Secrets of the Universe

by Paul Murdin

Between 1609 and 1621, Johannes Kepler formulated three laws of planetary motion, which he derived from the accurate measurements of the orbit of Mars made by his teacher, Tycho Brahe. In his First Law, Kepler found that the path of each planet around the Sun is not perfectly circular, but elliptical, with the Sun at one of the foci of the ellipse. He formulated a further two mathematical laws, describing the rate at which each planet moves along its orbit as dependent on the size of the orbit. The Second Law was that the line joining the Sun to the planet sweeps through equal areas of the orbit in equal periods of time. Finally, the Third Law was that the ratio of the squares of the orbital periods for two planets is equal to the ratio of the cubes of the semi-major axes (half the longest diameter of each orbital ellipse).

  In his 1687 treatise Principia, Isaac Newton showed that Kepler’s laws were underpinned by the more fundamental theories of dynamics and gravitation, and added an all-important principle about gravity to existing theories of dynamics: all bodies in the Universe attract one another across space, and the force of this attraction between any two bodies varies according to the inverse square of the distance between them (that is, 1 divided by the distance squared). The popular story of this discovery is that in 1666 Newton was musing in the garden of his mother’s house in Lincolnshire when he saw an apple fall. He began to think that the power of gravity was not limited to a certain height above the ground but extended to the Moon and beyond.

  The first record of the anecdote dates from the year before Newton died and it may be the poetic reminiscence of an old man, a story improved in the retelling; the echoes of the Genesis account of the Tree of Knowledge add to its repeatability. Whatever the circumstances, Newton had a flash of insight about gravity that sparked his investigation into how the Moon orbited the Earth and the Earth the Sun. His findings eventually would transform every branch of science and alter the fundamental human view of the Universe.

  Newton’s assertion that objects attracted each other across space was controversial. The concept was hard for everyone to grasp and was ridiculed by philosophers, particularly Cartesians, who thought that space was filled with a substance called the plenum, swirling in vortices that transmitted force from one body to another. But Newton’s theory of gravitation worked. It explained the motions of the planets and even the shape of the Earth. It enabled Edmond Halley to predict the return of his eponymous comet, and in due course led Urbain Le Verrier and John Couch Adams to the discovery of Neptune in 1846.

  Modern astronomers use Newton’s theory not only to calculate the motions of planets in the Solar System but also to calculate how satellites move in orbit. These calculations are essential in planning complicated orbital tours that loop a spacecraft via nearby planets to more distant ones using the ‘gravity-assist’ technique to make the spacecraft pick up (or lose) speed – for instance, when approaching the planet Mercury, which is perilously close to the Sun and notoriously difficult to reach. Although computers enable more complicated calculations to be made accurately, the essential theory has remained the same for over 300 years.

  Newton’s theory of gravitation was considered superior to Kepler’s laws of planetary motion, which only apply to cases where two bodies are interacting (usually the Sun and a planet, but also two stars or two galaxies). A comet may orbit through the Solar System controlled mainly by the Sun and obey Kepler’s laws, but it may pass close to a planet and be pulled from its elliptical orbit, at which point Kepler’s laws fail to predict its movements because three bodies are involved in the interaction. Moreover, the orbits of planets are actually more complex than the simple ellipses assumed by Kepler – for instance, there are perturbations in the Earth’s orbit that cause the Milankovič climate cycles. In principle, Newton’s theory could be applied to any type of orbit and any number of planets, stars and galaxies.

  In practice, however, the extension of Newton’s theory from two bodies to even just three proved difficult, indeed, intractable. Entering an 1887 competition to solve what by then had become known as the ‘Three-Body Problem’, the French mathematician Henri Poincaré found that he could not give an exact prediction for the orbits of three stars or planets mutually attracted by gravity. He was able to calculate the orbits numerically – we would nowadays do this by computer, he did it by hand – but the paths were ‘so tangled that I cannot even begin to draw them’. Moreover, he found that when the three bodies were started from lightly different initial positions, the orbits would be entirely different. ‘It may happen that small differences in the initial positions may lead to enormous differences in the final phenomena. Prediction becomes impossible.’ Poincaré had discovered a concept that we now term ‘chaos theory’.

