Philae comes in to land on 67P/Churyumov-Gerasimenko. It reached the comet using carefully calculated forces of attraction. Image: 2014 European Space Agency/Getty
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Wandering in the heavens: how mathematics explains Saturn’s rings

Ian Stewart shows how maths is changing cosmology, and explains why the best way to reach a comet near Mars is to go round the back of the sun.

The Enūma Anu Enlil, a series of 70 clay tablets, was found in the ruins of King Ashurbanipal’s library in Nineveh (on the eastern bank of the River Tigris, opposite modern-day Mosul in Iraq). The name means “in the days of Anu and Enlil”; Anu was the sky god, Enlil the wind god. The tablets, which date as far back as 1950BC, list 7,000 omens from Babylonian astrology: “If the moon can be seen on the first day, the land will be happy.” But tablet 63 is different: it gives the times when Venus first became visible, or disappeared, over a 21-year period. This Venus tablet of Ammisaduqa is the earliest known record of planetary observations.

The Babylonians were expert astronomers who produced star catalogues and tables of eclipses, planetary motion and changes in the length of day. They were also capable mathematicians, with a number system much like ours, but using base 60 rather than ten. They could solve quadratic equations and calculate the diagonal of a square with precision, and they applied their mathematical skills to the heavens. In those days, mathematics and astronomy were part and parcel of astrology and religion, and the whole package was intimately bound up with agriculture through the progression of the seasons.

The torch of astronomy passed by way of ancient Greece to India. In 6th-century India, mathematics was a sub-branch of astronomy, and astronomy still played second fiddle to reading omens in the stars. The Arab world made further advances in our understanding of the cosmos, and kept the ancient knowledge alive until Europe once more turned its attentions to the science of the heavens.

In 1601 Johannes Kepler became imperial mathematician to the Holy Roman emperor Rudolf II. Casting the emperor’s horoscope paid the bills, and it also left time for serious mathematics and astronomy. Kepler had inherited accurate observations of Mars from his former master Tycho Brahe, and from these he extracted three mathematical patterns, his laws of planetary motion. By then, thanks to Nicolaus Copernicus, it was known – though still controversial, to say the least – that the planets revolve round the sun, not the Earth. Their orbits were thought to be combinations of circles, but Kepler’s calculations showed that planets move in ellipses. His other two laws govern how quickly the planet moves and how long it takes to go round the sun.

In his epic Mathematical Principles of Natural Philosophy of 1687, Isaac Newton built on Kepler’s laws and deduced his law of universal gravitation: every body in the universe attracts every other body with a force that obeys a specific mathematical rule. These forces determine how moons, planets and stars move. Newton’s book paved the way to a rational scientific understanding of nature based on precise mathematical laws, and opened up the metaphor of the clockwork universe.

One of the great tests of Newtonian gravitation was Edmond Halley’s prediction about a comet. In ancient times comets, bright bodies with long curved tails that seemed to appear from nowhere, were seen as omens of disaster. From old records, Halley realised one particular comet was a repeat visitor, with an elliptical orbit that took it near the Earth every 76 years. He predicted its next return in 1758. By then Halley was dead, but his prediction proved correct.

Even today, Newton’s law remains vital to astronomy and space exploration; Einstein’s later refinements are seldom needed. A topical example concerns another comet, rejoicing in the name 67P/Churyumov-Gerasimenko, which takes about six and a half years to orbit the sun. In 2004 the European Space Agency (ESA) launched the Rosetta probe to visit the comet and find out what it looked like and what it was made of. Famously, it resembled a rubber duck: two round lumps joined by a narrow neck. On 12 November 2014 a small capsule, Philae, landed on the head of the duck, which was 480 million kilometres from Earth and travelling at over 50,000 kilometres per hour. Unfortunately Philae bounced and ended up on its side, but even so it had sent back vital and unprecedented data.

It’s worth visiting the ESA’s “Where is Rosetta?” web page to see an animation of the astonishing route the probe took. It wasn’t direct. The probe began by moving towards the sun, even though the comet was far outside the orbit of Mars, and moving away. Rosetta’s orbit swung past the sun, returned close to the Earth, and was flung outwards to an encounter with Mars. It then swung back to meet the Earth for a second time, then back beyond Mars’s orbit. By now the comet was on the far side of the sun and closer to it than Rosetta was. A third encounter with Earth flung the probe outwards again, chasing the comet as it now sped away from the sun. Finally, Rosetta made its rendezvous with destiny.

