A KAM Tori Algorithm for Earth Satellite Orbits
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A KAM Tori Algorithm for Earth Satellite Orbits William E. Wiesel1
© American Astronautical Society (Outside the U.S.) 2017
Abstract This paper offers a new approach for constructing Kolmogorov - Arnold - Moser (KAM) tori for orbits in the full potential for a non-spherical planet. The Hamilton - Jacobi equation is solved numerically by a Newton-Rhapson iteration, achieving convergence to machine precision, and still retaining literal variable dependence. Similar iteration methods allow correcting the orbital frequencies, and permit the calculation of the state transition matrix for the full problem. Some initial numerical examples are offered. Keywords KAM theorem · Earth satellite · Perturbations
Introduction The KAM theorem, named after Kolmogorov [1], Arnold [2] and Moser [3] has at last offered a theoretical answer to the well known “small divisor” problem. But while it is of major theoretical importance in celestial mechanics, practical applications have not materialized. This seems to be as true today as it was 50 years ago when M. Henon [4] first noticed it. In an earlier effort, Wiesel [5], the current author compared the Von Ziepel method to a KAM theorem derived method for a simple problem. There were several differences
William E. Wiesel
[email protected] 1
Department of Aeronautics and Astronautics, Air Force Institute of Technology, 2950 Hobson Way, Wright Patterson AFB, OH 45433, USA
J of Astronaut Sci
between the two methods. First, there do exist perturbations in the coordinates, not just the momenta as the KAM proof postulated. Second, the frequencies change when a degenerate system is used as the reference problem, an effect the KAM theorem rules out. These difficulties were overcome for a coupled harmonic oscillator, producing a perturbation algorithm that is at once numerically based, retains the literal dependence of the variables, and which can be iterated to convergence. In this paper, that method is further developed, and successfully applied to the problem of a satellite orbiting a non-spherical Earth. The methods of [5] are extended, eliminating the dependence on numerical partial derivatives. Rather, the first two orders of the Hamilton-Jacobi equation are solved by iterative techniques. This is sufficient to describe not only the KAM torus itself, but to also extract its state transition matrix. Frequency corrections can be found through integrations over the torus as convergence proceeds. The goal of this paper is to construct a specified KAM torus and its immediate environs in the full Earth geopotential problem. A KAM torus is more than an orbit. It is a three dimensional surface embedded in a six dimensional phase space. On this surface, all the momenta Ji are constant, and all three frequencies are constant. Three angle coordinates ϑi parameterize the surface, and all increment linearly with time. It is a local realization of the Hamilton-Jacobi theorem. The torus is a static, geometric object, here rotating with the Earth below. As all three frequencies are consta
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