Simulations of spheroidal systems with substructure: trees in fields

We present a hybrid technique of N -body simulation to deal with collisionless stellar systems having an inhomogeneous global structure. We combine a treecode and a self-consistent field code such that each of the codes models a different component of the system being investigated. The treecode is suited to treatment of dynamically cold or clumpy components, which may undergo significant evolution within a dynamically hot system. The hot system is appropriately evolved by the self-consistent field code. This combined code is particularly suited to a number of problems in galactic dynamics. Applications of the code to these problems are briefly discussed.     

Dynamical studies of cluster pairs in the Magellanic Clouds

We performed N -body simulations of star cluster encounters with Hernquist's TREECODE in a CRAY YMP-2E computer under different initial conditions (relative positions and velocities, cluster sizes, masses and concentration degrees). The total number of particles per simulation ranged from 1024 to 20480. These models are compared with a series of isodensity maps of cluster pairs in the Magellanic Clouds. Evidence is found that during the interactions, transient morphological effects such as an expanded halo, isophotal deformation and isophotal twisting can occur as a result of tidal effects and dynamical friction. The simulations also show that different outcomes are possible depending on the initial parameters: (i) long-standing changes of concentration degree can occur after the collision; (ii) one member can disaggregate; or (iii) the pair can coalesce into a single cluster with a distinct structure compared with the original ones. These simulations can reproduce a wide range of morphological structures in observed cluster pairs.     

Linear response of galactic haloes to adiabatic gravitational perturbations

We determine the response of a self-similar isothermal stellar system to small adiabatic gravitational perturbations. For odd spherical harmonics the response is identical to the response of the analogous isothermal fluid system. For even spherical harmonics the response can be regarded as an infinite series of wavetrains in log r , implying alternating compression and rarefaction in equal logarithmic radius intervals. Partly because of the oscillatory nature of the solutions, tidal fields from external sources are not strongly amplified by an intervening isothermal stellar system, except at radii ≲10^−3.5 times the satellite radius; at some radii the stellar system can even screen the external tidal field in a manner analogous to Debye screening. As Weinberg has pointed out, individual resonances in a stellar system can strongly amplify external tidal fields over a limited radial range, but we cannot address this possibility because we examine only adiabatic perturbations. We also discuss the application of our method to the halo response caused by the slow growth of an embedded thin disc.     

On a time-symmetric Hermite integrator for planetary N-body simulation

We describe a P(EC) ⁿ Hermite scheme for planetary N -body simulation. The fourth-order implicit Hermite scheme is a time-symmetric integrator that has no secular energy error for the integration of periodic orbits with time-symmetric time-steps. In general N -body problems, however, this advantage is of little practical significance, since it is difficult to achieve time-symmetry with individual variable time-steps. However, we can easily enjoy the benefit of the time-symmetric Hermite integrator in planetary N -body systems, where all bodies spend most of the time on nearly circular orbits. These orbits are integrated with almost constant time-steps even if we adopt the individual time-step scheme. The P(EC) ⁿ Hermite scheme and almost constant time-steps reduce the integration error greatly. For example, the energy error of the P(EC)² Hermite scheme is two orders of magnitude smaller than that of the standard PEC Hermite scheme in the case of an N  = 100,  m  = 10²⁵ g planetesimal system with the rms eccentricity 〈 e ²〉^1/2 ≲0.03.     

Orbit classification in arbitrary 2D and 3D potentials

A method of classifying generic orbits in arbitrary 2D and 3D potentials is presented. It is based on the concept of spectral dynamics introduced by Binney &38; Spergel that uses the Fourier transform of the time series of each coordinate. The method is tested using a number of potentials previously studied in the literature and is shown to distinguish correctly between regular and irregular orbits, to identify the various families of regular orbits (boxes, loops, tubes, boxlets, etc.), and to recognize the second-rank resonances that bifurcate from them. The method returns the position of the potential centre and, for 2D potentials, the orientation of the principal axes as well, should this be unknown. A further advantage of the method is that it has been encoded in a FORTRAN program that does not require user intervention, except for 'fine tuning' of search parameters that define the numerical limits of the code. The automatic character makes the program suitable for classifying large numbers of orbits.     

Frequency map analysis of the orbital structure in elliptical galaxies

We present an application of the frequency map analysis to an elliptical galaxy which is represented by a generalization of a double-power-law spherical mass model. The density distribution of this model varies as r ^−γ close to the centre and as r ⁻⁴ at large radii. We study the case with γ = 1, which is known as the 'weak-cusp' model and which represents well the density profile of the 'core' galaxies observed by the Hubble Space Telescope . The final objective of our work is to improve our understanding of the dynamics of elliptical galaxies in a similar way to Merritt &38; Fridman, finding the regions of stochasticity, looking for resonances that might play an important role in sustaining the triaxial morphology, and analysing the diffusion of orbits. To this end, we use the frequency map analysis of Laskar, which has been applied widely in the field of celestial mechanics but which is a relatively new technique in the area of galactic dynamics. Finally, we show some useful features of this method in understanding the global dynamical structure of the system.     

