mirrored file at http://SaturnianCosmology.Org/ For complete access to all the files of this collection see http://SaturnianCosmology.org/search.php ========================================================== [INLINE] [INLINE] [INLINE] DR. ROBERT BASS AND BODE'S LAW In the history of modern astronomy, few principles have drawn more attention than that of Bode's "Law", relating to the spatial relationships between planets. In more ways than one the principle has found its way into discussions of planetary stability and of dynamical considerations relating to hypothesized dramatic shifts in the planetary order in geologically recent times. Dr. Robert Bass, Rhodes Scholar and former professor of mathematics, physics and astronomy, has recently completed a mathematical "derivation" of Bode's Law, which (if correct) could have far-reaching impact on our understanding of planetary history. Kronia Communications A. INTRODUCTION B. Planetary Stability Dynamics (1) C. Planetary Stability Dynamics (2) D. Planetary Stability Dynamics (3) E. Dynamical Derivation of Bode's Law A . INTRODUCTION R. W. Bass Dave Talbott has asked me to post a summary of my recent work on Bode's Law in relationship to the subject of planetary stability dynamics, which I shall do before the end of October. However, I have decided that I would rather the reader see it in the context of the two papers that I published in Pensee in 1974, as well as a follow-up paper in Kronos in 1975, which were elaborated upon in my presentation at the Glasgow Symposium in 1978. Accordingly I am going to post the Abstracts for at least the first three of those papers, under the titles of "Planetary Stability Dynamics (1), (2) & (3)." The old-timers who have already seen my papers can delete these titles without loss. My fourth posting will be the summary that Dave requests, and my fifth posting will discuss my new "dynamical derivation of Bode's Law" in the context of recent papers by Chambers et al and by Gladwin. R. W. Bass B Pensee, Immanuel Velikovsky Reconsidered, No. 8, pp. 21-26; reprinted, Kronos II.2 (1976), pp.27-45, & SISR III:I (1978), pp. 16-22. Copyright 1974 by Robert W. Bass 'Proofs' of the Stability of the Solar System: An Examination of the Traditional Arguments ABSTRACT In 1773 Laplace published a theorem, later improved by Poisson, which was believed to show the stability of the solar system in the sense that the mean planetary distances would always remain bounded and that adjacent planets could not interchange their distances nor nearly collide. In 1784, utilizing work of Lagrange, Laplace published another theorem alleging that the planetary inclinations and eccentricities must always remain small (if the eccentricities could become large, near-collisions could occur). Experts have discounted these results since 1899, when Poincare 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, a finite number of terms could give only an approximation valid at most for a limited interval of time. Then informed interest shifted to the question of the length of time of validity. (Laplace had guessed 10^7 years, without proof.) In 1902 Moulton published an analysis of these theorems, showing that the 1784 work was fatally flawed (in a system of linear differential equations with time-varying coefficients, periodic terms were replaced by their mean values, now well known in simple examples, e.g. Hill's equation or Mathieu's equation, to give wholly fallacious results). In 1933, Brown and Shook stated that the theorems of Laplace and Poisson were "much overestimated." Unfortunately, even in 1973 authoritative publications quote Brown and Shook as advancing the unsupported opinion that the mean planetary distances are invariant over times of 10^7-10^8 years, but this is a misquotation of a very indirect statement in which they actually said only that they doubted the reliability of first and second order perturbation theory with respect to calculations of the eccentricities much more than with respect to the mean distances; furthermore, this indirect statement was a summary of an earlier summary by Brown, published in 1932 as his retiring Presidential address to the American Astronomical Society. Upon looking up what Brown actually said on that occasion, we find that he allowed for the possibility that the eccentricities could increase until some of the planets nearly collided, at which point the entire theory would immediately become invalid. Also he stated that there were no logical or mathematical reasons to doubt the possibility that some of the terrestrial planets (e.g. Earth and Mars) could have actually interchanged their mean distances. Immediately after stating this, he made a partial retraction in terms of his own private opinion, claiming that, for reasons which he had no time to enter into, he personally was inclined to believe (on the basis of intuition, without any quantitative