Planetary formation in motion; the birth of a planetary system was captured by the ALMA observatory in Chile in 2014.
Let’s return to the human towers. What sets the maximum height of a tower? Consider an artificial situation in which the tower is a vertical stack of humans, one on top of the other. If there are only two people in the stack, then the force on the person at the base is the weight of the person above. Let’s understand that sentence. What is weight? Your weight at the Earth’s surface is given by Newton’s equation; it is defined to be the force exerted on you by the Earth. What numbers should we put into the equation to calculate it? Your mass: 75kg. The mass of the Earth: 5.972 x 1024 kg.6 Newton’s gravitational constant, G: 6.6738 x 10-11 m3 kg-1s-2. What should we use for r? This is the distance from the centre of the Earth to the centre of you. That sounds a bit vague. More precisely, r is the distance between the centre of mass of the Earth and your centre of mass, but it’s a very good approximation to simply insert the radius of the Earth into Newton’s equation. This is because you are only around a couple of metres tall, and the average radius of the Earth is 6,371,000 metres, so moving your centre of mass around by a few tens of centimetres isn’t going to change the calculation much.
Plugging in the numbers, Newton’s equation tells us that the force on you at the Earth’s surface – your weight – is approximately 736 Newtons (a force of 1 Newton produces an acceleration of 1 m/sˆ2 on a 1kg mass).
We now need to introduce another of Newton’s laws – his third law of motion, also published in the Principia: To every action, there is an equal and opposite reaction. This says that the Earth exerts a force on you and you exert an equal and opposite force on the Earth. We can now understand what happens when the human towers get higher and higher. If one person stands on another’s shoulders, there is a downward force on the lower person of around 730 Newtons. If another person of the same mass climbs up, the force on the person at the base doubles to 1460 Newtons. If another two people climb up to form a tower five people high, the force on the base person is 2920 Newtons, and so on. Clearly, at some point, the person at the base isn’t strong enough to hold the tower up, and the whole thing will collapse. This is where the skill of the Castellars comes in. By having a base, made up of many individuals, the forces can be distributed across the human structure, and this allows the towers to get higher before catastrophe strikes. There is clearly a trade-off; a larger base can support a larger layer above, which in turn can support a larger layer above, and so on. But a larger layer weighs more, and exerts a larger force on the layer below. The ingenious geometrical solutions to this gravitational conundrum emerge through a combination of trial and error, instinct and skill, and this is what makes the Tarragona Castells competition so compelling. For our purposes, it is the principle that matters. As the tower gets higher, the forces on the base increase, and ultimately a limit will be reached.
Perhaps you can see where this is leading. High human towers are more difficult to sustain because the force on the base becomes increasingly large as the mass of the tower increases. This suggests that the size of structures that rise above the surface of a planet is limited by the structural strength of the rock out of which the planet is made, and the mass of the planet, which sets the gravitational pull and therefore the weight of the structure. On Earth, the tallest mountain as measured from its base on the sea floor is Mauna Kea, on the island of Hawaii. This dormant volcano is 10 kilometres high, over a kilometre higher than Mount Everest. Mauna Kea is sinking because its weight is so great that the rock beneath cannot support it. Mars, by contrast, is a less massive planet. At a mere 6.39 x 1023 kg, it is around 10 per cent of Earth’s mass and has a radius about half that of Earth. A quick calculation using the equation on here will tell you that an object on the surface of Mars weighs around 40 per cent of its weight at the Earth’s surface. Since Mars has a similar composition to Earth, its surface rock has a similar strength, and this implies that more massive mountains can exist on Mars because they weigh less – and this is indeed the case. The Martian mountain Olympus Mons is the highest mountain in the Solar System; at over 24 kilometres in altitude, it is close to the height of three Everests stacked on top of each other. Such a monstrous structure is impossible on Earth because of the immense weight – a result of the Earth’s greater mass and therefore stronger gravitational pull at the surface.
At the summit of Mauna Kea, at an altitude of 4,200 metres above sea level, sits one of the world’s largest observatories.
We see that there must be a limit to the height to which a structure can rise above the surface of a planet. The more massive the planet, the stronger the gravitational pull at its surface, and the lower the height of structures that the surface can support. As the planets get more and more massive, their surfaces will get smoother and smoother because of the stronger gravity. Less-massive planets can be more uneven. We are approaching an answer to our question; we have a mechanism for smoothing out the surface of a planet, but why should this mean that planets get smoothed into a sphere?
There must be a limit to the height to which a structure can rise above the surface of a planet, assuming that the height of a hill or mountain is proportional to its mass.
Imagine a mountain on the surface of a planet. Let’s say it is at the North Pole. Now, in your mind’s eye, imagine rotating the planet through, say, 90 degrees, so the mountain sits on the Equator. Has anything changed? All the arguments about the maximum height of the mountain still apply, because the gravitational force at the surface depends only upon the radius and mass of the planet and the mass of the mountain. There is no reference to any angle in Newton’s equation (here).
This computer-enhanced photo shows the vast scale of the Olympus Mons volcano on Mars. Such a huge landmass feature would be impossible on Earth because of its immense weight.
