Secular Evolution of Self-Gravitating Systems Over Cosmic Age by Jean-Baptiste Fouvry

Author:Jean-Baptiste Fouvry
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(4.50)

where, similarly to Eq. (4.34), stands for the ensemble average over the 32 different realisations with the same number of particles. The radii considered here are restricted to the range , where the active surface density of the disc is only weakly affected by the inner and outer tapers. Finally, to obtain the second equality in Eq. (4.50), we replaced the active surface density, , by a discrete sum over all the particles of the system. Here, the sum on n is restricted only to particles whose radius lies between and , and we noted their azimuthal phase as . The function aims at quantifying the strength of non-axisymmetric features within the disc and should therefore be seen as a way to estimate how much the disc has evolved. During the initial Balescu–Lenard collisional evolution of the system, one expects low values of , as such an evolution is an orbit-averaged evolution, i.e. we assumed that , so that the mean system’s DF should not depend on the angles . During this first slow collisional phase, still remains non-zero, because of both unavoidable Poisson sampling shot noise and the fact that the disc sustains transient spiral waves driving its secular evolution. On the long-term, this collisional evolution leads to a destabilisation of the system. The dynamical drivers of the system’s evolution are no more discrete distant resonant collisional encounters, but exponentially growing collisionless dynamical instabilities. Because of the appearance of strong non-axisymmetric features, in this collisionless regime, one expects much larger values of . Figure 4.13 illustrates this transition between the two regimes of diffusion, thanks to the behaviour of the function .3

This phase transition can also easily be seen by directly looking at the disc’s active surface density during these two regimes. This is illustrated in Fig. 4.14, where one notices that during the late time collisionless evolution, the disc becomes strongly non-axisymmetric.

Fig. 4.13Illustration of the behaviour of the function , as introduced in Eq. (4.50), as one varies the number of particles. The prefactor was added to mask Poisson shot noise allowing for the initial values of to be independent of N. This illustrates the out-of-equilibrium transition between the initial Balescu–Lenard collisional evolution, for which low values of are expected, and the collisionless Vlasov evolution, for which the system loses its mean axisymmetry and larger values of are reached. As expected, the larger the number of particles, the later the transition

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