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KRONOS Vol. I, Issue 3

CAN WORLDS COLLIDE?

Copyright (c) 1975 by ROBERT W. BASS

Are the "Loopholes" in a 50-Dimensional Sponge Large Enough for Velikovsky to Slip Through?

Dr. Immanuel Velikovsky outraged the astronomical establishment in 1950 with his hypotheses of planetary shufflings during a game of cosmic billiards which, he postulated, took place within the past four millennia. Mankind supposedly was so traumatized that the memory of these fearful events has, in part, been repressed in a collective amnesia, while objective records of the incidents have come to be regarded as mere myths and legends. Trained in the natural sciences, medicine, and psychoanalysis, Dr. Velikovsky has in effect psychoanalyzed the bulk of surviving ancient literature in a heroic effort -- he haunted Columbia University Library for ten years -- to fathom the collective unconscious and dredge up for inspection the nightmarish global catastrophes which our ancestors endured.

Is Velikovsky the world's greatest archaeoastronomer? Is he a prescient psycho-historian? Or is he, as Isaac Asimov has argued both in ANALOG and F & SF, a tragically self-deluded zealot, victim of his own Zionist yearnings, vainly seeking to rationalize the physically impossible?

Ironically, the "young Asimov", at least in his fiction, was far more tolerant of Velikovskian themes. In what an ANALOG reviewer (A. Budrys) has called Dr. Asimov's best novel, Pebble in the Sky, the Good Doctor in 1950 portrayed a lone archaeologist whose wild theories about Earth's past not only outraged the establishment but left him in the position of being quite literally the only thinker in the entire galaxy who had surmised the actual truth; furthermore, the essential clue to this truth had to be dredged out of ancient religious writings. Moreover, in what the Science Fiction Writers of America have voted the best science fiction short story of all time, Asimov's 1941 Nightfall, the Good Doctor in his early period treats us to the vivid spectacle of an unusual cosmic alignment (specifically, a rare multiple-sun/planetary configuration) which, once every 2050 years, brings about the destruction of an entire planetary civilization and simultaneously induces a collective amnesia!* Of course, now that Asimov has evolved into a pop-oracle who prolifically purveys received opinion on every subject from Astrophysics to Zoology, he has seen fit to modulate his youthful boldness and close ranks with orthodox academe.** In fact, Asimov has told ANALOG readers, in this context, that he is "ready to die" for NASA-Medal-winning astronomer, Carl Sagan, who at the AAAS Symposium on Velikovsky, claimed*** that the odds against multiple planetary near-collisions were 1023 to 1. When I asked him afterwards how he could have computed this without employing "ergodic theory", Sagan told me that the proof would appear as an Appendix to a forthcoming paper by him based on his AAAS presentation. He mentioned that he had followed a published method, used by such scientists as Öpik and Urey, to obtain apparently reasonable statistics about meteoritic collisions with the Moon, Mars, and Venus; but in such calculations it is assumed (as an approximation) that the collisions were statistically independent events. Because the planetary motions inherently tend under their mutual gravitational attractions toward some sort of quasi periodicity, in which future near-misses can be causally related to past near misses, this assumption is absolutely identical to the assumption that Newton's Law of Gravity may be ignored! (That is, the planets are regarded as non-interacting random billiard balls, an approximation used in the kinetic theory of gases. ) For the reason indicated ( planetary masses are comparable while the analogy with meteoritic collision seems of questionable applicability), I am skeptical of Professor Sagan's figure of 1023 to 1; however it seems premature to pursue the matter because I have not yet had a chance to study the (still unpublished) text of his discussion with its accompanying mathematical appendices.

[* For a succinct discussion of the two stories, see L. M. Greenberg, "Phobia, Amnesia, and the Psyche," KRONOS, I, 1 (Spring-1975), pp. 21-26- available for $2.50 from KRONOS PRESS, c/o Warner B. Sizemore, Glassboro State College, Glassboro, N. J. 08028.]

