Bode's Rule, Exoplanets, and an Un-clockwork Solar System?

After it catching my eye while perusing the shelves of the departmental library, I have recently finished reading Destiny or Chance Revisited by Stuart Ross Taylor. While I initially only too the book out with the plan of reading the final chapter about the possibility of extraterrestrial life, I ended up reading, and getting very invested in, the whole thing. The book walked through the origins of the Solar System and formation of the planets, touching on all topics from star formation and exoplanets, to the Great Oxidation Event and the Cambrian Explosion, with the aim of showing how the the events that have lead to the formation of the Earth and the existence of intelligent life upon the Earth were completely random, chance events, and that equally improbable events would have had to occur in other planetary systems in order for the conditions to arise in which a planet could develop and host intelligent life. Basically, the chances are low of this happening, but the universe is vast and just a quick glance at the Drake equation can tell you the probability is not zero. The book introduced me to a number of new concepts, one of them being Bode's rule in relation to not just the formation of our solar system but exoplanetary systems. I think it's a nice example of how random the formation of these planetary systems are and how we as humans like to try and put everything into neat patterns and boxes which is quite often just not possible in the natural sciences.

A clockwork model of the Solar System. Image credit to Aleks Marinkovic via Unsplash

Bode's rule, or more correctly Titius-Bode's rule, is a mathematical relationship that can predict the distances of the planets in our solar system from the sun. It was first noticed by Johann Daniel Titius in 1766, and then popularised by Johann Elert Bode in 1772.

The distances of the planets from the sun can be expressed:

D(n) = ( 3 x 2n + 4 ) / 10 AU, where n = - infinity, 0, 1, 2, 3, 4 ...

Or in other words the distance of the planets from the sun can be found by taking the series 0, 0.3, 0.6, 1.2, 2.4 (where after 0.3 each one is double the last) and adding 0.4 to it. This then gives you the distance of the planets from the Sun in astronomical units (AU), where 1 AU is the mean distance of the Earth from the Sun.

This is summarised in the table below:

Planet	Bode's Rule Prediction	Actual Distance from Sun
Mercury	0.4	0.387
Venus	0.7	0.723
Earth	1.0	1.0
Mars	1.6	1.52
(Ceres)	2.8	2.77
Jupiter	5.2	5.20
Saturn	10.0	9.55
Uranus	19.6	19.2
Neptune	38.8	30.1

Neither Uranus or Neptune had been discovered when this relationship was noted. Neither had the asteroid/dwarf planet Ceres which fits into the sequence at the 2.8AU spot. When Uranus was discovered in 1781 at the 'correct' distance of 19.2 AU this was seen to confirm Bode's rule, and even briefly promoted its status to Bode's law. Bode strongly encouraged looking for a planet at the 2.8AU mark which appeared to be missing, and in 1901 the asteroid Ceres was located at the spot, giving further support for the rule. This all, however, came crashing down upon the discovery of Neptune in 1846. It sits at 30.1 AU, over 8 AU closer to the Sun than Bode's rule predicts. The situation only got worse upon the discovery of Pluto which at 39 AU has no regard at all for the rule . Subsequently, today, Bode's rule is general just seen a nice near coincidence and not a rule of thumb for our solar system (or other star systems out there).

The development of Bode's rule is a example of humans wanting to understand the universe by making it make sense, often spotting rules and patterns where rules and patterns do not exist. Another example of this is the clockwork solar system model that was suggested after Newtonian mechanics was successfully able to predict the placing of Neptune. From this view, everything in the solar system is ordered and deliberately put in a specific place following the laws of Newtonian mechanics, much like a clockmaker puts everything in place to produce a regularly mowing clock. We now know this is not the case, but it was popular back in the day when the idea of some God like 'clockmaker' was able to compromise new scientific discoveries with the grip of religion. Bode's rule would also give order to the Solar System, and allow us to potentially predict where other terrestrial like exoplanets may be hidden in their own systems, but I'm getting ahead of myself.

Today, we understand that Bode's rule has nothing to do with the formation of solar systems creating regularly spaced planets as the nebula condenses, and that it comes about due to the gravitational interactions of the planets, their tidal forces locking them into this pattern. Interestingly, Bode's rule works for satellite systems of the giant planets in the Solar System, such as the inner moons of Jupiter, Saturn, and Uranus.

If Bode's rule had turned out to be a product of the formation of the Solar System, then maybe it would be true for other solar systems, and in principle we could use this to predict the location of exoplanets that would other wise be very laborious, or even not possible, to detect. Just to check this is not the case, numerous studies have been done looking at exoplanetary systems to see if Bode's Rule might work. One example is a paper by Heon-Young Chang, published back in May 2023, that attempted to see if the Titius-Bode rule was valid for exoplanetary systems consisting of planets hosted by a single star. The study found that Titius-Bode's rule does not hold for enough systems for it to be statistically significant. Although, other studies on the topic have had a range of results, with a study by Patricia Lara et al. in 2020 showing that Titius-Bode's rule did hold to a precision of 78% in systems with at least four planets. The planetary systems that the rule does work for are probably due to similar gravitational interactions that we see in our own solar system rather than it being a property of planetary system formation, and the varying results come from the particular system used in a given study.

So overall, what can Bode's Rule tell us? Not a lot other than the fact that we as scientists love to find patterns and use them to explain the world around us. In some cases this is great, but in others, such as so often in the natural sciences, it simply does not work. The universe is a complex place, and filled with random chance events that have led to us being here today. I highly recommend the book Destiny or Chance Revisited if you want to explore this further, it really does leave you thinking about the shear luck that has lead to our existence in the universe.