To specialists in relativity, the black holes in binary star systems and galactic nuclei signify places where the space of our Universe has been punctured by the accumulation and collapse of large masses – collapse to smooth geometrical entities that can be described exactly by simple formulae, the “Kerr solution” to Einstein's equations. As Roger Penrose has emphasized, “It is ironic that the astrophysical object which is strangest and least familiar, the black hole, should be the one for which our theoretical picture is most complete.” Black holes are, according to theory, just as “standardized” as elementary particles such as electrons and protons. All black holes of a given mass and spin are exactly alike.

It seems astonishing testimony to the power of mathematical physics that any entities of the real world can be so fully understood. This thought made a deep impression on Chandrasekhar. In a lecture given in 1975, he said: “In my entire scientific life … the most shattering experience has been the realization that an exact solution of Einstein’s equations of general relativity, discovered by the New Zealand mathematician Roy Kerr, provides the absolutely exact representation of untold numbers of massive black holes that populate the Universe.” It is now even clearer, as we’ve seen in earlier chapters, that massive collapsed objects indeed exist, and that the power supply in quasars and other kinds of active galactic nuclei is, ultimately, gravitational. But how confident can we be that these objects have exactly the properties that Einstein's theory predicts?

It will not have escaped the reader's attention that much of the evidence for black holes is still circumstantial. That in itself does not mean that the case is weak. After all, until the detection of solar neutrinos the evidence that stars like the Sun are fueled by nuclear fusion was also circumstantial, yet we were persuaded by the success that astronomers had achieved in predicting the brightnesses, colors, radii, and other properties of stars with different masses and ages, and in explaining how nuclear reactions in stellar cores and supernova explosions transmute nuclei up the periodic table to account for the relative abundances of the chemical elements. The quest to find black holes has also had many observational successes.

Gravity is the one truly universal force. No substance, no kind of particle, not even light itself, is free of its grasp. It was Isaac Newton who, over three hundred years ago, realized that the force holding us to the ground, and governing a cannon ball's trajectory, is the same force that holds the Moon in its orbit around the Earth. This force, which later came to be called gravity, is the mutual attraction among all bodies. Newton showed how the motion of each planet in the Solar System is the combined outcome of the gravitational pull of the Sun and all the other planets, each contributing according to its mass and distance from the others. Out of Newton's conceptually simple prescriptions came the calculations that have guided spacecraft to all the planets, told us a year in advance that Comet Shoemaker–Levy would hit Jupiter, and enabled us to determine the mass of the Milky Way Galaxy. On cosmic scales gravity dominates over every other kind of force. Every significant level of structure in the Universe – stars, clusters of stars, galaxies, and clusters of galaxies – is maintained by the force of gravity.

Nowhere is gravity stronger than near the objects we call black holes. The effect of gravity at the surface of a body intensifies if the mass of the body is increased, or its size is decreased. Gravity's strength can be characterized by how fast a rocket must be fired to escape from the body. For the Earth, the escape speed is 40 000 kilometers per hour; for the Sun it is nearly a hundred times greater. This tendency for the escape speed to become ever larger for more massive, or more compact, bodies raises an obvious conceptual problem. The gravitational pull of the Sun dominates the Solar System, yet its escape speed is only 1/500 the highest speed possible, the speed of light. What would be the effect of gravity around a body for which the escape speed was as high as that of light?

The Reverend John Michell , an underappreciated polymath of eighteenth-century science, puzzled over this question in 1784.

Narrow, fast streams of gas emerge from deep in the nuclei of radio galaxies and often project millions of light-years into space. Those galaxies are suspected to harbor black holes at their centers, and yet such jets of gas are just about the last thing you would expect a black hole to produce. Aren't black holes supposed to be (at least in popular imagination) “cosmic vacuum cleaners,” sucking in everything within their reach? It seems obvious that gas being drawn toward a black hole can become hot and luminous just before it falls in, thus accounting for the luminosity of an accreting black hole. But the idea that much of the in falling gas can be “turned around” and propelled outward, at 99 percent of the speed of light or more, strains our credulity. Rather, it would have strained our credulity had it been proposed before the advent of radio astronomy. As has happened so many times in the past, a new technology has turned our view of the Universe on its head.

