PROBLEM II
REPEATED COLLISIONS AMONG
THE EARTH, VENUS AND MARS

“THAT A COMET may strike our planet is not very probable, but the idea is not absurd” (page 40.) This is precisely correct: it remains only to calculate the probabilities, which Velikovsky has unfortunately left undone.

Fortunately, the relevant physics is extremely simple and can be performed to order of magnitude even without any consideration of gravitation. Objects on highly eccentric orbits, traveling from the vicinity of Jupiter to the vicinity of the Earth, are traveling at such high speeds that their mutual gravitational attraction to the object with which they are about to have a grazing collision plays a negligible role in determining the trajectory. The calculation is performed in Appendix 1, where we see that a single “comet” with aphelion (far point from the Sun) near the orbit of Jupiter and perihelion (near point to the Sun) inside the orbit of Venus should take at least 30 million years before it impacts the Earth. We also find in Appendix 1 that if the object is a member of the currently observed family of objects on such trajectories, the lifetime against collision exceeds the age of the solar system.

But let us take the number 30 million years to give the maximum quantitative bias in favor of Velikovsky. Therefore, the odds against a collision with the Earth in any given year is 3 × 10⁷ to 1; the odds against it in any given millennium are 30,000 to 1. But Velikovsky has (see, e.g., page 388) not one but five or six near-collisions among Venus, Mars and the Earth—all of which seem to be statistically independent events; that is, by his own account, there does not seem to be a regular set of grazing collisions determined by the relative orbital periods of the three planets. (If there were, we would have to ask the probability that so remarkable a play in the game of planetary billiards could arise within Velikovsky’s time constraints.) If the probabilities are independent, then the joint probability of five such encounters in the same millennium is on the short side of (3 × 10⁷/10⁸)⁻⁵ = (3 × 10⁴)⁻⁵ = 4.1 × 10⁻²³, or almost 100 billion trillion to 1 odds. For six encounters in the same millennium the odds rise to (3 × 10⁷/10³)⁻⁶ = (3 × 10⁴)⁻⁶ = 7.3 × 10⁻²⁸, or about a trillion quadrillion to 1 odds. Actually, these are lower limits—both for the reason given above and because close encounters with Jupiter are likely to eject the impacting object out of the solar system altogether, rather as Jupiter ejected the Pioneer 10 spacecraft. These odds are a proper calibration of the validity of Velikovsky’s hypothesis, even if there were no other difficulties with it. Hypotheses with such small odds in their favor are usually said to be untenable. With the other problems mentioned both above and below, the probability that the full thesis of Worlds in Collision is correct becomes negligible.