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.     

Evolution of massive binary black holes

Since many or most galaxies have central massive black holes (BHs), mergers of galaxies can form massive binary black holes (BBHs). In this paper we study the evolution of massive BBHs in realistic galaxy models, using a generalization of techniques used to study tidal disruption rates around massive BHs. The evolution of BBHs depends on BH mass ratio and host galaxy type. BBHs with very low mass ratios (say, ≲0.001) are hardly ever formed by mergers of galaxies, because the dynamical friction time-scale is too long for the smaller BH to sink into the galactic centre within a Hubble time. BBHs with moderate mass ratios are most likely to form and survive in spherical or nearly spherical galaxies and in high-luminosity or high-dispersion galaxies; they are most likely to have merged in low-dispersion galaxies (line-of-sight velocity dispersion ≲90 km s⁻¹) or in highly flattened or triaxial galaxies.
The semimajor axes and orbital periods of surviving BBHs are generally in the range  10^-3–10 pc  and  10–10⁵ yr;  they are also larger in high-dispersion galaxies than in low-dispersion galaxies, larger in nearly spherical galaxies than in highly flattened or triaxial galaxies, and larger for BBHs with equal masses than for BBHs with unequal masses. The orbital velocities of surviving BBHs are generally in the range  10²–10⁴ km s^-1  . The methods of detecting surviving BBHs are also discussed.
If no evidence of BBHs is found in AGNs, this may be either because gas plays a major role in BBH orbital decay or because nuclear activity switches on soon after a galaxy merger, and ends before the smaller BH has had time to spiral to the centre of the galaxy.     

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.     

Integrable models of galactic discs with double nuclei

We introduce a new class of 2D mass models, whose potentials are of Stäckel form in elliptic coordinates. Our model galaxies have two separate strong cusps that form double nuclei. The potential and surface density distributions are locally axisymmetric near the nuclei and become highly non-axisymmetric outside the nucleus. The surface density diverges toward the cuspy nuclei with the law     Our model is sustained by four general types of regular orbits: butterfly , nucleophilic banana , horseshoe and aligned loop orbits. Horseshoes and nucleophilic bananas support the existence of cuspy regions. Butterflies and aligned loops control the non-axisymmetric shape of outer regions. Without any need for central black holes, our distributed mass models resemble the nuclei of M31 and NGC 4486B. It is also shown that the self-gravity of the stellar disc can prevent the double nucleus to collapse.     

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.     

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.     

Can gas dynamics in centres of galaxies reveal orbiting massive black holes?

If supermassive black holes in centres of galaxies form by merging of black hole remnants of massive Population III stars, then there should be a few black holes of mass one or two orders of magnitude smaller than that of the central ones, orbiting around the centre of a typical galaxy. These black holes constitute a weak perturbation in the gravitational potential, which can generate wave phenomena in gas within a disc close to the centre of the galaxy. Here, we show that a single orbiting black hole generates a three-arm spiral pattern in the central gaseous disc. The density excess in the spiral arms in the disc reaches values of 3–12 per cent when the orbiting black hole is about 10 times less massive than the central black hole. Therefore, the observed density pattern in gas can be used as a signature in detecting the most massive orbiting black holes.     

Chaos in Hill's generalized problem: from the solar system to black holes

We generalize the well‐known Hill's circular restricted three‐body problem by assuming that the primary generates a Schwarzschild‐type field of the form U = A/r + B/r³. The term in B influences the particle, but not the far secondary. Many concrete astronomical situations can be modelled via this problem. For the two‐body problem primary‐particle, a homoclinic orbit is proved to exist for a continuous range of parameters (the constants of energy and angular momentum, and the field parameter B > 0). Within the restricted three‐body system, we prove that, under sufficiently small perturbations from the secondary, the homoclinic orbit persists, but its stable and unstable manifolds intersect transversely. Using a result of symbolic dynamics, this means the existence of a Smale horseshoe, hence chaotic behaviour. Moreover, we find that Hill's generalized problem (in our sense) is nonintegrable. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of the dynamical structure of galaxies using optical spectra

Galaxy spectra are a rich source of kinematical information since the shapes of the absorption lines reflect the movement of stars along the line-of-sight. We present a technique with which to build directly a dynamical model for a galaxy by fitting model spectra, calculated from a dynamical model, to the observed galaxy spectra. Using synthetic spectra from a known galaxy model we demonstrate that this technique indeed recovers the essential dynamical characteristics of the galaxy model. Moreover, the method allows a statistically meaningful error analysis on the resulting dynamical quantities.     

