# Study of the Open Clusters in Kepler Prime Field

Yüksel Karataş1, Hikmet Çakmak1, İnci Akkaya Oralhan2, Charles Bonatto 3, Raúl
Michel4and Martin Netopil5
1Department of Astronomy and Space Sciences, Science Faculty, İstanbul
University, 34116, Üniversite-Istanbul, Türkiye
2Department of Astronomy and Space Sciences, Faculty of Arts and Sciences,
Erciyes University, Talas Yolu, 38039, Kayseri, Türkiye
3Universidade Federal do Rio Grande do Su, Departamento de Astronomia, CP
15051, RS, Porto Alegre 91501-970, Brazil
4Observatorio Astronómico Nacional, Universidad Nacional Autónoma de México,
Apartado Postal 877, C.P. 22800, Ensenada, B.C., México
5Kuffner Observatory, Johann-Staud-Straße 10, A-1160 Wien, Austria E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a detailed study of NGC 6791, NGC 6811, NGC 6819 and NGC 6866, the
four open clusters that are located in the Kepler prime field. We use new CCD
$UBV(RI)_{KC}$ photometry, which was combined with Gaia EDR3
photometric/astrometric data, to derive the astrophysical parameters with two
independent methods - one of them the fitCMD algorithm. Furthermore, we
provide among others estimates of the mass and mass function, the cluster
structure, derive the cluster orbits, and discuss the cluster dynamics. All
objects belong to the older open cluster population ($\sim 1-7$ Gyr), are in
an advanced dynamical stage with signs of mass segregation, and are located
close to the solar circle, but show a large range in respect of radii, member
stars or observed cluster mass ($\sim 100-2000\,M_{\odot}$). For the three
younger objects we were also able to provide photometric metallicity
estimates, which confirms their status as clusters with a roughly solar
metallicity. The most outstanding object is clearly NGC 6791, a very old
cluster with a high metallicity at a distance of about 4.5 kpc from the Sun.
We estimate a probable radial migration by about 7 kpc, resulting in a birth
position close to the Galactic center.

###### keywords:

(Galaxy:) open clusters and associations:general - Galaxy: abundances -
Galaxy: evolution

††pubyear: 2022††pagerange: Study of the Open Clusters in Kepler Prime
Field–17

##  1 Introduction

In this paper, we present new CCD photometry of the open clusters (OCs) NGC
6791, NGC 6811, NGC 6819, and NGC 6866, which was combined with Gaia EDR3
photometric and astrometric data (Gaia Collaboration et al., 2021). These
objects are Kepler asteroseismic targets, so a detailed understanding of the
properties of these open clusters is important also for studies on
correlations with variable stars.

These four old-aged OCs represent objects of the first Galactic quadrant (see
Fig. 1 and Table 1). Their location criteria are important due to the survival
rate of the OC population. As discussed e.g. by Bonatto & Bica (2007) and
Güneş et al. (2017), the majority of OCs older than 1 Gyr lie outside the
solar circle. On the other hand, the OC population gets rare in direction to
the Galactic centre, because of the effects of strong absorption, crowding or
dissolution by Giant Molecular Clouds (GMCs).

A detailed understanding of the dynamical evolution of the four OCs depends on
the knowledge of the astrophysical parameters (reddening, distance, age),
structural parameters (core, cluster and tidal radii), overall masses, mass
function (MF), relaxation times and evolutionary parameters. The members of
the OCs undergo internal and external perturbations such as stellar evolution,
two-body relaxation, mass segregation, tidal interactions with the Galactic
disc and bulge, spiral arm shocks, Galactic tidal field, and collisions with
GMCs (Lamers & Gieles, 2006; Gieles et al., 2007).

Heggie & Hut (2003) theoretically interpreted the proportional relations
between the half-mass, core, and tidal radii. Later, there have been attempts
to explain the dynamic evolution from the observations of star clusters e.g.
by Baumgardt et al. (2010), Angelo et al. (2018, 2020, 2021) \- hereafter A18,
A20, A21, or Piatti et al. (2019). According to these studies, as a star
cluster expands to the point of being tidally filling, it is exposed to
internal dynamical evolution in its core region due to two-body relaxation,
mass segregation, and core-collapse. Due to mass segregation and core
collapse, the cores contract whereas the half mass radius remains almost
constant. The binaries and possible stellar black-holes in the central parts
of the clusters may be responsible for their expansion. This expansion is
accompanied by mass losses from the outer parts. By the effect of tidal
interaction, an OC heats and its stars gain kinetic energy, which leads to an
increase in the evaporation rate. Finally, they are dissolved in the Galactic
field.

In this paper we will investigate the role and degree of internal and external
dynamic effects of the four OCs. The astrophysical open cluster parameters
(the colour excess, the distance, and the age) are determined from CCD
$UBV(RI)_{KC}$ and Gaia EDR3 photometric data. For this, we adopt
spectroscopic metal abundances and employ the approach fitCMD, presented by
Bonatto (2019). Furthermore, Gaia EDR3 data are used to obtain structural
information, the mass and mass function, dynamical evolution parameters and
kinematics of the objects.

Our programme objects are covered by several individual studies, but are
certainly also included in larger scale surveys. For example, Cantat-Gaudin et
al. (2020) derived the distance, age and reddening for 1867 OCs using Gaia DR2
data, a sample similar in size was also investigated by Dias et al. (2021)
using the same data. Tarricq et al. (2022) adopt Gaia EDR3 data to study the
structural parameters of 389 OCs. They note that older OCs have on average
smaller core radii and that mass segregation operates more efficiently in
older OCs. Such catalogue provide also an unique opportunity for detailed
comparisons of the results for our sample OCs.

Furthermore, to understand that these objects move away from their birth
places (or migrate radially), their birth radii and radial migration distances
are also estimated. According to Anders et al. (2017), non- or inward
migrating OCs may be more prone to disruption, leading to an appearance of
metal-rich OCs (e.g. NGC 6791) in the solar vicinity.

This paper is organized as follows. The CCD $UBV(RI)_{KC}$ photometry of the
four OCs is presented in Section 2. The cluster membership technique is
discussed in Section 3. The derivation of the astrophysical parameters based
on fitCMD and the differential grid technique is presented in Section 4. The
obtained cluster dimensions, masses/mass function slopes, dynamical parameters
together with their indicators, kinematics, and orbital parameters are given
in Sections 5–7. A discussion/conclusion about the above topics is finally
presented in the last Section together with a comparison with the literature
and investigation of the dynamical evolution.

Figure 1: Spatial distribution of the four OCs (filled red circles) in
Galactocentric cartesian coordinates. The schematic projection of the Galaxy
with its spiral arms is seen from the North pole. The Sun is located at 8.2
kpc. The figure is adapted from the fig. 10 by Reid et al. (2019).

##  2 Observation and Data Reduction

The observations of NGC 6791, NGC 6811, NGC 6819 and NGC 6866 were carried out
at the San Pedro Martir Observatory (SPMO) during photometric nights (7-10
June 2013) with very good seeing (0$\aas@@fstack{\prime\prime}$6 in long V
exposures) using the 0.84-m (f/15) Ritchey-Chretien telescope equipped with
the Mexman filter wheel and the ESOPO CCD detector. The ESOPO detector, a
2048$\times$4608 13.5-$\mu m$ square pixels E2V CCD42-40, has a gain of 1.7
e-/ADU and a readout noise 3.8 e- at 2$\times$2 binning. The combination of
telescope and detector ensures an unvignetted field of view of 7.4$\times$9.3
arcmin2. The star charts of the OC areas are shown in Fig. 2. It is noticeable
that our observations are unfortunately restricted to the core of most objects
and do not cover cluster members in the outskirts, thus e.g. a structural
analysis of the objects (see Sect. 5) has to be based on Gaia data.

Figure 2: The star charts of the four OCs are produced using the chart tool at
https://www.aavso.org/apps/vsp/. The field of view of the SPM detector is
shown by the red rectangle ($7.4^{\prime}$ E-W $\times$ $9.3^{\prime}$ N-S),
the blue circles represent the radii obtained from the radial density profile
($R_{RDP}$) listed in Table 5. Big plus symbols show the central equatorial
coordinates.

Each OC was observed through the Johnson’s $UBV$ and the Kron-Cousins’ $RI$
filters with short and long exposure times in order to properly cover both,
bright and faint stars in the region. Standard star fields (Landolt, 2009)
were observed at the meridian and at about two airmasses to determine the
atmospheric extinction coefficients.

The log of the observations is shown in Table 1. It includes the object names,
centre coordinates of the observed fields, air mass range during the
observations, and exposure times in each band. The flat fields were taken at
the beginning and end of each night, and bias images were obtained between
cluster observations. Data reduction was carried out by Raul Michel with the
IRAF/DAOPHOT111IRAF is distributed by the National Optical Observatories,
operated by the Association of Universities for Research in Astronomy, Inc.,
under cooperative agreement with the National Science Foundation. package
(Stetson, 1987). The standard magnitude in a given filter $\lambda$ is
obtained using the following relation:

$M_{\lambda}=m_{\lambda}-[k_{1\lambda}-k_{2\lambda}C)]X+\eta_{\lambda}C+\zeta_{\lambda}$
(1)

where $m_{\lambda}$, $k_{1\lambda}$, $k_{2\lambda}$, $C$, and $X$ are the
observed instrumental magnitude, primary/secondary extinction coefficients,
colour index and air mass, respectively. $M_{\lambda}$, $\eta_{\lambda}$,
$\zeta_{\lambda}$ are standard magnitude, transformation coefficient and
photometric zero point, respectively. More details about the data reduction,
the extinction coefficients and zero points for the $UBVRI$ filters can be
found in the papers by Akkaya et al. (2010), Akkaya Oralhan et al. (2015) and
Akkaya Oralhan et al. (2019). The photometric errors in $V$ and the colours
$(R$–$I)$, $(V$–$I)$, $(B$–$V)$, $(U$–$B)$ of the four OCs are presented in
Fig. 16 and the mean errors in $V$-mag intervals are listed in Table 11 in
Appendix.

Table 1: Equatorial/Galactic coordinates and observation summary of the four OCs. Cluster | $\alpha(2000)$ | $\delta(2000)$ | $\ell$ | $b$ | Airmass | U | B | V | R | I
---|---|---|---|---|---|---|---|---|---|---
| (h m s) | $(^{\circ}\,{}^{\prime}\,{}^{\prime\prime})$ | $(^{\circ})$ | $(^{\circ})$ |  | Exp.Time (s) | Exp.Time (s) | Exp.Time (s) | Exp.Time (s) | Exp.Time (s)
NGC 6791 | 19 20 52.6 | 37 46 05.6 | 69.95 | 10.90 | 1.012-1.050 | 100,1200 | 20,30,800 | 25,50,500 | 10,30,300 | 15,50,300
NGC 6811 | 19 37 17.0 | 46 23 18.0 | 79.20 | 12.07 | 1.039-1.054 | 90,1200 | 20,500 | 6,10,200 | 6,150 | 6,150
NGC 6819 | 19 41 16.9 | 40 11 47.3 | 73.98 | 8.49 | 1.016-1.035 | 90, 900 | 20,300 | 5,50,200 | 4,30,120 | 20,120
NGC 6866 | 20 03 56.1 | 44 09 28.7 | 79.58 | 6.84 | 1.045-1.077 | 30, 900 | 20,500 | 15,300 | 15,200 | 15,200

##  3 Membership selection

In order to identify the cluster members of NGC 6791, NGC 6811, NGC 6819 and
NGC 6866, we have obtained Gaia EDR3 astrometric/photometric data (Gaia
Collaboration et al., 2021) from VizieR222http://vizier.u-strasbg.fr/viz-
bin/VizieR?-source=II/246. for a large area of 40-70 arcmin.