[** I am attempting to tweak Professor Asimov's conscience in response to his having called Dr. Velikovsky a "CP", even though I recognize myself that this rap on lsaac's knuckles my appear to some as being somewhat exaggerated. Actually I am a great Asimov fan and eagerly devour most of his writings with-enjoyment and profit. What I demur from is his tendency in recent years to portray holders of conclusions which are presently only minority views, such as those who have real scientific doubts about orthodox stochastic macro-evolutionary theory in biology or perception of evidences of catastrophism in geology, as academically deficient, incompetent scholars. Even though I regard academic snobbery and intellectual intolerance as ethically indefensible and (in view of many repeated warnings from the history of ideas) as pragmatically foolhardy, I have decided to take a charitable view of Isaac's excesses and to repeat to myself -- as often as necessary -- that when the world's leading science popularizer referred to Immanuel as "CP" he was unconsciously trying to acknowledge him as a "Champion Philosopher".]

[*** Quoted e.g. by Graham Chedd in New Scientist, vol. 61 (1974), No. 888, p. 625.]

Actually I believe that those astronomers who are not trained specialists in the mathematical aspects of celestial mechanics are correctly quoting received majority opinion about planetary dynamics and are quite sincere in doubting that either Venus or Mars can have had Earth crossing orbits within astronomically recent time; the problem, as I see it, is that there has been an unfortunate divergence in communication since about 1899 between mathematicians who study the problem rigorously and between "dynamical astronomers" who, as will be documented in the sequel, have traditionally relied upon a mixture of heuristic [purely "formal," i.e. non-rigorous] theories, numerical procedures, and attempts to justify their techniques empirically [by comparison with observations] rather than by the strict logic demanded in rigorous analyses. In this manner, to quote directly the retiring address of the President of the American Astronomical Association in 1931, by the late pre-eminent dynamical astronomer Ernest W. Brown, " . . . In my subject . . . statements which are wholly or partly erroneous have obtained currency . . ." [italics added] . Unfortunately the matter is so complicated and technical, and the convincing uprooting of opinions that have gained the status of dogma over centuries is so demanding, that it seems to be impossible to give a definitive treatment of the subject without reviewing some seventy references in the technical literature and without assembling some 20,000 words of relevant quotations, which I have published* elsewhere. Here I shall attempt merely to give a hopefully more accessible and briefer summary of my previous paper, including important evidence that has been published since then.

[* Pensee, Vol 4, No 3, Summer 1974, pp 8-26; available for $2 00 from Student Academlc Freedom Forum, P O Box 414, Portland, Oregon 97207. For British readers, write to Harold Tresman, 18 Fir Tree Court, Allum Lane, Elstree, Herts -- England]

If I may be forgiven for writing a review of my own work, I would like to mention that, after the cited paper appeared, a European archaeologist took it upon himself to show my work to an internationally recognized dynamical astronomer, author of important research papers and a book on astro-dynamics, who telephoned two weeks later to say that he had studied Bass' arguments with care, and was "very impressed", and thought that "if Bass is right, he has knocked out the main prop from under the anti-Velikovsky celestial mechanicians."

Two further reactions to my previous paper may be of interest. An internationally eminent mathematician, commonly considered by most specialists to be among the world's leading celestial mechanicians, without authorizing me to quote him by name for publication, told me privately that there is no doubt, that the problem at issue has not been solved rigorously and that ( apart from educated guesses or informed conjectures) true science cannot make any flat statements about the matter based upon presently available evidence. Also a dynamical astronomer now preparing a book on the subject wrote to ask permission to quote extensively from my previous paper regarding this matter and stated explicitly his own conclusion that at present the only possible truly scientific verdict would have to be "NOT PROVEN."

Certainly I do not expect the preceding ad hominem rhetoric to have convinced any reader that Velikovskian collisions may be mechanically possible. However, I hope to have aroused the reader's interest to the point where he would like to know enough to decide intelligently for himself.

