The Kepler map in the three-body problem

Abstract

The Kepler map was derived by Petrosky (1986) and Chirikov and Vecheslavov (1986) as a tool for description of the long-term chaotic orbital behaviour of the comets in nearly parabolic motion. It is a two-dimensional area-preserving map, describing the motion of a comet in terms of energy and time. Its second equation is based on Kepler's third law, hence the title of the map. Since 1980s the Kepler map has become paradigmatic in a number of applications in celestial mechanics and atomic physics. It represents an important kind of general separatrix maps. Petrosky and Broucke (1988) used refined methods of mathematical physics to derive analytical expressions for its single parameter. These methods became available only in the second half of the 20th century, and it may seem that the map is inherently a very modern mathematical tool. With the help of the Jacobi integral I show that the Kepler map, including analytical formulae for its parameter, can be derived by quite elementary methods. The prehistory and applications of the Kepler map are considered and discussed.

18 pages

References in corpus (4)

• Marginal resonances and intermittent behaviour in the motion in the vicinity of a separatrix
• On the recurrence and Lyapunov time scales of the motion near the chaos border
• On the relationship between instability and Lyapunov times for the 3-body problem
• Symbolic computation of the Birkhoff normal form in the problem of stability of the triangular libration points

Cited by in corpus (14)

• Chaotic zones around gravitating binaries
• Long-term orbital dynamics of trans-Neptunian objects
• Chaos and Lévy Flights in the Three-Body Problem
• The Stability Boundary of the Distant Scattered Disk
• Chaotic dynamics around cometary nuclei
• Chaotic zones around rotating small bodies
• Dynamics around Non-Spherical Symmetric Bodies: I. The case of a spherical body with mass anomaly
• Chaotic Dynamics of Trans-Neptunian Objects Perturbed by Planet Nine
• Dynamical environments of MU69 and similar objects
• Symplectic map description of Halley's comet dynamics
• Fractal structures for the Jacobi Hamiltonian of restricted three-body problem
• Dynamical environments of (486958) Arrokoth: prior evolution and present state
• Multi-Fidelity Modelling of Low-Energy Trajectories for Space Mission Design
• Diffusion and escape times in the open-leaky standard map