Abstract-In the primordial solar system, the most plausible sources of the water accreted by the Earth were in the outer asteroid belt, in the giant planet regions, and in the Kuiper Belt. We investigate the implications on the origin of Earth's water of dynamical models of primordial evolution of solar system bodies and check them with respect to chemical constraints. We find that it is plausible that the Earth accreted water all along its formation, from the early phases when the solar nebula was still present to the late stages of gas-free sweepup of scattered planetesimals. Asteroids and the comets from the Jupiter-Saturn region were the first water deliverers, when the Earth was less than half its present mass. The bulk of the water presently on Earth was carried by a few planetary embryos, originally formed in the outer asteroid belt and accreted by the Earth at the final stage of its formation. Finally, a late veneer, accounting for at most 10% of the present water mass, occurred due to comets from the Uranus-Neptune region and from the Kuiper Belt. The net result of accretion from these several reservoirs is that the water on Earth had essentially the D/H ratio typical of the water condensed in the outer asteroid belt. This is in agreement with the observation that the D/H ratio in the oceans is very close to the mean value of the D/H ratio of the water inclusions in carbonaceous chondrites.

Abstract. We extendÖpik's 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 bplane 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.

Celestial Mechanics and Dynamical Astronomy, 102, pp. 111-132, http://dx.doi.org./10.1007/s10569-008-9140-yInternational audienc

When a new Near Earth Asteroid is discovered, it is important to know whether or not there is the possibility of an impact with the Earth in the near future. In this paper, we describe the technical approaches employed by the two operational second-generation asteroid impact monitoring systems, CLOMON2 and Sentry, paying particular attention to the similarities and differences between these independent systems. The detection and characterization of a potential impact requires the propagation of the orbital probability density function from the time of discovery to the time of hypothetical impact. Since the N-body problem is not integrable, this can be done only by sampling the orbital elements space with a finite number of Virtual Asteroids (VAs), the orbit of each one being propagated numerically. Our methods, illustrated in this paper, use the Line Of Variation (LOV), a unidimensional subspace, to perform this sampling. The primary goal is to detect Virtual Impactors (VIs), which are regions in the initial conditions space leading to dynamically distinct collision solutions; then a probability integral needs to be computed on the volume of the VI. An important issue is how to assure completeness of such a search down to some impact probability threshold. This problem cannot be efficiently solved just by computing more VAs, but requires Email addresses: milani@dm.unipi.it (Andrea Milani), steve.chesley@jpl.nasa.gov (Steven R. Chesley), genny@pisces.eis.uva.es (Maria Eugenia Sansaturio), tommei@mail.dm.unipi.it (Giacomo Tommei), giovanni@rm.iasf.cnr.it (Giovanni B. Valsecchi). a geometric description of the behavior of the LOV in order to identify the critical segments of this curve. We have studied these behaviors on the Target Plane (TP) through our analytical theory and the output of many numerical tests. Assuming that the geometry is the simplest compatible with the data available from the sampling, we obtain a classification which allows us to use iterative methods, appropriate for each case, to find the closest approach distance possible along the LOV. After an LOV minimum has been identified, it is possible to use a probability density linearized at this point. However, when the cross section of the Earth is not crossed by the LOV, there is no guarantee that nonlinearity would be negligible in the direction on the TP transversal to the LOV. We describe how to test for such nonlinearity, and thus reduce or eliminate the possibility of spurious VIs. In this way, the performance of our impact monitoring systems has been significantly increased in comparison to the earlier and simpler solitary system. These more advanced systems have identified and then eliminated (through additional observations) more than one hundred cases of asteroids with VIs in the years 2002-2003.

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