Distance Between Two Arbitrary Unperturbed Orbits

In the paper by Kholshevnikov and Vassilie, 1999, (see also references therein) the problem of finding critical points of the distance function between two confocal Keplerian elliptic orbits (hence finding the distance between them in the sense of set theory) is reduced to the determination of all real roots of a trigonometric polynomial of degree eight. In non-degenerate cases a polynomial of lower degree with such properties does not exist. Here we extend the results to all possible cases of ordered pairs of orbits in the Two–Body–Problem. There are nine main cases corresponding to three main types of orbits: ellipse, hyperbola, and parabola. Note that the ellipse–hyperbola and hyperbola–ellipse cases are not equivalent as we exclude the variable marking the position on the second curve. For our purposes rectilinear trajectories can be treated as particular (not limiting) cases of elliptic or hyperbolic orbits.     

Natural Metrics in the Spaces of Elliptic Orbits

Different natural metrizations by Hölder type on the five dimensional space of Keplerian elliptic orbits are introduced. Certain applications of topological and metrical properties of the space of Keplerian elliptic orbits to several problems of Celestial Mechanics are discussed.     

An Algebraic Method to Compute the Critical Points of the Distance Function Between Two Keplerian Orbits

We describe an efficient algorithm to compute all the critical points of the distance function between two Keplerian orbits (either bounded or unbounded) with a common focus. The critical values of this function are important for different purposes, for example to evaluate the risk of collisions of asteroids or comets with the Solar system planets. Our algorithm is based on the algebraic elimination theory: through the computation of the resultant of two bivariate polynomials, we find a 16th degree univariate polynomial whose real roots give us one component of the critical points. We discuss also some degenerate cases and show several examples, involving the orbits of the known asteroids and comets.      

On the Relationship Between Fast Lyapunov Indicator and Periodic Orbits for Continuous Flows

It is already known (Froeschlé et al., 1997a) that the fast Lyapunov indicator (hereafter FLI), i.e. the computation on a relatively short time of a quantity related to the largest Lyapunov indicator, allows us to discriminate between ordered and weak chaotic motion. Using the FLI many results have been obtained on the standard map taken as a model problem. On this model we are not only able to discriminate between a short time weak chaotic motion and an ordered one, but also among regular motion between non resonant and resonant orbits. Moreover, periodic orbits are characterised by constant FLI values which appear to be related to the order of periodic orbits (Lega and Froeschlé, 2001). In the present paper we extend all these results to the case of continuous dynamical systems (the Hénon and Heiles system and the restricted three-body problem). Especially for the periodic orbits we need to introduce a new value: the orthogonal FLI in order to fully recover the results obtained for mappings.     

Some Recurrence Relations Between Hansen Coefficients

A new system of recurrence relations for Hansen coefficients is obtained. This system gives a connection between only those coefficients which are included in the disturbing function of planetary or satellite motion and allows to compute efficiently the Hansen coefficients for perturbations both from internal and external bodies. The recurrence process can be realized both from high to low and from low to high harmonical terms of the disturbing function. The corresponding algorithms of evaluation of Hansen coefficients are presented. The efficiency of the obtained system of recurrence relations is discussed.