Bode's Law

As Mark Littmann states in Planets Beyond, "Bode's Law is neither Bode's nor a law." Though it enjoyed quite a bit of credibility three hundred years ago, Bode's Law lives on today mostly as a mathematical curiosity.

In 1772, Johann Bode had been hired by the Berlin Academy of Sciences to work on the Berliner Astronomisches Jahrbuch, their almanac of astronomy. He was young and full of enthusiasm, and made the publication profitable by adding general science news to the usual array of numbers. That year, Bode published a relationship of planetary distance originally noted in a work by Swiss naturalist Charles Bonnet.

Bonnet's premise dealt with the orderliness of nature, using the solar system as an example. This was translated from French to German by Johann Titius in 1766, who added a paragraph to help mathematically prove Bonnet's case. It works out like this:

Start with the series 0, 3, 6, 12, 24, 48 . . ., where each succeeding number is twice its predecessor, and each number represents a planet, starting at Mercury. Now add 4 to each number. Finally, divide by 10 to give Earth the value of 1 (though Titius did not perform this last division).

With the distance from the Earth to the Sun being 1 Astronomical Unit (AU), the resulting series gives the distances of all the planets from the Sun expressed in AUs--at least for the planets known at the time. This correlation is shown below.

                  Distance from the Sun in AU
      Planet          Titius's Calculation      Actual
      Mercury                 .4                  .39 
      Venus                   .7                  .72
      Earth                  1.0                 1.0 
      Mars                   1.6                 1.52 
      ----                   2.8	
      Jupiter                5.2                 5.2 
      Saturn                10.0                 9.54

There is a remarkable correlation between what Titius's series gave and the actual distances of the planets. Bode published these results, without crediting Titius or Bonnet, and was the theory's main proponent. Thus it became known as Bode's Law, though it is now often referred to as the Titius-Bode Rule.

After Uranus was discovered in 1781, Bode pointed out that it fitted his rule about planetary distances. The next number in the series would be 19.6, Uranus came in at 19.18 AU.

The gap in the series after Mars was troubling, and as Bode did not believe that "the Founder of the Universe left this space empty," he proposed that an undiscovered planet lay at that distance. Other astronomers shared his belief, and on December 31, 1801, Franz von Zach found a small celestial body, observed briefly a year earlier and thought to be a comet. This object had a nearly circular orbit, unlike a comet's. Its distance by the Titius-Bode chart was at 27.7 AU, matching almost exactly the 28 figure. Other small bodies were soon found at this distance -- we know them as the asteroids, or minor planets.

The next planet to be discovered was Neptune, which completely disobeyed Bode's prediction. Whereas a planet should be at 38.8 AU, Neptune orbited at 30.06 AU. This pretty much discredited the rule -- until Pluto was discovered. Pluto, it turns out, orbits at 39.44 AU (though we know its orbit varies widely).

Leaving out Neptune from the series, Bode's Law gains credibility again. Here's the full chart:

	           Distance from the Sun in AU
      Planet            Titius-Bode Rule     Actual
      Mercury                 .4               .39 
      Venus                   .7               .72 
      Earth                  1.0              1.0 
      Mars                   1.6              1.52 
      Ceres (asteroid)       2.8              2.77 
      Jupiter                5.2              5.2 
      Saturn                10.0              9.54 
      Uranus                19.6             19.18 
      Neptune                                30.06 
      Pluto                 38.8             39.4

Still, the rule is based on an arbitrary series, with an arbitrary number added to each member of the series. After all, there was no physical basis for choosing a series starting 0, 3, 6 . . ., nor any laws of nature that suggest 4 as the magic number to add to the series.

The Titius-Bode Rule remains an interesting coincidence, for which no one has offered a satisfactory explanation.

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