A-1 THE MERCURY Mt-BASED EXPONENTIAL PLANETARY FRAMEWORK
Although the exponential function P(x) = Mt phi^x (x = -2 to 16, base Mt = 0.240842658 years ) generates successive mean sidereal and mean synodic periods from IMO to out beyond Pluto, the resulting exponential function is (naturally enough) based on the well-known Phi-Series., i.e.,

Fig 4. The Exponential Period Function P(x) = Mtk ^x ( x = - 2, -1, 0, 1, 2,..,16) and the Phi-Series

Table 1. The Phi-Series Exponential Planetary Framework

N	PLANETS	PERIODS	DISTANCE	INVERSES	VELOCITY	Vi DIFFS	DIFFERENCES
-2	IMO	0.091505	0.2030631	0.4506252	2.2191392	2.2191392	Next-Neptune
-1	Synodic	0.148059	0.2798698	0.5290272	1.8902620	1.8902620	Syn 17-Syn 15
0	MERCURY	0.239564	0.3857279	0.6210700	1.6101245	1.6101245	Neptune-Uranus
1	Synodic	0.387623	0.5316260	0.7291269	1.3715035	1.3715035	Syn 15-Syn 13
2	VENUS	0.627187	0.7327086	0.8559840	1.1682461	1.1682461	Uranus-Saturn
3	Earth/Synodic	1.014810	1.0098489	1.0049124	0.9951117	0.9951117	Syn 13-Syn 11
4	MARS	1.641996	1.3918149	1.1797520	0.8476357	0.8476357	Saturn-Jupiter
5	Synodic	2.656806	1.9182560	1.3850112	0.7220158	0.7220158	Syn 11-Syn 9
6	AST.BELT	4.298802	2.6438186	1.6259824	0.6150128	0.6150128	Jupiter-Ast.Belt
7	Synodic	6.955608	3.6438186	1.9088789	0.5238677	0.5238677	Syn 9-Syn 7
8	JUPITER	11.25441	5.0220594	2.2409952	0.4462303	0.4462303	Ast.Belt-Mars
9	Synodic	18.21002	6.9216070	2.6308947	0.3800988	0.3800988	Syn 7-Syn 5
10	SATURN	29.46443	9.5396410	3.0886309	0.3237680	0.3237680	Mars-Venus
11	Synodic	47.67445	13.147922	3.6260064	0.2757855	0.2757855	Syn 5-Syn 3
12	URANUS	77.13888	18.121002	4.2568771	0.2349140	0.2349140	Venus-Mercury
13	Synodic	124.8133	24.975104	4.9975098	0.2000997	0.2000997	Syn 3-Syn 1
14	NEPTUNE	201.9522	34.421707	5.8670016	0.1704448	0.1704448	Mercury-IMO
15	Synodic	326.7655	47.441400	6.8877718	0.1451848
16	NEXT	528.7177	65.385672	8.0861408	0.1236684

Table 2. The Mt3 Based Exponential Planetary Framework

B. THE PHI-SERIES AND THE SOLAR SYSTEM

B.1 THE EXPONENTIAL CONSTANTS
Up to this point the representations of the Solar System have been largely logarithmic, two-dimensional in form and generally static in nature, despite discussions concerning periods of revolution, lap cycles, planetary orbits and velocities. Yet the complexity of the Solar System, its endless and varying motions, its waxings and wanings, its growth and decay, its anomalies and its regularities all suggest it is something far beyond a mechanical clock or indeed anything that simplistic. But at least from the above analysis there appears to be some justification for suggesting that an exponential component exists in the structure of the Solar System, and moreover, that remnants of it remain in the two log-linear zones and the three inverse-velocity relationships discussed earlier.
But where does this leave us? According to the methodology applied to the mean periods of revolution and the intervening synodic periods, the suspected log-linearity in the Solar System largely translates into variants of the Phi-Series such that the mean periods (Sidereal and Synodic) increase sequentially by successive powers of Phi while the mean periods of the planets increase by Phi squared, i.e., relations 5a and 5b:

Relations 5a and 5b. The Fundamental Period Constants

Relations 6a and 6b. The Fundamental Distance Constants

Thus with relation 6b we obtain a constant increase in mean planetary distances of 1.88995476295... as opposed to the ad hoc multiples of 2 that belong to the Titius-Bode relationship. But even so, this still leaves an unaccounted "gap" between Mars and Jupiter. For more on this complex topic see Figure 6 and part F below.

C. THE PHI-BASED EQUIANGULAR PERIOD SPIRAL
Although the above digressions and what follows next impinge on matters discussed in Part IV and later sections, returning to the technical side of the matter there seems little doubt that the phi-based exponential planetary frameworks can (and likely should) be considered in terms of equiangular period spirals based on relation 5b expressed in the form:

Relation 9. The Exponential Period Function and Equiangular Period Spiral [ from Part IV ]

The resulting spiral (see Part IV) is predicated on the equiangular "square" dictated by relation 5b, i.e., the Phi-squared increase in mean planetary periods.
Thus for example, Figure 6c incorporates the Phi-Series mean sidereal and mean synodic periods from Mercury to Mars:

• Phi ^-3 = 0.236067978 = MERCURY, mean sidereal period
• Phi ^-2 = 0.381966011 = Mercury-Venus mean synodic period
• Phi ^-1 = 0.618033988 = VENUS, mean sidereal period
• Phi ⁰ = 1.000000000 = Venus-Mars mean synodic period; also the mean sidereal period of EARTH
• Phi ¹ = 1.618033988 = MARS, mean sidereal period

