A triangle of meetings that turns in eight centuries

On 21 December 2020 Jupiter and Saturn stood six arcminutes apart in the evening sky, closer than they had appeared since 1623. Conjunctions of the two planets are not rare — they happen every twenty years — but where they happen is not random, and the pattern they make over centuries was one of the first structures in the sky to be drawn as a geometric figure rather than listed as a sequence of dates.

Each meeting falls about 243° further round the sky than the previous one. Three meetings take the pair 728° round, which is two full turns plus eight degrees, so every third conjunction lands almost where the first did. The conjunctions sit at the corners of a triangle, and the triangle creeps round by eight degrees every sixty years.

The loop a planet traces in the sky is the difference between two orbits seen from one of them. A conjunction pattern is the same difference seen from the Sun, sampled once per lap, and what it samples is how nearly the two periods fit a ratio of small whole numbers.

The arithmetic of a meeting #

Two bodies moving round the Sun at mean angular rates $n_J$ and $n_S$ meet when the faster has gained a full turn on the slower, so the interval between conjunctions is

$$S = \frac{1}{1/P_J - 1/P_S}$$

With Jupiter’s period of 11.86 years and Saturn’s of 29.45, that is 19.86 years. In that time Saturn moves $360° \times S/P_S = 242.8°$ round its orbit, and the conjunction happens at Saturn’s new position.

A figure appears if some small number of those advances adds up to nearly a whole number of turns. Three advances make 728.5°, which is two turns and 8.5°. So the conjunctions visit three regions of the sky in rotation, a hundred and twenty degrees apart, and the whole pattern drifts forward by 8.5° every three conjunctions. At that rate it takes 42 sets of three — about 2,500 years — to go right round, and a third of that, 838 years, to bring the triangle back to looking the same, since a triangle with identical corners repeats itself after a third of a turn.

Kepler drew exactly this triangle in 1596, as a sequence of chords inscribed in a circle, and noticed that the circle inscribed within the triangle had a radius close to half the circle’s — near the ratio of Jupiter’s orbit to Saturn’s. That observation led him to the nested regular solids of his first book, which were wrong. The triangle itself was right, and it had a better explanation.

A chronology built on the drift #

The triangle’s slow turning was put to use long before anybody could explain it. Medieval astrologers in the Islamic world and later in Europe divided the zodiac into four triplicities — sets of three signs a hundred and twenty degrees apart, each associated with an element — and noticed that successive great conjunctions stay within one triplicity for a long time before moving to the next.

The numbers follow from the drift. Each sign is thirty degrees wide, the triangle turns 8.5° every 59.6 years, so the conjunctions take about 210 years to cross the width of a sign and move on to the next triplicity. Four triplicities make a cycle of about 840 years before the conjunctions return to the first — which is the triangle’s third of a turn, the 838 years of the figure. Historical chronologies were built on this cycle, with the passage from one triplicity to the next treated as a marker of historical change. The astrology was not physics, but the arithmetic underneath it was the arithmetic of the 5:2 near-commensurability, measured from centuries of observation by people who had no idea it had a dynamical meaning.

Why three, and why so slowly #

The drift is small because the number three is not an accident of the two periods. Three conjunctions take 59.6 years. In 59.6 years Jupiter completes 5.02 orbits and Saturn 2.02. Five of Jupiter’s years are 59.31 years and two of Saturn’s are 58.90: the periods nearly fit a ratio of five to two, and a near-fit of periods is a near-return of positions.

The period of the slow angle $2\lambda_J - 5\lambda_S$ is

$$T = \frac{1}{\left|2/P_J - 5/P_S\right|} = 838 \text{ years}$$

and it is the same number as the triangle’s third of a turn, because the two are the same statement. The triangle’s corners are where $\lambda_J = \lambda_S$; three conjunctions later the corners have moved by an amount proportional to how far $2\lambda_J - 5\lambda_S$ has moved in the meantime. A figure that turns slowly is a slow angle made visible.

The ordering of combinations by period is the useful part of the second figure. It says that among all the ways of combining the two planets’ positions with small integers, one combination is nearly stationary, and every conjunction pattern, every recurrence of configuration and every slowly accumulating gravitational effect between the two planets is dominated by it.

The same number as a perturbation #

That is where the triangle stops being a curiosity. Two planets tug on each other continually. Most of those tugs depend on angles that go round in a few decades, so they average away: a push one way is followed a few years later by a push the other way, and the orbit oscillates slightly about its mean. A tug that depends on a nearly stationary angle does not average away on any short timescale. It keeps pushing the same way for centuries.

