We have shown that in the inner belt the loss of asteroids from the ν6 secular resonance and the 3:1 Jovian mean motion resonance accounts for the observation that the mean size of the asteroids increases with increasing orbital inclination. We have used that observation to constrain the Yarkovsky loss timescale and to show that the family asteroids are embedded in a background population of old ghost families. We argue that all the asteroids in the inner belt originated from a small number of asteroids and that the initial mass of the belt was similar to that of the present belt. We also show that the observed size frequency distribution of the Vesta asteroid family was determined by the action of Yarkovsky forces, and that the age of this family is comparable to the age of the solar system.

This paper reviews the basic equations used in the study of the tidal variations of the rotational and orbital elements of a system formed by one star and one close-in planet as given by the creep tide theory and Darwin’s constant time lag (CTL) theory. At the end, it reviews and discusses the determinations of the relaxation factors (and time lags) in the case of host stars and hot Jupiters based on actual observations of orbital decay, stellar rotation and age, etc. It also includes a recollection of the basic facts concerning the variations of the rotation of host stars due to the leakage of angular momentum associated with stellar winds.

We give here a detailed description of the latest INPOP planetary ephemerides INPOP20a. We test the sensitivity of the Sun oblateness determination obtained with INPOP to different models for the Sun core rotation. We also present new evaluations of possible GRT violations with the PPN parameters β, γ and . With a new method for selecting acceptable alternative ephemerides we provide conservative limits of about 7.16 × 10-5 and 7.49 × 10-5 for β-1 and γ-1 respectively using the present day planetary data samples. We also present simulations of Bepi-Colombo range tracking data and their impact on planetary ephemeris construction. We show that the use of future BC range observations should improve these estimates, in particular γ. Finally, interesting perspectives for the detection of the Sun core rotation seem to be reachable thanks to the BC mission and its accurate range measurements in the GRT frame.

The Lidov-Kozai (LK) resonance is one of the most widely discussed topics since the discovery of exoplanets in eccentric orbits. It constitutes a secular protection mechanism for systems with high mutual inclinations, although large variations in eccentricity and inclination are observed. This review aims to illustrate how the LK resonance influences the dynamics of the three-body problem at different scales, namely i) for two-planet extrasolar systems where the orbital variations occur in a coherent way such that the system remains stable, ii) for inclined planets in protoplanetary discs where the LK cycles are produced by the gravitational force exerted by the disc on the planet, iii) for migrating planets in binary star systems, whose dynamical evolution is strongly affected by the LK resonance even without experiencing a resonance capture, and iv) for triple-star systems for which the migration through LK cycles combined with tidal friction is a possible explanation for the short-period pile-up observed in the distribution of multiple stars.

We revisit the problem of the existence of KAM tori in extrasolar planetary systems. Specifically, we consider the υ Andromedæ system, by modelling it with a three-body problem. This preliminary study allows us to introduce a natural way to evaluate the robustness of the planetary orbits, which can be very easily implemented in numerical explorations. We apply our criterion to the problem of the choice of a suitable orbital configuration which exhibits strong stability properties and is compatible with the observational data that are available for the υ Andromedæ system itself.

Perturbative analyses of planetary resonances commonly predict singularities and/or divergences of resonance widths at very low and very high eccentricities. We have recently re-examined the nature of these divergences using non-perturbative numerical analyses, making use of Poincaré sections but from a different perspective relative to previous implementations of this method. This perspective reveals fine structure of resonances which otherwise remains hidden in conventional approaches, including analytical, semi-analytical and numerical-averaging approaches based on the critical resonant angle. At low eccentricity, first order resonances do not have diverging widths but have two asymmetric branches leading away from the nominal resonance location. A sequence of structures called “low-eccentricity resonant bridges” connecting neighboring resonances is revealed. At planet-grazing eccentricity, the true resonance width is non-divergent. At higher eccentricities, the new results reveal hitherto unknown resonant structures and show that these parameter regions have a loss of some – though not necessarily entire – resonance libration zones to chaos. The chaos at high eccentricities was previously attributed to the overlap of neighboring resonances. The new results reveal the additional role of bifurcations and co-existence of phase-shifted resonance zones at higher eccentricities. By employing a geometric point of view, we relate the high eccentricity phase space structures and their transitions to the shapes of resonant orbits in the rotating frame. We outline some directions for future research to advance understanding of the dynamics of mean motion resonances.

