This research uses the third edition of the Gaia Data Release (DR3) to re-investigate the open star cluster NGC 2509. We employed the pyUPMASK Python package and HDBSCAN algorithms to identify the cluster member stars. The current analysis introduces a new method that connects the membership probability of stars in the cluster (using the pyUPMASK tool) with the number of stars predicted by the King model at different distances from the center of the cluster. This approach divides the cluster’s area into concentric rings, or shells, and calculates the membership probability for each shell based on its specific star count, rather than using one average probability for the entire cluster. We calculated all astrophysical parameters of NGC 2509-including center, cluster radius, radial density distribution, color-magnitude diagram, distance, age, and reddening-using the photometric and astrometric data of Gaia DR3. The cluster’s relaxation time, total mass, luminosity, and mass functions are computed. The components of the proper motions ( \(\mu _{\alpha }\text {cos}\delta\) , \(\mu _{\delta }\) ), and the parallax ( \(\varpi\) ) are found to be \(-2.710 \pm 0.142\) , \(0.802 \pm 0.146\) mas/yr and \(0.368 \pm 0.043\) mas, respectively. According to the King model and pyUPMASK membership, we obtained \(497 \pm 39\) stars with a total mass of \(491.2\pm 59.5\) \(M_{\odot }\) . Using the PARSEC stellar isochrones fit, the mean cluster age and its relaxation time are \(1.72\pm 12.3\) Gyr and \(93.5 \pm 24.5\) Myr, respectively. The cluster distance modulus and reddening are estimated to be \(11.68\pm 0.12\) , and \(0.13 \pm 0.04\) mag, resulting in a distance of \(2168.5\pm 261.5\) pc. The mass function MF for this cluster has been constructed using a piecewise powerlaw with two power laws, \(\alpha _1\) and \(\alpha _2\) , rather than the single power law as suggested by Salpeter (1955). The \(\alpha _1\) and \(\alpha _2\) are found to be \(-2.74 \pm 0.14\) and \(2.29 \pm 0.13\) , respectively. Moreover, the \(\alpha _2\) is closest to the Salpeter value. Also, we identified 20 member stars as red clump that have G magnitudes between 12.6 and 13.1 mag and slightly higher temperatures than typical giants. We found that 11 members are flagged as variable stars in Gaia DR3 archive. In addition, there are 88 stars with a radial velocity of around 57.6 ± 7.8 km. Then, we have used the galpy Python package to calculate the cluster’s kinematics and orbital parameters.
