Transversal ejection-collision orbits in Hill's problem forC≫1

In a recent paper [3], Lacomba and Llibre showed numerically the existence of two transversal ejection-collision orbits in Hill's problem for a valueC=5 of the Jacobian constant. This result can be used to prove the non-existence ofC ¹-extendable regular integrals for Hill's problem. Here we give an analytic proof of the existence of four ejection-collision orbits which are transversal for large enough values ofC.     

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.     

Effect of 3rd-degree gravity harmonics and Earth perturbations on lunar artificial satellite orbits

In a previous work we studied the effects of (I) the J ₂ and C ₂₂ terms of the lunar potential and (II) the rotation of the primary on the critical inclination orbits of artificial satellites. Here, we show that, when 3rd-degree gravity harmonics are taken into account, the long-term orbital behavior and stability are strongly affected, especially for a non-rotating central body, where chaotic or collision orbits dominate the phase space. In the rotating case these phenomena are strongly weakened and the motion is mostly regular. When the averaged effect of the Earth’s perturbation is added, chaotic regions appear again for some inclination ranges. These are more important for higher values of semi-major axes. We compute the main families of periodic orbits, which are shown to emanate from the ‘frozen eccentricity’ and ‘critical inclination’ solutions of the axisymmetric problem (‘J ₂ + J ₃’). Although the geometrical properties of the orbits are not preserved, we find that the variations in e, I and g can be quite small, so that they can be of practical importance to mission planning.     

Analytical and numerical study of the ground-track resonances of Dawn orbiting Vesta

The aim of Dawn mission is the acquisition of data from orbits around two bodies (4) Vesta and (1) Ceres, the two most massive asteroids.Due to the low thrust propulsion, Dawn will slowly cross and transit through ground-track resonances, where the perturbations on Dawn orbit may be significant. In this context, to safety go the Dawn mission from the approach orbit to the lowest science orbit, it is essential to know the properties of the crossed resonances.This paper analytically investigates the properties of the major ground-track resonances (1:1, 1:2, 2:3 and 3:2) appearing for Vesta orbiters: location of the equilibria, aperture of the resonances and period at the stable equilibria. We develop a general method using an averaged Hamiltonian formulation with a spherical harmonic approximation of the gravity field. If the values of the gravity field coefficient change, our method stays correct and applicable. We also discuss the effect of one uncertainty on the C₂₀ and C₂₂ coefficients on the properties of the 1:1 resonance. These results are checked by numerical tests. We determine that the increase of the eccentricity appearing in the 2:3 resonance is due to the C₂₂ and S₂₂ coefficients.Our method can be easily adapted to missions similar to Dawn because, contrarily to the numerical results, the analytical formalism stays the same and is valid for a wide range of physical parameters of the asteroid (namely the shape and the mass) as well as for different spacecraft orbits.Finally we numerically study the probability of the capture in resonance 1:1. Our paper reproduces, explains and supplements the results of Tricarico and Sykes (2010).     

Oblateness effect of Saturn on periodic orbits in the Saturn-Titan restricted three-body problem

We explore the effect of oblateness of Saturn (more massive primary) on the periodic orbits and the regions of quasi-periodic motion around both the primaries in the Saturn-Titan system in the framework of planar circular restricted three-body problem. First order interior and exterior mean motion resonances are located. The effect of oblateness is studied on the location, nature and size of periodic and quasi-periodic orbits, using the numerical technique of Poincare surface of sections. Some of the periodic orbits change to quasi-periodic orbits due to the effect of oblateness and vice-versa. The stability of the orbits around Saturn, Titan and both varies with the inclusion of oblateness. The centers of the periodic orbits around Titan move towards Saturn, whereas those around Saturn move towards Titan. For the orbit around Titan at C=2.9992, x=0.959494, the apocenter becomes pericenter. By incorporating oblateness effect, the orbit around Titan at C=2.99345, x=0.924938 is captured by Saturn, remains in various trajectories around Saturn, and as time progresses it spirals away around both the primaries.     

Newtonian analogue of force and motion of a free particle in the gravitational field of theC-metric

In the first part of the paper the Newtonian analogue of force for theC-metric has been investigated. To the first-order of approximation in the absence of acceleration of the particle generating theC-metric, one component of the force vector corresponds to the Newtonian analogue of force. In general there are relativistic correction terms due to acceleration term in theC-metric. In the second part of the paper the motion of a freely falling body has been investigated. It is found that plane orbits are not possible. Also the radial fall is not possible and in the equation of the orbit there are terms having no classical analogue. They can be interpreted as the effect of the dragging of the inertial frame produced by the rectilinear acceleration.     