  Poincaré’s work has been confirmed by modern computer techniques. The planetary orbits, especially those of the inner planets, are ‘chaotic’. If you displace one of the planets by just a single centimetre from its initial position, you might logically expect to find the same single centimetre difference in the planet’s final position in 10 million, or even 100 million years. But in practice the planet’s final position could be anywhere in its orbital range.

  In modern physics, ‘chaos’ is the word used to describe behaviour that is predictable in the short term but that in the long term depends so much on the starting conditions that minuscule changes can have enormous effects that are impossible to calculate. Weather can be predicted, more or less accurately, one day or one week ahead. However, since something as negligible as a butterfly flapping its wings can set off long-term changes in the air currents, and since there are millions of butterflies all over the world constantly flapping their wings, meteorologists cannot predict at any given moment whether a hurricane will strike Texas a year later. This phenomenon was discovered in 1963 by Edward Lorenz, an MIT meteorologist. He was testing a new computer and re-ran a weather model that he had run before on the old computer. For simplicity in running the test, he included fewer decimal places in the initial data. When he ran the simulation again, he found that the weather patterns generated were completely different. The marginal changes in the data simulated the uncertainties of the real measurements, so it was not a problem with his model, nor with his new computer: it was an innate practical limitation of mathematics. It showed why weather prediction was such an uncertain business. Lorenz called the problem the ‘butterfly effect’; later, American mathematician and physicist James Yorke coined the name ‘chaos’.

  In principle, Newton’s theory of mutual attraction could be used to calculate the future state of the entire Universe. The French mathematician Pierre-Simon Laplace therefore imagined a demon who would be able to predict everything that would happen, down to the movement of the tiniest atom:

  We may regard the present state of the Universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the Universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

  This was the first published articulation of scientific determinism. But because the Universe is so large and made up of so many bodies and particles, Newton’s theory – for all the astounding discoveries it has made possible – cannot predict the future. The ultimate secret of the Universe is still a secret.

  Relativity

  The nature of space and time

  Spacetime tells matter how to move; matter tells spacetime how to curve.

  J. A. Wheeler, A Journey into Gravity and Spacetime, 1990

  According to Galileo and Isaac Newton, who followed the classical Greek philosophers, space and time are separate from each other, and together form a framework within which events occur. But Albert Einstein thought space and time were more closely connected. H
e saw them as a single entity: spacetime. This spacetime is not simply the framework or stage on which events unfold – it affects how they unfold. Einstein’s theory would inform all major twentieth-century discoveries about the nature of the Universe.

  Einstein’s theory of Special Relativity is based on two principles. The first principle is that every law of nature has the same mathematical form. The second principle is that the speed of light (represented by the letter c in Einstein’s equations) is the same to all observers who move at constant velocity relative to one another.

  Relativity is a more important concept in astronomy than in everyday life. For example, because the distances between places in space are enormous, light takes years, perhaps millions of years, to move between them. Two events that are simultaneous for one observer may therefore occur at different times according to other observers. A double-star eclipse in the Milky Way and the explosion of a nova in another galaxy, which on Earth we see happening in the sky on the same night, may appear at completely different times for observers in another, distant galaxy.

  Time dilation is another effect of relativity that has immediate astronomical consequences. The faster an object moves, the slower it seems to experience time – at least to the observer. A clock that is moving (for instance, aboard a spacecraft in orbit) runs slow when compared with an identical clock that is stationary on the ground. Terrestrial clocks are used to time the ticks of pulsars; however, terrestrial clocks are not actually stationary, but run fast and slow depending on the speed of the Earth as it revolves around the Sun in its eccentric annual orbit. This time dilation – a difference in the measurable passage of time due to an object’s physical motion – must be taken into account when terrestrial clocks are used to measure astronomical processes. The effects of time dilation are not limited to man-made terrestrial clocks, but apply to any physical process of fixed duration. Supernovae have a ‘clock’, namely the time that it takes for their light to fade from maximum brightness. Because of time dilation, distant supernovae travelling at high speeds inside their parent galaxies, which are receding further from the Earth as the Universe expands, fade more slowly than nearby supernovae in our own Galaxy, which are travelling at the same speed as the Earth.

