Baixe Physics - Diffuse Matter in the Universe e outras Notas de estudo em PDF para Física, somente na Docsity! Michael A. Dopita & Ralph S. Sutherland Diffuse Matter in the Universe, Subtitle, Edition SPIN Springer’s internal project number, if known Physics – Monograph (Editorial W. Beiglböck) January 9, 2001 Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest ‘. . . Light is a confused aggregate of Rays indued with all sorts of Colours, as they are promiscuously darted from the various parts of luminous bodies. A natu- ralist would scarce expect to see ye science of those colours become mathematicall, and yet I dare affirm that there is as much certainty in it as in any other part of Opticks.’ Sir Isaac Newton, in a letter to the Royal Society, 1672 Contents 1. What is Diffuse Matter? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. Line Emission Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Atomic Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Resonance Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 Pure Recombination Lines . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.3 The Spectroscopic Notation . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.4 Intercombination and Forbidden Lines . . . . . . . . . . . . . . 22 2.2 Molecular Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.1 Rotating Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . 24 2.2.2 Rotational Wavefunction Symmetry . . . . . . . . . . . . . . . . 26 2.2.3 Rotating Diatomic Molecules with Identical nuclei . . . . 29 3. Collisional Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1 Collisional Excitation by Electron Impact . . . . . . . . . . . . . . . . . 34 3.1.1 Limiting Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 The Three–Level Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.1 Low Density Limit; E12 ∼ E23 . . . . . . . . . . . . . . . . . . . . . 39 3.2.2 Ions in which E23 E12 . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.3 Infrared Line Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 The General Multi–Level Atom . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4. Line Transfer Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1 Resonance Line Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.1 Resonance Line Absorption by Heavy Elements . . . . . . 50 4.1.2 Absorption Line Studies of the ISM . . . . . . . . . . . . . . . . . 52 4.1.3 Line Transfer in Emission Resonance Lines . . . . . . . . . . 56 4.1.4 Line Transfer in the Lyman Series . . . . . . . . . . . . . . . . . . 60 4.2 Fluorescent Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2.1 The Bowen Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2.2 O i Fluorescence with Lyβ. . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2.3 H2 Fluorescence with Lyα . . . . . . . . . . . . . . . . . . . . . . . . . 63 X Contents 4.2.4 Raman Scattering Fluorescence . . . . . . . . . . . . . . . . . . . . 63 4.3 Astrophysical Masers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3.2 Observations of Galactic Masers . . . . . . . . . . . . . . . . . . . . 69 4.3.3 Observations of Extragalactic Masers . . . . . . . . . . . . . . . 72 5. Collisional Ionisation Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.1 Collisional Ionisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.1.1 The Case of Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2.1 Radiative Recombination Rates . . . . . . . . . . . . . . . . . . . . 83 5.2.2 Di–electronic Recombination Rates . . . . . . . . . . . . . . . . . 84 5.3 Photoionisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3.1 From Outer Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3.2 From Inner Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3.3 The Milne Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.3.4 Photoionisation Cross–sections . . . . . . . . . . . . . . . . . . . . . 88 5.4 Charge–Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.5 Coronal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.5.1 The Case of a Pure Hydrogen Plasma . . . . . . . . . . . . . . . 95 5.5.2 Ionisation Equilibrium of Heavy Elements . . . . . . . . . . . 96 6. Continuum & Recombination Line Processes . . . . . . . . . . . . . 99 6.1 Free–Free Continuum Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.1.1 free–free Gaunt factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.2 The Free–Bound Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.3 The Two–Photon Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.4 Recombination Line Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.4.1 Recombination Line Spectra . . . . . . . . . . . . . . . . . . . . . . . 109 6.4.2 The Radio Recombination Lines . . . . . . . . . . . . . . . . . . . . 111 7. Cooling Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.1 The Cooling Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.2 Conditions for Non-Equilibrium Cooling . . . . . . . . . . . . . . . . . . . 118 7.3 Heat Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.3.1 Electron Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 7.3.2 Boundary Layer Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.4 Cold Clouds in Hot Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.5 Thermal Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.5.1 In a Stationary Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.5.2 In an Expanding Medium . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.6 Hot Galactic Coronae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.6.1 Early-Type Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.6.2 Clusters of Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 7.6.3 Disk Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Contents XIII 14. Thermal Phases of Diffuse Matter . . . . . . . . . . . . . . . . . . . . . . . . 299 14.1 Phase Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 14.2 Thermal Phases of Galactic Interstellar Gas . . . . . . . . . . . . . . . 303 14.2.1 Giant Molecular Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 14.2.2 The Atomic ISM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 14.2.3 The Warm Ionised Phases . . . . . . . . . . . . . . . . . . . . . . . . . 307 14.2.4 The Hot Ionised Component . . . . . . . . . . . . . . . . . . . . . . . 309 14.3 Feedback & Mass Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 14.3.1 Shells, Supershells & Interstellar Froth . . . . . . . . . . . . . . 310 14.3.2 Self-Propagating Star Formation . . . . . . . . . . . . . . . . . . . 313 14.3.3 Self-Regulated Star Formation . . . . . . . . . . . . . . . . . . . . . 315 14.3.4 Mass Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 14.3.5 Dust Evolution in a Multi-Phase Medium . . . . . . . . . . . 319 1. What is Diffuse Matter? “Now entertain conjecture of a time When creeping murmur and the pouring dark Fills the wide vessel of the Universe” — Shakespeare (Henry IV, Act4) Nowhere in the universe can we find a perfect vacuum. Around ordinary stars, a hot magnetized plasma seethes and bubbles outward in a thermally- powered wind. Within the disks of spiral galaxies, dusty clouds of molecular gas continually coalesce and collapse under their self gravity to form new stars. The light from these new-born stars heats and ionizes their placental cloud before finally dispersing it back into the galactic disk, ready to repeat the cycle. Wherever there are stars, some reach the end of their lives and explode as supernovae, hurling out the gas that transformed in their ther- monuclear furnaces to heavy elements such as iron. The shocks they produce rumble their way through space, heating the interstellar gas anew. Some is thrown up and far away from the galactic plane, while other parts are crushed into dense sheets and filaments which shine briefly as their shock energy is radiated away. In clusters of galaxies, the gaseous shells ejected by old stars collide one with another, heating the gas so that it glows softly in X–rays, cooling over billions of years before finally falling back into the bright galaxy cores to feed the massive monster black holes that lurk at their centres. In intergalactic space jets of relativistic plasma shot from the cores of active galaxies emit radio waves as charged particles circle and shed their energy in the magnetic fields. Even in the vast reaches of space between the clusters of galaxies, hot plasma can still be found, even though it is so tenuous that it can never cool again, and a hydrogen nucleus could travel a distance equal to the width of our galaxy before encountering another of its kind. This then is the broad canvas of diffuse matter in the universe, displaying a rich range of thermal plasma phenomena and covering a remarkable variety of conditions and chemical compositions. We work to understand the physics of such plasmas because, by gaining insight into their physics, we can hope to understand and interpret the observed phenomena, measure their physical parameters and determine their chemical composition. Since all the stars in all the galaxies have been ultimately formed from this gas, our study provides insight into the structure and evolution of the universe we live in. 4 1. What is Diffuse Matter? of the electrons. A plasma in which the Boltzmann equilibrium is not a good approximation is said to be in a non–LTE (or NLTE) condition. The study of the physics of the interstellar medium is therefore, in large part, the study of highly NLTE plasmas. Historically, diffuse astrophysical plasmas have been divided into broad environmental divisions or domains, of which the most familiar and the best studied is the medium between the stars in our galaxy. This medium, and the gas in between stars in other galaxies is generally referred to as the interstellar medium (ISM). It provides in its dense molecular phase the cradle and the birthplace of stars. In turn, it derives its complex phase structure and energy balance from the input of energy derived from nuclear burning occurring within these stars in the form of photons, stellar winds and outflows or stellar explosions. Within the ISM itself, the details of these interactions has provided a po- tent testing ground for theory, and has provided a good deal of insight into the evolution of stars, and the chemical and structural evolution of galax- ies. The dense molecular clouds have given us an understanding of molecular chemistry, interstellar dust physics, the diffusion of magnetic fields, and a detailed insight into the gravitational instabilities which lead to the forma- tion of stars. Star formation regions embedded within them have led to an understanding of how newly forming stars shed the angular momentum of their parent cloud through the formation of collimated outflows and jets, and the interaction of these jets with the clouds has enabled us to understand shock physics and chemistry. Radio observations of molecular lines enable us to study physical conditions in star–forming regions, and tell us about the isotope ratios of various elements, which are key data in understanding the sites of production of the heavier elements. The photoionized regions around massive young stars provide a means of probing the chemical composition of the atomic gas both in our Galaxy, and in distant galaxies. In particular, they provide us with estimates of the primordial helium abundance – a key parameter in cosmological models. We can also study the bubbles formed by the powerful radiation-pressure driven winds of their central stars. The so–called planetary nebula shells which have been ejected and pho- toionized by dying solar–like stars provide insight into the chemical processing and dredge–up which has occurred in their atmospheres, and give us an ob- servational test of stellar evolution in low mass stars, while the nova shells ejected from the surfaces of White Dwarf stars enable us to study explosive nuclear processing under electron–degenerate conditions. Finally, the material ejected in supernova explosions gives us a sample of the end–products of nucleosynthesis in stars, and the properties of the shock waves driven into the surrounding medium measures the kinetic energy input into the ISM by these explosions, while the properties of the radio synchrotron 1.1 Phases 5 spectrum generated in the shell provides insight into particle acceleration mechanisms and the origin of cosmic rays. With each generation of stars, some of the ISM is lost forever in the dying embers of stars – the White Dwarfs, in neutron stars formed during supernova explosions, or in Black holes formed in the collapse of the cores of massive stars. In addition, some matter is effectively lost in low–mass stars which frugally burn their nuclear fuel over time–scales much longer than the age of the universe, while yet another part is stored for long periods within normal stars like our own sun. In the solar neighborhood, the ISM now accounts for only about 15% of the total baryonic mass, and this figure is typical of spiral galaxies. However important the gas and dust in galaxies may be, we must not for- get other parts of the cosmos where we find components that are not moulded and controlled by stars alone, and indeed may not even located between stars as the word interstellar implies. In the cores of many galaxies lurk massive black holes which when fed by matter subject their environs to extremes of ionization or temperature. Here we find rings of gas which are irradiated by X–rays, massive outflows of material, or highly relativistic jets of gas shot out into intergalactic space. These plasmas could collectively be regarded as the Active Galactic Medium (AGM). Developing an understanding of the properties of the AGM is a cornerstone of research in active galaxies. Finally, on the largest of scales we have the intergalactic medium (IGM) or the hot gas found within whole clusters of galaxies, the intra-cluster medium (ICM). This material is detected by means of the X-rays it produces, by the effect it has upon the propagation of relativistic jets, by faraday rotation and depolarization of distant radio sources, by evidence of ram-pressure stripping of matter in galaxies, or through the absorption it can produce in the light of distant galaxies. Since the IGM is the most difficult to observe, it is also the least well studied of the diffuse astrophysical plasmas. 