Quasar Absorption Lines

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 wave model of hydrogenic ions naturally yields transition probabilities. These probabilities are written in terms of three Einstein coefficients, which are determined from ``overlap integrals” for spontaneous emission. Under the assumption that a simple dipole describes the moment between the charge densities of the initial and final stationary states of an electron transition, the transition probabilities yield selection rules, emission line intensities, and absorption cross sections. The former governs whether a transition is permitted or forbidden. The amplitudes of the latter two are often written as oscillator strengths. In this chapter, we describe the formalism for determining selection rules and oscillator strengths. We begin with the Schrödinger model and generalize to fine structure transitions for bound-bound transitions. We then address the oscillator strengths of bound-free transitions. Finally, we derive the line spread function describing the natural line width, which depends on the magnitude of the Einstein coefficients and is written in terms of the damping constant. Full expressions for the bound-bound and bound-free absorption cross sections are provided.

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.   

In this chapter we apply the formalism of hydrogenic and multi-electron atoms and build the periodic table of ground-state elements.  Examination of the table shows that all elements in given column share the same Russell-Saunders state symbol; they have identical orbital and total angular momentum states and valence electron multiplicities. These columns are formally grouped, and we show how each group shares the same spectral characteristics (the transition energies differ, but the relationships between transitions are identical from one element to another in a group). We then introduce the idea of iso-electronic sequences, which neatly explain the many lithium-like and sodium-like ions (CIV, NV, OVI, NeVIII, MgII, etc) that have hydrogenic-like spectral series, including zero-volt resonant fine-structure doublets. We then provide accurate tables of ionization potentials and describe the physical reasons for the ion-to-ion trends in these potentials. We conclude the chapter with a complete suite of Grotrian diagrams (visual representations of the energy states and allowed electron transitions) for ions commonly studied using quasar absorption lines.

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 constant, and four present-epoch cosmological parameters. Today, we live in an era known as precision cosmology, in which the Hubble 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. 

The fundamental quantity of the expansion dynamics of the Universe is the time dependent scale factor. However, neither time nor the scale factor is a measurable quantity. The measurable quantity due to universal expansion is the cosmological redshift of observed radiation. This redshift gives the ratio by which the Universe has expanded relative to the present epoch. In this chapter, we rewrite the expansion dynamics in terms of redshift and define proper and co-moving coordinates. Using the radial and traverse components of the Robertson-Walker metric, we derive relations for cosmic time and multiple useful distance measures as a function of redshift.  These include the radial and transverse proper and co-moving distances, the angular diameter distance, the luminosity distance, and the ``absorption” distance. We also derive the equations for the redshift dependence of the line-of-sight separations of gravitationally lensed quasars. The redshift path density is derived. Finally, the redshift dependence of line-of-sight peculiar velocities and cosmological recessional velocities are derived from the metric. 

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.

The depths, widths, and shapes of absorption lines are the code of optical depth profiles. Line depth is the amplitude of the optical depth, which is absorber column density. Line width and shape mirror the total cross section. This is the atomic cross section convolved with a wavelength redistribution function, usually a Gaussian attributable to thermal Doppler broadening. The resulting optical depth profile is a Voigt function. In this chapter, we quantitatively described Voigt profiles in detail. The total absorption is the equivalent width and its functional dependence on column density and Doppler broadening is called the curve of growth. Expressions are derived for its three major regimes: the linear, flat, and damped “parts.” The measured equivalent width increases with increasing absorption redshift, and this must be calibrated out. Inverting absorption line profiles yields apparent optical depth (AOD) profiles, which can be converted into integrable column density profiles. We also describe how to compute the covering factor from doublets showing signs of partial covering and conclude with an in-depth discussion of Lyman-limit ionization breaks from optically thick absorbers.

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.

In this chapter, we describe how blended multi-component absorption profiles can be modeled. Simple deblending that bypasses radiative transfer and atomic and gas physics can be performed using multi-component Gaussian fitting. We show how further sophistication can be added by tying doublets or multiplets and forcing Gaussian components to match known line spacings. To extract column densities and Doppler broadening parameters for each component, we use Voigt profile fitting. We begin with a general expression for a multi-component absorption profile for which each component has a unique column density and Doppler broadening parameter. We then discuss progressively more complex Voigt profile fitting, starting with multiple components for a single transition, then multiple components for a doublet (two transitions from a single ion), and then generalize to multi-component multi-transition multi-ion absorption systems. We also discuss methods for measuring the turbulent velocity component and approaches to multiphase decomposition for ions of different ionization levels. We conclude by discussing fitters and fitting philosophies. Optimized AOD column densities are also discussed.

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.

It is time to take a deep dive into several of the ``key quantities” introduced in Chapter 3. Above all are the population density functions, which describe the number of absorbers per unit redshift per unit column density (or equivalent width). In this chapter, we present practical equations for obtaining maximum likelihood estimates of the population parameters for commonly adopted distribution functions: the power law, the exponential, and Schechter. Summing absorber counts and/or integrating these parameterized distribution functions in absorber subspaces (i.e., bins of redshift and column density) --- or along one axis of the absorber survey space (i.e., across all equivalent widths at fixed redshift, etc.) --- allows absorber evolution to be quantified. Examples include the redshift path density, absorber cross sections, the column density and equivalent width distributions, and the mass density of absorbers. We derive these quantities from first principles and then show how they can be computed accounting for the detection completeness, the redshift path sensitivity, and the total redshift sensitivity path of the survey. 

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. 

The atomic physics of excitation, ionization, and recombination is the story of frolicking electrons— like space traveling aliens, these little leptons are busy jumping up and down when bound to their ``home planet” atom/ion. Launching freely into space, they adventure out and engage in a series of friendly energy exchanges with fellow particles and photons in an expansive plasma. Landing on and being captured by some other random “planet” atom/ion, the cavorting continues. In this chapter, we follow the dynamic lives of electrons, photons, and ions and present an abridged review of the physics of collisional excitation, ionization, and recombination. We describe photoionization, Auger ionization, direct collisional ionization, excitation auto-ionization, radiative and dielectronic recombination, and charge exchange. We show that detailed balancing and reaction cross sections, rates, and rate coefficients are the heart of chemical-ionization modeling of absorbers. We then present the cosmic photoionization rate of HI, HeII, MgII, CIV, and OVI as a function of redshift. We conclude with a comprehensive treatment of the heating a cooling functions of astrophysical gas. 

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.

This chapter covers the most challenging aspect of quasar absorption line studies--- estimating the densities, dynamic conditions, metallicities, ionization conditions, and general cloud properties (masses, sizes, stability) that match the observed data. The techniques have evolved from single-cloud single phase models that were simply constrained by the measure column densities, to kinematically complex, multi-cloud multiphase models that are constrained by absorption profile morphologies on a pixel-by-pixel basis. In this chapter, we cover the modeling methods by describing them in order of complexity and ambition. These methods are the chi-square method, the density-metallicity locus method, and Bayesian approaches, including Markov Chain Monte Carlo (MCMC) methods and profile-based multiphase Bayesian modeling. Methods are discussed and examples are provided, but modeling absorbers is a scientific artform that requires a deep intuition that can only be developed through lots of practice.