# A study of the NGC 1193 and NGC 1798 open clusters using CCD UBV photometric
and Gaia EDR3 data

Talar Yontan  11affiliation: Department of Astronomy & Space Sciences,
Istanbul University, Istanbul, Turkey. Hikmet Çakmak  11affiliationmark:
Selçuk Bilir  11affiliationmark:  Timothy Banks  22affiliation: Nielsen,
Chicago, USA. 33affiliation: Harper College, Illinois, USA. Michel Raúl
44affiliation: Observatorio Astronomico Naciona, Universidad Nacional Autonoma
de Mexico, Ensenada, Mexico. Remziye Canbay  55affiliation: Institute of
Graduate Studies in Science, Istanbul University, Istanbul, Turkey. Seliz Koç
55affiliationmark:  Seval Taşdemir  55affiliationmark:  Hülya Erçay
55affiliationmark:  Burçin Tanık Öztürk  55affiliationmark:  & Deniz Cennet
Dursun55affiliationmark:  Talar Yontan, Hikmet Çakmak & Selçuk Bilir: Istanbul
University, Faculty of Science, Department of Astronomy and Space Sciences,
34119, Beyazıt, Istanbul, Turkey<EMAIL_ADDRESS><EMAIL_ADDRESS>sbilir@istanbul.edu.tr). Timothy Banks: Nielsen, 200
W Jackson Blvd, Chicago, IL 60606, USA (tim.banks@nielsen.com); Department of
Physical Science & Engineering, Harper College, 1200 W Algonquin Rd, Palatine,
IL 60067, USA (tbanks@harpercollege.edu). Michel Raúl: Observatorio
Astronomico Naciona, Universidad Nacional Autonoma de Mexico, Ensenada, Mexico
(rmm@astro.unam.mx). Remziye Canbay, Seliz Koç, Seval Taşdemir, Hülya Erçay,
Burçin Tanık Öztürk, & Deniz Cennet Dursun: Istanbul University, Institute of
Graduate Studies in Science, Programme of Astronomy and Space Sciences, 34116,
Beyazıt, Istanbul, Turkey<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>denizcdursun@gmail.com).

###### Abstract

We present photometric, astrometric, and kinematic studies of the old open
star clusters NGC 1193 and NGC 1798. Both of the clusters are investigated by
combining data sets from Gaia Early Data Release 3 (EDR3) and CCD UBV
observational data. Analysis of the radial distribution of stars through the
cluster regions indicates that the cluster limit radii are $r_{\rm
lim}=8^{\prime}$ for both of the clusters. We determine the membership
probabilities of stars considering Gaia EDR3 proper motion and trigonometric
parallax data, resulting in 361 stars in NGC 1193 and 428 in NGC 1798 being
identified as most likely cluster members, having membership probabilities
greater than $P\geq 0.5$. Mean proper motion components are estimated as
($\mu_{\alpha}\cos\delta$, $\mu_{\delta})=(-0.207\pm 0.009,-0.431\pm 0.008$)
for NGC 1193 and ($\mu_{\alpha}\cos\delta$, $\mu_{\delta})=(0.793\pm
0.006,-0.373\pm 0.005$) mas yr-1 for NGC 1798. $E(B-V)$ color excesses were
derived for NGC 1193 as $0.150\pm 0.037$ and for NGC 1798 as $0.505\pm 0.100$
mag through the use of two-color diagrams. Photometric metallicities are also
determined from two-color diagrams with the results of [Fe/H] = $-0.30\pm
0.06$ dex for NGC 1193 and [Fe/H]=$-0.20\pm 0.07$ dex for NGC 1798. The
isochrone fitting distance and age of NGC 1193 are $5562\pm 381$ pc and
$4.6\pm 1$ Gyr, respectively. For NGC 1798, these parameters are $4451\pm 728$
pc and $1.3\pm 0.2$ Gyr. These ages indicate that NGC 1193 and NGC 1798 are
old open clusters. The overall present day mass function slopes for main-
sequence stars are found as $\Gamma=1.38\pm 2.16$ for NGC 1193 and
$\Gamma=1.30\pm 0.21$ for NGC 1798, which are in fair agreement with the value
of Salpeter (1955). Kinematic and dynamic orbital calculations indicate that
NGC 1193 and NGC 1798 belong to the thick-disk and thin-disk populations,
respectively. In addition, both of the clusters were born outside the solar
circle, and both orbit in the metal-poor region of the Galactic disk.

Galaxy: open cluster and associations - Individual: NGC 1193, NGC 1798 -
Galaxy: Disk stars - Hertzsprung-Russell (HR)  ——————————————————————————————–

##  0.1 General

Open clusters are identified as groupings of stars, beyond those found in a
single multiple star system, that are bound together by their weak self-
gravitational forces. As cluster stars are formed by the collapse of the same
molecular cloud, their basic astrophysical parameters such as color excess,
distance, metal abundance, and age are similar while their masses and
luminosities can range widely. This paper concentrates on open star clusters
inside our own galaxy’s disk, which are often called ‘galactic clusters’.
These properties make such open clusters important tools to investigate the
structure, formation, and evolution of the Galactic disk as well as to give
opportunities to enhance our understanding of stellar evolution models. In
particular, the study of old open clusters can give insight into the kinematic
properties and chemical structure of the Galactic disk (Friel, 1995).

###  0.1.1 NGC 1193

In 1786 William Herschel discovered the open cluster NGC 1193 ($\alpha=03^{\rm
h}\>05^{\rm m}\>56^{\rm s}\\!\\!.\,64$,
$\delta=+44^{\circ}\>22^{\rm{}^{\prime}}\>58^{\rm{}^{\prime\prime}}\\!\\!.\,80,\leavevmode\nobreak\
l=146^{\circ}\\!\\!.\,8143$, $b=-12^{\circ}\\!\\!.\,1624$), located in the
constellation of Perseus (Dreyer, 1888). Together with an angular size of
$2^{\prime}$, NGC 1193 has a dense central stellar concentration and is
classified as II3m (Ruprecht, 1966). King (1962) reported that NGC 1193 is
likely to be old. In the study of Janes & Adler (1982), NGC 1193 was
identified as a dense, poorly studied open cluster with the angular diameter
of $2^{\prime}$. Kaluzny (1988) presented the first CCD BV photometric study
of NGC 1193, identifying five possible blue struggler stars in the cluster. By
BV isochrone fitting to the color magnitude diagram (CMD), they determined the
color excess and distance to be $0.12\leq E(B-V)\leq 0.23$ mag and $4.2\leq
d\leq 4.9$ kpc, respectively. Additionally the metallicity, distance module,
and age of the cluster were adopted as $Z=0.01$, $(m-M)_{\rm V}=13.8$ mag, and
$t=8\times 10^{9}$ years. Through investigation of the cluster’s color-
magnitude diagram, Kaluzny (1988) indicated that the subgiant branch stars are
more populous than red giant branch stars. Utilizing spectroscopic
observations, Friel, Liu, & Janes (1989) calculated the first radial velocity
estimate for the cluster as $\langle V_{\rm r}\rangle=-82$ km s-1. Friel &
Janes (1993) performed medium resolution spectroscopic analyses and estimated
cluster metallicity as ${\rm[Fe/H]}=-0.50\pm 0.18$ dex from four giant
members. They also calculated radial velocities of these stars, whose values
lie within the $-64\leq V_{\rm r}\leq-103$ km s-1. Tadross (2005) used
photometric data of Kaluzny (1988) and astrometric data from the USNO-B1.0
catalog of Monet et al. (2003) to determine the cluster’s color excess
$E(B-V)=0.10\pm 0.06$, distance modulus $\mu=13.90\pm 0.10$ mag, distance
$d=5.25\pm 0.24$ kpc, age $t=8$ Gyr, and metallicity as $Z=0.008$. Moreover,
Tadross (2005) analysed the cluster with regard to the radial profile of van
den Bergh & Sher (1960) and estimated the core radius as ${r}_{\rm
c}=1^{\prime}.4$ and the limiting radius as $r_{\rm lim}=6^{\prime}.5$. Kyeong
et al. (2008) applied a fitting procedure of the theoretical isochrones of
Bertelli et al. (1994) to the color-magnitude diagrams based on CCD UBVI
photometric data of NGC 1193. They calculated the color excess as
$E(B-V)=0.19\pm 0.04$ mag, metallicity ${\rm[Fe/H]}=-0.45\pm 0.12$ dex, true
distance module $(m-M_{\rm V})_{\rm 0}=13.30\pm 0.15$ mag, and the cluster age
as $\log{t}(\rm yr)=9.7\pm 0.1$.

The Gaia mission (Gaia collaboration et al., 2016) has led to substantial
improvements in the quality and precision of astrometric, photometric, and
spectroscopic data. Gaia has provided precise astrometric, photometric, and
spectroscopic data of nearly 1.8 billion stars. Cantat-Gaudin et al. (2018)
identified 215 most likely cluster members, using astrometric and photometric
data of stars across locality of NGC 1193. In the study, they determined the
mean proper motion of the cluster as ($\mu_{\alpha}\cos\delta$,
$\mu_{\delta}$)=($-0.125\pm 0.023,-0.329\pm 0.019$) mas yr-1 and the
trigonometric parallax as $\varpi=0.159\pm 0.009$ mas. Soubiran et al. (2018)
used the second Gaia data release (Gaia DR2; Gaia collaboration et al., 2018)
spectroscopy to identify a radial velocity measurement for one member star of
NGC 1193, calculating its radial velocity as $\langle V_{\rm
r}\rangle=-83.24\pm 0.51$ km s-1. In addition, Carrera et al. (2019)
determined the mean radial velocity of the cluster as $\langle V_{\rm
r}\rangle=-85.16$ km s-1, based on APOGEE spectroscopic data for two member
stars of the cluster. Donor et al. (2020) analysed three cluster member stars
using APOGEE DR16 spectroscopic data and calculated the radial velocity and
metallicity of the NGC 1193 as $\langle V_{\rm r}\rangle=-84.7\pm 0.2$ km s-1
km s-1 and ${\rm[Fe/H]}=-0.34\pm 0.01$ dex, respectively. Using Gaia DR2 data,
they determined the mean proper motion components of the cluster as
($\mu_{\alpha}\cos\delta$, $\mu_{\delta}$)=($-0.22\pm 0.10,-0.36\pm 0.07$) mas
yr-1.