1.1 Phases Diffuse matter in the universe is found over an extraordinary range of scales; from structures smaller than the size of the solar system (∼ 1015 cm), up to regions encompassing whole clusters of galaxies (∼ 1024 cm). Fortunately, the characteristic densities decrease as the scale size increases (otherwise the mass of the diffuse medium would be infinite!). A consequence of this enormous dynamic range in parameter space is that the range of possible phenomena is very rich. However, provided that characteristic time–scales remain apprecia- bly shorter than the characteristic time–scale for the evolution of the universe (∼ 1010 yr), it is remarkable how often phenomena occurring on small scales find physical analogues on much larger scales. As an example, we might cite the bipolar outflows from very young stars which have characteristic scales 105 yr and 1017 cm, but which have very similar physics to the bipolar jets 6 1. What is Diffuse Matter? in active galactic nuclei (AGN), with characteristic scales of order 108 yr and 1022 cm. The diffuse medium has always represented an important component of the baryonic mass in the universe. Shortly after the recombination epoch in the early expansion of the universe, diffuse matter reigned supreme, ac- counting for all the baryonic mass, with the exception of a few primordial Black Holes. However, gravity soon took over, and the gas clouds started co- agulating and collapsing. It should always be remembered that gas is sticky stuff. When two gas clouds collide, their energy of motion is turned into heat through shocks, which is then radiated away into space. Thus cloud– cloud collisions can be considered as almost completely inelastic, conserving momentum, but losing all their kinetic energy. Shocks are also highly com- pressive, and aid the development of a cold dense phase in the ISM which in turn favors star formation when the densities become high enough. Early on in the universe, these processes led to the formation and the evolution of galaxies, and today the ISM in galaxies is kept in a dynamic, self-regulating equilibrium determined by the rate of star formation, balanced against the energy input these stars put back into the interstellar medium. In galaxies, the ISM forms a multi–phase structure in response to this feedback, and develops a heirachical fractal spatial structure. The multi–phase structure, discussed in detail in Chapter (14), develops as a consequence of the fact that a stable balance of heating and cooling at a given pressure can often be achieved at more than one temperature. Various names have been given to the most common phases of the ISM in galaxies. The molecular medium (MM), the cold neutral medium (CNM) and the warm neutral medium (WNM) are three such phases of the atomic gas in the ISM. In similar fashion, we may also find components due to a warm ionized medium (WIM) and a hot ionized medium (HIM). None of these components should regarded a static in time or space, and matter is constantly in flux between them. Dying stars constantly feed matter back into the ISM, which has been transformed into heavier elements of one kind or another (often labelled, in cavalier fashion, “metals” by astronomers). These processes of chemical evolu- tion are very clearly described by Pagel (1997). Much of the non-volatile frac- tion of heavy elements finds its way eventually into interstellar dust grains, which are important constituents of the ISM, absorbing and polarizing the light from distant stars, coupling gas and magnetic fields through photoelec- tric or collisional charging, playing an important role in the total energy bal- ance of the ISM, and providing on their surfaces sites for chemical reactions which allow complex molecules to form. If the plasma is hot, its lifetime in the hot phase depends on its heat content or internal energy, and how fast it can radiate this heat away. The rate of radiation is a complex function of temperature and of density. Thus, we can arrange the plasmas we meet in the ISM according to their characteristic scale 1.2 Observability 9 EM ∼107 pc cm−6. Still fainter are H ii regions, which are typically 10–100 pc across and are ionized by the UV light of massive, hot, and young stars a few million years old. In this case the density is as low as 10 cm−3, and so the EM is only 103−4 pc cm−6. Such nebulae are still easy to detect with modern telescopes, although some faint lines which are important to establish the density, temperature or abundances may be difficult to observe. Finally, consider the case of the diffuse galactic ISM which pervades the disks of spiral galaxies. Here the densities are as low as 0.1 cm−3, while the medium is limited by the scale height of the gas in the galactic plane, typically 150 pc. In this case the EM is only ∼ 1 pc cm−6, and specialist instruments are needed simply to detect it, let alone measure it accurately enough for analysis. Even when the plasma is too faint to be seen by its own emission, it may still be detected by the absorption it produces in a background source of continuum emission. This is because resonance transitions to higher states are excited by the continuum light in the beam, but the atom re-radiates this light in all directions when it returns to the ground state. Thus effectively the light has been scattered out of the line of sight. The absorptions are proportional to the column density ∫ nedl, and the cross–sections for absorption are large, comparable with the Bohr radius. Thus, species which have column densities as low as 1012−13 cm−2 along the line of sight may be detected with high dispersion spectrographs on large telescopes. Such absorption techniques are the only means whereby the hot, highly ionized, and very tenuous gas in our Galactic halo can be detected. This gas has typical densities of 0.001 cm−3, and columns of ∼ 3 kpc (1022 cm), so its emission measure would only be of order 10−2 pc cm−6, and therefore totally undetectable in its own emission. 2. Line Emission Processes “All Science is either physics or stamp collecting” — Ernest Rutherford In this chapter we will review the sometimes arcane, frequently confus- ing, notations that are used to classify atomic and molecular transitions and spectra. A basic understanding of these is essential, since it is through the atomic and molecular lines that we ultimately derive our understanding about the physical conditions in, and chemical abundances of, diffuse astrophysical plasmas. In what follows, we will assume that the reader is familiar with the basic concepts of quantum mechanics, usually developed during the second year of a physics major. If not, the reader is referred to the highly readable account of this and other topics in modern physics which is to be found in the book by Rohlf ( 1994), and a lucid introduction to molecular spectroscopy is to be found in Atkins (1983); see notes on this chapter. 2.1 Atomic Spectra 2.1.1 Resonance Lines Let us first consider a simple two–level atom, as in Figure (2.1). If the transition shown is a resonance line, it arises from a normal elec- tronic dipole radiative transitions. Such a line is called a permitted line. This means that the transitions follow the standard selection rules of quantum mechanics which require that: • Only one electron is involved in the transition. • The initial and final states have different parities. In addition, selection rules are imposed by the requirement of conservation of angular momentum: • The change in the magnetic quantum number has to be ∆ml = 0,±1. • The intrinsic angular momentum quantum number of the electron, ms, does not change; ∆ms = 0. • Since the photon carries one unit of angular momentum, the electron or- bital angular momentum must change by one unit, ∆l = ±1. 14 2. Line Emission Processes gf = g1f12 = g2f21 (2.6) here, f21 is called the Emission oscillator strength. The oscillator strengths for all transitions in the atom obey the Reiche–Thomas–Kuhn sum rule, namely, for an atom with Z optically–active electrons: ∞∑ n=1 fmn = Z (2.7) This follows from the definition of oscillator strength in terms of the equiva- lent numbers of classical electrons; clearly the total cannot exceed the number of electrons that are available to absorb. In nebulae generally, thanks to the NLTE conditions which prevail, most atoms are sitting in their ground state, and the excited states have only a very low population. Therefore, transitions between excited states appear strong only as a result of recombination and cascade down towards the ground level. Such recombination lines which are commonly observed are the Balmer Series, Hα, Hβ etc. or recombination lines of He i and He ii. These are seen in preference to the recombination lines of heavier elements thanks to the very large relative abundances of hydrogen and helium in the ISM. 2.1.2 Pure Recombination Lines Hydrogenic ions are ions with a single electron bound to a nucleus of charge Z. This simple atomic system is one of the best studied and understood of all atomic systems. Because it consists, in the case of hydrogen itself, of just two distinguishable particles, the quantum mechanical wavefunction description is quite soluble. Even though strictly speaking heavier hydrogenic ions, such as Cvi, have many particles because of complex nuclei, on the atomic scale the nucleus can be considers as a single particle to a good approximation. His- toricly, observations of the hydrogen spectrum in astrophysical plasmas have provided stringent tests of the quantum mechanical model, and predictions of high precision have been verified. In the limit of low density, the hydrogen spectrum is dominated by a pro- cess called recombination cascades. Electrons from the surrounding plasma have combined with a bare hydrogen nucleus to from a hydrogen atom in an excited state. The details of this recombination process are deferred until chapter 6, here we will concentrate on the processes of line emission after recombination as the recombined electron cascades down through the many possible hydrogen energy levels to the ground state through resonance lines. The principle difference between this emission line process and the two level resonance line model shown earlier is the nearly infinite number of levels in- volved and the fact that electrons are appearing from ‘above’. If we just con- sider the energy levels and their probability wavefunctions, we can calculate the essential parameters of the cascade process, the transition probabilities, 2.1 Atomic Spectra 15 to great precision. Precision is necessary because the final spectrum depends on many probabilities all multiplied together and errors could accumulate unless they are very small to begin with. It has been known for a long time that the hydrogen system with two particles orbiting each other with masses mp for the proton and me can be treated equivalently as a reduced mass system. We have an atom with two particles of mass m1 and m2 separated by a distance r spinning about the center of mass located at distance r1 and r2 from each atom, respectively. The moment of inertia of the atom is given by: I = m1r21 +m2r 2 2, (2.8) which can be expressed in terms of the reduced mass of the system as I = m1m2 m1 +m2 r2 = µr2 (2.9) where µ is the reduced mass at the radius r from the centre of rotation The reduced mass hydrogen atom system can be described in terms of spherical wavefunctions Ψ that satisfy the of the time–independent Schrödinger equation: HΨ = EΨ, (2.10) where H is the Hamiltonian operator and E is the energy. Wavefunctions that satisfy the Schrödinger equation for this system, using spherical coordinates can be separated into angular and radial functions. Ψ(r, θ, φ) = R(r)Θ(θ)Φ(φ) . (2.11) The functions Θ(θ) and Φ(φ) are shown in figure ?? in the next section. The eigenvalues or energy levels associated with each of the solution wave- functions are quantised in three variables; n, l, m and result in a series of discrete, degenerate, energy levels determined by n alone of E = −mee 4Z2 2h̄2n2 . (2.12) These energy levels are typically illustrated as shown in figure 2.2. Verti- cally we have energy with levels determined by the principle quantum number n, and horizontally the levels are spread out according to the angular momen- tum number l with no vertical displacement. In more complex atoms with many electrons, the repulsion between electrons results in vertical differences in the l levels. Here there is no repulsion and all the l levels are degenerate. The m levels are also degenerate because under spherical symmetry there is no prefered direction to put the axis around which m operates. The levels due to m are not shown, apart from needing a 3D figure the m levels are not distinguishable unless a strong external magnetic field is applied to the atom. 16 2. Line Emission Processes 3 2 1 4 5 … 1 2 3 4 5 …l n Hα Lyα Paα Brα e- S P D F G … Fig. 2.2. Figure Hydrogen n, l, Levels This can occur in extreme conditions such as compact star atmosphere, but it does not occur to any extent in diffuse plasmas. The wavefunctions associated with each energy level have analytical forms. The angular parts Θ(θ)Φ(φ) from the spherical harmonics, analagous to a vibrating string, but on the surface of a sphere. These functions are smooth and wrap around the sphere exactly as you might expect. They are described by a polynomial series in θ and φ known as the Legendre Polyno- mials. The radial functions also have analytical form As with the resonance, intercombination and forbidden transitions al- ready covered, the transitions between all these levels obey selection rules. The dominant one is the usual rule that the unit angular momentum of the emitted photon requires that a corresponding unit change of angular mo- mentum of ∆l = ±1 must take place. This rule is particularly strict in the hydrogen case and except for one special case, only resonant, permitted tran- sition are observed. In the figure the allowed transitions are shown and form a characteristic zig–zag pattern. An atom in a given level can usually go one of two ways, and the pro- portion of times is goes one way or the other depends on the transition probabilities between the levels considered. I most complex atoms where