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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Inside the minds of the Isis murderers

As pressure on the terror group who claimed responsiblity for the Manchester attack intensifies, the threat to Britain will only become more acute.

The police and security services had consistently warned that a significant terrorist attack in Britain was inevitable. Yet no warning could have prepared us for the horror of the suicide attack on the Manchester Arena on Monday night. Twenty-two people were killed and at least 60 were wounded as they were leaving a concert by Ariana Grande in what was the most deadly attack in Britain since the London bombings of 7 July 2005, in which 56 people died.

Like the London bombers, the Manchester suicide attacker, Salman Ramadan Abedi, was British. He was 22, lived in Manchester and studied business management at Salford University before dropping out. He worshipped at Didsbury Mosque. The son of Libyans, Abedi is said to have returned recently from a visit to the North African country, where Islamic State has a foothold.

Ariana Grande is a former children’s TV star who made her name on channels such as Nickelodeon. Her fan base is overwhelmingly young and female, and many of those killed or wounded were children, including Saffie Rose Roussos, an eight-year-old girl from Leyland, Lancashire.

Islamic State inevitably claimed responsibility for the massacre, dismissing the victims as “crusaders”, “polytheists” and “worshippers of the cross”. This is not the first time Islamist terrorists have targeted children.

A Chechen jihadist group calling itself ­Riyad-us Saliheen (meaning “Gardens of the Righteous”) took more than 1,100 hostages, including 777 children, in a school siege in Beslan, Russia, in September 2004. In the event, more than 330 were massacred, including 186 children. Gunmen from the Pakistani Taliban also stormed a school in 2014, killing 148.

For terrorist actors, these are neither whimsical nor irrational acts. Contemporary jihadist movements have curated a broad and expansive intellectual ecosystem that rationalises and directs their actions. What they want is to create an asymmetry of fear by employing indiscriminate barbarism to intimidate and subdue their opponents into submission.

We have grown accustomed to a wave of terrorist attacks being carried out in the name of the self-styled Islamic State ever since the group’s official spokesman Abu Muhammad al-Adnani began prioritising them in 2014. (He was killed in an American air strike on Aleppo province in Syria in August last year.)

The US-led coalition against Islamic State has weakened the terror group in its former strongholds of Mosul in Iraq and Raqqa in Syria. In response, IS has been forced to concentrate more on what it calls “external operations” – by which it means inspiring its sympathisers and operatives to carry out attacks on Western countries. Indeed, al-Adnani encouraged the group’s supporters not to migrate towards IS-held territory but rather to focus their efforts on attacks in their home countries.

“The tiniest action you do in the heart of their [Western] land is dearer to us than the biggest action by us,” he said in an audio statement released last year. “There are no innocents in the heart of the lands of the crusaders.”

Islamic State refers to its strategy as “just terror”. Its framing places culpability for attacks on Western states on these nations themselves by claiming that IS actions are a response to aggression or assault. That much has been outlined in the group’s literature. “When will the crusaders end their hostilities towards Islam and the Muslims? . . . When will they recognise that the solution to their pathetic turmoil is right before their blinded eyes?” the militants ask in the IS magazine Dabiq. “Until then, the just terror will continue to strike them to the core of their deadened hearts.”

IS offered a rationale of this sort as justification for its bombing of a Russian commercial aircraft – Metrojet Flight 9268, travelling from Sharm el-Sheikh in Egypt to St Petersburg. That attack in October 2015 killed 224. Similar reasoning was offered for the attacks in Paris the following month in which 137 people were killed, in a series of co-ordinated, commando-style gun and bomb outrages across the city.

“Revenge was exacted upon those who felt safe,” IS declared in Dabiq. “Let the world know that we are living today in a new era. Whoever was heedless must now be alert. Whoever was sleeping must now awaken . . . The [caliphate] will take revenge for any aggression against its religion and people, sooner rather than later. Let the ­arrogant know that the skies and the lands are Allah’s.”


Through my academic research at King’s College London, I have ­interviewed scores of Westerners who became foreign fighters in Syria and Iraq to quiz them about their motives. Last year, one man from High Wycombe who had joined IS told me that it wanted to attack British targets in response to the vote in the House of Commons to extend British air strikes against IS targets to include sites in Syria (the British had only been targeting the group in Iraq until that point). “Do they [the British government] expect us to sit back and do nothing? ­Idiots,” he said.

In this respect, IS frames its attacks as acts of “revenge” and predicates its response on the Islamic principle of qisas, which is comparable to lex talionis or the doctrine of “an eye for an eye”. Qisas was always intended to be a tool of private redress for an individual or his/her family to seek justice in matters relating to bodily harm. Typically, it relates to cases of murder and manslaughter, or acts involving physical mutilation (say, leading to loss of limbs). The principle creates a framework for retributive justice.

The contemporary Salafi-jihadi movement has adopted a particularly innovative approach to the concept of qisas in two ways. First, groups such as IS have taken the idea and construed it in a way that justifies indiscriminate terrorism, such as the attack in Manchester. They argue that qisas has a political dimension and that it can be applied to international affairs in a way that holds civilians responsible for the perceived crimes of their governments.

Second, qisas is normally applied only in cases where the aggressor is known. IS, by contrast, holds every citizen-stranger of an enemy state responsible for the actions of his or her government. Thus, when it released its statement claiming responsibility for the Manchester attack, it said that it had struck against a “gathering of the crusaders . . . in response to their transgressions against the lands of the Muslims”.

It is this militaristic construction of qisas that allows IS to rationalise the bombing of a venue where large numbers of young girls had gathered to watch a pop concert, dismissing them as “crusaders”.

This is not new. In 1997, Osama Bin Laden told CBS News that “all Americans are our enemies, not just the ones who fight us directly, but also the ones who pay their ­taxes”. His rationale was that all Americans, by virtue of citizenship alone, are vicariously liable for the actions of their government.

