Asteroid Encounter Enhancement

Just what you wanted to know: that the likelihood of an encounter between the earth and an asteroid is enhanced by the gravitational pull between the two bodies. The formula for the enhancement is worked out in a new paper by University of Iowa physicist James Van Allen, the same man who, a half century ago, predicted the existence (later confirmed) of plasma-particle radiation belts around the earth. The enhancement of the collision cross section, which Van Allen admits leaves out additional forces exerted by the sun and other planets in the solar system, equals 1 plus the square of the ratio of the escape velocity for the planet (Van Allen worked out the case Earth, Mars, Jupiter, and Saturn) to the approach velocity of the asteroid (starting far away).

Van Allen, American Journal of Physics, August 2006