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S.I. Ipatov (IAM, Moscow), L.M. Ozernoy (GMU)
Previous analytical calculations of characteristic times, T, for collisions of Earth-crossing objects (ECOs) and comets with the Earth were mainly based on Öpik's formulas. However, this approach becomes very inaccurate whenever the periods of a minor body and the planet are close to each other. While taking into account this situation, we also abandon the limitations of other approaches, which consider bodies moving in their orbots with an invariable velocity. When a minor body moving in an orbit with semimajor axis a collides with a planet at a distance r from the Sun, then T is larger by a factor of k = \sqrt{2a/r -1} compared to the approximation of a constant velocity. The obtained values of T (in Myr) and k (in parenthesis) are presented in the following table.
\begin{array}{lccccc}
& Atens & Apollos & Amors & ECOs & JFCs \\ Venus & 106~ (1.2) & 186~ (1.7) & & 154~ (1.5) & 2900~ (2.5) \\
Earth &15~ (0.9) & 164~ (1.4) & 211~ (2.0) & 67~ (1.1) &2200~ (2.3)\\
Mars &475~ (0.4) & 4250~ (0.9) &5810~ (1.1)&4710~ (1.0) &17000~ (1.8)
\end{array}
Note that relatively small values of T for Atens and all ECOs colliding with the Earth are due to just several Atens with small inclinations i, which were discovered during last two years. If for the Aten object having 0.1\circ one takes i=1\circ, then for collisions with Earth one gets 28 Myr for Atens and T=97 Myr for ECOs.
For Jupiter-family comets (JFCs) colliding with the terrestrial planets, k\ge2. For three Halley-type comets with periods between 71 and 76 yr, we obtained T=350 Gyr and k=6 (collision with Earth) and T=3500 Gyr and k=5 (collision with Mars). We have performed a series of runs of orbital evolution of resonant asteroids and JFCs and computed values of T for those runs. The results demonstrate that the effect of non-constant velocity must be taken into account, especially when semimajor axes of a minor body and a planet differ considerably.
We acknowledge support of this work by NASA grant NAG5-10776, the RFP ``Astronomy", RFBR~(01-02-17540), and INTAS~(00-240).