###  0.1.2 NGC 1798

The open cluster NGC 1798 ($\alpha=05^{\rm h}11^{\rm m}39^{\rm s}\\!\\!.\,36$,
$\delta=+47^{\circ}\,41^{\rm{}^{\prime}}27^{\rm{}^{\prime\prime}}\\!\\!.\,60,\leavevmode\nobreak\
l=160^{\circ}\\!\\!.\,7043$, $b=+04^{\circ}\\!\\!.\,8500$) was discovered in
1885 by Edward Barnard, located in the Auriga constellation (Dreyer, 1888).
With an angular size of about 5 arcmin, this cluster is classified as II2m
with a central dense stellar concentration (Ruprecht, 1966). Examination of
the cluster’s CMD reveals that the regions of the main sequence and red clump
(RC) stars are more distinct than the red giant branch (RGB). Based on this
morphological feature, Janes & Phelps (1994) gave the age of NGC 1798 as 1.5
Gyr and distance as 3.44 kpc.

The first CCD UBVI photometric observations of the NGC 1798 were made by Park
& Lee (1999). The angular diameter of the cluster was given as 8.3 arcmin
(10.2 pc), the color excess $E(B-V)=0.51\pm 0.04$ magnitude, distance
$d=4.2\pm 0.3$ kpc, metallicity ${\rm[Fe/H]}=-0.47\pm 0.15$ dex, and age
$t=1.4\pm 0.2$ Gyr. Lata et al. (2002) used the data of Park & Lee (1999) to
determine the absolute magnitude and color indices for the $I$ band as
$I(M_{\rm V})=-4.86$, $I(U-V)_{\rm 0}=0.97$, $I(B-V)_{\rm 0}=0.82$, and
$I(V-I)_{\rm 0}=1.14$ mag. Maciejewski & Niedzielski (2007) obtained the
structural and astrophysical parameters of 42 open clusters with CCD BV
photometry. They determined the cluster’s limiting radius $r_{\rm lim}=9$
arcmin, the core radius $r_{\rm c}=1.3\pm 0.1$ arcmin, central stellar density
$f_{\rm 0}=9.5\pm 0.28$ star arcmin-2, and the background stellar density
$f_{\rm bg}=3.14\pm 0.05$ stars arcmin-2. These researchers used the
isochrones of Bertelli et al. (1994), obtaining the color excess of the
cluster as $E(B-V)=0.37^{+0.10}_{-0.09}$, the distance modulus as
$(m-M)=13.90_{-0.63}^{+0.26}$ mag, distance as $d=3.55_{-1.22}^{+0.64}$ kpc,
and age being $\log t\leavevmode\nobreak\ (\rm yr)=9.2$. Ahumada & Lapasset
(2007) examined 1,887 blue struggler star (BSS) candidates in 427 open
clusters and identified 24 BSS in the direction of NGC 1798. They indicated
that six of these BSS are massive and 18 are low mass stars. Carrera (2012)
calculated the radial velocities of four open clusters including NGC 1798 by
analyzing spectroscopic data of their member stars. By measuring Ca II lines,
Carrera (2012) determined the mean radial velocity of NGC 1798 as $\langle
V_{\rm r}\rangle=2\pm 10$ km s-1. They used six member stars in total,
consisting of five RGB stars and one main sequence turn-off star. Oralhan et
al. (2015) analyzed CCD UBVRI photometric observations of 20 open clusters and
obtained their astrophysical parameters. They determined the reddening,
photometric metallicity, distance modulus, distance, and age of the NGC 1798
as $E(B-V)=0.47\pm 0.07$ mag, ${\rm[Fe/H]}=-0.50\pm 0.28$ dex, $(m-M)_{\rm
0}=12.70\pm 0.04$ mag, $d=3.47\pm 0.06$ kpc, and $t=1.78\pm 0.22$ Gyr.

Cantat-Gaudin et al. (2020) used photometric and astrometric data from the
Gaia DR2 (Gaia collaboration et al., 2018) to determine astrometric and
astrophysical parameters of 2,017 open clusters. They identified 218 member
stars in NGC 1798. Considering these members they calculated mean proper-
motion components and trigonometric parallaxes of the cluster as
$(\mu_{\alpha}\cos\delta$, $\mu_{\delta})=(0.913\pm 0.011$, $-0.318\pm 0.010$)
mas yr-1 and $\varpi=0.178\pm 0.005$ mas. Liu & Pang (2019) used astrometric
and photometric data of 78 member stars of NGC 1798 to calculate the mean
proper-motion components, trigonometric parallaxes, and age of the cluster as
($\mu_{\alpha}\cos\delta$, $\mu_{\delta}$)= ($0.903\pm 0.026$, $-0.400\pm
0.295$) mas yr-1, $\varpi=0.241\pm 0.026$ mas, and $t=1.7\pm 0.1$ Gyr.

A number of studies explored open clusters using ground-based telescopes
within the scope of spectroscopic survey programs (Gilmore et al., 2012;
Conrad et al., 2014; Maciejewski & Niedzielski, 2007; Kos et al., 2018).
Within the context of the APOGEE survey, Donor et al. (2018) utilized spectral
observations of 259 cluster member stars in 19 open clusters including NGC
1798 and obtained the radial velocity and different metal abundance values of
the stars. Analysing nine member stars in NGC 1798 Donor et al. (2018)
determined the mean radial velocity as $\langle V_{\rm r}\rangle=2\pm 1.7$ km
s-1 and the iron abundance ${\rm[Fe/H]}=-0.18\pm 0.02$ dex. Soubiran et al.
(2018) analysed Gaia DR2 spectroscopic data of four member stars in the
cluster and obtained the mean radial velocity as $\langle V_{\rm
r}\rangle=2.60\pm 0.41$ km s-1. Donor et al. (2020) analysed eight cluster
member stars using APOGEE DR16 spectroscopic data and calculated the radial
velocity and metallicity of the NGC 1798 as $\langle V_{\rm r}\rangle=2.7\pm
0.8$ km s-1 km s-1 and ${\rm[Fe/H]}=-0.27\pm 0.03$ dex, respectively. Using
Gaia DR2 data, they determined the mean proper motion components of the
cluster as ($\mu_{\alpha}\cos\delta$, $\mu_{\delta}$)=($0.83\pm 0.04,-0.31\pm
0.04$) mas yr-1.

Figure 1: Identification charts for NGC 1193 (left panel) and NGC 1798 (right
panel), taken from the Leicester database and archive service (LEDAS).

##  0.2 Observations and data reductions

###  0.2.1 CCD UBV Photometric Data

The observations of these two clusters, along with many others, were carried
out at the San Pedro Martir
Observatory,111https://www.astrossp.unam.mx/en/home/ as part of an ongoing
UBVRI photometric survey of Galactic stellar clusters. The 84-cm ($f/15$)
Ritchey-Chretien telescope was employed in combination with the Mexman filter
wheel.

NGC 1193 was observed on 2013-09-19 with the ESOPO CCD detector (a $2048\times
2048$ 13.5-$\mu$m square-pixels E2V CCD42-40 with a gain of
$1.65\>\mathrm{e^{-}}$/ADU and a readout noise of $3.8\>\mathrm{e^{-}}$ at the
$2\times 2$ binning employed, providing an unvignetted field of view of
$7.4\times 9.3$ arcmin2). Short and long exposures were taken to properly
measure both the bright and faint stars of the fields. Exposure times were 2,
12, 120s for both I and R; 6, 30, 200 for V; 30, 100, 700s for B; and 60 and
1800s for U.

NGC 1798 was observed on 2009-11-01 with the SITE3 detector (a Photometrics
$1024\times 1024$ 24-$\mu$m square-pixels with a gain of
$1.3\>\mathrm{e^{-}}$/ADU and a readout noise of $6.8\>\mathrm{e^{-}}$, giving
an unvignetted field of view of $6.8\times 6.8$ arcmin2). Exposure times for I
and R were 2, 12 and 120s in duration; 6, 30 and 200s for V; 30, 100 and 700s
for B; and 60 and 1800s for U.

Landolt’s standard stars (Landolt, 2009) were also observed in good sky
conditions, at the meridian and at about two airmasses, to properly determine
the atmospheric extinction coefficients. Flat fields were taken at the
beginning and the end of each night and bias images were obtained between
cluster observations. Data reduction with point spread function (PSF)
photometry was carried out by one of the authors (RM) with the IRAF/DAOPHOT
packages (Stetson, 1987) and employing the transformation equations
recommended, in their Appendix B, by Stetson et al. (2019).

Figure 2: Interval $G$ and $V$-band magnitude histograms of NGC 1193 (a, b)
and NGC 1798 (c, d): The red arrows show the faint limiting apparent
magnitudes in $G$ and $V$-bands.

##  0.3 Data Analysis

###  0.3.1 Gaia Astrometric and Photometric Data

The (early) third data release of Gaia (hereafter Gaia EDR3, Gaia
collaboration et al., 2021) provides high quality astrometric and photometric
data of nearly 1.5 billion celestial objects. Together with ground-based CCD
UBV photometry we took into account Gaia EDR3 astrometric and photometric data
to perform astrometric, photometric, and kinematic analyses of NGC 1193 and
NGC 1798. We extracted such EDR3 data for all stars within regions of 20
arcmins about the centres of each cluster, using the coordinates given by
Cantat-Gaudin et al. (2020) ($\alpha=03^{\rm h}05^{\rm m}56^{\rm
s}\\!\\!.\,64$,
$\delta=+44^{\circ}22^{\rm{}^{\prime}}58^{\rm{}^{\prime\prime}}\\!\\!.\,80$
for NGC 1193 and $\alpha=05^{\rm h}11^{\rm m}39^{\rm s}\\!.\,36$,
$\delta=+47^{\rm o}41^{\rm{}^{\prime}}27^{\rm{}^{\prime\prime}}\\!\\!.\,60$
for NGC 1798). Thus we reached 9,141 stars within the magnitude range $7<G<23$
mag for NGC 1193 and 14,834 stars within $8<G<21$ mag for NGC 1798,
respectively. 20 arcmin field of view optical images for the two clusters are
presented in Figure 1. To construct photometric and astrometric catalogues for
each cluster, we matched the UBV data to that from the Gaia EDR3 catalogue
using stellar equatorial coordinates considering distances less than 5 arcsec.
The mean difference in distances between the coordinates of stars in the
matched catalogues was $\sim 0.08$ arc seconds for both clusters. Both
resulting catalogues contain positions ($\alpha,\delta$), $UBV$ observational
data (apparent $V$ magnitudes, color indices $U-B$, $B-V$), Gaia EDR3
astrometric ($\mu_{\alpha}\cos\delta,\mu_{\delta}$, $\varpi$) and photometric
data ($G$, $G_{\rm BP}-G_{\rm RP}$), and membership probabilities ($P$) as
calculated in this study (Table 1). Catalogues of CCD UBV photometric as well
as Gaia photometric and astrometric data for all the detected stars in the
cluster regions are available electronically for NGC 1193 and NGC 1798222The
complete tables can be obtained from VizieR electronically. Errors of the UBV
and Gaia EDR3 photometric data were adopted as internal errors, being the
uncertainties in the determination of the instrumental magnitudes of the
stars. We calculated the mean photometric errors separately as functions of
$V$ and $G$ intervals, listing the results in Table 2 (on page 2). It can be
seen from the table that the mean internal UBV errors reach 0.08 mag for stars
brighter than $V=20$ mag for both clusters. The mean internal errors of Gaia
EDR3 photometry for stars brighter than $G=21$ mag reach 0.011 mag for NGC
1193 and 0.007 mag for NGC 1798.

