Spira Solaris Archytas-Mirabilis Part II. The Alternative

A. CONSIDERATIONS AND COMPLICATIONS
    1.   GENERAL CONSIDERATIONS
    2.   SOLAR SYSTEM PLANETS
    3.   THE INTER-MERCURIAL OBJECT

B. ORBITAL AND LINEAR REPRESENTATIONS
    1.   NORMAL AND LOGARITHMIC PLANVIEWS
    2.   LOGARITHMIC LINEAR REPRESENTATIONS

C. TIME AND MOTION
    1.   SYNODIC MOTION AND SYNODIC "ORBITS"
    2.   MEAN ORBITAL VELOCITIES
    3.   INVERSE VELOCITY RELATIONS

D. CONSTANTS OF LINEARITY
    1.   THE SYNODIC FORMULA AND PLANETARY SPACINGS
    2. DETERMINATION OF THE FUNDAMENTAL CONSTANT OF LINEARITY
    3.   RELATED CONSTANTS AND FORMULAS

E.   ADDITIONAL TABLES AND GRAPHICS
    1.   RELATED PHEIDIAN CONSTANTS AND FORMULAS

    A. CONSIDERATIONS AND COMPLICATIONS

    A1.   GENERAL CONSIDERATIONS
Although we are perhaps still too close to judge the matter effectively, the apparent lack of progress in coming to terms with the overall structure of the Solar System since the Titius-Bode era suggests that the latter's ad hoc approach was in all likelihood more of an impediment than a help, and a long-lived impediment at that. With this in mind it therefore seems necessary to seek a wider yet more critical approach to the matter--specifically an approach that: (a) avoids preconceptions and ad hoc methodology and (b), an approach that necessarily incorporates an increased planetary database beyond that restricted to mean heliocentric distances alone. But although the inventory of the Solar System has been enlarged and refined over the past two centuries, fundamental difficulties nonetheless remain. In particular, even if it should prove feasible to approximate the present planetary structure, it may still not be possible to account for the many anomalies that currently exist within the Solar System. Here one necessarily includes such things as origins, changes over time, and also the possibility of occasional catastrophic events, periodic or otherwise. But even without such considerations the absence of a standard frame of reference for planetary systems per se means that we currently have no way of knowing whether the Solar System represents the norm for planetary systems, an evolving or modified form of the latter, or (for whatever reason) even an exception to the rule.

    A2.   SOLAR SYSTEM PLANETS

Although oddities and exceptions abound throughout the Solar System and the mechanisms involved remain beyond our understanding, we can at least take stock of the System as we know it today. Thus we may begin with the generally accepted division of the nine planets into two main planetary groups. In due order outwards from the Sun then, are firstly the innermost solid planetary bodies known as the Terrestrial Planets, i.e., Mercury, Venus, Earth and Mars. Beyond the last named lies the Asteroid Belt and beyond this massive Jupiter, the largest of the nine planets and the innermost of the Four Gas Giants--Jupiter, Saturn, Uranus and Neptune. Lastly, out beyond Neptune in a class of its own and in an unusual orbit lies tiny Pluto--a body so small that even as a moon it would still rank no higher than the fifth largest moon in the Solar System.

Thus from a preliminary viewpoint there are essentially four distinct regions--the first occupied by the relatively small Terrestrial Planets, the second by the Asteroid Belt, the third by the four immense Gas Giants and lastly the remote and singular domain of tiny Pluto. Here one can at least begin to regroup the data, for it is questionable whether Pluto is necessarily a planet at all, even though it may be occupying a planetary "position" per se . But what constitutes a planetary position in this context anyway, given that we have no established planetary framework to guide us? About all that can be suggested at this stage is that Pluto, though still part of the planetary set represents an anomaly that may or may not be explained by further investigation. Next, a second anomaly of a different kind would seem to be the Asteroid Belt, but while there are over 5,000 asteroids in the region and others beyond it, their combined masses (Ceres included) are nevertheless far too small to account for a planet per se. But was there ever a planet between Mars and Jupiter? Again, we simply do not know; but then neither do we know what caused so many asteroids to be in this particular region in the first place, though various hypothetical scenarios involving collisions and/or the gravitational break-up of planets have been proposed (notably by Tom Van Flandern;¹see also the latter's Exploded Planet Hypothesis - 2000). But here matters increase in complexity, for orbital shifts and the redistribution of planetary masses necessarily affect the total angular momentum of the Solar System, rotational component of the Sun included. Which brings us to a third anomaly, namely that although the Sun has by far the greatest mass, it is the planets--predominately the Four Gas Giants--that possess almost all of the angular momentum. Thus postulating orbital changes and/or the break-up of hypothetical planets between Mars and Jupiter (or indeed anywhere in the System) involves mathematics of N-body proportions and complexity. Difficult enough for a single occurrence, what then of other events that may or may may not have been sequential, periodic, or alternatively, totally unrelated in both time and place? As for the early historical side of the matter, there are also problems and unanswered questions that pertain to the precedence of planetary formation itself, i.e., to what extent the Gas Giants may have preceded the Terrestrial planets, and whether the Asteroid Belt and/or the orbit of Pluto preceded or followed the formation of the Terrestrial Planets in turn, etc. Here once again the matter of origins comes to the fore, as does the possibility of catastrophic events and relatively large-scale changes within the Solar System itself. All of which suggest that the first and foremost requirement is an underlying planetary structure--not necessarily complete in its entirety either, but useful enough to provide a valid starting point and an initial frame of reference for further analysis.

