The Hell of Gravity

The chaos theory may have gone out of fashion, but be careful not to just whistle along pretending everything’s all right when you get out of bed in the morning to sip an instant caffè latte. Besides, what happened in L’Aquila should teach you something. But it is not of the immediate, concrete risks of vindictive Mother Nature that I wish to speak, for that you already have plenty of people getting indignant on Facebook.

The other side of the Moon, the one never visible from the Earth. On the left the Giordano Bruno crater. Courtesy NASA The other side of the Moon, the one never visible from the Earth. On the left the Giordano Bruno crater. Courtesy NASA

I wish instead to speak to you about the celestial spheres, about one of the possible Ends of the World, and about one end that has already taken place. Lift up your nose every once in a while from the asphalt and look up. There is the Sun, or there is the Moon (or both.) There they are, round, daily symbols of the calm of the cosmos – as a poet sang “what are you doing up there in heaven Moon, tell me what you’re doing silent Moon.” After all, what – aside from benzodiazepine – inspires more tranquility than a summer’s night under a full moon?

A guarantee of all this, one learns at grade school, is gravity, which controls the sound and regular movement of the planets’ orbits. The planets, those little dots – or pale dots as space age rhetoric called them – that, well-behaved, complete their ellipses, each holy revolution, from eternity to eternity, with Father Newton looking after everything, along with Grandpa Kepler and Grandpa Copernicus. The universe is like a cosmic watch, they said in the 1700s, of which God is the watchmaker. The cosmic watchmaker, in his goodness, was supposed to be there to permit us mortals to deduce something very fundamental: If we know where the stars are today, we will know where they are tomorrow. We just have to make the calculations. In 365 days and a handful of minutes, the Earth, in its orbit, will be in the same position that it is today, and the same goes for 2, 3, 4 … n years, with slow and reassuring monotony. Pierre-Simon de Laplace said that

An intellect which at a determined instant were to know all the forces that put nature in motion, and all the positions of all the objects of which nature is composed, if this intellect were furthermore sufficiently large to subject these data to analysis, it would enclose in a single formula the movements of the largest bodies of the universe and the smallest atoms; for such an intellect nothing would be uncertain and the future would like the past be evident before his eyes.

For such angelic being, curiously known as ‘Laplace’s demon,’ the past and the future are as present as the present itself: they are One1.

A world made by a cosmic watchmaker: The benevolent God makes the gears, winds them up, and all the rest ticks on its own, while we live suspended from the hands. Convenient, simple, but also quite boring. Today we are adults, and we know that there are not any cosmic watchmakers – or if there are, they are so fully hidden as to mind their own business. The equations of movement are however still around. And in the deceitfully simple litany of an equal and constant force by the product of the masses divided by the distance squared, there is a demon hiding, whose name alludes to sin: the problem of three bodies.

Chaotic behaviour due to resonance in a three-body system. From P. Hut and J.N. Bahcall, Binary-single star scattering I, ApJ 268, 319-41, 1983.  Courtesy AAS American Astronomical Society and WikipediaChaotic behaviour due to resonance in a three-body system. From P. Hut and J.N. Bahcall, Binary-single star scattering I, ApJ 268, 319-41, 1983. Courtesy AAS American Astronomical Society and Wikipedia

To put it another way. In high school they still teach, I hope, that when you put two balls in an empty space, they spin around the relative center of mass, forming an elliptical, parabolic, or hyperbolic trajectory. This is all well and good as long as there are just two balls (as is the norm.) Now let’s add a third ball. What happens?

This apparently innocent game of bowling does in fact quite literally sow the seeds of chaos. Laplace’s omniscient demon found himself dragged down to an even deeper pit of hell by another Frenchman, Henri Poincaré. Poincaré demonstrated that if you add a third body to the gravitational dance, the monogamous elegance of attraction between two bodies disintegrates into the orgiastic chaos of a threesome: A general solution to the movement of three bodies attracted by gravity does not, in fact, exist2. Or rather, there is no convenient and simple equation that we can write down once to tell us where on earth these objects will be in a billion years3. You can only insert Newton’s equations into a computer and make it patiently grind out their position, moment by moment. Which is not exactly the same thing.

caos1The Michel Hénon map (Nice observatory) is a geometric model to show the development of chaotic systems (like three-body ones), naturally generating fractals. The resulting picture is an example of chaotic attractor. In the image: an attraction basin of the Hénon map. Courtesy James P. Crutchfield and Scientific American

Good old Newton, however, had already thought that things were in fact probably less boring than they seemed. He wrote:

In fact, while comets move in all directions in very eccentric orbits, blind fate could never make all the planets move in the same direction in concentric orbits, with the exception of some not very relevant irregularities, which could derive from the mutual attraction that the comets and the planets exert on themselves, and that will tend to grow until this system needs to be reformed.

