Titius-Bode law

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

Contents

1 Discovery and history
2 Formulation
3 Theoretical explanation
4 See also
5 References

5.1 External links

Discovery and history

Johann Daniel Titius.

Johann Elert Bode

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

Formulation

The original formulation was

[a = \frac]

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:

[a = 0.4 + 0.3\times k]

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

For the outer planets, the first term becomes more and more negligible, and the interpretation is that each planet is roughly twice as far from the sun as the last one.

Here are the distances of planets calculated from the 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)¹ 8 2.8 2.77
Jupiter 16 5.2 5.20
Saturn 32 10.0 9.54
Uranus 64 19.6 19.2
Neptune 128 38.8 30.06
Pluto 256 77.2 39.44

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)¹	8	2.8	2.77
Jupiter	16	5.2	5.20
Saturn	32	10.0	9.54
Uranus	64	19.6	19.2
Neptune	128	38.8	30.06
Pluto	256	77.2	39.44

¹ 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, which was at one time considered a planet as well.

A plot of this law vs real planetary distances can be seen to the right of the table above, or in another version [here].

Theoretical explanation

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 celestial mechanics 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.

One plausible 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 Titius-Bode law are a natural consequence of planetary formation, according to the current theories in this area.

Dubrulle and Graner have shown that power-law distance rules are a natural consequence of collapsing-cloud models of planetary systems possessing two symmetries: rotational invariance (the cloud and its contents are axially symmetric) and scale invariance (the cloud and its contents look the same on all length scales), the latter being a feature of many phenomena considered to play a role in planetary formation, such as inverse-square force fields and turbulence.

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. [link]

Recent discoveries of extrasolar planetary systems do not yet provide enough data to test whether the rule applies to other solar systems.

References

[Articles from the Smithsonian/NASA Astrophysics Data System (ADS)]

"Titius-Bode laws in the solar system. Part I: Scale invariance explains everything". F. Graner, B. Dubrulle Astronomy and Astrophysics 282, 262-268 (1994).

"Titius-Bode laws in the solar system. Part II: Build your own law from disk models",B. Dubrulle, F. Graner Astronomy and Astrophysics 282, 269-276 (1994).

The ghostly hand that spaced the planets New Scientist 9 April 1994, p13

External links

[The Titius-Bode Number Sequence Deciphered]