Titius-Bode law

The Titius-Bode law (or Bode's law) is the observation that orbits of planets in the solar system closely follow a simple geometric rule.

It was discovered in 1766 by Johann Daniel Titius and "published" (without attribution) in 1772 by Johann Elert Bode, thus the name. However, some say it was first proposed by Christian Wolff in 1724.

The original formulation was

<math>a = \frac{n+4}{10}<math>

where n = 0, 3, 6, 12, 24, 48 ..., with each value of n twice the previous value.

The modern formulation is that the mean distance a of the planet from the Sun is, in astronomical units:

<math>a = 0.4 + 0.3\times k<math>

where k=0,1,2,4,8,16,32,64,128 (0 followed by the powers of two)

A weaker formulation with no geocentric point of view and less "ad-hocism" reads:

The distance of one planet to the innermost one is about twice as much as that of the previous one. (* See below for this formulation as table.)

Here are the distances of planets calculated from the original rule and compared with the real ones:

Planet	k	T-B rule distance	Real distance
Mercury	0	0.4	0.39
Venus	1	0.7	0.72
Earth	2	1.0	1.00
Mars	4	1.6	1.52
Asteroid Belt	8	2.8	2.77
Jupiter	16	5.2	5.20
Saturn	32	10.0	9.54
Uranus	64	19.6	19.2
Neptune	-	-	30.1
Pluto	128	38.8	39.5

We can see that two notes have to be made:

• The Asteroid Belt has to be considered a planet in order to make something satisfy k=8. Being spread out as it is, the number taken for the distance to the Sun (2.77 AU) is actually that of the Belt's biggest asteroid Ceres (at one time considered a planet as well).
• Neptune violates the law (by falling halfway between k=64 and k=128). Instead, Pluto takes up the place where the next planet after Uranus is expected to be based on the "rule" (which is quite ironic considering that Pluto's status of planet is actually under dispute). It has been suggested that maybe something strange happened to alter the orbits of the outermost two planets of the solar system, perhaps a passage of a large mass close to the system (see Nemesis) or something knocking Neptune out of its original orbit with the resulting debris becoming Pluto, but these are just unsubstantiated hypotheses.

Here is a plot of this law against real planet distances:

There is no solid theoretical explanation of the Titius-Bode law, and it is not known whether this is just a numerical coincidence or a more fundamental cosmological rule.

When originally published the law was satisfied by all the known planets -- Mercury through Saturn -- with a gap between the fourth and fifth planets. It was regarded as interesting, but of no great importance until the discovery of Uranus in 1781 which fit neatly into the series. Based on its new credibility, Bode urged a search for a fifth planet. Ceres, the largest of the asteroids in the Asteroid Belt, was found at the predicted position of the fifth planet. Bode's law was then widely accepted until Neptune was discovered (in 1846) and found not to satisfy it.

Currently the most likely explanation other than chance is that orbital resonance (see Planetary pairs) from major orbiting bodies creates regions around the Sun that are free of long-term stable orbits. Results from simulation of planetary formation seem to support the idea that laws like the Titius-Bode law are a natural consequence of planetary formation, according to the current theories in this area.

Given the limits of current telescopy, there are a decidedly limited number of systems on which Bode's law can be tested. Two of the solar planets have a number of large moons that appear possibly to have been created by a process similar to that which created the planets themselves. The four large satellites of Jupiter plus the largest inner satellite -- Amalthea -- adhere to a regular, but non-Bode, spacing with the four innermost locked into orbital periods that are each twice that of the next inner satellite. The whole lot are thought to be moving outward under the influence of tidal drag to lock to the period of the outermost large moon Callisto. The large moons of Uranus have a regular, but non-Bode, spacing. See http://www.floridastars.org/9605cohe.html

Recent discoveries of extrasolar planetary systems also indicate that some form of this rule may be present universally, but the evidence is still too weak to draw any strong conclusions. The law provides a challenge for statistical analysis as it falls into the category of uncomfortable science where inadequate data are available for formal validation of an hypothesis.

• A table showing the above, so-called "weaker formulation" is reproduced here. AU is Astronomical Unit; MU is a newly coined "Mercury Unit" (as distances represented are to "innermost one," or Mercury) to which all AU's have been recalculated so the proportions remain the same; MU from Mercury takes the distance of the planet to the orbit of Mercury, instead of the Sun, by subtracting 1 from the MU; and the final column takes the result over that of the preceding planet, showing a pattern coming pretty close to doubling. Considering the enormous distances involved, it's hard not to consider the results curious.

Planet	AU	MU	MU from Mercury	Proportion over Preceding Planet
Mercury	0.387	1	0	n/a
Venus	.723	1.868	.868	n/a
Earth	1.0	2.584	1.584	1.825
Mars	1.524	3.938	2.938	1.855
Asteroid Belt	2.77	7.158	6.158	2.096
Jupiter	5.203	13.444	12.444	2.021
Saturn	9.539	24.649	23.649	1.9
Uranus	19.18	49.561	48.561	2.053
Neptune	30.06	77.674	76.674	1.579
Pluto	39.44	101.912	100.912	2.078 of Uranus

With the ratios approximating 2:1, pure speculation might lead us to posit something on the order of orbital resonance, similar to that of the objects of the Kuiper Belt but on a much greater scale of space and time.

External Link

The Titius-Bode Number Sequence Deciphered (http://bellsouthpwp.net/j/o/josephconklin/jw/tb/titius.html)

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