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A first integral to the partially averaged Newtonian potential of the three-body problem
Full text linkPerihelion Librations in the Secular Three-Body Problem
Full text linkA normal form theory for non-quasiperiodic systems is combined with the special properties of the partially averaged Newtonian potential pointed out in Pinzari (Celest Mech Dyn Astron 131(5):22, 2019) to prove, in the averaged, planar three-body problem, the existence of a plenty of motions where, periodically, the perihelion of the inner body affords librations about one equilibrium position and its ellipse squeezes to a segment before reversing its direction and again decreasing its eccentricity (perihelion librations)
Perturbation theory and canonical coordinates in celestial mechanics
Full text linkKAM theory owes most of its success to its initial motivation: the application to problems of celestial mechanics. The masterly application was offered by Arnold in the 60s who worked out a theorem, that he named the “Fundamental Theorem” (FT), especially designed for the planetary problem. However, FT could be really used at that purpose only when, about 50 years later, a set of coordinates constructively taking the invariance by rotation and close-to-integrability into account was used. Since then, some progress has been done in the symplectic assessment of the problem, and here we review such results
Euler integral and perihelion librations
Full text linkWe discuss dynamical aspects of an analysis of the two–centre problem started in [15]. The perturbative nature of our approach allows us to foresee applications to the three–body problem
Four classical methods for determining planetary elliptic elements: A comparison
Full text linkThe discovery of the asteroid Ceres by Piazzi in 1801 motivated the development of a mathematical technique proposed by Gauss, (Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections, 1963) which allows to recover the orbit of a celestial body starting from a minimum of three observations. Here we compare the method proposed by Gauss (Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections, New York, 1963) with the techniques (based on three observations) developed by Laplace (Collected Works 10, 93-146, 1780) and by Mossotti (Memoria Postuma, 1866). We also consider another method developed by Mossotti (Nuova analisi del problema di determinate le orbite dei corpi celesti, 1816-1818), based on four observations. We provide a theoretical and numerical comparison among the different procedures. As an application, we consider the computation of the orbit of the asteroid Juno
Exponential stability of fast driven systems, with an application to celestial mechanics
Full text linkWe construct a normal form suited to fast driven systems. We call so systems including actions I, angles ψ, and one fast coordinate y, moving under the action of a vector-field N depending only on I and y and with vanishing I-components. In the absence of the coordinate y, such systems have been extensively investigated and it is known that, after a small perturbing term is switched on, the normalised actions I turn to have exponentially small variations compared to the size of the perturbation. We obtain the same result of the classical situation, with the additional benefit that no trapping argument is needed, as no small denominator arises. We use the result to prove that, in the three-body problem, the level sets of a certain function called Euler integral have exponentially small variations in a short time, closely to collisions
Dependence on the observational time intervals and domain of convergence of orbital determination methods
Full text linkIn the framework of the orbital determination methods, we study some properties related to the algorithms developed by Gauss, Laplace and Mossotti. In particular, we investigate the dependence of such methods upon the size of the intervals between successive observations, encompassing also the case of two nearby observations performed within the same night. Moreover we study the convergence of Gauss algorithm by computing the maximal eigenvalue of the jacobian matrix associated to the Gauss map. Applications to asteroids and Kuiper belt objects are considered
Euler integral as a source of chaos in the three–body problem
Full text linkIn this paper we address, from a purely numerical point of view, the question, raised in Pinzari (2019), Pinzari (2020), and partly considered in Pinzari (2020), Di Ruzza et al. (2020), Chen and Pinzari (2021), 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 Euler Integral (which are in fact small), we detect the existence of chaos closely to the unperturbed separatrix. The latter result is obtained through a careful use of the machinery of covering relations, developed in Gierzkiewicz and Zgliczyński (2019), Zgliczynski and Gidea (2004), Wilczak and Zgliczynski (2003)
Properly-degenerate KAM theory (following V.I. Arnold)
No full textArnold’s “Fundamental Theorem” on properly–degenerate systems
[3, Chapter IV] is revisited and improved with particular attention to the relation
between the perturbative parameters and to the measure of the Kolmogorov
set. Relations with the planetary many–body problem are shortly
discussed
Planetary Birkhoff normal forms
Full text linkBirkhoff normal forms for the (secular) planetary problem are investigated. Existence and uniqueness is discussed and it is shown the classical Poincare variables and the RPS-variables (introduced in [6]), after a trivial lift, lead to the same Birkhoff normal form; as a corollary the Birkhoff normal form (in Poincare variables) is degenerate at all orders (answering a question of M. Herman). Non-degenerate Birkhoff normal forms for partially and totally reduced cases are provided and an application
to long-time stability of secular action variables (eccentricities and inclinations) is discussed
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