Solution to the main problem of the artificial satellite by reverse normalization
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ORIGINAL PAPER
Solution to the main problem of the artificial satellite by reverse normalization Martin Lara
Received: 16 April 2020 / Accepted: 29 July 2020 / Published online: 7 August 2020 © Springer Nature B.V. 2020
Abstract The nonlinearities of the dynamics of Earth artificial satellites are encapsulated by two formal integrals that are customarily computed by perturbation methods. Standard procedures begin with a Hamiltonian simplification that removes non-essential shortperiod terms from the Geopotential, and follow with the removal of both short- and long-period terms by means of two different canonical transformations that can be carried out in either order. We depart from the tradition and proceed by standard normalization to show that the Hamiltonian simplification part is dispensable. Decoupling first the motion of the orbital plane from the in-plane motion reveals as a feasible strategy to reach higher orders of the perturbation solution, which, besides, permits an efficient evaluation of the long series that comprise the analytical solution. Keywords Main problem · Hamiltonian mechanics · Normalization · Canonical perturbation theory · Floating-point arithmetic
M. Lara (B) Scientific Computing Group–GRUCACI, University of La Rioja, Madre de Dios 53, 26006 Logroño, La Rioja, Spain e-mail: [email protected] M. Lara Space Dynamics Group, Polytechnic University of Madrid– UPM, Plaza Cardenal Cisneros 3, 28040 Madrid, Spain
1 Introduction The dynamics of close Earth satellites under gravitational effects are mostly driven by perturbations of the Keplerian motion induced by the Earth oblateness. For this reason, the approximation obtained when truncating the Legendre polynomials expansion of the Geopotential to the only contribution of the zonal harmonic of the second degree, whose coefficient is usually denoted J2 , is traditionally known as the main problem of artificial satellite theory. The J2 -truncation of the gravitational potential is known to give rise to non-integrable dynamics [5,8,12,26] that comprise short- and longperiod effects, as well as secular terms [31,36]. However, due to the smallness of the J2 coefficient of the Earth, the full system can be replaced by a separable approximation, which is customarily obtained by removing the periodic effects by means of perturbation methods [54]. When written in the action-angle variables of the Kepler problem, also called Delaunay variables, the main problem Hamiltonian immediately shows that the right ascension of the ascending node is a cyclic variable. In consequence, its conjugate momentum, the projection of the angular momentum vector along the Earth’s rotation axis, is an integral of the main problem, which, therefore, is just of two degrees of freedom. Then, following Brouwer [6], the main problem Hamiltonian is normalized in two steps. First, the short-period effects are removed by means of a canonical transfor-
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mation that, after truncation to some order of the perturbation approach, turns the conjugate momentum to the mean a
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