Showing papers in "Celestial Mechanics and Dynamical Astronomy in 2003"

Evolution of migrating planet pairs in resonance

Abstract: Numerical simulations of the evolution of planets or massive satellites captured in the 2/1 and 3/1 resonances, under the action of an anti-dissipative tidal force. The evolution of resonant trapped bodies show a richness of solutions: librations around stationary symmetric solutions with aligned periapses (Δϖ = 0) or anti-aligned periapses (Δϖ = π), librations around stationary asymmetric solutions in which the periapses configuration is fixed, but with Δϖ taking values in a wide range of angles. Many of these solutions exist for large values of the eccentricities and, during the semimajor axes drift, the solutions show turnabouts from one configuration to another. The presented results are valid for other non-conservative anti-dissipative forces leading to adiabatic convergent migration and capture into one of these resonances.

New families of periodic orbits in hill's problem of three bodies

Abstract: The search for symmetric periodic solutions of Hill's problem, begun in Henon (1996) for simple-periodic orbits, is extended to double- and triple-periodic orbits. Seven new families are found. We also complete the description of family g3, begun in Henon (1970). In all of these families, both ends consist of orbits of increasing dimensions with Γ → −∞. They can be continued into second species families, which are identified.

The Impact of the Static Part of the Earth's Gravity Field on Some Tests of General Relativity with Satellite Laser Ranging

Abstract: In this paper we calculate explicitly the classical secular precessions of the node Ω and the perigee ω of an Earth artificial satellite induced by the even zonal harmonics of the static part of the geopotential up to degree l = 20. Subsequently, their systematic errors induced by the mismodelling in the even zonal spherical harmonics coefficients Jl are compared to the general relativistic secular gravitomagnetic and gravitoelectric precessions of the node and the perigee of the existing laser-ranged geodetic satellites and of the proposed LARES. The impact of the future terrestrial gravity models from CHAMP and GRACE missions is discussed as well. Preliminary estimates with the recently released EIGEN-1S gravity model including the first CHAMP data are presented.

Stability of Surface Motion on a Rotating Ellipsoid

Abstract: The dynamical environment on the surface of a rotating, massive ellipsoid is studied, with applications to surface motion on an asteroid. The analysis is performed using a combi- nation of classical dynamics and geometrical analysis. Due to the small sizes of most asteroids, their shapes tend to differ from the classical spheroids found for the planets. The tri-axial ellips- oid model provides a non-trivial approximation of the gravitational potential of an asteroid and is amenable to analytical computation. Using this model, we study some properties of motion on the surface of an asteroid. We find all the equilibrium points on the surface of a rotating ellipsoid and we show that the stability of these points is intimately tied to the conditions for a Jacobi or MacLaurin ellipsoid of equilibria. Using geometrical analysis we can define global constraints on motion as a function of shape, rotation rate, and density, we find that some asteroids should have accumulation of material at their ends, while others should have accumulation of surface mate- rial at their poles. This study has implications for motion of a rover on an asteroid, and for the distribution of natural material on asteroids, and for a spacecraft hovering over an asteroid.

Disc–Planet Interactions: Migration and Resonances in Extrasolar Planetary Systems

Abstract: We consider orbital resonances in multiplanet systems. These are expected to arise during or just after formation in a gaseous disc. Disc–planet interaction naturally produces orbital migration and circularization through the action of tidal torques which in turn may lead to an orbital resonance. The mass and angular momentum content of the disc is likely to be comparable to that in the planets so that it is essential to fully incorporate the disc in the analysis.

Tidally Induced Volcanism

Abstract: The dissipation of tidal energy causes the ongoing silicate volcanism on Jupiter's satellite, Io, and cryovolcanism almost certainly has resurfaced parts of Saturn's satellite, Enceladus, at various epochs distributed over the latter's history. The maintenance of tidal dissipation in Io and the occurrence of the same on Enceladus depends crucially on the maintenance of the respective orbital eccentricities by the existence of mean motion resonances with nearby satellites. A formation of the resonances among the Galilean satellites by differential expansion of the satellite orbits from tides raised on Jupiter by the satellites means the onset of the volcanism on Io could be relatively recent. If, on the other hand, the resonances formed by differential migration from resonant interactions of the satellites with the disk of gas and particles from which they formed, Io would have been at least intermittently volcanically active throughout its history. Either means of assembling the Galilean satellite resonances lead to the same constraint on the dissipation function of Jupiter Q J ≲ 106, where the currently high heat flux from Io seems to favor episodic heating as Io's eccentricity periodically increases and decreases. Either of the two models might account for sufficient tidal dissipation in the icy satellite Enceladus to cause at least occasional cryovolcanism over much of its history. However, both models are assumption-dependent and not secure, so uncertainty remains on how tidal dissipation resurfaced Enceladus.

