0.009
(14, 15] | 111 | 0.002 | 0.006 | 0.002 | 0.003 | 0.005 | 131 | 0.001 | 0.003 | 0.002 | 0.003 | 0.010
(15, 16] | 163 | 0.001 | 0.008 | 0.002 | 0.003 | 0.006 | 184 | 0.002 | 0.007 | 0.003 | 0.003 | 0.011
(16, 17] | 313 | 0.002 | 0.014 | 0.004 | 0.003 | 0.006 | 366 | 0.003 | 0.013 | 0.005 | 0.003 | 0.011
(17, 18] | 587 | 0.004 | 0.028 | 0.007 | 0.003 | 0.010 | 584 | 0.006 | 0.025 | 0.009 | 0.003 | 0.186
(18, 19] | 965 | 0.008 | 0.046 | 0.014 | 0.003 | 0.020 | 1023 | 0.012 | 0.047 | 0.020 | 0.003 | 0.219
(19, 20] | 1435 | 0.018 | 0.056 | 0.029 | 0.003 | 0.040 | 1262 | 0.026 | 0.070 | 0.044 | 0.003 | 0.201
(20, 21] | 1431 | 0.041 | — | 0.070 | 0.004 | 0.082 | 176 | 0.043 | 0.073 | 0.072 | 0.004 | 0.060

Fig. 2 : Interval $V$ and $G$-band magnitude histograms of Czernik 2 (a, b)
and NGC 7654 (c, d): The arrows show the faint limiting apparent magnitudes in
$V$ and $G$-bands. Black lines indicate the star counts based on the stars
detected in the study, while red lines are star counts based on the stars
taken from Gaia EDR3 for the same cluster regions.

Fig. 3 : Comparisons of observational magnitude and colours with those
calculated from Phelps & Janes (1994) (a-b) and Maciejewski & Niedzielski
(2008) (c-d) for Czernik 2. The mean differences and their $\pm 1\sigma$
dispersions are represented with solid and dashed red lines, respectively.
Yellow solid lines indicate the trends of moving average of data

.

The exposure times used during the observations contribute toward (together
with crowding) the detection limit for a star in the observed star field. In
order to obtain this completeness limit, we constructed $V$ and $G$ magnitude
histograms for Czernik 2 and NGC 7654 in which the number of stars is a
function of $V$ and $G$ magnitude intervals (see Fig. 2). To check photometric
completeness limits we took the stars for the region of each cluster using
Gaia EDR3 data. Taking into account central equatorial coordinates (Cantat-
Gaudin et al., 2020) and using the same area as the CCD images
($21^{\prime}\times 21^{\prime}$) of two clusters, we identified the stars
within $8<V<24$ mag range for each cluster region. Using these stars we
constructed histograms of $G$-magnitude intervals, displayed as Figures 2b and
2d. In Figure 2 the solid black lines denote the observational values, while
the red lines (Figures 2b and 2d) represent the stars from Gaia EDR3. We can
clearly see from Figures 2b and 2d that the two distributions are in good
agreement before the completeness limit is met for our observational data.
Considering the turnover points from which the number of detected stars drops
with increasing magnitude, we adopted completeness limits as being $V=20$ mag
for both clusters. This limit corresponds to $G=20$ mag for Czernik 2 and
$G=19$ mag for NGC 7654 where the number of Gaia EDR3 stars starts to overrun
the number of observed ones. The use of different telescope-detector
combinations in observations affects the detected number of stars due to
changes in telescope detection limits, particularly at fainter magnitudes.
This could result in detecting more stars at fainter magnitudes ($G>20$) with
Gaia observations in the two cluster areas (Figs. 2b & 2d). It can be seen in
Fig. 2 that incompleteness (of stellar detections) becomes significant beyond
the completeness limits. Therefore further analyses did not consider stars
fainter than $V=20$ mag for both Czernik 2 and NGC 7654.

We compared our photometry with those from previous studies employing the same
filter system. We reviewed CCD UBVI data for NGC 7654 given by Pandey et al.
(2001) and CCD BV data for Czernik 2 presented in two studies (Phelps & Janes,
1994; Maciejewski & Niedzielski, 2008). 724 and 1,589 of the 5,077 stars
detected in the direction of Czernik 2 were matched (via equatorial
coordinates) with the stars in catalogues of Phelps & Janes (1994) and
Maciejewski & Niedzielski (2008), respectively. We plotted apparent $V$
magnitude versus differences between our $V$ and $(B-V)$ measurements and
those given in the two studies (see Fig. 3). As seen in the figure, there is
no bias with Phelps & Janes (1994) and up to $V=17$ mag with Maciejewski &
Niedzielski (2008). The mean difference and standard deviation of $V$
magnitudes and the $(B-V)$ colour index are derived as $\langle\Delta
V\rangle=-0.003$ mag, $\langle\sigma_{\rm V}\rangle=0.043$ mag,
$\langle\Delta(B-V)\rangle=0.012$ mag, and $\sigma_{\rm B-V}=0.044$ mag for
the comparison with Phelps & Janes (1994). Comparison with Maciejewski &
Niedzielski (2008)’s data led to $\langle\Delta V\rangle=-0.014$ mag,
$\langle\sigma_{\rm V}\rangle=0.152$ mag, and
$\langle\Delta(B-V)\rangle=-0.076$ mag, $\sigma_{\rm B-V}=0.155$ mag. We also
calculated a moving average from the comparison with each data to understand
the trend of mean differences, presented as yellow solid lines in Fig. 3. For
the comparison with Phelps & Janes (1994), the moving average trends are near
zero in $V$ magnitude and $B-V$ colour index (Fig. 3a, b). This trend for the
comparison of $V$ magnitudes is nearly zero up to $V=18$ mag (Fig. 3c).
Similarly, the moving average trend in the $(B-V)$ colour index is nearly zero
up to $V=15.5$ mag, while it goes towards nearly 0.2 mag at the fainter
magnitudes (Fig. 3d). Consequently, the mean differences in apparent magnitude
and colour indices, together with their small standard deviations, show that
our photometric data are well-matched with those of Phelps & Janes (1994).

