Stellar dynamics in razor-thin discs with massive nuclear black holes

The bifurcations of orbit-averaged dynamics are studied in a class of razor-thin discs with central black holes. The model used here consists of a perturbed harmonic oscillator Hamiltonian augmented with a GM r potential. Through a sequence of conformal and canonical transformations, we reduce the phase-space flows of the system to a set of non-linear differential equations on a sphere. Based on the critical points of the averaged system, we classify orbit families and reveal the existence of six types of periodic motions: circular , long - and short-axis elliptical , long - and short-axis radial and inclined radial orbits. Long-axis elliptical orbits and their surrounding tubes have significant features: whilst they keep stars away from the centre, they elongate in the same direction as the density profile. These properties are helpful in the construction of self-consistent equilibria.     

Three-integral models for axisymmetric galactic discs

We present new equilibrium component distribution functions that depend on three analytic integrals in a Stäckel potential, and that can be used to model stellar discs of galaxies. These components are generalizations of two-integral ones and can thus provide thin discs in the two-integral approximation. Their most important properties are the partly analytical expression for their moments, the disc-like features of their configuration space densities (exponential decline in the galactic plane and finite extent in the vertical direction) and the anisotropy of their velocity dispersions. We further show that a linear combination of such components can fit a van der Kruit disc.     

Rates of tidal disruption of stars by massive central black holes

There is strong evidence for some kind of massive dark object in the centres of many galaxy bulges. The detection of flares from tidally disrupted stars could confirm that these objects are black holes (BHs). Here we present calculations of the stellar disruption rates in detailed dynamical models of real galaxies, taking into account the refilling of the loss cone of stars on disruptable orbits by two-body relaxation and tidal forces in non-spherical galaxies. The highest disruption rates (one star per 10⁴ yr) occur in faint ( L ≲10¹⁰ L_⊙) galaxies, which have steep central density cusps. More luminous galaxies are less dense and have much longer relaxation times and more massive BHs. Dwarf stars in such galaxies are swallowed whole by the BH and hence do not emit flares; giant stars could produce flares as often as every 10⁵ yr, although the rate depends sensitively on the shape of the stellar distribution function. We discuss the possibility of detecting disruption flares in current supernova searches. The total mass of stars consumed over the lifetime of the galaxy is of the order of 10⁶ M_⊙, independent of galaxy luminosity; thus, disrupted stars may contribute significantly to the present BH mass in galaxies fainter than ∼10⁹ L_⊙.     

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.     

Unstable modes of Keplerian discs

We derive the softened analogue of the Laplace–Lagrange secular theory of planetary motion, and use it to show that a small fraction of counter-rotating stars is all it takes for a hot Keplerian disc to grow unstable lopsided modes.     

Analytical galactic models with mild stellar cusps

In the past two decades, it has been established by high-resolution observations of early-type galaxies that their nuclear surface brightness and corresponding stellar mass densities are characterized by cusps. In this paper, we present a new spherical analytical model family describing mild cuspy centres. We study isotropic and anisotropic models of Osipkov–Merritt type. It is shown that the associated distribution functions and intrinsic velocity dispersions can be represented analytically in a unified way in terms of hypergeometric series, allowing thus a straightforward comparison of these important global quantities for galaxies having underlying mass densities which may differ significantly in their degree of central cuspiness or radial falloff.     

Gauss's method for secular dynamics, softened

We show that the algorithm proposed by Gauss to compute the secular evolution of gravitationally interacting Keplerian rings extends naturally to softened gravitational interactions. The resulting tool is ideal for the study of the secular dynamical evolution of nearly Keplerian systems such as stellar clusters surrounding black holes in galactic nuclei, cometary clouds or planetesimal discs. We illustrate its accuracy, efficiency and versatility on a variety of configurations. In particular, we examine a secularly unstable system of counterrotating discs, and follow the unfolding and saturation of the instability into a global, uniformly precessing, lopsided  ( m = 1)  mode.     

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.     

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.     

Orbits supporting bars within bars

High-resolution observations of the inner regions of barred disc galaxies have revealed many asymmetrical, small-scale central features, some of which are best described as secondary bars. Because orbital time-scales in the galaxy centre are short, secondary bars are likely to be dynamically decoupled from the main kiloparsec-scale bars. Here we show that regular orbits exist in such doubly barred potentials, and that they can support the bars in their motion. We find orbits in which particles remain on loops : closed curves which return to their original positions after two bars have come back to the same relative orientation. Stars trapped around stable loops could form the building blocks for a long-lived, doubly barred galaxy. Using the loop representation, we can find which orbits support the bars in their motion, and the constraints on the sizes and shapes of self-consistent double bars. In particular, it appears that a long-lived secondary bar may exist only when an inner Lindblad resonance is present in the primary bar, and that it would not extend beyond this resonance.     

From kinematics to dynamics in thin galactic discs

We present a method for recovering the distribution functions of edge-on thin axisymmetric discs directly from their observable kinematic properties. The most generally observable properties of such a stellar system are the line-of-sight velocity distributions of the stars at different projected radii along the galaxy. If the gravitational potential is known, then the general two-integral distribution function can be reconstructed using the shapes of the high-velocity tails of these line-of-sight distributions. If the wrong gravitational potential is adopted, then a distribution function can still be constructed using this technique, but the low-velocity parts of the observed velocity distributions will not be reproduced by the derived dynamical model. Thus, the gravitational potential is also tightly constrained by the observed kinematics.