Newton declared therefore that one of God’s tasks was to preserve the universe from the chaos in which it would fall because of the very same laws conceived by Him: the Omnipotent puts the planets back in line, pushing them over here or pulling them over there, to prevent them from scattering every which way. This hypothesis was mocked by Leibniz:

Sir Isaac Newton and his followers have quite a curious opinion about God’s works. According to their doctrine, Omnipotent God must wind up His watch again from time to time, otherwise it would cease to tick. He did not have, it seems, enough farsightedness to make it tick eternally.

Since the idea of God getting up in the morning to recharge the movement of the planets was all in all a bit embarrassing, the problem was investigated by the usual Frenchmen, i.e. the above-mentioned Laplace and another team’s heavyweight, Lagrange. Aynway they somewhat brushed the problem under the rug: unable to find a general solution to the problem of three bodies – not by their own fault: as we have mentioned earlier such a solution does not, in fact, exist – they chose to use approximations known as ‘perturbative solutions.’ Perturbative theory, although approximate, worked quite well overall and explained minimal variations in the movement of the planets, thus one could answer resentfully that Poincaré could go nitpick elsewhere, everything was alright and de minimis non curat physica. Right?

No. The problem is that systems like those with three (or four, five … n) bodies, devoid of rigidly determined mathematical solutions, are chaotic by definition. Let it be pointed out that ‘chaotic’ in this case does not mean that they evolve completely randomly, ignoring lowly every law of nature; quite the contrary, they are chaotic precisely because they servilely follow those laws, yet in doing so they run into two peculiar characteristics.

nice_model_wiki2Simulation of evolution of Solar System along the Nice model, showing outer planets and planetesimal belt: a) Early configuration, before Jupiter and Saturn reach 2:1 resonance b) Scattering of planetesimals into the inner Solar System after the orbital shift of Neptune and Uranus c) After ejection of planetesimals by planets. Courtesy Mark Booth and Wikipedia

The first characteristic is the system’s practical unpredictability. The system is deterministic in theory, but there is no practical way to predict its behavior. The concept is simple. Take a system of three bodies, like that of the Earth-Moon-Sun, as an example. Imagine that you, a Laplacian angel, know with absolute exactness where these three bodies were at a given moment. Calculate in every instant Newton’s equations, and you will know where the bodies will be at any given moment in the future. Let’s say that, proceeding in this manner and making your computer sweat, you know where they are in a billion years.

Now, start again from the beginning, with the bodies in starting position, except for the fact that you have moved one of them, the Moon let’s say, just a little bit forward or behind on its orbit. When I say a little, I mean a real little: let’s say just a few meters. Where will these celestial bodies be in a billion years? One is tempted to say: practically the same place where we predicted them to be before, give or take a few meters. After all, what’s a few meters compared the infinite cosmos?

The umpteenth Frenchman, Jacques Laskar, demonstrated a few decades ago just how much a few meters count. Do you remember that something called the butterfly effect4? It is the same idea, bonly played out on a cosmic scale. Making simulations on his calculator, Laskar demonstrated that an error of just 150 meters in the determination of the Earth’s position would be enough to make it practically impossible to know where it will be in a few million years. Moreover, it becomes impossible to know not only where the Earth will be, but also where all the other bodies of the Solar System will be: The trajectory of one, in fact, influences that of all the rest. Imagine a tremendous spider web of invisible forces, in which if you pull lightly on the end of one thread, all the rest move and tangle themselves up with complete unpredictability. This nightmarish feature is typical of chaotic systems – like weather predictions, or Youtube comments – and is generally known as Lyapunov instability, after a man who, for a change, was not French5.

