Retrograde precession of stellar orbits and gravitational loss-cone instability in galactic centers

We consider disk and spherical subsystems of stars with nearly radial orbits under conditions when the well-known radial orbit instability is not possible. This requires that the precession of stellar orbits be retrograde, i.e., in the direction opposite to the orbital rotation of stars. We show that an instability that is an analogue of the loss-cone instability known in plasma physics can then develop in the presence of a “loss cone” in the angular momentum distribution of stars, which ensures a deficit or even absence of stars with low angular momenta. Examples of systems with a loss cone are the centers of galaxies or star clusters with massive black holes. The instability can produce a flux of stars onto the galactic center, i.e., it can serve as a mechanism of fueling the nuclear activity of galaxies. Mathematically, the problem is reduced to analyzing simple characteristic equations that describe small perturbations in a disk and a sphere of radially highly elongated stellar orbits. In turn, these characteristics equations are derived through a number of successive simplifications of the general linearized Vlasov equations (i.e., the system that includes the collisionless Boltzmann kinetic equation and the Poisson equation) in action—angle variables.     

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.     

Slow modes in stellar systems with nearly harmonic potentials: II. Gravitational loss-cone instability

Using a consistent perturbation theory for collisionless disk-like and spherical star clusters, we construct a theory of slow modes for systems having an extended central region with a nearly harmonic potential due to the presence of a fairly homogeneous (on the scales of the stellar system) heavy, dynamically passive halo. In such systems, the stellar orbits are slowly precessing, centrally symmetric ellipses (2: 1 orbits). We consider star clusters with monoenergetic distribution functions that monotonically increase with angular momentum in the entire range of angular momenta (from purely radial orbits to circular ones) or have a growing region only at low angular momenta. In these cases, there are orbits with a retrograde precession, i.e., in a direction opposite to the orbital rotation of the star. The presence of a gravitational loss-cone instability, which is also observed in systems of 1: 1 orbits in near-Keplerian potentials, is associated with such orbits. In contrast to 1: 1 systems, the loss-cone instability takes place even for distribution functions monotonically increasing with angular momentum, including those for systems with circular orbits. The regions of phase space with retrograde orbits do not disappear when the distribution function is smeared in energy. We investigate the influence of a weak inhomogeneity of a heavy halo with a density that decreases with distance from the center.     

Slow modes in stellar systems with nearly harmonic potentials: I. Spoke approximation,radial orbit instability

Using a consistent perturbation theory for collisionless disk-like and spherical star clusters, we construct a theory of slow modes for systems having an extended central region with a nearly harmonic potential due to the presence of a fairly homogeneous (on the scales of the stellar system) heavy, dynamically passive halo. In such systems, the stellar orbits are slowly precessing, centrally symmetric ellipses (2: 1 orbits). Depending on the density distribution in the system and the degree of halo inhomogeneity, the orbit precession can be both prograde and retrograde, in contrast to systems with 1: 1 elliptical orbits where the precession is unequivocally retrograde. In the first paper, we show that in the case where at least some of the orbits have a prograde precession and the stellar distribution function is a decreasing function of angular momentum, an instability that turns into the well-known radial orbit instability in the limit of low angular momenta can develop in the system. We also explore the question of whether the so-called spoke approximation, a simplified version of the slow mode approximation, is applicable for investigating the instability of stellar systems with highly elongated orbits. Highly elongated orbits in clusters with nonsingular gravitational potentials are known to be also slowly precessing 2: 1 ellipses. This explains the attempts to use the spoke approximation in finding the spectrum of slow modes with frequencies of the order of the orbit precession rate. We show that, in contrast to the previously accepted view, the dependence of the precession rate on angular momentum can differ significantly from a linear one even in a narrow range of variation of the distribution function in angular momentum. Nevertheless, using a proper precession curve in the spoke approximation allows us to partially “rehabilitate” the spoke approach, i.e., to correctly determine the instability growth rate, at least in the principal (O(α _T^−1/2) order of the perturbation theory in dimensionless small parameter α _T, which characterizes the width of the distribution function in angular momentum near radial orbits.     

Gravitational loss-cone instability: Monotonic and nonmonotonic angular momentum distributions of stars

Small perturbations of spherical star clusters around massive black holes are studied. The presence of a black hole gives rise to peculiar distributions that have no stars with low angular momenta (falling into the so-called “loss cone”). The stability of such a distribution has been found to depend significantly on whether it monotonically increases with angular momentum L (from the loss cone up to L = L _circ in circular orbits) or has a maximum at some intermediate L = L _*. In the case of spherical systems under consideration, the loss-cone instability is shown to be possible only for nonmonotonic distributions.     

