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

Abstract

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

Families of first kind periodic solutions of the spatial and planar elliptic restricted three-body problem with collision

“…We can find more information about this phenomenon in Breiter and Elipe (2006). Cors et al in Barrabés et al (2008) considered the problem of continuation of elliptic orbits in the ERTBP with collision, and they also found the same restriction about the inclination of the orbital plane, that is, they found that when the inclination is critical the elliptic orbits arise.…”

“…Here, the primaries move without collision, but the solutions are only approximated. The problem of the J 2 effect and elliptic inclined periodic orbits in the collision restricted three-body problem is considered in Barrabés et al (2008) and the existence of symmetric periodic solutions near a Keplerian elliptic orbit is proved.…”