We applied the Gaussian Mixture Model (GMM) and the scikit-learn package
(Pedregosa et al., 2011) to determine the membership probabilities
$P$(%)333$P$ is defined by $\Phi_{c}$ /$\Phi$. Here $\Phi=\Phi_{c}+\Phi_{f}$
is the total probability distribution. c and f are subscripts for cluster and
field parameters, respectively. Parameters for the estimation of $\Phi_{c}$
and $\Phi_{f}$ are $\mu_{\alpha}$, $\mu_{\delta}$, $\varpi$,
$\sigma_{\mu\alpha}$, $\sigma_{\mu\delta}$, $\sigma_{\varpi}$. of the cluster
stars. The GMM model considers that the distribution of proper motions of the
stars in a cluster region can be represented by two elliptical bivariate
Gaussians. The used expressions can be found in the papers by Balaguer-Nunez
et al. (1998), Wu et al. (2002), Sariya et al. (2012), Dias et al. (2018), or
Cakmak et al. (2021).

Figures 3 and 4 show the proper motion and membership distributions of the
cluster stars. Here we adopt the first significant rise in the distribution of
the membership probabilities ($P>90\%$) as the membership percentage limit. In
Fig. 3, the potential cluster members are clearly standing out compared to the
scatter caused by field stars.

The distances of the four OCs based on Gaia-EDR3 parallaxes are obtained from
the posterior probability density functions (Bailer-Jones et al., 2018, 2021).
For this, we use the global zero point of $-$0.017 mas (Lindegren et al.,
2021). The median equatorial coordinates, proper motion components, the median
parallaxes and the distances of the four OCs are listed in Table 2.

Cantat-Gaudin et al. (2020) and Dias et al. (2021) use UPMASK and maximum
likelihood methods, respectively, for the membership determination based on
Gaia DR2 data. These authors consider a membership probability higher than
$50\%$ as limit and their number of members is almost close to each other (see
Table 2). Our membership determination, on the other hand, is based on the GMM
technique and Gaia EDR3 data using P$>90\%$. This certainly leads to some
discrepancies in the number of the members. In particular for the more
populous clusters (NGC 6791 and NGC 6819), Gaia EDR3 data apparently reveal a
much higher number of members. However, our derived astrometric median values
are within the errors compatible with the results by these authors.

Figure 3: The ($\mu_{\alpha}$, $\mu_{\delta}$) diagrams of the four OCs. Black and gray dots represent the members and field stars, respectively. Figure 4: The distributions of membership probabilities for the four OCs according to GMM. The vertical dashed blue line shows the selected probability limit. Table 2: The median proper motion components and parallaxes/distances of the likely cluster members.  Cluster | $\mu_{\alpha}$ | $\mu_{\delta}$ | $\varpi$ | $d$ | N |
---|---|---|---|---|---|---
| (mas/yr) | (mas/yr) | (mas) | (kpc) |  |
NGC 6791 | -0.42$\pm$0.08 | -2.28$\pm$0.09 | 0.21$\pm$0.08 | 4.37$\pm$0.02 | 2923 |
NGC 6811 | -3.33$\pm$0.12 | -8.80$\pm$0.11 | 0.87$\pm$0.05 | 1.05$\pm$0.04 | 166 |
NGC 6819 | -2.90$\pm$0.03 | -3.88$\pm$0.04 | 0.37$\pm$0.03 | 2.37$\pm$0.20 | 2430 |
NGC 6866 | -1.38$\pm$0.02 | -5.79$\pm$0.02 | 0.69$\pm$0.04 | 1.30$\pm$0.03 | 67 |
NGC 6791 | -0.42$\pm$0.16 | -2.27$\pm$0.19 | 0.19$\pm$0.09 | 4.23 | 1629 |  Cantat (2020)
NGC 6811 | -3.40$\pm$0.12 | -8.81$\pm$0.12 | 0.87$\pm$0.04 | 1.16 | 296
NGC 6819 | -2.92$\pm$0.13 | -3.86$\pm$0.14 | 0.36$\pm$0.05 | 2.76 | 1527
NGC 6866 | -1.36$\pm$0.08 | -5.74$\pm$0.09 | 0.69$\pm$0.03 | 1.41 | 72
NGC 6791 | -0.43$\pm$0.18 | -2.27$\pm$0.22 | 0.19$\pm$0.09 | 4.23 | 1520 |  Dias (2021)
NGC 6811 | -3.40$\pm$0.15 | -8.80$\pm$0.16 | 0.87$\pm$0.04 | 1.16 | 302
NGC 6819 | -2.92$\pm$0.13 | -3.86$\pm$0.16 | 0.36$\pm$0.05 | 2.76 | 1535
NGC 6866 | -1.36$\pm$0.09 | -5.73$\pm$0.09 | 0.69$\pm$0.03 | 1.41 | 104

##  4 Determination of the Reddening, Distance, Age and Mass

We first apply fitCMD, an algorithm improved by Bonatto (2019), on the CCD
$UBV(RI)_{KC}$ and Gaia EDR3 $(G,G_{BP}-G_{RP})$ photometry of the probable
members in the four OCs.

The algorithm transposes theoretical initial mass function (IMF) properties
for the isochrones of given age and metallicity to their observational colour-
magnitude diagrams (CMDs) (see e.g. Bonatto, 2019; Cakmak et al., 2021). Based
on the IMF properties of the B12 PARSEC isochrones (Bressan et al., 2012)
444http://stev.oapd.inaf.it/cgi-bin/cmd., fitCMD searches for the values of
the cluster stellar mass $(m_{cl})$, Age, global metallicity ($Z$), foreground
reddening, distance modulus $(m-M)$, and magnitude-dependent photometric
completeness that produce the artificial and observational CMDs.

The observed CMD is converted into a Hess diagram, representing the density of
stars in a magnitude range. The mass in each Hess cell is then computed based
on the mass range of the cells’ magnitude range read from the best-fitting
isochrone. The theoretical IMF is used only for the purpose of estimating the
completeness-corrected mass by using the difference between the number of
actually detected stars at a given magnitude to the expected ones. Therefore,
all the other parameters are unaffected by this procedure. The fitCMD
algorithm was already succesfully applied to several OCs (see e.g. Cakmak et
al., 2021).

Spectroscopic metal abundances are available for the programme clusters by
Donor et al. (2020) \- see Table 9, reference 13. Thus, we adopt these
measured values as input for fitCMD to reduce the number of free parameters.
The obtained best-fitting astrophysical parameters from fitCMD \- the
reddening (colour excess), the distance modulus / distance in pc, and the Age
in Gyr) for all five investigated CMDs are listed in Table 3 and the observed
CMDs (Hess diagrams) are shown in Figs. 5-7. The B12 isochrones reproduce well
the main-sequence, turn-off and red giant/red clump regions in the individual
CMDs.

Figure 5: V-(U-B) CMDs of the four OCs. The solid red line represents the
best-fit PARSEC isochrones, and the coloured areas the Hess diagram.

Figure 6: The V-(B-V), V-(V-I) and V-(R-I) CMDs of the four OCs. The symbols
are the same as in Fig. 5.

Figure 7: GAIA CMDs of the four OCs. The symbols are the same as in Fig. 5. Table 3: The derived astrophysical parameters using fitCMD. CE is the colour excess (reddening). NGC 6791 | CE | $(V-M_{V})_{o}$ | d (pc) | Age (Gyr)
---|---|---|---|---
(U-B) |  $0.308$${}\pm{}$ | $0.042$ |  $12.40$${}\pm{}$ | $0.21$ |  $3023$${}\pm{}$ | $294$ |  $5.80$${}\pm{}$ | $0.60$
(B-V) |  $0.174$${}\pm{}$ | $0.031$ |  $13.16$${}\pm{}$ | $0.20$ |  $4294$${}\pm{}$ | $392$ |  $5.80$${}\pm{}$ | $0.90$
(V-I) |  $0.076$${}\pm{}$ | $0.030$ |  $13.40$${}\pm{}$ | $0.15$ |  $4777$${}\pm{}$ | $327$ |  $7.00$${}\pm{}$ | $0.90$
(R-I) |  $0.036$${}\pm{}$ | $0.027$ |  $13.41$${}\pm{}$ | $0.14$ |  $4808$${}\pm{}$ | $319$ |  $6.80$${}\pm{}$ | $0.90$
$(G_{BP}-G_{RP})$ |  $0.179$${}\pm{}$ | $0.024$ |  $13.21$${}\pm{}$ | $0.13$ |  $4390$${}\pm{}$ | $257$ |  $7.20$${}\pm{}$ | $0.90$
NGC 6811 |  |  |  |
(U-B) |  $0.030$${}\pm{}$ | $0.011$ |  $10.29$${}\pm{}$ | $0.23$ |  $1142$${}\pm{}$ | $122$ |  $1.45$${}\pm{}$ | $0.20$
(B-V) |  $0.010$${}\pm{}$ | $0.061$ |  $9.93$${}\pm{}$ | $0.33$ |  $970$${}\pm{}$ | $147$ |  $1.20$${}\pm{}$ | $0.20$
(V-I) |  $0.001$${}\pm{}$ | $0.077$ |  $10.03$${}\pm{}$ | $0.34$ |  $1014$${}\pm{}$ | $157$ |  $1.15$${}\pm{}$ | $0.20$
(R-I) |  $0.042$${}\pm{}$ | $0.035$ |  $9.84$${}\pm{}$ | $0.31$ |  $931$${}\pm{}$ | $133$ |  $1.25$${}\pm{}$ | $0.20$
$(G_{BP}-G_{RP})$ |  $0.023$${}\pm{}$ | $0.083$ |  $10.01$${}\pm{}$ | $0.30$ |  $1004$${}\pm{}$ | $138$ |  $1.20$${}\pm{}$ | $0.20$
NGC 6819 |  |  |  |
(U-B) |  $0.095$${}\pm{}$ | $0.014$ |  $11.85$${}\pm{}$ | $0.24$ |  $2343$${}\pm{}$ | $264$ |  $2.85$${}\pm{}$ | $0.30$
(B-V) |  $0.111$${}\pm{}$ | $0.024$ |  $12.04$${}\pm{}$ | $0.33$ |  $2557$${}\pm{}$ | $384$ |  $2.60$${}\pm{}$ | $0.40$
(V-I) |  $0.111$${}\pm{}$ | $0.002$ |  $11.72$${}\pm{}$ | $0.35$ |  $2212$${}\pm{}$ | $358$ |  $3.10$${}\pm{}$ | $0.30$
(R-I) |  $0.089$${}\pm{}$ | $0.001$ |  $11.57$${}\pm{}$ | $0.45$ |  $2057$${}\pm{}$ | $434$ |  $3.20$${}\pm{}$ | $0.40$
$(G_{BP}-G_{RP})$ |  $0.169$${}\pm{}$ | $0.011$ |  $11.91$${}\pm{}$ | $0.26$ |  $2407$${}\pm{}$ | $292$ |  $2.90$${}\pm{}$ | $0.30$
NGC 6866 |  |  |  |
(U-B) |  $0.066$${}\pm{}$ | $0.029$ |  $10.67$${}\pm{}$ | $0.34$ |  $1360$${}\pm{}$ | $214$ |  $0.90$${}\pm{}$ | $0.20$
(B-V) |  $0.061$${}\pm{}$ | $0.073$ |  $10.38$${}\pm{}$ | $0.38$ |  $1189$${}\pm{}$ | $208$ |  $1.00$${}\pm{}$ | $0.20$
(V-I) |  $0.087$${}\pm{}$ | $0.079$ |  $10.38$${}\pm{}$ | $0.35$ |  $1190$${}\pm{}$ | $190$ |  $1.10$${}\pm{}$ | $0.20$
(R-I) |  $0.066$${}\pm{}$ | $0.039$ |  $10.34$${}\pm{}$ | $0.37$ |  $1169$${}\pm{}$ | $198$ |  $0.95$${}\pm{}$ | $0.20$
$(G_{BP}-G_{RP})$ |  $0.105$${}\pm{}$ | $0.101$ |  $10.51$${}\pm{}$ | $0.35$ |  $1266$${}\pm{}$ | $206$ |  $1.00$${}\pm{}$ | $0.20$

