V. L. Polyachenko

Notes on the stability threshold for radially anisotropic polytropes

We discuss some contradictions found in the literature concerning the problem of stability of collisionless spherical stellar systems, which are the simplest anisotropic generalization of the well-known polytrope models. Their distribution function F(E, L) is a product of powerlaw functions of the energy E and angular momentum L, i.e. F ∝ L −s (− E) q . On the one hand, calculation of the growth rates in the framework of linear stability theory and N-body simulations shows that these systems become stable when the parameter s characterizing the velocity anisotropy of the stellar distribution is lower than some finite threshold value, s < scrit. On the other hand, Palmer & Papaloizou showed that the instability remains up to the isotropic limit s = 0. Using our method of determining the eigenmodes for stellar systems, we show that the growth rates in weakly radially anisotropic systems are indeed positive, but decrease exponentially as the parameter s approaches zero, i.e. γ ∝ exp(−s∗/s). In fact, for systems with a finite lifetime this means stability.

Stability of self-gravitating astrophysical disks

Criteria which guarantee the stability of self-gravitating gaseous and stellar disks toward any localized small perturbations are obtained. These criteria are formulated as inequalities of the form Q>Qc (separately for gas and stars). The latter should be satisfied by the “stability parameter” Q, which is equal, by definition, to unity on the stability boundary of radial perturbations. The critical value of the stability parameter Qc is appreciably greater than (although of the order of) unity, attesting to the great instability of nonaxially symmetric perturbations. It is shown that the stability criterion derived for gaseous disks is valid for disks rotating within a spheroidal component (as in spiral galaxies) or in the field of a central mass (planetary rings and accretion disks). Stellar disks are stabilized with significantly greater difficulty. This is attributable mainly to the anisotropy of the velocity distribution inherent to them, which is favorable for instability.

Equilibrium models of radially anisotropic spherical stellar systems with softened central potentials

We study a new class of equilibrium two-parametric distribution functions of spherical stellar systems with a radially anisotropic velocity distribution of stars. The models are less singular counterparts of the so-called generalized polytropes, widely used in the past in works on equilibrium and the stability of gravitating systems. The offered models, unlike the generalized polytropes, have finite density and potential in the centre. The absence of the singularity is necessary for proper consideration of the instability of the radial orbit, which is the most important instability in spherical stellar systems. We provide a comparison of the main observed parameters (i.e. potential, density, anisotropy) predicted by the present models and by other popular equilibrium models.

On the Instability of Weakly Radially Anisotropic Star Clusters

We discuss contradictions existing in the literature in the problem on the stability of collisionless spherical stellar systems, which are the simplest anisotropic generalization of the well-known polytropic models. On the one hand, calculations of the growth rates within the framework of a linear stability theory and N-body simulations suggest that these systems should become stable when the parameter s characterizing the degree of anisotropy of the stellar velocity distribution becomes lower than some critical value scrit > 0. On the other hand, according to Palmer and Papaloizou, the growth rate should be nonzero up to the isotropic limit s = 0. Using our method of determining the eigenmodes of stellar systems, we show that even though the mode growth rates in weakly radially anisotropic systems of this type are nonzero, they are exponentially small, i.e., decrease as γ ∝ exp(−a/s) when s → 0. For slightly radially anisotropic systems with a finite lifetime, this actually implies stability.

Slow modes in stellar systems with nearly harmonic potentials: I. Spoke approximation, radial orbit instability

Using a consistent perturbation theory for collisionless disk-like and spherical star clusters, we construct a theory of slow modes for systems having an extended central region with a nearly harmonic potential due to the presence of a fairly homogeneous (on the scales of the stellar system) heavy, dynamically passive halo. In such systems, the stellar orbits are slowly precessing, centrally symmetric ellipses (2: 1 orbits). Depending on the density distribution in the system and the degree of halo inhomogeneity, the orbit precession can be both prograde and retrograde, in contrast to systems with 1: 1 elliptical orbits where the precession is unequivocally retrograde. In the first paper, we show that in the case where at least some of the orbits have a prograde precession and the stellar distribution function is a decreasing function of angular momentum, an instability that turns into the well-known radial orbit instability in the limit of low angular momenta can develop in the system. We also explore the question of whether the so-called spoke approximation, a simplified version of the slow mode approximation, is applicable for investigating the instability of stellar systems with highly elongated orbits. Highly elongated orbits in clusters with nonsingular gravitational potentials are known to be also slowly precessing 2: 1 ellipses. This explains the attempts to use the spoke approximation in finding the spectrum of slow modes with frequencies of the order of the orbit precession rate. We show that, in contrast to the previously accepted view, the dependence of the precession rate on angular momentum can differ significantly from a linear one even in a narrow range of variation of the distribution function in angular momentum. Nevertheless, using a proper precession curve in the spoke approximation allows us to partially “rehabilitate” the spoke approach, i.e., to correctly determine the instability growth rate, at least in the principal (O(αT−1/2) order of the perturbation theory in dimensionless small parameter αT, which characterizes the width of the distribution function in angular momentum near radial orbits.

