$l=1$: Weinberg's weakly damped mode in an $N$-body model of a spherical stellar system (2002.00783v2)

Abstract: Spherical stellar systems such as King models, in which the distribution function is a decreasing function of energy and depends on no other invariant, are stable in the sense of collisionless dynamics. But Weinberg showed, by a clever application of the matrix method of linear stability, that they may be nearly unstable, in the sense of possessing {\sl weakly} damped modes of oscillation. He also demonstrated the presence of such a mode in an $N$-body model by endowing it with initial conditions generated from his perturbative solution. In the present paper we provide evidence for the presence of this same mode in $N$-body simulations of the King $W_0 = 5$ model, in which the initial conditions are generated by the usual Monte Carlo sampling of the King distribution function. It is shown that the oscillation of the density centre correlates with variations in the structure of the system out to a radius of about 1 virial radius, but anticorrelates with variations beyond that radius. Though the oscillations appear to be continually reexcited (presumably by the motions of the particles) we show by calculation of power spectra that Weinberg's estimate of the period (strictly, $2\pi$ divided by the real part of the eigenfrequency) lies within the range where the power is largest. In addition, however, the power spectrum displays another very prominent feature at shorter periods, around 5 crossing times.