the calculation of transition probabilities can be very difficult and we often have to resort to hard won experimental data which unfortunately often cannot cover all possible transitions. In the hydrogenic case the transition probabil- ities can be computed very precisely as follows: 2.1 Atomic Spectra 19 Table 2.1. Hydrogen transition probabilities between the first 5 levels. Upper Lower Radial Integral Probability Observed n l n′ l′ I(n, l, n′, l′) A(n, l, n′, l′)s−1 Line λ(Å) 2 1 1 0 ( 32768 19683 ) 6.258085E+08 Lyα 1215.67 3 1 1 0 ( 2187 8192 ) 1.670707E+08 Lyβ 1025.72 4 1 1 0 ( 113246208 1220703125 ) 6.811244E+07 Lyγ 972.54 5 1 1 0 ( 625000 14348907 ) 3.433791E+07 Lyδ 949.74 3 0 2 1 ( 214990848 244140625 ) 6.306708E+06 Hα 6562.80 3 1 2 0 ( 2293235712 244140625 ) 2.242385E+07 Hα 6562.80 3 2 2 1 ( 27518828544 1220703125 ) 6.458069E+07 Hα 6562.80 4 0 2 1 ( 2097152 14348907 ) 2.575344E+06 Hβ 4861.32 4 1 2 0 ( 2621440 1594323 ) 9.657538E+06 Hβ 4861.32 4 2 2 1 ( 41943040 14348907 ) 2.060275E+07 Hβ 4861.32 5 0 2 1 ( 1728000000000 33232930569601 ) 1.287221E+06 Hγ 4340.46 5 1 2 0 ( 19906560000000 33232930569601 ) 4.942930E+06 Hγ 4340.46 5 2 2 1 ( 221184000000000 232630513987207 ) 9.415105E+06 Hγ 4340.46 4 0 3 1 ( 198429099687936 33232930569601 ) 1.833427E+06 Paα 18751.0 4 1 3 0 ( 994117681152000 33232930569601 ) 3.061786E+06 Paα 18751.0 4 1 3 2 ( 56358560858112 33232930569601 ) 3.471578E+05 Paα 18751.0 4 2 3 1 ( 1902101428961280 33232930569601 ) 7.029945E+06 Paα 18751.0 4 3 3 2 ( 24346898290704384 232630513987207 ) 1.377295E+07 Paα 18751.0 5 0 3 1 ( 516849609375 549755813888 ) 9.037057E+05 Paβ 12818.1 5 1 3 0 ( 701719453125 137438953472 ) 1.635931E+06 Paβ 12818.1 5 1 3 2 ( 64072265625 274877906944 ) 1.493728E+05 Paβ 12818.1 5 2 3 1 ( 2421931640625 274877906944 ) 3.387776E+06 Paβ 12818.1 5 3 3 2 ( 12109658203125 1099511627776 ) 4.537200E+06 Paβ 12818.1 5 0 4 1 ( 85762416640000000000 4052555153018976267 ) 6.443709E+05 Brα 40512.0 5 1 4 0 ( 2646238780456960000000 36472996377170786403 ) 7.363837E+05 Brα 40512.0 5 1 4 2 ( 338249646080000000000 36472996377170786403 ) 1.882532E+05 Brα 40512.0 5 2 4 1 ( 493837352960000000000 4052555153018976267 ) 1.484167E+06 Brα 40512.0 5 2 4 3 ( 33554432000000000000 12157665459056928801 ) 5.042185E+04 Brα 40512.0 5 3 4 2 ( 2405181685760000000000 12157665459056928801 ) 2.581599E+06 Brα 40512.0 5 4 4 3 ( 34359738368000000000000 109418989131512359209 ) 4.249545E+06 Brα 40512.0 20 2. Line Emission Processes nsi npj ndk… 2S+1 LJ (o) Electronic Configuration: the electrons and their orbitals (i.e. 1s2 2s2 3p1) Total Term Spin Multiplicity: S is vector sum of electron spins (±1/2 each) Inner full shells sum to 0 Term Parity: o for odd, nothing for even The Number of levels in a term is the smaller of (2S+1) or (2L+1) Total Term Orbital Angular Momentum: Vector sum of contributing electron orbitals. Inner full shells sum to 0. Total Level Angular Momentum: Vector sum of L and S of a particular level in a term. Fig. 2.3. Spectral notation for an atomic term, comprised of 1 or more levels. ric potentials, it is possible to separate the eigenfunctions into their spatial parts. Thus in terms of the quantum numbers, the principal quantum num- ber n, the angular momentum quantum number l, and the magnetic quantum number m, the wavefunction can be written in terms of the polar coordinates as: Ψnlm(r, θ, φ) = Rnl(r)Θlm(θ)Φm(φ) (2.21) where the quantum numbers satisfy: n = 1, 2, 3, ..., l = 0, 1, 2, ..., (n− 1) and m = −1,−(l − 1)..., 0, ...(l − 1), l. (2.22) The fact that the each electron has a spin angular momentum s = 1/2 ensures that the electron has a magnetic moment (equal to one Bohr magneton, eh/4πmec) which will interact with the magnetic field due to orbital motion, producing a total angular momentum j = |l ± 1/2|. Thus, for an orbital angular momentum l = 0, the only possible levels associated with the binding of a single electron have j = 1/2;n = 1, 2, 3 . . .. In spectroscopic notation, these will be referred to as n 2S1/2 levels, meaning that they have principal quantum number n, the term is a doublet (although in this particular case the lower value of j would be negative, so one of the levels of the doublet cannot exist), the orbital angular momentum is zero (S state) and the total angular momentum is 1/2. Now consider the orbital 2.1 Atomic Spectra 21 angular momentum l = 1. Now the possible levels have j = 1/2, 3/2 and n = 1, 2, 3 . . .. In spectroscopic notation, these will be n 2P1/2 or n 2P3/2 levels. Continuing to higher orbital angular momentum, l = 2, j = 3/2, 5/2 and n = 1, 2, 3 . . .. so these are n 2D3/2 or n 2D5/2 levels, and so on to higher l states. Now, since the Pauli exclusion principle states that no two electrons can have identical quantum numbers, then the n = 1 state can be occupied by only two electrons (l = 0, s = ±1/2). This forms a closed shell of configu- ration 1s2. Here, the leading number is the shell number, (here the first), s refers to the angular momentum state of the electrons occupying this shell, and the superscript 2 refers to the number of electrons present. Thus the electron configuration of the normal state of magnesium (Z = 12) would be: 1s22s22p63s2. Ionised magnesium, (Mg+ or Mg ii), which is isoelectronic with sodium and has one optically active electron in its outer shell will have the ground state, defined by the electron configuration and the ground term; 1s22s22p63s 2S1/2. While one optically active electron only allows us one way of forming the total angular momentum by combining the spin and the orbital angu- lar momentum, with two or more electrons life gets much more complicated. Consider the case of two electrons. For light atoms, as first shown by Russell & Saunders, the angular momentum vectors are coupled by electrostatic in- teraction to form the total orbital angular momentum, and the spins are also coupled by electrostatic interaction to give the total spin: L = l1 + l2 = (l1 + l2), (l1 + l2 − 1), ... |l1 − l2| S = s1 + s2 = 1/2 ± 1/2 = 0, 1 (2.23) and the resultant L and S combine by magnetic interaction to give the total angular momentum: J = L + S = (L+ S), (L+ S − 1), ... |L− S| (2.24) This situation is known as (Russell–Saunders) LS coupling. In the case of LS coupling for our atom with two optically active electrons, the possible states are n 1S0, n 1P1 and n 3P0,1,2 . These are called the ground terms. In the case of neutral magnesium, Mg i, the ground state would be 1s22s22p63s2 1S0, and the two excited states of the ground term are therefore 1s22s22p63s2 1P1 and 1s22s22p63s3p 1P0,1,2. Clearly this is a rather clumsy notation, so usually the configuration of the closed shells is omitted, viz. 3s2 1S0, 3s2 1P1, and 3s3p 1P0,1,2, respectively. Other kinds of coupling are possible. For example, in heavier ions, the spin and orbit of individual electrons can be coupled, with the resulting total angular momenta of each electron being coupled together. This is called JJ coupling. This is merely indicative of the complexity that can be obtained with multi–electron atoms. 24 2. Line Emission Processes Table 2.3. Molcular Species Detected in the ISM (after Snyder(1997) and Bakes(1997)) Atoms Species 2 H2 HD C2 N2 OH CH CN CO CP CS NH NO NS SO PN AlF AlCl HCl KCl NaCl SiC SiN SiO SiS CH+ CO+ CN+ NO+ MgH+ SO+ CS+ 3 C3 C2H CH2 C2O C2S CO2 HCN HCO NH2 N2O H2O H2S HNC HNO MgCN MgNC NaCN OCS SO2 c-SiC2 HCO + HCS+ H+3 HOC + N2H + 4 c-C3H i-C3H C3N C3O C3S C2H2 HCCN HNCO HNCS H2CO H2CN CH2N H2CS NH3 CH2D + HCNH+ HOCO+ H3O + 5 C5 C4H C4D l-C3H2 c-C3H2 CH2CN CH4 HC3N HC2NC CH2C2 HCOOH H2CNH H2C2O H2NCN HNC3 SiH4 C4 Si H2COH + HC3NH + 6 C5H l-H2C4 C2H4 C5S? CH3CN CH3NC CH3OH CH3SH HC2CHO HCOC2H HCONH2 HC3NH + 7 C6H HC5N CH2CHCN CH3C2H HC5N HCOCH3 NH2CH3 c-C2H4O CH3CHO 8 C7H H2C6 CH3COOH CH3C3N HCOOCH3 9 C8H HC7N CH3C4H CH3CH2CN (CH3)2O CH3CH2OH 10 CH3C5N? (CH3)2CO 11 HC9N 12 13 HC11N 2.2.1 Rotating Diatomic Molecules The simplest molecular line spectra are due to diatomic molecules in rota- tional motion. These molecules emit in the infrared and microwave regions of the electromagnetic spectrum and produce a spectrum of emission or absorp- tion lines which are nearly equally spaced in frequency (or energy). These lines are formed by transitions between quantized rotational energy levels which are directly related to the masses and inter–atomic spacing of the atoms in the molecules. To a first approximation a slowly spinning diatomic molecule can be con- sidered as a rigid rotator – that is, the interatomic spacing is constant. In reality, as the molecules spin faster centrifugal effects can change the atomic spacing and hence the energy level spacing. Fortunately, many molecular 2.2 Molecular Spectra 25 species such as H2 detected in diffuse astrophysical plasmas are observed in the lowest spin levels where the rigid rotator model is a fair approximation to the truth. The rotation of a rigid diatomic rotator is equivalent to the motion of a single reduced mass µ on a sphere of radius r from the centre of rota- tion, analagous with the hydrogen atom discussed above. The rotator is a bit simpler though, because while the reduced system is spherical, it has a fixed radius. The wavefunctions can be separated into Θ(θ) and Φ(φ) angu- lar components in spherical coordinates as usual, but the radial component disappears and need not be included. The solutions are quantised in two variables, mj the quantum number of the z component of the angular momentum and J the rotational quantum number (analogous to l in one electron atoms). For the mj the simplest determining expression is: d2Φ dφ2 = −m2jΦ, (2.25) The quantum number J is related to the moment of inertia, µ and r2, and the rotational energy E by: 8π2µr2E h2 = J(J + 1) (2.26) and the wave equation constrains J to integral values and J ≥ |mj |, so there are 2J + 1 values of mj for each value of J . Some of the rigid rotator wavefunctions are represented in figure (2.3) for J = 0 up to J = 4. The complex wavefunction Ψ is represented as two rectan- gular φ - θ maps, analogous to a mercator projection world map. This is done in preference to the more usual polar line plots because it shows the sym- metry properties of the wavefunctions more clearly. The probability density Ψ∗Ψ is shown, and black represents the regions where the rotating particle spends the most time as it spins. The high J , high |mj | plots show the trend towards the classical high angular momentum result with the nuclei spinning in a well defined equatorial plane. The probability plots are also mapped onto a spherical representation to aid visualising the molecule’s motion. From equation (2.26), the possible rotational energy levels in the rigid rotator are: Erot = J(J + 1) h2 8π2I . (2.27) Transitions between different J levels are possible through absorption and emission of photons. However, since photons have a spin angular momentum of h/2π, and so the conservation of angular momentum limits the change in angular momentum of the molecule to ∆J = ±1. In collisional processes, the colliding particle can inject or carry away any amount of angular momentum, so here transitions with |∆J | > 1 are possible. The energy of a transition is given by, 26 2. Line Emission Processes E′ − E′′ = ∆E = h 2 8π2I [J ′(J ′ + 1) − J ′′(J ′′ + 1)]. (2.28) When J ′ = J ′′ + 1 = J + 1, for radiative transitions, this reduces to ∆E = h2 4π2I (J + 1), (2.29) and the lines frequencies are, ν = ∆E h = h 4π2I (J + 1) = 2B(J + 1), J = 0, 1, 2... (2.30) where B is the commonly used molecular rotation constant h/(8π2I). Equa- tion (2.30) implies that the successive J transitions are separated by a con- stant value in frequency; ∆ν = 2B. Astrophysical examples of such rota- tional ladders are presented by the CO or CS molecules. In CO the observed wavelengths of the J = 1 − 0, J = 2 − 1 and J = 3 − 2 transitions are 115271.204 MHz, 230537.974 MHz and 345795.900 MHz, respectively. If the separations were rigorously identical, then we would expect the J = 2 − 1 and J = 3 − 2 transitions to occur at 115271.204 MHz, 230542.408 MHz and 345813.612 MHz, respectively. The slight lowering of the observed frequency compared with that predicted is due to the centrifugal distortion mentioned above. This increases the separation of the molecules, and hence the moment of inertia, and so lowers the energy of the transition, as observed. The change in the moment of inertia of the molecule obtained by substituting a 13C for the 12C in the CO molecule shifts the frequency of the J = 1 − 0 transition to 110201.370 MHz ( c.f. 110204.338 MHz expected simply from the ratio of the reduced masses of the two molecules). 2.2.2 Rotational Wavefunction Symmetry Advanced Section In addition to angular momentum conservation, considerations of symme- try places important restrictions on the allowable rotational transitions. In any quantum wavefunction system, such as an atomic or molecular system, the wave equation (Eqn ??) should remain unchanged after reflection about the origin (x, y, z replaced by −x, −y, −z), that is, it should not depend on the choice of coordinate systems. However the eigenfunctions, or solutions, as shown in Figure (2.5) may be affected by reflection about the origin, and will be either unchanged or have a sign reversal after the reflection. To see this, take a wavefunction in Figure (2.4), Ψ(real) for J = 3 and |mj | = 2 for example, flip it vertically and then shift it left to right by half the width of the diagram with wrap around. Doing this results in the black and white pattern exactly inverting. This process is the graphical equivalent of reflection about the origin point, that is θ′ = π− θ and φ′ = π+ φ. Inspection will show that all the wavefunctions for even values of J are identical (symmetric) after this 2.2 Molecular Spectra 29 S be any symmetric function and A be any anti–symmetric one, then the following apply: SS → S SA → A AS → A AA→ S So, for the transition probability to be non–zero, with an anti–symmetric operator like D, the wavefunctions Ψ1 and Ψ2 must be of opposite symmetry. This can be confirmed by taking triplets of S s and As with the middle term being A and multiplying through with the above rules. In the rotator case, all the positive levels are either symmetric or anti– symmetric and similarly for the negative levels. So the only transitions that are allowed under the dipole operator are positive to negative and negative to positive transitions. This is more general but is still consistent with the ∆J = ±1 rule derived earlier. Other operators such as for magnetic dipole or quadrupole transitions can be symmetric under point reflection and thus have the opposite selection rule, however they are still restricted by angular momentum restrictions. 2.2.3 Rotating Diatomic Molecules with Identical nuclei If the two nuclei of the diatomic molecule are identical, such as in the astro- physically important H2 molecule, further more rigorous symmetry occurs. In addition to reflection symmetry about the origin point, there is symmetry under the exchange of the two nuclei – or reflection about a plane between the nuclei. The total wavefunction must remain unchanged or change sign after the exchange of the two nuclei. The states are either symmetric or anti–symmetric in the nuclei respectively. For a given electronic state, Ψe, either all the positive (in origin reflec- tion) rotational states are symmetric (in the nuclei) and the negative levels are anti–symmetric or the opposite is true. If Ψe does not change sign on exchange, and Λ = 0, then the states with positive symmetric and negative anti–symmetric levels are designated Σ+g and those with negative symmetric and positive anti–symmetric levels are designated Σ+u . If the Ψe does change sign on exchange, then the equivalent states are Σ−u and Σ − g . If the nuclei have zero spin, or nuclear spin is ignored, then there is a very strict selection rule in addition to the previous ones. Here, transitions between symmetric and anti–symmetric states are completely forbidden, for not only any radiative transitions, but also for all collisional and other processes as well. First consider dipole radiative transitions. Previously we had the dipole operator D in the matrix element integrals anti–symmetric under point origin 30 2. Line Emission Processes reflection, forcing the product of the two level wavefunctions Ψ1 and Ψ2 to be anti–symmetric under reflection so the integral would be symmetric and hence able to have a non–zero value. This could only be achieved if the levels were positive and negative or emphvise versa. Under nuclear exchange, or plane reflection symmetry, the dipole operator D is always symmetric, as is any other multipole. Therefore, for the integrand Ψ1DΨ2 to be symmetric, and the integral be able to have values other than 0, Ψ1 and Ψ2 must have the same symmetry. If they do not then the integral will be 0 and the transition forbidden. Even in collisional processes, any third particle interacting with the diatomic nucleus cannot distinguish between the exchange of the nuclei, so the collisional processes are symmetric in the nuclei, and so cannot perform transitions between symmetric and anti–symmetric states. Since symmetry alternates between symmetric and anti–symmetric levels in J , ∆J is never an odd number of levels. The implication of this is that systems of zero nuclear spin, like O2, and C2, could exist in two distinct populations, one with only symmetric rotation levels, and one with anti–symmetric rotation levels, depending on what sym- metry the molecules were formed with. These molecules would not be seen directly in dipole transitions between rotational levels, but may be seen in rotational fine structure of the vibration lines of electronic transitions. Obser- vationally only the odd J levels are occupied, which are symmetric, and no anti–symmetric rotational levels are present. In Raman spectra of O2, where transitions occur with ∆J = 0 or ±2 and thus are allowed, all the even num- bered lines are missing, supporting the hypothesis that while anti–symmetric levels are mathematically possible, in our universe only the symmetric ones are used. This is presumably related to the intrinsic symmetry of the nuclei before the molecule formation and the symmetry of the process that forms them. In the case of H2, the nuclei have non–zero spin – namely ±1/2. The nuclear spin of the molecule can take on values according to the vector sums of the two spins, either 1 (both +1/2, or parallel, anti–symmetric) or 0 ( one +1/2 and the other −1/2, anti–parallel , symmetric). More generally for various spins, the total nuclear spin can take on the values, N = 2I, 2I − 1, ..., 0 (2.33) where I is the spin number of each nucleus. Also each value of N has a statistical weight of 2N + 1 in the usual manner. For H2, I = 1/2 and the parallel state has a statistical weight of 3 and the anti–parallel a weight of 1. Furthermore, when the nuclear wavefunction is combined with the rest of the molecule wavefunction in eqn (2.31), the symmetry of the nuclear wavefunc- tion can change the symmetry of the overall wavefunction. For example, an anti–symmetric nuclear function multiplied by a symmetric molecular func- tion will become anti–symmetric overall. However this alone is not sufficient to allow symmetric to anti–symmetric transitions to occur. If the nuclear wavefunction Ω and molecular wavefunction Ψ are separable: 2.2 Molecular Spectra 31 Ψtot. = ΨΩ (2.34) In this case the transition matrix element integrals are also separable, and each part must still obey the symmetry laws and be strictly zero for sym- metric to anti–symmetric states. However if the nuclear function is coupled to the molecular wavefunction, so that some part 9 is not separable, Ψtot. = ΨΩ + 9. (2.35) Then the symmetry of the rotational components of the total wavefunction are not perfectly symmetrical and the transition probabilities will be non– zero. The nuclear coupling is usually extremely weak however, and the tran- sition rates are very small indeed. The mean lifetime of a molecule of one rotational symmetry before it makes a transition to a state of another sym- metry (∆J = ±1) can be years. As a result of this there are two types of H2 molecules; one with anti– symmetric (parallel) nuclear and anti–symmetric rotational wavefunctions (odd J numbers), and one with symmetric (anti–parallel) nuclear and sym- metric rotational wavefunctions (giving even J numbers). Because of their long lifetimes, these behave as two separate rotation systems which can only very slowly mix with one another. Given enough time - and that is a lot, they will come into statistical equilibrium with each other in the ratio of their statistical weights, that is 3 : 1. The most common type is known as the ortho- variety, ortho–hydrogen or o−H2, and the less common variety is para-hydrogen or p−H2. In homonuclear molecules with integral spins, i.e. deuterium, D2, with I = 1, a similar circumstance arises, with the most com- mon variety also known as ortho–deuterium, however in this case the ortho variety are the symmetric systems and the para variety is the less common anti–symmetric type. Notes on Chapter 2 • A very useful introductory summary of quantum physics is given by James William Rohlf “Modern Physics from α to Z0”, John Wiley & Sons: NY, 1994. • The quantum mechanics of molecules is fully treated in the book by P.W. Atkins “Molecular Quantum Mechanics” Second Edition, 1983, (Oxford University Press: Oxford), ISBN 0-19-855170-3. • The atomic and molecular physics relevant to IR and radio astronomy is clearly summarised by Reinhard Genzel in Saas–Fee Advanced Course 21, “The Galactic Interstellar Medium”, 1991, (Springer–Verlag: Berlin), ISBN 3-540-55805-5, whose work has been used in the preparation of this chapter and in Chapter 4. 34 3. Collisional Excitation E12 N2 N1g1 g2 E12 Electron Energy, E Threshold Energy C ro ss -s ec ti on , σ 12 Fig. 3.1. Above the threshold energy for collisional excitation, the cross section decreases approximately inversely as energy. radiative decay rates are comparable, the intensity of an emission line is de- termined both by the temperature and the density. Thus if the temperature is known, the density can be determined from the intensity ratio of two such lines. These simple considerations hold the key to the spectroscopic determi- nation of the physical conditions in any given parcel of interstellar plasma. Let us now consider the physics of collisional excitation in more detail. 3.1 Collisional Excitation by Electron Impact Consider an (idealised) atom with only two energy states, a ground state and an excited state which can radiate back down to the ground state. Electron impacts can collisionally excite an atom into an upper level. Once there, if left alone, it will return to the ground state by a radiative transition. If, on the other hand, it suffers another collision with an electron while still in the excited state, it may collisionally de–excite back down to the ground state. The collisional cross–section is a strongly varying function of energy, in general varying approximately inversely as the impact energy. Therefore, the collision strength, Ω12, defined in terms of the collisional cross–section, σ12(E): σ12(E) = ( h2 8πmeE )( Ω12 g1 ) cm2, (3.1) 3.1 Collisional Excitation by Electron Impact 35 is a more convenient quantity to use, since it removes the primary energy dependence for most atomic transitions. In this equation, me is the electron mass, E is the electron energy, and g1 is the statistical weight of the ground state. In addition Ω12 has another advantage; that of symmetry between the upper and the lower states. This is readily demonstrated. Consider the situation at very high density, where the population of the ground and the excited level are determined entirely by collisions, the radiative rate being negligible by comparison to either the collision excitation rate or the col- lisional de–excitation rate. These circumstances ensure that the atom is in Local Thermodynamic Equilibrium (LTE) and that the two levels are popu- lated according to the Boltzmann equilibrium (equation 1.1) at the electron temperature, T , (equation 1.2): N2 N1 = g2 g1 exp [−E12 kT ] (3.2) Consider the equation of detailed balance at high density (Boltzmann Equilibrium). However we can also compute the population ratio from equa- tion (3.1), using the principle of detailed balance. This states that, in equi- librium, the rate of population of the upper level per unit volume, through collisional excitation, R12 ( cm−3 s−1) is equal to the rate of depopulation through collisional de–excitation, R21 ( cm−3 s−1). If the electron density is ne ( cm−3), and these are distributed according to the Maxwell–Boltzmann energy distribution function f(E)dE; given by equation (1.2), then: R12 = neN1 ∞∫ E12 σ12(E).E.f(E)dE = neN1 α12 (3.3) = neN1 ( 2πh̄4 km3e )1/2 T−1/2 ( Ω12 g1 ) exp [−E12 kT ] cm−3 s−1, and R21 = neN2 ∞∫ 0 σ21(E).E.f(E)dE = neN2 α21 (3.4) = neN2 ( 2πh̄4 km3e )1/2 T−1/2 ( Ω21 g2 ) cm−3 s−1. (it is a useful exercise to derive these two equations). The alphas, α12 amd α21 are known as the collisional excitation and de–excitation coefficients, have units of cm3 s−1, and in general α12 = α21 because of the Boltzmann factor exp[−E12/kT ] and different statistical weights. 36 3. Collisional Excitation Applying detailed balance by setting the rate in equation (3.3) to that of equation (3.4) gives the ratio of the two level populations in terms of the rate corefficients: N2 N1 = α12 α21 = ( Ω12 g1 )( g2 Ω21 ) exp [−E12 kT ] (3.5) Comparing this with the Boltzmann Equation (eqn 3.2), it follows that the collision strength, Ω, has to be symmetric between the levels, i.e. Ω12 = Ω21. There is a simple quantum mechanical sum rule for collision strengths for the case that one term consists of a single level, and the second consists of a multiplet. This occurs, for example, in the case that one of the levels has had its degeneracy removed by spin–orbital interactions, as in, for example, the individual levels within the term 3P0,1,2. For such terms: Ω(L1,S1.J1:L2,S2.J2) = (2J2 + 1) (2S2 + 1)(2L2 + 1) Ω(L1,S1:L2,S2) (3.6) provided that either S = 0 or L = 0. Here, the factor (2J2+1) is the statistical weight of an individual level in the multiplet, and (2S2 + 1)(2L2 + 1) is the statistical weight of the multiplet. In the example of the term 3P0,1,2, we can regard the collision strength as being “shared out” amongst these levels in proportion to the statistical weights of the individual levels; gJ = (2J + 1). Thus, the 3P0 level will carry 1/9 of the total collision strength, the 3P1 level has a fraction 3/9, and the 3P2 level accounts for 5/9 of the total. Quantum mechanical calculations show that the resonance structure in the collision strengths is important, and that, for neutral species, the col- lision strength increases with energy. However, because of the property of resonances that they oscillate -ve and +ve over a small energy range, then the effect on the temperature averaged collision strength (average over the Maxwell–Boltzmann distribution ), Ωij is usually small, and can be usually fitted by a simple power law; Ωij = A+B(T/104K)n. For electric dipole tran- sitions, the collision strength and the gf value of the transition are related through the equation given by Seaton (1958): Ωij = ( 8π√ 3 ) E−1ij gfijG(T ) (3.7) where G(T ) is a Gaunt Factor, which is a numerical multiplication factor which changes the result of a “classical” calculation into one which is rigor- ously identical to the result produced by a full quantum mechanical calcu- lation ( often a ‘fudge’ factor or an empirical value when the full quantum solution is not solved yet). In this case, the G(T ) is a fairly complex function of temperature. This can be written (from Landini & Monsignori Fosse, 1991) in terms of the first exponential integral E1 as: G(x) = A+ exp[x] ( Bx− Cx2 +Dx3 + E ) E1(x) + (C +D)x−Dx2. (3.8) 3.2 The Three–Level Atom 39 number. As a consequence, many of the transitions that are involved are forbidden, and are only important at the low densities characteristic of the ISM. However, many of these transitions occur at optical wavelengths, and counted amongst the strongest lines in ground–based spectra of ionised nebu- lae. Such systems provide an instructive insight into the main way line ratios can be used for plasma diagnostics for the ISM, and have given us our basic understanding of the physical conditions applying in ionised regions in the ISM. Let Cij be the collision rate (Cij = neαij s−1) between any two levels defined by equations (3.3) and (3.4), above; depending on whether the rate represents a collisional excitation or a collisional de–excitation. If the Aij are the radiative transitional rates, then the equations of statistical equilibrium for a three–level atom are: N1C13 +N2C23 = N3(C31 + C32 +A32 +A31) N1C12 +N3(C23 +A32) = N2(C23 + C21 +A21) (3.15) N1 +N2 +N3 = 1 The last line normalises of the populations in all of the levels to sum to unity, so the populations of individual levels are given as a fraction per ion. Clearly, these are three linear equations, and can be readily solved. However, to illustrate the way in which such an atom can be used to determine the physical parameters of temperature and density, it is interesting to consider two special cases. 