Front matter

Contents

Front matter

Preface

This book is a revised, updated, and expanded edition of Astrophysics of Gaseous Nebulae and Active Galactic Nuclei (1989) by one of us (D. E. O.), and that book was based in turn on an earlier volume, Astrophysics of Gaseous Nebulae (W. H. Freeman, 1974) by the same author. Over the 30 years between the publication of that first book and the completion of the manuscript of the present one the subject of ionized-gas astrophysics has grown phenomenally. Astronomers have long studied gaseous nebulae, and the cooler, less intrinsically luminous dust and molecular clouds associated with them, for their intrinsic interest. In addition, stars are almost completely opaque, and it is impossible to observe their interiors directly, or to measure their internal temperatures, pressures, densities, or abundances of the elements within them except by indirect methods. Many nebulae, on the other hand, are relatively transparent, and we can measure these quantities directly.

Since Walter Baade's discovery of the two stellar populations, which turned out to be young stars and old, and the working out of the main nuclear reaction chains that release energy on stars by Hans Bethe, C. F. von Weizsäcker, and others, we have begun to understand the way in which most stars evolve. Much remains to be filled in, both observationally and theoretically, and we have learned how to trace star formation and evolution not only from stars themselves, but from their effect on the gas around them as well. We can estimate or measure the star formation rate not only with nebulae in our Galaxy, but throughout distant galaxies in the universe, from emission lines emitted in nebulae, clouds, associations, and spiral arms in galaxies, and even from entire galaxies.

Interpreting the measurements is not straightforward; the path between is through "nebular astrophysics", as explained in this book. Likewise we can measure abundances of the elements in all these objects with the same basic tools. We can directly observe, in some planetary nebulae and supernova remnants, the changed abundances of the elements that resulted within the stars themselves, leading to slow mass loss, shells thrown off from the star, or thermonuclear reactions at the surfaces of highly evolved stars.

Active galactic nuclei and their close relatives, quasars and other types of quasi-stellar objects, are a field of research only 40 years old, which has grown explosively. Almost all we know about them came from study of their spectra, and especially their emission-line spectra, very similar to but not identical with those of gaseous nebulae. Maarten Schmidt, J. Beverly Oke, and Jesse L. Greenstein's giant leap in understanding the basic nature of 3C 273 and 3C 48 depended on their knowledge of nebular

emission lines, and the subsequent study of all these objects has paralleled the study of gaseous nebulae.

Cosmology has advanced tremendously in the past decade or two, based on the availability of new, very large telescopes, either on high mountains like Keck I and II on Mauna Kea, Hawaii, and the VLT on Paranal, Chile, or in space, like the Hubble Space Telescope, and their fast, digital imaging systems and spectrometers. All the most distant objects we know have been identified and measured by their “nebular” emission lines emitted within them, from H I \text{L}\alpha to [O II] \lambda 3727, [O III] \lambda \lambda 4959, 5007, and H \alpha. How can we be sure it is one of these lines that an astronomer has observed? If it was, what other lines should be seen? If they are not seen, should that purported identification be rejected? If the identification is correct, what physical information (besides redshift and hence distance and look-back time) can be drawn from it? These are some of the questions which nebular astrophysics can help to answer.

In all these subjects, going beyond simply using a formula or canned interpretation from a previously published paper, is a necessity, in our opinion. We have tried to explain and illustrate the methods used, but also to express reasonable doubts about some far-reaching conclusions drawn from minimal data, without examining possible alternative interpretations. In the final chapter of the book we have outlined several questions about active galactic nuclei which shall await full explanations. The methods of the book will be useful in deciding the answers, we feel certain.

The fundamental outline and style of exposition of Astrophysics of Gaseous Nebula and Active Galactic Nuclei , often referred to as (AGN) ² , has been preserved in this new edition, but we have carefully gone through every chapter, weeding out the old and including many new results, measurements, and ideas. Each chapter was completely rewritten, and then revised several times. Furthermore, since its publication of the first edition, results from two fast-developing observational techniques, infrared astronomy and X-ray astronomy, have multiplied many fold. Hence we not only updated the material in these two fields for this edition, which we call AGN3, but have added much more new material, so it now includes two more chapters, one on each of these topics. In both these chapters, and in the rest of the book as well, we have emphasized the continuity of the physical ideas across the boundaries of “spectral regions” and the importance of observing and analyzing data over as wide a range of wavelengths as possible.

From comments of many recent students and current researchers who are using (AGN) ² , we are aware that beginning graduate students of astrophysics today generally know much more about applications of quantum mechanics to nuclear physics than the previous generation, but much less about its applications to atomic spectroscopy, so necessary in nebular astrophysics. Hence we have added a completely new appendix on “nebular quantum mechanics”, and also a shorter one on “molecular quantum mechanics” (which neither generation of astrophysics graduate students know very well) to help bring them in to the fold.

Most of the figures are new, based on recent published measurements and interpretations, and on images obtained with the most advanced telescopes and detecting systems of today. We are grateful to many colleagues who gave us permission to use them in AGN3.

All the chapters, after the first five, which deal with basic material, have been reviewed for us by colleagues and friends who are active research experts in the various fields. We are most grateful to the following for their efforts on our behalf, who either contributed material or reviewed chapters: Jack Baldwin, John Bally, Robert Bauman, Mark Bottorff, Eugene Capriotti, John Danziger, Kris Davidson, Reginald Dufour, Robert Fesen, Donald Garnett, William Henney, Richard B. Henry, Luis Ho, Roberta Humphries, George Jacoby, Sveneric Johansson, Steven Kahn, Kirk Korista, Steven Kraemer, Karen Kwitter, Sun Kwok, Xiaowei Liu, John Mathis, Jon Morse, C. Robert O'Dell, Manuel Peimbert, Richard Shaw, Gregory Shields, Joseph Shields, Lewis Snyder, Phillip Stancil, Barry Turner, Sidney van den Bergh, Peter van Hoof, Sylvain Veilleux, Nolan Walborn, Joseph Weingartner, Robin Williams, Robert E. Williams, Mark Wolfire, and Stan Woosley. We also are extremely grateful to Nick Abel who carefully read the entire manuscript. We thank Jeffrey Mallory and John Rickard for help in producing the final manuscript. Any errors which may remain are our responsibility, not theirs.

D. E. O.
G. J. F.

Front matter

Preface to the First Edition

Fifteen years ago I sent to the publisher my book on Astrophysics of Gaseous Nebulae . It was a graduate-level text and research monograph that evidently filled a need, for it soon became widely used and quoted. Over the years since then the book has found increasing use, not only in nebular research, but also in problems connected with quasars, Seyfert galaxies, quasistellar objects, and all the other fascinating types of active galactic nuclei whose emission-line spectra are similar, in general terms, to those of gaseous nebulae. My own research had turned in those directions since I came to Lick Observatory in 1973 and began obtaining data with its superbly instrumented 3-m Shane reflecting telescope, as it now is named.

Hence as AGN (for so my first book is often referred to) gradually became dated, particularly in its tables of observational results and theoretical calculations, it was natural for me to think of revising it, and of extending it to Astrophysics of Gaseous Nebulae and Active Galactic Nuclei at the same time. Many of my friends and colleagues urged me to do so. Thus the present ( AGN ) ² came about.

Like the earlier AGN , it is both a graduate-level text and an introduction to nebular and AGN research. The first nine chapters are based upon the first nine chapters of the earlier book, but have been heavily revised and updated. The last three chapters are completely new, one on nova and supernova remnants, and the final two chapters on active galactic nuclei. The emphasis is very strongly on the ionized gas in AGNs and the emission-line spectra they emit; their X-ray and radio-frequency radiations are only briefly mentioned.

The book is based upon graduate courses that I have given often at the University of California, Santa Cruz. It represents the material I consider necessary to understand research papers that are now being published in its fields. So much is known today, and so many new results are pouring out, that it is probably impossible to go straight from studying any book to doing frontier research oneself. But I believe that this book will enable the reader to get up to speed, so that he or she will be able to read and understand current research, and then begin to add to it.

The reader for whom ( AGN ) ² was written is assumed to have a reasonably good preparation in physics, and some knowledge of astronomy and astrophysics. The simplest concepts of radiative transfer are used without explanation, since the reader almost invariably has studied stellar atmospheres before gaseous nebulae and active galactic nuclei. Physical parameters, such as collision cross sections, transition probabilities, and energy levels, are taken as known quantities; no attempt is made to derive them. When I teach this material I usually include some of these derivations, linking them to the quantum-mechanics textbooks with which the students are most

familiar. Omitting this material from the book left room to include more interpretation and results on gaseous nebulae and active galactic nuclei.

References are given at the end of each chapter, in a separate section. They are not inserted in the text, partly so that they will not break up the continuity of the discussions, and partly because the text is a complicated amalgam of many papers, with no obvious single place at which to refer to many of them. Almost all the references are to the American, English, and European astronomical literature, with which I am most familiar; it is also the literature that will be most accessible to the readers of this book.

I would like to express my deep gratitude to my teachers at the University of Chicago, who introduced me to the study of gaseous nebulae: Thornton L. Page, S. Chandrasekhar, W. W. Morgan, and the late Bengt Strömberg. I am also very grateful to my colleagues and mentors at the Mount Wilson and Palomar Observatories, as it was then named, the late Walter Baade and the late Rudolph Minkowski, who encouraged me to apply what I knew of nebular astrophysics to the study of galaxies. I owe much to all these men, and I am grateful to them all for their continued encouragement, support, and stimulation.

I am extremely grateful to my colleagues and friends who read early drafts of various chapters in this book and sent me their suggestions, comments, and criticisms on them: Donald P. Cox, Gary J. Ferland, William G. Mathews, John S. Mathis, Manuel Peimbert, Richard A. Shaw, Gregory A. Shields, Sidney van den Bergh, Robert E. Williams, and Stanford E. Woosley. In addition, my two current graduate students, Richard W. Pogge and Sylvain Veilleux, carefully read the entire manuscript; their comments and corrections greatly improved it, as did those of Dieter Hartmann and Philip A. Pinto, both of whom carefully read the supernova material. I am most grateful to them all.

Though these readers found many misprints and errors, corrected many misstatements, and clarified many obscurities, the ultimate responsibility for the book is mine. I have tried very hard to find and remove all the errors, but some must surely remain, to be discovered only after publication. I can do no better than repeat once again the words of a great physicist, Richard P. Feynman, "Listen to what I mean, not to what I say." If the reader finds an error, I am sorry I did not catch it, but he or she will have proved his or her real understanding of the material, and I shall be very pleased to receive a correction.

I am greatly indebted to Gerri McLellan, who entered on the word processor the first drafts of all the chapters, and all the successive revisions of the manuscript, and to Pat Shand, who made the final editorial revisions and prepared the camera-ready copy for publication. I deeply appreciate the skill, accuracy, and dedication with which they worked on this book. I am also most grateful to my wife, Irene H. Osterbrock, who prepared the index for the book.

My research on gaseous nebulae and active galactic nuclei has been supported over the past fifteen years by the University of California, the John Simon Guggenheim Memorial Foundation, the University of Minnesota, the University of Chicago, the Institute for Advanced Study, the Ohio State University, and especially by the National Science Foundation. I am grateful to all of these organizations for their gen-

erous support. Much of my own research, and of the research of the graduate students and postdocs who have worked and are working with me, has gone into this book; I could never have written it without doing that research myself.

I am especially grateful to my friends George H. Herbig, Paul W. Hodge, Guido Munch, and Robert E. Williams, who provided original photographs included in this book. I am grateful to them and also to Palomar Observatory, Lick Observatory, and the National Optical Astronomy Observatories for permission to use the photographs (which are all credited individually) in this book. Publication of the photographs from NOAO does not imply the endorsement by NOAO, or by any NOAO employee, of this book! Many of the other figures are derived from published papers, and I am grateful to their authors for permission to modify and use their figures in this book.

Lastly, I wish to express my sincere thanks to my friends Bruce Armbruster, president of University Science Books, and Joseph S. Miller, my colleague, former student, and astronomy co-editor with me for USB, both of whom encouraged me time after time to go on with revising AGN and writing the additional new chapters for (AGN) ² . Bruce was the astronomy editor for W. H. Freeman and Company when I wrote the earlier book, and he helped me greatly with it then, as he has helped me with (AGN) ² now. It was a great pleasure for me to work with him on both these books. I am also grateful to W. H. Freeman and Company for releasing me from my obligation to them, and allowing me to publish this book with USB.

Donald E. Osterbrock

Chapter 1

General Introduction

1.1 Introduction

Many important topics in astrophysics involve the physics of ionized gases and the interpretation of their emission-line spectra. The subject is fascinating in itself. In addition, H II regions allow us to probe the evolution of the elements and the star-formation history of the far reaches of our own Galaxy, and of distant galaxies. Planetary nebulae let us see the outer remaining envelopes of dying stars. Supernova remnants allow us to observe material from the burned-out deep interiors of exploded, massive stars. Starburst galaxies, quasars, and QSOs are the most luminous objects in the universe, and hence the most distant that we can observe. Spectra can reveal details surrounding the first generations of star birth and the formation of the heavy elements in the young universe. All of these are subjects we shall cover in this book. Further applications, such as the properties of intergalactic material, X-ray flows, and primordial galaxies, though not treated here, are straightforward extensions of the physics that forms the spine of this volume.

1.2 Gaseous Nebulae

Gaseous nebulae are observed as bright extended objects in the sky. Those with the highest surface brightness, such as the Orion Nebula (NGC 1976) or the Ring Nebula (NGC 6720), are easily observed on direct images, or even at the eyepiece of a telescope. Many other nebulae that are intrinsically less luminous or that are more strongly affected by interstellar extinction are faint on ordinary images, but can be imaged on long exposures with filters that isolate a narrow wavelength region around a prominent nebular emission line, so that the background and foreground stellar and sky radiations are suppressed. The largest gaseous nebula in the sky is the Gum Nebula, which has an angular diameter of the order of 30^\circ, while many familiar nebulae have sizes of the order of one degree, ranging down to the smallest objects at the limit of resolution of the largest telescopes. The surface brightness of a nebula is independent of its distance, but more distant nebulae have (on the average) smaller angular size and greater interstellar extinction; so the nearest members of any particular type of nebula tend to be the most-studied objects.

Gaseous nebulae have emission-line spectra. Their spectra are dominated by forbidden lines of ions of common elements, such as [O III] \lambda\lambda 4959, 5007, the famous green nebular lines once thought to indicate the presence of the hypothetical element nebulium; [N II] \lambda\lambda 6548, 6583 and [S III] \lambda\lambda 9069, 9523 in the red; and [O II] \lambda\lambda 3726, 3729, the ultraviolet doublet which appears as a blended \lambda 3727 line on low-dispersion spectrograms of almost every nebula (Figure 1.1). In addition, the permitted lines of hydrogen, H\alpha\ \lambda 6563 in the red, H\beta\ \lambda 4861 in the blue, H\gamma\ \lambda 4340 in the violet, and so on, are characteristic features of every nebular spectrum, as is He I \lambda 5876, which is considerably weaker, while He II \lambda 4686 occurs only in higher-ionization nebulae. Long-exposure spectrophotometric observations extending to faint intensities, show progressively weaker forbidden lines, as well as faint permitted lines of common elements, such as C II, C III, C IV, O II, and so on. Nebular emission-line spectra, of course, extend into the infrared, where [Ne II] \lambda 12.8\ \mu\text{m} and [O III] \lambda 88.4\ \mu\text{m} are among the strongest lines measured, and into the ultraviolet, where Mg II \lambda\lambda 2796, 2803, C III] \lambda\lambda 1907, 1909, C IV \lambda\lambda 1548, 1551, and even L\alpha\ \lambda 1216 are also observed. It is seldom possible to observe all important stages of ionization in a particular spectral region. In these cases one must model the physical system or obtain spectra

outside of the traditional visible/near-IR bands to get an accurate picture of the system in question.

Gaseous nebulae have weak continuous spectra, consisting of atomic and reflection components. The atomic continuum is emitted chiefly by free-bound transitions, mainly in the Paschen continuum of H I at \lambda > 3646 \text{ \AA}, and the Balmer continuum at 912 \text{ \AA} < \lambda < 3646 \text{ \AA}. In addition, many nebulae have reflection continua consisting of starlight scattered by dust. The amount of dust varies from nebula to nebula, and the strength of this continuum fluctuates correspondingly. In the infrared, the nebular continuum is largely thermal radiation emitted by the dust.

In the radio-frequency region, emission nebulae have a reasonably strong continuous spectrum, mostly due to free-free emission or bremsstrahlung of thermal electrons accelerated on Coulomb collisions with protons. Superimposed on this continuum are weak emission lines of H, such as 109\alpha at \lambda = 6 \text{ cm}, resulting from bound-bound transitions between very high levels of H. Weaker radio recombination lines of He and still weaker lines of other elements can also be observed in the radio region, slightly shifted from the H lines by the isotope effect. In the infrared spectral region most nebulae have strong continuous spectra, emitted by heated dust particles within them. These emission continua have bands, some identified as resulting from silicate or graphite in the particles, others still not positively identified.

1.3 Observational Material

Nebulae emit a broad range of “light”, by which we mean the full range of electromagnetic radiation, although only a few wavelengths pass easily through the Earth’s atmosphere. Figure 1.2 shows the altitude above which atmospheric attenuation (“absorption”) is negligible. Visible light, and some infrared and radio radiation, can be studied from the ground, but most other wavelength regions can only be observed from high-altitude aircraft, balloons, or orbit.

The resolution that can be achieved is limited by the telescope and the detector, and also by the refractive effects of the air. Spatial resolution is measured by the apparent diameter of a point source, usually expressed in arcsec. It is influenced by the aperture of the telescope, the number of resolution elements per unit area on the detector, and “seeing”, the blurring of an image caused by refraction of light in the Earth’s atmosphere. Spectroscopic resolution specifies the smallest wavelength or energy interval that can be discerned, and is usually measured in the most convenient units for a particular form of light. The angstrom unit is commonly used for the wavelength resolution \delta\lambda in visible light. Resolution has the disadvantage that while 1 \text{ \AA} resolution is very low resolution for \lambda \approx 5 \text{ \AA} X-rays, it is very high resolution for infrared wavelengths near 10,000 \text{ \AA}. The resolving power, the ratio \lambda/\delta\lambda, is more convenient for many purposes. It is inversely proportional to the radial-velocity resolution \delta u since \lambda/\delta\lambda = c/\delta u for the Doppler effect. Resolving powers of 10^3 – 10^4 correspond to radial-velocity resolutions of 30 – 300 \text{ km s}^{-1}. Typically, a radial velocity can be measured with an accuracy that is roughly a tenth of the velocity resolution, for lines that have symmetric profiles.

Figure based on http://www.mpifr-bonn.mpg.de/div/hhertz/general_info4.html

The following paragraphs outline observational aspects of various regions of the electromagnetic spectrum.

1.3.1 Ground Based Optical

Investigations in the optical spectral region have a long and rich history. Modern telescopes must be large to be competitive, since light gathering power is proportional

to the area of the primary mirror. Consortia of universities or countries have built most of today's new research facilities. The leading examples, with 8-m or larger telescopes, include at present the Keck Observatory, the United States' National Optical Astronomy Observatories (NOAO), and the European Southern Observatory (ESO).

Spectroscopic resolving powers up to 10^5 can often be achieved on moderately bright sources, and \sim 10^3 on faint ones. Spatial resolution is limited to just under 1'' by atmospheric seeing . It can be partly overcome by adaptive optics, in which the optical surfaces of the telescope move to compensate for atmospheric distortions, but to date it has only been completely removed by observations from space.

Optical emission from nebulae include emission lines from warm ionized gas, and continua produced either by atomic processes in the nebula or by scattering of light from the photospheres of stars within it. Absorption lines are produced in the spectra of background stars or nebulae by ions or atoms of interstellar matter (ISM).

1.3.2 Ultraviolet

The vacuum ultraviolet (912\text{\AA} < \lambda < 3000\text{\AA}) can only be observed from space. Initially this spectral region was observed from high-altitude balloons and sounding rockets; today from long-lived orbital missions. The early successes of the Orbiting Astronomical Observatories (OAO) spacecraft, and the International Ultraviolet Explorer (IUE) were followed by the Far Ultraviolet Spectroscopic Explorer (FUSE) and the Hubble Space Telescope (HST). HST may be able to operate through much of the decade 2000–2010 with more modest ultraviolet missions expected to follow it. HST is also highly competitive in the optical region because of the freedom from seeing , despite its modest aperture. Spatial resolution of 0.1'' can be achieved routinely, and spectroscopic resolving powers similar to ground-based detectors (up to 10^5, but more typically 10^3 – 10^4) are possible.

Emission lines in the vacuum ultraviolet generally originate in higher-ionization species than the optical, and are produced in warm ionized gas. Interstellar matter (ISM) absorption lines from a broad range of species can also be found. Hotter stars and atomic processes in the emitting gas generally produce the continuum.

1.3.3 X-ray

X-rays have photon energies between 0.1 keV and 10 keV, corresponding to wavelengths between 100\text{\AA} and \sim 1\text{\AA}. (The region between the short-wavelength limit of the vacuum ultraviolet, 912\text{\AA}, and the long-wavelength limit of X-rays, 25\text{\AA}, is heavily absorbed by the ISM and hence is nearly unobservable.) Early observations were made from sounding rockets, but long-lived orbital missions, such as Uhuru , Einstein , ROSAT , ASCA , and at this writing Chandra and XMM , have had the greatest impact. Early missions were limited by the technology available then, and used small grazing incidence telescopes and proportional counting detectors. For instance, Einstein had two imagers, the Imaging Proportional Counter with 1' resolution and a High Resolution Imager with 3'' resolution. The spectral resolving power of the HRI was between 10–50. The current missions have, for the first time, achieved resolutions similar to optical observations.

X-ray continuum sources are either very hot material, radiating thermal spectra, or very energy-rich material, radiating by non-thermal processes such as synchrotron emission. They are often coronae of “normal” stars or accreting material very near a collapsed object in AGN, cataclysmic binaries, or pulsars. Emission lines are often produced by atomic processes involving inner shells of heavy-element ions, and can come from a wide range of ionization.

1.3.4 Infrared

The infrared spectral region covers wavelengths between 1\ \mu\text{m} and several hundred microns, although there is no agreed boundary between the far infrared and the sub-millimeter radio. Some wavelengths can be observed from the ground or from aircraft (Figure 1.2), while other wavelengths require high-altitude balloons or orbiting spacecraft. KAO, IRAS, and ISO were among the most prominent airborne and orbital missions. The lifetime of orbiting IR missions has been limited by the need to cool parts of the telescope and detectors cryogenically to very low temperatures, to minimize thermal emission. The resolution has been limited by technology to \sim 1' and a resolving power between 10 and 1000. The infrared is the spectral region in which, with SOFIA, the Spitzer Space Telescope, and the James Webb Space Telescope, the greatest technical advances will take place over the next several years.

Cool thermal emission from grains, atomic processes in nebulae, and photospheres of cooler stars are efficient sources of IR continua. Emission lines from ions and atoms with a wide range of ionization potentials are observed, along with molecular rotational and vibrational transitions. Infrared light can penetrate through dusty regions more easily than optical light, making it possible to use the IR to detect otherwise heavily obscured objects.

1.3.5 Radio

Many radio wavelengths can be studied from the ground; such investigations date back to the mid-twentieth century. Very long waves are reflected by the Earth's ionosphere and can only be detected from orbit (Figure 1.2), while other radio waves are absorbed by water vapor in the atmosphere and can only be studied from dry mountain-top sites, such as the Atacama Large Millimeter Array (ALMA). Radio telescopes are often interferometers because the long wavelengths result in very poor diffraction limits for single-dish telescopes. The spectroscopic and spatial resolutions that are possible in the radio region are the very best; spatial resolutions of 10^{-3''} and resolving powers of over 10^5 are relatively routine.

1.3.6 Returned Data

In nearly all types of observing, digital data is returned from the telescope. Most observatories have software reduction packages specifically designed to handle data from their instrument. The final product may be an image or the emission-line fluxes or measures of the continuum at selected energies. Most missions produce large data archives that form a rich data set. These archives are now so extensive that many pilot projects can be carried out solely using them. Although the biggest discoveries will

come from new observations, archival research is usually the first step in starting a large research program.