Since their discovery in radio galaxies, jets have proven to be a common phenomenon in the Universe. They often occur when gas with a lot of angular momentum swirls deep into a gravitational field, as, for example, at the center of an accretion disk. We might think of them as celestial waterspouts. They form in all kinds of systems: radio galaxies, X-ray binaries, and even ordinary stars in their infancy. So they are clearly not a manifestation of black holes alone. Yet the jets that apparently do form close to black holes carry the signature of the extreme conditions surrounding their birth – nowhere else in the Universe do we see evidence of matter propelled at such enormous speeds, sometimes to within a fraction of a percent of the speed of light.

Radio Astronomy and the Prediction of Jets

To be precise, radio astronomers rediscovered jets from active galactic nuclei. As long ago as 1917, Heber Curtis had noticed a jet of visible light emerging from the nucleus of the large elliptical galaxy M87, which lies in a cluster of galaxies in the constellation Virgo.

If you select a nearby galaxy at random and observe its nucleus, there is only one chance in a hundred that you will detect moderately vigorous activity attributable to a massive black hole. The chance of stumbling on a powerfully active galaxy – a quasar or giant double radio source – is even smaller. But this does not mean that massive black holes occur only in the nuclei of a small fraction of galaxies, for activity requires not only a massive black hole but also a supply of “fuel.” To power a quasar or luminous Seyfert nucleus by accretion there must be more than a solar mass per year of gas flowing into the black hole. Radio jets powered by a black hole flywheel might survive on a smaller yearly mass budget, but a substantial gas supply is nevertheless needed to maintain the magnetic fields in the hole's vicinity. And, without more vigorous episodes of accretion, the hole's rate of spin will slowly decline.

A solar mass of gas per year is a sizable fuel supply, even by galactic standards. It is unlikely that the nucleus of a galaxy maintains this level of gas inflow for more than a small fraction of its lifetime. Perhaps there is a time early in the galaxy's life when the stars and gas in the nucleus become so closely packed that some kind of runaway catastrophe occurs, dumping matter copiously into the black hole. From time to time, the galaxy might merge with a smaller, gas-rich galaxy or swallow an intergalactic gas cloud – we see examples of such encounters. For a while following such an event, matter might pour into the nucleus at a rate fast enough to power a quasar. But such episodes of violent activity most likely represent a short-lived phase in the life of a galaxy. When the fuel supply dwindles, the activity dies away but the black hole does not disappear. Black holes in “hibernation” – massive black holes now starved of fuel, and therefore quiescent – apparently lurk at the centers of virtually all large galaxies, including our own Milky Way. Their masses are correlated with properties of their hosts, indicating a link with the epoch of galaxy formation in the distant past. These objects can provide clues to the enigma of quasars, and remind us that even the benign-looking galaxies around us could reawaken and turn violent once again.

Ordinary stars like the Sun cannot last forever . The Sun is held in equilibrium by a balance between gravitation and the pressure in its hot interior. Take away that pressure and the Sun would go into free-fall, halving its size in less than an hour. If, on the other hand, gravity were magically switched off, the hot interior would just as suddenly explode and disperse. To us, the Sun looks like a steady beacon in the sky, and in fact its appearance has hardly changed for more than 4 billion years. But the heat of its interior is continuously ebbing. The rays of sunlight you catch on a beach or in a park are plain enough evidence that the Sun is losing energy. And although this heat is now being replenished by nuclear reactions at the Sun's center, these reactions cannot continue indefinitely at their present rate. Eventually the Sun, like any other star, will use up its nuclear fuel, and its finely tuned gravitational equilibrium will fail.

