Substellar companions and isolated planetary-mass objects from protostellar disc fragmentation

Self-gravitating protostellar discs are unstable to fragmentation if the gas can cool on a time-scale that is short compared with the orbital period. We use a combination of hydrodynamic simulations and N -body orbit integrations to study the long-term evolution of a fragmenting disc with an initial mass ratio to the star of   M _disc/ M _*= 0.1  . For a disc that is initially unstable across a range of radii, a combination of collapse and subsequent accretion yields substellar objects with a spectrum of masses extending (for a Solar-mass star) up to  ≈0.01 M_⊙  . Subsequent gravitational evolution ejects most of the lower mass objects within a few million years, leaving a small number of very massive planets or brown dwarfs in eccentric orbits at moderately small radii. Based on these results, systems such as HD 168443 – in which the companions are close to or beyond the deuterium burning limit – appear to be the best candidates to have formed via gravitational instability. If massive substellar companions originate from disc fragmentation, while lower-mass planetary companions originate from core accretion, the metallicity distribution of stars which host massive substellar companions at radii of ∼1 au should differ from that of stars with lower mass planetary companions.     

Dynamical friction in modified Newtonian dynamics

We have tested a previous analytical estimate of the dynamical friction time-scale in modified Newtonian dynamics (MOND) with fully non-linear N -body simulations. The simulations confirm that the dynamical friction time-scale is significantly shorter in MOND than in equivalent Newtonian systems, i.e. systems with the same phase-space distribution of baryons and additional dark matter. An apparent conflict between this result and the long time-scales determined for bars to slow and mergers to be completed in previous N -body simulations of MOND systems is explained. The confirmation of the short dynamical-friction time-scale in MOND underlines the challenge that the Fornax dwarf spheroidal poses to the viability of MOND.     

Scaling up tides in numerical models of galaxy and halo formation

The purpose of this article is to show that when dynamically cold, dissipationless self-gravitating systems collapse, their evolution is a strong function of the symmetry in the initial distribution. We explore with a set of pressureless homogeneous fluids the time evolution of ellipsoidal distributions and map the depth of potential achieved during relaxation as function of initial ellipsoid axis ratios. We then perform a series of N -body numerical simulations and contrast their evolution with the fluid solutions. We verify an analytic relation between collapse factor and particle number N in spherical symmetry, such that  ∝ N ^1/3  . We sought a similar relation for axisymmetric configurations, and found an empirical scaling relation such that  ∝ N ^1/6  in these cases. We then show that when mass distributions do not respect spherical or axial symmetry, the ensuing gravitational collapse deepens with increasing particle number N but only slowly: 86 per cent of triaxial configurations may collapse by a factor of no more than 40 as   N →∞  . For   N ≈10⁵  and larger, violent relaxation develops fully under the Lin–Mestel–Shu instability such that numerical N -body solutions now resolve the different initial morphologies adequately.     

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).     

Possible solution to the problem of the extreme mass ratio W UMa-type binaries

When the total angular momentum of a binary system is at a critical (minimum) value, a tidal instability occurs (Darwin's instability), eventually forcing the stars to merge into a single, rapidly rotating object. The instability sets in at some critical separation which in the case of contact binaries corresponds to a minimum mass ratio depending on dimensionless gyration radius k ₁. If one considers   n = 3  polytrope (fully radiative primary with  Γ₁= 4/3  ),   k ²₁= 0.075  and   q _min≈ 0.085–0.095  . There appears to be, however, some W UMa-type binaries with q values very close, if not below these theoretical limits, implying that primary in these systems is probably more centrally condensed. We try to solve the discrepancy between theory and observations by considering rotating polytropes. We show by deriving and solving a modified Lane–Emden equation for   n = 3  polytrope that including the effects of rotation does increase the central concentration and could reduce   q _min  to as low as 0.070–0.074, more consistent with the observed population.     

Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies

In the presence of a strong   m = 2  component in a rotating galaxy, the phase-space structure near corotation is shaped to a large extent by the invariant manifolds of the short-period family of unstable periodic orbits terminating at L ₁ or L ₂. The main effect of these manifolds is to create robust phase correlations among a number of chaotic orbits large enough to support a spiral density wave outside corotation. The phenomenon is described theoretically by soliton-like solutions of a Sine–Gordon equation. Numerical examples are given in an N -body simulation of a barred spiral galaxy. In these examples, we demonstrate how the projection of unstable manifolds in configuration space reproduces essentially the entire observed bar–spiral pattern.     

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.     

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.     

On the clustering phase transition in self-gravitating N-body systems

The thermodynamic behaviour of self-gravitating N -body systems has been worked out by borrowing a standard method from molecular dynamics. The link between dynamics and thermodynamics is made in the microcanonical ensemble of statistical mechanics. Through the computation of basic thermodynamic observables and of the equation of state in the     plane, the clustering phase transition appears to be of the second-order type. The dynamical–microcanonical averages are compared with their corresponding canonical ensemble averages, obtained through standard Monte Carlo computations. The latter seem to have completely lost any information about the phase transition. Finally, our results – obtained in a 'microscopic' framework – are compared with some existing theoretical predictions – obtained in a 'macroscopic' (thermodynamic) framework: qualitative and quantitative agreement is found, with an interesting exception.     