3.2.1 Low Density Limit; E12 ∼ E23 In this case, because of the low density, collisional de–excitations of the ex- cited levels can be safely ignored (Cij ∼ 0 for i > j). Also, because of the increasing threshold energies to excite each level, N3 N2 N1 so that equation (3.15) can be reduced to: N1C13 = N3(A32 +A31) N1C12 +N3A32 = N2A21 (3.16) N1 +N2 +N3 = 1 hence, N3 = N1C13/(A32 + A31) and N2 = N1C12/A21. If we now form the line intensity ratio for the 3 → 2 and 2 → 1 transitions, and substitute equations (3.3) and (3.4) in the resulting expression we have: F32 F12 = E32A32C13 E21(A32 +A31)C12 = E32A32Ω13 E21(A32 +A31)Ω12 exp [−E23 kT ] (3.17) 40 3. Collisional Excitation E12 N1 g1 N3 g3 N2 g2 hν12 hν13 hν23E23 E13 Fig. 3.3. A three–level atom with nearly equi–spaced excited states. Such a level configuration allows the ion to be used for temperature diagnostic purposes. Because this line ratio varies as exp [−E23/kT ], it can be used to measure the electron temperature in the plasma. Astrophysical examples of such temperature–sensitive emission line ratios are to be found in the forbidden lines of the p2 and p4 ions. These are typically excited by thermal electrons at nebular temperatures ∼10000K. An example of the temperature diagnostics is shown for the case of the [O iii] ion in Figure (??). The reason we can use the three level approximation for ions such as these is that the characteristic resonance lines have much higher threshold energies, and are unimportant as cooling transitions at these temperatures and densities. For these atoms, the ground term is a triplet, as a result of the spin–orbit interactions. However, the ground term splitting is small, so the atom can be approximated by a three–level system. In addition, the relative intensities of the individual transitions to the ground term can easily be worked out. Quantum mechanics shows that, apart from relativistic corrections; A(1D2 →3 P2) : A(1D2 →3 P1) ≈ 3 : 1 (3.18) and A(1S0 →3 P2) : A(1S0 →3 P1) ≈ 3 : 1. (3.19) In Table 3.1 we list the wavelengths of some important nebular lines in p2 and p2 ions. Note that in the p4 ions the order of the ground term energy levels is reversed. 3.2 The Three–Level Atom 41 [OIII]λ5007,4363 3.5 4.0 4.5 5.0 5.5 -3.0 -2.0 -1.0 0.0 log[Ratio] 3.0 4.0 5.0 6.0 7.0 -27.0 -26.0 -25.0 -24.0 -23.0 log[Flux] log[T (K)] [5007] [4363] log[T (K)] Fig. 3.4. The temperature dependence of the emissivity (left) and the ratio (right) of the two [ O iii] lines, λ 5007Å and λ 4363Å . Table 3.1. Some important nebular lines (Å) p2 Ions [N ii] [O iii] [Nev] [S iii] 1S0 → 1D2 5755 4363 2974 6312 1D2 → 3P2 6583 5007 3426 9532 1D2 → 3P1 6548 4959 3346 9069 p4 Ions [O i] [Ne iii] [Ar iii] 1S0 → 1D2 5577 3343 5192 1D2 → 3D2 6300 3869 7136 1D2 → 3P1 6363 3968 7751 3.2.2 Ions in which E23 E12 For three–level ions configuration like that shown in figure (3.5 ), and in the low density limit, we can neglect collisionally induced transitions between the higher levels. Hence, N1C12 = N2A21 and N1C13 = N3A31 . Therefore, in the low density case the line flux ratio is given by : F32 F12 = E32A32N3 E21A21N2 = E31C13 E21C12 = Ω31 Ω21 exp [−E23 kT ] ∼ Ω31 Ω21 . (3.20) using the quantum mechanical sum rule for collision strengths equation (3.6) yields the result: 44 3. Collisional Excitation [OIII]λ4363 / [OIII]λ5007 -3.0 -2.0 -1.0 0.0 -3.0 -2.0 -1.0 3.8 4.0 4.2 4.4 5.0 4.0 4.2 4.4 5.0 log[Te] @ log[ne] = 2.0 3.8 8.75 8.0 7.25 log[ne] [O III ]λ 43 63 / [ O III ]λ λ1 66 3 log[Te] @ log[ne] = 9.5 Fig. 3.7. The temperature and density dependence of the O iii forbidden line ratio λλ4363/5007Å and the forbidden to intercombination line ratio λλ4363/1663Å. At low densities both are temperature diagnostics, but at high enough densities, both density and temperature can be determined. This pair of ratios would be a useful diagnostic in objects such as young nova shells, and in active galactic nuclear emission line regions. other example is provided by the [S ii] λλ(4069+4076))/(6716+6731)Å and the λλ(6731)/(6716)Å line ratio. In practice, almost any two line ratios for a given ion can be used for such diagnostics. At successively higher densities, the ratio of a intercombination to a forbidden line, the ratio of a resonance to an intercombination or the even ratio of two resonance lines may be used. An example of the use of this is shown for the case of the O iii ion in Figure (3.7). Here we plot the ratio which is usually used as a temperature sensitive line ratio at low density, the O iii(λ 4363/λ 5007Å) line ratio against the ratio of the intercombination and forbidden line, the O iii(λ4363/λλ(1660+1666)Å)line ratio (which is also sensitive mainly to temperature at low densities). See Figure (2.4) for the terms involved. Note the way in which each of these ratios becomes sensitive to density in turn, once the critical density for one of the lines involved in the line ratio is exceeded. For forbidden lines, the range of density sensitivity is typically 100 – 107 cm−3, while intercombination lines extend the range up to ∼ 1010 cm−3, and resonance lines could, in principle, be used up to the point of transition to full LTE conditions. However, in practice, this is not usually possible, since 3.3 The General Multi–Level Atom 45 the radiative line transfer problems and radiative pumping of levels will tend to dominate. 3.2.3 Infrared Line Diagnostics In dense regions of rapid star formation, or in regions close to the centres of galaxies, the visible light is often blocked from view by the surrounding dust. In order to probe the conditions of such regions, we need to observe in the far–infrared, where the dust obscuration is low, allowing us to probe these dense regions. In addition, the dust itself displays a rich emission spectrum at these wavelengths, allowing us to identify the nature, composition, and size distribution of the dust component itself. The advent of the Infrared Astron- omy Satellite (IRAS) and the Infrared Space Observatory (ISO) opened up this far–infrared window of the electromagnetic spectrum to detailed study. It is necessary to observe in space because at most ground–based sites, the re- gion of the spectrum covering roughly 15-250µm is blocked from observation by atmospheric absorption, mainly caused by water vapour. The far–IR region of the spectrum contains most of the emission lines which are responsible for cooling plasmas with temperatures of between 100 – 3000 K, but these lines are still important in even hotter plasmas. The principal ions which give rise to lines in this region are the p2 and p4 ions, since these have the multiplet ground terms in which hyperfine transitions may occur. Because the transition probabilities are low in these transitions, the critical densities are also quite low (unless the species is highly ionised), so that ratios of lines from the same ion form useful density diagnostics. However, the mean thermal energy of the electrons or ions is usually ap- preciably higher than the excitation energies of the upper states, so nothing can usually be gleaned about the temperature. More useful is the fact that the emissivity in any line which is not affected by collisional de–excitation is simply proportional to the ionic abundance, thus these lines be used as abundance diagnostics, and ratios of lines of different ionisation may also be used to measure the excitation of the plasma. In Table (3.2), we list some of the most important lines detected with ISO. Here the critical density is given in terms of the collisions with atomic hydrogen. Collisions with molec- ular hydrogen may also be important for IR transitions involving un–ionised atomic species. 3.3 The General Multi–Level Atom It is easy to generalise the equations of statistical equilibrium given in equa- tion (3.15) up to an arbitrary number of levels. In equilibrium, the rate of collisional and radiative population of any level is matched by the collisional and radiative depopulation rates of that same level. When combined with the 46 3. Collisional Excitation Table 3.3. Important lines detected with ISO Species Transition λ(µm) ncrit. (cm −3) IP(eV) [C ii] 2P03/2− 2P01/2 157.74 3 × 103 11.26 [O i] 3P0− 3P1 145.50 9 × 104 . . . [O i] 3P1− 3P2 63.18 5 × 105 . . . [O iii] 3P1− 3P0 88.36 4 × 103 35.12 [O iii] 3P2− 3P1 51.82 5 × 102 35.12 [O iv] 2P3/2− 2P1/2 25.87 1 × 104 54.93 [N ii] 3P1− 3P0 203.5 5 × 101 14.53 [N ii] 3P2− 3P1 121.9 3 × 102 14.53 [N iii] 2P3/2− 2P1/2 57.32 3 × 103 29.60 [N iv] 3P1−3P2 . . . 1 × 106 47.45 [Ne ii] 3P1/2− 3P3/2 12.81 5 × 105 21.56 [Ne iii] 3P0−3P1 36.02 4 × 104 40.96 [Ne iii] 3P1−3P2 15.55 3 × 105 40.96 [Nev] 3P1−3P0 24.28 5 × 104 126.2 [Nev] 3P2−3P1 14.33 4 × 105 126.2 [S iii] 3P1−3P0 33.48 2 × 103 23.33 [S iii] 3P2−3P1 18.71 2 × 104 23.33 [S iv] 2P3/2− 2P1/2 10.51 6 × 104 34.83 [Si ii] 2P03/2− 2P01/2 34.81 3 × 105 8.15 [Ar ii] 2P1/2− 2P3/2 6.99 2 × 105 6.99 [Ar iii] 3P1− 3P2 8.99 3 × 105 27.63 [Ar iii] 3P0− 3P1 21.8 3 × 104 27.63 [Fe ii] 6D7/2− 6D9/2 25.99 2 × 106 16.18 [Fe ii] 6D5/2− 6D7/2 35.35 3 × 106 16.18 population normalisation equation (the sum of the populations of all levels must add up to the total number of ions, we have a linear set of simultaneous equations which may be solved in the standard way. Formally the equations of statistical equilibrium for all levels j and the population normalisation equation can be written: ∞∑ J =j NJCJj + ∞∑ J>j NJAJj −Nj   ∞∑ J =j CjJ + ∞∑ J<j AjJ   = 0 ∞∑ J=1 NJ = 1. (3.23) This can be expressed in a more convenient form by splitting the collisional excitation (E) rates from the collisional de–excitation (D) rates as: ∞∑ J NJ(CDJj + C E Jj +AJj) −Nj ∞∑ J ( CDjJ + C E jJ +AjJ ) = 0 4. Line Transfer Effects “There are no such things as applied sciences, only applications of science” — Louis Pasteur 4.1 Resonance Line Transfer The discussion of the previous chapter has established the way in which emis- sion lines may be produced by collisional excitation locally within a nebula. However, whether the observer can see the emission depends upon whether the nebula is optically thick or optically thin to the escape of this radiation. This in turn depends upon the optical depth of the nebula at the frequency, ν, considered; τν . For transmission of light through an absorbing screen of material, the optical depth is defined by the factor by which the intensity of the radiation has been reduced, I(ν)/I0(ν) = exp[− τν ]. The optical depth is the integral along the line of sight of the linear absorption coefficient (cm−1) κ(s, ν); τν = ∫ κ(s, ν)ds. However a more useful measure is the mass ab- sorption coefficient (cm2g−1), κν , which measures the effective cross section contributed per unit mass of matter at this frequency, κν = κ(s, ν)/ρ(s). This is also called the opacity of matter. Since matter can either scatter light (re– radiation at the same frequency), or absorb it one frequency to re–radiate at another, we have to recognise that the total opacity is the sum of these two contributions κν = κν(scat)+κν(abs). Absorption is produced by either dust, nebular continuum processes, or by fluorescence (degradation of the original photon into two or more photons, which occurs when the excited state of the atom has more than one permitted decay route). Scattering can be produced by free electrons, or by resonant scattering in lines. In the case of a scattering screen, a photon in the beam of light from a distant object is first removed from the beam, and then re–radiated in all directions, so is effectively lost from the beam. Thus, in resonance line absorption by the ISM, we see a series of dark lines superimposed on the intrinsic continuum spectrum of the distant object (star, galaxy or whatever). Let us first consider this process in a little more detail. 50 4. Line Transfer Effects 4.1.1 Resonance Line Absorption by Heavy Elements Consider an excited state, j = 2, and the corresponding resonance transition to the ground state, j = 1. The width in energy (or frequency) of an excited state is not a perfect delta function. This is because the lifetime ∆t, of the excited state is finite, determined by the transition probablility back down to the ground state, A21; given by equation (??) or (2.5):∆t = 1/A21. This finite lifetime implies finite energy width by the Heisenberg uncertainty principle of quantum mechanics; ∆E.