Analysis of periodic orbits in the Saturn-Titan system using the method of Poincare section surfaces

We explore the periodic orbits and the regions of quasi-periodic motion around both the primaries in the Saturn-Titan system in the framework of planar circular restricted three-body problem. The location, nature and size of periodic and quasi-periodic orbits are studied using the numerical technique of Poincare surface of sections. The maximum amplitude of oscillations about the periodic orbits is determined and is used as a parameter to measure the degree of stability in the phase space for such orbits. It is found that the orbits around Saturn remain around it and their stability increases with the increase in the value of Jacobi constant C. The orbits around Titan move towards it with the increase in C. At C=3.1, the pericenter and apocenter are 358.2 and 358.5 km, respectively. No periodic or quasi-periodic orbits could be found by the present method around the collinear Lagrangian point L ₁ (0.9569373834…).     

On mini-halo encounters with stars

We study, analytically and numerically, the energy input into dark matter mini-haloes by interactions with stars. We find that the fractional energy input in simulations of Plummer spheres agrees well with the impulse approximation for small and large impact parameters, with a rapid transition between these two regimes. Using the impulse approximation, the fractional energy input at large impact parameters is fairly independent of the mass and density profiles of the mini-halo; however, low-mass mini-haloes experience a greater fractional energy input in close encounters. We formulate a fitting function which encodes these results and use it to estimate the disruption time-scales of mini-haloes, taking into account the stellar velocity dispersion and mass distribution. For mini-haloes with mass     on typical orbits which pass through the disc, we find that the estimated disruption time-scales are independent of mini-halo mass, and are of the order of the age of the Milky Way. For more massive mini-haloes, the estimated disruption time-scales increase rapidly with increasing mass.     

Surface fitting of three-dimensional closed solutions of the restricted three-body problem

The orbits of a family of three-dimensional periodic orbits in the restricted problem of three bodies form a surface. In this paper we determine the equation of this surface in the case of the orbits of double symmetry of the family which emanates from the equilibrium pointL ₁. This equation is obtained numerically by a least squares approximation method.     

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.     

Instabilities and bifurcations of the families of collision periodic orbits in the restricted three-body problem

By using Birkhoff's regularizing transformation, we study the evolution of some of the infinite j-k type families of collision periodic orbits with respect to the mass ratio μ as well as their stability and dynamical structure, in the planar restricted three-body problem. The μ-C characteristic curves of these families extend to the left of the μ-C diagram, to smaller values of μ and most of them go downwards, although some of them end by spiralling around the constant point S* (μ=0.47549, C=3) of the Bozis diagram (1970). Thus we know now the continuation of the families which go through collision periodic orbits of the Sun-Jupiter and Earth-Moon systems. We found new μ-C and x-C characteristic curves. Along each μ-C characteristic curve changes of stability to instability and vice versa and successive very small stable and very large unstable segments appear. Thus we found different types of bifurcations of families of collision periodic orbits. We found cases of infinite period doubling Feigenbaum bifurcations as well as bifurcations of new families of symmetric and non-symmetric collision periodic orbits of the same period. In general, all the families of collision periodic orbits are strongly unstable. Also, we found new x-C characteristic curves of j-type classes of symmetric periodic orbits generated from collision periodic orbits, for some given values of μ. As C varies along the μ-C or the x-C spiral characteristics, which approach their focal-terminating-point, infinite loops, one inside the other, surrounding the triangular points L₄ and L₅ are formed in their orbits. So, each terminating point corresponds to a collision asymptotic symmetric periodic orbit for the case of the μ-C curve or a non-collision asymptotic symmetric periodic orbit for the case of the x-C curve, that spiral into the points L₄ and L₅, with infinite period. All these are changes in the topology of the phase space and so in the dynamical properties of the restricted three-body problem.     

A mapping for the study of the 1/1 resonance in a galactic type Hamiltonian

We derive an algebraic mapping for an autonomous, two-dimensional galactic type Hamiltonian in the 1/1 resonance case. We use the mapping to study the stability of the periodic orbits. Using the x — p _x Poincaré surface section, we compare the results of the mapping with those found by the numerical integration of the full equations of motion. For small values of the perturbation the results of the two methods are in very good agreement while satisfactory agreement is obtained for larger perturbations.     

Periodic orbits in the photogravitational restricted problem with the smaller primary an oblate body

In this paper, we have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when more massive body is a source of radiation and the smaller primary is an oblate body. We have determined periodic orbits for fixed values of μ, σ and different values of p and h (μ mass ratio of the two primaries, σ oblate parameter, p radiation parameter and h energy constant). These orbits have been determined by giving displacements along the tangent and normal to the mobile co-ordinates as defined by Karimov and Sokolsky (in Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of radiation pressure on the periodic orbits by taking some fixed values of μ and σ.     

