The dynamics of tadpole and horseshoe orbits: II. The coorbital satellites of saturn

The recently discovered coorbital satellites of Saturn, 1980S1 and 1980S3, are shown to be librating in horseshoe orbits. By considering the effects of tangential forces on the semimajor axes of the satellite orbits, we derive an accurate relation between the sum of the satellite masses and (a) their minimum angular separation, (b) the variation of their angular separation with time and (c) the libration period. Observations of (b) and (c) are the most practical methods of determining the satellite masses. The orbits of the coorbital satellites of Dione and Tethys are discussed. We demonstrate the possibility of calculating a new value for the mass of Dione and we show that one of the coorbital satellites of Tethys could be moving in a horseshoe orbit even though another satellite is librating in a tadpole orbit about the leading Lagrangian equilibrium point L₄. The origin of coorbital satellites and the stability of their orbits are discussed.     

Effects of the physical libration of the moon on lunar orbiters

Lunar physical libration, which is true oscillation of lunar equator in the space, alters the lunar gravitational field in the space coordinate system and affects the orbiting motion of lunar orbiters (hereafter called as lunar satellites) correspondingly. The effect is very similar to that of the precession and nutation on the earth satellites, and a similar treatment can be used. The variations in the gravitational force and in the orbit perturbation solution are clearly given in this paper together with numerical illustrations.     

Disturbance compensating control of orbit radially aligned two-craft Coulomb formation

This paper investigates the orbit radial stabilization of a two-craft virtual Coulomb structure about circular orbits and at Earth–Moon libration points. A generic Lyapunov feedback controller is designed for asymptotically stabilizing an orbit radial configuration about circular orbits and collinear libration points. The new feedback controller at the libration points is provided as a generic control law in which circular Earth orbit control form a special case. This control law can withstand differential solar perturbation effects on the two-craft formation. Electrostatic Coulomb forces acting in the longitudinal direction control the relative distance between the two satellites and inertial electric propulsion thrusting acting in the transverse directions control the in-plane and out-of-plane attitude motions. The electrostatic virtual tether between the two craft is capable of both tensile and compressive forces. Using the Lyapunov’s second method the feedback control law guarantees closed loop stability. Numerical simulations using the non-linear control law are presented for circular orbits and at an Earth–Moon collinear libration point.     

The rotation of Janus and Epimetheus

Epimetheus, a small moon of Saturn, has a rotational libration (an oscillation about synchronous rotation) of 5.9°±1.2°, placing Epimetheus in the company of Earth’s Moon and Mars’ Phobos as the only natural satellites for which forced rotational libration has been detected. The forced libration is caused by the satellite’s slightly eccentric orbit and non-spherical shape.Detection of a moon’s forced libration allows us to probe its interior by comparing the measured amplitude to that predicted by a shape model assuming constant density. A discrepancy between the two would indicate internal density asymmetries. For Epimetheus, the uncertainties in the shape model are large enough to account for the measured libration amplitude. For Janus, on the other hand, although we cannot rule out synchronous rotation, a permanent offset of several degrees between Janus’ minimum moment of inertia (long axis) and the equilibrium sub-Saturn point may indicate that Janus does have modest internal density asymmetries.The rotation states of Janus and Epimetheus experience a perturbation every 4 years, as the two moons “swap” orbits. The sudden change in the orbital periods produces a free libration about synchronous rotation that is subsequently damped by internal friction. We calculate that this free libration is small in amplitude (<0.1°) and decays quickly (a few weeks, at most), and is thus below the current limits for detection using Cassini images.     

The Characteristics and Related Problems of the Orbits Around the Earth-Moon Libration Points

Due to the specific dynamics, the probes located at the halo orbits or Lissajous orbits around the Earth-Moon collinear libration point L₁ or L₂ are always studied in the synodic system to understand their trajectories. In fact, they are also orbiting the Earth in a distant Keplerian ellipse. Because of their intrinsic orbital instability, in the orbit prediction the initial errors propagate more prominently than those of the normal orbiting satellites, this requires special attention in the orbit design, maneuver, and control. Despite of all this, they are similar to the normal orbiting satellites in orbit determination and hardly require other special attentions. In this paper, the quantitative results of error propagation under the unstable dynamics, together with the theoretical analysis are presented. The results of precise orbit determination and short-arc orbit predictions are also shown, and compared with the results from the Beijing Aerospace Control Center.     