Just a few years later, Bin Laden used the same idea to justify the 11 September 2001 attacks and also invoked it in reference to the Israeli-Palestinian conflict. “The blood pouring out of Palestine must be equally revenged,” he wrote. “You must know that the Palestinians do not cry alone; their women are not widowed alone; their sons are not orphaned alone.”

IS used the concept most dramatically in January 2015, when it burned alive a Royal Jordanian Air Force pilot, Muath al-Kasasbeh, whose plane had crashed in its territory. A video of the killing was circulated on the internet and social media. The group claimed his bombing raids had killed civilians and that it wanted to punish him with “equal retaliation”, in keeping with qisas.

What is well known about al-Kasasbeh’s murder is that he was burned alive inside a cage – but that is not the whole story. To understand how IS tethered this to the principle of qisas, it is the end of the gruesome video that is invested with most significance. After al-Kasasbeh has died, a truck emerges and dumps rubble over the cage. It was claimed this was debris from a site he had bombed, thus completing the “equal retaliation” of returning like for like. The idea was that IS had retaliated using the two principal forms in which a missile attack kills – by fire or debris.


The Manchester attack came on the fourth anniversary of the brutal murder of Fusilier Lee Rigby in Woolwich, south London. Rigby was killed by Michael Adebolajo and Michael Adebowale in the middle of the afternoon on a street outside a military barracks. That attack was in keeping with a pattern we have become increasingly accustomed to in Europe: an unsophisticated plot that employs ordinary, everyday items – a car, say, or a knife.

The consequences of such attacks have been seen across Europe, most notably in Nice on 14 July 2016, when 86 people were killed during Bastille Day celebrations after a jihadist drove a truck into crowds on the promenade. Similar attacks followed in Berlin, Westminster and Stockholm.

The security services find that these murderous attacks are extremely hard to disrupt because they typically involve lone actors who can mobilise quickly and with discretion. The Manchester attack was different. Explosives were used, which means the plot was inherently more sophisticated, requiring careful planning and preparation.

We know that two of the 7/7 bombers had previously trained in Pakistan’s lawless tribal regions, where they honed their skills. In other plots, such as the connected attacks in London and Glasgow Airport of 2007, the explosive devices failed mainly because the bomb-makers had found it difficult to travel abroad and develop their skills in safe environments. Whatever Abedi’s connections, the long war in Syria and Iraq has once again created a permissive environment for terrorist training and attack planning.

The devastating impact of this has already been felt across Europe. Since the Syrian uprising began in 2011, more than 800 Britons are believed to have travelled there to fight. From Europe as a whole, the figure is over 5,000, of which a significant number are believed to have joined IS. Of the British contingent, the security services estimate that about half have returned or become disengaged from the conflict. Of those who remained, a hundred are believed to be active, the rest having been killed.

It is improbable that Abedi acted alone in Manchester or that this plot had no international component. Indeed, he was already known to the authorities (and had returned recently from Libya). As pressure on IS intensifies across Syria and Iraq, the threat to Britain will only become more acute as the group’s sympathisers prepare for what they consider to be a fightback.

This speaks to the scale of the threat facing Britain, and Europe more generally. Our police and security services have been stretched and continuously tested in recent years. Just recently, in March, the Metropolitan Police assistant commissioner Mark Rowley told Radio 4’s Today programme that 13 plots had been thwarted since Lee Rigby’s murder in 2013. Put another way, the police have disrupted terrorist plots every four months for the past four years.

Naturally, Islamic State is not the only threat. On 13 May, one of Osama Bin Laden’s sons, Hamza, released a video, titled “Advice for martyrdom-seekers in the West”, on behalf of al-Qaeda. Hamza, 27, who was his father’s favoured successor to lead the group, called on its supporters to concentrate on attacks in the West rather than migrating to conflict zones in the Middle East and beyond. Scenes of previous ­terrorist attacks in Britain played throughout the video.

The central leadership of al-Qaeda is increasingly looking for opportunities to reassert itself after being eclipsed by Islamic State and losing control of its affiliates in Syria. It needs attacks and a cause in the West with which to revive itself. Hamza therefore cited the January 2015 Charlie Hebdo attack in Paris as a critical example, calling for the assassination of anyone deemed to have “insulted” Islam.

The Charlie Hebdo attack was especially important for al-Qaeda because it enabled the group to transcend the fratricidal conflicts that frequently define relations between the various jihadist groups. In Syria, for instance, al-Qaeda’s affiliates (when it had better control over them) and Islamic State have been in open war with each other.

Yet, the Charlie Hebdo attack brought warm praise from the group’s Islamist rivals because none of them wanted to appear ­unsupportive of an atrocity that had, as the terrorists proclaimed, “avenged” the Prophet Muhammad’s honour.

The British man from High Wycombe who joined IS told me the group had welcomed the attack for precisely those reasons. It was something that, in his view, had confirmed the “nobility” of the attackers, even if they had not been members of IS.

Is it too late for the West to save itself, I asked him. What if the West simply accepted all of Islamic State’s demands: would that provide respite?

The answer was as emphatic as it was stark: “We primarily fight wars due to ppl [sic] being disbelievers. Their drones against us are a secondary issue.”

He went on: “Their kufr [disbelief] against Allah is sufficient of a reason for us to invade and kill them. Only if they stop their kufr will they no longer be a target.”

In other words, we are all guilty, and we are all legitimate targets.

Shiraz Maher is a contributing writer for the New Statesman and a senior research fellow at King’s College London’s International Centre for the Study of Radicalisation.

This article first appeared in the 25 May 2017 issue of the New Statesman, Why Islamic State targets Britain

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