Long-term density evolution through semi-analytical and differential algebra techniques

Abstract

This paper introduces and combines for the first time two techniques to allow long-term density propagation in astrodynamics. First, we introduce an e cient method for the propagation of phase space densities based on Di↵erential Algebra (DA) techniques. Second, this DA density propagator is used in combination with a DA implementation of the averaged orbital dynamics through semi-analytical methods. This approach combines the power of orbit averaging with the e ciency of DA techniques.While the DA-based method for the propagation of densities introduced in this paper is independent of the dynamical system under consideration, the particular combination of DA techniques with averaged equations of motion yields a fast and accurate technique to propagate large clouds of initial conditions and their associated probability density functions very e ciently for long time. This enables the study of the long-term behavior of particles subjected to the given dynamics.To demonstrate the e↵ectiveness of the proposed approach, the evolution of a cloud of high area-to-mass objects in Medium Earth Orbit is reproduced considering the e↵ects of solar radiation pressure, the Earth's oblateness and luni-solar perturbations. The method can easily propagate 10, 000 random

Phase space description of the dynamics due to the coupled effect of the planetary oblateness and the solar radiation pressure perturbations

“…As far as we know, such derivation does not exist at the moment and it can represent a fundamental tool in different fields. In particular, the stationary configurations, the invariant curves in the libration regions corresponding to the elliptic points and the hyperbolic invariant manifolds associated with the hyperbolic points can have several applications in planetary dynamics (e.g., [1,2]) and in mission design (e.g., [3,4]).…”

Probabilistic Optical and Radar Initial Orbit Determination

“…More importantly, the O2S map (and the Taylor representation of its inverse, the S2O map) can be manipulated to analytically represent the probability density function (pdf) of the IOD solution for any arbitrary pdf of the measurements. This is achieved by expanding in Taylor series the determinant of the Jacobian of the O2S map and using well known formula for pdf transformation [28,29]. This approach is independent on the representation of the dynamics and thus the Keplerian approximation can be removed.…”

A new subdivision algorithm for the flow propagation using polynomial algebras

“…Jet Transport methods where introduced by M. Berz in 1986 ([7]) to study beam dynamics in particle accelerators problems and, since then, have been used in several other fields, for instance: in Astrodynamics to study the close approaches of Near Earth Asteroids ( [3,4]), to compute a Gaussian particle filter for spacecraft navigation ( [18]), or to detect structures in dynamical systems using several indicators ( [14]). JT methods have also been used to compute the time evolution of probability density functions (PDF) according to the flow associated to an ordinary differential equation ( [13,21,5]).…”