We note that Gaia EDR3 $G$-magnitudes require a slight correction depending on
the magnitude, colour, and astrometric solution
555https://www.cosmos.esa.int/web/gaia/edr3-known-issues. These corrections
might be up to only 0.01 mag in the colour range of the identified cluster
members. However, most of our members show a five-parameter astrometric
solution, which is generally unaffected. Only few stars would require a
minimal correction of up to 2 mmag. Thus, we have not applied a correction,
because of the negligible influence on the derived cluster parameters, in
particular if considering their errors.

Furthermore, we have also determined the astrophysical parameters metallicity,
the reddening, the distance modulus, and the age of our sample OCs by using
the differential grid (DG) technique developed by Pöhnl and Paunzen (2010),
and improved by Netopil & Paunzen (2013); Netopil et al. (2022). This method
was already applied to about 90 open clusters and provides well-scaled results
for the metallicity at reasonable acccuracy (see e.g. Netopil et al., 2016).
Thus, it also allows to verify the spectroscopic metallicities, which were
adopted as input for the fitCMD approach.For this, the photometric data of
main-sequence cluster stars were transformed to luminosities and mean
effective temperatures, the latter based on up to five colour indices using
2MASS (Skrutskie et al., 2006), Gaia EDR3, and our photometric data. These
were compared to zero-age main-sequence (ZAMS) normalised B12 PARSEC
isochrones. For details we refer to the paper by Netopil et al. (2022) and
references therein. The derived astrophysical parameters are presented in
Table 4 and the fits are shown in Fig. 17. The results agree well with the
ones obtained by fitCMD and the spectroscopic metallicities. We note that this
method was not applicable to NGC 6791, because of the very old cluster age and
the resulting small luminosity range of the main-sequence down to solar mass
stars.

Table 4: Derived astrophysical parameters using the differential grids. | Z | [Fe/H] | E(B-V) | $(V-M_{V})_{o}$ | d (pc) | $\log Age$ | Age (Gyr)
---|---|---|---|---|---|---|---
NGC 6811 | 0.014$\pm$0.002 | -0.03$\pm$0.06 | 0.02 | 9.90 | 955 | 9.10 | 1.26
NGC 6819 | 0.017$\pm$0.004 | 0.06$\pm$0.11 | 0.14 | 11.80 | 2291 | 9.45 | 2.82
NGC 6866 | 0.017$\pm$0.005 | 0.06$\pm$0.13 | 0.09 | 10.40 | 1202 | 9.00 | 1.00

##  5 Structural parameters and Mass Function

The structural parameters of the four OCs were derived using stellar radial
density profiles (RDPs). For this, we selected member stars that are brighter
than 20 mag in the G band and applied the relation
$\sigma(R)=\sigma_{bg}+\sigma_{0}/(1+(R/R_{core})^{2}$ by King (1966)666Here,
$\sigma_{bg}$, $\sigma_{0}$ and $R_{core}$ are the residual background and the
central densities of stars, and the core radius, respectively.. We obtained
the cluster radii $(R_{RDP})$ by comparing the RDP level with the background
(see Fig. 8). The low star content in the central parts of some OCs (NGC 6811
and NGC 6866) are responsible for the large uncertainties within
$R<1^{\prime}$ of the RDPs.

Figure 8: The radial density profiles of the four OCs. The solid lines show
the best-fitting King profile and the horizontal gray area indicates the
stellar background level measured in the comparison field. The $1\sigma$ King
fit uncertainty is shown by the shaded range.

Table 5 lists the derived structural parameters. The results for $R_{core}$
and $R_{RDP}$ in pc are almost consistent with the ones by Bukowiecki et al.
(2011), who presents data for all four objects. Tarricq et al. (2022), on the
other hand, list somewhat larger $R_{core}$ values for two objects in common
(NGC 6811 and NGC 6866), and Zhong et al. (2022) obtained for these by far the
largest radii.

For simplicity, the tidal radii $R_{t}$ in Table 5 were estimated using
$R_{t}~{}(pc)=1.54~{}R_{RDP}$ given by Piskunov et al. (2007). Notice that
Bonatto et al. (2005) indicate $R_{t}~{}(pc)=1.4-1.9~{}R_{RDP}$ based on
bright OCs. Gao (2020) and Tarricq et al. (2022), on the other hand, use the
three-parameter function ($R$, $R_{core}$, $R_{t}$) by King (1962). Our
$R_{t}$ estimate for NGC 6791 is in reasonable agreement with literature, but
for NGC 6811 and NGC 6866 our radii are much smaller. The differences for
$R_{t}$ and $R_{core}$ can be explained as follows: the three-parameter King
(1962) model describes well the outer parts of a cluster, while the two-
parameter King (1966) model describes the central region of the clusters
(Bonatto et al., 2005). Additionally, the tidal radius can only be derived
well for globular clusters or populous OCs at higher Galactic latitude, but
for sparse OCs close to the galactic plane an accurate determination of
$R_{t}$ becomes difficult.

Note that the results by Gao (2020), Tarricq et al. (2022) and Zhong et al.
(2022) are based on Gaia data, but the membership selection differs. Gao
(2020) and Zhong et al. (2022) use the GMM model with $P>80\%$ and UPMASK with
$P>70\%$, respectively. Tarricq et al. (2022) consider the clustering
algorithm named hierarchical density based spatial clustering of applications
with noise (HDBSCAN) in its python implementation with using 0.5 as a
probability cut-off for the membership. Platais et al. (2011) on the other
hand use the astrometric data from Lick and Kitt Peak National observatory by
adopting $P>80\%$, while Bukowiecki et al. (2011) and Güneş et al. (2012)
apply a King (1966) fit to 2MASS $JHK_{s}$ data and a photometric member
selection. Furthermore, Piskunov et al. (2007) and Kharchenko et al. (2013)
adopt stars with kinematic/photometric membership probabilities higher than
$60\%$ and apply a fit using the model by King (1962). All these factors
results in different cluster sizes and do not allow a proper direct
comparison, in particular for small samples in common.

Table 5: Structural parameters of the four objects.

Cluster | $(1^{\prime})$ | $\sigma_{0K}$ | $\sigma_{bg}$ | $R_{core}$ | $R_{RDP}$ | $\sigma_{0K}$ | $\sigma_{bg}$ | $R_{core}$ | $R_{RDP}$ | $R_{t}$ | CC
---|---|---|---|---|---|---|---|---|---|---|---
|  ($pc$) |  ($*\,\prime^{-2}$) |  ($*\,\prime^{-2}$) |  ($\prime$) |  ($\prime$) |  ($*\,pc^{-2}$) |  ($*\,pc^{-2}$) |  ($pc$) |  ($pc$) |  ($pc$) |
($1$) |  ($2$) |  ($3$) |  ($4$) |  ($5$) |  ($6$) |  ($7$) |  ($8$) |  ($9$) |  ($10$) |  ($11$) |  ($12$)
NGC 6791 | 1.27 |  $3.75$${}\pm{}$ | $0.36$ |  $50.6$${}\pm{}$ | $3.6$ |  $1.97$${}\pm{}$ | $0.08$ |  $11.27$${}\pm{}$ | $0.45$ |  $4.8$${}\pm{}$ | $0.4$ |  $64.4$${}\pm{}$ | $4.6$ |  $2.51$${}\pm{}$ | $0.11$ |  $14.33$${}\pm{}$ | $0.57$ |  | 22.07 | 0.998
NGC 6811 | 0.29 |  $3.44$${}\pm{}$ | $0.07$ |  $13.5$${}\pm{}$ | $0.4$ |  $0.50$${}\pm{}$ | $0.02$ |  $5.46$${}\pm{}$ | $0.28$ |  $1.0$${}\pm{}$ | $0.02$ |  $3.9$${}\pm{}$ | $0.1$ |  $0.15$${}\pm{}$ | $0.01$ |  $1.59$${}\pm{}$ | $0.08$ |  | 2.45 | 0.993
NGC 6819 | 0.69 |  $5.65$${}\pm{}$ | $0.34$ |  $32.9$${}\pm{}$ | $2.7$ |  $2.00$${}\pm{}$ | $0.11$ |  $13.05$${}\pm{}$ | $0.58$ |  $3.9$${}\pm{}$ | $0.2$ |  $22.7$${}\pm{}$ | $1.8$ |  $1.38$${}\pm{}$ | $0.07$ |  $8.99$${}\pm{}$ | $0.40$ |  | 13.84 | 0.993
NGC 6866 | 0.36 |  $4.38$${}\pm{}$ | $0.11$ |  $5.6$${}\pm{}$ | $0.6$ |  $3.15$${}\pm{}$ | $0.25$ |  $15.02$${}\pm{}$ | $0.57$ |  $1.6$${}\pm{}$ | $0.04$ |  $2.0$${}\pm{}$ | $0.2$ |  $1.15$${}\pm{}$ | $0.09$ |  $5.48$${}\pm{}$ | $0.21$ |  | 8.44 | 0.971
NGC 6791 |  |  |  |  |  |  |  |  $2.62$${}\pm{}$ | $0.23$ |  $24.61$${}\pm{}$ | $2.61$ |  |  | 1
|  |  |  |  |  |  |  |  $3.91$${}\pm{}$ | $0.68$ |  |  |  29.54$\pm$5.42 | 4
|  |  |  |  |  |  |  | 3.80 |  |  | 27.00 | 5
|  |  |  |  |  |  |  |  $4.40$${}\pm{}$ | $0.30$ |  |  |  20.30$\pm$2.70 | 6
NGC 6811 |  |  |  |  |  |  |  |  $0.36$${}\pm{}$ | $0.04$ |  $2.89$${}\pm{}$ | $0.42$ |  |  | 1
|  |  |  |  |  |  |  | 1.70 |  |  | 5.90 | 2
|  |  |  |  |  |  |  |  $1.81$${}\pm{}$ | $0.39$ |  |  |  6.78$\pm$1.22 | 4
|  |  |  |  |  |  |  |  $1.63$${}\pm{}$ | $0.27$ |  |  |  20.85$\pm$2.00 | 7
|  |  |  |  |  |  |  | 4.30 |  |  | 8.80 | 8
NGC 6819 |  |  |  |  |  |  |  |  $2.20$${}\pm{}$ | $0.20$ |  $17.67$${}\pm{}$ | $1.77$ |  |  | 1
|  |  |  |  |  |  |  |  $1.76$${}\pm{}$ | $0.12$ |  |  |  13.32$\pm$0.96 | 4
NGC 6866 |  |  |  |  |  |  |  |  $0.59$${}\pm{}$ | $0.09$ |  $2.94$${}\pm{}$ | $0.46$ |  |  | 1
|  |  |  |  |  |  |  |  $0.61$${}\pm{}$ | $0.11$ |  |  |  6.50$\pm$0.96 | 4
|  |  |  |  |  |  |  |  $1.97$${}\pm{}$ | $0.46$ |  $9.15$${}\pm{}$ | $0.27$ |  |  | 3
|  |  |  |  |  |  |  |  $1.96$${}\pm{}$ | $0.36$ |  |  |  25.66$\pm$4.70 | 7
|  |  |  |  |  |  |  | 4.70 |  |  | 12.90 | 8

The mass function relation between $\phi(m)(stars~{}M_{\odot}^{-1})$ and
m($M_{\odot}$) for the programme objects is presented in Fig. 9, the obtained
masses with the help of the B12 isochrones and fitCMD are listed in Table 6.
With the exception of the last two columns, which refer to the full simulation
of fitCMD (considering stars down to $>0.08~{}M_{\odot}$), the data represent
the simulation results of the actually observed cluster.