The interplanetary magnetic field may cause large amplitude changes in the orbital inclinations of charged dust particles. In order to study this effect in the case of dust grains moving in 1:1 mean motion resonance with planet Jupiter, a simplified semi-analytical model is developed to reduce the full dynamics of the system to the terms containing the information of the secular evolution dominated by the Lorentz force. It was found that while the planet causes variations in all orbital elements, the influence of the magnetic field most heavily impacts the long-term evolution of the inclination and the longitude of the ascending node. The simplified secular-resonant model recreates the oscillations in these parameters very well in comparison to the full solution, despite neglecting the influence of the magnetic field on the other orbital parameters.

Close encounters or resonances overlaps can create chaotic motion in small bodies in the Solar System. Approaches that measure the separation rate of trajectories that start infinitesimally near, or changes in the frequency power spectrum of time series, among others, can discover chaotic motion. In this paper, we introduce the ACF index (ACFI), which is based on the auto-correlation function of time series. Auto-correlation coefficients measure the correlation of a time-series with a lagged duplicate of itself. By counting the number of auto-correlation coefficients that are larger than 5% after a certain amount of time has passed, we can assess how the time series auto-correlates with each other. This allows for the detection of chaotic time-series characterized by low ACFI values.

We present a closed-form normalization method suitable for the study of the secular dynamics of small bodies inside the trajectory of Jupiter. The method is based on a convenient use of a book-keeping parameter introduced not only in the Lie series organization but also in the Poisson bracket structure employed in all perturbative steps. In particular, we show how the above scheme leads to a redefinition of the remainder of the normal form at every step of the formal solution of the homological equation. An application is given for the semi-analytical representation of the orbits of main belt asteroids.

With the success of the Cassini-Huygens mission, the dynamic complexity surrounding natural satellites of Saturn began to be elucidated. New ephemeris could be calculated with a higher level of precision, which made it possible to study in detail the resonant phenomena and, in particular, the 54:53 near mean-motion resonance between Prometheus and Atlas. For this task, we have mapped in details the domains of the resonance with dense sets of initial conditions and distinct ranges of parameters. Our initial goal was to identify possible regions in the phase space of Atlas for which some critical angles, associated with the 54:53 mean motion have a stable libration. Our investigations revealed that there is no possibility for the current Atlas orbital configuration to have any regular behavior since it is in a chaotic region located at the boundary of the 54:53 mean-motion resonance phase space. This result is in accordance with previous works (Cooper et al. 2015; Renner et al. 2016). In this work, we generalize such investigations by showing detailed aspects of the Atlas-Prometheus 54:53 mean-motion resonance, like the extension of the chaotic layers, the thin domain of the center of the 54:53 resonance, the proximity of other neighborhood resonances, among other secondary conclusions. In particular, we have also shown that even in the deep interior of the resonance, it is difficult to map periodic motion of the resonant pair for very long time spans.

The Laplace resonance is a configuration that involves the commensurability between the mean motions of three small bodies revolving around a massive central one. This resonance was first observed in the case of the three inner Galilean satellites, Io, Europa, and Ganymede. In this work the Laplace resonance is generalised by considering a system of three satellites orbiting a planet that are involved in mean motion resonances. These Laplace-like resonances are classified in three categories: first-order (2:1&2:1, 3:2&3:2, 2:1&3:2), second-order (3:1&3:1) and mixed-order resonances (2:1&3:1). In order to study the dynamics of the system we implement a model that includes the gravitational interaction with the central body, the mutual gravitational interactions of the satellites, the effects due to the oblateness of the central body and the secular interaction of a fourth satellite and a distant star. Along with these contributions we include the tidal interaction between the central body and the innermost satellite. We study the survival of the Laplace-like resonances and the evolution of the orbital elements of the satellites under the tidal effects. Moreover, we study the possibility of capture into resonance of the fourth satellite.

