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 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.     

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.     

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.     

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．     

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.     

The cosmological dependence of galactic specific angular momenta

Hydrodynamical simulations of galaxy formation in spatially flat cold dark matter (CDM) cosmologies with and without a cosmological constant (Λ) are described. A simple star formation algorithm is employed and radiative cooling is allowed only after redshift z =1 so that enough hot gas is available to form large, rapidly rotating stellar discs if angular momentum is approximately conserved during collapse. The specific angular momenta of the final galaxies are found to be sensitive to the assumed background cosmology. This dependence arises from the different angular momenta contained in the haloes at the epoch when the gas begins to collapse and the inhomogeneity of the subsequent halo evolution. In the Λ-dominated cosmology, the ratio of stellar specific angular momentum to that of the dark matter halo (measured at the virial radius) has a median value of ∼0.24 at z =0. The corresponding quantity for the Λ=0 cosmology is over three times lower. It is concluded that the observed frequency and angular momenta of disc galaxies pose significant problems for spatially flat CDM models with Λ=0 but may be consistent with a Λ-dominated CDM universe.     

The present status of periodic orbits

Recent results on periodic orbits are presented and it is shown that the periodic orbits can be used in the study of planetary systems and triple or multiple stellar systems. Triple stellar systems are stable even for close approaches of the three components. Also stable triple systems exist with nearly zero angular momentum. For the planetary systems a global view is obtained from which it is clear which configurations are stable or unstable and also what factors affect the stability. Also, the relation between resonance and instability is studied by making use of periodic orbits.     

The formation of the Hubble sequence of disc galaxies: the effects of early viscous evolution

We investigate a model of disc galaxies whereby viscous evolution of the gaseous disc drives material inwards to form a protobulge. We start from the standard picture of disc formation through the settling of gas into a dark halo potential well, with the disc initially coming into centrifugal equilibrium with detailed conservation of angular momentum. We derive generic analytic solutions for the disc–halo system after adiabatic compression of the dark halo, with free choice of the input virialized dark halo density profile and of the specific angular momentum distribution. We derive limits on the final density profile of the halo in the central regions. Subsequent viscous evolution of the disc is modelled by a variation of the specific angular momentum distribution of the disc, providing analytic solutions to the final disc structure. The assumption that the viscous evolution time-scale and the star formation time-scale are similar leads to predictions of the properties of the stellar components. Focusing on small 'exponential' bulges, i.e., ones that may be formed through a disc instability, we investigate the relationship between the assumed initial conditions, such as halo 'formation', or assembly, redshift z _f, spin parameter λ , baryonic fraction F , and final disc properties such as global star formation time-scale, gas fraction, and bulge-to-disc ratio. We find that the present properties of discs, such as the scalelength, are compatible with a higher initial formation redshift if the redistribution by viscous evolution is included than if it is ignored. We also quantify the dependence of final disc properties on the ratio F λ , thus including the possibility that the baryonic fraction varies from galaxy to galaxy, as perhaps may be inferred from the observations.     

On a Formation Scenario of Star Clusters

Most formation scenarios of globular clusters assume a molecular cloud as the progenitor of the stellar system. However, it is still unclear, how this cloud is transformed into a star cluster, i.e. how the destructive processes related to gas removal or low star formation effiency can be avoided. Here a scheme of supernova (SN) induced cluster formation is studied. According to this scenario an expanding SN shell accumulates the mass of the cloud. This is accompanied by fragmentation resulting in star formation in the shell. Provided the stellar shell expands sufficiently slow, its self-gravity stops the expansion and the shell recollapses, by this forming a stellar system. I present N-body simulations of collapsing shells which move in a galactic potential on circular and elliptic orbits. It is shown that typical shells (10⁵ M_⊙, 30 pc) evolve to twin clusters over a large range of galactocentric distances. Outside this range single stellar systems are formed, whereas at small galactocentric distances the shells are tidally disrupted. In that case many small fragments formed during the collapse survive as single bound entities. About 1/3 of the twin cluster systems formed on circular orbits merge within 400 Myr. On elliptic orbits the merger rate reduces to less than 4%. Thus, there could be a significant number of twin clusters even in our Galaxy, which, however, might be undetected as twins due to a large phase shift on their common orbit. This revised version was published online in September 2006 with corrections to the Cover Date.     

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.     

