Fans across the globe are now counting the days and minutes until the new Star Wars film hits the big screen. Professor Ian Stewart, a mathematician from the University of Warwick, is a self-confessed Star Wars enthusiast, but he does have one intergalactic-axe to grind. Here he explains – with maths – why one of the most iconic chases in space couldn’t happen.
I’ve been a fan of Star Wars ever since my family watched the first movie of the series in a tiny cinema in Connecticut in 1983. It worked because it took science fiction seriously; the special effects were brilliant, and it put in all the little details that made everything seem real, like the skeleton of a monster lying in the desert and X-wing fighters that looked like they’d actually been in a battle. Computer-Generated Imagery was in its infancy then. Star Wars was one of the pioneers — but it was still pretty good.
However, one thing has always bugged me: Asteroid belts.
I understand why George Lucas wanted to create excitement, and there’s no question that a rebel freighter fighting a running battle with an Imperial Star Cruiser, threading its precarious way through an ever-changing maze of gigantic spinning rocks, ticks the box. I’m also open to the view that some asteroid ‘belts’, somewhere in the universe, might even be like that (though probably not, to be honest). However, our asteroid belt, the only one we actually know about, is not a blizzard of tumbling rocks.
Yet whenever movies, or the TV, show asteroids belts, the rocks are crammed together like the M6 in rush hour. Even science programmes do it. But, it can’t possibly be like that. Why do we get it wrong? I reckon it’s all down to a badly chosen metaphor: Belt."
Call the Celestial Police
Once upon a time the solar system didn’t have a belt. Instead, it had a gap. According to a mathematical pattern called the Bode-Titius law, there ought to have been a planet between Mars and Jupiter. But there wasn’t. If there had been one, the ancients would have seen it long ago and associated yet another god with it. When the more distant planet Uranus was discovered, it neatly extended the mathematical pattern. This inspired astronomers to try to fill the gap between Mars and Jupiter. A group known as the Celestial Police (in German) shared out slices of the solar system to watch for planetary interlopers. But, wouldn’t you just know it, while they were sorting out a systematic search an Italian named Giuseppe Piazzi beat them all to it – by accident. He wasn’t even looking for a planet; he was on the trail of a star. When he noticed a different star that crawled across the sky from night to night, he realised he’d found a new planet. He named it Ceres after the Roman harvest goddess.
Ceres turned out to be too small to be a genuine planet. Then it was discovered that lots of smaller bodies also orbited in the same gap. Soon there were dozens, then hundreds. Today we know there are about 150 million of them more than 100 metres across. Initially they were called minor planets, but when William Herschel suggested calling them asteroids, the name stuck. It was a bit silly because ‘aster’ refers to a star, which they’re not. By 1850 there were so many of them that Elise Otté, in her translation of Alexander von Humboldt’s Cosmos, wrote that some meteorite showers “probably form part of a belt of asteroids” – and so the phrase asteroid belt was coined.
This is true. Diagrams of the asteroids show an awful lot of dots jammed into a small ring-shaped region, pretty much bumping into each other. But even the largest asteroid, Ceres, would be totally invisible if its dot were to scale. The asteroid belt spans an awful lot of space, but contains relatively few asteroids. A quick back-of-the-envelope calculation bears this out. The asteroid belt lies between 320 million and 480 million kilometres from the Sun. Like the planets, most asteroids orbit close to the same plane, the ecliptic. The total area of the ecliptic occupied by the asteroid belt is about 400 quadrillion square kilometres. If you share that among 150 million rocks, the typical distance between neighbouring asteroids turns out to be 30,000 kilometres. Roughly two and a half times the diameter of the Earth.
If you were in the asteroid belt, you wouldn’t see hundreds of rocks charging around. You probably wouldn’t see anything."
One thing Star Wars got right, mind you, is all that tumbling. In the films it’s too fast, of course, because that makes it all even more exciting, but only a few of the biggest asteroids are spheres, spinning regularly about an axis. Most asteroids are shaped like potatoes, and they tumble. Chaos theory predicts that if they’re sufficiently irregular, they tumble unpredictably. Another irregularly shaped body, Saturn’s moon Hyperion, has been observed over long periods, and that prediction turns out to be right.
There are all sorts of wonderful mathematical patterns in asteroids. Their orbits tend to concentrate near some distances from the Sun, while avoiding others. You don’t notice this in diagrams, because the orbits aren’t perfect circles. But statistically, asteroids ‘like’ some distances and ‘dislike’ others, as Daniel Kirkwood noticed in 1866. He even suggested the reason: Jupiter. It’s the biggest planet in the solar system, and its gravitational field tugs at the asteroids. The most obvious Kirkwood gap corresponds to an orbital distance at which the asteroid makes three revolutions of the Sun for every one made by Jupiter, a so-called 3:1 resonance. This repeated alignment makes the long-term effect of Jupiter’s gravity stronger. Other resonances explain most of the gaps and clumps.
In 1772 Joseph-Louis Lagrange proved that if a planet circles the Sun, a small body can stay at rest in their combined gravitational field in precisely five different places. Three of these ‘Lagrange points’ lie on the line joining planet and Sun. The other two are more interesting: they lie close to the planet’s orbit, 60 degrees ahead of it and 60 degrees behind it. It turns out that the asteroids know this, ands there are two ‘clumps’ around those points, further out than the main asteroid belt. Johann Palisa suggested naming these asteroids after participants in the Trojan war. Nearly all of the Greeks are 60 degrees ahead of Jupiter, and most Trojans are 60 degrees behind it. About 3,900 Greeks and 2,000 Trojans are known. Both of these Lagrange points are known as Trojan points — the Greeks don’t get any credit.
Closely associated with Jupiter’s Trojans and Greeks is another fascinating family of asteroids, the Hildas. These orbit the Sun three times while Jupiter goes round twice, a 3:2 resonance. They circulate slowly relative to the Trojans and Jupiter along an equilateral triangle that rotates at the same rate as Jupiter. Their orbits suggest that Jupiter initially formed about 10 per cent further from the Sun than it is now, and then migrated inwards.
So, unfortunately, our real life asteroids are too sedate to provide a stunning backdrop for a small band of courageous freedom fighters to weave skillfully through a blizzard of tumbling rocks, seeking refuge from a fleet of warships firing bolts of pure energy. What they can do is teach us a lot of fascinating maths.
8 Dec 2017
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