On a time-symmetric Hermite integrator for planetary N-body simulation

We describe a P(EC) ⁿ Hermite scheme for planetary N -body simulation. The fourth-order implicit Hermite scheme is a time-symmetric integrator that has no secular energy error for the integration of periodic orbits with time-symmetric time-steps. In general N -body problems, however, this advantage is of little practical significance, since it is difficult to achieve time-symmetry with individual variable time-steps. However, we can easily enjoy the benefit of the time-symmetric Hermite integrator in planetary N -body systems, where all bodies spend most of the time on nearly circular orbits. These orbits are integrated with almost constant time-steps even if we adopt the individual time-step scheme. The P(EC) ⁿ Hermite scheme and almost constant time-steps reduce the integration error greatly. For example, the energy error of the P(EC)² Hermite scheme is two orders of magnitude smaller than that of the standard PEC Hermite scheme in the case of an N  = 100,  m  = 10²⁵ g planetesimal system with the rms eccentricity 〈 e ²〉^1/2 ≲0.03.     

Eddy viscosity and turbulent Schmidt number by kink-type instabilities of toroidal magnetic fields

The potential of the non-axisymmetric magnetic instability to transport angular momentum and to mix chemicals is probed considering the stability of a nearly uniform toroidal field between conducting cylinders with different rotation rates. The fluid between the cylinders is assumed as incompressible and to be of uniform density. With a linear theory, the neutral-stability maps for   m = 1  are computed. Rigid rotation must be sub-Alfvénic to allow instability, while for differential rotation also an unstable domain with faster rotation exists [azimuthal magnetorotational instability (AMRI)]. The rotational quenching of the magnetic instability is strongest for magnetic Prandtl number of the order of unity.
The effective angular momentum transport by the instability is directed outwards for subrotation. The resulting magnetic-induced eddy viscosity exceeds the microscopic values by factors of 10–100. This is only true for AMRI; in the opposite case of Tayler instability, the viscosity results are very small.
The same instability also quenches concentration gradients of chemicals by dynamic fluctuations. The corresponding diffusion coefficient always remains smaller than the magnetic-generated eddy viscosity. A Schmidt number of the order of 30 is found as the ratio of the effective viscosity and the diffusion coefficient. For not too strong magnetic fields in the radiation zone of young solar-type stars, the magnetic instability transports much more angular momentum than that it mixes chemicals.     

Ram pressure stripping of spiral galaxies in clusters

We use three-dimensional SPH/ N -body simulations to study ram pressure stripping of gas from spiral galaxies orbiting in clusters. We find that the analytic expectation of Gunn & Gott, relating the gravitational restoring force provided by the disc to the ram pressure force, provides a good approximation to the radius at which gas will be stripped from a galaxy. However, at small radii it is also important to consider the potential provided by the bulge component. A spiral galaxy passing through the core of a rich cluster, such as Coma, will have its gaseous disc truncated to ∼4 kpc, thus losing ∼80 per cent of its diffuse gas mass. The time-scale for this to occur is a fraction of a crossing time ∼10⁷ yr. Galaxies orbiting within poorer clusters, or inclined to the direction of motion through the intracluster medium, will lose significantly less gas. We conclude that ram pressure alone is insufficient to account for the rapid and widespread truncation of star formation observed in cluster galaxies, or the morphological transformation of Sabs to S0s that is necessary to explain the Butcher–Oemler effect.     

Crash-testing the cauldron code for joint lensing and dynamics analysis of early-type galaxies

We apply the joint lensing and dynamics code for the analysis of early-type galaxies, 'Combined Algorithm for Unified Lensing and Dynamics ReconstructiON ( cauldron )', to a rotating N -body stellar system with dark matter halo which significantly violates the two major assumptions of the method, i.e. axial symmetry supported by a two-integral distribution function. The goal is to study how cauldron performs in an extreme case, and to determine which galaxy properties can still be robustly recovered. Three data sets, corresponding to orthogonal lines of sight, are generated from the N -body system and analysed with the identical procedure followed in the study of real lens galaxies, adopting an axisymmetric power-law total density distribution. We find that several global properties of the N -body system are recovered with remarkable accuracy, despite the fact that the adopted power-law model is too simple to account for the lack of symmetry of the true density distribution. In particular, the logarithmic slope of the total density distribution is robustly recovered to within less than 10 per cent (with the exception of the ill-constrained very inner regions), the inferred angle-averaged radial profile of the total mass closely follows the true distribution, and the dark matter fraction of the system (inside the effective radius) is correctly determined within ∼10 per cent of the total mass. Unless the line-of-sight direction is almost parallel to the total angular momentum vector of the system, reliably recovered quantities also include the angular momentum, the   V /σ  ratio and the anisotropy parameter δ. We conclude that the cauldron code can be safely and effectively applied to real early-type lens galaxies, also providing reliable information for the systems that depart significantly from the method's assumptions.     