Table 1:  The photometric and astrometric catalogues for NGC 1193 and NGC 1798. NGC 1193 |
---|---
ID | RA | DEC | $V$ | $U-B$ | $B-V$ | $G$ | $G_{\rm BP}-G_{\rm RP}$ | $\mu_{\alpha}\cos\delta$ | $\mu_{\delta}$ | $\varpi$ | $P$
| (hh:mm:ss.ss) | (dd:mm:ss.ss) | (mag) | (mag) | (mag) | (mag) | (mag) |  (mas yr-1) |  (mas yr-1) | (mas) |
001 | 03:05:36.67 | +44:22:00.62 | 19.194(0.062) | — | 1.155(0.120) | 21.929(0.049) | 1.693(0.116) | — | — | — | —
002 | 03:05:37.44 | +44:22:01.67 | 18.984(0.044) | — | 0.893(0.083) | 18.777(0.003) | 1.080(0.043) | -0.587(0.247) | -0.251(0.223) | -0.083(0.241) | 1.00
003 | 03:05:38.33 | +44:19:33.12 | 19.162(0.056) | — | 1.475(0.140) | 18.254(0.003) | 2.116(0.035) | -4.640(0.181) | -0.566(0.165) | 1.009(0.177) | 0.00
004 | 03:05:38.40 | +44:21:46.02 | 19.286(0.067) | — | 1.262(0.137) | 18.592(0.003) | 1.759(0.040) | -4.072(0.220) | -2.553(0.200) | 0.640(0.214) | 0.00
005 | 03:05:38.74 | +44:24:25.10 | 18.631(0.034) | 0.194(0.170) | 0.993(0.065) | 18.418(0.003) | 0.969(0.032) | -0.426(0.220) | -0.410(0.235) | 0.185(0.220) | 1.00
… | … | … | … | … | … | … | … | … | … | … | …
565 | 03:06:18.54 | +44:18:51.83 | 17.758(0.017) | 0.458(0.088) | 0.881(0.026) | 17.510(0.003) | 1.137(0.014) | 1.640(0.128) | -2.394(0.097) | 0.433(0.119) | 0.00
566 | 03:06:19.07 | +44:24:53.68 | 17.558(0.014) | 0.042(0.042) | 0.616(0.020) | 17.463(0.003) | 0.786(0.020) | -0.717(0.177) | -0.447(0.122) | 0.149(0.121) | 0.99
567 | 03:06:19.29 | +44:19:29.22 | 15.451(0.006) | 0.058(0.011) | 0.697(0.008) | 15.287(0.003) | 0.908(0.005) | -5.602(0.039) | -2.523(0.030) | 0.604(0.036) | 0.00
568 | 03:06:19.45 | +44:27:38.77 | 16.806(0.042) | 0.534(0.156) | 0.941(0.044) | 16.465(0.003) | 1.235(0.008) | 5.074(0.078) | -1.965(0.062) | 0.799(0.068) | 0.00
569 | 03:06:19.77 | +44:18:25.54 | 19.881(0.101) | — | 0.658(0.152) | 19.655(0.005) | 1.061(0.069) | 0.254(0.530) | 0.166(0.412) | 0.264(0.521) | 0.88
NGC 1798 |
ID | RA | DEC | $V$ | $U-B$ | $B-V$ | $G$ | $G_{\rm BP}-G_{\rm RP}$ | $\mu_{\alpha}\cos\delta$ | $\mu_{\delta}$ | $\varpi$ | $P$
| (hh:mm:ss.ss) | (dd:mm:ss.ss) | (mag) | (mag) | (mag) | (mag) | (mag) |  (mas yr-1) |  (mas yr-1) | (mas) |
001 | 05:11:19.70 | +47:39:52.01 | 20.254(0.135) | — | 0.563(0.192) | 19.499(0.004) | 1.573(0.056) | 0.968(0.426) | -0.986(0.343) | 0.270(0.344) | 0.95
002 | 05:11:19.73 | +47:38:35.08 | 19.765(0.089) | — | 1.182(0.171) | 19.278(0.004) | 1.575(0.048) | 0.919(0.372) | -0.611(0.283) | -0.051(0.289) | 0.91
003 | 05:11:19.79 | +47:41:26.68 | 18.623(0.041) | — | 1.046(0.074) | 18.215(0.003) | 1.471(0.024) | -0.020(0.175) | -0.596(0.137) | 0.083(0.140) | 0.29
004 | 05:11:20.00 | +47:43:15.08 | 20.046(0.151) | — | 0.920(0.208) | 19.369(0.004) | 1.604(0.057) | 1.623(0.331) | -2.917(0.270) | 0.418(0.279) | 0.00
005 | 05:11:20.44 | +47:41:06.65 | 17.635(0.016) | — | 1.442(0.039) | 17.001(0.003) | 1.683(0.012) | 3.835(0.152) | -3.762(0.118) | 0.584(0.116) | 0.00
… | … | … | … | … | … | … | … | … | … | … | …
486 | 05:11:58.43 | +47:39:28.67 | 16.202(0.024) | 0.423(0.045) | 0.494(0.032) | 16.057(0.003) | 0.747(0.006) | 0.920(0.051) | -0.354(0.042) | 0.229(0.042) | 1.00
487 | 05:11:58.51 | +47:38:31.88 | 17.366(0.020) | 0.807(0.267) | 1.321(0.046) | 16.766(0.003) | 1.623(0.008) | 0.843(0.073) | -1.241(0.059) | 0.119(0.059) | 0.97
488 | 05:11:58.73 | +47:38:58.99 | 17.628(0.024) | 0.286(0.162) | 0.913(0.040) | 17.158(0.003) | 1.322(0.008) | 0.805(0.098) | -3.354(0.077) | 0.354(0.074) | 0.00
489 | 05:11:58.88 | +47:40:05.24 | 15.629(0.040) | 0.281(0.038) | 0.551(0.043) | 15.582(0.003) | 0.870(0.005) | 0.701(0.056) | -0.518(0.043) | 0.289(0.043) | 1.00
490 | 05:11:59.15 | +47:42:42.85 | 20.344(0.191) | — | 0.931(0.298) | 19.339(0.004) | 1.461(0.057) | 0.414(0.358) | 0.057(0.294) | 0.906(0.305) | 0.18

Table 2:  The mean internal photometric errors and number of measured stars in the corresponding $V$ apparent-magnitude interval for each cluster. NGC 1193 | NGC 1798
---|---
$V$ | $N$ | $\sigma_{\rm V}$ | $\sigma_{\rm U-B}$ | $\sigma_{\rm B-V}$ | $N$ | $\sigma_{\rm V}$ | $\sigma_{\rm U-B}$ | $\sigma_{\rm B-V}$
(8, 12] | 1 | 0.004 | 0.007 | 0.007 | — | — | — | —
(12, 14] | 11 | 0.019 | 0.021 | 0.024 | 5 | 0.04 | 0.047 | 0.045
(14, 15] | 16 | 0.007 | 0.012 | 0.010 | 7 | 0.022 | 0.055 | 0.035
(15, 16] | 28 | 0.008 | 0.019 | 0.011 | 44 | 0.021 | 0.056 | 0.033
(16, 17] | 45 | 0.012 | 0.037 | 0.016 | 70 | 0.020 | 0.070 | 0.033
(17, 18] | 178 | 0.018 | 0.065 | 0.028 | 114 | 0.023 | 0.148 | 0.037
(18, 19] | 209 | 0.034 | 0.120 | 0.057 | 140 | 0.045 | 0.230 | 0.075
(19, 20] | 152 | 0.076 | 0.234 | 0.132 | 126 | 0.077 | — | 0.141
(20, 21] | 26 | 0.146 | — | 0.244 | 26 | 0.136 | — | 0.242
NGC 1193 | NGC 1798
$G$ | $N$ | $\sigma_{\rm G}$ | $\sigma_{G_{\rm BP}-G_{\rm RP}}$ |  | $N$ | $\sigma_{\rm G}$ | $\sigma_{G_{\rm BP}-G_{\rm RP}}$ |
(5, 10] | 5 | 0.003 | 0.006 |  | 6 | 0.003 | 0.005 |
(10, 12] | 21 | 0.003 | 0.005 |  | 32 | 0.003 | 0.005 |
(12, 13] | 52 | 0.003 | 0.005 |  | 59 | 0.003 | 0.005 |
(13, 14] | 101 | 0.003 | 0.005 |  | 125 | 0.003 | 0.005 |
(14, 15] | 196 | 0.003 | 0.006 |  | 315 | 0.003 | 0.005 |
(15, 16] | 366 | 0.003 | 0.006 |  | 630 | 0.003 | 0.006 |
(16, 17] | 634 | 0.003 | 0.010 |  | 1115 | 0.003 | 0.009 |
(17, 18] | 1219 | 0.003 | 0.019 |  | 2026 | 0.003 | 0.017 |
(18, 19] | 1588 | 0.004 | 0.044 |  | 2936 | 0.003 | 0.035 |
(19, 20] | 2616 | 0.005 | 0.155 |  | 3943 | 0.004 | 0.075 |
(20, 21] | 2144 | 0.011 | 0.232 |  | 3647 | 0.007 | 0.152 |
(21, 23] | 198 | 0.027 | 0.378 |  | — | — | — |

To obtain precise astrophysical parameters, we identified photometric
completeness limits for each cluster. Stars fainter that these limits were not
included in further analyses. $G$ and $V$ magnitude histograms were
constructed to determine the photometric completeness limits for each clusters
(see Fig. 2). Stellar counts decrease for magnitudes fainter than $G=20$ for
both NGC 1193 (Fig. 2a) and NGC 1798 (Fig. 2c). Stellar counts reduce for
magnitudes fainter than $V=19$ for NGC 1193 (Fig. 2b) and NGC 1798 (Fig. 2d),
indicating that incompleteness (of stellar recovery) has set in. Thus, for
both clusters, we adopted these values as the cluster photometric completeness
limits.

