  We performed detailed photometric and astrometric analyses of the open star
clusters Berkeley 68 and Stock 20. This was based on ground-based CCD UBV
photometric data complemented by space-based Gaia Data Release 3 photometry and
astrometry. 198 stars were identified as likely cluster members for Berkeley 68
and 51 for Stock 20. Two-color diagrams were used to derive the reddening and
photometric metallicity for each cluster. The reddening for Berkeley 68 is
$E(B-V)=0.520 \pm 0.032$ and $0.400 \pm 0.048$ mag for Stock 20. Photometric
metallicity [Fe/H] is $-0.13 \pm 0.08$ dex for Berkeley 68, and $-0.01 \pm
0.06$ dex for Stock 20. Keeping as constant reddening and metallicity, we
determined the distance moduli and ages of the clusters through fitting
isochrones to the UBV and Gaia based color-magnitude diagrams. Photometric
distances are $d=3003 \pm 165$ pc for Berkeley 69 and $2911 \pm 216$ pc for
Stock 20. The cluster ages are $2.4 \pm 0.2$ Gyr and $50 \pm 10$ Myr for
Berkeley 68 and Stock 20, respectively. Present-day mass function slopes were
found to be $\Gamma = 1.38 \pm 0.71$ and $\Gamma = 1.53 \pm 0.39$ for Berkeley
68 and Stock 20, respectively. These values are compatible with the value of
Salpeter (1955). The relaxation times were estimated as 32.55 Myr and 23.17 Myr
for Berkeley 68 and Stock 20, respectively. These times are less than the
estimated cluster ages, indicating that both clusters are dynamically relaxed.
Orbit integration was carried out only for Berkeley 68 since radial velocity
data was not available for Stock 20. Analysis indicated that Berkeley 68 was
born outside the solar circle and belongs to the thin-disc component of the
Milky Way.
