# A Dynamical Evolution Study of the Open Clusters: Berkeley 10, Berkeley 81,
Berkeley 89 and Ruprecht 135

Hikmet Çakmak<EMAIL_ADDRESS>Yüksel Karataş

###### Abstract

By utilising Gaia EDR3 photometric/astrometric data, we studied the dynamical
evolution from the obtained astrophysical, structural and dynamical parameters
of the open clusters (OCs), Berkeley 10 (Be 10), Berkeley 81 (Be 81), Berkeley
89 (Be 89), and Ruprecht 135 (Ru 135). The Gaia EDR3 photometric distances
from the isochrone fitting method are smaller than the ones of Gaia EDR2. The
relaxation times of four OCs are smaller than their ages, in this regard, they
are dynamically relaxed. Their steep overall mass function slopes mean that
their low mass stars outnumber their massive ones. Their large
$\tau$/relatively small $t_{rlx}$ values imply an advanced mass segregation.
Therefore, they seem to have lost their low-mass stars much to the field. Be
89’s outer parts indicate an expansion with time. However, Be 10 and Be 81
show the relatively shrinkage core/cluster radii due to dynamical evolution.
Ru 135 (1.0 Gyr) may have a primordial origin, instead of shrinking in size
and mass with time. Be 89’s tidal radius is less than its cluster radius. This
means that its member stars lie within its tidal radius, in the sense it is
gravitationally bound to the cluster. For the rest OCs, the cluster members
beyond their tidal radii are gravitationally unbound to the clusters, which
are more influenced by the potential of the Galaxy.

###### keywords:

(Galaxy:) open clusters and associations:general , (Galaxy:) open clusters and
associations: individual , Galaxy: stellar content

††journal: New Astronomy

[i1]organization=İstanbul University Faculty of Science, Department of
Astronomy and Space Sciences, addressline=İstanbul Üniversitesi Merkez
Kampüsü, city=İstanbul, postcode=34116, state=Türkiye, country=TR

##  1 Introduction

Cluster members of the OCs evolve dynamically due to internal (mass loss
during stellar evolution, mass segregation, evaporation) and external
perturbations such as encounters with the disk and Giant Molecular Clouds
(GMCs), spiral arm shocks, and tidal stripping due to the galactic potential.
The OCs lose a significant fraction of their primordial masses through via
these processes (Lamers & Gieles, 2006) and (Gieles et al., 2007). As
emphasized by Converse & Stahler (2011), inner cores of star clusters
contract, and thus their outer parts expand by transferring energy to the
outer skirts. In the sense mass segregation drives the low mass stars to the
outer parts of the clusters. Most of the original mass are lost to the field
because of dynamical effects along the several Myrs since their formation.

From the analysis of Gaia EDR3 data of 389 OCs, Tarricq et al. (2021) report
that the cluster dimensions and mass segregation depend on cluster ages, in
this regard, their core radii are on the average small for old OCs. They also
state that mass segregation is more effective in old OCs.

In this paper we study the dynamical evolution of Be 81, Ru 135, Be 10, and Be
89 from Gaia EDR3 photometric/astrometric data. The first two are located in
the inner galaxy and the other two are outside the solar circle. These old-
aged OCs have been selected as representatives of first and second galactic
quadrants in Fig. 1, where their positions relative to the spiral arms are
displayed. Location criterion is important due to the survival rate of the
OCs. As discussed by Bonatto & Bica (2007b) and Güneş et al. (2017), the
majority of OCs with $age>1$ Gyr lie outside the solar circle. The OCs inside
the solar circle are scarce in the direction of galactic centre, due to the
effects of strong absorption/crowding/dissolution by GMCs. Bonatto & Bica
(2007b) reported a deficiency of OCs older than 0.85 Gyr inside solar radius.
Within the framework of these issues, the astrophysical parameters (reddening,
distance, age), structural parameters $(R_{RDP},~{}R_{core})$, tidal radii
($R_{t}$), overall masses, mass function slopes ($\chi$, MFs), time scales
(relaxation time $t_{rlx}$, evolutionary parameter $\tau$, dissociation time
$t_{dis}$) of these OCs have been derived. Our sample OCs were studied before
in the literature with 2MASS JHKs and UBVRI photometries (Table 2). Cantat-
Gaudin et al. (2020) also provided their astrophysical parameters from Gaia
DR2 data.

The organization of this paper is as follows. The separation of the cluster
members is explained in Section 2. The determination of reddenings, distance
moduli/distances, ages, and structural parameters of four OCs is given in
Section 3. Masses/mass function slopes, time scales (relaxation and
dissolution times and evolutionary parameters), tidal radii, half-mass radii
of four OCs are presented in Sections 4-5. Kinematics and orbital parameters
are given in Section 6. A Discussion/Conclusion is presented in Section 7.

Figure 1: Spatial distribution of four OCs (filled red circles). The schematic
projection of the Galaxy with its spiral arms is seen from the North pole.
$(X,~{}Y)$ kpc show the galactosentric cartesian coordinates. The sun is at
8.2 kpc.

##  2 Separation of the Cluster Members

To separate the cluster members of our sample OCs from the field stars, their
Gaia EDR3 astrometric/photometric data Brown et al. (2018) have been
downloaded for a large-area (10 arcmin) from

Figure 2: $\mu_{\alpha}$ versus $\mu_{\delta}$ diagrams of four OCs. Filled
dots denote the likely members, which are determined by using GMM. The grey
points represent the field stars.

