Tesseral resonance effects on satellite orbits

Resonance effects on satellite orbits due to tesseral harmonics in the potential field have been studied by many authors. Most of these studies have been restricted to nearly circular 24-hour orbits and to the deep resonance regime, where there is exact commensurability between earth rotation and orbit period. Resonance effects have also been noted, however, on eccentric synchronous and subsynchronous orbits and on orbits with far from commensurate periods. These have received much less attention; the object of this paper is to study the whole spectrum of orbits with respect to resonance effects.     

An analytical treatment of resonance effects on satellite orbits

A first order analytical approximation of the tesseral harmonic resonance perturbations of the Keplerian elements is presented, and the mean elements (the Keplerian elements with the long period portions averaged out) will also be given in closed form. Finally the results of a numerical test, which compares the analytical solution against a numerical integration of the Lagrange equations of motion, will be summarized.This work was sponsored with the support of the Department of the Air Force under contract F19628-85-C-0002.The views expressed are those of the author and do not reflect the official policy or position of the U.S. Government.     

On periodic orbits and resonance in extrasolar planetary systems

We study the evolution of an extrasolar planetary system with two planets, for planar motion, starting from an exact resonant periodic motion and increasing the deviation from the equilibrium solution. We keep the semimajor axes and the eccentricities of the two planets fixed and we change the initial conditions by rotating the orbit of the outer planet by Δω. In this way the resonance is preserved, but we deviate from the exact periodicity and there is a transition from order to chaos as the deviation increases. There are three different routes to chaos, as far as the evolution of (ω ₂ ? ω ₁) is concerned: (a) Libration → rotation → chaos, with intermittent transition from libration to rotation in between, (b) libration → chaos and (c) libration → intermittent interchange between libration and rotation → chaos. This indicates that resonant planetary systems where the angle (ω ₂ ? ω ₁) librates or rotates are not different, but are closely connected to the exact periodic motion.     

Libration celestial mechanics experiment

The exploration of planet moons and minor bodies (Avduevskii et al., 1996) is a basic task for comprehending the nature of the processes occurring in our Solar System. Knowing the current state of the moons, we can better describe their past and look into the future. This knowledge is important, first of all, for understanding the origin of the Solar System. Interest in the Martian moon Phobos has been displayed during recent decades. The interest is caused by some questions to which there have been no answers up until now (Sagdeev et al., 1988; 1989). For example, there is a question regarding the origin of the moon: whether it is an asteroid captured by Mars’ gravitational field or it is an accumulated body in the Martian orbit. In connection with this, it is interesting to conduct studies aimed at answering this question. If Phobos appears to be an asteroid, then investigations regarding the chemical and isotopic compositions of the moon as the primary matter of the Solar System as well as its evolution are of great interest.     

Symmetric doubly-asymptotic periodic orbits at collinear equilibria

A systematic search for periodic orbits doubly-asymptotic to the collinear equilibrium points of the restricted three-body problem is carried out and many such orbits are found, each of them existing for a specific value of the mass parameter. These may be useful as reference orbits and seem to be special limit orbits representing period discontinuities in the evolution of the families of periodic orbits.     

On the periodic solutions and resonance of spinning satellites in near-circular orbits

Attitude motion of spinning axisymmetric satellites in presence of gravity-gradient and solar radiation pressure torques is studied analytically. The approximate closed-form solution developed for the nonlinear, nonautonomous, coupled fourth-order system proves to be an excellent tool in locating periodic solutions of the system in both circular and noncircular orbits. The variational stability of the periodic motion is examined using the Floquet theory. The resonance analysis suggests the existence of critical combinations of system parameters leading to large amplitude oscillations.

     

Analysis of 208 U.S. Navy orbits for the satellite 1970-47B at 14th-order resonance

The orbit of 1970-47B passed very slowly through 14th-order resonance, and the changes in orbital inclination and eccentricity have been analysed over a 4-year period, from January 1977 to January 1981, using 208 U.S. Navy orbits. The analysis has yielded values for three pairs of lumped harmonic coefficients of 14th order, which have accuracies equivalent to 0.4, 1.5 and 2.0 cm in geoid height. Three pairs of values of 28th-order lumped harmonic coefficients were also obtained, and the best of these has a standard deviation (S.D.) corresponding to an accuracy of 0.7 cm in geoid height. The lumped harmonic coefficients have been compared with the corresponding values from the latest geopotential models, and agreement is satisfactory.     

Luni-solar effects of geosynchronous orbits at the critical inclination

The luni-solar effects of a geosynchronous artificial satellite orbiting near the critical inclination is investigated. To tackle this four-degrees-of-freedom problem, a preliminary exploration separately analyzing each harmonic formed by a combination of the satellite longitude of the node and the Moon longitude of the node is opportune. This study demonstrates that the dynamics induced by these harmonics does not show resonance phenomena. In a second approach, the number of degrees of freedom is halved by averaging the total Hamiltonian over the two non-resonant angular variables. A semi-numerical method can now be applied as was done when considering solely the inhomogeneity of the geopotential (see Delhaise et Henrard, 1992). Approximate surfaces of section are constructed in the plane of the inclination and argument of perigee. The main effects of the Sun and Moon attractions compared to the terrestrial attraction alone are a strong increase in the amplitude of libration in inclination (from 0.6° to 3.2°) and a decrease of the corresponding libration period (from the order of 200 years to the order of 20 years).Research Assistant for the Belgian National Fund for Scientific Research     

Complements to Moons'1 Lunar Libration Theory

Analytical complements have been brought to Moons' lunar libration theory concerning tidal effects, direct perturbations due to the Earth's figure, and indirect non periodic perturbations. Comparisons to JPL numerical integrations DE245 and DE403 have been performed and the residuals treated by frequency analysis, allowing the determination of fitted free libration parameters and numerical complements. This revised version was published online in July 2006 with corrections to the Cover Date.     

