• Opik theory of close encounters
• Yarkovsky effect
• Completeness of Impact Monitoring
• Imminent impactors
• Impact Monitoring

The subject of this thesis is the Impact Monitoring (IM) problem: in particular, the mathematical theory developed over the years to deal with this problem have been improved and extended. Chapter 1 is an introduction to the state of the art about IM. Chapter 2 is devoted to imminent impactors, that is the IM problem applied to very short arcs (i.e., arcs for which is not possible to obtain a full least squares orbit): we developed an orbit determination technique based on systematic ranging and on a geometric object, the Manifold Of Variations, and we find an analytical formula for the computation of the impact probability of such an object. Chapter 3 concerns the detection of the Yarkovsky effect among the NEA population, via fit to the astrometry: in particular, we propose a new filtering criterion for the identification of spurious detections based on the Yarkovsky calibration used for the identification of asteroid families, and we present the results of this new method; moreover, we show that the inclusion of a non-gravitational model is crucial for asteroids' hazard analysis in case a deep planetary encounter takes place or in case the monitoring time horizon is very far. Chapter 4 explores the problem of the completeness of IM: a new sampling of the Line Of Variations (LOV), uniform in the probability carried by each sampling interval, is proposed and proved to be optimal once the generic completeness level has been fixed; furthermore, we perform a global statistical analysis to investigate whether the foreseen level of completeness has been actually achieved by the system and we show an analytical proof to justify the power-law that appears to be satisfied by the cumulative number of virtual impactors as a function of the time elapsed from initial conditions. Chapter 5 contains new research on Opik theory: in particular we investigate the effect of a close encounter on the LOV, modeled according to the wire approximation; we also compare the analytical results with the ones obtained by numerical integration of the circular restricted three-body problem. Chapter 6 contains on-going research about a semilinear method to predict the impact location of an asteroid that has a non-zero chance of impacting our planet. The thesis also includes four appendixes, containing technical tools used through the development of the chapters.