###  0.3.2 Structural Parameters of the Clusters

We utilized Radial Density Profile (RDP) analysis to determine the structural
parameters of the studied clusters. First, we specified many concentric rings
out from the cluster center, using the central coordinates given by Cantat-
Gaudin et al. (2020). Stellar densities ($\rho$) were estimated for each ring
by dividing the number of stars within the photometric completeness limit
($G\leq 20$ mag) in it by the ring area. The resulting RDPs were fitted with
King (1962) models via $\chi^{2}$ minimisation, giving estimates for the core,
limiting, and effective radii of each cluster. The King (1962) model is
described as $\rho(r)=f_{\rm bg}+[f_{\rm 0}/(1+(r/r_{\rm c})^{2})]$ where $r$
is the radius from the cluster centre, $f_{\rm bg}$ the background density,
$f_{\rm 0}$ the central density, and $r_{\rm c}$ the core radius. See Figure 3
for each cluster’s RDP together with the best fitting King (1962) model to it.
As a result of the fitting procedure, we inferred central stellar density,
core radius and background stellar density as $f_{\rm 0}=166.865\pm 1.573$
stars arcmin-2, $r_{\rm c}=0.526\pm 0.009$ arcmin and $f_{\rm bg}=5.225\pm
0.124$ stars arcmin-2 for NGC 1193 and $f_{\rm 0}=53.597\pm 3.789$ stars
arcmin-2, $r_{\rm c}=1.190\pm 0.056$ arcmin and $f_{\rm bg}=11.318\pm 0.321$
stars arcmin-2 for NGC 1798, respectively. At the $r=8$ arcmin limiting
radius, the stellar density becomes similar to the background density (a grey
horizontal line) as seen in Fig. 3a (NGC 1193) and Fig. 3b (NGC 1798).
Therefore, we concluded that the limiting radii for both clusters are $r_{\rm
lim}=8$ arcmin. We considered only the stars inside these limiting radii in
further analyses.

Figure 3: Radial density profiles for NGC 1193 (a) and NGC 1798 (b). Errors
were derived using the equation of $1/\sqrt{N}$, where $N$ represents the
number of stars used in the density estimation. The solid lines represents the
optimal King (1962) profiles. The background density level and its errors are
the horizontal grey bands. The King fit uncertainty ($1\sigma$) is shown by
the red shaded region.

###  0.3.3 CMDs and Membership Probabilities of Stars

The membership probabilities ($P$) of stars located in each of two cluster
regions were calculated applying the Unsupervised Photometric Membership
Assignment in Stellar Cluster program (upmask; Krone-Martins & Moitinho,
2014). upmask uses $k$-means clustering, where $k$ is the number of clusters,
to detect spatially concentrated groups and identify the most likely cluster
members. As an integer $k$-means is not adjusted directly by the user and the
best result from the upmask methodology is achieved when the $k$-means value
is within 6 to 25 (Krone-Martins & Moitinho, 2014; Cantat-Gaudin et al.,
2020). We applied upmask to calculate stellar membership probabilities by
considering each star’s five-dimensional astrometric parameters from Gaia EDR3
(Gaia collaboration et al., 2021), which contains equatorial coordinates
($\alpha$, $\delta$), proper motion components ($\mu_{\alpha}\cos\delta$,
$\mu_{\delta}$), trigonometric parallaxes ($\varpi$), and their uncertainties.
During application we scaled these five parameters to unit variance and ran
100 iterations for each clusters to assess cluster membership. The membership
probability of a star is defined by the frequency of the group in which it is
clustered. We reached the best results when $k$ were set to 12 for NGC 1193
and 15 for NGC 1798. We identified as possible cluster members those stars
brighter than $G=20$ mag with membership probabilities $P\geq 0.5$ that we
identified as possible members of clusters. This led to 735 possible members
for NGC 1193 and 1,536 for NGC 1798. Cantat-Gaudin et al. (2020) give the
number of stars brighter than $G=18$ mag with the membership probabilities
$P>0.5$ as 215 for NGC 1193 and 218 for NGC 1798. The dissimilarity can be
explained by lower precision in the astrometric Gaia DR2 data compared to Gaia
EDR3, as well as the $G$ magnitude limit of stars used in analyses. With the
release of Gaia EDR3 data, the precision of astrometric and photometric
measurements increased with reference to Gaia EDR2 data. For the Gaia EDR3
release accuracy of trigonometric parallaxes increased by 30 percent and
uncertainties reduced by nearly 40%, proper motion accuracy increased by a
factor of 2 and associated uncertainties improved by a factor $\sim 2.5$.
Moreover, the precision of photometric data and celestial positions are better
in terms of homogeneity (Gaia collaboration et al., 2021).

To take into consideration the impact of binary stars in the main-sequences of
the studied clusters, we plotted the $V\times(B-V)$ CMDs of the stars within
the cluster limiting radii ($r_{\rm lim}$) which we had obtained above for the
clusters and then fitted the Zero Age Main-Sequence (ZAMS) of Sung et al.
(2013) to these diagrams. The ZAMS fitting was by eye according to the stars
with the membership probability $P\geq 0.5$ and shifted 0.75 mag towards
brighter magnitudes in order to account for the most likely cluster binary
stars (4a and c). During the ZAMS fitting we made sure for each cluster that
the main-sequence, turn-off, and giant stars with the membership probabilities
$P\geq 0.5$ were selected. The process was resulted in 361 likely member stars
for NGC 1193 and 428 for NGC 1798 which lie between the fitted ZAMS curves and
are located inside the $r_{\rm lim}$ radii. We used these stars to determine
astrophysical parameters of the two clusters. Fig. 4 shows the $V\times(B-V)$
CMDs with the best fitted ZAMS (Figs. 4a and c) and $G\times(G_{\rm BP}-G_{\rm
RP}$) CMDs (Figs. 4b and d) with the background and most likely member stars.
Fig. 5 presents histograms of number of stars located through the two cluster
fields versus their membership probabilities. Vector-Point Diagrams (VPDs)
were plotted for the stars within the limiting radii and are shown as Fig. 6.
It can be seen from the figure that NGC 1193 (Fig. 6a) and NGC 1798 (Fig. 6b)
are affected by field stars but with the membership selection criteria, the
‘most likely’ cluster stars (shown as the color-scaled points in Fig. 6) can
be separated from field stars (grey dots in Fig. 6). The mean proper motion
components of the most likely cluster members are ($\mu_{\alpha}\cos\delta$,
$\mu_{\delta})=(-0.207\pm 0.009,-0.431\pm 0.008$) for NGC 1193 and
($\mu_{\alpha}\cos\delta$, $\mu_{\delta})=(0.793\pm 0.006,-0.373\pm 0.005$)
mas yr-1 for NGC 1798. Moreover, using these members we obtained mean
trigonometric parallaxes of NGC 1193 and NGC 1798 as $\varpi_{\rm
Gaia}=0.191\pm 0.157$ mas and $\varpi_{\rm Gaia}=0.203\pm 0.099$ mas,
respectively.

Figure 4: $V\times(B-V)$ and $G\times(G_{\rm BP}-G_{\rm RP})$ CMDs of NGC 1193
(a, b) and NGC 1798 (c, d). The blue dot-dashed lines represent the ZAMS (Sung
et al., 2013) including the binary star effect. The membership probabilities
of stars that lie within the fitted ZAMS are shown with different colors
according to the color scales shown to the right of the figure. These member
stars are located within $r_{\rm lim}=8$ arcmin of the cluster centres
calculated for NGC 1193 and NGC 1798. Grey dots indicate low probability
members ($P<0.5$), or field stars ($P=0$). Figure 5: Histograms of the
membership probabilities for NGC 1193 (a) and NGC 1798 (b). The red colored
shading denotes the stars that lie within the main-sequence band and effective
cluster radii ($\rm r_{lim}\leq 8^{\prime}$), while the white colored bars
indicate the membership probabilities of all stars in each cluster’s
direction.
Figure 6: VPDs of NGC 1193 (a) and NGC 1798 (b). Colored dots identify the
membership probabilities of the most likely cluster members according to color
scale shown on the right. The zoomed box in panels represents the region of
condensation for both clusters in the VPD. Dashed lines are the intersection
of the mean proper motion values.

##  0.4 Astrophysical Parameters of the Clusters

We summarize in this section the processes we performed to determine the
astrophysical parameters of NGC 1193 and NGC 1798 (for detailed descriptions
on the methodology see Yontan et al., 2015, 2019, 2021; Ak et al., 2016; Bilir
et al., 2006, 2010, 2016; Bostancı et al., 2015, 2018; Banks et al., 2020;
Akbulut et al., 2021; Koç et al., 2022). Color excesses and metallicities of
the clusters were derived using two-color diagrams (TCDs), whereas we obtained
distance moduli and ages individually by fitting theoretical models on CMDs.

