A spacing law that random orbits obey

The harmonic law relates each planet’s period to its distance and says nothing about which distances there should be. Kepler, who found it, also looked for a rule for the distances themselves — he tried the five Platonic solids nested between the planets’ spheres — and the search has continued ever since. The most famous answer was published by Johann Titius in 1766 and popularised by Johann Bode: take the sequence 0, 3, 6, 12, 24, 48, 96, add four to each and divide by ten. The result is 0.4, 0.7, 1.0, 1.6, 2.8, 5.2, 10.0, 19.6 — and in astronomical units the planets then known sat at 0.39, 0.72, 1.00, 1.52, nothing, 5.20 and 9.54.

That is a striking fit, and the rule’s history made it more striking still. Uranus, found by accident in 1781, turned out to lie at 19.2, close to the next term. The gap at 2.8 was searched deliberately, and Ceres was found there on the first night of the nineteenth century, followed by the rest of the asteroid belt. A rule that predicted two discoveries seemed to be telling something about how planetary systems are built.

The planet the rule could not hold #

Neptune was found in 1846 by one of the great calculations of celestial mechanics. Uranus was drifting from its predicted positions, the residuals were inverted to find the unseen planet pulling on it, and the position predicted was accurate to a degree. Both of the people who made the prediction assumed the new planet would lie at the distance the Titius–Bode rule gave, 38.8 astronomical units, because there was nothing else to assume. The planet was found where they said to look. It was not at the distance they had assumed. It lay at 30.1, and the prediction had worked because, over the few decades of Uranus’s arc that mattered, an orbit at the wrong distance with a compensating mass pulled in nearly the right direction.

Neptune’s distance was the first real test of the rule on an object not used to construct it, and the rule failed by more than a fifth. Pluto, found in 1930, sits at 39.5 — close to Neptune’s predicted slot — which was briefly taken as a sort of rescue and is now understood as the opposite: Pluto is locked in a 3:2 resonance with Neptune, placed where it is by Neptune’s migration, and its distance is a statement about Neptune rather than about any rule.

The rule also has freedoms that its fit hides. Mercury is given the first term, 0.4, by making the doubling sequence start from zero rather than 1.5 — a choice made to fit Mercury and no other planet. The gap at 2.8 was left empty for a planet that had not been found and is filled by a belt of small bodies whose total mass is about three per cent of the Moon’s. And the offset and the ratio, 0.4 and 0.3, were chosen with the answer known. A fit with three chosen constants, one skipped slot and one special case, to seven points, is not obviously significant, and the question is how to find out.

The ratios are not a ratio #

The first thing to check is whether the planets actually share a spacing ratio, since that is what the rule claims.

The ratios scatter. Venus to the Earth is 1.38; Mars to Jupiter is 3.41, with the asteroid belt in between; the outer planets manage 1.83, 2.01 and 1.57. Nothing in that list is a constant. What the list has is a floor: no pair is closer than a ratio of about 1.4, and each is far outside the distance at which two planets of their masses would disturb each other’s orbits into instability.

That floor has a physical origin. Two planets on circular orbits are stable against close encounters, as a pair, if they are separated by more than about three and a half of their mutual Hill radii — the radius of the region around the pair’s combined mass in which its own gravity dominates the Sun’s tide. For small planets that is a ratio just above one; for giant planets it is larger. Systems with more planets need more room, because the planets perturb each other collectively, and simulations of three or more planets find that systems packed closer than about ten mutual Hill radii tend to become unstable within the age of the solar system, after which they collide, eject a planet or reorganise into a wider arrangement.

So the planets are spaced by something and not by a ratio. They are spaced by the need to have survived. The question is what a set of distances that satisfies only that need looks like.

The floor is not only a condition for surviving; it is also what formation leaves behind. Planets grow from a disc of small bodies, and a growing protoplanet sweeps up the material within a few of its own Hill radii and gravitationally scatters the rest. Neighbouring protoplanets therefore settle into feeding zones of a characteristic width, about ten Hill radii apart, and grow until they have consumed what lay between them. The spacing that results is set in Hill radii, which scale with distance from the star, so on a logarithmic axis it is roughly even — geometric-looking, again, for a reason that has nothing to do with a doubling rule. The later phase of chaotic interactions among those embryos, which ends in giant impacts, widens the spacing further and makes it irregular, and that is the phase that produced the terrestrial planets.