But where then to begin? Perhaps with planet Earth itself, not least of all because it is the mean parameters of Earth that provide the fundamental frame of reference for Solar System distances, periods, velocities, and planetary masses, etc. Next, notwithstanding the recent (2006) demotion of Pluto and elevation of the asteroid Ceres to the status of "Dwarf Planets," the present analysis -- though occasionally including the latter pair -- will primarily concentrate on the following two major groupings (parameters relative to unity):

Additional data, i.e., physical composition, orientation of planetary axes and planes of revolution, densities and gravities, etc., could also have been included the above, but such data remain difficult to separate from the early formation of the planetary structure itself, with or without subsequent modifications. More in keeping with the present approach and the need to enlarge the available database we may on the other hand add to the attested planets an Inter-Mercurial Object (hereafter IMO) that owes its origins to orbital parameters determined by Leverrier. Reported in the journal Nature ⁴ in 1876, the "object" (mean period: 33.0225 days, corresponding mean distance 0.201438 A.U.) lies between Mercury and the Sun as the title implies. Thus for present purposes it serves to extend the range of planetary mean distances at the innermost extremity, which is not to suggest that the object is/was necessarily a planet per se, but rather (subject to reservations already stated above) an object that may or may not be occupying a planetary location. This step is in fact a minor addition that although helpful is not vital to the development of the final framework or the enlarged database; in fact the latter results largely from the inclusion of mean periods and mean velocities in critical contexts that will be discussed in detail later. However, before moving on to this stage further groundwork remains; specifically, the reevaluation of the manner in which planetary orbits and parameters are generally represented.

B. ORBITAL AND LINEAR REPRESENTATIONS

In retrospect, the suggestion of an exponential component in the structure of the Solar System implied in the Titius-Bode relationship might reasonably have been explored with far more rigor and forethought than appears to have been the case. The planview of the Solar System could surely have been examined more closely from a logarithmic viewpoint, especially since the deficiencies of normal planview representations have long been known. In fact, although it may not be immediately obvious, it is particularly beneficial to represent Solar System orbits logarithmically since the three innermost terrestrial planets (Earth, Venus and Mercury) can scarcely be discerned using normal values and normal means of presentation (see Figure 1a). On the other hand, logarithmic ranges in such applications effectively "compress" the outermost values and "expand" the inner, providing an informative and revealing view of the entire Solar System as shown in Figure 1b:

Fig 1a. The Solar System: EllipticalOrbits, Normal Scale

Figure 1a. Elliptical Orbits, Normal Scale. Figure 1b. Mean Distance Orbits, Logarithmic Scale

Clearly, from the second representation alone the presence of an exponential element might be suspected which in turn gives to an initial regrouping Apart from its convenience this representation of the mean value orbits also suggests that:

Figure 1c. Mean Distance Orbits IMO-NEPTUNE Figure 1d. The Two Log-Linear Zones

Continuing with the preliminary indications provided by Figure 1b and the suggestion made in Part I, the planet-to-planet increases for Earth and Mars may both be atypical, with Earth possibly occupying an "intermediate" position between Mars and Venus. In Part I it was not possible to discuss the matter in further, but here mean values provide further frames of reference, or more properly, indexes of the proximity of adjacent planets and a tentative reference frame for the mean distances. In other words, values that are greater than the mean may serve to indicate that the body in question is further out from the Sun than the "theoretical" norm (i.e., the position corresponding to the log-linear framework), while values that are lower are correspondingly nearer the Sun, etc. Thus in practice, Venus may be considered to be slightly closer to the Sun, while Mars (because of the Venus-Mars and Earth-Mars increases) considerably further out. Similarly, both Uranus and Pluto may also be considered to be beyond the "norm", while Neptune--already suspected of occupying an "intermediate" position--may be much closer in. Moreover, the increases for the two exceptions (Earth and Neptune) are not only similar in value, they also appear to be reversed. Here, of course, matters are complicated by the possibility (if not the fact) that the distances for the neighboring planets (i.e., Mars in the inner zone and Uranus in the outer) may also deviate somewhat from the "norm" for whatever cause or reason. About all that can be said at present is that these preliminary indications remain simply that.