  In the case of cause-and-effect events, however, the signal that triggers the effect cannot travel faster than the speed of light. As a result, the cause always precedes the effect, no matter how fast you are travelling when you see the two events. This rule is used in astronomy to estimate the maximum size of a source of variable light, such as a pulsar. If a star varies in intensity during a clearly defined period of time (T) – say, 1 minute – its size cannot be greater than cT (that is, 1 minute multiplied by c, the speed of light, or 1 light minute). This is because different parts of the star must be able to communicate with each other so that their brightnesses vary together. This argument was used to discover the small size of quasars, because some vary over only a day and therefore must be less than a light day (about the size of the Solar System) at their maximum dimension; pulsars vary in less than a second and therefore must be less than a light second in diameter – they cannot be bigger than a planet.

  The relationship between mass and energy in the theory of Special Relativity is fundamental to understanding the nature of stars. When a body’s mass reduces by a given amount (represented in equations by m), its energy (represented by E) reduces by an equivalent amount. This is the iconic E = mc2, equation, which was first presented by Einstein in 1905.

  The equation expresses the direct relationship between an object’s mass and its energy that underlies the whole of nuclear physics. The speed of light, c, is a big number, and c2 is even bigger, so a little mass makes a lot of energy. The equation explains why huge amounts of energy are released when hydrogen fuses to helium, and powers the stars, and why gravitational waves have been detected from the huge amounts of energy liberated by merging black holes, even though the gravitational waves are so weak.

  Einstein’s theory of General Relativity represents a finetuning of the work of Galileo and Isaac Newton. It incorporates the Principle of Equivalence discovered by Galileo, which Newton had incorporated into his theory of gravitation.

  Galileo knew that the gravitational force on a body is directly proportional to the mass, and that the body’s resistance to being moved by gravity (physicists call this ‘inertia’) is also proportional to the mass. The Principle of Equivalence claims that these two factors precisely cancel each other out. This is why all bodies falling in gravity fall together, no matter what their mass. Galileo is said to have dropped two weights of different sizes from the Leaning Tower of Pisa to demonstrate this: in the Earth’s gravity both weights fell at the same rate and hit the ground at the same time, so far as he could judge. The Apollo astronaut David Scott repeated the experiment on the airless Moon in 1971, when a feather dropped to the lunar surface without air resistance hit the ground at exactly the same time as a hammer.

  Newton’s theory of gravity works well enough for most calculations of the orbits of the planets but on rare occasions – for instance, in the case of the planet Mercury – its accuracy subtly and mysteriously fails. Einstein discovered the reason for this in 1915. He realized that General Relativity alters the orbit of Mercury by an extra 43 arc seconds (43⁄3600 of a degree) per century because of its close proximity to the Sun, which exerts a strong gravitational force. This discrepancy had confounded astronomers since the nineteenth century. The French astronomer Urbain Le Verrier had thought it might be caused by an undiscovered planet, which he called Vulcan and unsuccessfully tried to locate inside Mercury’s orbit. Vulcan faded from astronomy. When Albert Einstein discovered that the discrepancy could be explained by his theory of General Relativity, he reported that ‘for a few days I was beside myself with joyous excitement’. The realization that General Relativity could explain the longstanding mystery of Mercury’s orbit gave him confidence to publish his theory.

  General Relativity has, since then, facilitated countless astronomical discoveries. General Relativity is used to explain how black holes work, and to calculate the orbits of the binary pulsars, which test the theory to the limit. In the twenty-first century General Relativity is essential in solving the problems of dark energy and gravitational waves.