1.4 Physical Ideas

The source of energy that enables emission nebulae to radiate is, almost always, ultraviolet radiation from stars within or near the nebula. There are one or more hot stars, with effective surface temperature T_* \geq 3 \times 10^4 K, near or in almost every nebula; and the ultraviolet photons these stars emit transfer energy to the nebula by photoionization. In nebulae and in practically all astronomical objects, H is by far the most abundant element, and photoionization of H is thus the main energy-input mechanism. Photons with energy greater than 13.6 eV, the ionization potential of H, are absorbed in this process, and the excess energy of each absorbed photon over the ionization potential appears as kinetic energy of a newly liberated photoelectron. Collisions between electrons, and between electrons and ions, distribute this energy and maintain a Maxwellian velocity distribution with temperature T in the range 5,000 \text{ K} < T < 20,000 \text{ K} in typical nebulae. Collisions between thermal electrons and ions excite the low-lying energy levels of the ions. Downward radiation transitions from these excited levels have very small transition probabilities, but at the low densities (n_e \leq 10^4 \text{ cm}^{-3}) of typical nebulae, collisional deexcitation is even less probable; so almost every excitation leads to emission of a photon, and the nebula thus emits a forbidden-line spectrum that is quite difficult to excite under terrestrial laboratory conditions.

Thermal electrons are recaptured by the ions, and the degree of ionization at each point in the nebula is fixed by the equilibrium between photoionization and recapture. In nebulae in which the central star has an especially high temperature, T_*, the radiation field has a correspondingly high number of high-energy photons, and the nebular ionization is therefore high. In such nebulae collisionally excited lines up to [Ne V] and [Fe VII] may be observed, but the high ionization results from the high energy of the photons emitted by the star, and does not necessarily indicate a high nebular temperature T, defined by the kinetic energy of the free electrons.

In the recombination process, recaptures occur to excited levels, and the excited atoms thus formed then decay to lower and lower levels by radiative transitions, eventually ending in the ground level. In this process, line photons are emitted, and this is the origin of the H I Balmer- and Paschen-line spectra observed in all gaseous nebulae. Note that the recombination of \text{H}^+ gives rise to excited atoms of \text{H}^0 and thus leads to the emission of the H I spectrum. Likewise, \text{He}^+ recombines and emits the He I spectrum, and in the most highly ionized regions, \text{He}^{++} recombines and emits the He II spectrum, the strongest line in the ordinary observed region being \lambda 4686. Much weaker recombination lines of C II, C III, C IV, and so on, are also emitted; however, the main excitation process responsible for the observed strengths of such lines with the same spin or multiplicity as the ground term is resonance fluorescence by photons, which is much less effective for H and He lines because the resonance lines of these more abundant elements have greater optical depth.

In addition to the bright-line and continuous spectra emitted by atomic processes, many nebulae also have an infrared continuous spectrum emitted by dust particles heated to a temperature of order 100 K by radiation derived originally from the central star.

Gaseous nebulae may be classified into two main types, diffuse nebulae or H II regions, and planetary nebulae. Though the physical processes in both types are quite similar, the two groups differ greatly in origin, mass, evolution, and age of typical members; so for some purposes it is convenient to discuss them separately. Nova shells are rare but interesting objects: tiny, rapidly expanding, cool photoionized nebulae. Supernova remnants, an even rarer class of objects, differ greatly from both diffuse and planetary nebulae. We will briefly examine each of these types of object, and then discuss Seyfert galaxies and other active galactic nuclei, in which much the same physical processes occur, although with differences in detail because considerably higher-energy photons are involved.

1.5 Diffuse Nebulae

Diffuse nebulae or H II regions are regions of interstellar gas (Figure 1.3) in which the exciting star or stars are O- or early B-type stars. Figure 1.3 is an example. They are young stars, which use up their nuclear energy quickly. Often there are several exciting stars, a multiple star, or a galactic cluster whose hottest two or three stars are the main sources of ionizing radiation. These hot, luminous stars undoubtedly formed fairly recently from interstellar matter that would otherwise be part of the same nebula they now ionize and thus illuminate. The effective temperatures of the stars are in the range 3 \times 10^4 \text{ K} < T_* < 5 \times 10^4 \text{ K}; throughout the nebula, H is ionized, He is singly ionized, and other elements are mostly singly or doubly ionized. Typical densities in the ionized part of the nebula are of order 10 or 10^2 \text{ cm}^{-3}, ranging to as high as 10^4 \text{ cm}^{-3}, although undoubtedly small denser regions exist close to or even below the limit of resolvability. In many nebulae dense neutral condensations are scattered throughout the ionized volume. Internal motions occur in the gas with velocities of order 10 \text{ km s}^{-1}, approximately the isothermal sound speed. Bright rims, knots, condensations, and so on, are apparent to the limit of resolution. The hot, ionized gas tends to expand into the cooler surrounding neutral gas, thus decreasing the density within the nebula and increasing the ionized volume. The outer edge of the nebula is surrounded by ionization fronts running out into the neutral gas.

The spectra of these “H II regions,” as they are often called (because they contain mostly \text{H}^+), are strong in H I recombination lines and [N II] and [O II] collisionally excited lines, but the strengths of [O III] and [N II] lines may differ greatly, being stronger in the nebulae with higher central-star temperatures.

These H II regions are observed not only in our Galaxy but also in other nearby galaxies. The brightest H II regions can easily be seen on almost any large-scale images, but those taken in a narrow wavelength band in the red, including \text{H}\alpha and the [N II] lines, are especially effective in showing faint and often heavily

reddened H II regions in other galaxies. The H II regions are strongly concentrated to the spiral arms, and indeed are the best objects for tracing the structure of spiral arms in distant galaxies. Radial-velocity measurements of H II regions then give information on the kinematics of Population I (young) objects in our own and other galaxies. Typical masses of observed H II regions are of order 10^2 to 10^4 M_\odot, with the lower limit depending largely on the sensitivity of the observational method used.

1.6 Planetary Nebulae

Planetary nebulae are isolated nebulae, often (but not always) possessing a fair degree of bilateral symmetry, that are actually shells of gas that have been lost in the fairly recent past by their central stars (Figure 1.4). The name “planetary” is purely historical and refers to the fact that some of the bright planetaries appear as small, disk-like, greenish objects in small telescopes. The central stars of planetary nebulae are old stars, typically with T_\star \approx 5 \times 10^4 K, even hotter than galactic O stars, and often less luminous (M_V = -3 to +5). The stars are in fact rapidly evolving toward the white-dwarf stage, and the shells are expanding with velocities of order of several times the velocity of sound (25 \text{ km s}^{-1} is a typical expansion velocity). However, because they are decreasing in density, their emission is decreasing, and on a cosmic time scale they rapidly become unobservable, with mean lifetimes as planetary nebulae of a few times 10^4 years.

As a consequence of the higher stellar temperatures of their exciting stars, typical planetary nebulae are generally more highly ionized than H II regions, often including large amounts of \text{He}^{++}. Their spectra thus include not only the H I and He I recombination lines, but often also the He II lines; the collisionally excited lines of [O III] and [Ne III] are characteristically stronger in their spectra than those in diffuse nebulae, and [Ne V] is often strong. There is a wide range in the temperatures of planetary-nebula central stars, however, and the lower-ionization planetaries have spectra that are quite similar to those of H II regions.

The space distribution and kinematic properties of planetary nebulae indicate that, on the cosmic time scale, they are fairly old objects, usually called old Disk Population or old Population I objects. This indicates that the bulk of the planetaries we now see, though relatively young as planetary nebulae, are actually near-terminal stages in the evolution of quite old stars.

Typical densities in observed planetary nebulae range from 10^4 \text{ cm}^{-3} down to 10^2 \text{ cm}^{-3}, and typical masses are of order 0.1 M_\odot to 1.0 M_\odot. Many planetaries have been observed in other nearby galaxies, especially the Magellanic Clouds and M 31, but their luminosities are so much smaller than the luminosities of the brightest H II regions that they are more difficult to study in great detail. However, spectroscopic measurements of these planetaries give good information on velocities, abundances of the elements, and stellar evolution in these galaxies, and HST images show that they have forms quite similar to those of planetary nebulae in our Galaxy.

1.7 Nova and Supernova Remnants

Many recent novae are surrounded by small, faint shells with emission-line spectra. As we shall see, they are tiny photoionized nebulae. A few emission nebulae are known to be supernova remnants. The Crab Nebula (NGC 1952), the remnant of the supernova of A.D. 1054, is the best known example, and small bits of scattered nebulosity are the observable remnants of the much more heavily reddened objects, Tycho's supernova of 1572 and Kepler's supernova of 1604. All three of these supernova remnants have strong non-thermal radio spectra, and several other filamentary nebulae with appearances quite unlike typical diffuse or planetary

nebulae have been identified as older supernova remnants by the fact that they have similar non-thermal radio spectra. Two of the best known examples are the Cygnus Loop (NGC 6960–6992–6995) and IC 443. In the Crab Nebula, the non-thermal synchrotron spectrum observed in the radio-frequency region extends into the optical region, and extrapolation to the ultraviolet region indicates that this synchrotron radiation is probably the source of the photons that ionize the nebula. However, in the other supernova remnants no photoionization source is seen, and much of the energy is instead provided by the conversion of kinetic energy of motion into heat. In other words, the fast-moving filaments collide with ambient interstellar gas, and the energy thus released provides ionization and thermal energy, which later is partly radiated as recombination- and collisional-line radiation. Thus these supernova remnants are objects in which collisional ionization occurs, rather than photoionization. However, note that in all the nebulae, collisional excitation is caused by the thermal electrons that are energized either by photoionization or by collisional ionization.

1.8 Active Galactic Nuclei

Many galaxies, in addition to having H II regions and planetary nebulae, show characteristic nebular emission lines in the spectra of their nuclei. In most of these objects, the gas is evidently photoionized by hot stars in the nucleus, which is thus much like a giant H II region, or perhaps a cluster of many H II regions. The galactic nuclei with the strongest emission lines of this type are often called “extragalactic H II regions”, “star-forming regions”, or “starburst galaxies.” Besides these objects, however, a small fraction of spiral galaxies have ionized gas in their nuclei that emits an emission-line spectrum with a wider range of ionization than any H II region. Usually the emission-line profiles show a significantly greater range of velocities than in starburst galaxies. These galaxies, totaling a few percent of all spiral galaxies, are called Seyfert galaxies. Many of the most luminous radio galaxies, typically N, cD, D, or E galaxies in form, have nuclei with very similar emission-line spectra. Quasars (quasistellar radio sources) and QSOs (quasistellar objects) are radio-loud and radio-quiet analogues of radio galaxies and Seyfert galaxies; they have similar optical spectra and even greater optical luminosities, but are much rarer in space. All these objects together are called active galactic nuclei. Among them are the most luminous objects in the universe, quasars and QSOs with redshifts up to z \approx 6, corresponding to recession velocities of more than 0.9c.

Much if not all of the ionized gas in active galactic nuclei appears to be photoionized. However, the source of the ionizing radiation is not a hot star or stars. Instead, it is probably an extension to high energies of the blue “featureless continuum” observed in these objects in the optical spectral region. This is probably emitted by an accretion disk around a black hole, or by relativistic particles and perhaps a magnetic field associated with the immediate environs of the black hole. The spectrum of the ionizing radiation, whatever its source, certainly extends to much higher energies

than the spectra of the hot stars that ionize H II regions and planetary nebulae. Also, the particle and energy densities are much larger in some ionized regions in active galactic nuclei than in nebulae.

1.9 Star Formation in Galaxies

Newly born stars “appear” in or near the interstellar matter from which they and their neighbors formed. Generally, when stars are formed under conditions we can observe in other galaxies, they do so in large numbers and some of them are O and B stars. These hot stars immediately begin photoionizing the residual ISM around themselves, creating large emission-line diffuse nebulae or huge regions of nebulosity. Thus we observe many star-forming regions (in nearby galaxies), or star-forming galaxies (more distant objects). The galaxies with the strongest nebular emission lines (such as H \beta or H \alpha or P \alpha or Br \alpha) are called starburst galaxies, as unusually large numbers of stars are being formed within them within a short time interval.

The first survey of the infrared (5–500 \mu\text{m}) sky by the Infrared Astronomical Satellite (IRAS) discovered a class of galaxies that emit more energy in the infrared than in all the other pass bands combined. Previous optical surveys of galaxies had deduced a “luminosity function” , a description of the fraction of that population that has various luminosities. From low to high luminosities, the range went from dwarf irregular galaxies like the Magellanic Clouds, to normal spiral galaxies like the Milky Way, to Seyfert galaxies (mostly early-type or distorted spirals), giant ellipticals, and quasars, which have the greatest luminosities we know. Among the spirals are many star-forming galaxies and fewer starburst ones. All of these sources radiate most of their energy in the optical passband. The infrared luminous sources discovered by IRAS were too faint to be included in optical surveys but are among the most luminous infrared sources in the sky. When these are included, it appears that most of the very most luminous galaxies (L > 10^{11} L_{\odot}) emit the majority of their luminosity in the infrared.

These infrared luminous galaxies are among the most luminous starburst galaxies. The infrared continuum that carries most of their luminosity is emitted by interstellar dust heated to 30–60 K by hot stars and by the emission lines they produce in the gas, creating a peak in the energy distribution near 60 \mu\text{m}. The underlying source of the great luminosity is a “super starburst”, in which a large fraction of a galaxy’s mass is involved in vigorous star formation. Interstellar dust then absorbs the energy emitted by the hot stars, which is eventually reradiated in the infrared. The observations can be understood if a very large fraction of a galaxy’s mass, \sim 10^{10} M_{\odot}, is quickly converted into stars in a dusty environment.

The starburst phenomenon is thought to be the result of interactions and mergers of gas-rich spirals. The host galaxies tend to show signs of interacting with other galaxies or of being an otherwise disturbed system. During such galactic collisions some of the gas in the ISM loses its angular momentum and quickly falls towards the merger nuclei. Models suggest that as much as 10^{10} M_{\odot} of gas and dust can build up

within a few hundred parsecs of the center, resulting in vigorous star formation. Most of this star formation occurs in regions that are heavily obscured by dust, so many parts of the system can only be observed directly in the infrared. For this reason it is necessary to develop emission line diagnostics that use only infrared lines, the subject of a later chapter.

The basic physical principles that govern the structures and emitted spectra of active galactic nuclei are largely the same as those that apply in H II regions and planetary nebulae. However, because of the large proportion of high-energy photons in the ionizing flux, some new physical processes become important in active galactic nuclei, and these cause their structures to differ in important details from those of classical nebulae. These differences can best be analyzed after H II regions and planetary nebulae are well understood. Hence we treat nova shells, supernova remnants, and active galactic nuclei in the final chapters of this book.

References

General reviews on astronomical research are found in a variety of places. Three of the best series are

Annual Reviews of Astronomy and Astrophysics, by Annual Reviews Inc. These come out once per year and feature articles on a variety of research topics.

Astronomical Society of the Pacific Conference Series, by the Astronomical Society of the Pacific. Each book of this series summarizes a specialist conference on a chosen topic, many of them organized by the International Astronomical Union. The articles are a mix of longer reviews and shorter research summaries.

The Saas Fe Conference Series. These are the proceedings from summer schools held each year. Each book is on a chosen topic and features a few longer articles.

A brief historical sketch of the development of nebular astrophysics is contained in an article on “pioneer nebular theorists” (which also includes some of the main observational results on which they built):

Osterbrock, D. E. 2001, Revista Mexicana de Astronomia, Serie d. Conferencias , 12, 1 (in English).

The following are on-line data bases of astronomical sources. At this time most archives are limited to space-based data, but archives of ground-based are being developed. The main archives at the time of this writing are

ADS, the NASA Astrophysics Data System, has links to much of the published astronomical literature. http://adsabs.harvard.edu/

HEASARC, the High Energy Astrophysics Science Archive Research Center, is “a source of gamma-ray, X-ray, and extreme ultraviolet observations of cosmic (non-solar) sources”. http://heasarc.gsfc.nasa.gov/

SIMBAD, a database operated at CDS, France. This “brings together basic data, cross-identifications, observational measurements, and bibliography, for celestial objects outside the solar system: stars, galaxies, and non-stellar objects within our galaxy, or in external galaxies”. http://cdsweb.u-strasbg.fr/Simbad.html

The Hubble Data Archive provides access to data obtained with the Hubble Space Telescope. http://archive.stsci.edu/

MAST, the Multi-mission Archive at STScI, provides access to data from a variety of missions ranging from the extreme UV through near IR. http://archive.stsci.edu/mast.html

IPAC, the Infrared Processing and Analysis Center, has access to NASA's infrared program. http://www.ipac.caltech.edu/

NED, the NASA/IPAC Extragalactic Database at JPL, can be searched by object name, type of data, literature, or tools. http://nedwww.ipac.caltech.edu/

Additionally, meta-archives, collections of links to individual archives, are being developed. A good one is

Canadian Astronomy Data Centre, at http://cadwww.hia.nrc.ca/

The following gives an overview of various ways to find information in these databases:

Skiff, Brian A. 2002, Sky & Telescope , 103, 50.

Chapter 2

Photoionization Equilibrium

2.1 Introduction

Emission nebulae result from the photoionization of a diffuse gas cloud by ultraviolet photons from a hot “exciting” star or from a cluster of exciting stars. The ionization equilibrium at each point in the nebula is fixed by the balance between photoionizations and recombinations of electrons with the ions. Since hydrogen is the most abundant element, we can get a first idealized approximation to the structure of a nebula by considering a pure H cloud surrounding a single hot star. The ionization equilibrium equation is:

\begin{aligned} n(\text{H}^0) \int_{\nu_0}^{\infty} \frac{4\pi J_{\nu}}{h\nu} a_{\nu}(\text{H}^0) d\nu &= n(\text{H}^0) \int_{\nu_0}^{\infty} \phi_{\nu} a_{\nu}(\text{H}^0) d\nu = n(\text{H}^0) \Gamma(\text{H}^0) \\ &= n_e n_p \alpha(\text{H}^0, T) [\text{cm}^{-3} \text{ s}^{-1}] \end{aligned} \quad (2.1)

where J_{\nu} is the mean intensity of radiation (in energy units per unit area, per unit time, per unit solid angle, per unit frequency interval) at the point. Thus \phi_{\nu} = 4\pi J_{\nu}/h\nu is the number of incident photons per unit area, per unit time, per unit frequency interval, and a_{\nu}(\text{H}^0) is the ionization cross section for H by photons with energy h\nu (above the threshold h\nu_0); the integral [denoted by \Gamma(\text{H}^0)] therefore represents the number of photoionizations per H atom per unit time. The neutral atom, electron, and proton densities per unit volume are n(\text{H}^0), n_e, and n_p, and \alpha(\text{H}^0, T) is the recombination coefficient; so the right-hand side of the equation gives the number of recombinations per unit volume per unit time.

To a first approximation, the mean intensity J_{\nu} [see Appendix 1 for definitions of it and other observed (or measured) quantities connected with radiation] is simply the radiation emitted by the star reduced by the inverse-square effect of geometrical dilution. Thus

4\pi J_{\nu} = \frac{R^2}{r^2} \pi F_{\nu}(0) = \frac{L_{\nu}}{4\pi r^2} [\text{erg cm}^{-2} \text{ s}^{-1} \text{ Hz}^{-1}], \quad (2.2)

where R is the radius of the star, \pi F_\nu(0) is the flux at the surface of the star, r is the distance from the star to the point in question, and L_\nu is the luminosity of the star per unit frequency interval.

At a typical point in a nebula, the ultraviolet radiation field is so intense that the H is almost completely ionized. Consider, for example, a point in an H II region, with density 10 H atoms and ions per \text{cm}^3, 5 pc from a central O7.5 star with T_* = 39,700 K. We will examine the numerical values of all the other variables later, but for the moment, we can adopt the following very rough values:

Q(\text{H}^0) = \int_{\nu_0}^{\infty} \frac{L_\nu}{h\nu} d\nu \approx 1 \times 10^{49} \text{ [photons s}^{-1}\text{]};

a_\nu(\text{H}^0) \approx 6 \times 10^{-18} \text{ [cm}^2\text{]};

\int_{\nu_0}^{\infty} \frac{4\pi J_\nu}{h\nu} a_\nu(\text{H}^0) d\nu \approx 1 \times 10^{-8} = \tau_{ph}^{-1} \text{ [s}^{-1}\text{]};

\alpha(\text{H}^0, T) \approx 4 \times 10^{-13} \text{ [cm}^3 \text{ s}^{-1}\text{]}

where \tau_{ph} is the lifetime of the atom before photoionization. Substituting these values and taking \xi as the fraction of neutral H, that is, n_e = n_p = (1 - \xi)n(\text{H}) and n(\text{H}^0) = \xi n(\text{H}), where n(\text{H}) = 10 \text{ cm}^{-3} is the density of H, we find \xi \approx 4 \times 10^{-4}; that is, H is very nearly completely ionized.

On the other hand, a finite source of ultraviolet photons cannot ionize an infinite volume, and therefore, if the star is in a sufficiently large gas cloud, there must be an outer edge to the ionized material. The thickness of this transition zone between ionized and neutral gas, since it is due to absorption, is approximately one mean free path of an ionizing photon \{l \approx [n(\text{H}^0) \alpha_\nu]^{-1} \text{ cm}\}. Using the same parameters as before, and taking \xi = 0.5, we find the thickness

d \approx \frac{1}{n(\text{H}^0) \alpha_\nu} \approx 0.1 \text{ pc},

or much smaller than the radius of the ionized nebula. Thus we have the picture of a nearly completely ionized “Strömgren sphere” or H II region, separated by a thin transition region from an outer neutral gas cloud or H I region. In the rest of this chapter we will explore this ionization structure in detail.

First we will examine the photoionization cross section and the recombination coefficients for H, and then use this information to calculate the structure of hypothetical pure H regions. Next we will consider the photoionization cross section and recombination coefficients for He, the second most abundant element, and then calculate more realistic models of H II regions, that take both H and He into account. Finally, we will extend our analysis to other, less abundant heavy elements; these often do not strongly affect the ionization structure of the nebula, but are always quite important in the thermal balance to be discussed in the next chapter.

2.2 Photoionization and Recombination of Hydrogen

Figure 2.1 is an energy-level diagram of H; the levels are marked with their quantum numbers n (principal quantum number) and L (angular momentum quantum number), and with S, P, D, F, \dots standing for L = 0, 1, 2, 3, \dots in the conventional notation. Permitted transitions (which, for one-electron systems, must satisfy the selection rule \Delta L = \pm 1) are marked by solid lines in the figure. The transition probabilities A(nL, n'L') of these lines are of order 10^4 to 10^8 \text{ s}^{-1}, and the corresponding mean lifetimes of the excited levels,

\tau_{nL} = \frac{1}{\sum_{n' < n} \sum_{L' = L \pm 1} A_{nL, n'L'}}, \quad (2.3)

are therefore of order 10^{-4} to 10^{-8} s. The only exception is the 2^2S level, from which there are no allowed one-photon downward transitions. However, the transition 2^2S \rightarrow 1^2S does occur with the emission of two photons, and the probability of this process is A(2^2S, 1^2S) = 8.23 \text{ s}^{-1}, corresponding to a mean lifetime for the 2^2S level of 0.12 s. Even this lifetime is quite short compared with the mean lifetime of an H atom against photoionization, which has been estimated previously as \tau_{ph} \approx 10^8 s for the 1^2S level, and is of the same order of magnitude for the excited levels. Thus, to a very good approximation, we may consider that very nearly all the H^0 is in the 1^2S level, that photoionization from this level is balanced by recombination to all levels, and that recombination to any excited level is followed very quickly by radiative transitions downward, leading ultimately to the ground level. This basic approximation greatly simplifies calculations of physical conditions in gaseous nebulae.

