Lunisolar Apsidal Resonances at low Satellite Orbits

Eccentricity resonances between the secular motion of an Earth satellite's orbit and the longitudes of the Sun and the Moon are studied within a Hamiltonian framework. The problem is approximated in a traditional manner, with the Earth's potential including only the second zonal harmonic, and a Hill‐type approximation for perturbing bodies. For a family of 10 resonances, stable and unstable points are identified and libration widths are estimated. Numerical values are given for the maximum variation of eccentricity available at each resonance. The respective amplitudes of the perigee heights' librations range from 2 to 750 km. The resonances of the solar origin are generally stronger than their lunar counterparts. This revised version was published online in July 2006 with corrections to the Cover Date.     

Combinations of Satellite Crossovers to Study Orbit and Residual Errors in Altimetry

We define combinations of dual‐ and single‐satellite crossover differences to isolate both stationary orbit‐geopotential and non‐geopotential errors in altimetry data. Specifically two types of combinations are found useful. While no combination of differences can resolve the full radial error of single or paired satellites, an approximation of the mean or geographically correlated error of the generally dominant lower orbit of a pair can be found from one kind (substitutions). (The variable part of the error is always available from the single‐satellite crossover differences.) A second useful combination type is found to yield no geopotential orbit error (zeros): uniquely, these can reveal errors in altimetry from imperfect media corrections as well as oceanographic changes in sealevel. The later circumstance is particularly important when the missions for a pair of satellites are disparate in time. This revised version was published online in July 2006 with corrections to the Cover Date.     

Oscillatory orbits in the planar three‐body problem with equal masses

In the free‐fall three‐body problem, distributions of escape, binary, and triple collision orbits are obtained. Interpretation of the results leads us to the existence of oscillatory orbits in the planar three‐body problem with equal masses. A scenario to prove their existence is described. This revised version was published online in July 2006 with corrections to the Cover Date.     

Explicit Symplectic Algorithms For Time‐Transformed Hamiltonians

By Hamiltonian manipulation we demonstrate the existence of separable time‐transformed Hamiltonians in the extended phase‐space. Due to separability explicit symplectic methods are available for the solution of the equations of motion. If the simple leapfrog integrator is used, in case of two‐body motion, the method produces an exact Keplerian ellipse in which only the time‐coordinate has an error. Numerical tests show that even the rectilinear N‐body problem is feasible using only the leapfrog integrator. In practical terms the method cannot compete with regularized codes, but may provide new directions for studies of symplectic N‐body integration. This revised version was published online in July 2006 with corrections to the Cover Date.     

Some Analytical Results in Dynamics of Spheroidal Galaxies

The surfaces of section in a harmonic oscillator potential, perturbed by quartic terms, are obtained analytically. A succession of action‐angle, Lissajous and Lie transformations near the 1:1 commensurability, reduces the three‐dimensional motion to a one‐dimensional one. The latter is solved in terms of Jacobi's elliptic functions. Existence conditions for periodic orbits are found and two general families of such solutions are introduced. Two examples of regular motions in oblate and prolate spheroids are discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Improved nutation series for the non–rigid earth with a precise adjustment of parameters with nonlinear dependence

In this paper, the previous nutation series corresponding to the rotation of a non‐rigid earth composed of a rigid mantle and a liquid core obtained by Getino and Ferrándiz in 1997 are notably improved by using a high performance data fitting method. This method can be applied to many other problems presenting a non‐linear dependence on the free parameters. This revised version was published online in July 2006 with corrections to the Cover Date.     

Chaotic Scattering in the Restricted Three‐Body Problem II. Small mass parameters

We study the scattering motion of the planar restricted three‐body problem for small mass parameters μ. We consider the symmetric periodic orbits of this system with μ = 0 that collide with the singularity together with the circular and parabolic solutions of the Kepler problem. These divide the parameter space in a natural way and characterize the main features of the scattering problem for small non‐vanishing μ. Indeed, continuation of these orbits yields the primitive periodic orbits of the system for small μ. For different regions of the parameter space, we present scattering functions and discuss the structure of the chaotic saddle. We show that for μ < μc and any Jacobi integral there exist departures from hyperbolicity due to regions of stable motion in phase space. Numerical bounds for μc are given. This revised version was published online in July 2006 with corrections to the Cover Date.     

