Applying the Titius-Bode Rule to Exoplanet Systems

One of the key methods employed in the practice of the sciences is the search for patterns. Their discovery often hints at something important to which we should pay attention if we want to understand a principle. This can be from simple things like the cycles of the sky throughout the year that trace out our motion in the solar system to the patterns of spectral lines that allow astronomers to measure the universe. Back on our solar system scale, one such apparent pattern that stood steadfast until 1846, was the Titius-Bode rule. This rule noted that the distance of the planets from the sun seemed to follow a pattern described by the equation a = 0.4 + 0.3 × 2ⁿ where n was the planet number in order of distance from the Sun. This pattern held very well for the first 7 planets, so long as one included the asteroid Ceres, or the asteroid belt itself, as planet #5. Yet the discovery of Neptune and Pluto discredited this pattern as a mere coincidence, mathematical happenstance and numerology, as the Titius-Bode rule severely underpredicted their distances.

Some still wonder if there wasn’t something more to the rule and orbital resonances didn’t have some sort of subtle effect that was being overlooked and made the rule more of a law, at least for innermost planets. With the rapid discovery of planets around other stars, astronomers are once again looking to see if there might just be some sort of truth to this pattern.

One of the most well populated and well studied exo-planetary systems is 55 Cancri. In 2008, a paper was published in the Mexican Journal of Astronomy and Astrophysics attempting to apply the Titius-Bode rule to this system. In that study, the classical rule could not fit, but, from the five planets known at the time, the researchers were able to fit a similar exponential function to the system. With their fit, they found that, much like our own solar system, there was a “missing planet” for what should be the 5th from the parent star. The fit predicted it should lie at a distance of roughly two AU. However, since the paper was published, the orbital characteristics of the system have been revised significantly, throwing off the predictions of the 2008 study.

However, another paper was recently written, updating the fit for the 55 Cnc system. This time, to make the fit work well, the author was forced to assume the possibility of four undiscovered planets. If they were to exist, one of them should exist at a distance of 1.5 AU which, for that system may place it in the habitable zone.

But what of other planetary systems? Presently, there have been few other systems that are sufficiently explored to begin to explore such potential relations. One paper, released in 2010, noted that, at that time, only 15 systems were known with three or more planets. While some appeared, superficially, to have some sort of patterning, the authors declined to speculate on whether or not there was any deeper meaning since, with so little data, a line would be quite easy to fit.

So for now, it’s another game of patience as astronomers continue probing more systems and discovering more planets. If, at some point, a planet were discovered that was predicted by a Titius-Bode relation, it would support the underlying principle that something was sorting the planets in a regular manner. But then again, that’s what they said when Ceres and Neptune were discovered.