Optimization of interplanetary trajectories for impulsive and continuous asteroid deflection

The chaotic dynamic of near earth objects and our limited knowledge on the asteroid population makes it impossible to predict the next impact of an asteroid with our planet. The collision at Jupiter of the comet Shoemaker-Levy 9, the more recent discovery of the asteroid 99442 Apophis, and other similar events, have stimulated the discussion on our capabilities of deflecting. an asteroid in collision route with Earth. We propose a simple expression to evaluate the merit of a given deflection strategy. We then specialize the derived analytical expression in the special cases of a kinetic impactor and of a long duration thrust deflection. We then study two examples. The first deals with a massive spacecraft equipped with an advanced low-thrust propulsion system. We show, fictitiously, that it is convenient to send such a spacecraft to impact the asteroid rather than to rendezvous and push it. In the second example, we solve the optimal interplanetary trajectory problem to deflect the asteroid 99942 Apophis. We find that several optimal interplanetary trajectories exist that allow a 750-Kg spacecraft to be launched from 2015-2026, and impact 99942 Apophis allowing prekeyhole deflections of orders of magnitude larger than the likely required amount.