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Open letter to science editors




Editor's note: This article is an expanded and modified version of a letter that first appeared in Physics Today (Sept. 1980), pp. 97-98. LMG

When Carl Sagan presented his paper "An Analysis of Worlds in Collision" at the AAAS Annual Meeting of 1974, he claimed to have calculated odds of 1023 to one (later changed to 1027 to one*) against the planetary encounters described in Worlds in Collision. These odds were widely quoted and publicized in the scientific and general press. However, scientists who may have wanted to check Sagan's calculations could not do so, for the mathematical appendices to which he referred in his paper were not supplied at the time. They were not distributed until two years later, in mimeographed form, and were finally made public in Scientists Confront Velikovsky.

* [NB: a probability of 7.3 x 10 -28 ~ odds of 1.4 x 1027 against. LMG]

Since Sagan's criticism is considered by many as a "quantitative refutation" of Worlds in Collision, it is only fair to subject these appendices which include most of the quantitative and scientific arguments, and which are usually taken on good faith to a brief but rigorous scrutiny. The appearance of Sagan's paper, in print, no less than five times is also sufficient reason for additional critical examination.(1)

* * *

Appendix 1 deals with the probability of an encounter between the Earth and a comet. In the text (pp. 62-63), Sagan writes:

"The calculation is performed in Appendix 1, where we see that a single 'comet' with aphelion (far point from the sun) near the orbit of Jupiter and perihelion (near point to the sun) inside the orbit of Venus should take at least thirty million years before it impacts the Earth [and therefore] the odds against it in any given millennium are thirty thousand to one. But Velikovsky has (see, e.g., page 388) not one but five or six near collisions.... If the probabilities are independent, then the joint probability of five such encounters in the same millennium is on the short side of (3 x 107 / 103 )-5 = (3 x 104) -5 = 4.1 x 10 -23 . . . . For six encounters in the same millennium the odds rise to . . . 7.3 x 10 -28, or about a trillion quadrillion to one."

Notice that Sagan's calculations(2) only apply to actual impact collisions. Yet, in the above quoted text, while Sagan admits that his odds apply to impact, he also writes that Velikovsky has "five or six near collisions".(3) He then takes the odds against an impact collision and raises them to the fifth and sixth powers. It is obvious that the odds for a near collision are much better than for an impact collision. It is also obvious to those who read Worlds in Collision that the consecutive encounters described there were not independent but interdependent events; its author explicitly stressed the interdependency of the collisions.(4)

"Each collision between two planets in the past caused a series of subsequent collisions, in which other planets became involved" (Worlds in Collision, p. 373).

Towards the end of Appendix 1, after pages of calculations where Sagan shows how he found the odds against actual impact collision to be thirty thousand to one per millennium,(5) he writes (p. 98):

"Note that . . . 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." (Ten Earth radii = 1/6 the distance to the Moon.)(6) This would considerably improve the probability. But, instead, Sagan proceeds to give his reasons for insisting upon keeping impact rather than close approach probability (p. 98) which he will then raise to the 6th power.

1) "The book, after all, is called Worlds in Collision."

2) "Also it is claimed (page 87 [of W in C]) that, as a result of the passage of Venus by the Earth, the oceans were piled to a height of sixteen hundred miles. From this it is easy to calculate backwards from simple tidal theory . . . that Velikovsky is talking about a grazing collision: the surfaces of Earth and Venus scrape!"

Sagan's first "reason" needs no comment. The second "reason", phrased as it is, can mislead the reader into thinking that Velikovsky claimed such a figure. Consider, therefore, Worlds in Collision, the section titled "The Tide" (p. 87 of the Pocket Book ed.). There we find:

"The Midrashim contain the following description: 'The waters were piled up to the height of sixteen hundred miles, and they could be seen by all the nations of the earth' [Ginzberg, Legends, III, 22]. The figure in this sentence intends to say that the heap of water was tremendous."

So, not only did Velikovsky not claim this number cited in Ginzberg's translation of the Midrash, he clearly explained it as meaning "tremendous". (In the original Aramaic text, three hundred "milin" are cited which Ginzberg translated into sixteen hundred miles. Three hundred is explicitly noted in Jewish tradition as a number used for exaggeration.) And really, was there any way for the witnesses to measure the tides even if only one mile in height?

