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Hello great conjunction, my old friend

Jupiter, and three of its moons, left, and Saturn, right, are seen in the sky, Tuesday, Dec. 22, 2020, above Chicago. The two planets are in their closest observable alignment since 1226, appearing in the alignment known as the ''Great Conjunction,'' which has also been called the ''Christmas Star.'' (AP Photo/Charles Rex Arbogast) (AP)Premium
Jupiter, and three of its moons, left, and Saturn, right, are seen in the sky, Tuesday, Dec. 22, 2020, above Chicago. The two planets are in their closest observable alignment since 1226, appearing in the alignment known as the ''Great Conjunction,'' which has also been called the ''Christmas Star.'' (AP Photo/Charles Rex Arbogast) (AP)

Saturn, Jupiter seemed so close in the sky on 21 Dec but the two planets didn’t actually overlap

This was the week of the Great Conjunction; I’m sure you know. On the evening of 21 December, Jupiter and Saturn appeared so close together in the sky that they looked like one bright “star". Given that it was only four days before today, several reports couldn’t resist labelling it a “Christmas Star".

But what, really, was this conjunction? It’s a simple phenomenon, but to make it clearer, here’s something else that happened as I stared at the two planets that evening: a plane that had just taken off from Mumbai flew past. So for a fleeting moment, Saturn, Jupiter and a red light on the plane were aligned—or nearly so—in the sky. That is, they were in conjunction.

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Strictly, the positions of the two planets in the sky didn’t actually coincide. They were very close but still separated just enough to discern even without binoculars or a telescope. And of course, even this was just a visual coincidence. The two planets came nowhere near one another, just like the plane was nowhere close to either of them.

In fact, there’s no way Jupiter and Saturn, traipsing along in their very different orbits around the Sun, could have drawn closer together. In reality, on average Jupiter is about 890 million km from us, and Saturn almost twice that distance, at 1.6 billion km.

It’s as if you’re sitting somewhere beside Mumbai’s suburban rail tracks, looking across them. To your left, you see a train approaching on the farthest track. A couple of seconds later, you see a second train approaching on a closer track. When you first see them, they are separated—the first train is “ahead" of the second one. But a few seconds later, the second has caught up and, as they pass directly in front of you, you can see only one train.

But of course, it’s not just one train passing you by. It’s not just one planet in that conjunction.

Yet there’s something intriguing going on here. Sitting on the side of the tracks watching the trains roll by, you will no doubt do a sort of quick, intuitive calculation as the two trains approach. That’s how you deduce that they will “merge" into one when they pass in front of you. You may even realize that the second, closer, train is not necessarily travelling faster than the further one. It might actually be slower, but it seems faster because it’s that much closer to you.

Again, think of that plane that was briefly conjoined with the planets. In reality, it is moving almost laughably slowly compared to them. And yet it zips past, while you can’t discern any movement in the planets.

The sort of intuition that underlies all this may be easy, you’d think, when the distances are relatively small: the trains, that plane. By comparison, Jupiter and Saturn are unimaginably far from us. How do scientists calculate with astonishing precision when this conjunction will happen?

And they have been doing this for thousands of years. The ancient Greeks, for example, were fascinated by the movements of celestial bodies. In 1900, divers found a shipwreck dating from about 100 BC, off the coast of the Greek island Antikythera. Among various treasures they lifted out of the ancient ship was a most intriguing machine—or really, the remnants of one. This so-called “Antikythera mechanism" likely had 30 gears, the largest with over 200 teeth. The Greeks must have used it to calculate and predict planetary motions, and even celestial conjunctions, years into the future.

In more recent times, computers and mathematics do the job the Antikythera mechanism did. Will you try walking through the process with me here?

With years of close observation, we have a pretty good idea of how Jupiter and Saturn—all the planets in our solar system, in fact—move through space. Take Jupiter. On average, it is about 780 million km from the Sun. “Average" because its orbit is not a perfect circle—but it’s close enough that we can use that number as the radius of the orbit. So the length of the orbit is—calling on that formula for the circumference of a circle that you remember from school— 2πr (r=780 million), or about 4.9 billion km.

Jupiter takes nearly 12 years to travel that distance. Dividing, we find it’s moving at about 408 million km/year, or 13 km/second. Compare to the plane, which crawls along at about 0.2 km/s, some 65 times slower than Jupiter. (Aside: Since the orbit is a circle, it is sometimes convenient to note and use its angular speed instead. It covers the 360 degrees of the orbit in 12 years. That’s 30 degrees/year.)

We can do the same calculations for Saturn and the Earth (why the Earth? I’m coming to that). Saturn is some 1.43 billion km from the Sun, and it completes one orbit in 29 years. That translates to a speed of nearly 10km/s. And our Earth, in comparison to these huge tortoises of our Solar System, is a veritable hare: we’re soaring through the Solar System at 30km/s. So since we know how fast each planet is moving and the path it takes, we can actually write equations to describe their motions.

At any given moment, these three planets form a gigantic triangle in space. We can calculate the area of the triangle and express it in terms of the planets’ motions. Now when all three form a straight line, that area is zero. But it’s precisely when the Earth, Jupiter and Saturn are in a straight line that Jupiter and Saturn will be in conjunction, as seen from the Earth. So the question we must answer, then, is: given the motion of each point on this triangle, when will its area become zero, collapsing the triangle into a straight line? (Now you know why we need to know how the Earth moves.)

Answering that question is a matter of setting the right constraints and then solving the equations “simultaneously"—in essence, the kind of thing college students learn to do in an early Linear Algebra course. True, it is a little more complex with these planets. For one thing, for reasons that include the effect of other celestial objects and the planes of each individual orbit, the equations are not likely to describe the planets’ motions precisely. For another, they don’t have to line up perfectly for us to recognize the sight as conjunction. Remember that this week they never did line up perfectly. Saturn and Jupiter seemed so close in the sky on 21 December as to appear as one bright “star", but Jupiter never actually overlapped Saturn. The straight line, you see, doesn’t have to be ramrod straight.

So when solving the equations, we have to account for such “near misses", knowing that we on Earth will still see them as actual conjunctions. Still, that’s just one more constraint we have to work with while solving our simultaneous equations.

Naturally, we can project such calculations into the past too. This is how we know that there was a conjunction of Jupiter and Saturn about 2000 years ago. In fact, we know there was even brighter conjunction about then too—of Jupiter and Venus.

So is there any basis in reality to the legend about an earlier Christmas Star, a bright one about 2,000 years ago that heralded Jesus’s birth? I’ll leave that for you to wonder about.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun

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