Visible spectroscopic and photometric survey of L5 Trojans: investigation of dynamical families

In this paper we present results obtained in the framework of a visible spectroscopic and photometric survey of Trojan asteroids. We concentrated on bodies orbiting at the L5 Lagrangian point of Jupiter that are also members of dynamical families. Spectroscopy is a crucial tool that allows us to characterize the mineralogical composition of families and their parent bodies, gives evidence of ongoing space weathering, and confirms family membership. We have observed 18 objects belonging to the Aneas, Astyanax, Sarpedon, and Phereclos families as defined by Beaugé and Roig (2001, Icarus 53, 391). In addition, we have determined the spectroscopic properties of 8 background Jupiter Trojans. The observed spectra are reddish with a dominance of D-type asteroids. As expected, the spectra of the non-family members are more heterogeneous compared to the spectra of family members, with the exception of the members of the Aneas family. We also confirm the lack of absorption features in the visible region, as already reported by other authors.     

Stationary Orbits in Resonant Extrasolar Planetary Systems

We present a catalog of stable and unstable apsidal corotation resonance (ACR) for the resonant planar planetary three-body problem, including both symmetric and asymmetric solutions. Calculations are performed with a new approach based on a numerical determination of the averaged Hamiltonian function. It has the advantage of being very simple to use and, with the exception of the immediate vicinity of the collision curve, yields precise results for any values of the eccentricities and semimajor axes. The present catalog includes results for the 3/2, 3/1, and 4/1 mean-motion resonances. The 5/1 and 5/2 commensurabilities are also discussed briefly. These results complement our previous results for the 2/1 (Beaugé et al. 2006, MNRAS, 365, 1160–1170), and give a broad picture of the structure of many important planetary resonances.     

Numerical investigation on the orbital evolution of 1:2 resonant asteroids between Jupiter and Saturn

According to some investigations (Lecar and Franklin, 1973; Franklin et al., 1989; Soper et al., 1990) asteroids cannot remain for along time between Jupiter and Saturn. But as it is well known there is a near 5:2 commensurability between Jupiter and Saturn. So there might be a possibility that asteroids between Jupiter and Saturn could be trapped in a resonant relation.In order to investigate this possibility, the changes of orbital elements of an asteroid whose initial value of semi-major axis corresponds to that of a 1:2 resonant orbit were investigated by means of a double precision Cowell method. The integration routine was kindly supplied by Dr Yoshikawa.We considered first a planar restricted problem of three bodies, Sun-Jupiter-Asteroid, then a four body problem, Sun-Jupiter-Asteroid-Saturn. When integrating the equations of motion, short periodic terms were not eliminated and in the second test the interactions between Jupiter and Saturn were retained. Whether a close approach occured or not was not investigated. In every case a _j = 5.20, a _s = 9.54 and a = 8.26 were adopted as initial values of the semi-major axis of Jupiter, Saturn and Asteroid respectively.     

On the Relationship Between Fast Lyapunov Indicator and Periodic Orbits for Continuous Flows

It is already known (Froeschlé et al., 1997a) that the fast Lyapunov indicator (hereafter FLI), i.e. the computation on a relatively short time of a quantity related to the largest Lyapunov indicator, allows us to discriminate between ordered and weak chaotic motion. Using the FLI many results have been obtained on the standard map taken as a model problem. On this model we are not only able to discriminate between a short time weak chaotic motion and an ordered one, but also among regular motion between non resonant and resonant orbits. Moreover, periodic orbits are characterised by constant FLI values which appear to be related to the order of periodic orbits (Lega and Froeschlé, 2001). In the present paper we extend all these results to the case of continuous dynamical systems (the Hénon and Heiles system and the restricted three-body problem). Especially for the periodic orbits we need to introduce a new value: the orthogonal FLI in order to fully recover the results obtained for mappings.     