###  0.4.1 Reddening

The $E(U-B)$ and $E(B-V)$ color excesses for NGC 1193 and NGC 1798 were
derived using $(U-B)\times(B-V)$ TCDs. We selected the main-sequence stars for
which simultaneous $U$, $B$, and $V$ magnitudes are available as well as with
membership probabilities $P\geq 0.5$. As shown in Fig. 7, we constructed TCDs
for these stars and compared their positions by fitting the solar metallicity
de-reddened ZAMS of Sung et al. (2013). The ZAMS was fitted according to the
equation $E(U-B)=0.72\times E(B-V)+0.05\times E(B-V)^{2}$ (Garcia et al.,
1988) by applying $\chi^{2}$ optimisation with steps of 0.001 mag. The best
solutions for $E(B-V)$ and $E(U-B)$ values are those corresponding to the
minimum $\chi^{2}$, being $E(B-V)=0.150\pm 0.037$ mag for NGC 1193 and
$E(B-V)=0.505\pm 0.100$ mag for NGC 1798. The errors of color excesses are
determined as $\pm 1\sigma$ deviations, and are presented as the green lines
in Fig. 7. When we compared the reddening estimated for NGC 1193, we concluded
that it is in a good agreement within the errors with the values ($0.10\leq
E(B-V)\leq 0.19$ mag) given by different authors (Kaluzny, 1988; Tadross,
2005; Kyeong et al., 2008). For NGC 1798, our finding result is compatible
with the values given by Park & Lee (1999, $E(B-V)=0.51\pm 0.04$ mag) and
Oralhan et al. (2015, $E(B-V)=0.47\pm 0.07$ mag).

Figure 7: Two-color diagrams of the most probable member main-sequence stars
in the regions of NGC 1193 (a) and NGC 1798 (b). Red dashed and green solid
curves represent the reddened ZAMS given by Sung et al. (2013) and $\pm
1\sigma$ standard deviations, respectively.  Figure 8: Two-color diagrams
(upper panels) and the distributions of normalised $\delta_{0.6}$ (lower
panels) for NGC 1193 (a) and NGC 1798 (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.  Table 3: Metallicities
calculated for two clusters. $N$ is the number of member stars used in the
analyses.

NGC 1193 | NGC 1798
---|---
$\langle[{\rm Fe/H}]\rangle$ | $N$ | Survey/ | Ref | $\langle[{\rm Fe/H}]\rangle$ | $N$ | Survey/ | Ref
(dex) |  | Catalog/ |  | (dex) |  | Catalog/ |
|  | Telescope |  |  |  | Telescope |
$-0.51\pm 0.09$ | 4 | KPNO | (01) | $-0.18\pm 0.02$ | 4 | KPNO | (10)
$-0.22\pm 0.14$ | 2 | HET | (02) | $-0.165$ | 4 | KPNO | (11)
$-0.17\pm 0.13$ | 1 | HET | (03) | $-0.18\pm 0.01$ | 4 | KPNO | (06)
$-0.22\pm 0.01$ | 1 | PASTEL | (04) | $-0.294$ | 4 | KPNO | (12)
$-0.17$ | 1 | HET | (05) | $-0.34\pm 0.01$ | 4 | KPNO | (13)
$-0.25\pm 0.01$ | 2 | APOGEE DR14 | (06) | $-0.200\pm 0.006$ | 4 | KPNO | (14)
$-0.34\pm 0.01$ | 3 | APOGEE DR16 | (07) | $-0.30\pm 0.02$ | 4 | KPNO | (15)
$-0.320\pm 0.012$ | 1 | GALAH DR3 | (08) | $-0.27\pm 0.03$ | 4 | KPNO | (07)
$-0.30\pm 0.06$ | 12 | SPMO | (09) | $-0.267\pm 0.007$ | 4 | KPNO | (08)
|  |  |  | $-0.20\pm 0.07$ | 7 | SPMO | (09)

(01) Friel et al. (2002), (02) Friel, Jacobson, & Pilachowski (2010), (03)
Jacobson & Friel (2013), (04) Heiter et al. (2014), (05) Overbeek, Friel, &
Jacobson (2016), (06) Carrera et al. (2019), (07) Donor et al. (2020), (08)
Spina et al. (2021), (09) This study, (10) Donor et al. (2018), (11) Ting,
Hawkins, & Rix (2018), (12) Ting & Rix (2019), (13) Hasselquist et al. (2020),
(14) Sit & Ness (2020), (15) Olney et al. (2020)

###  0.4.2 Metallicities

The determination of photometric metallicities of the two clusters employed
the method given by Karaali et al. (2003a, b, 2011). This method is based on F
and G type main-sequence stars and their UV-excesses as well as the stars
whose color index range correspond to $0.3\leq(B-V)_{0}\leq 0.6$ mag (Eker et
al., 2018, 2020). We selected F-G type main-sequence stars within the range
$0.3\leq(B-V)_{0}\leq 0.6$ mag after calculating the intrinsic $(B-V)_{0}$ and
$(U-B)_{0}$ colors of the most likely cluster member ($P\geq 0.5$) stars. To
determine the difference between the $(U-B)_{0}$ color indices of cluster
stars and Hyades main sequence which corresponds to the same $(B-V)_{0}$ color
indices, we constructed $(U-B)_{0}\times(B-V)_{0}$ TCDs. This difference
between cluster and Hyades stars is defined as the UV-excess which is
expressed by the equation of $\delta=(U-B)_{\rm 0,H}-(U-B)_{\rm 0,S}$, where H
and S denote the Hyades and cluster stars respectively, which implies the same
$(B-V)_{0}$ color indices. By calibrating $(B-V)_{0}$ of stars to
$(B-V)_{0}=0.6$ mag (i.e., $\delta_{0.6}$) we normalised the UV excess and
plotted the histogram of normalised $\delta_{0.6}$ values. To calculate the
mean $\delta_{0.6}$, we fitted a Gaussian to the distribution. Taking into
account the Gaussian peak, the photometric metallicities of the studied
clusters are obtained from the equation given by Karaali et al. (2011):

$\displaystyle{\rm[Fe/H]}=-14.316(1.919)\delta_{0.6}^{2}-3.557(0.285)\delta_{0.6}+0.105(0.039).$
(1)

We identified 12 and 7 F-G type main-sequence stars to calculate the
photometric metallicity of NGC 1193 and NGC 1798, respectively. TCDs and the
distributions of normalised $\delta_{0.6}$ UV excesses for two clusters are
shown in Fig. 8. The calculated mean $\delta_{0.6}$ values of NGC 1193 and NGC
1798 are 0.085$\pm$0.010 mag and 0.068$\pm$0.011 mag, respectively. The
photometric metallicity [Fe/H] value for NGC 1193 is ${\rm[Fe/H]}=-0.30\pm
0.06$ dex and for NGC 1798 ${\rm[Fe/H]}=-0.20\pm 0.07$ dex, which correspond
to their peak values in the $\delta_{0.6}$ distribution.

The [Fe/H] metallicities were transformed to the mass fraction $Z$ to derive
ages of the clusters. For this, the analytic equations of
Bovy333https://github.com/jobovy/isodist/blob/master/isodist/Isochrone.py,
444The equations are given in lines between 199 and 207 in the code. for
parsec (Bressan et al., 2012) models were used, namely:

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

and

$z=\frac{(z_{\rm x}-0.2485\times z_{\rm x})}{(2.78\times z_{\rm x}+1)}.$ (3)

$z$ and $z_{\rm x}$ are the elements heavier than helium and the intermediate
operation function, respectively. $z_{\odot}$ is the solar metallicity which
was adopted as 0.0152 (Bressan et al., 2012). We calculated $z=0.008$ for NGC
1193 and $z=0.010$ for NGC 1798.

Many authors obtained spectroscopic metallicities of NGC 1193 and NGC 1798
based on ground-based observations, as listed in Table 3. Photometric
metallicities calculated in this study are well supported by the spectroscopic
studies presented in literature. We conclude that our metallicity findings are
reliable. Thus, we adopted our results for the determination of distance
moduli and age.

###  0.4.3 Distance Moduli and Age Estimation

We used parsec isochrones (Bressan et al., 2012), which contain UBV filters as
well as Gaia pass-bands, to obtain the distance moduli and ages of the studied
clusters simultaneously. To do this, we selected the parsec models considering
the mass fractions ($z$) estimated for each cluster and compared them to the
$V\times(U-B)$, $V\times(B-V)$, and $G\times(G_{\rm BP}-G_{\rm RP})$ CMDs
according to member stars ($P\geq 0.5$). Selected isochrones were fitted to
CMDs visually by attaching importance to ‘most likely’ member stars which make
up the main-sequence, turn-off and giant regions of each cluster. During the
fitting process of parsec models to the $UBV$ data, we used the $E(B-V)$
values derived above by this study, while for the Gaia EDR3 data we considered
the equation of $E(G_{\rm BP}-G_{\rm RP})=1.41\times E(B-V)$ (Sun et al.,
2021). We obtained the error of the distance moduli and distances using the
relation given by Carraro et al. (2017). We fitted two more isochrones to
estimate age uncertainties considering the spread of the most likely member
stars in the turn-off and sub-giant regions of the cluster. The age of such
selected isochrones give the higher and lower acceptable values for the
estimated cluster ages. The best fit with $z=0.008$ gave the distance moduli
and age of NGC 1193 as $\mu=14.191\pm 0.149$ mag and $t=4.6\pm 1.0$ Gyr. For
NGC 1798, the best fit of $z=0.010$ gave these values as $\mu=14.808\pm 0.332$
mag and $t=1.3\pm 0.2$ Gyr, respectively. The distances of the clusters
corresponding to the estimated distance moduli are also $d_{\rm iso}=5562\pm
381$ pc for NGC 1193 and $d_{\rm iso}=4451\pm 728$ pc for NGC 1798. The
$V\times(U-B)$, $V\times(B-V)$, and $G\times(G_{\rm BP}-G_{\rm RP})$ CMDs with
the best fit isochrones and associated errors are shown in Fig. 9.

The isochrone-based distance for NGC 1193 as estimated by this study is
compatible with the result given by Tadross (2005, $d=5.25\pm 0.24$ kpc). As
well, the estimated age of the cluster is in a good agreement with the value
of Kyeong et al. (2008, $t=5.0\pm 1.3$ Gyr). For NGC 1798, the derived
distance matches well within the errors with the result of Park & Lee (1999,
$d=4.2\pm 0.3$ kpc). The age of the cluster is coherent with the findings
given by Park & Lee (1999, $t=1.4\pm 0.2$ Gyr) and Maciejewski & Niedzielski
(2007, $t=1.6$ Gyr).

Applying the linear equation of $\varpi\>({\rm mas})=1000/d\>({\rm pc})$, we
converted isochrone distances to trigonometric parallaxes for the two
clusters. This indicated that the parallax distances of NGC 1193 and NGC 1798
are $\varpi_{\rm iso}=0.180\pm 0.012$ mas and $\varpi_{\rm iso}=0.225\pm
0.037$ mas, respectively. It is concluded that these values are in good
agreement with the Gaia EDR3 trigonometric parallax distances for both
clusters.

