Abstract
Short-term satellite onboard orbit propagation is required when GPS position measurements are unavailable due to an obstruction or a malfunction. In this paper, it is shown that natural intermediary orbits of the main problem provide a useful alternative for the implementation of short-term onboard orbit propagators instead of direct numerical integration. Among these intermediaries, Deprit’s radial intermediary (DRI), obtained by the elimination of the parallax transformation, shows clear merits in terms of computational efficiency and accuracy. Indeed, this proposed analytical solution is free from elliptic integrals, as opposed to other intermediaries, thus speeding the evaluation of corresponding expressions. The only remaining equation to be solved by iterations is the Kepler equation, which in most of cases does not impact the total computation time. A comprehensive performance evaluation using Monte-Carlo simulations is performed for various orbital inclinations, showing that the analytical solution based on DRI outperforms a Dormand–Prince fixed-step Runge–Kutta integrator as the inclination grows.
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Notes
While some authors consider von Zeipel’s method to be a simple application of Poincaré’s theory, the latter applies to non-degenerate Hamiltonians, whereas the main problem Hamiltonian is degenerate. Improvements in Poincaré’s theory due to von Zeipel, and following successful applications by Brouwer in the field of artificial satellites, renders the “von Zeipel–Brouwer theory” different from Poincaré’s theory (see details in Ferraz-Mello 2007).
Alternatively, adequate units of length and time can be chosen to show that Eq. (1) depends only on \(C_{2,0}\).
Note that the equatorial Hamiltonian \(\mathcal {H}_\mathrm {E}=-\frac{1}{2}(\mu /a)+\frac{1}{2}(\mu /r)\,(\alpha /r)^2\,C_{2,0}\), in spite of being a particular integrable case of the main problem (Jezewski 1983), does not comply with the traditional requirements on intermediary orbits because, as easily checked, its secular terms differ from those of the main problem in first order effects.
According to common practice, the prime notation is suppressed to simplify notation, as long as there is no risk of confusion.
The DOPRI5 code is publicly available at http://www.unige.ch/hairer/prog/nonstiff/dopri5.f.
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This research was supported by the Government of Spain (Projects AYA 2009-11896, AYA 2010-18796) and by the European Research Council Starting Independent Researcher Grant 278231: Flight Algorithms for Disaggregated Space Architectures (FADER).
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Gurfil, P., Lara, M. Satellite onboard orbit propagation using Deprit’s radial intermediary. Celest Mech Dyn Astr 120, 217–232 (2014). https://doi.org/10.1007/s10569-014-9576-1
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DOI: https://doi.org/10.1007/s10569-014-9576-1