VizieR111http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=II/246.. On the
($\mu_{\alpha}$, $\mu_{\delta}$) diagrams (Fig. 2), the potential cluster
members show a more concentrated structure, whereas field stars have a more
scattered distribution. We have applied Gaussian Mixture Model (GMM)
(Pedregosa et al., 2011) to the cluster stars. During GMM model trials, more
than two suitable components were selected to clearly obtaion the central
density in the vector point diagrams. Thus, the membership probabilities P(%)
of each cluster have been determined appropriately. The first significant rise
in Fig. 3 is taken into account as the membership percentage (vertical dashed
blue line). From Fig. 3, the probable members of four OCs seem to have greater
than 85$\%$. The GMM 222$P$ is defined $\Phi_{c}$ /$\Phi$. Here
$\Phi=\Phi_{c}+\Phi_{f}$ is the total probability distribution. c and f
subscripts for cluster and field parameters, respectively. The used parameters
for estimation of $\Phi_{c}$ and $\Phi_{f}$ are $\mu_{\alpha}$,
$\mu_{\delta}$, $\varpi$, $\sigma_{\mu\alpha}$, $\sigma_{\mu\delta}$,
$\sigma_{\varpi}$. model considers that the distribution of proper motions of
the stars in a cluster’s region can be represented by two elliptical bivariate
Gaussians. For the used parameters, see the papers of Balaguer-Nunez et al.
(1998), Wu et al. (2002), Sariya et al. (2012) and Dias et al. (2018).

Table 1: Equatorial/Galactic coordinates, the median proper motion components,
parallaxes/distances, and the numbers of the probable members of four OCs (top
rows) and Cantat-Gaudin et al. (2018, 2020) (bottom rows).

Cluster $\alpha_{2000}(h\,m\,s)$
$\delta_{2000}(^{\circ}\,{}^{\prime}\,{}^{\prime\prime})$ $\ell(^{\circ})$
$b(^{\circ})$ $\mu_{\alpha}$ (mas/yr) $\mu_{\delta}$ (mas/yr) $\varpi$ (mas)
$d(kpc)$ N Be 10 3 39 30.9 66 29 08.7 138.61 8.88 $-1.50$${}\pm{}$ $0.03$
$1.01$${}\pm{}$ $0.06$ $0.39$${}\pm{}$ $0.06$ $2.45$${}\pm{}$ $0.36$ 197 Be 81
19 01 41.2 -00 27 25.7 33.69 -2.49 $-1.18$${}\pm{}$ $0.08$ $-1.93$${}\pm{}$
$0.07$ $0.24$${}\pm{}$ $0.06$ $3.70$${}\pm{}$ $1.01$ 171 Be 89 20 24 24.9 46
02 03.1 83.13 4.84 $-1.98$${}\pm{}$ $0.06$ $-2.32$${}\pm{}$ $0.06$
$0.28$${}\pm{}$ $0.06$ $3.35$${}\pm{}$ $0.64$ 221 Ru 135 17 57 59.4 -11 39
59.8 16.38 6.27 $1.44$${}\pm{}$ $0.07$ $-1.46$${}\pm{}$ $0.05$ $0.89$${}\pm{}$
$0.06$ $1.11$${}\pm{}$ $0.07$ 119 Be 10 $-1.47$${}\pm{}$ $0.07$
$1.05$${}\pm{}$ $0.11$ $0.37$${}\pm{}$ $0.05$ $2.37$${}\pm{}$ $0.34$ 179 Be 81
$-1.20$${}\pm{}$ $0.16$ $-1.85$${}\pm{}$ $0.16$ $0.25$${}\pm{}$ $0.09$
$3.31$${}\pm{}$ $1.17$ 127 Be 89 $-2.01$${}\pm{}$ $0.13$ $-2.25$${}\pm{}$
$0.14$ $0.24$${}\pm{}$ $0.08$ $3.51$${}\pm{}$ $1.10$ 176 Ru 135
$1.42$${}\pm{}$ $0.12$ $-1.42$${}\pm{}$ $0.07$ $0.90$${}\pm{}$ $0.05$
$1.05$${}\pm{}$ $0.05$ 48

Figure 3: The membership probability histogram P(%). The vertical blue lines
indicate the selected lower limit of the probability.

The median proper motion components, the median parallaxes/distances, and the
member numbers of four OCs are presented in Table 1. Instead of the inverse of
Gaia EDR3 parallaxes, we averaged the parallaxes of cluster stars and then get
the posterior probability density functions(PDFs) from those reduced errors,
given by Bailer-Jones et al. (2018, 2021). For this, the global zero point is
taken as $-$0.017 mas (Lindegren et al., 2021). Within the uncertainties, all
the values are consistent with each other within the uncertainties, as
compared to Cantat-Gaudin et al. (2018, 2020) (bottom row). The estimated
median equatorial coordinates from the likely members of our OCs and their
corresponding galactic coordinates are listed in Table 1. These coordinates
are similar to those of WEBDA (Mermilliod, 1992).

##  3 Astrophysical and Structural Parameters

Figure 4: The Gaia EDR3 CMDs of Be 10 and Be 81. The coloured solid lines
represent the B12 isochrones in terms of Age (top panels) and $Z$-heavy
element abundance (bottom panels). The used ages and heavy element abundances
for the fitted isochrones are indicated in the insets. The isochrones of
$(Z=0.008,~{}1.35~{}Gyr)$ for Be 10 and of $(Z=0.025,~{}1.6~{}Gyr)$ for Be 81
provide satisfactory fits. For Be 81, the dashed isochrone shows the fit of
Cantat-Gaudin et al. (2020).

The astrophysical parameters from the probable members of four OCs have been
determined by fitting the observational $(G,G_{BP}-G_{RP})$ colour-magnitude
diagrams (CMDs) to the PARSEC isochrones of Bressan et al. (2012) (hereafter
B12). Magrini et al. (2015) give the spectroscopic metal abundance of Be 81 as
$[Fe/H]=+0.23\pm 0.08$. This corresponds to heavy element abundance $Z=0.025$.
For the rest OCs, their spectroscopic metal abundances are not available in
the literature. Therefore, several iterations by taking the isochrones of
three heavy element abundances ($Z=0.008,0.0152,0.025$) together with an
appropriate age are performed on the Gaia CMDs. The isochrones with
$(Z,Age~{}(Myr))=$ (0.008, 1350) for Be 10, (0.025, 1600) for Be 81, (0.0152,
3200) for Be 89, and (0.0152, 1000) for Ru 135, respectively provide
satisfactory fits (Figs. 4-5). The PARSEC isochrones are shifted both
vertically and horizontally on the Gaia CMDs to obtain the best fit to the
observed main sequence, as well as the RG/RC (Red Giant/Red Clump) sequence,
according to the interstellar reddening values of $E(G_{BP}-G_{RP})$ (Col. 2
of Table 2). Thus, distance modulus $DM=(V_{0}-M_{V})$ and age are obtained.
The distance moduli $(V_{0}-M_{V})$, distances d(pc) and ages (Myr) of four
OCs are given in Cols. 5-7 of Table 2. Their $E(B-V)$’s are converted from
their $E(G_{BP}-G_{RP})$’s via $E(B-V)=0.775E(G_{BP}-G_{RP})$ (Wang & Chen,
2019).