New Challenges to Trajectory Design by the Use of Electric Propulsion and Other New Means of Wandering in the Solar System

Abstract: The design of spacecraft trajectories is a crucial part of a space mission design. Often the mission goal is tightly related to the spacecraft trajectory. A geostationary orbit is indeed mandatory for a stationary equatorial position. Visiting a solar system planet implies that a proper trajectory is used to bring the spacecraft from Earth to the vicinity of the planet. The first planetary missions were based on conventional trajectories obtained with chemical engine rockets. The manoeuvres could be considered 'impulsive' and clear limitations to the possible missions were set by the energy required to reach certain orbits. The gravity-assist trajectories opened a new way of wandering through the solar system, by exploiting the gravitational field of some planets. The advent of other propulsion techniques, as electric or ion propulsion and solar sail, opened a new dimension to the planetary trajectory, while at the same time posing new challenges. These 'low thrust' propulsion techniques cannot be considered 'impulsive' anymore and require for their study mathematical techniques which are substantially different from before. The optimisation of such trajectories is also a new field of flight dynamics, which involves complex treatments especially in multi-revolution cases as in a lunar transfer trajectory. One advantage of these trajectories is that they allow to explore regions of space where different bodies gravitationally compete with each other. We can exploit therefore these gravitational perturbations to save fuel or reduce time of flight. The SMART-1 spacecraft, first European mission to the Moon, will test for the first time all these techniques. The paper is a summary report on various activities conducted by the project team in these areas.

Double Material Segment as the Model of Irregular Bodies

Abstract: We propose a new, simple model to describe the gravity field of irregular, nonspherical celestial bodies, like small moons or minor asteroids. The simple idea of Duboshin to use a material straight segment for such bodies is extended by combining two perpendicular segments of different lengths and masses. In typical situations, when the longest axis of the body coincides with one segment, the remaining segment must have an imaginary length. The potential remains a real function even if one segment is imaginary. The new model is confronted with the exact form of an ellipsoid's potential and with two alternative simple models for a two-axial and a three-axial ellipsoid.

Solution of the Sitnikov Problem

Abstract: A new analytic expression for the position of the infinitesimal body in the elliptic Sitnikov problem is presented. This solution is valid for small bounded oscillations in cases of moderate primary eccentricities. We first linearize the problem and obtain solution to this Hill's type equation. After that the lowest order nonlinear force is added to the problem. The final solution to the equation with nonlinear force included is obtained through first the use of a Courant and Snyder transformation followed by the Lindstedt–Poincare perturbation method and again an application of Courant and Snyder transformation. The solution thus obtained is compared with existing solutions, and satisfactory agreement is found.

The Apsidal Antialignment of the HD 82943 System

Abstract: We perform numerical simulations to explore the dynamical evolution of the HD 82943 planetary system. By simulating diverse planetary configurations, we find two mechanisms of stabilizing the system: the 2:1 mean motion resonance (MMR) between the two planets can act as the first mechanism for all stable orbits. The second mechanism is a dynamical antialignment of the apsidal lines of the orbiting planets, which implies that the difference of the periastron longitudes Θ3 librates about 180° in the simulations. We also use a semi-analytical model to explain the numerical results for the system under study.

Repeat Ground Track Orbits of the Earth Tesseral Problem as Bifurcations of the Equatorial Family of Periodic Orbits

Abstract: Orbits repeating their ground track on the surface of the earth are found to be members of periodic-orbit families (in a synodic frame) of the tesseral problem of the Earth artificial satellite. Families of repeat ground track orbits appear as vertical bifurcations of the equatorial family of periodic orbits, and they evolve from retrograde to direct motion throughout the 180 degrees of inclination. These bifurcations are always close to the resonances of the Earth's rotation rate and the mean motion of the orbiter.

On the Stability of Planar Oscillations and Rotations of a Satellite in a Circular Orbit

Abstract: We deal with the stability problem of planar periodic motions of a satellite about its center of mass. The satellite is regarded a dynamically symmetric rigid body whose center of mass moves in a circular orbit.

