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KRONOS Vol II, No. 2 "PROOFS" OF THE STABILITY

s_{n} =  1  x^(n+1) (1x)  (1 < x < 1). 
S = 1 / (1x),  (1 < x < 1). 
S= 1 / (1½) = 1/(½) = 2,
It should be clear that an infinite series cannot possibly converge unless its nth term tends to zero as n increases to infinity.
However, the mere decrease to zero of the nth term is insufficient to guarantee convergence. In fact, the series
Elsewhere,* we noted that for planetary configurations involving small eccentricities and inclinations Simon Newcomb had in 1874 proved the existence of formal (possibly divergent), infiniteseries, exact, almostperiodic solutions of Newton's equations of gravitational motion for the osculating elements of the planetary orbits. But the work of Poincaré and Arnol'd already discussed shows that these series converge in general only for nonresonant motions. For resonant and nearresonant motions, these series in general may diverge: sometimes they do converge, such as for certain particular periodic resonant motions, which may be either stable (Trojan asteroids) orunstable (Kirkwood gaps), but they may equally well fail to converge, as they must do in the case of "wild motions" (5). The difficulty is that there are always resonant motions (uncertain convergence) arbitrarily close to any particular nonresonant motion (certain convergence). Hence, an infinitesimally small change in the initial conditions of a given motion may cause a discontinuous change in the convergence properties of the associated Newcomb series (from convergence to divergence) and likewise a discontinuous change in the qualitative physical properties of the motion (from almostperiodic to wild) (2: p. 218).
[*"Did Worlds Collide?", loc. cit 33]
Prior to the 1961 proof by Arnol'd that Newcomb's series converges for nonresonant motions, it was known only that the series diverged "in general," as shown in 1899 by Poincaré (6).
In the theory of perturbations, one computes successive approximations to the first, second, and third terms in Newcomb's series. The first term corresponds to firstorder perturbation theory, the second term to secondorder perturbation theory, etc. In the days of Laplace, Lagrange and Poisson it was hoped that the series converged rapidly and that the terms higher than the third order would be negligibly small.
Unfortunately, Poincaré proved that "in general" the series had the following frustrating property. The terms would decrease until a certain term, say the nth, was reached, and then the terms would start to increase again, and the final result would be a divergent series! Poincaré called such series asymptotic and showed that if the series, say s, was replaced by its nth approximate s_{n}, the result would give an answer differing from the actual solution by a small amount, over a suitably restricted interval of time.
This rigorous result of Poincaré' saved the classical perturbation theories from total abandonment. One could always fall back on the fact that a first or secondorder perturbation theory would have some approximate validity for a limited interval of time. Unfortunately, Laplace and his immediate successors had guessed that the interval of validity of their "expansions" (infinite series) was "a few million years." But there is not a shred of rigorous evidence for this guess.
Laplace and his successors could argue, as physical scientists rather than mathematicians, that their theories accurately represented the observed facts for a few decades or (at most) a few centuries, and so should be granted some empirical validity even in the absence of rigour. While this is a justifiable attitude, the extrapolation from centuries to millennia was wholly unwarranted. Indeed, these theories represented all celestial motions as quasiperiodic; but it is now known that aperiodic wild motions may, in phase space, lie arbitrarily close to quasiperiodic motions. Since wild motions need not be of a recurrent nature for more than a few years or decades, prior to near collisions or escapes, the associated perturbation expansions cannot possibly be valid (in general) for more than a few years or decades. In 1974 it is known (5) that wild behaviour occurs both in hypothetical simulated planetary systems and in actual stellar orbits in our galaxy. Therefore Laplace's belief in the general validity of his expansions over millions of years is unquestionably false. (For an explicit repudiation of Laplace's theorem by Brown and Shook (2) see their page 202, where they call it "much overestimated.")
The reader will probably accept this last conclusion as logically watertight, if he has confidence that I have not misrepresented the situation. Accordingly I shall give now a number of detailed quotations from recognised sources regarding the situation summarised above.
Authoritative Testimony
According to H. C. Plummer (1918), ". . . to the second order in the masses there is no secular inequality in the mean distance .... This is Poisson's theorem, an extension of Laplace's corresponding theorem for the first order, and it is the most important elementary result bearing on the stability of the solar system" (7: p. 190).