The photoionization cross section for the 1^2S level of H^0, or, in general, of a hydrogenic ion with nuclear charge Z, may be written in the form

a_\nu(Z) = \frac{A_0}{Z^2} \left( \frac{\nu_1}{\nu} \right)^4 \frac{\exp \{ 4 - [(4 \tan^{-1} \varepsilon) / \varepsilon] \}}{1 - \exp(-2\pi/\varepsilon)} [\text{cm}^2] \text{ for } \nu \geq \nu_1 \quad (2.4)

where

A_0 = \frac{2^9 \pi}{3e^4} \left( \frac{1}{137.0} \right) \pi a_0^2 = 6.30 \times 10^{-18} [\text{cm}^2], \varepsilon = \sqrt{\frac{\nu}{\nu_1} - 1},

and

h\nu_1 = Z^2 h\nu_0 = 13.6Z^2 \text{ eV}

is the threshold energy. This cross section is plotted in Figure 2.2, which shows that it drops off rapidly with energy, approximately as \nu^{-3} not too far above the threshold, which, for H, is at \nu_0 = 3.29 \times 10^{15} \text{ s}^{-1} or \lambda_0 = 912 \text{ Å}, so that the higher-energy photons, on the average, penetrate further into neutral gas before they are absorbed.

The electrons produced by photoionization have an initial distribution of energies that depends on J_\nu a_\nu / h\nu. However, the cross section for elastic scattering collisions between electrons is quite large, of order 4\pi(e^2/mu^2)^2 \approx 10^{-13} \text{ cm}^2, and these collisions tend to set up a Maxwell–Boltzmann energy distribution. The recombination cross section, and all the other cross sections involved in the nebulae, are so much smaller that, to a very good approximation, the electron-distribution function is Maxwellian, and therefore all collisional processes occur at rates fixed by the local

temperature defined by this Maxwellian. Therefore, the recombination coefficient to a specified level n^2L may be written

\alpha_{n^2L}(H^0, T) = \int_0^\infty u \sigma_{n^2L}(H^0, u) f(u) du \text{ [cm}^3 \text{ s}^{-1}] \quad (2.5)

where

f(u) = \frac{4}{\sqrt{\pi}} \left( \frac{m}{2kT} \right)^{3/2} u^2 \exp(-mu^2/kT) \quad (2.6)

is the Maxwell–Boltzmann distribution function for the electrons, and \sigma_{n^2L}(H^0, u) is the recombination cross section to the term n^2L in H^0 for electrons with velocity u. These cross sections vary approximately as u^{-2}, and the recombination coefficients, which are proportional to u\sigma, therefore vary approximately as T^{-1/2}. A selection of numerical values of \alpha_{n^2L} is given in Table 2.1. Since the mean electron velocities at

the temperatures listed are of order 5 \times 10^7 \text{ cm s}^{-1}, it can be seen that the recombination cross sections are of order 10^{-20} \text{ cm}^2 or 10^{-21} \text{ cm}^2, much smaller than the geometrical cross section of an H atom.

In the nebular approximation discussed previously, recombination to any level n \, ^2L quickly leads through downward radiative transitions to 1 \, ^2S, and the total recombination coefficient is the sum over captures to all levels, ordinarily written

\begin{aligned} \alpha_A &= \sum_{n,L} \alpha_{n \, ^2L}(\text{H}^0, T) \text{ [cm}^3 \text{ s}^{-1}] \\ &= \sum_n \sum_{L=0}^{n-1} \alpha_{nL}(\text{H}^0, T) \\ &= \sum_n \alpha_n(\text{H}^0, T) \end{aligned} \tag{2.7}

where \alpha_n is thus the recombination coefficient to all the levels with principal quantum number n. Numerical values of \alpha_A are also listed in Table 2.1. A typical recombination time is \tau_r = 1/n_e \alpha_A \approx 3 \times 10^{12}/n_e \text{ s} \approx 10^5/n_e \text{ yr}, and deviations from ionization equilibrium are ordinarily damped out in times of this order of magnitude.

2.3 Photoionization of a Pure Hydrogen Nebula

Consider the simple idealized problem of a single star that is a source of ionizing photons in a homogeneous static cloud of H. Only radiation with frequency \nu \geq \nu_0 is effective in the photoionization of H from the ground level, and the ionization equilibrium equation at each point can be written

n(\text{H}^0) \int_{\nu_0}^{\infty} \frac{4\pi J_{\nu}}{h\nu} a_{\nu} d\nu = n_p n_e \alpha_A(\text{H}^0, T) [\text{cm}^{-3} \text{ s}^{-1}]. \quad (2.8)

The equation of transfer for radiation with \nu \geq \nu_0 can be written in the form

\frac{dI_{\nu}}{ds} = -n(\text{H}^0) a_{\nu} I_{\nu} + j_{\nu} \quad (2.9)

where I_{\nu} is the specific intensity of radiation and j_{\nu} is the local emission coefficient (in energy units per unit volume, per unit time, per unit solid angle, per unit frequency) for ionizing radiation.

It is convenient to divide the radiation field into two parts, a "stellar" part, resulting directly from the input radiation from the star, and a "diffuse" part, resulting from the emission of the ionized gas:

I_{\nu} = I_{\nu s} + I_{\nu d}. \quad (2.10)

The stellar radiation decreases outward because of geometrical dilution and absorption, and since its only source is the star, it can be written

4\pi J_{\nu s} = \pi F_{\nu s}(r) = \pi F_{\nu s}(R) \frac{R^2 \exp(-\tau_{\nu})}{r^2} [\text{erg cm}^{-2} \text{ s}^{-1} \text{ Hz}^{-1}], \quad (2.11)

where \pi F_{\nu s}(r) is the standard astronomical notation for the flux of stellar radiation (per unit area, per unit time, per unit frequency interval) at r, \pi F_{\nu s}(R) is the flux at the radius of the star R, and \tau_{\nu} is the radial optical depth at r,

\tau_{\nu}(r) = \int_0^r n(\text{H}^0, r') a_{\nu} dr', \quad (2.12)

which can also be written in terms of \tau_0, the optical depth at the threshold:

\tau_{\nu}(r) = \frac{a_{\nu}}{a_{\nu_0}} \tau_0(r)

The equation of transfer for the diffuse radiation I_{\nu d} is

\frac{dI_{\nu d}}{ds} = -n(\text{H}^0) a_{\nu} I_{\nu d} + j_{\nu} \quad (2.13)

and for kT \ll h\nu_0 the only source of ionizing radiation is recaptures of electrons from the continuum to the ground 1^2S level. The emission coefficient for this radiation is (erg cm ^-2 s ^-1 Hz ^-1 sr ^-1 )

j_\nu(T) = \frac{2h\nu^3}{c^2} \left( \frac{h^2}{2\pi mkT} \right)^{3/2} a_\nu \exp[-h(\nu - \nu_0)/kT] n_p n_e \quad (\nu > \nu_0) \quad (2.14)

which is strongly peaked to \nu = \nu_0, the threshold. The total number of photons generated by recombinations to the ground level is given by the recombination coefficient

4\pi \int_{\nu_0}^{\infty} \frac{j_\nu}{h\nu} d\nu = n_p n_e \alpha_1(H^0, T) [\text{cm}^{-3} \text{s}^{-1}], \quad (2.15)

and since \alpha_1 = \alpha_{1s} < \alpha_A, the diffuse field J_{\nu d} is smaller than J_{\nu s} on the average, and may be calculated by an iterative procedure. For an optically thin nebula, a good first approximation is to take J_{\nu d} \approx 0.

On the other hand, for an optically thick nebula, a good first approximation is based on the fact that no ionizing photons can escape, so that every diffuse radiation-field photon generated in such a nebula is absorbed elsewhere in the nebula:

4\pi \int \frac{j_\nu}{h\nu} dV = 4\pi \int n(H^0) \frac{a_\nu J_{\nu d}}{h\nu} dV, \quad (2.16)

where the integration is over the entire volume of the nebula. The so-called —on-the-spot— approximation amounts to assuming that a similar relation holds locally:

J_{\nu d} = \frac{j_\nu}{n(H^0) a_\nu} \quad (2.17)

This, of course, automatically satisfies (2.16), and would be exact if all photons were absorbed very close to the point at which they are generated (—on the spot—). This is not a bad approximation because the diffuse radiation-field photons have \nu \approx \nu_0, and therefore have large a_\nu, and correspondingly small mean free paths before absorption.

Making this on-the-spot approximation and using (2.11) and (2.15), we find that the ionization Equation (2.8) becomes

\frac{n(H^0) R^2}{r^2} \int_{\nu_0}^{\infty} \frac{\pi F_\nu(R)}{h\nu} a_\nu \exp(-\tau_\nu) d\nu = n_p n_e \alpha_B(H^0, T) \quad (2.18)

where

\begin{aligned} \alpha_B(H^0, T) &= \alpha_A(H^0, T) - \alpha_1(H^0, T) \\ &= \sum_2^{\infty} \alpha_n(H^0, T) \end{aligned}

The physical meaning is that in optically thick nebulae, the ionizations caused by stellar radiation-field photons are balanced by recombinations to excited levels of

H, while recombinations to the ground level generate ionizing photons that are absorbed elsewhere in the nebula but have no net effect on the overall ionization balance.

For any stellar input spectrum \pi F_\nu(R), the integral on the left-hand side of (2.18) can be tabulated as a known function of \tau_0, since a_\nu and \tau_\nu are known functions of \nu. Thus, for any assumed density distribution,

n_{\text{H}}(r) = n(\text{H}^0, r) + n_p(r)

and temperature distribution T(r), equations (2.18) and (2.12) can be integrated outward to find n(\text{H}^0, r) and n_p(r) = n_e(r). Two calculated models for homogeneous nebulae with constant density n(\text{H}) = 10 H atoms plus ions cm ^-3 and constant temperature T = 7,500 K are listed in Table 2.2 and graphed in Figure 2.3. For one of these ionization models, the assumed \pi F_\nu(R) is a blackbody spectrum with T_* = 40,000 K, chosen to represent approximately an O7.5 main-sequence star, while for the other, the \pi F_\nu(R) is a computed model stellar atmosphere with T_* = 37,450 K. The table and graph clearly show the expected nearly complete ionization out to a critical radius r_1, at which the ionization drops off abruptly to nearly zero. The central ionized zone is often referred to as an "H II region" ("H ⁺ region" would be a better name), and it is surrounded by an outer neutral H ⁰ region, often referred to as an "H I region".

The radius r_1 can be found from (2.18), substituting from (2.12),

\frac{d\tau_\nu}{dr} = n(\text{H}^0) a_\nu

and integrating over r:

\begin{aligned} R^2 \int_{\nu_0}^{\infty} \frac{\pi F_\nu(R)}{h\nu} d\nu \int_0^{\infty} d[-\exp(-\tau_\nu)] &= \int_0^{\infty} n_p n_e \alpha_B r^2 dr \\ &= R^2 \int_{\nu_0}^{\infty} \frac{\pi F_\nu(R)}{h\nu} d\nu \end{aligned}

Using the result that the ionization is nearly complete [n_p = n_e \approx n(\text{H})] within r_1, and nearly zero (n_p = n_e \approx 0) outside r_1, this becomes

\begin{aligned} 4\pi R^2 \int_{\nu_0}^{\infty} \frac{\pi F_\nu}{h\nu} d\nu &= \int_{\nu_0}^{\infty} \frac{L_\nu}{h\nu} d\nu \\ &= Q(\text{H}^0) = \frac{4\pi}{3} r_1^3 n_{\text{H}}^2 \alpha_B. \end{aligned} \tag{2.19}

Here 4\pi R^2 \pi F_\nu(R) = L_\nu is the luminosity of the star at frequency \nu (in energy units per time per unit frequency interval), and the physical meaning of (2.19) is that the total number of ionizing photons emitted by the star just balances the total number of recombinations to excited levels within the ionized volume \frac{4\pi}{3} r_1^3, often called the

Strömgren sphere. Numerical values of radii calculated by using the model stellar atmospheres discussed in Chapter 5 are given in Table 2.3.

2.4 Photoionization of a Nebula Containing Hydrogen and Helium

The next most abundant element after H is He, whose relative abundance (by number) is of order 10 percent, and a much better approximation to the ionization structure of an actual nebula is provided by taking both these elements into account. The ionization potential of He is h\nu_2 = 24.6 eV, somewhat higher than H; the ionization potential of \text{He}^+ is 54.4 eV, but since even the hottest O stars emit practically no photons with h\nu > 54.4 eV, second ionization of He does not occur in ordinary H II regions. (The situation is quite different in planetary nebulae, as we shall see later in this chapter.) Thus photons with energy 13.6 \text{ eV} < h\nu < 24.6 \text{ eV} can ionize H only, but photons with energy h\nu > 24.6 \text{ eV} can ionize both H and He. As a result, two different types of ionization structure are possible, depending on the spectrum of ionizing radiation and the abundance of He. At one extreme, if the spectrum is

concentrated to frequencies just above 13.6 eV and contains only a few photons with h\nu > 24.6 eV, then the photons with energy 13.6 \text{ eV} < h\nu < 24.6 \text{ eV} keep the H ionized, and the photons with h\nu > 24.6 \text{ eV} are all absorbed by He. The ionization structure thus consists of a small central \text{H}^+, \text{He}^+ zone surrounded by a larger \text{H}^+, \text{He}^0 region. At the other extreme, if the input spectrum contains a large fraction of photons with h\nu > 24.6 \text{ eV}, then these photons dominate the ionization of both H and He, the outer boundaries of both ionized zones coincide, and there is a single \text{H}^+, \text{He}^+ region.

The \text{He}^0 photoionization cross section a_{\nu}(\text{He}^0) is plotted in Figure 2.2, along with a_{\nu}(\text{H}^0) and a_{\nu}(\text{He}^+) calculated from Equation (2.4). The total recombination coefficients for He to configurations L \geq 2 are, to a good approximation, the same as for H, since these levels are hydrogen-like, but because He is a two-electron system, it has separate singlet and triplet levels and

\left. \begin{aligned} \alpha_{n \, ^1L}(\text{He}^0, T) &\approx \frac{1}{4} \alpha_{n \, ^2L}(\text{H}^0, T) \\ \alpha_{n \, ^3L}(\text{He}^0, T) &\approx \frac{3}{4} \alpha_{n \, ^2L}(\text{H}^0, T) \end{aligned} \right\} L \geq 2 \quad (2.20)

For the P and particularly the S terms there are sizeable differences between the He and H recombination coefficients. Representative numerical values of the recombination coefficients are included in Table 2.4.

The ionization equations for H and He are coupled by the radiation field with h\nu > 24.6 \text{ eV}, and are straightforward to write down in the on-the-spot approximation,

though complicated in detail. First, the photons emitted in recombinations to the ground level of He can ionize either H or He, since these photons are emitted with energies just above h\nu_2 = 24.6 eV. The fraction absorbed by H is

y = \frac{n(\text{H}^0) a_{\nu_2}(\text{H}^0)}{n(\text{H}^0) a_{\nu_2}(\text{H}^0) + n(\text{He}^0) a_{\nu_2}(\text{He}^0)}, \tag{2.21}

and the remaining fraction, 1 - y, is absorbed by He. Second, following recombination to excited levels of He, various photons are emitted that ionize H. Of the recombinations to excited levels of He, approximately three-fourth are to the triplet levels and approximately one-fourth are to the singlet levels. All the captures to triplets lead ultimately through downward radiative transitions to 2^3S, which is highly metastable, but which can decay by a one photon forbidden line at 19.8 eV to 1^1S, with transition probability A(2^3S, 1^1S) = 1.26 \times 10^{-4} \text{ s}^{-1}. Competing with this mode of depopulation of 2^3S, collisional excitation to the singlet levels 2^1S, and 2^1P^o can also occur with fairly high probability, while collisional transitions to 1^1S or to the continuum are less probable. Since the collisions leading to the singlet levels involve a spin change, only electrons are effective in causing these excitations, and the transition rate per atom in the 2^3S level is

n_e q(2^3S, 2^1L) = n_e \int_{\frac{1}{2}mu^2=\chi}^{\infty} u \sigma(2^3S, 2^1L, u) f(u) du \tag{2.22}

where the \sigma(2^3S, 2^1L) are the electron collision cross sections for these excitation processes, and the \chi are their energy thresholds. These rate coefficients are listed in Table 2.5, along with the critical electron density n_c(2^3S), defined by

n_c(2^3S) = \frac{A(2^3S, 1^1S)}{q(2^3S, 2^1S) + q(2^3S, 2^1P^o)} \tag{2.23}

a. Critical density in \text{cm}^{-3}.

at which collisional transitions are equally probable with radiative transitions. In typical H II regions, the electron density n_e \leq 10^2 \text{ cm}^{-3}, is considerably smaller than n_c, so practically all the atoms leave 2^3S by emission of a 19.8 eV-line photon. In contrast, in typical bright planetary nebulae, n_e \approx 10^4 \text{ cm}^{-3}, somewhat larger than n_c, and therefore many of the atoms are transferred to 2^1S or 2^1P^o before emitting a line photon. From the ratio of excitation rates, it can be seen that, for instance, at T = 10^4 \text{ K}, a fraction 0.83 of the transitions lead to 2^1S, and 0.17 to 2^1P^o. If the less probable collisional deexciting collisions to 1^1S are also included, these fractions become 0.78 and 0.16, respectively.

Of the captures to the singlet-excited levels in He, approximately two-thirds lead ultimately to population of 2^1P^o, while approximately one-third lead to population of 2^1S. Atoms in 2^1P^o decay mostly to 1^1S with emission of a resonance-line photon at 21.2 eV, but some also decay to 2^1S (with emission of 2^1S - 2^1P^o at 2.06 \mu\text{m}) with a relative probability of approximately 10^{-3}. The resonance-line photons are scattered by \text{He}^0, and therefore, after-approximately 10^3 scatterings, a typical photon would, on the average, be converted to a 2.06\text{-}\mu\text{m} line photon and thus populate 2^1S. However, it is more likely that before a resonance-line photon is scattered this many times, it will photoionize an H atom and be absorbed. He atoms in 2^1S decay by two-photon emission (with the sum of the energies 20.6 eV and transition probability 51.3 \text{ s}^{-1}) to 1^1S. From the distribution of photons in this continuous spectrum, the probability that a photon is produced that can ionize H is 0.56 per radiative decay from \text{He}^0 2^1S.

All these He bound-bound transitions produce photons that ionize H but not He, and they can easily be included in the H ionization equation in the on-the-spot approximation. The total number of recombinations to excited levels of He per unit volume per unit time is n(\text{He}^+)n_e\alpha_B(\text{He}^0, T), and of these a fraction p generate ionizing photons that are absorbed on the spot. As shown by the preceding discussion, in the low-density limit n_e \ll n_c

p \approx \frac{3}{4} + \frac{1}{4} \left[ \frac{2}{3} + \frac{1}{3} (0.56) \right] = 0.96,

but in the high-density limit n_e \gg n_c,

p \approx \left[ \frac{3}{4} (0.78) + \frac{1}{4} \cdot \frac{1}{3} \right] (0.56) + \left[ \frac{3}{4} (0.16) + \frac{1}{4} \cdot \frac{2}{3} \right] = 0.66.

Thus, in the on-the-spot approximation, the ionization equations become

\begin{aligned} \frac{n(\text{H}^0)}{r^2} \int_{\nu_0}^{\infty} \frac{\pi F_{\nu}(R)}{h\nu} a_{\nu}(\text{H}^0) \exp(-\tau_{\nu}) d\nu + \gamma n(\text{He}^+) n_e \alpha_1(\text{He}^0, T) \\ + p n(\text{He}^+) n_e \alpha_B(\text{He}^0, T) = n_p n_e \alpha_B(\text{H}^0, T); \end{aligned} \quad (2.24)

\frac{n(\text{He}^0) R^2}{r^2} \int_{\nu_2}^{\infty} \frac{\pi F_{\nu}(R)}{h\nu} a_{\nu}(\text{He}^0) \exp(-\tau_{\nu}) d\nu + (1 - y) n(\text{He}^+) n_e \alpha_1(\text{He}^0, T') \tag{2.25} = n(\text{He}^+) n_e \alpha_A(\text{He}^0, T'),

with

\frac{d\tau_{\nu}}{dr} = n(\text{H}^0) a_{\nu}(\text{H}^0) \quad \text{for} \quad \nu_1 < \nu < \nu_2

and

\frac{d\tau_{\nu}}{dr} = n(\text{H}^0) a_{\nu}(\text{H}^0) + n(\text{He}^0) a_{\nu}(\text{He}^0) \quad \text{for} \quad \nu_2 < \nu, \tag{2.26}

and

n_e = n_p + n(\text{He}^+)

These equations again can be integrated outward step-by-step, and sample models for a diffuse nebula [with n(\text{H}) = 10 \text{ cm}^{-3}, n(\text{He})/n(\text{H}) = 0.15] excited by stars with T_* = 40,000 \text{ K} and 30,000 \text{ K}, respectively, are plotted in Figure 2.4. It can be seen that the hotter star excites a coincident \text{H}^+, \text{He}^+ zone, while the cooler star has an inner \text{H}^+, \text{He}^+ zone and an outer \text{H}^+, \text{He}^0 zone.

Although the exact size of the \text{He}^+ zone can be found only from the integration because of the coupling between the H and He ionization by the radiation with \nu > \nu_2, the approximate size can easily be found by ignoring the absorption by H in the \text{He}^+ zone. This corresponds to setting y = 0 in Equation (2.25) and n(\text{H}^0) = 0 in the second of Equations (2.26), and we then immediately find, in analogy to Equation (2.19), that

\int_{\nu_2}^{\infty} \frac{L_{\nu}}{h\nu} d\nu = Q(\text{He}^0) = \frac{4\pi}{3} r_2^3 n(\text{He}^+) n_e \alpha_B(\text{He}^0), \tag{2.27}

where r_2 is the radius of the \text{He}^+ zone. Furthermore, since according to the preceding discussions, p \approx 1, the absorptions by He do not greatly reduce the number of photons available for ionizing H; therefore, to a fair approximation,

\int_{\nu_0}^{\infty} \frac{L_{\nu}}{h\nu} d\nu = Q(\text{H}^0) = \frac{4\pi}{3} r_1^3 n(\text{H}^+) n_e \alpha_B(\text{H}^0). \tag{2.19}

If we suppose that the \text{He}^+ zone is much smaller than the \text{H}^+ zone, then throughout most of the \text{H}^+ zone the electrons come only from ionization of H, but in the \text{He}^+ zone, the electrons come from ionization of both H and He. With this simplification,

\left(\frac{r_1}{r_2}\right)^3 = \frac{Q(\text{H}^0)}{Q(\text{He}^0)} \frac{n_{\text{He}}}{n_{\text{H}}} \left(1 + \frac{n_{\text{He}}}{n_{\text{H}}}\right) \frac{\alpha_B(\text{He}^0)}{\alpha_B(\text{H}^0)} \text{ if } r_2 < r_1. \quad (2.28)

A plot of r_2/r_1, calculated according to this equation for n(\text{He})/n(\text{H}) = 0.15 and T = 7,500 \text{ K}, is shown in Figure 2.5, and it can be seen that, for T_* \geq 40,000 \text{ K}, the \text{He}^+ and \text{H}^+ zones are coincident, while at significantly lower temperatures, the \text{He}^+ zone is much smaller. The details of the curve, including the precise effective temperature at which r_2/r_1 = 1, are not significant, because of the simplifications made, but the general trends it indicates are correct. For a smaller relative helium abundance, for instance n(\text{He})/n(\text{H}) = 0.10 instead of 0.15, r_2/r_1 is larger by approximately 16% at corresponding values of T_*, up to r_2/r_1 = 1.

2.5 Photoionization of \text{He}^+ to \text{He}^{++}

Although ordinary O stars of Population I do not radiate any appreciable number of photons with h\nu > 54.4 \text{ eV} (hence galactic H II regions do not have a \text{He}^{++} zone), the situation is quite different for the central stars of planetary nebulae. Many of these stars are much hotter than even the hottest O3 stars, and do radiate high-energy photons

that produce central \text{He}^{++} zones, which are observed by the He II recombination spectra they emit.