Incredibly, it can be shown from Sagan's own text that he actually knows that Velikovsky never thought in terms of a grazing or an impact collision. For, in another connection (purporting to prove Velikovsky wrong on terrestrial-tidal problems), Sagan unwittingly writes (p. 67):

"Velikovsky believes that the close passage of Venus (or Mars) to the Earth would have produced tides at least miles high (page 70 and 71 [W in C]); in fact if these planets were ever tens of thousands of kilometers away, as he seems to think, the tides, both of water and of the solid body of our planet, would be hundreds of miles high. This is easily calculated from the height of the present water and body lunar tide, since the tide height is proportional to the mass of the tide-producing object and inversely proportional to the cube of the distance" (emphasis added).(7)

Therefore according to Sagan's own understanding it is "easily calculated" that, when Velikovsky wrote of tides "miles high", a distance of approach of two or three hundred thousand kilometers not tens of thousands of kilometers would have been sufficient to raise such tides. However, using Sagan's N2 rule to calculate the probability of an approach to within two or three hundred thousand kilometers, we find that, for 190,000 km (1/2 the distance to the Moon = 30 Earth radii), it is 1 in 33.3; and for 320,000 km (5/6 the distance to the Moon = 50 Earth radii), it is 1 in 12 per millennium.(8)

If Sagan still insists on raising the probabilities to the 6th power, he should first correct for those factors he admits neglecting, e.g., gravitational attraction, planetary motion, etc. Yet, no matter what the statistically meaningful number would be, statistics, almost by definition, are irrelevant once a particular event happens.

Therefore, even though Sagan knew that no grazing or impact collision was ever described in Worlds in Collision, he took the high improbability of an impact collision (leaving in a few factors he admits should have been corrected for) and then unabashedly raised it all to the 5th and 6th powers on the basis of "If the probabilities are independent. . .". The results were then allowed to appear in the scientific journals without any qualification.

Is this an example of the "scientific method"? Will scientists stand behind such methods?

* * *

In Appendix 2, Sagan demonstrates that the criticism of his predecessors (Payne-Gaposchkin, et al.) who claim that, if the Earth were to slow down, everything not attached would fly off, has all along been wrong. In a short calculation he shows that, were the Earth to stop rotating within a little over an hour not even stalactites would break, let alone things fly off the Earth. Yet, these objections are still being raised by such people as Asimov, Krupp, and Carlson the editor of Archaeoastronomy speaking at the Smithsonian in July 1980 and earlier by Sagan himself.

Next, Sagan calculates that the heat generated would not be sufficient to melt the Earth but would cause an average temperature rise of 100K. Sagan concludes (p. 64): "The oceans would have been raised to the boiling point of water, an event which seems to have been overlooked by Velikovsky's ancient sources."

Turning to Worlds in Collision, we find an entire section titled "Boiling Earth and Sea". By overlooking even the "Contents" of the book he was analysing, Sagan unwittingly helped to demonstrate that these ancient writings were probably based on events that could and did happen; and thus supplied an answer to what he considered the "nub" of the whole issue. For, in the introduction to "An Analysis of Worlds in Collision", Sagan wrote (p. 48):

"In the 4.5-billion-year history of the solar system, many collisions must have occurred. But have there been major collisions in the last thirty-five hundred years, and can the study of ancient writings demonstrate such collisions? That is the nub of the issue."

* * *

In Appendix 3, Sagan assumes "the heating of Venus by a presumed close passage by the sun," and calculates "the planet's subsequent cooling by radiation to space", as if a close passage by the Sun were the only reason cited in Worlds in Collision for predicting Venus to be hot. Of course, Sagan then comes up with a ridiculously low temperature 79K for present-day Venus.

But in the relevant section of Worlds in Collision, "The Thermal Balance of Venus", on p. 371, we find:

"Venus experienced in quick succession its birth and expulsion under violent conditions; an existence as a comet on an ellipse which approached the sun closely; two encounters with the earth . . . with a thermal effect caused by conversion of momentum into heat . . . the core of the planet Venus must still be hot."

We can only assume that Sagan did not read this relevant section, for otherwise he would not have gone to the trouble of the completely irrelevant calculation of Appendix 3, nor would he have written in the text (p. 79):

"I find it odd that Velikovsky does not attribute the temperature of Venus to its ejection from Jupiter . . . but he does not."(9)

And, earlier, after considering the possible temperature Venus would have been heated to if ejected from Jupiter, Sagan observed: "Incidentally, this would appear to be a good Velikovskian argument for the high temperature of the surface of Venus; but . . . this is not the argument" (p. 61).

As we have seen, this very argument "birth and expulsion" is brought by Velikovsky as the first among three or four others, only one of them being a close passage by the Sun.

