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We review two initial orbit determination methods for too short arcs (TSAs) of optical observations of a solar system body. These methods employ the conservation laws of Kepler's problem, and allow to attempt the linkage of TSAs referring to quite far epochs, differing by even more than one orbital period of the observed object. The first method ({\tt Link2}) concerns the linkage of 2 TSAs, and leads to a univariate polynomial equation of degree 9. An optimal property of this polynomial is proved using Gr\"obner bases theory. The second method ({\tt Link3}) is thought for the linkage of 3 TSAs, and leads to a univariate polynomial equation of degree 8. A numerical test is shown for both algorithms.

We investigate the planar motion of a mass particle in a force field defined by patching Kepler's and Stark's dynamics. This model is called Sun-shadow dynamics, referring to the motion of an Earth satellite perturbed by the solar radiation pressure and considering the Earth shadow effect. The existence of periodic orbits of brake type is proved, and the Sun-shadow dynamics is investigated by means of a Poincare'-like map defined by a quantity that is not conserved along the flow. We also present the results of our numerical investigations on some properties of the map. Moreover, we construct the invariant manifolds of the hyperbolic fixed points related to the periodic orbits of brake type. The global picture of the map shows evidence of regular and chaotic behaviour.

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 figures

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.

The modern optical telescopes produce a huge number of asteroid observations, that 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, refer 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 orbit computations to be performed 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. The equations introduced in (Gronchi et al. 2015) can be reduced to a univariate polynomial of degree 9: the unknown is the topocentric distance $\rho$ of the observed body at the mean epoch of one of the VSAs. Using elimination theory we show an optimal property of this polynomial: it has the least degree among the univariate polynomials in the same variable that are consequence of the algebraic conservation laws and are obtained without squaring operations, that can be used to bring these algebraic equations in polynomial form. In this paper we also introduce a procedure to join three VSAs belonging to different nights: from the conservation of angular momentum at the three mean epochs of the VSAs, we obtain a univariate polynomial equation of degree 8 in the topocentric distance $\rho_2$ at the intermediate epoch. This algorithm has the same computational complexity as the classical method by Gauss, but uses more information, therefore we expect that it can produce more accurate results. For both methods, linking two and three VSAs, we also discuss how to select the solutions, making use of the full two-body dynamics, and show some numerical tests comparing the results with the ones obtained by Gauss' method., Comment: 20 pages, 1 figure

We study the long term evolution of the distance between two Keplerian confocal trajectories in the framework of the averaged restricted 3-body problem. The bodies may represent the Sun, a solar system planet and an asteroid. The secular evolution of the orbital elements of the asteroid is computed by averaging the equations of motion over the mean anomalies of the asteroid and the planet. When an orbit crossing with the planet occurs the averaged equations become singular. However, it is possible to define piecewise differentiable solutions by extending the averaged vector field beyond the singularity from both sides of the orbit crossing set. In this paper we improve the previous results, concerning in particular the singularity extraction technique, and show that the extended vector fields are Lipschitz-continuous. Moreover, we consider the distance between the Keplerian trajectories of the small body and of the planet. Apart from exceptional cases, we can select a sign for this distance so that it becomes an analytic map of the orbital elements near to crossing configurations. We prove that the evolution of the 'signed' distance along the averaged vector field is more regular than that of the elements in a neighborhood of crossing times. A comparison between averaged and non-averaged evolutions and an application of these results are shown using orbits of near-Earth asteroids., Comment: 29 pages, 8 figures