Figure 9: CMDs for the NGC 1193 (panels a, b, and c) and NGC 1798 (panels d,
e, and f). The differently colored dots represent the membership probabilities
according to the color scales shown on the right side of the diagrams. Grey
dots indicate low probability members ($P<0.5$), or field stars ($P=0$). The
blue lines show the parsec isochrones, while the shaded areas surrounding
these lines are their associated errors.

##  0.5 Kinematics and Galactic Orbit Parameters of Clusters

The MWPotential2014 (Bovy, 2015) algorithm, one of the potential functions
defined in galpy (the galactic dynamics library (Bovy, 2015)555See also
https://galpy.readthedocs.io/en/v1.5.0/), was applied to calculate the space
velocity components and galactic orbital parameters for NGC 1193 and NGC 1798.
The algorithm assumes an axisymmetric potential for the Milky Way galaxy. We
adopted the galactocentric distance to be $R_{\rm GC}=8$ kpc, the Solar
circular velocity of $V_{\rm rot}=220$ km s-1 (Bovy, 2015; Bovy & Tremaine,
2012), and the Solar distance from the galactic plane as $27\pm 4$ pc (Chen et
al., 2000). Since the MWPotential2014 code comprises bulge, disk, and halo
potentials of the Milky Way, we assumed that it well represents the Galaxy.

Bovy (2015) defined the bulge component as a spherical power law density
profile, given as follow:

$\rho(r)=A\left(\frac{r_{\rm
1}}{r}\right)^{\alpha}\exp\left[-\left(\frac{r}{r_{\rm c}}\right)^{2}\right]$
(4)

where $r_{\rm 1}$ is the present reference radius, $r_{\rm c}$ the cut-off
radius, $A$ the amplitude that is applied to the potential in mass density
units, and $\alpha$ is the inner power. We adopted the potential presented by
Miyamoto & Nagai (1975) for the galactic disk component:

$\Phi_{\rm disk}(R_{\rm GC},Z)=-\frac{GM_{\rm d}}{\sqrt{R_{\rm
GC}^{2}+\left(a_{\rm d}+\sqrt{Z^{2}+b_{\rm d}^{2}}\right)^{2}}}$ (5)

$R_{\rm GC}$ is the distance from the galactic centre, $Z$ the vertical
distance from the galactic plane, $G$ the universal gravitational constant,
$M_{\rm d}$ the mass of the galactic disk, and $a_{\rm d}$ and $b_{\rm d}$
being the scale-length and scale-height of the disk, respectively.

The potential for the halo component was obtained by Navarro et al. (1996) as:

$\Phi_{\rm halo}(r)=-\frac{GM_{\rm s}}{R_{\rm GC}}\ln\left(1+\frac{R_{\rm
GC}}{r_{\rm s}}\right)$ (6)

where $M_{\rm s}$ is the mass of the dark matter halo of the Milky Way and
$r_{\rm s}$ is its radius.

To determine the spacial velocities and galactic orbit parameters of NGC 1193
and NGC 1798, we used the equatorial coordinates, proper motion components,
distances, and radial velocity data with their uncertainties in the
calculations. These values are listed in Table 4 (on page 4). We performed
kinematic and dynamic analyses with 1 Myr steps over a 3.5 Gyr integration
time. We considered the proper motion components and distances of the two
clusters as derived by this study (see Section 0.3.3), while for the radial
velocities we used the data of Donor et al. (2020) who gave $\langle V_{\rm
r}\rangle=-84.7\pm 0.2$ km s-1 for NGC 1193 and $\langle V_{\rm
r}\rangle=2.7\pm 0.8$ km s-1 for NGC 1798. As a result, we obtained for both
clusters estimates of apogalactic distance $R_{\rm a}$, perigalactic distance
$R_{\rm p}$, eccentricity $e$, maximum vertical distance from galactic plane
$Z_{\rm max}$, galactic space velocity components ($U$, $V$, $W$), and orbital
period $T$. These estimates are listed in Table 4. The space velocity
components $(U,V,W)$ were calculated as $(70.95\pm 0.16,-47.62\pm 0.10,5.56\pm
0.59)$ for NGC 1193 and $(-7.18\pm 1.50,-14.64\pm 2.27,9.13\pm 1.69)$ for NGC
1798. In their study based on Gaia DR2 astrometric data (Gaia collaboration et
al., 2018), Soubiran et al. (2018) derived the space velocity components for
NGC 1193 as $(U,V,W)$ = ($68.84\pm 0.53$, $-46.77\pm 0.54$, $9.00\pm 0.65$)
and for NGC 1798 as $(U,V,W)$ = ($7.50\pm 0.41$, $-16.63\pm 0.50$, $12.85\pm
0.39$) km s-1. These results are in good agreement with the values calculated
in the study. The correction to the local standard of rest (LSR), given by
Coşkunoǧlu et al. (2011) as ($U,V,W$) = $(8.83\pm 0.24$, $14.19\pm 0.34$,
$6.57\pm 0.21)$ km s-1, was applied to the space velocity components. The
derived LSR corrected space velocity components are $(U,V,W)_{\rm LSR}$ =
($79.78\pm 0.29$, $-33.43\pm 0.35$, $12.13\pm 0.62$) for NGC 1193 and
$(U,V,W)_{\rm LSR}$ = ($1.65\pm 1.51$, $-0.45\pm 2.30$, $15.70\pm 1.70$) km
s-1 for NGC 1798. Moreover, the space velocities of NGC 1193 were calculated
to be $87.35\pm 0.77$ km s-1 and $15.79\pm 3.23$ km s-1 for NGC 1798.

Considering the space velocity components of stars in different Galactic
populations, Schuster et al. (2012) divided the stars into thin disk
($-50<V_{\rm LSR}$ km/s), thick disk ($-180<V_{\rm LSR}\leq-50$ km/s) and halo
($V_{\rm LSR}\leq 180$ km/s) groups. Figure 10 shows the positions of the
clusters according to Schuster et al. (2012)’s kinematic criteria. According
to these criteria, the open clusters NGC 1193 and NGC 1798 appear to be
members of the thick disk and thin disk populations, respectively. Considering
the metal abundance range of NGC 1193 ($-0.51\leq[{\rm Fe/H]}\leq-0.17$ dex,
see Table 3), it is concluded that the cluster belongs to the metal-rich side
of the thick-disk population.

Figure 10: Toomre diagram for NGC 1193 and NGC 1798. Blue, pink and yellow
regions show thin disk, thick disk and halo populations, respectively.

Figure 11 presents the orbits of NGC 1193 (Fig. 11a) and NGC 1798 (Fig. 11c)
as functions of distance from the galactic center and the galactic plane
($Z\times R_{\rm GC}$ and $R_{\rm GC}\times t$). The birth and present-day
locations for the two clusters are marked with yellow triangles and circles in
sub-figures 11b and 11d. Figures 11a and 11c show that both of the clusters
entirely orbit outside the solar circle. The orbital eccentricities of NGC
1193 and NGC 1798 are smaller than 0.15, thus their orbits are close to
circular. The results of orbital integrations imply that NGC 1193 reaches its
maximum vertical distance from the galactic plane at $Z_{\rm max}=1342\pm 77$
pc with an orbital period $T=370\pm 12$ Myr, and these values correspond to
$Z_{\rm max}=725\pm 148$ pc and $T=381\pm 23$ Myr for NGC 1798. Considering
the age values determined in this study for the clusters, we ran the galpy
program backwards in time and examined the resulting birth–places. The program
indicated that the birth-place radial distances are 10.86 kpc and 11.82 kpc
for NGC 1193 and NGC 1798, respectively, meaning that the clusters were born
in the metal-poor region outside the solar circle.

Figure 11:  The galactic orbits and birth radii of NGC 1193 (a,b) and NGC 1798
(c,d) in the $Z\times R_{\rm GC}$ and $R_{\rm GC}\times t$ planes. The filled
yellow circles and triangles show the present day and birth positions,
respectively. Red arrows are the motion vectors of OCs.

##  0.6 Luminosity and Present-day Mass Functions

The distribution of stars according to their brightness is defined as the
luminosity function (LF). We used Gaia EDR3 photometric data to determine LFs
for the two clusters. For this, main-sequence stars located inside the 8
arcmin limiting radii, as derived above, were selected for the two clusters.
The magnitude ranges of the chosen stars are within the $17.25\leq G\leq 20$
mag for NGC 1193 and $16.5\leq G\leq 20$ mag for NGC 1798. We converted the
$G$ magnitudes of the selected stars to absolute magnitudes with the equation
$M_{\rm G}=G-5\times\log d+5+A_{\rm G}$, where $G$ is the apparent magnitude
and $d$ the distance derived earlier in this study. $A_{G}$ is extinction for
$G$ magnitudes and represented by $A_{G}=0.84\times A_{V}$ (Sun et al., 2021)
(here $A_{V}$ is the extinction for $V$ magnitudes). This led to the absolute
magnitude ranges being limited within the $2.5<M_{\rm G}<5.5$ and $0.5<M_{\rm
G}<4.5$ mag for NGC 1193 and NGC 1798, respectively. We constructed LF
histograms with the step-size 0.5 mag, as shown as Fig. 12 for both clusters.

Figure 12:  The luminosity functions of NGC 1193 (a) and NGC 1798 (b). The
histograms show the absolute magnitudes of the main-sequence stars belonging
to the clusters.

