Effect of angular momentum distribution on gravitational loss-cone instability in stellar clusters around a massive black hole

We study the small perturbations in spherical and thin disc stellar clusters surrounding a massive black hole. Because of the black hole, stars with sufficiently low angular momentum escape from the system through the loss cone. We show that the stability properties of spherical clusters crucially depend on whether the distribution of stars is monotonic or non-monotonic in angular momentum. It turns out that only non-monotonic distributions can be unstable. At the same time, instability in disc clusters is possible for both types of distribution.     

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.     

Self-gravitating warped discs around supermassive black holes

We consider warped equilibrium configurations for stellar and gaseous discs in the Keplerian force field of a supermassive black hole, assuming that the self-gravity of the disc provides the only acting torques. Modelling the disc as a collection of concentric circular rings and computing the torques in the non-linear regime, we show that stable, strongly warped precessing equilibria are possible. These solutions exist for a wide range of disc-to-black-hole mass ratios   M _d/ M _bh  , can span large warp angles of up to  ±∼120°  , have inner and outer boundaries, and extend over a radial range of a factor of typically two to four. These equilibrium configurations obey a scaling relation such that in good approximation     where     is the (retrograde) precession frequency and Ω is a characteristic orbital frequency in the disc. Stability was determined using linear perturbation theory and, in a few cases, confirmed by numerical integration of the equations of motion. Most of the precessing equilibria are found to be stable, but some are unstable. The main result of this study is that highly warped discs near black holes can persist for long times without any persistent forcing other than by their self-gravity. The possible relevance of this to galactic nuclei is briefly discussed.     

Regular and chaotic motion in softened gravitational systems

The stability of the dynamical trajectories of softened spherical gravitational systems is examined, both in the case of the full N -body problem and that of trajectories moving in the gravitational field of non-interacting background particles. In the latter case, for   N 10 000  , some trajectories, even if unstable, had exceedingly long diffusion times, which correlated with the characteristic e-folding time-scale of the instability. For trajectories of   N ≈100 000  systems this time-scale could be arbitrarily large – and thus appear to correspond to regular orbits. For centrally concentrated systems, low angular momentum trajectories were found to be systematically more unstable. This phenomenon is analogous to the well-known case of trajectories in generic centrally concentrated non-spherical smooth systems, where eccentric trajectories are found to be chaotic. The exponentiation times also correlate with the conservation of the angular momenta along the trajectories. For N up to a few hundred, the instability time-scales of N -body systems and their variation with particle number are similar to those of the most chaotic trajectories in inhomogeneous non-interacting systems. For larger N (up to a few thousand) the values of the these time-scales were found to saturate, increasing significantly more slowly with N . We attribute this to collective effects in the fully self-gravitating problem, which are apparent in the time variations of the time-dependent Liapunov exponents. The results presented here go some way towards resolving the long-standing apparent paradoxes concerning the local instability of trajectories. This now appears to be a manifestation of mechanisms driving evolution in gravitational systems and their interactions – and may thus be a useful diagnostic of such processes.     

The gravitational wave background from star–massive black hole fly-bys

Stars on eccentric orbits around a massive black hole (MBH) emit bursts of gravitational waves (GWs) at periapse. Such events may be directly resolvable in the Galactic Centre. However, if the star does not spiral in, the emitted GWs are not resolvable for extragalactic MBHs, but constitute a source of background noise. We estimate the power spectrum of this extreme mass ratio burst background (EMBB) and compare it to the anticipated instrumental noise of the Laser Interferometer Space Antenna (LISA). To this end, we model the regions close to an MBH, accounting for mass segregation, and for processes that limit the presence of stars close to the MBH, such as GW inspiral and hydrodynamical collisions between stars. We find that the EMBB is dominated by GW bursts from stellar mass black holes, and the magnitude of the noise spectrum  ( fS _GW)^1/2  is at least a factor of ∼10 smaller than the instrumental noise. As an additional result of our analysis, we show that LISA is unlikely to detect relativistic bursts in the Galactic Centre.     

