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)     

Effects of stellar wind,dynamical friction,and star mergers on the dynamical evolution of multiple stars

The dynamical evolution of small stellar groups composed of N=6 components was numerically simulated within the framework of a gravitational N-body problem. The effects of stellar mass loss in the form of stellar wind, dynamical friction against the interstellar medium, and star mergers on the dynamical evolution of the groups were investigated. A comparison with a purely gravitational N-body problem was made. The state distributions at the time of 300 initial system crossing times were analyzed. The parameters of the forming binary and stable triple systems as well as the escaping single and binary stars were studied. The star-merger and dynamical-friction effects are more pronounced in close systems, while the stellar wind effects are more pronounced in wide systems. Star-mergers and stellar wind slow down the dynamical evolution. These factors cause the mean and median semimajor axes of the final binaries as well as the semimajor axes of the internal and external binaries in stable triple systems to increase. Star mergers and dynamical friction in close systems decrease the fraction of binary systems with highly eccentric orbits and the mean component mass ratios for the final binaries and the internal and external binaries in stable triple systems. Star mergers and dynamical friction in close systems increase the fraction of stable triple systems with prograde motions. Dynamical friction in close systems can both increase and decrease the mean velocities of the escaping single stars, depending on the density of the interstellar medium and the mean velocity of the stars in the system.     

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)     

Three‐body problem periodic orbits with vanishing angular momentum

Periodic solutions of the general three‐body problem are investigated in the shape space. Two different solutions are considered: the first is an extension of the well‐known figure‐eight orbit, and the second one is from the free‐fall problem. Using the shape space, we reduce the dimension of the problem. These orbits are obtained numerically and described on the Euclidean plane and on the shape sphere. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Classification of orbits in the plane isosceles three-body problem

The general plane isosceles three-body problem is considered for different ratios of the central body mass to the masses of other bodies. The central body goes through the middle of the segment connecting the other bodies along the perpendicular to this segment. The initial conditions are chosen by two parameters: the virial ratio k and the parameter     , where r˙ is the relative velocity of the 'outer' bodies, and R˙ is the velocity of the 'central' body with respect to the mass centre of the 'outer' bodies. The equations of motion are numerically integrated until one of three times: the time of escape of the central body, its time of ejection with   R >100 d   , or 1000 τ (here d is the mean size, and τ is the mean crossing time of the triple system). The regions corresponding to escapes of the central body after different numbers of triple approaches are found at the plane of parameters   k ∈(0,1)  and   μ ∈(-1,1)  . The regions of stable motions are revealed. The zones of regular and stochastic orbits are outlined. The fraction of stochastic trajectories increases with the central mass. The fraction of stable orbits is highest for equal masses of the bodies.     

Two-integral distribution functions for axisymmetric systems

Some formulae are presented for finding two-integral 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. They come from a combination of the ideas of Eddington and Fricke and they are also an extension of those shown by Jiang and Ossipkov for finding anisotropic DFs for spherical galaxies. The density of the system is required to be expressed as a sum of products of functions of the potential and of the radial coordinate. The solution corresponding to this type of density is in turn a sum of products of functions of the energy and of the magnitude of the angular momentum about the axis of symmetry. The product of the density and its radial velocity dispersion can be also expressed as a sum of products of functions of the potential and of the radial coordinate. It can be further known that the density multiplied by its rotational velocity dispersion is equal to a sum of products of functions of the potential and of the radial coordinate minus the product of the density and the square of its mean rotational velocity. These formulae can be applied to the Binney and the Lynden-Bell models. An infinity of the odd DFs for the Binney model can be also found under the assumption of the laws of the rotational velocity.     

Unified analytical solutions to two-body problems with drag

The two-body problem with a generalized Stokes drag is discussed. The drag force is proportional to the product of the velocity vector and the inverse square of the distance. The generalization consists of allowing two different proportionality constants for the radial and the transverse components of the force. Under the 'generalized Robertson transformation', the equation of the orbit takes the form of the Lommel equation and admits solutions in terms of Bessel and Lommel functions. The exact, analytical solutions for this type of drag reveal a paradoxical effect of increasing eccentricity for all trajectories. The Poynting–Robertson drag and Poynting–Plummer–Danby problems are discussed as particular cases of the general solution.     

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.     