Figure 2b. Planetary positions in the Two Log-Linear Zones (IMO omitted)

Apart from its convenience this representation of the mean value orbits also suggests that:

A degree of linear separation (i.e., log-linearity) may exist in the spacings of the majority of the planets in the Solar System.
In particular, log-linearity appears to be present among the three adjacent planets Jupiter, Saturn and Uranus in the Gas Giant Zone.
Further, log-linearity also appears to be present among the three Terrestrial Planets Mercury, Venus and Mars in the Terrestrial Zone, IMO included.
Although no planet exists in the Mars-Jupiter Gap, the Mars-Jupiter (and/or Ceres) mean matches the spacing between the two log-linear zones.
Possible deviations from both log-linear sequences are suggested by the orbits of Earth in the inner zone and that of Neptune in the outer.

With respect to the latter possibilities, the "kink" in Bode's Law discussed in Part I has already engendered the suggestion that Earth may be occupying a "synodic" (i.e., intermediate) position between Venus and Mars. Similarly, from Figure 1b Neptune may perhaps be occupying an "intermediate" or "synodic" position between Pluto and Uranus. If so, then Pluto (though still an errant moon rather than a planet per se) may in turn be occupying the next planetary position beyond Uranus.

Although the preliminary possibilities suggested by the log-linear representation of the Solar System are little more than that at present, the latter approach nevertheless provides further avenues for investigation. In particular, the mean distances can now be reexamined in terms of the Solar System planet-to-planet increments discussed in Part I, but from a narrower viewpoint with a correspondingly sharper focus. Thus the planet-to-planet multiplication factors shown below concentrate primarily on the two suggested log-linear zones while the distances and attendant multiplication factors for Earth and Neptune provide similar exceptions.

What is required next is the introduction of synodic motion and orbital velocity, after which the suspected deviations and anomalous locations of Earth, Neptune and the Mars-Jupiter Gap may be revisited and examined in further detail.

What is required is something that binds the planets together, and for this purpose synodic periods and planetary velocities now enter into the discussion, i.e., if the planets are indeed ordered, then the manner in which they move with respect to each other should also be ordered, and any ordering that involves the distances necessarily also involves both the periods and the orbital velocities. Which means, because of the exponential relationships that exist between all three, that the two latter sets of parameters should also be available for present purposes.

C2. THE GENERAL SYNODIC FORMULA
In more detail, although rarely described in the following form, for any pair of co-orbital bodies (where T₂denotes the period of revolution of an outer body, T₁ the period of revolution of an inner body, and T₂ > T₁) the general synodic period (or "lap" time) may be expressed as the product of the two sidereal periods (T₁ and T₂) divided by their difference:

Here the mean parameters of Earth provide the standard frame of reference (unity). The synodic periods of the planets are more commonly given with respect to the relative motion of Earth using simpler formulas. The latter variants are, however, merely special cases of Relation 1a with unity (the sidereal period of Earth) replacing T₁ or T₂ according to which group of planets is under consideration. Relation 1a is also further simplified by implicit multiplication (i.e., by unity: 1 x T₂ = T₂, etc.) such that the standard synodic formulas are usually expressed separately.

C3. MEAN SYNODIC CYCLES

With the motion and mean parameters of Earth providing the fundamental frame of reference for planetary mean distances, mean periods and mean velocities it now proves possible to extend the investigation to include synodic cycles between each adjacent planetary pairing in the Solar System. For example, in simple terms, in the cases of the first three gas giants the synodic cycle of Jupiter with respect to Saturn (i.e., Jupiter's lap-cycle) is approximately 20 years while that of Saturn with respect to the next planet (Uranus) is approximately 45 years. Thus the two synodic periods lie neatly between the sidereal periods of the bracketing planets in both cases. But what of the synodic cycle of Uranus with respect to Neptune, the latter planet already suspected of occupying a synodic location? In this case the synodic cycle turns out to be greater than that of Neptune itself. It is natural, therefore to wonder if the same situation prevails in the case of the other suspected anomaly, i.e., the position of Earth and therefore the synodic cycle of Earth with respect to Mars. Here once again the synodic period turns out to be greater than that of the outermost planet in the pairing. But this is not all, for the two suspect planets Earth and Neptune now show a further anomaly, namely that the synodic cycles on either side of these two planets are larger than the sidereal period of the planet itself (see Table 3). A double coincidence? Perhaps, perhaps not, especially when the remaining synodic cycles are all found to follow a consistent yet different pattern, i.e, for the Terrestrial planets from IMO to Venus and the first three Gas Giants the intervening synodic periods each lie between their associated bracketing planets. Moreover, if we do consider that Earth might be occupying the synodic position between Venus and Mars, then the Terrestrial Planets sequence can be extended by two additional steps with the inclusion of Mars. But even without this extension and IMO it seems that two similar log-linear regions might exist in the present Solar System--the first involving the three successive periods from Mercury to Venus (five with the inclusion of Mars), and the second from Jupiter to Uranus once again involving five successive periods.

