I’ve been seeing Saturn and Jupiter coming closer to each other in the sky lately. Jupiter passes by Saturn every 19.6 years, and it’s called a great conjunction. But I just learned that on December 21st they’ll look closer than they have since March 1226! They’ll be just 0.1 degrees apart: 6.1 arcminutes, to be precise. That’s less than a fifth of the Moon’s apparent width.

Here’s the expected view from New York on December 16th, 45 minutes after sunset, when there will also be a crescent Moon:

Jupiter and Saturn were even closer on July 17, 1623—just 5.2 arcminutes apart—but the glare from the the Sun made them invisible from Earth. There will be another close great conjunction on March 15, 2080. Jupiter and Saturn will be just 6.0 arcminutes apart then! If you’re young, maybe you can see that one. Not me.

On February 16, 7541, Jupiter will actually pass in front of part of Saturn! This called a transit. But if you can wait that long, you might as well wait for June 17, 7541, when Jupiter will completely block the view of Saturn. This is called an occultation.

So yes, Jupiter passes by Saturn more than once that year! In fact it’ll do it three times: this is called a triple conjunction. Because the Earth moves around the Sun much faster than Jupiter or Saturn, these planets sometimes seem to move backwards in the sky, and thanks to this, there are some great conjunctions where Jupiter and Saturn come close to each other in the sky three times in rapid succession, like in 1682–1683:

You can have a lot of fun reading this. Since Jupiter and Saturn are in a 5:2 orbital resonance—that is, Jupiter orbits the Sun 5 times in the time it takes Saturn to go around twice—the great conjunctions are not random. Instead, they follow interesting patterns!

Puzzle. Why are triple conjunctions more common than double conjunctions?

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I know you’re probably not into astrology too much, but this great conjunction is quite rare and also signifies “high science”. Galileo, Newton, Heisenberg and Fermi had this conjunction. Newton in particular wasn’t born so far from the winter solstice, so let’s hope this conjunction will bring another first class scientist as them :)

Why are triple conjunctions more common than double conjunctions

The inner planet has an angular velocity which is faster than the outer planet. Note that the black curve starts out lower and ends up higher.

So the curves CROSS each other an odd number of times.

The only way a double conjunction can happen is if one of the conjunctions involves the two curves approaching, kissing, than then retreating. This counts as a conjunction, but is not really a cross.

Right! It’s theoretically possible to have a double conjunction; the probability of a perfect double conjunction is zero, but if Saturn and Jupiter cross by each other twice in very rapid succession that’s one conjunction for all practical purposes, so we can get a double conjunction that way… or another way, where once Saturn and Jupiter don’t quite cross each other, but come so close it counts as a conjunction for all practical purposes.

All of this can be thought of as math involving the space of pairs of cubic curves , and how they can intersect. Pairs with a single intersection form an open set Pairs with a triple intersection form an open set But the set

I believe that I know the answer to the puzzle, but rather than answer it outright, I’ll just give this hint: In these conjunctions, Jupiter seems always to be overtaking Saturn. That makes sense; Jupiter travels around faster than Saturn, so from time to time, it will pass Saturn, and we’re just considering when exactly that happens, as seen from our sightly off-centre perspective. (From here we can go on to intersection theory, the difference between tangents and secants, and the derivative test for double roots of a polynomial, but I’ll stop for now.)

What you say is true in the main, and will be so this year, as both Jupiter and Saturn are far from their respective oppositions. These happened a month or so apart last summer. However near opposition superior planets have a period of apparent retrograde movement due to the parallax effect of Earth’s faster motion. This can result in so-called triple conjunctions of the same pair of planets within a few months. With Jupiter and Saturn this happened last in 1980-81 and will next in 2223. Source:

The “great conjunction” of 1980-81 signified some pretty low science, of the type “all the planets are lined up ==> earthquakes ==> end of the world as we know it”.

Ps. I see John already knows all this. The answer to the puzzle is, if you win a race from behind but the other guy was briefly ahead, you have to overtake, be overtaken, overtake again. Or more cycles but it’s always an odd number.