Explanation of dominant oblique radio emission at Jupiter and comparison to the terrestrial case

The Io-Jupiter S-bursts are series of quasi-periodic impulsive decameter radio emissions from the magnetic flux tube connecting Jupiter to its closest galilean satellite Io. This paper discusses the possibility, suggested by previous works by Hess et al., that the S-bursts are triggered by upgoing electrons accelerated (downward) by trapped Alfvén waves, that have mirrored above the Jupiter ionosphere. According to this theory, the S-bursts would correspond to wave modes that propagate at oblique angles with respect to the magnetic field. Oblique propagation is also inferred for the more slowly varying components of Io-Jupiter radio emissions. Previous works, mainly based on observations of the terrestrial AKR, whose generation process is closely related to those of S-bursts, showed that these waves are emitted on perpendicular wave modes. This discrepancy between the Jovian and Terrestrial cases has led to a controversy about the credibility of the S-bursts model by Hess et al. In the present paper, we show that indeed, the most unstable wave modes for Earth AKR, and Io-Jupiter S-bursts, as they are seen from ground based radio-telescopes, are not the same. Several causes are evaluated: observational bias, the different degree of plasma magnetization above Earth and Jupiter, the role of a cold plasma component and of plasma auroral cavities. Furthermore, we make predictions about what kind of radiation modes a probe crossing the low altitude Io-Jupiter flux tube will see.     

Planetary perturbations on Mercury’s libration in longitude

Two space missions dedicated to Mercury (MESSENGER and BepiColombo) aim at understanding its rotation and confirming the existence of a liquid core. This double challenge requires much more accurate models for the spin-orbit resonant rotation of Mercury. The purpose of this paper is to introduce planetary perturbations on Mercury’s rotation using an analytical method and to analyse the influence of the perturbations on the libration in longitude. Applying a perturbation theory based on the Lie triangle, we were able to re-introduce short periodic terms into the averaged Hamiltonian and to compute the evolution of the rotational variables. The perturbations on Mercury’s forced libration in longitude mainly come from the orbital motion of Mercury (with an amplitude around 41 arcsec that depends on the momenta of inertia). It is completed by various effects from Jupiter (11.86 and 5.93 year-periods), Venus (with a 5.66 year-period), Saturn (14.73 year-period), and the Earth (6.58 year-period). The amplitudes of the oscillations due to Jupiter and Venus are approximately 33% and 10% of those from the orbital motion of Mercury and the amplitudes of the oscillations due to Saturn and the Earth are approximately 3% and 2%. We compare the analytical results with the solution obtained from the spin-orbit numerical model SONYR.     

A planetary system with an escaping Mars

The chaotic behaviour of the motion of the planets in our Solar System is well established. In this work to model a hypothetical extrasolar planetary system our Solar System was modified in such a way that we replaced the Earth by a more massive planet and let the other planets and all the orbital elements unchanged. The major result of former numerical experiments with a modified Solar System was the appearance of a chaotic window at κ _E ∈ (4, 6), where the dynamical state of the system was highly chaotic and even the body with the smallest mass escaped in some cases. On the contrary for very large values of the mass of the Earth, even greater than that of Jupiter regular dynamical behaviour was observed. In this paper the investigations are extended to the complete Solar System and showed, that this chaotic window does still exist. Tests in different ‘Solar Systems’ clarified that including only Jupiter and Saturn with their actual masses together with a more ‘massive’ Earth (4 < κ _E < 6) perturbs the orbit of Mars so that it can even be ejected from the system. Using the results of the Laplace‐Lagrange secular theory we found secular resonances acting between the motions of the nodes of Mars, Jupiter and Saturn. These secular resonances give rise to strong chaos, which is the cause of the appearance of the instability window. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochasticity of two comets in resonance with Jupiter

The orbital stochasticity of comets P/Ciffréo (1985 XVI) and P/Maury (1985 VI), at the present time near the 5/3 and 4/3 resonances with Jupiter, is investigated using Lyapunov Characteristic Indicators. First results indicate a strong stochastic behaviour for the two comets, mainly induced by encounters with Jupiter, which looks roughly like the behaviour of the group of comets in 1/1 resonance with Jupiter.     

Evidence for Lévy Random Walks in the Evolution of Comets from the Oort Cloud

With a simple map model, derived within the framework of the planar circular restricted three-body problem (SunuuJupiteruucomet), we study the dynamical evolution of near-parabolic comets under the perturbation of Jupiter. The commonly adopted random walk assumption about the energy evolution of the comets is examined. Numerical results show that for the comets on Jupiter-crossing orbits, due to the large energy changes with Jupiter per passage, the statistical evolution of the cometary energy follows a Lévy random walk, thus statistically the final energy parameter that a comet reaches is linked to the number of passages by a power law K _f n _f. The mechanism that generates the Lévy random walk is explained in this model.     

