Equilibria, periodic orbits around equilibria, and heteroclinic connections in the gravity field of a rotating homogeneous cube

This paper investigates the dynamics of a particle orbiting around a rotating homogeneous cube, and shows fruitful results that have implications for examining the dynamics of orbits around non-spherical celestial bodies. This study can be considered as an extension of previous research work on the dynamics of orbits around simple shaped bodies, including a straight segment, a circular ring, an annulus disk, and simple planar plates with backgrounds in celestial mechanics. In the synodic reference frame, the model of a rotating cube is established, the equilibria are calculated, and their linear stabilities are determined. Periodic orbits around the equilibria are computed using the traditional differential correction method, and their stabilities are determined by the eigenvalues of the monodromy matrix. The existence of homoclinic and heteroclinic orbits connecting periodic orbits around the equilibria is examined and proved numerically in order to understand the global orbit structure of the system. This study contributes to the investigation of irregular shaped celestial bodies that can be divided into a set of cubes.Comment: 29 pages, 16 figures, accepted for publication in Astrophysics & Space Scienc

Mission design through averaging of perturbed Keplerian systems: the paradigm of an Enceladus orbiter

International audiencePreliminary mission design for planetary satellite orbiters requires a deep knowledge of the long term dynamics that is typically obtained through averaging techniques. The problem is usually formulated in the Hamiltonian setting as a sum of the principal part, which is given through the Kepler problem, plus a small perturbation that depends on the specific features of the mission. It is usually derived from a scaling procedure of the restricted three body problem, since the two main bodies are the Sun and the planet whereas the satellite is considered as a massless particle. Sometimes, instead of the restricted three body problem, the spatial Hill problem is used. In some cases the validity of the averaging is limited to prohibitively small regions, thus, depriving the analysis of significance. We find this paradigm at Enceladus, where the validity of a first order averaging based on the Hill problem lies inside the body. However, this fact does not invalidate the technique as perturbation methods are used to reach higher orders in the averaging process. Proceeding this way, we average the Hill problem up to the sixth order obtaining valuable information on the dynamics close to Enceladus. The averaging is performed through Lie transformations and two different transformations are applied. Firstly, the mean motion is normalized whereas the goal of the second transformation is to remove the appearance of the argument of the node. The resulting Hamiltonian defines a system of one degree of freedom whose dynamics is analyzed

Periodic orbits in the gravity field of a fixed homogeneous cube

In the current study, the existence of periodic orbits around a fixed homogeneous cube is investigated, and the results have powerful implications for examining periodic orbits around non-spherical celestial bodies. In the two different types of symmetry planes of the fixed cube, periodic orbits are obtained using the method of the Poincar\'e surface of section. While in general positions, periodic orbits are found by the homotopy method. The results show that periodic orbits exist extensively in symmetry planes of the fixed cube, and also exist near asymmetry planes that contain the regular Hex cross section. The stability of these periodic orbits is determined on the basis of the eigenvalues of the monodromy matrix. This paper proves that the homotopy method is effective to find periodic orbits in the gravity field of the cube, which provides a new thought of searching for periodic orbits around non-spherical celestial bodies. The investigation of orbits around the cube could be considered as the first step of the complicated cases, and helps to understand the dynamics of orbits around bodies with complicated shapes. The work is an extension of the previous research work about the dynamics of orbits around some simple shaped bodies, including a straight segment, a circular ring, an annulus disk, and simple planar plates.Comment: 23 pages, 10 figures, accepted for publication in Astrophysics & Space Scienc

Symplectic coordinates on S2 × S2 for perturbed Keplerian problems: Application to the dynamics of a generalised Størmer problem

In order to analyse the dynamics of a given Hamiltonian system in the space defined as the Cartesian product of two spheres, we propose to combine Delaunay coordinates with Poincaré-like coordinates. The coordinates are of local character and have to be selected accordingly with the type of motions one has to take into consideration, so we distinguish the following types: (i) rectilinear motions; (ii) circular and equatorial motions; (iii) circular non-equatorial motions; (iv) non-circular equatorial motions; and (v) non-circular and non-equatorial motions. We apply the theory to study the dynamics of the reduced flow of a generalised Størmer problem that is modelled as a perturbation of the Kepler problem. After using averaging and reduction theories, the corresponding flow is analysed on the manifold S2×S2, calculating the occurring equilibria and their stability. Finally, the flow of the original problem is reconstructed, concluding the existence of some families of periodic solutions and KAM tori. © 2010 Elsevier Inc

The Keplerian regime of charged particles in planetary magnetospheres

The dynamics of a charged particle orbiting around a rotating magnetic planet is studied. The system is modelled by the two-body Hamiltonian perturbed by an axially-symmetric function which goes to infinity as soon as the particle approaches the planet. The perturbation consists in a magnetic dipole field and a corotational electric field. When it is weak compared to the Keplerian part of the Hamiltonian, we average the system with respect to the mean anomaly up to first order in terms of a small parameter defined by the ratio between the magnetic and the Keplerian interactions. After dropping higher-order terms, we use invariant theory to reduce the averaged system by virtue of its continuous and discrete symmetries, determining also the successive reduced phase spaces. Then, we study the flow of the resulting system in the most reduced phase space, describing all equilibria and their stability, as well as the different classes of bifurcations. Finally, we connect the analysis of the flow on these reduced phase spaces with the one of the original system. © 2004 Published by Elsevier B.V

FRIPON: a worldwide network to track incoming meteoroids

Context: Until recently, camera networks designed for monitoring fireballs worldwide were not fully automated, implying that in case of a meteorite fall, the recovery campaign was rarely immediate. This was an important limiting factor as the most fragile – hence precious – meteorites must be recovered rapidly to avoid their alteration. Aims: The Fireball Recovery and InterPlanetary Observation Network (FRIPON) scientific project was designed to overcome this limitation. This network comprises a fully automated camera and radio network deployed over a significant fraction of western Europe and a small fraction of Canada. As of today, it consists of 150 cameras and 25 European radio receivers and covers an area of about 1.5 × 106 km2. Methods: The FRIPON network, fully operational since 2018, has been monitoring meteoroid entries since 2016, thereby allowing the characterization of their dynamical and physical properties. In addition, the level of automation of the network makes it possible to trigger a meteorite recovery campaign only a few hours after it reaches the surface of the Earth. Recovery campaigns are only organized for meteorites with final masses estimated of at least 500 g, which is about one event per year in France. No recovery campaign is organized in the case of smaller final masses on the order of 50 to 100 g, which happens about three times a year; instead, the information is delivered to the local media so that it can reach the inhabitants living in the vicinity of the fall. Results: Nearly 4000 meteoroids have been detected so far and characterized by FRIPON. The distribution of their orbits appears to be bimodal, with a cometary population and a main belt population. Sporadic meteors amount to about 55% of all meteors. A first estimate of the absolute meteoroid flux (mag < –5; meteoroid size ≥~1 cm) amounts to 1250/yr/106 km2. This value is compatible with previous estimates. Finally, the first meteorite was recovered in Italy (Cavezzo, January 2020) thanks to the PRISMA network, a component of the FRIPON science project