Speaker
Mr
Davide Amato
(Technical University of Madrid)
Description
In Solar System dynamics, a close encounter with a major body is the only natural phenomenon capable of modifying the orbital elements of a body on a very short timescale. If not properly taken into account during orbit propagation close encounters may heavily degrade the quality of a solution, or even completely compromise it. When numerically integrating the equations of motion of the body in a heliocentric reference frame, a close encounter will introduce an impulsive perturbation which has to be dealt with either by decreasing the step size close to the perturbing body or by some other device. Also, a close encounter introduces *gravitational scattering*: trajectories which are close before the close encounter may diverge afterwards due to different post-encounter major axes and therefore different orbital periods [1].
The accumulation of numerical error can be reduced by integrating regularized equations of motion, whose characteristics are more advantageous for numerical integrations. When perturbations are weak it is especially convenient to integrate regularized equations which describe the variation of orbital elements, since they will follow an almost-linear behaviour [2]. At first, this fact seems to rule out orbital elements for the propagation of close encounters, which by definition introduce a strong perturbation in heliocentric orbits. Here, we circumvent this problem by switching the primary body from the Sun to the planet during the close encounter phase, along the lines of the patched conics method of preliminary orbit design [3]. Thus, the propagation is split in three weakly-perturbed legs: heliocentric pre-encounter, planetocentric, and heliocentric post-encounter. A particularly delicate aspect is the definition of the point at which to switch primary bodies during the propagation. Ideally, this has to be chosen according to a criterion
which minimizes the final propagation error and the computational cost.
We tested our approach by performing large-scale numerical simulations of close encounters in the planar Sun-Earth CR3BP. Each simulation is parametrized in the conditions at the point of minimum approach distance. The propagation performance is evaluated for encounters taking place in a wide range of asymptotic velocities, and for different kinds of heliocentric orbits. As an additional parameter we choose the geocentric distance at which the switch between the dynamics is executed, as to study the influence of the switch point on the propagation efficiency. We employ different formulations of the Dromo family of element methods [4, 5], and we compare them against the integration of the equations of motion in Cartesian coordinates (Cowell’s method) and the Kustaanheimo-Stiefel method [2]. The integrator used is an implicit multistep with variable step size and order, which automatically alternates between Adams-Bashforth-Moulton and BDF numerical schemes [6]. For each propagation, the accuracy is estimated with respect to a reference solution computed in quadruple precision using Cowell’s method, while the computational effort is measured by the number of calls to the right-hand side of the equations.
Adopting regularized element methods and switching the primary bodies increases propagation efficiency, especially for relatively low minimum approach distances. Figure 1 depicts results for close encounters with a minimum approach distance of 5.03 Earth radii. Even with a sophisticated integration scheme with variable step size and order, regularized element methods guarantee a gain of up to three orders of magnitude in accuracy for about the same computational cost as Cowell’s method. Varying the geocentric distance at which the dynamics are switched does not have an effect on the final propagation error, but it does influence the number of function calls. An optimal range of switch distances exists in which the function calls reach a minimum. Preliminary tests with different integrators for a limited set of initial conditions have shown that the existence and extension of an optimal switch distance range depend on the characteristics of the numerical scheme.
Work which is currently being carried out includes a comprehensive study aimed at the definition of a criterion for switching between heliocentric and planetocentric dynamics, with the objective of maximizing the propagation efficiency with regularized element methods. The simulations will be extended and re-parametrized for the 3D case, and the propagation efficiency will be estimated for case studies modelled on objects of particular significance for Space Situational Awareness activities, such as 99942 Apophis. Open-source software tools for the propagation of close encounters with regularized element methods will be made available through an online repository.
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**References**
[1] Valsecchi G. B., Milani A., Gronchi G. F., Chesley S. R., “Resonant returns to close approaches: Analytical
theory”. Astron. Astrophys., v. 408, pp. 1179-1196. 2003.
[2] Stiefel E. L., Scheifele G., “Linear and Regular Celestial Mechanics”. Springer-Verlag. 1971.
[3] Amato D., Bombardelli C., Baù G., “Mitigation of propagation error in interplanetary trajectories”. Adv. Astronaut. Sci., v. 155. 2015.
[4] Baù G., Bombardelli C., Peláez J. et al., “Non-singular orbital elements for special perturbations in the two-body
problem”. Mon. Not. R. Astron. Soc., v. 454(3), pp. 2890-2908. 2015.
[5] Peláez J., Hedo J. M., Rodríguez de Andrés P., “A special perturbation method in orbital dynamics”. Celest. Mech.
Dyn. Astr., v. 97(2), pp. 131-150. 2007.
[6] Radhakrishnan K., Hindmarsh A. C., “Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations”. LLNL Report, UCRL-ID-113855. 1993.
| Applicant type | First author |
|---|
Primary author
Mr
Davide Amato
(Technical University of Madrid)
Co-authors
Dr
Claudio Bombardelli
(Space Dynamics Group, Technical University of Madrid)
Dr
Giulio Ba'
(University of Pisa)