The MF slopes are the observed ones, i.e., those computed directly from the
observed CMDs. The theoretical IMF is used only for the purpose of estimating
the completeness-corrected mass (by estimating the difference in the number of
stars actually detected at a given magnitude with respect to the expected
one). Therefore, all the other parameters are unaffected by this procedure.

The overall MF of NGC 6791 (Fig. 9) shows a break followed by a flatter slope.
This cluster is the most distant one in our sample, thus the mass range below
$m\sim 0.93M_{\odot}$ is clearly affected by incompleteness. Therefore, we use
the MF slope based on stars $m>0.93M_{\odot}$ as representative for the
overall cluster. The MFs of NGC 6811, NGC 6819 and NGC 6866 show on the other
hand rather flat slopes $(\chi=-0.82\pm 0.24)$, $(\chi=-1.09\pm 0.13)$ and
$(\chi=-0.32\pm 0.27)$, respectively.

Figure 9: $\phi(m)$ versus $m~{}(M_{\odot})$ for the overall regions of the
four OCs. Here $\phi(m)=dN/dm$ (stars $m_{\odot}^{-1}$).

Figure 10: The cumulative mass versus radius of the four OCs. The vertical lines represent the half-mass radii correspond to the half-masses (horizontal dotted lines). The blue dotted lines show the total masses. Table 6: Mass information for the overall regions of the four OCs. Cluster |  | Mass range | MF slope | Observed mass | Mean Mass | Members | Total mass | Members
---|---|---|---|---|---|---|---|---
|  | $M_{\odot}$ | $\chi$ | $m_{obs}(M_{\odot})$ | $(M_{\odot})$ | N1 | $m_{tot}(M_{\odot})$ | N2
NGC 6791 | 1 | 0.81 $-$ 0.93 |  $-18.55$${}\pm{}$ | $0.84$ |  $298.2$${}\pm{}$ | $9.1$ |  $0.90$${}\pm{}$ | $0.001$ |  $333$${}\pm{}$ | $10$ | 60500 | 186362
| 2 | 0.94 $-$ 1.11 |  $-5.26$${}\pm{}$ | $0.22$ |  $1759.2$${}\pm{}$ | $17.9$ |  $1.03$${}\pm{}$ | $0.004$ |  $1700$${}\pm{}$ | $14$ | 60500 | 186362
NGC 6811 |  | 0.62 $-$ 1.86 |  $-0.82$${}\pm{}$ | $0.24$ |  $198.1$${}\pm{}$ | $18.9$ |  $1.23$${}\pm{}$ | $0.021$ |  $161$${}\pm{}$ | $14$ | 2480 | 6502
NGC 6819 |  | 0.82 $-$ 1.39 |  $-1.09$${}\pm{}$ | $0.13$ |  $2111.9$${}\pm{}$ | $40.0$ |  $1.11$${}\pm{}$ | $0.004$ |  $1901$${}\pm{}$ | $33$ | 23100 | 68595
NGC 6866 |  | 0.62 $-$ 2.10 |  $-0.32$${}\pm{}$ | $0.27$ |  $112.6$${}\pm{}$ | $21.0$ |  $1.28$${}\pm{}$ | $0.131$ |  $88$${}\pm{}$ | $12$ | 556 | 1436

##  6 Dynamical parameters

The relaxation time $t_{rlx}$ is obtained from the relation by Spitzer & Hart
(1971)

$t_{rlx}=\frac{8.9\times
10^{5}\sqrt{N}\times{R_{h}}^{3/2}}{log(0.4N)\times\sqrt{m}}$ (2)

where $m$ and $N$ refers to the mean mass and the number of the cluster
members, respectively, adopted from Table 6. $R_{h}$ is defined as the radius
from the core that contains half the total mass of the cluster. This parameter
was inferred from Fig. 10, showing the cumulative mass in dependence of the
radius. The half-mass radii of NGC 6791, NGC 6811, NGC 6819 and NGC 6866 are
2′.71, 5′.41, 2′.37, and 1′.96, respectively. A conversion into pc is listed
in Table 7.

As an indicator of the dynamical evolution, the evolutionary parameter is
estimated from the relation $\tau=Age/t_{rlx}$ by using the age obtained from
the Gaia CMDs given in Table 3.

The disruption times of the four OCs have been determined from the equation by
Binney & Tremaine (2008), as given in Converse & Stahler (2011).

$t_{dis}=250~{}Myr\left(\frac{M}{300M_{\odot}}\right)^{1/2}\times\left(\frac{R_{h}}{2~{}pc}\right)^{-3/2}$
(3)

The Jacobi tidal radius $R_{J}$ and the Galactic mass $M_{G}$ inside a
Galactocentric radius $R_{GC}$ are estimated from the equations given by Kim
et al. (2000),

$R_{J}=\left(\frac{M}{2M_{G}}\right)^{1/3}\times R_{GC}$ (4)

$M_{G}=2\times 10^{8}M_{\odot}\left(\frac{R_{GC}}{30pc}\right)^{1.2}$ (5)

where $M$ is the observed cluster mass taken from Table 6 and the
Galactocentric distances of the clusters are given in Table 8. All the derived
dynamical parameters are provided in Table 7.

Table 7: The dynamical parameters of the four OCs. Cluster | $M_{G}$ | $R_{J}$ | $R_{h}$ | $t_{rlx}$ | $t_{dis}$ | $\tau$
---|---|---|---|---|---|---
| $10^{9}\,M_{\odot}$ | pc | pc | Myr | Myr |
NGC 6791 |  $160.27$${}\pm{}$ | $0.11$ |  $13.91$${}\pm{}$ | $0.06$ |  $3.44$${}\pm{}$ | $0.07$ |  $81.4$${}\pm{}$ | $2.7$ |  $268$${}\pm{}$ | $9$ |  $88$${}\pm{}$ | $11$
NGC 6811 |  $164.80$${}\pm{}$ | $0.06$ |  $6.82$${}\pm{}$ | $0.09$ |  $1.49$${}\pm{}$ | $0.03$ |  $11.0$${}\pm{}$ | $0.8$ |  $294$${}\pm{}$ | $20$ |  $109$${}\pm{}$ | $20$
NGC 6819 |  $160.31$${}\pm{}$ | $0.08$ |  $14.79$${}\pm{}$ | $0.05$ |  $1.64$${}\pm{}$ | $0.10$ |  $26.8$${}\pm{}$ | $1.1$ |  $895$${}\pm{}$ | $38$ |  $108$${}\pm{}$ | $12$
NGC 6866 |  $164.67$${}\pm{}$ | $0.01$ |  $5.64$${}\pm{}$ | $0.12$ |  $0.71$${}\pm{}$ | $0.05$ |  $3.0$${}\pm{}$ | $0.4$ |  $683$${}\pm{}$ | $77$ |  $330$${}\pm{}$ | $77$

###  6.1 Indicators of Dynamical Evolution

As an indicator for the internal dynamical evolution, the mass segregation
degree (small/mild/large) provides a proxy if higher or lower mass stars are
dominant in the cluster. In the sense, the MF slopes ($\chi$) also reflect the
ratio between massive and low-mass stars. Low-mass stars are transferred from
the core to the halo, and are then lost to the field. Mass segregation is
directly related to $t_{rlx}$ and $\tau$, where $t_{rlx}$ gives the time it
takes for a star to move from one end of the cluster to the other. The smaller
$t_{rlx}$, the sooner the star can leave the cluster. Thus, with time the
cluster loses stars and dynamically evolves. A high $\tau$ value of an OC
implies an advanced dynamical evolution, showing the degree to which it has
lost its low-mass stars to the field. There is a negative relationship between
$t_{rlx}$ and $\tau$ \- higher $\tau$ and lower $t_{rlx}$ values indicate
advanced mass segregation.

As to external dynamical evolution, the three-parameter King (1962) model
describes well the outer parts of a cluster and provides the tidal radius
$R_{t}$. Internal relaxation and the stripping of stars from the cluster by
the Galactic tidal field depend on $R_{t}$. $R_{J}$ (Eq. 4) is also the
distance from the cluster center at which the external gravitation of the
Galaxy has more influence on the cluster stars than the cluster itself (von
Hoerner, 1957). Since the OCs are mostly in nearly circular orbits, $R_{J}$
should be close to $R_{t}=R_{RDP}$ (tidally filling).

As noted by A18, OCs are exposed to significant mass loss processes towards
their final disruption in case the $R_{t}$ radii are larger than the $R_{J}$
radii. In the opposite case, stars within the Roche lobe are gravitationally
bound to the OCs, and therefore such OCs keep their stellar contents within
their $R_{J}$ radii. The cluster members outside the $R_{J}$ radii are more
influenced by the external potential of the Galaxy. Clusters that show
comparatively equal radii are in the transitional phase towards their final
disruption stage.

According to Piatti et al. (2017a, b), the concentration parameter
$c=\log(R_{t}/R_{core})$ is the efficiency of heating on cluster stars caused
by dynamic interactions in the center of the clusters. This parameter is
almost negatively correlated with $R_{h}/R_{t}$ and increases with age as a
result of their dynamic evolution.

According to Heggie & Hut (2003), Piatti et al. (2019), A20 and A21,
$R_{h}/R_{t}$ is as an indicator of the tidal influence of the Galaxy on the
dynamical evolution of the OCs. Lower $R_{h}/R_{t}$ ratios indicate a more
compact cluster, which is less subject to tidal stripping/disruption caused by
Galactic gravitational forces. $R_{core}/R_{h}$ is a measure of the
compactness of a cluster in its inner regions. Note that a small half-mass
radius indicates a dense core relative to the overall size, and that clusters
with low $R_{core}/R_{h}$ ratios are also less subject to tidal disruption.