According to Moulton (1902), "In 1773 Laplace . . . proved his celebrated theorem that . . . the major axes . . . have no secular terms .... Poisson proved in 1809 that the major axes have no purely secular terms in the perturbations of the second order with respect to the masses. Haretu proved . . . in 1878 that there are secular variations in the expressions for the major axes in the terms of third order with respect to the masses .... Lagrange began the study of the secular terms in 1774 .... The investigations were carried on by Lagrange and Laplace, each supplementing and extending the work of the other, until 1784 when the work became complete by Laplace's discovery of his celebrated equations. These equations were derived by using only the linear terms in the differential equations .... The process of breaking up a differential equation in this manner is not permissible except as a first approximation, and any conclusions based on it are open to suspicion. Equations (112) [Moulton's equations] give the celebrated theorems of Laplace that the eccentricities and inclinations cannot vary, except within very narrow limits. Although the demonstration lacks complete rigour, yet the results must be considered as remarkable and significant .... Newcomb .... has established the more farreaching results that it is possible, in the case of the planetary perturbations, to represent the elements by purely periodic functions of the time which formally satisfy the differential equations of motion. If these series were [uniformly] convergent, the stability of the solar system would be assured; but Poincaré has shown that they are in general divergent . . ." (8: pp. 432, 423, 425).
The recent astounding triumphs of Arnol'd, which rest on two centuries of theoretical toil, show that the attainment of mathematical rigour is not an impossible goal. However, the mathematician's attitude is that he would rather regard the answer as unknown until a rigorous answer is available, even if that requires waiting for decades or centuries of patient progress. In contrast the dynamical astronomer oriented to the comparison of observations with computational predictions is often not so patient. To quote Brown and Shook: "The mathematical processes which are used in developing the theories of the planets . . . are largely formal. While mathematical rigour is desirable when it can be attained, nearly all progress . . . would be stopped if complete justification of every step in the process were demanded. The use of formal processes is justified whenever experience shows that the results . . . are useful for the prediction of physical phenomena. Thus when calculating with an infinite series whose convergence properties are not known, one has to be guided by the results obtained; if the series appears to be converging with sufficient rapidity to yield the needed degree of accuracy, there is no choice save that of using the numerical values which it gives. We have not attempted to deal with convergence questions, but have retained throughout the practical point of view ...." (2: p. x).
In a similar vein, Brown admitted that "Those of us who, like myself, are mainly concerned with the applications of the laws of motion and gravitation to the actual problems presented by nature are often quite justly accused of a lack of logic in the processes we follow. Sometimes this apparent lack of logic has been explained, as in the case of divergent series, which Poincaré showed could give a limited degree of numerical accuracy. In other cases no explanation is forthcoming and we fall back on the unproved thesis that since those processes coordinate the phenomena with high accuracy, the probability that they are valid is great .... All progress would be stopped if the rigid standards of the pure mathematician had to be applied at every stage of the work. The point of view I take is similar to that of the physicist who generalises from a limited number of experiments" (4: p. 33).
However, Brown, in his retiring AAS Presidential Address, admitted that dynamical astronomy is rife with false dogmas:
"A considerable portion of what I have to say is . . . quite frankly critical in the destructive sense. Owing, perhaps, to the fact that the number of those who work in my field is very small, a certain amount of dust has gathered in the corners. In any subject in which there is considerable activity, correction of errors, fundamental or otherwise, is often rightly regarded as unnecessary: the results are soon forgotten. In my subject it has not been so. Statements which are wholly or partly erroneous have obtained currency and I have found no solution except that of giving the reasons why such statements cannot be accepted....
"The validity of the proofs of many of the earlier theorems which have been quoted has disappeared with the increase of the estimate of the time scale to some 20 times its former value. As long as the interval required for the attainment of the present configuration of the solar system was thought to be less than 10^{8} years, there was some hope that an approximate idea of initial conditions could be obtained from the present configuration. With an interval of at least 10^{9} years, most of these deductions appear to have lost all meaning....
"Laplace . . . showed that the eccentricities . . . would oscillate between certain limits .... An examination of the methods seems to indicate that in the absence of other forces they gave approximate indications for 10^{7} years and that these limits might not be greatly exceeded even in 10^{8} years. Beyond this time the linear differential equations on which they were based and which gave oscillating solutions have to be modified by the inclusion of terms which can no longer be neglected. This modification introduces into the solution periodic terms whose coefficients become larger whenever [near resonance] exists. In the majority of such cases the limits of variation of the unknowns are considerably increased."
Brown then proceeds to note, as quoted above, that this leads to nearcollisions, and that there is no reason why bodies could not have interchanged their mean distances. He then backtracks on this last sensational admission by hastily adding: "However, there are other circumstances which I cannot stop to develop here which render such interchange improbable. The present order of distances from the sun of the greater planets is, I think, the same as the initial order, though the relative magnitudes of some of their distances may have been considerably changed." (Emphasis added.)
