Orbital approach to studying the slow dynamics of stellar systems

Abstract

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.