Accurate orbit propagation in the presence of planetary close encounters

Abstract

We present an efficient strategy for the numerical propagation of small Solar System objects undergoing close encounters with massive bodies. The trajectory is split in several phases, each of them being the solution of a perturbed two-body problem. Formulations regularized with respect to different primaries are employed in two subsequent phases. In particular, we consider the Kustaanheimo-Stiefel regularization and a novel set of non-singular orbital elements pertaining to the Dromo family. In order to test the proposed strategy, we perform ensemble propagations in the Earth-Sun CR3BP using a variable step-size and order multi-step integrator and an improved version of Everhart's radau solver of 15 th order. By combining the trajectory splitting with regularized equations of motion in short-term propagations (one year), we gain up to six orders of magnitude in accuracy with respect to the classical Cowell's method for the same computational cost. Moreover, in the propagation of asteroid (99942) Apophis through its 2029 Earth encounter, the position error stays within 100 meters after 100 years. In general, as to improve the performance of regularized formulations, the trajectory must be split between 1.2 and 3 Hill radii from the Earth. We also devise a robust iterative algorithm to stop the integration of regularized equations of motion at a prescribed physical time. The results rigorously hold in the CR3BP, and similar considerations may apply when considering more complex models. The methods and algorithms are implemented in the Naples Fortran 2003 code, which is available online as a GitHub repository.

Revisiting high-order Taylor methods for astrodynamics and celestial mechanics

“…This suggests that, for this specific case, the use of EDROMO orbital elements, KS regularization and trajectory splitting methods is not necessary to achieve satisfactory results. The computational time of our approach is also significantly better than that reported by Amato et al (2017) on their approach. While it is always unfair to compare CPU times obtained in heterogeneous setups, we note how this observation is compatible with what reported in Figure 3, considering that in this case the number of planets is 𝑁 = 3 and the numerical techniques we compare to are both very similar to IAS15…”

“…Interestingly, two different numerical integration schemes are considered, XRA15 and LSODAR, both based on the Everhart-Radau numerical scheme (Everhart 1985) improved with the adaptive step size strategy proposed by Rein & Spiegel (2015), and thus resulting in numerical schemes similar to IAS15 4 . In Table 2 we summarize the comparison between our Taylor scheme and the results reported in Amato et al (2017). Interestingly, our approach is able to reach better or equivalent precision even if based on the straight forward Cowell's method.…”

“…The results, in terms of absolute position and velocity errors, 9. While the close encounter clearly degrades the integration accuracy, Apophis position, some time after the Earth flyby, is still determined within an error significantly smaller than 100 m. For comparison, we look into the results reported in Amato et al (2017), where a similar Apophis close encounter trajectory is discussed at length. A special set of orbital elements based on DROMO (Urrutxua et al 2016) and of the Kustaanheimo-Stiefel (KS) regularization (Kustaanheimo et al 1965) is reported to allow for the best precision on the same, 100 year long, numerical propagation centered around the Apophis close encounter to the Earth.…”

Non-averaged regularized formulations as an alternative to semi-analytical orbit propagation methods

“…Symplectic integration methods, which are based on the rigorous conservation of an approximate Hamiltonian of the problem, are commonly used in astrophysical research to perform extremely long integrations and may seem like a feasible candidate for the study. Nevertheless, we do not take them into account in this work since previous research has shown that the conservation of the symplectic structure does not necessarily imply the reduction of errors in position and velocity (Amato et al, 2017).…”

Uniform formulation for orbit computation: the intermediate elements

“…The special perturbation method generated from the new elements can exhibit a substantial advantage with respect to Dromo. Numerical investigations conducted by Amato et al (2017Amato et al ( , 2019 show its excellent behaviour in the propagation of both asteroids and artificial satellites of the Earth. An analogous formulation was derived independently by Roa and Peláez (2015) and Baù et al (2016) for positive values of the Keplerian and total energy, respectively.…”