l = 1: Weinberg’s weakly damped mode in an N-body model of a spherical stellar system

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 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, 2π 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.

Linear response theory and damped modes of stellar clusters

“…4 of Weinberg (1994), in the context of the King sphere (a cluster with a finite radial extension) and recovered numerically in Fig. 6 of Heggie et al (2020).…”

Noise and waves: a unified kinetic theory for stellar systems

“…They tend to have characteristic spatial scales comparable to the size of the cluster/galaxy itself. They are easily excited by external perturbations such as a massive flyby encounter (Murali 1999;Vesperini & Weinberg 2000), or they can be self-excited by the Poisson noise inherent in the system's gravitational potential (Weinberg 2001;Heggie et al 2020). When discussing relaxation it is usually assumed that 1 Collective behaviour can also lead to instabilities -see e.g.…”

“…Since the operator ˆ (and hence its inverse ˆ −1 ) is calculated purely from f , waves are a 'collisionless' phenomenon in the sense that ωg and φg(x) are independent of N , and would be the same even in a system with N → ∞. The calculation of {ωg, φg(x)} is not the subject of this paper but is a standard, if often difficult, numerical task (Weinberg 1994;Binney & Tremaine 2008;Heggie et al 2020).…”

Probabilistic distribution functions

“…In the case of an isolated, non-rotating cluster O f would then be a random variable with vanishing mean and a dispersion that would be independent of the azimuthal quantum number m. The variance of O f would be a prediction that could be tested from observations of a sufficient number of real or simulated clusters (e.g. Lau & Binney 2019;Heggie et al 2020). This programme requires a rule for assigning a priori probabilities to possible fields f (w).…”