Thus, while misrepresenting Worlds in Collision by calculating an irrelevant appendix, Sagan verified Velikovsky's foremost reason for predicting Venus to be hot by indicating, in his text, Venus' high temperature of ejection.*

* [Interestingly, it has recently been shown that Sagan's cooling computation is not only wrong but that the observed surface temperature of Venus is consistent with its having been "candescent" 3,500 years ago. In the Fall 1978 issue of KRONOS (IV:2), Dr. George R. Talbott showed that Sagan's 79K was derived by using an irrelevant ratioing procedure and ignoring the essential parameters of mass, specific heat, and surface area. Talbott then showed that a Venus-like body could cool by radiation, given sufficient volcanism, from a "candescent" state to 750K in 3,500 years. LMG]

* * *

In Appendix 4, Sagan calculates the electromagnetic field that would have been necessary in order to circularize Venus' orbit. Then he claims that the evidence of rock magnetization on Earth shows no such external field existed in recent geological time.(10)

It is interesting to note that, here, Sagan accepts the idea itself as a respectable possibility. Yet, when Velikovsky first suggested that electromagnetic fields should not be excluded from playing a role in celestial mechanics, he was ridiculed by astronomers. Since then, electromagnetic fields have been "admitted" into astronomy; and only recently Jupiter's electromagnetic field was found by Voyager to be much stronger than expected, and definitely affecting its satellites (11) However, Sagan does not concede that this was one of Velikovsky's important scientific insights;(12) he only tries to show that rock magnetization evidence does not indicate such a field existed in the vicinity of the Earth a few thousand years ago.

But though Velikovsky suggested electromagnetic fields as a possible solution, he maintained that nothing described in Worlds in Collision violated the laws of Newtonian celestial mechanics (W in C, pp. 384-387). As he emphasized in 1968, "Only the mechanism was upset; not the mechanics." And this can actually be seen from Sagan's own text.

On page 98 of Scientists Confront Velikovsky, Sagan observes 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 . . . Therefore, the present existence of the planet Venus must imply . . . that its orbit was circularized rapidly."(13) He concludes: "That there seems to be no way to accomplish such rapid [<1000 years] circularization is discussed in the text."

Yet, in the text (p. 85), in the section of his paper dealing with the circularization of the orbit of Venus, Sagan makes the following admission: "The idea that Venus could have been converted, in a few thousand years, from an object in a highly eccentric orbit to its present orbit . . . is at odds with what we know about the three-body problem in celestial mechanics. However, it must be admitted that this is not a completely solved problem, and that, while the odds are large, they are not absolutely overwhelming, against Velikovsky's hypothesis on this score."(14) In attempting to discredit Velikovsky, Sagan has been K.O.'d by none other than Carl E. Sagan.