Gravitational loss-cone instability in stellar systems with retrograde orbit precession

We study spherical and disc clusters in a near-Keplerian potential of galactic centres or massive black holes. In such a potential orbit precession is commonly retrograde, that is, the direction of the orbit precession is opposite to the orbital motion. It is assumed that stellar systems consist of nearly-radial orbits. We show that if there is a loss-cone at low angular momentum (e.g. due to consumption of stars by a black hole), an instability similar to loss-cone instability in plasma may occur. The gravitational loss-cone instability is expected to enhance black hole feeding rates. For spherical systems, the instability is possible for the number of spherical harmonics   l ≥ 3  . If there is some amount of counter-rotating stars in flattened systems, they generally exhibit the instability independent of azimuthal number m . The results are compared with those obtained recently by Tremaine for distribution functions monotonically increasing with angular momentum.
The analysis is based on simple characteristic equations describing small perturbations in a disc or a sphere of stellar orbits highly elongated in radius. These characteristic equations are derived from the linearized Vlasov equations (combining the collisionless Boltzmann kinetic equation and the Poisson equation), using the action-angle variables. We use two techniques for analysing the characteristic equations: the first one is based on preliminary finding of neutral modes, and the second one employs a counterpart of the plasma Penrose–Nyquist criterion for disc and spherical gravitational systems.     

Feast and Famine: regulation of black hole growth in low-redshift galaxies

We analyse the observed distribution of Eddington ratios  ( L / L _Edd)  as a function of supermassive black hole mass for a large sample of nearby galaxies drawn from the Sloan Digital Sky Survey. We demonstrate that there are two distinct regimes of black hole growth in nearby galaxies. The first is associated with galaxies with significant star formation [   M _*/star formation rate (SFR) ∼  a Hubble time] in their central kiloparsec regions, and is characterized by a broad lognormal distribution of accretion rates peaked at a few per cent of the Eddington limit. In this regime, the Eddington ratio distribution is independent of the mass of the black hole and shows little dependence on the central stellar population of the galaxy. The second regime is associated with galaxies with old central stellar populations (   M _*/SFR ≫  a Hubble time), and is characterized by a power-law distribution function of Eddington ratios. In this regime, the time-averaged mass accretion rate on to black holes is proportional to the mass of stars in the galaxy bulge, with a constant of proportionality that depends on the mean stellar age of the stars. This result is once again independent of black hole mass. We show that both the slope of the power law and the decrease in the accretion rate on to black holes in old galaxies are consistent with population synthesis model predictions of the decline in stellar mass loss rates as a function of mean stellar age. Our results lead to a very simple picture of black hole growth in the local Universe. If the supply of cold gas in a galaxy bulge is plentiful, the black hole regulates its own growth at a rate that does not further depend on the properties of the interstellar medium. Once the gas runs out, black hole growth is regulated by the rate at which evolved stars lose their mass.     

The black hole mass–stellar velocity dispersion correlation: bulges versus pseudo-bulges

We investigate the correlation between the supermassive black holes (SMBHs) mass ( M _bh) and the stellar velocity dispersion  (σ_*)  in two types of host galaxies: the early-type bulges (disc galaxies with classical bulges or elliptical galaxies) and pseudo-bulges. In the form  log ( M _bh/M_⊙) =α+β log (σ_*/200 km s⁻¹)  , the best-fitting results for the 39 early-type bulges are the slope  β= 4.06 ± 0.28  and the normalization  α= 8.28 ± 0.05  ; the best-fitting results for the nine pseudo-bulges are  β= 4.5 ± 1.3  and  α= 7.50 ± 0.18  . Both relations have intrinsic scatter in  log  M _bh  of ≲0.27 dex. The   M _bh–σ_*  relation for pseudo-bulges is different from the relation in the early-type bulges over the 3σ significance level. The contrasting relations indicate the formation and growth histories of SMBHs depend on their host type. The discrepancy between the slope of the   M _bh–σ_*  relations using different definition of velocity dispersion vanishes in our sample, a uniform slope will constrain the coevolution theories of the SMBHs and their host galaxies more effectively. We also find the slope for the 'core' elliptical galaxies at the high-mass range of the relation appears steeper  (β≃ 5–6)  , which may be the imprint of their origin of dissipationless mergers.     

Influence of a stellar cusp on the dynamics of young stellar discs and the origin of the S-stars in the Galactic Centre

Observations of the Galactic Centre show evidence of one or two disc-like structures of very young stars orbiting the central supermassive black hole within a distance of a few 0.1 pc. A number of analyses have been carried out to investigate the dynamical behaviour and consequences of these discs, including disc thickness and eccentricity growth as well as mutual interaction and warping. However, most of these studies have neglected the influence of the stellar cusp surrounding the black hole, which is believed to be one to two orders of magnitude more massive than the disc(s).
By means of N -body integrations using our bhint code, we study the impact of stellar cusps of different compositions. We find that although the presence of a cusp does have an important effect on the evolution of an otherwise isolated flat disc, its influence on the evolution of disc thickness and warping is rather mild in a two-disc configuration. However, we show that the creation of highly eccentric orbits strongly depends on the graininess of the cusp (i.e. the mean and maximum stellar masses). While Chang recently found that full cycles of Kozai resonance are prevented by the presence of an analytic cusp, we show that relaxation processes play an important role in such highly dense regions and support short-term resonances. We thus find that young disc stars on initially circular orbits can achieve high eccentricities by resonant effects also in the presence of a cusp of stellar remnants, yielding a mechanism to create S-stars and hypervelocity stars.
Furthermore, we discuss the underlying initial mass function (IMF) of the young stellar discs and find no definite evidence for a non-canonical IMF.     