The structure of these central \text{He}^{++} zones is governed by equations that are very similar to those for pure \text{H}^+ zones discussed previously, with the threshold, absorption cross section, and recombination coefficient changed from \text{H}^0 (Z = 1) to \text{He}^+ (Z = 2). This \text{He}^{++} zone is, of course, also an \text{H}^+ zone, and the ionization equations of \text{H}^0 and \text{He}^+ are therefore, in principle, coupled—but in practice they can be fairly well separated. The coupling results from the fact that, in the recombination of \text{He}^{++} to form \text{He}^+, photons are emitted that ionize \text{H}^0. Three different mechanisms are involved, namely: recombinations that populate 2^2P^o, resulting in He II \text{L}\alpha emission with h\nu = 40.8 eV; recombinations that populate 1^2S, resulting in He II 2^2S \rightarrow 1^2S two-photon emission for which h\nu' + h\nu'' = 40.8 eV (the spectrum peaks at 20.4 eV, and, on the average, 1.42 ionizing photons are emitted per decay); and recombinations directly to 2^2S and to 2^2P^o, resulting in He II Balmer-continuum emission, which has the same threshold as the Lyman limit of H and therefore emits a continuous spectrum concentrated just above h\nu_0. The He II \text{L}\alpha photons are scattered by resonance scattering, and therefore diffuse only slowly away from their point of origin before they are absorbed, while the He II Balmer-continuum photons are concentrated close to the \text{H}^0 ionization threshold and therefore have a short mean free path. Both these sources tend to ionize \text{H}^0 in the \text{He}^{++} zone, and at a “normal” abundance of He, the number of ionizing photons generated in the \text{He}^{++} zone by these two processes is nearly sufficient to balance the recombinations of \text{H}^+ in this zone and thus to maintain the ionization of \text{H}^0. This is shown in Table 2.6, which lists the ionizing photon generation rates relative to the recombination rates for two temperatures. Thus, to a good approximation, the He II \text{L}\alpha and Balmer-continuum photons are absorbed by and maintain the ionization of \text{H}^0 in the \text{He}^{++} zone, but the stellar radiation with 13.6 \text{ eV} < h\nu < 54.4 \text{ eV} is not significantly absorbed by the \text{H}^0 in the \text{He}^{++}, \text{H}^+ zone, and that with h\nu > 54.4 \text{ eV} is absorbed only by the \text{He}^+. The He II two-photon continuum is an additional source of ionizing photons for H; most of these photons escape from the \text{He}^{++} zone and therefore must be added to the stellar radiation field with h\nu > 54.4 \text{ eV} in the \text{He}^+ zone. Of course, a more accurate calculation may be made, taking into account the detailed frequency dependence of

NOTE: Numerical values are calculated assuming that n(\text{He}^{++})/n(\text{H}^+) = 0.15.

each of the emission processes, but since normally the helium abundance is small, only an approximation to its effects is usually required.

Some sample calculations of the ionization structure of a model planetary nebula, with the radiation source a blackbody at T_* = 10^5 K are shown in Figure 2.6. The sharp outer edge of the \text{He}^{++} zone, as well as the even sharper outer edges of the \text{H}^+ and \text{He}^+ zones, can be seen in these graphs. There is, of course, an equation that is exactly analogous to (2.19) and (2.27) for the “Strömgren radius” r_3 of the \text{He}^{++} zone:

Q(\text{He}^+) = \int_{4\nu_0}^{\infty} \frac{L_\nu}{h\nu} d\nu = \frac{4\pi}{3} r_3^3 n(\text{He}^{++}) n_e \alpha_B(\text{He}^+, T) \quad (2.29)

Thus stellar temperatures T_* \geq 10^5 K are required for r_3/r_1 \approx 1.

2.6 Further Iterations of the Ionization Structure

As described previously, the on-the-spot approximation may be regarded as the first approximation to the ionization and, as will be described in Chapter 3, to the temperature distribution in the nebula. From these a first approximation to the emission coefficient j_\nu may be found throughout the model: from j_\nu a first approximation to I_{\nu,d} and hence J_\nu at each point, and from J_\nu a better approximation to the ionization and temperature at each point. This iteration procedure can be repeated as many times as desired (given sufficient computing time) and actually converges quite rapidly, but except where high accuracy is required, the first (on-the-spot) approximation is usually sufficient. The higher approximations show that the degree of ionization nearest the star is so high that ionizing photons emitted there are not really absorbed on the spot. There is thus a net flow of these photons from the center to the outer regions. However, the ionization calculated from the on-the-spot approximation is reasonably close to the final exact result.

2.7 Photoionization of Heavy Elements

Finally, let us examine the ionization of the heavy elements, of which O, C, Ne, N, Si, and Fe, with abundances (by number) of order 10^{-3} to 10^{-4} that of H, are the most abundant. The ionization–equilibrium equation for any two successive stages of ionization i and i + 1 of any element X may be written

\begin{aligned} n(X^{+i}) \int_{\nu_i}^{\infty} \frac{4\pi J_\nu}{h\nu} a_\nu(X^{+i}) d\nu &= n(X^{+i}) \Gamma(X^{+i}) \\ &= n(X^{+i+1}) n_e \alpha_G(X^{+i}, T), \end{aligned} \quad (2.30)

where n(X^{+i}) and n(X^{+i+1}) are the number densities of the two successive stages of ionization; a_\nu(X^{+i}) is the photoionization cross section from the ground level of X^i

with the threshold \nu_i; and \alpha_G(X^{+i}, T) is the recombination coefficient of the ground level of X^{+i+1} to all levels of X^{+i}. These equations, together with the total number of ions of all stages of ionization,

n\left(X^0\right) + n\left(X^{+1}\right) + n\left(X^{+2}\right) + \dots + n\left(X^{+n}\right) = n(X)

(presumably known from the abundance of X), completely determine the ionization equilibrium at each point. The mean intensity J_\nu, of course, includes both the stellar and diffuse contributions, but the abundances of the heavy elements are so small that their contributions to the diffuse field are negligible, and only the emission by H, He, and \text{He}^+ mentioned previously needs to be taken into account.

The heavy elements do not usually make an appreciable contribution to the optical depth, but in some situations they can, particularly at frequencies just below the \text{He}^0 threshold. They can always be included by simple generalizations of Equations (2.26),

\frac{d\tau_\nu}{dr} = n\left(\text{H}^0\right) a_\nu\left(\text{H}^0\right) + \sum_{X,i} n\left(X^{+i}\right) a_\nu\left(X^{+i}\right) \text{ for } \nu_0 < \nu < \nu_2 \quad (2.31)

and

\begin{aligned} \frac{d\tau_\nu}{dr} = & n\left(\text{H}^0\right) a_\nu\left(\text{H}^0\right) + n\left(\text{He}^0\right) a_\nu\left(\text{He}^0\right) \\ & + \sum_{X,i} n\left(X^{+i}\right) a_\nu\left(X^{+i}\right) \text{ for } \nu_2 < \nu \end{aligned}

In both tbese equations all stages of ionization of all heavy elements that have thresholds helow the frequency for which the optical depth is being calculated are included in the sums over X and i.

Many ions of the heavy elements have one or more shells of electrons below the outer valence shell. Electrons within any shell can be removed when the source of ionizing radiation extends to high enough energies. The vacancies left by removal of inner electrons can be filled by outer electrons, producing emission lines at very high energies, or by the Auger effect, a multi-electron process where some electrons cascade down to fill the inner vacancy while others are ejected from the atom. Both processes are important contributors to the X-ray spectrum, and detailed discussion is presented in a later chapter, since here we are mainly concerned with nebulae photoionized by starlight.

The photoionization cross section of a particular shell behaves somewhat like the hydrogen and helium cross sections described above. The largest values are \sim 10^{-18} – 10^{-17} \text{ cm}^2 and often occur near the threshold, tending to decrease with increasing energy. Unlike H and He, there can also be many strong resonance features produced by interactions between various electrons.

The recombination coefficients for complex ions may be divided into radiative and dielectronic parts,

\alpha_G(X^{+i}, T) = \alpha_R(X^{+i}, T) + \alpha_d(X^{+i}, T). \quad (2.32)

The radiative part represents simple bound–free recaptures. Just as in H and He, captures to any level are followed by downward radiative transitions, leading ultimately to the ground level. Thus the radiative recombination coefficient is a sum over all levels, and is dominated by the excited levels, which to a good approximation are hydrogen-like (Table 2.7 gives these).

In the best calculations, the deviations due to departures of the energy levels from the exact hydrogen-like energies must be taken into account. The recombinations to the terms resulting from recaptures of an electron to the innermost unfilled shell, for instance,

\begin{aligned} \left.\begin{array}{l} \mathrm{O}^{++}(2p^2\,{}^3P)+ks\\ \mathrm{O}^{++}(2p^2\,{}^3P)+kd \end{array}\right\}&\rightarrow \mathrm{O}^{+}(2p^3\,{}^4S^\circ)+h\nu,\\[0.4em] \left.\begin{array}{l} \mathrm{O}^{++}(2p^2\,{}^3P)+ks\\ \mathrm{O}^{++}(2p^2\,{}^3P)+kd \end{array}\right\}&\rightarrow \mathrm{O}^{+}(2p^3\,{}^2D^\circ)+h\nu,\\[0.4em] \left.\begin{array}{l} \mathrm{O}^{++}(2p^2\,{}^3P)+ks\\ \mathrm{O}^{++}(2p^2\,{}^3P)+kd \end{array}\right\}&\rightarrow \mathrm{O}^{+}(2p^3\,{}^2S^\circ)+h\nu. \end{aligned}

are strongly affected by the presence of the other bound electrons, through the Pauli principle, and are far from hydrogen-like. These recombination coefficients can be found from the computed photoionization cross sections, using the Milne relation as explained in Appendix 2.

The additional dielectronic part of the recombination coefficient is larger than the radiative part for many, but not all, heavy ions at nebular temperatures. It results from resonances (at specific energies) in the total recombination cross sections, which are related to resonances (at related specific frequencies) in the corresponding photoionization cross section. Physically, these occur at energies at which the incoming free electron can give up nearly all its kinetic energy to exciting a bound level of the ion, thus creating a short-lived doubly excited level of the next lower stage of ionization. This level can then often decay to a singly excited bound level, and then by further successive radiative transitions downward to the ground level.

An example is the recombination of C^{++} to form C^+. A free electron with kinetic energy 0.41 eV colliding with a C^{++} ion in the ground 2s^2\ ^1S level makes up a five-electron system with the same total energy as a C^+ ion in the 2s2p3d\ ^2F^o term. There is thus a strong resonance centered at this energy, representing the high probability of collisional excitation of one of the bound 2s electrons to 2p together with capture of the free electron into the 3d shell. This term can then emit a photon decaying with non-zero transition probability to 2s2p^2\ ^2D, a bound term, which decays ultimately to the ground 2s^22p\ ^2P_{1/2} level. This dielectronic transition process may be written

\begin{aligned} \mathrm{C}^{++}(2s^2\,{}^1S)+kf &\rightarrow \mathrm{C}^{+}\!\left[2s2p({}^3P^\circ)3d\right],{}^2F^\circ\\ &\downarrow\\ &\mathrm{C}^{+}(2s2p^2\,{}^2D)+h\nu\\ &\downarrow\\ &\mathrm{C}^{+}(2s^22p\,{}^2P^\circ)+h\nu \end{aligned}

This is actually the main recombination process for C^{++} at nebular temperatures. Note that spin, orbital angular momentum, and parity must all be conserved in the first radiationless transition; for this reason, only free f (l = 3) electrons are involved in this specific dielectronic recombination process.

In addition, there is a third contribution, the integrated effect of many higher-energy resonances, which is ordinarily small at nebular temperatures. It can have a non-zero effect, and should be included in the most accurate calculations.

Selected calculated values of \alpha(X^{+i}, T) are listed in Appendix 5 in Tables A5.1 (for ions present in nebulae) and A5.2 (for ions present in X-ray sources such as AGN). The temperature dependence is not simple, because the resonance effects introduce an exponential behavior, so the values are given at three representative temperatures. Dielectronic recombination is more important in some situations, but radiative recombination is more important at others. For most but not all of these ions the importance of dielectronic recombination grows with increasing temperature. Both recombination processes must be taken into account in all calculations.

Calculations have been made of the ionization of heavy elements in many model H II regions and planetary nebulae. In H II regions, the common elements, such

as O^+ and N^+, tend to be mostly singly ionized in the outer parts of the nebulae, although near the central stars there are often fairly large amounts of O^{++}, N^{++}, and Ne^{++}. Most planetary nebulae have hotter central stars, and the degree of ionization is correspondingly higher. This is shown in Figure 2.6, where the ionization of O is plotted for a calculated model planetary nebula. Note that in this figure the outer edge of the He^{++} zone is also the outer edge of the O^{+3} zone and the inner edge of the O^{++} zone, since O^{++} has an ionization potential 54.9 eV, nearly the same as He^+.

Again, in actual nebulae, density condensations play an important role in complicating the ionization structure; these simplified models do, however, give an overall picture of the ionization.

One other atomic process is important in determining the ionization equilibrium of particular light elements, especially near the outer boundaries of radiation-bounded nebulae. This process is charge exchange in two-body reactions with hydrogen. As an example, consider neutral oxygen, which has the charge-exchange reaction with a proton

\mathrm{O}^{0}(2p^{4}\,{}^{3}P)+\mathrm{H}^{+} \rightarrow \mathrm{O}^{+}(2p^{3}\,{}^{4}S^{\circ}) +\mathrm{H}^{0}(1s^{2}\,{}^{2}S)\tag{2.33}

This reaction converts an originally neutral O atom into an O^+ ion, and thus is an ionization process for O. There is an attractive polarization force between O^0 and H^+; in addition, the ionization potentials of O and H are very nearly the same, so that the reaction is very nearly a resonance process. For both of these reasons the cross section for this charge-exchange reaction is relatively large. The reaction rate per unit volume per unit time for the reaction can be written

n_O n_p \delta(T) \text{ [cm}^{-3} \text{ s}^{-1}\text{]}, \quad (2.34)

where \delta(T) is expressed in terms of the reaction cross section \sigma(u) by an integral analogous to Equation (2.5),

\delta(T) = \int_0^\infty u \sigma(u) f(u) du. \quad (2.35)

Here it should be noted that f(u) is the Maxwell-Boltzmann distribution function for the relative velocity u in the OH^+ center-of-mass system, and thus involves their reduced mass.

A selection of computed values of \delta(T) for many ions is given in Tables A5.3 and A5.4. We concentrate on O because of its importance. For instance, in an H II region with n_p = 10 \text{ cm}^{-3}, the ionization rate per O atom per unit time is about 10^{-8} \text{ s}^{-1}, comparable with the photoionization rate for the typical conditions adopted in Section 2.1. Likewise, the rate for the inverse reaction

\mathrm{O}^{+}({}^{4}S)+\mathrm{H}^{0}({}^{2}S) \rightarrow \mathrm{O}^{0}({}^{3}P)+\mathrm{H}^{+}\tag{2.36}

can be written

n(O^+)n(H^0)\delta'(T).

Numerical values of \delta'(T), which of course is related to \delta(T) through an integral form of the Milne relation

\frac{\delta'}{\delta} = \frac{9}{8} \exp(\Delta E/kT) \quad (2.37)

where \Delta E = 0.19 eV is the difference in the ionization potentials of O^0 and H^0, are given for O^+ and several other ions in Table A5.4. Comparison of Table A5.4 with Table A5.1 shows that charge exchange has a rate comparable with recombination in converting O^+ to O^0 at the typical H II region conditions, but at the outer edge of the nebula, charge exchange dominates because of the higher density of H^0. At the higher radiation densities that occur in planetary nebulae, charge exchange is not important in the ionization balance of O except near the outer edge of the ionized region. The charge-exchange reactions do not appreciably affect the ionization equilibrium of H, because of the low O and O^+ densities.

At temperatures that are high compared with the difference in ionization potentials between O^0 and H^0, the charge-exchange reactions (2.33) and (2.36) tend to set up an equilibrium in which the ratio of species depends only on the statistical weights, since \delta'/\delta \rightarrow 9/8. It can be seen from Tables A5.3 and A5.4 that this situation is closely realized at T = 10,000 K. Thus in a nebula, wherever charge-exchange processes dominate the ionization balance of O, its degree of ionization is locked to that of H by the equation

\frac{n(O^0)}{n(O^+)} = \frac{9}{8} \frac{n(H^0)}{n_p} \quad (2.38)

Thus high densities of O^0 and free electrons exist together only in the edges or transition zone of the ionized region, as shown in Figure 2.6. Dense neutral condensations within planetary nebulae therefore show strong [OI] emission, as seen in Figure 2.7.

Another type of charge-exchange reaction that can be important in the ionization balance of heavy ions is exemplified by

\mathrm{O}^{++}(2s^{2}2p^{2}\,{}^{3}P) +\mathrm{H}^{0}(1s^{2}\,{}^{2}S) \rightarrow \mathrm{O}^{+}(2s\,2p^{4}\,{}^{4}P)+\mathrm{H}^{+}\tag{2.39}

Note that the O^+ is left in an excited state. This reaction is strongly exothermic, with an energy difference \Delta E = 6.7 eV. In addition, there are almost no O^+(2s 2p^4 \ ^4P) ions (with excitation energy 14.9 eV) present in the nebula. Hence the inverse reaction, which has this threshold, essentially does not proceed at all at nebular temperatures. Reaction (2.39) and some others like it, have large cross sections because it is a two-body process for which the strong Coulomb repulsion of the products speeds up the process. For the general reaction

X^{+i}+\mathrm{H}^{0} \rightarrow X^{+(i-1)}+\mathrm{H}^{+}\tag{2.40}

the rate per unit volume per unit time may be written

n(X^{+i})n(H^0)\delta'(T),

and calculated values of \delta'(T) are listed for three temperatures in Table A5.4. As the table shows, several of these rate coefficients are quite large. Thus even though the density of H^0 is small in comparison with the electron density, these charge-exchange reactions can be as important as recombination, or even more so, in determining the ionization equilibrium for these heavy ions. Similar charge reactions with He^0, and even between two heavy elements, can also make appreciable contributions to the ionization balance, and these should be included in any complete calculation.

References

Much of the early work on gaseous nebulae was done by H. Zanstra, D. H. Menzel, L. H. Aller, and others. The very important series of papers on physical processes in gaseous nebulae by D. H. Menzel and his collaborators is collected in Selected Papers on Physical Processes in Ionized Nebulae (New York: Dover) 1962. The treatment in this chapter is based on ideas that were, in many cases, given in these pioneering papers. The specific formulation and the numerical values used in this chapter are largely based on the references listed here.

Basic papers on ionization structure:

Strömgren, B. 1939, ApJ , 89, 529.

Hummer, D. G., & Seaton, M. J. 1963, MNRAS , 125, 437.

Hummer, D. G., & Seaton, M. J. 1964, MNRAS , 127, 217.

The name “Strömgren sphere” comes from the first of these papers, the pioneering basic treatment that began the theoretical understanding of H II regions.

Numerical values of H recombination coefficient:

• Seaton, M. J. 1959, MNRAS , 119, 81.
• Burgess, A. 1964, Mem. RAS , 69, 1.
• Pengelly, R. M. 1964, MNRAS , 127, 145.
• Hummer, D. G., & Storey, P. J. 1987, MNRAS , 224, 80.
• Storey, P. J., & Hummer, D. G. 1995, MNRAS , 272, 41 (on the web at http://adc.gsfc.nasa.gov/adc-cgi/cat.pl?/catalogs/6/6064/).

(Tables 2.1, 2.6, and 2.7 are based on these references.)

Numerical values of He recombination coefficient:

• Burgess, A., & Seaton, M. J. 1960, MNRAS , 121, 471.
• Robbins, R. R. 1968, ApJ , 151, 497.
• Robbins, R. R. 1970, ApJ , 160, 519.
• Brown, R. L., & Mathews, W. G. 1970, ApJ , 160, 939.
• Verner, D. A., & Ferland, G. J. 1996, ApJS , 103, 467.
• Smits, D. P. 1996, MNRAS , 278, 683.
• Benjamin, R. A., Skillman, E. D., & Smits, D. P. 1999, ApJ , 514, 307.

(Table 2.4 is based on the last two references.)

The calibration of the spectral classes in terms of effective temperature and number of ionizing photons is discussed by

• Morton, D. C. 1969, ApJ , 158, 629.
• Panagia, N. 1973, AJ , 78, 929.
• Vacca, W. D., Garmany, C. D., & Shull, J. M. 1996, ApJ , 460, 914.
• Schaerer, D., & de Koter, A. 1997, A&A , 322, 598.
• Sternberg, A., Hoffman, T. L., & Pauldrach, A. W. A. 2003, ApJ , 599, 1333.

Table 2.3 comes from the third of these references. There are only minor differences between them and the somewhat later numerical values in the last reference. There differences are especially small for the hottest and most luminous objects.

Numerical values of He (2^3S \rightarrow 2^1S and 2^1P) collisional cross sections:

• Bray, I., Burgess, A., Fursa, D. V., & Tully, J. A. 2000, A&AS , 146, 481.

Table 2.5 is based on this reference. It also gives collisional excitation rates to levels n = 3, 4, and 5, which are negligible at low temperatures and make at most a 10 percent correction at the highest temperature in Table 2.5.

Numerical values of the H and He ⁺ photoionization cross sections:

• Hummer, D. G., & Seaton, M. J. 1963, MNRAS , 125, 437.

(Figure 2.2 is based on this reference and the following ones.)

Numerical values of the He photoionization cross section:

Bell, K. L., & Kingston, A. E. 1967, Proc. Phys. Soc. , 90, 31.

Brown, R. L. 1971, ApJ , 164, 387.

Hummer, D. G., & Storey, P. J. 1998, MNRAS , 297, 1073.

Numerical values of He (2\ ^1S \rightarrow 1\ ^1S) and He (2\ ^3S \rightarrow 1\ ^1S) transition probabilities, including the frequency distribution in the 2\ ^1S \rightarrow 1\ ^1S two photon spectrum:

Lin, C. D., Johnson, W. R., & Dalgarno, A. 1977, Phys Rev A , 15, 154.

Drake, G. W. F. 1979, Phys Rev A , 19, 1387.

Hata, J., & Grant, I. P. 1981, J Phys B , 14, 2111.

Lach, G., & Pachucki, K. 2001, Phys Rev A , 64, 42510.

Early calculations of model H II regions:

Hjellming, R. M. 1966, ApJ , 143, 420.

Rubin, R. H. 1968, ApJ , 153, 761.

Rubin, R. H. 1983, ApJ , 274, 671.

(Figure 2.4 is based on the second of these references. The last emphasizes the importance of heavy-element opacity in some situations.)

Early calculations of model planetary nebulae:

Harrington, J. P. 1969, ApJ , 156, 903.

Flower, D. R. 1969, MNRAS , 146, 171.

Harrington, J. P. 1979, Planetary Nebulae, Observations and Theory (IAU Symposium No. 76), ed. Y. Terzian (Dordrecht: Reidel), p. 15.

(Figure 2.6 is based on the second of these references.)

References to other later, still more detailed models of H II regions and planetary nebulae are given in Chapter 5.

Dense neutral condensations in planetary nebulae are discussed by

O'Dell, C. R., Henney, W. J., & Burkert, A. 2000, AJ , 119, 2910.

O'Dell, C. R., 2000, AJ , 119, 2311.

Model atmospheres for hot stars are discussed in more detail in Chapter 5. In the present chapter, it is easiest to say that the simplest models are blackbodies; a much better approximation is provided by models in which the continuous spectrum is calculated; and the best models are those that also include the effects of the absorption lines, since they are strong and numerous in the ultraviolet, as well as winds. ^§

Chapter 3

Thermal Equilibrium

3.1 Introduction

The temperature in a static nebula is fixed by the equilibrium between heating by photoionization and cooling by recombination and by radiation from the nebula. When a photon of energy h\nu is absorbed and causes an ionization of H, the photoelectron produced has an initial energy \frac{1}{2}mu^2 = h(\nu - \nu_0), and we may think of an electron being "created" with this energy. The electrons thus produced are rapidly thermalized, as indicated in Chapter 2, and in ionization equilibrium these photoionizations are balanced by an equal number of recombinations. In each recombination, a thermal electron with energy \frac{1}{2}mu^2 disappears, and an average of this quantity over all recombinations represents the mean energy that "disappears" per recombination. The difference between the mean energy of a newly created photoelectron and the mean energy of a recombining electron represents the net gain in energy by the electron gas per ionization process. In equilibrium this net energy gain is balanced by the energy lost by radiation, chiefly by electron collisional excitation of bound levels of abundant ions, followed by emission of photons that can escape from the nebula. Free-free emission, or bremsstrahlung, is another, less important radiative energy-loss mechanism.

