Collisional Probability of Periodic Comets with the Terrestrial Planets: an Invalid Case of Analytic Formulation
Orbits of 228 known periodic comets (P < 1000 yr) were numerically integrated over +/-30,000 yr, and thereby mean collision rates (CRs) of those comets with the planets from Venus through Neptune were calculated. For Venus through Mars, past calculations of CRs by analytic methods such as Opik's are shown to overestimate by factors of 20-45, whereas for the outer planets agreement between numerical and analytic estimates is fairly good (within a factor of ~2). This is because analytic methods calculate CRs of comets as if they occupied their current orbits for an unlimited period of time, while in actuality observable periodic comets (the majority of which are of low inclination and interact with Jupiter) stay in the inner planetary regions for much shorter durations than the orbital evolution timescale of periodic comets. It is argued that, assuming a steady state population, the ratio between analytic and numerical estimates of collision rates for periodic comets with the terrestrial planets should be approximately equal to the ratio of observable to unobservable (but still Jupiter-interacting) comets. Implications of our CRs for impact frequencies on the terrestrial and outer planets are also briefly discussed. In particular, we show that, for Jupiter-interacting comets of greater than 1 km diameter, a Jupiter impact takes place every 500-1000 yr, and an Earth impact every 2-4 Myr.