The Titius-Bode Law, aka: Bode’s Law
Published on Sep 5, 2007 at 11:43 pm.
6 Comments.
Filed under asteroids.
In my last posting, I mentioned something called Bode’s Law. I sometimes get questions about Bode’s Law, so I thought that I’d say something about it.
First of all, it isn’t really a “law” at all, in the scientific sense of the word. It is at best a “rule.” There is no physical basis for it, and not even any solid theoretical basis. It is a mathematical formulation that isn’t even mathematically sound. But, it gets a lot of attention. The relationship is called Bode’s Law because it was popularized by Johann Bode in 1772. However, the relationship that Bode published and then trumpeted was identical to one proposed just six years earlier by Johann Titius. Bode almost certainly knew of Titius’ work, yet he published the relationship as his own creation. For many years, therefore, it was simply known as Bode’s Law. In the last decade or two, however, I have seen the relationship referred to more and more often as the Titius-Bode Law, or sometimes as the Titius-Bode Rule (see the beginning of this paragraph).
The Titius-Bode Law is a mathematical curiosity, and it has been presented as such in introductory astronomy textbooks. It has historical significance. But, all too often people take it to be more than it really is. That is one reason that I think many newer textbooks that I’ve seen have dropped discussion of it. Another reason, perhaps, is that there is simply a lot more to put in the textbooks today, and the new material is pushing out a lot of the historical material.
So, what is the fuss all about? What is Bode’s Law? What is this relationship that Titius came up with? Simply put, you take a series of numbers:
Now, as you can see this series is not really a proper series. Each number is double the previous number, with the exception of the leading zero. That leading zero is why I say that this is not a proper series, and that the relationship is not on firm mathematical footing. Next, you add four to each of the numbers in the series:
Then, you divide each of these numbers by 10. That gives a series of numbers ranging from 0.4 to 38.8. OK, so what? Well, as it turn out that the numbers 0.4, 0.7, 1.0, 1.6, 5.2, and 10.0 (six of the first seven numbers) turn out to be rather close to the semi-major axis distances (measured in Astronomical Units) for the planets Mercury, Venus, Earth, Mars, Jupiter and Saturn. This was an interesting curiosity, and it was treated as such when Bode published the series. Then, in 1771, William Herschel discovered the planet Uranus at a distance of about 19.2 AU from the Sun. This was quite close to the 19.6 that was the next of the series in the relationship. This created quite a stir. Bode went on a crusade to begin to promote the relationship extensively, claiming it to have more significance than it really did. The only sticking point was that the relationship predicted a planet should be located at a spot between Mars and Jupiter, and no planet was known there. Bode continually urged astronomers to search the heavens for a planet in that orbit. Baron Franz Xavier von Zach took up the challenge and began an exhaustive search for the planet. After about a decade of searching, he realized that he needed help and he solicited the help of several other astronomers. Quickly, they realized that the search was going to be too big for just the few of them, so they recruited more astronomers. Together, in 1800 they formed a group that called themselves the “celestial police.” The celestial police took on the self assigned task of searching the heavens for the missing planet.
In 1801, only the year after the celestial police began in earnest to search for the missing planet, Giuseppi Piazzi, an Italian astronomer who was not a member of the celestial police, found a body located near 2.8 AU from the Sun. This object, which received the name Ceres, was hailed as the “missing planet” predicted by Bode’s Law. Baron von Zach hurriedly invited Giuseppi to join the celestial police. But, matters soon became very complicated. In 1802, Heinrich Olbers, a founding member of the celestial police, was looking for Ceres again when he discovered another “planet” at nearly the same semi-major axis, but in a very inclined orbit. This body was called Pallas. 
Soon astronomers also found two more bodies in similar orbits, Juno and Vesta. Now, there were four planets in the location that Bode’s Law called for there being one planet. Astronomers were at a loss to explain this, suggesting that perhaps a single planet may have somehow broken up. Within a few years, though, more “planets” were found in the area. Astronomers began to realize that there were a lot of bodied located between Mars and Jupiter, and that these were all very small bodies — significantly smaller, even, than our own Moon. So, Ceres, Pallas, Juno, and Vesta were demoted from planet status to simply “asteroids.” Eventually, the term “minor planet” was applied to these bodies. Now, the largest of them are called dwarf planets.
