In Chapter 2 we saw the basic workings of the tree algorithm. Now we will discuss some methods that can be used to optimise the performance of this type of code. Although most of these techniques are not specific to tree codes, they are not always straightforward to implement within a hierarchical data structure. It therefore seems worthwhile to reconsider some of the common tricks of the N-body trade, in order to make sure that the tree code is optimised in every sense – not just in its N log N scaling.

There are basically two points of possible improvement:

• Improvement of the accuracy of the particle trajectory calculation by means of higher order integration schemes and individual timesteps. This is especially important for problems involving many close encounters of the particles, that is, ‘collisional’ problems.

• Speedup of the computation time needed to evaluate a single timestep by use of modern software and hardware combinations, such as vectorisation, and special-purpose hierarchical or parallel computer architecture.

Individual Timesteps

For most many-body simulations one would like the total simulated time T = ntΔt (where nt is the number of timesteps) to be as large as possible to approach the hydrodynamic limit. However, the choice of the timestep Δt has to be a compromise between this aim and the fact that as Δt increases, the accuracy of the numerical integration gets rapidly worse.

The hierarchical tree method can not be adapted only for Monte Carlo applications: It can also be modified to perform near-neighbour searches efficiently. This means that the tree algorithm could also have applications for systems with short-range or contact forces. Hernquist and Katz (1989) first showed how the tree structure can be used to find near neighbours through range searching. Following their method, the near-neighbour search is performed the following way.

Consider a system in which only neighbours lying within a distance h will interact with the particle i in question. For the near-neighbour search this sphere is enclosed in a cube whose sides are of length 2h. The tree is built the usual way and the tree search starts at the root. The tree search is performed in a very similar way to the normal force calculation of Section 2.2 by constructing an interaction list. The main difference is that the s/d criterion is substituted by the question: ‘Does the volume of the search cube overlap with the volume of the pseudoparticle presently under consideration?’

If there is no overlap, this branch of the tree contains no near neighbours and is not searched any further. However, if there is an overlap, the cell is subdivided into its daughter cells and the search continues on the next highest level. If the cell is a leaf – meaning there is only one particle in the cell – one has to check whether it actually lies within the radius h of particle i.

THE RESTLESS UNIVERSE

From ancient Hindu mythology comes this story about the Pole Star: King Uttanapada had two wives. The favourite, Suruchi, was haughty and proud, while the neglected Suniti was gentle and modest. One day Suniti's son Dhruva saw his co-brother Uttama playing on their father's lap. Dhruva also wanted to join him there but was summarily repulsed by Suruchi, who happened to come by. Feeling insulted, the five-year-old Dhruva went in search of a place from where he would not have to move. The wise sages advised him to propitiate the god Vishnu, which Dhruva proceeded to do with a long penance. Finally Vishnu appeared and offered a boon. When Dhruva asked for a place from where he would not have to move, Vishsnu placed him in the location now known as the Pole Star – a position forever fixed.

Unlike other stars and planets, the Pole Star does not rise and set; it is always seen in the same part of the sky. This immovability of the Pole Star has proved to be a useful navigational aid to mariners from ancient to modern times. Yet, a modern-day Dhruva could not be satisfied with the Pole Star as the ultimate position of rest. Let us try to find out why.

The Pole Star does not appear to change its direction in the sky because it happens to lie more or less along the Earth's axis of rotation. As the Earth rotates about its axis, other stars rise over the eastern horizon and set over the western horizon.

FUSION

It is often argued that man's growing energy needs will be met if he succeeds in making fusion reactors. In a fusion reactor, energy is generated by fusing together light atomic nuclei and converting them into heavier ones. The primary fuel for such a fusion reactor on the Earth would be heavy hydrogen, whose technical name is deuterium. Through nuclear fusion, two nuclei of deuterium are brought together and converted to the heavier nucleus of helium, and in this process nuclear energy is released.

The following is the recipe for a fusion reactor. First, heat a small quantity of the fusion fuel, deuterium, above its ignition point – to a temperature of some 100 million degrees Celsius. Second, maintain this fuel in a heated condition long enough for fusion to occur. When this happens, the energy that is released exceeds the heat input, and the reactor can start functioning on its own. The third and final part of the operation involves the conversion of the excess energy to a useful form, such as electricity.

