The Universe as revealed in state of the art surveys appears organized with large scale clustering of galaxies in filaments bounding large scale voids, as shown in Fig. 1.1. This distribution was discovered by the Center for Astrophysics (CfA) Redshift Survey [364], and has now been mapped in great detail by the Two-Micron All Sky Survey (2MASS) [542] and the Sloan Digital Sky Survey (SDSS) [541].

The large scale structure emerged out of embryonic inhomogeneities in the early evolution of the Universe [263] as imprinted, in pattern and amplitude, in the cosmic microwave background (CMB). These fluctuations in the CMB can be seen as tiny temperature variations with an amplitude of about 10 microkelvin, roughly 10−5 of the average CMB temperature, 2.724 K [308]. The present day low CMB temperature results from adiabatic cooling in the cosmological expansion over 13.75 Gyr [308], since radiation decoupled from matter, when the Universe was a mere 400 kyr of age and about one thousand times smaller in linear size. By Newtonian attraction, the associated local inhomogeneities in the (dark) matter distribution gave rise to the large scale structure of the Universe, as presently observed. The evolution of this structure is accompanied by violent processes and entropy creation on scales of ~1 Mpc and less [558, 263, 644], in addition to entropy of possibly cosmological origin (e.g., [238, 190]).

Accretion disks play a central role in essentially all compact astrophysical systems, from active galactic nuclei on galactic scales, to X-ray binaries, microquasars and gamma-ray bursts on stellar scales. They represent the accumulation of angular momentum in the attraction of matter from the host environment – the ionized medium around supermassive black holes provided by stellar winds and/or tidally disrupted stars, Roche lobe overflow from a companion star in compact binaries, and fall back matter from the envelope of a collapsed star in GRBs. Their fluid dynamical properties are key to their radiative signatures, stability, wave modes and outflows. To leading order, an accretion disk assumes a largely Keplerian motion, subject to inflow of matter from larger radii and a consequent outflow of angular momentum. Dissipation of their rotational energy and the resulting radial motion are mediated by some form of macroscopic viscosity, as will be discussed below. A fraction of the binding energy may be released as disk winds, further complicating the system. Moreover, for certain configurations gravitational-wave emissions may become appreciable, opening another window into the physics of black holes and their accretion disks.

On large scales, accretion disks can be observed directly, such as the disk in NGC 4256 with its rotational motion measured by Doppler shifts of its maser emissions. X-ray spectroscopy and other techniques can be exploited to probe rotational motion of accreted matter around supermassive black holes and stellar mass compact objects down to the innermost regions, on angular scales much smaller than directly accessible by current instruments.

Some of us only rarely stop to stare at the night sky, with the naked eye, let alone with binoculars or a telescope. And when we do, the heavens may seem to be majestic, peaceful, and eternal. This impression, however, is deceptive. The Universe is a magnificently violent place. Gigantic clouds contract and ignite, producing the large and fiercely burning globes that we call stars; these stars, in turn, can explode in flashes that are more luminous than millions of suns, and they can do this in a multitude of ways. Pairs of stars may coalesce, again giving rise to unimaginable outbursts of energy. Black holes may form, whose gravitational attracting force is so huge that neighboring stars, planets and gases may be accelerated to reach velocities nearing that of light, being torn apart in the process, unless they are black holes themselves.

At larger distance scales, events take place at much slower rates: galaxies devour smaller galaxies, black holes millions or even billions of times heavier than our Sun devour other objects in the central regions of galaxies. And the most catastrophic happening of all is the creation process of the Universe itself, the big bang.

Conversely, in other cosmic events, and at the smallest distance scales, atomic nuclei and subatomic particles are blown away and reach kinetic energies so enormous that no man-made laboratory, such as the Large Hadron Collider at CERN, will ever be able to match them.

The Transient Universe is abundant with sources that serve as exceptional laboratories for studying radiation processes in multiple windows, as evidenced by UHECRs, black holes in AGN and microquasars, SN1987A, Cygnus X-1, PSR 1913+16, CC-SNe and GRBs. As general relativity is becoming a genuine experimental science with the LAGEOS and Gravity Probe B experiments beyond mere redshift effects, and gravitational-wave and neutrino experiments are advancing to next generation sensitivity, this decade is expected to bring major new discoveries with an inevitable transformation of our understanding of their astronomical origin and the physics of their radiation processes.

This pursuit requires a concerted effort on effective observational strategies, theory, advanced data analysis and high-performance computing, including integration of the three different areas of electromagnetic, hadronic and gravitational radiation processes. We hope that the present book offers a useful introduction to this exciting development, for those who wish to pick up this challenge.

In the electromagnetic spectrum, radiation processes tend to extend far out, while neutrino and gravitational-wave emissions often tend to be confined to or nearby the energy source driving these emissions. The relation between these various windows of observations thereby tends to be non-trivial, also because the most relativistic sources are transient. It generally calls for time-dependent models that elucidate correlations in energies, time scales, light curves, the associated scaling relations and, possibly, normalizations.

UHECRs and GRBs discussed in the previous chapters are some of the most mysterious discoveries of the last century. Their astronomical origin has only recently been constrained by the PAO and various satellite missions since the discovery of X-ray afterglows by Beppo-SAX.

Black holes are natural candidates for powering these emissions. Frame dragging around rotating black holes acts universally on particles and fields alike, which opens a broad range of channels in non-thermal emissions. Furthermore, black holes are scale free, with no intrinsic reference to a particular mass, in sharp contrast to degenerate compact objects, i.e., neutron stars and white dwarfs (see Chapter 1).

