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# Titius-Bode law

The Titius-Bode Law or Rule is the observation that orbits of planets in the solar system follow a simple arithmetic rule quite closely.

It was discovered in 1766 by Johann Daniel Titius[?] and "published" (without attribution) in 1772 by Johann Elert Bode, thus the name.

The original formulation was

$a = \frac{n+4}{10}$

where n=0,3,6,12,24,48 ...

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

$a = 0.4 + 0.3\times k$

where k'=0,1,2,4,8,16,32,64,128 (sequence of powers of two and 0)

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, was found at the predicted position of the fifth planet.

Bode's Law was widely accepted until Neptune was discovered and found not to satisfy it.

Given the limits of current teloscopy, 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

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

PlanetnT-B rule distance Real distance
Mercury00.40.39
Venus10.70.72
Earth21.01.00
Mars41.61.52
-82.8-
Jupiter165.25.20
Saturn3210.09.54
Uranus6419.619.2
Neptune--30.1
Pluto12838.839.5

We can see that there are two exceptions:

• There is no planet between Mars and Jupiter. However there exists the Asteroid Belt between Mars and Jupiter. (The first Asteroid, Ceres, was discovered by Piazzi in 1801 with a mean distance of 2.77 a.u.)
• There is no place for Neptune. It has been suggested that something strange happened to alter the orbits of the outermost three planets of the solar system, perhaps a passage of a large mass close to the system (see Nemesis), but this is just a hypothesis without any further evidence.

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.

Currently the most likely explanation other than chance is that orbital resonance 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 Titus-Bode law are a natural consequence of planetary formation, according to the current theories in this area.

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.

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