Lyapunov-fractalGeneralized Lyapunov logistic fractal known as Zircon Zity. It is an abstract model of chaotic systems: chaotic solutions of the model are in the blue section of space, stable solutions in the yellow section. Courtesy Wikipedia

The problem is not only the position of the Earth – or of the Moon, or Pluto; any kind of approximation of the positions and of the forces in play brings to the same embarrassing consequences. Laplace’s omniscient demon, to be able to say something about our celestial watch, would have to know exactly the position, velocity, mass, and destiny of every atom of the Universe. The movement of the Earth depends no longer just on the movement of the Sun, Moon, and other planets, but also on that of the asteroids, the stars nearby, those far away, as far as the remotest galaxies. Technically speaking, none of these factors are negligible, and each of them depends in turn on all of the others. If I change place in this room, I move the center of gravity of the Earth-Moon-Sun system; I am a imperceptible but not negligiblefactor: Moving the center of gravity completely changes the system destiny. In this humble sense, each one of us is an artificer of the Solar System’s destiny6.

Yet what does it matter to my twenty five readers if in a billion years Pluto will be farther or closer to Jupiter? It will be a bit more over here or over there, you may say, yet more or less still on the same damned orbit. Maybe in five million years the summer solstice will fall in January, but what difference does it make for you now, while you’re munching away on sushi in Milan? They will just have to adjust the horoscopes.

Aside from the fact that you are a person to be pitied because you are munching away at your fake Japanese food not caring about the cool wonder of the cosmos that surrounds you – shame on you, such changes matter because it is not just a question of positions. Calculations show that the planets’ orbits, pulling and releasing each other here and there, change form. They oscillate, making orbits narrower or wider, they get unstable. Sometimes too much so.

In some of the simulations, in a few hundreds millions of years Mercury’s orbit begins to oscillate violently: The planet moves first near to and then away from the sun, rhythmically, driven forwards and backwards like a swing due to the combined attraction of Jupiter and Venus. At a certain point these oscillations become so violent that Mercury’s destiny is unavoidable: Either it will plunge into the Sun or crash against Venus or the Earth. And as we have said above all is contained within the gravitational spiderweb: in Laskar’s simulations, when Mercury gets restless all the other planets feel it as well: Newton’s demons are out of control. The silent tempest spreads itself: After Mercury has disappeared or given itself over to a career as a kamikaze against our pleasant blue world, Mars will, in the space of a few million years – that is to say no time at all, on the cosmic scale – float away out of the Solar System, taking with it Rover, the martians and everything into the eternal cold of interstellar space.

Up to this point, we are talking about models on a calculator – hypotheses, toys. But let’s go back to where we started, the Moon. It is in fact proof the chaos of the orbits has already caused cataclysms in our Solar System, not just once, but twice.

Meanwhile, what is the Moon doing up there? The question has intrigued astronomers for centuries because the other three planets nearest to the Earth (Mercury, Venus, and Mars) do not have moons – the pair of tiny stones borrowed by Mars do not count. Theories do not foresee the formation of large and stable moons around small planets near the Sun, as is the case of the Earth. Furthermore the Moon’s chemical composition too closely resembles the Earth’s, as if it had been in part detached from our planet. What is the Moon doing up there?

converted PNM fileThe highest resolution topography map to date of the Moon’s rugged South Pole. Generated by scientists at NASA’s Jet Propulsion Laboratory, Pasadena. The new map provides contiguous topographic details over a region approximately 500 kilometers by 400 kilometers / Courtesy NASA/JPL

The answer is that the Moon is an enormous monument to a cosmic September 11th. At the birth of the Solar System there were not four internal planets, but (at least) five: Mercury, Venus, Theia, Earth, and Mars. Theia was a planet about the same size as Mars, that had the misfortune to form itself too close to the Earth and to Venus. Its orbit was torn to pieces by its two neighboring planets, until it crashed into the Earth. Part of the debris drifted off into space, and part of them slowly clotted together around the Earth, eventually reassembling themselves as a single body.

Not that these things only happen to us. We have just recently learned that 300 light years from here, the small double star BD+20307 features a huge and curious disc of dust: the still warm cadaver of a past collision between two planets, some tens of thousands of years ago.7

But the Moon is not just a ball of debris: it is also a ball of debris that was then violently struck by stones. Many lunar craters come from a time that astronomers call Late Heavy Bombardment. Basically around 3.5 billion years ago the entire Solar System, as if it had not already gone through enough with the destruction of Theia, found itself submerged for hundreds of millions of years in quite a remarkable asteroid storm, the effects of which can be seen just by looking up at the spots on the Moon8.

