by Carl Sagan
SAGAN, C., “The Planet Venus.” Science, 133: 849 (1961).
SAGAN, C., The Cosmic Connection. New York, Doubleday, 1973.
SAGAN, C., “Erosion of the Rocks of Venus.” Nature, 261: 31 (1976).
SAGAN, C., and PAGE T., eds., UFOs: A Scientific Debate. Ithaca, N. Y., Cornell University Press, 1973; New York, Norton, 1974.
SILL, G., “Sulfuric Acid in the Venus Clouds.” Communications Lunar Planet Lab., University of Arizona, 9: 191-198 (1972).
SPITZER, LYMAN, and BAADE, WALTER, “Stellar Populations and Collisions of Galaxies.” Ap. J., 113: 413 (1951).
UREY, H. C., “Cometary Collisions and Geological Periods.” Nature, 242: 32-33 (1973).
UREY, H. C., The Planets. New Haven, Yale University Press, 1951.
VELIKOVSKY, I., Worlds in Collision. New York, Dell, 1965. (First printing, Doubleday, 1950.)
VELIKOVSKY, I., “Venus, a Youthful Planet.” Yale Scientific Magazine, 41: 8-11 (1967).
VITALLIANO, DOROTHY B., Legends of the Earth: Their Geologic Origins. Bloomington, Indiana University Press, 1973.
WILDT, R., “Note on the Surface Temperature of Venus.” Ap. J., 91: 266 (1940).
WILDT, R., “On the Chemistry of the Atmosphere of Venus.” Ap. J., 96: 312-314 (1942).
YOUNG, A. T., “Are the Clouds of Venus Sulfuric Acid?” Icarus, 18: 564-582 (1973).
YOUNG, L. D. G., and YOUNG, A. T., Comments on “The Composition of the Venus Cloud Tops in Light of Recent Spectroscopic Data.” Ap. J., 179: L39 (1973).
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 must be just one planet.
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 Velikovsk
ian 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
Consequences of a Sudden Deceleration of Earth’s Rotation
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 Scopes Trial, 1925
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
Present Temperature of Venus If Heated by a Close Passage to the Sun
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.”
The conclusion would not be altered materially were there to have been several close passes, rather than just one, through the solar photosphere. The source of the high temperature of Venus cannot be one or a few heating events, no matter how dramatic. The hot surface requires a continuous source of heat-which could be either endogenous (radioactive heating from the planetary interior) or exogenous (sunlight). It is now evident, as suggested many years ago (see Wildt, 1940; Sagan, 1960), that the latter is the case: it is the present radiation of the Sun, continuously falling on Venus, which is responsible for its high surface temperature.
APPENDIX 4
Magnetic Field Strengths Necessary to Circularize an Eccentric Co
metary Orbit
ALTHOUGH VELIKOVSKY has not, we can calculate approximately the order of magnitude of the magnetic field strength necessary to make a significant perturbation on the motion of a comet. The perturbing field might be from a planet, such as Earth or Mars, to which the comet is about to make a close approach, or from the interplanetary magnetic field. For this field to play an important role, its energy density must be comparable to the kinetic energy density of the comet. (We do not even worry about whether the comet has a distribution of charges and fields which will permit it to respond to the imposed field.) Thus, the condition is
where B is the magnetic field strength in gauss, R is the radius of the comet, m its mass, v its velocity and ρ its density. We note that the condition is independent of the mass of the comet. Taking a typical cometary velocity in the inner solar system of about 25 km/sec, and ρ as the density of Venus, about 5 gm/cm3, we find that a magnetic field strength of over 10 million gauss is required. (A similar value in electrostatic units would apply if the circularization is electrical rather than magnetic.) Earth’s equatorial surface field is about 0.5 gauss. The fields of Mars and Venus are less than 0.01 gauss. The Sun’s field is several gauss, ranging up to several hundred gauss in sunspots. Jupiter’s field as measured by Pioneer 10 is less than 10 gauss. Typical interplanetary fields are 10−5 gauss. There is no way to generate anything approaching a 10 megagauss field on a large scale in the solar system. And there is no sign that such a field was ever experienced in the vicinity of Earth. We recall that the magnetic domains of molten rock in the course of refreezing are oriented by the prevailing field. Had Earth experienced, even fairly briefly, a 10 Mg field 3,500 years ago, rock magnetization evidence would show it clearly. It does not.