541 research outputs found
The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities
We study the long term evolution of the distance between two Keplerian
confocal trajectories in the framework of the averaged restricted 3-body
problem. The bodies may represent the Sun, a solar system planet and an
asteroid. The secular evolution of the orbital elements of the asteroid is
computed by averaging the equations of motion over the mean anomalies of the
asteroid and the planet. When an orbit crossing with the planet occurs the
averaged equations become singular. However, it is possible to define piecewise
differentiable solutions by extending the averaged vector field beyond the
singularity from both sides of the orbit crossing set. In this paper we improve
the previous results, concerning in particular the singularity extraction
technique, and show that the extended vector fields are Lipschitz-continuous.
Moreover, we consider the distance between the Keplerian trajectories of the
small body and of the planet. Apart from exceptional cases, we can select a
sign for this distance so that it becomes an analytic map of the orbital
elements near to crossing configurations. We prove that the evolution of the
'signed' distance along the averaged vector field is more regular than that of
the elements in a neighborhood of crossing times. A comparison between averaged
and non-averaged evolutions and an application of these results are shown using
orbits of near-Earth asteroids.Comment: 29 pages, 8 figure
Symmetrization in Geometry
The concept of an -symmetrization is introduced, which provides a
convenient framework for most of the familiar symmetrization processes on
convex sets. Various properties of -symmetrizations are introduced and the
relations between them investigated. New expressions are provided for the
Steiner and Minkowski symmetrals of a compact convex set which exhibit a dual
relationship between them. Characterizations of Steiner, Minkowski and central
symmetrization, in terms of natural properties that they enjoy, are given and
examples are provided to show that none of the assumptions made can be dropped
or significantly weakened. Other familiar symmetrizations, such as Schwarz
symmetrization, are discussed and several new ones introduced.Comment: A chacterization of central symmetrization has been added and several
typos have been corrected. This version has been accepted for publication on
Advances in Mathematic
Orbit Determination with the two-body Integrals
We investigate a method to compute a finite set of preliminary orbits for
solar system bodies using the first integrals of the Kepler problem. This
method is thought for the applications to the modern sets of astrometric
observations, where often the information contained in the observations allows
only to compute, by interpolation, two angular positions of the observed body
and their time derivatives at a given epoch; we call this set of data
attributable. Given two attributables of the same body at two different epochs
we can use the energy and angular momentum integrals of the two-body problem to
write a system of polynomial equations for the topocentric distance and the
radial velocity at the two epochs. We define two different algorithms for the
computation of the solutions, based on different ways to perform elimination of
variables and obtain a univariate polynomial. Moreover we use the redundancy of
the data to test the hypothesis that two attributables belong to the same body
(linkage problem). It is also possible to compute a covariance matrix,
describing the uncertainty of the preliminary orbits which results from the
observation error statistics. The performance of this method has been
investigated by using a large set of simulated observations of the Pan-STARRS
project.Comment: 23 pages, 1 figur
Platonic polyhedra, periodic orbits and chaotic motions in the N-body problem with non-Newtonian forces
We consider the -body problem with interaction potential
for alpha>1. We assume
that the particles have all the same mass and that is the
order of the rotation group of one
of the five Platonic polyhedra. We study motions that, up to a relabeling
of the particles, are invariant under
. By variational techniques we prove the existence of periodic
and chaotic motions
Long term dynamics for the restricted N-body problem with mean motion resonances and crossing singularities
We 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
On the possible values of the orbit distance between a near-Earth asteroid and the Earth
We consider all the possible trajectories of a near-Earth asteroid (NEA), corresponding to the whole set of heliocentric orbital elements with perihelion distance q ≤ 1.3 au and eccentricity e ≤ 1 (NEA class). For these hypothetical trajectories, we study the range of the values of the distance from the trajectory of the Earth (assumed on a circular orbit) as a function of selected orbital elements of the asteroid. The results of this geometric approach are useful to explain some aspects of the orbital distribution of the known NEAs. We also show that the maximal orbit distance between an object in the NEA class and the Earth is attained by a parabolic orbit, with apsidal line orthogonal to the ecliptic plane. It turns out that the threshold value of q for the NEA class (qmax = 1.3 au) is very close to a critical value, below which the above result is not valid
Maps of secular resonances in the NEO region
Context. From numerical simulations, it is known that some secular resonances
may affect the motion of near-Earth objects (NEOs). However, the specific
location of the secular resonance inside the NEO region is not fully known,
because the methods previously used to predict their location can not be used
for highly eccentric orbits and when the NEOs cross the orbits of the planets.
Aims. In this paper, we aim to map the secular resonances with the planets
from Venus to Saturn in the NEO region, even for high values of the
eccentricity.
Methods. We used an averaged semi-analytical model that can deal with orbit
crossing singularities for the computation of the secular dynamics of NEOs,
from which we can obtain suitable proper elements and proper frequencies. Then,
we computed the proper frequencies over a uniform grid in the proper elements
space. Secular resonances are thus located by the level curves corresponding to
the proper frequencies of the planets.
Results. We determined the location of the secular resonances with the
planets from Venus to Saturn, showing that they appear well inside the NEO
region. By using full numerical N-body simulations we also showed that the
location predicted by our method is fairly accurate. Finally, we provided some
indications about possible dynamical paths inside the NEO region, due to the
presence of secular resonances.Comment: Accepted for publication in A&
Orbit Determination with the two-body Integrals. II
The 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 radar observations. We write polynomial equations
for this problem, that we can solve using the powerful tools of computational
Algebra. An algorithm to decide if the linkage of two short arcs is successful,
i.e. if they belong to the same observed body, is proposed and tested
numerically. In this paper we continue the research started in [Gronchi,
Dimare, Milani, 'Orbit determination with the two-body intergrals', CMDA (2010)
107/3, 299-318], where the angular momentum and the energy integrals were used.
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.Comment: 15 pages, 4 figure
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