I have studied many of Dr. Irving Michelson's challenging and original research publications in theoretical mechanics, and I regard him as a soundly well-informed and penetrating thinker on these matters. Unfortunately, in his trenchant ANALOG review of the controversy, I fear that Dr. Michelson has been unduly modest in assessing the weight of his own work on this matter, particularly the as-yet unresolved celestial mechanics anomalies which he documented in his brilliant and stimulating AAAS presentation; if he had tried less hard to be objective, and had not discounted his own discoveries, Professor Michelson might have been a trifle less pessimistic than in his conclusion that "The GOOD NEWS for Dr. Velikovsky is not here yet," although from a rigorous point of view that is a fair statement and I agree with virtually every scientific point in his article. For present purposes the part of Michelson's conclusion which bears emphasis is his verdict that there remains an as yet inadequately explored loophole: "Orbital stability mathematics: MAYBE."

It is on an inspection tour of this loophole that I now invite the reader. You may need to stretch your mind to conceive of a 50-dimensional sponge, but what prize ever came easily?

Neolithic and megalithic man saw great regularity in the nightly motions of the stars, which moved in circular arcs around the celestial North pole, but he saw less regularity in the motions of the planets or "wanderers".

Did late-bronze-age man witness greater irregularities in the planetary motions than we see? Velikovsky's alleged historical evidence is too extensive to be reviewed here, but I for one find some of his uncannily apposite ancient quotations hard to shrug off.

The astronomical establishment in 1950, and again in 1974, has labeled Velikovskian scenarios as dynamically "impossible". Is this correct? We shall examine this claim from two separate viewpoints:

a) history of claims of alleged proof of "stability" of the solar system; and

b) mathematically rigorous contemporary celestial mechanics.

From either viewpoint, there is well-nigh overwhelming evidence that the astronomers who have opined "impossible" are speaking without adequate information.

I am grateful to Asimov for reminding me that the received opinion about the solar system's stability rests upon Laplace's celebrated claims of 1773 and 1784.

The orthodox account of these results is that presented in Asimov's classic Guide to Science: ".... there had been some feeling in the early days of work with the theory of gravitation that perturbations arising from the shifting pull of one planet on another might eventually act to break up the delicate balance of the solar system.... however, the French astronomer Pierre Simon Laplace showed that the solar system was not as delicate as all that. The perturbations were all cyclic, and orbital irregularities never increased to more than a certain amount in any direction. In the long run, the solar system is stable . . ."

The word "stability" is often used loosely; in fact, it has acquired different meanings over the centuries. When Lagrange said "stable", he meant merely bounded (no escape to infinity). When Laplace or our contemporary, Littlewood, said "stable", he meant that each planet is permanently restricted to its own non-intersecting spherical annulus (region between two concentric spheres). When Poisson said "stable", he meant actually recurrent, in the sense that the planetary configuration returns arbitrarily often, arbitrarily close, to its original configuration. (However, the return times can get longer and longer apart.)

If the motion is stable in the sense of Poisson, and if, moreover, the return times have a certain property of boundedness, then we speak of almost periodicity. (It is well known, since Kepler, that the motion of just two bodies is strictly periodic; it is this strong and simple result which has misled generations of dynamical astronomers regarding the situation wherein there are N bodies, with N not less than three). If the almost periodicity is so regular that it can be represented as the composition of a finite number of strictly periodic motions, then it is called multiply periodic or quasiperiodic. It is the fallacious idea that the solar system's motion is necessarily quasiperiodic which we intend to expose here.

If we approximate a planetary orbit by the closest-fitting circular orbit, then the angular velocity appropriate to the latter is called the planet's mean motion.

