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.     

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.     

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: 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.     

振动Kuzmin盘中恒星的运动性质

本文研究振动盘中恒星的运动性质．所采用的势模型为它由一种具简单径向振动模态的Kuzmin盘和一种对数晕共同产生．得到的主要结论是：（1）恒星存在稳定且有序的近圆轨道；（2）盘振动对角动量较小的恒星及远离近圆轨道的恒星影响较大；（3）盘中大部分恒星的运动是有序的；（4）远离近圆轨道的恒星一般作混沌运动，并且最终可能逃逸，但在一个Hubble时间内实际逃逸的恒星比例较小；（5）盘振动可能是振动Kurmin盘中某些星团形成并长期维持的机制之一，盘振动幅度越大，盘中星团数目可能越多；在同一个星系盘中，角动量越大的星团数目可能越少．     

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 Morphological Evolution of Galaxy Satellites

We study the evolution of galaxy satellites with high resolutionN-body simulations. Satellites are modelled as replicas of typical low and high surface brightness galaxies (LSBs and HSBs). Encounters on high eccentricity orbits (as typical in hierarchical models of galaxy formation) strip LSBs of most of their stars and tend to decrease their surface brightness. In contrast, bar instability in HSBs leads to substantial loss of angular momentum of the stellar component and to an increase in central surface brightness. In both cases the remnant resembles a spheroidal galaxy with an exponential surface brightness profile. A simple modelling of colour evolution and interaction-driven star formation gives M/L ratios for the remnants that are roughly consistent with observations. These results suggest an evolutionary scenario for the dwarf galaxies in our Local Group, faint dSphs being the descendants of LSBs and brighter dSphs/dEs being the final state of HSB satellites. This revised version was published online in July 2006 with corrections to the Cover Date.     

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_⊙.     

自行与银河系研究的前沿

简述了精确测定相对自行的方法,特别介绍了在用2～3个历元的底片和CCD观测结果推导恒星自行的过程中如何消除光学视场畸变、星等差和色差的具体办法;并介绍了用星系把相对自行推算绝对自行的方法。还介绍了用自行资料研究银河系结构和演化的一些前沿课题,其中包括星团研究、与银河系兼并的矮星系的发现、暗物质的检测、外星行星的探测和银河系中心黑洞的质量估算等。最后评价了自行在研究银河系中的重要性,论述了我国研制4m光学/近红外望远镜的重要意义。     

Expansion of the stellar subsystems of disk galaxies

Based on archival Hubble Space Telescope images, we have performed stellar photometry for eight edge-on spiral and irregular galaxies. We have identified stars with ages of 20, 50, 80, 160, and 500 Myr in the derived Hertzsprung-Russell diagrams and constructed their number density distributions perpendicularly to the plane of the galactic disk. We have determined the sizes of the stellar subsystems and constructed the size-age diagrams for the stars constituting these subsystems. The stellar subsystems have been found to expand in all of the investigated galaxies within the range of ages studied (from 20 to 500 Myr). The expansion velocity of the subsystems decreases as one recedes from the galactic plane. The subsystems with ages of 1.5 and 6 Gyr also exhibit an increase in their sizes with age. The sizes of these subsystems approach those of the thick disk consisting of red giants. Our results confirm the model of thick-disk formation in irregular and low-mass spiral galaxies through thin-disk expansion.     

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.     

The kinematic properties of stars in an oscillating Kuzmin disk

We study the kinematic properties of stars under the combined potential of a Kuzmin disk with a simple radial oscillation and a logarithmic halo. The results are: 1) There exist stable, ordered and near-circular orbits. 2) The effect of the oscillating disk is greater on orbits with smaller angular momenta and on that departly greatly from the near-circular orbits. 3) Most of the motion in the disk is ordered motion. 4) Orbits that depart greatly from the near-circular orbits generally have chaotic motion and may eventually escape. But the actual fraction escaped in one Hubble time is small. 5) Disk oscillation may be one of the mechanisms for the formation and long-term maintenance of some star clusters; the larger the amplitude, the greater may be the number of clusters; for a given disk galaxy, there may be more clusters with small than with large angular momenta.     

Noncircular outer disks in unbarred S0 galaxies: NGC 502 and NGC 5485

Highly noncircular outer stellar disks have been detected in two SA0 (unbarred) galaxies by comparing the spectroscopic data on the rotation of stars and the photometric data on the shape and orientation of isophotes. In NGC 502, the oval distortion of the disk is manifested in the shape of the inner and outer elliptical rings occupying wide radial zones between the bulge and the disk and at the outer disk edge; such a structure can be a consequence of the so-called “dry minor merger,” multiple cannibalization of gas-free satellites. In NGC 5485, the stellar kinematics is absolutely unrelated to the orientation of isophotes in the disk region, and for this galaxy the conclusion about its global triaxial structure is unavoidable.     

On the Rectilinear Non-Collision Motion

We systematically investigate the rectilinear non-collision motion, i.e., the rectilinear one with C>0, where C is the angular momentum integral. This kind of motion appears in stellar dynamics when considering encounters of stars. For a short enough segment of a star's path, it is a very good approximation to the real motion of the star. Moreover, we derive also an analogue to Kepler's equation for this motion, and, considering the barycentric orbits of the stars, we find their minimal mutual distance.     