We cross-matched our catalogue with the catalogue of Pandey et al. (2001) to
compare the photometric measurements. 3,365 of the equatorial coordinate data
of 3,847 stars detected in our NGC 7654 area were matched with the stars in
the catalogue presented by Pandey et al. (2001). Plots of apparent $V$
magnitude versus differences between our $V$, $(U-B)$, $(B-V)$ measurements
and those given by Pandey et al. (2001) are shown in Fig. 4. There is no bias
up to $V=18$ mag with the study of Pandey et al. (2001). A small trend can be
seen for the magnitude fainter than $V=18$. Based on this comparison the mean
difference and standard deviation in $V$ are derived as $\langle\Delta
V\rangle=0.046$ mag and $\langle\sigma_{\rm V}\rangle=0.148$ mag,
respectively. While the mean differences up to $V=13.5$ mag in the $(B-V)$
colour index are about zero, these differences increase towards approximately
1.0 mag at the fainter magnitudes (Fig. 4b). The mean difference and standard
deviation of the $(B-V)$ colour index are derived as
$\langle\Delta(B-V)\rangle=0.059$ mag and $\sigma_{\rm B-V}=0.215$ mag,
respectively. Fig. 4c gives a comparison of the $(U-B)$ colour index. The mean
difference and standard deviation of the comparison of the $(U-B)$ colour
index are derived as $\langle\Delta(U-B)\rangle=0.101$ mag and $\sigma_{\rm
U-B}=0.269$ mag, respectively. Moving average trends are about zero for each
comparison. Although the mean errors of the $(U-B)$ colour index are small,
their standard deviations increase towards the fainter magnitudes as expected.
Overall, our photometric data are in good agreement with the study of Pandey
et al. (2001).

Fig. 4 : Comparison of photometric measurements in this study with data of
Pandey et al. (2001) (a-c) as a function of $V$ magnitudes for NGC 7654. The
mean differences and their $\pm 1\sigma$ dispersions are represented with
solid and dashed red lines, respectively. Yellow solid lines indicate the
trends of moving average of data.
Fig. 5 : The radial density profiles for the Czernik 2 (a) and NGC 7654 (b)
clusters. Errors were derived using equation of $1/\sqrt{N}$, where $N$
represents the number of stars used in the density estimation. King profile
and background stellar density are presented with solid red curve and blue
dashed line, respectively.

###  3.2 Spatial Structure of the Clusters

The morphologies of Czernik 2 and NGC 7654 are different to each other: NGC
7654 shows a spherical structure with a heavy central concentration, whereas
the irregular shape and sparse stellar distribution of Czernik 2 make it
difficult to separate from the surrounding star field. Taking central
coordinates from the study of Cantat-Gaudin et al. (2020) and utilising the
radial density profile (RDP) as defined by King (1962), we determined the
structural parameters of two clusters. We divided a given cluster’s area into
a series of concentric circles of one arcmin centred on the cluster centre.
Additionally we estimated the stellar density ($\rho$) by taking the ratio of
the number of stars in a given annulus to the area of that annulus. For each
cluster we plotted density versus radius, fitting the RDP (King, 1962) using a
$\chi^{2}$ minimisation method. King (1962) described the central density
$\rho(r)$ of a cluster as $\rho(r)=f_{bg}+[f_{0}/(1+(r/r_{c})^{2})]$ where $r$
and $r_{c}$ are a given radius from the cluster centre and core radius
respectively, $f_{0}$ and $f_{bg}$ are the central and background densities.
The best fits to the RDPs of Czernik 2 and NGC 7654 are shown in Fig. 5. The
solid red curves denote the King profiles and the blue dashed lines represent
the background densities. The levels of the modeled number density profiles
equals the background densities $r=5$ arcmin from the centre of Czernik 2 and
$r=8$ arcmin from the centre of NGC 7654 (see Fig. 5). We considered the stars
within these radius values for further analysis. These adopted radius levels
for the two clusters are in good agreement with the study of Cantat-Gaudin et
al. (2018), who presented the clusters’ radii ($r_{50}$) which contain half of
the stars in the clusters. Applying the formulae of King (1962), we found the
central stellar densities to be $f_{0}=1.426~{}\pm 0.663$ and
$f_{0}=3.024~{}\pm 0.274$ stars arcmin-2, background stellar densities to be
$f_{bg}=0.424~{}\pm 0.266$ and $f_{bg}=1.680~{}\pm 0.257$ stars arcmin-2 and
core radii to be $r_{c}=0.988\pm 0.879$ and $r_{c}=4.663\pm 0.699$ arcmin for
Czernik 2 and NGC 7654, respectively (Table 6).

Table 6 : Results of RDP fitting for the two clusters: $f_{0}$, $f_{bg}$ and $r_{c}$ represent central stellar density, background stellar densities and the core radius, respectively. Cluster | $f_{0}$ | $f_{bg}$ | $r_{c}$
---|---|---|---
| (stars arcmin-2) | (stars arcmin-2) | (arcmin)
Czernik 2 | 1.426$\pm$0.663 | 0.424$\pm$0.266 | 0.988$\pm$0.879
NGC 7654 | 3.024$\pm$0.274 | 1.680$\pm$0.257 | 4.663$\pm$0.699

###  3.3 CMDs and Membership Probabilities of Stars

It is important in cluster studies to estimate the field star contamination.
In order to get reliable parameters for an open cluster, it is necessary to
identify physical member stars of it accurately. Sharing the same origin
allows member stars of the cluster to have co-movements together in the sky,
concentrated in the proper motion space unlike the neighbouring field stars.
These features make it possible to separate cluster members from field star
contamination using their proper motion values (Angelo et al., 2019; Angelo,
Santos, & Corradi, 2020).

Fig. 6 : $V\times(B-V)$ and $G\times(G_{\rm BP}-G_{\rm RP})$ CMDs of Czernik 2
(a, b) and NGC 7654 (c, d). The blue dot-dashed lines represent the ZAMS (Sung
et al., 2013) including binary star effect. The membership probabilities of
stars that lie within the fitted ZAMS are shown with different colours, these
member stars are located within $r\leq 5^{\prime}$ and $r\leq 8^{\prime}$ of
the cluster centres calculated for Czernik 2 and NGC 7654, respectively. Grey
dots indicate field stars.

