Astrophysical spectral lines offer two important insights into the workings of our Universe. First, they are probes of the fundamental (QM) nature of matter because they originate from subatomic, atomic and molecular systems. Second, they provide, via the Doppler effect, critical dynamical information on astrophysical systems ranging in scale from planetary systems to superclusters of galaxies. Examples of major contemporary problems in astrophysics that can be addressed through spectral line studies and the associated quantum mechanics include.

Missing mass and the halos of galaxies The most common element in the Universe is hydrogen and much of it is in a cold state. Given the 10 eV gap between the ground state and the first excited state of the simple Bohr atom, we should have little direct knowledge of this gas, yet it is the best studied gaseous component of the Universe. The reason is the 21 cm line corresponding to the hyperfine splitting of the ground state. The extremely low transition probability of this transition and the consequently narrow width of this line have led to its widespread use in measuring galaxy dynamics and kinematics. Studies of galaxy rotation have shown evidence for missing matter and point to the possibility of dark-matter halos. The nature of the dark matter and the implication on the long-term fate of the Universe remain contentious issues in astrophysics. The nature of this line and its use in these studies is discussed.

We are convinced of a genuine need for a monograph describing the many facets and new developments in numerical relativistic hydrodynamics. Such calculations are crucial to several areas of current research in the physics of stellar collapse, supernovae, and black hole formation, as well as the merging of the final orbits of coalescing binary neutron stars. Both problems are only now entering the level of sophistication where three-dimensional relativistic hydrodynamics simulations are both possible and necessary. In the former problem such calculations are crucial to understand the explosion mechanism. In the latter problem, a great deal of interest in such calculations has recently been inspired by the development of next-generation gravity wave detectors to search for such events, and as a possible explanation of the physics underlying observed astrophysical γ-ray bursts.

The field of numerical relativistic hydrodynamics has developed over the past 30 years, but there has not been written a technical text explaining the many techniques relevant to this discipline, many of which are much different than standard general relativity textbook approaches. This book will present such a review of techniques for numerical general relativistic hydrodynamics developed by one of the pioneers of this field over the past three decades.

We begin by developing the equations and differencing schemes for special relativistic hydrodynamics as an introduction to the metric formulation of the problems. Here, the basic numerical techniques and a number of test problems and applications will be discussed.

Following this, the formalism for matter flows in the curved spacetime of general relativity will be presented in the usual (3+1) formalism.

Progress in computing full general relativistic hydrodynamics in three spatial dimensions has been slow. The problem is not in the hydrodynamics, but in the solution of the field equations. The equation for extrinsic curvature, Kij, is particularly unstable. For example, at the writing of this book no strong field fully relativistic calculation has computed more than two orbits of a neutron star binary without becoming unstable. Currently, there seems to be some promise, however, in a modified version of the ADM equations based upon a conformal decomposition as originally proposed by Shibata and Nakamura and later reinvented by Baumgarte and Shapiro. At the end of this chapter we briefly summarize this method for completeness. First, however, we summarize a useful alternative which the authors have developed for solving strong field systems which avoids the nonlinearities of the full Einstein equations by reducing the problem to an implementation of constraint equations.

The conformally flat approximation

For most gravitating systems studied so far (e.g.), only a relatively small amount of energy is emitted by gravitational waves. Even for the merger of two black holes it is expected that only a few tenths of a percent of the rest mass will be radiated away in gravitation. For the case of two neutron stars we would not expect any more radiation to be emitted during the last few orbits than for a two black hole merger, i.e. during the inspiral, the radiated energy per orbit is a minuscule fraction of the energy in orbital motion. Furthermore, an explicit treatment of the radiation reaction is exceedingly difficult.

As discussed in Chapter 3, the treatment of the spherical collapse of a massive star to produce a supernova is sufficiently complex to warrant a separate chapter. It is believed that a Type II supernova arises from the delicate balance between energy deposited by escaping neutrinos from the core and the gravitational energy of collapsing outer layers. Thus, the relativistic energy and mass transport must be considered with high numerical accuracy to obtain a believable simulation. The model discussed herein includes the experience of about 30 years of development and should be of some guidance to those who wish to understand this fascinating phenomenon.