Figure 6c. The Phi-Series Equiangular Period Spiral from Mercury to Mars

Delineated on the vertical axis, the mean planetary periods increase by Phi squared per sidereal revolution of 360 degrees while the synodic periods occur at the 180-degree half-cycle points. Exactly the same configuration could be given for the Phi-Series periods for Jupiter, Saturn and Uranus (or indeed any such segment of the Phi-Series) since the periods increase in the same manner, whereas a uniform (i.e., log-linear) representation necessarily requires logarithmic data in addition, as shown in the inset. But there is far more to this equiangular spiral, for although the above represents Solar System mean periods, i.e., Time, it turns out that to produce corresponding equiangular distance and velocity spirals would be entirely redundant, for both sets of parameters are already integral features of period spiral itself. The details are discussed further in Part IV, but small wonder that Jacob Bernoulli should have called the equiangular spiral "Spira Mirabilis" and included it on his tombstone, or that part of the title is retained here, albeit shared with Archytas for reasons that will become apparent in the next few sections.
On a more recent historical note, investigation reveals that research concerning the spiral form in related astronomical contexts includes the work of Lothar Komp in 1996 (see F1. below) and William M. Malisoff in 1929. For the latter's inclusion of velocities, distances, periods and the logarithmic spiral see paragraph (7) in his 1929 letter to the editor of the Science ("Some New Laws of the Solar System".)

D. PHYLLOTAXIS AND THE EXPONENTIAL PLANETARY FRAMEWORKS
Although the present treatment has concentrated on Time, it now becomes necessary to consider the results in terms of the relationship between Phi, the Fibonacci Series and natural growth. In other words, physical considerations concerning growth itself, with time, "distance" and speed (i.e., rate of growth) integral components. So far the generated exponential planetary frameworks have largerly concerned the mean periods, i.e., time, but as understood from the outset, this was to obtain more data, new methodology and a more productive approach to the structure of the Solar System. Carried though all this, however, were still the inter-relationships between Time, Distance and Velocity provided by the velocity expansions to the third law of planetary motion and the third law itself. Moreover, as can be seen in Figure 6c, the obvious complexities of the Phi Series in this specific astronomical context reveal that the exact values for the mean periods also occur elsewhere in the table among the mean Velocities (e.g., the mean sidereal period of Mars and the mean velocity of Mercury; see also Table 1) in a complex, if not distinctly "ourobotic" context that will be discussed in later Sections. As for the occurrence of the Phi Series in the present context, those unfamilar with the subject might wish to bear in mind that Phi, the Fibonacci, Lucas and related series, far from being confined to plant and animal growth alone, occur in numerous diverse contexts over an enormous range that extends from the structure of quasi-crystals out to the very structure of spiral galaxies. And this being so, should there really be any great surprise if Phi should also prove to be an underlying element in the structure of planetary systems? It has long been recognized that although Phi and the Fibonacci Series are intimately related to the subject of natural growth that they are hardly limited to these two fields alone. Remaining with the Phi-Series, Jay Kappraff ³ points out that the French architext Le Corbusier "developed a linear scale of lengths based on the irrational number (phi), the golden mean, through the double geometric and Fibonacci (phi) series" for his Modular System. The latter's interest in the topic is explained further in the following informative passage from Jay Kappraff's CONNECTIONS : The Geometric Bridge between Art and Science:

With respect to the Phi-series and the exponential planetary frameworks under consideration, accepting (a) that an exponential component does exist in the structure of the Solar System, and (b) that the inverse-velocity relationships are indeed an integral feature of the latter, then it becomes possible to consider phyllotaxis in this explicit context, especially since the spiral form can be considered to be operating here also. At which point it may be recalled that in seeking to reduce the common minor deviations in the inverse-velocity relationships in the Phi-based planetary frameworks a substitute base period for Mercury (Mt3 = 0.2395640) years was applied and Phi retained as the constant of linearity. However, although the determinatiyon of the new base period Mt3 was necessary in terms of the initial framework, there was nevertheless another way that the common deviations could have been reduced to zero, namely the substitution of a slightly different value for the major constant Phi itself. Or, if one wishes, the establishment of a practical ratio similar to those discussed above that nevertheless reduced all inverse-velocity errors to zero. This requirement is readily achieved by back-solving, resulting in the retention of the present day estimate for the mean sidereal period of Mercury (Mt = 0.240827 years) as the base period but the substitution of a a new, slightly lower value of 1.6171413367027 for the constant of linearity. With this substitution the minor deviations in the inverse-velocity relationships are still reduced to zero while the resulting exponential planetary framework is found to differ only marginally from the other three (see Table 3 below).

The question that now arises is of considerable interest, for how does this new constant of linearity compare with the Fibonacci and Lucas ratios discussed above in association with natural growth? Although not entirely comparable, it turns out that the zeroing constant is indeed close to some of the phyllotaxic ratios in question, slightly lower, in fact, than the Sunflower's 89:55 phyllotaxis. In other words, the value in question--1.617141336703--is closest to the Lucas Series ratio 76 / 47 followed by the Fibonacci Series ratio of 55 / 34. The occurrence of the Lucas ratio in this context is perhaps the least surprising given the well-known relationship that exists between the Phi Series and the Lucas Series, namely that the difference between the two is the value obtained from reciprocal exponent of the generating power applied in the former. For example, in the Phi-Series exponential planetary framework the theoretical mean sidereal period of Uranus (76.0131556174.. years) is generated by Phi raised to the ninth power, while Lucas number 76 is less than this by exactly Phi to the minus ninth power, i.e., 0.0131556174.., and the same applies in the case of the eighth powers and the 47-year period, and so on. But is it pure coincidence that the 76 and 47-year periods correspond to the respective Phi Series periods for Uranus and the Saturn-Uranus synodic? And does the Lucas Series predominate here, or is there a Fibonacci component as suggested by proximity of the 55:34 ratio? Either way, there is little variance between the new exponential periods of the Lucas-Fibonacci (MtLF) framework and those provided by Mt3 and the two previous frameworks as shown in Table 3, which features the modern estimate for the mean sidereal period of Mercury for the initial exponential planetary framework (Mt-based) and also the last variant that employs the modified constant of linearity. Noteworthy in the MtLF-based data (but possibly coincidental) is the unforced correlation between the value for the mean sidereal period of Saturn of 29.45867 years in the latter and the modern estimate of 29.45252 years.