The reader should note that in discussing the timescale of LaplacePoissontype stability theorems, this eminent planetary theorist is continually driven to use such words as "supposed," "appeared," "hope," "seems," "might," and "think," all of which are indicative of subjective intuitions rather than objective evidence. I have tried diligently but have not found anywhere any suggestion that Laplace's figure of 10^{7} years is supported by credible evidence. Furthermore, the hard evidence cited elsewhere* (e.g. observed behaviour of Oterma III and simulated behaviour of wild motions) indicates clearly that Laplace and Brown's timeintenal guesses of less than 10^{7} 10^{8} years must be replaced, in general, by timeintervals of less than 10^{2}  10^{3} years, which replacement demolishes all claim of the LaplacePoisson theorems to have any relevance to the stability of the solar system.
[*"Did Worlds Collide?", loc. cit. 37]
Brown's 1931 address culminates in two conclusions which contain a remarkable prognostication of the recent leastinteraction planetary orbit results of Bass, Hills, and Ovenden:
"When the [time] interval has reached a certain length, small non gravitational forces have to be included .... When, with this fact, is placed the observational evidence of distribution of the orbits [in a resonant configuration], the first conclusion is that the original configuration cannot be deduced from gravitational methods alone from the present configuration, if the interval is long enough. The second is that the present configuration might have been substantially developed from a large number of quite different initial configurations. The general effect of the mutual gravitational forces appears to be, not a recurrent motion, but a development [i.e. evolution ] into types of configurations which are least affected by small internal or external forces. If this is the case it follows also that we can predict the future course of the system for a given interval more accurately than we can learn of the past during the same length of interval.
"The extension of the time scale has had, in fact, a disastrous result in the new limitations it has placed on the possibilities of obtaining information concerning the past history of the solar system.... If to this is added the opinion of Jeans, that the discovery of the effects of radiation requires 'nothing less than a complete recasting of the theory of configurations assumed by rotating masses; all our hardly won knowledge, both of the configurations and of their stability, must be cast into the melting pot,' one wonders to what extent the present speculations as to the origin of the solar system rest on a basis of observation and calculation."
Brown's revised estimate
Our demurrer from E. W. Brown's comments about the time interval of validity of LaplacePoisson type calculations is bolstered by the fact that shortly after he delivered the remarks quoted above, he published a very similar statement in which (with no explanation) he had lowered all of his estimates by a factor of ten or one hundred.
This statement was contained in Bulletin of the National Research Council, No. 80, June 1931, pp. 460466, in an article on "The Age of the Earth from Astronomical Data." Brown says:
"These deductions, which can be justified as long as we have to deal with periods of time of the order of 10^{6} or possibly 10^{7} years, are not necessarily valid when the interval . . . is of the order 10^{8} years or longer. Proofs of the statements made below are not available at the present time .... They must be regarded as results that have been reached through accumulated experience in dealing with the problems of celestial mechanics: in other words, the balance of evidence appears to the writer to be in favour of them.
"One of the best known and most frequently quoted results is that of the stability of the mean distances, of the eccentricities, and of the inclinations .... For a limited degree of approximation this is true. In celestial mechanics a limited degree of approximation is equivalent to a limited interval of time; the longer the interval, the less accurate are the results produced by the approximation .... the only mathematical methods we know . . . are not capable of giving greater accuracy. The result of this mathematical deficiency on the stabilities is that the statement can only be regarded as valid over a limited interval of time of the order of 10^{6} or perhaps 10^{7} years at most .... The chief reason for this insufficiency is ... due to the phenomena of resonance."
W. M. Smart
In 1953, Dr. W. M. Smart, Regius Professor of Astronomy in the University of Glasgow, published an authoritative calculation (see his Celestial Mechanics, Longmans Green and Co. Ltd. and John Wiley and Sons Inc., 1953, pp. 4, 9495, 198) indicating that the maximum timeinterval over which LaplaceLagrangePoissontype "stability" calculations can be trusted is 300 years.
On page 4 he defines the mean solar day as the unit of time t for planetary theory.
On page 94 he says: "At first sight it would seem that the stability of the solar system is not assured. Such a conclusion would be hasty, for it would mean that we had forgotten the assumption . . . that the squares . . . of these quantities can be neglected. But owing to the appearance of secular terms . . . this restriction is only binding provided that the value of t is severely circumscribed, to the order, for example, of a century or two. The analysis . . . will then give . . . reasonable accuracy over the restricted interval concerned. . . but this analysis will not shed any light on the problem of the stability, or instability, of the planetary system."
