A comprehensive analysis of NGC 2158 in Gaia era: photometric
  parameters, apex and orbit
  We present an investigation of NGC 2158 using Gaia-DR2 data. We identified
800 most likely cluster members with membership probability higher than 90%.
The mean proper motions of this object are determined as (mu_{x}=-0.203 \pm
0.003, mu_{y}=-1.99 \pm 0.004) mas yr^{-1}. The limiting radius, log(age), and
distance of the cluster are obtained as 23.5 arcmin, 9.38 \pm 0.04 Gyr and 4.69
\pm 0.22 kpc, respectively. The overall mass function slope (0.93 \pm 0.14) is
flatter than the Salpeter value (1.35) within the mass range 1.17-1.44 M_\odot.
This cluster also shows the mass-segregation effect and our study demonstrates
that NGC 2158 is a dynamically relaxed open cluster. Using the AD-diagram, the
apex coordinates of the cluster are obtained in different ways and examined
using (\mu_{U},\mu_{T}) diagram. The best value of apex coordinates is
determined as A=87.24^\circ \pm 1.60^\circ, D=-36.61^\circ \pm 5.30 ^\circ. We
also determined the orbit of the cluster and found that NGC 2158 moves almost
in the Solar antapex direction. The resulting spatial velocity of NGC 2158 is
51 km s^{-1}. A significant oscillations along the Z-coordinate up to 529~pc is
detected. Various scenarios regarding the origin of this cluster are also
discussed.
