Orbit classification in the Hill problem: I. The classical case

Abstract

The case of the classical Hill problem is numerically investigated by performing a thorough and systematic classification of the initial conditions of the orbits. More precisely, the initial conditions of the orbits are classified into four categories: (i) non-escaping regular orbits; (ii) trapped chaotic orbits; (iii) escaping orbits; and (iv) collision orbits. In order to obtain a more general and complete view of the orbital structure of the dynamical system our exploration takes place in both planar (2D) and the spatial (3D) version of the Hill problem. For the 2D system we numerically integrate large sets of initial conditions in several types of planes, while for the system with three degrees of freedom, three-dimensional distributions of initial conditions of orbits are examined. For distinguishing between ordered and chaotic bounded motion the Smaller ALingment Index (SALI) method is used. We managed to locate the several bounded basins, as well as the basins of escape and collision and also to relate them with the corresponding escape and collision time of the orbits. Our numerical calculations indicate that the overall orbital dynamics of the Hamiltonian system is a complicated but highly interested problem. We hope our contribution to be useful for a further understanding of the orbital properties of the classical Hill problem.

Nonlinear librations of distant retrograde orbits: a perturbative approach—the Hill problem case

“…Corresponding initial conditions of the test orbits are provided in Table 1, which also includes the theoretical values of the orbital and librational periods predicted by Eqs. ( 47) and (48), respectively. Table 1: Test cases for the sixth order solution…”

“…The non-integrable problem admits two symmetric equilibria, the so-called libration points, as well as a variety of periodic orbit families. Further than these particular solutions, the global dynamics of the Hill problem must unavoidably be approached with numerical techniques [14,44,48].…”

Beyond-Newtonian dynamics of a planar circular restricted three-body problem with Kerr-like primaries

“…There is a comprehensive body of work on such "open" or "leaking" Hamiltonian systems (e.g. Barrio, Blesa & Serrano 2009;Contopoulos, Harsoula & Lukes-Gerakopoulos 2012;Ernst & Peters 2014;Kandrup et al 1999;Lai & Tél 2011;Navarro & Henrard 2001;Siopis, Contopoulos & Kandrup 1995a;Siopis et al 1995bSiopis et al , 1996Zotos 2014aZotos ,b, 2015aZotos ,b, 2016aZotos , 2017a. However, it is needless to say that this list of citations is neither complete nor exhaustive.…”

“…The Hill problem was proved to be non-integrable by Meletlidou, Ichtiaroglou & Winterberg (2001), and is chaotic, as shown by Si ḿo & Stuchi (2000). Subsequently, thorough numerical investigations of this problem were performed by carrying out a systematic classification of the initial conditions of the orbits (Zotos 2017a). More precisely, the initial conditions of the orbits were classified into four categories: (i) non escaping regular orbits; (ii) trapped chaotic orbits; (iii) escaping orbits; and (iv) collisional orbits.…”

Integrability of the generalised Hill problem

“…Intensive numerical investigations of the classical Hill problem were performed in [10]. Among other things the author gives a classification of orbits in the considered problem.…”