There’s probably a refinement to this. Was John’s “more common” (triple than double) the mathematical “more” (including “double never happens”) or the common person’s “more” (double happens but not that often) ?

If Jupiter and Saturn were point particles, then double would almost never happen (by the mathematicians’ meaning of ‘almost never’, which means 0% of the time). This is what my comment about tangents is about. But since they have some angular width, double should be possible. But that doesn’t mean that it ever happens in practice.

Toby created some nice pictures answering the Puzzle. Let me show them here, though it’s better to click on these pictures and go to his pages!

When studying some basic aspects of great conjunctions, we might as well be studying the space of pairs of curves in the plane where is a cubic polynomial. There’s an open set of pairs that intersect in 3 points:

These are triple conjunctions.

There’s also a set of pairs that intersect in 2 points:

These are double conjunctions. But this set has empty interior, so double conjunctions are rare.

John was too kind to mention, the case of a real cubic crossing the x-axis zero times actually never happens.

It spurs the reflection that according to Bezout’s theorem, two cubics (in the broader sense i.e f(x,y)=0, g(x,y)=0 where f and g are bivariate polynomials of order 3) cross 3×3=9 times. Some of the crossings might be imaginary, some might be points at infinity, some might be repeated roots. A well-known example is two circles. These can have at most two real crossings but also share two imaginary points at infinity. In homogeneous coordinates these are (1,i,0) and (1,-i,0).

So: if we approximate a planet’s apparent path in a small part of the sky as a cubic curve, do 9 associated conjunctions ever occur?

I don’t know. But you’re making me wonder if there’s anything interesting that comes of letting time be complex, analytically continuing the planet’s paths to functions defined on some subset of the complex plane, and considering “complex conjunctions”—i.e. conjunctions that happen at complex times?

Letting time be complex is a big deal in statistical mechanics and quantum field theory, but I haven’t thought about “complex celestial mechanics”. Probably someone has!

Yesterday, 21 December 2021, Lisa and I sat out in the driveway, drinking wine and eating chili almonds while admiring the Great Conjunction. Here’s a picture I took of it with an ordinary camera at 7:22 Pacific Time:

The elliptical shape of Saturn is actually due to the rings of Saturn, though it’s hard to tell.

Here’s a much better picture:

The #GreatConjunction of #Jupiter and #Saturn thru my telescope just after 6pm. 4 of Jupiter's moons; Europa, Ganymede, Io & Callisto, and Saturn's Titan moon visible. Stacked many images for more clarity and color. Nexstar Celestron 6SE with Nikon D750 attached. #scwx#ncwxpic.twitter.com/vzP2IAuFnS

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I know you’re probably not into astrology too much, but this great conjunction is quite rare and also signifies “high science”. Galileo, Newton, Heisenberg and Fermi had this conjunction. Newton in particular wasn’t born so far from the winter solstice, so let’s hope this conjunction will bring another first class scientist as them :)

Looks like Earth in the centre goes fast and Saturn outside slow, essentially one has to wait until Jupiter at medium speed catches up with Saturn.

The inner planet has an angular velocity which is faster than the outer planet. Note that the black curve starts out lower and ends up higher.

So the curves CROSS each other an odd number of times.

The only way a double conjunction can happen is if one of the conjunctions involves the two curves approaching, kissing, than then retreating. This counts as a conjunction, but is not really a cross.

Right! It’s theoretically possible to have a double conjunction; the probability of a

perfectdouble conjunction is zero, but if Saturn and Jupiter cross by each other twice in very rapid succession that’s one conjunction for all practical purposes, so we can get a double conjunction that way… or another way, where once Saturn and Jupiter don’tquitecross each other, but come so close it counts as a conjunction for all practical purposes.All of this can be thought of as math involving the space of pairs of cubic curves , and how they can intersect. Pairs with a single intersection form an open set Pairs with a triple intersection form an open set But the set

contains pairs that intersect twice.

