On some long time dynamical features of the Trojan asteroids of Jupiter

The equation of motion of long periodic libration around the Lagrangian point $L_4$ L 4 in the restricted three-body problem is investigated. The range of validity of an approximate analytical solution in the tadpole region is determined by numerical integration. The predictions of the model of libration are tested on the Trojan asteroids of Jupiter. The long time evolution of the orbital eccentricity and the longitude of the perihelion of the Trojan asteroids, under the effect of the four giant planets, is also investigated and a slight dynamical asymmetry is shown between the two groups of Trojans at $L_4$ L 4 and $L_5$ L 5 .     

A simple analytical model for the secular resonance ν6 in the asteroidal belt

When asteroids are in the secular resonance ₆, the variation of the eccentricity becomes very large. In this paper, the dynamics of this secular resonance ₆ is investigated by a simple analytical model, in which the third degree terms of the eccentricity and inclination are taken into account. The eccentricity variations of asteroids located near this resonance are represented clearly by the diagrams of equi-Hamiltonian curves on the plane of versuse ( the longitude of perihelion of asteroids and Saturn,e: the eccentricity of asteroids). These diagrams predict that the eccentricity of these asteroids suffers a large increase or decrease, and that the secular resonance argument librates about 0° and 180°. In order to confirm these predictions, numerical integrations are carried out over one million years. By these integrations, it is found that the eccentricity of secular resonant asteroids becomes more than 0.8, and that the libration about 0° also exists, as well as the libration about 180°. The strongly depopulated region in the asteroidal belt, which corresponds to the position of the secular resonance ₆, is also explained well by this analytical model.     

Resonant asteroids between the main belt and Jupiter's orbit

Selected orbits of asteroids close to the 3/2 and 4/3 resonances in the outer belt are studied by extended numerical integrations of the four-body problem Sun-Jupiter-Saturn-asteroid. Jupiter's variable eccentricity causes strong or dominant effects in the asteroidal eccentricity, rate of perihelion, and critical argument. However, a suitable transformation removes most of these effects by the transition to respective new quantities. Cases of circulation can change to permanent libration in the new critical argument. This happens in the case of (1256) Normannia, one of the two objects found to be circulating near the 3/2 resonance in former work. The other object, (334) Chicago, remains a case of circulation and shows significant deviations from quasi-periodic motion, contrary to all the other studied Hildas (3/2 cases) and to (279) Thule (4/3). Temporary libration with respect to a resonance of high order is visible in the long-period evolution of Chicago's orbit. There are cases of some analogy further inward in the belt: (903) Nealley changes to permanent libration at the 2/1 resonance by the use of the new critical argument, and (522) Helga's orbit shows a non-quasi-periodic behaviour.     

Expansion of the disturbing function for high eccentricity and large amplitude of libration

Following some ideas, developed by Woltjer (1928), Message (1989), Yokoyama (1988, 1989) and Duriez (1990) an expansion of the disturbing function is given for high values of the eccentricity and large amplitude of libration. The classical expansion can be obtained as a particular case of the present model. Several asteroids with high eccentricity and large amplitude of libration are tested and the results are much better than those obtained from the classical theory.     

Resonant structure of the outer asteroid belt

An analysis of ordered and chaotic regions of motion in the outer asteroid belt has shown that once the eccentricity of Jupiter is introduced the chaotic regions of the circular model are quite easily depleted. This suggests that also objects in neighbouring regions must be strongly perturbed. Therefore it is not surprising that many outer belt asteroids have been reported in the literature as resonant or anyway dynamically protected. By using the planar elliptic restricted 3-body model we have investigated the motion of outer belt asteroids which had not been suspected to librate. We find 3 cases of libration and 11 cases of e, coupling that can be explained within the theory of secular resonances. It is thus established that in the outer belt only resonant and dynamically protected asteroids can have lifetimes of the same order as the age of the Solar System.     

Dynamical study of the Hilda asteroids

Schubart's model of a planar, elliptic restricted three-body problem is used to study the orbital motion of the Hilda asteroids from thePalomar-Leiden Survey. The 3:2 resonant coupling to Jupiter of some of these small asteroids are found to be stable. However, some of the small asteroids with absolute magnitudeg>15 have large amplitude of variation in their orbital elements in one libration period. Since the lifetime scales against catastrophic collision of the Hilda asteroids are estimated to be several times larger than those of the main belt objects, a significant portion of these resonant asteroids could be the original members of the Hilda group. From this point of view, it is suggested that such size-dependence of resonant orbital motions might be the result of the cosmogonic effects ofjet stream accretion.     