Efficient Symplectic Integration of Satellite Orbits

The use of the extended phase space and time transformations for constructing efficient symplectic methods for computing the long term behavior of perturbed two‐body systems are discussed. Main applications are for artificial satellite orbits. The methods suggested here are efficient also for large eccentricities. This revised version was published online in July 2006 with corrections to the Cover Date.     

Unified Symbolic Algorithm of Gauss Method for Near-Parabolic Orbits

In this paper, a unified algorithm of Gauss method for near‐parabolic orbits that is valid for both elliptic and hyperbolic cases is established symbolically and numerically. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamical Behavior of Asteroids Near Resonance: the 4:1 gap and the 7:2 Group

A comparative study of the evolution of the Sun–Jupiter–Asteroid system near the 4:1 and 7:2 resonances is performed by means of two techniques that proceed differently from the Hamiltonian corresponding to the planar restricted elliptic three-body problem. One technique is based on the classical Schubart averaging while the other is based on a mapping method in which the perturbing part of the Hamiltonian is expanded and the resulting terms are ordered according to a weight function that depends on the powers of eccentricities and the coefficients of the terms. For the mapping method the effect of Saturn on the asteroidal evolution is introduced and the degree of chaos is estimated by means of the Lyapunov time. Both methods are shown to lead to similar results and can be considered a suitable tool for describing the evolution of asteroids in the Kirkwood gap and the group corresponding to the 4:1 and 7:2 Jovian resonances, respectively. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stability of relative equilibria in arbitrary axisymmetric gravitational and magnetic fields

Relative equilibria occur in a wide variety of physical applications, including celestial mechanics, particle accelerators, plasma physics, and atomic physics. We derive sufficient conditions for Lyapunov stability of circular orbits in arbitrary axisymmetric gravitational (electrostatic) and magnetic fields, including the effects of local mass (charge) and current density. Particularly simple stability conditions are derived for source‐free regions, where the gravitational field is harmonic (∇²U = 0) or the magnetic field irrotational (∇ × B = 0). In either case the resulting stability conditions can be expressed geometrically (coordinate‐free) in terms of dimensionless stability indices. Stability bounds are calculated for several examples, including the problem of two fixed centers, the J₂ planetary model, galactic disks, and a toroidal quadrupole magnetic field. This revised version was published online in July 2006 with corrections to the Cover Date.     

Study of a Particular Model of Encounter Between a Comet and a Planet

The objective of this paper is to develop a simple model of an encounter between a comet and a planet, with a subsequent capture or an escape, and to study the potential consequences. The hypothetical scenario is as follows: a comet with a conic orbit meets close to one of its vertices (located near the ecliptic plane), a jovian planet, and transforms its orbit. There are two hypotheses which are made for the shock: this vertex becomes one of the final vertices and the orbital plane of the comet is unchanged during the encounter as it was the case for Brooks 2 in 1886. In this model, it was able to find an equation which was then used to obtain the pre‐ and post‐encounter orbits elements and the kind of orbit (ellipse, hyperbola, parabola) with respect to the initial inclination. The numerical experiments with the observed comets often provide pre‐encounter orbits with an aphelion point located near another jovian planet farther from the Sun, and so on with sometimes several planets, or with an aphelion point located beyond the Pluto orbit. This revised version was published online in July 2006 with corrections to the Cover Date.     

The limiting case of the double-averaged parabolic restricted three-body problem

The planar case of the parabolic restricted three-body problem is considered. The equations of motion are integrated within the framework of the double-averaged problem taking into account only the first term in the expansion of the perturbing function. It is demonstrated that, at moderate approaches to the central body, the size and the shape of the orbit of the perturbing body are invariable and only the orientation of the orbit changes.     

Effects on the nutation of C4,m and S4,m Harmonics

In this paper, we make a study about the influence of the coefficients of the geopotential C_4,m and S_4,m, (m=1,2,3,4) on the nutation, starting from the Hamiltonian theory as developed by Kinoshita (1977).We obtain ten coefficients larger than 0.05 μ as for the nutation in longitude and six for the nutation in obliquity. The present results are included in the reconstruction of the theory of nutation (REN‐2000) at the level of truncation of 0.1 μ as (Souchay et al., 1997). This revised version was published online in July 2006 with corrections to the Cover Date.     