Table 3. Comparison between Solar System Periods and the four exponential Frameworks.

As explained above, the MtLF exponential planetary framework also provides error-free inverse-velocity relationships, which perhaps suggests that it should provide the preferred planetary framework. The following log-linear representation of the latter as the diagonal reference line is applied to compress of the range of the periods and facilitate the comparison between the exponential frameworks and Solar system parameters. Here with the diagonal providing the reference frame, deviations above and below the line represent longer and shorter periods respectively and thus also deviations in heliocentric distance, i.e., the greater distance above the line the further out for from the Sun, and below the line, the closer in with respect to the frame of reference. Thus the expected deviations for Pluto, Neptune, Mars and to a lesser extent Uranus are all evident, as is the suggested location of Earth in the synodic position between Venus and Mars. Also included in the comparison is the Mars-Jupiter Mean and associated synodics on either side. One other point of interest is the suggestion of oscillatory "quenching" among the gas giants (Neptune and Uranus especially) from the outer regions inwards towards the lower.

Figure 5. The MtLF Mean Periods and the Solar System: Mars-Jupiter Mean included

E. SIMILARITIES AND DIFFERENCES
A visual comparison between the twelve mean periods of the Mt3-based planetary framework and Solar System mean data is provided in Figure 5 (for data concerning Neptune and Pluto see Table 3). The next and outermost theoretical planetary position (period: 526.8669 years; mean distance: 65.233 A.U.) provides the inverse-velocity data for IMO although there is no known planet in the region. However, it is relevant to note here that Clyde W. Tombaugh (the discoverer of Pluto) wrote in 1980 that the search for a tenth Solar System planet occasioned a number of reports, mostly arising from observed irregularities in the orbits of known objects. Although it remained unconfirmed, a planet with a mean distance of 65.5 A.U. was in fact proposed by Joseph L. Brady of the Lawrence Livermore Laboratory, University of California in 1972.⁴ Since that time further proposals concerning a possible planet in the outer regions have been made by Van Flandern and Harrington, (50 A.U.-100 AU.),⁵ Whitmire and Matese (80 A.U.),⁶ Anderson (78-100 A.U.),⁷ and Powell (60.8 A.U., later modified to 39.8 A.U.)⁸ To date no tenth planet has been found, but most of these proposals require planets with highly inclined orbits, large eccentricities, and relatively long intervals between returns, all of which complicate confirmation, especially for small objects. However, further delineation on a wider scale may eventually be forthcoming from the gravity-based analyses of Aleksandr N. Timofeev, Vladimir A. Timofeev and Lubov G. Timofeeva;⁹ see also Aleksandr Timofeev's: Two fundamental laws of nature in the gravity field.
   In terms of departures from the norm perhaps the most difficult anomaly to accept is that Earth may currently be occupying a resonant synodic location between Venus and Mars. The establishment of the heliocentric concept notwithstanding, it would still appear inordinately difficult to perceive the position of Earth as anything other than an immutable and unquestioned constant. However, the relatively recently advent of Chaos Theory, its application in astronomical contexts and the investigations carried out by Sussman,¹⁰ Wisdom,¹¹ Kerr,¹² Milani,¹³ Laskar,^{14,16, 17} and others have now changed matters irrevocably. The Solar System can now no longer be wound backwards or forwards indefinitely like some well-oiled and well understood mechanical device, as Ivars Petersen¹⁸ recounted in Newton's Clock: Chaos in the Solar system. Nor can the positions of any of its various members be considered sacrosanct, not even that of Earth.
Whether Earth has always been in the synodic location between Venus and Mars and in such complex resonant relationships is uncertain, but the zone of habitability is generally defined by the orbits of the latter pair of planets, and it is an open question whether life would necessarily have developed at either extremity, or if it had, whether it would have necessarily flourished, given the large-scale periodic extinctions which appear to have taken place at Earth's more advantageous central synodic location. This even suggests that a fortuitous element may have played a role in the continuance, if not the very development of life here on Earth, and that while life may still abound in the universe, it may not be quite as common-place as previously supposed. Whether this has a direct bearing on the negative results obtained over the last four decades by the Search for Extra-Terrestrial Intelligence (SETI) is, however, another matter altogether. (For an up-to-date analysis of the search and an alternative, see Gerry Zeitlin's 1997-1999 essay: OPEN SETI: Rethinking the SETI Paradigm and the Future of SETI ).
For present purposes it may be noted that deviations exist between the Solar System and the exponential planetary frameworks, and that depending on the degree of confidence assigned to the latter, it may be feasible to quantify these anomalies in terms of planetary masses, mean distances, and the conservation of angular momentum, etc. This still leaves the anomalous position of Neptune, but it is possible to suggest a number of scenarios based on mass-distance changes that might include a further belt of asteroids and/or cometary material at approximately 65 astronomical units from the sun periodically perturbed by an object or objects in a eccentric polar orbit, etc. Apart from the exponential framework itself, very little of this is actually new, though scenarios based on total angular momentum might well remain problematic owing to uncertainties concerning the complete inventory and total mass of the Solar System itself. On the other hand, new avenues and new insights concerning the structure of the Solar System have already begun to surface; e.g., the Fibonacci-related paper by Aleksandr N Timofeev entitled: "Sprouts of New Gravitation Without Mathematical Chimeras of XX Century."

RE-ASSESSMENT FOR THE NEW MILLENNIUM
 In looking back over the three years that have elapsed since this third section of Spira Solaris was first uploaded in 1997 I have come to realize that I have been somewhat slow to react and even slower to change in spite of a number of positive inputs concerning the present subject that arrived one way or another via the Internet. This was partly because of an increasing interest in the historical side of the matter but also my own inability to absorb and act on the various inputs received.