The primary fuel for this process, the heavy hydrogen, is chemically similar to but a rarer version of the commonly known hydrogen. An atom of ordinary hydrogen is made up of a charged electrical particle called the proton at the nucleus with a negatively charged particle, the electron, going round it. The nucleus of heavy hydrogen carries an additional particle called the neutron in its nucleus. The neutron has no electric charge so the total charge of the nucleus of heavy hydrogen is the same as that of ordinary hydrogen.

It is often said that modern theoretical physics began with Newton's law of gravitation. There is a good measure of truth in this remark, especially when we take into account the aims and methods of modern physics – to describe and explain the diverse and complex phenomena of nature in terms of a few basic laws.

Gravity is a basic force of the Universe. From the motions of ocean tides to the expansion of the Universe, a wide range of astronomical phenomena are controlled by gravity. Three centuries ago Newton summed up gravity in his simple inverse-square law. Yet, when asked to say why gravity follows such a law, he declined to hazard an opinion, saying ‘Non fingo hypotheses’ (I do not feign hypotheses). A radically new attempt to understand gravity was made in the early part of this century by Einstein, who saw in it something of deeper significance that linked it to space and time. The modern theoretical physicist is trying to accommodate it within a unified theory of all basic forces. Yet, gravity remains an enigma today.

In this book I have attempted to describe the diversity, pervasiveness, and importance of this enigmatic force. It is fitting that I have focused on astronomical phenomena, because astronomy is the subject that first provided and continues to provide a testing ground for the study of gravity. These phenomena include the motions of planets, comets, and satellites; the structure and evolution of stars; tidal effects on the Earth and in binary star systems; gigantic lenses in spaced highly dense objects, such as neutron stars, black holes, and white holes; and the origin and evolution of the Universe itself.

THE BLACK HOLE IN HISTORY AND ASTRONOMY

The oldest mention of a black hole is found not in books of physics or astronomy but in books of history. In the summer of the year 1757, Nawab Siraj-Uddaula, the ruler of Bengal in eastern India, marched on Calcutta to settle a feud with the British East India Company. The small garrison stationed in Fort William at Calcutta was hardly a match for the Nawab's army of 50 000. In the four-day battle that ensued, the East India Company lost many lives, and a good many, including the company's governor, simply deserted. The survivors had to face the macabre incident now known as the Black Hole of Calcutta.

The infuriated Nawab, whose army had lost thousands of lives in the battle, ordered the survivors to be imprisoned in what came to be known as the Black Hole, a prison cell in Fort William. In a room 18 feet by 14 feet, normally used for housing three or four drunken soldiers, the 146 unfortunate survivors were imprisoned. The room had only two small windows (see Figure 7–1). During the 10 hours of imprisonment, from 8 p.m. on 20 June to 6 a.m. on 21 June in the hottest part of the year, 123 prisoners died. Only 22 men and 1 woman lived to tell the tale.

Apart from its macabre aspect, the Black Hole of Calcutta did bear some similarity to its astronomical counterpart, involving as it did a large concentration of matter in a small space from which there was no escape.

WHEN NEWTON AND EINSTEIN AGREE

Einstein's general theory of relativity and Newton's law of gravitation offer radically different interpretations of the phenomenon of gravity. Yet, in practical terms, the difference between their predictions seem to be very small. In Chapter 5 we saw two examples of observations in the solar system: the precession of the orbit of Mercury and the bending of light rays from a distant star by the Sun. In both cases the differences in the predictions of Newton and Einstein are very small and are measurable only with very patient and sophisticated astronomical observations. Is it just a coincidence that these two approaches give almost the same answer?

A mathematical analysis of Einstein's equations tells us that the agreement between the two approaches is not coincidental. It can be shown that, in all phenomena of weak gravitational effects and where the gravitating bodies are moving slowly compared to light, the two theories must almost agree. In our discussion of the escape speed in Chapter 3, we saw how to measure the relative strength of gravity. We use the criterion of the escape speed in the present context to understand the difference between ‘weak’ and ‘strong’ gravity. The rule is simple: compare the escape speed V with the speed of light c. If the ratio V/c is very small compared to 1, the gravitational effects are weak. If the ratio is comparable to 1, say between 0.1 and 1, the gravitational effects are strong (see Figure 6–1). Referring back to Table 3–2, we see that the gravitational effects are weak in all cases except on the surface of neutron stars.