3.2 Energy Input by Photoionization

Let us first examine the energy input by photoionization. As in Chapter 2, it is simplest to begin by considering a pure H nebula. At any specific point in the nebula, the energy input (per unit volume per unit time) is

G(\text{H}) = n(\text{H}^0) \int_{\nu_0}^{\infty} \frac{4\pi J_{\nu}}{h\nu} h(\nu - \nu_0) a_{\nu}(\text{H}^0) d\nu \text{ [erg cm}^{-3} \text{s}^{-1}]. \quad (3.1)

Furthermore, since the nebula is in ionization equilibrium, we may eliminate n(\text{H}^0) by substituting Equation (2.8), giving

\begin{aligned} G(\text{H}) &= n_e n_p \alpha_A(\text{H}^0, T) \frac{\int_{\nu_0}^{\infty} \frac{4\pi J_\nu}{h\nu} h(\nu - \nu_0) a_\nu(\text{H}^0) d\nu}{\int_{\nu_0}^{\infty} \frac{4\pi J_\nu}{h\nu} a_\nu(\text{H}^0) d\nu} \\ &= n_e n_p \alpha_A(\text{H}^0, T) \frac{3}{2} kT_i \end{aligned} \tag{3.2}

From this equation it can be seen that the mean energy of a newly created photoelectron depends on the form of the ionizing radiation field, but not on the absolute strength of the radiation. The rate of creation of photoelectrons depends on the strength of the radiation field, or, as Equation (3.2) shows, on the recombination rate. The quantity \frac{3}{2}kT_i represents the initial temperature of the newly created photoelectrons. For assumed blackbody spectra with J_\nu = B_\nu(T_*), it is easy to show that T_i \approx T_* so long as kT_* < h\nu_0. For any known J_\nu (for instance, the emergent spectrum from a model atmosphere), the integration can be carried out numerically; a short list of representative values of T_i is given in Table 3.1. Note that the second column in the table, \tau_0 = 0, corresponds to photoionization by the emergent model-atmosphere spectrum. At larger distances from the star, the spectrum of the ionizing radiation is modified by absorption in the nebula, the radiation nearest the series limit being most strongly attenuated because of the frequency dependence of the absorption coefficient. Therefore, the higher-energy photons penetrate further into the gas, and the mean energy of the photoelectrons produced at larger optical depths from the star is higher. This effect is shown for a pure H nebula in the columns labeled with values of \tau_0, the optical depth at the ionization limit.

3.3 Energy Loss by Recombination

The kinetic energy lost by the electron gas (per unit volume per unit time) in recombination can be written

L_R(\text{H}) = n_e n_p k T \beta_A(\text{H}^0, T), \quad [\text{erg cm}^{-3} \text{s}^{-1}] \quad (3.3)

where

\beta_A(\text{H}^0, T) = \sum_{n=1}^{\infty} \beta_n(\text{H}^0, T) = \sum_{n=1}^{\infty} \sum_{L=0}^{n-1} \beta_{nL}(\text{H}^0, T) \quad [\text{cm}^3 \text{s}^{-1}] \quad (3.4)

with

\beta_{nL}(\text{H}^0, T) = \frac{1}{kT} \int_0^{\infty} u \sigma_{nL}(\text{H}^0, T) \frac{1}{2} m u^2 f(u) du. \quad (3.5)

The left-hand side of Equation (3.5) is thus effectively a kinetic energy averaged recombination coefficient. Note that since the recombination cross sections are approximately proportional to u^{-2}, the electrons of lower kinetic energy are preferentially captured, and the mean energy of the captured electrons is somewhat less than \frac{3}{2}kT. Calculated values of \beta_1 and \beta_A are listed in Table 3.2.

In a pure H nebula that had no radiation losses, the thermal equilibrium equation would be

G(\text{H}) = L_R(\text{H}), \quad (3.6)

and the solution for the nebular temperature would give a T > T_i because of the "heating" due to the preferential capture of the slower electrons.

The radiation field J_\nu in Equation (3.1) should, of course, include the diffuse radiation as well as the stellar radiation modified by absorption. This can easily be included in the on-the-spot approximation, since, according to it, every emission of an ionizing photon during a recombination to the level n = 1 is balanced by absorption of the same photon at a nearby spot in the nebula. Thus production of photons by the diffuse radiation field and recombinations to the ground level can simply be omitted from the gain and loss rates, leading to the equations

\begin{aligned} G_{OTS}(\text{H}) &= n(\text{H}^0) \int_{\nu_0}^{\infty} \frac{4\pi J_{\nu s}}{h\nu} h(\nu - \nu_0) a_\nu(\text{H}^0) d\nu \\ &= n_e n_p \alpha_B(\text{H}^0, T) \frac{\int_{\nu_0}^{\infty} \frac{4\pi J_{\nu s}}{h\nu} h(\nu - \nu_0) a_\nu(\text{H}^0) d\nu}{\int_{\nu_0}^{\infty} \frac{4\pi J_{\nu s}}{h\nu} a_\nu(\text{H}^0) d\nu} \end{aligned} \quad (3.7)

and

L_{OTS}(\text{H}) = n_e n_p kT \beta_B(\text{H}^0, T), \quad (3.8)

with

\beta_B(\text{H}^0, T) = \sum_{n=2}^{\infty} \beta_n(\text{H}^0, T). \quad (3.9)

The on-the-spot approximation is not as accurate for the thermal equilibrium as it is in the ionization equation, because of the fairly large difference in h(\nu - \nu_0) between the ionizing photons in the stellar and diffuse radiation fields, but it may be improved by further iterations if necessary.

The generalization to include He in the heating and recombination cooling rates is straightforward to write, namely,

G = G(\text{H}) + G(\text{He}), \quad (3.10)

where

G(\text{He}) = n_e n(\text{He}^+) \alpha_A(\text{He}^0, T) \frac{\int_{\nu_2}^{\infty} \frac{4\pi J_\nu}{h\nu} h(\nu - \nu_2) a_\nu(\text{He}^0) d\nu}{\int_{\nu_2}^{\infty} \frac{4\pi J_\nu}{h\nu} a_\nu(\text{He}^0) d\nu} \quad (3.11)

and

L_R = L_R(\text{H}) + L_R(\text{He}), \quad (3.12)

with

L_R(\text{He}) = n_e n(\text{He}^+) kT \beta_A(\text{He}^0, T) \quad (3.13)

and so on.

It can be seen that the heating and recombination cooling rates are proportional to the densities of the ions involved, so the contributions of the heavy elements, which are much less abundant than H and He, can, to a good approximation, be omitted from these rates.

3.4 Energy Loss by Free-Free Radiation

Next we will examine cooling by radiation processes that do not involve recombination. In most circumstances such cooling is far more important than the recombination cooling, and therefore dominates the thermal equilibrium. A minor contributor to the cooling rate, which nevertheless is important because it can occur even in a pure H nebula, is free-free radiation or bremsstrahlung, in which a continuous spectrum is emitted. The rate of cooling by this process by ions of charge Z, integrated over all frequencies, is, to a fair approximation,

\begin{aligned} L_{FF}(Z) &= 4\pi j_{ff} \\ &= \frac{2^5 \pi e^6 Z^2}{3^{3/2} h m c^3} \left( \frac{2\pi kT}{m} \right)^{1/2} g_{ff} n_e n_+ \text{ [erg cm}^{-3} \text{ s}^{-1}] \\ &= 1.42 \times 10^{-27} Z^2 T^{1/2} g_{ff} n_e n_+ \end{aligned}

where n_+ is the number density of the ions. Again \text{H}^+ dominates the free-free cooling, because of its abundance, and \text{He}^+ can be included with \text{H}^+ (since both have Z = 1) by writing n_+ = n_p + n(\text{He}^+). The numerical factor g_{ff} is called the mean Gaunt factor for free-free emission; it is a slowly varying function of n_e and T, generally for nebular conditions in the range 1.0 < g_{ff} < 1.5, and a good average value to adopt is g_{ff} \approx 1.3.

3.5 Energy Loss by Collisionally Excited Line Radiation

A far more important source of radiative cooling is collisional excitation of low-lying energy levels of common ions, such as \text{O}^+, \text{O}^{++}, and \text{N}^+. These ions make a significant contribution in spite of their low abundance because they have energy levels with excitation potentials of the order of kT, but all the levels of H and He have much higher excitation potentials, and therefore are usually not important as collisionally excited coolants. Let us therefore examine how an ion is excited to level 2 by electron collisions with ions in the lower level 1. The cross section for excitation

\sigma_{12}(u) is a function of electron velocity u and is zero below the threshold \chi = h\nu_{21}. Not too far above the threshold, the main dependence of the excitation cross section is \sigma \propto u^{-2} (because of the focusing effect of the Coulomb force); so it is convenient to express the collision cross sections in terms of the energy-specific collision strength \Omega(1, 2) defined by

\sigma_{12}(u) = \frac{\pi \hbar^2}{m^2 u^2} \frac{\Omega(1, 2)}{\omega_1} \text{ for } \frac{1}{2} m u^2 > \chi, \quad (3.14)

where \Omega(1, 2) is a function of electron velocity (or energy) but is often approximately constant near the threshold, and \omega_1 is the statistical weight of the lower level.

There is a relation between the cross section for deexcitation, \sigma_{21}(u), and the cross section for excitation, namely

\omega_1 u_1^2 \sigma_{12}(u_1) = \omega_2 u_2^2 \sigma_{21}(u_2) \quad (3.15)

where u_1 and u_2 are related by

\frac{1}{2} m u_1^2 = \frac{1}{2} m u_2^2 + \chi. \quad (3.16)

Equation (3.15) can easily be derived from the principle of detailed balancing, which states that in thermodynamic equilibrium each microscopic process is balanced by its inverse. Thus in this particular case, the number of excitations caused by collisions with electrons in the velocity range u_1 to u_1 + du_1 is just balanced by the deexcitations caused by collisions that produce electrons in the same velocity range. Thus

n_e n_1 u_1 \sigma_{12}(u_1) f(u_1) du_1 = n_e n_2 u_2 \sigma_{21}(u_2) f(u_2) du_2,

and using the Boltzmann equation of thermodynamic equilibrium,

\frac{n_2}{n_1} = \frac{\omega_2}{\omega_1} \exp(-\chi/kT),

we derive the relation (3.15). Combining Equations (3.14) and (3.15), so that the deexcitation cross section can be expressed in terms of the collision strength \Omega(1, 2),

\sigma_{21}(u_2) = \frac{\pi \hbar^2}{m^2 u_2^2} \frac{\Omega(1, 2)}{\omega_2}; \quad (3.17)

that is, the collision strengths are symmetrical in 1 and 2.

The total collisional deexcitation rate per unit volume per unit time is

\begin{aligned} n_e n_2 q_{21} &= n_e n_2 \int_0^\infty u \sigma_{21} f(u) du \\ &= n_e n_2 \left( \frac{2\pi}{kT} \right)^{1/2} \frac{\hbar^2}{m^{3/2}} \frac{\Upsilon(1, 2)}{\omega_2} [\text{cm}^{-3} \text{ s}^{-1}] \\ &= n_e n_2 \frac{8.629 \times 10^{-6}}{T^{1/2}} \frac{\Upsilon(1, 2)}{\omega_2} \end{aligned} \tag{3.18}

where \Upsilon(1, 2) is the velocity-averaged collision strength

\Upsilon(1, 2) = \int_0^\infty \Omega(1, 2; E) \exp(-E/kT) d\left(\frac{E}{kT}\right) \tag{3.19}

with E = \frac{1}{2} m u^2. Likewise, the collisional excitation rate per unit volume per unit time is n_e n_1 q_{12}, where

\begin{aligned} q_{12} &= \frac{\omega_2}{\omega_1} q_{21} \exp(-\chi/kT) \\ &= \frac{8.629 \times 10^{-6}}{T^{1/2}} \frac{\Upsilon(1, 2)}{\omega_1} \exp(-\chi/kT) [\text{cm}^3 \text{ s}^{-1}] \end{aligned} \tag{3.20}

The collision strengths must be calculated quantum-mechanically, and some of the most important numerical values are listed in Tables 3.3 through 3.7. Each collision strength in general consists of a part that varies slowly with energy, on which, in many cases, there are superimposed resonance contributions that vary rapidly with energy; but when the cross sections are integrated over a Maxwellian distribution, as in almost all astrophysical applications, the effect of the exact positions of the resonances tends to be averaged out. The resulting averaged collision strengths, calculated from Equation (3.19), are given in Tables 3.3 to 3.7, evaluated at T = 10,000 K, a representative nebular temperature. It is convenient to remember that,

Cochrane, D. M., & McWhirter, R. W. P. 1983, PhyS , 28, 25.

McWhirter, R. W. P. 1994, ADNDT , 57, 39.

C ⁺² Berrington, K. A., Burke, P. G., Dufton, P. L., & Kingston, A. E. 1985, ADNDT, 33, 195; N ⁺³ , O ⁺⁴ Ramsbottom, C. A., Berrington, K. A., Hibbert, A., & Bell, K. L. 1994, Physica Scripta, 50, 246; O ⁺⁴ McKenna, R. C., et al. 1997, ApJ, 486, 571; Al ⁺¹ Aggarwal, K. M., & Keenan, F. P. 1998, J. Phys. B, 31, 4545, and Aggarwal, K. M., & Keenan, F. P. 1994, J. Phys. B, 27, 5321; Si ⁺² Dufton, P. L., & Kingston, A. E. 1994, ADNDT, 57, 273; S ⁺⁴ Dufton, P. L., & Kingston, A. E. 1984, J. Phys. B, 17, 3321 (extrapolated).

B-like ions from Blum, R. D., & Pradhan, A. K. 1992, ApJS 80, 425; F-like ions from Saraph, H. E. & Tully, J. A. 1994, A&AS, 107, 29; Si ⁺ Dufton, P. L., & Kingston, A. E. 1994, ADNDT, 57, 273, S ⁺³ Tayal, S. S. 2000, ApJ, 530, 1091; Ar ⁺⁵ Saraph, H. E., & Storey, P. J. 1996, A&AS, 115, 151; Ar ⁺ , Ca ⁺³ Pelan, J., & Berrington, K. A. 1995, A&AS, 110, 209.

for an electron with the mean energy at a typical nebular temperature, T \approx 7,500 K, the cross sections for excitation and deexcitation are \sigma \approx 10^{-15} \Upsilon / \omega \text{ cm}^2.

Note that there is a simple relation for the collision strengths between a term consisting of a single level and a term consisting of various levels, namely,

\Upsilon(SLJ, S'L'J') = \frac{(2J' + 1)}{(2S' + 1)(2L' + 1)} \Upsilon(SL, S'L') \quad (3.21)

N ⁺ , O ⁺² , and Ne ⁺⁴ from Lennon, D. J., & Burke, V. M. 1994, A&AS, 103, 273; Ne ⁺² from Butler, K., & Zeippen, C. J. 1994, A&AS, 108, 1; S ⁺² from Tayal, S. S., and Gupta, G. P. 1999 ApJ 526, 544; Ar ⁺² , Ar ⁺⁴ from Galavis, M. E., Mendoza, C., & Zeippen, C. J. 1995, A&AS, 111, 347.

O ⁺ Pradhan, A. K. 1976, MNRAS, 177, 31, 1998, and J Phys B, 31, 4317; Ne ⁺³ , Giles, K. 1981, MNRAS, 195, 63, and Ramsbottom, C. A., Bell, K. L., & Keenan, F. P. 1998, MNRAS, 293, 233; S ⁺ Ramsbottom, C. A., Bell, K. L., Stafford, R. P. 1996, ADNDT, 63, 57; Ar ⁺³ Ramsbottom, C. A., & Bell, K. L. 1997, ADNDT, 66, 65.

if either S = 0 or L = 0. The factors (2J' + 1) and (2S' + 1)(2L' + 1) are the statistical weights of the level and of the term, respectively. On account of this relation, the rate of collisional excitation in p^2 or p^4 ions (such as O ⁺⁺ ) from the ground ^3P term to the excited (singlet) ^1D and ^1S levels is very nearly independent of the distribution of ions among ^3P_0, ^3P_1, and ^3P_2.

N IV], Flemming, J., Brage, T., Bell, K. L., Vaecck, N., Hibbert, A., Godefroid, M., & Froese Fischer, C. 1995, ApJ, 455, 758; C III], OV], Flemming, J., Bell, K. L. K., Hibbert, A., Vaecck, N., & Godefroid, M. 1996, MNRAS, 279, 1289.

Galavis, M. E., Mendoza, C., & Zeippen, C. J. 1998, A&AS, 131, 499; Wiese, W. L., Fuhr, J. R., & Deters, T. M. 1996, J. Chem Phys Ref Data Monograph No. 7; Kaufman, J., & Sugar, J. 1986, J. Chem Phys Ref Data 15, 321.

For all the low-lying levels of the ions listed in Tables 3.3 through 3.5, the excited levels arise from the same electron configurations as the ground level. Radiative transitions between these excited levels and the ground level are forbidden by the electric-dipole selection rules, but can occur by magnetic-dipole and/or electric-quadrupole transitions. These are the well known forbidden lines, many of which are observed in nebular spectra, in the optical, infrared, or ultraviolet regions. Transition probabilities, as well as wavelengths for the observable lines, are listed in Tables 3.8 through 3.14.

For an ion with a single excited level, in the limit of very low electron density, every collisional excitation is followed by the emission of a photon, and the cooling rate per unit volume is therefore

L_C = n_e n_1 q_{12} h\nu_{21}. \quad (3.22)

Si II, Dufton, P. L., & Kingston, A. E. 1991, MNRAS, 248, 827; S IV; \lambda and A(\text{UV}), Gupta, G. P., & Msezane, A. Z. 2000, ApJS, 130, 227; S^{+4} A(\text{IR}), Johnson, C. T., Kingston, A. E., Dufton, P. L. 1986, MNRAS 220. 155.

Kaufman, J., & Sugar, J. 1986, J. Chem. Phys. Ref. Data 15, 321.

However, if the density is sufficiently high, collisional deexcitation is not negligible and the cooling rate is reduced. The equilibrium equation for the balance between the excitation and deexcitation rates of the excited level is, in general,

n_e n_1 q_{12} = n_e n_2 q_{21} + n_2 A_{21}, \quad (3.23)

and the solution is

\frac{n_2}{n_1} = \frac{n_e q_{12}}{A_{21}} \left[ \frac{1}{1 + \frac{n_e q_{21}}{A_{21}}} \right], \quad (3.24)

C-like: Galavis, M. E., Mendoza, C., & Zeippen, C. J. 1997, A&AS, 123, 159; Mendoza, C., Zeippen, C. J., & Storey, P. J. 1999, A&AS, 135, 159; Storey, P. J. & Zeippen, C. J. 2000, MNRAS, 312, 813; Si-like: Tayal, S. S. 1997, ADNDT, 67, 331; Biemont, E., & Bromage, G. E. 1983, MNRAS, 205, 1085.

[O II] Zeippen, C. J. 1987, A&A, 173, 410; [Ne IV] Becker, S. R., Butler, K., & Zeippen, C. J. 1989, A&A, 221, 375; P-like from Mendoza, C., & Zeippen, C. J. 1982, MNRAS, 198, 127; [N I] wavelengths from Slanger, T. G., Huestis, D. L., Cosby, P. C., Osterbrock, D. E. 2000, J. Chem. Phys. 113, 8514.

[O I], [Ne III] Galavis, M. E., Mendoza, C., & Zeippen, C. J. 1997, A&AS, 123, 159; Storey, P. J., & Zeippen, C. J. 2000, MNRAS, 312, 813; [Ar III] Mendoza, C., & Zeippen, C. J. 1983, MNRAS, 202, 981.

so the cooling rate is

L_C = n_2 A_{21} h\nu_{21} = n_e n_1 q_{12} h\nu_{21} \left[ \frac{1}{1 + \frac{n_e q_{21}}{A_{21}}} \right] \quad (3.25)

It can be seen that as n_e \rightarrow 0, we recover Equation (3.22), but as n_e \rightarrow \infty,

L_C \rightarrow n_1 \frac{\omega_2}{\omega_1} \exp(-\chi/kT) A_{21} h\nu_{21}, \quad (3.26)

the thermodynamic-equilibrium cooling rate.

Some ions have only two low-lying levels and may be treated by this simple formalism, but most ions have more levels, and all ions with ground configurations p^2, p^3, or p^4 have five low-lying levels. Examples are O^{++} and N^+, whose energy-level diagrams are shown in Figure 3.1. For such ions, collisional and radiative transitions can occur between any of the levels, and excitation and deexcitation cross sections and collision strengths exist between all pairs of the levels.

The equilibrium equations for each of the levels i = 1, 5 thus become

\sum_{j \neq i} n_j n_e q_{ji} + \sum_{j > i} n_j A_{ji} = \sum_{j \neq i} n_i n_e q_{ij} + \sum_{j < i} n_i A_{ij} \quad (3.27)

which, together with the total number of ions

\sum_j n_j = n, \quad (3.28)

can be solved for the relative population in each level, and then for the collisionally excited radiative cooling rate

L_C = \sum_i n_i \sum_{j < i} A_{ij} h\nu_{ij} \text{ [erg cm}^{-3} \text{ s}^{-1}\text{]}. \quad (3.29)

In the low-density limit, n_e \rightarrow 0, this becomes a sum of terms like (3.22), but if

n_e q_{ij} > \sum_{k < i} A_{ik}

for any i, j, collisional deexcitation is not negligible and the complete solution must be used. In fact, for any level i, a critical density n_c(i) may be defined as

n_c(i) = \sum_{j < i} A_{ij} / \sum_{j \neq i} q_{ij} \quad (3.30)

NOTE: All values are calculated for T \approx 10,000 K.

so that for n_e < n_c(i), collisional deexcitation of level i is negligible, but for n_e > n_c(i) it is important. Critical densities for levels that are most important in radiative cooling are listed in Table 3.15.

3.6 Energy Loss by Collisionally Excited Line Radiation of H

H^+, the most abundant ion in nebulae, has no bound levels and no lines, but H^0, although its fractional abundance is low, may affect the radiative cooling in a nebula. The most important excitation processes from the ground 1^2S term are to 2^2P^o, followed by emission of a L\alpha photon with h\nu = 10.2 eV, and to 2^2S, followed by emission of two photons in the 2^2S \rightarrow 1^2S continuum with h\nu' + h\nu'' = 10.2 eV and transition probability A(2^2S, 1^2S) = 8.23 \text{ s}^{-1}. Cross sections for excitation of neutral atoms by electrons do not vary as u^{-2}, but rise from zero at the threshold, peak at energies several times the threshold, and then decline at high energies, often with superimposed resonances. Nevertheless the mean collision strengths, integrated over the Maxwellian velocity distribution of the electrons as defined by (3.18) and (3.19), for these transitions and for 1^2S \rightarrow 3^2S, 3^2P^o, and 3^2D are quantities that vary fairly slowly, as Table 3.16 shows.

3.7 Resulting Thermal Equilibrium

The temperature at each point in a static nebula is determined by the equilibrium between heating and cooling rates, namely,

G = L_R + L_{FF} + L_C. \quad (3.31)

Anderson, H., Balance, C. P., Badnell, N. R., &amp; Summers, H. P. 2000, J.Phys.B, 33, 1255.

The collisionally excited radiative cooling rate L_C is a sum (over all transitions of all ions) of individual terms like (3.22), (3.25), or (3.29). In the low-density limit, since all the terms in G, L_R, L_{FF}, and L_C are proportional to n_e and to the density of some ion, Equation (3.31) and therefore the resulting temperatures are independent of the total density, but do depend on the relative abundances of the various ions. When collisional deexcitation begins to be important, the cooling rate at a given temperature is decreased, and the equilibrium temperature for a given radiation field is therefore somewhat increased.

