Two-integral distribution functions for axisymmetric stellar systems with separable densities

We show different expressions of distribution functions (DFs) which depend only on the two classical integrals of the energy and the magnitude of the angular momentum with respect to the axis of symmetry for stellar systems with known axisymmetric densities. The density of the system is required to be a product of functions separable in the potential and the radial coordinates, where the functions of the radial coordinate are powers of a sum of a square of the radial coordinate and its unit scale. The even part of the two-integral DF corresponding to this type of density is in turn a sum or an infinite series of products of functions of the energy and of the magnitude of the angular momentum about the axis of symmetry. A similar expression of its odd part can be also obtained under the assumption of the rotation laws. It can be further shown that these expressions are in fact equivalent to those obtained by using Hunter & Qian's contour integral formulae for the system. This method is generally computationally preferable to the contour integral method. Two examples are given to obtain the even and odd parts of their two-integral DFs. One is for the prolate Jaffe model and the other for the prolate Plummer model.
It can be also found that the Hunter–Qian contour integral formulae of the two-integral even DF for axisymmetric systems can be recovered by use of the Laplace–Mellin integral transformation originally developed by Dejonghe.     

The error ellipsoid in phase space of stellar encounters and its evolution in time

We systematically investigate stellar encounters in their six‐dimensional phase space, including also the error ellipsoid and its evolution in time. It allows to give not only a mathematical model of stellar encounters but also an error model for this process. On that occasion we derive fundamental formulae for the positional error in ℝⁿ: the probability of a position X falling into the error ellipsoid, the standard deviation in an arbitrary direction, and we briefly discuss the mean positional error in ℝⁿ. The case of the close encounter of the star GL 710 (HIP 89825) is an illuminating example of applying the derived results. Moreover, we show that the error ellipsoid can also be successfully applied to approximate the confidence region (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of the dynamical structure of galaxies using optical spectra

Galaxy spectra are a rich source of kinematical information since the shapes of the absorption lines reflect the movement of stars along the line-of-sight. We present a technique with which to build directly a dynamical model for a galaxy by fitting model spectra, calculated from a dynamical model, to the observed galaxy spectra. Using synthetic spectra from a known galaxy model we demonstrate that this technique indeed recovers the essential dynamical characteristics of the galaxy model. Moreover, the method allows a statistically meaningful error analysis on the resulting dynamical quantities.     

Degenerate equilibrium states of collisionless stellar systems

We study spherically symmetrical equilibrium states of collisionless stellar systems confined to a spherical box. These equilibrium states correspond to the statistics introduced by Lynden-Bell in his theory of 'violent relaxation', and are described by a Fermi–Dirac distribution function. We compute the corresponding equilibrium diagram and show that a global entropy maximum exists for any accessible control parameter. This equilibrium state shows a pronounced separation between a degenerate core and a halo. We therefore check that degeneracy is able to stop the gravitational collapse (of a collisionless system), and we propose a simple model for the 'core–halo' structure. We also discuss the relevance of our study for real galaxies or other astrophysical systems such as massive neutrinos.     

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.     

Dynamical friction for compound bodies

In the framework of the fluctuation–dissipation approach to dynamical friction, we derive an expression giving the orbital energy exchange experienced by a compound body, as it moves, interacting with a non-homogeneous discrete background. The body is assumed to be composed of particles endowed with a velocity spectrum and with a non-homogeneous spatial distribution. The Chandrasekhar formula is recovered in the limit of a point-like satellite with zero velocity dispersion and infinite temperature moving through a homogeneous infinite medium. In this same limit, but dropping the zero satellite velocity dispersion (σ_S) condition, the orbital energy loss is found to be smaller than in the σ_S=0 case by a factor of up to an order of magnitude in some situations.     

Orbital migrations in planetesimal discs: N-body simulations and the resonant dynamical friction

We have performed N -body numerical simulations of the exchange of angular momentum between a massive planet and a 3D Keplerian disc of planetesimals. Our interest is directed at the study of the classical analytical expressions of the lineal theory of density waves, as representative of the dynamical friction in discs 'dominated by the planet' and the orbital migration of the planets with regard to this effect. By means of a numerical integration of the equations of motion, we have carried out a set of numerical experiments with a large number of particles  ( N ≥10 000)  , and planets with the mass of Jupiter, Saturn and one core mass of the giant planets in the Solar system  ( M _c=10 M_⊕)  . The torque, measured in a phase in which a 'steady forcing' is clearly measurable, yields inward migration in a minimum-mass solar disc  (Σ∼10 g cm^-2  ), with a characteristic drift time of ∼ a few 10⁶ yr. The planets predate the disc, but the orbital decay rate is not sufficient to allow accretion in a time-scale relevant to the formation of giant planets. We found reductions of the measured torque on the planet, with respect to the linear theory, by a factor of 0.38 for M _c, 0.04 for Saturn and 0.01 for Jupiter, due to the increase in the perturbation on the disc. The behaviour of planets whose mass is larger than M _c is similar to the one of type II migrators in gaseous discs. Our results suggest that, in a minimum mass, solar planetesimals disc, type I migrations occur for masses smaller than M _c, whereas for this mass value it could be a transition zone between the two types of migration.     

