Titius–Bode law

The Titius–Bode law (sometimes termed simply Bode's law) is a formulaic prediction of spacing between planets in any given planetary system. The formula suggests that, extending outward, each planet should be approximately twice as far from the Sun as the one before. The hypothesis correctly anticipated the orbits of Ceres (in the asteroid belt) and Uranus, but failed as a predictor of Neptune's orbit. It is named after Johann Daniel Titius and Johann Elert Bode.

Later work by Mary Adela Blagg and D. E. Richardson significantly revised the original formula, and made predictions that were subsequently validated by new discoveries and observations. It is these re-formulations that offer "the best phenomenological representations of distances with which to investigate the theoretical significance of Titius–Bode type Laws".

The law relates the semi-major axis ${\displaystyle a_{n}}$ of each planet's orbit outward from the Sun in units such that the Earth's semi-major axis is equal to 10:$${\displaystyle a=4+x}$$where ${\displaystyle x=0,3,6,12,24,48,96,192,384,768\ldots }$ such that, with the exception of the first step, each value is twice the previous value. There is another representation of the formula:$${\displaystyle a=4+3\times 2^{n}}$$where ${\displaystyle n=-\infty ,0,1,2,\ldots ~.}$ The resulting values can be divided by 10 to convert them into astronomical units (AU), resulting in the expression:$${\displaystyle a=0.4+0.3\times 2^{n}.}$$For the far outer planets, beyond Saturn, each planet is predicted to be roughly twice as far from the Sun as the previous object. Whereas the Titius–Bode law predicts Saturn, Uranus, Neptune, and Pluto at about 10, 20, 39, and 77 AU, the actual values are closer to 10, 19, 30, 40 AU.

The "Classical" (Canonical) form of the Titius-Bode Law is ${\displaystyle a_{n}=0.4+0.3\times 2^{n}}$. This Formula also has a "Recursive" form: ${\displaystyle a_{n+1}=2\times a_{n}-0.4}$, where ${\displaystyle a_{0}=0.55.}$

The first mention of a series approximating Bode's law is found in a textbook by D. Gregory (1715):

... supposing the distance of the Earth from the Sun to be divided into ten equal Parts, of these the distance of Mercury will be about four, of Venus seven, of Mars fifteen, of Jupiter fifty two, and that of Saturn ninety five.

A similar sentence, likely paraphrased from Gregory (1715), appears in a work published by C. Wolff in 1724.

In 1764, C. Bonnet wrote: