30,793 research outputs found
Long term dynamics for the restricted N-body problem with mean motion resonances and crossing singularities
No full textWe consider the long term dynamics of the restricted N -body problem, modeling in a
statistical sense the motion of an asteroid in the gravitational field of the Sun and the solar
system planets. We deal with the case of a mean motion resonance with one planet and
assume that the osculating trajectory of the asteroid crosses the one of some planet, possibly
different from the resonant one, during the evolution. Such crossings produce singularities in
the differential equations for the motion of the asteroid, obtained by standard perturbation
theory. In this work we prove that the vector field of these equations can be extended to
two locally Lipschitz-continuous vector fields on both sides of a set of crossing conditions.
This allows us to define generalized solutions, continuous but not differentiable, going beyond
these singularities. Moreover, we prove that the long term evolution of the ’signed’ orbit
distance (Gronchi and Tommei 2007) between the asteroid and the planet is differentiable in
a neighborhood of the crossing times. In case of crossings with the resonant planet we recover
the known dynamical protection mechanism against collisions. We conclude with a numerical
comparison between the long term and the full evolutions in the case of asteroids belonging to
the ’Alinda’ and ’Toro’ classes (Milani et al. 1989). This work extends the results in (Gronchi
and Tardioli 2013) to the relevant case of asteroids in mean motion resonance with a planet
Symmetric Constellations of Satellites Moving Around a Central Body of Large Mass
No full textWe consider a (1 + N) -body problem in which one particle has mass m≫ 1 and the remaining N have unitary mass. We can assume that the body with larger mass (central body) is at rest at the origin, coinciding with the center of mass of the N bodies with smaller masses (satellites). The interaction force between two particles is defined through a potential of the form U∼1rα,where α∈ [1 , 2) and r is the distance between the particles. Imposing symmetry and topological constraints, we search for periodic orbits of this system by variational methods. Moreover, we use Γ -convergence theory to study the asymptotic behaviour of these orbits, as the mass of the central body increases. It turns out that the Lagrangian action functional Γ -converges to the action functional of a Kepler problem, defined on a suitable set of loops. In some cases, minimizers of the Γ -limit problem can be easily found, and they are useful to understand the motion of the satellites for large values of m. We discuss some examples, where the symmetry is defined by an action of the groups Z4 , Z2× Z2 and the rotation groups of Platonic polyhedra on the set of loops
On the stability of periodic N-body motions with the symmetry of Platonic polyhedra
No full textIn Fusco et al (2011 Inventiones Math. 185 283-332) several periodic orbits of the Newtonian N-body problem have been found as minimizers of the Lagrangian action in suitable sets of T-periodic loops, for a given T > 0. Each of them share the symmetry of one Platonic polyhedron. In this paper we first present an algorithm to enumerate all the orbits that can be found following the proof in Fusco et al (2011 Inventiones Math. 185 283-332). Then we describe a procedure aimed to compute them and study their stability. Our computations suggest that all these periodic orbits are unstable. For some cases we produce a computer-assisted proof of their instability using multiple precision interval arithmetic
On the Sun-shadow dynamics
No full textWe investigate the planar motion of a mass particle in a force field defined by patching Kepler's and Stark's dynamics. This model is called Sun-shadow dynamics, referring to the motion of an Earth satellite perturbed by the solar radiation pressure and considering the Earth shadow effect. The existence of periodic orbits of brake type is proved, and the Sun-shadow dynamics is investigated by means of a Poincaré map defined by a quantity that is not conserved along the flow. We also present the results of our numerical investigations on some properties of the map. Moreover, we construct the invariant manifolds of the hyperbolic fixed points related to the periodic orbits of brake type. The global picture of the map shows evidence of regular and chaotic behaviour
Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem
No full textWe prove the existence of a number of smooth periodic motions u(*) of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group R of one of the five Platonic polyhedra. The number N coincides with the order vertical bar R vertical bar of R and the particles have all the same mass. Our approach is variational and u(*) is a minimizer of the Lagrangian action A on a suitable subset K of the H-1 T-periodic maps u : R -> R-3N. The set K is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group R. There exist infinitely many such cones K, all with the property that A vertical bar(K) is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric-kinematic structure
Orbit identification for large sets of data: Preliminary results
No full textWe propose a strategy to attack the problems of orbit determination arising from the large number of short arcs. The method uses a solution of the linkage problem depending on the first integrals of the Keplerian motion
Backward-looking and Forward-looking NDC Pension Schemes
No full textIn order to spread notional capital accrued at retirement by members of a cohort
over their own life expectancies, pay-as-you-go notional-defined-contribution (payg-ndc)
scheme uses multipliers (different by retirement age) called conversion coefficients.