An integer is a whole number (0, 1, 2, 3, . . . or-l, -2, -3, . . .). The ratio of two integers is a rational number; it can always be expressed as a repeating decimal (1/3 = 0.333 . . . ). Otherwise, a number is irrational (such as [PI], which can only be expressed as an infinite non-repeating decimal 3.14157 . . .). Two mean motions are commensurable or in resonance if their ratio is a rational number; otherwise, they are incommensurable. This concept can be extended to cover three or more mean motions. In a technical sense, which we cannot enter into here, all possible motions can be divided into two disjoint classes: nearly commensurable, and very incommensurable. These same classes can likewise be labeled: nearly resonant and very non-resonant. By "motions", in this context, we mean initial conditions, or initial configuration and velocities, which uniquely determine all past and future states. It turns out that very non-resonant initial conditions lead to quasiperiodic motions (which are necessarily stable in the senses of Lagrange, Laplace/Littlewood, and Poisson). However, the nearly resonant initial conditions pose an unresolved enigma.

In the case of a resonant or nearly resonant configuration, there are three possibilities, the first two of which were the only cases conceivable to the classical astronomers: (1) the motion could be quasiperiodic and also orbitally stable (to be defined below); or it could be (2) quasiperiodic and orbitally unstable; or it could be (3) a so-callod wild motion, which exhibits behavior not possible in the two-body problem, such as collisions or escapes (to infinity). It is case(3) which would constitute good news for Dr. Velikovsky.

Fortunately for Velikovsky, the present solar system is in a nearly resonant configuration, as demonstrated in independent publications by Roy and Ovenden in 1954 and by Molchanov in 1968 (though some astronomers have tried to argue that the quantitative "nearness" to resonance is inadequate for the above-cited results to apply).

Consider a trajectory of a point-particle moving under Newton's laws of dynamics and gravitation. The trajectory is said to be orbitally stable if small deviations in the initial conditions lead merely to nearby trajectories which remain close to the original trajectory as geometrical paths, not necessarily as time-parameterized motions. (That is, what engineers call phase-shifts are allowed; if phase-shifts are not allowed, we have what in electromechanical engineering is called Liapunov stability, a concept too strong to be applicable to celestial mechanics.)

Certain orbits in the asteroid belt are in resonance with Jupiter's motion. There is a standard calculation whose results are always ambiguous in the case of possible stability or wild motion, but which can unambiguously and reliably select orbitally unstable motions. When this calculation is performed, it predicts certain zones of instability, which coincide perfectly with the observed Kirkwood gaps.

Are any examples of orbital stability known? Most readers will have heard of the Trojan Asteroids, which orbit in the same path as Jupiter, but 60 ahead or behind, so that they are always equidistant from both the Sun and Jupiter, thus constituting Lagrange's famous equilateral triangle solutions. A particle placed near such a point will (in an appropriately rotating coordinate system) seem to merely rock back and forth or tremble slightly; such points are called libration points and denoted by L4 and L5. Analogous positions L4 and L5 occur in the Earth -- Moon system, which we may expect to see increasingly in the news. For example, the long-delayed-echoes of the 1928 radio pioneers, which some have construed in 1973 as evidence of an alien interstellar monitor in the solar system, have been more prosaically explained in 1974 by A. T. Lawton as reflections from plasma trapped at the lunar libration points L4 or L5. As another example, Princeton physicist Dr. G. K. O'Neill has lately been gathering great support, both from his professional peers and in the popular media, for his ambitious proposals to place city-sized space-habitats at Lunar L5.

In 1773, Laplace published an alleged theorem, later improved by Poisson, purporting to show that the solar system was stable in the Laplace/Littlewood sense. In 1784, utilizing work of Langrange, Laplace published another "theorem", alleging that the planetary inclinations and eccentricities must always remain small. Experts have discounted these results since 1899, when Poincaré proved that "Newcomb's series", of which Laplace, Poisson and Lagrange were utilizing only the first two terms, generally failed to converge and so, at the very best, could only give an approximation valid at most for a limited interval of time.

