On the Dynamics Around the Collinear Points in the Sun-Jupiter System

Abstract

This work aims the study of the Rapid Transition Mechanism that explains some properties of orbits of some spatial objects, as for instance, comet 39P/Oterma, which will be the main object of this research.

Considering Sun and Jupiter are the masses that more influence the considered object, this mechanism describes a transition which makes the object to change from an orbit which is outside the Jupiter's one from one inside of it or viceversa.

This mechanism is observed, in particular, in the phase space of the considered models: the Restricted Three-Body Problem, both Planar Circular and Planar Elliptic. In these models three bodies are considered, two of them, named primaries, have positive mass and their orbits evolve according to the solution of the Two-Body Problem, i.e., they are circles, ellipses, parabolas or hyperbolas, having as focus (or centre) the centre of mass of both masses. The third body (which movement is to be described) is considered to have zero mass, hence it does not influence the movement of the primaries but it is under their gravitational influence. We will present the cases on which the orbit of the primaries is a circle or an ellipse and that the orbit of the third body is confined to the same plane of movement of the primaries.

Having chosen the models to study this type of transition, we proceed to the study of the skeletons of these systems, i.e., which invariant objects are the more important and responsible to describe Oterma's dynamics. This methodology is general in the study of the phase space of dynamical system: these objects are equilibrium points, periodic orbits, tori, manifolds, atractors, repulsors, among others, based on each problem.

To compute the equilibrium points L1 and L2 in the circular model (which will be also used in the elliptic one) it is enough to numerically solve a polynomial equation of 5th degree, known as Euler's quintic. Afterwards, the periodic orbits around them are computed via two approaches: a semi-analytical one (which also permit the compute a good initial approximation of their stable and unstable invariant manifolds) using Birkhoff Normal Forms at the equilibirum points and a numerical one. In the elliptic model, the tori around L1 and L2 are computed using numerical techniques, approximating a parameterization using Fourier series. In fact, it is considered the mapping as integrating a period of Jupiter and an invariant curve can be computed. Due to the strong instability of the region around the equilibirum points, we consider a representation using more than 1 section in the independent variable and the 1-period-integration is done in smaller steps - this approach is called parallel shooting.

Finally, we visualize Oterma in this context. Changes of variable are done in order to fit its real data in both model. This lets us read Oterma's positions and velocities from JPL-Horizons system and represent them in synodical coordinates. Approximating the initial coordinates (projecting them in the primaries plane of movement) and integrating them in the planar elliptic model we obtain a good hint that this model is suitable to reproduce, at least partially, Oterma's dynamics. With this, we are able to visualize Oterma inside the phase space and how it interacts with the considered invariant objects. In particular, making sections in the true anomaly and in the x coordinate at the same time, it is possible to compute invariant tori around L1 and around L2 which invariant manifolds are closer to Oterma's orbit. In addition, still in these double sections we are able to visualize the heteroclinic connections between these tori near Oterma's orbit.