Direct numerical integrations of the Fokker-Planck equation in energy-angular momentum space are carried out for stars orbiting a supermassive black hole (SBH) at the center of a galaxy. The algorithm, which was described in detail in an earlier pape
r, includes diffusion coefficients that describe the effects of both random (classical) and correlated (resonant) encounters. Steady-state solutions are similar to the Bahcall-Wolf solution but are modified at small radii due to the higher rate of diffusion in angular momentum, which results in a low-density core. The core radius is a few percent of the influence radius of the SBH. The corresponding phase-space density f(E,L) drops nearly to zero at low energies, implying almost no stars on tightly-bound orbits about the SBH. Steady-state rates of stellar disruption are presented, and a simple analytic expression is found that reproduces the numerical feeding rates with good accuracy. The distribution of periapsides of disrupted stars is also computed. Time-dependent solutions are also computed, starting from initial conditions similar to those produced by a binary SBH. In these models, feeding rates evolve on two timescales: rapid evolution during which the region evacuated by the massive binary is refilled by angular-momentum diffusion; and slower evolution as diffusion in energy causes the density profile at large radii to attain the Bahcall-Wolf form.
An algorithm is described for evolving the phase-space density of stars or compact objects around a massive black hole at the center of a galaxy. The technique is based on numerical integration of the Fokker-Planck equation in energy-angular momentum
space, f(E,L,t), and includes, for the first time, diffusion coefficients that describe the effects of both random and correlated encounters (resonant relaxation), as well as energy loss due to emission of gravitational waves. Destruction or loss of stars into the black hole are treated by means of a detailed boundary-layer analysis. Performance of the algorithm is illustrated by calculating two-dimensional, time-dependent and steady-state distribution functions and their corresponding loss rates.
Supermassive black holes can capture or disrupt stars that come sufficiently close. This article reviews the dynamical processes by which stars or stellar remnants are placed onto loss-cone orbits and the implications for feeding rates. The capture r
ate is well defined for spherical galaxies with nuclear relaxation times that are shorter than the galaxys age. However, even the dense nucleus of the Milky Way may be less than one relaxation time old, and this is certainly the case for more massive galaxies; the capture rate in such galaxies is an initial-value problem with poorly-known initial conditions and the rate can be much higher, or much lower, than the rate in a collisionally relaxed nucleus. In nonspherical (axisymmetric, triaxial) galaxies, torquing of orbits by the mean field can dominate perturbations due to random encounters, leading to much higher capture rates than in the spherical geometry, particularly in (massive) galaxies with long central relaxation times. Relativistic precession plays a crucial role in mediating the capture of compact remnants from regions very near to the black hole, by destroying the orbital correlations that would otherwise dominate the torques. The complex dynamics of relativistic loss cones are not yet well enough understood for accurate estimates of compact-object (EMRI) capture rates to be made.
The spin angular momentum S of a supermassive black hole (SBH) precesses due to torques from orbiting stars, and the stellar orbits precess due to dragging of inertial frames by the spinning hole. We solve the coupled post-Newtonian equations describ
ing the joint evolution of S and the stellar angular momenta Lj, j = 1...N in spherical, rotating nuclear star clusters. In the absence of gravitational interactions between the stars, two evolutionary modes are found: (1) nearly uniform precession of S about the total angular momentum vector of the system; (2) damped precession, leading, in less than one precessional period, to alignment of S with the angular momentum of the rotating cluster. Beyond a certain distance from the SBH, the time scale for angular momentum changes due to gravitational encounters between the stars is shorter than spin-orbit precession times. We present a model, based on the Ornstein-Uhlenbeck equation, for the stochastic evolution of star clusters due to gravitational encounters and use it to evaluate the evolution of S in nuclei where changes in the Lj are due to frame dragging close to the SBH and to encounters farther out. Long-term evolution in this case is well described as uniform precession of the SBH about the clusters rotational axis, with an increasingly important stochastic contribution when SBH masses are small. Spin precessional periods are predicted to be strongly dependent on nuclear properties, but typical values are 10-100 Myr for low-mass SBHs in dense nuclei, 100 Myr - 10 Gyr for intermediate mass SBHs, and > 10 Gyr for the most massive SBHs. We compare the evolution of SBH spins in stellar nuclei to the case of torquing by an inclined, gaseous accretion disk.
Chandrasekhars most important contribution to stellar dynamics was the concept of dynamical friction. I briefly review that work, then discuss some implications of Chandrasekhars theory of gravitational encounters for motion in galactic nuclei.
In spherical galaxies, binary supermassive black holes (SMBHs) have difficulty reaching sub-parsec separations due to depletion of stars on orbits that intersect the massive binary - the final-parsec problem. Galaxies that form via major mergers are
substantially nonspherical, and it has been argued that the centrophilic orbits in triaxial galaxies might provide stars to the massive binary at a high enough rate to avoid stalling. Here we test that idea by carrying out fully self-consistent merger simulations of galaxies containing central SMBHs. We find hardening rates of the massive binaries that are indeed much higher than in spherical models, and essentially independent of the number of particles used in the simulations. Binary eccentricities remain high throughout the simulations. Our results constitute a fully stellar-dynamical solution to the final-parsec problem and imply a potentially high rate of events for low-frequency gravitational wave detectors like LISA.
