Speaker
Dr
Giulio Bau'
(University of Pisa)
Description
Close encounters with massive bodies, such as planets or Jupiter/Saturn's satellites, make the orbit of any asteroid or spacecraft chaotic. Moreover, in the case of subsequent encounters the Lyapunov time can become very short. Accurate propagation is required in the orbit determination of chaotic bodies, because it mitigates the exponential divergence of nearby orbits. For example, the impact monitoring of natural and artificial objects with the Earth, and the pianification of space missions with several fly-bys, have to be done with mathematical tools that are able to deal with chaos. One of these tools is a reliable and accurate orbit propagator.
We propose new methods to accurately compute elliptic and hyperbolic motion in the Solar System. Our approach roots in the regularization of the two-body equation, which is transformed into a set of linear differential equations with constant coefficients. This result is obtained by introducing a new independent variable (also called fictitious time) and new spatial coordinates in place of the position and velocity.
In the Burdet-Ferrándiz linearization the fictitious time is the true anomaly, and the new state variables are the inverse of the orbital radius, the radial direction and the angular momentum. In this way the motion is decomposed into the radial displacement and the rotation of the radial unit vector. We show that a new linearization of the two-body equation can be obtained with a similar decomposition when either the eccentric or the hyperbolic anomaly is the independent variable. Then, by applying the variation of parameters we introduce six variables that can be used to describe the perturbed motion of the propagated object. The new quantities, together with the total energy and the physical time, constitute the state vector of the special perturbation methods proposed here (the method that works with negative total energy is described in ref. 1). We also investigate the geometrical and physical meaning of the six parameters: they are all related to an intermediate frame which shares with the local-vertical local-orizontal frame the direction of the angular momentum. This slowly moving frame recalls the ideal frame discovered by the Danish astronomer P. A. Hansen in 1857, which plays a key role in Deprit's (ref. 2) and Peláez's (ref. 3) sets of orbital elements.
The performance of the new formulation has been evaluated for geocentric motion and for trajectories with close encounters. We found a considerable advantage with respect to the traditional integration in Cartesian coordinates.
(1) G. Baù, C. Bombardelli, J. Peláez and E. Lorenzini. Non-singular orbital elements for special perturbations in the two-body problem. Monthly Notices of the Royal Astronomical Society, Vol. 454, No. 3, pp. 2890-2908, 2015
(2) A. Deprit, A. Elipe and S. Ferrer. Ideal Elements for Perturbed Keplerian Motions. Journal of research of the National Bureau of Standards, 79B:1–15, 1975
(3) J. Peláez, J. M. Hedo and P. R. de Andrés. A special perturbation method in orbital dynamics. Celestial Mechanics and Dynamical Astronomy, 97(2):131–150, 2007
| Applicant type | First author |
|---|
Primary author
Dr
Giulio Bau'
(University of Pisa)
Co-authors
Prof.
Andrea Milani Comparetti
(University of Pisa)
Dr
Claudio Bombardelli
(Technical University of Madrid)
Dr
Davide Amato
(Technical University of Madrid)
