Giovanni F. Gronchi

Resonant returns to close approaches: Analytical theory ?

We extend ¨ Opiks theory of close encounters of a small body (either an asteroid or a comet) by explicitly introducing the nodal distance and a time coordinate. Assuming that the heliocentric motion between consecutive close encounters is Keplerian, or given by an explicit propagator, we can compute the initial conditions for an encounter as functions of the outcomes of a previous one; in this way it is possible to obtain a completely analytical theory of resonant returns. It is found that the initial conditions of a close encounter that lead to a resonant return must lie close to easily computable circles on the b- plane of the first encounter. By further assuming that the nodal distance varies uniformly with time, due to secular perturbations, and considering the derivatives of the coordinates on the b-plane of the second encounter with respect to those on the b-plane of the first encounter, we compute in the latter the location, shape and size of collision keyholes.

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.

On the Stationary Points of the Squared Distance between Two Ellipses with a Common Focus

In this paper we introduce an effective algebraic method for the computation of all the stationary points of the squared distance

Searching for the first near-Earth object family

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THE DISTRIBUTION OF ENERGY PERTURBATIONS AT PLANETARY CLOSE ENCOUNTERS

between a point on one ellipse and a point on a second ellipse with a focus in common with the first one. This problem comes from celestial mechanics, in which the minima between two elliptic Keplerian orbits are relevant to study the probability of collision; some applications of our algorithm in this field are shown. This algorithm is based on the use of the fast Fourier transform to obtain the coefficients of the resultant of the two bivariate components of the gradient of

Generalized Averaging Principle and the Secular Evolution of Planet Crossing Orbits

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Classical and modern orbit determination for asteroids

with respect to one variable and relies on specific tools in symbolic computation. An upper bound to the total number of stationary points that we have to expect in this problem is also given; this is done using some tools from algebraic geometry.

Unbiased orbit determination for the next generation asteroid/comet surveys

Abstract We report on our search for genetically related asteroids amongst the near-Earth object (NEO) population—families of NEOs akin to the well known main belt asteroid families. We used the technique proposed by Fu et al. (Fu, H., Jedicke, R., Durda, D.D., Fevig, R. Binzel, R.P. [2005]. Icarus 178(2), 434–449) supplemented with a detailed analysis of the statistical significance of the detected clusters. Their significance was assessed by comparison to identical searches performed on 1000 ‘fuzzy-real’ NEO orbit distribution models that we developed for this purpose. The family-free ‘fuzzy-real’ NEO models maintain both the micro and macro distribution of five orbital elements (ignoring the mean anomaly). Three clusters were identified that contain four or more NEOs but none of them are statistically significant at ⩾ 3 σ . The most statistically significant cluster at the ∼ 2 σ level contains four objects with H 20 and all members have long observational arcs and concomitant good orbital elements. Despite the low statistical significance we performed several other tests on the cluster to determine if it is likely a genetic family. The tests included examining the cluster’s taxonomy, size–frequency distribution, consistency with a family-forming event during tidal disruption in a close approach to Mars, and whether it is detectable in a proper element cluster search. None of these tests exclude the possibility that the cluster is a family but neither do they confirm the hypothesis. We conclude that we have not identified any NEO families.

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

AbstractÖpik’s theory of close encounters allows us to deduce an analytic expression for the distribution of energy perturbations at close encounters between small bodies and planets. The derivation of this expression highlights a simple geometric structure in the plane containing the planet and perpendicular to the unperturbed planetocentric velocity vector. The analytic formulation reproduces well the results of the numerical integrations that first pointed out the asymmetries in the distribution of energy perturbations.

Low solar elongation searches for NEO: a deep sky test and its implications for survey strategies

Planet crossing orbits give rise to mathematical singularities that make it not possible to apply the classical averaging principle to study the qualitative evolution of Near Earth Asteroids (NEAs). Recently this principle has been generalized to deal with crossings in a mathematical model with the planets on circular coplanar orbits. More accuracy is needed to compute the averaged evolution of planet crossing orbits for different purposes: computing reliable crossing times for the averaged motion, writing more precise proper elements and frequencies for NEAs, etc. In this paper we present the generalization of the averaging principle using a model where the eccentricity and the inclination of the planets are taken into account.