Tidal Stress Patterns on Europa's Crust

Abstract: The ice crust of Europa probably floats over a deep liquid-water ocean, and has been continually resurfaced by tectonic and thermal processes driven by tides. Tidal working causes rotational torque, surface stress, internal heating, and orbital evolution. The stress patterns expected on such a crust due to reorientation of the tidal bulge by non-synchronous rotation and due to orbital eccentricity, which introduces periodic ('diurnal') variations in the tide, are shown as global maps. By taking into account the finite rate of crack propagation, global maps are generated of cycloidal features and other distinctive patterns, including the crack shapes characteristic of the wedges region and its antipode on the sub-Jovian hemisphere. Theoretical maps of tidal stress and cracking can be compared with observed tectonics, with the possibility of reconstructing the rotational history of the satellite.

Non-Integrability of the Problem of a Rigid Satellite in Gravitational and Magnetic Fields

Abstract: In this paper we analyse the integrability of a dynamical system describing the rotational motion of a rigid satellite under the influence of gravitational and magnetic fields. In our investigations we apply an extension of the Ziglin theory developed by Morales-Ruiz and Ramis. We prove that for a symmetric satellite the system does not admit an additional real meromorphic first integral except for one case when the value of the induced magnetic moment along the symmetry axis is related to the principal moments of inertia in a special way.

Symplectic mapping for Trojan-type motion in the elliptic restricted three-body problem

Abstract: A symplectic mapping for Trojan-type motion has been developed in the secularly changing elliptic restricted three-body problem. The mapping describes well the characteristics of Trojan-type dynamics at small eccentricities. By using this mapping the boundary of the stability region has been studied for different values of the initial eccentricities of hypothetical Jupiter's Trojans. It has been found that in the secularly changing elliptic case the chaotic diffusion at the border of the stability region is stronger than simply in the elliptic case. An explanation of this observation might be the destruction of the chain of islands of the 13:1 secondary resonance between the short and long period component of the Trojan-like motion, caused possibly by the indirect perturbations of Saturn.

Periodic Orbits of a Collinear Restricted Three-Body Problem

Abstract: In this paper we study symmetric periodic orbits of a collinear restricted three-body problem, when the middle mass is the largest one. These symmetric periodic orbits are obtained from analytic continuation of symmetric periodic orbits of two collinear two-body problems.

Dynamical Evolution of Planets in Disks

Abstract: We study the evolution of a system consisting of two protoplanets still embedded in a protoplanetary disk. Results of two different numerical approaches are presented. In the first kind of model the motion of the disk material is followed by fully viscous hydrodynamical simulations, and the planetary motion is determined by N-body calculations including exactly the gravitational potential from the disk material. In the second kind we only solve the N-body part and add additional analytically given forces which model the effect of the torques of the disk. This type of modeling is of course orders of magnitudes faster than the full hydro-model. Another advantage of this two-fold approach is the possibility of adjusting the otherwise unknown parameters of the simplified model.

Energy change in a hard binary due to distant encounters

Abstract: An approximate analytical technique for computing the change in the binding energy of a binary due to an incoming third star moving in a distant parabolic orbit is presented This is an example of a tidal encounter since we assume that the distance of the third star always considerably exceeds the size of the binary The perturbation is also adiabatic, varying on a time scale much exceeding the binary period, and the change has an exponential form Different cases arise depending on the choice of the masses and the angle of inclination of the plane in which the star moves Some numerical experiments are performed as a means of checking the analytical theory

Extension of Cassini's Laws

Abstract: We investigate the Cassini's laws which describe the rotational motion in a 1:1 spin-orbit resonance. When this rotational motion follows the conventional Cassini's laws, the figure axis coincides with the angular momentum axis. In this case we underline the differences between the rotational Hamiltonian for a 'slow rotating' body like the Moon and for a 'fast rotating' body like Phobos. Then, we study a more realistic rotational Hamiltonian where the angle J between the figure axis and the angular momentum axis could be different from zero. This Hamiltonian has not been studied before. We have found a new particular solution for this Hamiltonian which could be seen as an extension of the Cassini's laws. In this new solution the angle J is constant, which is not zero, and the precession of the angular momentum plane is equal to the mean motion of the argument of pericenter of the rotating body. This type of rotational motion is only possible when the orbital eccentricity of the rotating body is not zero. This new law enables describing in particular, the Moon mean rotational motion for which the mean value of the angle J is found to be equal to 103.9±0.7 s of arc.

