Detecting Globular Cluster Tidal Extensions with Bayesian Inference: I. Analysis of $\omega$ Centauri with Gaia EDR3

The peripheral regions of globular clusters (GCs) are extremely challenging to study due to their low surface brightness nature and the dominance of Milky Way contaminant populations along their sightlines. We have developed a probabilistic approach to this problem through utilising a mixture model in spatial and proper motion space which separately models the cluster, extra-tidal and contaminant stellar populations. We demonstrate the efficacy of our method through application to Gaia EDR3 photometry and astrometry in the direction of NGC 5139 ($\omega$ Cen), a highly challenging target on account of its Galactic latitude ($b\approx 15^{\circ}$) and low proper motion contrast with the surrounding field. We recover the spectacular tidal extensions, spanning the $10^{\circ}$ on the sky explored here, seen in earlier work and quantify the star count profile and ellipticity of the system out to a cluster-centric radius of $4^{\circ}$. We show that both RR Lyrae and blue horizontal branch stars consistent with belonging to $\omega$ Cen are found in the tidal tails, and calculate that these extensions contain at least $\approx 0.1$ per cent of the total stellar mass in the system. Our high probability members provide prime targets for future spectroscopic studies of $\omega$ Cen out to unprecedented radii.