RDAN97: An Analytical Development of Rigid Earth Nutation Series Using the Torque Approach

New series of rigid Earth nutations for the angular momemtum axis, the rotation axis and the figure axis, named RDAN97, are computed using the torque approach. Besides the classical J2 terms coming from the Moon and the Sun, we also consider several additional effects: terms coming from J3 and J4 in the case of the Moon, direct and indirect planetary effects, lunar inequality, J2 tilt, planetary‐tilt, effects of the precession and nutations on the nutations, secular variations of the amplitudes, effects due to the triaxiality of the Earth, new additional out‐of‐phase terms coming from second order effect and relativistic effects. Finally, we obtain rigid Earth nutation series of 1529 terms in longitude and 984 terms in obliquity with a truncation level of 0.1 μ (microarcsecond) and 8 significant digits. The value of the dynamical flattening used in this theory is HD=(C-A)/C=0.0032737674 computed from the initial value pa=50′.2877/yr for the precession rate. These new rigid Earth nutation series are then compared with the most recent models (Hartmann et al., 1998; Souchay and Kinoshita, 1996, 1997; Bretagnon et al., 1997, 1998. We also compute a benchmark series (RDNN97) from the numerical ephemerides DE403/LE403 (Standish et al., 1995) in order to test our model. The comparison between our model (RDAN97) and the benchmark series (RDNN97) shows a maximum difference, in the time domain, of 69 μas in longitude and 29 μas in obliquity. In the frequency domain, the maximum differences are 6 μas in longitude and 4 μ as in obliquity which is below the level of precision of the most recent observations (0.2 mas in time domain (temporal resolution of 1 day) and 0.02 mas in frequency domain). This revised version was published online in July 2006 with corrections to the Cover Date.     

Tidal effects in the earth–moon system and the earth's rotation

Analytical expressions for tidal torques induced by a tide‐arising planet which perturbs rotation of a nonrigid body are derived. Corresponding expressions both for secular and periodic perturbations of the Euler's angles are given for the case of the earth's rotation. Centennial secular rates of the nutation angle θ and of the earth's angular velocity ω, as well as the centennial logarithmic decrement ν of the Chandler wobble are evaluated:  mas, . In the Universal Time (UT) a large out‐of‐phase (sine) dissipative term with the period 18.6 years and the amplitude 2.3 ms is found. Corrections to nutation coefficients, which presumably have not been taken into account in IAU theory, are given. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the expansion of the secular part of the perturbing function of mutual attraction in the satellite systems of planets

We propose a special representation for the secular part of the perturbing function describing the mutual attraction of satellites. In contrast to the known representations, it has a single analytical form for any ratio between the semimajor axes of the perturbed and perturbing satellites. The resulting expression is a partial sum of a power series with respect to the small eccentricities and planet-equatorial inclinations of the satellites’ orbits. This sum includes terms up to and including the fourth degree with respect to these small parameters. The proposed expansion is compared with one of the known expansions for the secular part of the perturbing function.     

A new solution to the lagrange's case of the three‐body problem

The motion of three particles, interacting by gravitational forces, is studied in a new coordinate system given by the principal axes of inertia, as determined by Euler angles, and using the inertia principal moments and an auxiliar angle as coordinates. The solution to the particular Lagrange case of the three‐body problem is reviewed and solved in these new coordinates. This revised version was published online in July 2006 with corrections to the Cover Date.     

Two‐Body Problems with Drag or Thrust: Qualitative Results

We consider two‐body problems in which the drag is proportional to the velocity divided by the square of the distance and whose radial and tangential components have distinct coefficients. For all parameters, we study the flow of the system obtained by suitable coordinate and time transformations and draw conclusions about the qualitative behavior of solutions. In each case, we examine the existence of collision–ejection, collision–escape, capture–collision, capture–escape, and oscillatory rectilinear orbits, study the motion near collision, and show that if periodic orbits exist they must be limit cycles. This revised version was published online in July 2006 with corrections to the Cover Date.     

Perturbation Theory of Tori Without Complete Fourier Expansion

Perturbation theory of tori involves solving a first order partial differential equation, the so‐called homological equation. The usual complete Fourier expansions employed in its solution can be replaced by partial ones. The alternative solution so obtained is expressed in terms of integrals and without any explicit reference to trigonometric functions. Application to second order averaging is also discussed. This revised version was published online in July 2006 with corrections to the Cover Date.