Reduction of Arbitrary Orbit Perturbation Problems to a Standard Form

Details Reduction of Arbitrary Orbit Perturbation Problems to a Standard Form Reduction of Arbitrary Orbit Perturbation Problems to a Standard Form

Jul 2007

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2007dda....38.1308h&link_type=abstract

American Astronomical Society, DDA meeting #38, #13.08

Statistics

Computation

Scientific paper

A novel coefficient-based method is developed to evaluate the trajectory dynamics of a perturbed particle. The perturbation acceleration vector components are represented as Fourier series in eccentric anomaly. Assuming low force magnitude such that the orbit does not change significantly over any individual revolution, Gauss's variational equations are averaged over one orbit with respect to mean anomaly. The averaged Gauss equations are shifted to eccentric anomaly as an independent parameter, resulting in expressions for the average rate of change of the six orbital elements in terms of the perturbation vector components. Substituting the Fourier series for these perturbation components, orthogonality conditions eliminate all but the 0th, 1st, and 2nd coefficients of each Fourier series. The resulting set of secular equations is a function of only 14 of the original Fourier coefficients, regardless of the order of the original Fourier series.
The set of equations resulting from this analysis is sufficient to determine the secular evolution of perturbed motion with significantly reduced computational requirements as compared to integration of the full Newtonian problem. It also provides a standard set of equations to be solved for analytical investigations of perturbed motion. One implication of this result is that arbitrary continuous time perturbations acting on a particle's dynamics over one orbit period can be effectively represented with a simple Fourier series with at most 14 coefficents. Application of this result to spacecraft trajectory control problems allows for continuous time controls to be rigorously described by only 14 constant coefficients. Examples and applications of these reductions will be presented.
This material is based upon work supported under a National Science Foundation Graduate Research Fellowship.

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