Non-existence of new meromorphic first integrals in the planar three-body problem

Abstract: We consider the planar three-body problem and prove that, apart from some exceptional cases, there is no additional first integral meromorphic with respect to positions, mutual distances and momenta

Stability of the Triangular Libration Points in the Unrestricted Planar Problem of a Symmetric Rigid Body and a Point Mass

Abstract: In papers (Goździewski and Maciejewski, 1998a, b, 1999), we investigate unrestricted, planar problem of a dynamically symmetric rigid body and a sphere. Following the original statement of the problem by Kokoriev and Kirpichnikov (1988), we assume that the potential of the rigid body is approximated by the gravitational field of a dumb-bell. The model is described in terms of a 2D Hamiltonian depending on three parameters.

Geometric Derivation of the Delaunay Variables and Geometric Phases

Abstract: We derive the classical Delaunay variables by finding a suitable symmetry action of the three torus T3 on the phase space of the Kepler problem, computing its associated momentum map and using the geometry associated with this structure. A central feature in this derivation is the identification of the mean anomaly as the angle variable for a symplectic S1 action on the union of the non-degenerate elliptic Kepler orbits. This approach is geometrically more natural than traditional ones such as directly solving Hamilton–Jacobi equations, or employing the Lagrange bracket. As an application of the new derivation, we give a singularity free treatment of the averaged J2-dynamics (the effect of the bulge of the Earth) in the Cartesian coordinates by making use of the fact that the averaged J2-Hamiltonian is a collective Hamiltonian of the T3 momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J2 effect as geometric phases.

An Abridged Model of the Precession–Nutation of the Celestial Pole

Abstract: Precise astrometric observations show that significant systematic differences of the order of 10 milliarcseconds (mas) exist between the observed position of the celestial pole in the International Celestial Reference Frame (ICRF) and the position determined using the International Astronomical Union (IAU) 1976 Precession (Lieske et al., 1977) and the IAU 1980 Nutation Theory (Seidelmann, 1982). The International Earth Rotation Service routinely publishes these 'celestial pole offsets', and the IERS Conventions (McCarthy, 1996) recommends a procedure to account for these errors. The IAU, at its General Assembly in 2000, adopted a new precession/nutation model (Mathews et al., 2002). This model, designated IAU2000A, which includes nearly 1400 terms, provides the direction of the celestial pole in the ICRF with an accuracy of ±0.1 mas. Users requiring accuracy no better than 1 mas, however, may not require the full model, particularly if computational time or storage are issues. Consequently, the IAU also adopted an abridged procedure designated IAU2000B to model the celestial pole motion with an accuracy that does not result in a difference greater than 1 mas with respect to that of the IAU2000A model. That IAU2000B model, presented here, is shown to have the required accuracy for a period of more than 50 years from 1995 to 2050.

Co-Orbital Motion with Slowly Varying Parameters

Abstract: We consider the dynamics of a test particle co-orbital with a satellite of mass m s which revolves around a planet of mass M 0 ≫ m s with a mean motion n s and semi-major axis a s. We study the long term evolution of the particle motion under slow variations of (1) the mass of the primary, M 0, (2) the mass of the satellite, m s and (3) the specific angular momentum of the satellite J s. The particle is not restricted to small harmonic oscillations near L 4 or L 5, and may have any libration amplitude on tadpole or horseshoe orbits. In a first step, no torque is applied to the particle, so that its motion is described by a Hamiltonian with slowly varying parameters. We show that the torque applied to the satellite, as measured by ∈s = js/(n s J s) induces an distortion of the phase space which is entirely described by an asymmetry coefficient α = ∈s/μ, where μ = m s/M. The adiabatic invariance of action implies furthermore that the long term evolution of the particle co-orbital motion depends only on the variation of m s a s with time. Applying a constant torque to the particle, as measured by ∈s = js/(n s J p) is then merely equivalent to replacing α = ∈s/μ by α = (∈s − ∈p)/μ. However, if the torque acting on the particle exhibits a radial gradient, then the action is no more conserved and the evolution of the particle orbit is no more controlled by m s a s only. We show that even mild torque gradients can dominate the orbital evolution of the particle, and eventually decide whether the latter will be pulled towards the stable equilibrium points L 4 or L 5, or driven away from them. Finally, we show that when the co-orbital bodies are two satellites with comparable masses m 1 and m 2, we can reduce the problem to that of a test particle co-orbital with a satellite of mass m 1 + m 2. This new problem has then parameters varying at rates which are combinations, with appropriate coefficients, of the changes suffered by each satellite.

