ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: energy. Figure 7. Drift along time of the CoM of the bound core with respect to the CoM of the whole system, for one of the simulations of set 1A. The Lagrangian radii are evaluated in percentage of the total mass of the cluster and so all the bodies, including possibly escaping IMBHs and growing (in mass) objects, are taken into account. Of course, the escaping IMBHs lead to a natural increase of the high-percentage Lagrangian radii so that in Fig. 5 we display only up to the 50 percent Lagrangian radius ( R 1/2 ). Note that growing massive objects, although they remain inside the bound core of the system and close to its CoM, do not a priori coincide with this CoM (see Fig. 8 that shows, also, how the growing BH movement is since the beginning well within the half-mass radius) and could, consequently, induce sharp variations in the latter evolution of the low-percentage Lagrangian radii. Figure 8. Black line gives the distance ( r smo ) of the growing supermassive object to the CoM of the bound ‘core’ of the system, for one arbitrarily chosen simulation of set 1A, taken as example. For comparison, we also give in purple the half-mass radius of the system. The curves are plotted starting from the time when the SMBH ‘seed’ is already grown by 42 IMBH mergers. The late time evolution of the average half-mass radius R 1/2 (50 percent Lagrangian radius), which is a good definition of the system radial scale, is well fitted by a linear relation: For set 1 and t > 15 yr, the values of the parameters are leading to a root mean square error of the fit equal to 0.8 mpc for R 1/2 . For set 2 and t > 500 yr, the values are giving a root mean square error equal to 5 mpc. The average half-mass radius at the final simulation time for set 1 is equal to 24 mpc (37 mpc for set 2), that is 50 times (8 times for set 2) the initial half-mass radius, R 1/2 (0) = 2 −1/3 R 0 . Even with our original safety margin of one order of magnitude for the initial cluster radial size, at the end of the simulation the core of the cluster extends much farther out than the area allowed for our purpose of mimicking the presence of an SMBH by a dense cluster of IMBHs. This is true for every simulation of set 1 and not only on average. This is even more the case for set 2, where no safety margin was taken. This result is not surprising: We actually expected that the extreme conditions required (a stable system of IMBHs of total mass and maximal size ≤5 mpc) were very unlikely to be reached. As we see in the next section, in both set 1 and set 2 the IMBH cluster undergoes various merger episodes. This has relevant consequences, whose main result is the formation of a very massive BH as coming out from the dominant object growing up after successive merger events. Hence, we conclude that, as expected on basic theoretical understanding, the answer to the first of the issues we raised in introduction is negative: A cluster of IMBHs dense enough to mimic the dynamical role of an