Resonant forcing of Mercury's libration in longitude

The period of free libration of Mercury's longitude about the position it would have had if it were rotating uniformly at 1.5 times its orbital mean motion is close to resonance with Jupiter's orbital period. The Jupiter perturbations of Mercury's orbit thereby lead to amplitudes of libration at the 11.86 year period that may exceed the amplitude of the 88 day forced libration determined by radar. Mercury's libration in longitude may be thus dominated by only two periods of 88 days and 11.86 years, where other periods from the planetary perturbations of the orbit have much smaller amplitudes.     

Analytical model of the long-period forced longitude librations of Mercury

The shaking of Mercury’s orbit by the planets forces librations in longitude in addition to those at harmonics of the orbital period that have been used to detect Mercury’s molten core. We extend the analytical formulation of Peale et al. (Peale, S.J., Margot, J.L., Yseboodt, M. [2009]. Icarus 199, 1-8) in order to provide a convenient means of determining the amplitudes and phases of the forced librations without resorting to numerical calculations. We derive an explicit relation between the amplitude of each forced libration and the moment of inertia parameter (B-A)/C_m. Far from resonance with the free libration period, the libration amplitudes are directly proportional to (B-A)/C_m. Librations with periods close to the free libration period of ∼12 years may have measurable (∼arcsec) amplitudes. If the free libration period is sufficiently close to Jupiter’s orbital period of 11.86 years, the amplitude of the forced libration at Jupiter’s period could exceed the 35 arcsec amplitude of the 88-day forced libration. We also show that the planetary perturbations of the mean anomaly and the longitude of pericenter of Mercury’s orbit completely determine the libration amplitudes.While these signatures do not affect spin rate at a detectable level (as currently measured by Earth-based radar), they have a much larger impact on rotational phase (affecting imaging, altimetry, and gravity sensors). Therefore, it may be important to consider planetary perturbations when interpreting future spacecraft observations of the librations.     

Theory of satellite orbit-orbit resonance

On the basis of the strong mathematical and physical parallels between orbit-orbit and spin-orbit resonances, the dynamics of mutual orbit perturbations between two satellites about a massive planet are examined, exploiting an approach previously adopted in the study of spin-orbit coupling. The satellites are assumed to have arbitrary mass ratio and to move in non-intersecting orbits of arbitrary size and eccentricity. Resonances are found to exist when the mean orbital periods are commensurable with respect to some rotating axis, which condition also involves the apsidal and nodal motions of both satellites. In any resonant state the satellites are effectively trapped in separate potential wells, and a single variable is found to describe the simultaneous librations of both satellites. The librations in longitude are 180° out-of-phase, with fixed amplitude ratio that depends only on their relative masses and semimajor axes. At the same time the stroboscopic longitude of conjunction also librates about the commensurate axis with the same period. The theory is applicable to Saturn's resonant pairs Titan-Hyperion and Mimas-Tethys, and in these cases our calculated libration periods are in reasonably good agreement with the observed periods.This research supported under a grant from the California Institute of Technology President's Fund and NASA Contract NAS 7-100.     

The problem of the critical inclination revisited

The behaviour of the argument of the pericentre is investigated for the orbit of an artificial satellite which is moving under the potential when the inclination of the orbit is close to thecritical value tan^?1 2. The theory is developed to first order and it is applicable to all values of the eccentricity, with the exception of those in the neighbourhood of zero and unity. Four principal types of behaviour are noted and these are illustrated in appropriate phase-plane diagrams. It is shown that the two types which exhibit double libration in the argument of the pericentre are restricted to a relatively small domain in the (a, e)-plane of possible motions. Moreover, it is demonstrated that for double libration to occur it is necessary, but not sufficient, that \(e > \sqrt 6/13\) . The ranges of values of the inclination for which libration of the pericentre is a possibility are given for the more important cases. The general results are applied to the specific case of artificial Earth satellites whose orbits are inclined to the equator at angles close to the value of the critical inclination.     