1. "An Analysis of Worlds in Collision" was first presented at the AAAS Meeting in February 1974, though not in its entirety because of time constraints. It was officially published as part of Scientists Confront Velikovsky in November 1977 while being excerpted simultaneously in the November/December 1977 issue of The Humanist. In 1979, a slightly re-edited and up-dated version was reprinted in Sagan's book Broca's Brain. This version was then excerpted by Biblical Archaeology Review for its January/February 1980 issue. Along with an earlier mimeographed version that was made available in February 1976, containing the appendices for the first time this totals five times to date that Sagan's paper has appeared in print (not counting the xerox copies of the 1974 version that were passed out at the AAAS meeting). On some of these occasions, it was singled out by many reviewers as the best criticism of Worlds in Collision. The magazine versions never included the appendices.
2. [In spite of Sagan's unexplained statement (p. 98) that he obtains "about one-third the mean free path lifetime" when Öpik's more exact calculation is used instead of his own "simpler argument", the two results are virtually identical when the Öpik calculation is carried out properly. Sagan's reduction of Öpik's equation to the form
T/P ~ (p sin i)/Q2
seems valid for this application, and his value of P (5 years) seems appropriate. His value of R is not appropriate; it must be equal to the radius of the target body (i.e., Earth) plus the radius of the test body (i.e., Venus) if all possible impact collisions are to be included. This means that R must be approximately twice the radius of Earth, so that Q = 8.5x10 -5 and Q2 = 7.3x10 -9. Furthermore, the value of sin i must be weighted according to the probability of each value of i over the domain 1.2 >= i >= 0; the resulting value of sin i is approximately 0.014. Using these values in the equation one finds that T = 3x107 years, which is the same value that Sagan derived on p. 97 using his own method. Thus, either method suggests that the odds against an actual impact collision occurring during one millennium are thirty thousand to one. RCV]
3. Sagan's text is ambiguous for he uses "impacts", "near collisions", and "such encounters" interchangeably. That his calculations apply only to actual impact collisions is made clear only in the appendix.
4. [Dr. Robert Jastrow, founder and director of NASA's Goddard Institute for Space Studies, and a major establishment astronomer, has also noted Sagan's mistake. Writing in the New York Times (12/2/79, p. 22E), Jastrow had this to say: ". . . Dr. Velikovsky had his day when he spotted a major scientific boner in Professor Sagan's argument. Calculating the probability of several collisions involving Venus, Mars and the Earth, Dr. Sagan estimated 1 chance in 1023 (10 followed by 22 zeros) that the collisions could occur. This number was widely quoted by reporters as proof of the absurdity of Velikovsky's thesis. Professor Sagan's error lay in the assumption that the collisions were independent of one another, so that the probability of a series of collisions would be the product of separate probabilities for each collision. Dr. Velikovsky pointed out that the collisions are not independent; in fact, if two bodies orbiting the sun under the influence of gravity collide once, that encounter enhances the chance of another, a fact well known in celestial mechanics. Professor Sagan's calculations, in effect, ignore the law of gravity. Here Velikovsky was the better astronomer." Despite Sagan's later protestations, Jastrow stood firm and pointed out Sagan's error once again in Science Digest (Special Edition) Sept./Oct. 1980, p. 96. LMG]
5. The fact that the odds apply only to actual impact collisions, and the qualifying statement that these odds increase with the square of the distance, should have been stated openly and clearly in the text, not at the end of an appendix first made available two years after the fact.
6. [Note that a center-to-center distance of about 2 Earth radii is sufficient for an impact collision between Earth and Venus, as pointed out in note 2. Thus, Sagan should have said that an approach to within N Earth radii has 0.25N2 times the probability of a physical collision. For N = 10, the mean free time T would be reduced by a factor of 25. RCV]
7. Actually, the closest approach described in Worlds in Collision could conceivably be several lunar distances away (see W in C, pp. 84-85 and the remarks by Velikovsky in the Nov./Dec. 1977 issue of The Humanist, p. 23). Velikovsky also believed that the planets, because of their magnetic fields, would avoid "an actual crushing collision of the lithosphere" during a close encounter (see W in C, p. 372). But already for N = 100 (less than twice the distance to the Moon), we find according to Sagan's statistical methods a probability greater than 1.0, which is of course absurd. Sagan's method is only a good approximation when short periods of time and/or small distances of approach are considered. But Sagan makes no such qualifications; and actually on page 97 he simply divides 3x104 by as large a number as 3x107, and on the next page suggests multiplying by N2, giving N = 10 as an example Thus, he would divide 3x104 x 102 by 3x107. It seems that Sagan handles statistics almost as shoddily as he does Worlds in Collision.
8. [See note 6; Cf. the remarks by Ralph E. Juergens in Velikovsky's and Establishment Science (KRONOS III:2), p. 85. RCV/LMG]
9. [Sagan is once again incorrect. Reiterating his earlier thesis in the Yale Scientific Magazine (April, 1967, p. 21), Velikovsky had this to say: writing in W. in C. that Venus is very hot and gives off heat, I explained the mechanism of the origin of its heat. It can be traced to the natal heat of recent birth by explosion from Jupiter, [etc.]" LMG]
10. But see H. Manley, "Paleomagnetism, "ScienceNews, July, 1949; Earth in Upheaval, pp. 282-283. [While there is as yet no definitive evidence for a 10 megagauss field, new discoveries involving paleomagnetism may prove quite revelatory see the commentary by McCreery on paleomagnetism elsewhere in this issue of KRONOS. LMG]
11. See for example Science News (3/ 10/79), p. 149; Nature, 282 (12/20/79) , pp. 811-813; Scientific American (Jan. 1980), p. 91.
12. Interestingly, in 1952, Velikovsky wrote: "the views expounded in Worlds in Collision were appropriated piecemeal by those who first opposed them, though not with frankness and candor, but rather under the guise of showing how wrong the author of that heretical book is" (from the "Foreword" of Ages in Chaos, February, 1952).
13. [In a recent issue of Star & Sky a catastrophic origin for the rings of Saturn, having to do with a "collision-catastrophe concept", was proposed. "In such a cataclysmic impact, material would likely be ejected into orbits with varying inclinations and eccentricities. Those in retrograde orbits would soon fall directly into Saturn. As randomly inclined orbits crossed, repeated collisions at high velocity would subsequently grind down the material. This process would have the effect of very rapidly reducing the average size of the particles, as well as the average orbital inclination (in only a few hundred thousand years by one estimate). A planar ring would soon form automatically.... Even if the original collision left several hundred-mile-wide fragments in widely inclined and eccentric orbits, the severe tidal forces exerted by Saturn would lead to a complete circularization of orbits and a flattening of inclinations (to essentially the plane of Saturn's equator) in less than a million years. The excess energy would be dissipated internally as heat a much smaller analog to Io" (Nov. 1980, p. 64, emphasis in original). LMG]
14. See Chris S. Sherrerd, "The Electromagnetic Circularization of Planetary Orbits," KRONOS IV:4 (June, 1979), pp. 55-58; Shane Mage, Velikovsky and his Critics (Grandhaven, 1978), pp. 11-12.

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