PART I. BODE'S FLAW
PART I. BODE'S FLAW


spacer.gif More correctly called the Titius-Bode relationship, the ad hoc scheme to account for the mean distances of the planets from the Sun known as "Bode's Law" was originally conceived by Johann Titius in 1766 and then relegated by the latter (perhaps because of its deficiencies) to the status of a page-note. Subsequently revived and popularized by Johannes Bode in 1772, the "Law" correlated well enough with the mean distance of the newly discovered major planet Uranus (1781) and also that of the largest asteroid Ceres (discovered in 1801), but it later failed completely when applied to the mean distances of Neptune and Pluto. Nevertheless, apart from latter-day variants1-7 of Bode's Law it would seem that the underlying structure of the Solar System still remains essentially undetermined. Yet there are obvious objections to Bode's Law over and above its ad hoc nature and the fact that it is no law all. In reality, the scheme was a dubious combination of two separate ad hoc relationships that was flawed from the outset, as Titius himself was perhaps aware. In particular, with reference to unity and the ad hoc constants: A = 4, B = 3, C = 10, the first part of the "Law" applies only to the mean heliocentric distance of Mercury, the first planet from the Sun, i.e.,

Relation 1: The Titius-Bode Relationship for Mercury

whereas the rest of the mean distances are obtained from the application of a second ad hoc relationship expressed in terms of exponents as follows:

Relation 2: The Titius-Bode Relationship for all Planets except Mercury)

Relation [2] either begins with the mean distance of Venus for n = 0, or requires the ultimate limit of minus infinity to commence with the mean distance of Mercury, as Nieto5 and others have noted. For exponents 0 through 5 the second relationship nevertheless approximated the mean distances of five of the six planets then known, and also suggested (albeit for no good reason), that a planetary position might exist between Mars and Jupiter. It is true that the largest asteroid Ceres was subsequently found near the position in question, but even so this relatively minuscule object hardly possesses suffficent mass to qualify as a planet per se. Actually, relation [2] is quite misleading, especially with respect to the apparent doubling associated with the component Exp. Rel 2. . In this relation planet-to-planet increases in distance undoubtedly approach a limiting value of 2 as n becomes large, but only at the expense of the ad hoc parameter A, which becomes increasingly less significant. A plot of the increase in mean distances between adjacent planets (Figure 1) also emphasizes that even before the discoveries of Neptune and Pluto that Bode's Law was no smoothly increasing function, with or without Ceres.

More specifically, the ratio of the increase in distance between Mercury and Venus is 1 : 1.75 , but instead of being greater, the ratios for the next two planetary positions (Venus : Earth and Earth : Mars respectively) are in fact lower, i.e., 1 : 1.42857 and 1 : 1.6 respectively. Moreover, the situation is actually worse with respect to the mean distances of the Solar System itself. Beyond Mars the Bode's Law ratios rise to 1.75 again, and thereafter they increase steadily towards the limiting value. But even so, the actual distances and the ratios for the Solar System diverge radically from Bode's Law beyond Uranus, as Figure 1 shows. Thus although relation [2] at first glance appears to represent a reasonable sequential expansion, it is far from that, and perhaps more the result of judicious ad hoc adjustments than anything else. As a result, there is a contrived 1 : 1 correlation between the "law" and the Solar System with respect to the mean distance of Earth. On examination, however, it turns out that this position actually possesses one of the poorest correlations of all with respect to the planet-to-planet increases.

Thus there is not only a definite kink in the "law," there is also a major question mark concerning the position of Earth itself. In fact, it appears that the ratios and the resulting distances for Earth and Mars are both atypical, those pertaining to Earth especially so. The "law" implies that planet-to-planet distances increase exponentially towards a ratio of 2 : 1, but in the case of the Venus-Earth increase it is closer to the square root of two, or, in general astronomical terms, Earth's position appears to correspond more to the synodic difference cycle between Mars and Venus than a planetary position per se.

