The Great Conjunction

Near the end of December 2020, I saw Jupiter and Saturn very close in the sky just after sunset. I didn’t know this was called a great conjunction. The next one will happen in November 2040. And it will happen in a very different part of the sky: close to 120° away.

This is how it always works. People have known this for millennia. They just forgot to teach me about it in school. The time between great conjunctions is always roughly 20 years, and if you keep track of them, each one is roughly 120° to the east of the last. Thus, they trace out enormous equilateral triangles in the sky.

Almost equilateral. These triangles drift slightly over time! This picture, drawn by Johannes Kepler in 1606, shows how it works:

After three great conjunctions, 60 years later, we get back to a great conjunction in almost the same place in the sky, but it’s shifted east by roughly 7¼ degrees.

Kepler was not just an astronomer: he earned money working as an astrologer. Astrologers divide the sky into 12 zones called houses of the zodiac. These 12 houses are divided into 4 groups of 3 called triplicities. Successive great conjunctions usually loop around between 3 corners of a triplicity—but due to the gradual drifting they eventually move on to the next triplicity.

Astrologers connected these triplicities to the 4 classical elements:

• Fire — Aries, Leo, Sagittarius — hot, dry • Earth — Taurus, Virgo, Capricorn — cold, dry • Air — Gemini, Libra, Aquarius — hot, wet • Water — Cancer, Scorpio, Pisces — cold, wet

Yeah, now things are getting weird. They were taking solid facts and running too far with them, kinda like string theorists.

Anyway: the great conjunctions stay within one triplicity for about 260 years… but then they drift to the next triplicity. This event is called a greater conjunction, and naturally astrologers thought it’s a big deal.

(They still do.)

Here’s a nice picture of the triplicities. The 12 houses of the zodiac are arbitrary human conventions, as is their connection to the 4 classical elements (earth, fire, water and air). But the triangles have some basis in reality, since the great conjunctions of Saturn and Jupiter approximately trace out equilateral triangles.

The actual triangles formed by great conjunctions drift away from this idealized pattern, as I noted. But if you’re a mathematician, you can probably feel the charm of this setup, and see why people liked it!

People look for patterns, and we tend to hope that simple patterns are ‘the truth’. If we make a single assumption not adequately grounded by observation, like that the motions of the planets affect human affairs, or that every elementary particle has a superpartner we haven’t seen yet, we can build a beautiful framework based which may have little to do with reality.

Since ancient times, ancient astrologers actually knew about the gradual drift of the triangles formed by great conjunctions. And they realized these triangles would eventually come back to same place in the sky!

In fact, based on section 39d of Plato’s Timaeus, some thought that after some long period of time all the planets would come back to the exact same positions. This was called the Great Year.

A late 4th-century Neoplatonist named Nemesius got quite excited by this idea. In his De natura hominis, he wrote:

The Stoics say that the planets, returning to the same point of longitude and latitude which each occupied when first the universe arose, at fixed periods of time bring about a conflagration and destruction of things; and they say the universe reverts anew to the same condition, and that as the stars again move in the same way everything that took place in the former period is exactly reproduced. Socrates, they say, and Plato, will again exist, and every single man, with the same friends and countrymen; the same things will happen to them, they will meet with the same fortune, and deal with the same things.

My hero the mathematician Nicole Oresme argued against this ‘eternal recurrence of the same’ by pointing out it could only happen if all the planet’s orbital periods were rational multiples of each other, which is very unlikely. I would like to learn the details of his argument. He almost seems to have intuited that rational numbers are a set of measure zero!

But as the historian J. D. North wrote, the subtle mathematical arguments of Oresme had about as much effect on astrologers as Zeno’s arguments had on archers. Only slightly more impactful was Étienne Tempier, the Bishop of Paris, who in his famous Condemnation of 1277 rejected 219 propositions, the sixth being

That when all the celestial bodies return to the same point, which happens every 36,000 years, the same effects will recur as now.

For him, an eternal recurrence of endless Jesus Christs would have been repugnant.

The figure of 36,000 years was just one of many proposed as the length of the Great Year. Some astrologers thought the triangle formed by three successive great conjunctions rotates a full turn every 2400 years. If so, this would happen 15 times every Great Year.