Its effect on the longitude is larger still, because a small change in orbital energy changes the mean motion, and the change in mean motion accumulates into a change in position for as long as the push persists. The displacement goes as the perturbation divided by the square of the slow frequency. A frequency ten times slower than the others, squared, makes an effect a hundred times larger than its raw size would suggest.

By the late seventeenth century, comparisons of ancient observations with modern ones showed Jupiter apparently speeding up and Saturn slowing down. Halley tabulated the effect and treated it as a steady acceleration. If it were steady, Jupiter would eventually fall into the Sun and Saturn escape the solar system, and Newton’s law would imply a solar system that was not stable. Euler and Lagrange tried to explain it with the gravitational interaction of the two planets and could not.

Laplace, in 1785, found the explanation in the 5:2 near-commensurability. The terms in the mutual perturbation that depend on $2\lambda_J - 5\lambda_S$ are of third order in the eccentricities and inclinations, which makes them small in amplitude, and they have a period of about nine hundred years, which makes their effect on the longitude enormous. Jupiter runs ahead of and behind its mean position by about twenty arcminutes and Saturn by nearly fifty, in opposite senses, over that cycle. What Halley had seen was one leg of a nine-hundred-year oscillation. The solar system was not falling apart; it was near a resonance.

The 838 years drawn here and the “about nine hundred” of the Great Inequality are the same quantity measured two ways. The figure uses the mean motions from a modern fit; the Great Inequality’s period is affected by the very perturbation it describes, since the planets’ mean motions are themselves slightly shifted by it, and published values run from about 880 to 920 years depending on which terms are included. The agreement to within ten per cent between a pattern of dates and a term in celestial mechanics is the point.

The same slow angle round other stars #

The Great Inequality has a modern counterpart that is measured routinely, and it is measured through exactly the quantity the triangle makes visible.

When two planets round another star both transit, each transit is a clock tick, and a second planet near a commensurability makes the first planet’s ticks run early and late. Planets have been found from a transit running late in hundreds of systems this way. The timing variations are not random: they oscillate with a period set by how far the pair is from exact commensurability,

$$P_{\rm super} = \frac{1}{\left| j/P_2 - (j-1)/P_1 \right|}$$

for a pair near $j{:}(j-1)$, which is the period of the slow angle — the same formula as the Great Inequality’s, applied to a first-order pair rather than a third-order one. A pair of planets with periods of 10.0 and 20.4 days, two per cent outside 2:1, has a super-period of 510 days, and its transit times swing back and forth over that interval by minutes to hours.

The difference in scale is instructive. Jupiter and Saturn are a third-order near-commensurability with eccentricities of five per cent, so the perturbation is small in amplitude and shows itself only because its period is nine centuries. Compact exoplanet systems are often first-order near-commensurabilities, whose perturbation terms are first order in eccentricity and correspondingly strong, and their slow angles turn in a year or two. The physics is identical; what differs is the order of the combination and the patience of the observer. Halley needed two thousand years of records to see one leg of the oscillation. A transit survey sees several full cycles in its lifetime, and a chain of such pairs records not only masses but how the chain was assembled.

A pentagram in the evening sky #

The same arithmetic applied to Venus and the Earth produces a figure that has been noticed for far longer than the Jupiter–Saturn triangle, because Venus’s meetings with the Sun are the most conspicuous events in the planetary sky.

Five synodic periods of Venus take 7.99 years; in that time the Earth goes round eight times and Venus thirteen. Venus therefore returns to nearly the same place in the sky, at nearly the same date, every eight years — the regularity the Maya built into their Venus tables, where five Venus cycles were matched against eight solar years. The five conjunction points, joined in order, trace a pentagram, because each advance of 215.5° is a little less than three-fifths of a turn: the conjunctions skip round the pentagon two corners at a time.

The near-fit is closer than Jupiter and Saturn’s, and of higher order. Thirteen against eight is order five, so any gravitational perturbation tied to it is of fifth order in the small quantities — eccentricities of 0.007 and 0.017 — and correspondingly weak. There is such a term in the motions of Venus and the Earth, with a period of about 240 years, and it is one of the larger terms in precise ephemerides of the inner planets; but it displaces the planets by seconds of arc rather than minutes. The pattern in the sky and the perturbation have the same source, and the perturbation’s size is set by the order rather than by the closeness of the fit.