evidence) that no interchange of mean distances had actually taken place in our solar system, even though it might be theoretically possible in some other solar system. In the present paper we set forth explicit evidence that whenever these allegedly authoritative statements about time intervals of validity have been made, they are without exception accompanied by words like "supposed," "appeared," "hope," "seems," "might," and "think," revealing clearly that the writer was relying on his personal intuition rather than quantitative evidence. Here, citing Oterma III's behavior as well as the recently discovered wild motions (predicted by Poincare in 1899), we provide quantitative evidence that the time interval of validity in general cannot be more than 10^2-10^3 years, and in particular cases could be less. Brown himself later, without explanation, reduced his estimate from 10^8 years to 10^6 years. Most contemporary astronomers are aware, from an authoritative 1961 report by Hagihara, that Newcomb's original 1895 estimate was 10^11 years; hence they must admit that in lowering this traditional figure by a factor of 10^5, Brown was engaged in a drastic retreat. The retreat became a rout when the Regius Professor of Astronomy of the University of Glasgow, in a 1953 treatise on dynamical astronomy, spelled out the explicit quantitative details of Brown's doubts and demonstrated unquestionably that the interval of assured reliability of the Laplace-Lagrange perturbation equations is at most some interval "small" relative to 3 x 10^2 years; Prof. W. M. Smart's exact words are "one or two centuries." In conclusion we combine these results with the consequences of an accompanying paper to provide proof that the astronomers who have asserted that Velikovsky's central hypothesis is incompatible with Newtonian dynamics have been laboring under a radical misapprehension of the objective facts. ---------------------------------------- The 1974 paper went on to contain about 20,000 words of quotations from the above-mentioned eminent authorities in demonstration that the above summary is a fair report and does not rely on quotation out of context or other misrepresentation. --------------------------------------- NOTE ADDED in October, 1996: I sent a copy of my two 1974 Pensee papers to the late Prof. Michael Ovenden, who referred to them (and to my citation of his Principle of Least Interaction Action in the context of the Velikovsky controversy) in his paper on that principle printed on pages 295-305 of "Longtime Predictions in Dynamics," edited by V. Szebehely & B.D. Tapley, and published by Reidel (Dordrecht-Holland) in 1976. Citing a paper by J.Birn, "Astron. & Astrophys.", vol. 24 (1973), p. 283, Ovenden [after generously acknowledging that Bass's 1958 Principle of Least Mean Absolute Potential Energy had anticipated his own principle by an actual mathematical theorem "which has a strong resemblance" to his own empirical conjecture] goes on to say: "...systems with relatively large perturbations may exhibit rapid evolution, as shown by numerical integration. In the case of Birn (1973), drastic evolution in the Jupiter-Saturn-Uranus- Neptune system was obtained if the perturbations were magnified by starting the system far from minimum interaction. Significant changes in the major semi-axes (including a 'switch' of Uranus & Neptune [!]) occurred in times as short as ~ 10^3 years. Indeed in this time the system has already reached a quasi-steady state far from the original configuration." [Emphases added.] R. W. Bass C Pensee, Immanuel Velikovsky Reconsidered, Number 8, pages 8-20, 1974. Copyright 1974 by Robert W. Bass Can Worlds Collide? ABSTRACT 1) The subtle but fatal flaw in the received opinion regarding the alleged immutability of the planetary distances is the following inadequately recognized fact: whether or not the Solar System is stable in any of the senses defined by Laplace, Lagrange, Poisson, or Littlewood, or is quasi periodic, it need not be "orbitally stable." 2) As demonstrated in the text in considerable detail, it is perfectly possible, according to Newton's Laws of Dynamics and Gravitation when three or more bodies are involved, for planets to nearly collide and then relax into an apparently stable Bode's Law type of configuration within a relatively short time; therefore Velikovsky's historical evidence cannot be ignored. 3) If one started Venus in an orbit lying entirely between the orbits of Jupiter and Saturn, with precisely the appropriate initial position and velocity, it would within less than two decades work its way inward into an orbit lying entirely between the orbits of Mars and Jupiter. (This follows from observations of the comet Oterma III and the fact that, in the restricted problem of three bodies, the mass of the smallest body is irrelevant.) 