The ratio $R_{h}/R_{J}$ is a useful measure for the degree of tidal filling
that an OC experiences in the tidal field of the Galaxy. In the sense it is
defined as the Roche volume filling factor, and characterizes the impact of
the tidal field - lower $R_{h}/R_{J}$ ratios indicate a weaker tidal field
impact. We note that the tidal field strength itself weakens with increasing
Galactocentric distance.

##  7 Kinematics and Orbital Parameters

Based on Gaia EDR3 radial velocities ($V_{R}$) for the bright giants in NGC
6791 ($N=46$), NGC 6811 ($N=5$), NGC 6819 ($N=121$), and NGC 6866 ($N=2$) we
derived weighted cluster averages (Table 8). The heliocentric velocities ($U$,
$V$, $W$) in the right-hand system have been obtained using the radial
velocities, the median proper motion components, the cluster distances and the
algorithm by Johnson & Soderblom (1987). For the distance we adopt the result
based on the Gaia EDR3 CMD (Table 3). These space velocities were transformed
to the components $U^{\prime}$, $V^{\prime}$, $W^{\prime}$ by correcting for
the solar motion $(U,V,W)_{\odot}=(+11.10,+12.24,+7.25)$ km s-1 (Schönrich and
Binney, 2010) with respect to the local standard of rest (LSR). For this we
adopt $R_{\odot}=8.2\pm 0.1$ kpc (Bland-Hawthorn et al., 2016) and $V_{LSR}$ =
239 km s-1 (Brunthaler et al., 2011). The heliocentric cartesian distances
($x^{\prime}$, $y^{\prime}$, $z^{\prime}$) in kpc and LSR-velocity components
($U^{\prime}$, $V^{\prime}$, $W^{\prime}$) have been converted to the Galactic
Rest of Frame (GSR) i.e., ($x$, $y$, $z$) and ($V_{x}$, $V_{y}$, $V_{z}$)
using the equations by Kepley et al. (2007). The azimuthal velocity
($V_{\Phi}$) in km s-1 is estimated using

$V_{\Phi}=\frac{xV_{y}-yV_{x}}{R}$

We note that $V_{\Phi}<0$ means prograde. From the "MWPotential2014" code in
the galpy-code library 111http://github.com/jobovy/galpy by Bovy (2015), peri-
and apo-galactic distances $(R_{min},~{}R_{max})$ and the maximum height above
the Galactic plane (zmax) in kpc are obtained. MWPotential2014 is an
axisymmetric Galactic potential which includes a spherical Galactic bulge, a
Miyamoto-Nagai disc, and a halo with a Navarro-Frenk-White profile. For
details about the parameters and properties of the Galactic components we
refer to Bovy (2015).

For the orbital eccentricity (ecc) we adopt

$ecc=\frac{R_{max}-R_{min}}{R_{max}+R_{min}}$

The current mean Galactocentric radius $R_{m}=(R_{min}+R_{max})/2$ is also
known as the guiding or mean orbital radius. The orbits have been integrated
using the obtained kinematic parameters for the ages of the four OCs within
the Galactic potential. Furthermore, their orbital angular momentum components
$J_{z}$ (kpc km s-1) are calculated from the equation by Kepley et al. (2007).
All these parameters are listed in Table 8.

The rotational velocities, the eccentricities, and the orbital angular
momentum values indicate that the four OCs have Galactic thin disc properties.
Figure 11 shows the Galactic orbits of the objects. The x-y (kpc) plane is
known as projected on to the Galactic plane, whereas z-R (kpc) is the
meridional plane. They follow a circular path around the Galactic center with
eccentricities in the range of 0.04$-$0.29 and are orbiting near the Galactic
disc. Therefore, they might be affected by the tidal forces of the disc.
According to their revolution periods T (Myr) around the Galactic center, NGC
6791 has made 31 revolutions around the Galactic center, and the remaining
objects show 5 to 14 revolutions (see Table 8).

The orbits in the z-R plane show boxy-like type properties, thus the four OCs
move in the meridional planes within the confined spaces and are oscillating
along the z-axis. The orbit of NGC 6791 is confined in the range of $\sim
7.2<R_{GC}\leq 7.8$ kpc, therefore it is interacting with the inner region of
the Galaxy. The confined spaces of the others are as follows: $\sim
8.0<R_{GC}\leq 8.6$ kpc for NGC 6811, $\sim 7.82<R_{GC}\leq 8.00$ kpc for NGC
6819, $\sim 8.0<R_{GC}\leq 8.9$ kpc for NGC 6866.

Their initial and present day positions in our Galaxy are shown in Fig. 11
with filled blue and red dots, respectively. For this, the times were adopted
as zero (initial) and the cluster age (present day). Their closest approaches
to the current solar position are determined as $(d~{}(kpc),~{}t(Gyr))$ =
(0.46, 4.86) for NGC 6791), (0.99, 1.19) for NGC 6811, (0.55, 2.74) for NGC
6819, (0.97, 0.97) for NGC 6866, respectively.

The derived radial / rotational velocities and the orbital parameters of the
objects in Table 8 are in reasonable agreement with literature values (Carrera
et al., 2022; Tarricq et al., 2021). We note that this paper and Tarricq et
al. (2021) use the same Galactic components for the Galactic potential, but
Carrera et al. (2022) also consider the bar and spiral arms in addition to the
three main components.

Figure 11: Galactic orbits of the four OCs based on the galpy-code package and "MWPotential2014". The trajectories represent the paths traveled by the OCs during their age. The filled blue/red dots show their initial/present day positions. The red circle shows the location of the Sun. Table 8: Velocities and orbital parameters of the objects. | NGC 6791 | NGC 6811 | NGC 6819 | NGC 6866 | Ref.
---|---|---|---|---|---
$V_{R}$ | -47.20$\,\pm\,$0.01 | 5.40$\,\pm\,$8.1 | 2.50$\,\pm\,$0.01 | 9.5$\,\pm\,$26.0 | 1
| -47.75$\,\pm\,$0.17 | 7.17$\,\pm\,$0.13 | 2.80$\,\pm\,$0.14 | 12.44$\,\pm\,$0.34 | 2
| -46.49$\,\pm\,$0.53 | 6.92$\,\pm\,$0.16 | 2.96$\,\pm\,$0.26 | – | 3
$U$ | 28.33 | 44.87 | 52.01 | 34.86 | 1
$V$ | -57.13 | -2.13 | -13.56 | 4.49 | 1
$W$ | -21.49 | -3.50 | 8.15 | -9.82 | 1
$V_{\Phi}$ | -185.80 | -253.92 | -245.56 | -259.70 | 1
| -189.04 | -256.75 | -246.78 | -263.77 | 2
| 189.50 | 256.70 | 247.00 | – | 3
ecc | 0.29 | 0.11 | 0.04 | 0.10 | 1
| 0.28 | 0.12 | 0.05 | 0.11 | 2
| 0.35 | 0.17 | 0.13 | – | 3
Rmin | 4.71 | 7.80 | 7.82 | 8.05 | 1
| 4.83 | 7.91 | 7.94 | 8.19 | 2
| 4.12 | 6.69 | 6.44 | – | 3
Rmax | 8.62 | 9.67 | 8.50 | 9.79 | 1
| 8.62 | 10.02 | 8.70 | 10.23 | 2
| 8.42 | 9.50 | 8.42 | – | 3
Rm | 6.67 | 8.74 | 8.16 | 8.92 | 1
zmax | 1.01 | 0.27 | 0.46 | 0.20 | 1
| 0.95 | 0.31 | 0.52 | 0.23 | 2
| 1.26 | 0.30 | 0.55 | – | 3
Rin | 7.28 | 8.52 | 8.43 | 8.78 | 1
RGC | 7.84 | 8.07 | 7.88 | 8.07 | 1
| 7.92 | 8.20 | 8.02 | 8.20 | 2
| 7.94 | 8.20 | 8.03 | – | 3
$J_{z}$ | -1465.32 | -2050.79 | -1937.37 | -2095.60 | 1
$T$ | 235 | 213 | 204 | 214 | 1
$N_{Rev}$ | 30.7 | 5.6 | 14.2 | 4.7 | 1

##  8 Discussion and Conclusion

###  8.1 Comparison of astrophysical parameters

In Sect. 4 we derived the astrophysical parameters of the clusters based on
various colours and methods (see also Tables 3 and 4. A comparison of our
findings for $E(B-V)$, $d~{}(kpc)$, and $Age$ (Gyr) of the four OCs to
literature is given in Table 9 and Fig. 13. For the Fig. 13(a) the individual
colour excesses are converted to $E(B-V)$ using the reddening ratios given by
Bessell et al. (1998) and $E(B-V)=0.775E(G_{BP}-G_{RP})$ by Bragaglia et al.
(2018).

Our individual results mostly lie within the error range based on the $(B-V)$
colour. However, e.g. the reddening obtained for NGC 6791 from the $(R-I)$ and
$(V-I)$ colours is somewhat lower. These colours also provide closer distances
and a somewhat older age of NGC 6819.

A broad age range (4.4–12 Gyr) can be found in the literature in particular
for NGC 6791, though some results adopt solar metallicity isochrones for this
metal-rich object. Also our results show some deviations, the results based on
the $V/(U-B)$ and $V/(B-V)$ CMDs are about 1.0-1.4 Gyr younger than using the
other colours. These two CMDs also provide a younger age compared to e.g.
Carraro et al. (2013) and Kaluzny & Rucinski (1995) who indicate an age of 7.0
and 7.2 Gyr from CCD $UB$ and CCD $UBV$ data. On the other hand, the results
by these authors agree well with our finding using Gaia photometry. To
investigate the age difference, Fig. 12 compares our photometry for NGC 6791
to the CCD $UB$ photometry by Carraro et al. (2013) and CCD $UBV$ photometry
by Kaluzny & Rucinski (1995). The mean differences are up to
$\Delta(U-B)=+0.80$ and $\Delta(B-V)=0.50$, respectively. The $(U-B)$ colours
by Carraro et al. (2013) and Kaluzny & Rucinski (1995) are systematically
bluer than our $(U-B)$ and $(B-V)$ colours. However, the agreement between the
reddening results based on our $(B-V)$ and the Gaia colour gives a hint for a
correct $(B-V)$ scale, also an additional comparison with the homogeneous
photometry by Stetson et al. (2003), which gives $\Delta(B-V)=0.006\pm 0.097$
over the complete magnitude range. We are unfortunately unable to clarify the
difference in $(U-B)$, a colour for which a standard transformation is
challenging for CCDs anyway (see e.g. Sung & Bessell, 2000).

Our astrophysical parameters based on Gaia EDR3 are within the uncertainties
almost compatible with the ones by Cantat-Gaudin et al. (2020) (reference 14)
or Dias et al. (2018) (reference 15 in Table 9 and Fig. 13), except e.g. for
the reddening of NGC 6791. Our distances from the Gaia EDR3 parallaxes also
lie in the given error range based on the $(B-V)$ colour. Figure 13 indicates
that some literature values present an agglomeration close to our results, but
there are certainly also discrepancies that might stem from the usage of
different membership techniques, isochrone sets, metal abundances, reddening
determination and various photometric data, as discussed e.g. by Moitinho
(2010). However, even homogeneous methods do not necessarily produce agreeing
results for close OCs, mean intrinsic errors of about 0.2 dex for the age or
0.35 mag for the distance modulus are quite common (see e.g. Netopil et al.,
2015). In this context Gaia parallaxes might provide reasonable cluster
distances for closer objects, whereas isochrone fits are favourable for more
distant clusters (Monteiro et al., 2020).