The same eight-year recurrence governs the transits of Venus across the Sun’s disc, which come in pairs eight years apart — 2004 and 2012 — when a conjunction falls close enough to one of the nodes where Venus’s orbit crosses the ecliptic. Between pairs the gap is more than a century, because the conjunction points drift past the nodes at the pentagram’s slow rate, and the full pattern of transit pairs repeats every 243 years — a closer approximation to the same ratio, 395 orbits of Venus against 243 of the Earth, with the nodes’ positions included.

The 2020 meeting, and what it could not show #

The closeness of a particular conjunction on the sky is a separate matter from its longitude. Jupiter’s orbit is inclined 1.3° to the ecliptic and Saturn’s 2.5°, and their nodes are in different places, so two planets at the same longitude can be separated by up to a few degrees in latitude. A very close conjunction needs the meeting to fall near where the two orbit planes intersect. The heliocentric figure has no information about that, and a conjunction’s geocentric date and separation also depend on where the Earth is, which shifts the apparent meeting by up to several months and can make the planets pass each other three times in one apparition when the conjunction falls near opposition.

The meetings of 1623 and 2020 were both close for the same reason — a conjunction near the nodal line — and the 397 years between them is twenty synodic periods, during which the triangle turned nearly 57°. The next equally close meeting, in 2080, is three conjunctions later again.

The resonances that did capture #

A near-commensurability is a resonance that has not closed. Jupiter and Saturn are not locked at 5:2; their slow angle circulates, turning once every eight or nine centuries rather than oscillating about a fixed value. Elsewhere in the solar system, angles like it are locked. Pluto’s orbital period is exactly three-halves of Neptune’s, and the angle $3\lambda_P - 2\lambda_N - \varpi_P$ librates about 180°, so Pluto never comes close to Neptune even though their orbits cross. Io, Europa and Ganymede are locked in a 4:2:1 chain whose combined angle has not circulated in the history of the Jovian system, and Saturn’s moons show how a resonance can clear a gap in a ring and lock a moon with the same mathematics at a thousand times the speed.

The difference between a pattern that turns slowly and one that holds still is capture, and capture is a direction, not a strength: two bodies whose periods converge through a commensurability can be caught in it, and two that diverge through it cannot. Jupiter and Saturn are thought to have crossed commensurabilities during the early migration of the giant planets, and whether they were ever held in one is part of the argument about how the outer solar system reached its present shape. What remains now is a near-miss, visible as a triangle that turns and as a nine-hundred-year wobble in the longitudes of both planets.

What the figures leave out #

The conjunction longitudes come from Keplerian ellipses turning at mean rates, which is to say from a solar system in which the Great Inequality does not exist. The real conjunction times depart from those drawn by up to several days because of it, and more over the centuries-long spans. The heliocentric view also discards everything about what is seen: apparent separations, geocentric dates, and the retrograde loops that turn a single conjunction into a triple one.

The commensurability search is a purely kinematic ranking. It says which angle is slowest; it does not compute the amplitude of any perturbation tied to that angle, which depends on masses, eccentricities, inclinations and the order of the combination in ways the figure only gestures at by printing the order.

A pattern of recurring events is a slow frequency drawn on the sky #

A figure traced by repeated events is a slow frequency made spatial. Wherever two periodic motions are sampled at their mutual recurrence — conjunctions, eclipses, oppositions, calendars — the pattern of samples drifts at the rate of the slowest integer combination of the two frequencies, and that rate is the most important number describing how the two motions interact. A calendar is a fraction chosen near that combination; an eclipse cycle is three frequencies nearly commensurate at once; a planetary resonance is the same near-fit with gravity attached.

The half-month asymmetry that defeated Aristarchus was also a beat, between the Moon’s motion and a small geometric offset. It failed because the Moon’s own irregularity was twenty times the signal. The Jupiter–Saturn triangle succeeds because the signal it carries — a slow angle — is the very thing that also drives the largest irregularity, so the irregularity is not noise on the measurement but a second measurement of the same number.

Still open: when the brightest configuration is not the nearest #

The conjunction pattern samples where two bodies meet. What an observer on the Earth sees at each point of the cycle is a different question, and for an inner planet it has an answer that the geometry alone does not predict. Venus is nearest at inferior conjunction and fully lit at superior conjunction, and it is brightest at neither: it peaks as a crescent, weeks from inferior conjunction, where the loss of its lit face and the gain from its approach balance. Whether that balance has an interior winner at all depends on the size of the orbit — and where Venus’s real peak falls, compared with where a matte planet’s would, is a measurement of what its clouds do to light.