Then, in 1846, Neptune was discovered, and it lay nowhere near where the Titius-Bode law predicted that it should be found. Astronomers again began to think of the relationship as simply a mathematical curiosity: basically, a numbers game. Today, most astronomers don’t believe that there is really any physical basis for the relationship. However, there are always some people who are enthralled by numbers, and these people don’t want to give up on the relationship. These are the people who insist that Bode’s Law really is a physical law, and they go to great lengths to come up with some series that fits the planets. There are several versions of Bode’s Law and revisions of Bode’s Law out there. But, most of these are desperate attempts to come up with a series of numbers. All of these have the fatal flaw that they are derived by people trying to come up with a complex series to fit a handful of observed planetary semi-major axes. That is not how you do science. Rather, a theoretical basis should be developed. Then, from that theoretical basis, there should be a way to compute the locations of the planets. Anyone with just a bit of mathematical prowess, if they work hard enough, can come up with a series that will nearly fit a handful of numbers, even if that handful of numbers is randomly picked. That isn’t science, and it isn’t a real physical law. It is just playing games with numbers. That is what Bode’s Law is seen to be. Still, some people refuse to accept that. But, most major astronomical and astrophysical journals will not now even look at papers submitted trying to come up with a new revision of Bode’s Law from simply finding a series of numbers without any physical basis.
Now, I should point out that there is a physical basis for why the inner planets have orbits that are closer together than the outer planets. But, that is a story for another blog entry. And, it doesn’t mean that there really is anything to Bode’s Law. As I said, anyone can come up with a series that comes up with numbers that fit a handful of random, or nearly random, numbers.
-Astroprof
Astrogeek on September 6, 2007 at 1:26 am: 1
But wait! I have a mathematical proof of Bode’s Law that is derived from a complex equation involving pi, i, e, (and sometimes y), the Fibonacci sequence, my mother’s red bicycle, and months ending in ‘er’!
Ed Minchau on September 6, 2007 at 9:01 am: 2
How’s this for a physical basis:
The planet Jupiter is very massive, more than twice the mass of all the other planets combined. When the solar system was being formed, the vast majority of the mass was located where our sun is, and a large amount of the remainder was located in Jupiter’s orbit.
Since Jupiter is orbiting the sun in a regular orbit, its mass pulls on all the other mass in the solar system on a recurring basis. There are zones within the solar system that Jupiter’s mass pulls back and forth in such a way that there never can be enough mass accumulated to form a planet, and zones where there is a resonance between Jupiter and any mass orbiting the sun at that distance. In other words, Jupiter acted as a sort of “shepherd” in the early solar system, guiding where planets could be formed and where they could not.
The next most massive planet after Jupiter is Saturn, which is itself more than twice as massive as the remainder of the planets combined. Saturn orbits almost exactly twice for every five orbits of Jupiter, so it is in a 5:2 resonance with Jupiter. So, Jupiter’s mass guided the accumulation of mass in Saturn’s orbit.
So then there are two “shepherds”, and the situation becomes much more complicated. Jupiter’s mass still dominates the location of possible orbits, but Saturn’s mass contributes as well. Further complicating matters, the planets are all largely forming at roughly the same time (at least on the time scale of the age of the solar system), and planets with similar mass are forming in “adjacent” orbits (Neptune and Uranus are fairly close in mass, as are Venus and the Earth) and each planet is pulling on all the others.
OK, so this isn’t as neat and tidy as Titius-Bode, and there is a lot of math that I simply cannot be bothered to do right now - but it does appear at first approximation that there is a mathematical relationship between the orbits of Jupiter, Saturn, and the other planets.
Astroprof on September 6, 2007 at 9:17 am: 3
Ed,
I said that there is not a physical basis for the Titius-Bode Law. But, I also said that there is a physical basis for why the outer planets have orbits spaced farther apart than then inner ones. You just gave a nice summary of that! Good job!
Yes, the orbits do interact, but the Titius-Bode Law is a numbers game. There are a number of possible stable configurations. The one that we’ve got is not the only one, which is what the T-B Law would seem to imply.
A Ler…-- Rastos de Luz on September 7, 2007 at 9:47 am: 4
[…] The Titius-Bode Law, aka: Bode’s Law no Astroprof’s Page […]
Tom’s Astronomy Blog » Blog Archive » 3 Juno on September 19, 2008 at 2:05 pm: 5
[…] why 3 Juno, and not just Juno, well Juno was the third such body found in the location where Bode’s Law predicted there should be a planet, and for a time it was considered a planet. Planets have […]
Frank Hatch on December 4, 2008 at 12:40 am: 6
The Pluto/Neptune anomaly of the Titius-Bode progression is predicted in the Initial Mass Displacements.
FrankHatchiii.com