In this study, we used Gaia EDR3 astrometric data (Gaia collaboration et al.,
2021) and utilised the UPMASK (Unsupervised Photometric Membership Assignment
in Stellar Clusters; Krone-Martins & Moitinho, 2014) method to calculate
cluster membership probabilities of the observed stars. UPMASK describes an
open star cluster as a group of stars with a common origin taking into account
their astrometric data (proper motion components and trigonometric parallaxes)
and defines the field stars as spatially ‘scattered’ objects with different
origins. This assumption by UPMASK is key for the method to determine the
membership probabilities of stars and to separate cluster members from field
stars. The method uses a machine-learning algorithm, $k$-means clustering, to
find spatially concentrated groups and identify member stars of the cluster.
Here, $k$ is the number of clusters and is not adjusted directly by the user.
The number of the clusters is identified by dividing the total number of stars
by the number of stars per cluster, converting the result to an integer in
each iteration (Krone-Martins & Moitinho, 2014). As mentioned by Krone-Martins
& Moitinho (2014) and Cantat-Gaudin et al. (2018), the best result from this
technique is achieved when the integer value is within 10 to 25. Recently,
Cantat-Gaudin & Anders (2020) used Gaia DR2 astrometric data together with the
UPMASK methodology on 1,481 open clusters, successfully calculating stellar
membership probabilities in these clusters as a continuation of the study of
Cantat-Gaudin et al. (2018). In the studies of Cantat-Gaudin et al. (2018) and
Cantat-Gaudin & Anders (2020), the mean $k$-means values across the clusters
were both 10.

In this study, we utilised the UPMASK method according to a five-dimensional
astrometric space ($\alpha$, $\delta$, $\varpi$, $\mu_{\alpha}\cos\delta$,
$\mu_{\delta}$) and applied it to calculate the membership probabilities ($P$)
of stars for the two clusters. We scaled these five observable measures to
unit variance and ran for each cluster 100 iterations assessing cluster
membership. The membership probability is defined as the frequency with which
a star is marked as a part of a clustered group. Best results were reached
when $k$ was set to 15 for both clusters. Stars with membership probabilities
$P\geq 0.5$ were chosen as the most likely member of clusters. We found 154
and 530 potential member stars for Czernik 2 and NGC 7654, respectively, for
subsequent analysis. Examining the two open clusters, Cantat-Gaudin et al.
(2018) and Liu & Pang (2019) give the number of stars with cluster membership
probabilities $P\geq 0.5$, as 102 and 95 for Czernik 2, and 1,063 and 1,061
for NGC 7654 respectively. Considering both studies, Cantat-Gaudin et al.
(2018) used the UPMASK method with Gaia DR2 data, while Liu & Pang (2019)
applied the SHiP method considering Gaia DR2 astrometric data in the
calculation of the cluster membership probabilities of the stars. The reason
of the differences between these studies and the current one is that in this
study we used current Gaia EDR3 astrometric data during membership
determinations, while Cantat-Gaudin et al. (2018) and Liu & Pang (2019) took
into account the astrometric data of Gaia DR2. It is worth noting that while
UPMASK determines the membership probabilities of stars, it assumes that stars
are single and therefore cannot be used to identify probable binary stars. For
this reason, beside the membership probability criteria, membership selection
additional information was taken into account to account for binaries. We
plotted the $V\times(B-V)$ CMDs for two clusters and fitted the zero age main-
sequence (ZAMS, Sung et al., 2013) to these diagrams. The lower base of the
cluster main-sequence was determined by fitting the ZAMS by eye according to
the most probable member stars ($P\geq 0.5$), then to make it possible to
select the most probable binary stars we shifted the fitted ZAMS 0.75 mag to
the brighter magnitudes (Fig. 6). To determine basic parameters of the two
clusters, we considered the most probable member stars to be those that lie
between the fitted ZAMS lines and are located within the cluster limiting
(outer) radii estimated in the study (mentioned in Section 3.2). Thus, we
arrived at 28 and 369 “member” stars for Czernik 2 and NGC 7654, respectively.
Probability histograms for the stars (by cluster) are shown in Fig. 7. Vector-
point diagrams (VPDs) were prepared to visualise the distributions of most
probable member stars in each cluster (see Fig. 8). Additionally, we estimated
mean proper motion components of selected member stars, showing the
intersection of these values on Fig. 8 with dashed lines.

Fig. 7 : The histogram of membership probabilities versus number of stars for
Czernik 2 (a) and NGC 7654 (b). The green coloured shading denotes the stars
that lie within the main-sequence band and effective cluster radii.
Fig. 8 : Vector points diagrams of Czernik 2 (a) and NGC 7654 (b). Coloured
dots identify the membership probabilities of the cluster stars according to
colour scale shown on the right. Dashed lines are the intersection of the mean
proper motion values. The colour scale shows the membership probabilities of
the most likely cluster members. Fig. 9 : $(U-B)\times(B-V)$ two-colour
diagrams of the most probable member main-sequence stars in the regions of
Czernik 2 (a) and NGC 7654 (b). Red dashed and green solid curves represent
the reddened ZAMS given by Sung et al. (2013) and $\pm 1\sigma$ standard
deviations, respectively.

##  4 Astrophysical Parameters of the Clusters

In this section we summarised the techniques applied during the cluster
analyses (for detailed analyses on methods used see Bilir et al. (2006, 2010,
2016); Yontan et al. (2015, 2019, 2021); Bostancı et al. (2015, 2018); Ak et
al. (2016); Banks et al. (2020)). We estimated reddening and photometric
metallicities from TCDs on an individual basis, then keeping these parameters
as a constants we fitted theoretical isochrones on CMDs to determine distance
moduli and age simultaneously.

###  4.1 Reddening

In order to estimate reddening through the regions of two clusters, we
selected the most probable main-sequence members whose $V$ magnitude ranges
are within $12.5\leq V\leq 18.5$ and $11.5\leq V\leq 19$ mag for Czernik 2 and
NGC 7654, respectively. Using these sample stars, we constructed
$(U-B)\times(B-V)$ diagrams and fitted an intrinsic ZAMS (Sung et al., 2013)
on the observational data. The fitting procedure made steps of 0.001 mag
utilising the $\chi^{2}$ minimisation technique until improvement was within a
predefined limit. During the fitting process we adopted the slope of the
reddening $E(U-B)/E(B-V)$ as 0.72 (Johnson & Morgan, 1953) for both clusters.
This procedure provided mean values of reddening as $E(B-V)=0.46\pm 0.02$ and
$E(B-V)=0.57\pm 0.04$ mag for Czernik 2 and NGC 7654, respectively. Errors for
the reddening are $\pm 1\sigma$ deviations. Best fit results with TCDs are
shown in Fig. 9.