To understand better the concepts here, let us consider an example, namely, an H II region with “typical” abundances of the elements. We will adopt n(O)/n(H) = 7 \times 10^{-4}, n(Ne)/n(H) = 9 \times 10^{-5}, and n(N)/n(H) = 9 \times 10^{-5}, and neglect other elements for simplicity. Let us suppose that O, Ne, and N are each 80 percent singly ionized and 20 percent doubly ionized, and n(H^0)/n(H^+) = 1 \times 10^{-3}. Some of the individual contributions to the radiative cooling (in the low-density limit) and the total radiative cooling L_C + L_{FF} are shown in Figure 3.2. For each level the contribution is small if kT \ll \chi, then increases rapidly and peaks at kT \approx \chi, and then decreases slowly for kT > \chi. The total radiative cooling, composed of the sum of the individual contributions, continues to rise with increasing T as long as there are levels with excitation energy \chi > kT. It can be seen that, for the assumed composition and ionization, O^{++} dominates the radiative cooling contribution at low temperatures, and O^+ at somewhat higher temperatures. At all temperatures shown, the contribution of collisional excitation of H^0 is small.

It is convenient to rewrite Equation (3.31) in the form

G - L_R = L_{FF} + L_C,

where G - L_R is then the “effective heating rate”, representing the net energy gained in photoionization processes, with the recombination losses already subtracted. This effective heating rate is also shown in Figure 3.2, for model stellar atmospheres with various temperatures. Notice in the figure that the calculated nebular temperature at which the curves cross and at which Equation (3.31) is satisfied is rather insensitive to the input stellar radiation field. Typical nebular temperatures are T \approx 7,000 K, according to Figure 3.2, with somewhat higher temperatures for hotter stars or larger optical depths.

At high electron densities, collisional deexcitation can appreciably modify the radiative cooling rate and therefore the resulting nebular temperature. For instance, at n_e \approx 10^4 \text{ cm}^{-3}, a density that occurs in condensations in many H II regions, the [O II] ^4S-^2D and [O III] ^3P_0-^3P_1 and ^3P_0-^3P_2 transitions are only about 20 percent effective, [N II] ^3P_0-^3P_1 and ^3P_0-^3P_2 are only about 1 percent effective, and [N III] ^2P_{1/2}-^2P_{3/2} is about 20 percent effective, as Table 3.15 shows. Figure 3.3 shows the effective cooling rate for this situation, with the abundances and ionization otherwise as previously described, and demonstrates that appreciably higher temperatures occur at high densities. Similarly, lower abundances of the heavy elements tend to decrease the cooling rate and thus to increase the resulting equilibrium temperature.

Under conditions of very high ionization, however, as in the central part of a planetary nebula, the ionization is high enough that there is very little \text{H}^0, \text{O}^+, or \text{O}^{++}, and then the radiative cooling is appreciably decreased. Under these conditions the main coolants are \text{Ne}^{+4} and \text{C}^{+3}, and the nebular temperature may be T \lesssim 2 \times 10^4 \text{ K}. Detailed results obtained from models of both H II regions and planetary nebulae are discussed in Chapter 5.

References

The basic papers on thermal equilibrium are:

Spitzer, L. 1948, ApJ , 107, 6.

Spitzer, L. 1949, ApJ , 109, 337.

Spitzer, L., & Savedoff, M. P. 1950, Ap. J. 111, 593.

Some additional work, including the on-the-spot approximation and the effects of collisional deexcitation, is described in

Burbidge, G. R., Gould, R. J., and Pottasch, S. R. 1963, ApJ , 138, 945.

Osterbrock, D. E. 1965, ApJ , 142, 1423.

Numerical values of recombination coefficients, \beta, are given in

Hummer, D. G., & Seaton, M. J. 1963, MNRAS , 125, 437.

Seaton, M. J. 1959, MNRAS , 119, 81.

Ferland, G. J., Petersou, B. M., Horne, K., Welsh, W. F., & Nahar, S. 1992, ApJ , 387, 95.

Hummer, D. G. 1994, MNRAS , 268, 109.

LaMothe, J., & Ferland, G. J. 2001, PASP , 113, 165.

(Table 3.2 is based on these references.)

Extensive tables of the free-free Gaunt factor are given by

Karzas, W. J., & Latter, R. 1961, ApJS , 6, 167.

Carson, T. R. 1988, A&A , 189, 319.

Kylafis, N. D., & Lamb, D. Q. 1982, ApJS , 48, 239.

Basic papers on collisional excitation and the methods used to calculate the collision strengths are:

Seaton, M. J. 1968, Advances in Atomic and Molecular Physics 4, 331.

Seaton, M. J. 1975, Advances in Atomic and Molecular Physics 11, 83.

This material is further elucidated in the book

Burke, P. G., Eissner, W. B., Hummer, D. G., & Percival, I. C., eds. 1983, Atoms in Astrophysics (New York: Plenum Press).

Numerical values of collision strengths are widely scattered through the physics literature, but mostly in the journals Atomic Data Nuclear Data Tables (ADNDT), Journal of Physics B (J. Phys. B), Physics Review A (Phys Rev A), and Physica Scripta (PhysSc). Volume 57 of ADNDT consists of a number of reviews of sources for collision strengths.

Today many data bases are located on the web. Among others, check the

Chianti: http://www.solar.nrl.navy.mil/chianti.html

TopBase, the Opacity Project database http://heasarc.gsfc.nasa.gov/topbase/

ADfA database http://www.pa.uky.edu/~verner/atom.html

The best wavelengths come from nebular measurements made with high-dispersion spectrographs. The following give extensive emission line lists.

Kaufman, V., & Sugar, J. 1986, J. Phys. Chem. Ref. Data , 15, No. 1, 321 (forbidden lines).

House, L.L. 1969, ApJS , 155, 21 (X-ray).

Hyung, S., & Aller, L. H. 1997, ApJ , 491, 242 (ultraviolet and optical).

Feuchtgruber, H., Lutz, D., et al., 1997, ApJ , 487, 962 and Feuchtgruber, H., Lutz, D., & Beuterna, D. A., 2001, ApJS , 136, 221 (infrared).

http://www.pa.uky.edu/~peter/atomic/ is a web-based list of wavelengths.

The following is the most recent in a series of papers that give a bibliography of atomic line identifications:

Adelman, S. J. 2001, PASP , 113, 344.

Chapter 4

Calculation of Emitted Spectrum

4.1 Introduction

The radiation emitted by each element of volume in a gaseous nebula depends upon the abundances of the elements, determined by the previous evolutionary history of the gas, and on the local ionization, density, and temperature, determined by the radiation field and the abundances as described in the preceding two chapters. The most prominent spectral features are the emission lines, and many of these are the collisionally excited lines described in the preceding chapter on thermal equilibrium. The formalism developed there to calculate the cooling rate, and thus the thermal equilibrium, may be taken over unchanged to calculate the strength of these lines. If we could observe all the lines in the entire spectral region from the extreme ultraviolet to the far infrared, we could measure directly the cooling rate at each observed point in the nebula. Many of the most important lines in the cooling, for instance [O II] \lambda\lambda 3726, 3729 and [O III] \lambda\lambda 4959, 5007, are in the optical region and are easily measured. Other lines that are also important in the cooling, such as [O III] 2p^2\ ^3P_0 - 2p^2\ ^3P_1 \lambda 88.4\ \mu\text{m}, and ^3P_1 - ^3P_2 \lambda 51.8\ \mu\text{m}, are in the far infrared region, while still others, such as C IV \lambda\lambda 1548, 1551, are in the vacuum ultraviolet.

For historical reasons, astronomers tend to refer to the chief emission lines of gaseous nebulae as forbidden lines. Actually, it is better to think of the bulk of the lines as collisionally excited lines, which arise from levels within a few volts of the ground level, and which therefore can be excited by collisions with thermal electrons. In fact, in the ordinary optical region all these collisionally excited lines are forbidden lines, because in the abundant ions all the excited levels within a few volts of the ground level arise from the same electron configuration as the ground level itself, and thus radiative transitions are forbidden by the parity selection rule. However, at wavelengths just slightly below the ultraviolet cutoff of the Earth's atmosphere, collisionally excited lines begin to appear that are not forbidden lines. Among others, many nebulae have strong permitted lines of Mg II 3s\ ^2S - 3p\ ^2P^\circ \lambda\lambda 2796, 2803, C IV 2s\ ^2S - 2p\ ^2P^\circ \lambda\lambda 1548, 1551, and Si IV 3s\ ^2S - 3p\ ^2P^\circ \lambda\lambda 1394, 1403.

In addition to the collisionally excited lines, the recombination lines of H I, He I, and He II are characteristic features of the spectra of gaseous nebulae. They are

emitted by atoms undergoing radiative transitions in cascading down to the ground level following recombinations to excited levels. In the remainder of this chapter, these recombination emission processes will be discussed in more detail. Finally, the continuum-emission processes, which are the bound-free and free-free analogues of the bound-bound transitions emitted in the recombination-line spectrum, will be examined.

4.2 Optical Recombination Lines

The recombination-line spectrum of H I is emitted by H atoms that have been formed by captures of electrons into excited levels and that are cascading by downward radiative transitions to the ground level. In the limit of very low density, the only processes that need be considered are captures and downward-radiative transitions. Thus the equation of statistical equilibrium for any level nL may be written

n_p n_e \alpha_{nL}(T) + \sum_{n' > n} \sum_{L'} n_{n'L'} A_{n'L', nL} = n_{nL} \sum_{n''=1}^{n-1} \sum_{L''} A_{nL, n''L''}. \quad (4.1)

Note in general A_{n'L', n''L''} \neq 0 only if L' = L'' \pm 1.

It is convenient to express the population in terms of the dimensionless factors b_{nL} that measure the deviation from thermodynamic equilibrium at the local T, n_e, and n_p. In thermodynamic equilibrium the Saha equation

\frac{n_p n_e}{n_{1S}} = \left( \frac{2\pi m k T}{h^2} \right)^{3/2} \exp(-h\nu_0/kT), \quad (4.2)

and the Boltzmann equation

\frac{n_{nL}}{n_{1S}} = (2L + 1) \exp(-\chi_n/kT), \quad (4.3)

apply. The factor 2L + 1 is the ratio of statistical weights of the nL and 1S levels. Then the population in the level nL in thermodynamic equilibrium may be written

n_{nL} = (2L + 1) \left( \frac{h^2}{2\pi m k T} \right)^{3/2} \exp(X_n/kT) n_p n_e, \quad (4.4)

where

X_n = h\nu_0 - \chi_n = \frac{h\nu_0}{n^2} \quad (4.5)

is the ionization potential of the level nL. Therefore, in general, the population may be written

n_{nL} = b_{nL}(2L + 1) \left( \frac{h^2}{2\pi mkT} \right)^{3/2} \exp(X_n/kT)n_p n_e, \quad (4.6)

and b_{nL} = 1 in thermodynamic equilibrium.

Substituting this expression in (4.1),

\begin{aligned} & \alpha_{nL} \frac{1}{(2L + 1)} \left( \frac{2\pi mkT}{h^2} \right)^{3/2} \exp(-X_n/kT) \\ & + \sum_{n' > n} \sum_{L''} b_{n'L'} A_{n'L', nL} \left( \frac{2L' + 1}{2L + 1} \right) \exp[(X_{n'} - X_n)/kT] \\ & = b_{nL} \sum_{n''=1}^{n-1} \sum_{L''} A_{nL, n''L''}, \end{aligned} \quad (4.7)

it can be seen that the b_{nL} factors are independent of density as long as recombination and downward-radiative transitions are the only relevant processes. Furthermore, it can be seen that the Equations (4.7) can be solved by a systematic procedure working downward in n, for if the b_{nL} are known for all n \geq n_K, then the n Equations (4.7), with L = 0, 1, \dots, n - 1 for n = n_K - 1, each contain a single unknown b_{nL} and can be solved immediately, and so on successively downward.

It is convenient to express the solutions in terms of the cascade matrix C(nL, n'L'), which is the probability that population of nL is followed by a transition to n'L' via all possible cascade routes. The cascade matrix can be generated directly from the probability matrix P(nL, n'L'), which gives the probability that population of the level nL is followed by a direct radiative transition to n'L',

P_{nL, n'L'} = \frac{A_{nL, n'L'}}{\sum_{n''=1}^{n-1} \sum_{L''} A_{nL, n''L''}} \quad (4.8)

which is zero unless L' = L \pm 1.

Hence, for n' = n - 1,

C_{nL, n-1L'} = P_{nL, n-1L'};

for n' = n - 2,

C_{nL, n-2L'} = P_{nL, n-2L'} + \sum_{L''=L' \pm 1} C_{nL, n-1L''} P_{n-1L'', n-2L'};

and for n' = n - 3,

C_{nL, n-3L'} = P_{nL, n-3L'} + \sum_{L''=L' \pm 1} [C_{nL, n-1L''} P_{n-1L'', n-3L'} + C_{nL, n-2L''} P_{n-2L'', n-3L'}]

so that if we define

C_{nL, nL''} = \delta_{LL''}, \quad (4.9)

then in general

C_{nL, n'L'} = \sum_{n'' > n'}^n \sum_{L''=L' \pm 1} C_{nL, n''L''} P_{n''L'', n'L'}. \quad (4.10)

The solutions of the equilibrium Equations (4.1) may be immediately written down, for the population of any level nL is fixed by the balance between recombinations to all levels n' \geq n that lead by cascades to nL and downward radiative transitions from nL:

n_p n_e \sum_{n'=n}^{\infty} \sum_{L'=0}^{n'-1} \alpha_{n'L'}(T) C_{n'L', nL} = n_{nL} \sum_{n''=1}^{n-1} \sum_{L''=L \pm 1} A_{nL, n''L''}. \quad (4.11)

It is convenient to express the results in this form because once the cascade matrix has been calculated, it can be used to find the b_{nL} factors or the populations n_{nL} at any temperature, or even for cases in which the population occurs by other non-radiative processes, such as collisional excitation from the ground level or from an excited level. To carry out the solutions, it can be seen from (4.11) that it is necessary to fit series in n, n', L, and L' to C_{nL, n'L'} and \alpha_{n'L'}(T), and extrapolate these series as n \rightarrow \infty. Once the populations n_{nL} have been found, it is simple to calculate the emission coefficient in each line:

j_{nn'} = \frac{h\nu_{nn'}}{4\pi} \sum_{L=0}^{n-1} \sum_{L'=L \pm 1} n_{nL} A_{nL, n'L'}. \quad (4.12)

The situation we have been considering is commonly called Case A in the theory of recombination-line radiation, and assumes that all line photons emitted in the nebula escape without absorption and therefore without causing further upward transitions. Case A is thus a good approximation for gaseous nebulae that are optically thin in all H I resonance lines, but in fact such nebulae can contain only a relatively small amount of gas and are mostly too faint to be easily observed.

Nebulae that contain observable amounts of gas generally have quite large optical depths in the Lyman resonance lines of H I. This can be seen from the equation for the central line-absorption cross section,

a_0(Ln) = \frac{3\lambda_{n1}^3}{8\pi} \left( \frac{m_H}{2\pi kT} \right)^{1/2} A_{nP,1S} [\text{cm}^2], \quad (4.13)

where \lambda_{n1} is the wavelength of the line. Thus, at a typical temperature T = 10,000 K, the optical depth in L\alpha is about 10^4 times the optical depth at the Lyman limit \nu = \nu_0 of the ionizing continuum, and an ionization-bounded nebula with \tau_0 \approx 1 therefore has \tau(L\alpha) \approx 10^4, \tau(L\beta) \approx 10^3, \tau(L\gamma) \approx 10^2, and \tau(L\delta) \approx 10. In each scattering there is a finite probability that the Lyman-line photon will be converted to a lower-series photon plus a lower member of the Lyman series. Thus, for instance, each time an L\beta photon is absorbed by an H atom, raising it to the 3^2P level, the probability that this photon is scattered is P_{31,10} = 0.882, while the probability that it is converted to H\alpha is P_{31,20} = 0.118, so after nine scatterings an average L\beta photon is converted to H\alpha (plus two photons in the 2^2S \rightarrow 1^2S continuum) and cannot escape from the nebula. Likewise, an average L\gamma photon is transformed, after a relatively few scatterings, either into a P\alpha photon plus an H\alpha photon plus an L\alpha photon, or into an H\beta photon plus two photons in the 2^2S-1^2S continuum. Thus, for these large optical depths, a better approximation than Case A is the opposite assumption that every Lyman-line photon is scattered many times and is converted (if n \geq 3) into lower-series photons plus either L\alpha or two-continuum photons. This large optical depth approximation is called Case B, and is more accurate than Case A for most nebulae. However, it is clear that the real situation is intermediate, and is similar to Case B for the lower Lyman lines, but progresses continuously to a situation nearer Case A as n \rightarrow \infty and \tau(Ln) \rightarrow 1.

Under Case B conditions, any photon emitted in an n^2P^o \rightarrow 1^2S transition is immediately absorbed nearby in the nebula, thus populating the n^2P^o level in another atom. Hence, in Case B, the downward radiative transitions to 1^2S are simply omitted from consideration, and the sums in the equilibrium Equations (4.1), (4.7), (4.8), and (4.11) are terminated at n'' = n_0 = 2 instead of at n_0 = 1 as in Case A. The detailed transition between Cases A and B will be discussed in Section 4.5.

Selected numerical results from the recombination spectrum of H I are listed in Tables 4.1 and 4.2 for Cases A and B, respectively. Note that, in addition to the emission coefficient j_{42} = j_{H\beta} and the relative intensities of the other lines, it is also sometimes convenient to use the effective recombination coefficient, defined by

n_P n_e \alpha_{nn'}^{eff} = \sum_{L=0}^{n-1} \sum_{L'=L\pm 1} n_{nL} A_{nL,n'L'} = \frac{4\pi j_{nn'}}{h\nu_{nn'}}. \quad (4.14)

For hydrogen-like ions of nuclear charge Z, all the transition probabilities A_{nL,n'L'} are proportional to Z^4, so the P_{nL,n'L'} and C_{nL,n'L'} matrices are independent of Z. The recombination coefficients \alpha_{nL} scale as

a_{nL}(Z, T) = Z \alpha_{nL}(1, T/Z^2);

the effective recombination coefficients scale in this same way, and since the energies h\nu_{nn'} scale as

\nu_{nn'}(Z) = Z^2 \nu_{nn'}(1),

the emission coefficient is

j_{nn'}(Z, T) = Z^3 j_{nn'}(1, T/Z^2). \quad (4.15)

Thus the calculations for H I at a temperature T can also be applied to He II at T' = 4T. In Table 4.3 some of the main features of the He II recombination line spectrum are listed for Case B, with the strongest line in the optical spectrum, \lambda 4686 (n = 4 \rightarrow 3), as the reference line. Note that the Fowler series (n \rightarrow 3) except for \lambda 4686 and \lambda 3203, and the entire "Balmer" series (n \rightarrow 2) are in the vacuum ultraviolet spectral region.

Next let us return to the H I recombination lines and examine the effects of collisional transitions at finite nebular densities. The largest collisional cross sections involving the excited levels of H are for transitions nL \rightarrow nL \pm 1 which have essentially zero energy difference. Collisions with both electrons and protons can cause these angular-momentum-changing transitions, but because of the small energy difference, protons are more effective than electrons; for instance, representative values of the mean cross sections for thermal protons at T \approx 10,000 K are \sigma(2^2S \rightarrow 2^2P^o) \approx 3 \times 10^{-10} \text{ cm}^2, \sigma(10^2L \rightarrow 10^2L \pm 1) \approx 4 \times 10^{-7} \text{ cm}^2, and \sigma(20^2L \rightarrow 20^2L \pm 1) \approx 6 \times 10^{-6} \text{ cm}^2. (Both of the latter are evaluated for L \approx n/2.) These collisional transitions must then be included in the equilibrium equations, which are modified from (4.1) to read

\begin{aligned} n_p n_e \alpha_{nL}(T) + \sum_{n' > n} \sum_{L' = L \pm 1}^{\infty} n_{n'L'} A_{n'L', nL} + \sum_{L' = L \pm 1} n_{n'L'} n_p q_{nL', nL} \\ = n_{nL} \left[ \sum_{n'' = n_0}^{n-1} \sum_{L'' = L \pm 1} A_{nL, n''L''} + \sum_{L'' = L \pm 1} n_p q_{nL, nL''} \right] \end{aligned} \quad (4.16)

where n_0 = 1 or 2 for Cases A and B, respectively, and

q_{nL, n'L'} \equiv q_{nL, n'L'}(T) = \int_0^{\infty} u \sigma(nL \rightarrow n'L') f(u) du \text{ [cm}^3 \text{ s}^{-1}] \quad (4.17)

is the collisional transition probability per proton per unit volume. For sufficiently large proton densities, the collisional terms dominate, and because of the principle of detailed balancing, they tend to set up a thermodynamic equilibrium distribution of the various L levels within each n; that is, they tend to make the populations proportional to the statistical weights,

\frac{n_{nL}}{n_{nL'}} = \frac{\omega_{nL}}{\omega_{nL'}} = \frac{(2L+1)}{(2L'+1)}

or

n_{nL} = \frac{(2L+1)}{n^2} n_n, \quad (4.18)

which is equivalent to b_{nL} = b_n, independent of L, where

n_n = \sum_{L=0}^{n-1} n_{nL}

is the total population in the levels with the same principal quantum number n. Since the cross sections \sigma_{nL \rightarrow nL \pm 1} increase with increasing n, but the transition probabilities A_{nL, n' L \pm 1} decrease, Equations (4.18) become increasingly good approximations with increasing n, and there is therefore (for any density and temperature) a level n_{cL} (for coupled angular momentum or "well L -mixed") above which they apply. For H at T \approx 10,000 K, this level is approximately n_{cL} \approx 15 at n_p \approx 10^4 \text{ cm}^{-3}, n_{cL} \approx 30 at n_p \approx 10^2 \text{ cm}^{-3} and n_{cL} \approx 45 at n_p \approx 1 \text{ cm}^{-3}.

Exactly the same type of effect occurs in the He II spectrum, because it also has the property that all the levels nL with the same n are degenerate. The He II lines are emitted in the \text{H}^+, \text{He}^{++} zone of a nebula, so both protons and \text{He}^{++} ions (thermal \alpha particles) can cause collisional, angular momentum-changing transitions in excited levels of \text{He}^+. The cross sections \sigma_{nL \rightarrow nL \pm 1} actually are larger for the \text{He}^{++} ions than for the \text{H}^+ ions, and both of them must be taken into account in the \text{He}^{++} region. The principal quantum numbers above which (4.18) applies for He II at T \approx 10,000 K are approximately n_{cL} \approx 22 for n_p \approx 10^4 \text{ cm}^{-3}, and n_{cL} \approx 32 for n_p \approx 10^2 \text{ cm}^{-3}.

After the angular-momentum-changing collisions at fixed n, the next largest collisional transition rates occur for collisions in which n changes by \pm 1, and of these the strongest are those for which L also changes by \pm 1. For this type of transition, collisions with electrons are more effective than collisions with protons, and representative cross sections for thermal electrons at T \approx 10,000 K are of order \sigma(nL \rightarrow n \pm \Delta n, L \pm 1) \approx 10^{-16} \text{ cm}^2. The effects of these collisions can be incorporated into the equilibrium equations by a straightforward generalization of (4.16). Indeed, since the cross sections for collisions \sigma(nL \rightarrow n \pm \Delta n, L \pm 1) decrease with increasing \Delta n (but not too rapidly), collisions with \Delta n = 1, 2, 3, \dots must all be included. The computational work required to set up and solve the equilibrium equations numerically becomes increasingly complicated and lengthy, but is straightforward in principle. It is clear that the collisions tend to couple levels with \Delta L = \pm 1 and small \Delta n, and that this coupling increases with increasing n_e (and n_p) and with increasing n. With collisions taken into account, the b_{nL} factors and the resulting emission coefficients are no longer independent of density.

Some calculated results for H I, including these collisional effects, are given in Table 4.4, which shows that the density dependence is rather small. Therefore this table, together with Table 4.2, which applies in the limit n_e \rightarrow 0, enables the H-line

emission coefficients to be evaluated over a wide range of densities and temperatures. Similarly, Table 4.5 shows calculated results for the He II recombination spectrum at finite densities and may be used in conjunction with Table 4.3, which applies in the same limit.