Normal form methods allow one to compute quasi-invariants of a Hamiltonian system, which are referred to as proper elements. The computation of the proper elements turns out to be useful to associate dynamical properties that lead to identify families of space debris, as it was done in the past for families of asteroids. In particular, through proper elements we are able to group fragments generated by the same break-up event and we possibly associate them to a parent body. A qualitative analysis of the results is given by the computation of the Pearson correlation coefficient and the probability of the Kolmogorov-Smirnov statistical test.

We consider four- and five-body problems with symmetrical masses (Caledonian problems). Families of periodic orbits originate from the collinear Schubart orbits. We present and discuss some of these periodic orbits.

A powerful tool to investigate the stability of the orbits of natural and artificial bodies is represented by perturbation theory, which allows one to provide normal form estimates for nearly-integrable problems in Celestial Mechanics. In particular, we consider the orbital stability of point-mass satellites moving around the Earth. On the basis of the J2 model, we investigate the stability of the semimajor axis. Using a secular Hamiltonian model including also lunisolar perturbations, the so-called geolunisolar model, we study the stability of the other orbital elements, namely the eccentricity and the inclination. We finally discuss the applicability of Nekhoroshev’s theorem on the exponential stability of the action variables. To this end, we investigate the non-degeneracy properties of the J2 and geolunisolar models. We obtain that the J2 model satisfies a “three-jet” non-degeneracy condition, while the geolunisolar model is quasi-convex non-degenerate.

By means of numerical simulations we study the radial-orbit instability in anisotropic self-gravitating N–body systems under the effect of noise. We find that the presence of additive or multiplicative noise has a different effect on the onset of the instability, depending on the initial value of the orbital anisotropy.

In the spin-orbit resonances, we assume that the orbit of the secondary asteroid around the primary is invariant, which is a reasonable assumption at first glance. Owing to the irregularity of asteroids’ geometry and their effect on the mutual orbit, this assumption should be revised. Therefore, we focus on a binary asteroid with a spherical primary and a secondary with an irregular shape. When the shape of a secondary asteroid is not a sphere, the gravitational interaction is important, and we should consider the interaction of orbit and spin. We generate fast Lyapunov indicator (FLI) maps for both spin-orbit resonance and spin-orbit coupling problems and investigate the effect of orbit alternation on the structure of phase space.

We have studied the probabilistic evolution of four candidates for young pairs of trans-Neptunian objects: 2003 QL91 – 2015 VA173, 1999 HV11 – 2015 VF172, 2002 CY154 – 2005 EW318 and 2013 SD101 – 2015 VY170 over 10 Myr in the past. All pairs belong to cold Classical Kuiper Belt objects. We concluded that the age of the considered pairs exceeds 10 Myr.

The objective of this paper is to carry out periodic orbital propagation and bifurcations detection around asteroid 433 Eros. Specifically, we propose to exploit a frequency-domain method, the harmonic balance method, as an efficient alternative to the usual time integration. The stability and bifurcations of the periodic orbits are also assessed thanks to the Floquet exponents. Numerous periodic orbits are found with various periods and shapes. Different bifurcations, including period doubling, tangent, real saddle and Neimark-Sacker bifurcations, are encountered during the continuation process. Resonance phenomena are highlighted as well.

Based on observations by Bailer-Jones et al. (2018) who propose a close fly-by of the K-type star Gliese 710 in approximately 1.36 Myr we investigate the immediate influence of the stellar passage on trajectories of Oort cloud objects. Using a newly developed GPU-based N-body code (Zimmermann (2021)) we study the motion of 3.6 million testparticles in the outer Solar system where the comets are distributed in three different “layers” around the Sun and the 4 giant planets. We study the immediate influence of Gliese 710 at three passage distances of 12000, 4300, and 1200 au. Additionally, different inclinations of the approaching star are considered. Depending on the passage distance a small number of comets (mainly from the disk and flared disk) is scattered into the observable region (< 5 au) around the Sun. In addition, a huge number of comets (mainly the ones directly in the path of the passing star) shows significant changes of their perihelia. But, they will enter the inner Solar system a long time after the stellar fly-by depending on their dynamical evolution.

This paper provides a study on the weak stability transition region in the framework of the planar elliptic restricted three-body problem. We define the lower boundary curve of the weak stability transition region and as a particular case, we determine this curve in the Sun-Earth system. The orbit of the Moon is near the lower boundary of the weak stability transition region.