The need for a central power source to replenish the energy lost by a star was recognized in the nineteenth century. If there were no power source at the center, the heat loss would cause the Sun to deflate, at a rate first calculated by Lord Kelvin . The deflation would be far slower than free-fall, because the energy takes a long time to leak out of the Sun's interior – indeed, the process would take about 10 million years. Kelvin knew, however, that 10 million years was much shorter than the that biologists and geologists had estimated for the age of the Earth and its life forms. Some energy source within the Sun must prolong its life, and Kelvin readily showed that chemical energy would be utterly inadequate. To reconcile his theory with the Earth's estimated age would require, as Kelvin wrote, “some unknown source of energy laid down in the storehouse of creation.” Not until the 1930s, when the energetic potential of nuclear fusion was recognized, was there a solution to Kelvin's paradox. The Sun's center is hot enough for hydrogen to fuse into helium at just the rate needed to compensate losses; moreover, the energy release is so high that the supply of hydrogen could keep the Sun shining for several billions of years.

How would you go about searching for the black hole remnant of a massive star in our Galaxy? When this first became a serious question among astronomers early in the 1960s, surprisingly few of the experts on black holes bothered to ask it. The reasons, in part, were “cultural.” C. P. Snow had written about “two cultures” – scientific and humanistic – diverging in language and habits of thought, and becoming less and less able to communicate with each another. But cultural divergences had occurred within the physical sciences as well. Black holes, with their elegant mathematical properties, were primarily the intellectual toys of a highly specialized group of relativists – that is, scientists specializing in the subtleties of general relativity. In the tradition of Einstein, relativists liked to rely on pure thought and mathematical elegance to deduce deep truths about the nature of the Universe. On the other hand, astrophysics had a strong empirical tradition. Theory was important, but it was (and remains) observation that drove progress in the field.

The discovery of black holes in nature required synergism between astronomers and relativists; fortunately, just such an alliance had developed by the mid-1960s. Perhaps the “send-off” was the First Texas Symposium on Relativistic Astrophysics, which was held in Dallas in December 1963 in the aftermath of the discovery of the extraordinarily luminous and distant objects known as quasars. There the astrophysicist Thomas Gold (who in 1968 became the first to identify pulsars as rotating magnetized neutron stars) heralded the collaboration in his famous after-dinner speech, suggesting “that the relativists with their sophisticated work are not only magnificent cultural ornaments but might actually be useful to science! Everyone is pleased: the relativists who feel that they are being appreciated and are experts in a field [i.e., relativistic astrophysics] they hardly knew existed; the astrophysicists for having enlarged their domain, their empire, by the annexation of another subject – general relativity. It is all very pleasing, so let us hope that it is right. What a shame it would be if we had to go and dismiss all the relativists again.”

Perhaps, too, the observational prospects for detecting black holes had been largely ignored because success seemed unlikely. But the possibility of failure did not deter the Soviet theorist Yakov Zel’dovich, the American Edwin Salpeter, and a few others from boldly speculating on possible observational signatures.

Once planetesimals have formed, the dominant physical process that controls further growth is their mutual gravitational interaction. Conventionally the only further role the gas disk plays in terrestrial planet formation is to provide a modest degree of aerodynamic damping of protoplanetary eccentricity and inclination. In this limit the physics involved – Newtonian gravity – is simple and the problem of terrestrial planet formation is well posed. It is not, however, easy to solve. It would take 4 × 109 planetesimals with a radius of 5 km to build the Solar System's terrestrial planets, and it is infeasible to directly simulate the N-body evolution of such a system for long enough (and with sufficient accuracy) to watch planets form. Instead a hybrid approach is employed. For the earliest phases of terrestrial planet formation a statistical approach, similar to that used in the kinetic theory of gases, is both accurate and efficient. When the number of dynamically significant bodies has dropped to a manageable number (of the order of hundreds or thousands), direct N-body simulations become feasible, and these are used to study the final assembly of the terrestrial planets. Using this two-step approach has known drawbacks (for example, it is difficult to treat the situation where a small number of protoplanets co-exist with a large sea of very small bodies), but nevertheless it provides a reasonably successful picture for how the terrestrial planets formed.

Planets form from protoplanetary disks of gas and dust that are observed to surround young stars for the first few million years of their evolution. Disks form because stars are born from relatively diffuse gas (with particle number density n ~ 105 cm−3) that has too much angular momentum to collapse directly to stellar densities (n ~ 1024 cm−3). Disks survive as well-defined quasi-equilibrium structures because once gas settles into a disk around a young star its specific angular momentum increases with radius. To accrete, angular momentum must be lost from, or redistributed within, the disk gas, and this process turns out to require time scales that are much longer than the orbital or dynamical time scale.

