Platonic polyhedra, periodic orbits and chaotic motions in the N-body problem with non-Newtonian forces

We consider the $N$-body problem with interaction potential $U_alpha=rac{1}{ert x_i-x_jert^alpha}$ for alpha>1. We assume that the particles have all the same mass and that $N$ is the order $ertmathcal{R}ert$ of the rotation group $mathcal{R}$ of one of the five Platonic polyhedra. We study motions that, up to a relabeling of the $N$ particles, are invariant under $mathcal{R}$. By variational techniques we prove the existence of periodic and chaotic motions

On the nodal distance between two Keplerian trajectories with a common focus

We study the possible values of the nodal distance $\delta_{\rm nod}$ between two non-coplanar Keplerian trajectories ${\cal A}, {\cal A}'$ with a common focus. In particular, given ${\cal A}'$ and assuming it is bounded, we compute optimal lower and upper bounds for $\delta_{\rm nod}$ as functions of a selected pair of orbital elements of ${\cal A}$, when the other elements vary. This work arises in the attempt to extend to the elliptic case the optimal estimates for the orbit distance given in (Gronchi and Valsecchi 2013) in case of a circular trajectory ${\cal A}'$. These estimates are relevant to understand the observability of celestial bodies moving (approximately) along ${\cal A}$ when the observer trajectory is (close to) ${\cal A}'$.Comment: 34 pages, 34 figure

On the stability of periodic N-body motions with the symmetry of Platonic polyhedra

In (Fusco et. al., 2011) several periodic orbits of the Newtonian N-body problem have been found as minimizers of the Lagrangian action in suitable sets of T-periodic loops, for a given T>0. Each of them share the symmetry of one Platonic polyhedron. In this paper we first present an algorithm to enumerate all the orbits that can be found following the proof in (Fusco et. al., 2011). Then we describe a procedure aimed to compute them and study their stability. Our computations suggest that all these periodic orbits are unstable. For some cases we produce a computer-assisted proof of their instability using multiple precision interval arithmetic

On the possible values of the orbit distance between a near-Earth asteroid and the Earth

We consider all the possible trajectories of a near-Earth asteroid (NEA), corresponding to the whole set of heliocentric orbital elements with perihelion distance q ≤ 1.3 au and eccentricity e ≤ 1 (NEA class). For these hypothetical trajectories, we study the range of the values of the distance from the trajectory of the Earth (assumed on a circular orbit) as a function of selected orbital elements of the asteroid. The results of this geometric approach are useful to explain some aspects of the orbital distribution of the known NEAs. We also show that the maximal orbit distance between an object in the NEA class and the Earth is attained by a parabolic orbit, with apsidal line orthogonal to the ecliptic plane. It turns out that the threshold value of q for the NEA class (qmax = 1.3 au) is very close to a critical value, below which the above result is not valid

Orbit determination with the two-body integrals. III

We present the results of our investigation on the use of the two-body integrals to compute preliminary orbits by linking too short arcs of observations of celestial bodies. This work introduces a significant improvement with respect to the previous papers on the same subject: citet{gdm10,gfd11}. Here we find a univariate polynomial equation of degree 9 in the radial distance $ho$ of the orbit at the mean epoch of one of the two arcs. This is obtained by a combination of the algebraic integrals of the two-body problem. Moreover, the elimination step, which in (Gronchi et al. 2010, 2011) was done by resultant theory coupled with the discrete Fourier transform, is here obtained by elementary calculations. We also show some numerical tests to illustrate the performance of the new algorithm

Keplerian integrals, elimination theory and identification of very short arcs in a large database of optical observations

Modern asteroid surveys produce an increasingly large number of observations, which are grouped into very short arcs (VSAs) each containing a few observations of the same object in one single night. To decide whether two VSAs collected in different nights correspond to the same observed object we can attempt to compute an orbit with the observations of both arcs: this is called the linkage problem. Since the number of linkages to be attempted is very large, we need efficient methods of orbit determination. Using the first integrals of Kepler’s motion we can write algebraic equations for the linkage problem, which can be put in polynomial form. In Gronchi et al. (Celest Mech Dyn Astron 123(2):105–122, 2015) these equations are reduced to a polynomial equation of degree 9: the unknown is the topocentric distance of the observed body at the mean epoch of one VSA. Here we derive the same equations in a more concise way, and show that the degree 9 is optimal in a sense that will be specified in Sect. 3.3. We also introduce a procedure to join three VSAs: from the conservation of angular momentum we obtain a polynomial equation of degree 8 in the topocentric distance at the mean epoch of the second VSA. For both identification methods, with two and three VSAs, we discuss how to discard solutions. Finally, we present some numerical tests showing that the new methods give satisfactory results and can be used also when the time separation between the VSAs is large. The low polynomial degree of the new methods makes them well suited to deal with the very large number of asteroid observations collected by the modern surveys

On the computation of preliminary orbits for Earth satellites with radar observations

We introduce a new method to perform preliminary orbit determination for satellites on low Earth orbits (LEO). This method works with tracks of radar observations: each track is composed by $nge 4$ topocentric position vectors per pass of the satellite, taken at very short time intervals. We assume very accurate values for the range $ho$, while the angular positions (i.e. the line of sight, given by the pointing of the antenna) are less accurate. We wish to correct the errors in the angular positions already in the computation of a preliminary orbit. With the information contained in a pair of radar tracks, using the laws of the two-body dynamics, we can write 8 equations in 8 unknowns. The unknowns are the components of the topocentric velocity orthogonal to the line of sight at the two mean epochs of the tracks, and the corrections $Delta$ to be applied to the angular positions. We take advantage of the fact that the components of $Delta$ are typically small. We show the results of some tests, performed with simulated observations, and compare this method with Gibbs' and the Keplerian integral

Preliminary orbits with line-of-sight correction for LEO satellites observed with radar

In Fusco et al (2011 Inventiones Math. 185 283–332) several periodic orbits of the Newtonian N-body problem have been found as minimizers of the Lagrangian action in suitable sets of T-periodic loops, for a given T  >  0. Each of them share the symmetry of one Platonic polyhedron. In this paper we first present an algorithm to enumerate all the orbits that can be found following the proof in Fusco et al (2011 Inventiones Math. 185 283–332). Then we describe a procedure aimed to compute them and study their stability. Our computations suggest that all these periodic orbits are unstable. For some cases we produce a computer-assisted proof of their instability using multiple precision interval arithmetic

Periodic orbits of the N-body problem with the symmetry of platonic polyhedra.

We review some recently discovered periodic orbits of the N-body problem, whose existence is proved by means of variational methods. These orbits are minimizers of the Lagrangian action functional in a set of $T$-periodic loops, equivariant for the action of a group $G$ and satisfying some topological constraints. Both the group action and the topological constraints are defined using the symmetry of Platonic polyhedra