To convert these LFs to present day mass functions (PDMFs) we employed the
parsec isochrones (Bressan et al., 2012), which gives the ages and metal
abundances ($z$) of the clusters. We utilized a high degree polynomial
equation between $G$-band absolute magnitudes and masses of theoretical main-
sequence stars. The resulting absolute magnitude-mass relation was used to
transform the observational absolute $G$ band magnitudes to masses. The
number, mass range, and mean mass of main-sequence stars that transformed are
212, $0.85\leq M/M_{\odot}\leq 1.2$, and 0.99$M_{\rm\odot}$ for NGC 1193, and
226, $1.1\leq M/M_{\odot}\leq 2$, and 1.53 $M_{\rm\odot}$ for NGC 1798. The
mass function PDMF can be approximated by a power law defined as by Salpeter
(1955):

${\log(\frac{dN}{dM})=-(1+\Gamma)\times\log(M)+{\rm constant}}$ (7)

Here $dN$ is the number of stars in a mass bin of width $dM$ with a central
mass $M$ and $\Gamma$ being the slope of the PDMF. We estimated the slope of
the PDMF in both clusters for apparent $G\leq 20$ mag, which corresponds to
stars more massive than $0.85\>M_{\odot}$ in NGC 1193 and $1.1\>M_{\odot}$ for
NGC 1798. The resulting PDMFs with the best fits are presented in Fig. 13. We
calculated the slope values to be $\Gamma=1.38\pm 2.16$ for NGC 1193 and as
$\Gamma=1.30\pm 0.21$ for NGC 1798. Since the NGC 1193 cluster is about 5.5
kpc from the Sun, the magnitudes of the main-sequence stars are within a
narrow range. This causes the mass range of the main-sequence stars to be
limited and the distribution of the mass function to show a large scatter.
While the PDMF of the NGC 1193 is compatible with Salpeter (1955)’s result of
$\Gamma=1.35$, the error of the PDMF is large. This situation is not similar
for NGC 1798. Considering the value and error of the PDMF for NGC 1798, it is
in agreement with Salpeter’s result.

Figure 13:  Present-day mass functions of NGC 1193 (a) and NGC 1798 (b)
derived from all sample (red circle). The blue and dashed lines represent the
mass functions of the open clusters and Salpeter (1955)’s mass function,
respectively. The purple dashed lines show $\pm 1\sigma$ prediction levels.

##  0.7 Summary and Conclusion

We performed photometric, astrometric, and kinematic studies of two old age
open clusters, NGC 1193 and NGC 1798, using CCD UBV and Gaia EDR3 data. We
examined the cluster structure, obtaining basic astrophysical parameters as
well as properties of galactic orbits for two clusters. Outcomes of the study
are listed in Table 4 (on page 4) and summarised as follows:

  1. 1.

Performing the RDP analyses, we determined the limiting radii $r_{\rm lim}=8$
arcmin for both clusters. This value corresponds to limiting radii of 12.95 pc
and 10.36 pc for NGC 1193 and NGC 1798, respectively. We considered the stars
within these limiting radii as potential cluster members and restricted
subsequent analysis to this set of stars.

  2. 2.

The calculation of membership probabilities of stars was made using the upmask
program together with a five-dimensional parameter space containing the stars’
proper motion components, trigonometric parallaxes, and their uncertainties.
We considered the stars with probabilities $P\geq 0.5$ to be cluster members.
Additionally we adopted two more criteria to to clarify cluster membership:

Table 4: Fundamental parameters of NGC 1193 and NGC 1798. Parameter | NGC 1193 | NGC 1798
---|---|---
($\alpha,\leavevmode\nobreak\ \delta)_{\rm J2000}$ (Sexagesimal) |  03:05:56.64, $+$44:22:58.80 |  05:11:39.36, $+47$:41:27.60
($l,b)_{\rm J2000}$ (Decimal) |  148.8143, $-12.1624$ |  160.7043, $+04$.8500
$f_{0}$ (stars arcmin-2) | $166.865\pm 1.573$ | $53.597\pm 3.789$
$r_{\rm c}$ (arcmin) | $0.526\pm 0.009$ | $1.190\pm 0.057$
$f_{\rm bg}$ (stars arcmin-2) | $5.225\pm 0.124$ | $11.318\pm 0.321$
$r_{\rm lim}$ (arcmin) | 8 | 8
$r$ (pc) | 12.95 | 10.36
$\mu_{\alpha}\cos\delta$ (mas yr-1) | $-0.207\pm 0.009$ | $0.793\pm 0.006$
$\mu_{\delta}$ (mas yr-1) | $-0.431\pm 0.008$ | $-0.373\pm 0.005$
Cluster members ($P\geq 0.5$) | 361 | 428
$\varpi$ (mas) | $0.191\pm 0.157$ | $0.203\pm 0.099$
$E(B-V)$ (mag) | $0.150\pm 0.037$ | $0.505\pm 0.100$
$E(U-B)$ (mag) | $0.109\pm 0.027$ | $0.376\pm 0.073$
$A_{\rm V}$ (mag) | $0.465\pm 0.084$ | $1.566\pm 0.310$
$[{\rm Fe/H}]$ (dex) | $-0.30\pm 0.06$ | $-0.20\pm 0.07$
Age (Gyr) | $4.6\pm 1.0$ | $1.3\pm 0.2$
Distance modulus (mag) | $14.191\pm 0.149$ | $14.808\pm 0.332$
Isochrone distance (pc) | $5562\pm 381$ | $4451\pm 728$
$(X,Y,Z)_{\odot}$ (pc) |  ($-4651$, 2815, 1172) |  ($-4186$, 1466, 376)
$R_{\rm GC}$ (kpc) | 12.96 | 12.27
PDMF slope | $-1.38\pm 2.16$ | $-1.30\pm 0.21$
$U_{\rm LSR}$ (km/s) | $79.78\pm 0.29$ | $1.65\pm 1.51$
$V_{\rm LSR}$ (km/s) | $-33.43\pm 0.35$ | $-0.45\pm 2.30$
$W_{\rm LSR}$ (km/s) | $12.13\pm 0.62$ | $15.70\pm 1.70$
$S_{{}_{\rm LSR}}$ (km/s) | $87.35\pm 0.77$ | $15.79\pm 3.23$
$R_{\rm a}$ (kpc) | $14.44\pm 0.34$ | $14.11\pm 0.30$
$R_{\rm p}$ (kpc) | $10.80\pm 0.43$ | $11.72\pm 0.50$
$z_{\rm max}$ (pc) | $1342\pm 77$ | $725\pm 148$
$e$ | $0.144\pm 0.008$ | $0.092\pm 0.011$
$P$ (Myr) | $370\pm 12$ | $381\pm 23$
Birthplace (kpc) | 10.86 | 11.82

    1. (a)

binary star contamination in the cluster main-sequences which was interpreted
by the de-reddened ZAMS fitted to $V\times(B-V)$ CMDs with a shift of $+0.75$
mag in the $V$ band, and

    2. (b)

within the limiting radii determined in the study (as per step 1).

Consequently, we selected the stars inside the clusters’ limiting radii,
within best-fitting ZAMS and with the membership probability $P\geq 0.5$ as
‘real’ members of two clusters. Thus we identified 361 and 428 stars as most
likely members of NGC 1193 and NGC 1798, respectively.

  3. 3.

The reddening and photometric metallicities of the two clusters were derived
separately using CCD UBV TCDs. The reddening analyses were performed by
fitting de-reddened ZAMS to main sequence member stars. Photometric
metallicity was based on the comparison of F-G type main sequence members with
the Hyades main-sequence. The reddening and photometric metallicity for NGC
1193 are $E(B-V)=0.150\pm 0.037$ mag and [Fe/H]=$-0.30\pm 0.06$ dex,
respectively. The corresponding values for NGC 1798 are $E(B-V)=0.505\pm
0.100$ mag and [Fe/H]=$-0.20\pm 0.07$ dex.

  4. 4.

The distance moduli, distance, and age of the NGC 1193 were derived as
$\mu_{\rm V}=14.191\pm 0.149$ mag, $d=5562\pm 381$ pc, and $t=4.6\pm 1$ Gyr,
respectively. Similarly $\mu_{\rm V}=14.808\pm 0.332$ mag, $d=4451\pm 728$ pc,
and $t=1.3\pm 0.2$ Gyr were calculated for NGC 1798. These results were
obtained by simultaneously fitting parsec isochrones on the UBV and Gaia EDR3
photometric CMDs utilizing the most likely member stars according to reddening
and metallicities derived in the study.

  5. 5.

Mean proper motion components were calculated as
($\mu_{\alpha}\cos\delta,\mu_{\delta})=(-0.207\pm 0.009,-0.431\pm 0.008$) mas
yr-1 for NGC 1193 and ($\mu_{\alpha}\cos\delta,\mu_{\delta})\\\ =(0.793\pm
0.006,-0.373\pm 0.005$) mas yr-1 for NGC 1798.

  6. 6.

We estimated mean trigonometric parallaxes using Gaia EDR3 data of most likely
members for two clusters. The results are $\varpi_{\rm Gaia}=0.191\pm 0.157$
mas for NGC 1193 and $\varpi_{\rm Gaia}=0.203\pm 0.099$ mas for NGC 1798. We
also converted isochrones distances to trigonometric parallaxes by applying
the linear equation $\varpi\>({\rm mas})=1000/d\>({\rm pc})$ and found
$\varpi_{\rm iso}=0.180\pm 0.012$ mas for NGC 1193 and $\varpi_{\rm
iso}=0.225\pm 0.037$ mas for NGC 1798. For both clusters our derived
trigonometric parallaxes values calculated from isochrone fitting distances
are well supported by the values determined from Gaia EDR3 trigonometric
parallaxes of member stars.

  7. 7.

Space velocities and galactic orbital parameters show that NGC 1193 belongs to
the thick-disk population, whereas NGC 1798 is a member of the thin-disk
population. Moreover, both of the clusters orbit completely outside the solar
circle.

  8. 8.

We found NGC 1193 and NGC 1798 were born outside the solar circle with the
birth radii of 10.86 and 11.82 kpc from the Galactic centre, respectively.
These birth radii indicate the metal-poor formation region and support the
metallicities calculated in the study for the two clusters.

  9. 9.

Present day mass function slopes of $\Gamma=1.38\pm 2.16$ and $\Gamma=1.30\pm
0.21$ were derived for NGC 1193 and NGC 1798, respectively. While the results
for two clusters are in good agreement with the value of Salpeter (1955), that
for NGC 1193 possesses a large uncertainty. We concluded that because of its
distance, the main-sequence stars of NGC 1193 are limited within narrow
magnitudes.

## Acknowledgments

The observations of this publication were made at the National Astronomical
Observatory, San Pedro Mártir, Baja California, México, and the authors wish
to thank the staff of the Observatory for their assistance during these
observations. This research has made use of the WEBDA database, operated at
the Department of Theoretical Physics and Astrophysics of the Masaryk
University. This research has made use of NASA’s Astrophysics Data System. We
also made use of VizieR and Simbad databases at CDS, Strasbourg, France as
well as data from the European Space Agency (ESA) mission _Gaia_
666https://www.cosmos.esa.int/gaia, processed by the _Gaia_ Data Processing
and Analysis Consortium
(DPAC)777https://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.
We thank the University of Queensland for collaboration software.