Cantat-Gaudin et al. (2020) calculated the $V$ band absorptions of Czernik 2
and NGC 7654 open clusters as 1.41 and 1.85 mag, respectively. With the ratio
of the $V$ band absorptions to the value of 3.1 belonging to the normal
interstellar medium, the $E(B-V)$ colour excess for Czernik 2 and NGC 7654
were determined to be 0.45 and 0.60 mag, respectively. Considering the colour
excess values of 0.46 and 0.57 mag calculated for Czernik 2 and NGC 7654 in
this study, it seems that our results are consistent with those of Cantat-
Gaudin et al. (2020).

Fig. 10 : TCDs (upper panel) and the distributions of normalised
$\delta_{0.6}$ (lower panel) for Czernik 2 (a) and NGC 7654 (b). The solid
blue lines in the upper and lower panels represent the main-sequence of Hyades
and Gaussian models which were fitted to the histograms, respectively.

###  4.2 Estimation of Metallicities

We estimated photometric metallicities for the two clusters using their
observational CCD UBV photometric data. The metallicities of Czernik 2 and NGC
7654 have not been estimated before. Applying the calibration given by Karaali
et al. (2011) on de-reddened TCDs of the two clusters, the metallicities were
derived using the normalised UV excesses ($\delta(U-B)_{0.6}$) of the most
probable member ($P\geq 0.5$) F-G type main-sequence stars within the colour
index of $0.3\leq(B-V)_{0}\leq 0.6$ mag (Eker et al., 2018, 2020).
$(U-B)_{0}\times(B-V)_{0}$ TCDs were plotted using F-G type cluster member
stars and compared to the main-sequence of Hyades cluster. $(U-B)_{0}$
differences between the cluster and Hyades stars corresponding the same
$(U-B)_{0}$ colour indices were calculated. These difference values are
defined as UV excesses for each star, which can be written as
$\delta=(U-B)_{\rm 0,H}-(U-B)_{\rm 0,S}$ ($H$ and $S$ are the Hyades and
cluster star, respectively, corresponding to an $(B-V)_{0}$ colour index, see
also Karaali et al., 2003). We normalised each UV excess ($\delta$) based on
$(B-V)_{0}=0.6$ mag (i.e. $\delta_{\rm 0.6}$) and plotted the histogram of
$\delta_{0.6}$ values. The mean $\delta_{0.6}$ was calculated by fitting a
Gaussian to the distribution. We considered the mean $\delta_{0.6}$ value
which represents the peak of optimal, fitted Gaussian. Based on
$\delta_{0.6}$, we calculated the metallicities of each cluster using the
equation improved by Karaali et al. (2011) which is given as follow:

${\rm[Fe/H]}=0.105-3.557\times\delta_{0.6}-14.316\times\delta_{0.6}^{2}.$ (1)

TCDs and histograms of normalised $\delta_{0.6}$ are shown in Fig. 10. The
statistical uncertainty is the $\pm 1\sigma$ width of the optimal Gaussian
model. We derived the metallicity for Czernik 2 as $\rm[Fe/H]=-0.08\pm 0.02$
dex from five member stars and for NGC 7654 as $\rm[Fe/H]=-0.05\pm 0.01$ dex
from 47 member stars.

The fractional abundance by mass, known as $Z$, is required to determine the
age of a cluster. Bovy666
https://github.com/jobovy/isodist/blob/master/isodist/ Isochrone.py
analytically denoted an equation which transforms the [Fe/H] metallicities to
the mass fraction $Z$, using PARSEC777PAdova and TRieste Stellar Evolution
Code models:

$Z_{\rm X}={10^{{\rm[Fe/H]}+\log\left(\frac{Z_{\odot}}{1-0.248-2.78\times
Z_{\odot}}\right)}},$ (2)

$Z=\frac{(Z_{X}-0.2485\times Z_{\rm X})}{(2.78\times Z_{\rm X}+1)}.$ (3)

$Z_{\rm X}$ and $Z$ represent the intermediate operation function and all
elements heavier than helium, respectively. $Z_{\odot}$ denotes the solar mass
fraction, which we adopted as 0.0152 (Bressan et al., 2012). Using the
equations given above, we measured $Z=0.013$ and $Z=0.014$ for Czernik 2 and
NGC 7654, respectively.

###  4.3 Distance Moduli and Age Estimation

We derived distance moduli and age estimation (for each cluster)
simultaneously. We used classical isochro-nes fitting methods (e.g. Phelps &
Janes, 1994; Sharma et al., 2006, 2017). We fitted PARSEC isochrones (Bressan
et al., 2012) to the observational $V\times(U-B)$, $V\times(B-V)$, and
$G\times(G_{\rm BP}-G_{\rm RP}$) CMDs consisting of the most probable member
stars with $P\geq 0.5$.

In the CMD morphology of the two clusters, the turn-off and main-sequence
stars are significant, in addition, NGC 7654 has a star (BD +60 2532) in an
advanced evolutionary stage. This star is important in estimating the age of
NGC 7654. For this reason, it is necessary to examine the cluster membership
status of BD +60 2532 by considering its different parameters. According to
astrometric data of BD+60 2532 in the Gaia EDR3 catalogue, its membership
probability was determined to be $P=0.9$ and its distance to the cluster
center as $r=3.81$ arcmin. In this study, the effective radius of the cluster
is also determined to be $r=8$ arcmin. Moreover, Luck (2014) gives the
atmosphere model parameters of the BD+60 2532 as $T_{\rm eff}=6114$ K, $\log
g=3.75$ (cgs) and [Fe/H] = $-0.06$ dex. In our study we obtained the
metallicity of the NGC 7654 as [Fe/H] = $-0.05\pm 0.01$ dex, which is very
well compatible with the result of Luck (2014). These results show that the
BD+60 2532 is most probably a member star of the cluster.