Before proceeding with the next stage is seems necessary to emphasize that although the mean synodic periods represent difference or lap cycles between successive pairs of adjacent co-orbital planets, such cycles nevertheless represent complete revolutions of 360 degrees per mean synodic period. The difference between this type of orbital motion and planetary revolutions per se is that the latter take place with respect to a fixed sidereal reference point, whereas the former take place with respect to a moving point of reference. Even so, for every such mean synodic period the concept of an equivalent sidereal period and equivalent synodic "orbit" can be applied. With this device applied consistently throughout, the equivalent synodic orbits may then be included in log-scale representations of the Solar System. Here the results serve to emphasize the log-linear aspect far more effectively than the numeric representation provided in Table 2. In fact, omitting both Neptune and Pluto for the time being, the suggestion of log-linearity appears to be quite pronounced whether planet-to-planet, synodic-to synodic, or indeed sequentially (i.e. planet-synodic-planet) in the two log-linear zones under consideration (see Figures 2c, 2d and 2e). Although no planet can be assigned to the Asteroid Belt per se, an orbit that corresponds to the geometric mean between Mars and Jupiter (2.8156896 A.U., sidereal period: 4.7247945 years) also provides bracketing synodic periods similar to those of the Terrestrial and Gas Giant zones, if not the continuation from the latter to the former. Not included in the table but shown in Figures 2c and 2d is the Venus-Mars synodic of 0.914222 years that lies just inside to the mean orbit of Earth. Omitted for clarity from the Figures 2c and 2d are the Earth-Mars and Mars-Jupiter synodics with mean periods of 2.135375 years and 2.234902 years respectively that lie just beyond the orbit of Mars.

Relations 2a-2c: Velocity Expansions of the Harmonic Law

Once again such relationships serve to emphasize that if the mean distances of the planets are indeed ordered, then the manner in which the planets move with respect to one another, and thus the mean periods and the mean velocities should also be ordered. Which in a sense means that with these new additions the major orbital parameters available for analysis in the present context have been effectively quadrupled. The use of the Inverse Velocity in this context may appear unusual at first acquaintance, but it is a useful device nevertheless, not least of with respect to the computation of angular momentum. Moreover, as will be seen next, the inverse velocities also provide a linkage between the two log-linear zones.

C5. INVERSE VELOCITY RELATIONS

The inclusion of Earth in this context--synodic location notwithstanding--thus serves to augment the linkage between the terrestrial planets of the lower log-linear zone and three adjacent gas giants of the outer zone, ( Jupiter, Saturn and Uranus.) One or two other inverse-velocity relationship also appear to exist that are almost sequential--a necessary qualifier in so much as the latter appear to incorporate synodics and planetary inverse velocities. Although mean values are applied in the above relationships, in real time such functions will obviously vary according to the elliptical natures of the associated orbits. Real-time investigation of the Mars-Jupiter-Saturn relationship with frames of reference provided by the mean orbital velocity of Earth of 29.7859 kilometers per second and 24.1309 kilometers per second for that of Mars revealed that the real-time maxima and minima for Relation [4b] range between 19.66 and 28.3 kilometers per second, thus well exceeding the extremal velocities of Mars itself. However, utilizing the methods of Bretagon and Simon⁷ adapted to generate sequential data for 5-day intervals from 1700 to 2000 A. D., the mean value nevertheless still turns out to be 24.0938 kilometers per second.

C6. VARYING INVERSE VELOCITY
Similarly, the data for the real-time function based on Relation [4s1] reveals that although there is an even wider swing in extremal values, the mean value is also comparable to that obtained from Relation [4s1] directly. All of which is further complicated by the proximity of the Mars-Jupiter synodic to the Earth-Mars synodic and various resonances known to exist in the Solar System -- complications that at this stage no doubt intrude rather than enlighten and as such will be deferred until later.