The influence of reactive torques on comet nucleus rotation

Abstract: Reactive torques, due to anisotropic sublimation on a comet nucleus surface, produce slow variations of its rotation. In this paper the secular effects of this sublimation are studied. The general rotational equations of motion are averaged over unperturbed fast rotation around the mass center (Euler-Poinsot motion) and over the orbital comet motion. We discuss the parameters that define typical properties of the rotational evolution and discover different classifications of the rotational evolution. As an example we discuss some possible scenarios of rotational evolution for the nuclei of the comets Halley and Borrelly.

The Anisotropic Schwarzschild-Type Problem, Main Features

Abstract: The two-body problem associated to an anisotropic Schwarzschild-type field is being tackled. Both the motion equations and the energy integral are regularized via McGehee-type transformations. The regular vector field exhibits nice symmetries that form a commutative group endowed with an idempotent structure. The physically fictitious flows on the collision and infinity manifolds, as well as the local flows in the neighbourhood of these manifolds, are fully described. Homothetic, spiral, and oscillatory orbits are pointed out. Some features of the global flow are depicted for all possible levels of energy. For the negative-energy case, few things have been done. The positive-energy global flow does not have zero-velocity curves; every orbit is of the type ejection – escape or capture – collision. In the zero-energy case, the collision and infinity manifolds have a very similar structure. The existence of eight trajectories that connect the equilibria on these manifolds is proved. The projectability of the zero-energy global flow completes the full understanding of the problem in this case.

Numerical Stabilization of Orbital Motion

Abstract: Mainly, the author focuses on Baumgarte's method and its applications in satellite, asteroid, stellar and planetary problems. In the paper arguments are given for the use of energy relations for stabilization in the elliptical two-body problem. Stabilizing properties of Baumgarte's equations and others are discussed. A simple approach is proposed for stabilizing the equations of almost circular motion. By using Baumgarte's technique, the author derives stabilized equations of perturbed restricted three-body problem. It is shown experimentally that stabilization in the problems mentioned above can raise the accuracy of numerical integration by several orders.

Extended Fundamental Model of Resonance

Abstract: Fundamental models are the simplest, one degree of freedom Hamiltonians that serve as a tool to understand the qualitative effects of various resonances. A new, extended fundamental model (EFM) is proposed in order to improve the classical, Andoyer type, second fundamental model (SFM). The EFM Hamiltonian differs from the SFM by the addition of a term with the third power of momentum; it depends on two free parameters. The new model is studied for the case of a first-order resonance, where up to five critical points can be present. Similarly, to the respective SFM, it admits only the saddle-node bifurcations of critical points, but its advantage lies in the capability of generating the separatrix bifurcations, known also as saddle connections. The reduction of parameters for the EFM has been performed in a way that allows the use of the model in the case of the so-called abnormal resonance.

Special Solutions in a Simplified Restricted Three-Body Problem with Gravitational Radiation Taken into Account

Abstract: The distinctive feature of the relativistic restricted three-body problem within the c−5 order of accuracy (2\(\frac{1}{2}\) post-Newtonian approximation) is the presence of the gravitational radiation To simplify the problem the motion of the massive binary components is assumed to be quasi-circular In terms of time these orbits have linearly changing radii and quadratically changing phase angles By substituting this motion into the Newtonian-like equations of motion one gets the quasi-Newtonian restricted quasi-circular three-body problem sufficient to take into account the main indirect perturbations caused by the binary radiation terms Such problem admits the Lagrange-like quasi-libration solutions and rather simple quasi-circular orbits lying at large distance from the binary

Controlled Solar Sail Transfers into Near-Sun Regions Combined with Planetary Gravity-Assist Flybys

Abstract: Results of numerical simulations of 'local-optimal' (or 'instantaneously optimal') trajectories of a space probe with a flat solar sail which moves from the circular Earth orbit to near-Sun regions are presented. We examine planar (ecliptic) solar sail transfer with gravity-assist flybys of Earth, Venus and Mercury. Several complex control modes of the sail tilt orientation angle for near-Sun orbits and for some 'falling onto the Sun' trajectories are investigated. The numerical simulations are used to examine the flight duration of some sail missions and to investigate the evolution of osculating elliptical orbits.