Maximum feedback and dark matter profiles of dwarf galaxies

The observed rotation curves of dark matter-dominated dwarf galaxies indicate low-density cores, contrary to the predictions of CDM models. A possible solution of this problem involves stellar feedback. A strong baryonic wind driven by vigorous star formation can remove a large fraction of the gas, causing the dark matter to expand. Using both numerical and analytical techniques, we explore the maximum effect of the feedback with an instantaneous removal of the gaseous disc. The energy input depends on the compactness of the disc, hence the specific angular momentum of the disc. For the plausible cosmological parameters and a wide range of the disc angular momenta, the feedback is insufficient to destroy the central halo cusp, while the inner density is lowered only by a modest factor of 2 to 6. Any realistic modelling of the feedback would have even lesser impact on dark matter. We find that no star formation effect can resolve the problems of CDM cusps.     

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.     

Effect of partial mixing of matter on the hydrodynamic angular momentum transport processes in massive main-sequence stars

We consider the evolution of a rotating star with a mass of 16M _⊙ and an angular momentum of 3.25 × 10⁵² g cm² s^?1, along with the hydrodynamic transport of angular momentum and chemical elements in its interiors. When the partial mixing of matter of the turbulent radiative envelope and the convective core is taken into account, the efficiency of the angular momentum transport by meridional circulation in the stellar interiors and the duration of the hydrogen burning phase increase. Depending on the Schmidt number in the turbulent radiative stellar envelope, the ratio of the equatorial rotational velocity to the circular one increases with time in the process of stellar evolution and can become typical of early-type Be stars during an additional evolution time of the star on the main sequence. Partial mixing of matter is a necessary condition under which the hydrodynamic transport processes can increase the angular momentum of the outer stellar layer to an extent that the equatorial rotational velocity begins to increase during the second half of the evolutionary phase of the star on the main sequence, as shown by observations of the brightest stars in open star clusters with ages of 10–25 Myr. When the turbulent Schmidt number is 0.4, the equatorial rotational velocity of the star increases during the second half of the hydrogen burning phase in the convective core from 330 to 450 km s^?1.     

Close encounters of a rotating star with planets in parabolic orbits of varying inclination and the formation of hot Jupiters

In this paper we extend the theory of close encounters of a giant planet on a parabolic orbit with a central star developed in our previous work (Ivanov and Papaloizou in MNRAS 347:437, 2004; MNRAS 376:682, 2007) to include the effects of tides induced on the central star. Stellar rotation and orbits with arbitrary inclination to the stellar rotation axis are considered. We obtain results both from an analytic treatment that incorporates first order corrections to normal mode frequencies arising from stellar rotation and numerical treatments that are in satisfactory agreement over the parameter space of interest. These results are applied to the initial phase of the tidal circularisation problem. We find that both tides induced in the star and planet can lead to a significant decrease of the orbital semi-major axis for orbits having periastron distances smaller than 5?C6 stellar radii with tides in the star being much stronger for retrograde orbits compared to prograde orbits. Assuming that combined action of dynamic and quasi-static tides could lead to the total circularisation of orbits this corresponds to observed periods up to 4?C5 days. We use the simple Skumanich law to characterise the rotational history of the star supposing that the star has its rotational period equal to one month at the age of 5 Gyr. The strength of tidal interactions is characterised by circularisation time scale, t _ev , which is defined as a typical time scale of evolution of the planet??s semi-major axis due to tides. This is considered as a function of orbital period P _obs , which the planet obtains after the process of tidal circularisation has been completed. We find that the ratio of the initial circularisation time scales corresponding to prograde and retrograde orbits, respectively, is of order 1.5?C2 for a planet of one Jupiter mass having P _obs ~ 4 days. The ratio grows with the mass of the planet, being of order five for a five Jupiter mass planet with the same P _orb . Note, however, this result might change for more realistic stellar rotation histories. Thus, the effect of stellar rotation may provide a bias in the formation of planetary systems having planets on close orbits around their host stars, as a consequence of planet?Cplanet scattering, which favours systems with retrograde orbits. The results reported in the paper may also be applied to the problem of tidal capture of stars in young stellar clusters.     

Formation of trans-Neptunian satellite systems at the stage of condensations

The formation of trans-Neptunian satellite systems at the stage of rarefied preplanetesimals (i.e., condensations of dust and/or objects less than 1 m in diameter) is discussed. It is assumed that trans-Neptunian objects (including those with satellites) could form as a result of compression of parental rarefied preplanetesimals. The formulas for calculating the angular momentum of two colliding condensations with respect to their center of mass, which were applied earlier in (Ipatov, 2010) in the comparison of such momenta with the angular momenta of observed satellite systems, are used to estimate the angular momenta of condensations needed to form satellite systems. It is demonstrated that the angular velocities of condensations used in (Nesvorny et al., 2010) as the initial data in the computer simulation of compression of rarefied preplanetesimals and subsequent formation of trans-Neptunian satellite systems may be obtained in collisions of preplanetesimals with their radii comparable to the corresponding Hill radii. For example, these angular velocities are in the range of possible values of angular velocities of a parental rarefied preplanetesimal formed as a result of a merger of two colliding rarefied preplanetesimals that moved in circular heliocentric orbits before a collision. Some rarefied preplanetesimals formed as a result of collision of preplanetesimals in the region of formation of solid small bodies acquire such angular momenta that are sufficient to form satellite systems of small bodies. It is likely that the ratio of the number of rarefied preplanetesimals with such angular momenta to the total number of rarefied preplanetesimals producing classical trans-Neptunian objects with diameters larger than 100 km was 0.45 (the initial fraction of satellite systems among all classical trans-Neptunian objects).     

