by Marcus Chown
This solution to the puzzle of planetary motion is in fact a big con. With enough circles within circles within circles it is possible to mimic absolutely any motion whatsoever. Not only that but the solution is complex and messy. And a key characteristic of modern scientific explanations is that they are simple and economical.
A better explanation of the peculiar planetary motion was proposed by the Polish astronomer Nicolaus Copernicus in 1543. Say the centre of everything is not the Earth but the Sun, and that all of the planets, including the Earth, actually go around the Sun? In this case, Copernicus pointed out in On the Revolutions of the Heavenly Spheres, the motion of planets is easy to explain. As it circles the Sun, the Earth regularly catches up and overtakes a planet like Mars, which is orbiting more slowly in its orbit. From the point of view of the Earth, the planet drops behind, appearing briefly to travel backwards against the fixed stars.12
Copernicus’s explanation of the motion of the planets came at a cost. There were now two bodies about which other bodies circle – the Sun, which ensnares the planets, including the Earth, and the Earth, which holds onto the Moon. And things got even worse when the Italian scientist Galileo zoomed in on the heavens with his new-fangled astronomical telescope. Not only did he see stars invisible to the naked eye, mountains on the Moon and the phases of Venus but, in 1610, he was amazed to find that Jupiter is orbited by four moons. There are not two bodies acting as centres in the Solar System: there are at least three.
Ancient ideas were crumbling. According to the Greeks, the most important factor for understanding our world and the Universe was location. Each of the four ‘fundamental elements’ – earth, fire, air and water – seeks out its allotted place. And all are related to the Earth, with earth, not surprisingly, desiring to get as close to the centre of the Earth as possible. But, in the new view, there was nothing at all special about location. How could there be when there are at least three locations about which other celestial bodies revolve?
The lesson from observing our Solar System is that massive bodies orbit other massive bodies. Location is not the important thing.13 Mass is the key.
Nature’s lonely hearts club force
The pressing question is: how does one mass enslave another? A clue came from magnetism. Lodestones are naturally magnetised chunks of the mineral magnetite. One lodestone attracts another lodestone with a mysterious ‘force’ that reaches across the empty space between them. As early as the sixth century BC their unusual properties had been remarked upon by the father of Greek philosophy, Thales of Miletus.
In 1600, the English scientist William Gilbert suggested that magnetism might be the force holding together the Solar System. He demonstrated experimentally that the attraction exerted on a piece of iron by a lodestone is bigger the bigger the mass of the lodestone. He also showed that the attraction is mutual – that is, the force of attraction exerted by a lodestone on a piece of iron is exactly as strong as the force of attraction exerted by the iron on the lodestone.
Others such as Robert Hooke, the man who would become Newton’s greatest rival, were much taken by Gilbert’s findings. But the Sun is a hot body and lodestones heated until red hot were known to lose their magnetism. Hooke therefore saw magnetism as merely a model for the force that is orchestrating the motion of the bodies of the Solar System. Like magnetism, gravity reaches out from one mass across empty space and grabs another mass. Like magnetism, the force is bigger the bigger the masses involved. And, like magnetism, it is a mutual force.
Gravity pulls masses together. It attempts to break their terrible isolation. It is truly nature’s lonely hearts club force.
This was the state of play in the plague year of 1666 as Newton sat deep in thought at his desk at Woolsthorpe Manor and began to ponder the nature of the force between massive bodies. He had no more idea what the force of ‘gravity’ is than what the magnetic force of a lodestone is. But not knowing what the force is did not hamper him. In the words of the twentieth-century physicist Niels Bohr: ‘It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.’
Newton knew this truth instinctively. Just because he did not know what gravity is did not mean he could not ask: how does gravity behave?
Reading the book of nature (Kepler’s laws)
The vital clues to gravity’s behaviour had been discovered by the German mathematician Johann Kepler. Between 1609 and 1619, he had built on the work of the Danish astronomer Tycho Brahe, famous among other things for having a prosthetic nose made of brass after his real one was sliced off in a duel. Brahe had made precise naked-eye observations of the planets from his observatory on the island of Hven, now part of Sweden. After poring long and hard over Brahe’s records, Kepler deduced three laws that govern the behaviour of the planets.
Kepler’s first law states that the orbit of a planet is an ellipse, with the Sun at one focus. An ellipse is a very specific closed curve, not simply an oval. It can be drawn by pinning two tacks to a flat surface, stretching a loop of string over them, pulling the loop of string taut with a pencil, and moving the pencil point in a complete circuit around them. The two tacks mark the foci of the ellipse. In mathematical terms, wherever a point is located on the ellipse, the sum of the distances to the two foci is the same.
Kepler’s recognition that the orbit of a planet is an ellipse was a decisive and significant break with the past. The Greek conviction that circles are perfect had caused them to impose circles on the cosmos. But nature is a book to be read not a book to be written. Realising this, Kepler, and the scientists who followed him, demonstrated more humility than their Greek predecessors. They studied nature and looked to see what it was telling them. And what nature was telling Kepler, through the medium of Brahe’s painstaking observations, was that the planets are orbiting the Sun not in circles but in egg-shaped ellipses.