Planetary perturbations on the Libration of the Moon

A theory of the libration of the Moon, completely analytical with respect to the harmonic coefficients of the lunar gravity field, was recently built (Moons, 1982). The Lie transforms method was used to reduce the Hamiltonian of the main problem of the libration of the Moon and to produce the usual libration series p₁, p₂ and . This main problem takes into account the perturbations due to the Sun and the Earth on the rotation of a rigid Moon about its center of mass. In complement to this theory, we have now computed the planetary effects on the libration, the planetary terms being added to the mean Hamiltonian of the main problem before a last elimination of the angles. For the main problem, as well as for the planetary perturbations, the motion of the center of mass of the Moon is described by the ELP 2000 solution (Chapront and Chapront-Touze, 1983).     

Additional effects of resonance in Hilda-asteroid orbits by the combined action of Jupiter and Saturn

Inside the 3/2 mean motion resonance some Hilda-type orbits show effects of a three-body resonance that includes the frequency of the libration due to the 3/2 resonance. A graphical method presents numerical results for such orbits and demonstrates in 6 cases a process of temporary libration, that is ruled by the additional resonance together with the secular period of the eccentricities of Jupiter and Saturn.     

Non-symmetric doubly-asymptotic orbits at the outer collinear equilibrium pointL 3

In the framework of the solar system case (with only the larger primary radiating) of the photogravitational restricted three-body problem we compute and present some non-symmetric asymptotic orbits connecting the outer collinear equilibrium pointL ₃ with the neighbourhood of one of the triangular equilibrium pointsL _{4, 5}. Such orbits have not been found previously in the restricted problem.     

Periodic orbits

Recent results on periodic orbits are presented. Planetary systems can be studied by the model of the general 3-body problem and also some satellite systems and asteroid orbits can be studied by the model of the restricted 3-body problem. Triple stellar systems and planetary systems with two Suns are close to periodic systems. Finally, the motion of stars in various types of galaxies can be studied by finding families of periodic orbits in several galactic models.     

Libration of Laplace's argument in the Galilean satellites theory

In the Galilean satellites motion, the Laplace argument ₁3₂+2₃ librates around the value . The amplitude of libration is very small so that the classical theories have not been set up to take into account large librations. On the other hand large librations have to be considered when we describe possible scenarii of capture into resonance by tidal effects. The aim of this paper is to present a new way of applying Hamiltonian perturbation methods to the problem of the Galilean satellites in such a way that the theory is valid for large librations. Preliminary results from such a theory are discussed.     

月球物理天平动对环月轨道器运动的影响

月球物理天平动是月球赤道在空间真实的摆动，会导致月球引力场在空间坐标系中的变化，从而引起环月轨道器(以下称为月球卫星)的轨道变化，这与地球的岁差章动现象对地球卫星轨道的影响类似．采用类似对地球岁差章动的处理方法，讨论月球物理天平动对月球卫星轨道的影响，给出相应的引力位的变化及卫星轨道的摄动解，清楚地表明了月球卫星轨道的变化规律，并和数值解进行了比对，从定性和定量方面作一讨论．     

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.     

Frozen orbits at high eccentricity and inclination: application to Mercury orbiter

We hereby study the stability of a massless probe orbiting around an oblate central body (planet or planetary satellite) perturbed by a third body, assumed to lay in the equatorial plane (Sun or Jupiter for example) using a Hamiltonian formalism. We are able to determine, in the parameters space, the location of the frozen orbits, namely orbits whose orbital elements remain constant on average, to characterize their stability/unstability and to compute the periods of the equilibria. The proposed theory is general enough, to be applied to a wide range of probes around planet or natural planetary satellites. The BepiColombo mission is used to motivate our analysis and to provide specific numerical data to check our analytical results. Finally, we also bring to the light that the coefficient J ₂ is able to protect against the increasing of the eccentricity due to the Kozai-Lidov effect and the coefficient J ₃ determines a shift of the equilibria.     

Resonance in regular variables I: Morphogenetic analysis of the orbits in the case of a first-order resonance

Morphogenetic analysis of the orbits of the ideal first-order resonance problem in the neighbourhood of the origin. It is shown that for problems involving central and near-central resonance it is necessary to consider as parameter the cube root of the perturbation instead of the square root used in classical non-central resonance problems.     

New Trends in the Development of the Lunar Physical Libration Theory

A review of the modern state of the lunar libration theory is presented. A significant progress in the lunar investigation is achieved due to the simultaneous processing of results of the satellite Doppler tracing and of the lunar laser ranging. The data evidencing existence of a small iron core in the Moon are discussed. In this connection, the further development of the theory of rotation of the Moon presents the study of internal structure and dynamics of a lunar body. A model of a two-layer Moon can have a very advanced application to explain some observed phenomena and to be as a first approach in the modelling of internal processes determining the lunar rotation.