The Science of Discworld Revised Edition

by Terry Pratchett

  Between 1600 and 1900 three theories of the formation of the Moon came into vogue and out again. One was that the Moon had formed at the same time as the Earth when the dustcloud condensed to form the solar system – Sun, planets, satellites, the whole ball of wax … or rock, anyway. This theory, like early theories of the solar system’s formation, falls foul of angular momentum. The Earth is spinning too fast, and the moon is revolving too fast, to be consistent with the Moon condensing from a dustcloud. (We misled you earlier when we said that the dustcloud theory explained the satellites too. Mostly it does, but not our enigmatic Moon. Lies-to-children, you see – now you’re ready for the next layer of complication.)

  Theory two was that the Moon is a piece of the Earth that broke away, maybe when the Earth was still completely molten and spinning rather fast. That theory bounced into the bin because nobody could find a plausible way for a spinning molten Earth to eject anything that would remotely resemble the Moon, even if you waited a bit for things to cool down.

  According to theory three, the Moon formed elsewhere in the solar system, and was wandering along when it happened to come within the Earth’s gravitational clutches and couldn’t get out again. This theory was very popular, even though gravitational capture is distinctly tricky to arrange. It’s a bit like trying to throw a golfball into the hole so that it goes round and round just inside the rim. What usually happens is that it falls to the bottom (collides with the Earth) or does what every golfer has experienced to their utter horror, and goes in for a split second before climbing back out again (escapes without being captured).

  The rock samples from Apollo missions added to the mystery of the Moon’s origins. In some respects, Moon rock is astonishingly similar to Earth rock. If they were similar in most respects, this would be evidence for a common origin, and we’d have to take another look at the theory that they both condensed from the same dustcloud. But Moon rock doesn’t resemble all Earth rock, only the mantle. The current theory, which dates from the early 1980s, is that the Moon was once part of the Earth’s mantle. It wasn’t ejected as a result of the Earth’s spin: it was knocked into space about four billion years ago when a giant body, about the size of Mars, struck the early Earth a glancing blow. Computer calculations show that such an impact can, if conditions are right, strip a large chunk of mantle from the Earth, and sort of smear it out into space. This takes about 13 minutes (aren’t computers good?). Then the ejected mantle, which is molten, begins to condense into a ring of rocks of various sizes. Some of it forms a big lump, the proto-Moon, and this quickly sweeps up most of the rest. What’s left doesn’t go away so easily, however, but over 100 million years nearly all of it crashes into either the Moon or the Earth, because of gravity.

  The first simulations to support this theory of the Moon’s formation suffered from some problems; in particular, they dated the impact very early in the Earth’s formation, in order to get the Moon’s angular momentum right. But if the collision had occurred that early, then the Moon would have accumulated a lot of iron from later impacts, just as the Earth did. However, there is little iron on (or in) the Moon. More recent work shows that a rather later impact could also give the Moon the correct angular momentum, and avoids this difficulty. However, it predicts that about 80% of the impacting body would have ended up on the Moon. In order for the Moon to resemble Earth’s mantle as closely as it does, the impacting body would also have had to resemble the Earth’s mantle.

  This line of theorising may be losing the plot, though, because it was that very resemblance that needed explanation in the first place, and gave rise to the giant impact theory of the Moon’s formation. Anything that explains why the impactor was like the mantle, such as ‘it formed at the same distance from the Sun as the Earth did,’ can probably explain why the Moon is like the mantle, without an impactor being needed. Maybe both the Moon and the Earth’s mantle were splashed off something else by an impact.

  Because Earth has weather – especially back then, oh boy, did it have weather then – the resulting impact craters all got eroded away; but because the Moon has no weather, the lunar impact craters didn’t get eroded away, and a lot of them are still there now. The great charm of this theory is that it explains many different features of the Moon in one go – its similarity to the Earth’s mantle, the fact that its surface seems to have undergone a sudden and extreme amount of heating about 4 billion years ago, its craters, its size, its spin – even those sea-like maria, released as the proto-Moon slowly cooled. The early solar system was a violent place.