  A little over a year ago (mid-December 1999) I received an E-Mail and later a written follow-up concerning a series of Phi-associated planetary relationships arrived at by Mr. John Shanahan of Berlin, Germany. The latter provided a modern source for the mean planetary data applied in his relationships, but was concerned in part that the results for some varied according to differences in modern parameters. In particular, based on mean dista nces of 1.523691 A.U., 0.387099 A.U. and 9.575616 A.U. for Mars, Mercury and Saturn respectively, although he arrived at a close approximation for Phi via the following route:

his concern was indeed justified, for only minor differences in the associated mean distances produced departures from the excellent value for Phi shown in relation [S1]. But be that as it may, on examination it turns out that the relationship is nevertheless directly applicable to the Phi-Series planetary framework, and moreover, it not only applies to the mean distances, but also the mean periods throughout. Actually, it does lead to Phi when the Phi-Series Mean Periods are applied, and also the corresponding value for the distances (1.378240772 ) when the distances are applied in turn. The reason for this becomes clear when one reduces the periods involved to their associated exponents although it is unnecessary to repeat the analysis here. The initial derivation by Mr. Shanahan was therefore in one way erroneous, but in another, it might well have resulted in an awareness of the Phi-Series planetary framework. But this was not the end of the matter anyway, for the latter had forged ahead and produced further exponential relationships, including the following involving the mean distances of Mercury, Mars, Jupiter and Neptune (the mean distances used were 0.387099, 1.523691, 5.204829, 30.068963 A.U. respectively.

The agreement here is again excellent based on modern values for the mean distances, but the reader will recall that Jupiter, Mars and Neptune all deviate from the exponential planetary framework, the latter especially. Thus one would hardly expect the exponential equivalent of this relationship to show any real correlation. But surprisingly, it does, in fact it produces perfect agreement using the the Phi-Series mean distances. But how can this be, with so much divergence between the mean distances of the latter framework and the modern estimates? In the case of Neptune, for example the Phi-Series equivalent is 34.0859 A.U. compared to the modern mean distance used above of 30.0689 A.U. and there is also a marked difference between the distances for Mars in addition. Then there is the difference for Jupiter with a mean distance of 5.203336 A.U. compared to the Phi-Series value of 4.97308025 A.U., and also Uranus at 19.191264 A.U. versus 17.9442719 A.U. A puzzle certainly, but also an opportunity, for the differences could be considered in terms of adjustments made with respect to the exponential planetary framework. Why should there be adjustments of this kind in the Solar System? The most likely cause, if not the most obvious, would be some change or other in planetary masses and/or positions. If so, there are certainly places in the Solar System that immediately come to mind where this might have taken place--not only the Asteriod Belt, but also perhaps, in the region occupied by slightly anomalous Uranus. Or should one say highly anomalous Uranus with its axis tilted almost ninety degrees off the "vertical"--a planet that is truly rolling along its orbit; or barreling around it, if one wishes. Just how Uranus came to be in this unusual situation has not been established, although there have been a number of theories proposed, as Eric Burgess explains below, adding a few more points of interest for good measure:¹⁵

The axis of Uranus is tilted so as to lie almost in the plane of its orbit ... one speculation is that there was a catastrophic collision with another planetary body early in the planet's history. . . What could have caused this axial tilt? One possibility is soon after the planet's formation it was hit by an Earth-sized body. An impacting speed of about 64,000 kph might be sufficient, according to some calculations, to push the planet on its side.. . A major mystery about Uranus is the low heat flux from the interior ... .all the small (Uranian) satellites have . . . . extremely dark surfaces. . . .What is this surface material? All the Uranian satellites are darker than the satellites of Saturn .. Also the colour of the surfaces is grey, whereas many other Solar System satellites have a tendency towards redness. (Eric Burgess, Uranus and Neptune, Columbia University Press, New York, 1988:51)