Collapse supernovae

A brief review of the scenario is as follows. Massive stars (i.e. 10 M/M⊙ 30) evolve until the iron core exceeds ∼1–1.3 M⊙. At this point there can be no more nuclear energy generation in the core. Neutrino emission, electron capture, and photodisintegration cool the inner ∼1 M⊙ and remove pressure support from the core. The central density then rises. When the central density ρc approaches ρc 109 g cm−3, neutrino emission is so large that collapse becomes supersonic, i.e vmax > cs, where cs is the speed of sound. As the core collapses, the inner ∼0.7 M⊙ collapses homologously (e.g.). Once the core density exceeds nuclear density, ρc > 2.5×1014 g cm−3, the pressure rises rapidly and collapse is halted. Matter continues to fall inward, however, so an outward moving shock wave is formed. This is referred to as the core bounce.

Most cosmology calculations utilize a homogeneous background spacetime in either a Friedmann–Robertson–Walker (FRW) metric or an equivalent extension. Calculations of local physical variables are treated as a perturbation on the background homogeneous expansion.

A natural question, however, is whether strong gravity waves in the early universe can cause a significant departure from this FRW treatment. There are also other cosmological questions which may require significant deviations from a FRW-plus-perturbations approach, such as the formation of primordial black holes, or the development of an inflating spacetime from inhomogeneous initial conditions. All such problems require the ability to solve for cosmological evolution in a metric more general than that of a simple perturbed FRW. Here we describe some attempts to model such cosmologies numerically.

Planar cosmology

In a linear numerical cosmology program was developed to study how strong waves might affect the physics of the early universe. In particular, the paradigms for inflation, nucleosynthesis, and microwave anisotropy have been analyzed in this context, along with the question of whether strong gravity waves steepen in the early universe. That is, the nonlinear nature of general relativity could cause colliding waves to produce an even stronger superposition wave. In spherical symmetry such superpositions can even form black holes.

The simplest cosmology to study is that of planar symmetry. A system with planar symmetry can have gravity waves. In one-dimensional simulations, physical quantities are restrained to vary in one direction, say z. All fluid quantities are then functions of z and t only.

Relativistic numerical hydrodynamics is currently a field of intense interest. On the one hand, the development of next-generation laser interferometric and cryogenic gravity wave detectors is opening a new window of astronomy, one which will peer into a world of multidimensional rapidly varying matter and gravity fields such as occur in and around neutron stars, black holes, supernovae, compact binary systems, dense clusters, collapsing stars, the early universe, etc. At the same time, X-ray and γ-ray observatories are providing (or will soon provide) a wealth of data on the evolution of matter in and around X-ray and γ-ray emitting compact objects such as accreting black holes and neutron stars. Such systems can only be realistically analyzed by a detailed numerical study of the spacetime and matter fields.

A quantitative understanding of these systems as well as a host of other astrophysical phenomena such as stellar collapse leading to supernovae, the evolution of massive stars, and the origin of γ-ray bursts, the origin and evolution of relativistic jets, all require multidimensional complex relativistic numerical simulations in three spatial dimensions. Since analytic and post-Newtonian methods are only applicable for systems of special symmetry and/or relatively weak fields, numerical relativistic hydrodynamics is the only viable method to model such highly dynamical asymmetrical strong field systems.

Much of the language of atomic physics was inspired by early studies of hydrogen and other atoms of low Z to moderate Z, where the dynamics are dominated by the electrostatic Coulomb interaction. Thus the n-dependent Balmer energy splittings, which are proportional to Z2, are called the “gross energy,” and the J -dependent Sommerfeld energy splittings, which are proportional to Z4, are called the “fine structure.” E1 processes, which are the primary radiative coupling between gross structure levels, are called “allowed” transitions, and M1 processes, which are the primary radiative coupling among fine structure levels, are called “forbidden” transitions. Clearly this Z-scaling causes the situation to enter a new domain for highly ionized heavy atoms, where “fine structure” can exceed “gross structure” and “forbidden” transition rates can exceed “allowed” transition rates. Isoelectronic studies can provide a fine-tuning mechanism, whereby the interactions can be studied at values of Z where they are strong to elucidate their contributions at values of Z where they are weak.

