Periodic orbits of the planar collision restricted 3-body problem

Llibre, Jaume; Paşca, Daniel

doi:10.1007/s10569-006-9013-1

Cited by 10 publications

(17 citation statements)

References 9 publications

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“…The results of this paper are philosophically close to the ones of [8], but three main differences must be mentioned. The first one is that in our case the parameter used in the prolongation of the periodic orbits is related with the mass of the primaries and in [8] this parameter is related with the distance between the primaries.…”

Section: Introduction and Statement Of The Main Resultssupporting

confidence: 75%

“…Poincaré started the study of the periodic orbits of the nbody problem studying the periodic orbits of the planar circular restricted 3-body problem, see [11,14]. There is an extensive literature on the existence of the periodic solutions of the n-body problem, especially for the restricted 3-body problems, see the books [6,10,13] and the papers [1][2][3][4][5]8], etc.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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Pseudo periodic orbits of the planar collision restricted 3-body problem in rotating coordinates

Llibre

Roberto

2009

Nonlinear Analysis: Theory, Methods & Applications

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Section: Introduction and Statement Of The Main Resultssupporting

confidence: 75%

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Pseudo periodic orbits of the planar collision restricted 3-body problem in rotating coordinates

Llibre

Roberto

2009

Nonlinear Analysis: Theory, Methods & Applications

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“…If k = p/q is an irreducible rational then similar arguments show that in q complete orbits of the infinitesimal the primaries undergo p complete orbits. 7 .…”

Section: Continuation Of Symmetric Periodic Solutionsmentioning

confidence: 99%

“…We overcome the difficulty by using Arenstorf's theorem, where weaker assumptions of differentiability are needed (see [1]). A planar configuration of this problem is studied in [7].…”

mentioning

confidence: 99%

$J_2$ Effect and Elliptic Inclined Periodic Orbits in the Collision Restricted Three-Body Problem

Barrabés¹,

Cors²,

Pinyol³

et al. 2008

SIAM J. Appl. Dyn. Syst.

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Abstract. The existence of a new class of inclined periodic orbits of the collision restricted three-body problem is shown. The symmetric periodic solutions found are perturbations of elliptic kepler orbits and they exist only for special values of the inclination and are related to the motion of a satellite around an oblate planet.Key words. Collision restricted three-body problem, periodic orbits, symmetric orbits, critical inclination, continuation method.AMS subject classifications. 70F07, 70F15, 70H09, 70H12, 70M20.1. Introduction. The launch of the Sputnik in October 1957 opened the space age. The use of circular, elliptic, and synchronous orbits, combined with dynamical effects due to the Earths equatorial bulge gives rise to an array of orbits with specific properties to support various mission constraints. One example is the Molniya orbit: a highly elliptic 12-hour-period orbit the former USSR originally designed to observe the northern hemisphere. The orbital plane makes an angle of about 63 degrees with the equatorial plane of the Earth, and this is the only value that prevents the orbit itself from rotating slowly within its plane and around the focus.In the lines that follow we will introduce briefly a few common notions of orbital dynamics, together with the current terminology (sometimes a few centuries old), and state the aim of the paper.The position of a body on a Keplerian elliptic orbit can be completely characterized by six parameters. One such set of parameters are the classical orbital elements. As the orbital plane is fixed in any inertial frame and passes through the origen, one should give first of all the position of this plane. In a cartesian frame with axes xyz this is given by the inclination i with respect to the xy-plane and the angle Ω from the positive x-axis to the intersection of the orbital plane with the xy-plane. In the classical terminology of Astronomy this line is known as the line of nodes (the nodes of the orbit being the two points of intersection with the xy-plane, and the ascending node that in which the body crosses from z < 0 to z > 0), and Ω is called the longitude of the ascending node.Then we need the position of the ellipse on its plane. One focus is at the origen, and the line joining pericenter and apocenter (classically the line of apsides) forms an angle ω with the line of nodes which gives the position of the ellipse. Usually the half-line from origin to pericenter and the half-line from origin to the ascending node are taken, and then we say that ω is the longitude of the pericenter.The size and shape of the ellipse are given by the semi-major axis a (directly related to the energy) and the eccentricity e (related to the energy and the angular

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“…For further details the reader is referred to the mentioned work, which has been uploaded to arXiv.org as an accompanying article to this paper. It is worth mentioning that there are several recent works dealing with the search for families of POs in the RTBP and the study of their properties (Bolotin and MacKay 2006;Llibre and Paşca 2006;Perdios 2007). However, as far as we know the only work making reference to periodic transfer orbits is the one by Bruno and Varin (2006), which although dealing only with symmetric orbits in the RTBP, explores the full range of the mass parameter µ, from 0 to 1/2.…”

Section: Introductionmentioning

confidence: 99%

Extension of fast periodic transfer orbits from the Earth–Moon RTBP to the Sun–Earth–Moon Quasi-Bicircular Problem

Leiva

Briozzo

2008

Celest Mech Dyn Astr

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Starting from 80 families of low-energy fast periodic transfer orbits in the Earth-Moon planar circular Restricted Three Body Problem (RTBP), we obtain by analytical continuation 11 periodic orbits and 25 periodic arcs with similar properties in the Sun-Earth-Moon Quasi-Bicircular Problem (QBCP). A novel and very simple procedure is introduced giving the solar phases at which to attempt continuation. Detailed numerical results for each periodic orbit and arc found are given, including their stability parameters and minimal distances to the Earth and Moon. The periods of these orbits are between 2.5 and 5 synodic months, their energies are among the lowest possible to achieve an EarthMoon transfer, and they show a diversity of circumlunar trajectories, making them good candidates for missions requiring repeated passages around the Earth and the Moon with close approaches to the last.

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Periodic orbits of the planar collision restricted 3-body problem

Cited by 10 publications

References 9 publications

Pseudo periodic orbits of the planar collision restricted 3-body problem in rotating coordinates

Pseudo periodic orbits of the planar collision restricted 3-body problem in rotating coordinates

$J_2$ Effect and Elliptic Inclined Periodic Orbits in the Collision Restricted Three-Body Problem

Extension of fast periodic transfer orbits from the Earth–Moon RTBP to the Sun–Earth–Moon Quasi-Bicircular Problem

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