∆t ≥ h̄. Thus, this energy uncertainty trans- lates to a natural line width in terms of frequency (full width half maximum; FWHM), ∆νN =A21/2π. Since the frequency dependence of the absorption profile can be represented as the Fourier transform of an exponentially de- caying harmonic oscillator, the effective absorption coefficient at frequency ν has a natural line shape which is Lorentzian, centered at the frequency of the transition ν0 and having a FWHM ∆ν . In terms of the population of the ground level, N1, and the absorption oscillator strength f12, the cross section is given by: σ(ν) = πe2 mec f12N1 [ 1 − g1N2 g2N1 ] ∆ν/2π (ν − ν0)2 + (∆ν/2)2 (4.1) Here, the term in square brackets contains the factor which corrects for the stimulated emission, and the frequency function has been normalised so that its integral over the full line profile is unity. This function declines as a power law with frequency for frequencies which lie far away from the line core. These are often referred to as the damping wings of the profile. If we neglect the correction for stimulated emission, and assume that at the low density limit all of the atoms are in their ground state, then at the line centre the cross section per atom is σ(ν0) = 2e2(mec)−1f1j∆ν−1N ∼ 0.0169f1j∆ν−1 cm2. In most cases, near the line core, the line broadening is not dominated by the natural width, but by the Doppler broadening caused by the thermal motions of the atoms along the line of sight, vx. The Doppler shift in frequency produced by this velocity relative to the line center is given by (ν − ν0) = ν0vx/c. At a given ionic temperature T , the fraction of ions, with mass M , in the velocity range vx to vx + dvx is given by the one dimensional Maxwell distribution: dN(vx) = ( M 2πKT )1/2 exp [−Mv2x 2kT ] dvx. (4.2) This will produce an absorption cross section profile which is Gaussian: σ(ν) = πe2 mec f12N1 [ 1 − g1N2 g2N1 ]( M 2πKT )1/2 exp [ −Mc2 (ν − ν0)2 2kTν20 ] . (4.3) In this case the line width of the Doppler profile, ∆νD, (FWHM) is given by ∆νD = 2(ln 2)1/2(2kT/M)1/2(ν0/c). Again, neglect the correction for stimu- lated emission, and assume that at the low density limit all of the atoms are 4.1 Resonance Line Transfer 51 in their ground state, then at the line centre, the cross section per atom is σ(ν0) = 2(π ln 2)1/2e2(mec)−1f1j∆ν−1D ∼ 0.0249f1j∆ν−1D cm2. In addition to these natural and thermally–broadened line profiles, the absorption line will also be broadened by the internal turbulence of the gas cloud in which it is formed, and shifted to higher or lower frequencies by the Doppler shift due to bulk motion of the cloud with respect to the observer. What is the effect of these absorbers on the light of a continuum source of intensity I0 located behind the cloud? At any frequency, the fraction of light which gets through is determined by the optical depth at that frequency, τ(ν) = Nσ(ν), where N is the total column density of absorbing atoms through the cloud. Thus: I(ν) = I0 exp [−Nσ(ν)] (4.4) It is convenient to define an equivalent width, W , for the line, which measures the net flux removed from the incident beam by the absorption line. This quantity is useful because it can be measured even when the spectrograph used to observe it does not have the resolution needed to resolve the details of the actual line profile. By the definition of equivalent width, the total flux removed from the beam by the line is I0W . We can also write this product as: I0W = I0 ∫ dν − ∫ I(ν)dν (4.5) that is, the flux that was in the beam before absorption less the flux that remains in the beam after absorption. However, from this equation, and equa- tion (4.4), it is clear that we can write W as: W = ∫ (1 − exp [−Nσ(ν)]) dν (4.6) When the optical depth of the cloud is small at all frequencies, τ(ν) = Nσ(ν) 1, it follows that W = N ∫ σ(ν)dν so that the equivalent width is directly proportional to the number of atoms on the line of sight. If this con- dition is satisfied, then we refer to the line as lying on the “linear” portion of its curve of growth. From observational usage, the equivalent width is often presented in terms of wavelength rather than frequency. In this convention, a very convenient form for the column density in the linear portion of the curve of growth is: [ N/cm−2 ] = 1.13 × 1017f−112 [ λ/Å ]−2 [ Wλ/mÅ ] (4.7) This equation is extensively used in observational studies. When the line becomes optically thick in its core, then the residual flux in these regions is very small, and so the equivalent width can only increase as a result of absorption in the Doppler wings of the profile in equation (4.3). In this case, the effective width of the line core is determined by the condition 54 4. Line Transfer Effects -50.0 0.0 -50.0 0.0 0.0 1.0 1.0 1.0 Relative Velocity (kms-1) C II* λ1336 Fe II λ2374 Fe II λ2599Si II λ1527 Si II λ1808 S II λ1254 N o rm al is ed F lu x Fig. 4.1. Interstellar line profiles observed towards the reddened halo star HD 93521. Many velocity components are evident in this montage which covers both weak and strong lines. Analyses of such data give the physical conditions and chem- ical abundances in the different clouds (after Spitzer & Fitzpatrick, 1993). Along other lines of sight, or even in other cloud components along the same line of sight, we see different patterns of depletion. These are connected with the temperature, density and thermal history of the ISM within them, and collectively they give insight into the way in which energetic processes in the ISM, such as shocks, or heating by radiation fields, can destroy the interstellar dust. The ratio of deuterium to hydrogen in the cosmos is one of the key obser- vational parameters which probe the nature of the Big Bang. This is because deuterium is made in the Big Bang nucleosynthesis, and the amount that is made is very sensitive to the cosmological parameters. The deuterium Lyα line lies 0.331Å to the shortward of the hydrogen Lyα line, thanks to the isotope shift which is due the difference in the reduced mass of the electron in deuterium as compared with hydrogen. Provided that the absorption line due to deuterium is strong enough to be detected in the wings of the hydro- gen Lyα line, then the deuterium abundance can be accurately determined because the deuterium line lies on the linear section of the curve of growth, when equation (4.7) applies; whereas the hydrogen line is firmly in the damp- ing portion of its curve of growth, for which equation (4.6) applies. This is evident from Figure (4.3). The GHRS observations for Capella yield (D/H) = 1.60 × 10−5 (±0.09 × 10−5);(+0.05; −0.10 × 10−5) (Linsky et al. 1995), 4.1 Resonance Line Transfer 55 500 1000 1500 -4.0 -3.0 -2.0 -1.0 Condensation Temperature (K) D ep le ti on o f El em en t (w .r .t . S un ) 0.0 1.0 Ar C Kr N O Tl Pb B Zn S Se Sc Cl Ge Ga Na K Cu P As Mn Si Mg Li Cr Fe Co Ni Ti Ca Fig. 4.2. The depletion of the gas–phase abundances of various elements, plotted as a function of their condensation temperatures for dust formation. This is derived from resonance line absorption measurements made along the line of sight to ζ Oph, which passes through a dense cool interstellar cloud (after Savage & Sembach, 1996). which is currently one of the most accurate value yet determined for this ratio. During the process of nucleosynthesis, deuterium is easily destroyed in stars, measured interstellar abundances only provide an upper limit to the primordial deuterium abundance. Nonetheless, this is still sufficient to place a strong upper limit on the local baryon density of ΩBh75 <∼ 0.055; where ΩB is the ratio of the local baryon density to the critical density needed to close the universe, and h75 is the Hubble Constant in units of 75 km s−1Mpc−1. Even tighter limits can in principle be obtained by using lines of sight to distant QSOs. These probe the intergalactic medium present in the early universe, in which very little of the deuterium should have been destroyed. In these distant reaches of space, a much greater proportion of the gas is located in clouds between the galaxies. Deep observations with large tele- scopes have revealed these from the Lyα absorption line which they produce. These enable us to detect H column densities as small as 1014 cm−2. In the denser clouds, absorption lines of heavier elements such as Fe or Zn are also detected, which allow us to investigate the chemical evolution of the inter- galactic medium (IGM) of the early universe. 56 4. Line Transfer Effects Fig. 4.3. The Lyα line profile in β Cas measured using the GHRS instrument on HST. The intrinsic line profile of the star is shown as a smooth curve, as is the model fit including absorption. The D/H ratio can be accurately measured because the deuterium line is detected in its linear portion of the curve of growth, while the H i line is in the damping portion (after Dring et al. 1997). Finally, absorption line studies have been used to probe the hot phase of gas, and the nature of the diffuse plasma in the halo of our galaxy and in the Magellanic Clouds, using such ions as Si iv, C iv, and Nv. In the local ISM this hot phase was also studied by the Copernicus satellite in the Ovi ion. With the advent of the FUSE satellite, we will obtain a much deeper insight into this phase, and, linking these data with observations in the soft X–ray region of the spectrum, we should be able to finally understand the energy balance of hot galactic halos. Some of these applications will be discussed in more detail in section (7.6). 4.1.3 Line Transfer in Emission Resonance Lines When the plasma is ionised, and hot enough that electron excitation becomes important, the equation of transfer in the line has to be modified to include the local emissivity, jν , of the plasma in the resonance line: dI(ν) ds = −κνI(ν) + jν (4.9) Now, suppose that we have a slab of emitting material illuminated by some background source with intensity I0(ν), then, substituting for the optical 4.1 Resonance Line Transfer 59 Place Holder Figure Fig. 4.4. The escape probability for a sphere, needs a better caption. Hence, the total outward flux per unit area and per unit time is given by integrating over all possible angles: πF (τT , ν) = 2π ∫ π/2 0 I(θ, ν) cos θ sin θdθ = πjν 2κντ2T ( 2τ2T − 1 + (2τT + 1) exp[−2τT ] ) (4.15) Comparing (4.15) with the flux which emerges when τT = 0, for which πF (0, ν) = (4π/3)jνR, we can estimate the escape probability, P (τT ) as: P (τT ) = πF (τT , ν) πF (0, ν) = 3 4τT ( 1 − 1 2τ2T + ( 1 τT + 1 2τ2T ) exp[−2τT ] ) (4.16) Note that as τT → 0; P (τT ) → 1 and, as τT → ∞; P (τT ) → 3/4τT . In a plane parallel nebula, of optical thickness τ , a reasonable approxi- mation to the escape probability is: P (τ) = 1 τ (1 − exp[−τ ]) ; τ ≤ e; = 1 τ(ln τ)1/2 (1 − exp[−τ ]) ; τ > e; (4.17) In this case therefore: as τ → 0; P (τ) → 1 and, as τT → ∞; P (τT ) → 1/τ(ln τ)1/2. 60 4. Line Transfer Effects log τ 0 2 4 6 8 n=109 cm-3 n=1011 cm-3 lo g P (τ ) 0 -2 -4 -6 Fig. 4.5. The escape probability for the Lyα line as a function of the optical depth in the line and of the electron density. After Drake & Ulrich (1980). In both these cases, and as also proves to be the case in other geometries, the escape probability at high optical depth varies approximately inversely as the optical depth, rather than as exp[−τ ], as we might have näively expected. The reason for this is that the photon can only escape if it is produced within the optically thin zone at the boundary of the nebula or at the boundary of the line profile, and the probability of that varies as 1/τ . In detailed line transfer computations, the escape probability from the line in enhanced at high densities by pressure broadening of the natural width of the line, as is shown in Figure (4.4). Such densities are appropriate for the broad line regions around active galactic nuclei. 4.1.4 Line Transfer in the Lyman Series As Figure (4.4) makes clear, enormous optical depths can be achieved in the Lyα line. In H ii regions which are ionised by hot stars, the Lyα line optical depths are of order 104 while in collisionally excited and ionised plasmas found in the accretion disks near active galactic nuclei, Lyα line optical depths are estimated to rise to several million. In these conditions, all the other lines in the Lyman series become optically thick as well. Consider, as the simplest example, the fate of Lyβ photons in such a nebula. Although the Lyα line itself is condemned to be emitted and re– absorbed for as long as it is trapped in the emitting region, or as long as the 4.1 Resonance Line Transfer 61 b c a Lyβ Hα 6562.80 Å Lyα 1215.67Å 3 2 1 2P2S 2D a b c collisions 1025.71 Å Fig. 4.6. Decay paths from the 3 2P state of hydrogen. Lyβ photons will eventually be degraded into an Hα and a Lyα photon. photon is not destroyed by some other means, the same is not true for the higher members of the Lyman series. However, it is clear from figure (4.5) that there exists another route for the radiative decay from the n = 3 level. For hydrogen, the relative probabilities of radiative decay from the excited (n = 3) level are: P (32P − 12S) = 0.882; P (32P − 22S) = 0.118. Thus, the Lyβ photon has roughly a 12% chance of being destroyed in each scattering. After a few scatterings, the survival probability for a Lyβ photon becomes negligibly small, as Table 4.1.4 shows. Table 4.2. Lyβ photon scattering and survival. No. of Scatterings 1 5 10 100 1000 Survival Probability 0.882 0.534 0.285 3.5 × 10−6 2.9 × 10−55 All similar cascades resulting from the fluorescent conversion of higher members of the Lyman series must eventually produce a Lyα photon, since even were they to produce another Lyman series photon with lower n, this would quickly be converted in its turn. Thus, in nebulae, each photon belong- ing to a higher member of the Lyman series is rapidly converted to a Lyα photon plus members of other series (Balmer, Paschen, Pfund etc.). 64 4. Line Transfer Effects Σ1 + Σ1 + g u ν = 7 4 3 2 1 0 J = 5 J = 4 5 61216 Å 32S , 2P, 2D H I H2 22S , 2P Fig. 4.8. Fluoresence of H2 with Lyα. In order for this process to occur, the molecular hydrogen has to be both vibrationally and rotationally excited,which requires a temperature above about 1000 K 1270 1272 1274 1292 1294 1296 (1,3) P(8) (1,3) P5 (1,3) R6 Wavelength (Å) F lu x U ni ts HH47 HH47 Fig. 4.9. Some fluorescent lines of molecular hydrogen observed in the Herbig– Haro object HH47 using the GHRS instrument on the Hubble Space Telescope (after Curiel et al. 1995) 4.3 Astrophysical Masers 65 enhanced, producing a rapid increase of scattering cross section as the virtual state approaches the bound state. This virtual state can then fluorescently decay by radiation to other excited states of the scattering atom or ion. The best example of this phenomenon is the Raman scattering of the Ovi reso- nance lines by H i which is seen in Symbiotic stars (see 4.10). The theory was given by Schmid (1989). If the incoming photon has frequency νi and the scattered photon as frequency νf then the cross–section for Raman Scattering, σR, is given in terms of the Thompson scattering cross–section, σT = (8π/3)(e2/mec2)2 = 6.65 × 10−25 cm2, by: σR = σT νi νf |M |2 (4.18) with: |M | = ∑ m νiνf 4 ( gf1mgffm ν1mνfm )1/2 (ν1m + vfm) (ν1m − vi) (νfm + vi) (4.19) Since the Doppler width of the incoming line (in energy space) is un- changed in the scattering, but the scattered photon has much lower en- ergy, the apparent Doppler line width of the outgoing photons at λ6825Åand λ6825Åare larger in the ratio: (∆λ/λ)f = (∆λ/λ)i(λf/λi). (4.20) This ratio is about 6.7 in the case of the Ovi resonance. This increase in apparent line width is clearly visible in Figure (4.11). Both the identification of these two lines, and their unusual width were a great mysteries when these were first observed in the spectra of symbiotic stars (Allen, 1980). 