PARSEC models (Bressan et al., 2012) were selected with reference to the mass
fractions ($Z$) measured for each cluster. The fitting procedure was done by
eye keeping reddening and metallicity as constants. To obtain parameters
carefully, we fitted PARSEC models considering main-sequence, turn-off and
giant likely member stars in the cluster CMDs. To determine the errors in
cluster ages, we fitted two more isochrones on CMDs at closer values (one
younger and the other older than the best fit age) to the adopted cluster ages
by considering the distribution of likely member stars in CMDs. Besides the
best fit isochrones which give the expected ages of the clusters, these other
closing fitting isochones provided insight into the likely errors in estimated
ages. In addition to this, we calculated errors in distance moduli and
distances using the relations presented in Carraro et al. (2017). When
utilising the isochrone fitting procedure, we used reddening values, $E(B-V)$,
determined earlier (see Sec. 4.1) for $V\times(U-B)$ and $V\times(B-V)$ CMDs.
While estimating ages for the $G\times(G_{\rm BP}-G_{\rm RP})$ CMDs, the
PARSEC isochrones were reddened considering the equation of $E(G_{\rm
BP}-G_{\rm RP})=1.2803\times E(B-V)$ and extinction of $A_{\rm G}/A_{\rm
V}=0.789$ whose the coefficients were inferred from the equations presented by
Wang & Chen (2019). The best fit $Z=0.013$ isochrone gives an age $t=1.2\pm
0.2$ Gyr and a distance modulus $\mu=12.80\pm 0.07$ mag for Czernik 2. The
best fit $Z=0.014$ isochrone gives $t=120\pm 20$ Myr and a distance modulus
$\mu=13.20\pm 0.16$ mag for NGC 7654. The distance modulus values correspond
to the cluster distances being $d=1883\pm 63$ pc and $d=1935\pm 146$ pc for
Czernik 2 and NGC 7654, respectively (See Table 1). Although 37 Myr represents
the NGC 7654’s main-sequence stars well, it seems that this model does not
match the stars at the turn-off point. When BD+60 2532 is excluded from the
statistics and the 120 Myr isochrone fitted according to cluster’s most likely
member stars, it is seen that this age better represents the turn-off and main
sequence stars of the NGC 7654. In this case, it has been concluded that the
position of the BD+60 2532 on CMDs may be due to the binary star effect and
the radial velocity measurements of the star should be examined to elucidate
this. Consequently, in this study we adopted the age of the NGC 7654 as
$t=120\pm 20$ Myr. The CMDs of two clusters with best fit isochrones are shown
in Fig. 11.

Fig. 11 : $V\times(U-B)$, $V\times(B-V)$ and $G\times(G_{\rm BP}-G_{\rm RP})$
CMDs for the Czernik 2 (sub-figures a, b, and c) and NGC 7654 (sub-figures d,
e, and f) clusters. The differently coloured dots represent the membership
probabilities according to the colour scales shown on the right side of the
diagrams. Grey coloured dots identify the field stars. The blue dashed lines
indicate the isochrones (of the ages noted in the diagram). The green shaded
areas surrounding these lines are their associated errors. The blue dashed
lines in the panels d, e, and f are isochrones corresponding to 37 and 120 Myr
ages. The best fitting isochrone for NGC 7654 corresponds to a 120 Myr age for
the cluster, while that for Czernik 2 is 1.2 Gyr.

###  4.4 Comparison of Astrometric Results

Using Gaia EDR3 astrometric data (Gaia collaboration et al., 2021), we
calculated both mean proper motion components and trigonometric distances from
Gaia for each clusters. Measurement of mean astrometric values was based on
the most probable cluster members. To obtain trigonometric distances from Gaia
($d_{\rm Gaia~{}EDR3}$), we first converted the trigonometric parallaxes into
distances for each member star via the linear expression of $d$(pc) =
1000/$\varpi$ (mas), then we plotted the histograms of distances versus number
of stars before finally fitting Gaussian models to the distributions (Fig.
12).

Fig. 12 : The distance histograms of Czernik 2 (a) and NGC 7654 (b). Black and
red dashed lines represent the best Gaussian fits to the distances calculated
from trigonometric parallaxes (Gaia collaboration et al., 2021) and Bayesian
approach method. (Bailer-Jones et al., 2021).

Just after the second data release of Gaia (Gaia collaboration et al., 2018),
various researchers deduced numerous values of zero-point offsets for
trigonometric parallaxes (i.e. Lindegren et al., 2018; Riess et al., 2018;
Arenou et al., 2018). Arenou et al. (2018) estimated several zero-point
offsets using parallaxes of different objects (dwarf galaxies, classical
variable stars, open and globular clusters). They concluded that zero-point
offset values are within the $-0.01$ to $-0.1$ mas depending on the type of
object and also are a function of systematic errors, positions on the sky, and
magnitude/colour of the objects. In addition to this, there are many zero-
point offsets whose values are among the $-0.029$ mas (Lindegren et al., 2018)
and $-0.082$ mas (Stassun & Torres, 2018). These were calculated by taking
into account trigonometric parallaxes of quasars, classical cepheids,
eclipsing binaries samples, RR Lyrae stars, Kepler red giant branch and red
clump stars (Lindegren et al., 2018; Stassun & Torres, 2018; Riess et al.,
2018; Khan et al., 2019; Hall et al., 2019). Moreover, some researchers
reported that the offset correction of parallaxes in Gaia DR2 may increase for
distances in excess of 1 kpc (Stassun & Torres, 2018; Lohr et al., 2018). The
commonly accepted value for the zero-point offset correction was $-0.029$ mas
in Gaia DR2 which was determined from the quasars (Lindegren et al., 2018).
Recently, some research groups investigated parallax zero-point values from
Gaia EDR3 data taking into account parallaxes of classical cepheids, blue RR
Lyrae stars and eclipsing binaries, quasars and red clump stars (Riess et al.,
2021; Bhardwaj et al., 2020; Lindegren et al., 2021; Stassun & Torres, 2021;
Huang et al., 2021). According to these studies, the zero-point offset values
range from $-0.037$ mas (Stassun & Torres, 2021) to $0.026$ mas (Huang et al.,
2021). As mentioned in previous studies involving Gaia DR2 parallaxes
corrections, the zero-point offset values are functions of factors such as
colour index, magnitude, and position on the sky (Lindegren et al., 2018). In
this study, we transformed the distances estimated via fitting PARSEC
isochrones and used the Gaia EDR3 data to parallaxes linear expression,
comparing the values with each other to understand how these change. For
Czernik 2 the isochrone distance ($d_{\rm iso}$) and Gaia EDR3 distance
($d_{\rm Gaia~{}EDR3}$) parallaxes are subtending to $\varpi_{\rm
iso}=0.531\pm 0.018$ and $\varpi_{\rm Gaia~{}EDR3}=0.521\pm 0.051$ mas,
respectively. The difference between two trigonometric parallax values is
$\varpi_{\rm iso}-\varpi_{\rm Gaia~{}EDR3}=0.010\pm 0.054$ mas. For NGC 7654
the distances correspond to $\varpi_{\rm\rm iso}=0.517\pm 0.039$ and
$\varpi_{\rm Gaia~{}EDR3}=0.610\pm 0.031$ mas from which the difference is
$\varpi_{\rm iso}-\varpi_{\rm Gaia~{}EDR3}=-0.093\pm 0.050$ mas.