The effectiveness of the Kozai mechanism in the Galactic Centre

I examine the effectiveness of Kozai oscillations in the centres of galaxies and in particular the Galactic Centre (GC) using standard techniques from celestial mechanics. In particular, I study the effects of a stellar bulge potential and general relativity on Kozai oscillations, which are induced by stellar discs. Löckmann et al. recently suggested that Kozai oscillations induced by the two young massive stellar discs in the GC drive the orbits of the young stars to large eccentricity  ( e ≈ 1)  . If some of these young eccentric stars are in binaries, they would be disrupted near pericentre, leaving one star in a tight orbit around the central supermassive black hole and producing the S-star population. I find that the spherical stellar bulge suppresses Kozai oscillations, when its enclosed mass inside a test body is of the order of the mass in the stellar disc(s). Since the stellar bulge in the GC is much larger than the stellar discs, Kozai oscillations due to the stellar discs are likely suppressed. Whether Kozai oscillations are induced from other non-spherical components to the potential (e.g. a flattened stellar bulge) is yet to be determined.     

Regular black hole motion and stellar orbital resonances

The motion of a black hole about the centre of gravity of its host galaxy induces a strong response from the surrounding stellar population. We treat the case of a harmonic potential analytically and show that half of the stars on circular orbits in that potential shift to an orbit of lower energy, while the other half receive a positive boost and recede to a larger radius. The black hole itself remains on an orbit of fixed amplitude and merely acts as a catalyst for the evolution of the stellar energy distribution function f ( E ). We show that this effect is operative out to a radius of approximately three to four times the hole's influence radius, R _bh. We use numerical integration to explore more fully the response of a stellar distribution to black hole motion. We consider orbits in a logarithmic potential and compare the response of stars on circular orbits, to the situation of a 'warm' and 'hot' (isotropic) stellar velocity field. While features seen in density maps are now wiped out, the kinematic signature of black hole motion still imprints the stellar line-of-sight mean velocity to a magnitude ≃13 per cent the local rms velocity dispersion σ. A study in three dimensions suggests a reduced effect for polar orbits.     

The tidal disruption rate in dense galactic cusps containing a supermassive binary black hole

We consider the problem of tidal disruption of stars in the centre of a galaxy containing a supermassive binary black hole with unequal masses. We assume that over the separation distance between the black holes, the gravitational potential is dominated by the more massive primary black hole. Also, we assume that the number density of stars is concentric with the primary black hole and has a power-law cusp. We show that the bulk of stars with a small angular-momentum component normal to the black hole binary orbit can reach a small value of total angular momentum through secular evolution in the gravitational field of the binary, and hence they can be tidally disrupted by the larger black hole. This effect is analogous to the so-called Kozai effect well known in celestial mechanics. We develop an analytical theory for the secular evolution of the stellar orbits and calculate the rate of tidal disruption. We compare our analytical theory with a simple numerical model and find very good agreement.
Our results show that for a primary black hole mass of  ∼10⁶–10⁷ M_⊙  , the black hole mass-ratio   q > 10⁻²  , cusp size ∼1 pc, the tidal disruption rate can be as large as  ∼10⁻²–1 M_⊙ yr⁻¹  . This is at least 10²–10⁴ times larger than estimated for the case of a single supermassive black hole. The duration of the phase of enhanced tidal disruption is determined by the dynamical-friction time-scale, and it is rather short: ∼10⁵ yr. The dependence of the tidal disruption rate on the mass ratio, and on the size of the cusp, is also discussed.     

Kinematics of hypervelocity stars in the triaxial halo of the Milky Way

Hypervelocity stars (HVSs) ejected by the massive black hole at the Galactic Centre have unique kinematic properties compared to other halo stars. Their trajectories will deviate from being exactly radial because of the asymmetry of the Milky Way potential produced by the flattened disc and the triaxial dark matter halo, causing a change of angular momentum that can be much larger than the initial small value at injection. We study the kinematics of HVSs and propose an estimator of dark halo triaxiality that is determined only by instantaneous position and velocity vectors of HVSs at large Galactocentric distances ( r ≳ 50 kpc). We show that, in the case of a substantially triaxial halo, the distribution of deflection angles (the angle between the stellar position and velocity vector) for HVSs on bound orbits is spread uniformly over the range 10°–180°. Future astrometric and deep wide-field surveys should measure the positions and velocities of a significant number of HVSs, and provide useful constraints on the shape of the Galactic dark matter halo.     

