Synchronous Locking of Tidally Evolving Satellites (original) (raw)
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Effective stability around the Cassini state in the spin-orbit problem
Celestial Mechanics and Dynamical Astronomy, 2014
We investigate the long-time stability in the neighborhood of the Cassini state in the conservative spin-orbit problem. Starting with an expansion of the Hamiltonian in the canonical Andoyer-Delaunay variables, we construct a highorder Birkhoff normal form and give an estimate of the effective stability time in the Nekhoroshev sense. By extensively using algebraic manipulations on a computer, we explicitly apply our method to the rotation of Titan. We obtain physical bounds of Titan's latitudinal and longitudinal librations, finding a stability time greatly exceeding the estimated age of the Universe. In addition, we study the dependence of the effective stability time on three relevant physical parameters: the orbital inclination, i, the mean precession of the ascending node of Titan orbit, Ω, and the polar moment of inertia, C.
LONGITUDINAL LIBRATIONS OF A SATELLITE
Electronic Journal of Differential Equations
Furi, Martelli and Landsberg gave a theoretical explanation of the chaotic longitudinal librations of Hyperion, a satellite of Saturn. The analysis was made under the simplifying assumption that the spin axis remains perpen-dicular to the orbit plane. Here, under the same assumption, we investigate the behavior of the longitudinal librations of any satellite. Also we show that they are possibly chaotic depending on two parameters: a constant k related to the principal moments of inertia of the satellite, and the eccentricity e of its orbit. We prove that the plane k-e contains an open region Ω with the property that the longitudinal librations of any satellite are possibly chaotic if the point (k, e) belongs to this region. Since Hyperion's point is inside Ω, the results of this paper are more general than those obtained previously.
On the stability of spinning satellites
Acta Astronautica, 2011
We study the directional stability of rigid and deformable spinning satellites in terms of two attitude angles. The linearized attitude motion of a free system about an assumed uniform-spin reference solution leads to a generic MGK system when the satellite is rigid or deformable. In terms of Lyapunov's stability theory, we investigate the stability with respect to a subset of the variables. For a rigid body, the MGK system is 6-dimensional, i.e., 3 rotational and 3 translational variables. When flexible parts are present the system can have any arbitrary dimension. The 2 Â 2 McIntyre-Myiagi stability matrix gives sufficient conditions for the attitude stability. A further development of this method has led to the Equivalent Rigid Body method. We propose an alternative practical method to establish sufficiency conditions for directional stability by using the Frobenius-Schur reduction formula. As practical applications we discuss a spinning satellite augmented with a spring-mass system and a rigid body appended with two cables and tip masses. In practice, the attitude stability must also be investigated when the spinning satellite is subject to a constant axial thrust. The generic format becomes MGKN as the thrust is a follower force. For a perfectly aligned thrust along the spin axis, Lyapunov's indirect method remains valid also when deformable parts are present. We illustrate this case with an apogee motor burn in the presence of slag. When the thrust is not on the spin axis or not pointing parallel to the spin axis, the uniform-spin reference motion does not exist and none of the previous methods is applicable. In this case, the linearization may be performed about the initial state. Even when the linearized system has bounded solutions, the non-linear system can be unstable in general. We illustrate this situation by an instability that actually happened in-flight during a station-keeping maneuver of ESA's GEOS-I satellite in 1979.
Celestial Mechanics and Dynamical Astronomy, 2018
Comprehensive analysis of space debris rotational dynamics is vital for active debris removal missions that require physical capture or de-tumbling of a target. We study the attitude motion of used rocket bodies acknowledgedly belonging to one of the categories of large space debris objects that pose an immediate danger to space operations in low Earth orbits. Particularly, we focus on Sun-synchronous orbits (SSO) with altitudes in the interval 600 ÷ 800 km, where the density of space debris is maximal. Our mathematical model takes into account the gravity gradient torque and the torque due to eddy currents induced by the interaction of conductive materials with the geomagnetic field. Using perturbation techniques and numerical methods we examine the deceleration of the initial fast rotation and the subsequent transition to a relative equilibrium with respect to the local vertical. A better understanding of the latter phase is achieved owing to a more accurate model of the eddy currents torque than in most prior research. We show that SSO precession is also an important factor influencing the motion properties. One of its effects is manifested at the deceleration stage as the angular momentum vector oscillates about the direction to the south celestial pole. Keywords space debris • attitude dynamics • eddy currents torque • Cassini cycles 1 Introduction This paper presents a study of rotational dynamics of large space debris objects in Sun-synchronous orbits. SSO are characterized by 600-800 km altitude and inclination of about 90 • (Vallado (2007)). These orbits are best suited for the Earth' observation from space, because of consistent lighting conditions in their subsatellite points for all satellite passes. Throughout the last few decades SSO have been in use, there
Rotation of Synchronous Satellites Application to the Galilean Satellites
Celestial Mechanics & Dynamical Astronomy, 2004
An analytical theory of the rotation of a synchronous satellite is developed for the application to the rotation of the Galilean satellites. The theory is developed in the framework of Hamiltonian mechanics, using Andoyer variables. Special attention is given to the frequencies of libration as functions of the moments of inertia of the satellite.