An impacting speed of about 64,000 kph translates into some 17.778 k/sec--an orbital velocity that corresponds to a distance of 2.7992 A.U., which places it in the central region of the Asteriod Belt, another enigma and a complex one at that. Again there are various theories that might be considered with respect to the latter, including either the breakup of a planetary body in the region and/or the ejection of a sizable amount mass from it. So perhaps there are two areas that may have required adjustment--the first being a loss of mass within the Asteriod Belt itself, and the second involving a possible loss of mass from Uranus, with the event in the latter region perhaps responsible for it. In other words, the collision may not only have knocked Uranus on its side, it may also have been of such force that it chipped off perhaps sufficient mass to require adjustments into the bargain. Either way the loss of mass would necessarily influence the total angular momentum and compensation would be required to maintain the conservation of energy. And since orbital angular momentum is a function of velocity, mass and distance, the slack would have to be taken up by an outward movement of one more of the other planets or Uranus itself.
Apart from their theoretical nature, there are unfortunately far too many variables in such scenarios, yet we do know the masses of the planets, we do have the parameters of the present Solar System and also the mean values from the exponential planetary frameworks. As a consequence the total angular momentum can at least be calculated in both cases and the difference between the two examined. Now at this juncture it is necessary to recognise that it is most difficult to obtain any guidance regarding what may or may not have happened to the Solar System in the past, even with the frame of reference provided by the exponential planetary framework. Nevertheless, Relation [S1] does provide an initial avenue to explore. But if comparisons with the Solar System are required, then it would perhaps be better to employ the MtLF-based planetary framework in so much as it not only retains the modern sidereal period for Mercury for its base period, it also gives the closest correlation with Saturn, i.e., a mean distance of 9.53842 A.U. compared to the modern value of 9.53707 A.U. This means for the same mass there is only a slight difference in angular momentum for this planet, whereas the difference between the Solar System and the exponential framework are considerable for both Jupiter and Neptune and to a lesser extent, also Uranus.
In seeking the simplest of initial tests it is possible to immediately focus attention on Uranus, not only because of the possibility of a catastrophic event, but also because relation of [2S], which suggests that both Jupiter and Neptune may have been two of the planets that underwent adjustments. A further simplification is to consider that the event (or events) took place before the formation of the four terrestrial planets and Pluto. This leaves us the four major superior planets, and with Saturn virtually unchanged, just the two adjusting planets (Jupiter and Neptune) plus Uranus, the possible prime casualty. All that remains is the Asteriod Belt, and that too may not have formed to any great extent at this hypothetical stage. Various scenarios can be suggested, but it still most difficult to assess what changes may have resulted, or which planet may or may not have played a role in the readjustment. But there is one thing that can be done above all else, and that is compare the total angular momentum for the known planetary masses applied to both sets of planetary distances.
For a total of 446.83728 Earth masses the angular momentum for the nine Solar System planets relative to unity is 1179.2179; for the same total mass and the last mentioned exponential planetary framework the angular momentum is in turn 1171.4148, thus smaller by 7.807458. The breakdown of the Solar System surplus is: +12.52712 for Jupiter, +1.830719 for Uranus, a mere -0.02073 for Saturn and -6.52966 for Neptune. These discrepancies are the result of the differences in the mean distances, i.e, the suspected adustments, but if so, then what caused them? Still focussing on Uranus, we know from the comparison with the exponential framework that this planet may have moved further out, possibly an adjustment of its own. In other words, to perhaps maintain angular momentum after losing a certain percentage of its mass, which would not be entirely surprising if it had indeed been hit by a sizable chuck streaking out of the inner solar system faster than a high-velocity bullet. But even so, what then of the other "adjustees"? Thus the problem expands, and with it the number of variables. But there remains one further possibility that can at least be checked. If, as suggested earlier, the event in question took place before the formation of the four terrestrial planets, then where did their respective masses come from? From the collision with Uranus, perhaps, and this too can be investigated, merely by assigning the total Solar System mass to the four superior planets in the exponential planetary framework and finding what change in mass for Uranus brings the surplus in the angular momentum back in line. Recomputed in this manner, the mass of the four terrestrial planets and Pluto (1.980278) increases the present mass of Uranus from 14.559 to 16.5393, and somewhat surprisingly, balance is achieved at 16.8348.with only a further increase in mass of 0.29554.
All of which leaves far too many gaps and other possibilities; but thanks to Mr Shanahan, it at least serves to emphasize that the exponential planetary frameworks discussed here may have some additional validity, if not additional uses. For more on Mr. Shanahan's continuing research see the latter's Web Site.

F. LUCAS AND FIBONACCI RESONANCES IN THE SOLAR SYSTEM

F.1 LOTHAR KOMP
Up to the present point the discussion has concentrated on the baseline provided by mean periods of revolution (sidereal and synodic), mean heliocentric distances and mean orbital velocities. In the Solar System itself, however, matters are far more complicated owing to the elliptical natures of the planetary orbits and differences (although slight) in the inclination of the orbital planes. Added to these complications are the various resonances that occur between the planets themselves, both real-time and mean motion resonances. As it turns out, many of the planetary resonances in the Solar System are either Fibonacci or Lucas Series related--resonances that not only incorporate the ratios 1:2, 2:3, 3:5, 5:8, 8:13 and higher, but also triples and in some case quadruple resonances.

It cannot be said that such occurrences have gone unnoticed, nor can it be said that the relationship between the Fibonacci Series and related resonances have not been equated with the structure of the Solar System in the past. But even with such qualifiers it is difficult to understand how recent research on the subject carried out by Lothar Komp²⁰ during the last decade has languished basically unheralded and unappreciated. First published in 1996 in the German-language magazine Fusion and the English-language magazine 21st Century the following year (1997) in an article entitled: "The Keplerian Harmony of the Planets and Their Moons," Lothar Komp's work was far more Science than History though both subjects were interwoven in a dense, highly informative and original discourse. But more to the present point, Lothar Komp's researches dealt in considerable detail with the Fibonacci Series in specific astronomical contexts and spiral forms that included the following:

Systems of Planets and Moons as "Barred Spirals."
Spiral Galaxies, Logarithmic Spirals and Ratios
A precise hyperbolic cosine distance formula, i.e.,

A Barred Spiral and the Moons of Neptune
A Barred Spiral and the Moons of Uranus
A Barred Spiral and the Moons Jupiter
A Barred Spiral and the Moons of Saturn
A Barred Spiral and the Outer Planets
A Barred Spiral and the Inner Planets
Synodic and Sidereal motion
Planetary and lunar resonances
Resonances that included rotation rates
Resonances among the Asteriods
Solar Activity and Planetary motion

F.2 MEAN MOTION RESONANCES

Without going into too much detail, in addition to Lothar Komp's treatment of the 1:2, 2:3, 3:5, 5:8, 8:13 resonances there also exist other perhaps lesser-known mean motion resonances, e.g., an interval of 34 years for Earth and Mars that corresponds to 18 sidereal revolutions for the latter, while further complexities concerning Venus arise when various multiples of 8 years are investigated, e.g., the 64-year period with 104 sidereal periods of Venus, 64 sidereal revolutions of Earth, 40 Venus-Earth synodics and additionally 34 sidereal revolutions of Mars. Remaining with the latter but using attested Babylonian period relations, in 47 years there will be almost 25 sidereal revolutions and 22 Earth-Mars synodic periods, to which may also be added that in 76 years there are 34 Mars-Jupiter synodic cycles. The Babylonians possessed far more period relationships than the few given here; including a related 79-year period for Mars, 29 and 59-year periods for Saturn, and 12, 71,83, 95, 166, 261 and 427 years for Jupiter (for further details concerning these periods and their application see Babylonian Planetary Theory and the Heliocentric concept) in each instance, while some of the above periods--especially those of 34, 47 and 76 years are also reflected--perhaps coincidentally--in the two ratios of primary interest--the Lucas 76:47 and Fibonacci 55:34 ratios.
Then there are the further complexities associated with resonances in the Asteriod Belt, including the 1:1 mean motion resonances of Jupiter-associated asteroids, known mean-motion 3:1, 5:2, 7:3, 2:1 resonance gaps and 3:2, 4:3, 1:1 concentrations. Such resonances within the Asteroid Belt may also be considered with respect to the mean sidereal periods of 1.880751 years and 11.868991 years of Mars and Jupiter respectively and the resulting geometric mean (MJM) between the two of 4.724682 years which is comparable to known 5:2 mean motion resonances. Secondly, the Mars-MJM synodic period stands in the ratio of 5:3 with respect to Mars, while the MJM-Jupiter synodic stands in a 3:2 ratio with respect to Jupiter, and a number of further 5:3:2 resonances also occur. Although obvious, it may be overlooked at times that all integer period relations expressed in years necessarily include the sidereal revolution of Earth and hence the resonances of Earth itself