4) There is no plausible explanation for the anomalous (retrograde) rotation of Venus, other than that it originally had prograde spin and was later flipped upside down by a near collision with some other planet. 5) The fact that the spin rate of Venus is now mysteriously locked in resonance with the rate of revolution of Venus relative to the Earth (so that Venus presents the same face to Earth at every inferior conjunction) may provide a dynamical clue as to which planet Venus encountered. 6) Laplace's theorem allegedly proving stability of the solar system (1773) was shown to be fallacious in 1899 by Poincare; in 1953 dynamical astronomer W. M. Smart proved that the maximum interval of reliability of the perturbation equations of Laplace and Lagrange was not 10^11 years, as stated in 1895 by S. Newcomb, but actually at most a small multiple of 10^2 years. 7) The eminent dynamical astronomer E. W. Brown, in his retiring speech as President of the American Astronomical Society in 1931, quite explicitly stated that there is no quantitative reason known to celestial mechanics why Mars, Earth and Venus could not have nearly collided in the past. A SUMMARIZING NOTE In these two articles I have not sought, as yet, to demonstrate that Velikovsky's central hypothesis is true, so much as to prove that it is not forbidden by Newtonian dynamics. In a resonant, orbitally unstable or 'wild' motion, the eccentricities of one or more of the terrestrial planets can increase in a century or two until a near collision occurs. Subsequently the Principle of Least Interaction Action predicts that the planets will rapidly "relax" into a configuration very near to a (presumably orbitally stable) resonant, Bode's Law type of configuration. Near such a configuration, small, non- gravitational effects such as tidal friction can in a few centuries accumulate effectively to a discontinuous "jump" from the actual phase-space path to a nearby, truly orbitally stable, path. Subsequently, observations and theory would agree that the solar system is in a quasi-periodic motion stable in the sense of Laplace and orbitally stable. Also, numerical integrations backward in time would show that no near collision had ever occurred. Yet in actual fact this deduction would be false. I arrived independently at the preceding scenario before learning that dynamical astronomer, E.W. Brown, President of the American Astronomical Society, had already outlined the same possibility in 1931. R. W. Bass D. KRONOS I.3 (1975), pp. 59-71 Copyright 1975 by Robert W. Bass CAN WORLDS COLLIDE? Are the "Loopholes" In a 50- Dimensional Sponge Large Enough to Allow Velikovsky to Slip Through? EXCERPTS 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. ... 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. ... 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 Godlike 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 planet, idealized as a Newtonian point-particle, requires 3 coordinates of position and 3 coordinates of velocity to specify its current state, so N particles require 6N coordinates to specify their complete state; but the 10 integrals of Conservation of Energy and of Conservation of Angular Momentum reduce the 6N dimensional state- space by 10 dimensions.] 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 hairsplitting 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 upon 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 lying 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 single "hole," and it is this "hole" which constitutes the loophole through which I am proposing to drive Velikovsky's bandwagon. R.W. Bass E. The Titus-Bode Law from 1766 to 1996 Copyright 1996 by Robert W. Bass Some 230 years ago, when only six planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn) were known, Johann Daniel Titius von Wittenberg published the comment that, very roughly speaking, each planet was about twice as far from the Sun than its predecessor. More precisely, he noted that if one takes as the unit distance that between the Sun and the Earth, then the progression 0, 1, 2, 4, 8, 16, 32, 64, ... can be manipulated by multiplying by 3, adding 4 and dividing by 10 to give 0.4, 0.7, 1.0, 1.6, 2.8, 5.2, 10.0, 19.6 ... which predicted the mean distance from the Sun of all 6 known planets within an accuracy of 5%, provided that one presumed that God had not left the position (at radius 2.8) empty. According to the fascinating book on this subject by Michael Nieto, the exact words of Titius, regarding this "praiseworthy relation" (bewundernswuerdige Verhaeltniss), are: "And shall the Builder have left this place empty? Never!" In 1772, Johann Elert Bode came upon the Titius observation and included it in the Second Edition of his Astronomy book (without mention of Titius), which popularized the "Law" and has borne the name of "Bode's Law" ever since. (In a later Edition Bode did give credit to Titius, but by then it was too late.) In 1781 Sir William Herschel