Title: Long term dynamics for the restricted N-body problem with mean motion resonances and crossing singularities.
Author(s): Stefano Marò (ICMAT) and Giovanni Federico Gronchi (Università di Pisa).
Source: SIAM Journal on applied dynamical systems SIAM J. Appl. Dyn. Syst., 17(2), 1786–1815.
Date of publication: 19 June 2018.
Abstract: Celestial mechanics is concerned with the study of the motions of celestial bodies under the action of gravitational forces. Despite possessing a clear physico-astronomical connotation, this discipline is closely related to mathematics. Firstly, celestial mechanics employs the formalism and the ideas of mathematics, for which furthermore is an inexhaustible source of inspiration.
For example, Gauss for the first time used with success the least squares method to determine the orbit of the first asteroid (Ceres), and the restricted three-body problem led Poincaré to establish the foundations of modern chaos theory.
Differential equations are used to describe the motion of planets and asteroids; specifically, differential equations with a Hamiltonian structure. The study of these systems is a classical subject in applied mathematics, and celestial mechanics makes great use of these techniques. For example, thanks to perturbation theory it is possible to find simple systems that approximate well to what one observes in the sky. Furthermore, the results of the numerical integrations of the equations can be interpreted in the light of the theory of Hamiltonian systems.
One of the greatest problems in current celestial mechanics concerns the study of the dynamics of asteroids. Most of these small celestial bodies in the solar system follow elliptical orbits in the region between the orbits of Mars and Jupiter, which is known as the asteroid main belt. However, the perturbations arising from the gravitational influence of these two major planets, especially Jupiter, have caused a considerable number of asteroids to leave the main asteroid belt and enter into orbits quite close to the Earth. These asteroids are known as Near Earth Asteroids (NEA).
From a mathematical point of view, the motion of an asteroid is modelled as a restricted N-body problem; the dynamics of the asteroid is given by the gravitational influence produced by the Sun and the Planets. The corresponding system of differential equations falls within the framework of near-integrable non-autonomous Hamiltonian systems with three degrees of freedom, the position of the planets being a known time function. The influence of the planets constitutes a small perturbation of the dynamics due to the gravitational influence of the Sun, and this latter system (Sun-asteroid) behaves in accordance with the Kepler model: the asteroid describes an elliptical path with the Sun at one of the foci, and the position on the ellipse is given by the Kepler equation.
The position of the asteroid and the planets in a heliocentric system are described by six coordinates divided into two groups: five geometric coordinates that describe the trajectories at each moment, and a sixth coordinate that determines the position throughout the trajectory. In Kepler’s problem, the five geometric variables remain constant, since the elliptical orbit does not change. However, if the influence of the planets is also taken into account, these variables cease to be constant, although they evolve very slowly (in periods of time much longer than the evolution of the sixth coordinate).
For this reason, the geometric variables are known as “slow” and the sixth as “fast”. The Hamiltonian structure of the problem then consists of two parts; one main part that corresponds to the Kepler problem and depending only on the slow variables, and a remainder that is proportional to a small parameter corresponding to the perturbations of the planets and containing dependencies of the fast coordinates of the asteroid and the planets.
From an intuitive point of view, this description by coordinates shows that the asteroid moves along an ellipse that deforms very slowly. This is a mechanism by means of which an asteroid may leave the main belt and approach the Earth as its trajectory changes.
It is then interesting to study the evolution of the asteroid’s trajectory, disregarding its position throughout that trajectory. The equations describing the slow coordinates are deduced by the Hamiltonian perturbation theory. The idea is to eliminate the fast coordinates (of both the asteroid and the planets) from the expression of the Hamiltonian by means of an average. On a highly intuitive level, we contemplate a system in which the asteroid and the planets are scattered all along their trajectories.
Formally, this is achieved by a canonical change of variable. The result is a new Hamiltonian consisting of three parts: the main part, then a first remainder proportional to the small parameter, and a second remainder proportional to the square of the small parameter. The effect of changing the variable is to move the dependency of the fast variable from the first remainder to the second. The Hamiltonian tought is found by ignoring the second remainder and is often referred to as the normal form. The execution of this plan is very complicated if a proportionality exists between the period of revolution of the asteroid and a planet. In this case, mean motion resonance is said to exist. However, it is possible to obtain a normal form that is known as the resonant normal form.
In either case, with or without resonances, theorems exist which state that the evolution according to the normal form is a good approximation to the real evolution. For more information on this question, one should consult the following reference: Morbidelli A., Modern celestial mechanics, Taylor & Francis, 2002.
These theorems are valid providing that there are no singularities corresponding to intersections between the asteroid and a planet. In that case, the normal Hamiltonian form is not differentiable, so the corresponding vector field is not continuous and it is not possible to arrive at a solution in the classical sense. Nevertheless, asteroids that close to the Earth often present intersections between their trajectories, which in principle makes the study of the dynamics by the normal form impossible.
In recent years, a possible solution has been advanced in the case without resonance (Gronchi, G.F., Tardioli, C: Secular evolution of the orbit distance in the double average restricted three-body problem with crossing singularities, Discrete Contin. Dyn. Syst. Ser. B 18 (2013), 1323-1344). This shows that it is possible to define a generalized solution that passes through the singularities. These solutions are not regular with respect to time at the moment of the intersection, but it can be shown that they are Lipschitz-continuous. Moreover, numerical experiments suggest that they may constitute a good approximation of the real evolution of the slow variables.
Stefano Marò (ICMAT) and Giovanni Federico Gronchi (Università di Pisa) have recently extended this theory to chaos with resonance, taking into account the substantial differences that it presents. For example, if one assumes a resonance with Jupiter, it is necessary to distinguish whether the orbit of the asteroid intersects the orbit of Jupiter itself or that of another planet (e.g. the Earth). Likewise in this case, numerical experiments also suggest that the generalized solution is a good approximation of the real evolution of the slow variables. A formal demonstration of these phenomena poses a challenge for future work.