Table 7 : Comparison of distances estimated from isochrone fitting and trigonometric parallaxes of Gaia DR2 ($\varpi_{\rm Gaia~{}DR2}$) and Gaia EDR3 ($\varpi_{\rm Gaia~{}EDR3}$). Here, $\varpi_{\rm iso}$ is trigonometric parallax derived from isochrone distance ($d_{\rm iso}$), $\Delta\varpi_{1}$ and $\Delta\varpi_{2}$ indicate the differences between ($\varpi_{\rm iso}$-$\varpi_{\rm Gaia~{}DR2}$) and ($\varpi_{\rm iso}$\- $\varpi_{\rm Gaia~{}EDR3}$), respectively. Cluster | $d_{\rm iso}$ | $d_{\rm Gaia~{}DR2}$ | $d_{\rm Gaia~{}EDR3}$ | $\varpi_{\rm iso}$ | $\varpi_{\rm Gaia~{}DR2}$ | $\varpi_{\rm Gaia~{}EDR3}$ | $\Delta\varpi_{1}$ | $\Delta\varpi_{2}$ | Reference
---|---|---|---|---|---|---|---|---|---
| (pc) | (pc) | (pc) | (mas) | (mas) | (mas) | ($\mu$mas) | ($\mu$mas) |
ASCC 115 | 732$\pm$69 | 755$\pm$14 | 732$\pm$30 | 1.366$\pm$ 0.128 | 1.325$\pm$ 0.025 | 1.366$\pm$ 0.056 | +41 | 0 | Yontan et al. (2019)
Collinder 421 | 1245$\pm$103 | 1220$\pm$99 | 1224$\pm$70 | 0.803$\pm$ 0.067 | 0.820$\pm$ 0.066 | 0.817$\pm$ 0.047 | -17 | -14 | Yontan et al. (2019)
Czernik 2 | 1883$\pm$63 | 1998$\pm$179 | 1919$\pm$189 | 0.531$\pm$ 0.018 | 0.501$\pm$ 0.045 | 0.521$\pm$ 0.051 | +30 | +10 | This study
Frolov 1 | 2864$\pm$254 | 2800$\pm$136 | 3098$\pm$275 | 0.349$\pm$ 0.031 | 0.357$\pm$ 0.017 | 0.323$\pm$ 0.029 | -8 | +26 | Yontan et al. (2021)
Melotte 105 | 2078$\pm$78 | 2460$\pm$180 | 2384$\pm$134 | 0.481$\pm$ 0.018 | 0.407$\pm$ 0.030 | 0.419$\pm$ 0.023 | +74 | +62 | Banks et al. (2020)
NGC 6793 | 610$\pm$40 | 607$\pm$37 | 590$\pm$34 | 1.639$\pm$ 0.108 | 1.647$\pm$ 0.100 | 1.695$\pm$ 0.098 | -8 | -56 | Yontan et al. (2019)
NGC 7031 | 1212$\pm$146 | 1404$\pm$81 | 1370$\pm$86 | 0.825$\pm$ 0.099 | 0.712$\pm$ 0.041 | 0.730$\pm$ 0.046 | +113 | +95 | Yontan et al. (2019)
NGC 7039 | 743$\pm$64 | 767$\pm$41 | 751$\pm$52 | 1.346$\pm$ 0.116 | 1.304$\pm$ 0.069 | 1.332$\pm$ 0.092 | +42 | +14 | Yontan et al. (2019)
NGC 7086 | 1618$\pm$182 | 1684$\pm$140 | 1673$\pm$116 | 0.618$\pm$ 0.069 | 0.594$\pm$ 0.049 | 0.598$\pm$ 0.041 | +24 | +20 | Yontan et al. (2019)
NGC 7510 | 2818$\pm$247 | 3450$\pm$477 | 3222$\pm$24 | 0.355$\pm$ 0.031 | 0.290$\pm$ 0.040 | 0.310$\pm$ 0.002 | +65 | +45 | Yontan et al. (2021)
NGC 7654 | 1935$\pm$146 | 1694$\pm$120 | 1640$\pm$82 | 0.517$\pm$ 0.039 | 0.590$\pm$ 0.042 | 0.610$\pm$ 0.031 | -73 | -93 | This study
Roslund 1 | 836$\pm$48 | 883$\pm$54 | 881$\pm$32 | 1.196$\pm$ 0.068 | 1.133$\pm$ 0.069 | 1.135$\pm$ 0.041 | +63 | +61 | Yontan et al. (2019)
Stock 21 | 1931$\pm$27 | 1934$\pm$159 | 1981$\pm$294 | 0.518$\pm$ 0.007 | 0.517$\pm$ 0.042 | 0.505$\pm$ 0.075 | +1 | +13 | Yontan et al. (2019)
Table 8 : Mean proper motion components and distances ($d_{\rm iso}$, $d_{\rm Gaia~{}EDR3}$, $d_{\rm BJ21}$) as estimated in this study. The results of Cantat-Gaudin & Anders (CA20, 2020) and (BJ21, Bailer-Jones et al., 2021) are also listed in the table. | This Study | CA20 | BJ21
---|---|---|---
Cluster | $d_{\rm iso}$ | $d_{\rm Gaia~{}EDR3}$ | $\mu_{\alpha}\cos\delta$ | $\mu_{\delta}$ | $d$ | $\mu_{\alpha}\cos\delta$ | $\mu_{\delta}$ | $d_{\rm BJ21}$
| (pc) | (pc) | (mas yr-1) | (mas yr-1) | (pc) | (mas yr-1) | (mas yr-1) | (pc)
Czernik 2 | 1883$\pm$63 | 1919$\pm$189 |  $-$4.03$\pm$0.04 |  $-$0.99$\pm$0.05 | $1907^{+449}_{-305}$ | $-4.06\pm 0.01$ | $-0.90\pm 0.01$ | 1844$\pm$102
NGC 7654 | 1935$\pm$146 | 1640$\pm$82 |  $-$1.89$\pm$0.03 |  $-$1.20$\pm$0.03 | $1600^{+305}_{-221}$ | $-1.94\pm 0.01$ | $-1.13\pm 0.01$ | 1569$\pm$71

