Broca's Brain - Carl Sagan [175]
Wedge-shaped volume occupied by Velikovsky’s comet.
But 1/n is also the volume containing one planet. Therefore, the two quantities are equal and
T is called the mean free time.
In reality, of course, the comet will be traveling on an elliptical orbit, and the time for collision will be influenced to some degree by gravitational forces. However, it is easy to show (see, for example, Urey, 1951) that for typical values of v and relatively brief excursions of solar system history such as Velikovsky is considering, the gravitational effects are to increase the effective collision cross section σ by a small quantity, and a rough calculation using the above equation must give approximately the right results.
The objects which have, since the earliest history of the solar system, produced impact craters on the Moon, Earth and the inner planets are ones in highly eccentric orbits: the comets and, especially, the Apollo object—which 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/(nσv) = 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, n applies to the abundance of planetary targets rather than Velikovskian comets, and T = 1/[(10−40 cm−3) × (5 × 1018 cm2) × (2 × 106 cm sec−1)] = 1018 secs 3 × 107 years. Thus, the chance of Velikovsky’s “comet” making a single full or grazing collision with Earth within the last few thousand years is (3 × 104)/(3 × 107) = 10−3, or one chance in 1,000—if it is independent of the other debris populations. If it is part of such populations, the odds rise to (3 × 104)/1014 = 3 × 10−10, or one chance in 3 billion.
A more exact formulation of orbital-collision theory can be found in the classic