Simon Newcomb was probably America's greatest scientific genius of the nineteenth century. In 1874, for an initial configuration closely resembling that of the solar system (i.e. a massive central body surrounded by nearly coplanar smaller bodies in nearly concentric circular orbits) he made an amazing discovery: a quasiperiodic trigonometric series which is an exact (though merely "formal") solution of the N-body problem. By merely formal, we mean that when these infinite series are inserted into Newton's equation of motion and the left-hand-side is compared algebraically with the right-hand-side, there is perfect term-by-term agreement; but this says nothing about whether or not the infinitely many (increasingly smaller) terms add up in the limit to a finite number, i.e. converge, or merely diverge. In the case of convergence, the actual planetary motion is quasiperiodic. In the case of divergence, it is a wild motion.

Newcomb and most dynamical astronomers (as opposed to mathematicians specializing in celestial mechanics) assumed in 1875, and still in 1975 assume, that Newcomb's series always converge; by analogy with the case N=2, they suppose that the N-body problem is quasiperiodic when the initial conditions are of the Newcomb type. If this were so, the solar system would be stable in the senses of Lagrange, Laplace/Littlewood, and Poisson. Furthermore, if the series not only converged, but converged "uniformly" (with respect to small parameter variations), then the solar system would also be orbitally stable, and (at least, on a Newtonian basis) the astronomers' claims of utter impossibility of near-collisions would be correct.

Fortunately for Velikovsky, Poincaré (the greatest mathematician of 1899) proved rigorously that, in general, Newcomb's series either diverge or, at best, fail to converge uniformly.

In the early 1960's, the great Russian mathematician Kolmogorov and his inspired student Arnol'd succeeded, during work of almost superhuman difficulty, subtlety, and complexity, in clarifying the status of Newcomb's series, at least for the "majority" of planetary motions (in an appropriate statistical sense). For all very non-resonant initial states, Newcomb's series converge (non-uniformly), and so these motions are quasiperiodic; but they are not orbitally stable, and so arbitrarily small perturbations in the initial conditions can (so far as we know) yield wild motions. For resonant or nearly resonant motions the series can converge uniformly (orbitally stable quasiperiodic motion), or converge non-uniformly (orbitally unstable quasiperiodic motion), or diverge (wild motion).

The late, great dynamical astronomer E. W. Brown, who retired as President of the American Astronomical Association in 1931, did not have the benefit of the results just explained. He knew that Poincare had shown that, in general (including the case of divergence) Newcomb's series could be regarded as at best "asymptotic series" which furnish some validity for limited intervals of time. Simon Newcomb had guessed 10^11 years. In work widely believed today by astronomers, Brown advanced the guess that 108 would be more prudent; later, with no explanation, Brown retreated to 106 years. In explaining why he had abandoned Newcomb's guess in favor of a 100,000-fold reduction, Brown pointed out that if the eccentricity of, say, Venus increased to the point where it nearly collided with the Earth, then the conditions under which Newcomb had proved the (formal) existence of his (seemingly) quasiperiodic series solutions would be violated, and one would have to start a fresh solution at that point.

I am grateful to Sagan for pointing out to me why the majority of dynamical astronomers (even in this post-Arnol'd era) still believe in Newcomb-type estimates of 1011 years (or rather, since the Sun's lifetime may be measured in billions of years, in estimates of 109 years). Recently a masterfully exhaustive five-volume treatise on celestial mechanics by Yusuke Hagihara has been appearing. In 1961, Hagihara published, in an important astronomical handbook, a relatively short preview of this magnum opus, which contains no incorrect statements but can be somewhat misleading when scanned hastily by non-dynamically-oriented astronomers. Hagihara stated explicitly that he was merely reporting received opinion regarding Newcomb's assuredly nonrigorous 1011-year guess, but Arnold's 1962-1963 papers had not yet appeared, and Hagihara did express puzzlement over why Brown had retreated from 1011 to 105 years. This puzzlement seems due to the fact that, in an otherwise remarkably complete bibliography, Hagihara had failed to cite only three papers by Brown, namely the 1931 retiring Presidential address and two related papers, in which Brown made the confessions and explained the doubts cited above.