We investigated the zero-point offsets for Gaia trigonometric parallaxes by
making a comparison of both the isochrone and the Gaia EDR3 distances
estimated for the 13 open clusters (also including Czernik 2 and NGC 7654)
that we had previously analysed in earlier studies with similar methods
(Yontan et al., 2019, 2021; Banks et al., 2020). Also, we derived Gaia DR2
distances of these 13 open clusters and compared the values with both
isochrone and Gaia EDR3 distances to understand how the zero-point offsets
change between Gaia DR2 and EDR3 data. Trigonometric parallaxes corresponding
to isochrone and Gaia EDR3 distances of these open clusters (whose isochrone-
based distances are within the range 610 and 2864 pc) were calculated by the
linear method. The differences between the methods are listed in Table 7. It
can be seen that the median value of the differences in trigonometric
parallaxes for Gaia DR2 and Gaia EDR3 are 30 and 14 $\mu$mas, respectively,
which are an indication of the zero-point offset for the distances of the open
clusters. These results show that the uncertainty in the zero-points
calculated from Gaia EDR3 is approximately half of Gaia DR2. Also, the zero-
point offset of 14 $\mu$mas inferred in this study is in the range of values
given for Gaia EDR3 in the literature for zero-point offsets (see discussion
above).

Recently Bailer-Jones et al. (2021, BJ21) presented a Bayesian approach taking
into account a prior with Galactic model parameters, calculating distances
with errors for nearly 1.47 billion stars using Gaia EDR3 trigonometric
parallaxes and their uncertainties. We also took into consideration the
distances of stars presented by Bailer-Jones et al. (2021) to understand the
changes between distances calculated by the trigonometric parallax and
Bayesian approaches methods. We retrieved the distance data of the most likely
cluster member stars from the catalogue of Bailer-Jones et al. (2021) and
constructed the histogram of distance versus number of stars (Fig. 12). Then
we fitted the distribution with a Gaussian model and determined the mean
distances ($d_{\rm BJ21}$) from the data of Bailer-Jones et al. (2021). Errors
in distance distribution for both clusters were taken as the standard
deviations of Gaussian fits (see Table 8). Two comparisons were made on each
histogram as shown in Fig. 12. The distances were estimated as ($d_{\rm
Gaia~{}EDR3}$, $d_{\rm BJ21}$) = ($1919\pm 189,1844\pm 102$) pc and ($d_{\rm
Gaia~{}EDR3}$, $d_{\rm BJ21}$) = ($1640\pm 82,1569\pm 71$) pc for Czernik 2
and NGC 7654, respectively. These results are compatible with those distances
derived from isochrone fitting (see Table 8).

Cantat-Gaudin & Anders (2020) estimated mean proper motions and parallaxes of
more than 1,200 open clusters using the Gaia DR2 (Gaia collaboration et al.,
2018) astrometric and photometric data, and utilising maximum likelihood
procedure. We compared mean proper motion components and distances of the two
clusters with those reported by Cantat-Gaudin & Anders (2020). This comparison
of mean proper motions and distances ($d_{\rm iso}$, $d_{\rm Gaia~{}EDR3}$,
$d_{\rm BJ21}$) is given in Table 8. For the Czernik 2 we estimated the proper
motion components as ($\mu_{\alpha}\cos\delta$, $\mu_{\delta}$) = ($-4.03\pm
0.04$, $-0.99\pm 0.05$) mas yr-1 and for NGC 7654 as
($\mu_{\alpha}\cos\delta$, $\mu_{\delta}$) = ($-1.89\pm 0.03$, $-1.20\pm
0.03$) mas yr-1. These are in general agreement with the results of Cantat-
Gaudin & Anders (2020) and Liu & Pang (2019), being inside the $2\sigma$
ranges for both clusters.

Fig. 13 : Present day mass functions of Czernik 2 (a) and NGC 7654 (b) derived from member stars. Solid blue lines are the mass functions of the two clusters. Table 9 : Present day mass function slopes ($X$) of Czernik 2 and NGC 7654: $N$ is number of stars. The modelled mass range interval is also given. Cluster | $N$ | $X$ | Mass Range
---|---|---|---
Czernik 2 | 32 | -1.37$\pm$0.24 | 0.80 $<M/M_{\odot}<$ 2
NGC 7654 | 401 | -1.39$\pm$0.19 | 0.75 $<M/M_{\odot}<$ 4

###  4.5 Present Day Mass Functions

To obtain estimates for the present mass functions of the two clusters we
considered 32 and 401 main-sequence stars with probability $P>0$ in Czernik 2
and NGC 7654, respectively. We derived the absolute magnitude of each star
using the equation $M_{V}=V-\mu_{V}+E(B-V)$ (where $V$ is apparent magnitude
of star, $\mu_{V}$ the distance modulus, and $E(B-V)$ is the colour excess of
the cluster). Masses of cluster member stars were calculated from PARSEC
models (Bressan et al., 2012). A polynomial expression was constructed by
correlating a relation between theoretical absolute magnitude and the masses
presented in PARSEC models. Using this expression we transformed the absolute
magnitudes of stars to their mass values. As a result, we confirmed that the
mass ranges of the main sequence stars are lying within $0.80<M/M_{\odot}<2$
and $0.75<M/M_{\odot}<4$ for Czernik 2 and NGC 7654, respectively. To
calculate present day mass functions of the clusters, one should first
construct the mass distributions. We therefore tabulated the stars within 0.2
and 0.5 $M/M_{\odot}$ mass ‘steps’ for Czernik 2 and NGC 7654, respectively.
We identified the number of stars in each mass range and calculated their
logarithmic values. Equation (4) was used to determine present day mass
functions of two clusters.