Figure 1. The Kink in Bode's Law

Figure 1. The Kink in Bode's Law

But what are the norms for planetary systems in any case, and where might the Solar System fit in the general scheme of things? We now know that this planetary system embodies a number of anomalies, including major variations in planetary composition and size, exceptions in both orbital and rotational inclination, marked differences in eccentricity, an asteroid belt of debatable origin, as-yet unexplained periodic changes in solar output, planetary resonances, and that the System is also subject to chaotic orbital behaviour. Moreover, although it is generally accepted that the Sun and planets owe their origins to the process of accretion, there still remains an apparent lack of primordial residue, an unusual mass/momentum distribution between the Sun and planets, and a slower than expected solar rotational speed to be factored into the equation. Because of these anomalies it would seem unlikely that the Solar System is perfect, unchanging, and readily represented by ordinal numbers and simple expansions, etc. In fact, recent chaos theory-related investigations carried out by Sussman,8 Wisdom,9 Kerr,10-11 Milani,12 Laskar et al,13-16 and others have now established that the Solar System can no longer be considered to be immutable or stable in perpetuity.
Thus the situation is not only as described by Ivars Petersen17 in his 1993 publication: Newton's Clock: Chaos in the Solar System, it also appears--as Heraclitus generalized it so very long ago--that "all things flow and nothing stands." Because of these considerations it therefore makes very little sense to base any detailed analysis of the structure of the Solar System on ad hoc functions, especially those that are neither continuous nor sequential in the first place.

How then should we proceed? And what are the viable alternatives? These questions and partial answers provide the subject matter for Part Two.


REFERENCES

  1. Caswell, A.E., "A Relation Between the Mean Distances of the Planets from the Sun," Science 69 (1929):384.
  2. Malisoff, William M, "Some New Laws for the Solar System," Science 70 (1929):328-329. The latter is included here mainly because of his response to Caswell above and also his initial treatment of integers in the present context.The remaining parts of his paper, however, concern neither Bode nor variants of the "Law", but concepts that include logarithmic spirals in the same context; see Part III for further details.
  3. Richardson, D.E., "Distances of Planets from the Sun," Popular Astronomy 53, (1945) 14-26.
  4. Ovenden, M. W., "Bode's Law and the Missing Planet," Nature 239 (1972) 508-509.
  5. Nieto, M.M., "The Titius-Bode Law and the Evolution of the Solar System," Icarus 25 (1974) 171-174.
  6. Bass, R. "The Titius-Bode Law from 1766 to 1996," (Kronia Communications).
  7. Blagg, M.A., "On a Suggested Substitute for Bode's Law," Monthly Notices of the Royal Astronomical Society, Vol. LXXIII 6, April 1913:414:422. Noteworthy here are M.A. Blagg's early and valid criticisms of the "law" itself, and also the application of logarithmic ratios in the analysis.
  8. Sussman, G. and Wisdom, J. "Chaotic Evolution of the Solar System," Science 257, 3 Jul 1992: 56-62.
  9. Wisdom, Jack. "Chaotic Dynamics in the Solar System, Icarus 72 (Nov 1987):241-275.  
  10. Kerr, Richard, A. "From Mercury to Pluto, chaos pervades the Solar System," Science 257 (Jul 1992):33.
  11. _________  "Does Chaos permeate the Solar System?" Science 244 (14 Apr 1989). 
  12. Milani, A. "Emerging stability and chaos." Nature 338 (16 March 1989):207-208.
  13. Laskar, J. "A numerical experiment on the chaotic behavior of the Solar System" Nature, 338 (16 Mar 1989):237-238. 
  14. _________  "The chaotic motion of the Solar System: a numerical estimate of the size of the chaotic zones." Icarus 88 (Dec 1990):266-291. 
  15. _________  Laskar, J., Joutel, F. and Robutel, P. "The chaotic obliquity of the planets" Nature 361 (18 Feb 1993):608-612.
  16. _________  Laskar, J. Thomas Quinn, and Scott Tremaine. "Confirmation of resonant structure in the Solar System". Icarus 95 (Jan 1992):148-152.
  17. Petersen, I. Newton's Clock: Chaos in the Solar System , W.H. Freeman, New York, 1993.

Copyright © 1997. John N. Harris, M.A.(CMNS). Last Updated on November 16, 2001.


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II The Alternative
http://www.spirasolaris.ca/sbb4b_07.html
Describes an alternative approach to the structure of the Solar System that employs logarithmic data, orbital velocity, synodic motion, and mean planetary periods in contrast to ad hoc methodology and the use of mean heliocentric distances alone.

III The Exponential Order
http://www.spirasolaris.ca/sbb4c_07.html

The constant of linearity for the resulting planetary framework is the ubiquitous constant Phi known since antiquity. Major departures from the theoretical norm are the ASTEROID BELT, NEPTUNE, and EARTH in a resonant synodic position between VENUS and MARS. Fibonacci/Golden Section Resonances in the Solar System.

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