On the character of evolution of orbits in the vicinity of the 1 : 3 Kirkwood gap

The character of orbital evolution for bodies moving near the if 1 : 3 commensurability with Jupiter was studied by model calculations for the time interval of ～500 years. A comparison of oscillations of the orbital elements a, e, q and q′ is made for ensembles of bodies along three starting orbits in the vicinity of the sharp commensurability with Jupiter. These orbits are eccentric ones of low inclinations having perihelia near the Earth's orbit. Examples of a deceleration of the rate of orbital evolution near the sharp commensurability are revealed. The existence of a group of asteroids connected with the Kirkwood gap, i.e., being in a resonant motion with Jupiter, is suggested. A connection of asteroids 887 Alinda and 1915 Quetzalcoatl with this gap is confirmed.     

Additions to the Theory of the Rotation of Europa

In a previous paper (The Rotation of Europa, Henrard, Celest. Mech. Dyn. Astr., 91, 131–149, 2005) we have developed a semi-analytical theory of Europa, one of the Galilean satellites of Jupiter. It is based on a synthetic theory of the orbit of Europa and is developed in the framework of Hamiltonian formalism. It was assumed that Europa is a rigid body and Jupiter a point mass. Several additional effects should be investigated in order to complete the theory. The present contribution considers the effect of the shape of Jupiter and of the gravitational pull of Io. The sensitivity of the main theory to a change in the values of the moments of inertia of Europa is also considered.     

Secular perturbations of asteroids in secular resonance

When the precessional rate of the orbital plane of an asteroid is nearly equal to that of Jupiter, the orbital inclination of the asteroid changes quite largely due to this near equality of their precessional rates, which is called a secular resonance. In the vicinity of the exact resonance the difference of their longitudes of nodes librates with quite a long period of order of 1×10⁶ yr. In this paper we treat this secular resonance by a method of semianalytical secular perturbations with use of numerical averaging for both non-resonant and resonant asteroids and show that the results by the semi-analytical treatment agrees qualitatively with those obtained by direct numerical integrations of asteroid's orbits.     

Rotational distortion on the surfaces of Jupiter and Saturn

Having formulated the Clairaut second-order differential equations up to the fourth order in superficial distortion due to Hensen's coefficients in the previous article (El-Sharawyet al., 1989 III, hereafter denotes by SM3), we are now in a position to solve them. In this paper we shall discuss the methods of solving the Clairaut theory, to give an explicit form about the distortion of the surfaces of Jupiter and Saturn, numerically up to the fourth-order.     

Massive identification of asteroids in three-body resonances

An essential role in the asteroidal dynamics is played by the mean motion resonances. Two-body planet–asteroid resonances are widely known, due to the Kirkwood gaps. Besides, so-called three-body mean motion resonances exist, in which an asteroid and two planets participate. Identification of asteroids in three-body (namely, Jupiter–Saturn–asteroid) resonances was initially accomplished by Nesvorný and Morbidelli (Nesvorný D., Morbidelli, A. [1998]. Astron. J. 116, 3029–3037), who, by means of visual analysis of the time behaviour of resonant arguments, found 255 asteroids to reside in such resonances. We develop specialized algorithms and software for massive automatic identification of asteroids in the three-body, as well as two-body, resonances of arbitrary order, by means of automatic analysis of the time behaviour of resonant arguments. In the computation of orbits, all essential perturbations are taken into account. We integrate the asteroidal orbits on the time interval of 100,000 yr and identify main-belt asteroids in the three-body Jupiter–Saturn–asteroid resonances up to the 6th order inclusive, and in the two-body Jupiter–asteroid resonances up to the 9th order inclusive, in the set of ～250,000 objects from the “Asteroids – Dynamic Site” (AstDyS) database. The percentages of resonant objects, including extrapolations for higher-order resonances, are determined. In particular, the observed fraction of pure-resonant asteroids (those exhibiting resonant libration on the whole interval of integration) in the three-body resonances up to the 6th order inclusive is ≈0.9% of the whole set; and, using a higher-order extrapolation, the actual total fraction of pure-resonant asteroids in the three-body resonances of all orders is estimated as ≈1.1% of the whole set.     