Two scenarios of the radial orbit instability in spherically symmetric collisionless stellar systems

The stability of a two-parameter family of radially anisotropic models with a nonsingular central density distribution is considered. Instability takes place at a sufficiently strong radial anisotropy (the so-called radial orbit instability, ROI). We show that the character of instability depends not only on the anisotropy but also on the energy distribution of stars. If this distribution is such that the highly eccentric orbits responsible for the instability are “trapped” in the radial direction near the center, then the instability develops with a characteristic growth time that exceeds considerably the Jeans and dynamical times of the trapped particles. In this case, the instability takes place only for even spherical harmonics and is aperiodic. If, however, almost all of the elongated orbits reach the outer radius of the sphere, then both even and odd harmonics turn out to be unstable. The unstable modes corresponding to odd harmonics are oscillatory in nature with characteristic frequencies of the order of the dynamical ones. The unstable perturbations corresponding to even harmonics contain only one aperiodic mode and several oscillatory modes, with the aperiodic mode being always the most unstable one. Two main interpretations of the ROI available in the literature have been analyzed: the “classical” Jeans instability related to an insufficient stellar velocity dispersion in the transversal direction and the “orbital” approach relying heavily on the analogous Lynden-Bell bar formation mechanism in disk galaxies. The assumptions that the perturbations are slow (compared to the orbital frequency of stars) and that the shape of the perturbed potential is symmetric are inherent integral conditions for the applicability of the latter. Our solutions show that the orbital approach cannot be considered as a universal one.

Radially anisotropic models of collisionless spherical stellar systems without central singularity

A new class of the simplest equilibrium two-parameter distribution functions for spherical stellar systems with a radially anisotropic stellar velocity distribution is investigated. The models under consideration are a less singular counterpart of the so-called generalized polytropes, which in the past were among the most popular models in works on the equilibrium and stability of gravitating systems. In contrast to the well-known generalized polytropes, the proposed models have finite density and potential at the center. The absence of a singularity is necessary for a proper consideration of the radial orbit instability, which is the most important instability of spherical stellar systems. The main observed parameters of the proposed models (potential, density, anisotropy) are compared with those in well-known equilibrium models.

Slow modes in stellar systems with nearly harmonic potentials: II. Gravitational loss-cone instability

Using a consistent perturbation theory for collisionless disk-like and spherical star clusters, we construct a theory of slow modes for systems having an extended central region with a nearly harmonic potential due to the presence of a fairly homogeneous (on the scales of the stellar system) heavy, dynamically passive halo. In such systems, the stellar orbits are slowly precessing, centrally symmetric ellipses (2: 1 orbits). We consider star clusters with monoenergetic distribution functions that monotonically increase with angular momentum in the entire range of angular momenta (from purely radial orbits to circular ones) or have a growing region only at low angular momenta. In these cases, there are orbits with a retrograde precession, i.e., in a direction opposite to the orbital rotation of the star. The presence of a gravitational loss-cone instability, which is also observed in systems of 1: 1 orbits in near-Keplerian potentials, is associated with such orbits. In contrast to 1: 1 systems, the loss-cone instability takes place even for distribution functions monotonically increasing with angular momentum, including those for systems with circular orbits. The regions of phase space with retrograde orbits do not disappear when the distribution function is smeared in energy. We investigate the influence of a weak inhomogeneity of a heavy halo with a density that decreases with distance from the center.