Figure 5: The Gaia EDR3 CMDs of Be 89 and Ru 135. The symbols are the same as
Fig. 4. The solar abundance isochrones, $(Z=0.0152,~{}3.2~{}Gyr)$ for Be 89
and $(Z=0.0152,~{}1.0~{}Gyr)$ for Ru 135 provide good fits to the cluster
members. Table 2: Comparison of the astrophysical parameters of four OCs to
the literature.

Cluster $E(G_{BP}-G_{RP})$ $E(B-V)$ Z $(V-M_{V})_{0}$ d(pc) Age (Myr) Phot.
Isochrone Reference Be 10 1.03$\pm$0.03 0.75$\pm$0.02 0.008 11.40$\pm$0.12
1909$\pm$96 1350$\pm$100 Gaia EDR3 Bressan et al.(2012) This work solar 11.87
2365 1202 Gaia DR2 Bressan et al.(2012) (1) 0.80 solar 11.97 2201 646 2MASS
Marigo et al.(2008) (2) 0.52 solar 11.92 1970 1000 2MASS Girardi et al.(2002)
(3) 0.75 0.006 11.20 1700 1100 UBVRI Marigo et al.(2008) (4) 0.87 0.008 11.80
2290 631 UBVRI Girardi et al.(2002) (5) 0.71 solar 11.26 1790 1000 BV Bertelli
et al. (1994) (6) Be 81 1.16$\pm$0.03 0.85$\pm$0.02 0.025 12.12$\pm$0.10
2648$\pm$119 1600$\pm$100 Gaia EDR3 Bressan et al.(2012) This work solar 12.60
3313 1148 Gaia DR2 Bressan et al.(2012) (1) 1.06 solar 13.00 3401 1000 2MASS
Marigo et al.(2008) (2) 0.90 solar 13.55 3601 891 2MASS Girardi et al.(2002)
(3) 1.00 solar 15.60 3000 1000 BVI Bertelli et al.(1994) (7) 0.91 solar 12.40
3020 900 BVI Bressan et al. (1993) (8) Be 89 0.79$\pm$0.001 0.58$\pm$0.001
0.0152 11.93$\pm$0.11 2433$\pm$118 3200$\pm$100 Gaia DR2 Bressan et al.(2012)
(1) 0.73 solar 12.50 2839 1585 2MASS Marigo et al.(2008) (2) 0.49 solar 12.99
3272 1585 2MASS Girardi et al.(2002) (3) 0.60 0.008 11.90 2400 3600 UBVRI
Marigo et al.(2008) (4) 1.03 solar 13.30 3005 850 2MASS Bonatto et al.(2004)
(9) 1.05 solar 14.80 2040 1122 BVI Girardi et al.(2000) (10) Ru 135
0.74$\pm$0.05 0.54$\pm$0.04 0.0152 9.76$\pm$0.06 897$\pm$60 1000$\pm$100 Gaia
EDR3 Bressan et al.(2012) This work solar 10.12 1054 646 Gaia DR2 Bressan et
al.(2012) (1) 0.87 solar 13.00 3497 3162 2MASS Marigo et al.(2008) (2) 1.14
solar 12.66 2172 562 2MASS Girardi et al.(2002) (3) 0.63 0.004 9.50 750 4000
UBVRI Marigo et al.(2008) (4) 1.10 solar 11.33 1850 501 2MASS Bonatto et
al.(2004) (11)

References of the last column

1 | Cantat-Gaudin et al. (2018, 2020) | 7 | Sagar & Griffiths (1998)
---|---|---|---
2 | Kharchenko et al. (2013) | 8 | Donati et al. (2014)
3 | Bukowiecki et al. (2011) | 9 | Tadross (2008a)
4 | Akkaya et al. (2010) | 10 | Subramaniam et al. (2010)
5 | Lata et al. (2004) | 11 | Tadross (2008b)
6 | Maciejewski & Niedzielski (2007) |  |

The stellar radial density profiles (RDPs) of four OCs are constructed from
the Gaia EDR3 photometric/astrometric data for cluster members and the field
within $10^{\prime}.0$ down to $G=19$ mag. From the RDPs, their structural
parameters are derived. The fitted RDPs of four OCs to the relation of King
(1966) are given in Fig. 6. For this, the three-parameter function (King,
1966),

$\sigma(R)=\sigma_{bg}+\frac{\sigma_{0}}{1+(\frac{R}{R_{core}})^{2}}$

is considered. For this equation, $\sigma_{bg}$ is the residual background
density. $\sigma_{0}$ and $R_{core}$ are the central

Figure 6: The radial density profiles of four OCs. Solid line shows the best
fit King profile and the horizontal gray bar denotes the stellar background
level measured in the comparison field. The $1\sigma$ King fit uncertainty is
shown by the red shaded domain.

density of stars and the core radius, respectively. Thus, the $R_{core}$,
$\sigma_{bg}$, and $\sigma_{0}$ of four OCs have been derived by fitting King
profile to the observational RDPs (Fig. 6). The meanings of solid line,
horizontal bar and the shaded area are mentioned in caption of Fig. 6.

