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Open letter to science editors



The Titius-Bode Law
and the Evolution of the Solar System

M. M. Nieto
Copyright 1974 by M. M. Nieto

Dr.  Nieto is a staff member at Los Alamos Scientific Laboratory, Los Alamos, New Mexico, in Group T-8, "Elementary Particles and Field Theory."  He is author of the widely reviewed volume, The Titius-Bode Law of Planetary Distances--Its History and Theory.  For a review of this book, see p. 44.

I have been asked to discuss the Titius-Bode Law of Planetary Distances: does it shed any light on whether or not there has been any recent large-scale evolution in the composition of the solar system?  I will aim to stick directly to this question--even though it may not be a question.  Let me explain that enigmatic statement.

Before we can know whether the subject of this talk means anything, we have to know whether the "Titius-Bode Law of Planetary Distances" is indeed a "law." If it is not a "law," then clearly the "law" says nothing about anything, we can skip the rest of this talk, drop the discussion, and concentrate on something useful, like Watergate.

In modern notation, the original law can be stated this way: the distance to the nth planet is

Rn = (4) + (3) (2)n, n = -, 0, 1, 2....                                       (1)

where n = for Mercury, 0 for Venus, 1 for Earth, etc.  The law is normalized to 10 for the Earth.  It "predicted" the discovery of the asteroid belts, Uranus, and was used as the basis for computations to discover Neptune, even though it failed badly for Neptune and Pluto (see Table 1).

The question thus arises, were the early successes of the law just coincidences, the true worth of the law being the "failures" for Neptune and Pluto?

As I emphasized in my recent book on the subjectl, before one can begin to answer that question one must realize that equation (1), with its ad hoc first term, should not be thought of as the only possible parametrization, but only as a first guess.  In fact, if one looks at the actual distances of the planets from the Sun and searches for the best pure geometric progression fit, one finds that the geometric progression

Rn = 3.34 (1.73)n, n = 0, 1, 2....          (2)

gives the fit shown in Figure 1. The actual distances oscillate about the pure geometric progression, so that one can represent the "true" curve as

Rn = 3.34 (1.73)n f (n),                  (3)

where f (n) is a function which oscillates about 1. (Similar distributions can be made for the satellite systems of the major planets.)

Planet       n     Distance   Titius-Bode Law

Mercury      -            3.9              4
Venus           0             7.2              7
Earth             1           10.0            10
Mars             2           15.2            16
(Ceres)         3           (27.7)          28
Jupiter           4            52.0           52
Saturn           5             95.5         100
(Uranus)       6         (192.0)          196
(Neptune)     7          (300.9)         388
(Pluto)          8            (395)          772

A comparison of the original Titius-Bode Law with observation.  The planets in parentheses were not known in 1776.

Table I

From all this, can we decide whether or not the "law" is a "law"?  The answer, unfortunately, is "No."  The "predictions" of the law are tantalizingly close to the observations (especially given the early history of the law, when it predicted Uranus and the asteroids).  But they are not right on top of the planets.  Further, by the time you add in the periodic oscillation about the geometric progression one is left with "intuition" as to the over all validity, and "intuition" presently goes both ways.  One needs a theory that predicts (in whatever mathematical format) both the progression and the oscillations about it.

My own feelings arel that if the law is a "law," then the split into the geometric progression and the oscillation come from two separate processes.  The  geometric progression, I feel, would come from the early history of the solar system, when the material that would eventually form the planets was in the form of a rotating disk of gas and dust around the proto-Sun.  Then, as in a theory like Hoyle's, energy and angular momentum would be transferred to the disk by the "magnetic brake" mechanism.  This mechanism is possible because the whole system would be a plasma, so that a magnetic coupling could be made between the rapidly rotating proto-Sun and the disk.  This would slow down the Sun and speed up the disk and make it larger.

Figure 1. Distribution of planets as distance versus number.

In such a system, turbulence would be set up.  Further, it is known that such systems can set up regular turbulence patterns, which in turn would lead to preferred condensation at the edges of the patterns.  This, as explained elsewhere,l could in principle set up the geometric progression.

The point I have madel is that I do not see how point-gravitational and/or tidal interactions, either as or after the planets were formed, could set up the geometric progression.  The problem there is that such interactions tend to set up resonances and/or stable patterns that have preferred or antipreferred distances which involve ratios of periods of orbits--not a geometric progression of 1.73. (Examples of this phenomenon are the rings of Saturn and Mimas, or the Trojan asteroids and Jupiter.)

The oscillation about the geometric progression represents, I believe, a point-gravitational or tidal evolution starting from after the planets were formed, and slowly reaching equilibrium.  Calculations that have been performed on the time scales of such evolutions all seem to indicate that this evolution would have been completed long ago.

