Bounded Motion Design in the Earth Zonal Problem using Differential Algebra based Normal Form Methods
AbstractEstablishing long-term relative bounded motion between orbits in perturbed dynamics
is a key challenge in astrodynamics to enable cluster flight with minimum propellant
expenditure. In this work, we present an approach that allows for the design of long-term
relative bounded motion considering a zonal gravitational model. Entire sets of
orbits are obtained via high-order Taylor expansions of Poincarč return maps about
reference fixed points. The high-order normal form algorithm is used to determine a
change in expansion variables of the map into normal form space, in which the phase
space behavior is circular and can be easily parameterized by action-angle
coordinates. The action-angle representation of the normal form coordinates is then
used to parameterize the original Poincarč return map and average it over a full phase
space revolution by a path integral along the angle parameterization. As a result, the
averaged nodal period and drift in the ascending node are obtained, for which the
bounded motion conditions are straightforwardly imposed. Sets of highly accurate
bounded orbits are obtained, extending over several thousand kilometers, and valid for
A. Weisskopf, R. Armellin, M. Berz,
Celestial Mechanics and Dynamical Astronomy, 132 (2020) 14.
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