Figure 4. The Jupiter-Saturn Vid Cycle and Orbital Velocities of Mars

Finally, suffice it to note here that although only a few inverse-velocity relationships are readily apparent in the Solar System and this scarcity might suggest such relations have little to do with the log-linear sequences, it turns out that they are in fact an integral feature with corresponding values for all planetary and synodic positions. The reason for there being so few obvious relations would appear to lie in the fact that the inverse-velocity relationships are influenced in no small way by deviations in the planetary structure. Thus with three suspected deviations in the Solar System to contend with it is perhaps fortunate that those that were evident were sufficient to connect the two log-linear zones in the manner discussed above.

The fundamental question that remains to be answered is whether an exponential component exists in the structure of the Solar System. This is certainly suggested by the near log-linear spacings of the sidereal and synodic periods for the inferior planets (Mercury and Venus) and similar spacings further out among the adjacent superior planets Jupiter, Saturn and Uranus. Moreover, a related question that arose above was whether the trio of unexpected inverse-velocity relationships linking the outer and inner log-linear regions were residuals of the same suspected exponential structure, the possibly anomalous locations of Earth, Neptune and Pluto notwithstanding. It is fortunate in one sense yet unfortunate in another that the suggestion of log-linearity in two separate regions of the Solar System and the complex linkage provided by the inverse-velocity relationships impose rigid requirements on any exponential function that might connect the two.

In particular, an exponential function for the mean periods (both sidereal and synodic) in the form F (x) = Mtk ^x (where Mt is a base constant provided by the mean sidereal period of Mercury) must not only produce the three inverse-velocity relationships in the same order, it must also generate a complete exponential planetary framework that commences with the mean sidereal period of Mercury for x = 0, thereafter sequentially generate the Mercury-Venus synodic (Ts) for x = 1, followed by the mean sidereal period of Venus for x = 2, and so on.

Furthermore, if as suggested in the first two parts, Earth is indeed occupying the synodic location between Venus and Mars, then the position for Earth must necessarily be obtained from the next exponent (3), followed by exponent 4 for Mars. Moreover, from the log-linear representations demonstrated in Part II it is clear that sufficient spacing exists to incorporate a planetary position in the Mars-Jupiter Gap itself and also synodic locations on either side. Together these three positions should therefore account for exponents 5, 6 and 7. This leaves the successive mean sidereal and mean synodic periods for Jupiter through Uranus to be generated in turn by exponents 8 through 12, with results that should also produce inverse-velocity relations comparable to those exhibited in the Solar System. Lastly, if IMO (the Inter-Mercurial Object) is indeed occupying a valid planetary position, then the function should also be expandable in the opposite direction and thus include both the latter's mean sidereal period and the IMO-Mercury synodic for exponents -2 and -1 respectively. Or, (and even simpler) since the latter is uncertain, a continuous exponential function for the mean periods commencing with Mercury (Mt) that connects the two log-linear zones utilizing sequential exponents that run from 0 through 14:

the first expansion Mk¹is directly obtainable from The product of the mean sidereal periods of Mercury ( Mk ⁰ = M ) and Venus ( Mk ²) divided by their difference:

This leads directly to the determination that the value of k for the successive sidereal and synodic periods of the exponential Solar System is none other than the constant Phi = 1.6180339887949, the "Golden Section" known and revered since antiquity, defined in turn (Livio, 2002) ⁸ by the quadratic equation that results from Relation 1b.

D3. RELATED PHEIDIAN CONSTANTS AND FORMULAS
From the above result it is apparent that the mean periods of revolution and intervening synodic periods increase sequentially by successive powers of Phi itself, (essentially the Phi-Series) while the mean periods of the planets increase in turn by Phi squared with corresponding constants for the mean heliocentric distances and mean orbital velocities both readily obtained from the Harmonic Law and the velocity variants discussed earlier, i.e.,

REFERENCES

PREVIOUS
I Bode's Flaw
http://www.spirasolaris.ca/sbb4a.html
Bode's "Law" - more correctly the Titius-Bode relationship - was an ad hoc scheme for approximating mean planetary distances that was originated by Johann Titius in 1866 and popularized by Johann Bode in 1871. The " law " later failed in the cases of the outermost planets Neptune and Pluto, but it was flawed from the outset with respect to distances of both MERCURY and EARTH, as Titius was perhaps aware.

Relations 2a-2c: Velocity Expansions of the Harmonic Law

Figure 2g. The Exponential Period Function F(x) = Mtk x ( x = 0, 1, 2,..,14 )

REFERENCES

Figure 2g. The Exponential Period Function F(x) = Mtk ^x ( x = 0, 1, 2,..,14 )