tokens: Plummer profile reads while the normalized projected number of stars for the same model is where φ( X ) is the angle corresponding to the effective circular section where we analyse the data (see Fig 1 ), and whose analytical expression for small cone apertures (i.e. R max ≪ 1 radian) is where R max is the maximum radius of the original cone search (usually two degrees) and d is the distance between the fitted centre and the centre from the original cone search (), set as the source centre on SIMBAD 2 by the automatic Gaia advanced query. The projected radius R is defined, in spherical trigonometry, as where () is the centre of the galactic object. The likelihood function is therefore written as Figure 1. Geometry of the centre estimation: The original cone search, centred in () is shown as the grey circle of radius R max , while the new fitted centre is (). The angles φ 1 and φ 2 represent the effective circular sections (of respective radii R 1 and R 2 ) where we consider circular symmetry. The distance between the two centres is labeled as d , and we consider this value to approach zero for the cone searches around the centres from the SIMBAD data base. We minimize with the differential_evolution routine from the scipy.optimize method and find the centres that best fit the data, later displaying them in Table C1 . For simplicity, since we expect the quantity d to approach zero, given the reliable previous measurements of the centres in SIMBAD, one can assume the case where , and thus derive Notably, for more general cases, the BALRoGO routine allows to use the more general representation of φ( X ), and thus to account for a more complicated expression for , presented in Appendix A . 2.2 Surface density The next step of our algorithm is to fit the surface density of the galactic object plus interlopers once again, using a maximum likelihood estimation (MLE), but this time with fixed centres from the previous step. It may seem as an unnecessary step given the previous one, but it actually allows the fitting routine to better explore the other parameters ranges like the scale radius and the interloper fraction, and thus derive more robust results. In this step, we also reduced the maximum projected radius whenever there was a parasite galactic object on the field or when the MW contamination was too intense, disturbing the fits. In this step, we not only tested Plummer profiles, but also Sérsic models (Sérsic 1963 , 1968 ) that can be useful for core-collapsed clusters such as NGC 6397, and the Kazantzidis model that is motivated from dynamical simulations of repeated tidal encounters (Kazantzidis et al. 2004 ). Equations ( 2 ), ( 6 ), and ( 7 ) remain valid for these two other models, while the normalized surface density and projected number of stars become for Sérsic, where is the lower incomplete gamma function and X = R / R e , R e being the effective projected radius. And for Kazantzidis, where K 0 and K 1 are modified Bessel functions of
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