We present a comprehensive study of the young open cluster King 13 using photometric and astrometric data from Gaia DR3 and 2MASS. Our analysis refines the cluster’s fundamental parameters, including its structure, kinematics, and evolutionary status. To assess membership, we employed the pyUPMASK Python package with the HDBSCAN algorithm. The primary emphasis of this study is our new approach to assign a membership probability at each radius, rather than applying a single value to the entire cluster. These probabilities are calculated based on the number of stars deduced from the King model. This revealed a dense core with an elongated halo aligned with the cluster’s tangent velocity. Cluster orbital analysis suggests the cluster moves in the Galactic plane toward the Galactic center, with its tidal tail aligned with orbital motion-likely due to Galactic tidal effects. We identified 1571 \(\pm\) 41 member stars with a total mass of 2658.4 \(\pm\) 61.5 M \(\odot\) . The mass function (MF) for the cluster has been constructed using a step function with two power lows, \(\alpha _1\) and \(\alpha _2\) , rather than the single power low suggested by Salpeter. In this cluster, the \(\alpha _1\) and \(\alpha _2\) are found to be -3.7 \(\pm\) 0.4 and 2.3 \(\pm\) 0.15 , respectively. . The cluster’s physical parameters were derived using PARSEC stellar isochrones, estimating an age of 310 \(\pm\) 28 Myr and a relaxation time of 134 \(\pm\) 13 Myr, indicating dynamical stability. The proper motions ( \(\mu {\alpha } \cos \delta\) , \(\mu _{\delta }\) ) and parallax ( \(\varpi\) ) were measured as -2.64 \(\pm\) 0.36 mas yr \(^{-1}\) , -0.89 \(\pm\) 0.25 mas yr \(^{-1}\) , and 0.245 \(\pm\) 0.05 mas, respectively. The corresponding distance of the cluster, derived from the parallax, is 4082 \(\pm\) 231 pc. The derived distance modulus is 13.11 \(\pm\) 1.03 mag ( 4187 \(\pm\) 262 pc), with color excess values of 1.17 \(\pm\) 0.07 mag (Gaia) and 0.44 \(\pm\) 0.03 mag (2MASS), further validating our results. Additionally, 46 member stars with radial velocity data allowed us to compute the cluster’s orbit using the galpy package. Our findings highlight the presence of a tidal tail directed toward the center of the Galaxy and underscore the role of Galactic tidal forces in shaping King 13’s morphology, reinforcing its importance in the evolution of open clusters.