Dynamics for galactic archaeology

Our Galaxy is a complex machine in which several processes operate simultaneously: metal-poor gas is accreted, is chemically enriched by dying stars, and then drifts inwards, surrendering its angular momentum to stars; new stars are formed on nearly circular orbits in the equatorial plane and then diffuse through orbit space to eccentric and inclined orbits; the central stellar bar surrenders angular momentum to the surrounding disc and dark halo while acquiring angular momentum from inspiralling gas; the outer parts of the disc are constantly disturbed by satellite objects, both luminous and dark, as they sweep through pericentre. We review the conceptual tools required to bring these complex happenings into focus. Our first concern must be the construction of equilibrium models of the Galaxy, for upon these hang our hopes of determining the Galaxy’s mean gravitational field, which is required for every subsequent step. Ideally our equilibrium model should be formulated so that the secular evolution of the system can be modelled with perturbation theory. Such theory can be used to understand how stars diffuse through orbit space from either the thin gas disc in which we presume disc stars formed, or the debris of an accreted object, the presumed origin of many halo stars. Coupling this understanding to the still very uncertain predictions of the theory of stellar evolution and nucleosynthesis, we can finally extract a complete model of the chemodynamic evolution of our reasonably generic Galaxy. We discuss the relation of such a model to cosmological simulations of galaxy formation, which provide general guidance but cannot be relied on for quantitative detail.     

Gas-driven evolution of stellar orbits in barred galaxies

We carry out a detailed orbit analysis of gravitational potentials selected at different times from an evolving self-consistent model galaxy consisting of a two-component disc (stars+gas) and a live halo. The results are compared with a pure stellar model, subject to nearly identical initial conditions, which are chosen so as to make the models develop a large-scale stellar bar. The bars are also subject to hose-pipe (buckling) instability which modifies the vertical structure of the disc. The diverging morphological evolution of both models is explained in terms of gas radial inflow, the resulting change in the gravitational potential at smaller radii, and the subsequent modification of the main families of orbits, both in and out of the disc plane.   We find that dynamical instabilities become milder in the presence of the gas component, and that the stability of planar and 3D stellar orbits is strongly affected by the related changes in the potential — both are destabilized, with the gas accumulation at the centre. This is reflected in the overall lower amplitude of the bar mode and in the substantial weakening of the bar, which appears to be a gradual process. The vertical buckling of the bar is much less pronounced and the characteristic peanut shape of the galactic bulge almost disappears when there is a substantial gas inflow towards the centre. Milder instability results in a smaller bulge, the basic parameters of which are in agreement with observations. We also find that the overall evolution in the model with a gas component is accelerated because of the larger central mass concentration and the resulting decrease in the characteristic dynamical time.     

Alignment time-scale of the microquasar GRO J1655−40

The microquasar GRO J1655−40 has a black hole with spin angular momentum apparently misaligned to the orbital plane of its companion star. We analytically model the system with a steady-state disc warped by Lense–Thirring precession and find the time-scale for the alignment of the black hole with the binary orbit. We make detailed stellar evolution models so as to estimate the accretion rate and the lifetime of the system in this state. The secondary can be evolving at the end of the main sequence or across the Hertzsprung gap. The mass-transfer rate is typically 50 times higher in the latter case but we find that, in both the cases, the lifetime of the mass-transfer state is at most a few times the alignment time-scale. The fact that the black hole has not yet aligned with the orbital plane is therefore consistent with either model. We conclude that the system may or may not have been counter aligned after its supernova kick but that it is most likely to be close to alignment rather than counter alignment now.     

Influence of the r-mode instability on hypercritically accreting neutron stars

We have investigated the influence of the r-mode instability on hypercritically accreting neutron stars in close binary systems during their common envelope phases, based on the scenario proposed by Brown et al. On the one hand, neutron stars are heated by the accreted matter at the stellar surface, but on the other hand they are also cooled down by the neutrino radiation. At the same time, the accreted matter transports its angular momentum and mass to the star. We have studied the evolution of the stellar mass, temperature and rotational frequency.
The gravitational-wave-driven instability of the r-mode oscillation strongly suppresses spinning up of the star, the final rotational frequency of which is well below the mass-shedding limit, in fact typically as low as 10 per cent of that of the mass-shedding state. On a very short time-scale the rotational frequency tends to approach a certain constant value and saturates there, as long as the amount of accreted mass does not exceed a certain limit to collapse to a black hole. This implies that a similar mechanism of gravitational radiation to that in the so-called 'Wagoner star' may work in this process. The star is spun up by accretion until the angular momentum loss by gravitational radiation balances the accretion torque. The time-integrated dimensionless strain of the radiated gravitational wave may be large enough to be detectable by gravitational wave detectors such as LIGO II.     