Trojan orbits with mass exchange of the primaries

The librational motion round the Lagrangian triangular points L₄, L₅ with mass exchange of the primaries is investigated according to Brown's theory. The results are the same as in the case of isotropic mass variation studied earlier (Horedt, 1974a): (i) The extrema of the elongations with respect to the small mass are unaffected by mass exchange. (ii) The equations for the extrema of the Trojan's distance from the Sun and for the libration period are formally the same as in the constant mass problem, but with the understanding that the masses are now time dependent quantities. A Trojan cannot leave the libration domain due to a mass variation of the primaries obeying the constraints from Equation (2.4), with a mass ratio of the primaries m/M≤0.0401.     

Numerical investigation of motions of resonant asteroids in the three-dimensional space

Motions of asteroids in mean motion resonances with Jupiter are studied in three-dimensional space. Orbital changes of fictitious asteroids in the Kirkwood gaps are calculated by numerical integrations for 10⁵ – 10⁶ years. The main results are as follows: (1) There are various motions of resonant asteroids, and some of them are very complicated and chaotic and others are regular. (2) The eccentricity of some asteroids becomes very large, and the variation of the inclination is large while the eccentricity is large. (3) In the 3:1 resonance, there is a long periodic change in the variation of the inclination, when (7 : ) is a simple ratio (7: longitude of perihelion, : longitude of node). (4) In the 7:3 resonance, the variation of the inclination of some resonant asteroids is so large that prograde motion becomes retrograde. Some asteroids in the 7:3 resonance can collide with the Sun as well as with the inner planets.     

The Mars 1:2 resonant population

An excess of around 400 asteroids in the distribution of the semimajor axes of the asteroids is identified by means of numerical integrations as generated by a population of approximately 1000 asteroids evolving inside the exterior resonance 1:2 with Mars. Approximately 200 asteroids are librating around the asymmetric libration centers and their evolution in a time-scale of 1 million years appears stable but with a strong influence of Mars' eccentricity. The biggest Mars 1:2 resonant asteroid is (142) Polana.     

Capture of dust grains in exterior resonances with planets

The temporary capture of the dust grains in the exterior resonances with planets is studied in the frames of the planar circular three-body problem with Poynting-Robertson (PR) drag. For the Earth and particles ~ 10 m the resonances 4/5, 5/6, 6/7, 7/8 are shown to be most effective. The capture is only temporary (of order 10⁵ years) and the position of resonance may be calculated from semi-analytical model using averaged disturbing function. These semi-analytical results are confirmed by numerical integration. For various planet this picture changes as with increasing planetary mass the more exterior resonances become more important. We showed that for Jupiter (at least in the space between Jupiter and Saturn) the resonance 1/2 plays the dominant role. The capture time is here several myr but again eccentricity is evolving to eccentricity e ₀ ~ 0.48 of libration point for this resonance.     

Libration effects for retrograde satellites in the restricted three-body problem

Numerical explorations of the restricted problem have shown that for stable large nonperiodic retrograde satellite orbits, the motion can be decomposed into a fast reference motion and a slow libration aroundB ₂ We study here this libration in the circular plane Hill's case, for which the reference motion is elliptic. We establish the equations of motion for the coordinates of the centre of this ellipse. We find two integrals of motion: the first is the semi-major axis of the ellipse; the second is essentially Jacobi's integral, translated into the new coordinates. We give a formula for the period of the libration and we find its limiting value for small libration amplitudes. A numerical verification gives very good agreement for all these results.     

Sun–Earth L_{1} and L_{2} to Moon transfers exploiting natural dynamics

This paper examines the design of transfers from the Sun–Earth libration orbits, at the \(L_{1}\) and \(L_{2}\) points, towards the Moon using natural dynamics in order to assess the feasibility of future disposal or lifetime extension operations. With an eye to the probably small quantity of propellant left when its operational life has ended, the spacecraft leaves the libration point orbit on an unstable invariant manifold to bring itself closer to the Earth and Moon. The total trajectory is modeled in the coupled circular restricted three-body problem, and some preliminary study of the use of solar radiation pressure is also provided. The concept of survivability and event maps is introduced to obtain suitable conditions that can be targeted such that the spacecraft impacts, or is weakly captured by, the Moon. Weak capture at the Moon is studied by method of these maps. Some results for planar Lyapunov orbits at \(L_{1}\) and \(L_{2}\) are given, as well as some results for the operational orbit of SOHO.     