The existence of families of periodic orbits in theN-body problem

It is proved that monoparametric families of periodic orbits of theN-body problem in the plane, for fixed values of all masses, exist in a rotating frame of reference whosex axis contains always two of the bodiesP ₁ andP ₂. A periodic motion of theN-body problem is obtained as a continuation ofN–2 symmetric periodic orbits of the circular restricted three-body problem whose periods are in integer dependence, by increasing the masses of the originallyN–2 massless bodiesP ₃, ...,P _k. The analytic continuation, for sufficiently small values of theN–2 bodiesP ₃ ...P _k and finite values for the masses ofP ₁ andP ₂ has been proved by the continuation method and the solution itself has been found explicitly to a linear approximation in the small masses. Also, the possible application of the above periodic orbits to the study of the Solar system and of stellar systems is mentioned.     

The rectilinear three-body problem using symbol sequence II: role of the periodic orbits

We study the change of phase space structure of the rectilinear three-body problem when the mass combination is changed. Generally, periodic orbits bifurcate from the stable Schubart periodic orbit and move radially outward. Among these periodic orbits there are dominant periodic orbits having rotation number (n − 2)/n with n ≥ 3. We find that the number of dominant periodic orbits is two when n is odd and four when n is even. Dominant periodic orbits have large stable regions in and out of the stability region of the Schubart orbit (Schubart region), and so they determine the size of the Schubart region and influence the structure of the Poincaré section out of the Schubart region. Indeed, with the movement of the dominant periodic orbits, part of complicated structure of the Poincaré section follows these orbits. We find stable periodic orbits which do not bifurcate from the Schubart orbit.     

Asymptotic Orbits at the Triangular Equilibria in the Photogravitational Restricted Three-Body Problem

We study numerically the asymptotic homoclinic and heteroclinic orbits associated with the triangular equilibrium points L ₄ and L ₅, in the gravitational and the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these critical points, are also presented. Hundreds of asymptotic orbits for equal mass of the primaries and for various values of the radiation pressure are computed and the most interesting of them are illustrated. In the Copenhagen case, which the problem is symmetric with respect to the x- and y-axis, we found and present non-symmetric heteroclinic asymptotic orbits. So pairs of heteroclinic connections (from L ₄ to L ₅ and vice versa) form non-symmetric heteroclinic cycles. The termination orbits (a combination of two asymptotic orbits) of all the simple families of symmetric periodic orbits, in the Copenhagen case, are illustrated.     

Periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries is an oblate body

We have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries is an oblate body. We have determined the periodic orbits for different values of μ, h and A (h is energy constant, μ is mass ratio of the two primaries and A is an oblateness factor). These orbits have been determined by giving displacements along the tangent and normal to the mobile coordinates as defined by Karimov and Sokolsky (Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of oblateness by taking some fixed values of μ, A and h. As starters for our method, we use some known periodic orbits in the classical restricted three body problem.     

Stability analysis of the martian obliquity during the Noachian era

We performed numerical simulations of the obliquity evolution of Mars during the Noachian era, at which time the giant planets were on drastically different orbits than today. For the preferred primordial configuration of the planets we find that there are two large zones where the martian obliquity is stable and oscillates with an amplitude lower than 20°. These zones occur at obliquities below 30° and above 60°; intermediate values show either resonant or chaotic behaviour depending on the primordial orbits of the terrestrial planets.     

Determination of the orbits of near-Earth asteroids from observations at the first opposition

Observations at the first opposition are used to determine the orbits of 16 near-Earth asteroids with two or more observed oppositions. The orbits are improved by the differential method. This paper considers two modifications of the improvement technique, which are compared to the classical method based on the principle of the least square method (LSM). The first modification uses the principle of least absolute deviations (LAD). In the second modification, the differences O - C (between the observed and calculated positions) are transformed to fit into a new coordinate system whereby the axes go parallel and perpendicular to the asteroid’s apparent path (the modified differential method (MDM)). The orbits determined from one opposition by the classical LSM, LAD, and MDM are compared to a more accurate orbit calculated by the LSM from all the available oppositions. The calculations show that in 13 cases out of 16, the asteroid orbits calculated by LAD are more accurate than those calculated by the classical LSM. The improvement by the modified differential method, which includes the O - C transformation, does not produce any perceptible increase in accuracy when compared to the orbits calculated by the classical method.     

Families of Periodic Orbits Emanating From Homoclinic Orbits in the Restricted Problem of Three Bodies

We describe and comment the results of a numerical exploration on the evolution of the families of periodic orbits associated with homoclinic orbits emanating from the equilateral equilibria of the restricted three body problem for values of the mass ratio larger than μ ₁. This exploration is, in some sense, a continuation of the work reported in Henrard [Celes. Mech. Dyn. Astr. 2002, 83, 291]. Indeed it shows how, for values of μ. larger than μ ₁, the Trojan web described there is transformed into families of periodic orbits associated with homoclinic orbits. Also we describe how families of periodic orbits associated with homoclinic orbits can attach (or detach) themselves to (or from) the best known families of symmetric periodic orbits. This revised version was published online in July 2006 with corrections to the Cover Date.