Hagihara and the mainstream astronomers had also failed to notice that, in 1953, the Regius Professor of Astronomy of the University of Glasgow, W. M. Smart, in a text on dynamical astronomy, had quantified Brown's doubts and shown that the equation for secular change of a planetary eccentricity cannot, in general, be trusted for more than 105 days or about 300 years. Smart stated explicitly that he would not rely on Newcomb's series for more than "one or two centuries".

In summary, the traditional astronomical estimate of invariability of the order of planets from the Sun has to be lowered from 1011 years to 102 years! Surely the anti-Velikovsky astronomers should be sobered by this revelation that received opinion was in error by a factor of billions.

Recently, D.G. Saari has proved that the set of initial conditions leading to (exact) collisions has negligible probability of occurrence. He has also proved that if the solar system is stable in the sense of Lagrange, then (with the exception of a set of initial conditions having negligible probability of occurrence) it is also stable in the sense of Poisson. When combined with Arnol'd's results, this implies that the set of initial conditions leading to wild motions has only a relatively small probability of occurrence; but small is not zero, and (as foreseen in 1899 by Poincare) recent computer simulations by J. M. A. Danby, L. Martinet, and others, have shown that wild motions can and do occur.

So much for generalities. But in particular, can the precise wild motions postulated by Velikovsky actually occur? Velikovsky's friends, philosopher L. E. Rose and his collaborator R. C. Vaughan have dived in where traditionalists fear to swim, and have emerged with the pearls displayed by physicist C. J. Ransom and his collaborator L. H. Hoffee in Figure 1. Speaking as a trained celestial mechanician, I regard their work as magnificently heroic, for three reasons:

(a) they are nominally amateurs in this field;

(b) they had no access to giant computers but worked laboriously under adverse conditions;

(c) as any reader can see for himself, they have come within what Sagan would call "a gnat's eyelash" of driving the final nail into the anti-Velikovsky camp's coffin.

Although I am the one presently making noise by quoting advanced results in an effort to shame the astronomical establishment into taking an objective look at Velikovsky's hypotheses, if he is ever vindicated (in this context), primary credit for triumphantly dedicated resourcefulness should go to the four men just mentioned.

Happily not all dynamical astronomers are closed-minded about Velikovskian ideas. Recently, one West-coast group and another East-coast group, who have access to CDC 6600's, have independently manifested an interest in trying a computer simulation with the Rose-Vaughan initial conditions, to see if Venus, starting near Jupiter, could have relaxed into its present near-circular orbit within the past four millennia. Another computer test, which I would accept as decisive, would be to integrate the presently observed planetary configuration backwards in time for 4,000 years, while simultaneously computing the running average of a certain 50-dimensional square array of numbers know as the "fundamental matrix of the equations of variations", which allows one at negligible further expense to survey not just the given orbital path but an entire "tube" of all nearby orbital paths. Also, tidal friction should be included. Now if, going backwards in time, this matrix should turn out to be a "contraction matrix" (i.e. the tube gets smaller), then I would have to regard this as devastatingly bad news for Dr. Velikovsky.* On the other hand, if going back a mere four millennia is not contractive, then the astronomical establishment will have no choice but to concede that planetary wild motions could have occured within historic times.

[* within the framework of Newtonian mechanics- for heterodox views about the possibly large role played by electromagnetic forces in ways not now recognized, see references to papers by Michelson and Juergens in my Pensee paper; also see App. III of The Science of High Explosives by Nitro-Nobel Medallist M.A. Cook, A.C.S. No. 139, reprinted 1971 by Krieger Pub. Co., Huntington, N. Y. Also, as stated in print by archaeoastronomer Dr. Peter Huber prior to his authorship of a AAAS paper on the alleged cuneiform evidence against Velikovsky's hypotheses, and as Dr. Lynn Rose has repeatedly reminded me, if a large (N+l)th body, now unknown, had intruded into our solar system from inter-stdlar space and then passed on, this would totally vitiate the relevance of the N-body calculation under discussion.]

