# Titius-Bode Law

## Physical Law

The **Titius-Bode law** is the observation that the semi-major axes of successive planets of the solar system are governed by the equation:

- $a = 4 + \floor {2^{n - 1} } \times 3$

where:

- $a$ is the measure of distance such that the Earth's semi-major axes is defined as $10$
- $n = 0, 1, 2, \ldots$ for successive planets
- $\floor {2^{n - 1} }$ denotes the floor of $2^{n - 1}$.

That is, $a$ is defined by the sequence:

- $4, 7, 10, 16, 28, 52, 100, 196, \ldots$

This sequence is A003461 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

While accurate for the planets out as far as Uranus, the law breaks down for Neptune and beyond.

## Also known as

The **Titius-Bode Law** is also seen named as **Bode's Law**.

## Source of Name

This entry was named for Johann Daniel Titius and Johann Elert Bode.

## Historical Note

It Titius-Bode Law first appears in David Gregory's *Astronomiae physicae et geometricae elementa* of $1702$.

It was later picked up on by Johann Daniel Titius, who inserted it into his $1766$ translation of Charles Bonnet's *Contemplation de la Nature*.

It was noticed that, were this law to hold, there was a missing planet between Mars and Jupiter.

Johann Elert Bode followed this up initially in $1772$, in a footnote to the second edition of his *Anleitung zur Kenntniss des gestirnten Himmels*.

After the discovery of Uranus in $1781$, which also appeared to obey the same law, the gap between Mars and Jupiter was taken more seriously.

Bode proposed a systematic search for the planet which was expected to exist. In due course, Ceres was discovered in $1801$.

The observational data on Ceres was sparse, and it was soon lost when it passed into the area of the sky occupied by the sun.

However, Gauss succeeded in working out, using the method of least squares, where it was most likely to be, and when it was once more in an area of sky where it could be seen, it was found again.

The Titius-Bode Law was then widely accepted until Neptune was discovered in $1846$ and found not to satisfy it, causing the law to be discredited.

It needs to be borne in mind that the Titius-Bode Law was no more than a pattern which had been noticed, and no scientific reason had been offered for it.

In more recent times, however, it has been suggested that the gravitational influence of Jupiter and Saturn may be the source of a resonance effect of some kind.