Energy relaxation in galaxies induced by an external environment and/or incoherent internal pulsations

This paper explores the phenomenon of energy relaxation for stars in a galaxy embedded in a high density environment that is subjected continually to perturbations reflecting the presence of other nearby galaxies and/or incoherent internal pulsations. The analysis is similar to earlier analyses of energy relaxation induced by binary encounters between nearby stars and between stars and giant molecular clouds in that the perturbations are idealised as a sum of near-random events which can be modeled as diffusion and dynamical friction. However, the analysis differs in one important respect: because the time scale associated with these perturbations need not be short compared with the characteristic dynamical time t_D for stars in the original galaxy, the diffusion process cannot be modeled as resulting from a sequence of instantaneous kicks, i.e., white noise. Instead, the diffusion is modeled as resulting from random kicks of finite duration, i.e., coloured noise characterised by a nonzero autocorrelation time t_c. A detailed analysis of coloured noise generated by sampling an Ornstein-Uhlenbeck process leads to a simpling scaling in terms of t_c and an effective diffusion constant D. Interpreting D and t_c following early work by Chandrasekhar (1941) (the `nearest neighbour approximation') implies that, for realistic choices of parameter values, energy relaxation associated with an external environment and/or internal pulsations could be important on times short compared with the age of the Universe.