The age-old riddle — “Which came first, the chicken or the egg?”– applies to the architecture of the Solar System as well. For the average person, the spacing of the planets in the Solar System just reflects random numbers, yet the spacings have a deeper mathematical relationship which has captivated many an astronomer over the years.
Introductory physics is often presented as being completely deterministic. In the classroom, physics often begins with the study of projectile motion. This means that if one is given initial conditions of the particle, its trajectory can be calculated exactly and for all time. This determinism completely informs the sub-discipline of classical mechanics.
So, what about the spacing of the planetary orbits? Is there some kind of underlying mathematical scheme that can be applied to the orbits?
The truth of the matter is that there is an equation/algorithm that predicts the orbital radii (or more precisely, the semi-major axes) of the planets to a high degree of accuracy. This “Law” of spacing is known as “Bode’s Law.” The numerical formulation for Bode’s Law is given in equation 1.
Equation 1: R = (3*2n + 4)/10,
where n = -infinity, 0, 1, 2, 3, 4, 5, 6 — going from -infinity for Mercury through 6 for Uranus. This formulation gives the semi-major axis for each planet’s orbit in astronomical units, where one astronomical unit (AU) is the distance from the Earth to the Sun or 93 million miles.
Table 1: calculated and observed values for a planet’s semi-major axis of orbital motion.
Planet Bode (AU) Observed (AU)
Mercury 0.400 0.387
Venus 0.700 0.723
Earth 1.000 1.000
Mars 1.600 1.524
Ceres 2.800 2.768
Jupiter 5.200 5.203
Saturn 10.000 9.537
Uranus 19.600 19.189
Neptune – 30.070
Pluto 38.800 39.482
It is intriguing to see that Bode’s Law predicts the existence of a planet right in the middle of the asteroid belt where the minor planet Ceres is located (see Table 1 and Figure 1). It was this very prediction that spurred a search for a planet at 2.8 AU. Ceres is not a planet but is an asteroid and is much, much less massive than Earth or even Mars. Nevertheless, Bode’s Law predicts something in that area of the Solar System. In similar fashion, Bode’s Law predicts a planet to be localized at a semi-major axis of 38.800 AU. That is where we find Pluto, one of the largest “dwarf” planets that orbit within the so-called Kuiper Belt. Meanwhile, Neptune persists as a not-so-subtle reminder that not all planets adhere to Bode’s Law of orbital spacing. We are further reminded that what we see today is but a snapshot in the 4.6-billion-year-old history of the Solar System. Planets have been provoked to alter their orbital radii in the past and will continue to respond to future provocations.
But the harmonies present in equation 1 prompt us to seek an understanding of their source. We see that as n increases, the term 3*2n outstrips the constant term of 4 and the spacings begin growing as 2n. We have Jupiter at five AU, Saturn at ten and Uranus at twenty. What exactly is happening here? It is interesting to note that as planets change from the rocky orbs that characterize the inner Solar System into the gas/ice giants farther out, the spacing goes to 2n. This encourages us to look more closely at the physics of gas/ice giant planet formation.
When we consider computer simulations of the Solar System’s formation, we find that the physics of a forming planet is a very complex problem. It requires the existence of a seed, something called a planetesimal which is massive enough to accrete gas and dust and grow into a planet. Once formed, such a planet would have been subject to all sorts of environmental influences. Indeed, it is possible that the primordial birth sites of the planets were significantly different than their current orbital locations. Models such as the popular “Grand Tack” endeavor to explain what happened over cosmic time. In the Grand Tack scenario, the planets were subject to viscous forcing by debris in the remnant protoplanetary disk. This forcing caused Jupiter to perform an inward-outward do-si-do, with Saturn, Uranus, and Neptune moving outward to their current orbital radii. Perhaps we are seeing today some sort of resonance effect among the orbiting outer planets that is keeping their orbits relatively stable.
There are planets with moons in the inner Solar System and in the outer Solar System, but these respective systems of moons differ. The most popular account of the Earth’s moon is that a Mars sized object (1/10 mass of Earth) smashed into the Earth and produced a ring of orbiting debris that gravitationally congealed into the Moon. Mars’ two moons are thought to be captured asteroids. These two scenarios are considered to be highly likely, and the astronomical community has adopted them as the best possible explanations for these two moon systems.
The story for the moons of the gas/ice giants is entirely different. Here, the major moons formed along with the formation of the planets. Jupiter has long been labeled a miniature solar system because of its array of bright, massive moons known as the Galilean satellites. Astronomers have attempted to find an equation akin to Bode’s Law for these four moons but to no avail. The ratios of the spacings of the moons just do not correlate with the relative spacings of the planets.
The beauty and mystery of Saturn’s rings have captivated astronomers ever since Galileo first observed them through a telescope in 1610. In the beginning there were many theories as to what the rings were made of. With much better scientific instruments we now know that the rings are composed of icy gravel and small planetesimals in orbit around the gas giant.
It is interesting to ask the question Why is there not a moon in orbit where the rings are located? After all the raw material is there. It all comes down to something called the “Roche limit.” It says that inside a certain radius of orbit, a moon cannot form because of tidal forces which act to break it apart. Truth be known, all of the giant planets have rings inside their Roche limit but for some reason none of them are as pronounced as Saturn’s rings.
We return to the original question: Is Bode’s Law the chicken or the egg? Bode’s Law, which seems to be pulled out of thin air, gives us a form to work with. This equation/algorithm suggests that we will find spacings that go as 2n in the outer Solar System and that it works in the inner Solar System as well. There are definite interrelations between the spacings of the planets – they are much more than giant rocks randomly flying through space. There is order and a scheme, a cosmic dance of the planets as some romantics like to say.
What about extra-solar planets and their respective orbital spacings? Can we expect to see agreement between the observed and the 2n planetary spacing of Jupiter-like planets in other solar systems? Right now, there are more than 5,600 confirmed extrasolar planets orbiting other stars. Most of these have masses akin to those of Jupiter, Uranus/Neptune, or a new intermediate category of super-Earths. In systems with several exoplanets, the orbital spacings have yet to be codified in any meaningful way. Perhaps we will find analogues to Bode’s Law, or perhaps we will find a fascinating diversity of orbital spacings that defy one simple algorithm. Meanwhile, the ever-increasing number of exoplanetary systems being discovered should soon help us to resolve the possibilities.
William Stride – December, 2024