Why such a complicated route? The ESA didn’t just point its rocket at the comet and blast off. That would have required far too much fuel, and by the time it got there the comet would have been somewhere else. Instead, Rosetta performed a carefully choreographed cosmic dance, tugged by the combined gravitational forces of the sun, the Earth, Mars and other relevant bodies. Its route was designed for fuel efficiency; the price paid was that it took Rosetta ten years to get to its destination. Each close fly-by with Earth and Mars gave the probe a free boost as it borrowed energy from the planet. An occasional small burst from four thrusters kept the craft on track. And every kilometre of the trip was governed by Newton’s law of gravity.

Complex trajectories such as this one have now become standard in many unmanned space missions. They originated in mathematical studies of chaotic dynamics in the motion of three gravitating bodies, and go back to pioneering work by Edward Belbruno at the Jet Propulsion Laboratory in California in 1990. He realised that these techniques could put a Japanese probe, Hiten, into lunar orbit after a failure of its parent craft, even though there was hardly any fuel available.

Mathematics has always enjoyed a close relationship with astronomy; not just in the technology of space missions but in understanding planets, stars, galaxies – even the entire universe. How, for example, did the solar system form? We can’t go back to take a look, so we have to do some celestial archaeology, inferring what happened from the evidence that remains. Our main tool is mathematical modelling, which lets us test whether hypothetical scenarios make sense.

When Galileo first spied Saturn in 1610, he took it to be a trinity of planets. Image: Nasa/Eyevine

Observations and theoretical astrophysics tell us that the sun came into being about 4.8 billion years ago, and the planets of the solar system formed at much the same time. Everything condensed out of the solar nebula, a huge cloud of gas – mainly hydrogen and helium, the two commonest elements in the universe. The cloud was about 65 light years across, 15 times the distance to the nearest star today. One fragment, about four light years across, gave rise to the solar system; other fragments became other stars – many of which, we now know, have their own planets. As our fragment collapsed under its own gravitational field, most of the gas collected at the centre, where enormous pressures ignited nuclear reactions to create the sun. Much of the remaining gas clumped into smaller, but still gigantic, bodies: the planets. The rest either got swept away or remains as various items of clutter – asteroids; centaurs (small bodies with characteristics of both comets and asteroids); Kuiper Belt objects, in the debris field beyond Neptune; comets in the Oort Cloud, which is a quarter of the way to the next-nearest star.

This scenario, minus the nuclear physics, was first proposed in the 18th century, but fell out of favour in the 20th because it seemed not to account for the sun’s low angular momentum (a measure of how much rotation it has, taking into account its mass and speed) compared to that of the planets. But in the 1980s astronomers observed gas clouds round young stars, and mathematical modelling of the collapsing clouds showed plausible, and very dramatic, mechanisms that fitted the observations.

According to these ideas, the early solar system was very different from the sedate one we see today. The planets formed not as single clumps, but by a chaotic process of accretion. For the first 100,000 years, slowly growing “planetesimals” swept up gas and dust, and created circular rings in the nebula by clearing out gaps between them. Each gap was littered with millions of these tiny bodies. At that point the planetesimals ran out of new matter to sweep up, but there were so many of them that they kept bumping into each other. Some broke up, but others merged; the mergers won and planets built up, piece by tiny piece.

Late in 2014 dramatic evidence for this process was found: an image of a proto-planetary disc around the young star HL Tau, 450 light years away in the Taurus
constellation. This image showed concentric bright rings of gas, with dark rings in between. The dark rings are almost cer­tainly caused by nascent planets sweeping up dust and gas.

Until very recently, astronomers thought that once the solar system came into being it was very stable: the planets trundled ponderously along preordained orbits and nothing much changed. No longer: it is now thought that the larger worlds – the gas giants Jupiter and Saturn and the ice giants Uranus and Neptune – first appeared outside the “frost line” where water freezes, but subsequently reorganised each other in a lengthy gravitational tug of war.

In the early solar system, the giants were closer together and millions of planetesimals roamed the outer regions. Today the order of the giants, outwards from the sun, is Jupiter, Saturn, Uranus, Neptune. But in one likely scenario it was originally Jupiter, Neptune, Uranus, Saturn. When the solar system was about 600 million years old, this cosy arrangement came to an end. All of the planets’ orbital periods were slowly changing, and Jupiter and Saturn wandered into a 2:1 resonance – Saturn’s “year” became twice that of Jupiter. Repeated alignments of these two worlds then pushed Neptune and Uranus outwards, with Neptune overtaking Uranus. This disturbed the planetesimals, making them fall towards the sun. Chaos erupted in the solar system as planetesimals played celestial pinball among the planets. The giant planets moved out, and the planetesimals moved in. Eventually the planetesimals took on Jupiter, whose huge mass was decisive. Some were flung out of the solar system altogether, while the rest went into long, thin orbits stretching out to vast distances. After that, it mostly settled down.