Long-period forcing of Mercury's libration in longitude

Planetary perturbations of Mercury's orbit lead to forced librations in longitude in addition to the 88-day forced libration induced by Mercury's orbital motion. The forced librations are a combination of many periods, but 5.93 and 5.66 years dominate. These two periods result from the perturbations by Jupiter and Venus respectively, and they lead to a 125-year modulation of the libration amplitude corresponding to the beat frequency. Other periods are also identified with Jupiter and Venus perturbations as well as with those of the Earth, and these and other periods in the perturbations cause several arc second fluctuations in the libration extremes. The maxima of these extremes are about 30″ above and the minima about 7″ above the superposed ∼^60″, 88-day libration during the 125-year modulation. Knowledge of the nature of the long-period forced librations is important for the interpretation of the details of Mercury's rotation state to be obtained from radar and spacecraft observations. We show that the measurement of the 88-day libration amplitude for the purposes of determining Mercury's core properties is not compromised by the additional librations, because of the latter's small amplitude and long period. If the free libration in longitude has an amplitude that is large compared with that of the forced libration, its ∼10-year period will dominate the libration spectrum with the 88-day forced libration and the long-period librations from the orbital perturbations superposed. If the free libration has an amplitude that is comparable to those of the long-period forced libration, it will be revealed by erratic amplitude, period and phase on the likely time span of a series of observations. However, a significant free libration component is not expected because of relatively rapid damping.     

The Effect of GPS Ephemeris on the Accuracy of Precise Orbit Determination of LEO Satellite-borne GPS

The satellite-borne GPS receivers dedicated to precise orbit determination are now being carried by more and more low earth orbit (LEO) satellites and the satellite-borne GPS has become one of the main means for the precise orbit determination of low earth orbit satellites. The accuracy of satellite-borne GPS precise orbit determination depends on the accuracies of the GPS ephemeris and the clock error. Based on the orbit determination function of SHORDEIII zero-difference dynamics and using the observational data obtained by the GRACE satellites for the week from 2005 August 1 to 7 as an example, three versions of GPS ephemerides (igs, igr and igu) are used to carry out orbit determination under the same conditions and to estimate the effect of the GPS ephemeris accuracy on the accuracy of orbit determination of low earth orbit satellites. Our calculated results show that the two ephemerides, igs and igr, are equivalent to each other in orbit determination accuracy (about 9.5 cm), while igu is slightly less accurate, at about 10.5 cm. The effect produced by the data of the high frequency GPS satellite clock error on the accuracy of orbit determination is 1–6 cm.     

The spin state of 433 Eros and its possible implications

In this paper, we show that Asteroid (433) Eros is currently residing in a spin-orbit resonance, with its spin axis undergoing a small-amplitude libration about the Cassini state 2 of the proper mode in the nonsingular orbital element sinI/2exp(?Ω), where I the orbital inclination and Ω the longitude of the node. The period of this libration is ?53.4 kyr. By excluding these libration wiggles, we find that Eros' pole precesses with the proper orbital plane in inertial space with a period of ?61.4 kyr. Eros' resonant state forces its obliquity to oscillate with a period of ?53.4 kyr between ?76° and ?89.5°. The observed value of ?89° places it near the latter extreme of this cycle. We have used these results to probe Eros' past orbit and spin evolution. Our computations suggest that Eros is unlikely to have achieved its current spin state by solar and planetary gravitational perturbations alone. We hypothesize that some dissipative process such as thermal torques (e.g., the so-called YORP effect) may be needed in our model to obtain a more satisfactory match with data. A detailed study of this problem is left for future work.     

The vector approach to the problem of physical libration of the Moon. II. The nonlinear problem

By the new vector method in a nonlinear setting, a physical libration of the Moon is studied. Using the decomposition method on small parameters we derive the closed system of nine differential equations with terms of the first and second order of smallness. The conclusion is drawn that in the nonlinear case a connection between the librations in a longitude and latitude, though feeble, nevertheless exists; therefore, the physical libration already is impossible to subdivide into independent from each other forms of oscillations, as usually can be done. In the linear approach, ten characteristic frequencies and two special invariants of the problem are found. It is proved that, taking into account nonlinear terms, the invariants are periodic functions of time. Therefore, the stationary solution with zero frequency, formally supposing in the linear theory a resonance, in the nonlinear approach gains only small (proportional to e) periodic oscillations. Near to zero frequency of a resonance there is no, and solution of the nonlinear equations of physical libration is stable. The given nonlinear solution slightly modifies the previously unknown conical precession of the Moon’s spin axis. The character of nonlinear solutions near the basic forcing frequency Ω₁, where in the linear approach there are beats, is carefully studied. The average method on fast variables is obtained by the linear system of differential equations with almost periodic coefficients, which describe the evolution of these coefficients in a nonlinear problem. From this follows that the nonlinear components only slightly modify the specified beats; the interior period T ≈ 16.53 days appears 411 times less than the exterior one T ≈ 18.61 Julian years. In particular, with such a period the angle between ecliptic plane and Moon orbit plane also varies. Resonances, on which other researches earlier insisted, are not discovered. As a whole, the nonlinear analysis essentially improves and supplements a linear picture of the physical libration.     