Table 3: Structural parameters. Row 1: arcmin to parsec scale. The symbol
$*\,\prime^{-2}$ in rows 2–3 denotes $stars~{}arcmin^{-2}$. $\delta_{c}$ in
row 4 means the density contrast parameter. Core and cluster radii are listed
in rows. 5–7 in units of arcmin and pc, respectively. CC (last row) means the
correlation coefficient.

Parameters Be 10 Be 81 Be 89 Ru 135 $(1^{\prime})\,(pc)$ 0.58 0.67 0.87 0.26
$\sigma_{0}\,(*\,\prime^{-2})$ $8.32$${}\pm{}$ $0.98$ $20.81$${}\pm{}$ $2.76$
$10.90$${}\pm{}$ $0.75$ $7.02$${}\pm{}$ $0.09$ $\sigma_{bg}\,(*\,\prime^{-2})$
$2.53$${}\pm{}$ $0.08$ $6.69$${}\pm{}$ $0.24$ $4.66$${}\pm{}$ $0.13$
$8.11$${}\pm{}$ $0.82$ $\delta_{c}$ $4.28$${}\pm{}$ $0.52$ $4.11$${}\pm{}$
$0.56$ $3.34$${}\pm{}$ $0.25$ $2.17$${}\pm{}$ $1.05$ $R_{core}\,(\prime)$
$1.90$${}\pm{}$ $0.15$ $0.63$${}\pm{}$ $0.03$ $2.22$${}\pm{}$ $0.20$
$0.96$${}\pm{}$ $0.08$ $R_{RDP}\,(\prime)$ $11.23$${}\pm{}$ $0.43$
$4.51$${}\pm{}$ $0.29$ $13.02$${}\pm{}$ $0.57$ $6.51$${}\pm{}$ $0.30$
$R_{core}\,(pc)$ $1.10$${}\pm{}$ $0.09$ $0.42$${}\pm{}$ $0.02$ $1.93$${}\pm{}$
$0.17$ $0.25$${}\pm{}$ $0.02$ $R_{RDP}\,(pc)$ $6.50$${}\pm{}$ $0.25$
$3.02$${}\pm{}$ $0.19$ $11.38$${}\pm{}$ $0.50$ $1.69$${}\pm{}$ $0.08$ CC 0.991
0.995 0.988 0.989 Bukowiecki et al. (2011) $R_{core}\,(pc)$ $1.36$${}\pm{}$
$0.18$ $0.63$${}\pm{}$ $0.08$ $0.95$${}\pm{}$ $0.12$ $0.45$${}\pm{}$ $0.07$
$R_{RDP}\,(pc)$ $7.96$${}\pm{}$ $1.30$ $5.88$${}\pm{}$ $0.94$ $8.23$${}\pm{}$
$1.18$ $3.42$${}\pm{}$ $0.62$ Kharchenko et al. (2013) $R_{core}\,(pc)$ 0.80
1.28 1.67 1.71 Maciejewski & Niedzielski (2007) $R_{core}\,(pc)$
$0.70$${}\pm{}$ $0.25$ $R_{RDP}\,(pc)$ $4.30$${}\pm{}$ $1.50$ Sagar &
Griffiths (1998) $R_{RDP}\,(pc)$ 2.35

Their cluster radii $(R_{RDP})$ by comparing the RDP level with background and
by measuring the distance from the centre (Bonatto & Bica, 2007a) are
determined. The $R_{RDP}$ is taken as an observational truncation radius,
whose value depends both on the radial distribution of member stars and the
field density. The large uncertainties within $R<1^{\prime}$ in the RDPs are
due to their low star contents in their central parts. The density contrast
parameter ($\delta_{c}$) to estimate the compactness of the clusters is
obtained from the relation,

$\delta_{c}=1+\frac{\sigma_{0}}{\sigma_{bg}}$

As emphasized by Bica & Bonatto (2005), the $\delta_{c}$ values of the OCs
depend on the Galactic latitude than the cluster mass, accordingly, the
populous and high Galactic latitude OCs have large density contrast
parameters, i.e.: NGC 188 $(b=+22.^{\circ}39,\sim 3800M_{\odot})$ and M 67
$(b=+31.^{\circ}89,\sim 990M_{\odot})$. Different Galactic latitudes have
different properties and also suffer from differences in density contrast,
since low-Galactic latitude OCs tend to be projected against denser
fore/backgrounds. As compared to the range $(7\leq\delta_{c}\leq 23)$, which
is found from compact star clusters by Bonatto (2009), our $\delta_{c}$ values
are lower. This indicates that our OCs with low-latitude are sparse. These
structural parameters and their meanings together with the literature values
are listed in rows 1-9 of Table 3.

As compared to literature values (bottom rows of Table 3), for the
$R_{core}$(pc)’s of Be 10, Be 81 and Ru 135, there is in a reasonable
agreement with the ones of Bukowiecki et al. (2011) within the uncertainties.
Be 89’s $R_{core}$ value is somewhat larger than Bukowiecki et al. (2011).
Kharchenko et al. (2013) give a large $R_{core}$ (pc) value for Be 81 and Ru
135. For Be 10 and Be 89, their $R_{core}$ (pc) values are in good agreement
with our findings within the error bars. Our $R_{RDP}$ values for Be 89 and Ru
135 are quite larger than the 8.23 pc value of Bukowiecki et al. (2011).
Within the error limits, our $(R_{core},~{}R_{RDP})$ values for Be 10 are in
consistent with the ones of Maciejewski & Niedzielski (2007). For Be 81, Sagar
& Griffiths (1998) give a slightly small $R_{RDP}$ (pc) value.

