Then remember the sequence 0, 3, 6, 12, 24, 48, … ; each number greater than 3 in the sequence is twice the previous number.
To get the distance each planet lies from the sun, relative to the distance between the Earth and the sun, add 4 plus n and divide the result by 10.
A common yardstick used to measure distances in the solar system is the astronomical unit (AU). So 1 AU is the distance between the sun and Earth (about 93 million miles, or 150 million kilometers). For example, to find the distance between the sun and Mercury, the first planet in the solar system, use n equals zero, the first number in the sequence.
For Mercury, add 0 and 4, which equals 4, and divide by 10 to get 0.4 — meaning Mercury is 0.4 AU from the sun, or four-tenths of the distance between the sun and Earth. Right on.
For Earth (I skipped Venus so you can try it yourself), the third planet in the solar system, use n equals 6, the third number in the sequence. Add 4 and 6, which equals 10, and divide by 10 to get 1. Obviously, Earth is 1 AU from the sun!
How about the fifth number in the sequence (here I skipped Mars for you): 4 plus 24 equals 28, divided by 10 equals 2.8.
What is found orbiting the sun 2.8 AU out? Unfortunately, not Jupiter, the fifth planet in the solar system.
For Jupiter, use the sixth number in the sequence to get 4 plus 48 equals 52, divided by 10 equals 5.2 AU, which is right on the nose.
It turns out, however, that there is something orbiting the sun between the orbits of Mars, at 1.5 AU and Jupiter at 5.2 AU: the asteroid belt.
The asteroid belt is a ring of rocky debris orbiting the sun, material that would have formed into a planet if it weren’t for the fact that massive Jupiter was in the vicinity.
The largest body in the asteroid belt, a dwarf planet named Ceres that is about 600 miles in diameter — about half the size of Pluto — and which was discovered in 1801, orbits the sun at a distance of 2.8 AU, just as predicted by the curious law we’re exploring.
This curious law is known as the Titius-Bode rule (or law, if you like). The sequence of numbers leading out to the planets’ distances was homed in on in the 18th century by several scientists, most notably the two German astronomers whose names the rule now bears.
In the 18th century, all the planets — except for the asteroid belt — out to Saturn were known. Imagine how satisfying it must have been to find Uranus in 1781 at 19 AU from the sun, right where the Titius-Bode rule predicted, and to discover the asteroid/dwarf planet Ceres at 2.8 AU in 1801.
What is nowadays most curious about the Titius-Bode rule is that it is not based on any form of physical or mathematical reasoning: The rule just happens to work.
Alas, the Titius-Bode rule does not work perfectly, for it fails to predict the position of anything in the solar system beyond the orbit of Uranus.
— By Doug Furton, a member of the physics faculty at GVSU. Send questions and suggestions to firstname.lastname@example.org. An archive of some of his “What’s up” columns is available at http://gegenschein.wordpress.com.