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