As already noted in Sect. 4, the results based on the DG method are also
consistent with the ones obtained by fitCMD and agree well with available
spectroscopic metallicities.

Figure 12: The comparison of our CCD $UB$ / CCD $UBVI$ data for NGC 6791 to
literature data: panel (a) (Carraro et al., 2013), and panels (b)-(c) (Kaluzny
& Rucinski, 1995). The difference $\Delta$ means our data minus literature.
Table 9: Literature comparison of the astrophysical parameters.

E(B-V) | $(V-M_{V})_{0}$ | d (kpc) | Z | [Fe/H] | Age (Gyr) | Isochrone | Photometry | Reference
---|---|---|---|---|---|---|---|---
NGC 6791 |  |  |  |  |  |  |  |
0.174$\pm$0.031 | 13.16$\pm$0.20 | 4.29$\pm$0.39 | 0.030 | 0.30 | 5.80$\pm$0.90 | Bressan et al. (2012) | CCD UBVRI | This paper
0.18 | 13.60 | 5.25 | 0.04 | – | 7.00$\pm$1.00 | Bertelli (2008) | CCD UB | 1
0.09$\pm$0.01 | 13.07$\pm$0.05 | 4.30 | 0.046 | 0.39$\pm$0.05 | 8.00$\pm$2.00 | Girardi et al.(2000) | CCD BVI | 2
0.125 | 13.07 | 4.11 | 0.04 | 0.40 | 7.00$\pm$1.00 | Bertelli et al.(2008) | CCD UB | 3
0.16$\pm$0.025 | 13.60$\pm$0.15 | 5.25 | – | 0.45$\pm$0.04 | 7.00$\pm$1.00 | Girardi et al. (2002) | ubyCa$H_{b}eta$ | 4
0.09 | 12.79 | 3.61 | 0.035 | 0.30 | 12.00 | VandenBerg (2002) | CCD BVI | 5
0.14 | – | 5.04 | – | 0.38$\pm$0.01 | 8.32 | Marigo et al. (2008) | 2MASS JH$K_{s}$ | 6
0.117 | 13.02$\pm$0.08 | 4.02$\pm$0.15 | – | – | – | Red Clump | SDSS DR8/2MASS | 7
0.17$\pm$0.01 | 13.48 | 4.97 | 0.039 |  | 7.20 | VandenBerg and Poll (1989) | CCD UBVI | 8
0.13$\pm$0.03 | 14.00$\pm$0.02 | 6.31$\pm$0.06 | 0.02 | – | 7.00 | Ciardullo and Demarque (1977) | Photoelectric UBV | 9
0.105$\pm$0.014 | 13.04$\pm$0.08 | 4.06$\pm$0.02 |  | 0.42$\pm$0.07 | 9.50$\pm$0.30 | Delahaye and Pinsonneault (2015) | BVIc-2MASS JH$K_{s}$ | 10
– | – | – | – | 0.32$\pm$0.02 | – | – | Spectroscopy | 11
0.117 | 13.50 | 4.93 | solar | solar | 4.42 | Girardi et al. (2002) | 2MASS JH$K_{s}$ | 12
– | – | – | 0.030 | 0.35$\pm$004 | – | – | Spectroscopy | 13
0.23(Av=0.70) | 13.13 | 4.23 | solar | solar | 6.31 | Bressan et al. (2012) | Gaia DR2 | 14
0.10$\pm$0.01(Av=0.313$\pm$0.028) | – | 4.46$\pm$0.10 | – | 0.399$\pm$0.024 | 7.23$\pm$0.62 | Bressan et al. (2012) | Gaia DR2 | 15
NGC 6811 |  |  |  |  |  |  |  |
0.010$\pm$0.061 | 9.93$\pm$0.33 | 0.97$\pm$0.15 | 0.014 | –0.04 | 1.20$\pm$0.20 | Bressan et al. (2012) | CCD UBVRI | This paper
0.07$\pm$0.02 | 10.37$\pm$0.03 | 1.19$\pm$0.02 | 0.0147 | 0.04 | 1.00$\pm$0.05 | Bressan et al. (2012) | BVIc | 16
0.074$\pm$0.024 | 10.26$\pm$0.18 | 1.13$\pm$0.09 | 0.012 | –0.19 | 1.00$\pm$0.17 | Yale-Yonsei/Padova | CCD UBVRI | 17
0.12$\pm$0.02 | 10.42$\pm$0.03 | 1.21$\pm$0.02 | – | – | 0.70 | Castellani et al. (1992) | photoelectric UBVRI | 18
0.05$\pm$0.02 | 10.29$\pm$0.14 | 1.14$\pm$0.07 | 0.012 | 0.04$\pm$0.01 | 1.00$\pm$0.10 | Bressan et al. (2013) | BV of Janes et al. (2013) | 19
0.14 | – | 1.64 | – | – | 0.19 | Girardi et al. (2003) | uvby | 20
0.16 | – | 1.22 | 0.025 | –0.02 | 0.63 | Marigo et al. (2008) | 2MASS JH$K_{s}$ | 6
| – | – | 0.014 | –0.05$\pm$0.02 | – | – | Spectroscopy | 13
0.21 | 10.51 | 1.23 | solar | solar | 0.64 | Girardi et al. (2002) | 2MASS JH$K_{s}$ | 12
0.12$\pm$0.05 | 10.59$\pm$0.09 | 1.31$\pm$0.05 | 0.019 | – | 0.58$\pm$0.12 | Girardi et al. (2000) | CCD BV | 21
0.03 (Av= 0.09) | 10.33 | 1.16 | solar | solar | 1.07 | Bressan et al. (2012) | Gaia DR2 | 14
0.07$\pm$0.01(Av=0.213$\pm$0.033) | – | 1.10$\pm$0.01 | – | 0.032$\pm$0.015 | 1.01$\pm$0.05 | Bressan et al. (2012) | Gaia DR2 | 15
NGC 6819 |  |  |  |  |  |  |  |
0.11$\pm$0.02 | 12.04$\pm$0.33 | 2.56$\pm$0.38 | 0.017 | 0.05 | 2.60$\pm$0.40 | Bressan et al. (2012) | CCD UBVRI | This paper
0.21 | 12.00 | 2.51 | 0.04 | – | 3.00 | Bertelli et al. (2008) | CCD UB | 1
0.10 | 12.30$\pm$0.12 | 2.50 | – | – | 2.50 | Hyades main sequence | CCD BVR | 22
0.15 | 11.75$\pm$0.09 | 2.24 | – | – | – | – | – | 23
0.16$\pm$0.007 | 12.40$\pm$0.12 | 3.02 | – | –0.06$\pm$0.04 | 2.30$\pm$0.20 | Demarque et al. (2004) | uvby | 24
0.14 | – | 2.43 | – | 0.07$\pm$0.01 | 2.57 | Marigo et al. (2008) | 2MASS JH$K_{s}$ | 6
0.16 | 12.35 | 2.95 | – | 0.00/-0.10 | 2.40 | VandenBerg et al. (1998) | BV | 25
0.14 | 11.93$\pm$0.10 | 2.43 | – | 0.09 | 2.60 | Dotter et al. (2008) | CCD VI | 26
| – | – | 0.014 | 0.05$\pm$0.03 | – | – | Spectroscopy | 13
0.24 | 11.94 | 2.36 | solar | solar | 1.62 | Girardi et al. (2002) | 2MASS JH$K_{s}$ | 12
0.12 | 12.20 | 2.75 | 0.02 | 0.09 | 2.00 | Bressan et al. (1993) | CCD VI | 27
0.13 (Av= 0.40) | 12.21 | 2.76 | solar | solar | 2.24 | Bressan et al. (2012) | Gaia DR2 | 14
0.16$\pm$0.02(Av=0.487$\pm$0.051) | – | 2.44$\pm$0.05 | – | 0.093$\pm$0.006 | 2.63$\pm$0.18 | Bressan et al. (2012) | Gaia DR2 | 15
NGC 6866 |  |  |  |  |  |  |  |
0.06$\pm$0.07 | 10.38$\pm$0.38 | 1.19$\pm$0.21 | 0.016 | 0.02 | 1.00$\pm$0.20 | Bressan et al. (2012) | CCD UBVRI | This paper
0.16$\pm$0.04 | 10.98$\pm$0.24 | 1.57$\pm$0.02 | 0.014 | – | 0.70$\pm$0.17 | Bressan et al. (2012) | CCD UBVRI | 28
| – | – | – | 0.016 | 1.48$\pm$0.21 | Paxton et al. (2013) | KIC 8263801 RG star | 29
0.204$\pm$0.002 | 10.47$\pm$0.02 | 1.24$\pm$0.01 | solar | solar | 0.65$\pm$0.10 | Bertelli et al. (2008) | Kepler Input Catalogue | 30
0.27 | 10.70 | 1.33 | solar | solar | 0.44 | Girardi et al. (2002) | 2MASS JH$K_{s}$ | 12
| – | – | 0.016 | 0.01$\pm$0.01 | – | – | Spectroscopy | 13
0.19$\pm$0.06 | 11.08$\pm$0.11 | 1.64$\pm$0.08 | solar | solar | 0.80$\pm$0.10 | Girardi et al. (2002) | 2MASS JH$K_{s}$ | 31
0.06$\pm$0.05 | 10.61$\pm$0.02 | 1.32$\pm$0.01 | 0.015 | –0.10 | 0.75$\pm$0.04 | Marigo et al. (2008) | CCD UBVRI | 32
0.15 (Av= 0.48) | 10.74 | 1.41 | solar | solar | 0.65 | Bressan et al. (2012) | Gaia DR2 | 14
0.16$\pm$0.02(Av=0.498$\pm$0.074) | – | 1.35$\pm$0.05 | – | 0.047$\pm$0.018 | 0.73$\pm$0.08 | Bressan et al. (2012) | Gaia DR2 | 15

Figure 13: Comparison of the astrophysical parameters obtained in this work
with literature (Table 9). The vertical line represent our findings based on
the $(B-V)$ (blue), $(R-I)$ (red), $(V-I)$ (black), and $(G_{BP}-G_{RP})$
(yellow) colours, respectively. The shaded regions show the uncertainties of
the parameters for $(B-V)$. The filled red and blue dots represent the
distances from Gaia EDR3 parallaxes, and the astrophysical parameters of the
DG method, respectively. In panel (a) the colour excess in the individual
colours are converted to $E(B-V)$ according to Sect. 8.1

###  8.2 Dynamical evolution

There appears to be an increasing trend between $R_{core}$ and $R_{RDP}$.
Güneş et al. (2017) obtained the relation $R_{RDP}=(4.69\pm
0.35)R_{core}^{(0.56\pm 0.11)}$ and all our objects follow this relation as
seen in Fig. 14(a).