M1 transitions

In the zeroth-order nonrelativistic limit there is only electrostatics, and thus no magnetic dipole moment. In the first-order relativistic correction to the electrostatic problem, the magnetic dipole moment is given by μ = μB(L + geS), and contains only angular factors. Since the Schrödinger picture separates radial and angular portions completely, the orthogonality of the radial wave functions restricts M1 processes in this approximation to occur only in transitions between levels within the same configuration.

Historical development

The study of optical radiation, dispersed to reveal its frequency content, has a long and venerable history. However, the fact that this radiation consists of a continuous distribution of colors when emitted by free ions in a dense plasma or solid, and of a discrete distribution of lines of color when emitted by an atomic gas, was long unnoticed. The first recorded observation of the dispersed solar (ark) spectrum is usually attributed to Noah, who beheld the rainbow after the flood. In Genesis 9:13, God is reported to have said “I have placed my rainbow in the clouds.” Regrettably, no revelation of the Fraunhofer lines was reported.

The first published observation of a dispersed solar spectrum using a slit and a prism was by Isaac Newton in his 1666 treatise on optics. Again, Newton made no mention of observing dark lines superimposed on the continuous “Phænomena of Colours.” The first recorded observation of a line spectrum was by Thomas Melvill in 1752. Melvill inserted a piece of sea-salt into a flame and allowed the emitted radiation to pass through a slit onto a prism. He noted a “constancy of refrangeability” of the bright yellow sodium light.

The observation of seven dark lines superimposed on the solar spectrum was noted by William Wollaston in 1802, and that number was increased to several hundred by Joseph Fraunhofer in 1814. In 1859 Robert Bunsen and Gustav Kirchhoff combined the experiments of Melvill and Fraunhofer to launch the field of laboratory astrophysics.

Spectroscopic notation

The nomenclature that is used to describe measured quantities in atomic spectroscopy is very much governed by the approximations inherent in the Schrödinger equation. Two theoretical approximations are particularly important. One is the central-field approximation, in which a many-electron atom is described by wave functions that are constructed from products of one-electron states. Another is the nonrelativistic approximation, which leads to a separation of the space and spin portions of the wave function.

A one-electron atomic state is defined by the hydrogenic basis state of quantum numbers |n l ml ms〉, where states with a common value of n are denoted as a “shell” and states with common values of n and l are denoted as a subshell. Since the electron–electron interaction is treated in an averaged manner by the central-field approximation, and the spin and space portions are treated as independent by the nonrelativistic approximation, electrons with the same value of n and l are treated as “equivalent.” As in the case of hydrogen, each electron is assigned a set of n and l quantum numbers, to yield a “configuration.” Here the numerical values of l are replaced by letters according to the code s, p, d for l = 0, 1, 2 and f, g, h, etc., for l = 3, 4, 5 etc., (alphabetically from f with the letter j omitted), with a superscript to describe the number of equivalent electrons in each subshell. This notation was originally formalized in a 1929 meeting that was attended by most of the leading spectroscopists of that era.

Since the time of Kirchhoff it has been known that, when light is passed through an atomic gas, those wavelengths are observed that would be emitted if the gas were incandescent. If the gas is sufficiently cold, then the wavelengths observed are limited to ground-state transitions. As the temperature of the sample is elevated, wavelengths corresponding to transitions between excited states become absorbing, and balances between emission and absorption occur.

The study of the central wavelengths of emission lines or absorption notches is known as first-order spectroscopy, and it provides information on the energy-level structure of the atom. The study of the shape of these lines in frequency space is known as second-order spectroscopy, and provides information on the lifetime of the level and the collision rates and temperature of the gas. Thus, whereas first-order spectroscopy shows that emission and absorption measurements yield the same central wavelengths, second-order spectroscopy shows that the natural linewidth for emission and absorption are both specified by the level lifetime, and that the intensity of emission and absorption features both involve the linestrength factor (through the emission transition probability rate and the absorption oscillator strength).