  In fact, the Dean’s mis-designed sun might have done us some good after all …

  The Moon affects life on Earth in at least two or three ways that we know of, probably dozens more that we haven’t yet appreciated.

  The most obvious effect of the Moon on the Earth is the tides – a fact that the wizards are stumbling towards. Like most of science, the story of the tides is not entirely straightforward, and only loosely connected to what common sense, left to its own devices, would lead us to expect. The common sense bit is that the Moon’s gravity pulls at the Earth, and it pulls more strongly on the bit that is closest to the Moon. When that bit is land, nothing much happens, but when it’s water – and more than half our planet’s surface is ocean – it can pile up. This explanation is a lie-to-children, and it doesn’t agree with what actually happens. It leads us to expect that at any given place on Earth, high tide occurs when the Moon is overhead, or at least at its highest point in the sky. That would lead to one high tide every day – or, allowing for a little complexity in the Earth–Moon system, one high tide every 24 hours 50 minutes.

  Actually, high tides occur twice a day, 12 hours and 25 minutes apart. Exactly half the figure.

  Not only that: the pull of the Moon’s gravity at the surface of the Earth is only one ten millionth of the Earth’s surface gravity; the pull of the Sun is about half that. Even when combined together, these two forces are not strong enough to lift masses of water through heights of up to 70 feet (21 m) – the biggest tidal movement on Earth, occurring in the Bay of Fundy between Nova Scotia and New Brunswick.

  An acceptable explanation of the tides eluded humanity until Isaac Newton worked out the law of gravity and did the necessary calculations. His ideas have since been refined and improved, but he had the basics.

  For simplicity, ignore everything except the Earth and the Moon, and assume that the Earth is completely made of water. The watery Earth spins on its axis, so it is subjected to centrifugal force and bulges slightly at the equator. Two other forces act on it: the Earth’s gravity and the Moon’s. The shape that the water takes up in response to these forces depends on the fact that water is a fluid. In normal circumstances, the surface of a standing body of water is horizontal, because if it wasn’t, then the fluid on the higher bits would slosh sideways into the lower bits. The same kind of thing happens when there are extra forces acting: the surface of the water settles at right angles to the net direction of the combined forces.

  When you work out the details for the three forces we’ve just mentioned, you find that the water forms an ellipsoid, a shape that is close to a sphere but very slightly elongated. The direction of elongation points towards the Moon. However, the centre of the ellipsoid coincides with the centre of the Earth, so the water ‘piles up’ on the side furthest from the Moon as well as on the side nearest it. This change of shape is only partly caused by the Moon’s gravity ‘lifting’ the water closest to it. Most of the motion, in fact, is sideways rather than upwards. The sideways forces push more water into some regions of the oceans, and take it away from others. The total effect is tiny – the surface of the sea rises and falls through a distance of 18 inches (half a metre).

  The coast, where land meets sea, is what creates the big tidal movements. Most of the water is moving sideways (not up) and its motion is affected by the shape of the coastline. In some places the water flows into a narrowing funnel, and then it piles up much more t
han it does elsewhere. This is what happens in the Bay of Fundy. This effect is made even bigger because coastal waters are shallow, so the energy of the moving water gets concentrated into a thinner layer, creating bigger and faster movements.

  Finally, let’s put the sun back. This has the same kind of effect as the Moon, but smaller. When Sun and Moon are aligned – either both on the same side of the Earth, in which case we see a new moon, or both on opposite sides (full moon) – their gravitational pulls reinforce each other, leading to so-called ‘spring tides’ in which high tide is higher than normal and low tide is lower. These have nothing to do with the season Spring. When the Sun and Moon are at right angles as seen from Earth, at half moon, the Sun’s pull cancels out part of the Moon’s, leading to ‘neap tides’ with less movement than normal (these presumably have nothing to do with the season Neap …).

  By putting all these effects together, and keeping good records of past tides, it is possible to predict the times of high and low tide, and the amount of vertical movement, anywhere on Earth.

