Exploring membership and variability in NGC 7419: An open cluster rich in super giants and Be type stars

NGC 7419 is a young open cluster notable for hosting five Red Supergiants and a much higher abundance of Classical Be stars (CBe) than typical open clusters. We perform a membership analysis using GAIA DR3 data and machine learning techniques like Gaussian Mixture Models (GMM) and Random Forest (RF) and determine the cluster's mean distance to be 3.6 $\pm$ 0.7 kpc. We identify 499 GAIA-based members with a mass above $\sim$ 1.5 M$\odot$, and estimate the cluster's age to be $21.1 ^{+1.6}_{-0.6}$ Myr. Using our revised $H\alpha$ excess-based analysis, we find 42 CBe stars containing many known CBe stars, bringing the total number of CBe stars in NGC 7419 to 49 and the fraction of CBe to (B+CBe) members to 12.2 %. We investigate the variability of the candidate members from ZTF and NEOWISE data using standard deviation, median absolute deviation, and Stetson Index (J), and their periodicity using the Generalized Lomb Scargle Periodogram variability. We find that 66 % of CBe stars are variable: 23 % show periodic signals from pulsation or rotation, 41 \% exhibit variability from disk dynamics or binarity, and 14 % have long-term variations due to disk dissipation/formation. We also find that all pulsating CBe stars are early-type, while 50 \% of stars with long-term variations are early-type, and the other 50 % are mid-type. Our results agree with previous findings in the literature and confirm that CBe stars display variability through multiple mechanisms across different timescales.