These theories are not idle speculation. They are supported by huge computer calculations of the solar system’s dynamics over billions of years, carried out in particular by the research groups of Jack Wisdom of the Massachusetts Institute of Technology and Jacques Laskar of CNRS, the French national centre for scientific research. Some cunning mathematics is required even to set up these simulations: the deep structure of the laws of motion must not be disturbed by the unavoidable numerical approximations that occur. This structure includes the laws of conservation of energy and angular momentum, whose totals cannot change. Amazingly, the planetary migrations not only keep these quantities in balance, but happen because they balance.

Another playground for mathematicians and astronomers investigating Newtonian gravitation is the rings of Saturn. The most distant of the planets known to the ancients, Saturn is about 1.3 billion kilometres from Earth. In 1610, when Galileo looked at Saturn through his telescope, he sent his fellows a Latin anagram: smaismrmilmepoetaleumibunenugttauiras. This was a standard way to preannounce a discovery without giving it away. Kepler deciphered it as reading – in translation – “Be greeted, double knob, offspring of Mars,” and thought Galileo was claiming Mars had two moons (as Kepler had predicted, and rightly so). But Galileo later explained that his anagram actually meant: “I have observed the most distant of planets to have a triple form.” That is, Saturn consists of three bodies.

So much for anagrams.

Galileo’s image of the planet was blurred. Using a better telescope, the Dutch mathematician Christiaan Huygens realised that the middle body was the planet and the others were parts of a gigantic system of rings. Mathematics proves – contrary to an early suggestion by the French scholar Pierre-Simon Laplace – that the rings cannot be solid. In fact, they are made up of ice particles, ranging in size from fine dust to lumps ten metres across. There are several current theories for the rings’ formation: the break-up of a moon, or perhaps leftovers from Saturn’s own primordial nebula. Mathematics is being used to try to find out which explanation, if any, is correct.

Mathematical studies also explain many puzzling features of Saturn’s rings. For one thing, the rings are dense in some regions, but so thin in others that at first sight there seem to be gaps. Some of these gaps come from resonances between the rings and the periods of Saturn’s 62 moons, which can systematically disturb gas in orbits related to that of the moon itself. Other gaps are organised by “shepherd moons” that hustle out any sheepish moonlet that strays into the gap. When the spacecraft Voyager 1 flew past in 1980, some rings appeared to be braided. We now know that they are kinked and lumpy, another subtle consequence of Newtonian gravity in this complex system.

Mathematics has illuminated many other cosmic puzzles: the formation of Earth’s moon, the future of the solar system, the formation and dynamics of galaxies – even the origin of the universe itself in the Big Bang. In ancient India, mathematics was a sub-branch of astronomy. Today, if anything, it is the other way round. Mathematicians are making discoveries and inventing methods; astronomers and cosmologists are making ever greater use of the latest mathematical tools and concepts to advance this utterly fascinating subject. Mathematical thinking teaches us more about humanity’s place in the universe. And it helps us to seek out new places.

Ian Stewart is an emeritus professor of mathematics at the University of Warwick

This article first appeared in the 19 December 2014 issue of the New Statesman, Christmas Issue 2014

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The Okay Place: the psychological value of mediocre TV

Why do we watch comedies that don’t make us laugh?

I’ve been watching Brooklyn 99 on the train. The comedy cop show makes me laugh roughly once an episode, but nonetheless I watch it compulsively. I watch it on my commute, and I watch it while cooking dinner. It’s in the background when I’m paying my bills. I consumed so many episodes last night, Netflix sent me its most notoriously judgemental pop-up: “Are you still watching?”

Yes, Netflix, I was still watching. The real question was: why?

Brooklyn 99 doesn’t really make me laugh, and it’s far from the most critically-acclaimed show available on the streaming service right now. It’s not technically mediocre – the sitcom has won two Golden Globes – but it is to me*. It provokes the same feelings in me as Netflix’s The Good Place, a kitsch sitcom set in the afterlife. I am compelled to watch at all costs, but on the whole unamused and occasionally frustrated by formulaic storylines. (Sometimes, The Good Place even makes me cringe.)

I enjoy both shows, sure, but I don’t love them. So why am I wasting my time?

(*Because this is the internet, it's a good time to specify that "mediocre" here means in the view of the person being quoted, not objectively.)

“To understand why people are drawn to certain shows, it’s helpful to look at the type of feelings the shows elicit,” says Elizabeth Cohen, a media psychologist and assistant professor at West Virginia University. Cohen says media often has a “mood management function”, in that we use it to make ourselves feel better.