Figure 6. Log-linear representation of the Asteriod Belt with Mars-Jupiter Synodic and MJM (Mean)

Further out among the anomalous planets Neptune and Pluto there are additional resonances, and with respect to the former it is also known that both Earth and Neptune are locked in similar resonant relationships. Denoting synodic periods by Ts, inner and outer mean sidereal periods by T1 and T2 and resonant relationships by: T1 : Ts : T2, both planets are in fact in 2:1:1 resonant relationships with adjacent bodies (Earth with Mars; Neptune with Uranus) while Neptune is also locked in a further 3:2:1 resonant relationship with Pluto. The latter's mean period produces poor results throughout as a base parameter for the exponential frameworks, but in comparison to its neighbors the gas giants, this small planet is already anomalous on a number of counts. Undoubtedly problems exist with the location of Pluto in the present context, but it is nevertheless still Neptune that represents the major discrepancy in the outer regions of the Solar System. Whether resonances among the four major superior planets will shed any light on the matter remains to be seen, but there is far more to this whole matter than mean motion resonances in any case, since real-time resonances in the Solar System must also be addressed. The question that now arises is how best to investigate these resonances on one hand and display them effectively on the other.

F.3. REAL-TIME RESONANCES: THE INFERIOR PLANETS
For this purpose the methodology of Bretagnon and Simon²¹ adapted to time-series analysis is particularly useful, especially the power series data and formulas for deriving heliocentric distances. The adaptation (given in Times Series Analysis) will be explained in more detail later but for present purposes it is sufficient to note that for any part of a planet's orbit at any point in time the instantaneous value of the radius vector can be treated as the mean value of an equivalent mean distance orbit and consequently also provide corresponding periods and velocities for the same. In other words, each planetary orbit may be considered in terms of successive mean motion orbits extending outwards from the shortest distance at perihelion to the longest at aphelion. In this way not only the varying distances, but also the velocities and periods may be treated as continuous functions over successive intervals. Instantaneous values of successive radius vectors may then be used to generate corresponding periods that serve to illustrate some of the better-known resonances among the inferior and superior planets. With respect to the former, particularly the adjacent planets Venus, Earth and Mars there seems little doubt that from a dynamic viewpoint Earth's location between Venus and Mars is highly complex. In addition to the resonances listed by Lothar Komp it may be noted that although the Venus-Mars mean synodic period is 0.914224 years, in practice the elliptical nature of the orbits of the three planets cause the instantaneous sidereal and synodic velocities to vary widely and also periodically coincide. But Earth is not only locked in a 2:1:1 resonance with Mars, but also in a 13:5:8 resonant relationship with Venus, which is itself linked to Mars by a further 3:2:1 resonance. Moreover, a plot of the true varying sidereal and synodic motion in the form of time-series data reveals the existence of even more complex resonant relationships as seen in see Figure 7 below:

Figure 7. The Venus-Earth-Mars Resonances and the Lucas Series Numbers

This actual example computed a number of years ago remains part of a relatively inconclusive but not entirely negative investigation of planetary resonances and their possible inter-relationship with solar activity. At that time even the more obvious feature--that all the numbers involved belong to the Lucas Series, i.e., 1,3,4,7,11 was not noted; nor were the other resonances encountered examined in terms of the Fibonacci Series per se. The present example (which repeats after almost thirty-two years) is however but one of a number of approaches that can be applied to the problem. It may be further noted here that in addition to occupying a resonant intermediate synodic location between Venus and Mars, that the corresponding inverse-velocity function for Earth may also defined in terms of the inverse-velocities of the three adjacent gas giants (the Uranus-Saturn and the Saturn-Jupiter synodics respectively) which are in turn subject to real-time periodic variations of their own.
But there still remains the unexplained occurrence of the Lucas 76:47 and Fibonacci 55:34 ratios and why the former gives the better correction for the inverse-velocity functions in question. On the other hand, there is the apparent linkage between the major superior and the terrestrial planets provided by the inverse-velocity functions and the undoubted Fibonacci relationships that exist among the more massive group of planets, Jupiter and Saturn especially.

F.4. REAL-TIME RESONANCES: THE MAJOR SUPERIOR PLANETS
Perhaps the best known resonance in the Solar System involves the relative motion of Jupiter with respect to Saturn. But before examining this example in detail it is necessarily to emphasize the predominance of this pair of planets above all others, including the adjacent major superior planets Uranus and Neptune. Alone Jupiter accounts for 71% of the planetary mass in the Solar System and more than half of the total angular momentum. Saturn comes next with 21% of the mass and and 25% of the angular momentum; taken together Jupiter and Saturn thus account for 92% of the mass and more than 85% of the angular momentum. The further inclusion of Uranus lifts the totals to 95% and 92% respectively, while all four major superior planets account for more than 99% of the planetary mass and more than 99% of the total angular momentum in the entire Solar System.
Of the four major planets, the heliocentric positions of the first three not only compare to successive positions on the exponential planetary frameworks, they also permit the generation of the three inverse-velocity relationships discussed in Part II. But there are other considerations to be factored into this complex equation, for Jupiter is not only the largest planet by far in terms of size and mass, it is also the swiftest moving major planet, followed in due order by Saturn (the next most massive) and then Uranus. Neptune at present represents an anomaly though it obviously cannot be ignored. But if one is going to concentrate on the major planets then it would be logical to expect that the influence of Jupiter and Saturn would predominate, followed next by Uranus. In other words, the three adjacent planets that belong to the five successive sidereal and synodic periods from Jupiter out to Uranus from the original log-linear segment. But since the sidereal and synodic relationships between Jupiter, Saturn and Uranus have long been known and to some extent researched, whatever it is that remains to be determined must be more than this alone, or even perhaps entirely different. Then again, perhaps it is something relatively simple but difficult to check exhaustively. Now at least the exponential planetary frameworks provide bases for comparison, as do the inverse-velocity relationships. Finally, the phyllotaxic Fibonacci/Lucas ratios at least permit the narrowing of the inquiry to an investigation of real-time resonances among the the four most massive objects in the Solar System.