I personally do not expect the computed matrix to be a contraction. In fact, as pointed out by C.J. Ransom and myself, if one removed Venus from its present orbit and gave it the initial conditions of the comet Oterma III, the initial orbit of Venus would lie entirely between the orbits of Saturn and Jupiter, while in less than two decades, Venus would work itself inward into an orbit Iying entirely between the orbits of Mars and Jupiter! (For detailed proof, see my Pensee article.) Thus we can be quite sure that the solar system is not stable in the sense of Laplace and Littlewood.

One problem that will have to be carefully watched, though, in computer simulations, is that (as demonstrated beyond doubt by R. H. Miller during the past decade) the planetary N-body problem is tremendously unstable numerically; in fact, particularly when near collisions occur, it is as unstable as any problem ever tried on a computer, which leads Miller to conclude that in a computer simulation, the same causal relation between initial and final conditions that occurs in physical reality no longer exists! It is for this and related reasons that E.W. Brown cautioned that very different initial conditions could all lead by computation to the presently observed final conditions (within experimental error). For this and related reasons, Brown opined that an integration backwards in time is inherently less reliable than an integration forwards in time.

Readers of George Harper's provocative article (ANALOG, Nov. 1973) hardly need reminding that, roughly speaking (and disregarding a constant displacement), according to Bode's Law, each planet, from Mercury to Neptune, is about twice as far from the Sun as its predecessor. Astronomer L. Motz, one of the few establishment figures to treat Velikovsky with respect and seriousness, has cited Bode's Law as one of his own reasons for doubting that Venus is a planetary newcomer. But C. J. Ransom has countered this argument by nimbly noting that a trivially modified Bode type of law applies to the solar system without Venus, just as well as Bode's original law applies when Venus is present.

However, science is beginning to realize that Bode's Law is just a superficial and none-too-accurate manifestation of a much deeper underlying principle. As E. W. Brown had predicted with remarkable intuition in 1931, several 1970 computer simulations of the N-body problem by J. G. Hills showed empirically that arbitrary planetary configurations, started with purely random initial positions and velocities, tend, during a few thousand to a few hundred thousand subsequent years, to "relax" into a Bode's Law type of nearly resonant configuration. More accurately, the random initial configuration repeatedly comes close to such a configuration, but then departs and repeatedly returns, on the average spending most of its time near such a configuration. If the resonant configuration were orbitally stable, the evolving configuration couldn't get arbitrarily close to it (for then orbital stability would imply that it had always been close); however, we can evade this difficulty by noting that the hitherto neglected tidal friction effect could be added on by making a suitable discontinuous "jump" in the configuration's motion, which might bring it within the region of true orbital stability. Subsequently, both observations and theory would agree that the solar system is in a quasiperiodic motion stable in the sense of Laplace. Also, numerical integrations of the observed configuration backwards in time would show that no near-collision had ever occurred; yet in actual fact this seemingly water-tight deduction would be false!

Is there any reason to surmise that a Velikovskian wild motion might evolve (,as just outlined) into an orbitally stable nearly resonant configuration? Following upon J. G. Hills' pioneering numerical simulation work, together with some earlier theoretical research of himself and A. E. Roy, and additional independent Hills-type simulations, dynamical astronomer M. W. Ovenden was led in 1971 to formulate a far-reaching Principle of Least-Interaction Action as specifying the kind of motion into which planetary systems naturally tend to evolve. Personally, I was delighted to learn of Ovenden's work, for a major portion of his principle (as a purely abstract theorem with no specific evidence of its concrete importance in our solar system) had already been announced independently by myself under the label Principle of Least Mean Absolute Potential Energy (Edinburgh, 1958; Stockholm, 1960).