In more sophisticated language, we can say that Newton’s law of gravitation possesses a rotational symmetry. By that, we mean that it gives the same results for the gravitational force between any two objects regardless of their orientation. This is an example of what physicists and mathematicians mean when they speak of the symmetries of an equation or law of Nature, and it means that the calculation for the maximum height of a mountain at any place on the Earth’s surface must give the same answer irrespective of the position of the mountain because Newton’s law of gravitation is symmetric under rotations. The symmetry of the law of gravity is reflected in the symmetry of the objects it forms. Gravity smooths mountains democratically, symmetrically, with the result that lumps of matter with a gravitational pull strong enough to overcome the rigidity of the substance out of which they are made end up being spherical. This is the reason why the Earth is spherically symmetric.
Artist Gordon Legg created this image of Olympus Mons using photos taken by the Viking 1 orbiter in 1975 to show its vast scale.
There is a deep idea lurking here that lies at the heart of modern theoretical physics. Thinking of things in terms of symmetry is extremely powerful, and perhaps fundamental. Consider the possibility that the laws of Nature possess certain symmetries, which are the fundamental properties of the Universe. This would be reflected in the physical objects they create. For example, imagine a Universe in which only laws of Nature that are symmetric under rotations through 90 degrees are allowed. In such a Universe, objects that remain the same under rotations through 90 degrees are created; cubes exist but spheres are forbidden. This isn’t quite as crazy as it sounds. As far as we can tell, our Universe does possess a set of extremely restrictive symmetries, and the subatomic particles that exist and the forces that act between them are determined by these underlying symmetries.7 In fact, all of the laws of Nature we regard as fundamental today can be understood by thinking in this way. There is certainly a strong case to be made that Nature’s symmetries can be regarded as truly fundamental. The Nobel Prize-winning physicist Steven Weinberg wrote, ‘I would like to suggest something here that I am not real
ly certain about but which is at least a possibility: that specifying the symmetry group of Nature may be all we need to say about the physical world, beyond the principles of quantum mechanics.’ Nobel laureate Philip Anderson wrote, ‘It is only slightly overstating the case to say that physics is the study of symmetry.’ Nobel laureate David Gross wrote, ‘Indeed, it is hard to imagine that much progress could have been made in deducing the laws of Nature without the existence of certain symmetries … Today we realise that symmetry principles are even more powerful – they dictate the form of the laws of nature.’ The complexity we perceive when casually glancing at the Universe masks the underlying symmetries, and it is one of the goals of modern theoretical physics to strip away the complexity and reveal the underlying simplicity and symmetry of the laws of Nature.
Returning to the task in hand, this reasoning leads to a prediction about the size and shape of planets and moons that can be checked: they should be spherical if they are large enough, and therefore massive enough, for their gravitational pull to overcome the structural strength of the rock out of which they are made. The strength of rock is ultimately related to the strength of the force of Nature that holds the constituents of rock together – molecules of silicon dioxide, iron and so on. This is the electromagnetic force; what other force could it be? There are only four forces, and the two nuclear forces are confined within the atomic nuclei themselves. Big things like planets are shaped by the interplay between gravity, trying to squash them into spheres; and electromagnetism, trying to resist the squashing. We can perform a calculation to estimate the minimum size that a lump of matter must be to form into a near-spherical shape by equating the weight of a mass of rock near its surface to the structural strength of the rocks below.8 Our answer is approximately 600 kilometres.
We can check this by direct observation of the Solar System. Phobos fits with our prediction; with a mean radius of just over 11 kilometres and a mass of only 1016 kilograms, the gravitational force at its surface is far too weak to overcome the rigidity of rock and act to flatten the surface and sculpt Phobos into a sphere. At around 550 kilometres across, the asteroid Pallas is the largest known non-spherical object. Saturn’s moon, Mimas, with a radius of just under 200 kilometres, is the smallest known body in the Solar System that is spherical. It is made mostly of ice, which is much easier to deform than rock – this is why it is so small and still round. Our estimate is certainly in the right ballpark.
As an important aside, ‘back-of-the-envelope’ estimates such as these are very important in physics; they tell us that we are on the right track, without overcomplicating things unnecessarily. We could have refined our calculation by taking into account the different compositions of different objects, and by computing the gravitational pull at different depths more carefully. We could even have tried to use General Relativity instead of Newton’s laws, but we wouldn’t have learnt a lot by doing so. Learning what to ignore and what to include is part and parcel of becoming a professional scientist – one might call it physical intuition. There is no precise size above which a body will be spherical; the limit depends on the object’s composition; a mixture of rock and ice is easier to deform than solid rock. As a general rule, any icy moon over 400 kilometres in diameter will be a sphere. Objects made of rock need to be larger, because the gravitational force needed to deform rock is greater. If a rocky moon has an internal heat source, perhaps as a result of the presence of large amounts of radioactive material in its core or tidal heating, the body is easier to deform and may be spherical at a smaller size than would be expected for a less active object. The Solar System is full of examples of this interplay of rigidity and gravity in action, but, very roughly, we have deduced that anything with a radius in excess of a few hundred kilometres must be spherical because the gravitational forces will overcome the strength of the rock.