So there are two floors, a formation floor and a survival floor, and both are measured in the same units. Neither is a ratio. Both make any set of planets look more regularly spaced on a logarithmic axis than a set of random numbers would, and the question the rule raises is whether the planets are more regular than that.

A null hypothesis with no rule in it #

The test that settles it is to generate planetary systems with no spacing law at all, apart from a minimum separation, and ask how often they fit a geometric law as well as the solar system does.

The construction is deliberately crude. The distances are uniform in the logarithm, which is the least informative choice for a quantity spanning two orders of magnitude, and the only constraint is a minimum ratio between neighbours. The fit is the same one the historical rule used in effect: a geometric sequence with its ratio and starting point chosen to fit, and the freedom to skip one slot wherever that helps most. The solar system’s eight major planets scatter about their best geometric law by 0.048 in the logarithm — about eleven per cent in distance. Forty per cent of the random systems do at least that well.

A result that forty per cent of random systems match is not a result. The solar system’s fit is typical of what orbits that merely avoid each other produce, and there is no evidence in it for any law beyond the avoidance.

The floor does the work #

The fraction depends on how strong the floor is, and that dependence is the real content of the test.

Raising the minimum spacing forces the random systems towards regularity. Eight planets that must each be at least a factor of 1.5 from their neighbours, fitted between 0.3 and 35 astronomical units, have used most of the available logarithmic range on the minimum gaps alone, and whatever is left over can only be spread thinly. The result is a set of distances that is nearly geometric by necessity. A system packed to its stability limit would look like a Titius–Bode sequence however it was assembled.

This is the general shape of the argument against the rule, and it was made carefully in the 1990s and 2000s, with more realistic random systems — planets placed according to stability criteria that depend on their masses, and fits that used the rule’s full freedoms. The conclusion in every version was the same: geometric-looking spacing is what the logarithm does to any sequence of positive numbers with a floor on their ratios, and the solar system’s degree of regularity is not unusual among systems built with no rule at all. Which of the tests is most appropriate is arguable; that none of them finds the solar system significantly more regular than chance is not.

A spacing law with a mechanism #

There are real spacing laws in the solar system, and comparing one with the Titius–Bode rule shows what a real one looks like.

The inner three Galilean moons are in the Laplace resonance: for every orbit of Ganymede, Europa makes two and Io four, and a combination of their three longitudes is held fixed by their mutual perturbations. The harmonic law turns a period ratio of two into a distance ratio of $2^{2/3}$, and the moons obey it to a quarter of a per cent. That is a spacing law, and it is one with every property the Titius–Bode rule lacks. It is exact rather than approximate; it has a mechanism, the same one that clears gaps and locks moons elsewhere; and it fails exactly where the mechanism does not reach. Callisto, outside the lock, is spaced by 1.759 — a ratio that nothing requires.

The large moons of Uranus fit a geometric law almost as well, with a scatter of 0.016 in the logarithm, and have no resonance among them now. Their fit is of the Titius–Bode kind: good, and explained by the floor. A spacing that is geometric only on average, with no exact ratio anywhere and no failure where a mechanism stops, is the signature of the floor rather than of a law.

Why the rule looked like physics #

The rule’s history deserves a fairer account than its failure suggests, because the reason it seemed compelling is instructive.

It predicted the asteroid belt, and the prediction was real: there is a gap between Mars and Jupiter, and the rule put something in it. But the gap is exactly what the floor argument predicts too. Mars and Jupiter are spaced by a ratio of 3.41, which is unusually wide, and a random system would have been as likely to show one wide gap somewhere as the solar system is. Any rule fitted to the other planets would have put a slot in the widest gap. The belt is there for a different reason — Jupiter’s resonances kept the material from assembling into a planet — and the rule found it only because the gap was conspicuous.