“Sometimes we are looking to be emotionally stimulated, so we might choose to watch something that we think will thrill us,” she says. “But other times we might decide to forego the dark cerebral drama on our DVR and opt for a safe sitcom instead. That could be because we need something that will help us wind down, relax, and boost our mood.”

Photo: Netflix

A desire to unwind is one of the reasons Oliver Savory, a 30-year-old grad student from London, watches The Big Bang Theory, a comedy that has inspired much ire.

“It fills a niche of something to watch while eating, when you can’t focus fully, or you’ve just got in and want to unwind without thinking too hard,” he explains. Oliver says “average” TV comforts him more than “good” TV because he doesn’t have to worry about keeping up to date. “Good TV you have to make time for, average TV can fit around your own schedule without imposing itself.”

Cohen says this is referred to as “comfort food TV”, the entertainment equivalent of eating boxed mac and cheese even if you have the recipe for mum’s spaghetti. “These are shows that people watch not because they are exceptional in quality, but because they are simple, predictable, or nostalgic.”

Sometimes, we watch “okay” shows because we feel they have the potential to be great. Karen Dill-Shackleford is a media psychologist who explains this was her experience with The Good Place.

“I love The Good Place, but there was a stretch when I thought it was poor,” she says. “I kept waiting for it to right itself because I thought it had real potential.”

The potential many of us see in the show is its fresh premise, and its engagement with moral philosophy. As Dill-Shackleford puts it: “[the show] is a palatable way to ponder life’s biggest questions. So, even if the jokes are lame, the potential for real value is still there.”

Charlotte Mullin, a 23-year-old illustrator, says she doesn't laugh at the jokes either. “But what keeps me watching is the premise, and the characters. I’m a sucker for good character development, and The Good Place has it in spades,” she says. (Cohen tells me she does laugh at The Good Place, once again illustrating that mediocrity is in the eye of the beholder.)

Photo: Netflix

Ross McCafferty is a 27-year-old journalist from Glasgow who couldn’t tell you anything about NBC’s Parks and Recreation, even though he’s seen every episode. During a difficult time at work, he consumed the entire show.

“It’s actually quite a derivative, even mediocre show,” he says. “But I still ate it up, because at the time it was oddly comforting to me, self-contained and uncomplicated and unobtrusive, like so little in my life at that time.”

The reasons McCafferty liked the show, he says, is because it was “nice”, “brightly lit”, “nonthreatening” and “so sweet it was cloying”.

Bright lights and pretty colours certainly feel like one of the reasons I keep going back to mediocre sitcoms, but I also find comfort in certain characters: Chidi in The Good Place and Boyle in Brooklyn 99 are comfortingly familiar – I almost switch on to keep up to date with them, as if they were friends.

George Clarke is a 25-year-old management consultant who finds similar comfort in Seinfeld characters, even though the show doesn’t make him laugh much. “Some days I might fancy Netflix’s latest psychological thriller, but most of the time I’d just prefer to sit and watch Kramer doing something ridiculous or George stuff it up with the girl of his dreams for the fourth time that season,” he says.

But couldn’t Clarke and I find our televisual buds in prestige dramas?

“I find the idea of watching prestige shows non-stop to be exhausting,”  says David Renshaw, a 30-year-old news editor, who jokes it can feel like you “need a map” to keep up with Game of Thrones. When he finishes watching something acclaimed, such as Breaking Bad, he “cleanses the palette” with shows like Masterchef or Gogglebox. “They are much lower maintenance… especially if you’re switching between TV and phone as often as I do.”

Photo: Netflix

The comfort of the mediocre is so powerful that it can often override other emotions, such as the cringing I experience during some of The Good Place’s more strained jokes. Lizzie Roberts is a 25-year-old masters student who enjoys Gilmore Girls even though she dislikes the character Lorelai’s “painfully unfunny monologues”.

“It’s my way of letting my brain reset,” she says of the show, as well as reality TV such as Towie and I’m A Celeb. “It’s not taxing, it’s tolerable.”

“Not taxing and tolerable” are perhaps the words that best sum up the complex psychological reasons we continue to watch mediocre TV during the Golden Age of Television. Streaming services like Netflix are also designed to keep us watching, with episodes auto-playing one after the other (plus it's easier to find a show you've essentially already paid for on the Netflix homepage than go out and hunt for something better suited to your tastes).

Although watching mediocre TV can feel like a waste of time, it does seem to have a psychological purpose. When we're stressed, busy, or tired, it can be exactly the entertainment we need. Nothing is more stressful, busy, or tiring than a commute – so I'll be watching Brooklyn 99 on the train home.

Amelia Tait is a technology and digital culture writer at the New Statesman.

This article first appeared in the 19 December 2014 issue of the New Statesman, Christmas Issue 2014