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Register a new userRegular and chaotic orbits in axisymmetric stellar systems
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The gravitational potentials of realistic galaxy models are in general non-integrable, in the sense that they admit orbits that do not have three independent isolating integrals of motion and are therefore chaotic. However, if chaotic orbits are a small minority in a stellar system, it is expected that they have negligible impact on the main dynamical properties of the system. In this paper we address the question of quantifying the importance of chaotic orbits in a stellar system, focusing, for simplicity, on axisymmetric systems. Chaotic orbits have been found in essentially all (non-Stackel) axisymmetric gravitational potentials in which they have been looked for. Based on the analysis of the surfaces of section, we add new examples to those in the literature, finding chaotic orbits, as well as resonantly trapped orbits among regular orbits, in Miyamoto-Nagai, flattened logarithmic and shifted Plummer axisymmetric potentials. We define the fractional contributions in mass of chaotic ($xi_{rm c}$) and resonantly trapped ($xi_{rm t}$) orbits to a stellar system of given distribution function, which are very useful quantities, for instance in the study of the dispersal of stellar streams of galaxy satellites. As a case study, we measure $xi_{rm c}$ and $xi_{rm t}$ in two axisymmetric stellar systems obtained by populating flattened logarithmic potentials with the Evans ergodic distribution function, finding $xi_{rm c}sim 10^{-4}-10^{-3}$ and $xi_{rm t}sim 10^{-2}-10^{-1}$.
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Modelling the chaotic states in terms of the Gaussian Orthogonal Ensemble of random matrices (GOE), we investigate the interaction of the GOE with regular bound states. The eigenvalues of the latter may or may not be embedded in the GOE spectrum. We derive a generalized form of the Pastur equation for the average Greens function. We use that equation to study the average and the variance of the shift of the regular states, their spreading width, and the deformation of the GOE spectrum non-perturbatively. We compare our results with various perturbative approaches.
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We present the theoretical framework to efficiently solve the Jeans equations for multi-component axisymmetric stellar systems, focusing on the scaling of all quantities entering them. The models may include an arbitrary number of stellar distributions, a dark matter halo, and a central supermassive black hole; each stellar distribution is implicitly described by a two- or three-integral distribution function, and the stellar components can have different structural (density profile, flattening, mass, scale-length), dynamical (rotation, velocity dispersion anisotropy), and population (age, metallicity, initial mass function, mass-to-light ratio) properties. In order to determine the ordered rotational velocity and the azimuthal velocity dispersion fields of each component, we introduce a decomposition that can be used when the commonly adopted Satoh decomposition cannot be applied. The scheme developed is particularly suitable for a numerical implementation; we describe its realisation within our code JASMINE2, optimised to maximally exploit the scalings allowed by the Poisson and the Jeans equations, also in the post-processing procedures. As applications, we illustrate the building of three multi-component galaxy models with two distinct stellar populations, a central black hole, and a dark matter halo; we also study the solution of the Jeans equations for an exponential thick disc, and for its multi-component representation as the superposition of three Miyamoto-Nagai discs. A useful general formula for the numerical evaluation of the gravitational potential of factorised thick discs is finally given.
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