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Discovering Planets: The Titius- Bode Law ohann Daniel Titius noticed in the mid-18th century that if the planets are numbered beginning with Mercury = 0, Venus = 3, and doubling thereafter, so Earth = 6, Mars = 12, and so on, and then four is added to each of the planets' numbers and each is divided by 10, a series is created that very closely approximates the planets' distances from the Sun in astronomical units, or AU. (AU is defined as the distance from the Earth to the Sun; see glossary.) The final series he came up with is 0.4, 0.7, 1.0, 1.6, 2.8, 5.2, 10. In the following table, the planets are listed with their distances from the Sun and the Titius-Bode rule prediction. Remember that these calculations are in astronomical units: Mercury is 0.4 AU on average away from the Sun, and the Earth is 1 AU away. The original formulation by Titius, published by Bode, stated mathematically, was

where a is the average distance of the planet from the Sun in AU and n = 0,3,6,12,24,48. . . . Though now the rule is often written, with the same results, as r = 0.4 + 0.3 (2n), where r is the orbital radius of the plan, and n is the number of the planet.

planets from the Sun. Why are the inner planets closer together than the outer planets? In 1766, he calculated the average distance that each planet lies from the Sun, and he noted that each planet is about 1.5 times farther out than the previous planet. This rule was published and made famous in 1772 (without attribution to Titius) by Johann Elert Bode, a German astronomer and director of the Berlin Observatory, in his popular book Anleitung zur Kenntnis des gestirnten Himmels (Instruction for the knowledge of the starry heavens). Bode did so much to publicize the law, in fact, that it is often known simply as Bode's Law. Both scientists noticed that the

This series actually predicted where Uranus was eventually found, though, sadly, it completely fails for Neptune and Pluto. There is no physical basis for the formation of this rule, and there are still no good theories for why the rule works so well for the planets up through Uranus.

Titius-BoDE Rule Predictions

Planet

Average AU from the Sun

Titius-Bode rule |

Mercury

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