Exactly the same formalism can be applied to He I recombination lines, approximately treating the singlet and triplets as separate systems since all transition probabilities between them involving S , P , and D levels are quite small. The He I triplets always follow Case B, because downward radiative transitions to 1 ¹ S essentially do not occur. For the singlets, Case B is ordinarily a better approximation than Case A for observed nebulae, though the optical depths are lower for all lines than for the corresponding lines of H by a factor of approximately the abundance ratio. An extra complication is that He I 1 ¹ S – n^1P^o line photons can photoionize H ⁰ , and thus may be destroyed before they are converted into lower-energy photons. Calculated (Case B) results for the strongest He I lines are summarized in Table 4.6, with \lambda 4471 (2 ³ P ^o –4 ³ D ) as the reference line. Note that only H itself and the ions of its isoelectronic sequence have energy levels with the same n but different L degenerate, so for He I, Table 4.6 lists the j(n^{(2S+1)}L, n'^{(2S+1)}L), rather than j(n, n'), as for H. The radiative-transfer effects on the He I triplets, discussed in Section 4.6, and the collisional-excitation effects, discussed in Section 4.8, are not included in this table.

4.3 Optical Continuum Radiation

In addition to the line radiation emitted in the bound–bound transitions previously described, recombination processes also lead to the emission of rather weak continuum radiation in free–bound and free–free transitions. Because hydrogen is the most abundant element, the H I continuum, emitted in the recombination of protons with electrons, is the strongest, and the He II continuum may also be significant if He is mostly doubly ionized, but the He I continuum is always weaker. In the ordinary optical region the free–bound continua are stronger, but in the infrared and radio regions the free–free continuum dominates. In addition, there is a continuum resulting from the two-photon decay of the 2 ² S level of H, which is populated by recombinations and subsequent downward cascading. In this section we will examine each of these sources of continuous radiation.

The H I free–bound continuum radiation at frequency \nu results from recombinations of free electrons with velocity u to levels with principal quantum number n \geq n_1, and ionization potential X_n, where

h\nu = \frac{1}{2}u^2 + X_n \tag{4.19}

and

h\nu \geq X_{n_1} = \frac{h\nu_0}{n_1^2}, \tag{4.20}

so its emission coefficient per unit frequency interval per unit solid angle per unit time per unit volume is therefore

j_\nu = \frac{1}{4\pi} n_p n_e \sum_{n=n_1}^{\infty} \sum_{L=0}^{n-1} u \sigma_{nL}(\text{H}^0, u) f(u) h\nu \frac{du}{dv}. \quad (4.21)

The recombination cross sections \sigma_{nL}(\text{H}^0, u) can be calculated from the photoionization cross sections a_\nu(\text{H}^0, nL) by the Milne relation, as shown in Appendix 2.

The free-free (or bremsstrahlung) continuum emitted by free electrons accelerated in Coulomb collisions with positive ions (which are mostly \text{H}^+, \text{He}^+, or \text{He}^{++} in nebulae) of charge Z has an emission coefficient

j_\nu = \frac{1}{4\pi} n_+ n_e \frac{32Z^2 e^4 h}{3m^2 c^3} \left( \frac{\pi h \nu_0}{3kT} \right)^{1/2} \exp(-h\nu/kT) g_{ff}(T, Z, \nu), \quad (4.22)

where g_{ff}(T, Z, \nu) is a Gaunt factor. Thus the emission coefficient for the H I recombination continuum, including both bound-free and free-free contributions, may be written

j_\nu(\text{H I}) = \frac{1}{4\pi} n_p n_e \gamma_\nu(\text{H}^0, T). \quad (4.23)

Rather than tabulating and plotting this quantity itself, it is more convenient, especially for visualization, to use the quantity \nu j_\nu(\text{H I}), which is proportional to the emission coefficient per unit logarithmic frequency interval d \ln \nu = d\nu/\nu. Numerical values for \nu\gamma_\nu, as calculated from Equations (4.21) and (4.22), are given in Table 4.7. Likewise, the contributions to the continuum-emission coefficient from He I and He II may be written

j_\nu(\text{He I}) = \frac{1}{4\pi} n(\text{He}^+) n_e \gamma_\nu(\text{He}^0, T), j_\nu(\text{He II}) = \frac{1}{4\pi} n(\text{He}^{++}) n_e \gamma_\nu(\text{He}^{++}, T), \quad (4.24)

and numerical values of \nu\gamma_\nu are listed in Tables 4.8 and 4.9. The calculation for He II is exactly analogous to that for H I, while for He I the only complication is that there is no L degeneracy and Equations (4.19), (4.20), and (4.21) must be appropriately

generalized. Figure 4.1 shows these calculated values \nu\gamma_\nu, and also shows the large discontinuities at the ionization potentials of the various excited levels. Note that for a typical He abundance of approximately 10 percent of that of H, if the He is mostly doubly ionized, then the He II contribution to the continuum is roughly comparable to that of H I, but if the He is mostly singly ionized, the He I contribution to the continuum is only about 10 percent of the H I contribution.

An additional important source of continuum emission in nebulae is the two-photon decay of the 2^2S level of H I, which is populated by direct recombinations and by cascades following recombinations to higher levels. The transition probability for this two-photon decay is A_{2^2S,1^2S} = 8.23 \text{ s}^{-1}, and the sum of the energies of the two photons is h\nu' + h\nu'' = h\nu_{12} = h\nu(\text{L}\alpha) = (3/4)h\nu_0. The probability distribution

of the emitted photons is therefore symmetric around the frequency (1/2)\nu_{12} = 1.23 \times 10^{15} \text{ s}^{-1}, corresponding to \lambda = 2431 \text{ \AA}. The emission coefficient in this two-photon continuum may be written

j_\nu(2q) = \frac{1}{4\pi} n_{2^2S} A_{2^2S, 1^2S} 2hy P(y), \quad (4.25)

where P(y)dy is the normalized probability per decay that one photon is emitted in the range of frequencies y\nu_{12} to (y + dy)\nu_{12}.

To express this two-photon continuum-emission coefficient in terms of the proton and electron density, it is necessary to calculate the equilibrium population of n(2^2S) in terms of these quantities. In sufficiently low-density nebulae, two-photon decay is the only mechanism that depopulates 2^2S, and the equilibrium is given by

n_p n_e \alpha_{2^2S}^{eff}(\text{H}^0, T) = n_{2^2S} A_{2^2S, 1^2S} \quad (4.26)

where \alpha_{2^2S}^{eff} is the effective recombination coefficient for populating 2^2S by direct recombinations and by recombinations to higher levels followed by cascades to 2^2S. However, at finite densities, angular-momentum-changing collisions of protons and electrons with H atoms in the 2^2S level shift the atoms to 2^2P^o and thus remove them

from 2^2S. The protons are more effective than electrons, whose effects, however, are not completely negligible, as can be seen from the values of the collisional transition rates per 2^2S atom, in Table 4.10. With these collisional processes taken into account, the equilibrium population in 2^2S is given by

n_p n_e \alpha_{2^2S}^{\text{eff}}(\text{H}^0, T) = n_{2^2S} \left( A_{2^2S, 1^2S} + n_p q_{2^2S, 2^2P^o}^p + n_e q_{2^2S, 2^2P^o}^e \right). \quad (4.27)

From Table 4.10, it can be seen that collisional deexcitation of 2^2S via 2^2P^o is more important than two-photon decay for n_p \geq 10^4 \text{ cm}^{-3}; so at densities approaching this value, Equation (4.27) must be used instead of Equation (4.26). Thus combining Equations (4.25) and (4.27), we can write the emission coefficient as

j_\nu(2q) = \frac{1}{4\pi} n_p n_e \gamma_\nu(2q) \quad (4.28)

where

\gamma_\nu(2q) = \frac{\alpha_{2^2S}^{eff}(\text{H}^0, T)g_\nu}{1 + \left[ \frac{n_p q_2^p \text{ } ^2S, 2^2P + n_e q_2^e \text{ } ^2S, 2^2P}{A_{2^2S, 1^2S}} \right]} \quad (4.29)

The quantity \alpha_{2^2S}^{eff} is tabulated in Table 4.11 and \nu g_\nu is tabulated in Table 4.12. The two-photon continuum is also plotted in Figure 4.1 for T = 10,000 \text{ K} and in the low-density limit n_p \approx n_e \ll 10^4 \text{ cm}^{-3}. It can be seen that this continuum is quite significant in comparison with the H I continua, particularly just above the Balmer limit at \lambda 3646 \text{ \AA}. Note that although the two-photon continuum is symmetric about \nu_{12}/2 if expressed in photons per unit frequency interval, it is not symmetric about \lambda 2431 if expressed per

unit wavelength interval, nor is either symmetric if expressed in energy units rather than photons. From Equation (4.25), if \nu > \nu_{12}/2,

g_\nu = \frac{\nu}{\nu'} g_{\nu'},

where \nu' = \nu_{12} - \nu.

4.4 Radio-Frequency Continuum and Line Radiation

The line and continuous spectra described in Sections 4.2 and 4.3 extend to arbitrarily low frequency, and in fact give rise to observable features in the radio-frequency spectrum region. Though this “thermal” radio-frequency radiation is a natural extension of the optical line and continuous spectra, it is somewhat different, in detail, because in the radio-frequency region h\nu \ll kT, and stimulated emission, which is proportional to \exp(-h\nu/kT), is therefore much more important in that region than in the ordinary optical region. We will examine the continuous spectrum first, and then the recombination-line spectrum.

In the radio-frequency region, the continuum is due to free-free emission, and the emission coefficient is given by the same Equation (4.22) that applies in the optical region. However, in the radio-frequency region, the Gaunt factor g_{ff}(T, Z, \nu) \neq 1, as in the optical region, but rather

g_{ff}(T, Z, \nu) = \frac{\sqrt{3}}{\pi} \left[ \ln \left( \frac{8k^3 T^3}{\pi^2 Z^2 e^4 m \nu^2} \right)^{1/2} - \frac{5\gamma}{2} \right], \quad (4.30)

where \gamma = 0.577 is Euler's constant. Numerically, this is approximately

g_{ff}(T, Z, \nu) = \frac{\sqrt{3}}{\pi} \left( \ln \frac{T^{3/2}}{Z\nu} + 17.7 \right),

with T in K and \nu in Hz, and thus at T \approx 10,000 K, \nu \approx 10^3 MHz, g_{ff} \approx 10.

The free-free effective absorption coefficient can then be found from Kirchhoff's law, and is

\kappa_\nu = n_+ n_e \frac{16\pi^2 Z^2 e^6}{(6\pi m k T)^{3/2} \nu^2 c} g_{ff} \quad (4.31)

per unit length. Note that this effective absorption coefficient is the difference between the true absorption coefficient and the stimulated emission coefficient, since the stimulated emission of a photon is exactly equivalent to a negative absorption process; in the radio-frequency region (h\nu \ll kT) the stimulated emissions very nearly balance the true absorptions and the correction for stimulated emission [1 - \exp(h\nu/kT)] \approx h\nu/kT \ll 1.

Substituting numerical values and fitting powers to the weak temperature and frequency dependence of g_{ff},

\begin{aligned} \tau_\nu &= \int \kappa_\nu ds \\ &= 8.24 \times 10^{-2} T^{-1.35} \nu^{-2.1} \int n_+ n_e ds \\ &= 8.24 \times 10^{-2} T^{-1.35} \nu^{-2.1} E_c. \end{aligned} \quad (4.32)

In this formula T is measured in K, \nu in GHz, and E_c, the so-called continuum emission measure, in \text{cm}^{-6} pc. It can be seen from Equations (4.31) or (4.32) that at sufficiently low frequency all nebulae become optically thick; for example, an H II region with n_e \approx n_p \approx 10^2 \text{ cm}^{-3} and a diameter 10 pc has \tau_\nu \approx 1 at \nu \approx 200 MHz, and a planetary nebula with n_e \approx 3 \times 10^3 \text{ cm}^{-3} and a diameter of 0.1 pc has \tau_\nu \approx 1 at \nu \approx 600 MHz. Thus, in fact, many nebulae are optically thick at observable low frequencies and optically thin at observable high frequencies. The equation of radiative transfer,

\frac{dI_\nu}{ds} = -\kappa_\nu I_\nu + j_\nu \quad (4.33)

or

\frac{dI_\nu}{ds} = -I_\nu + \frac{j_\nu}{\kappa_\nu} = -I_\nu + B_\nu(T), \quad (4.34)

has the solution for no incident radiation

I_\nu = \int_0^{\tau_\nu} B_\nu(T) \exp(-\tau_\nu) d\tau_\nu. \quad (4.35)

In the radio-frequency region,

B_\nu(T) = \frac{2h\nu^3}{c^2} \frac{1}{\exp(h\nu/kT) - 1} \approx \frac{2\nu^2 kT}{c^2} \quad (4.36)

is proportional to T, so it is conventional in radio astronomy to measure intensity in terms of brightness temperature, defined by T_{bv} = c^2 I_\nu / 2\nu^2 k. Hence (4.35) can be rewritten

T_{bv} = \int_0^{\tau} T \exp(-\tau_\nu) d\tau_\nu, \quad (4.37)

and for an isothermal nebula, this becomes

T_{bv} = T[1 - \exp(-\tau_\nu)] \begin{cases} \rightarrow T\tau_\nu \text{ as } \tau_\nu \rightarrow 0 \\ \rightarrow T \text{ as } \tau_\nu \rightarrow \infty \end{cases}.

Thus the radio-frequency continuum has a spectrum in which T_{bv} varies approximately as \nu^{-2} at high frequency and is independent of \nu at low frequency.

The H I recombination lines of very high n also fall in the radio-frequency spectral region and have been observed in many gaseous nebulae. Some specific examples of observed lines are H 109 \alpha (the transition with \Delta n = 1 from n = 110 to n = 109) at \nu = 5008.89 MHz, \lambda = 5.99 cm, H 137 \beta (the transition with \Delta n = 2 from n = 139 to n = 137) at \nu = 5005.0 MHz, \lambda = 6.00 cm, and so on. The emission coefficients in these radio recombination lines may be calculated from equations similar to those described in Section 4.2 for the shorter wavelength optical recombination lines. For all lines observed in the radio-frequency region, n > n_{cL} defined there, so that at a fixed n, n_{nL} \propto (2L + 1), and only the populations n_n need be considered. One additional process, in addition to those described in Section 4.2, must also be taken into account, namely collisional ionization of levels with large n and its inverse process, three-body recombination,

\mathrm{H}^{0}(n)+e\rightleftharpoons\mathrm{H}^{+}+e+e

The rate of collisional ionization per unit volume per unit time from level n may be written

n_n n_e \overline{\sigma_{ionization}(n)} = n_n n_e q_{n,i}(T), \quad (4.38)

and the rate of three-body recombination per unit time per unit volume may be written n_p n_e^2 \phi_n(T); so, from the principle of detailed balancing,

\phi_n(T) = n^2 \left( \frac{h^2}{2\pi m k T} \right)^{3/2} \exp(X_n/kT) q_{n,i}(T). \quad (4.39)

Thus the equilibrium equation that is analogous to Equation (4.16) becomes, at high n,

\begin{aligned} n_p n_e [\alpha_n(T) + n_e \phi_n(T)] + \sum_{n' > n}^{\infty} n_{n'} A_{n',n} + \sum_{n'=n_0}^{\infty} n_{n'} n_e q_{n',n} \\ = n_n \left[ \sum_{n'=n_0}^{n-1} A_{n,n'} + \sum_{n'=n_0}^{\infty} n_e q_{n,n'}(T) + n_e q_{n,i}(T) \right], \end{aligned} \quad (4.40)

where

A_{n,n'} = \frac{1}{n^2} \sum_{L,L'} (2L+1) A_{nL,n'L'} \quad (4.41)

is the mean transition probability averaged over all the L levels of the upper principal quantum number. These equations can be expressed in terms of b_n instead of n_n, and the solutions can be found numerically by standard matrix-inversion techniques. Note that since the coefficients b_n have been defined with respect to thermodynamic equilibrium at the local T, n_e, and n_p, the coefficient b for the free electrons is identically unity, and therefore b_n \rightarrow 1 as n \rightarrow \infty. Some calculated values of b_n for T = 10,000 K and various n_e are plotted in Figure 4.2, which shows that the increasing importance of collisional transitions as n_e increases makes b_n \approx 1 at lower and lower n.

To calculate the emission in a specific recombination line, it is again necessary to solve the equation of transfer, taking account of the effects of stimulated emission. In this case, for an n_i \Delta n line between the upper level m = n + \Delta n and the lower level n, if k_{\nu l} is the true line-absorption coefficient, then the line-absorption coefficient, corrected for stimulated emission, to be used in the equation of transfer is

k_{\nu l} = k_{\nu l} \left( 1 - \frac{b_m}{b_n} \exp(-h\nu/kT) \right), \quad (4.42)

since it is the net difference between the rates of upward absorption processes and of downward-induced emissions. If we expand (4.42) in a power series, it becomes

k_{\nu L} = k_{\nu l} \left( \frac{b_m}{b_n} \frac{h\nu}{kT} - \frac{d \ln(b_n)}{dn} \Delta n \right). \quad (4.43)

Since b_m/b_n \approx 1 and h\nu \ll kT, the line-absorption coefficient can become negative, implying positive maser action, if (d \ln b_n)/dn is sufficiently large. Calculated values of this derivative are therefore also shown in Figure 4.2. Since, for typical observed lines h\nu/kT \approx 10^{-5}, it can be seen from this figure that the maser effect is in fact often quite important. We will again use these concepts and expressions to calculate the strengths of the radio-frequency recombination lines in Chapter 5.

4.5 Radiative Transfer Effects in H I

For most of the emission lines observed in nebulae there is no radiative-transfer problem; in most lines the nebulae are optically thin, and any line photon emitted simply escapes. However, in some lines, especially the resonance lines of abundant atoms, the optical depths are appreciable, and scattering and absorption must be taken into account in calculating the expected line strengths. Two extreme assumptions, Case A, a nebula with vanishing optical thickness in all the H I Lyman lines, and Case B, a nebula with large optical depths in all the Lyman lines, have already been discussed in

Section 4.2; and although these two cases do not require a detailed radiative-transfer solution, in the intermediate cases a more sophisticated treatment is necessary. Other radiative-transfer problems arise in connection with the He I triplets, the conversion of He II L\alpha and H I L\beta into observable O III or O I line radiation, respectively, by the Bowen resonance-fluorescence processes, and fluorescence excitation of other lines by stellar continuum radiation. In this section some general concepts about the escape of line photons from nebulae will be discussed in the context of the H I Lyman and Balmer lines, and then in succeeding sections these same concepts will be applied to the other problems mentioned.

In a static nebula the only line-broadening mechanisms are thermal Doppler broadening and radiative damping, and in the cores of the lines, where radiative damping can be neglected, the line-absorption coefficient has the Doppler form

\kappa_{\nu l} = k_{0l} \exp \left[ - (\Delta\nu / \Delta\nu_D)^2 \right] = k_{0l} \exp(-x^2) \quad [\text{cm}^2], \quad (4.44)

where

k_{0l} = \frac{\lambda^2}{8\pi^{3/2}} \frac{\omega_j}{\omega_i} \frac{A_{j,i}}{\Delta\nu_D} = \frac{\sqrt{\pi} e^2 f_{ij}}{mc \Delta\nu_D} \quad [\text{cm}^2] \quad (4.45)

is the line-absorption cross section per atom at the center of the line,

\Delta\nu_D = \sqrt{\frac{2kT}{m_H c^2}} \nu_0 \quad [\text{Hz}]

is the thermal Doppler width (Hz), and f_{ij} is the absorption oscillator strength between the lower and upper level i, j. The full-width at half-maximum (FWHM) of the line is 2\sqrt{\ln 2} times larger than \Delta\nu_D. Small-scale micro-turbulence can be taken into account as a further source of broadening of the line-absorption coefficient by adding the thermal and turbulent velocity terms in quadrature, \Delta\nu_D^2 = \Delta\nu_{\text{thermal}}^2 + \Delta\nu_{\text{turbulent}}^2. Larger scale turbulence and expansion of the nebula can be treated by considering the frequency shift between the emitting and absorbing volumes.

In a static nebula, a photon emitted at a particular point in a particular direction and with a normalized frequency x from the center of the line has a probability \exp(-\tau_x) of escaping from the nebula without further scattering and absorption. Here \tau_x is the optical depth from the point to the edge of the nebula in this direction and at this frequency. Averaging over all directions gives the mean escape probability from this point and at this frequency, and further averaging over the frequency profile of the emission coefficient gives the mean escape probability from the point.

For all the forbidden lines and for most of the other lines, the optical depths are so small in every direction, even at the center of the line, that the mean escape probabilities from all points are essentially unity. However, for lines of larger optical depth we must examine the probability of escape quantitatively.

Consider an idealized spherical homogeneous nebula, with optical radius in the center of line \tau_{0l}. So long as \tau_{0l} < 10^4, only the Doppler core of the line-absorption cross section need be considered. The photons are emitted with the same Doppler profile, and the mean probability of escape must therefore be averaged over this profile. If, at a particular normalized frequency x, the optical radius of the nebula is \tau_x, the mean probability of escape averaged over all directions and volumes is

p(\tau_x) = \frac{3}{4\tau_x} \left[ 1 - \frac{1}{2\tau_x^2} + \left( \frac{1}{\tau_x} + \frac{1}{2\tau_x^2} \right) \exp(-2\tau_x) \right]. \quad (4.46)

When we average over the Doppler profile, the mean escape probability for a photon emitted in the line is

\varepsilon(\tau_{0l}) = \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} p(\tau_x) \exp(-x^2) dx \quad (4.47)

where \tau_{0l} is the optical radius in the center of the line. This integral must be evaluated numerically, but for optical radii (\tau_{0l} \leq 50) that are not too large, the results can be fitted fairly accurately with \varepsilon(\tau_{0l}) = 1.72/(\tau_{0l} + 1.72).

If we consider a Lyman line L_n, photons emitted in this line that do not escape from the nebula are absorbed by another hydrogen atom, and each absorption process represents an excitation of the n^2 P^o level of H I. This excited level very quickly undergoes a radiative decay, and the result is either resonance scattering or resonance fluorescence excitation of another H I line. If the photon emitted from the n^2 P level decays in a 1^2 S - n^2 P^o transition, the process is resonance scattering of an L_n photon. If it is emitted in the 2^2 S - n^2 P^o transition, the process is conversion of L_n into H_n plus excitation of 2^2 S, leading to emission of two photons in the continuum. If it is emitted in the 3^2 S - n^2 P^o transition, the process is conversion of L_n into P_n plus excitation of 3^2 S, leading to emission of H\alpha plus L\alpha, and so on. The probabilities of each of these processes may be found directly from the probability matrices C_{nL,n'L'} and P_{nL,n'L'}, defined in Section 4.2. If we define P_n(Lm) and P_n(Hm) as the probabilities that absorption of an L_n photon results in emission of an Lm photon and of an Hm photon, respectively, then

P_n(Lm) = C_{n1,m1} P_{m1,10} \quad (4.48)

and

P_n(Hm) = C_{n1,m0} P_{m0,21} + C_{n1,m1} P_{m1,20} + C_{n1,m2} P_{m2,21}. \quad (4.49)

We can now use these probabilities to calculate the emergent Lyman-line spectrum emitted from a model nebula. It is easiest to work in terms of numbers of photons emitted. If we write R_n for the total number of L_n photons generated in the nebula per unit time by recombination and subsequent cascading, and A_n as the total number of L_n photons absorbed in the nebula per unit time, then J_n, the total number of

L_n photons emitted in the nebula per unit time, is the sum of the contributions from recombination and from resonance fluorescence plus scattering:

J_n = R_n + \sum_{m=n}^{\infty} A_m P_m(L_n). \quad (4.50)

Since each L_n photon emitted has a probability \varepsilon_n of escaping, the total number of L_n photons escaping the nebula per unit time is

E_n = \varepsilon_n J_n = \varepsilon_n \left[ R_n + \sum_{m=n}^{\infty} A_m P_m(L_n) \right]. \quad (4.51)

Finally, in a steady state the number of L_n photons emitted per unit time is equal to the sum of the numbers absorbed and escaping per unit time,

J_n = A_n + E_n = A_n + \varepsilon_n J_n. \quad (4.52)

Thus, eliminating J_n between (4.51) and (4.52),

A_n = (1 - \varepsilon_n) \left[ R_n + \sum_{m=n}^{\infty} A_m P_m(L_n) \right], \quad (4.53)

and since the R_n and P_m(L_n) are known from the recombination theory and the \varepsilon_n are known from the radiative-transfer theory, Equation (4.53) can be solved for the A_n by a systematic procedure, working downward from the highest n at which \varepsilon_n differs appreciably from unity. Then from these values of R_n, the E_n may be calculated from (4.51), giving the emergent Lyman-line spectrum.

