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Orbit propagation in Minkowskian geometry. (English) Zbl 1322.70026
Summary: The geometry of hyperbolic orbits suggests that Minkowskian geometry, and not Euclidean, may provide the most adequate description of the motion. This idea is explored in order to derive a new regularized formulation for propagating arbitrarily perturbed hyperbolic orbits. The mathematical foundations underlying Minkowski space-time \(\mathbb M\) are exploited to describe hyperbolic orbits. Hypercomplex numbers are introduced to define the rotations, vectors, and metrics in the problem: the evolution of the eccentricity vector is described on the Minkowski plane \(\mathbb R_1^2\) in terms of hyperbolic numbers, and the orbital plane is described on the inertial reference using quaternions. A set of eight orbital elements is introduced, namely a time-element, the components of the eccentricity vector in \(\mathbb R_1^2\), the semimajor axis, and the components of the quaternion defining the orbital plane. The resulting formulation provides a deep insight into the geometry of hyperbolic orbits. The performance of the formulation in long-term propagations is studied. The orbits of four hyperbolic comets are integrated and the accuracy of the solution is compared to other regularized formulations. The resulting formulation improves the stability of the integration process and it is not affected by the perihelion passage. It provides a level of accuracy that may not be reached by the compared formulations, at the cost of increasing the computational time.

MSC:
70M20 Orbital mechanics
70F15 Celestial mechanics
37D05 Dynamical systems with hyperbolic orbits and sets
37N05 Dynamical systems in classical and celestial mechanics
Software:
ODEPACK
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