F.5. THE JUPITER-SATURN CYCLES AND URANUS
As mentioned earlier, the present methods were first adapted a decade or more ago to generate real-time data to investigate the possible influence of planetary motion on Solar Activity cycles--an investigation that included resonances, but not exhaustively. Here the same methods can be directed towards more specific goals, though it is as well to be aware of the complexities in attempting to come to terms with interactions that involve multiple elliptical orbits and varying motion. A real-time period function for Jupiter will vary on either side of the mean sidereal period by the range permitted by the planet's eccentricity, in this case approximately 11 to 12.75 years and a similar situation prevails in the case of Saturn with a range of approximately 27 to 32 years. The more complex Jupiter-Saturn synodic cycle on the other hand has a somewhat wider theoretical range (approximately 17 to 24 years) with corresponding data derived from the synodic formula and periods obtained from the Jupiter-Saturn radius vectors. Time-series results in this case provide sinusoidal period functions that follow the variations of the respective radius vectors over time. Thus over approximately 59 years the 5:3:2 resonances of Jupiter and Saturn will be displayed as five sinusoidal waveforms for the former (i.e., 5 sidereal cycles), two sinusoidal cycles for Saturn, plus a three-cycle synodic waveform that maps the relative but varying motion of Jupiter with respect to Saturn over the same interval. "Resonances" occur when all three waveforms coincide--three times in the present example. But before proceeding there are two further matters that require explanation and emphasis. The first is that as long as the basic 5:3:2 relationship for Jupiter and Saturn holds, multiplications need not stop at the approximate 59 year period; nor for that matter, need the well-known 1:1:2 resonance of Uranus with respect to Neptune necessarily remain with unity (the latter provided by the mean period of Neptune), i.e.,

Figure 8. Jupiter-Saturn and Uranus-Neptune Resonances and the Fibonacci Series, 1940-1990.

Figure 8b. Multiple Jupiter-Saturn and Uranus-Neptune Resonances, 1890-1990.

At this juncture the matter begins to focus more firmly on the Fibonacci and Lucas Series, for in seeking to embrace the latter it seems that while it is still necessary to concentrate on the relative motion of Jupiter with respect to Saturn, the relative motion of Jupiter with respect to Uranus also has a significant role to play. The mean value of this period is readily obtained from the the mean sidereal periods of Jupiter and Uranus by way of the general synodic formula. Given to the sixth decimal the mean synodic period of Jupiter with respect to Uranus thus turns out to be 13.820371 years. What follows next is perhaps surprising, for in dealing with multiple harmonics--which is essentially what is under consideration here--it is one thing to invoke Fibonacci variants of the basic 5:3:2 resonant relationship between Jupiter and Saturn, and quite another to expect that the Jupiter-Uranus harmonics would relate to the Lucas Series in this precise context, especially in an opposite sense. Nor for that matter is it likely that one would anticipate that while it is necessary to reverse the order of the fibonacci triples to maintain the resonant relationship between Jupiter and Saturn (i.e., 5:3:2 to obtain 5 cycles of Jupiter, 3 Synodics and 2 cycles of Saturn in approximately 59 years, and so on), that the Lucas harmonic expansion would follow its normal order, i.e., 4, 7, 11, 18, 29, ... etc. But this being said, we are at least familiar with the Phi-Series planetary frameworks, the relationship between the latter and the Lucas Series and we are already dealing with the mean periods of revolution and synodic cycles expressed in years in both contexts. Again, however, bearing in mind the variance that results from the true orbital motions of the three planets in question, the relationship between the reversed Fibonacci triples and the Lucas harmonics is still not immediately apparent. One of the main reasons for this is that it only becomes clear after the multiple periods of the Jupiter-Saturn triples are averaged, and then only with the longer intervals is the relationship easily detectable. For example, based on a mean sidereal period of 11.869237 years for Jupiter, a corresponding mean synodic period 19.881324 years and mean sidereal period for Saturn of 29.452520 years, the fifth, third and second multiples (i.e., the 5:3:2 resonance) occur after 59.346 years, 59.644 years and 58.905 years respectively, whereas the average for all three products is 59.298 years. The fourth (4) Lucas augmentation of the Jupiter-Uranus mean synodic period on the other hand occurs after 55.282 years--a loose correlation easily dismissed as a chance occurrence. However, further investigation reveals that the 5:3:2 Jupiter-Saturn and Jupiter-Uranus Lucas multiple 4 are seemingly co-associated, for the next Lucas number (7) is similarly associated with the next reversed Fibonacci triple after 5:3:2, and as the two sets both proceed to their larger numbers, the difference between the averages of the Fibonacci triples and the Lucas multipliers becomes increasingly less. Thus by the time the 89:55:34 Fibonacci triple is reached the average of 1050.41 years is more closely approximated by the 1050.38 years obtained from the 76th multiple of the mean Jupiter-Uranus synodic cycle. In other words, the Fibonacci and Lucas assignments proceed sequentially, side-by-side in strict order. Thus the harmonic Fibonacci triples of the Jupiter-Saturn triad are related to the Lucas harmonics of the Jupiter-Uranus synodic cycle in the following manner for the given periods (rounded here to the nearest year for clarity and convenience):

5
3 -Lucas 4 ( 59 Years )
2

8
5 -Lucas 7 ( 94 Years )
3

13
8 -Lucas 11 ( 153 Years )
5

21
13 -Lucas 18 ( 248 Years )
8

34
21 -Lucas 29 ( 401 Years )
13

55
34 -Lucas 47 ( 649 Years )
21

89
55 -Lucas 76 ( 1050 Years )
34

Figure 9. The convergence towards the limit Phi by the Fibonacci and Lucas Series Ratios.