The structure of motion in a 4-component galaxy mass model

We use a composite galaxy model consisting of a disk-halo, bulge, nucleus and dark-halo components in order to investigate the motion of stars in ther-z plane. It is observed that high angular momentum stars move in regular orbits. The majority of orbits are box orbits. There are also banana-like orbits. For a given value of energy, only a fraction of the low angular momentum stars — those going near the nucleus — show chaotic motion while the rest move in regular orbits. Again one observes the above two kinds of orbits. In addition to the above one can also see orbits with the characteristics of the 2/3 and 3/4 resonance. It is also shown that, in the absence of the bulge component, the area of chaotic motion in the surface of section increases, significantly. This suggests that a larger number of low angular momentum stars are in chaotic orbits in galaxies with massive nuclei and no bulge components.     

A unified theory of the formation of galactic bars

We give arguments for a basically unified formation mechanism of slow (Lynden-Bell) and fast (common) galactic bars. This mechanism is based on an instability that is akin to the well-known instability of radial orbits and is produced by the mutual attraction and alignment of precessing stellar orbits (so far, only the formation of slow bars has been explained in this way). We present a general theory of the low-frequency modes in a disk that consists of orbits precessing at different angular velocities. The problem of determining these modes is reduced to integral equations of moderately complex structure. The characteristic pattern angular velocities Ω_p of the low-frequency modes are of the order of the mean orbital precession angular velocity \(\bar \Omega _{pr}\). Bar modes are also among the low-frequency modes; while \(\Omega _p \approx \bar \Omega _{pr}\) for slow bars, Ω_p for fast bars can appreciably exceed even the maximum orbital precession angular velocity in the disk Ω _pr ^max (however, it remains of the order of these precession angular velocities). The possibility of such an excess of Ω_p over Ω _pr ^max is associated with the effect of “repelling” orbits. The latter tend to move in a direction opposite to the direction in which they are pushed. We analyze the pattern of orbital precession in potentials typical of galactic disks. We note that the maximum radius of an “attracting” circular orbit r_c can serve as a reasonable estimate of the bar length l_b. Such an estimate is in good agreement with the available results of N-body simulations.     

Relativistic and Newtonian diskoseismology

The normal mode oscillations of thin accretion disks around black holes and other compact objects are analyzed and contrasted with those in stars. For black holes, the most robust modes are gravitationally trapped near the radius at which the radial epicyclic frequency is maximum. Their eigenfrequencies depend mainly on the mass and angular momentum of the black hole. The fundamental g-mode has recently been seen in numerical simulations of black hole accretion disks. For stars such as white dwarfs, the modes are trapped near the inner boundary (magnetospheric or stellar) of the accretion disk. Their eigenfrequencies are approximately multiples of the (Keplerian) angular velocity of the inner edge of the disk. The relevance of these modes to the high frequency quasi-periodic oscillations observed in the power spectra of accreting binaries will be discussed. In contrast to most stellar oscillations, most of these modes are unstable in the presence of viscosity (if the turbulent viscosity induced by the magnetorotational instability acts hydrodynamically).     

Formation of the spiral structure in SB galaxies

We investigate regions near the ends of the principal spiral arms in SB galaxies, where the non-axially symmetric part of the gravitational potential rapidly transforms to its asymptotic quadrupole form. The galactic disk responds to this transformation of the potential by forming nearly circular spirals with an angular extent of the order of π/2 (quarter-turn spirals). We consider the resonance mechanism for the formation of principal spirals. Expressions are derived for the resonance responses of disks with circular and nearly circular stellar orbits.     

On one classical problem in the radial orbit instability theory

Antonov’s classical problem of stability of a collisionless sphere with a purely radial motion of stars is considered as a limit of the problem in which stars move in nearly radial orbits. We provide the proper limiting equations that take into account the singularity in the density distribution at the sphere center and give their solutions. We show that there is instability for even and odd spherical harmonics, with all unstable modes being not slow. The growth rates of aperiodic even modes increase indefinitely when approaching purely radial models. The physics of the radial orbit instability is discussed.     

The origin of the density distribution of disc galaxies: a new problem for the standard model of disc formation

We present new models for the formation of disc galaxies that improve upon previous models by following the detailed accretion and cooling of the baryonic mass, and by using realistic distributions of specific angular momentum. Under the assumption of detailed angular momentum conservation, the discs that form have density distributions that are more centrally concentrated than an exponential. We examine the influence of star formation, bulge formation, and feedback on the outcome of the surface brightness distributions of the stars. Low angular momentum haloes yield disc galaxies with a significant bulge component and with a stellar disc that is close to exponential, in good agreement with observations. High angular momentum haloes, on the other hand, produce stellar discs that are much more concentrated than an exponential, in clear conflict with observations. At large radii, the models reveal distinct truncation radii in both the stars and the cold gas. The stellar truncation radii result from our implementation of star formation threshold densities, and are in excellent agreement with observations. The truncation radii in the density distribution of the cold gas reflect the maximum specific angular momentum of the gas that has cooled. We find that these truncation radii occur at H  i surface densities of roughly 1 M_⊙ pc⁻², in conflict with observations. We examine various modifications to our models, including feedback, viscosity, and dark matter haloes with constant-density cores, but show that the models consistently fail to produce bulge less discs with exponential surface brightness profiles. This signals a new problem for the standard model of disc formation: if the baryonic component of the protogalaxies out of which disc galaxies form has the same angular momentum distribution as the dark matter, discs are too compact.