Orbit Determination with the two-body Integrals

We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results from the observation error statistics. The performance of this method has been investigated by using a large set of simulated observations of the Pan-STARRS project.Comment: 23 pages, 1 figur

Orbit Determination with the two-body Integrals. II

The first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of a celestial body, which may be obtained either by optical or radar observations. We write polynomial equations for this problem, that we can solve using the powerful tools of computational Algebra. An algorithm to decide if the linkage of two short arcs is successful, i.e. if they belong to the same observed body, is proposed and tested numerically. In this paper we continue the research started in [Gronchi, Dimare, Milani, 'Orbit determination with the two-body intergrals', CMDA (2010) 107/3, 299-318], where the angular momentum and the energy integrals were used. A suitable component of the Laplace-Lenz vector in place of the energy turns out to be convenient, in fact the degree of the resulting system is reduced to less than half.Comment: 15 pages, 4 figure

Euler configurations and quasi-polynomial systems

In the Newtonian 3-body problem, for any choice of the three masses, there are exactly three Euler configurations (also known as the three Euler points). In Helmholtz' problem of 3 point vortices in the plane, there are at most three collinear relative equilibria. The "at most three" part is common to both statements, but the respective arguments for it are usually so different that one could think of a casual coincidence. By proving a statement on a quasi-polynomial system, we show that the "at most three" holds in a general context which includes both cases. We indicate some hard conjectures about the configurations of relative equilibrium and suggest they could be attacked within the quasi-polynomial framework.Comment: 21 pages, 6 figure

Classical Methods of Orbit Determination

N/

On the uncertainty of the minimal distance between two confocal Keplerian orbits

We introduce a regularization for the minimal distance maps, giving the locally minimal values of the distance between two points on two confocal Keplerian orbits. This allows to define a meaningful uncertainty for the minimal distance also when orbit crossings are possible, and it is useful to detect the possibility of collisions or close approaches between two celestial bodies moving approximatively on these orbits, with important consequences in the study of their dynamics. An application to the orbit of a recently discovered near--Earth asteroid is also given

Proper elements for Earth crossing asteroids

Proper elements for 987 known near-Earth asteroids (NEAs) have been computed using a generalized averaging principle, valid also for planet-crossing orbits. This allows us to assess the maximum and minimum eccentricities and inclinations resulting from secular perturbations, to determine which planetary orbits can be crossed, and to find the encounter conditions, which can also be used to search for parent bodies of meteor streams. Proper frequencies for these asteroids have also been computed; it is then possible to draw a map of the NEAs affected by the main secular resonances. This paper describes the methods used in this computation, discusses the main qualitative properties of the results, and announces the availability of the proper elements online

Mutual geometry of confocal Keplerian orbits: uncertainty of the MOID and search for virtual PHAs

The Minimum Orbit Intersection Distance (MOID) between two confocal Keplerian orbits is a useful tool to know if two celestial bodies can collide or undergo a very close approach. We describe some results and open problems on the number of local minimum points of the distance between two points on the two orbits and the position of such points with respect to the mutual nodes. The errors affecting the observations of an asteroid result in uncertainty in its orbit determination and, consequently, uncertainty in the MOID. The latter is always positive and is not regular where it vanishes; this prevents us from considering it as a Gaussian random variable, and from computing its covariance by standard tools. In a recent work we have introduced a regularization of the maps giving the local minimum values of the distance between two orbits. It uses a signed value of the distance, with the sign given to the MOID according to a simple orientation property. The uncertainty of the regularized MOID has been computed for a large database of orbits. In this way we have searched for Virtual PHAs, i.e. asteroids which can belong to the category of PHAs (Potentially Hazardous Asteroids) if the errors in the orbit determination are taken into account. Among the Virtual PHAs we have found objects that are not even NEA, according to their nominal orbit