Figure 14: $R_{RDP}$ versus $R_{core}$ (panel a), $\tau$ versus $t_{rlx}$
(panel b), $t_{diss}$ versus $t_{rlx}$ (panel c), $\chi$ versus $t_{rlx}$
(panel d), and $(R_{core},~{}R_{RDP})$ versus Age (panels e-f). The relation
and its 1$\sigma$ uncertainty as shaded area in panel (a), the regions $R1-R4$
in panels (e)-(f), and the comparison objects as filled grey dots are from
Güneş et al. (2017). Figure 15: $R_{core}$ versus $\log(\tau)$ (panel a), $c$
versus $R_{h}/R_{t}$ (panel b), $\log(R_{h})$ versus $\log(R_{core})$ (panel
c), $R_{core}/R_{h}$ versus $R_{h}/R_{t}$ (panel d), $R_{h}/R_{t}$ and
$R_{h}/R_{J}$ versus $R_{GC}$ (panel e). The grey filled dots and the dashed
line in panels (a)-(e) are from A20 and A21. Black and blue filled dots in
panel (e) represent the ratios with King-profile $R_{t}$ and Jacobi $R_{J}$
tidal radii, respectively.

The relaxation times ($t_{rlx}$) of the four OCs are much smaller than $\tau$
(see panel b of Fig. 14). Therefore, they can be considered as dynamically
relaxed. On the other hand, the disruption times of the four OCs are higher
than their relaxation times (see Fig. 14(c) and Table 7). This necessarily
implies that the four OCs break up in the initial phase of rapid expansion
because of the death of their most massive (bright) stars (Converse & Stahler,
2011).

Following the works by Camargo et al. (2009); Güneş et al. (2017); Cakmak et
al. (2021), the four OCs are plotted in a $(R_{core},~{}R_{RDP})$ versus Age
diagram in Fig. 14 (e)-(f). This relationship is related to survival and
dissociation rates of the OCs (Camargo et al., 2009). The horizontal and
vertical dotted lines in these panels define the regions $R1$ to $R4$ used by
Güneş et al. (2017) to separate small and large sized OCs.

The very steep negative MF of NGC 6791 $(\chi=-5.26)$ for stars
$m>0.93m_{\odot}$ and the value of $\tau=88$ indicates advanced dynamical
evolution. While its massive stars move into inner regions, its low mass stars
are transferred to its outer regions. Therefore, the $\chi$ value is highly
negative relative to its large $R_{RDP}=14$ pc. However, there is an
incompatibility between $\chi$ and $\tau=88$ of this OC. In this context,
there may be a large number of primordial massive stars in this cluster. From
panels (e)-(f) in Fig. 14, the location in the $R2$ region indicates an
expansion, which is caused by possible stellar black-holes and binaries, and
mass segregation in its core region.

The overall negative/steep MF of NGC 6819 $(\chi=-1.09)$ indicates mild scale
mass segregation due to the relatively large values of
$(t_{rlx},~{}\tau)=(27~{}Myr,108)$. The outer parts of NGC 6819 show an
expansion with time because of the transport of low-mass stars into the halo.
This is the reason why it occupies the $R2$ region in Fig. 14 (f). Due to its
sparse structure and mass segregation, it has a small core, resulting in the
position in the $R4$ region of Fig. 14 (e). Its steep $\chi$ points out that
the high mass stars slightly outnumber the low-mass stars in its central part
and that low-mass stars are transferred to its outer part. It seems to lose
few stars to the field, owing to the internal and external perturbations.

NGC 6811 with a negative/flat overall MF $(\chi=-0.82)$ and NGC 6866 with a
positive/flat MF $(\chi=-0.32)$ present signs of small scale mass segregation,
in the sense that their low mass stars slightly outnumber their high mass
stars. Their large $\tau$ values are linked to their advanced dynamical
evolution. The position of NGC 6811 in the $R4$ regions indicates a possible
shrinkage of the core/cluster radii. This means that it shrunk in size and
mass with time as it lost its star content because of the presence of massive
GMCs, and tidal effects from disc and Bulge crossings as external
perturbations. Instead of shrinking in size and mass with time, NGC 6811 may
also have a primordial origin in conjunction with high molecular gas density
in the Galactic directions (Camargo et al., 2009; van den Berg et al., 1991).
Note that NGC 6866 lies on the border of the $R2$ and $R4$ regions. This
object with a small mass will undergoe a dynamic evolution towards the $R4$
region as a result of mass segregation and core collapse.

To present the internal and external dynamical evolutionary context for our
sample OCs, we have plotted in Fig. 15 the relations $R_{core}$ versus
$\log(\tau)$ (panel a), $c$ versus $R_{h}/R_{t}$ (panel b), $\log(R_{h})$
versus $\log R_{core}$ (panel c), $R_{core}/R_{h}$ versus $R_{h}/R_{t}$ (panel
d), $R_{h}/R_{t}$ and $R_{h}/R_{J}$ versus $R_{GC}$ (panel e).

The four OCs show a decreasing trend between $R_{core}$ and $\log(\tau)$. As
shown by A20 and A21 (their data included as grey points in Fig. 15), this
trend points out that our OCs are dynamically evolved and that they are losing
their star content to the field due to internal/external dynamical processes.

Based on the concentration parameter $c$ one can infer that the four objects
are more compact clusters (panel b of Fig. 15). Three OCs are close to the OC
data by A20 and A21, but NGC 6811 stands far away. Furthermore, except NGC
6866, the objects follow the relation $R_{h}>R_{core}$ as shown in panel (c)
of the figure. The deviating OC has a small mass (113 $M_{\odot}$), and its
$R_{h}$ determination may also be affected by the small number of member stars
and their sparse distribution. A similar behaviour can be noticed also e.g.
for NGC 6573 studied by A18. However, we note that the data by A20 and A21
shown in Fig. 15 are based on the simple relation $R_{h}=1.3~{}a$ by using
Plummers’ $a$ parameter (Plummer, 1911) from the profile fit.

The $R_{h}/R_{t}$ ratios of NGC 6791 and NGC 6819 fall in the range of
0.10-0.23 according to the figures 2 in chapter 33 by Heggie & Hut (2003). In
this context they are tidally filled OCs. Hence, they are exposed to more
intense tidal effects due to their large tidal radii. With the relatively high
$R_{core}/R_{h}$ ratios, these OCs expand the point of being tidally filling,
and they undergo two-body relaxation and mass segregation in their core
regions. Their steep negative MFs and relatively large $\tau$ values support
this. These findings are also consistent with their expanding core/cluster
radii due to the likely presence of stellar black holes and binaries (Fig.
14e-f) and their relatively large $t_{rlx}$ and $t_{dis}$ values (see panel c
of Fig. 14).

NGC 6866, which shows $R_{core}/R_{h}=1.62$, and NGC 6811 with
$R_{h}/R_{t}=0.61$ deviate from the trend presented in Fig. 15(d). NGC 6811 is
also beyond the tidally limit value $R_{h}/R_{t}=0.40$ given by Heggie & Hut
(2003). Note that Collinder 110 with low $c$ and large $R_{GC}$ has a high
$R_{h}/R_{t}$ as is evident from fig.14(d) of A20. The high $R_{h}/R_{t}$
ratio of NGC 6811 suggests that it feels less the tidal effects by the
Galactic gravitational field without being tidally disrupted. Furthermore, its
low value $R_{core}/R_{h}=0.10$ as an indicator of its compactness implies
that its central part is in advanced evolutionary stage without being tidally
disrupted. The internal processes such as mass segregation can move its stars
into to the outskirts of this OC. The large $\tau$ value and the consequent
loss of low mass results in a small cluster size and mass. The high
$R_{core}/R_{h}$ ratio of NGC 6866 can be explained by that its core region is
exposed to mass segregation and the ow ratio $R_{h}/R_{t}=0.08$ indicates that
it is less subject to tidal effects. Note that low $R_{h}/R_{t}$ imply its
survival against tidal disruption.

All these features of NGC 6811 and NGC 6866 are also supported by their
contracting cores due to mass segregation and core-collapse (Fig. 14e-f) and
their small $t_{rlx}$ and $t_{diss}$ values (see panel c of Fig. 14). The
differences between the OCs that occupy almost the same age and $R_{GC}$
depend on the initial conditions at cluster formation and their environments
(Schilbach et al., 2006; Angelo et al., 2018). In the sense NGC 6811 might
have been born as a small size cluster.

The $R_{h}/R_{J}$ ratios of the four OCs in Fig. 15(e) show no trend with
$R_{GC}$. They are potentially exposed to stronger Galactic tidal field
because of their locations close to the solar circle. Our sample OCs do not
show a ratio $R_{h}/R_{J}>0.5$, such clusters would be disrupted with small
dissolution time depending on the strength of tidal perturbations. The
$R_{h}/R_{J}$ ratios of NGC 6791 and NGC 6811 fall in the range $[0.20,0.35]$
given by A18, indicating that they are tidally affected. The dynamical
evolution of NGC 6819 and NGC 6866 with a more compact $R_{h}/R_{J}$ ratio is
primarily driven by their internal relaxation.

The theoretical masses of the four OCs are obtained from the relation of King
(1962) with the help of the results for $R_{min}$ and $R_{J}$:

$M_{theo}=3.5M_{G}\left(\frac{R_{J}}{R_{min}}\right)^{3}$

Here, $M_{G}$, $R_{J}$, and $R_{min}$ are taken from Tables 7 and 8,
respectively. With these values, their theoretical masses have been calculated
as 14449 $M_{\odot}$ (NGC 6791), 386 $M_{\odot}$ (NGC 6811), 3796 $M_{\odot}$
(NGC 6819), and 198 $M_{\odot}$ (NGC 6866), respectively. By taking into
account their total masses (Table 6), for instance, the total mass of NGC 6791
(60500 $M_{\odot}$) decreases to $\sim 14449_{\odot}$ due to $R_{min}=4.71$
kpc, which corresponds to 76% of its initial total mass. Similarly, NGCC 6811,
NGC 6819 and NGC 6866 appear that they lost 84%, 83%, and 64% of their initial
total masses, respectively.

The fact that NGC 6791 indicates a tidal radius much larger than the Jacobi
radius might be linked to significant mass loss that occurs since its
formation about 7 Gyr ago. This can be also concluded for NGC 6819 and NGC
6866 with $R_{t}/R_{J}\gtrsim 1$. NGC 6811, on the other hand, with
$R_{t}/R_{J}=0.36$ keeps it stellar content within its Jacobi radius.

In summary from Figs. 14-15, the degree of mass segregation and the strength
of two-body relaxation played some role in shaping the inner parts of the OCs.
Their position in the first Galactic quadrant, different initial formation
conditions, strength of Galactic tidal perturbations from spiral arms and
Galactic disc/Bulge, Galactic tidal field, and encounters with GMCs are
responsible for their mass losses. These old OCs survived despite the
internal/external dynamical processes and 5 to 30 revolutions around the
Galactic center.

###  8.3 Radial migration and cluster orbits

We estimate the birth radii ($R_{birth}$) of the objects following Netopil et
al. (2022) by using their result for the current metallicity gradient based on
young open clusters, the model by Minchev et al. (2018) for the time evolution
of the Galactic Interstellar Medium (ISM) metallicity gradient, the
spectroscopic metallicities listed in Table 10, and distances and ages
obtained from the Gaia EDR3 CMD (Table 3). For details we refer to Netopil et
al. (2022), the results are given in Table 10.

