1,721,008 research outputs found
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
Full text linkIn 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
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
Plenty of fish in the sea: Divorce choice and the quality of singles
Full text linkIn the literature of marriage, divorce choices are usually assumed to not affect the distribution of types in the pool of singles. The scope of the present paper is to overcome this assumption. We analyse divorce choices when separation decision influences the distribution of singles and, thus, their expected quality. We consider a three-period model where heterogeneous individuals may unilaterally experience divorce and return to the marriage market. The choices of individuals are based on the change in the distribution of singles and the cost of waiting and divorcing, taking into consideration the individual’s eligibility in the marriage market. There are two main findings: Firstly, positive assortative matching dissolves with divorce for some intermediate types. Therefore, the endogenous positive assortative matching that usually emerges in models with nontransferable utility is weakened when matches can dissolve. Secondly, the existence of ranges where divorce emerges among individuals with positive assortative matching implies the existence of two disconnected classes of types. If matchings in the first period were to occur between individuals of different classes, such matches would be dissolved later
On the Sun-shadow dynamics
Full text linkWe 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
Regularization of the two-body problem via smoothing the potential
No full textWe investigate the existence of global solutions for the two-body problem, when the particles interact with a potential of the form 1/r(alpha), for alpha > 0. Our solutions are pointwise limits of approximate solutions u(alpha)(epsilon(k,)nu(k)) which solve the equation of motion with the regularized potential 1/(r(2)+epsilon(k)(2))(alpha/2),and with an initial condition nu(k); (epsilon(k,)nu(k))(k) is a sequence converging to (0,(ν) over bar) as k --> +infinity, where (ν) over bar is an initial condition leading to collision in the non-regularized problem. We classify all the possible limits and we compare them with the already known solutions, in particular with those obtained in the paper [9] by McGehee using branch regularization and block regularization. It turns out that when alpha > 2 the double limit exist, therefore in this case the problem can be regularized according to a suitable definition
Long term dynamics for the restricted N-body problem with mean motion resonances and crossing singularities
Full text linkWe 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
Modeling the overalternating bias with an asymmetric entropy measure
Full text linkPsychological research has found that human perception of randomness is biased. In particular, people consistently show the overalternating bias: they rate binary sequences of symbols (such as Heads and Tails in coin flipping) with an excess of alternation as more random than prescribed by the normative criteria of Shannon's entropy. Within data mining for medical applications, Marcellin proposed an asymmetric measure of entropy that can be ideal to account for such bias and to quantify subjective randomness. We fitted Marcellin's entropy and Renyi's entropy (a generalized form of uncertainty measure comprising many different kinds of entropies) to experimental data found in the literature with the Differential Evolution algorithm. We observed a better fit for Marcellin's entropy compared to Renyi's entropy. The fitted asymmetric entropy measure also showed good predictive properties when applied to different datasets of randomness-related tasks. We concluded that Marcellin's entropy can be a parsimonious and effective measure of subjective randomness that can be useful in psychological research about randomness perception
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
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