THE PROBLEMS OF LARGE-SCALE STRUCTURE

More than seven decades have elapsed since Friedmann proposed his mathematical models that describe the expanding Universe. As we saw in Chapter 9, these models lead to the conclusion that the Universe was created some 10–15 billion years ago in a big explosion (the so-called big bang) after which it has been expanding but more and more slowly because of brakes applied by gravity. This model also tells us that the Universe was very hot to begin with, and dominated by radiation, but with expansion it has cooled down and the temperature of the radiation background today is 2.73 kelvin (see Figure 9–10) as measured by the COBE satellite and other groundbased detectors. And one other set of relics of the hot era, namely the light nuclei like deuterium, helium, etc., are found in the right amount all over the Universe. Thus, we concluded the last chapter with a fair degree of confidence in the big–bang scenario.

However, over the last quarter of a century astronomical observations have become more sophisticated and the views of the largescale structure of the Universe they present go well beyond the simplified assumptions of a ‘homogeneous and isotropic Universe’. We shall see, for example, in Figure 10–1 how galaxies are distributed over the sky in depth. The dots in the figure represent galaxies and their distribution is clearly not smooth, as a homogeneous Universe would have us believe.

WAS THERE REALLY A BIG BANG?

The big-bang cosmology described in the last two chapters has a large following amongst the astronomical community. The models of Friedmann are able to account for the observed expansion of the Universe, for the smooth background of microwave radiation, and for the abundance of light nuclei that cannot be generated inside stars. Are these not good enough reasons for believing in the overall picture?

Playing the devil's advocate in this chapter, let me voice a few dissenting views. First, a scientific theory, howsoever successful it may be, must always be vulnerable to checks of facts and conceptual consistency. Even a well established theory like Newton's had to give way to Einstein's when it was found wanting under these checks (see Chapter 5). The formidable facade of big-bang cosmology is likewise developing cracks that can no longer be plastered over.

The first crack has actually been there right from the beginning and may have been noticed by the reader of Chapter 9. He or she may ask the questions, ‘What preceeded the big bang? How did the matter and radiation in the Universe originate in the first place? Does it not contradict the law of conservation of matter and energy?’

These questions cannot be answered within the framework of Einstein's general theory of relativity, which was used to construct the Friedmann models. The moment of ‘big bang’ is a singular epoch, according to the theory, just as the end of a collapsing object, described in Chapter 7, is in a singularity.

Our discussion of gravity began with the falling apple and has taken us from ocean tides to the planets, comets, and satellites of the solar system, to the different stages in the evolution of a star, to the curved spacetime of general relativity, to the illusions of gravitational lensing, to the weird effects associated with black holes and white holes, and finally to the large-scale structure of the Universe itself. None of the other basic forces of physics has such a wide range of applications. Although gravity is by far the weakest of the four known basic forces, its effects are the most dramatic.

Indeed, it would be an amusing exercise to speculate on the state of the world if there were no gravity at all! Would atoms and molecules be affected? As far as we know, the presence or absence of gravity does not play a crucial role in the existence and stability of the microworld. The strong, weak, and electromagnetic forces are the main forces at this level. Even at the macroscopic level of the objects we see around us in our daily lives, gravity does not appear to play a crucial role in their constitution or equilibrium. After all, even astronauts have demonstrated that they can live in simulated conditions of weightlessness. Neither the astronauts nor their spacecraft come apart in such circumstances. The basic binding force at this level is the force of electricity and magnetism.

But we can go no further in dispensing with gravity. If we eliminate gravity on a bigger scale, disasters lie in store.

WHY DID THE APPLE FALL?

Apples have played a prominent role in many legends, myths, and fairytales. It was the forbidden apple that became the source of temptation to Eve and ultimately brought God's displeasure upon Adam. It was the apple of discord that led to the launching of a thousand ships and the long Trojan War. It was a poisoned apple that nearly killed Snow White, and so on.