A plausible cause of the bombardment is the instability of the newly formed Solar System. According to this model – which, again speaks French: it is known as the Nice model – at the beginning of time Uranus and Neptune orbited much closer to the Sun, just a little outside Saturn’s orbit. A little ways beyond these planets there was at that time a very dense ring of asteroids and comets. The asteroids gradually disturbed the movement of the planets, slowly but surely, lapidem cavat planetam, until the orbits of the planets were in resonance with one another9. At this point the orbits of the external planets change rapidly and violently. In particular, Uranus and Neptune move away from the Sun and take their revenge, inserting themselves straight into the ring of asteroids and caroming into their midst like a couple of billiards balls. The asteroids scatter away in every direction, and 99 out of 100 are thrown outwards, many into the inner Solar System, giving to the young Earth, Moon and company a few hundred million years of cataclysms.

meliesGeorge Méliès, Le Voyage dans la lune, 1902

But this blind and chaotic dance does not just happen here. It is a known fact that the majority of the stars present in the Galaxy have planets around them. If in one star system in ten a planet gets expelled sooner or later by the gravitational attraction of its sun, it means that in this moments there are billions of worlds in our galaxy that wander without a destination in interstellar space: millions of gloomy and frozen worlds immersed in a night without end, invisible and elusive, orphans rejected by their mothers, that permeate the space between stars. Yet such worlds are not necessarily dead, and the astronomer David Stevensen has shown that, paradoxically, it is precisely the interstellar cold that allows them to sustain life. If as large as the Earth and young enough (and it is probable that the majority of these derelicts are cast away at the beginning of the formation of a solar system) these will in fact still maintain an atmosphere rich in hydrogen, which otherwise would evaporate into space if heated by the sun. This atmosphere would act as a thermos, stopping the heat of the lava inside the planet from dispersing. The planet remains dark and warm for millions of years, under a thick blanket of hydrogen, its oceans lit only by lighting and the lava of volcanoes.

So, in the end, even when our Solar System is able to get by in the midst of all this and remain stabler than the average planet, there is nothing that we can do to to escape the fate decreed by statistics. Even if we exclude the possibility that the Earth will be disintegrated when our Sun swells up to hundreds of times its present size, and assume that the burned remainder of the Earth will continue to orbit around the miniscule white star remaining after the explosion of the nova, the orbits of the worlds will nevertheless collapse slowly but inexorably, or they will be swept away by an encounter with another star, which would destroy the already precarious equilibrium of the orbits of the Solar System. An encounter that is improbable per se, but which is bound to happen sooner or later, in the arc of some billions of billions of years.

Our fate is written. The Moon and its infinitely ancient craters are the eternal symbol that there are no watchmakers in our heavens to save us, and our seasons are not forever.

1. Laplace is notable hero of atheism. When Napoleon summoned him about his book about natural philosophy asking: «How is it that in this book you do not give space to the action of the Creator?», he responded simply: «I had no need of that hypothesis.» Let us cherish him.
2. A classic example of a nerd ante litteram (his records from secondary school report that he was as bad in gym as he was good in mathematics), not to mention one of the last great all-purpose scientists, Henri Poincaré had a special penchant for demonstrating theorems as elegantly abstract as they were concretely disturbing: an example is the theorem of recurrence, which demonstrates that every possible state of the Universe, given infinite time, will repeat itself infinitely often. We will have to speak well of him sometimes.
3. If you what demonstrate to colleagues that you are veritable fonts of scientific knowledge, you can summarize the concept saying that “the problem of the three bodies is not generally integrable”.
4. The classic image of chaos theory, for which the wing beat of a butterfly in Hawaii can eventually bring about a storm in Paris (or vice versa).
5. Too bad.
6. The tombstone on the hopes of Laplace’s demon is quantum physics: even in the case that we wanted to measure all of these factors, to measure them with exactness is both formally and literally impossible.
7. Hot in the literal sense: they have measured the temperature of the dust and it is still hot from the explosion!
8. I thought for a moment about leaving you with the Star Wars image where millions of brown stones hurl through space spattering on the moon all at once. In reality you should imagine a cataclysm capable of sweeping away life on Earth, around every 100 years, for a few million years.
9. Resonance means that when ball A makes an orbit, ball B completes two, so that A and B are close to one another always more or less at the same point. The influences of the two planets therefore constantly add up with one another, and the reciprocal perturbation on the orbit is in the end much more violent.