But we’ll see that figure of 2400 years is a bit off. I get something closer to 2650 years.

The math

Why does the triangle formed by great conjunctions rotate a full turn in the sky every 2650 years? For that matter, why is there one great conjunction roughly every 20 years? Let’s see if we can work this stuff out. It turns out we only need two numbers to do it:

• how long it takes for Jupiter to orbit the Sun (about 12 years),

and

• how long it takes for Saturn to orbit the Sun (about 29 years).

But we need to know these numbers much more precisely! In fact the orbital period of Jupiter is 4332.59 days, while that of Saturn is 10759.22 days. So, Jupiter is moving around the Sun at a rate of

1/4332.59 orbits per day

while Saturn is moving more slowly, at

1/10759.22 orbits per day

Thus, relative to Saturn, Jupiter is moving around at

(1/4332.59 – 1/10759.22) orbits per day

Thus, Jupiter makes a full orbit relative to Saturn, coming back to the same location relative to Saturn, every

1/(1/4332.59 – 1/10759.22) days

This idea works for any pair of planets, and it’s called the synodic period formula. So now we just calculate! It takes

1/(1/4332.59 – 1/10759.22) days ≈ 7253.46 days                                                   ≈ 19 years, 313 days and 17 hours

for Jupiter to make a complete orbit relative to Saturn, coming back to the same place relative to Saturn. So this is the time between great conjunctions: somewhat less than 20 years.

How many orbits around the Sun does Jupiter make during this time? About

7253.46/4332.59 ≈ 1.67416

This is close to 1⅔. Nobody knows why. It means there’s a near-resonance between the orbits of Jupiter and Saturn—but their orbits don’t seem to be locked into this resonance by some physical effect, so most astronomers think it’s a coincidence.

Since ⅔ of 360° is 240°, and the planets are moving east, meaning counterclockwise when viewed looking down from far above the Earth’s north pole, each great conjunction is displaced roughly 240° counterclockwise from the previous one—or in other words, 120° clockwise. If you’re confused, look at Kepler’s picture!

But 1.67416 is not exactly 1⅔. The difference is

1.67416 – 1⅔ ≈ 0.00750

of an orbit. In terms of degrees, this is

0.00750 × 360° ≈ 2.7°

After three great conjunctions we get another one in almost the same place, but it’s shifted by

3 × 2.7° ≈ 8.1°

Hmm, this doesn’t match the figure of 7¼ that I quoted earlier. I got that, and a lot of this fascinating material, from a wonderful essay called ‘Astrology and the fortunes of churches’ in this book:

• J. D. North, Stars, Minds and Fate: Essays in Ancient and Medieval Cosmology, The Hambledon Press, London, 1989.

I don’t know why the figures don’t match.

Anyway, I am trying to figure out how many great conjunctions are required for the near-equilateral triangle in Kepler’s picture to turn all the way around. If it really makes 0.00750 of a full turn every 7253.46 days, as I’ve calculated, then it makes a full turn after

(7253.46 / 0.00750) days ≈ 96700 days ≈ 2650 years

This actually matches pretty well the estimate of 2600 years given by the great Persian astrologer Abū Ma‘shar al-Balkhi, who worked in the Abbasid court in Baghdad starting around 830 AD. It was his works, under the Latinized name of Albumasar, that brought detailed knowledge of the great conjunction to Europe.

So, I think I did okay. I have ignored various subtleties. Most importantly, the timing of great conjunctions as seen from Earth, rather than from the Sun, are complicated by the motion of the Earth around the Sun, which significantly affects what we see. Instead of happening at equally spaced intervals of 19 years, 313 days and 17 hours, the duration between great conjunctions as seen here ranges from roughly 18 years 10 months to 20 years 8 months.

You can see a listing of all great conjunctions from 1200 AD to 2400 AD here. Kepler witnessed one in December of 1603. He theorized that the Star of Bethlehem was a great conjunction, and computed that one occurred in 7 BC.

For more, see:

• D. V. Etz, Conjunctions of Jupiter and Saturn, Journal of the Royal Astronomical Society of Canada 94 (2000), 174–178.

Here is a movie of the December 21, 2020 great conjunction taken by ProtoSlav:

Click on this picture and others to see where I got them.