##  4 Mass and Mass Function Slope

The masses of the main sequence members of four OCs are determined from the
B12-isochrones. $G$ magnitudes of main sequence members have been converted
into the absolute magnitudes from their reddenings and distance moduli (Cols.
2 and 5 of Table 2). Mass function (MF) is defined as the distribution of
masses of cluster stars per unit volume. The mass functions in terms of 0.05
bin sizes are shown in Fig. 7. The MF slope is fitted to a power-law given by,

$\log\frac{dN}{dM}=-(1+x)\log(M)+\text{constant}$

Where $dN$ is the probable cluster members in a mass bin $dM$ with central
mass $M$ and $x$ is mass function slope. Since Gaia data ($G$ mag) is not
complete below $G=19$ mag (Arenou et al., 2018), stars brighter than this
limit are taken, which correspond to stars more massive than 1 $M_{\odot}$.
The obtained MF slopes are $x=1.46\pm 0.60$, $2.88\pm 0.93$, $1.24\pm 0.90$
and $1.99\pm 0.44$ for Be 10, Be 81, Be 89, and Ru 135, respectively. The MFs
of Be 10 and Be 89 are consistent with $\chi=1.3\pm 0.7$ of Kroupa (2001) with
the uncertainties. Total masses are obtained from the MF slopes of four OCs.
The mass range, MF slope, total mass, mean mass, and member number are listed
in Cols. 2-6 of Table 4. In order to see the mass-segregation effects of four
OCs, the normalized cumulative radial stellar distributions of member stars
for three mass ranges against cluster radius (arcmin) have been displayed in
Fig. 8. For Be 10, bright stars (high mass ones) slightly outnumber low
mass/fainter counterparts throughout the inner parts. Towards its outer parts,
low-mass

Table 4: The mass information of four OCs calculated from their mass functions. Cluster | Mass range | MF slope | Total mass | Mean Mass | Members
---|---|---|---|---|---
| $M_{\odot}$ | $\chi$ | $M_{\odot}$ | $M_{\odot}$ | N
Be 10 | 0.75 $-$ 1.13 |  $1.46$${}\pm{}$ | $0.60$ |  $109.6$${}\pm{}$ | $13.1$ |  $0.90$${}\pm{}$ | $0.15$ |  $122$${}\pm{}$ | $14$
Be 81 | 0.91 $-$ 1.21 |  $2.88$${}\pm{}$ | $0.93$ |  $134.5$${}\pm{}$ | $14.9$ |  $1.03$${}\pm{}$ | $0.16$ |  $131$${}\pm{}$ | $14$
Be 89 | 0.80 $-$ 1.04 |  $1.24$${}\pm{}$ | $0.90$ |  $154.7$${}\pm{}$ | $17.5$ |  $0.90$${}\pm{}$ | $0.14$ |  $171$${}\pm{}$ | $19$
Ru 135 | 0.69 $-$ 1.51 |  $1.99$${}\pm{}$ | $0.44$ |  $98.5$${}\pm{}$ | $16.8$ |  $0.90$${}\pm{}$ | $0.20$ |  $109$${}\pm{}$ | $16$

Figure 7: The mass functions of four OCs. The vertical error bars are from
$\frac{1}{\sqrt{N}}$.

stars dominate. For Be 81 and Be 89, high mass stars outnumber low mass
counterparts throughout the entire cluster radius. For Ru 135’core region,
high mass stars slightly outnumber low-mass stars. For $1<r(arcmin)<4$,
intermediate mass stars seem to be dominant over low mass ones. Towards its
skirts, high mass stars slightly outnumber the low mass ones. During the
lifetime of the OCs, encounters between its member stars lead to an increased
degree of energy equipartition. Thus, the high mass stars sink towards the
cluster center and transfer their kinetic energy to the low-mass stars, thus
leading to mass segregation.

Figure 8: The normalized cumulative distributions against cluster radius
(arcmin) of four OCs in three different mass ranges ($m(m_{\odot})$). The
vertical errors are from $\sqrt{N}$.

Some factors such as age, distance and selection of bright (high mass) stars
are effective in these trends. As stated by Tarricq et al. (2021), old OCs are
mass segregated than young OCs. Old OCs with a significant number of stars
show strong mass segregation ratio. These old OCs are on the average older
than the OCs with few stars which show strongly mass segregation. Cluster
stars in the outer skirts of old OCs are on average lower than for young ones.
In the sense the proportion of stars in the outer parts of the OCs decreases
towards old ages. Besides this, bright/high mass stars are more likely to be
selected than faint stars when distinguishing cluster members. In accordance
with that, especially in the core region, there is a possibility that faint
stars will not be selected as cluster members, due to the high density of
stars. Therefore, there may be a bias in favor of bright stars. Distance is a
problem for observing faint stars, due to dense dust.

##  5 Time-Scale Parameters and Tidal Radius

The relaxation time, $t_{rlx}$ is defined as the time required for the stars
in the core/halo of an open cluster to travel from one end of these regions to
the other. Encounters between the cluster stars lead to an increased degree of
energy equipartition. $t_{rlx}$ is obtained from the relation of Spitzer &
Hart (1971)

$t_{rlx}=\frac{8.9\times
10^{5}\sqrt{N}\times{R_{h}}^{3/2}}{log(0.4N)\times\sqrt{m}}$ (1)

where $N$, $R_{h}$, $m$ denote the cluster members (Col. 6 of Table 4), the
radius containing half the cluster mass, the mean mass of the cluster members
(Col. 5 of Table 4), respectively. Here $R_{h}$ is estimated from the equation
of Larsen (2006),

$R_{h}=0.547\times R_{c}\times(\frac{R_{t}}{R_{c}})^{0.486}$ (2)

where $R_{c}$ and $R_{t}$ are the core and tidal radii, respectively. $R_{t}$
is from equation (4). The dynamical evolutionary parameters of four OCs are
estimated from

$\tau=\frac{age}{t_{rlx}}$

The ages are from Col. 7 of Table 2. The dissociation times of four OCs have
been determined from the equation of Binney & Tremaine (2008), as given in the
paper of Converse & Stahler (2011).

