The Titius-Bode Number Sequence Deciphered

by Joseph Conklin

PART I

Introduction: The Astronomical Unit (AU)

In the 18th century, two astronomers, Johann Titius and Johann Bode, reported a numerical sequence into which the sizes of the planetary orbits fit.

By using the size of the earth's orbit as a standard, called an Astronomical Unit, or AU, astronomers recalculate the other orbital sizes in proportion to it. Titius and Bode noticed the following pattern- they started with 0, then took 3 and began doubling- 0, 3, 6, 12, 24, 48, 96, 192, 384, 768. Then they added 4 to each number and divided by 10, and the result approximated the planetary orbits in AU's. Table 1 shows the average distances from the sun of the 9 planets and the asteroid belt between Mars and Jupiter.
(Note that these are modern figures. In Titius' and Bode's time, the asteroid belt, Uranus, Neptune and Pluto were as yet undiscovered.)

Textbooks usually remark on the Titius-Bode sequence casually, noting that some astronomers dismiss it as a chance mathematical pattern. In any event, note the following points:
The Titius-Bode 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. Neptune doesn't fit the pattern, but Pluto does if we substitute it for Neptune.

When we speak of AU's, we speak of the average radius of each elliptical orbit. As Johannes Kepler pointed out in his 2nd and 3rd laws of planetary motion, a line from the planet to the sun will sweep over an equal area in an equal amount of time (so that the distance covered will be less as the planet is further away), and the planet's orbital time period squared is proportional to its AU cubed (so that cubing Jupiter's AU of 5.203 [140.852] yields the square of its orbital period [11.868 years]).

We usually assume a rotating cloud around the newly forming sun eventually "congealed" into the planets. What we can speculate on, however, is why this "congealing" formed the planets in their orbits in regular patterns, and with one planet to each orbit. Why are there not two orbits running closely together like runners on a track, or two planets in one orbit, and why is Neptune an apparent misfit?

This is a rethinking of the Titius-Bode Rule. To do this, let's try (ironically) to take it apart as we have taken apart our notions of the universe, historically. First, geocentrically, and then, heliocentrically.

Deconstruction: Geocentric

There are 2 elements to the Titius-Bode Rule which are apparently random and therefore generate the notion that there is a mathematical trick involved. The numbers 3 and 4. Why does it go from 0 to 3 (instead of 2 or 5), and why do we add 4 (instead of 3 or 20)? The answer is- our traditional, and understandable, desire to simplify things with AU's. The earth's orbit is 1 AU, of course, because we live here. Why should we let the standard fall with the Neptunians? But we know that the earth is no longer the center of the universe we once thought it was. By rights, we should treat it as just another element in the system. But it is the 3rd planet from the sun, and so that's where Titius-Bode's 3 comes from.

The 4 is there because of Mercury's orbit. Mercury is .387 AU. So, if we start at 0, add 4 and divide by 10, of course we get .4, which is close to .387. There is nothing arbitrary about it. What we should do, though, to be more accurate, is to try to get that .387 into the equation somehow. We can do this most easily by using Mercury's orbit, instead of Earth's, as the standard. It's difficult to throw away our traditions, I agree, but let's try it in the interests of science. We'll call Mercury an MU- a Mercury Unit. We recalculate the orbital sequence, then, in MU's (Table 2).

Deconstruction: Heliocentric

We know that the Sun is the center of our solar system, so of course we know it's significant, but it's not a planet, so let's leave it out of the equation just for a moment. Let's try calculating the distances of the planets, not from the Sun, but from the orbit of Mercury, since it's orbits we're concerned with. Since Mercury is 0 MU's from itself, we'll make it 0, and Venus is .868 MU's from Mercury. In fact, all we need do is subtract 1 from each MU (Table 3).

Now, from Earth outwards, we get a pattern approximating 1.5, 3, 6, 12, 24, 48, 75 (halfway) and 100. That's pretty close to double.

What Does it Mean?

First, let's state what cannot be said: It cannot be said that any kind of pattern, doubling or otherwise, is applicable to other planetary orbits in other systems, simply because we only have our own system to investigate. As far as speculation goes, we would have to make the assumption, as pure theory, that planets of other star systems do have a pattern or even a similar pattern. The key speculative question seems to be: Is it possible that there is something in the nature of planetary orbits that enforces a necessary distance between them? The objects of the Kuiper Belt operate in orbital resonance with Neptune, meaning basically that they are in an integral ratio and so have a gravitational influence on each other. Perhaps the Titius-Bode "effect" is simply this, spread out over a great distance and time. Is it also possible that there is a hitherto unknown force, on the mass scale of gravity, that is a repulsive rather than an attractive force? Could the components of such a force be of a similarly unknown nature as dark matter? These are just questions.

If you have thoughts you'd like to share on this, please send email by (this is to foil the spambots) putting together my first and last names, Joseph Conklin, with no space between them, as the name portion in the usual email format at bellsouth.net.