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.     

Relativistic Precession of Elliptical Orbits of a Circumbinary Planet can be Cancelled

In publications presenting analytical results on the non-coplanar motion of a circumbinary planet it was shown that the unperturbed elliptical orbit of the planet undergoes simultaneously two kinds of the precession: the precession of the orbital plane and the precession of the orbit in its own plane. It is also well-known that there is also the relativistic precession of the planetary orbit in its own plane. In the present paper we study a combined effect of the all of the above precessions. For the general case, where the planetary orbit is not coplanar with the stars orbits, we analyzed the dependence of the critical inclination angle i_c, at which the precession of the planetary orbit in its own plane vanishes, on the angular momentum L of the planet. We showed that the larger the angular momentum, the smaller the critical inclination angle becomes. We presented the analytical result for i_c(L) and calculated the value of L, for which the critical inclination value becomes zero. For the particular case, where the planetary orbit is not coplanar with the stars orbits, we demonstrated analytically that at a certain value of the angular momentum of the planet, the elliptical orbit of the planet would become stationary: no precession. In other words, at this value of the angular momentum, the relativistic precession of the planetary orbit and its precession, caused by the fact that the planet revolves around a binary (rather than single) star, cancel each other out. This is a counterintuitive result.     

Mass and spin co-evolution during the alignment of a black hole in a warped accretion disc

In this paper, we explore the gravitomagnetic interaction of a black hole (BH) with a misaligned accretion disc to study BH spin precession and alignment jointly with BH mass M _BH and spin parameter a evolution, under the assumption that the disc is continually fed, in its outer region, by matter with angular momentum fixed on a given direction     . We develop an iterative scheme based on the adiabatic approximation to study the BH–disc co-evolution: in this approach, the accretion disc transits through a sequence of quasi-steady warped states (Bardeen–Petterson effect) and interacts with the BH until the spin   J _BH  aligns with     . For a BH aligning with a corotating disc, the fractional increase in mass is typically less than a few per cent, while the spin modulus can increase up to a few tens of per cent. The alignment time-scale     is of  ∼10⁵–10⁶ yr  for a maximally rotating BH accreting at the Eddington rate. BH–disc alignment from an initially counter-rotating disc tends to be more efficient compared to the specular corotating case due to the asymmetry seeded in the Kerr metric: counter-rotating matter carries a larger and opposite angular momentum when crossing the innermost stable orbit, so that the spin modulus decreases faster and so the relative inclination angle.     

On various approaches to investigating the instability of stellar systems with highly elongated stellar orbits

We study the various approximations used to investigate the eigenmode spectrum for systems with highly elongated stellar orbits. The approximation in which the elongated orbits are represented by thin rotating spokes, with the rotation imitating the precession of real orbits, is the simplest and most natural one. However, we show that using this pictorial approximation does not allow the picture of stability to be properly presented. We show that for stellar systems with a plane disk geometry, this approach does not allow unstable spectral modes to be obtained even in the leading order in small parameter, which characterizes the spread of nearly radial orbits in angular momentum. For spherical systems, where the situation is more favorable, the spectrum can be determined but only in the leading order in this parameter. A rigorous approach based on the solution of more complex integral equations given here should be used to properly investigate the stability of stellar systems.     

The dynamical stability of W Ursae Majoris-type systems

Theoretical study indicates that a contact binary system would merge into a rapidly rotating single star due to tidal instability when the spin angular momentum of the system is more than a third of its orbital angular momentum. Assuming that W Ursae Majoris (W UMa) contact binary systems rigorously comply with the Roche geometry and the dynamical stability limit is at a contact degree of about 70 per cent, we obtain that W UMa systems might suffer Darwin's instability when their mass ratios are in a region of about 0.076–0.078 and merge into the fast-rotating stars. This suggests that the W UMa systems with mass ratio   q ≤ 0.076  cannot be observed. Meanwhile, we find that the observed W UMa systems with a mass ratio of about 0.077, corresponding to a contact degree of about 86 per cent would suffer tidal instability and merge into the single fast-rotating stars. This suggests that the dynamical stability limit for the observed W UMa systems is higher than the theoretical value, implying that the observed systems have probably suffered the loss of angular momentum due to gravitational wave radiation (GR) or magnetic stellar wind (MSW).