4.3 Astrophysical Masers 4.3.1 Theory Recall equation (4.3), which gives the absorption cross–section as a function of frequency for a line broadened by Doppler motions: σ(ν) = πe2 mec f12N1 [ 1 − g1N2 g2N1 ]( M 2πKT )1/2 exp [ −Mc2 (ν − ν0)2 2kTν20 ] (4.21) Consider the term in square brackets, which represents the correction due to stimulated emission. When the level populations are in their Boltzmann ratio, this term equals (1− exp[−hν/kT ]). For transitions giving rise to lines in the optical, the correction factor due to stimulated emission is negligible. 66 4. Line Transfer Effects 3 P 2 S 1 S 1032, 1038Å 1026Å 6825, 7082Å Scattering Cross- Section OVI Fig. 4.10. Raman scattering of Lyα by Ovi. This occurs in the interacting at- mospheres of ‘symbiotic’ stars, which are close binary systems consisting of a hot white dwarf and a cool red giant star. However, for lines in the microwave region, and at the temperatures com- monly encountered in molecular clouds (10-100K), the correction factor may be quite large. Now, suppose that there is a third transition which is being pumped by collisions or by photons, such as is shown in Figure (4.12). In general, transition rates vary as the cube of the frequency. Thus, if the pumping transition (1-3) and the cascade transition (3-2) are at a much higher frequency than the transition (2-1), then in most cases the transition rates in the pump (1-3) and (3-2) will be very much larger than the transition rate back to ground (2-1). This process therefore tends to build up the population in level (2) at the expense of level (1). If a population inversion (g1N2 > g2N1) can occur through this process, then 1 − −g1N2 > g2N1 < 0, which drives the effective absorption coefficient in equation (4.21) negative. That is to say, the light in the line is amplified rather than attenuated along the direction of the light propagation. Provided that the pumping rate is rapid, the light intensity increases exponentially over a distance determined by this negative absorption coefficient, called the maser gain G, viz. I(x) = I0 exp[Gx]. Such a maser is termed unsaturated. Interstellar masers therefore operate as one– pass travelling–wave masers without feedback. In such a cloud, the gain is a function of frequency: G = G(ν) = G0 exp [−Mc2∆ν2 2kTν20 ] (4.22) 4.3 Astrophysical Masers 69 N or m al is ed I nt en si ty Thermal e-Folding Widths Fig. 4.13. Narrowing of the thermal line profile such as can occur for an unsatu- rated maser. Here the gain factors at the line centre, G0x, are given for 5, 10, 20 and 30. 4.3.2 Observations of Galactic Masers Advanced Topic Maser emission has been observed in eight different molecular species, OH, H2O, SiO, HCN, CH, CH3OH, H2CO and NH3, although not all of these are true interstellar masers. Here we will only discuss the OH and H2O masers in detail. Of these, the OH sources show the highest brightness temperatures (as high as TB ∼ 1015K!). Such brightness temperatures are only possible because they are produced in the outer atmospheres of mass–losing luminous old stars, where both the matter and radiation field densities are very high, and much higher than the interstellar medium in general. Maser sources are classified according to their isotropic luminosity, that is the luminosity that they would have if Ω = 4π. Needless to say, this overestimates the true luminosity of the source by factors of possibly several thousand. Only a small fraction of the luminosity of the pumping source is converted to maser luminoisty. For example, a typical galactic OH source has an isotropic luminosity of only 10−3L, although the central star may be as luminous as 104L. The mechanisms which drive the OH maser emission have been explained in detail by Elitzur (1992), see notes. Because the OH molecule is symmetric about the inter–nuclear axis, projections of the internal angular momenta on this axis (the z-axis) are conserved quantities. Since the projection of the end over end rotational angular momentum on this axis is always zero, 70 4. Line Transfer Effects Jz = Lz + Sz. The electronic spin S = 1/2, and the ground electronic state is a Π-state (Lz = 1), so Jz = 1 ± 1/2 which gives rise to two rotational ladders, 2Π1/2 and 2Π3/2. For each of these levels, the electronic interaction with the next electronic configuration, the Σ-state, removes the degeneracy between each parity state in the doublet, producing level splitting, the so– called Λ−doubling. The hyperfine interaction with the nuclear spin I further splits each member of the doublet into two further levels according to total angular momentum F = J + I. The allowed transitions follow the dipole selec- tion rules which require a parity change and∆F = 0,±1 but with F = 0 → 0. The ∆F = 0 lines at 1665 and 1667 MHz are called the main lines, while the ∆F = 1 lines at 1612 and 1720 MHz are the satellite lines. Maser emission has been observed in all four levels as well as in the lines of some low–lying excited rotational states. OH masers are found to be associated with the dusty mass–loss regions of evolved stars – the so–called OH/IR stars which are stars on the asymptotic giant branch stage of evolution, the last before the planetary nebular stage of evolution. The OH/IR stars are classified as Type I if they emit in the main lines, or as Type II if the 1612 MHz satellite lines are strongest. Type II sources generally emit in the main lines as well, but the 1720 MHz satellite line is never seen. This difference reflects the difference in the physical conditions and pumping in the maser excitation region in the two classes. Because variations in the 1612 MHz emission in the Type II sources follow the variations in the radiation of the central star in lockstep, Harvey et al. (1974) were able to prove that this class of sources is pumped by rotational excitations caused by IR photons. This result is consistent with the absence of the 1720 MHz satellite line, which Elitzur (1976) showed can only be pro- duced by collisions in a plasma with temperature less than about 200 K. This proportionality of maser luminosity and IR (pumping) luminosity also implies that the 1612 MHz OH masers are running in the saturated regime. The Type II masers show a “two–horn” velocity structure, consistent with the maser emission being produced in a relatively narrow, non–accelerating but expanding shell about the star. Shell radii are typically inferred to be in the range 1016 to 1017 cm. The pumping mechanism of the main line Type I sources is more complex. To generate the required inversion of the ground–state Λ− doublet requires preferential excitation of the upper Λ− doublet components of the rotational ladders. This can be accomplished (Elitzur, 1978) by a radiation field whose photon ocupation number increases with frequency, and this can be generated by warm, optically thin dust emission. Detailed calculations show that dust temperatures in excess of 100 K are sufficient to produce main–line population inversion. Observationally, the 1667 MHz emission line is stronger than, and occurs more often than, the 1665 MHz emission. This places limits on the dust tem- perature TD in the range 150 K < TD < 280 K. The models imply that every 4.3 Astrophysical Masers 71 OH/IR star should exhibit main–line emission close to the central star, but that the Type II sources with the 1612 MHz line are produced only when the mass–loss rates from the central star are high. OH maser activity is also seen in star–formation regions, in the vicinity of ultra–compact ionised hydrogen regions associated with newly–born massive stars. The masering regions have ages of only a few hundred thousand years. The individual spots of maser emission are only a few 1014 cm across, and their separations are similar, so it is probable that their observed size is their real physical size, although it is likely that they are filamentary along the line of sight to provide favourable conditions for amplification. The masering spots have individual velocity dispersions of order 1 km s−1, but the ensemble of spots seems to be formed in a dense expanding shell of compressed molecular gas surrounding the ionised region. This has an expansion velocity of a few km s−1. Extensive masering activity is also seen in the excited states of OH. The details of the excitation mechanism in star fomring regions remains to be fully worked out. Unlike the case in OH/IR stars, conditions are not so suitable for radiative pumping – both the intensity of the radiation field and the dust temperatures are lower, but the maser luminosities are higher. This would be impossible if the maser is saturated as is the case in the OH/IR stars. It is likely therefore that collisional excitation is playing a key role in the masers seen in ultra–compact H ii regions. Masering activity is seen in the H2O molecule in the OH/IR stars and is also associated with star formation in the vicinity of the ultra–compact H ii regions. The pumping process which excites the H2O masers is definitely collisional, and it requires excitation temperatures of order 1000 K. The H2O molecule is planar with an axis of symmetry passing through the O atom and between the two H atoms. It is clear that the moment of inertia in this axis and in the two orthogonal axes are all different. In rotation the molecule acts neither as a symmetric prolate rotator not as a symmetric oblate rotator – it is intermediate between these two limits. However, the rotational states are quantised according to the total angular momentum quantum number (J) and its projection on these two axes (K− and K+), and labelled as JK−K+. For example, in order of increasing energy, the lowest rotational level is split in two levels, 101 and 110, the J = 2 state is split into two levels, 212 and 221, the J = 3 state is split into four levels, 303, 312, 321, and 330, in order of increasing excitation energy and the J = 4 state is likewise split into four levels, 414, 423, 432, and 441. The radiative selection rules require that K− and K+ must change their parity, and that ∆J = 0,±1. Thus, in collisional excitation of the molecule, within each J multiplet, radiative cascade down to the lowest–lying state is permitted, and this tends to build up a larger than LTE population in the 110, 212, 303, 414 and higher states. This creates a population inversion between these states, and the radiatively accessible state of next lower J , leading to maser activity between these levels. The radiative feeding of this 74 4. Line Transfer Effects a isotropic luminosity of 350 L. This compares with the galactic star for- mation region W49 – the most luminous in the Galaxy – which can reach an isotropic luminosity ∼ 1L, at best. At the time of writing, some sixteen of these sources are known (Braatz, Wilson & Henkel, 1994, 1996; Koekemoer et al. 1995). The nature of the H2O maser emission in the AGN sources was recently reviewed by Maloney (1997). In many of them the emission arises in a dense, shocked molecular ring embedded in the accretion disk around the central ob- ject, near to the point at which the H2O molecules become photo–dissociated. This ring is in Keplerian rotation within about one parsec of the central en- gine. The maser emission is seen in edge–on systems in regions where the velocity shear along the line of sight is low. This condition is satisfied di- rectly in front of the central engine, where the masering amplitude is also assisted by the radio emission produced of the central engine and its asso- ciated radio emission regions. Low velocity shear also occurs at the tangent point in the orbiting ring of material, where the column densities may be much larger. These masers amplify the weaker radio emsision in the disk of the Galaxy. Masers produced here are either red–shifted or blue–shifted (by several thousand km s−1) by the Keplerian orbital motion, vorb = √ GM/R, where R is the radius of the molecular ring, and M is the mass interior to it. The size of the observed orbital motion can be used to infer that there ex- ists a massive Black Hole in the centre of such galaxies. For example, Miyoshi et al. (1995) and Greenhill et al. (1995) find that the mass enclosed within 0.2 pc of the centre of NGC 4258 is 2.1 × 107 M. The central object inthis galaxy must be a Black Hole because the matter density implied by these numbers is in excess of 109 M pc−3, and a cluster this dense cannot possibly remain stable for a timescale comparable to the age of the galaxy. (Figure 4.14) The Seyfert NGC 1068 provides a similar case, with the Black Hole mass inferred to be ∼ 107 M (Greenhill et al. 1996). For the AGN sources, not only can the mass of the Black Hole be estimated, but the distance to the galaxy can be estimated to a remarkable accuracy (∼ 4%) from the proper motion of the individual spots of H2O maser activity as they pass in front of the central continuum source. At the same time, these features march steadily in velocity through the central emission complex (up to 600 km s−1) thanks to the variation in vrad = vorb sin θ, where θ is the (small) angle measured along the orbit from the line of sight. Notes on Chapter 4 • An excellent introduction to absorption line techniques and to atomic physics in the interstellar medium in general is that by Lyman Spitzer, Jr. 1978, Physical Processes in the Interstellar Medium, (Wiley:New York), ISBN 0-471-02232-2, which we have made extensive use of here. This book has recently been republished in the Wiley Classics Library Series. 