When gravity wins, the shape of the objects it creates reflects the underlying symmetry of the physical law, and this is why large single objects such as planets and stars are always spherical.
At larger scales, however, things change. The picture below shows our nearest galactic neighbour, Andromeda, which contains around 400 billion stars, formed and bound together by gravity. It is disc-shaped, not spherical. The Solar System itself is also a disc and not a sphere. Why?
Phobos, Mars’s largest moon, has a gravitational force at the surface that is too weak to sculpt its craters into a smooth sphere, giving it a pockmarked appearance.
Why are there discs as well as spheres in the Universe?
We argued above that planets and large moons are spheres because, if the gravitational forces are large enough to overcome the electromagnetic forces that keep matter rigid, the underlying symmetry of the gravitational force is made manifest in the objects it creates. Because there is no special direction in Newton’s Law of Gravitation, there will be no special direction in the objects that it creates. This is not entirely true, however, even for planets, because they spin.
The Andromeda Galaxy M31 – Earth’s nearest neighbour.
Our planet turns on its axis once every 24 hours. The spin axis marks out a special direction, which means that all points on the Earth are not the same. Someone standing on Earth’s Equator is rotating at a speed of 1670 km/hour, whilst someone in Minnesota is rotating at a speed of 1180 km/hour. The spherical symmetry is broken – the two points are different. As we’ll see in Chapter Two, this difference leads to observable effects such as the rotation of storm systems and the deflection of artillery shells in flight. It also leads to a very slight flattening of the Earth – the equatorial circumference is 40,075 kilometres and the polar circumference is 40,008 kilometres. The Earth is not spherical, but an oblate spheroid, because of its spin. If it were spinning faster, the Earth would be more oblate. When our Solar System formed, the spin – or more correctly angular momentum – was ‘exported’ outwards from the newly forming Sun, primarily through collisions and magnetic interactions in the protoplanetary disc, resulting in the system becoming flattened into a disc.
Some of the many properties that have resulted in partial loss of the symmetry in the Solar System disc.
The transfer of spin outwards from the centre also results in the flattening of some galaxies; for example, Andromeda. Globular star clusters, such as the spectacular Messier 80, remained spherical because they were too diffuse for angular momentum to be transferred outwards. The shapes of objects that are bound together by gravity are therefore dependent on the amount and location of the ‘spin’. For the experts, the ratio of angular momentum L to gravitational potential energy E is the figure of merit. Large L/E = disc. Small L/E = spherical.
There is a very important idea hiding here. We used the term ‘symmetry breaking’ to describe how the presence of a spin axis marks out a particular direction, resulting in an object deviating from the ‘perfect’ spherical shape that reflects the symmetry of the underlying law of Nature – in this case gravity. The disc of our solar system is less symmetric than a sphere because it only remains the same when rotated about a particular axis in space – the spin axis. The symmetry has been partially lost. We might say that the symmetry of the law of gravity that created our solar system has been hidden by the presence of a special direction in space – the spin axis. The spin itself came from the precise details of the collapse of the initial dust cloud almost 5 billion years ago, and the distribution of the spin between the Sun and planets depended on the precise speed of collapse, the density of the protoplanetary disc and myriad other subtle details over the history of the Solar System’s formation. This highlights one of the central challenges in modern science: which properties of the structures we see in Nature are reflections of the underlying laws of Nature, and which properties are determined by the history of formation or other influences? This is particularly difficult to answer when the physical systems in question are complicated. The shapes of planets, solar systems and galaxies, whilst astronomical in size, are easier to explain
than the shapes of more mundane objects that we encounter every day. Let’s jump from simple planets to the most complex of all physical structures – living things. By exploring the symmetries and structures of living organisms, we can further explore the idea that the shape and form of physical objects are the result of a complex interplay between deep physical principles and the history of their formation.
Messier 80, in the constellation of Scorpius, a spherical cluster of hundreds of thousands of stars 30,000 light years from the Sun.
Why does life come in so many shapes and sizes?
The competition between the force of gravity and the electromagnetic force is responsible for smoothing the surface of planets and moons into spheres and limiting the maximum size of mountains on their surfaces. One of the central ideas in this book, which we will expand on in Chapter Three, is that there is no fundamental difference between inanimate things, such as planets, and living things, such as bacteria or human beings; all objects in the Universe are made of the same ingredients and are shaped by the same forces of Nature. We should therefore expect to see limits on the form and function of living things imposed by the laws of Nature. Basic physics is not the only driver of the structure of organisms, of course; there is also the undirected hand of evolution by natural selection, which moulds living things over time in response to their changing environment, their interaction with other living things, and the myriad available environmental niches. This creative interplay between the relentless determinism of physical laws and the seething, infinitely intertwined, ever-shifting genetic database of life on Earth is beautifully captured in Darwin’s closing lines of On the Origin of Species;
Forces of Nature Page 5