Thus (always keeping in mind that we don't know whether the Titius-Bode law is a law) if one can believe in the law, then this would indicate that there has been no large-scale evolution in the composition of the solar system in the recent past.

Now let me come to two discussions of the Titius-Bode Law which would take exception to these conclusions:

The first was put forward by Bailey,2 just before the first Apollo landing on the Moon in 1969.  It raised an old question which dates back centuries, the idea of a missing planet in the interior of the solar system.  Bailey proposed that perhaps the Moon is a lost planet, "Luna," which perhaps once existed in an unstable orbit between Mercury and Venus, and which was captured by the Earth.  Bailey did not date this process, except to state that capture should have occurred in the last half of the age of the solar system.

The connection with the Titius-Bode Law comes about by considering the original form of equation (1) and then figuring out what changes in Mercury's orbit would have occurred by the transfer and capture.  Bailey fits the original form with the "original location" of Luna and the "original location" of Mercury.  The problems are what I consider to be the ad hoe nature of the first term in the Titius-Bode Law, and the detailed problems of how and when capture occurred.  If one were to choose, one could like the capture to have been recent, but one just cannot show it.

The other theory involving the Titius-Bode Law was put forward recently by Michael Ovenden.3  He placed in a new perspective the idea of Olbers that the asteroid belts are the remnants of an ancient planet which was disrupted, for some reason.  Ovenden tried to explain the Titius-Bode Law by looking at the gravitational evolution of the solar system from a set of initial conditions to its present configuration.  He found that he could do that if he assumed that there used to be a planet of 90 Earth masses where the asteroid belts now exist, and that 16 million years ago the planet, for some unexplained reason, disrupted, leaving the remaining system to evolve gravitationally to the present configuration.

      Bode's Law Without Venus

Acting as a respondent to Dr. Nieto at the McMaster University symposium, Dr. C. J. Ransom offered the revelation that Bode's law, which is normally stated

r = 0.4 + (0.3)2n (n= -,0,1,2,3...      ),

need only be stated in a slightly modified form and it describes the Solar System without Venus fully as well as the usual statement of the law describes the present system.  Ransom's reformulation of the law:

r = 0.4 + (0.6)2n (n =-,0,1,2,3...     ).

According to Ransom, "It is apparent that the outermost planet in the Solar System can be removed and not affect the equation, or that one planet can be added to the outer extreme of the system without requiring a major reformulation of the Bode equation.  However, it is not generally recognized that the number of objects in the interior of the system can be increased or decreased without altering the basic form of the equation."

Ransom asserts that the modified equation de-fuses the oft-stated objection to Velikovsky based on Bode's law and on the assumption that the law reflects conditions in the Solar System at the time of planetary formation.  Nevertheless, at the symposium he made it clear that he has no great faith in one's right to deduce anything about physical reality from his revision of Bode's law.  It just "shows what you can do when you start playing with these equations."

Dr.  Ransom, a plasma physicist, is employed by the Convair Aerospace Division of General Dynamics Corporation, Fort Worth.  Texas.

The critical point, of course, is what could have caused the explosion, and left the asteroid belts with only 0.1 Earth masses, all essentially in the plane of the ecliptic?  Napier and Dodd (4) looked into this question and, given the energy and angular momentum constraints, decided that the event was almost impossible to reconstruct using gravitational, nuclear, or chemical interactions.

So there you have it.  The Titius-Bode Law, since its validity as a physical law is by no means proven, cannot give you a definite answer as to implications for possible recent, large-scale evolution of the solar system.  But I think you can make the statement that if you argue that the law is indeed a law, and that the solar system obeys this law, then it speaks against recent large-scale evolution.

Let me put it this way: If God were to come and tell us that the law is a law and that the planets obey it, then the present configuration of the planets is something which "had to come about." On the other hand, since recent large-scale changes in the configuration of the solar system would have to be a random or accidental event (something like just happening to be under a rock when it fell), how could this chance happening have finally brought the planets into agreement with a law which they already were supposed to obey independently?

To sum up, I think you can believe in the Titius-Bode Law, or recent large-scale evolution, or neither of them.  But I don't think you can believe in both of them.


(1)        M. M. Nieto, The Titius-Bode Law of Planetary Distances: Its History and Theory (Oxford: Pergamon Press, 1972).

(2)        J. Martyn Bailey, "The Moon may be a Former Planet," Nature 223 (1969): 251-53.

(3)        M. W. Ovenden, "Bode's Law and the Missing Planet," Nature 239 (1972): 508-9.

(4)        W. McD.  Napier and R. J. Dodd, "The Missing Planet," Nature 242 (1973): 250-51.


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