Broca's Brain - Carl Sagan [174]
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APPENDICES TO
CHAPTER 7
APPENDIX 1
Simple Collision Physics Discussion of the Probability of a Recent Collision with the Earth by a Massive Member of the Solar System
WE HERE CONSIDER the probability that a massive object of the sort considered by Velikovsky to be ejected from Jupiter might impact Earth. Velikovsky proposes that a grazing or near-collision occurred between this comet and the Earth. We will subsume this idea under the designation “collision” below. Consider a spherical object of radius R moving among other objects of similar size. Collision will occur when the centers of the objects are 2R distant. We may then speak of an effective collision cross section of σ = π(2R)2 = 4πR2; this is the target area which the center of the moving object must strike in order for a collision to occur. Let us assume that only one such object (Velikovsky’s comet) is moving and that the others (the planets in the inner solar system) are stationary. This neglect of the motion of the planets of the inner solar system can be shown to introduce errors smaller than a factor of 2. Let the comet be moving at a velocity v and let the space density of potential targets (the planets of the inner solar system) be n. We will use units in which R is in centimeters (cm), σ is in cm2, v is in cm/sec, and n is in planets per cm3; n is obviously a very small number.
While comets have a wide range of orbital inclinations to the ecliptic plane, we will be making the most generous assumptions for Velikovsky’s hypothesis if we assume the smallest plausible value for this inclination. If there were no restriction on the orbital inclination of the comet, it would have equal likelihood of moving anywhere in a volume centered on the Sun and of radius r = 5 astronomical units (1 a.u. = 1.5 × 1012 cm), the semi-major axis of the orbit of Jupiter. The larger the volume in which the comet can move, the less likely is any collision of it with another object. Because of Jupiter’s rapid rotation, any object flung out from its interior will have a tendency to move in the planet’s equatorial plane, which is inclined by 1.2° to the plane of the Earth’s revolution about the Sun. However, for the comet to reach the inner part of the solar system at all, the ejection event must be sufficiently energetic that virtually any value of its orbital inclination, i, is plausible. A generous lower limit is then i = 1.2°. We therefore consider the comet to move (see diagram) in an orbit contained somewhere in a wedge-shaped volume, centered on the Sun (the comet’s orbit must have the Sun at one focus), and of half-angle i. Its volume is then (4/3)πr3 sin i = 4 × 1040 cm3, only 2 percent the full volume of a sphere of radius r. Since in that volume there are (disregarding the asteroids) three or four planets, the space density of targets relevant for our problem is about 10−40 planets/cm8. A typical relative velocity of a comet or other object moving on an eccentric orbit in the inner solar system might be about 20 km/sec. The radius of the Earth is R = 6.3 × 108 cm, which is almost exactly the radius of the planet Venus as well.
Now let us imagine that the elliptical path of the comet is, in our mind’s eye, straightened out, and that it travels for some time T until it impacts a planet. During that time it will have carved out an imaginary tunnel behind it of volume σvT cm3, and in that volume there