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

$M$ and $R_{h}$ are the total cluster mass and half mass radius of the
clusters, respectively. Here, tidal radius is the distance from the cluster
center where gravitational acceleration caused by the cluster becomes equal to
the tidal acceleration due to parent Galaxy (von Hoerner, 1957). As the
cluster goes near its perigalacticon, a star at a large distance from the
cluster center will be detached by galactic tidal forces whereas a star at a
small distance will not. In other words, the tidal radius is linked to the
stripping of stars from the cluster by the Galactic tidal field. Tidal
interactions are essential to interprete the initial structure and dynamical
evolution of the OCs (Chumak et al., 2010).

Tidal radius $R_{t}$ is estimated from the equation of Kim et al. (2000)

$R_{t}=(\frac{M}{2M_{G}})^{1/3}\times R_{G}$ (4)

Here, $M$ is the total mass of the clusters (Col.4 of Table 4). The Galactic
mass $M_{G}$ inside a galactocentric radius $R_{GC}$ (row 12 of Table 5) is
obtained from the equation of Genzel & Townes (1987),

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

The tidal radius of a $10^{4}M_{\odot}$ cluster which is located at 30 pc from
the globular cluster is $\sim$ 1 pc (Kim et al., 2000). $M_{G}$, $R_{t}$,
$R_{h}$, and time scale parameters are listed in Cols. 2-7 of Table 6.

Table 5: The weighted average radial velocities, ($V_{R}$) km s-1 (row 1), space velocity components and rotational velocity ($U$, $V$, $W$, $V_{\Phi}$) km s-1 (rows 2-5), eccentricity (ecc) and peri and apogalactic distances, birth and present day distances (Rmax, Rmin, Rm, zmax, Rbirth, RGC) (kpc) (rows 6-12). The orbital angular momentum ($J_{z}$) (kpc km s-1) (row 13). $T$ is the time of one revolution around the Galactic center as Myr (row 14). $N_{Rev}$ is the number of the revolution over the age of the cluster (last row). | Be 10 | Be 81 | Be 89 | Ru 135
---|---|---|---|---
$V_{R}$ | -36.7$\,\pm\,$1.2 | 49.1$\,\pm\,$0.9 | -44.1$\,\pm\,$4.0 | -13.7$\,\pm\,$1.4
$U$ | 37.90 | 56.59 | 29.66 | -6.62
$V$ | -11.61 | 3.74 | -48.17 | -8.98
$W$ | -6.62 | 0.23 | 0.15 | -11.76
$V_{\Phi}$ | -243.83 | -263.72 | -206.08 | -241.84
ecc | 0.07 | 0.09 | 0.16 | 0.04
Rmin | 9.45 | 6.17 | 6.05 | 7.32
Rmax | 10.96 | 7.33 | 8.39 | 7.91
Rm | 10.31 | 6.81 | 7.36 | 7.62
zmax | 0.35 | 0.13 | 0.25 | 0.30
Rbirth | 10.08 | 6.55 | 6.80 | 7.66
RGC | 9.69 | 6.18 | 8.27 | 7.54
$J_{z}$ | -2365.21 | -1628.94 | -1704.42 | -1824.18
$T$ | 260 | 159 | 220 | 194
$N_{Rev}$ | 5.2 | 10.1 | 14.6 | 5.2
Table 6: Dynamical parameters. See Sect. 5 for their meanings. Cluster | $M_{G}$ | $R_{t}$ | $R_{h}$ | $t_{rlx}$ | $t_{diss}$ | $\tau$
---|---|---|---|---|---|---
| $10^{9}\,M_{\odot}$ | pc | pc | Myr | Myr |
Be 10 |  $207.9$${}\pm{}$ | $0.9$ |  $6.28$${}\pm{}$ | $0.02$ |  $1.37$${}\pm{}$ | $0.06$ |  $9.9$${}\pm{}$ | $1.1$ |  $86.1$${}\pm{}$ | $7.5$ |  $137$${}\pm{}$ | $18$
Be 81 |  $121.8$${}\pm{}$ | $0.9$ |  $5.15$${}\pm{}$ | $0.04$ |  $0.84$${}\pm{}$ | $0.05$ |  $4.5$${}\pm{}$ | $0.6$ |  $45.3$${}\pm{}$ | $5.0$ |  $357$${}\pm{}$ | $50$
Be 89 |  $172.0$${}\pm{}$ | $0.5$ |  $6.41$${}\pm{}$ | $0.02$ |  $1.70$${}\pm{}$ | $0.05$ |  $14.8$${}\pm{}$ | $1.4$ |  $140.5$${}\pm{}$ | $10.3$ |  $217$${}\pm{}$ | $22$
Ru 135 |  $149.6$${}\pm{}$ | $0.5$ |  $5.14$${}\pm{}$ | $0.02$ |  $0.60$${}\pm{}$ | $0.02$ |  $2.8$${}\pm{}$ | $0.4$ |  $23.4$${}\pm{}$ | $2.4$ |  $363$${}\pm{}$ | $60$

##  6 Kinematics and Orbital Parameters

The Gaia radial velocities of the bright giants of Be 10 ($N=11$), Be 81
($N=32$), Be 89 ($N=24$), and Ru 135 ($N=1$) have been taken from Soubiran et
al. (2018, 2019). Their weighted averages are listed in row 1 of Table 5.
Their heliocentric velocities ($U$, $V$, $W$) in the right-hand system have
been obtained from the Gaia EDR3 radial velocities, the median proper motion
components and the distances of these bright giants by using the algorithm of
Johnson & Soderblom (1987). Here the photometric distances (Col. 6 of Table 7
in it) are adopted. These space velocities have been 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 with respect to
the local standard of rest (LSR) Schönrich and Binney (2010). Here,
$R_{\odot}=8.2\pm 0.1$ kpc (Bland-Hawthorn et al., 2016) and $V_{LSR}$ = 239
km s-1 (Brunthaler et al., 2011) are adopted. The heliocentric cartesian
distances ($x^{\prime}$, $y^{\prime}$, $z^{\prime}$) (kpc) and LRS-velocity
components ($U^{\prime}$, $V^{\prime}$, $W^{\prime}$) have been converted to
Galactic Rest of Frame (GSR) i.e., ($x$, $y$, $z$) (kpc) and ($V_{x}$,
$V_{y}$, $V_{z}$) from the equations of Kepley et al. (2007). The
galactocentric velocity component ($V_{\Phi}$) (km s-1) (or azimuthal
velocity) in a cylindrical frame is estimated via