Nonlinearity effects on resonant absorption of surface Alfvén waves in incompressible plasmas

The resonant absorption of small amplitude surface Alfvén waves is studied in nonlinear incompressible MHD for a viscous and resistive plasma. The reductive perturbation method is used to obtain the equation that governs the spatial and temporal behaviour of small amplitude nonlinear surface Alfvén waves. Numerical solutions to this equation are obtained under the initial condition that att = 0 the spatial variation is purely sinusoidal. The numerical results show that nonlinearity accelerates the wave damping due to resonant absorption. Resonant absorption is a more efficient wave damping mechanism than can be anticipated on the basis of linear theory.     

The structure of the planets Jupiter and Saturn

Planetary models for Jupiter and Saturn are computed using a fourth-order theory and a new molecular equation of state. The equation of state for the molecular hydrogen and helium planetary envelopes is taken from the Monte Carlo calculations of Slattery and Hubbard [Icarus 29, 187–192 (1976)]. Models for Jupiter are found that have a small amount of heavy elements either mixed with hydrogen and helium throughout the interior of the planet or concentrated in a small dense core. Saturn is modeled with a solar-composition hydrogen and helium envelope and a small derse core. We conclude that the molecular equation of state linked with suitable interior equations of state can produce Jovian models which satisfy the observational data. The planetary models show that the enrichment of heavy elements (relative to solar composition) is approximately 3 times for Jupiter and 10 times for Saturn.     

Electron Pitch-Angle Diffusion by ECH Waves in Earth and Jupiter Magnetospheres: Contribution to Diffuse Auroral Precipitation

Pitch-angle diffusion coefficients of electrons have been calculated for resonant interaction with electrostatic electron-cyclotron harmonic (ECH) waves using quasi linear diffusion theory. Calculations have been performed for the planets Earth and Jupiter at three radial distances for each planet. Electron precipitation fluxes have also been calculated and compared with observed fluxes. At Earth, electrons of energy ≤200 eV may be put on strong diffusion at L = 10. At lower L values, observed ECH wave amplitudes are insufficient to put electrons on strong diffusion. At Jupiter, electrons can be put on strong diffusion at all L values. However, the energy of electrons which may be put on strong diffusion decreases from about 1 keV at L = 7 to ~100 eV at L = 17. It is concluded that ECH waves may be partly responsible for diffuse auroral precipitation of low energy electrons at Jupiter for lower L values. At Earth contribution of ECH waves to diffuse aurora is quite small.     

Dynamics of the Jupiter Trojans with Saturn’s perturbation in the present configuration of the two planets

The dynamics of the two Jupiter triangular libration points perturbed by Saturn is studied in this paper. Unlike some previous works that studied the same problem via the pure numerical approach, this study is done in a semianalytic way. Using a literal solution, we are able to explain the asymmetry of two orbits around the two libration points with symmetric initial conditions. The literal solution consists of many frequencies. The amplitudes of each frequency are the same for both libration points, but the initial phase angles are different. This difference causes a temporary spatial asymmetry in the motions around the two points, but this asymmetry gradually disappears when the time goes to infinity. The results show that the two Jupiter triangular libration points should have symmetric spatial stable regions in the present status of Jupiter and Saturn. As a test of the literal solution, we study the resonances that have been extensively studied in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006). The resonance structures predicted by our analytic theory agree well with those found in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006) via a numerical approach. Two kinds of chaotic orbits are discussed. They have different behaviors in the frequency map. The first kind of chaotic orbits (inner chaotic orbits) is of small to moderate amplitudes, while the second kind of chaotic orbits (outer chaotic orbits) is of relatively larger amplitudes. Using analytical theory, we qualitatively explain the transition process from the inner chaotic orbits to the outer chaotic orbits with increasing amplitudes. A critical value of the diffusion rate is given to separate them in the frequency map. In a forthcoming paper, we will study the same problem but keep the planets in migration. The time asymmetry, which is unimportant in this paper, may cause an observable difference in the two Jupiter Trojan groups during a very fast planet migration process.