Orbital approach to studying the slow dynamics of stellar systems

We develop new approaches to the numerical simulations of slowly evolving stellar systems with characteristic times of the order of the precession period for a typical orbit. This period is assumed to be long compared to the characteristic oscillation periods of individual stars in their orbits. For such processes, the standard numerical simulations using various N-body methods become inadequate, since the bulk of the computational time is spent on the repeated calculations of almost invariable orbits. We suggest a new N-orbit approach (called so by analogy and by contrast with N-body methods) that takes into account the specifics of the problems under consideration, in which whole orbits take the place of individual stars in N-body methods. Accordingly, the stellar system is represented by a set of N orbits the changes in the spatial orientation and shape of which lead to a slow evolution of the system. We derive the equations governing the nonlinear dynamics of orbits separately for 2D (disk) and 3D systems. These equations have the form of Hamiltonian equations for canonically conjugate pairs of variables. In the 2D case, one pair of such equations will suffice: for the angular momentum L and for the angle of direction to the apocenter Ψ. In the 3D case, there are two such pairs. The first pair of equations is for the modulus of the angular momentum L and the angle of direction to the apocenter in the orbital plane Ψ, while the second pair is for Lz (the component of the angular momentum vector L along the z axis) and the orientation angle of the line of nodes W. Together with the energy E, which is an adiabatic invariant, these two (or four) parameters completely define the orbit (in the 2D and 3D cases, respectively). The evolution of the system is traced by solving these equations within the framework of the suggested N-orbit approach. We have in mind two versions of this approach. In the first version, a separate orbit corresponds to each star along which the mass of this star is “smeared.” In this version, the number of orbits Norb is equal to the total number of stars N in the system under consideration. This version is a complete analogue of the N-body approach, except that the motion of each star is averaged over the orbit and we consider not the behavior of the star but the behavior of its orbit. In the second version, all stars from one small cell in the phase space of orbit parameters correspond to the orbit. In fact, this version of the N-orbit approach represents the method of solving the collisionless Boltzmann kinetic equation for the distribution function of orbit parameters. The number of orbits Norb in this approach is equal to the chosen number of cells. There exist two types of objects to the description of which N-orbit methods can be applied. First, these include the central regions of galaxies containing no large point masses. The stars in these regions move in symmetric (about the center) elliptical orbits that slowly precess due to the small deviation of the self-consistent potential from an exactly quadratic form (when all orbits are closed, so that the precession velocity is exactly equal to zero). Second, these include the star clusters around massive black holes at the centers of these clusters. The orbit of a star revolving around a central mass is a closed Keplerian ellipse and, consequently, has no precession. Slow precession appears when the relatively weak (compared to the attraction of the massive black hole) self-consistent gravitational field produced by cluster stars is taken into account. In this paper, to be specific, we will mainly deal precisely with the latter, near-Keplerian systems.

Gravitational Loss-Cone Instability

We study physical corollaries of the existing analogy between the simplest plasma traps (mirror traps) and star clusters surrounding massive black holes or dense galactic nuclei. There is a loss cone in the system through which plasma particles with low velocities transverse to the trap axis or, similarly, stars with low angular momenta (destroyed or absorbed by the central body) escape. The consequences of the “beam-like” deformation of the plasma distribution function in a trap are well known: a peculiar loss-cone instability producing a plasma flow into the loss cone develops as a result. We show that a similar gravitational loss-cone instability can also arise under certain conditions in the galactic case of interest to us. This instability is related to the slow precessional motions of highly elongated (nearly radial) stellar orbits and the main condition for its growth is that the precession of such orbits be retrograde (in the direction opposite to the orbital rotation of stars). Only under this condition do oscillations that can become unstable in the presence of a loss cone arise instead of the radial orbit instability (a variety of the Jeans instability in systems with highly elongated orbits) that takes place in the case of prograde precession. The instability produces a stream of stars onto the galactic center, i.e., serves as a mechanism of fueling the nuclear activity of galaxies. For a mathematical analysis, we have obtained relatively simple characteristic equations that describe small perturbations in a sphere of radially highly elongated stellar orbits. These characteristic equations are derived through a number of successive simplifications from a general linearized system of equations, including the collisionless Boltzmann kinetic equation and the Poisson equation (in action-angle variables). The central point of our analysis of the characteristic equations is preliminary detection of neutral modes (or proof of their absence in the case of stability).