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

Here, $V_{\Phi}<0$ means prograde. From the "MWPotential2014" code in the
galpy-code library 111http://github.com/jobovy/galpy written by Bovy (2015),
peri- and apo-galactic distances $(R_{min},~{}R_{max})$ (kpc) and the maximum
height distance (zmax) (kpc) are obtained. The orbital eccentricity (ecc) is
estimated via the relation

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

Five evolved member’s orbits have been integrated for the ages of four OCs
(Col. 7 of Table 2) within the galactic potential. The galactic potential as
sum of the galactic components is explained by Bovy (2015). Their orbital
angular momentum components $J_{z}$ (kpc km s-1) are calculated from the
equation of Kepley et al. (2007). These all parameters are given in Table 5.

Rotational velocities $V_{\Phi}=206-264$ km s-1, the eccentricities,
0.04$-$0.16, and orbital angular momentum values (Table 5) indicate that four
OCs have Galactic thin disk properties. x-y (kpc) plane is known as projected
on to the Galactic plane, whereas z-R (kpc) is the meridional plane. On x-y
(kpc) plane (top panels of Fig. 9), they follow a circular path around the
Galactic center with the eccentricities, 0.04$-$0.16. From Fig. 9 (top
panels), four OCs are orbiting near the Galactic disk, therefore they may be
affected by the tidal forces of the disk. According to their revolution
periods T (Myr) around the Galactic center (row 14 of Table 5), Be 10 and Ru
135 have made nearly five revolutions around the center of the Galaxy. For Be
81 and Be 89, their tours are nearly 10 to 15 (row 15 of Table 5).

The orbits in z-R (kpc) plane (bottom panels of Fig. 9) have boxy-like type
properties. Four OCs move in the meridional planes within the confined spaces.
They are oscillating along z-axis. Be 10’s orbit is confined in a box of $\sim
9.5<R_{gc}\leq 11.5$ kpc, therefore it is not interacting with the inner
region of the Galaxy. The confined spaces are as the following,
$\sim$$6.5<R_{gc}\leq 8.0$ kpc for Be 81, $\sim 6.0<R_{gc}\leq 8.7$ kpc for Be
89, $\sim 7.3<R_{gc}\leq 7.8$ kpc for Ru 135.

Their birth and present day positions in our Galaxy are shown with the filled
blue and red dots , respectively (Fig. 9). These positions have been
calculated by using the "MWPotential2014" in the galpy-code package. For this,
time is taken as zero (birth) and cluster age (present day). Their closest
approaches to the Sun are determined as $(d~{}(kpc),~{}t(Gyr))$ = (1.91, 1.35)
(Be 10), (1.85, 0.16) (Be 81), (1.09, 0.09) (Be 89), (0.90, 1.00) (Ru 135),
respectively.

Figure 9: Galactic orbits of four OCs. Trajectories represent the paths
traveled by the OCs through their ages. The filled blue/red dots show their
birth/present day positions. The red circle shows the Sun’s location. Figure
10: The differences between this paper and literature for $E(B-V)$,
$d~{}(pc)$, and $Age$ (Myr) in Table 2.

##  7 Discussion and Conclusion

###  7.1 Astrophysical parameters

The astrophysical values ($E(B-V)$, $d~{}(pc)$, and $Age$ (Myr)) together with
the literature comparison are listed in Table 2. The comparisons are also
displayed in Fig. 10. Reddening differences are larger, up to 0.45$-$0.47
($2MASS$/$CCD~{}UBVRI$) for Be 89 and 0.33$-$0.60 ($2MASS$) for Ru 135,
respectively. Except this, our $E(B-V)$ values are in agreement with the
literature within the discrepancies, $<$ 0.20 mag. The distance discrepancies
to the literature are 61$-$456 pc (Be 10), 352$-$953 pc (Be 81), 33$-$1075 pc
(Be 89), and 147$-$2600 pc (Ru 135), respectively. Our distances are small, as
compared to the ones of $2MASS$. There does not seem to be agreement even
among the authors for $2MASS$/$CCD~{}UBVRI$ findings. Our distances for Be 89
and Ru 135 are larger but small for Be 81, as compared to the ones of the CCD-
based $UBVRI$. For Be 10, the distance of Lata et al. (2004) is larger than
our value. The distances from the median parallaxes of four OCs (Table 1) are
somewhat larger than the Gaia EDR3-isochrone fitting distances.

Note that the Gaia DR2 photometric distances of Cantat-Gaudin et al. (2020)
are larger than our distances. The discrepancy with the literature of the
distance and age values of Be 81 is due to the fact that Sagar & Griffiths
(1998) and Donati et al. (2014) fit the isochrones at turn-off value, $V\sim
16.5$. Cantat-Gaudin et al. (2020) give a weight at turn-off point, $G\sim
15.53$ (dashed isochrone of right panel of Fig. 4), which is slightly fainter
than our $G\sim 15.17$.

From Table 2/Fig. 10(c), the ages of CCD-based UBVRI for four OCs are older
than our ages. However, for Be 89, our age is older than the one of
Subramaniam et al. (2010). The old ages of Ru 135 and Be 89 given by Akkaya
Oralhan et al. (2010) are due to usage of the metal poor isochrones while
fitting in their CMDs. Our ages of four OCs are older than the ones of
$2MASS$, except for the old age value for Ru 135 of Kharchenko et al. (2013).
From Fig. 10(c), the age differences (yellow symbols) between this paper and
the ones of Cantat-Gaudin et al. (2020) are small at a level of 148$-$452 Myr
for Be 10, Be 81 and Be 89. For Ru 135, the discrepancy is somewhat larger, up
to $\sim 1000$ Myr. The potential sources of discrepancies between our
findings and the literature are of the adopted isochrones and the reddening
values, the ground-based astrometric data, and different procedures for the
selection of member stars, respectively, as stated by Paunzen & Netopil (2006)
and Moitinho (2010).

