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tokens: parameter c = r c / r t , given by the ratio between the core ( r c ) and tidal ( r t ) radius. On the other hand, the tidal radius depends on the cluster orbit through the relation where Ω is the GC angular velocity, and d 2 Φ/d r 2 the second order derivative of the background gravitational field. 1 In the simplest approximation that the BHB separation is much smaller than its distance from the GC and that it dominates the dynamics, the equation reduces to being r p the GC periastron. Although equation ( 2 ) was originally derived for point-like mass moving on circular orbits, it approximate reasonably well the tidal radius of an extended object moving on an eccentric orbit during the passage at periastron (King 1962 ; Arca-Sedda et al. 2016 ). Once the GC orbit is assigned, the tidal radius can be calculated and combined with the value of W 0 chosen in order to get r c . Another degree of freedom is represented by the GC pericentre ( r p ) and eccentricity ( e ), as well as the orbit inclination with respect to the BHB orbital plane, i , and the mutual orientation of the two orbital ascending nodes ω. Note that the definition of the inclination angle i is as the one between the BHB and the GC orbital angular momentum vectors, where i = 0 corresponds to prograde and i = π to retrograde configurations. Finally, the galaxy background can significantly alter the first part of the GC orbital evolution. According to the widely used Dehnen ( 1993 ) models, the galactic density profile can be described as where M g , r g represent the galaxy mass and length scale, respectively, whereas γ represents the inner slope of the density profile. Therefore, a lower limit to the total number of parameters needed to characterize such a system is given by the combination of the BHB parameters (4), the GC structural and orbital properties (6) and the galaxy parameters (3), namely: BHB parameters: semi-major axis a BHB , eccentricity e BHB , total mass M BHB and mass ratio q BHB ; GC structural parameters: Mass M , potential well W 0 , concentration c ; GC orbital parameters: semi-major axis a , eccentricity e and pericentre r p ; galaxy parameters: galaxy mass M g , typical radius r g and density profile slope γ. Assuming only 3 values for each parameter would imply then a total number of ∼1.6 × 10 6 different models. In order to keep the number of models reasonable in terms of computational request and human times, we limited the number of degrees of freedom in our problem as follows: we assumed an equal mass BHB and kept fixed its apocentre; we fixed the GC W 0 , M and r p ; we assumed the GC and BHB orbits in a coplanar, counter-rotating configuration; we fixed the galaxy background. This choices reduced our problem to only three parameters: the BHB mass M BHB and its eccentricity, e BHB , and the GC eccentricity e . We assumed two different values for M BHB = 10 7 − 10 8 M ⊙ , three values for e BHB = 0, 0.5, 0.9 and two for e = 0.5, 0.9, thus