On the co-orbital asteroids in the solar system: medium-term timescale analysis of the quasi-coplanar objects
The focus of this work is the current distribution of asteroids in co-orbital motion with Venus, Earth and Jupiter, under a quasi-coplanar configuration and for a medium-term timescale of the order of 900 years. A co-orbital trajectory is a heliocentric orbit trapped in a 1:1 mean-motion resonance with a given planet. As such, to model it this work considers the Restricted Three-Body Problem in the planar circular case with the help of averaging techniques. The domain of each co-orbital regime, that is, the quasi-satellite motion, the horseshoe motion and the tadpole motion, can be neatly defined by means of an integrable model and a simple two-dimensional map, that is invariant with respec…
Euler integral as a source of chaos in the three–body problem
In this paper we address, from a purely numerical point of view, the question, raised in [20, 21], and partly considered in [22, 9, 3], whether a certain function, referred to as "Euler Integral", is a quasi-integral along the trajectories of the three-body problem. Differently from our previous investigations, here we focus on the region of the "unperturbed separatrix", which turns to be complicated by a collision singularity. Concretely, we reduce the Hamiltonian to two degrees of freedom and, after fixing some energy level, we discuss in detail the resulting three-dimensional phase space around an elliptic and an hyperbolic periodic orbit. After measuring the strength of variation of the…
A New Analysis of the Three-Body Problem
In the recent papers [5, 18], respectively, the existence of motions where the perihelions afford periodic oscillations about certain equilibria and the onset of a topological horseshoe have been proved. Such results have been obtained using, as neighbouring integrable system, the so-called two-centre (or Euler) problem and a suitable canonical setting proposed in [16, 17]. Here we review such results.
Periodic and quasi-periodic orbits of the dissipative standard map
We present analytical and numerical investigations of the dynamics of the dissipative standard map. We first study the existence of periodic orbits by using a constructive version of the implicit function theorem; then, we introduce a parametric representation, which provides the interval of the drift parameter ensuring the existence of a periodic orbit with a given period. The determination of quasi--periodic attractors is efficiently obtained using the parametric representation combined with a Newton's procedure, aimed to reduce the error of the approximate solution provided by the parametric representation. These methods allow us to relate the drift parameter of the periodic orbits to th…
Numerical studies to detect chaotic motion in the full planar averaged three-body problem
AbstractIn this paper, the author deals with a well-known problem of Celestial Mechanics, namely the three-body problem. A numerical analysis has been done in order to prove existence of chaotic motions of the full-averaged problem in particular configurations. Full because all the three bodies have non-negligible masses and averaged because the Hamiltonian describing the system has been averaged with respect to a fast angle. A reduction of degrees of freedom and of the phase-space is performed in order to apply the notion of covering relations and symbolic dynamics.
Symbolic dynamics in a binary asteroid system
We highlight the existence of a topological horseshoe arising from a a--priori stable model of the binary asteroid dynamics. The inspection is numerical and uses correctly aligned windows, as described in a recent paper by A. Gierzkiewicz and P. Zgliczy\'nski, combined with a recent analysis of an associated secular problem.
The radio science experiment with BepiColombo mission to Mercury
BepiColombo is a joint ESA/JAXA mission to Mercury with challenging objectives regarding geophysics, geodesy and fundamental physics. The Mercury Orbiter Radio science Experiment (MORE) is one of the on-board experiments, including three different but linked experiments: gravimetry, rotation and relativity. Using radio observables (range and range-rate) performed with very accurate tracking from ground stations, together with optical observations from the on-board high resolution camera (SIMBIO-SYS) and accelerometer readings from the on-board accelerometer (ISA), MORE will be able to measure with unprecedented accuracy the global gravity field of Mercury and the rotation state of the plane…
Classical and relativistic n-body problem: from Levi-Civita to the most advanced interplanetary missions
The n-body problem is one of the most important issue in Celestial Mechanics. This article aims to retrace the historical and scientific events that led the Paduan mathematician, Tullio Levi-Civita, to deal with the problem first from a classic and then a relativistic point of view. We describe Levi-Civita's contributions to the theory of relativity focusing on his epistolary exchanges with Einstein, on the problem of secular acceleration and on the proof of Brillouin's cancellation principle. We also point out that the themes treated by Levi-Civita are very topical. Specifically, we analyse how the mathematical formalism used nowadays to test General Relativity can be found in Levi-Civita'…
The relativity experiment of MORE: Global full-cycle simulation and results
BepiColombo is a joint ESA/JAXA mission to Mercury with challenging objectives regarding geophysics, geodesy and fundamental physics. In particular, the Mercury Orbiter Radio science Experiment (MORE) intends, as one of its goals, to perform a test of General Relativity. This can be done by measuring and constraining the parametrized post-Newtonian (PPN) parameters to an accuracy significantly better than current one. In this work we perform a global numerical full-cycle simulation of the BepiColombo Radio Science Experiments (RSE) in a realistic scenario, focussing on the relativity experiment, solving simultaneously for all the parameters of interest for RSE in a global least squares fit …
High order normal form construction near the elliptic orbit of the Sitnikov problem
We consider the Sitnikov problem; from the equations of motion we derive the approximate Hamiltonian flow. Then, we introduce suitable action–angle variables in order to construct a high order normal form of the Hamiltonian. We introduce Birkhoff Cartesian coordinates near the elliptic orbit and we analyze the behavior of the remainder of the normal form. Finally, we derive a kind of local stability estimate in the vicinity of the periodic orbit for exponentially long times using the normal form up to 40th order in Cartesian coordinates.
Il problema degli n-corpi in relatività generale
Questa riproposizione in italiano dell’ultimo lavoro di Tullio Levi-Civita (1873- 1941) nasce da una esigenza culturale forte. Nel 2018 Padova e la sua Università hanno celebrato il loro grande scienziato con un importante convegno, accompagnato dalla ristampa di una selezione di suoi lavori, comprendente la sua tesi di laurea autografa. L’ultimo suo lavoro scientifico, “Le problème des n corps en relativité générale”, fu redatto in francese, e per le note vicende risalenti all’abominia fascista delle leggi razziali, fu stampato solo postumo, nel 1950, a cura dell’Accademia delle Scienze di Parigi.