## References

  * Ahumada & Lapasset (2007) Ahumada, J. A., Lapasset, E., 2007, A&A, 463, 789
  * Ak et al. (2016) Ak, T., Bostancı, Z. F., Yontan, T., et al., 2016, Ap&SS, 361, 126
  * Akbulut et al. (2021) Akbulut, B., Ak, S., Yontan, T., et al., 2021, Ap&SS, 366, 68
  * Banks et al. (2020) Banks, T., Yontan, T., Bilir, S., Canbay, R., 2020, JApA, 41, 6
  * Bressan et al. (2012) Bressan, A., Marigo, P., Girardi, L., Salasnich, B., Dal Cero, C., Rubele, S., Nanni, A., 2012, MNRAS, 427, 127
  * Bertelli et al. (1994) Bertelli, G., Bressan, A., Chiosi, C., Fagotto, F., Nasi, E., 1994, A&AS, 106, 275
  * Bilir et al. (2006) Bilir, S., Güver, T., Aslan, M., 2006, AN, 327, 693
  * Bilir et al. (2010) Bilir, S., Güver, T., Khamitov, I., Ak, T., Ak, S., Coşkunoğlu, K. B., Paunzen, E., Yaz, E., 2010, Ap&SS, 326, 139
  * Bilir et al. (2016) Bilir, S., Bostancı, Z. F., Yontan, T., et al., 2016, AdSpR, 58, 1900
  * Bostancı et al. (2015) Bostancı, Z. F., Ak, T., Yontan, T., et al., 2015, MNRAS, 453, 1095
  * Bostancı et al. (2018) Bostancı, Z. F., Yontan, T., Bilir, S., et al., 2018, Ap&SS, 363, 143
  * Bovy & Tremaine (2012) Bovy, J., Tremaine, S., 2012, ApJ, 756, 89
  * Bovy (2015) Bovy, J., 2015, ApJS, 216, 29
  * Cantat-Gaudin et al. (2018) Cantat-Gaudin, T., Jordi, C., Vallenari, A., et al., 2018, A&A, 618, 93
  * Cantat-Gaudin et al. (2020) Cantat-Gaudin, T., Anders, F., Castro-Ginard, A., et al., 2020, A&A, 640, A1
  * Carrera (2012) Carrera, R., 2012, A&A, 544, A109
  * Carrera et al. (2019) Carrera, R., Bragaglia, A., Cantat-Gaudin, T., et al., 2019, A&A, 623, A80
  * Carraro et al. (2017) Carraro, G., Sales Silva, J. V., Moni Bidin, C., Vazquez, R. A., 2017, ApJ, 153, 20
  * Chen et al. (2000) Chen, Y. Q., Nissen, P. E., Zhao, G., Zhang, H. W., Benoni, T., 2000, A&AS, 141, 491
  * Conrad et al. (2014) Conrad, C., Scholz, R.-D., Kharchenko, N. V., et al., 2014, A&A, 562, A54
  * Coşkunoǧlu et al. (2011) Coşkunoǧlu, B., Ak S., Bilir, S., et al., 2011, MNRAS, 412, 1237
  * Dreyer (1888) Dreyer, J. L. E., 1888, MNRAS, 49, 1
  * Donor et al. (2018) Donor, J., Frinchaboy, P. M., Cunha, K., et al., 2018, AJ, 156, 142
  * Donor et al. (2020) Donor, J., Frinchaboy, P. M., Cunha, K., et al., 2020, AJ, 159, 199
  * Eker et al. (2018) Eker, Z., Bakış, V., Bilir, S., et al., 2018, MNRAS, 479, 5491
  * Eker et al. (2020) Eker, Z., Soydugan, F., Bilir, S., et al., 2020, MNRAS, 496, 3887
  * Friel, Liu, & Janes (1989) Friel, E. D., Liu, T., Janes, K. A., 1989, PASP, 101, 1105
  * Friel & Janes (1993) Friel, E. D., Janes, K. A., 1993, A&A, 267, 75
  * Friel (1995) Friel, E. D., 1995, ARA&A, 33, 381
  * Friel et al. (2002) Friel, E. D., Janes, K. A., Tavarez, M., et al., 2002, AJ, 124, 2693
  * Friel, Jacobson, & Pilachowski (2010) Friel, E. D., Jacobson, H. R., Pilachowski, C. A., 2010, AJ, 139, 1942
  * Gaia collaboration et al. (2016) Gaia Collaboration, Brown A. G. A., Vallenari A., et al., 2016, A&A, 595, A1
  * Gaia collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., Prusti, T., et al., 2018, A&A, 616, 22
  * Gaia collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., Prusti, T., et al., 2021, A&A, 694, 1
  * Garcia et al. (1988) Garcia, B., Claria, J. J., Levato, H., 1988, Ap&SS, 143, 317
  * Gilmore et al. (2012) Gilmore, G., Randich, S., Asplund, M., et al., 2012, Msngr, 147, 25
  * Hasselquist et al. (2020) Hasselquist, S., Zasowski, G., Feuillet, D. K., et al., 2020, ApJ, 901, 109
  * Heiter et al. (2014) Heiter, U., Soubiran, C., Netopil, M., Paunzen, E., 2014, A&A, 561, A93.
  * Jacobson & Friel (2013) Jacobson, H. R., Friel, E. D., 2013, AJ, 145, 107
  * Janes & Adler (1982) Janes, K., Adler, D., 1982, ApJS, 49, 425
  * Janes & Phelps (1994) Janes, K. A., Phelps, R. L., 1994, AJ, 108, 1773
  * Kaluzny (1988) Kaluzny, J., 1988, AcA, 38, 339
  * Karaali et al. (2003a) Karaali, S., Bilir, S., Karataş, Y., Ak, S. G., 2003a, PASA, 20, 165
  * Karaali et al. (2003b) Karaali, S., Ak, S. G., Bilir, S., Karataş, Y., Gilmore, G., 2003b, MNRAS, 343, 1013.
  * Karaali et al. (2011) Karaali, S., Bilir, S., Ak, S., Yaz, E., Coşkunoğlu, B., 2011, PASA, 28, 95
  * King (1962) King, I., 1962, AJ, 67, 471
  * Koç et al. (2022) Koç, S., Yontan, T., Bilir, S., et al., 2022, AJ, 163, 191
  * Kos et al. (2018) Kos, J., de Silva, G., Buder, S., et al., 2018, MNRAS, 480, 5242
  * Krone-Martins & Moitinho (2014) Krone-Martins, A., Moitinho, A., 2014, A&A, 561, 57
  * Kyeong et al. (2008) Kyeong, J., Kim, S. C., Hiriart, D., Sung, E.-C., 2008, JKAS, 41, 147
  * Landolt (2009) Landolt, A. U., 2009, AJ, 137, 4186
  * Lata et al. (2002) Lata, S., Pandey, A. K., Sagar, R., Mohan, V., 2002, A&A, 388, 158
  * Liu & Pang (2019) Liu, L., Pang, X., 2019, ApJS, 245, 32
  * Maciejewski & Niedzielski (2007) Maciejewski, G., Niedzielski, A., 2007, A&A, 467, 1065
  * Miyamoto & Nagai (1975) Miyamoto, M., Nagai, R., 1975, PASJ, 27, 533
  * Monet et al. (2003) Monet, D. G., Levine, S. E., Canzian, B., et al., 2003, AJ, 125, 984
  * Navarro et al. (1996) Navarro, J. F., Frenk, C. S., White, S. D. M., 1996, ApJ, 462, 563
  * Olney et al. (2020) Olney, R., Kounkel, M., Schillinger, C., et al., 2020, AJ, 159, 182
  * Oralhan et al. (2015) Oralhan, İ. A., Karataş, Y., Schuster, W. J., Michel, R., Chavarría C., 2015, NewA, 34, 195
  * Overbeek, Friel, & Jacobson (2016) Overbeek, J. C., Friel, E. D., Jacobson, H. R., 2016, ApJ, 824, 75
  * Park & Lee (1999) Park, H. S., Lee, M. G., 1999, MNRAS, 304, 883
  * Ruprecht (1966) Ruprecht, J., 1966, BAICz, 17, 33
  * Salpeter (1955) Salpeter, E. E., 1955, ApJ, 121, 161
  * Schuster et al. (2012) Schuster, W. J., Moreno, E., Nissen, P. E., Pichardo, B., 2012, A&A, 538, A21
  * Sit & Ness (2020) Sit, T., Ness, M. K., 2020, ApJ, 900, 4
  * Soubiran et al. (2018) Soubiran, C., Cantat-Gaudin, T., Romero-Gómez, M., et al., 2018, A&A, 619, A155
  * Spina et al. (2021) Spina, L., Ting, Y.-S., De Silva, G. M., et al., 2021, MNRAS, 503, 3279
  * Stetson (1987) Stetson, P. B., 1987, PASP, 99, 191
  * Stetson et al. (2019) Stetson, P. B., Pancino, E., Zocchi, A., Sanna, N., Monelli, M., 2019, MNRAS, 485, 3042
  * Sun et al. (2021) Sun, M., Jiang, B., Yuan, H., Li, J., 2021, ApJS, 254, 38
  * Sung et al. (2013) Sung, H., Lim, B., Bessell, M. S., Kim, J. S., Hur, H., Chun, M., Park, B., 2013, JKAS, 46, 1
  * Tadross (2005) Tadross, A. L., 2005, AN, 326, 19
  * Ting, Hawkins, & Rix (2018) Ting, Y.-S., Hawkins, K., Rix, H.-W., 2018, ApJL, 858, L7
  * Ting & Rix (2019) Ting, Y.-S., Rix, H.-W., 2019, ApJ, 878, 21
  * van den Bergh & Sher (1960) van den Bergh, S., Sher, D., 1960, PDDO, 2, 203
  * Yontan et al. (2015) Yontan, T., Bilir, S., Bostancı, Z. F., et al., 2015, Ap&SS, 355, 267
  * Yontan et al. (2019) Yontan, T., Bilir, S., Bostancı, Z. F., et al., 2019, Ap&SS, 364, 20
  * Yontan et al. (2021) Yontan, T., Bilir, S., Ak, T., et al., 2021, AN, 342, 538