Exact density–potential pairs from the holomorphic Coulomb field

We show how the complex-shift method developed by Appell to study the gravitational field of a point mass (and used in electrodynamics by, among others, Newman, Carter, Lynden-Bell, and Kaiser to determine some remarkable properties of the electromagnetic field of rotating charged configurations) can be extended to obtain new and explicit density–potential pairs for self-gravitating systems departing significantly from spherical symmetry. The rotational properties of two axisymmetric baroclinic gaseous configurations derived with the proposed method are illustrated.     

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.     

Exact density–potential pairs from complex-shifted axisymmetric systems

In a previous paper, the complex-shift method has been applied to self-gravitating spherical systems, producing new analytical axisymmetric density–potential pairs. We now extend the treatment to the Miyamoto–Nagai disc and the Binney logarithmic halo, and we study the resulting axisymmetric and triaxial analytical density–potential pairs; we also show how to obtain the surface density of shifted systems from the complex shift of the surface density of the parent model. In particular, the systems obtained from Miyamoto–Nagai discs can be used to describe disc galaxies with a peanut-shaped bulge or with a central triaxial bar, depending on the direction of the shift vector. By using a constructive method that can be applied to generic axisymmetric systems, we finally show that the Miyamoto–Nagai and the Satoh discs, and the Binney logarithmic halo cannot be obtained from the complex shift of any spherical parent distribution. As a by-product of this study, we also found two new generating functions in closed form for even and odd Legendre polynomials, respectively.     

Constraining black hole masses from stellar kinematics by summing over all possible distribution functions

When faced with the task of constraining a galaxy's potential given limited stellar kinematical information, what is the best way of treating the galaxy's unknown distribution function (DF)? Using the example of estimating black hole (BH) masses, I argue that the correct approach is to consider all possible DFs for each trial potential, marginalizing the DF using an infinitely divisible prior. Alternative approaches, such as the widely used maximum-penalized likelihood method, neglect the huge degeneracies inherent in the problem and simply identify a single, special DF for each trial potential.
Using simulated observations of toy galaxies with realistic amounts of noise, I find that this marginalization procedure yields significantly tighter constraints on BH masses than the conventional maximum-likelihood method, although it does pose a computational challenge which might be solved with the development of a suitable algorithm for massively parallel machines. I show that in practice the conventional maximum-likelihood method yields reliable BH masses with well-defined minima in their χ² distributions, contrary to claims made by Valluri, Merritt & Emsellem.     

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.     

Mass profiles and anisotropies of early-type galaxies

We discuss the problem of using stellar kinematics of early-type galaxies to constrain the orbital anisotropies and radial mass profiles of galaxies. We demonstrate that compressing the light distribution of a galaxy along the line of sight produces approximately the same signature in the line-of-sight velocity profiles as radial anisotropy. In particular, fitting spherically symmetric dynamical models to apparently round, isotropic face-on flattened galaxies leads to a spurious bias towards radial orbits in the models, especially if the galaxy has a weak face-on stellar disc. Such face-on stellar discs could plausibly be the cause of the radial anisotropy found in spherical models of intermediate luminosity ellipticals such as NGC 2434, 3379 and 6703.
In the light of this result, we use simple dynamical models to constrain the outer mass profiles of a sample of 18 round, early-type galaxies. The galaxies follow a Tully–Fisher relation parallel to that for spiral galaxies, but fainter by at least 0.8 mag ( I -band) for a given mass. The most luminous galaxies show clear evidence for the presence of a massive dark halo, but the case for dark haloes in fainter galaxies is more ambiguous. We discuss the observations that would be required to resolve this ambiguity.