Titius-Bode Relation

A mnemonic for the sizes of the orbits of the planets is known as the Titius-Bode Rule.

orbit size = [3N+4]/10 A.U.

where we set N = 0, 1, 2, 4, 8, 16, 32, and so on. This series of numbers has a rather straightforward definition, but is not a completely natural sequence. Evaluating the Titius-Bode Rule we find,

Planet

Titius-Bode Rule (A.U.)

Actual (A.U.)

Mercury

0.4

0.387

Venus

0.7

0.723

Earth

1

1

Mars

1.6

1.524

asteroids

2.8

2.8

Jupiter

5.2

5.203

Saturn

10

9.554

Uranus

19.6

19.22

Neptune

38.8

30.11

Pluto

77.2

39.54



For Mercury, N = 0, and so a = 0.4 A.U.. Pretty close to the correct answer. For Venus, N = 1 and so a = 0.7 A. U., hmmmm. For Jupiter, N = 16, and so a = 5.2 A.U. The law works well for some planets. Further, an interesting prediction was made when the law was proposed. There was no planet for N = 8, i.e. for a = 2.8 A.U. However, subsequent work showed that there was a belt of small rocky objects near a = 2.8 A.U. (the asteroid belt). Is this the planet predicted by the Titius-Bode relation?

Astronomers tend to ignore the Titius-Bode Rule, as only an interesting quirk of numerics. There is no physical explanation for the Rule, but there may be physical content in the law. It is conceivable that the Titius-Bode Rule contains information on the stability properties of the orbital arrangement of the planets.