Kepler’s second law says that a planet does not go around the Sun at a uniform speed but moves more quickly when it is nearer the Sun and more slowly when it is further from the Sun. Actually, the law is a bit more precise than this. It states that an imaginary line joining a planet to the Sun sweeps out equal areas in equal times. Take, for instance, a time interval of 10 days. Two points on a planet’s orbit that are 10 days apart can be joined to the Sun to make a triangle. The area of the triangle is always the same irrespective of whether the planet is close to the Sun in its orbit or far from the Sun. It is impossible not to admire the sheer ingenuity of Kepler in teasing out such an odd law from Brahe’s observations.
Newton, ensconced at Woolsthorpe, thought long and hard about Kepler’s second law. And thinking long and hard was the secret of his genius. Yes, he could build complex things and carry out complex experiments, and he could do both of these things far better than most. But what truly set him apart from all others was his phenomenal, almost unearthly power of concentration. This was the secret of his success. This was his thing.
Newton took no exercise, indulged in no amusements, and worked incessantly, often spending eighteen or nineteen hours a day writing.14 The clockwork of his mind whirred incessantly. Every hour spent not studying he considered an hour lost. While others could hold an abstract problem in their mind’s eye for fleeting minutes, Newton could focus on a problem for hours, weeks, whatever it took, until finally, he burned through to its inner core and it yielded its precious secret. ‘I keep the subject constantly in mind before me and wait ’til the first dawnings open slowly, by little and little, into full and clear light,’ wrote Newton.15
Newton applied the laser beam of his intellect to Kepler’s second law. And eventually, inevitably, he saw what it was telling him about the force of gravity experienced by a planet. And the thing it was telling him has nothing to do with the strength of that force or the way in which that strength changes with distance from the Sun or any other detail like that. A planet sweeps out equal areas in equal times, Newton realised, on one condition and one co
ndition only: that the force it is experiencing is always directed towards the Sun.16
Kepler’s third law of planetary motion is subtly different from the first two. Instead of describing the individual orbits of planets, it describes how the orbits of different planets relate to each other. It states that the further a planet is from the Sun, the slower it moves and so the longer it takes to complete an orbit. This is a clear indication that the force of gravity experienced by a planet is weaker the further the planet is from the Sun. But there is more in the law than this. Kepler was a mathematical genius. His third and last law actually says that the square of the orbital periods of the planets goes up in step with the cube of their distances from the Sun. So, for instance, a planet that is 4 (that is, 22) times as far from the Sun as another takes 8 (that is, 23) times as long to complete an orbit.
Kepler’s third law is even more esoteric than his second. But do not get hung up on the detail. The key thing is that it is a precise mathematical relation. And that indicates that the force that gives rise to the law – the force between the Sun and the planets – must also be mathematical. This in itself is a revelation. Evidently, nature obeys mathematics. Or, as Kepler might have seen it, God is a mathematician.17 So the question Newton, frowning at his desk at Woolsthorpe, asked was: what is the mathematical law of gravity?
Newton was in a unique position to answer this question because he alone had defined the concept of a force, transforming it from a hand-waving, nebulous notion into a thing of rapier-sharp scientific precision. In this Newton was indebted to Galileo, who died a year before Newton was born.
Explaining the book of nature (Newton’s laws)
Bodies falling under gravity plummet too fast for their fall to be timed precisely with the primitive methods available to Galileo. But he found an ingenious way to dilute gravity and so brake the motion of falling bodies. He set balls rolling down an inclined plane on a table top. The shallower the slope, the more gravity is diluted and the slower a ball rolls. But – and this was a key observation by Galileo – when a ball reaches the bottom of the slope, it continues rolling at constant speed until it falls off the edge of the table.
On the table top, which is flat with no slope, gravity is diluted to zero and there is no force on the ball. So Galileo concluded that in the absence of a force bodies move at constant speed.
This result is totally counterintuitive. In the everyday world nothing moves at unvarying speed. Kick a stone along the ground and it quickly comes to rest. But the explanation for this, Newton reasoned, is that the stone is subject to a retarding force – the force of friction with the ground. In the absence of such a force – if the stone, for instance, is kicked across a perfectly slippery, ice-covered pond – it will keep on going.
That the natural motion of a body is to keep coasting explains a puzzle which had stumped people ever since they realised that the stars are not actually turning around the Earth but the Earth is instead spinning. Knowing the size of the Earth and that the Earth turns once every 24 hours, it follows that at the equator the Earth’s surface is moving at 1,670 kilometres an hour! Why do people living there not notice it? Why, if a ball is dropped there, does the Earth not rotate under it as it falls so that the ball hits the ground far to the east? The answer is that we and the ball and the air around us were all born into a moving world, and continue to move around with the Earth as it turns because that is what moving things do.
Even today no one knows why the natural motion of a body is to keep on coasting. But Newton latched onto Galileo’s extraordinary insight and encapsulated it in the first of his three ‘laws of motion’.
