Broca's Brain (53 page)

here; U is the relative velocity “at infinity” and U
_x
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 N
²
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/r
²
, 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

Consequences of a Sudden Deceleration of Earth’s Rotation

The Scopes Trial, 1925

THE GRAVITATIONAL
acceleration which holds us to the Earth’s surface has a value of 10
³
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 × 10
⁹
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 c
_p
= 8 × 10
⁶
erg gm
⁻¹
deg
⁻¹
, the stopping and restarting of Earth in one day would have imparted an
average
temperature increment of ΔT
2E/c
_p
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
v
²
/c
_p
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.