Programmed motion in the presence of homogeneity

In the framework of the inverse problem of dynamics, we face the following question with reference to the motion of one material point: Given a region T_orb of the xy plane, described by the inequality g (x, y) ≤ c₀, are there potentials V = V (x, y) which can produce monoparametric families of orbits f (x, y) = c (also to be found) lying exclusively in the region T_orb? As the relevant PDEs are nonlinear, an answer to this question (generally affirmative, but not with assurance) can be given by the procedure of the determination of certain constants specifying the pertinent functions. In this paper we ease the mathematics involved by making certain simplifying assumptions referring to the homogeneity of both the function g (x, y) (describing the boundary of T_orb) and of the slope function γ(x, y) = f_y/f_x (representing the required family f (x, y) = c). We develop the method to treat the so formulated problem and we show that, even under these restrictive assumptions, an affirmative answer is guaranteed provided that two algebraic equations have in common at least one solution (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dark halo mergers and the formation of a universal profile

We argue that a universal density profile for dark matter haloes arises as a natural consequence of hierarchical structure formation: it is a fixed point in the process of repeated mergers. We present analytic and numerical arguments for the emergence of a particular form of the central cusp profile. At small radii, the density should vary as r ^−α, with α determined by the way in which the characteristic density of haloes scales with their mass. If small haloes are dense, then α is large. The mass–density relation can be related to the power spectrum of initial fluctuations, P ( k ), through 'formation time' arguments. Early structure formation leads to steep cusps. For P ( k ) ∼  k ⁿ we find α ≃ 3(3 +  n )/(5 +  n ). The universal profile is generated by tidal stripping of small haloes as they merge with larger objects.     

Studies of multiple stellar systems – III. Modulation of orbital elements in the triple-lined system HD 109648

The triple-lined spectroscopic triple system HD 109648 has one of the shortest periods known for the outer orbit in a late-type triple, 120.5 d, and the ratio between the periods of the outer and the inner orbits is small, 22:1. With such extreme values, this system should show orbital element variations over a time-scale of about a decade. We have monitored the radial velocities of HD 109648 with the CfA Digital Speedometers for 8 yr, and have found evidence for modulation of some orbital elements. While we see no definite evidence for modulation of the inner binary eccentricity, we clearly observe variations in the inner and the outer longitudes of periastron, and in the radial velocity amplitudes of the three components. The observational results, combined with numerical simulations, allow us to put constraints on the orientation of the orbits.     

Eccentricity evolution in hierarchical triple systems with eccentric outer binaries

We develop a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular, while the outer one is eccentric. We consider coplanar systems with well-separated components and comparable masses. The derivation of short-period terms is based on an expansion of the rate of change of the Runge–Lenz vector. Then, the short-period terms are combined with secular terms, obtained by means of canonical perturbation theory. The validity of the theoretical equations is tested by numerical integrations of the full equations of motion.     

Families of orbits in planar anisotropic fields

The aim of the planar inverse problem of dynamics is: given a monoparametric family of curves f(x, y) = c, find the potential V (x, y) under whose action a material point of unit mass can describe the curves of the family. In this study we look for V in the class of the anisotropic potentials V(x, y) = v(a²x² + y²), (a=constant). These potentials have been used lately in the search of connections between classical, quantum, and relativistic mechanics. We establish a general condition which must be satisfied by all the families produced by an anisotropic potential. We treat special cases regarding the families (e. g. families traced isoenergetically) and we present certain pertinent examples of compatible pairs of families of curves and anisotropic potentials. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hill stability of a triple system with an inner binary of large mass ratio

We determine the maximum dimensionless pericentre distance a third body can have to the barycentre of an extreme mass ratio binary, beyond which no exchange or ejection of any of the binary components can occur. We calculate this maximum distance, q '/ a , where q ' is the pericentre of the third mass to the binary barycentre and a is the semimajor axis of the binary, as a function of the critical value of   L ²  E   of the system, where L is the magnitude of the angular momentum vector and E is the total energy of the system. The critical value is obtained by calculating   L ²  E   for the central configuration of the system at the collinear Lagrangian points. In our case we can make approximations for the system when one of the masses is small. We compare the calculated values of the pericentre distance with numerical scattering experiments as a function of the eccentricity of the inner orbit, e , the mutual inclination i and the eccentricity of the outer orbit, e '. These show that the maximum observed value of   q '/ a   is indeed the critical q '/ a , as expected. However, when   e '→1  , the maximum observed value of q '/ a is equal to the critical value calculated when   e '=0  , which is contrary to the theory, which predicts exchange distances several orders of magnitude larger for nearly parabolic orbits. This does not occur because changes in the binding energy of the binary are exponentially small for distant, nearly parabolic encounters.     

Two‐dimension potentials which generate spatial families of orbits

We consider the following case of the 3D inverse problem of dynamics: Given a spatial two‐parametric family of curves f (x, y, z) = c₁, g (x, y, z) = c₂, find possibly existing two‐dimension potentials under whose action the curves of the family are trajectories for a unit mass particle. First we establish the conditions which must be fulfilled by the family so that potentials of the form w (y, z) give rise to the curves of the family, and we present some applications. Then we examine briefly the existence of potentials depending on (x, z), respectively (x, y), which are compatible with the given family (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The dynamical evolution of stellar superclusters

Recent images taken with the Hubble Space Telescope ( HST ) of the interacting disc galaxies NGC 4038/4039 (the Antennae) reveal clusters of many dozens and possibly hundreds of young compact massive star clusters within projected regions spanning about 100 to 500 pc. It is shown here that a large fraction of the individual star clusters merge within a few tens to a hundred Myr. Bound stellar systems with radii of a few hundred parsecs, masses ≲ 10⁹ M⊙ and relaxation times of 10¹¹ − 10¹² yr may form from these. These spheroidal dwarf galaxies contain old stars from the pre-merger galaxy and much younger stars formed in the massive star clusters, and possibly from later gas accretion events. The possibility that star formation in the outer regions of gas-rich tidal tails may also lead to superclusters is raised. The mass-to-light ratio of these objects is small, because they contain an insignificant amount of dark matter. After many hundred Myr such systems may resemble dwarf spheroidal satellite galaxies with large apparent mass-to-light ratios, if tidal shaping is important.