Regular and chaotic motion in softened gravitational systems

The stability of the dynamical trajectories of softened spherical gravitational systems is examined, both in the case of the full N -body problem and that of trajectories moving in the gravitational field of non-interacting background particles. In the latter case, for   N 10 000  , some trajectories, even if unstable, had exceedingly long diffusion times, which correlated with the characteristic e-folding time-scale of the instability. For trajectories of   N ≈100 000  systems this time-scale could be arbitrarily large – and thus appear to correspond to regular orbits. For centrally concentrated systems, low angular momentum trajectories were found to be systematically more unstable. This phenomenon is analogous to the well-known case of trajectories in generic centrally concentrated non-spherical smooth systems, where eccentric trajectories are found to be chaotic. The exponentiation times also correlate with the conservation of the angular momenta along the trajectories. For N up to a few hundred, the instability time-scales of N -body systems and their variation with particle number are similar to those of the most chaotic trajectories in inhomogeneous non-interacting systems. For larger N (up to a few thousand) the values of the these time-scales were found to saturate, increasing significantly more slowly with N . We attribute this to collective effects in the fully self-gravitating problem, which are apparent in the time variations of the time-dependent Liapunov exponents. The results presented here go some way towards resolving the long-standing apparent paradoxes concerning the local instability of trajectories. This now appears to be a manifestation of mechanisms driving evolution in gravitational systems and their interactions – and may thus be a useful diagnostic of such processes.     

On the relationship between instability and Lyapunov times for the three-body problem

In this study we consider the relationship between the survival time and the Lyapunov time for three-body systems. It is shown that the Sitnikov problem exhibits a two-part power-law relationship as demonstrated by Mikkola & Tanikawa for the general three-body problem. Using an approximate Poincaré map on an appropriate surface of section, we delineate escape regions in a domain of initial conditions and use these regions to analytically obtain a new functional relationship between the Lyapunov time and the survival time for the three-body problem. The marginal probability distributions of the Lyapunov and survival times are discussed and we show that the probability density function of Lyapunov times for the Sitnikov problem is similar to that for the general three-body problem.     

Stability of fictitious Trojan planets in extrasolar systems

Our work deals with the dynamical possibility that in extrasolar planetary systems a terrestrial planet may have stable orbits in a 1:1 mean motion resonance with a Jovian like planet. We studied the motion of fictitious Trojans around the Lagrangian points L₄/L₅ and checked the stability and/or chaoticity of their motion with the aid of the Lyapunov Indicators and the maximum eccentricity. The computations were carried out using the dynamical model of the elliptic restricted three‐body problem that consists of a central star, a gas giant moving in the habitable zone, and a massless terrestrial planet. We found 3 new systems where the gas giant lies in the habitable zone, namely HD99109, HD101930, and HD33564. Additionally we investigated all known extrasolar planetary systems where the giant planet lies partly or fully in the habitable zone. The results show that the orbits around the Lagrangian points L₄/L₅ of all investigated systems are stable for long times (10⁷ revolutions). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Chaos in Hill's generalized problem: from the solar system to black holes

We generalize the well‐known Hill's circular restricted three‐body problem by assuming that the primary generates a Schwarzschild‐type field of the form U = A/r + B/r³. The term in B influences the particle, but not the far secondary. Many concrete astronomical situations can be modelled via this problem. For the two‐body problem primary‐particle, a homoclinic orbit is proved to exist for a continuous range of parameters (the constants of energy and angular momentum, and the field parameter B > 0). Within the restricted three‐body system, we prove that, under sufficiently small perturbations from the secondary, the homoclinic orbit persists, but its stable and unstable manifolds intersect transversely. Using a result of symbolic dynamics, this means the existence of a Smale horseshoe, hence chaotic behaviour. Moreover, we find that Hill's generalized problem (in our sense) is nonintegrable. (© 2005 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.     

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.     

On the transfer of radiation at asteroidal surfaces in relation to their orbit deflection — II

The efficiency of absorption of X-rays generated by a nuclear explosion at the surface of an asteroid, estimated earlier, is used to calculate the explosion yield needed to deflect the orbit of an asteroid. Following the work of Ahrens &38; Harris, it is shown that a recoil velocity of 1 cm s⁻¹ is required to deflect an asteroid from a collision course with the Earth, and the necessary yield of explosion energy is estimated. If it is assumed that the scaling law between the energy and the diameter of the resulting crater, obtained from experiments carried out on the Earth, remains valid on the asteroid surface, where gravity is much weaker, an explosion energy of 8 and 800 megaton (Mton) equivalent of TNT would be required for asteroids of diameter 1 and 10 km respectively. If, on the other hand, the crater diameter is proportional to a certain power of the gravity g , the power being determined from a dimension analysis, 130 kton and 12 Mton would be required to endow asteroids of diameters 1 and 10 km with the required velocity, respectively. The result indicates that in order to estimate the required explosion energy, a better understanding of cratering under gravity much weaker than on the Earth would be required.     

Exact density–potential pairs from the holomorphic Coulomb field

We show how the complex-shift method developed by Appell to study the gravitational field of a point mass (and used in electrodynamics by, among others, Newman, Carter, Lynden-Bell, and Kaiser to determine some remarkable properties of the electromagnetic field of rotating charged configurations) can be extended to obtain new and explicit density–potential pairs for self-gravitating systems departing significantly from spherical symmetry. The rotational properties of two axisymmetric baroclinic gaseous configurations derived with the proposed method are illustrated.