The connection between the lifetime and the linewidth can be made plausible by a simple semiclassical model. When an electron is excited to a specific orbit in an atom, its binding energy is established through the exchange of virtual photons with the effective central core. If the perturbations that eventually cause the electron to make a transition from that orbit are weak, the meanlife will be long.

Atomic physics is more than quantum mechanics

With the stirring testimonial above from one of the foremost scientific minds of our time, why is it that the subject of atomic structure is relegated to a chapter near the end of most elementary physics textbooks? Introductory physics texts tend to discuss gravitational interactions extensively, yet most of the examples treated are atomic in nature. Since “weightlessness” occurs when there is no floor to provide atomic charge polarizations to oppose a gravitational attraction, weight must be considered an atomic phenomenon. Barring the remote possibility of experiencing the huge gravitational gradients predicted near a black hole, no one is ever directly injured by a gravitational force, but rather by the atomic polarization that ultimately opposes it. Why is so important a topic as atomic physics not given an early and thorough conceptual presentation?

Atomic interactions are usually described in terms of three different types of interactions. The gross structure refers to the predictions of the Kepler–Coulomb–Schrödinger nonrelativistic electrostatic model in which the electron moves in a simple 1/r central potential. The fine structure refers to the relativistic correction to that picture due to: the relativistic momentum; various interactions between the magnetic moment of the electron with that of other electrons; and the relative motion of the static charge of the nucleus. The quantum electrodynamic corrections due to the interaction of the electron with the radiation field are often included with the fine structure. The hyperfine structure refers to a general class of interactions that arise as a result of the finite mass, size, charge distribution, and charge circulation of the nucleus.

The origins of hyperfine structure observations

Hyperfine structure was discovered by Albert A. Michelson in what might be called the second disappointment of the Michelson interferometer. Precision optical measurements were Michelson's lifelong passion, as evidenced by his pursuit of additional significant digits in the value for the speed of light. He began this quest in 1878, and by 1882 had a value good to within 0.02 percent. In 1926 he improved that measurement to just over one part in 105.

In 1881 Michelson began the construction of his “interferential refractometer” (the Michelson interferometer) in the hope of making a precision measurement of the motion of the Earth through the imagined luminiferous aether. The results of this attempt (jointly with Edward W. Morley) were declared a “failure” in 1887.

The study of atomic spectroscopy was central to the development of modern quantum mechanical theory. Thus, applications to the field of atomic physics are an important feature of any course in quantum mechanics. However, the converse is not necessarily true – a comprehensive course in atomic physics is not simply a study of quantum mechanics. The aspects of atomic physics that are most useful as illustrative examples for a quantum mechanics course usually involve either hydrogen or helium, and the methods used for these systems are very specialized and not particularly exemplary of the methods used for the study of complex atoms and ions. Graduate atomic physics courses often substitute for increased complexity of the atomic system studied an increased elegance in the theoretical representation of the one-electron system. Thus, a course on the Schrödinger theory of hydrogen is followed by a course on the Dirac theory of hydrogen, and that in turn is followed by a course on the quantum electrodynamic theory of hydrogen.

In the study of complex, many-electron spectra, the precision of the optical measurements greatly exceeds the accuracy that can be obtained with even the most sophisticated of currently available theoretical codes. Therefore, predictions based on these very high precision measurements usually rely on semiempirical methods, often utilizing simple semiclassical or parametrized single-particle models.

The approach adopted here will be to provide conceptual and intuitive insights into quantum mechanical phenomena, drawing on measured data, semiclassical models, and semiempirical parametrizations that reveal unexpected regularities among various atomic systems. While quantum mechanics has delegitimized the hope of ab initio quantitative predictability based on conceptual pictures, there is more to physics than mathematics.