###  7.2 Dynamical Evolution

Four OCs are in harmony with the increasing trend between $R_{core}$ and
$R_{RDP}$ (Fig. 11a) (Güneş et al., 2017; Camargo et al., 2009). On average,
larger cluster radii have larger core radii. Note that our OCs with old ages
are prone to have small core radii, as shown by Tarricq et al. (2021). Here as
an approximation, a value of $(R_{core},~{}R_{RDP})=(1.5,~{}7)$ pc of Güneş et
al. (2017) is taken into account in classifying the cluster dimensions as
small and large.

Figure 11: $R_{RDP}$ versus $R_{core}$ (panel a), $\tau$ versus $t_{rlx}$
(panel b), $t_{diss}$ versus $t_{rlx}$ (panel c), $\chi$ versus $t_{rlx}$
(panel d), $(R_{core},~{}R_{RDP})$ versus Age (Myr) (panels e–f). The relation
and its shaded area (1$\sigma$ uncertainty) in panel (a), the labels $R1$-$R4$
and filled grey points of panels (e-f) are from Güneş et al. (2017).

The dissociation times of four OCs are higher than their relaxation times
(Fig. 11c/Col. 6 of Table 6). This necessarily mean that four OCs disintegrate
very early in the initial phase of quick expansion that accompanies the death
of their most massive (bright) stars, as stated by Converse & Stahler (2011).
The relaxation times of four OCs are smaller than their ages (Col. 5 of Table
4 and Col. 7 of Table 2), from this respective, they are dynamically relaxed.
The steep overall MF slopes of four OCs (Fig. 11d) mean that their low mass
stars outnumber their massive ones (Fig. 7 and Fig. 11d). Their large $\tau$
and relatively small $t_{rlx}$ values imply an advanced mass segregation. On
account of this, their low mass stars are being lost to the field.

Following the works of Camargo et al. (2009); Güneş et al. (2017); Cakmak et
al. (2021), four OCs are plotted on $(R_{core},~{}R_{RDP})$ versus Age (Myr)
(panels (e-f) of Fig. 11). This relationship is related to survival and
dissociation rates of clusters (Camargo et al., 2009). The meanings of the
symbols are explained in caption of Fig. 11. Because Be 89 locates in $R2$
region (panels e$-$f), its core/outer parts show an expansion with time.
However, Be 10 and Be 81 indicate the relatively shrinkage core/cluster radii
($R4$). Be 81 and Ru 135 locate in solar circle, and they are survived
although they expose much to the external dynamical effects. Ru 135 (1.0 Gyr)
with the small dimensions shrunk in size and mass with time, as it lost its
star content because of the presence of massive GMCs, and tidal effects from
disk and Bulge crossings as external perturbations, taking care its direction
$(\ell=16^{\circ},~{}R_{GC}=7.5~{}kpc)$. Instead of shrinking in size and mass
with time, Ru 135 may have a primordial origin which may be related to high
molecular gas density in Galactic directions (Camargo et al., 2009; van den
Berg et al., 1991).

Figure 12: $R_{core}$ versus $\log(\tau)$ (panel a), $R_{t}$ versus $R_{RDP}$
(panel b). The grey filled dots and the dashed line in panel (a) are from
Angelo et al. (2021).

For our sample OCs, the relations, $R_{core}$ versus $\log~{}(\tau)$ and
$R_{t}$ versus $R_{RDP}$ (pc), respectively are presented in Fig. 12(a)$-$(b).
Four OCs are in concordance with the decreasing trend between $R_{core}$ and
$\log(\tau)$ (panel a). As stated by Angelo et al. (2020, 2021), this trend
points out that our OCs with old ages are dynamically evolved, and they are
losing their star contents to the field, due to internal/external dynamical
processes. Be 89’s tidal radius is less than its cluster radius (Fig.
12b/Tables 3 and 5). This means that its member stars lie within its tidal
radius, in the sense it is gravitationally bound to the cluster. For Be 10, Be
81 and Ru 135, their tidal radii are larger than their cluster radii. Thus,
the cluster members beyond their tidal radii are gravitationally unbound to
the cluster. The cluster members outside their tidal radii are more influenced
by external potential of the Galaxy.

The expected masses of four OCs based on its peri-galacticon distance/its
tidal radius are obtained from the relation (King, 1962),

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

where $R_{min}$ and $M_{G}$ is the peri-galacticon distance and the Galaxy
mass. $R_{t}$, $M_{G}$, and $R_{min}$, respectively, are taken from Tables
4$-$5\. With these values, their expected masses have been calculated as 214
$M_{\odot}$ (Be 10), 248 $M_{\odot}$ (Be 81), 716 $M_{\odot}$ (Be 89), and 181
$M_{\odot}$ (Ru 135), respectively. For instance, the expected mass 214
$M_{\odot}$ in Be 10 decreases to $\sim 110M_{\odot}$ (observed stellar total
mass), because mass-loss processes such as evaporation and the external
dynamical effects which have been acting on Be 10 for about 1.3 Gyr.

The orbits of Be 10 (Fig. 8) indicate that it passed a part of its time at
galactosentric radius, $R_{GC}=9.69$ kpc, and then it was born at $R=10.08$
kpc, which would explain its solar abundance. According to the ($[M/H]$)
versus R (kpc) of Lepine et al. (2011), Be 10 occupies a region of $R>9$ kpc
(co-rotation gap at 9 kpc). In the sense its origin may be attributed to
different galactic radius or different star formation region (Lepine et al.,
2011).

##  8 Acknowledgments

We thank the anonymous referee for the comments and suggestions. We sepically
thank to Charles Bonatto and Orhan Güneş for the valuable comments. 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 the Institute for Astronomy of the University of Vienna. This
publication also makes use of SIMBAD database-VizieR
(http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=II/246.).

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