Titius–Bode law - How Ceres was found

The Titius–Bode law (sometimes termed just Bode's law) is a hypothesis that the bodies in some orbital systems, including the Sun's, orbit at semi-major axes in a function of planetary sequence. The hypothesis correctly predicted the orbits of Ceres and Uranus, but failed as a predictor of Neptune's orbit.

The formula

The best way to see the law is to write down the sequence 0, 3, 6, 12 and so on, where each number is obtained by doubling its predecessor. Next, add 4 to each number, and divide the result by 10.

For example, Venus = (3+4)/10 = 0.7

Now get out any astronomy textbook and look up the distances of the planets from the Sun in astronomical units, the Earth-Sun distance being defined as 1. The distances are virtually identical to the terms in the number sequence for all but the outermost planets.

Planet   Titius–Bode law distance   Actual distance
Mercury            0.4                 0.39
Venus              0.7                 0.72
Earth              1.0                 1.0
Mars               1.6                 1.52
Ceres              2.8                 2.77
Jupiter            5.2                 5.2
Saturn            10.0                 9.74
Uranus            19.6                19.2
Neptune           38.8                30.06

When originally published, the law was approximately 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 happens to fit neatly into the series.

Based on this discovery, Bode urged a search for a planet between Mars and Jupiter. Ceres, the largest object in the asteroid belt, was found at Bode's predicted position in 1801.

Bode's law was then widely accepted until Neptune was discovered in 1846 and found not to satisfy Bode's law.

Theoretical explanations

There is no solid theoretical explanation of the Titius–Bode law, but if there is one it is possibly a combination of orbital resonance and shortage of degrees of freedom: any stable planetary system has a high probability of satisfying a Titius–Bode-type relationship. Because of this, it has been called a "rule" rather than a "law".

Orbital resonance from major orbiting bodies creates regions around the Sun that are free of long-term stable orbits. Results from simulations of planetary formation support the idea that a randomly chosen stable planetary system will likely satisfy a Titius–Bode law.

Still, Titius–Bode law may be just a coincidence. And we really do not have other planetary systems to test it.