55 Cancri. Image credit: NASA/JPL

Applying the Titius-Bode Rule to Exoplanet Systems

Article Updated: 24 Dec , 2015


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.

11 Responses

  1. jorgecandeias says:

    Oh dear. It’s such a bad idea to start fiddling with that old empyrical handful of nothing called Titus-Bode “Law”. SUCH a bad idea…

  2. IVAN3MAN_AT_LARGE says:

    Referring to the third sentence:

    This can be from simple things like the cycles of the sky throughout the year that trace out our motion in the solar to the patterns of spectral lines that allow astronomers to measure the universe.

    … I think that underlined bit should be Solar System.

  3. Anonymous says:

    Terrible science. From most series of numbers, such as orbital radii, you can devise some rule that describes them. But in the absence of any actual underlying physics to justify it, it’s just numerology. These rules tend to break down the first chance they get, just like the Titius-Bode law did when Neptune was discovered.

    The Titius-Bode law is like this: if I look up my phone number and the numbers of the people directly before and after me in the phone book, I can easily find some rule that fits the numbers. But if I try to use my rule to predict the phone numbers of Mr. Aaron Aardvark or Ms. Zelda Zygote it almost certainly wouldn’t work.

    • Torbjörn Larsson says:

      Agreed. The pattern writ large is the seemingly regular varying orbital radius; that lights up our greedy pattern detector.

      The pattern hypothesis is the rule that predicts what we believe we see. An ad hoc hypothesis isn’t wrong but it doesn’t need to connect with physics either. Chances are it doesn’t.

      But as Jon says, we have to try. The problem is when we insist on trying the same pattern despite repeated failure. (That is a decent ad hoc definition of insanity, btw. (O.o))

  4. Ray Fowler says:

    Please, the Titus-Bode Law has been so thoroughly debunked that I can’t believe anyone still gives it credence. It doesn’t even fit our system very well.

  5. Torbjörn Larsson says:

    A while back another pattern, with naive physics behind, was in some papers as I remember it.

    Roughly it went that multiplanet systems had orbital packings that didn’t permit placing another sizable body in between. The same general principle that lies behind the planet definition in our own system, “clearing its orbit”, extrapolated as far as it goes. It looked good elsewhere too, albeit with bad statistics at the time.

    With Kepler that pattern went away, since extremely flat systems like Kepler-11 has orbits where planets affect each other. Still a reasonable extrapolation from “clearing” but as constrained as it gets. Or conversely, as packed as you can get.

    I don’t know if you can derive a Bode type of law from that. Such a pattern looks more like the dynamical system pattern you would expect.

    And I don’t know how these types of patterns fares with Kepler data. With Jupiters pushing and pulling on orbits as they migrate inwards (hot Jupiters) or resonate to move outwards (solar system), you would naively expect deviations.

    • William Sparrow says:

      My initial thought upon reading this article was how could this “law” possibly apply when you consider hot Jupiters near their host stars. You summed it up perfectly with your statement in the last paragraph above. I think we’ll find that many of our prior assumptions comparing our own Solar System’s orbital dynamics don’t fit well with extrasolar systems.

  6. Anonymous says:

    I wonder whether Neptune’s orbit might be affected by possible gravitational effects of Kuiper and Oort objects?

    • Torbjörn Larsson says:

      On that topic, Earth is likely pushed around – by the asteroids!

      “Laskar et al. incorporated both Ceres and Vesta into their gravitational simulations, along with the other main asteroids Pallas, Iris, and Bamberga. Although they had attempted this before, this time they dug further into the details of the gravitational contribution of these bodies over very long time scales and found something quite remarkable.

      Not only are the orbits of the larger asteroids like Ceres and Vesta more chaotic than previously understood (limiting any predictability of orbital parameters to a window of less than 500,000 years – a veritable drop in the ocean of time), but their presence has a significant influence on the Earth’s orbital ellipticity (eccentricity). These tiny worlds perturb us.”

      Btw, they claim that present a cutoff to climate fits to ~ 60 My, at least if eccentricity is a major factor. Maybe, maybe not: in the comments Scharf notes that similar work shows that close enough Jupiters would push an Earth type planet in and out of ice age analog events.

  7. Your first reference to the 2008 paper links to the 2011 paper. Could you have meant arXiv:0803.2240 ( http://arxiv.org/abs/0803.2240 )?

Comments are closed.