Inspiral of compact stellar remnants into massive black holes (MBHs) is accompanied by the emission of gravitational waves at frequencies that are potentially detectable by space-based interferometers. Event rates computed from statistical (Fokker-Pl
anck, Monte-Carlo) approaches span a wide range due to uncertaintities about the rate coefficients. Here we present results from direct integration of the post-Newtonian N-body equations of motion descrbing dense clusters of compact stars around Schwarzschild MBHs. These simulations embody an essentially exact (at the post-Newtonian level) treatment of the interplay between stellar dynamical relaxation, relativistic precession, and gravitational-wave energy loss. The rate of capture of stars by the MBH is found to be greatly reduced by relativistic precession, which limits the ability of torques from the stellar potential to change orbital angular momenta. Penetration of this Schwarzschild barrier does occasionally occur, resulting in capture of stars onto orbits that gradually inspiral due to gravitational wave emission; we discuss two mechanisms for barrier penetration and find evidence for both in the simulations. We derive an approximate formula for the capture rate, which predicts that captures would be strongly disfavored from orbits with semi-major axes below a certain value; this prediction, as well as the predicted rate, are verified in the N-body integrations. We discuss the implications of our results for the detection of extreme-mass-ratio inspirals from galactic nuclei with a range of physical properties.
A supermassive black hole ejected from the center of a galaxy by gravitational wave recoil carries a retinue of bound stars - a hypercompact stellar system (HCSS). The numbers and properties of HCSSs contain information about the merger histories of
galaxies, the late evolution of binary black holes, and the distribution of gravitational-wave kicks. We relate the structural properties of HCSSs to the properties of their host galaxies, in two regimes: collisional, i.e. short nuclear relaxation times; and collisionless, i.e. long nuclear relaxtion times. HCSSs are expected to be similar in size and luminosity to globular clustersbut in extreme cases their stellar mass can approach that of UCDs. However they differ from all other classes of compact stellar system in having very high internal velocities. We show that the kick velocity is encoded in the velocity dispersion of the bound stars. Given a large enough sample of HCSSs, the distribution of gravitational-wave kicks can therefore be empirically determined. We combine a hierarchical merger algorithm with stellar population models to compute the rate of production of HCSSs over time and the probability of observing HCSSs in the local universe as a function of their apparent magnitude, color, size and velocity dispersion, under two assumptions about the star formation history prior to the kick. We predict that roughly 100 should be detectable within 2 Mpc of the center of the Virgo cluster and that many of these should be bright enough that their high internal velocity dispersions could be measured with reasonable exposure times.
Two-body relaxation times of nuclear star clusters are short enough that gravitational encounters should substantially affect their structure in 10 Gyr or less. In nuclear star clusters without massive black holes, dynamical evolution is a competitio
n between core collapse, which causes densities to increase, and heat input from the surrounding galaxy, which causes densities to decrease. The maximum extent of a nucleus that can resist expansion is derived numerically for a wide range of initial conditions; observed nuclei are shown to be compact enough to resist expansion, although there may have been an earlier generation of low-density nuclei that were dissolved. An evolutionary model for NGC 205 is presented which suggests that the nucleus of this galaxy has already undergone core collapse. Adding a massive black hole to a nucleus inhibits core collapse, and nuclear star clusters with black holes always expand, due primarily to heat input from the galaxy and secondarily to heating from stellar disruptions. The expansion rate is smaller for larger black holes due to the smaller temperature difference between galaxy and nucleus when the black hole is large. The rate of stellar tidal disruptions and its variation with time are computed for a variety of initial models. The disruption rate generally decreases with time due to the evolving nuclear density, particularly in the faintest galaxies, assuming that scaling relations derived for luminous galaxies can be extended to low luminosities.
We consider the consequences of gravitational wave recoil for unified models of active galactic nuclei (AGNs). Spatial oscillations of supermassive black holes (SMBHs) around the cores of galaxies following gravitational wave (GW) recoil imply that t
he SMBHs spend a significant fraction of time off-nucleus, at scales beyond that of the molecular obscuring torus. Assuming reasonable distributions of recoil velocities, we compute the off-core timescale of (intrinsically type-2) quasars. We find that roughly one-half of major mergers result in a SMBH being displaced beyond the torus for a time of 30 Myr or more, comparable to quasar activity timescales. Since major mergers are most strongly affected by GW recoil, our results imply a deficiency of type 2 quasars in comparison to Seyfert 2 galaxies. Other consequences of the recoil oscillations for the observable properties of AGNs are also discussed.