$\log(dN/dM)=-(1+X)\times\log M+C.$ (4)

$M$ is the mean mass value; $X$ is the slope of the mass function and $C$
expresses the constant of the linear fit. Errors in the mean masses were
calculated according to Poisson statistics. The distribution of mass versus
logarithmic numbers is shown in Fig. 13, and results are listed in Table 9.

##  5 Conclusion

In the present paper we derived fundamental astrophysical (reddening,
metallicity, distance, and age) and astrometric (mean proper motion
components) parameters of Czernik 2 and NGC 7654. These estimates were based
on CCD UBV photometric and Gaia EDR3 (Gaia collaboration et al., 2021)
photometric and astrometric data.

To obtain cluster membership probabilities we used stellar proper motion
components provided by the Gaia EDR3 catalogue (Gaia collaboration et al.,
2021). However such membership probabilities cannot identify binary star
contamination in the main-sequence of a cluster, we therefore also considered
photometric criteria to identify real cluster members. To do this, the ZAMS of
Sung et al. (2013) was fitted on CMDs taking into account the 0.75 mag
increase in brightness caused by binary stars. In this way we specified lower
and upper bases of the main-sequences for both open clusters. Thus we coupled
the membership probabilities with the photometric selection criteria and
strengthened the separation of real cluster members. We used those resulting
members to determine the fundamental parameters of the clusters.

  1. 1.

We estimated structural parameters of the two clusters through fitting the RDP
of King (1962) (see Table 6). The radii that the cluster stellar densities
equal the background stellar density were taken as the physical sizes of
clusters, being $r=5^{\prime}$ and $r=8^{\prime}$ for Czernik 2 and NGC 7654,
respectively.

  2. 2.

We refined the selection of cluster member stars by taking into account
together the membership probabilities, the effect of binary stars, and the
physical sizes of the clusters. Thus, for Czernik 2 and NGC 7654 we found 28
and 369 member stars, respectively, with $P\geq 0.5$.

  3. 3.

Based on the most likely member stars, we estimated the mean proper motions of
the Czernik 2 and NGC 7654 as ($\mu_{\alpha}\cos\delta$, $\mu_{\delta}$) =
($-4.03\pm 0.04,-0.99\pm 0.05$) mas yr-1 and ($\mu_{\alpha}\cos\delta$,
$\mu_{\delta}$) = ($-1.89\pm 0.03,-1.20\pm 0.03$) mas yr-1, respectively.
These values are in very good agreement with the study of Cantat-Gaudin &
Anders (2020).

  4. 4.

By comparing the $(U-B)\times(B-V)$ TCDs with the ZAMS of Sung et al. (2013),
we estimated the colour excesses of Czernik 2 and NGC 7654 as $E(B-V)=0.46\pm
0.02$ mag and $E(B-V)=0.57\pm 0.04$ mag, respectively. Both values are
consistent with the normal interstellar extinction law. These results are
compatible with the study of Cantat-Gaudin et al. (2020).

  5. 5.

Photometric metallicities of Czernik 2 and NGC 7654 were based on the
calculation of UV excesses of F-G type main-sequence member stars with respect
to the main-sequence of Hyades cluster on the TCDs. The metallicites of
Czernik 2 and NGC 7654 were estimated as ${\rm[Fe/H]}=-0.08\pm 0.02$ dex and
${\rm[Fe/H]}=-0.05\pm 0.01$ dex, respectively.

  6. 6.

Keeping constant both reddening and metallicity, we derived distance moduli
and ages of two clusters by fitting PARSEC models which have a metallicities
$Z=0.013$ for Czernik2 and $Z=0.014$ for NGC 7654 on CMDs. The distance
modulus for Czernik 2 was calculated to be $\mu_{V}=12.80\pm 0.07$ mag and the
age $t=1.2\pm 0.2$ Gyr. For NGC 7654 the distance modulus is $\mu_{V}=13.20\pm
0.16$ mag and its age $t=120\pm 20$ Myr.

  7. 7.

In the study, we calculated distances using three methods: by fitting
isochrones on CMDs, using Gaia EDR3 trigonometric parallaxes, and distances of
member stars which were presented Bailer-Jones et al. (2021). The distances
obtained via three methods are in very good agreement within the errors (see
Table 8). We can infer that the distance whose determination depends on
isochrone fitting gives reliable results.

  8. 8.

It has been shown that there is a zero-point offset of 14 $\mu$mas between
isochrone distances and Gaia distances of 13 open clusters examined by
homogeneous methods. Moreover, it is seen that the value of 30 $\mu$mas in
Gaia DR2 decreased to 14 $\mu$mas in Gaia EDR3, which complies with the low
zero-point values predicted in Gaia EDR3.

  9. 9.

The slopes of the present day mass functions were estimated as $X=-1.37\pm
0.24$ and $X=-1.39\pm 0.19$ for Czernik 2 and NGC 7654, respectively. Results
are in a good agreement with the value $-1.35$ presented by Salpeter (1955).

The authors are grateful to the anonymous referee for his/her considerable
contributions to improve the paper. This study has been supported in part by
the Scientific and Technological Research Council (TÜBİTAK) 119F014. We thank
TÜBİTAK for partial support towards using the T100 telescope via project
15AT100-738. We also thank the on-duty observers and members of the technical
staff at the TÜBİTAK National Observatory for their support before and during
the observations. This research made use of VizieR and Simbad databases at
CDS, Strasbourg, France. 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 DPAC has
been provided by national institutions, in particular the institutions
participating in the _Gaia_ Multilateral Agreement. IRAF was distributed by
the National Optical Astronomy Observatory, which was operated by the
Association of Universities for Research in Astronomy (AURA) under a
cooperative agreement with the National Science Foundation. PyRAF is a product
of the Space Telescope Science Institute, which is operated by AURA for NASA.

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