discovered a planet whose distance was only discrepant by 2.1% from the unfulfilled prediction of 19.6, and which Bode named Uranus. This seemed such a striking confirmation of "Bode's Law" that a systematic search was planned and on January 1, 1801, the asteroid Ceres was discovered at a distance which Gauss computed to be 2.767, in perfect agreement with expectations. But the 'praiseworthy relation' broke down badly when Neptune was discovered in 1847. However, more sophisticated formulations of the "Law" (such as that of Blagg & Richardson) have accommodated all 9 planets by taking as the base of the geometric progression the number beta = 1.728 . If now one plots the logarithm of the mean distance versus the ordinal number n of the planet, the deviation from a straight line is slight. Excellent illustrations and a full discussion will be found on pages 342-345 of Herbert Shaw's remarkable book "Craters, Cosmos & Chronicles," Stanford U. Press, 1994, which also has a full discussion of the history of Bode's Law on pages 440-446; on page 444, Shaw quotes Kaula (1968) as stating that the ratio of planetary semimajor axes is "rather constant: if we count the asteroids as a planet and disregard Pluto, we get for this ratio about beta = a[n+1]/a[n] = 1.75 +/- 0.20 . Biblical scholar Zecharia Sitchin, whose 7 books are based upon his interpretation of Sumerian writings, claims that in the 3rd millennium B.C. the Sumerians knew of all 9 planets, and knew that a 10th planet (Tiamat) had once resided between Mars and Jupiter. As evidence he cites a cylinder seal which depicts 10 planets (plus the Moon) arrayed in roughly correct sizes and in correct order around a central (Copernican) Sun; in his latest book ("Divine Encounters", Avon, Jan. 1996, pp. 70-71) he displays the drawings on an Assyrian cylinder seal which shows an Asteroid Belt between Mars and Jupiter, and shows Saturn with RINGS! In his 1990 book "Genesis Revisited," on page 39 he says that "Bode's law which was arrived at empirically, thus uses Earth as its arithmetic starting point. But according to the Sumerian cosmogony, at the beginning there was Tiamat between Mars and Jupiter, whereas Earth had not yet been formed. Dr. Amnon Sitchin has pointed out that if Bode's Law is stripped of its arithmetical devices and only the geometric progression is retained, the formula works just as well if Earth is omitted thus confirming Sumerian cosmogony." He then presents the following table [in miles]: PLANET DISTANCE from SUN RATIO of INCREASE Mercury 36,250,000 ---- Venus 67,200,000 1.85 Mars 141,700,000 2.10 Asteroids 260,400,000 1.84 Jupiter 484,000,000 1.86 Saturn 887,100,000 1.83 Uranus 1,783,900,000 2.01 The best fit for a value of beta of about 1.8 can be seen in the 4 pairs (Mars-Asteroids), (Asteroids-Jupiter), (Jupiter-Saturn), and (Saturn-Uranus). The obvious question is whether this distal ratio beta is of cosmogonical or dynamical origin, a matter discussed in Nieto's book at great length. As recounted in former Naval Observatory astronomer Tom Van Flandern's "Dark Matter, Missing Planets, & New Comets," (North Atlantic, 1993, pp. 158-162, 192, 373, 412), the late Scottish astronomer Michael Ovenden ["Nature", vol. 239 (1972), pp. 508- 509], cited computer simulations of planetary configurations which caused him to conjecture that Newtonian point-particles may evolve under mutual gravitational perturbations in such a way as to spend as much time as possible as far away from each other as possible [consistent with conservation of total energy and total angular momentum]. He called this the Principle of Planetary Claustrophobia [or "least interaction action"], and conjectured that a Bode's law type of configuration will eventually evolve from random initial conditions which do not lead to such close encounters that one body is ejected to infinity. Moreover Ovenden believed that our present Solar System is evolving toward a new minimum-interaction action configuration, after the sudden disappearance of a planet in what is now the Asteroid belt. (Van Flandern cites many evidences in favor of this idea, and predicts that future astrophysical discoveries, which he lists as tests of the theory, will tend to confirm it.) Earlier Archie Roy and Ovenden (1954) had noted that to a fair approximation all of the planets in our Solar System are in resonance with adjacent planets. A resonance occurs when the orbital periods of two planets are "commensurable" or expressible as a ratio of small integers. The most famous resonance is the (2,5) resonance between Jupiter and Saturn. Specifically, if PJ and PS are the periods of Jupiter and Saturn, respectively, then 2.PS = 5.PJ . In the case of 3 bodies, already in 1899 Poincare had noticed a connection between resonance and average action. I myself reviewed these matters extensively in papers in "Pensee" and "Kronos" during 1974-75. Readers of those papers