Law of Titius-Bode

The law of Titius-Bode , or law of Bode , is an empirical rule connecting the rays of the Orbite S of planets of the Solar system. It shows that the distance from planets to the Sun follows a geometric progression of reason 1,75. Highlighted by the mathematician max Wolf as of 1741, it was redécouverte and formalized mathematically in 1766 by Johann Daniel Tietz, said Titius. But it is with Johann Elert Bode that a long time of this law by making known it near the community of the astronomers in 1778 returned the paternity.

Recently, one sought to integrate the new bodies discovered in the continuation of Titus-Bode, without knowing if they would be finally regarded as planets. Today, the objects having the highest numbers in the continuation of Titus-Bode are not regarded as planets.

Mathematical expression

The law of Titius-Bode makes it possible to calculate an approximation of the ray R of the orbit of a planet according to its row N , starting from the formula:

$r = 0,4 + 0,3 \ times 2^n$

where R is expressed in astronomical unit (UA)

and N is worth - ∞ for Mercure, 0 for Venus, 1 for the Ground, 2 for Mars, 3 for the belt of Astéroïde S, etc

Another expression of this law consists starting from the continuation of number: 0,3,6,12… (which is connected with the geometrical Suite u_n+1 = 2*u_n). Á each number (0 for Mercury, 3 for Venus, etc) one adds 4, then one divides the sum by 10. The result of the operation is the same one as the mathematical expression of Titius, it gives an approximate value of the distance Sun-planet in UA. This law of distribution was improved by Otto Schmidt by applying a function for telluric planets and another for gas planets.

Discovered belt of asteroids

During its original publication, the law was checked by all known planets, of Mercure to Saturn, with a gap between the fourth ( N = 2) and fifth ( N = 4) planets. This law was then considered interesting but without much importance. The discovery of Uranus whose orbit respects the law, will validate it with the eyes of most of the scientific community. Making profitable this new credibility, Bode will push with the research of missing planet intermediate ( N = 3), which will lead to discovered Cérès, largest of the asteroids.

Urbain the Glassmaker and John Couch Adams will use then this law to envisage the orbit of a new planet Neptune which could explain the disturbances of Uranus. This value of 38,8 is rather bad and corresponds rather to the orbit of Pluton.

To explain this exception (Neptune), it was suggested that something had deteriorated the orbit of the three external Planets of the solar system, can be the passage of a bulky mass like that of the assumption of Némésis.

The last theories explain the law of Titius-Bode as the consequence of mechanisms of resonance which would create stable orbital zones during the creation of the solar systems.

Alternative

In the same spirit one can notice that the distribution of planets follows more one minus a logarithmic law.

One obtains the same kind of alignment with the satellites galiléens of Jupiter.

Note: There is proportionality, i.e. a line passing by the origin, by giving to mercury number 7.

Notes and references of the article

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