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Bode Titius Rule

 Johann Elert Bode 1747 -1826 Johann Daniel Titius 1729 - 1796
In the 18th century, two astronomers, Johann Bode and Johann Titius, reported a numerical sequence into which the sizes of the planetary orbits fit. At the time the rule was recognized, there were only six known planets; Mercury, Venus, Earth, Mars, Jupiter and Saturn.

Textbooks usually remark on the Bode-Titius sequence casually, noting that many astronomers dismiss it as mathematical slight of hand. In any event, astronomers note the following points, which we must keep in mind: The rule hypothesized a planet between Mars and Jupiter, which turned out to be where the asteroid belt is. It also hypothesized a planet out from Saturn, which turned out to be Uranus, but Neptune and Pluto don't fit the pattern very well. Pluto has an especially irregular orbit coming in closer than Neptune at one point and it is slightly askew from the plane on which the other orbits closely lie (the ecliptic).

Generating the Bode-Titius Sequence:
1. Orbit: From left to right, make ten columns and number them.
2. Scaling: Under orbit #1 write 0, under #2 write 3. For the rest, double the previous (6, 12, 24...)
3. Add 4: In the next row, add 4 to the number above it for each column.
4. Divide by 10: In the next row, divide the number above by 10.

 Orbits 1 2 3 4 5 6 7 8 9 10 Scaling Factor 0 3 6 12 24 48 96 192 384 768 Add 4 to Each 4 7 10 16 28 52 100 196 388 772 Divide by 10 0.4 0.7 1 1.6 2.8 5.2 10 19.6 38.8 77.2

5. Compare: compare the numbers in the last row to the known orbital radii of the planets (in AU's).

 Orbit Merc. Venus Earth Mars Aster. Jupiter Saturn Uranus Nept. Pluto Known AU's 0.387 0.723 1.000 1.524 2.770 5.203 9.539 19.18 30.06 39.52 Bode Titius 0.400 0.700 1.000 1.600 2.800 5.200 10.00 19.60 38.80 77.20

The numbers match the spacing of the inner planets very well. The correlation diminishes slightly in the outer solar system. Pluto represents a great discrepancy.