The energy structures and transition energies of single-electron atoms and ions are presented. Five Nobel Prizes in Physics were awarded for the theories discussed in this chapter. We first review the Bohr model, which was based on quantized angular momentum and classical circular orbits. The wave model of Schrödinger followed, in which spherical boundary conditions quantized polar and azimuthal standing waves. The energies were identical to Bohr’s, but transition selection rules dictated the change in angular momentum of the system during absorption and emission. Dirac incorporated electron spin and relativistic energies, resulting in energy shifts and fine structure splitting of the energy levels for non-zero angular momentum states. Feynman and Swinger incorporated quantization of the electric vector potential. This physics broke energy degeneracies in the Dirac model and correctly predicted the famous Lamb shift. In this chapter, each of these models are described in detail. The final full characterization of the energy levels and transitions are presented. The chapter ends with a discussion on isotope shifting and transitions to the continuum (ionization/recombination).

The “many electrons problem” for determining atomic energy levels cannot be solved analytically. It must be solved numerically using approximation techniques applied to each ion for each element. The industry standard approach is called the Hartree-Fock method, which incorporates a three-tiered Hamiltonian approximation. In this chapter, we describe how these approximations yield the Russell-Saunders vector model, for which we describe quantized vector addition. We then summarize the Russell-Saunders term and state symbols so commonly used to precisely notate atomic transitions. It is through this formalism that we come to understand that a given energy structure/transition does not describe a single active or optical electron but applies to the full bound multi-electron ionic system. We also describe intermediate coupling schemes and the j-j coupling scheme for heavier nuclei. We then derive the line strengths and oscillator strengths for both term averaged and fine structure transitions. Line emission power and line absorption cross sections are derived and the dipole selection rules for multi-electron ions are presented.   

After a series of observational and theoretical breakthrough in the 1960s, the Steady State theory was discarded, whereas the Big Bang cosmological paradigm remained viable. This model is described by the Friedmann equations with a Robertson-Walker metric. The metric describes the dynamic spacetime intervals and the Friedmann equations describe the expansion dynamics. The latter are derived from Einstein’s field equations of General Relativity assuming an isotropic and homogeneous medium, conservation of energy density, and an equation of state known as the “continuity equation.” Friedmann’s equations are conveniently written in terms of a time-dependent scale factor, the Hubble-Lemaître constant, and four present-epoch cosmological parameters. Today, we live in an era known as precision cosmology, in which the Hubble-Lemaître constant and cosmological parameters are measured with 1% or better uncertainties. In this chapter, we present an abridged derivation of the Friedmann equations and discuss the cosmological parameters and their temporal evolution in detail. The Robertson-Walker metric is then rewritten in terms of radial and transverse components suitable for convenient practical application. 

Astronomer collect light, nothing more. The formalism of radiative transfer is a macroscopic treatment of microscopic interplay between light and matter; it employs macroscopic variables that parameterize microscopic interactions. In this chapter, we describe the radiation and photon field and define the fundamental macroscopic quantity-- the specific intensity. The geometry of radiative transfer is key as it involves an origin and an observer defined line-of-sight perspective. The observed solid angle is expressed for a cosmologically distant observer, from which flux vectors and the observed flux are derived. The equation of radiative transfer is introduced, including the macroscopic parameters known as the emission and extinction coefficients and the optical depth and mean free path. The solution for pure absorption is given including illustrations of the anatomy of an absorption line in terms optical depth. The details hidden within a beam-averaged astronomical absorption spectrum are described, followed by a treatment of partial covering, from which the covering factor is derived.  Finally, a formal definition of column density is provided.