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_⊙.     

On the number of young stellar discs in the Galactic Centre

Observations of the Galactic Centre show evidence of disc-like structures of very young stars orbiting the central supermassive black hole within a distance of a few 0.1 pc. While it is widely accepted that about half of the stars form a relatively flat disc rotating clockwise on the sky, there is a substantial ongoing debate on whether there is a second, counter-clockwise disc of stars.
By means of N -body simulations using our bhint code, we show that two highly inclined stellar discs with the observed properties cannot be recognized as two flat circular discs after 5 Myr of mutual interaction. Instead, our calculations predict a significant warping of the two discs, which we show to be apparent among the structures observed in the Galactic Centre. While the high eccentricities of the observed counter-clockwise orbits suggest an eccentric origin of this system, we show the eccentricity distribution in the inner part of the more massive clockwise disc to be perfectly consistent with an initially circular disc in which stellar eccentricities increase due to both non-resonant and resonant relaxation.
We conclude that the relevant question to ask is therefore not whether there are two discs of young stars, but whether there were two such discs to begin with.     

Fluctuations in finite-N equilibrium stellar systems

Gravitational amplification of Poisson noise in stellar systems is important on large scales. For example, it increases the dipole noise power by roughly a factor of 6 and the quadrupole noise by 50 per cent for a King model profile. The dipole noise is amplified by a factor of 15 for the core-free Hernquist model. The predictions are computed by summing over the wakes caused by each star in the system — the dressed-particle formalism of Rostoker & Rosenbluth — and are demonstrated by N -body simulation.   This result implies that a collisionless N -body simulation is impossible; the fluctuation noise which causes relaxation is an intrinic part of self-gravity. In other words, eliminating two-body scattering at interparticle scales does not eliminate relaxation altogether.   Applied to dark matter haloes of disc galaxies, particle numbers of at least 10⁶ will be necessary to suppress this noise at a level that does not dominate or significantly affect the disc response. Conversely, haloes are most likely far from phase-mixed equilibrium and the resulting noise spectrum may seed or excite observed structure such as warps, spiral arms and bars. For example, discreteness noise in the halo, similar to that caused by a population of 10⁶-M⊙ black holes, can produce observable warping and possibly excite or seed other disc structure.     

Galactic satellite systems: radial distribution and environment dependence of galaxy morphology

We have studied the radial distribution of the early (E/S0) and late (S/Irr) types of satellites around bright host galaxies. We made a volume-limited sample of 4986 satellites brighter than   M _r=−18.0  associated with 2254 hosts brighter than   M _r=−19.0  from the Sloan Digital Sky Survey Data Release 5 sample. The morphology of satellites is determined by an automated morphology classifier, but the host galaxies are visually classified. We found segregation of satellite morphology as a function of the projected distance from the host galaxy. The amplitude and shape of the early-type satellite fraction profile are found to depend on the host luminosity. This is the morphology–radius/density relation at the galactic scale. There is a strong tendency for morphology conformity between the host galaxy and its satellites. The early-type fraction of satellites hosted by early-type galaxies is systematically larger than that of late-type hosts, and is a strong function of the distance from the host galaxies. Fainter satellites are more vulnerable to the morphology transformation effects of hosts. Dependence of satellite morphology on the large-scale background density was detected. The fraction of early-type satellites increases in high-density regions for both early- and late-type hosts. It is argued that the conformity in morphology of galactic satellite system is mainly originated by the hydrodynamical and radiative effects of hosts on satellites.     

Regular motions in double bars – I. Double-frequency orbits and loops

Bars in galaxies are mainly supported by particles trapped around stable periodic orbits. These orbits represent oscillatory motion with only one frequency, which is the bar driving frequency, and miss free oscillations. We show that a similar situation takes place in double bars: particles get trapped around parent orbits, which in this case represent oscillatory motion with two frequencies of driving by the two bars, and which also lack free oscillations. Thus the parent orbits, which constitute the backbone of an oscillating potential of two independently rotating bars, are the double-frequency orbits. These orbits do not close in any reference frame, but they map on to closed curves called loops. Trajectories trapped around the parent double-frequency orbit map on to a set of points confined within a ring surrounding the loop.