In this chapter we discuss the structure of protoplanetary disks. Anticipating the fact that angular momentum transport is slow, we assume here that the disk is a static structure. This approximation suffices for a first study of the temperature, density, and composition profiles of protoplanetary disks, which are critical inputs for models of planet formation. It also permits investigation of the predicted emission from disks that can be compared to a large body of astronomical observations. We defer for Chapter 3 the tougher question of how the gas and solids within the disk evolve with time.

The formation of terrestrial planets from micron-sized dust particles requires growth through at least 12 orders of magnitude in size scale. It is conceptually useful to divide the process into three main stages that involve different dominant physical processes:

• Planetesimal formation. Planetesimals are defined as bodies that are large enough (typically of the order of 10 km in radius) that their orbital evolution is dominated by mutual gravitational interactions rather than aerodynamic coupling to the gas disk. With this definition it is self-evident that aerodynamic forces between solid particles and the gas disk are of paramount importance in the study of planetesimal formation, since these forces dominate the evolution of particles in the large size range that lies between dust and substantial rocks. The efficiency with which particles coagulate upon collision – loosely speaking how “sticky” they are – is also very important.

• Terrestrial planet formation. Once a population of planetesimals has formed within the disk their subsequent evolution is dominated by gravitational interactions. This phase of planet formation, which yields terrestrial planets and the cores of giant planets, is the most cleanly defined since the basic physics (Newtonian gravity) is simple and well-understood. It remains challenging due to the large number of bodies – it takes 500 million 10 km radius planetesimals to build up the Solar System's terrestrial planets – and long time scales involved.

• Giant planet formation and core migration. Once planets have grown to about an Earth mass, coupling to the gas disk becomes significant once again, though now it is gravitational rather than aerodynamic forces that matter. […]

The study of planet formation has a long history. The idea that the Solar System formed from a rotating disk of gas and dust – the Nebula Hypothesis – dates back to the writings of Kant, Laplace, and others in the eighteenth century. A quantitative description of terrestrial planet formation was already in place by the late 1960s, when Viktor Safronov published his now classic monograph Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets, while the main elements of the core accretion theory for gas giant planet formation were developed in the early 1980s. More recently, a wealth of new observations has led to renewed interest in the problem. The most dramatic development has been the identification of extrasolar planets, first around a pulsar and subsequently in large numbers around main-sequence stars. These detections have furnished a glimpse of the Solar System's place amid an extraordinary diversity of extrasolar planetary systems. The advent of high resolution imaging of protoplanetary disks and the discovery of the Solar System's Kuiper Belt have been almost as influential in focusing theoretical attention on the initial conditions for planet formation and the role of dynamics in the early evolution of planetary systems.

My goals in writing this text are to provide a concise introduction to the classical theory of planet formation and to more recent developments spurred by new observations. Inevitably, the range of topics covered is far from comprehensive.

Planets can be defined informally as large bodies, in orbit around a star, that are not massive enough to have ever derived a substantial fraction of their luminosity from nuclear fusion. This definition fixes the maximum mass of a planet to be at the deuterium burning threshold, which is approximately 13 Jupiter masses for Solar composition objects (1 MJ = 1.899 × 1030 g). More massive objects are called brown dwarfs. The lower mass cut-off for what we call a planet is not as well defined. Currently, the International Astronomical Union (IAU) requires a Solar System planet to be massive enough that it is able to clear the neighborhood around its orbit of other large bodies. Smaller objects that are massive enough to have a roughly spherical shape but which do not have a major dynamical influence on nearby bodies are called “dwarf planets.” It is likely that some objects of planetary mass exist that are not bound to a central star, either having formed in isolation or following ejection from a planetary system. Such objects are normally called “planetary-mass objects” or “free-floating planets.”

Complementary constraints on theories of planet formation come from observations of the Solar System and of extrasolar planetary systems. Space missions to all of the planets have yielded exquisitely detailed information on the surfaces (and in some cases interior structures) of the Solar System's planets, satellites, and minor bodies.