Or is there more to this matter in any case, including wider horizons with additional degrees of complexity?
Consider, for example, the detailed arguments presented in the following:

Table 4. The Jupiter-Saturn, Jupiter-Uranus Resonances and the Fibonacci/Lucas Series

Returning to the matter at hand, however, we have now arrive at the 76:47 Lucas ratio in true consort with the 55:34 Fibonacci ratio, with Lucas harmonics always occupying the position between the highest and next highest values in the associated Fibonacci triple. And here, as can be seen in Figure 10 -- real-time 89:55:34 multiples of the Jupiter-Saturn cycles and the 76^th Jupiter-Uranus cycle -- the latter component also moves towards the nexus of the Jupiter-Saturn cycles, and this increasingly so with time. At which point it seems both relevant and useful to redirect the reader to Kurt Papke's Animation of Binet's Formula available on the Internet since 1998--a presentation that (speaking for myself, at least) only now comes into clearer focus.

Figure 10. The 89:55:34 Jupiter-Saturn and 76 Jupiter-Uranus Cycles, 1940-2000.

As a first approximation it therefore seems that the relative motions of Jupiter, Saturn and Uranus, and predominantly that of the first--the largest, swiftest and most massive of the three are intimately associated with the Golden Ratio. Not included here, yet likely also involved are the relative motions of Saturn with respect to Uranus, the motion of Saturn with respect to Neptune, and additional complications arising out the dominance provided by Jupiter with respect to all three. Nevertheless, the situation may be summarised at this initial stage in terms of the relative motions of the three major superior planets Jupiter, Saturn and Uranus as follows:

Figure 11. The Fundamental Fibonacci and Lucas Resonances; Jupiter-Saturn, Jupiter-Uranus

Where does this leave us? Is it that surprising that the relative motion of Jupiter--firstly with respect to Saturn, and secondly with respect to Uranus--should predominate in the Solar System? And also, that the Golden Ratio should underly so much order and proportion in addition? Where does it all originate? That question no doubt belongs to philosophical debates concerning the first of things, but if it is the product of the relative motions of Jupiter, Saturn and Uranus, then it is little wonder that such ordering occurs if it is indeed resonating throughout the Solar System.

The above somewhat limited discussion necessarily concerns complex waveforms and motions for the mean, varying and extremal values dictated by elliptical orbits.
Although one could suggest that both the Fibonacci and the Lucas Series are embedded in the Solar System, it might be more accurate to say that they are in fact pulsating through it, and perhaps have been since time immemorial. 

GOLDEN DIGRESSIONS
1. Babylonian.
2. Egyptian

REFERENCES

1. West, John Anthony, Serpent in the Sky: The High Wisdom of Ancient Egypt, Quest Books, Wheaton,1993.
2. Gillings, Richard J. Mathematics in the Time of the Pharohs, Dover Books, New York, 1982.
3. Kappraff, Jay. CONNECTIONS : The Geometric Bridge between Art and Science, McGraw-Hill, Inc. New York, 1991.
4. Brady, J., cited in Out of the Darkness: the Planet Pluto, by Clyde Tombaugh and Patrick Moore, Stackpole Books, Harrisburg, Pa., 1980:201.
5. Littman, M. Planets Beyond, John Wiley and Sons, New York, 1988:198.
6. Op.cit., p.202.
7. Op.cit., p.204.
8. Op.cit, pp.205-206.
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10. Sussman, G. and Wisdom, J. "Chaotic Evolution of the Solar System," Science 257, 3 July 1992: 56-62.
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12. Kerr, Richard, A. "Does Chaos permeate the Solar System?" Science 244 (14 April 1989). 10b. Kerr, Richard, A. "From Mercury to Pluto, chaos pervades the Solar System," Science 257 (July1992):33.
13. Milani, A. "Emerging stability and chaos." Nature 338 (16 March 1989):207-208.
14. Laskar, J. "A numerical experiment on the chaotic behavior of the Solar System" Nature, 338 (16 March 1989):237-238.
15. Laskar, J., "The chaotic motion of the Solar System: a numerical estimate of the size of the chaotic zones." Icarus 88 (December 1990):266-291.
16. Laskar, J., Joutel, F. and Robutel, P. "The chaotic obliquity of the planets" Nature 361 (18 February 1993):608-612.
17. Laskar, J., Thomas Quinn, and Scott Tremaine. "Confirmation of resonant structure in the Solar System". Icarus 95 (January 1992):148-152.
18. Petersen, I. Newton's Clock: Chaos in the Solar System, W.H. Freeman, New York, 1993.
19. Eric Burgess, Eric. Uranus and Neptune, Columbia University Press, New York, 1988:51
20. Komp, Lothar, "The Keplerian Harmony of the Planets and Their moons," 21st Century, Spring 1997:28-41, translated by Rick Sanders and David Cherry from the original article first published in FUSION, April-May-June 1996.
21. Bretagnon, P and Jean-Louis Simon, Planetary Programs and Tables from -4000 to +2800, Willman-Bell, Inc. Richmond, 1986.
    

Copyright © 1997. John N. Harris, M.A.(CMNS). Last Updated on March 2, 2004.

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