  There are similar tidal effects (large) on the Earth’s atmosphere, and (small) on the planet’s land masses. Tidal effects occur on other bodies in the solar system, and beyond. It is thought that Jupiter’s moon Io, whose surface is mostly sulphur and which has numerous active volcanoes, is heated by being ‘squeezed’ repeatedly by tidal effects from Jupiter.

  Another effect of the Moon on the Earth, discovered in the mid-’90s by Jaques Laskar, is to stabilize the Earth’s axis. The Earth spins like a top, and at any given moment there is a line running through the centre of the Earth around which everything else rotates. This is its axis. The Earth’s axis is tilted relative to the plane in which the Earth orbits the Sun, and this tilt is what causes the seasons. Sometimes the north pole is tilted closer to the sun than the south pole is, and six months later it’s the other way round. When the northern end of the axis is tilted towards the Sun, more sunlight falls on the northern half of the planet than on the southern half, so the north gets summer and the south gets winter. Six months later, when the axis points the other way relative to the sun, the reverse applies.

  Over longer periods of time, the axis changes direction. Just as a top wobbles when it spins, so does the Earth, and over 26,000 years its axis completes one full circle of wobble. At every stage, however, the axis is tilted at the same angle (23°) away from the perpendicular to the orbital plane. This motion is called precession, and it has a small effect on the timing of the seasons – they slowly shift by a total of one year in 26,000. Harmless, basically. However, the axes of most other planets do something far more drastic: they change their angle to the orbital plane. Mars, for example, probably changes this angle by 90° over a period of 10–20 million years. This has a dramatic effect on climate.

  Suppose that a planet’s axis is at right angles to the orbital plane. Then there are no seasonal variations at all, but everywhere except the poles there is a day/night cycle, with equal amounts of day and night. Now tilt the axis a little: seasonal variations appear, and the days are longer in summer and shorter in winter. Suppose that the axis tilts 90°, so that at some instant the north pole, say, points directly at the sun. Half a year later, the south pole points at the Sun. At either pole, there is a ‘day’ of half a year followed by a ‘night’ of half a year. The seasons coincide with the day/night cycle. Regions of the planet bake in high heat for half a year, then freeze for the other half. Although life can survive in such circumstances, it may be harder for it to get going in the first place, and it may be more vulnerable to extremes of climate, vulcanism, or meterorite impacts.

  The Earth’s axis can change its angle of tilt over very long periods of time, much longer than the 26,000 year cycle of precession, but even over hundreds of millions of years the angle doesn’t change much. Why? Because, as Laskar discovered when he did the calculations, the Moon helps keep the Earth’s axis steady. So it is at least conceivable that life on Earth owes quite a lot to the calming influence of its sister world, however much it may madden us individually.

  A third influence of the Moon was discovered in 1998: a clear association between tides and the rate of growth of trees. Ernst Zürcher and Maria-Giulia Cantiani measured the diameters of young spruce trees grown in containers kept at constant light levels. Over periods of several days the diameters changed in step with the tides. The scientists interpret this as an effect of the Moon’s gravity on the transport of water within the tree. It can’t be variations in moonlight, which would perhaps affect photosynthesis, because the trees were grown in darkness. But the effect may be similar to one that occurs with creatures that live on the seashore. Because they evolved to live there, they have to respond to the tides, and evolution sometimes achieves this by creating an internal dynamic that runs in step with the tides. If you remove the creatures to the laboratory, this internal dynamic makes them continue to ‘follow’ the tides.

  The Moon has been important in another way. The Babylonians and Greeks knew that the Moon is a sphere; the phases are obvious, and there is also a slight wobble which means that, over time, humans see rather more than one half of the Moon’s surface. There it was, hanging in the sky – a big ball, not a disc like the sun, and a hint that perhaps ‘big balls in space’ is a much better way of thinking about the Earth and its neighbours than ‘lights in the sky’.

  All this is a long way from lance-constable Angua – even a long way from the female menstrual cycle. But it shows how much we are creatures of the universe. Things Up There really do affect us Down Here, every day of our lives.