4.3 Astrophysical Masers 75 -5.00 0.00 5.00 -1000 -500 0 500 1000 8.0 4.0 0.0 -4.0 -8.0 8.0 4.0 0.0 -4.0 -8.0v ( km s -1 ) offset (mas) 4.1 mas 8.0 mas -0.25 0.00 0.25 -60 -40 -20 0 20 40 60 offset (mas) v Fig. 4.14. NGC 4258 Rotation curve. (after Nakai et al 1995) • An excellent recent review of interstellar line abundance analyses using the Hubble Space Telescope is by Savage, B.D., and Sembach, K.R. 1996, Ann. Rev. A&Ap, 34, 279. • Recent compilations of oscillator strengths are to found in Verner, D, Barthel, P., and Tytler, D. 1994, ApJS, 62, 109 and Cardelli, J.A., Fe- derman, S.R., Lambert, D.L., and Theodosiou, C.E. 1993, ApJL, 416, L41. • A coherent account of the whole field of maser emission processes is that by Moshe Elitzur, 1992, Astronomical Masers, Astrophysics & Space Sciences Library v.170, Kluwer: Dordrecht, ISBN 0-7923-1216-3. This represents es- sential further reading for those interested in the field. A fairly recent review of extragalactic masers is by Henkel, C., Baan, W.A. & Mauersberger, R. 1991, Astronomy and Astrophysics Review, 3, 47. Progress in the whole field was also reviewed by Moshe Elitzur, 1992, Ann. Rev. A&Ap, 30, 75. Exercises Exercise 4.3.1. What happens to the local radiation intensity at a fre- quency corresponding to the centre of a strong, optically-thick resonance line produced by an excited state of an ion radiating to the ground state: 76 4. Line Transfer Effects a. Within a nebula in which the upper level population is determined by collisions to and from the ground state and by the resonance line radiation itself (assume that the nebula is very optically-thick in the line). b. Outside the nebula, where the radiation density in the resonance line is very low, but where the ion involved in the transition is the most abundant ion c. Outside the nebula, where the radiation density in the resonance line is very low, but where the temperature is too low or too high to produce this ion in appreciable abundance Exercise 4.3.2. A particular ion has an a line at 1548Å which has an os- cillator strength of 3.0 (change to fit C IV?). Along a particular sight line, it is observed as an unsaturated absorption line with an equivalent width of 255mÅ. a. What is the column density of this ion along the line of sight? b. The line is broadened by turbulent motions to a velocity width of 60 km.s−1. What then is the optical depth at the line centre? Exercise 4.3.3. Assuming that the J = 1 − 0 transition in 12CO at 115271.204 MHz is optically thick in a molecular cloud of radius 1.0pc, that the cloud has a temperature of 200 K, and an internal microturbulence of 1.3 km.s−1 , then what is the luminosity of the cloud in this CO line? Exercise 4.3.4. An active galaxy is thought to have a massive Black Hole associated with it. An H2O maser source associated with this nucleus shows three components, one at the (known) redshift of the centre of the galaxy, and the other two at ±2200 km.s−1 with respect this velocity. a. Where do you think each of these components are formed? The central component shows a complex structure, with subcomponents appearing at −50 km.s−1with respect to the central velocity, moving in ve- locity linearly with time, and eventually disappearing at +50 km.s−1, about 1000 days after they first appeared. b. How would you explain these observations? c. In the light of your interpretation, what would you estimate the mass of the central black hole to be, in solar masses (1.998 × 1033 g)? 5.1 Collisional Ionisation 79 ionisation stage is given in terms of the ionisation cross–section σcoll(E) and the number density of electrons, ne, and ions, nA,i, as (c.f. equation 3.3): RA,icoll = nenA,iα A,i coll = nenA,i ∞∫ I σcoll(E).E.f(E)dE cm3 s−1, (5.3) where f(E) is the Maxwell distribution of the electrons in energy. The cross–section is best calculated (Arnaud & Rothenflug 1985; Suther- land & Dopita 1993) using the methods of Younger (1981, 1982, 1983), based on a five parameter fit to each channel of the collision cross–section. These parameters can be derived either directly from collision cross–section exper- iments, or from theoretical calculations. The fits are made on a channel–by– channel basis which separates the contribution due to each electron config- uration in the more highly ionised species. The cross–section for the mth. channel is expressed as (c.f. equation 3.8): uI2A,iσcoll(m,u) = A ( 1 − 1 u ) +B ( 1 − 1 u )2 + C ln [u] + D u ln [u] (5.4) with u = E/IA,i. Integrating the collisional ionisation cross–section over the Maxwell distribution at temperature Te gives the collisional ionisation rate: αA,icoll(n) = 6.69 × 10−7x3/2 I 3/2 A,i ∫ ∞ 1 uI2A,iσcoll(m,u) exp [−ux] du cm3 s−1. (5.5) with xe being the ratio of the ionisation energy of the ion to the thermal energy of the electron gas, IA,i/kTe. The actual rate of collisional ionisations per unit volume is then given by the product RA,icoll = nenA,iα A,i coll cm −3 s−1. (5.6) Using equation (5.4), equation (5.5) can be integrated explicitly over all the channels (m = 1 → mmax) which contribute to the total ionisation cross– section: αA,icoll = 6.69 × 10−7 kT 3/2 m=mmax∑ m=1 exp [−xm] xm Φ (xm) (5.7) with : Φ (x) = A+B (1 + x) − ( C +Ax−B(2x− x2) ) E1(x) exp [x] +DE2(x) exp [x] where E1 and E2 are the first and second Exponential integrals. For impact energies close to the threshold, kTe IA,i, the contributions of higher energy channels (n > 1) can be neglected. 80 5. Collisional Ionisation Equilibrium A major advantage of this method of solution for the collisional ionisation rate is that the coefficients show a regular progression along isoelectronic sequences. Thus, the coefficients may be estimated with good accuracy in those cases where laboratory of theoretical estimates might be lacking. At sufficiently high electron impact energies, more than one electron of the target nucleus may be excited, leaving the atom in an unstable state, which is stabilised by the radiationless ejection of an electron, possibly followed by a radiative decay of the ionised atom back to its ground state: Ai+ + e− → Ai+∗ + e− − E1 followed by : Ai+∗ → A (i+1)+ ∗ + e− + E2 : A (i+1)+ ∗ → A(i+1)+ + hν This process is known as excitation–autoionisation, and is favoured in heavy elements which have a large number of inner shell electrons and only one or two electrons in the outer shell. The complexity of this process means that is is often more difficult to give simple fitting formulae. However, for many ions excitation–autoionisation cross–sections of the following form provide an adequate fit: σEA(u) = a u ( 1 − 1 u3 ) . (5.8) Here u = E/IEA and the effective excitation–autoionisation potential along a given iso–electronic sequence is given by IEA = E0(Z−N)p where Z is the atomic number, N is the number of inner–shell electrons in the isoelectronic sequence, and p is some power. 5.1.1 The Case of Hydrogen The collisional cross–section for hydrogen according to equation (5.4) is shown in Figure 5.2. In this case there is, by definition, only one channel for ionisation. In fact, a reasonable approximation to this cross section can be obtained using the older semi–empirical Lotz formula, which was extensively used before good–quality theoretical and experimental cross sections were available. The Lotz formula is obtained by setting A,B and D in equation (5.4) equal to zero, and adopting C = 4×10−14 cm2 eV2. An even simpler (but still useful) approximation is obtained by a linear fit to the collisional ioni- sation cross–section in the region of the threshold: σcoll,H = σ0(E − IH)/IH with σ0 ∼ 10−16 cm2 for hydrogen. With this linear approximation, the collisional ionisation rate ( cm3 s−1) is then: αHcoll(Te) = 2.4849 × 106 σ0 IH ∫ ∞ IH (E − IH)E exp [−IH/kTe] (5.9) where the numerical constant is 27/2(πme)−1/2(kTe)−3/2. After integration by parts, and substitution for σ0 this gives: 5.2 Recombination 81 Place Holder Figure Fig. 5.2. The collisional ionisation cross–section for hydrogen calculated from equa- tion (5.4) as a function of electron energy (measured in units of the ionisation potential, 13.6eV) αHcoll(Te) = 2.5 × 10−10 ( 1 + T 78, 945 ) T 1/2e exp [−157, 890/Te] , (5.10) and as before the actual ionisations occuring per second per unit volume is the product of of this rate and the electron and hydrogen densities RHcoll = nenHα H coll. (5.11) The similarity of a collisional ionisation rate equation such as (5.10) and a collisional excitation rate equation such as (3.11) is obvious, the difference in the power–law dependence on temperature being simply related to the difference in the behaviour of the excitation and the ionisation cross–sections with energy above their respective thresholds. [Table of collisional ionisation rate coeffs for some ions] 5.2 Recombination Radiative recombination is the process of capture of an electron by an ion with the excess energy being radiated away in a photon. In most cases the electron is captured into an excited state, and usually into a state of large principal quantum number and high angular momentum state, so that the 84 5. Collisional Ionisation Equilibrium rate to the excited states using the hydrogenic approximation. The result is then often fitted to a simple power law with temperature: αA i rad(T ) = Arad[T/10 4K]−η cm3 s−1. (5.20) [Table of recombination rate coeffs for some ions] 5.2.2 Di–electronic Recombination Rates As mentioned above, di–electronic recombination is the inverse of collisional ionisation. The main pathway for di–electronic recombination is through an excitation of a core electron with capture of the passing electron; Ai+(1s, . . .) + e− → A(i−1)+∗ (n1l1;n2l2) (5.21) Here, one of the electrons is in an autoionising state, n1l1, and the other is in an excited state, n2l2.The ion first stabilises itself by one of the valence electrons radiating back to the ground state: A (i−1)+ ∗ (n1l1;n2l2) → A(i−1)+∗ (n3l3;n2l2) + hν (5.22) and then the ion is free to return to its ground state by radiative cascade: A (i−1)+ ∗ (n3l3;n2l2) → A(i−1)+(n3l3;n4l4) + hν1 + hν2 + . . . .. (5.23) At high temperatures, core relaxation is the most important stabilising pro- cess, while at low temperatures the electron is captured via low–lying res- onance states. There are thus two contributions to the total di–electronic recombination rate: αA,idiel.(Te) = α A,i Lo−T (Te) + α A,i Hi−T (Te) cm 3 s−1. (5.24) A fitting formula for low temperatures is given by Nussbaumer & Storey (1983): αA,iLo−T (t) = 10 −12t−3/2 (a t + b+ ct+ dt2 ) exp [ −f t ] cm3 s−1, (5.25) with t = Te/104 K; a, b, c, d and f being fitting constants for each ion. At high temperatures the Shull & Van Steenberg (1982) form can be used: αA,iHi−T (t) = Adielt −3/2 ( 1 +Bdiel exp [−t0 t ]) exp [ − t1 t ] cm3 s−1, (5.26) with Adiel, Adiel, t0 and t1 being fitting constants for each ion. In general, the low–temperature di–electronic recombination contribution is important at temperatures of about 1000–3000 K and the high temperature term is dominant above 20000 K. Single electron recombination dominates at very low temperatures and is usually an important contributor in the region 5000–20000 K. This is illustrated in Figure (5.3) for the case of carbon ions, taken from the fully self–consistent quantum mechanical calculations of Nahar & Pradhan (1997). [Table of dielectronic rate coeffs for some ions] 5.3 Photoionisation 85 . Fig. 5.3. Total recombination rate coefficients for C ii, C iii, C ivand Cv. The dashed line is the radiative recombination from a fit of the form (5.14 ), the dotted line represents the low temperature di–electronic term (5.17), and the dot–dashed line is the high temperature part fitted by (5.18). The fitted circles are another calculation of this same contribution (after Nahar & Pradhan, 1997). 5.3 Photoionisation 5.3.1 From Outer Shells Photoionisation is the inverse process to radiative recombination. As a conse- quence, there is an intimate connection between the radiative recombination cross–sections and the photoionisation cross section. This will be discussed later. Photoionisation, as the name suggests, is the ionisation of an atomic species by the absorption of a photon: Ai + hν → A(i+1)+ + e− +∆E (5.27) If the incoming photon has sufficient energy, it may leave the more highly ionised species in an excited state which subsequently decays by a radiative cascade back to down to the ground state: Ai + hν → A(i+1)∗ + e− +∆E A (i+1) ∗ → A(i+1)∗ + hν1 + hν2 . . . In this way, an appreciable number of additional channels to the photoionisa- tion process become energetically available at higher energy, thus increasing the photoionisation cross–section. For example: 86 5. Collisional Ionisation Equilibrium O0(2p4 3P) + hν → O0(2p3 4S) + e− : hν > 13.6eV O0(2p4 3P) + hν → O0(2p3 2D) + e− : hν > 16.9eV O0(2p4 3P) + hν → O0(2p3 2P) + e− : hν > 18.6eV in this example, the ionisation occurs to different terms of the same electron configuration. 5.3.2 From Inner Shells If we increase the energy of the incoming photon still further, it becomes possible to remove one of the inner shell electrons by inner shell photoionisa- tion which also results in a change the electron configuration in the excited species, i.e. : O0(1s22s22p4 3P)+hν → O+(1s2s22p4 2P or 4P)+e− : hν > 544 eV (5.28) This may be followed by a radiative re–adjustment back to the ground state. However, in this particular case another mode of photoionisation becomes not only energetically possible, but indeed more probable; that of Auger ion- isation. This is a photoionisation from an inner K– or L–shell, followed by a radiationless autoionisation, and is completed by radiative cascade back down to the ground state: Ai + hν → A(i+1)+∗∗ + e− +∆E1 A (i+1) ∗∗ → A(i+m+1)∗ +me− +∆E2 A (i+m+1) ∗ → A(i+m+1) + hν1 + hν2 . . . For example, in the case of the oxygen inner–shell photoionisation given above, the Auger ionisation path is (mostly) into the 3P excited state of O iii: O+(1s2s22p4 2P or4P) → O++(1s22s22p2 3P ) + e− (5.29) followed by radiative transitions to the ground state. In some cases these radiative transitions can affect the intensity of the weak lines which are nor- mally used as temperature or density diagnostics in objects such as Active Galactic Nuclei. This was pointed out by Aldovandi & Gruenwald (1985). This process has recently been investigated for carbon, nitrogen and oxygen lines in a series of papers by Petrini and his co–workers. (Petrini & Da Silva 1997, Petrini & Farras 1994, Petrini and Da Aranjo 1994) Note that the radiationless autoionisation may produce more than one electron. In general, m becomes greater than unity above some threshold energy, and increases as more channels for the Auger process become ener- getically accessible. In inner shell photoionisation followed by Auger ionisation, two high en- ergy electrons are produced, the first from the primary photoionisation, with