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The present work is divided into two stages: 1. By using large numbers (several millions) of accurate orbit integrations with the K-S regularization, probability distributions for changes in the orbital elements of comets during encounters with planets are evaluated. 2. These distributions are used in a Monte Carlo simulation scheme which follows the evolution of orbits under repeated close encounters. The method is typically about 10000 times faster than the usual integration of orbits, and produces results which agree statistically with earlier work. We have calculated the orbits of several million Oort Cloud comets after they have been perturbed by Galactic tides (or other perturbations) and have entered the inner Solar System. We find that enough comets are captured from the Oort Cloud by planetary perturbations in order to explain the present population of short period comets. If the active lifetime of a comet is not much greater than 400 orbital revolutions, the median value of cos(i) of the Jupiter family comets is 0.98, in agreement with observations. Also the inclination distribution of the Halley type comets, as well as the relative numbers of Jupiter family, Halley type and long period comets are predicted correctly by the model. After comets have lost their volatiles they frequently remain in the inner Solar System as dark bodies and some of them collide with the planets. The collision rates have been evaluated by \"Opik's method. We find 0.3--0.9 impacts/Myr by km-size bodies on the Earth. The rate may be compared with the rate of cratering (20 km or larger in diameter) which has been estimated to be 1 - 3 per Myr. Thus dead comets originating from the Oort Cloud may contribute significantly to the terrestrial cratering rate. Galactic tides cause a periodic modulation of the flux of Oort cloud comets into the inner Solar System, and we show that the same periodicity is reflected also in the cratering rate in our results, though with a time delay and with added noise.