Stellar dynamics in a galactic centre surrounded by a massive accretion disc — I. Newtonian description

The long-term evolution of stellar orbits bound to a massive centre is studied in order to understand the cores of star clusters in central regions of galaxies. Stellar trajectories undergo tiny perturbations, the origins of which are twofold: (i) the gravitational field of a thin gaseous disc surrounding the galactic centre, and (ii) cumulative drag arising from successive interactions of the stars with the material of the disc. Both effects are closely related because they depend on the total mass of the disc, assumed to be a small fraction of the central mass. It is shown that, in contrast to previous works, most of the retrograde (with respect to the disc) orbits are captured by the central object, presumably a massive black hole. Initially prograde orbits are also affected, so that statistical properties of the central star cluster in quasi-equilibrium may differ significantly from those deduced in previous analyses.     

Orbital motion of a massive object in a dense stellar system

In this paper we show the positional oscillation of a massive object in a dense stellar system by numerical N -body simulations. We found that the central massive object, which at first is placed at rest at the centre of the surrounding spherical stellar system, promptly departs from the centre and rotates in accordance with the rotation of the stellar system, if the stellar system has an appreciable rotation. This oscillatory motion continues for a long time because of the absence of dynamical friction. Such a long-lasting oscillation may explain the asymmetric structure observed in the centres of M31 and NGC 4486B, may cause the secular flow of gaseous elements distributed in the central regions of galaxies on to the massive object, and may ignite activity in the centres of galaxies.     

Resonant tidal disruption in galactic nuclei

It has recently been shown by Rauch 38 Tremaine that the rate of angular momentum relaxation in nearly Keplerian star clusters is greatly increased by a process termed 'resonant relaxation'; it was also argued, via a series of scaling arguments, that tidal disruption of stars in galactic nuclei containing massive black holes could be noticeably enhanced by this process. We describe here the results of numerical simulations of resonant tidal disruption which quantitatively test the predictions made by Rauch 38 Tremaine. The simulation method is based on an N -body routine incorporating cloning of stars near the loss cone and a semirelativistic symplectic integration scheme. Normalized disruption rates for resonant and non-resonant nuclei are derived at orbital energies both above and below the critical energy, and the corresponding angular momentum distribution functions are found. The black hole mass above which resonant tidal disruption is quenched by relativistic precession is determined. We also briefly describe the discovery of chaos in the Wisdom–Holman symplectic integrator applied to highly eccentric orbits and propose a modified integration scheme that remains robust under these conditions. We find that resonant disruption rates exceed their non-resonant counterparts by an amount consistent with the predictions; in particular, we estimate the net tidal disruption rate for a fully resonant cluster to be about twice that of its non-resonant counterpart. No significant enhancement in rates is observed outside the critical radius. Relativistic quenching of the effect is found to occur for hole masses M  >  M _Q  = (8 ± 3) × 10⁷  M . The numerical results combined with the observed properties of galactic nuclei indicate that for most galaxies the resonant enhancement to tidal disruption rates will be very small.     

Stability of power-law discs — II. The global spiral modes

This paper reports on the in-plane normal modes in the self-consistent and the cut-out power-law discs. Although the cut-out discs are remarkably stable to bisymmetric perturbations, they are very susceptible to one-armed modes. For this harmonic, there is no inner Lindblad resonance, thus removing a powerful stabilizing influence. A physical mechanism for the generation of the one-armed instabilities is put forward. Incoming trailing waves are reflected as leading waves at the inner cut-out, thus completing the feedback for the swing-amplifier. Growing three-armed and four-armed modes occur only at very low temperatures. However, neutral m  = 3 and m  = 4 modes are possible at higher temperatures for some discs. The rotation curve index β has a marked effect on stability. For all azimuthal wavenumbers, any unstable modes persist to higher temperatures and grow more vigorously if the rotation curve is rising (β < 0) than if the rotation curve is falling (β > 0). If the central regions or outer parts of the disc are carved out more abruptly, any instabilities become more virulent. The self-consistent power-law discs possess a number of unusual stability properties. There is no natural time-scale in the self-consistent disc. If a mode is admitted at some pattern speed and growth rate, then it must be present at all pattern speeds and growth rates. Our analysis — although falling short of a complete proof — suggests that such a two-dimensional continuum of non-axisymmetric modes does not occur and that the self-consistent power-law discs admit no global non-axisymmetric normal modes whatsoever. Without reflecting boundaries or cut-outs, there is no resonant cavity and no possibility of unstable growing modes. The self-consistent power-law discs certainly admit equi-angular spirals as neutral modes, together with a one-dimensional continuum of growing axisymmetric modes.