One of the things scientists sometimes do is consider a wild idea to see if it might work. These “what if” scenarios often take a physical principle beyond the limits of known physics to explore possible consequences, such as ideas about wormholes or time travel. Another approach is to take a pattern observed in one situation and try applying it to others. There’s no underlying physics to presume it might be true, but it might prove useful. These approaches are sometimes referred to as throwing spaghetti on the wall to see what sticks. Usually you’re left with a theoretical mess on the floor, but every now and then something seems to stick. Such is the case with a pattern known as the Titius-Bode relation (or Bode’s law).

The Titius-Bode relation for the solar system.

This relation was first noted in the 1700s, and stated that the distance of planets followed a pattern of d = 4 + n, where n was 3, 6, 12, 24, etc.If you let the Earth’s distance be 10, then Bode’s law matched the distances of the (then) known planets to within a couple percent, including Ceres. Like Kepler’s laws, it was originally stated as an observed pattern, with no clear reason as to why it worked. But while Kepler’s laws were later shown to be a consequence of Newton’s laws of motion and gravity, the Titius-Bode relation remained and interesting pattern. When Neptune was discovered, and it was found that it didn’t match the relation, it fell out of style. Like many ideas that “kind of” work, the Titius-Bode relation has become popular among some alternative physics fans, who try to derive some meaning out of the pattern. In mainstream physics the relation is mainly seen as interesting historical numerology.

The modified TB-relation “works” for some systems.

But with the discovery of exoplanetary systems, the Titius-Bode relation has inspired some theoretical astronomers to throw it against the wall to see if it sticks, and in some ways it seems to. Take, for example, a recent paper on the idea published in MNRAS. In this work the team noted that the core idea of the TB-relation was the geometric progression in planetary distances. This gives a similar relation in the periods of their orbits (which is an aspect of exoplanets that is more easily measured). So they proposed a modified TB-relation with a geometric progression that could be fit to different planetary scales. Looking at 151 exoplanetary systems with at least three planets, they then tried to fit their relation them. What they found was that 124 of the systems seemed to fit the pattern reasonably well. From this they used the relation to predict 98 planets in those systems with a reasonable chance of being detected by the Kepler satellite. After going through the Kepler data, 5 new exoplanets were found.

So this modified Titius-Bode relation seems to make a correct prediction 5% of the time (less if you add in the systems it doesn’t fit). So it’s interesting, but not particularly compelling. In a broad sense there does seem to be a geometric progression to planetary distances, but nothing as specific as the TB-relation. For now, at least, it is just spaghetti on the wall.

Paper: T. Bovaird, et al. Using the inclinations of Kepler systems to prioritize new Titius–Bode-based exoplanet predictions. MNRAS 448 (4): 3608-3627. doi: 10.1093/mnras/stv221 (2015)

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Astrophysics often follows a pattern of gathering lots of data and then developing models to explain the data.  Groundbreaking ideas such as Newton’s law of gravity or Einstein’s theory of relativity are rare, so often we’re reduced to trying to find patterns in the data that may (or may not) have a clear cause.  Sometimes this method works quite well.  Perhaps the most famous one is Kepler’s laws of planetary motion.  When Kepler proposed his laws, they were presented as a clear and simple description of what actually happens.  They were a breakthrough because they could be used to predict the motion of planets with great accuracy.

Despite their accuracy, Kepler’s laws were simply a concise description of an observed pattern.  Kepler gave no underlying physical principle to explain why his laws worked.  The underlying principle was only later discovered by Newton, who was able to derive Kepler’s laws from his law of gravity.

Looking for patterns in data is useful because such patterns are often caused by a simple underlying principle, but this isn’t always the case.  As an example, consider the story of the the Titus-Bode law.  This law was first noted by Bode in 1772, and it stated that the distance of planets followed a pattern of d = 4 + n, where n was 3, 6, 12, 24, etc.  Like Kepler’s laws, it was originally stated as an observed pattern, with no clear reason as to why it worked.

If you let the Earth’s distance be 10, then Bode’s law matched the distances of the (then) known planets to within a couple percent, with one exception.  Bode’s law predicted a planet between Mars and Jupiter that didn’t exist.  In 1781, Uranus was discovered, and agreed with Bode’s law to within a couple percent, so there began a search for the missing “fifth planet” between Mars and Jupiter.  In 1801, the minor planet Ceres was discovered, and it was at exactly the distance Bode predicted.  It would seem that Bode’s law was a valid model, despite its lack of an underlying cause.

Two things eventually killed Bode’s law.  The first was the discovery of other “planets” (now known as asteroids) between Mars and Jupiter which did not follow the law, and the second was the discovery of Neptune in 1846, which wildly disagreed with the law.  Of course now we know our solar system is far more complex than Bode’s law implies.  We also know of a few other stars with planetary systems, and they don’t seem to follow similar rules very well.

Still, the Titus-Bode law works really well for most of the planets in our system. I’ve plotted the law in red in the figure below, and when you compare it to observed distances (in blue) you can see how well it works up until Neptune.

Such a strong agreement would seem to imply that there must be some underlying cause.  It’s hard to make a compelling argument with only a handful of known solar systems, but computer models suggest that the pattern is driven by resonances between orbital periods.  For example, Earth and Mars have a ratio of 1 to 2, while Venus and Mars have a ratio of 1 to 3.  Such ratios are common for planets, asteroids and moons, because they tend to reinforce the stability of orbits.  When these ratios are common in a planetary system (as they are in ours), a Titus-Bode law can be found for the distances.

In modern times, the Titus-Bode relation is seen more as an interesting pattern than a true law.  As we discover more planetary systems, the relation may rise to prominence once again, but for now it’s an interesting cautionary tale that we should never limit our understanding to patterns alone.

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