For physicists, however, the most important apple legend concerns the apple that fell in an orchard in Woolsthorpe in Lincolnshire, England, in the year 1666. This particular apple was seen by Isaac Newton, who ‘fell into a profound meditation upon the cause which draws all bodies in a line which, if prolonged, would pass very nearly through the centre of the earth.’

The quotation is from Voltaire's Philosophie de Newton, published in 1738, which contains the oldest known account of the apple story. This story does not appear in Newton's early biographies, nor is it mentioned in his own account of how he thought of universal gravitation. Most probably it is a legend.

It is interesting to consider how rare it is to see an apple actually fall from a tree. An apple may spend a few weeks of its life on the tree, and after its fall it may lie on the ground for a few days. But how long does it take to fall from the tree to the ground? For a drop of, say, 3 metres, the answer is about three-quarters of a second.

THE MASS OF THE EARTH

Although with Newton's pioneering discoveries, gravity was the first basic force of nature to be described and studied quantitatively, it is the weakest of all known basic forces of nature. The other basic forces are the forces of electricity and magnetism and the forces of ‘strong’ and ‘weak’ interaction which act on subatomic particles. It is a measure of the success achieved to date that physicists are able to explain all observed natural and laboratory phenomena in terms of these four basic forces. As we shall see in later chapters, many physicists hope that one day they will be able to bring all the basic forces under the umbrella of one unified force.

Although atomic physicists consider gravity to be the weakest of the four known basic forces of nature, for astronomers gravity is the most dominant force in the celestial environment. How do we assess the strength of gravity in any given situation? We will try to answer this question with a few examples in this chapter.

All of us on the Earth are conscious of gravity. The feeling of weight that we have results from the gravitational pull the Earth exerts on us. Newton's inverse-square law of gravitation described in Chapter 2 tells us how strong this force is on any given body on the Earth's surface. Let m be the mass of the body and M the mass of the Earth. The distance between the body and the Earth is denoted by d.

THE COSMIC ENERGY PROBLEM

We here on Earth are constantly reminded by experts that with advancing technology our energy needs are growing, and that we need to worry about stocks of oil, coal, nuclear fuel, etc. that are needed to generate energy to meet these demands all over the world. How long will the supplies last? Can we extend that period by conserving energy? If so, how? These questions are being debated by experts and lay people alike.

Astronomers face the ‘energy problem’ in their investigations of cosmic sources of radiation. The age-old problem, where the Sun gets its energy to shine so brightly and steadily, has been solved. In Chapter 4 we saw that the key to solar energy lies in the nuclear fusion going on in the central core of the Sun.

But in the 1950s new problems with far greater magnitude began to confront the astronomers. The radio astronomers began to find sources of radio emission whose total energy reservoir exceeded that of the Sun by several billion. Where did the source of this energy lie? The problem was exacerbated in the early 1960s with the discovery of quasi-stellar sources, commonly called quasars. Initially mistaken for stars, quasars turned out to be far more energetic, and far more dramatic in spending their energy.

A typical quasar radiates in visible light as much as a galaxy of hundred billion stars. It also radiates in X-rays and possibly other wavebands.

IS NEWTON's LAW PERFECT?

We left Chapter 2 with the impression that Newton's law of gravitation gave a successful account of the diverse nature of phenomena in which gravity is believed to play a leading role. Not only is this law able to account for motions of such celestial bodies as planets, comets, and satellites, it also helps us in understanding the complex problem of the structure and evolution of the Sun and other stars. Modern scientists use the same law in determining the rocket thrusts, spacecraft trajectories, and the timing of space encounters. That a good scientific law should be basically simple but universal in application is epitomized in Newton's law of gravitation. What more could one ask for?

Yet science by nature is perfectionist. The laws and theories of science are accepted as long as they are able to fulfil its primary purpose of explaining natural phenomena. Any law of science, despite a history of past successes, is inevitably discarded if it fails in even one particular instance. To the scientist, such an event brings mixed feelings. Disappointment and confusion that an old, well established idea has to be given up or modified are coupled with excitement and expectation that nature is about to reveal a new mystery.

Newton's law of gravitation was no exception to this rule. By the beginning of the present century, cracks were beginning to appear in the impressive facade of physics erected on the Newtonian ideas of motion and gravitation.