Although, naturally, I regard Ovenden's Principle as irrefutably valid, it has become somewhat controversial and has been sharply questioned by several astronomers, mainly, I would assume, because Ovenden has used the Principle to "prove" that there must have once been a Saturn type planet (90 Earth masses) in the asteroid belt which somehow has been destroyed. (The Russians call this hypothetical planet Phaeton; Ovenden calls it Aztex. But that is another story!)

Ovenden has deduced from this work that the solar system's present configuration can tell us nothing about its earlier configurations.

Meanwhile, Isaac Asimov reports that he is still laughing at the Velikovskians. But, to quote Ben Bova, Editor of ANALOG, "The last laugh . . . has yet to be heard on the Velikovsky matter."

To set the stage for the last laugh, we must unavoidably consider some mind-stretching concepts. The total solution of the N-body problem, for all possible initial conditions, can be comprehended rigorously in a rather God-like or omniscient overview, by formulation as the flow of an incompressible fluid in a (6N-10)-dimensional "phase space" or state space. Here each stationary streamline represents the entire motion of an N-body planetary configuration. Each point on such a streamline requires 6N coordinates to specify it (i.e. three position coordinates and three velocity coordinates, for each of the N bodies. ) Now conservation of energy requires the streamlines to lie on a lower-dimensional "hypersurface", where each such (6N-1 )-dimensional surface corresponds to a different initial total energy. Furthermore, by conservation of linear momentum, we can replace N by (N-1), i.e. we can eliminate one body (simply by placing the system's center of mass at the coordinate origin of 3-dimensional configuration space). Hence the problem is actually (6N-7)-dimensional. Similarly, use of conservation of angular momentum reduces the problem to a (6N-10)-dimensional one. For N = 10, 6N-10 = 50. Hence we must consider a stationary incompressible flow in a 50-dimensional state space.

Note that the detailed nature of different regions of this flow depends basically upon such a hair-splitting abstraction as whether a number is rational or irrational. No wonder that magnetic bottles for controlling thermonuclear fusion plasmas exhibit radically different experimental properties depending on whether certain external controlling knobs are twisted by a rational or an irrational angle!

To further describe this numinous noumenon, we need an expert guide. The profound work of Kolmogorov and Arnol'd already cited was inspired by pioneering work in the 1930's by C. L. Siegel, and one of Siegel's students, J. Moser, now Director of the Courant Institute of Mathematical Sciences and recipient of the G.D. Birkhoff Prize, shall be our mentor. Writing in a recent Symposium on the Stability of the Solar System (IAS Symposium No. 62, ed. by Y. Kozai, pub. by Reidel, 1974) Moser, page 1, describes the system of stationary streamlines or "flow" in the (6N-10)-dimensional state space as follows: "It is the conventional view that one can delineate some 'blobs' or open regions on phase space in which the solutions remain bounded or stable, while outside such regions, the solutions escape or behave unboundedly.

. . . The recent mathematical work in this area has shown that for Hamiltonian systems this crude picture has to be replaced by another model: One finds complicated Cantor sets, which we may compare with a sponge, in which the solutions are stable and bounded for all time while the solutions Iying in the many fine holes of the sponge may gradually seep out and become unstable. The filament of these holes is connected and gives rise to a slow diffusion while the majority of the solutions belong to the solid part of the sponge consisting of stable solutions."

By now the reader should realize that the solid part of Moser's sponge is where Newcomb's series converge; the hole-region is where the series diverge, corresponding to wild motions!

To continue the metaphor: the "filament of the holes" in Moser's 50-dimensional sponge, being connected, is more properly referred to as a single "hole", and it is this "hole" which constitutes the loophole through which I am proposing to drive Velikovsky's bandwagon.

[*!* Image] INSERT KI3_72.TIF HERE

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