Newton’s first law says that every body either stays at rest or keeps moving forward at constant speed in a straight line unless compelled to change by an external force. (This should not be confused with the Law of Cat Inertia, which states: ‘A cat at rest will tend to remain at rest unless acted upon by some outside force such as the opening of cat food, or a nearby scurrying mouse.’)18 According to Newton, a ‘force’ is something that budges a body from its natural motion – changing its speed or its direction or both. This idea Newton encapsulated in his second law, which says that a body responds to a force by accelerating – that is, changing its speed – in the direction of the force and by an amount that is inversely related to its mass. In other words, a small mass accelerates more than a big mass in response to a given force.
More precisely, Newton’s second law states: ‘The rate of change of momentum of a body is equal to the force applied’. Newton defined ‘momentum’ as the product of a body’s ‘mass’ and its ‘velocity’, which in turn is defined as its speed in a particular direction. Newton, here, was laying the foundations of ‘dynamics’, the mathematical theory of motion.
That the natural behaviour of bodies is to move in straight lines at constant speed told Newton everything he needed to know about a planet orbiting the Sun. First, no force is required to push a planet around the Sun. This is fortunate circumstance since, as already mentioned, Newton’s interpretation of Kepler’s second law is that the force of gravity is directed solely towards the Sun, with no component along the path of a planet. A planet simply keeps moving for no other reason than that is what massive bodies naturally do.19
Think what an extraordinary revelation this is. Pretty much everyone who had ever thought about the problem of the motion of the planets imagined that some kind of force is necessary to push them around in their orbits. Some imagined invisible angels flying alongside and blowing the planets along or chivvying them with their beating wings. Kepler envisaged magnetic ‘spokes’ extending from the Sun and impelling the planets as the Sun turned. The French mathematician René Descartes favoured a solar ‘vortex’ swirling the planets around like cosmic flotsam. But Newton consigned all of these ideas to the dustbin of history. Kepler’s second law, he realised, is definitive proof that no force is driving the planets around in their orbits.
That the natural behaviour of bodies is to move in straight lines also told Newton what the force of gravity holding a planet in orbit around the Sun is doing. It is constantly changing its path from a natural straight line to a circle.
Of course, Newton knew from Kepler’s first law that the planets orbit the sun not in circles but in ellipses. But ellipses are more complicated figures than circles and, since the elliptical orbits of the planets are pretty close to circles, Newton felt justified in considering them to a first approximation as circular.
The question Newton asked himself was: what is the force required to a keep a body moving in a circle – that is, what is the force needed to continually bend its path away from its natural straight-line trajectory? The answer had already been obtained by others, including Hooke. But Newton did not know this.
Newton sat down with a piece of parchment and drew a circle of radius, r, with a dot, representing a mass, m, on its circumference. The mass, he assumed, is moving at a velocity, v. Now it was just a matter of geometry to work out the force necessary to continually deflect the mass from a straight-line path. The force turns out to be mass times velocity-squared divided by the radius, or mv2/r.
The formula for this ‘centripetal force’ actually encapsulates everyday intuition. Say, you swing a stone tied to the end of a length of string around your head. Common sense says the more massive the stone the harder you will have to pull on the string – that is, the bigger the force you will have to apply – to stop the stone flying off on a tangent, trailing the string behind it. Common sense also says that the faster you swing the stone, the bigger the necessary restraining force. And the shorter the string the harder you will have to pull.20 Gravity is the invisible string that holds onto the planets and stops them flying off to the stars.
Now, Newton asked the following crucial question: if the centripetal force on a planet is provided by gravity, how must that gravity vary with distance from the Sun in order to yield Kepler’s third law – that the square of the orbital period goes up in
step with the cube of its distance from the Sun? The answer, he discovered, is that the force must weaken with the square of the distance from the Sun. In other words, if a planet is twice as far away as another, the gravitational force it experiences from the Sun is four times as weak; if it is three times as far away, nine times as weak, and so on.21
There was one other place in the heavens where Newton could check this ‘inverse-square law’ of gravity. Jupiter’s four moons – Io, Europa, Ganymede and Callisto — had been observed whirling around the planet ever since Galileo first spotted them from Padua in 1610. Astronomers had measured the relative distances of these ‘Galilean’ moons from Jupiter and timed how long each takes to complete an orbit. They had discovered that the moons orbit Jupiter exactly as the planets orbit the Sun, with their orbital periods varying with their distances from Jupiter as predicted by Kepler’s third law. The hard work had been done for Newton. Kepler’s third law is an inevitable consequence of a gravitational force that weakens with distance according to an inverse-square law.22
The Moon is falling
Kepler’s third law, operating in the lofty realm of the heavens, was far removed from the everyday world of sheep grazing in the fields of Woolsthorpe, of hay-filled wagons bumping and jouncing along rutted tracks, of cocks crowing in the cold grey dawn. But Newton was harbouring a revolutionary, heartstopping thought. What if the force of gravity at work in the heavens is the same force of gravity at work on Earth? What if – and nobody in the history of the world had thought this thought before – what if there is not one law for the heavenly realm and one law for the everyday world? What if gravity is a universal force – that operates between every last mote of matter and every other last mote of matter?
Newton was the ultimate pragmatist. He knew that his insight meant nothing unless he could make it count – unless he could use it to calculate something.