  We estimated the rate of comet and asteroid collisions with the terrestrial
planets by calculating the orbits of 13000 Jupiter-crossing objects (JCOs) and
1300 resonant asteroids and computing the probabilities of collisions based on
random-phase approximations and the orbital elements sampled with a 500 yr
step. The Bulirsh-Stoer and a symplectic orbit integrator gave similar results
for orbital evolution, but may give different collision probabilities with the
Sun. A small fraction of former JCOs reached orbits with aphelia inside
Jupiter's orbit, and some reached Apollo orbits with semi-major axes less than
2 AU, Aten orbits, and inner-Earth orbits (with aphelia less than 0.983 AU) and
remained there for millions of years. Though less than 0.1% of the total, these
objects were responsible for most of the collision probability of former JCOs
with Earth and Venus. We conclude that a significant fraction of near-Earth
objects could be extinct comets that came from the trans-Neptunian region.