On page 198 he says that, in the first approximation, the eccentricity e of a planet is given by
To spell out the details left implicit, note that (at)^{2} < (at) only if (at) < 1, i.e.,if
t < T = (1/a) = 10^{5} days ~ 300 years.
This maximum interval of 300 years is an independent confirmation, by a noted astronomer, of the quantitative estimate of 100 to 1000 years given on the basis of different arguments (Oterma III and "wild motions") in "Did Worlds Collide?"
Yasuke Hagihara
The reference which is most widely considered by astronomers to contain a proof of the invariability of the planetary mean distances over periods of 10^{11} years, is Yusuke Hagihara's "The Stability of the Solar System," Chapter 4, volume III, The Solar System, ed. by G. P. Kuiper, University of Chicago Press, 1961. Hagihara's entire five volume treatise on celestial mechanics has not yet appeared; volumes I and II have recently been published by MIT Press; volumes III and IV are available from a Japanese publisher. From the one volume that I have seen, and the announced outline, it is evident that this will be the most exhaustively thorough and definitive treatment of celestial mechanics for decades to come.
Evidently (in a 1944 Japanese journal which I have been unable to obtain as yet) Hagihara presented a very elegant and possibly definitive proof of the theorem of Poisson discussed above. However, this still leaves the theorem of Poisson itself subject to the various criticisms explained above, which convinced, e.g., Brown and Shook that the importance of the theorem has been "much overrated."
Hagihara notes explicitly that the procedures of Laplace, Lagrange, and Poisson are special cases of Newcomb's series, which was shown in 1899 by Poincaré to be nonuniforrnly convergent in general and actually divergent for many particular nearresonant motions. Therefore any conclusions drawn from a finite number of terms in this series lack mathematical rigour and may be false.
However, as an empirical procedure, to be validated by experiment, Hagihara advocates use of the first two or three terms, though he admits that agreement with the results of observation of the planets (or more accurate "special perturbations" numerical integration) is only "fair." He then quotes the numerical work of Simon Newcomb (done in 1895 when Newcomb was still labouring under the illusion that he had proved the eternal stability of the solar system) in asserting Newcomb's result that secular terms in the mean distances appear only in the third approximation and then decrease the mean motion by one part in 3 x 10^{11} per year; "accordingly about 10^{11} years would be needed" for the distances to change by appreciable amounts. Hagihara then notes that E. W. Brown had given the figure of 10^{8} years (a discrepancy by a factor of one thousand) and complains that "the computations leading to this estimate have not been published." Fortunately, we can readily supply this computation.
The meandistance equation is only one of six coupled perturbation equations, for the six astronomical elements, and for rigour they must be considered as intrinsically coupled; that is, the meandistance equation cannot be considered separately from the others (for that would only be valid if the eccentricities were essentially constants; but in the present context, the eccentricities are extremely rapidly varying; see Moulton's discussion of this point above). However, the computation by W. M. Smart summarised above shows that the equation for the eccentricity remains valid for at most "one or two centuries." Thus, the system of six coupled equations is not valid for more than a few centuries (for the second approximation fails after 300 years and so a fortiori the third approximation cannot be considered over a longer interval). In particular, then, the third approximation of the distance equation used by Newcomb cannot be considered after 300 years, and so his estimate of billions of years is meaningless.
If Simon Newcomb had known that Poincaré would soon disprove the hypothesised convergence of his series, he would never have published the opinion that the third approximation could be trusted for billions of years. We have already seen that Brown, aware of Poincaré's results, retreated from 10^{11} years to 10^{8} years, and then to 10^{6} years. I leave it to the reader to judge whether or not, if Brown had lived to see Smart's 1953 calculation, he would have further retreated to 10^{2} years.
In fairness to Hagihara, it must be mentioned that he is not aware that Newcomb's 1895 result can be disproved (i.e., shown not to be a strictly logical consequence of the laws of Newton) by arguments of the type just presented; indeed, on the first page of his survey of the subject, Hagihara warns the reader as follows:
"Will the present configuration of the solar system be preserved without radical changes for a long interval of time? . . . The question of the stability of the solar system is closely related to the form of the solution and to the behaviour of the series employed. The problem can be put as follows: What is the interval of time, at the end of which the configuration deviates from the present by a given small amount? Present mathematics hardly permits this question to be answered satisfactorily for the actual solar system. We must limit ourselves to a description of the present status of the solution of this difficult problem. "
Hagihara also admits, quite explicitly, that even if the mean distances were invariant, no conclusion could be drawn about the long time invariance of the orbital eccentricities. His equations (Eqs. 16) are identical to the equations of Moulton (Eqs. 112) discussed above, i.e., they are the "famous integrals of Laplace and Lagrange." Now, says Hagihara, "because the value of the product mna^{2} for the Earth is only 1/700 of that of Jupiter, the contribution to the sums in [Eqs. 16] by the terrestrial planets is unimportant, and no conclusions can be derived for these planets from [Eqs. 16]."
