Exploring the Mass-Metallicity Relation in Open Clusters inside the Milky Way S M Rafee Adnan1,2, Ahmad Al- Imtiaz1, Sumit Roy Pronoy1,3, Anock Somaddar1, Syed Badiuzzaman Faruque1 1 Shahjalal University of Science and Technology, Bangladesh, 2 University of Louisville, Kentucky, 3 Daffodil International University, Bangladesh Background & Objective Open clusters (OCs) provide a controlled setting to investigate the interplay be- tween mass and metallicity on a more localized level. However, the specific anal- ysis of this relation within individual open clusters has been sparsely investigated, despite its significance. In this study, we aim to fill this research gap by conducting a comprehensive analysis of 75-star clusters, with the objective of examining the trends and probing the underlying mechanisms governing their mass-metallicity relationship. Data Collection The majority of the OCs reside within ∼1700 pcs thus belonging to the Milky Way galaxy. Hence, most of the selected cluster members will be lying in the main sequence of the HR diagram. APSIS utilizes the extensive data collected by the Gaia spacecraft, including mean RVS spectra and mean BP/RP, positional measurements, parallaxes, and photometric data for up to 470 million sources. Seabroke et al. (2022) offer a comprehensive and in-depth explanation of the RVS data and its processing methodology. APSIS derives a range of astrophysical parameters for stars, including stellar spectroscopic and evolutionary parameters, such as mass (M) (140 million), metallicity [M/H] (6 million using RVS, 470 million using BP/RP). We mainly used the following data for our analysis. parallax, pmra, pmdec We used the parallax and proper motions for determining the members of OCs using HDBScan. mass_flame We used the star’s mass, obtained from FLAME. This value was derived by comparing the teff_gspphot and lum_flame measurements with the BASTI stellar evolution models for solar metallicity, as described in the study by Hidalgo et al. (2018). mh_gspphot We also employed the decimal logarithm of the iron-to-hydrogen number abundance ratio, compared to the corresponding ratio of solar abundances determined by GSP-Phot Aeneas. This determination was based on analyzing BP/RP spectra, apparent G magnitude, and parallax, assuming the source to be a single star. Methodology: Member Selection HDBSCAN is a clustering method that considers all values and constructs a hier- archical tree, integrating multiple results. HDBSCAN was run on each dataset that came back from the ADQL Gaia DR3 query. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 GBP GRP 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 G (mag) All sources NGC_2682 Figure 1. HR diagram of NGC 2682 130 131 132 133 134 135 (deg) 9 10 11 12 13 14 (deg) All sources NGC_2682 Figure 2. Spatial distribution of NGC 2682 In cases where HDBSCAN failed to identify a cluster, we employed an iterative approach by modifying the parameters and reattempting the clustering process. Methodology: Curve fitting After determining the members of an Open Cluster (OC), we plotted metallicity vs. mass and fitted a model for each of the OCs. A non-linear curve fitting was performed using MCMC with the emcee library. A model representing the relationship between the mass and metallicity variables was defined as: y = x + d |x + d| (x + d)b + c (1) Here, x and y represent for mass and metallicity, respectively. And b, c, d are constants. Multiple walkers were initialized with random parameter values and moved through the parameter space, guided by the likelihood function. Param- eter estimates and uncertainties were obtained from the converged walker po- sitions. The equation we used above is based on the findings of Leethochawalit et al. 2018. They found that the mass metallicity relation of galaxies follows a power-law relationship. Here is the following equation: [Fe/H] = [Fe/H]0 log[1 + M∗/M0]γ (2) where [Fe/H] is the metallicity, M∗is the mass of the star, [Fe/H]0 is a normal- ization con