Implications of the Galilean satellites ice envelope explosions III. The origin of the Trojans and of some comets

Secondary explosions of the primary ice fragments ejected in the explosion of the electrolyzed massive ice envelopes of the Galilean satellites are capable of imparting velocities of up to ~5km s^–1 to the secondary fragments. As a result, the secondary fragments can enter the orbits of the irregular satellites (Agafonova and Drobyshevski, 1984b) and the Trojan libration orbits. In the latter case a perturbation velocity of V 0.3–2 km s^–1 is sufficient.The primary fragments ejected by the gravitational perturbations due to the Galilean satellites sunward from Jupiter's sphere of action move faster relative to the Sun than Jupiter does and therefore reach their first aphelion ahead of Jupiter in the neighborhood of L ₄. At the same time the fragments propelled from Jupiter's sphere of action beyond the planet's orbit approach it again in their perihelia behind Jupiter in the region of L ₅. The concentration of the fragments and, hence, the frequency of their collisions and explosions at L ₄ turn out to be much greater than those at L ₅. As a result, the number of the secondary fragments of diameter 15 km captured into libration orbits ahead of Jupiter can be as high as many hundreds and should exceed by more than a factor 3.5 that captured behind Jupiter.Since the icy mix of the fragments contains hydrocarbons and particulate material (silicates and the like), after ice sublimation from the surface layers the Trojans should reveal type C and RD spectra typical for Jupiter's irregular satellites, comet nuclei and other distant ice bodies of similar origin. Among the Trojans there cannot be rocky or metallic objects which are known to exist in the main asteroid belt.It is shown that a velocity perturbation of 150–200 m s^–1 resulting from a purely mechanical impact of two bodies may be sufficient to move collision fragments from the orbits of the Trojans to horseshoe-shaped trajectories with a subsequent transfer to the cometary orbits of Jupiter's family.     

Resonance and Capture of Jupiter Comets

A number of Jupiter family comets such as Otermaand Gehrels 3make a rapid transition from heliocentric orbits outside the orbit of Jupiter to heliocentric orbits inside the orbit of Jupiter and vice versa. During this transition, the comet can be captured temporarily by Jupiter for one to several orbits around Jupiter. The interior heliocentric orbit is typically close to the 3:2 resonance while the exterior heliocentric orbit is near the 2:3 resonance. An important feature of the dynamics of these comets is that during the transition, the orbit passes close to the libration points L ₁and L ₂, two of the equilibrium points for the restricted three-body problem for the Sun-Jupiter system. Studying the libration point invariant manifold structures for L ₁and L ₂is a starting point for understanding the capture and resonance transition of these comets. For example, the recently discovered heteroclinic connection between pairs of unstable periodic orbits (one around the L ₁and the other around L ₂) implies a complicated dynamics for comets in a certain energy range. Furthermore, the stable and unstable invariant manifold tubes associated to libration point periodic orbits, of which the heteroclinic connections are a part, are phase space conduits transporting material to and from Jupiter and between the interior and exterior of Jupiter's orbit.     