  1 Moreover, until the last few decades of human history, most women did not cycle. Nearly all the time, they were either pregnant or lactating. And for the great apes, the cycle is a week or so longer than for humans, and for gibbons it’s shorter. So it looks as though the relation with the Moon in coincidental.

  TWENTY-ONE

  THE LIGHT YOU SEE THE DARK BY

  THERE WAS NO Dark. This came as such a shock to Ponder Stibbons that he made HEX look again. There had to be Dark, surely? Otherwise, what was there for the light to show up against?

  Eventually, he reported this lack to the other wizards.

  ‘There should be lots of Dark and there isn’t,’ he said flatly. ‘There’s just Light and … no light. And it’s a pretty strange light, too.’

  ‘In what way?’ said the Archchancellor.

  ‘Well, sir, as you know,1 there’s ordinary light, which travels at about the same speed as sound …’

  ‘That’s right. You’ve only got to watch shadows across a landscape to realize that.’

  ‘Quite, sir … and then there’s meta-light, which doesn’t really travel at all because it is already everywhere.’

  ‘Otherwise we wouldn’t even be able to see darkness,’ said the Senior Wrangler.

  ‘Exactly. But the Project universe has just got the one sort of light. HEX thinks it moves at hundreds of thousands of miles a second.’

  ‘What use is that?’

  ‘Er … in this universe, that’s as fast as you can go.’

  ‘That’s nonsense, because –’ Ridcully began, but Ponder held up a hand. He had not been looking forward to this one.

  ‘Please, Archchancellor. It’s doing the best it can. Just trust me on this one. Please? Yes, I can see all the reasons why it’s impossible. But, in there, it seems to work. HEX has written pages of stuff about it, if anyone’s interested. Just don’t ask me about any of it. Please, gentlemen? It’s all supposed to be logical but you’ll find your brain squeaking around until the ends point out of your ears.’

  He placed his hands together and tried to look wise.

  ‘It really is almost as if the Project is aping the real universe –’

  ‘Ook.’

  ‘I beg your pardon,’ said Ponder. ‘A figure of speech.’

  The Librarian nodded at him and knuckled his way across the floor. The wizards watched him carefully.

  ‘You re
ally believe that that thing,’ said the Dean, pointing, ‘with its moon-hating water and worlds that go around suns –’

  ‘As far as I can see from this,’ interrupted the Senior Wrangler, who’d been reading HEX’s write-out on the more complex physics of the Project, ‘if you were travelling in a cart at the speed of light, and threw a ball ahead of you …’he turned over the page, read on silently for a moment, creased his brows, turned the page over to see if any enlightenment was to be found on the other side, and went on ‘… your twin brother would … be fifty years older than you when you got home … I think.’

  ‘Twins are the same age,’ said the Dean, coldly. ‘That’s why they are twins.’

  ‘Look at the world we’re working on, sir,’ said Ponder. ‘It could be thought of as two turtle shells tied together. It’s got no top and bottom but if you think of it as two worlds, bent around, with one sun and moon doing the work of two … it’s similar.’

  He fried in their gaze.

  ‘In a way, anyway,’ he said.

  Unnoticed by the others, the Bursar picked up the write-out on the physics of the Roundworld universe. After making himself a paper hat out of the title page, he began to read …

  1 A phrase meaning ‘I’m not sure you know this’

  TWENTY-TWO

  THINGS THAT AREN’T

  LIGHT HAS A speed – so why not dark?

  It’s a reasonable question. Let’s see where it leads.

  In the 1960s a biological supply company advertised a device for scientists who used microscopes. In order to see things under a microscope, it’s often a good idea to make a very thin slice of whatever it is you’re going to look at. Then you put the slice on a glass slide, stick it under the microscope lens, and peer in at the other end to see what it looks like. How do you make the slice? Not like slicing bread. The thing you want to cut – let’s assume it’s a piece of liver for the sake of argument – is too floppy to be sliced on its own.