Extracting evidence for black holes as inner engines powering these emissions requires detailed analysis of all their radiation channels, taking into account a large diversity in phenomenology as expressed by supermassive and stellar mass black holes in view of their scale-free behavior. Scale-free behavior may also be expressed in ensembles of specific types of sources, provided the ensembles are sufficiently large in number.

Alfvén waves in transient capillary jets

When the magnetosphere around rotating black holes is intermittent, e.g., due to instabilities in the disk or the inner torus magnetosphere [599], magnetic outflows produce terminal Alfvén fronts propagating along their spin axis out to large distances. The approximation of ideal MHD discussed in Chapter 6 assumes negligible dissipation of the electromagnetic field in the fluid, corresponding to an infinite magnetic Reynolds number, which applies to extragalactic radio jets [203].

The aim of this book is to bridge the considerable gap that exists between standard undergraduate mechanics texts, which rarely cover topics in celestial mechanics more advanced than two-body orbit theory, and graduate-level celestial mechanics texts, such as the well-known books by Moulton (1914), Brouwer and Clemence (1961), Danby (1992), Murray and Dermott (1999), and Roy (2005). The material presented here is intended to be intelligible to an advanced undergraduate or beginning graduate student with a firm grasp of multivariate integral and differential calculus, linear algebra, vector algebra, and vector calculus.

The book starts with a discussion of the fundamental concepts of Newtonian mechanics, as these are also the fundamental concepts of celestial mechanics. A number of more advanced topics in Newtonian mechanics that are needed to investigate the motions of celestial bodies (e.g., gravitational potential theory, motion in rotating reference frames, Lagrangian mechanics, Eulerian rigid body rotation theory) are also described in detail in the text. However, any discussion of the application of Hamiltonian mechanics, Hamilton-Jacobi theory, canonical variables, and action-angle variables to problems in celestial mechanics is left to more advanced texts (see, for instance, Goldstein, Poole, and Safko 2001).

Celestial mechanics (a term coined by Laplace in 1799) is the branch of astronomy that is concerned with the motions of celestial objects—in particular, the objects that make up the solar system—under the influence of gravity.

Introduction

This chapter describes an elegant reformulation of the laws of Newtonian mechanics that is due to the French-Italian scientist Joseph Louis Lagrange (1736–1813). This reformulation is particularly useful for finding the equations of motion of complicated dynamical systems.

Generalized coordinates

Let the qi, for i = 1, ℱ, be a set of coordinates that uniquely specifies the instantaneous configuration of some dynamical system. Here, it is assumed that each of the qi can vary independently. The qi might be Cartesian coordinates, angles, or some mixture of both types of coordinate, and are therefore termed generalized coordinates. A dynamical system whose instantaneous configuration is fully specified by ℱ independent generalized coordinates is said to have ℱ degrees of freedom. For instance, the instantaneous position of a particle moving freely in three dimensions is completely specified by its three Cartesian coordinates, x, y, and z. Moreover, these coordinates are clearly independent of one another. Hence, a dynamical system consisting of a single particle moving freely in three dimensions has three degrees of freedom. If there are two freely moving particles then the system has six degrees of freedom, and so on.

Suppose that we have a dynamical system consisting of N particles moving freely in three dimensions. This is an ℱ = 3 N degree-of-freedom system whose instantaneous configuration can be specified by F Cartesian coordinates. Let us denote these coordinates the xj, for j = 1, ℱ.

Introduction

The orbital motion of the planets around the Sun is fairly accurately described by Kepler's laws. (See Chapter 3.) Similarly, to a first approximation, the orbital motion of the Moon around the Earth can also be accounted for via these laws. However, unlike the planetary orbits, the deviations of the lunar orbit from a Keplerian ellipse are sufficiently large that they are easily apparent to the naked eye. Indeed, the largest of these deviations, which is generally known as evection, was discovered in ancient times by the Alexandrian astronomer Claudius Ptolemy (90 BCE–168 CE) (Pannekoek 2011). Moreover, the next largest deviation, which is called variation, was first observed by Tycho Brahe (1546–1601) without the aid of a telescope (Godfray 1853). Another non-Keplerian feature of the lunar orbit, which is sufficiently obvious that it was known to the ancient Greeks, is the fact that the lunar perigee (i.e., the point of closest approach to the Earth) precesses (i.e., orbits about the Earth in the same direction as the Moon) at such a rate that, on average, it completes a full circuit every 8.85 years. The ancient Greeks also noticed that the lunar ascending node (i.e., the point at which the Moon passes through the fixed plane of the Earth's orbit around the Sun from south to north) regresses (i.e., orbits about the Earth in the opposite direction to the Moon) at such a rate that, on average, it completes a full circuit every 18.6 years (Pannekoek 2011).

Introduction

Newtonian mechanics is a mathematical model whose purpose is to account for the motions of the various objects in the universe. The general principles of this model were first enunciated by Sir Isaac Newton in a work titled Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). This work, which was published in 1687, is nowadays more commonly referred to as the Principia.

Until the beginning of the twentieth century, Newtonian mechanics was thought to constitute a complete description of all types of motion occurring in the universe. We now know that this is not the case. The modern view is that Newton's model is only an approximation that is valid under certain circumstances. The model breaks down when the velocities of the objects under investigation approach the speed of light in a vacuum, and must be modified in accordance with Einstein's special theory of relativity. The model also fails in regions of space that are sufficiently curved that the propositions of Euclidean geometry do not hold to a good approximation, and must be augmented by Einstein's general theory of relativity. Finally, the model breaks down on atomic and subatomic length scales, and must be replaced by quantum mechanics. In this book, we shall (almost entirely) neglect relativistic and quantum effects. It follows that we must restrict our investigations to the motions of large (compared with an atom), slow (compared with the speed of light) objects moving in Euclidean space. Fortunately, virtually all the motions encountered in conventional celestial mechanics fall into this category.