Table 10: The migration distances $(d_{mig}=R_{guide}-R_{birth})$.  Cluster | [Fe/H] | Age | $R_{min}$ | $R_{max}$ | $R_{guide}$ | $R_{birth}$ | $d_{mig}$
---|---|---|---|---|---|---|---
| dex | Gyr | kpc | kpc | kpc | kpc | kpc
NGC 6791 | 0.35 | 7.20 | 4.71 | 8.62 | 6.67 | –0.45 | 7.12
NGC 6811 | –0.05 | 1.20 | 7.80 | 9.67 | 8.74 | 8.34 | 0.40
NGC 6819 | 0.05 | 2.90 | 7.82 | 8.50 | 8.16 | 5.79 | 2.37
NGC 6866 | 0.01 | 1.00 | 8.05 | 9.79 | 8.92 | 7.41 | 1.51

We note that in this paper we adopt a solar distance of $R_{GC}=8.2$ kpc and a
slightly different determination of the guiding radius ($R_{guide}$) than used
by Netopil et al. (2022). However, the derived $R_{birth}$ distances are in
good agreement with their results, suggesting that NGC 6791 has its origin in
or close to the Galactic center. We estimate that the cluster could have
radially migrated by about 7 kpc. The migration distances
$(d_{mig}=R_{guide}-R_{birth})$ of the objects generally increase with age,
thus they follow the trend of the cluster population as shown by Netopil et
al. (2022).

NGC 6791, an old metal-rich OC (7–8 Gyr, $[Fe/H]=0.35-0.47$), is an intriguing
system and therefore target of numerous works that deal with its astrophysical
parameters and origin (Peterson & Green, 1998; Anthony-Twarog et al., 2007;
Carretta et al., 2007; Casamiquela et al., 2017a; Donor et al., 2020).

Twarog et al. (2011) evaluate that NGC 6791 might show indication of an
external origin because of its position in the solar circle, the high metal
abundance, and the quite eccentric orbit. According to Jilkova et al. (2012),
a strong bar and spiral arm effects are responsible for its migration from the
inner disc $(R=3-5~{}kpc)$ to the current position.

Carrera et al. (2022) list for NGC 6791 an eccentricity of 0.35, which is
close to our result (see Table 8). The large eccentricity could indicate that
the cluster was formed in inner Galactic regions, as its metal content
suggests, but its orbit has been perturbed in such a way that it acquires a
higher eccentricity, thus spending a significant fraction of time at larger
radii. Anyway, it is not uncommon that the oldest clusters show larger
eccentricities (see e.g. Tarricq et al., 2021).

The large eccentricity $(ecc=0.59)$ obtained by Carraro et al. (2006) was also
interpreted as a core of a large system, which is exposed to strong tidal
stripping. Another scenario suggested by them is that this OC was formed in
the inner side of the Galaxy, close to the metal-rich bulge.

Linden et al. (2017) suggest that NGC 6791 is either an original member of the
thick disc or a former member of the Galactic bulge, because of the high metal
$([Fe/H]=0.28-0.34)$ and high-$\alpha$ $(0.08-0.10)$ abundances which are
obtained for five members from APOGEE DR13 data.

Martinez-Medina et al. (2018) report that NGC 6791 formed in the inner thin
disc or in the bulge, and then migrated to its current location. Also
Villanova et al. (2018) conclude based on the location of NGC 6791
$(z,R_{GC})=(1,8)$ kpc that is spatially member of the Galactic disc. From the
spectroscopic findings $[Fe/H]=0.313\pm 0.005$ and $[\alpha/Fe]=+0.06\pm 0.05$
for the giant sample, they support a scenario of a Galactic bulge origin with
radial migration to its current position.

Our migration and eccentricity values ($d_{mig}=7.12$ kpc and $ecc=0.29$) for
NGC 6791 provide a support in favor of the radial migration scenarios.

NGC 6811 seems to be still very close to its birth position and shows by far
the lowest migration rate (migration distance in dependence of the age). For
the remaining objects (NGC 6819 and NGC 6866) we estimate migration rates of
about 0.8-1.5 kpc/Gyr. These values are quite consistent with the results by
Netopil et al. (2022). They note that objects up to about 2 Gyr show mean
migration rates of about 1 kpc/Gyr and that there is a decrease of the
migration rate with age, because older clusters also tend to be dynamically
hotter objects. Their study shows a significant scatter of the migration
distances, which they attribute among others to the merge of objects at
different Galactic locations. However, the individual dynamical stage of the
OCs might contribute to the scatter as well. An analysis of the overall mean
migration distances in dependence on Galactic location and dynamical stage
will certainly require a much larger sample in respect of both, OC
metallicities and detailed dynamical knowledge for these objects.

###  8.4 Concluding remarks

In this paper we provide a detailed study of the open clusters NGC 6791, NGC
6811, NGC 6819 and NGC 6866 in respect of the “classical” astrophysical
parameters like the age and the distance, but also for the mass and mass-
function, cluster radii, orbital parameters, dynamical evolution, and radial
migration.

Such information is of general importance to trace the evolution of the OC
population in the Galactic context. The studied objects are covered also by
the Kepler prime field, thus the results can be furthermore used in the
context of variable stars - e.g. for studies related to their evolution or
dependency on the various OC parameters.

## Acknowledgments

This work was supported by Scientific Research Projects Coordination Unit of
Istanbul University. Project Number: FBA-2017-23599. We thank our referee for
her/his valuable suggestions/comments. We thank M. Angelo and J. Santos for
the valuable discussion on Fig. 15. O. Günes is thanked for his interpretation
on the dynamical evolution. G. Carraro is thanked for providing the CCD $UB$
data of NGC 6791 for the comparison. We thank R. Carrera for providing the
orbital parameters for Table 8 and his comments on the origin of NGC 6791. The
open cluster data is based upon the observations carried out at the
Observatorio Astronómico Nacional on the Sierra San Pedro Mártir (OAN-SPM),
Baja California, México. This paper has made use of results from the European
Space Agency (ESA) space mission Gaia, the data from which were processed by
the Gaia Data Processing and Analysis Consortium (DPAC). Funding for the DPAC
has been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. The Gaia mission website is
http: //www.cosmos.esa.int/gaia. This paper has also made use of the WEBDA
database, operated at Department of Theoretical Physics and Astrophysics of
the Masaryk University, Brno. This publication also makes use of SIMBAD
database-VizieR (http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=II/246.).

## Data Availability

The photometric data of the four OCs can be requested from Raul Michel
(rmm@astro.unam.mx).

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##  Appendix A Photometric Errors and Differential Grids Fits

Table 11: The mean photometric errors of $V$, $(R-I)$, $(V-I)$, $(B-V)$ and $(U-B)$ for NGC 6791, NGC 6811, NGC 6819 and NGC 6866 in terms of $V$ mag. |  |  | NGC 6791 |  |
---|---|---|---|---|---
V | $\sigma_{V}$ | $\sigma_{R-I}$ | $\sigma_{V-I}$ | $\sigma_{B-V}$ | $\sigma_{U-B}$
11 - 12 | 0.005 | 0.007 | 0.007 | 0.006 | 0.009
12 - 13 | 0.004 | 0.008 | 0.007 | 0.006 | 0.008
13 - 14 | 0.005 | 0.009 | 0.008 | 0.009 | 0.014
14 - 15 | 0.005 | 0.007 | 0.007 | 0.008 | 0.019
15 - 16 | 0.006 | 0.009 | 0.008 | 0.011 | 0.027
16 - 17 | 0.009 | 0.013 | 0.013 | 0.017 | 0.050
17 - 18 | 0.014 | 0.018 | 0.019 | 0.030 | 0.082
18 - 19 | 0.027 | 0.031 | 0.036 | 0.056 | 0.122
19 - 20 | 0.056 | 0.050 | 0.067 | 0.108 | –
20 - 21 | 0.104 | 0.055 | 0.107 | – | –
|  |  | NGC 6811 |  |
V | $\sigma_{V}$ | $\sigma_{R-I}$ | $\sigma_{V-I}$ | $\sigma_{B-V}$ | $\sigma_{U-B}$
10 - 11 | 0.031 | 0.044 | 0.032 | 0.036 | 0.024
11 - 12 | 0.015 | 0.030 | 0.017 | 0.019 | 0.013
12 - 13 | 0.010 | 0.022 | 0.012 | 0.015 | 0.012
13 - 14 | 0.008 | 0.018 | 0.010 | 0.012 | 0.011
14 - 15 | 0.006 | 0.012 | 0.009 | 0.011 | 0.012
15 - 16 | 0.007 | 0.009 | 0.009 | 0.010 | 0.012
16 - 17 | 0.010 | 0.014 | 0.014 | 0.015 | 0.023
17 - 18 | 0.018 | 0.020 | 0.022 | 0.027 | 0.040
18 - 19 | 0.035 | 0.036 | 0.042 | 0.051 | 0.053
|  |  | NGC 6819 |  |
V | $\sigma_{V}$ | $\sigma_{R-I}$ | $\sigma_{V-I}$ | $\sigma_{B-V}$ | $\sigma_{U-B}$
10 - 11 | 0.024 | 0.009 | 0.008 | 0.027 | –
11 - 12 | 0.010 | 0.033 | 0.031 | 0.028 | 0.030
12 - 13 | 0.013 | 0.031 | 0.028 | 0.021 | 0.019
13 - 14 | 0.009 | 0.022 | 0.019 | 0.015 | 0.017
14 - 15 | 0.014 | 0.014 | 0.017 | 0.017 | 0.013
15 - 16 | 0.011 | 0.012 | 0.014 | 0.014 | 0.013
16 - 17 | 0.012 | 0.016 | 0.015 | 0.015 | 0.018
17 - 18 | 0.014 | 0.022 | 0.020 | 0.020 | 0.029
18 - 19 | 0.021 | 0.034 | 0.031 | 0.033 | 0.046
19 - 20 | 0.029 | 0.045 | 0.041 | 0.045 | 0.056
|  |  | NGC 6866 |  |
V | $\sigma_{V}$ | $\sigma_{R-I}$ | $\sigma_{V-I}$ | $\sigma_{B-V}$ | $\sigma_{U-B}$
10 - 11 | 0.020 | 0.016 | 0.035 | 0.031 | 0.027
11 - 12 | 0.027 | 0.019 | 0.030 | 0.042 | 0.039
12 - 13 | 0.019 | 0.012 | 0.021 | 0.029 | 0.025
13 - 14 | 0.013 | 0.010 | 0.015 | 0.019 | 0.017
14 - 15 | 0.008 | 0.009 | 0.011 | 0.013 | 0.013
15 - 16 | 0.008 | 0.011 | 0.011 | 0.011 | 0.016
16 - 17 | 0.010 | 0.012 | 0.013 | 0.014 | 0.025
17 - 18 | 0.013 | 0.016 | 0.017 | 0.019 | 0.036
18 - 19 | 0.020 | 0.024 | 0.026 | 0.030 | 0.050

Figure 16: The distribution of the photometric errors of V, (R-I), (V-I),
(B-V) and (U-B) against V mag for NGC 6791, NGC 6811, NGC 6819 and NGC 6866.

Figure 17: The thick line represents the isochrone for the parameters given in
Table 4, the dotted line for $Z=0.006$, and the dashed line for $Z=0.02$.
$T_{N}$ is the temperature difference in dex between the star and the ZAMS at
solar metallicity using the mean temperature based on up to five colours.