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 paper by Ernst Öpik (1951). He considers a target body of mass m0 with orbital elements a0, e0 = i0 = 0 in orbit about a central body of mass M. Then, a test body of mass m with orbital elements a, e, i and period P has a characteristic time T before approaching within distance R of the target body, where
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 Öpik’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 Worlds in Collision. Also, it is claimed (page 72) that, as a result of the passage of Venus by the Earth, the oceans were piled to a height of 1,600 miles. From this it is easy to calculate backwards from simple tidal theory (the tide height is proportional to M/r2, where M is the mass of Venus and r the distance between the planets during the encounter) that Velikovsky is talking about a grazing collision: the surfaces of Earth and Venus scrape! But note that even a 63,000-km miss does not extricate the hypothesis from the collision physics problems as outlined in this appendix.
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 Worlds in Collusion.
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.
THE GRAVITATIONAL acceleration which holds us to the Earth’s surface has a value of 103 cm sec−2 = 1 g. A deceleration of a = 10−2 g = 10 cm sec−2 is almost unnoticeable. How much time, τ, would Earth take to stop its rotation if the resulting deceleration were unnoticeable? Earth’s equatorial angular velocity is Ω = 2π/P = 7.3 × 10−5 radians/sec; the equatorial linear velocity is RΩ = 0.46 km/sec. Thus, τ = RΩ/a = 4600 secs, or a little over an hour.
The specific energy of the Earth’s rotation is
where I is the Earth’s principal moment of inertia. This is less than the latent heat of fusion for silicates, L 4 × 109 erg gm−1. Thus, Clarence Darrow was wrong about the Earth melting. Nevertheless, he was on the right track: thermal considerations are in fact fatal to the Joshua story. With a typical specific heat capacity of cp = 8 × 106 erg gm−1 deg−1, the stopping and restarting of Earth in one day would have imparted an average temperature increment of ΔT 2E/cp 100°K, enough to raise the temperature above the normal boiling point of water. It would have been even worse near the surface and at low latitudes; with v RΩ, ΔT v2/cp 240°K. It is doubtful that the inhabitants would have failed to notice so dramatic a climatic change. The deceleration might be tolerable if gradual enough, but not the heat.
APPENDIX 3
THE HEATING of Venus by a presumed close passage by the Sun, and the planet’s subsequent cooling by radiation to space are central to the Velikovskian thesis. But nowhere does he calculate either the amount of heating or the rate of cooling. However, at least a crude calculation can readily be performed. An object which grazes the solar photosphere must travel at very high velocities if it originates in the outer solar system: 500 km/sec is a typical value at perihelion passage. But the radius of the Sun is 7 × 1010 cm. Therefore a typical time scale for the heating of Velikovsky’s comet is (1.4 × 1011cm) / (5 × 107 cm/sec) 3000 secs, which is less than an hour. The highest temperature the comet could possibly reach because of its close approach to the Sun is 6,000° K, the temperature of the solar photosphere. Velikovsky does not discuss any further sun-grazing events by his comet; subsequently it becomes the planet Venus, and cools to space-events which occupy, say, 3,500 years up to the present. But both heating and cooling occur radiatively, and the physics of both events is controlled in the same way by the Stefan-Boltzmann law of thermodynamics, according to which the amount of heating and the rate of cooling both are proportional to the temperature to the fourth power. Therefore the ratio of the temperature increment experienced by the comet in 3,000 secs of solar heating to its temperature decrement in 3,500 yrs of radiative cooling is (3 × 103 secs/1011 secs)1/4 = 0.013. The present temperature of Venus from this source would then be at most only 6000 × 0.013 = 79° K, or about the temperature at which air freezes. Velikovsky’s mechanism cannot keep Venus hot, even with very generous definitions of the word “hot.”