Asteroid motion near the 3 : 1 resonance

A mapping model is constructed to describe asteroid motion near the 3 : 1 mean motion resonance with Jupiter, in the plane. The topology of the phase space of this mapping coincides with that of the real system, which is considered to be the elliptic restricted three body problem with the Sun and Jupiter as primaries. This model is valid for all values of the eccentricity. This is achieved by the introduction of a correcting term to the averaged Hamiltonian which is valid for small values of the ecentricity.We start with a two dimensional mapping which represents the circular restricted three body problem. This provides the basic framework for the complete model, but cannot explain the generation of a gap in the distribution of the asteroids at this resonance. The next approximation is a four dimensional mapping, corresponding to the elliptic restricted problem. It is found that chaotic regions exist near the 3 : 1 resonance, due to the interaction between the two degrees of freedom, for initial conditions close to a critical curve of the circular model. As a consequence of the chaotic motion, the eccentricity of the asteroid jumps to high values and close encounters with Mars and even Earth may occur, thus generating a gap. It is found that the generation of chaos depends also on the phase (i.e. the angles andv) and as a consequence, there exist islands of ordered motion inside the sea of chaotic motion near the 3 : 1 resonance. Thus, the model of the elliptic restricted three body problem cannot explain completely the generation of a gap, although the density in the distribution of the asteroids will be much less than far from the resonance. Finally, we take into account the effect of the gravitational attraction of Saturn on Jupiter's orbit, and in particular the variation of the eccentricity and the argument of perihelion. This generates a mixing of the phases and as a consequence the whole phase space near the 3 : 1 resonance becomes chaotic. This chaotic zone is in good agreement with the observations.     

The problem of the Eulerian oscillations: A weakness of numerical versus analytical methods

The application of computer technology has permitted more and more problems of dynamical astronomy to be solved more easily, quickly, and accurately. In this area, numerical integration is often very efficient, and sometimes essential. There is often, however, a temptation to choose numerical integration simply because it is the easiest way to attack the problem. Sometimes this works to the detriment of a satisfactory understanding of the physics of the problem under study. It is particularly the case for the free, or Eulerian, oscillations. The forces that create such a motion are not of gravitational origin and are not even conservative. The theory can only specify the frequencies of oscillation, not their amplitudes nor phases. The case is complicated when the free oscillations interact with gravitationally-forced oscillations, a situation that is almost inevitable, since nothing is isolated in the Universe. The first author has particularly studied this problem in the case of the rotation of the Moon, and published the first credible determinations of the lunar free libration. In this kind of problem, the observations have to be used and care must to be taken to create no spurious free librations in the results by using numerical integrations to describe the other related motions. A differential correction of the starting conditions to fit the observations does not necessarily give any valid information on the real free oscillation contained in the data. An analytical model is necessary, if the goal of the research is to understand the origins and characteristics of an Eulerian oscillation in such a system.     

A Review on Co-orbital Motion in Restricted and Planetary Three-body Problems

The 1:1 mean motion resonance may be referred to as the lowest order mean motion resonance in restricted or planetary three-body problems. The five well-known libration points of the circular restricted three-body problem are five equilibriums of the 1:1 resonance. Coorbital motion may take different shapes of trajectory. In case of small orbital eccentricities and inclinations, tadpole-shape and horseshoe-shape orbits are well-known. Other 1:1 libration modes different from the elementary ones can exist at moderate or large eccentricities and inclinations. Coorbital objects are not rare in our solar system, for example the Trojans asteroids and the coorbital satellite systems of Saturn. Recently, dozens of coorbital bodies have been identified among the near-Earth asteroids. These coorbital asteroids are believed to transit recurrently between different 1:1 libration modes mainly due to orbital precessions, planetary perturbations, and other possible effects. The Hamiltonian system and the Hill’s three-body problem are two effective approaches to study coorbital motions. To apply the perturbation theory to the Hamiltonian system, standard procedures involve the development of the disturbing function, averaging and normalization, theory of ideal resonance model, secular perturbation theory, etc. Global dynamics of coorbital motion can be revealed by the Hamiltonian approach with a suitable expansion. The Hill’s problem is particularly suitable for the studies on the relative motion of two coorbital bodies during their close encounter. The Hill’s equation derived from the circular restricted three-body problem is well known. However, the general Hill’s problem whose equation of motion takes exactly the same form applies to the non-restricted case where the mass of each body is non-negligible, namely the planetary case. The Hill’s problem can be transformed into a “canonical shape” so that the averaging principle can be applied to construct a secular perturbation theory. Besides the two analytical theories, numerical methods may be consulted, for example the approach of periodic orbit, the surface of section, and the computation of invariant manifolds carried by equilibriums or periodic orbits.     

Non-linear stability of L 4 in the restricted problem when the primaries are finite straight segments under resonances

The non-linear stability of L ₄ in the restricted three-body problem when both primaries are finite straight segments in the presence of third and fourth order resonances has been investigated. Markeev’s theorem (Markeev in Libration Points in Celestial Mechanics and Astrodynamics, 1978) is used to examine the non-linear stability for the resonance cases 2:1 and 3:1. It is found that the non-linear stability of L ₄ depends on the lengths of the segments in both resonance cases. It is also found that the range of stability increases when compared with the classical restricted problem. The results have been applied in the following asteroids systems: (i) 216 Kleopatra–951 Gaspara, (ii) 9 Metis–433 Eros, (iii) 22 Kalliope–243 Ida.     