But, as Brown pointed out in his 1931 retiring presidential address to the American Astronomical Society (apparently the only reference by Brown which Hagihara does not list in his otherwise remarkably complete bibliography), if the eccentricities of the terrestrial planets increased sufficiently to bring about nearcollisions, the entire theory discussed above, including Newcomb's 1895 estimate for the time of invariability of the mean distances, would become meaningless and irrelevant, and the possibility that some of the terrestrial planets might have even interchanged their mean distances from the Sun cannot be logically excluded. Furthermore, Brown's argument, supplemented by Smart's calculation of the time of assured validity of the Newcomb type results, shows that such a "shuffling" of the planets Mars, Earth and Venus might have taken place in a matter of centuries rather than eons.
Expert Consensus
The central issue in the Velikovsky controversy since 1950 has been the question of whether or not it is dynamically possible, according to Newton's laws of motion and gravitation, for there to have been near encounters and perhaps even a "shuffling" of the mean distances of the planets Mars, Earth and Venus within historical times.
The late, great astronomer, Harlow Shapley, proclaimed in 1950 that the laws of mechanics had been thoroughly tested and that if Velikovsky's hypothesis were correct, then "the rest of us are crazy." In 1974 the dynamical astronomer J. Derral Mulholland called this same point the "fatal flaw" in Velikovsky's theory, which otherwise, he graciously stated, he found entertaining; Mulholland even went so far as to say that mechanics "absolutely" forbids the possibility of Earthcrossing orbits of Venus within astronomically recent times. Another of Velikovsky's most severe critics, astronomer Carl Sagan, stated forcefully and repeatedly in 1974 that of all classes of evidence pertaining to the controversy, dynamical evidence should be accorded the greatest weight. (In other words, even if some of the peripheral points in Velikovsky's archaeoastronomical theories should be mistaken, they would not themselves completely disprove the central assumptions of his theory; but dynamical evidence would be overriding.)
We have attempted here to review all pertinent dynamical evidence.The question of what conclusions follow with strict logic from Newton's laws is a question that can only be answered by rigorous mathematics and/or numerical computations with either rigorously theoretical or empirically controlled error bounds. Hence it is necessary to rely chiefly upon testimony by mathematicians. The so called "general perturbations" calculations used by dynarnical astronomers must be assessed with the greatest caution, for at least three reasons:
1 ) These theories have never yielded numerical accuracy that is at all satisfactory in comparison with either observations or "special perturbations" (numerical integrations).
2) They are based upon series which are probably at best merely "asymptotic" rather than uniformly convergent to correct answers, and in certain important cases (e.g., the "wild motions") give wholly false and misleading results, since they imply (when convergence is ignored) that all planetary motions are quasiperiodic, a "result" known since 1899 to be false.
3)The leading dynamical astronomers have admitted freely that their methods of calculation could not be justified by strict logic (which is required in all other branches of theoretical physics) and are open to the just accusation of being more an art, based upon "hope," than a strict science based upon strict logic.
Three of the greatest contemporary mathematical celestial mechanicians have stated explicitly and recently that nothing known to them forbids Velikovsky's hypothesis.
Only one dynamical astronomer has been found in recent times to have reviewed Newcomb's 1895 eonslong estimate with apparent approval, and even he complained explicitly that he could not follow E. W. Brown's reasons for lowering Newcomb's estimate from 10^{11} years to 10^{6} years; evidently he had missed Brown's retirement address, quoted in a preceding paragraph. Equally unfortunately, his recital of Newcomb's 1895 conceptions was printed in a very authoritative fivevolume astronomical handbook on the solar system, from which many astronomers have derived the misconception that celestial mechanicians accept Newcomb's ideas; but in fact, the author in question warned at the outset that "satisfactory" rigour was absent from the published statements upon which he was reporting.
The life's work of a sincere and dedicated scholar, who has published all of his sources for critical scrutiny by everyone, should not be dismissed hastily upon mere "group consensus" about the validity of obsolete ideas, which true experts have long ago dismissed as illusions.
REFERENCES
1. Isaac Asimov, Asimov's Guide to Science ( Basic Books, 1972), p. 357.
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