Cassini's motions of the Moon and Mercury and possible excitations of free librations
Geodesy and Geodynamics, 2018
On the basis of conditionally-periodic solutions of Hamiltonian systems at resonance of main frequencies Cassini's motions, their stability, Cassini's angle and periods of free librations of the Moon and Mercury have been recently studied and determined. The generalized formulations of Cassini's laws for the motion of the Moon and Mercury, that are considered as absolutely rigid non-spherical bodies, have been determined. The study of the second approximation equations of the desired quasi-periodic solutions in the case of the Moon allows us to determine the constant components of the first order for six Andoyer variables and the constant component of the second order for the angular velocity of the Moon. These effects are caused by the influence of the third harmonic of selenopotential. In this paper, these effects are described by analytical formulas, the dynamic and geometric interpretations are given, and a new interpretation of Mercury's motion under the generalized Cassini's laws has been proposed. Predictions of the existence of free librations of significant amplitude in the Mercury longitude, that are confirmed by the radar measurements data of the Mercury angular velocity, and in its pole motion in the body and in space have been made. The mechanism describing free librations of celestial bodies and their pole oscillations has been proposed due to the forced relative oscillations and wobble of the core-mantle system of celestial bodies (Moon, Mercury, Earth and other bodies in the solar system) under gravitational action of the external celestial bodies. The paper shows that the ascending node of equator of Mercury (and the intermediate plane orthogonal to the angular momentum) of epoch 2000.0 on the ecliptic does not coincide with the ascending node of orbital plane of Mercury on the same plane, and is ahead of it at an angle 23º4'. Angular momentum vector of the rotational motion of Mercury forms a constant angle r G ¼ 4 0 1±1 0 1 with normal to the moveable plane of its orbit. The observed inclination of the angular velocityr u ¼ 2 0 1±0 0 1, can be considered as a possible evidence of a significant amplitude of the poles free motion of the Mercury rotation axis (c amplitude of about 2 0-4 0).
Spin-Orbit Coupling and Chaotic Rotation for Coorbital Bodies in Quasi-Circular Orbits
The Astrophysical Journal, 2013
Coorbital bodies are observed around the Sun sharing their orbits with the planets, but also in some pairs of satellites around Saturn. The existence of coorbital planets around other stars has also been proposed. For close-in planets and satellites, the rotation slowly evolves due to dissipative tidal effects until some kind of equilibrium is reached. When the orbits are nearly circular, the rotation period is believed to always end synchronous with the orbital period. Here we demonstrate that for coorbital bodies in quasi-circular orbits, stable non-synchronous rotation is possible for a wide range of mass ratios and body shapes. We show the existence of an entirely new family of spin-orbit resonances at the frequencies n ± kν/2, where n is the orbital mean motion, ν the orbital libration frequency, and k an integer. In addition, when the natural rotational libration frequency due to the axial asymmetry, σ, has the same magnitude as ν, the rotation becomes chaotic. Saturn coorbital satellites are synchronous since ν σ, but coorbital exoplanets may present non-synchronous or chaotic rotation. Our results prove that the spin dynamics of a body cannot be dissociated from its orbital environment. We further anticipate that a similar mechanism may affect the rotation of bodies in any mean-motion resonance.
Steady state obliquity of a rigid body in the spin–orbit resonant problem: application to Mercury
Celestial Mechanics and Dynamical Astronomy, 2017
We investigate the stable Cassini state 1 in the p:q spin-orbit resonant problem. Our study includes the effect of the gravitational potential up to degree and order 4 and p:q spin-orbit resonances with p, q ≤ 8 and p ≥ q. We derive new formulae that link the gravitational field coefficients with its secular orbital elements and its rotational parameters. The formulae can be used to predict the orientation of the spin axis and necessary angular momentum at exact resonance. We also develop a simple pendulum model to approximate the dynamics close to resonance and make use of it to predict the libration periods and widths of the oscillatory regime of motions in phase space. Our analytical results are based on averaging theory that we also confirm by means of numerical simulations of the exact dynamical equations. Our results are applied to a possible rotational history of Mercury.
Applied Physics Research, 2020
This study identifies the unique features accompanying the phenomenon of synchronous rotation of the major (proximal) satellites of the gas giants and the earth’s moon, and the special features leading to the ‘negative’ rotation of Venus, Uranus and Pluto, as well as the most peripheral small satellites of the gas giants. Such features help us understand how these phenomena occur but also, by combining all of the observations help explain other (regular) planetary motions as well. In the synchronously rotating satellites, the salient features are the satellites’ low axial tilts and both the orbital speed and the axial rotation speed increasing with proximity to the mother body. In “negative” rotation, axial tilts are in excess of 120° and the axial rotation speeds are significantly delayed; this delay is most pronounced in Venus, which has an axial tilt of -174°. A scrutiny of the orbital parameters of all the satellites of the gas giants alone will yield sufficient data to propose ...
Spin axis behavior of the LAGEOS satellites
Journal of Geophysical Research, 2004
1] The satellites LAGEOS-I and LAGEOS-II are essential for the scientific study of various (geo)physical phenomena, such as geocenter motion and absolute scale. The high quality of such science products strongly depends on the absolute quality of the SLR observations and that of the orbit description. Therefore all accelerations experienced by the spacecraft need to be modeled as accurately as possible, the thermal radiation forces being one of them. Traditionally, this is done by estimating so-called empirical accelerations. However, the rotational dynamics of LAGEOS-I in particular no longer allows such a simple approach: a full modeling of the spin behavior, the temperature distribution over the spacecraft surface and the resulting net force prove necessary to achieve the best results. As a first step, a new model, Lageos Spin Axis Model (LOSSAM) has been developed. It is unique in its combination of analytical theory and empirical observations. Its mathematics is taken after previous investigators, although flaws have been corrected. LOSSAM describes the full spin behavior of LOSSAM based on the following phenomena: (1) the geomagnetic field, (2) the Earth's gravity field, (3) the satellite center of pressure offset, and (4) the effective difference in reflectivity between the satellite hemispheres. Its accuracy has been demonstrated by an improvement of about a 50% in the RMS residual of the Yarkovsky-Schach effect signal (as shown by Lucchesi et al. [2004]). Such a high-quality model for rotational behavior is indispensable for a proper force modeling, and hence also for the quality of typical LAGEOS science products.