These are backward-looking (b.l.) in that they relay on survival rates observed for
previous cohorts in the past. Under increasing longevity, b.l. coefficients undervalue life
expectancies, thus preventing full implementation of actuarial fairness (benefits
equivalent to contributions) which is the main objective of ndc scheme. They also
engender chronic deficits.
Forward-looking (f.l.) coefficients, relaying on forecast survival rates, can
improve actuarial fairness. Nevertheless, they face a rather serious political difficulty in
that forecasting tools are fallible. This explains why switching to f.l. coefficients is
unable to gain social consensus.
Apart from this, the paper shows that f.l. coefficients produce ‘overshooting’. In
fact, they generate chronic surpluses. The paper also shows that frontloading pension
profile helps sustainability because it reduces both surpluses and deficits generated,
respectively, by f.l. and b.l. approaches
Orbit determination with the two-body integrals. II
No full textInternational audienceThe first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of a celestial body, which may be obtained either by optical or by radar observations. We write polynomial equations for this problem, which can be solved using the powerful tools of computational Algebra. An algorithm to decide if the of two short arcs is successful, i.e. if they belong to the same observed body, is proposed and tested numerically. This paper continues the research started in Gronchi et al. (Celest. Mech. Dyn. Astron. 107(3):299-318, 2010), where the angular momentum and the energy integrals were used. The use of a suitable component of the Laplace-Lenz vector in place of the energy turns out to be convenient, in fact the degree of the resulting system is reduced to less than half
Numerical behaviour of the Keplerian Integrals methods for initial orbit determination
No full textWe investigate the behaviour of two recent methods for the computation of preliminary orbits. These methods are based on the conservation laws of Kepler’s problem, and enable the linkage of very short arcs of optical observations even when they are separated in time by a few years. Our analysis is performed using both synthetic and real data of 822 main belt asteroids. The differences between computed and true orbital elements have been analysed for the true linkages, as well as the occurrence of alternative solutions. Some metrics have been introduced to quantify the results, with the aim of discarding as many of the false linkages as possible and keeping the vast majority of true ones. These numerical experiments provide thresholds for the metrics which take advantage of the knowledge of the ground truth: the values of these thresholds can be used in normal operation mode, when we do not know the correct values of the orbital elements and whether the linkages are true or false
Sparse multi-apparition linkages in large datasets
No full textWe present a new procedure to identify observations of known objects in large data sets of unlinked detections. It begins with a Keplerian integrals method that allows us to link two tracklets, computing preliminary orbits, even when the tracklets are separated in time by a few years. In the second step, we represent the results in a ‘graph’ where the tracklets are the nodes and the preliminary orbits are the edges. Then, acceptable ‘3-cycles’ are identified and a least squares orbit is computed for each of them. Finally, we construct sequences of n=4 tracklets by searching through the orbits of nearby 3-cycles and attempting to attribute the remaining tracklets. We calculate the technique’s efficiency at identifying unknown objects using real detections that attempt to mimic key parameters of the Minor Planet Center’s (MPC) Isolated Tracklet File (ITF) and then apply the procedure to the ITF. This procedure enables the recovery of several orbits, despite some having few tracklets per apparition. The MPC accepted >95% of our linkages and most of the non-accepted linkages are 2-apparition linkages even when those linkages contained more than half a dozen tracklets.This work was supported by the Spanish State Research Agency through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M) and through the H2020 MSCA ETN Stardust-Reloaded, Grant Agreement Number 813644. Ó. Rodríguez was supported by the Spanish grant PID2021-123968NB-I00 (AEI/FEDER/UE). G. F. Gronchi and G. Baù acknowledge the Italian project MIUR-PRIN 20178CJA2B “New frontiers of Celestial Mechanics: theory and applications” and the GNFM-INdAM, Italy .Peer ReviewedPostprint (published version
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