are either dead comets or asteroids. Using simple equations for the mean free time, astronomers are able to account to good accuracy for, say, the number of craters on the Moon, Mercury or Mars produced since the formation of these objects: they are the results of the occasional collision of an Apollo object or, more rarely, a comet with the lunar or planetary surface. Likewise, the equation predicts correctly the age of the most recent impact craters on Earth such as Meteor Crater, Arizona. These quantitative agreements between observations and simple collision physics provide some substantial assurance that the same considerations properly apply to the present problem.
We are now able to make some calculations with regard to Velikovsky’s fundamental hypothesis. At the present time there are no Apollo objects with diameters larger than a few tens of kilometers. The sizes of objects in the asteroid belt, and indeed anywhere else where collisions determine sizes, are understood by comminution physics. The number of objects in a given size range is proportional to the radius of the object to some negative power, usually in the range of 2 to 4. If, therefore, Velikovsky’s proto-Venus comet were a member of some family of objects like the Apollo objects or the comets, the chance of finding one Velikovskian comet 6,000 km in radius would be far less than one-millionth of the chance of finding one some 10 km in radius. A more probable number is a billion times less likely, but let us give the benefit of the doubt to Velikovsky.
Since there are about ten Apollo objects larger than about 10 km in radius, the chance of there being one Velikovskian comet is then much less than 100,000-to-1 odds against the proposition. The steady-state abundance of such an object would then be (for r = 4 a.u., and i = 1.20) n = (10 ? 10?5)/4 ? 1040 = 2.5 ? 10?45 Velikovskian comets/cm3. The mean free time for collision with Earth would then be T = 1/(n?v) = 1/[(2.5 ? 10?45 cm?3) ? (5 ? 1018 cm2) ? (2 ? 106 cm sec?1)] = 4 ? 1021 secs 1014 years which is much greater than the age of the solar system (5 ? 109 years). That is, if the Velikovskian comet were part of the population of other colliding debris in the inner solar system, it would be such a rare object that it would essentially never collide with Earth.
But instead, let us grant Velikovsky’s hypothesis for the sake of argument and ask how long his comet would require, after ejection from Jupiter, to collide with a planet in the inner solar system. Then,
A more exact formulation of orbital-collision theory can be found in the classic paper by Ernst Opik (1951). He considers a target body of mass
here; U is the relative velocity “at infinity” and Ux is its component along the line of nodes.
If R is taken as the physical radius of the planet, then
For application of Opik’s results to the present problem, the equations reduce to the following approximation:
Using P 5 years (a 3 a.u.), we have
T 9 ? 10 9 sin i years,
or about 1/3 the mean free path lifetime from the simpler argument above.
Note that in both calculations, an approach to within N Earth radii has N2 times the probability of a physical collision. Thus, for N = 10, a miss of 63,000 km, the above values of T must be reduced by two orders of magnitude. This is about 1/6 the distance between the Earth and the Moon.
For the Velikovskian scenario to apply, a closer approach is necessary: the book, after all, is called
Finally, we observe that an orbit which intersects those of Jupiter and Earth implies a high probability of a close reapproach to Jupiter which would eject the object from the solar system before a near-encounter with Earth-a natural example of the trajectory of the Pioneer 10 spacecraft. Therefore, the present existence of the planet Venus must imply that the Velikovskian comet made few subsequent passages to Jupiter, and therefore that its orbit was circularized rapidly. (That there seems to be no way to accomplish such rapid circularization is discussed in the text.) Accordingly, Velikovsky must suppose that the comet’s close encounter with Earth occurred soon after its ejection from Jupiter-consistent with the above calculations.
The probability, then, that the comet would have impacted the Earth only some tens of years after its ejection from Jupiter is between one chance in 1 million and one chance in 3 trillion, on the two assumptions on membership in existing debris populations. Even if we were to suppose that the comet was ejected from Jupiter as Velikovsky says, and make the unlikely assumption that it has no relation to any other objects which we see in the solar system today-that is, that smaller objects are never ejected from Jupiter-the mean time for it to have impacted Earth would be about 30 million years, inconsistent with his hypothesis by a factor of about 1 million. Even if we let his comet wander about the inner solar system for centuries before approaching the Earth, the statistics are still powerfully against Velikovsky’s hypothesis. When we include the fact that Velikovsky believes in several statistically independent collisions in a few hundred years (see text), the net likelihood that his hypothesis is true becomes vanishing small. His repeated planetary encounters would require what might be called
APPENDIX 2
Q. Now, Mr. Bryan, have you ever pondered what would have happened to the Earth if it had stood still?
A. No. The God I believe in could have taken care of that, Mr. Darrow.
Q. Don’t you know that it would have been converted into a molten mass of matter?
A. You testify to that when you get on the stand. I will give you a chance.