The role of tidal interactions in star formation

Nearly all of the initial angular momentum of the matter that goes into each forming star must somehow be removed or redistributed during the formation process. The possible transport mechanisms and the possible fates of the excess angular momentum are discussed, and it is argued that transport processes in discs are probably not sufficient by themselves to solve the angular momentum problem, while tidal interactions with other stars in forming binary or multiple systems are likely to be of very general importance in redistributing angular momentum during the star formation process. Most, if not all, stars probably form in binary or multiple systems, and tidal torques in these systems can transfer much of the angular momentum from the gas around each forming star to the orbital motions of the companion stars. Tidally generated waves in circumstellar discs may contribute to the overall redistribution of angular momentum. Stars may gain much of their mass by tidally triggered bursts of rapid accretion, and these bursts could account for some of the most energetic phenomena of the earliest stages of stellar evolution, such as jet-like outflows. If tidal interactions are indeed of general importance, planet-forming discs may often have a more chaotic and violent early evolution than in standard models, and shock heating events may be common. Interactions in a hierarchy of subgroups may play a role in building up massive stars in clusters and in determining the form of the upper initial mass function (IMF) . Many of the processes discussed here have analogues on galactic scales, and there may be similarities between the formation of massive stars by interaction-driven accretion processes in clusters and the buildup of massive black holes in galactic nuclei.     

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).     

Orbits of charged bodies

An investigation into the possibility that material drawn out of a star goes into orbit around that star if electromagnetic effects are included, has been made. It is found that if the body has an initial charge of some 10³⁷ e.s.u., and decreasing with time then sufficient angular momentum can be transferred to make orbits not intersecting the stellar surface possible.     

A two-parameter scheme for the evolution of symmetrical galaxies

In the present paper, a general evolutionary scheme for axisymmetrical rotationally supported equilibrium models for galaxies is considered. Its main phases are: an expansion phase of the initial protogalaxy, assumed to consist into an homogeneous gas sphere structured into clouds, from recombination to maximum expansion, during which it is surmized that angular momentum is acquired by tidal interactions by the expanding configuration; then a violent relaxation collapse phase, following maximum expansion and ending into a virialized deformed polytropic configuration; the reaching of virialization is considered as an adequate initial state for the new phase of virialized contraction of the gaseous component, due to the collisions of the constituent gas clouds, while the stellar component, due to the stars already formed according to a generalized Schmidt-type law during the early expansion and violent relaxation phases, is assumed to have reached a stabilized situation.The initial mean density and radius for both galaxy and component clouds expressed as functions of the density fluctuation spectrum at recombination, act as physical parameters determining the characteristics of the system at maximum expansion, together with the total amount of angular momentum acquired during the expansion phase. The main physical parameters at virialization are then completely specified when the initial distribution of the clouds inside the galaxy is assigned and the constants appearing in it are derived by normalization with the observed data.We find for systems of given mass that the larger the angular momentum per unit mass is: (1) the larger are the equatorial semiaxis at maximum expansion and at virialization and the lower the mean density; (2) the larger is the time elapsed up the maximum expansion and to virialization; while for systems of different mass, we obtain that to the larger mass correspond the larger time elapsed up to maximum expansion and to virialization, and the lower mean density.For the contraction phase following virialization, two limiting cases are considered: (A) either the star component already present at virialization is entirely neglected; (B) or it is thought to contract as the gas component. In such cases, it is found for systems of equal mass that lower angular momenta lead to final configurations characterized by no or small flat gaseous components (which may correspond to lenticulars and early type spirals) while the contrary is true for large angular momenta (corresponding to late type spirals and irregulars). As mass and angular momentum per unit mass decrease, according to an assumed lawj M, the allowed configurations on the late type side of the morphological sequence tend towards earlier and earlier types, until for masses low enough (10¹⁰ m ), only halo type configurations seem to exist. According to this view, the observed lack of spirals with masses below 10¹⁰ m and the wide mass range exibited by the stellar halo type galaxies might be interpreted. In general, it appears that in the limit of the approximations made, a morphological sequence of galaxies can be described by two parameters, mass and angular momentum.