On the oscillations in Mercury's obliquity

Mercury's capture into the 3:2 spin–orbit resonance can be explained as a result of its chaotic orbital dynamics. One major objective of MESSENGER and BepiColombo spatial missions is to accurately measure Mercury's rotation and its obliquity in order to obtain constraints on internal structure of the planet. Analytical approaches at the first-order level using the Cassini state assumptions give the obliquity constant or quasi-constant. Which is the obliquity's dynamical behavior deriving from a complete spin–orbit motion of Mercury simultaneously integrated with planetary interactions? We have used our SONYR model (acronym of Spin–Orbit N-bodY Relativistic model) integrating the spin–orbit N-body problem applied to the Solar System (Sun and planets). For lack of current accurate observations or ephemerides of Mercury's rotation, and therefore for lack of valid initial conditions for a numerical integration, we have built an original method for finding the libration center of the spin–orbit system and, as a consequence, for avoiding arbitrary amplitudes in librations of the spin–orbit motion as well as in Mercury's obliquity. The method has been carried out in two cases: (1) the spin–orbit motion of Mercury in the 2-body problem case (Sun–Mercury) where an uniform precession of the Keplerian orbital plane is kinematically added at a fixed inclination (S_2K case), (2) the spin–orbit motion of Mercury in the N-body problem case (Sun and planets) (S_n case). We find that the remaining amplitude of the oscillations in the S_n case is one order of magnitude larger than in the S_2K case, namely 4 versus 0.4 arcseconds (peak-to-peak). The mean obliquity is also larger, namely 1.98 versus 1.80 arcminutes, for a difference of 10.8 arcseconds. These theoretical results are in a good agreement with recent radar observations but it is not excluded that it should be possible to push farther the convergence process by drawing nearer still more precisely to the libration center. We note that the dynamically driven spin precession, which occurs when the planetary interactions are included, is more complex than the purely kinematic case. Nevertheless, in such a N-body problem, we find that the 3:2 spin–orbit resonance is really combined to a synchronism where the spin and orbit poles on average precess at the same rate while the orbit inclination and the spin axis orientation on average decrease at the same rate. As a consequence and whether it would turn out that there exists an irreducible minimum of the oscillation amplitude, quasi-periodic oscillations found in Mercury's obliquity should be to geometrically understood as librations related to these synchronisms that both follow a Cassini state. Whatever the open question on the minimal amplitude in the obliquity's oscillations and in spite of the planetary interactions indirectly acting by the solar torque on Mercury's rotation, Mercury remains therefore in a stable equilibrium state that proceeds from a 2-body Cassini state.     

A web of secondary resonances for large <Emphasis Type="Italic">A</Emphasis>/<Emphasis Type="Italic">m</Emphasis> geostationary debris

The dynamics of space debris with very high A/m near the geostationary orbit is dominated by the gravitational coefficient C ₂₂ and the solar radiation pressure. An analysis of the stability of the orbits by the chaos indicator MEGNO and frequency analysis map FAM shows chaotic layers around the separatrix and reveals a web of sub-structures associated to resonances with the annual period of the Sun. This succession of stable thin islands and chaotic layers can be reproduced and explained by a quite simple toy model, based on a pendulum approach, perturbed, through the eccentricity, by the external (Sun) frequency. The use of suitable action-angle variables in the circulation and libration regions of the pendulum allows to point out new resonances between the geostationary libration angle and the Sun’s longitude. They correspond very well (positions, shape, width) to the structures visible on the FAM representations.     

Attitude stability of a symmetric satellite at the equilibrium points in the restricted three-body problem

A problem of attitude motion of the smallest body for the restricted three-body problem is analyzed. Axial symmetry is assumed for the body, and attention is focused on the case in which the symmetry axis is normal to the orbit plane. For libration point satellites, results are similar to those for a satellite in orbit about a single body. However, for orbit equilibrium points lying on the line joining the two larger bodies, attitude stability results depart markedly from those for the two-body problem.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and Space Administration.     

Discovery of Earth's quasi‐satellite

Abstract— The newly discovered asteroid 2003 YN₁₀₇ is currently a quasi‐satellite of the Earth, making a satellite‐like orbit of high inclination with apparent period of one year. The term quasi‐satellite is used since these large orbits are not completely closed, but rather perturbed portions of the asteroid's orbit around the Sun. Due to its extremely Earth‐like orbit, this asteroid is influenced by Earth's gravity to remain within 0.1 AU of the Earth for approximately 10 years (1997 to 2006). Prior to this, it had been on a horseshoe orbit closely following Earth's orbit for several hundred years. It will re‐enter such an orbit, and make one final libration of 123 years, after which it will have a close interaction with the Earth and transition to a circulating orbit. Chaotic effects limit our ability to determine the origin or fate of this object.