It predicted Uranus within two per cent, and that too is less remarkable than it looks. Uranus was found by a telescope survey, not by the rule; the rule was checked against it afterwards; and with one planet beyond Saturn and a doubling sequence already fitted to six, landing within a few per cent of the next term happens often in random systems of the kind drawn above. The rule was confirmed twice and failed once, and the failure was the only test in which it was used before the answer was known.

The rule also had successors that fitted better by adding freedom. In 1913 Mary Blagg published a formula with a geometric ratio multiplied by a periodic correction function, and with its extra parameters it fitted the planets and the major satellite systems more closely than the Titius–Bode rule, including Neptune. A fit that improves as parameters are added is the expected behaviour of any flexible function, and Blagg’s formula, like the rule it refined, has never predicted a distance that was not already known. It is a careful piece of curve-fitting and a useful warning: the more closely a numerical rule matches the data it was built from, the less that match says.

That asymmetry — confirmation by objects found independently, failure on the one object predicted using it — is the pattern of a coincidence. A statistical signal is only as strong as the test that could have refuted it, and the Titius–Bode rule was only ever tested once.

The rule was also not alone. Similar rules were written for the satellite systems of Jupiter, Saturn and Uranus, in the form that successive moons’ periods stand in a constant ratio, and they fit their systems about as well as the Titius–Bode rule fits the planets — with different constants for each system and with the same freedoms. A spacing law that needs different constants for every system it is applied to, and fits each about as well as random systems do, is a description of the floor rather than of a mechanism, and the satellite versions have been retired for the same reason as the planetary one.

Spacing in other planetary systems #

The question has come back with the discovery of thousands of multi-planet systems around other stars, and it has come back in a better form.

Compact systems of several small planets found by transit surveys have spacings that are, if anything, more regular than the solar system’s: neighbouring planets in the same system tend to have similar sizes and similar period ratios. That regularity is real and has been measured, with the selection effects of transit surveys modelled. But the period ratios cluster near the values at which the systems are just stable — ten to twenty mutual Hill radii — rather than at any universal number, and they depend on the planets’ masses in the way the stability floor requires. The regularity is the floor, measured in systems packed much closer to it than the solar system is.

The resonances show up too. Period ratios just wide of 3:2 and 2:1 are over-represented among neighbouring planets, and some systems are chains of resonances that cannot have been assembled in place. Those are the exoplanet equivalents of the Laplace resonance — exact ratios with a mechanism, convergent migration in a gas disc — and they sit alongside the looser, floor-driven spacing in the same catalogues. The two kinds of order are distinguishable exactly as they are in the solar system: one is exact and caused, the other approximate and statistical.

What the test leaves out #

The random systems drawn here are cruder than the published tests. They place planets uniformly in the logarithm of distance, which is a choice; a different prior — uniform in distance, or following the surface density of a protoplanetary disc — would change the fractions. They use a single minimum ratio for every pair regardless of mass, where a real stability criterion depends on the masses and allows small planets to sit closer than giant ones. And they ask only whether a geometric law fits as well, not whether the particular constants 0.4 and 0.3 come out, which is a different and more specific question the rule’s defenders sometimes preferred.

None of those refinements rescues the rule, because each of them makes random systems more varied in some ways and more regular in others, and the solar system stays inside the distribution. What they change is the percentage, not the conclusion.

Still open: whether the floor has a sharper edge #

The one real question left over is not about the Titius–Bode rule but about the floor that explains it. Is there a characteristic spacing at which planetary systems settle — a separation, measured in mutual Hill radii, at which they are just stable over their lifetimes and to which they are pushed by the instabilities that remove planets packed more closely? The compact exoplanet systems suggest a distribution of spacings peaked at a particular value with a sharp lower edge, which would be the modern, physical version of a spacing rule. If the peak is sharp, it is a statement about how instability sculpts systems over billions of years; if it is broad, it is a statement about how they formed. The period ratios of neighbouring planets in the systems already catalogued carry that information, once the planets the surveys missed are accounted for, and the surveys now reaching longer periods and smaller planets will say which.