I believe that I know the answer to the puzzle, but rather than answer it outright, I’ll just give this hint: In these conjunctions, Jupiter seems always to be overtaking Saturn. That makes sense; Jupiter travels around faster than Saturn, so from time to time, it will pass Saturn, and we’re just considering when exactly that happens, as seen from our sightly off-centre perspective. (From here we can go on to intersection theory, the difference between tangents and secants, and the derivative test for double roots of a polynomial, but I’ll stop for now.)

What you say is true in the main, and will be so this year, as both Jupiter and Saturn are far from their respective oppositions. These happened a month or so apart last summer. However near opposition superior planets have a period of apparent retrograde movement due to the parallax effect of Earth’s faster motion. This can result in so-called triple conjunctions of the same pair of planets within a few months. With Jupiter and Saturn this happened last in 1980-81 and will next in 2223. Source:

https://en.wikipedia.org/wiki/Triple_conjunction

The “great conjunction” of 1980-81 signified some pretty low science, of the type “all the planets are lined up ==> earthquakes ==> end of the world as we know it”.

Hmm, maybe the Betelgeuse dimming was a portent

Ps. I see John already knows all this. The answer to the puzzle is, if you win a race from behind but the other guy was briefly ahead, you have to overtake, be overtaken, overtake again. Or more cycles but it’s always an odd number.

There’s probably a refinement to this. Was John’s “more common” (triple than double) the mathematical “more” (including “double never happens”) or the common person’s “more” (double happens but not that often) ?

If Jupiter and Saturn were point particles, then double would almost never happen (by the mathematicians’ meaning of ‘almost never’, which means 0% of the time). This is what my comment about tangents is about. But since they have some angular width, double should be possible. But that doesn’t mean that it ever happens in practice.

Here’s a graph that I drew based on the picture of the 1682/1683 triple conjunction: https://www.desmos.com/calculator/4tppmmwndy

And here’s a similar graph showing a potential double conjunction: https://www.desmos.com/calculator/efu3lzzxfr

The curves in the double conjunction are tangent.

That’s great Toby. They look like real cubics, they

usuallycross the x-axis 3 times, 1 or none. Tangent case — rareToby created some nice pictures answering the Puzzle. Let me show them here, though it’s better to click on these pictures and go to his pages!

When studying some basic aspects of great conjunctions, we might as well be studying the space of pairs of curves in the plane where is a cubic polynomial. There’s an open set of pairs that intersect in 3 points:

These are triple conjunctions.

There’s also a set of pairs that intersect in 2 points:

These are double conjunctions. But

thisset has empty interior, so double conjunctions are rare.John was too kind to mention, the case of a real cubic crossing the x-axis zero times actually never happens.

It spurs the reflection that according to Bezout’s theorem, two cubics (in the broader sense i.e f(x,y)=0, g(x,y)=0 where f and g are bivariate polynomials of order 3) cross 3×3=9 times. Some of the crossings might be imaginary, some might be points at infinity, some might be repeated roots. A well-known example is two circles. These can have at most two real crossings but also share two imaginary points at infinity. In homogeneous coordinates these are (1,i,0) and (1,-i,0).

So: if we approximate a planet’s apparent path in a small part of the sky as a cubic curve, do 9 associated conjunctions ever occur?

I don’t know. But you’re making me wonder if there’s anything interesting that comes of letting time be complex, analytically continuing the planet’s paths to functions defined on some subset of the complex plane, and considering “complex conjunctions”—i.e. conjunctions that happen at complex times?

Letting time be complex is a big deal in statistical mechanics and quantum field theory, but I haven’t thought about “complex celestial mechanics”. Probably someone has!

The Great Conjunction looks really great tonight!

Yesterday, 21 December 2021, Lisa and I sat out in the driveway, drinking wine and eating chili almonds while admiring the Great Conjunction. Here’s a picture I took of it with an ordinary camera at 7:22 Pacific Time:

The elliptical shape of Saturn is actually due to the rings of Saturn, though it’s hard to tell.

Here’s a much better picture: