Possibility of collision between co-orbital asteroids and the Earth

Considering hypothetical co-orbital asteroids with the Earth, Tatum (1997) presented a study about the necessary radial distance in order to produce collision. He adopted an analysis considering a composition of circular two-body systems, Sun-Earth and Sun-Asteroid. He found that asteroids with radial positions between 0.9943 and 1.0057 AU would necessarily collide with the Earth. In the present work, we show that this problem cannot be treated in this way. Adopting the circular restricted three-body problem, we verified that asteroids in the supposed region of collision with the Earth have stable horseshoe trajectories. Therefore, in the co-orbital region the dynamical system has to be studied as a restricted three-body problem since the composition of two-body problems gives a misleading result. Mathematical subject classification: M50.