Development of planetary ephemerides EPM and their applications

This paper outlines the progress in development of the numerical planet ephemerides EPM—Ephemerides of Planets and the Moon. EPM was first created in the 1970s in support of Russian space flight missions and constantly improved at IAA RAS. Comparison between various available EPM ephemerides (EPM2004, EPM2008, EPM2011) is shown. The first results of the updated EPM2013 version which takes into account the two-dimensional annulus of small asteroids are presented. Currently two main factors drive the progress of planet ephemerides: dynamical models of planet motion and observational data, with the crucial role of spacecraft ranging. EPM ephemerides are the basis for the Russian Astronomical and Nautical Astronomical Yearbooks, are planned to use in the GLONASS and LUNA-RESOURCE programs, and are being used for determination of physical parameters: masses of asteroids, planet rotation parameters and topography, the \(GM_\odot \) and its secular variation, the PPN parameters, and the upper limit on the mass of dark matter in the Solar System. The files containing polynomial approximation for EPM ephemerides (EPM2004, EPM2008, EPM2011) along with TT–TDB and ephemerides of Ceres, Pallas, Vesta, Eris, Haumea, Makemake, and Sedna are available from ftp://quasar.ipa.nw.ru/incoming/EPM/. Files are provided in IAA’s binary and ASCII formats, as well as in the SPK format.     

The three-dimensional motion of Trojan asteroids

The problem is considered within the framework of the elliptic restricted three-body problem. The asymptotic solution is derived by a three-variable expansion procedure. The variables of the expansion represent three time-scales of the asteroids: the revolution around the Sun, the libration around the triangular Lagrangian pointsL ₄,L ₅, and the motion of the perihelion. The solution is obtained completely in the first order and partly in the second order. The results are given in explicit form for the coordinates as functions of the true anomaly of Jupiter. As an example for the perturbations of the orbital elements the main perturbations of the eccentricity, the perihelion longitude and the longitude of the ascending node are given. Conditions for the libration of the perihelion are also discussed.     

A Perturbative Treatment of The Co-Orbital Motion

We develop a formalism of the non-singular evaluation of the disturbing function and its derivatives with respect to the canonical variables. We apply this formalism to the case of the perturbed motion of a massless body orbiting the central body (Sun) with a period equal to that of the perturbing (planetary) body. This situation is known as the co-orbital motion, or equivalently, as the 1/1 mean motion commensurability. Jupiter's Trojan asteroids, Earth's co-orbital asteroids (e.g., (3753) Cruithne, (3362) Khufu), Mars' co-orbital asteroids (e.g., (5261) Eureka), and some Jupiter-family comets are examples of the co-orbital bodies in our solar system. Other examples are known in the satellite systems of the giant planets. Unlike the classical expansions of the disturbing function, our formalism is valid for any values of eccentricities and inclinations of the perturbed and perturbing body. The perturbation theory is used to compute the main features of the co-orbital dynamics in three approximations of the general three-body model: the planar-circular, planar-elliptic, and spatial-circular models. We develop a new perturbation scheme, which allows us to treat cases where the classical perturbation treatment fails. We show how the families of the tadpole, horseshoe, retrograde satellite and compound orbits vary with the eccentricity and inclination of the small body, and compute them also for the eccentricity of the perturbing body corresponding to a largely eccentric exoplanet's orbit.This revised version was published online in October 2005 with corrections to the Cover Date.     

On the Asteroidal Population of the First-Order Jovian Resonances

The frequency map analysis was applied to the fairly realistic models of the 2/1, 3/2, and 4/3 jovian resonances and the results were compared with the asteroidal distribution at these commensurabilities. The presence of the Hecuba gap at the 2/1 and of the Hilda group in the 3/2 is explained on the basis of different rates of the chaotic transport (diffusion) in these resonances. The diffusion in the most stable 2/1-resonant region is almost two orders in magnitude faster than the diffusion in the region which accommodates the Hildas. In the 2/1 commensurability there are two possible locations for long-surviving asteroids: the one centered at an eccentricity of 0.3 near the libration stable centers with small libration amplitude and the other at a slightly lower eccentricity with a moderate libration amplitude (∼90°). Surprisingly, all asteroids observed in the 2/1 resonance (8 numbered and multi-opposition objects in Bowell's catalog from 1994) occupy the moderate-libration area and avoid the area in a close vicinity of the libration stable centers. Possible explanations of this fact were discussed. Concerning the 4/3 resonance, the only asteroid in the corresponding stable region is 279 Thule, in spite of the fact that this region is almost as regular (although not as extensive) as the one where the Hilda group in the 3/2, with 79 members, is found.