Every recorded quasar spectrum is a blemished version of an otherwise pure light beam. It is blurred by the atmosphere and suffers interference and scattering when reflected off optical elements. It is imperfectly collimated, impurely dispersed, iteratively refocused, and inefficiently discretized when recorded. It is then converted to analog and re-digitized, which introduces “read” errors to an already noise-ridden Poissonian process of photon counting. To understand spectra, one needs to understand its recording device, the spectrograph. In this chapter, a range of long-slit low-resolution spectrographs and high-resolution echelle spectrographs are described. Grating equations, blaze functions, and cross dispersers are examined in detail. The equations for resolving power and instrumental resolution are derived from first principles, followed by illustrations showing the impact of CCD pixelization and line broadening on recorded absorption lines. We present quantitative models for the recorded counts in observed spectra. Flux calibration is also derived from first principles of telescope characteristics and spectrograph design.  Finally, integrated field units are described.

Handed a spectrum, the work begins. In this chapter, we explain how one takes a spectrum and objectively locates and quantifies the statistically significant absorption features peppered throughout. We describe a continuum normalization method that is objective and provides an error model. Multiple spectra may be co-added to improve signal to noise. For objectively locating absorption features, we present a scanning algorithm weighted by the line spread function and optimized for weak lines. Multiple absorption lines arising in rich absorption system can be found using autocorrelation methods, and one such method is described. To analyze absorption systems, a systemic absorber redshift is determined, and the wavelength scale of all absorption profiles are converted to and aligned in the absorber’s rest-frame velocity. For high-resolution profiles, methods are presented for measuring equivalent widths and quantifying kinematics directly from pixel flux decrements. These include velocity spreads containing 90% of total optical depth and other flux decrement weighted velocity moments. We conclude with detailed methods for building composite two-point velocity correlation functions.

Surveys answer the big science questions, but they are trickier than one might think. Designing a survey requires careful planning fraught with technical limitations, uncontrolled variables, and implicit sample biasing. Analyzing a large number of individual quasar spectra presents many challenges. In this chapter, we outline the fundamental attributes of a survey, which define its breadth, depth, and completeness over the domain of the survey space. Large surveys require automated algorithms for objectively identifying absorption lines; their success rates for finding true absorption and erroneously identifying false positives must be both objectively and subjectively assessed. We outline a comprehensive strategy, including automated routines, human inspection, and Monte Carlo simulations, for obtaining the best estimate of the number of true absorbers in the spectra. Other key quantities include the redshift path sensitivity and the total redshift sensitivity path of the survey. These can be computed in binned survey subspaces (redshift, etc.) and will be central to estimating absorption population statistics. We conclude with a summary of these complex survey assessment methods.

Astrophysical gases are characterized by their macro variables, which describe the radiation field and the particle field. Radiation can be described by its frequency dependent energy density or photon number density. Both these quantities can be integrated over frequency, yielding total radiative energy density and/or photon number density. Particles are described by their number and/or mass densities, thermal motions, partial pressures, and the charge density of ions and free electrons. Ion number densities depend on the abundances of elements in the gas and their ionization fractions. In this chapter, we describe the formalisms of the quantities describing the radiation and particle fields in astrophysical gases. We describe the cosmic ultraviolet ionizing background and starburst galaxies ionizing spectra. The principles of particle density and charge density conservation are derived, and the equation of state is presented. This chapter provides the fundamental formalism for studying the micro processes of ionization and the detailed balancing of partially ionized astrophysical gases key to modeling the ionization balance of these gases. 

In this chapter, we begin by writing out the full reaction rate matrix accounting for the radiative and collisional processes presented in Chapter 34. The radiation field is assumed to originate externally and is thus not in equilibrium with the gas. We then derive the closed-form equilibrium solution for a gas with no metals. Important to achieving equilibrium are the photoionization and recombination timescales. The industry standard ionization code is Cloudy; we describe how one uses this code to create model clouds. Important concepts such as the ionization parameter, cloud ionization structure, and shelf shielding of ionizing photons are discussed in detail. The building of grids of models is explained and example grids showing predictions of ionic column densities and ionization corrections are presented for commonly observed ions. Non-equilibrium collisional ionization models are described, and grids are presented. Sensitivities of the models to variations in the ionizing spectrum are explored. Finally, homology relationships useful for scaling cloud models to infer cloud densities, sizes, masses, and cloud stability are derived.