know that I left the subject in 1975 with what Roy (in his authoritative book on "Orbital Motion") has called my 'concise summary': a resonant motion can be either orbitally stable or unstable and no general rule is known, i.e. each case must be investigated individually. After meeting Roy in the Glasgow conference on Velikovsky in 1978, I was diverted by career exigencies to forget about this subject for 15 years, until Dave Talbott and Charles Ginenthal encouraged me to participate in the Velikovsky Centennial Symposium. While preparing for that memorable occasion, I built up such momentum in a renewed consideration of dynamical astronomy that afterwards I continued to work on the resonant stability problem with such zeal that in an utterly straightforward calculation whose only novelty is its discouragingly forbidding complexity I somehow bulldozed through to the arduously bitter end and discovered an amazing result: if two adjacent planets (whose masses are much smaller than the central Sun) are started on "generating orbits" (which are the circles that one would obtain if the ratios of both masses to that of the Sun are shrunk to zero) that are in resonance, then the motion of the outer planet will prevent the motion of the inner planet from closing upon itself, i.e. from remaining periodic when the actual mass ratios are raised from the limiting values of zero, unless the generating radii have a specific ratio beta which consists of a constant term plus small additive correction terms (which vanish as the ratios shrink to zero). I was able to calculate beta rigorously, and the result is that the constant term is a new universal constant (like pi or the basis e of natural logarithms), namely beta = 1/( (3/2)^[2/3] - 1)^[1/2] = 1.80 ! It is well accepted (e.g. the book of Kurth) that planetary dynamics may be approximated by taking just two adjacent planets at a time. I can therefore assert that if two planets start in resonance, the outer will "destabilize" the inner one (in the sense of preventing strict periodicity) UNLESS their distal ratio happens to satisfy a Bode's Law type of relationship. Adding a fourth planet will destabilize the third unless another Bode's Law relationship is satisfied. (The small observed deviations from exactly 1.80 are caused by the relevant mass-ratios being different from exactly zero.) In this way one may continue to build up a stable multiply-periodic configuration until the final planet is reached. But the final planet is "tied" to the inner ones by conservation of total angular momentum and conservation of total energy and so it cannot depart very far from its own initial [idealized] generating orbit. Therefore the entire planetary system can be stable if it is in a resonant configuration and if a Bode's Law relationship obtains; however, if any adjacent pair of planets does not satisfy a Bode's Law type of relationship, then the inner of the two will have its "miss distance" upon each return to its original position "pumped up resonantly" in exactly the same way that synchronous impulses to a child's swing will increase the amplitude to a point of danger. (Or think of hitting a pendulum with a hammer each time that it reaches its maximum amplitude, which will then increase indefinitely until a catastrophic instability occurs.) This latter assertion comes from the fact that (instead of relying on computer simulations as do all other investigators of this point known to me) I have calculated the "Poincare Map" ANALYTICALLY, exactly and rigorously, which is defined as follows. Imagine a line-segment cutting an orbit perpendicularly. Perturb the initial position of a body slightly so that it starts out slightly displaced on the line-segment from the unperturbed position. Now trace the motion one full revolution around the Sun, and see where it cuts the line again. (There are higher-dimensional complications glossed over here; actually, one needs a three-dimensional "surface of section" in a four-dimensional "state-space" in which the body's two position and two velocity coordinates define a 4-dimensional point.) If each next arrival point on the line is farther from the initial position, then the initial position is unstable; but if each next arrival point coincides with the starting point, the motion is periodic. Since I have derived the best possible approximation to Bode's Law in the form of beta = 1.80 when the mass-ratios vanish, and explained the observed discrepancies from exactly 1.80 in terms of the small additive correction terms (which depend upon the particular mass-ratios in a particular problem), it seems to me that my "dynamical derivation of Bode's Law" (whose main result is independent of the Newtonian gravitational constant G and of the masses of the planets [so long as they are sufficiently small relative to the mass of the Sun]) is a genuinely fundamental contribution to the subject. R.W. Bass