High-velocity stars from decay of small stellar systems

In this paper we present numerical results on the decay of small stellar systems under different initial conditions (multiplicity 3 ≤  N  ≤ 10, and various mass spectra, initial velocities and initial configurations). The numerical treatment uses the CHAIN1 code (Mikkola &38; Aarseth). Particular attention is paid to the distribution of high-velocity escapers: we define these as stars with velocity above 30 km s⁻¹. These numerical experiments confirm that small N -body systems are dynamically unstable and produce cascades of escapers in the process of their decay. It is shown that the fraction of stars that escape from small dense stellar systems with an escape velocity greater than 30 km s⁻¹ is ∼1 per cent for all systems treated here. This relatively small fraction must be considered in relation to the rate of star formation in the Galaxy in small groups: this could explain some moderately high-velocity stars observed in the Galactic disc and possibly some young stars with relatively high metallicity in the thick disc.     

The fine-grained phase-space structure of cold dark matter haloes

We present a new and completely general technique for calculating the fine-grained phase-space structure of dark matter (DM) throughout the Galactic halo. Our goal is to understand this structure on the scales relevant for direct and indirect detection experiments. Our method is based on evaluating the geodesic deviation equation along the trajectories of individual DM particles. It requires no assumptions about the symmetry or stationarity of the halo formation process. In this paper we study general static potentials which exhibit more complex behaviour than the separable potentials studied previously. For ellipsoidal logarithmic potentials with a core, phase mixing is sensitive to the resonance structure, as indicated by the number of independent orbital frequencies. Regions of chaotic mixing can be identified by the very rapid decrease in the real-space density of the associated DM streams. We also study the evolution of stream-density in ellipsoidal NFW haloes with radially varying isopotential shape, showing that if such a model is applied to the Galactic halo, at least 10⁵ streams are expected near the Sun. The most novel aspect of our approach is that general non-static systems can be studied through implementation in a cosmological N -body code. Such an implementation allows a robust and accurate evaluation of the enhancements in annihilation radiation due to fine-scale structure such as caustics. We embed the scheme in the current state-of-the-art code gadget -3 and present tests which demonstrate that N -body discreteness effects can be kept under control in realistic configurations.     

Monte Carlo simulations of star clusters — I. First Results

A revision of Stodoíkiewicz's Monte Carlo code is used to simulate evolution of star clusters. The new method treats each superstar as a single star and follows the evolution and motion of all individual stellar objects. The first calculations for isolated, equal-mass N -body systems with three-body energy generation according to Spitzer's formulae show good agreement with direct N -body calculations for N  = 2000, 4096 and 10 000 particles. The density, velocity, mass distributions, energy generation, number of binaries, etc., follow the N -body results. Only the number of escapers is slightly too high compared with N -body results, and there is no level-off anisotropy for advanced post-collapse evolution of Monte Carlo models as is seen in N -body simulations for N  ≤ 2000. For simulations with N  > 10 000 gravothermal oscillations are clearly visible. The calculations of N   2000, 4096, 10 000, 32 000 and 100 000 models take about 2, 6, 20, 130 and 2500 h, respectively. The Monte Carlo code is at least 10⁵ times faster than the N -body one for N  = 32 768 with special-purpose hardware. Thus it becomes possible to run several different models to improve statistical quality of the data and run individual models with N as large as 100 000. The Monte Carlo scheme can be regarded as a method which lies in the middle between direct N -body and Fokker–Planck models and combines most advantages of both methods.     

The evolutionary status of W Ursae Majoris-type systems

Well-determined physical parameters of 130 W Ursae Majoris (W UMa) systems were collected from the literature. Based on these data, the evolutionary status and dynamical evolution of W UMa systems are investigated. It is found that there is no evolutionary difference between W- and A-type systems in the   M – J   diagram, which is consistent with the results derived from the analysis of observed spectral type and of   M – R   and   M – L   diagrams of W UMa systems.   M – R   and   M – L   diagrams of W- and A-type systems indicate that a large amount of energy should be transferred from the more massive to the less massive component, so that they are not in thermal equilibrium and undergo thermal relaxation oscillation. Moreover, the distribution of angular momentum, together with the distribution of the mass ratio, suggests that the mass ratio of the observed W UMa systems decreases with decreasing total mass. This could be the result of the dynamical evolution of W UMa systems, which suffer angular momentum loss and mass loss as a result of the magnetic stellar wind. Consequently, the tidal instability forces these systems towards lower q values and finally to rapidly rotating single stars.     

Gauss's method for secular dynamics, softened

We show that the algorithm proposed by Gauss to compute the secular evolution of gravitationally interacting Keplerian rings extends naturally to softened gravitational interactions. The resulting tool is ideal for the study of the secular dynamical evolution of nearly Keplerian systems such as stellar clusters surrounding black holes in galactic nuclei, cometary clouds or planetesimal discs. We illustrate its accuracy, efficiency and versatility on a variety of configurations. In particular, we examine a secularly unstable system of counterrotating discs, and follow the unfolding and saturation of the instability into a global, uniformly precessing, lopsided  ( m = 1)  mode.