  Radio Waves

  A new window on the Universe

  How fathomless the mystery of the Unseen is! We cannot plumb its depths with our feeble senses – with eyes which cannot see the infinitely small or the infinitely great, nor anything too close or too distant, such as the beings who live on a star or the creatures which live in a drop of water…Ah! If we had other senses which would work other miracles for us, how many more things would we not discover around us!

  Guy de Maupassant, Le Horla et autres contes fantastiques, 1887

  Radio waves are a type of invisible radiation at the extreme end of the electromagnetic spectrum. Stars and galaxies emit radio waves of various lengths, along with the full spectrum of visible and invisible light. When a radio engineer in New Jersey heard a faint, mysterious static on his home-built antenna, a window began to open on this unseen universe.

  In 1928 radio engineer Karl Jansky started work at Bell Telephone Laboratories in northeast New Jersey. His job was to investigate the sources of interference that might affect transatlantic telephony, such as electrical equipment and automobile ignition systems, but also natural sources, such as thunderstorms. To this end he built an antenna that was sensitive to emissions at a wavelength of 15 metres (the sort of radio waves that you can pick up on a shortwave radio) and had a certain amount of directional discrimination. The antenna consisted of an open rectangular wooden frame with aerial wires strung over it, which rotated around a track on wheels, earning it the name ‘the Merry-Go-Round’.

  By 1932 Jansky had found three natural sources of ‘static’, or radio noise. The first type of interference was clearly associated with local thunderstorms. A second type had similar c
haracteristics but was weaker and steadier, with occasional peaks. He realized that this static corresponded to distant thunderstorms in the tropics, and that it reached New Jersey via radio waves that bounced off the ionosphere (the layer of electrically charged plasma in the Earth’s upper atmosphere). But the third type of natural static was a mystery. Jansky described it as ‘a very steady hiss-type static’. It was so weak that it had little practical effect on radio telephony, but Jansky’s curiosity was aroused and he decided to investigate it further.

  As he rotated the Merry-Go-Round, Jansky noticed that the mysterious static at first seemed to peak in intensity when the antenna was directed towards the Sun, but as the year progressed this correspondence broke down. Jansky began to study astronomy textbooks and concluded that the source of the hiss was not the Sun, but an unknown object fixed in space. In 1933 he presented a paper to fellow radio engineers called ‘Electrical Disturbances Apparently of Extraterrestrial Origin’. A press release on the subject by Bell Laboratories led to immense publicity, but astronomers who picked up the discovery puzzled fruitlessly over the precise origin of the static, and it was Jansky himself who discovered that the steady hiss came from a band along the Milky Way, peaking in the constellation Sagittarius, which is the location of the centre of the Galaxy. However, the economic stress of the Great Depression put a stop to Jansky’s research and forced him to turn to work of more practical benefit to his employer.

  Grote Reber, a radio engineer who pursued astronomy as a hobby, was one of the few of Jansky’s colleagues who continued to investigate the static. In a suburban lot in Wheaton, Illinois, Reber built a parabolic reflector 9 metres in diameter, which was able to measure the strength of the mysterious radio emissions from the Milky Way at metre and centimetre wavelengths. Reber’s reflector attracted much curiosity; when a light plane circling the radio telescope suffered an engine failure and had to make an emergency landing, a rumour circulated that he was transmitting a death ray. Reber was the first to map the Milky Way using radio waves. Like Jansky, he found a peak in Sagittarius at the position of the Galactic Centre, but also saw other bright radio sources in Cygnus and Cassiopeia. They were not accurately enough located to identify by Reber’s pioneering work but proved to be an exploding galaxy and a supernova remnant. Later, Reber pioneered investigations into very long-wavelength radio astronomy, and emigrated to Tasmania, where, because of the island’s position in the magnetic field of the Earth, it is easier to study this type of radiation.