Next we will investigate the Balmer-line spectrum, which requires further analysis. Let us write S_n for the number of H_n photons generated in the nebula per unit time by recombination and subsequent cascading. Suppose that there is no absorption of these Balmer-line photons, so that K_n, the total number of H_n photons emitted in the nebula per unit time, is the sum of contributions from recombination and from resonance fluorescence due to Lyman-line photons,

K_n = S_n + \sum_{m=n}^{\infty} A_m P_m(H_n).

Then, since the S_n and P_m(H_n) are known from the recombination theory and the A_m are known from the previous Lyman-line solution, the K_n can be calculated immediately to obtain the emergent Balmer-line spectrum. Note that R_n, S_n, J_n, K_n, and A_n are proportional to the total number of photons; the equations are linear in these quantities; and the entire calculation can therefore be normalized to any S_n, for instance, to S_4, the number of H\beta photons that would be emitted if there were no absorption effects. The results, in the form of calculated ratios of H\alpha/H\beta and H\beta/H\gamma intensities, are shown in Figure 4.3 as a function of \tau_{0l}(L\alpha), the optical radius of

the nebula at the center of L\alpha; and the transition from Case A (\tau_{0l} \rightarrow 0) to Case B (\tau_{0l} \rightarrow \infty) can be seen clearly.

Although in most nebulae, the optical depths in the Balmer lines are small, there could be situations in which the density n(H^0, 2^2S) is sufficiently high that some self-absorption does occur in these lines. The optical depths in the Balmer lines can again be calculated from Equation (4.45), and since they are proportional to n(H^0, 2^2S), the radiative-transfer problem is now a function of two variables, \tau_{0l}(L\alpha), giving the optical radius in the Lyman lines, and another, say, \tau_{0l}(H\alpha), giving the optical radius in the Balmer lines. Although the equations are much more complicated, since now Balmer-line photons may be scattered or converted into Lyman-line photons and vice versa, there is no new effect in principle, and the same general type of formulation developed previously for the Lyman-line absorption can still be used. We will not examine the details here, but will simply discuss physically the calculated results shown in Figure 4.3. For \tau_{0l}(H\alpha) = 0, the first effect of increasing \tau_{0l}(L\alpha) is that L\beta is converted into H\alpha plus the two-photon continuum. This increases the H\alpha/H\beta ratio of the escaping photons, corresponding to a move of the representative point to the right in Figure 4.3. However, for slightly larger \tau_{0l}(L\alpha), L\gamma photons are also converted into P\alpha, H\alpha, H\beta, L\alpha, and two-photon continuum photons, and since the main effect is to increase the strength of H\beta, this corresponds to a move downward and to the left in Figure 4.3. For still larger \tau_{0l}(L\alpha), as still higher Ln photons are

converted, H\gamma is also strengthened, and the calculation, which takes into account all of these effects, shows that the representative point describes the small loop of Figure 4.3 as the conditions change from Case A to Case B. For large \tau_{0l}(L\alpha), the effect of increasing \tau_{0l}(H\alpha) is that, although H\alpha is merely scattered (because any L\beta photons it forms are quickly absorbed and converted back to H\alpha), H\beta is absorbed and converted to H\alpha plus P\alpha. This increases H\alpha/H\beta and decreases H\beta/H\gamma, as shown quantitatively in Figure 4.3.

4.6 Radiative Transfer Effects in He I

The recombination radiation of He I singlets is very similar to that of H I, and Case B is a good approximation for the He I Lyman lines. However, the recombination radiation of the He I triplets is modified by the fact that the \text{He}^0\ 2^3S term is considerably more metastable than \text{H}^0\ 2^2S, and as a result self-absorption effects are quite important (as is collisional excitation from 2^3S, to be discussed later). As the energy-level diagram of Figure 4.4 shows, 2^3S is the lowest triplet term in He, and recaptures to triplets tend to cascade down to it. Depopulation occurs only by photoionization, especially by H I L\alpha, by collisional transitions to 2^1S and 2^1P^o, or by the strongly forbidden 2^3S-1^1S radiative transition, as discussed in Section 2.4. As a result n(2^3S) is large, which in turn makes the optical depths in the lower 2^3S-n^3P^o lines significant. Figure 4.4 shows that \lambda 10830\ 2^3S-2^3P^o photons are simply scattered, but that absorption of \lambda 3889\ 2^3S-3^3P^o photons can lead to their conversion to \lambda 4.3\ \mu\text{m}\ 3^3S-3^3P^o, plus 2^3P-3^3S\lambda 7065, plus 2^3S-2^3P^o\lambda 10830. The probability of this conversion is

\frac{A_{3^3S, 3^3P^o}}{A_{3^3S, 3^3P^o} + A_{2^3S, 3^3P^o}} \approx 0.10

per absorption. At larger \tau_{0l}(\lambda 10830), still higher members of the 2^3S-n^3P^o series are converted into longer wavelength photons.

The radiative-transfer problem is very similar to that for the Lyman lines discussed in Section 4.5, and may be handled by the same kind of formalism. Calculated ratios of the intensities of \lambda 3889 (which is weakened by self-absorption) and of \lambda 7065 (which is strengthened by resonance fluorescence) relative to the intensity of \lambda 4471\ 2^3P^o-4^3D (which is only slightly affected by absorption) are shown for spherically symmetric homogeneous model nebulae in Figure 4.5.

The thermal Doppler widths of He I lines are smaller than those of H I lines, because of the larger mass of He, and therefore whatever turbulent or expansion velocity there may be in a nebula is relatively more important in broadening the He I lines. The simplest example to consider is a model spherical nebula expanding with a linear velocity of expansion,

U_{\text{exp}}(r) = \omega r; \quad 0 \leq r \leq R; \quad (4.54)

for then, between any two points r_1 and r_2 in the nebula, the relative radial velocity is

u(r_1, r_2) = \omega s, \quad (4.55)

where s is the distance between the points and \omega is the constant velocity gradient. Thus photons emitted at r_1 will have a line profile centered about the line frequency \nu_L in the reference system in which r_1 is at rest. However, they will encounter at r_2 material absorbing with a profile centered on the frequency

\nu'(r_1, r_2) = \nu_L \left( 1 + \frac{\omega s}{c} \right), \quad (4.56)

and the optical depth in a particular direction to the boundary of the nebula for a photon emitted at r_1 with frequency \nu may be written

\tau_\nu(r_1) = \int_0^{r_2=R} n(2^3S) k_{0l} \exp \left\{ - \left[ \frac{\nu - \nu'(r_1, r_2)}{\Delta \nu_D} \right]^2 \right\} ds. \quad (4.57)

It can be seen that increasing velocity of expansion tends, for a fixed density n(2^3S), to decrease the optical depth to the boundary of the nebula and thus to decrease the self-absorption effects. This effect can be seen in Figure 4.5, where some calculated results are shown for various ratios of the expansion velocity u_{exp}(R) = \omega R to the thermal velocity u_{th} = (2kT/m_{He})^{1/2}, as functions of \tau_{0l}(\lambda 3889) = n(2^3S)\kappa_{0l}(\lambda 3889)R, the optical radius at the center of the line for zero expansion velocity. Note that the calculated intensity ratios for large u_{exp}/u_{th} and large \tau_0 are quite similar to those for smaller u_{exp}/u_{th} and smaller \tau_0.

4.7 The Bowen Resonance-Fluorescence Mechanisms for O III and O I

There is an accidental coincidence between the wavelength of the He II L\alpha line at \lambda 303.78 and the O III 2p^2\ ^3P_2-3d\ ^3P_2^o line at \lambda 303.80. As we have seen, in the He^{++} zone of a nebula there is some residual He^+, so the He II L\alpha photons emitted by

recombination are scattered many times before they escape. As a result, there is a high density of He II \lambda\alpha photons in the He ⁺⁺ zone, and since O ⁺⁺ is also present in this zone, some of the He II \lambda\alpha photons are absorbed by it and excite the 3d\ ^3P_2^o level of O III. This level then quickly decays by a radiative transition, most frequently (relative probability 0.74) by resonance scattering in the 2p^2\ ^3P_2-3d\ ^3P_2^o line—that is, by emitting a photon. The next most likely decay process (probability 0.24) is emission of \lambda 303.62\ 2p^2\ ^3P_1-3d\ ^3P_2^o, which may then escape or may be reabsorbed by another O ⁺⁺ ion, again populating 3d\ ^3P_2^o. Finally (probability 0.02), the 3d\ ^3P_2^o level may decay by emitting one of the six longer wavelength photons 3p^3L_J-3d\ ^3P_2^o indicated in Figure 4.6. These levels 3p\ ^3L_J then decay to 3s and ultimately back to 2p^2\ ^3P, as shown in the figure (or to 2p^3 and then back to 2p^2\ ^3P, emitting two far-ultraviolet line photons, as shown in the figure). This is the Bowen resonance-fluorescence mechanism, the conversion of He II \lambda\alpha to those lines that arise from 3d\ ^3P_2^o or from the levels excited by its decay. These lines are observed in many planetary nebulae, and their interpretation, requiring the solution of the problem of the scattering, escape, and destruction of He II \lambda\alpha with the complications introduced by the O ⁺⁺ scattering and resonance fluorescence, has been worked out years ago. Some of the most important references to it are given at the end of the chapter for those who wish to study it further.

A second accidental near-coincidence occurs between the wavelength of the H I \lambda\beta line at \lambda 1025.72\ \text{\AA} and the O I 2p^4\ ^3P_2-2p^33d\ ^3D_3^o line at \lambda 1025.76\ \text{\AA}, which excites the 2p^33d\ ^3D_3^o level. Some atomic oxygen exists in the H ⁺ zone, due to rapid charge transfer between oxygen and hydrogen, so the situation is analogous to that for O III and He II \lambda\alpha. Excitations of 2p^33d\ ^3D_3^o by this process are followed by successive decays producing 2p^33p\ ^3P_2-2p\ ^33d\ ^3D_3^o\ \lambda 11286.9\ \text{\AA}, 2p^33s\ ^3S_1^o-2p\ ^33p\ ^3P_2\ \lambda 8446.36\ \text{\AA}, and the three lines of the multiplet 2p^4\ ^3P_{2,1,0}-2p^33s\ ^3S_1^o\ \lambda\lambda 1302.17, 1304.86, 1306.03\ \text{\AA}. In the simplest case each excitation produces a cascade through the first two of these lines, followed by one of the last three, in the predicted ratio of their transition probabilities, 3.4:2.0:0.7. The relative intensities of the first two lines and the sum of the intensities of the multiplet are thus all equal in photon units, so their predicted relative intensities in energy units are inversely proportional to their wavelengths.

4.8 Collisional Excitation in He I

Collisional excitation of H is negligible in comparison with recombination in populating the excited levels in planetary nebulae and H II regions, because the threshold for even the lowest level, n = 2 at 10.2 eV, is large in comparison with the thermal energies at typical nebular temperatures. This can be confirmed quantitatively using collision strengths listed in Table 3.16. However, in He ⁰ the 2\ ^3S level is highly metastable, and collisional excitation from it can be important, particularly in exciting 2\ ^3P^o and thus leading to emission of He I \lambda 10830\ \text{\AA}. To fix our ideas, let us consider a nebula sufficiently dense (n_e \gg n_c) that the main mechanism for depopulating 2\ ^3S is collisional transitions to 2\ ^1S and 2\ ^1P^o, as explained in Section 2.4. The

equilibrium population in 2^3S is then given by the balance between recombinations to all triplet levels, which eventually cascade down to 2^3S, and collisional depopulation of 2^3S,

n_e n(\text{He}^+) \alpha_B(\text{He}^0, n^3L) = n_e n(2^3S) [q_{2^3S, 2^1S} + q_{2^3S, 2^1P^o}]. \quad (4.58)

The rate of collisional population of 2^3P^o is thus

n_e n(2^3S) q_{2^3S, 2^3P^o} = \frac{n_e n(\text{He}^+) q_{2^3S, 2^3P^o}}{[q_{2^3S, 2^1S} + q_{2^3S, 2^1P^o}]} \alpha_B(\text{He}^0, n^3L) \quad (4.59)

so the relative importance of collisional to recombination excitation of \lambda 10830 is given by the ratio

\frac{n_e n(2^3S) q_{2^3S, 2^3P^o}}{n_e n(\text{He}^+) \alpha_{\lambda 10830}^{eff}} = \frac{q_{2^3S, 2^3P^o}}{[q_{2^3S, 2^1S} + q_{2^3S, 2^1P^o}]} \frac{\alpha_B(\text{He}^0, n^3L)}{\alpha_{\lambda 10830}^{eff}} \quad (4.60)

Computed values for the collision strengths \Upsilon that are used to compute q_{2^3S, 2^3P^o} are listed in Table 4.13; they are much larger than those for q_{2^3S, 2^1S} and q_{2^3S, 2^1P^o} (listed in Table 2.5), because the cross section for the strong allowed 2^3S-2^3P transition is much larger than the exchange cross sections to the singlet levels. At a representative temperature T = 10,000 K, the first factor in Equation (4.60) has the numerical value 6.0; the second, 1.4; and the ratio of collisional to recombination excitation is thus about 8. In other words, collisional excitation from 2^3S completely dominates the emission of \lambda 10830, and the factor by which it dominates depends only weakly on T, and can easily be seen to decrease with n_e below n_c.

Though the collisional transition rates from 2^3S to 2^1S and 2^1P^o are smaller than to 2^3P^o, the recombination rates of population of these singlet levels are also smaller, and the collisions are therefore also important in the population of 2^1S and 2^1P^o. The cross sections for collisions to the higher singlets and triplets are smaller but not negligible; from the best available cross sections it appears likely that collisional population of 3^3P^o is significant and somewhat affects the strength of \lambda 3889. The available atomic data indicates that there is a non-negligible collisionally excited

component in the observed strength of \lambda 5876 in planetary nebulae. Some collision strengths for the excitation of He I levels with n = 3 are also listed in Table 4.13.

Similar collisional-excitation effects occur from the metastable \text{He}^0\ 2\ ^1S and \text{H}^0\ 2\ ^2S levels, but they decay so much more rapidly than \text{He}^0\ 2\ ^3S that their populations are much smaller and the resulting excitation rates are negligibly small.

References

A good general summary of the emission processes in gaseous nebulae is given by

Seaton, M. J. 1960, Reports on Progress in Physics 23, 313.

The theory of the recombination-line spectrum of H I goes back to the early 1930s and was developed in papers by H. H. Plaskett, G. G. Cillie, D. H. Menzel, L. H. Aller, L. Goldberg,

and others. In more recent years it has been refined and worked out more accurately by M. J. Seaton, A. Burgess, R. M. Pengelly, M. Brocklehurst, D. G. Hummer, P. J. Storey, and others. The treatment in Chapter 4 follows most closely the following definitive references:

Seaton, M. J. 1959, MNRAS, 119, 90.

Pengelly, R. M. 1964, MNRAS, 127, 145.

The second reference treats the low-density limit for H I and He II in detail. (Tables 4.1, 4.2, and 4.3 are derived from it.)

Pengelly, R. M., & Seaton, M. J. 1964, MNRAS, 127, 165.

The effects of collisions in shifting L at fixed n are discussed in this reference. Table 4.10 comes from it.

Brocklehurst, M. 1971, MNRAS, 153, 471.

Hummer, D. G., & Storey, P. J. 1987, MNRAS, 224, 801.

Hummer, D. G., & Storey, P. J. 1992, MNRAS, 254, 277.

Storey P. J., & Hummer D. G. 1995, MNRAS, 272, 41 (on the web at http://adc.gsfc.nasa.gov/adc-cgi/cat.pl?catalogs/6/6064/).

These references include the definitive results for H I and He II at finite densities in the optical region, taking full account of the collisional transitions. Tables 4.4 and 4.5 are based upon the last reference, which includes full electronic access.

Robbins, R. R. 1968, ApJ, 151, 497; and ApJ, 151, 511.

Robbins, R. R. 1970, ApJ, 160, 519.

Robbins, R. R., & Robinson, E. L. 1971, ApJ, 167, 249.

Brocklehurst, M. 1972, MNRAS, 157, 211.

Ferland, G. J. 1980, MNRAS, 191, 243.

Almog, U., & Netzer, H. 1989, MNRAS, 238, 57.

Smits, D. P. 1996, MNRAS, 278, 683.

Kingdon, J. B., & Ferland, G. J. 1996, MNRAS, 282, 723.

Hummer, D. G., & Storey, P. J. 1998, MNRAS, 297, 1073.

Porter, R. L., Bauman, R. P., Ferland, G. J., & MacAdam, K. B. 2005, ApJ, 622, 73L.

The first two papers by Robbins work out the theory in detail for the He I triplets; the third is concerned with the singlets (for Case A only). The paper by Ferland analyzes the applicability of Case B to the He I singlets. The Porter et al. article is complete, for it describes triplet and singlet results, including collisions from 2^3S. (Table 4.6 is based on it.)

Brown, R. L., & Mathews, W. G. 1970, ApJ, 160, 939.

This reference collects previous references and material on the H I and He II continuum, and includes the most detailed treatment of the He I continuum. (Tables 4.7–4.9 and 4.11 are taken from this reference.)

Ferland, G. J. 1980, PASP, 92, 596.

Martin, P. G. 1988, ApJS, 66, 125.

These references give the continuous spectrum of the H I continuum (for specific wavelengths and filters), and recombination coefficients for Cases A and B over a very wide range in temperature (500 \text{ K} \leq T \leq 2,000,000 \text{ K}).

Scheuer, P. A. G. 1960, MNRAS 120, 231.

Hummer, D. G. 1988, ApJ , 327, 477.

The first is a very complete reference on free-free emission in the radio-frequency region, while the rest do the general problem.

The first published prediction that the radio-frequency recombination lines of H I would be observable was made by

Kardashev, N. S. 1959, Astron. Zhurnal , 36, 838 (English translation, 1960, Soviet Astronomy AJ , 3, 813).

The key reference on the importance of maser action and on the exact variation of b_n with n is

Goldberg, L. 1966, ApJ , 144, 1225.

The radiative transfer treatment in this chapter essentially follows this reference.

The equilibrium equations for the populations of the high levels are worked out in

Seaton, M. J. 1964, MNRAS , 127, 177.

Sejnowski, T. J., & Hjellming, R. H. 1969, ApJ , 156, 915.

Brocklehurst, M. 1970, MNRAS , 148, 417.

The last of these three references is the definitive treatment and makes full allowance for all collisional effects. (Figure 4.2 is based on it.)

A good deal of theoretical work has been done by several authors on radiative transfer problems in nebulae. The portion of this research used in this chapter is summarized (with complete references) in

Osterbrock, D. E. 1971, JQSRT , 11, 623.

Some of the key references concerning the H I lines are

Capriotti, E. R. 1964, ApJ , 139, 225, and 140, 632.

Capriotti, E. R. 1966, ApJ , 146, 709.

Cox, D. P., & Mathews, W. G. 1969, ApJ , 155, 859.

(Figure 4.3 is based on the last reference.)

The radiative transfer problem of He I lines was worked out earlier in

Pottasch, S. R. 1962, ApJ , 135, 385.

Robbins, R. R. 1968, ApJ , 151, 511.

A more recent treatment is

Almog, U., & Netzer, H. 1989, MNRAS , 238, 57.

(Figure 4.5 is derived from calculations from these references.)

The Bowen resonance-fluorescence mechanism was first described by

Bowen, I. S. 1924, ApJ , 67, 1.

The radiative transition probabilities necessary for tracing all the downward radiative decays following excitation of O III and 3d\ ^3P_1^0 are given by

Saraph, H. E., & Seaton, M. J. 1980, MNRAS, 193, 617.

Bhatia, A. K., & Kastner, S. O. 1993, ADNDT, 54, 133.

Froese Fischer, C. 1994, Phys. Scripta, 49, 51.

Solutions are described in

Weymann, R. J., & Williams, R. E. 1969, ApJ, 157, 1201.

Harrington, J. P., 1972, ApJ, 176, 127.

Kallman, T., & McCray, R. 1980, ApJ, 242, 615.

Elitzur, M., & Netzer, H. 1985, ApJ, 291, 646.

Netzer, H., Elitzur, M., & Ferland, G. J. 1985, ApJ, 299, 752.

Kastner, S. O., & Bhatia, A. K. 1996, MNRAS, 279, 1137.

Observations of the lines in various objects are described by

Grandi, S. A. 1980, ApJ, 238, 10.

Liu, X.-w., & Danziger, J. 1993, MNRAS 261, 463, and 262, 699.

O'Dell, C. R., & Miller, C. O. 1992, ApJ, 390, 219.

Rudy, R. J., Mazuk, S., Puetter, R. C., & Hamann, F. 2000, ApJ, 539, 166.

The He I collision strengths listed in Table 4.13 are from

Bray, I., Burgess, A., Fursa, D. V., & Tully, J. A. 2000, A&AS, 146, 481.

Chapter 5

Comparison of Theory
with Observations

5.1 Introduction

In the preceding three chapters much of the available theory on gaseous nebulae has been discussed, so that we are now in a position to compare it with the available observations. The temperature in a nebula may be determined from measurements of ratios of intensities of particular pairs of emission lines—those emitted by a single ion from two levels with considerably different excitation energies. Although the relative strengths of H recombination lines vary only extremely weakly with T, the ratio of the intensity of a line to the intensity of the recombination continuum varies more rapidly and can be used to measure T. Further information on the temperature may be derived from radio observations, combining long- and short-wavelength continuum observations (large and small optical depths, respectively) or long-wavelength continuum and optical-line observations. The electron density in a nebula may be determined from measured intensity ratios of other pairs of lines—those emitted by a single ion from two levels with nearly the same energy but with different radiative-transition probabilities. Likewise, measurements of relative strengths of the radio recombination lines give information on both the density and the temperature in nebulae. These methods, as well as the resulting information on the physical parameters of characteristic nebulae, are discussed in the first sections of this chapter.

In addition, information on the involved stars that provide the ionizing photons may be derived from nebular observations. For, if a nebula is optically thick to a particular type of ionizing radiation (for instance, in the H Lyman continuum), then the total number of photons of this type emitted by the star can be determined from the properties of the nebula. By combining these nebular observations, which basically measure the far-ultraviolet-ionizing radiation from the involved stars, with optical measurements of the same stars, a long base-line color index that gives information on the temperature of the stars can be determined. This scheme and the information

derived from it, about main sequence O stars and about planetary-nebula central stars, are discussed in Section 5.10.

Once the temperature and density in a nebula are known, it is fairly clear that the observed strength of a line gives information on the total number of ions in the nebula responsible for the emission of that line. Thus information is derived on the abundances of the elements in H II regions and planetary nebulae.

Each of the next ten sections of this chapter discusses a particular kind of observational analysis or diagnostic measurement of a nebula. Each method gives some specific detailed information, integrated through whatever structure there may be along the line of sight through the nebula, and also over whatever area of the nebula is covered by the analyzing device used for the observations, such as the spectrograph slit or the radio-telescope beam pattern. A more detailed comparison, in integrated form, may be made by calculating models of nebulae intended to represent their entire structure and comparing the properties of these models with observations. A discussion of these types of models, and of the progress that has been made with them, closes the chapter.

5.2 Temperature Measurements from Emission Lines

A few ions, of which [O III] and [N II] are the best examples, have energy-level structures that result in emission lines from two different upper levels with considerably different excitation energies occurring in the observable wavelength region. The energy-level diagrams of these two ions are shown in Figure 3.1, where it can be seen that, for instance, [O III] \lambda 4363 occurs from the upper ^1S level, while \lambda 4959 and \lambda 5007 occur from the intermediate ^1D level. (^3P_0 - ^1D_2 \lambda 4931, which can occur only by an electric-quadrupole transition, has much smaller transition probability and is so weak that it can be ignored.) It is clear that the relative rates of excitation to the ^1S and ^1D levels depend very strongly on T, so the relative strength of the lines emitted by these levels may be used to measure electron temperature.

An exact solution for the populations of the various levels, and for the relative strengths of the lines emitted by them, may be carried out along the lines of the discussion in Section 3.5. However, it is simpler and more instructive to proceed by direct physical reasoning. In the low-density limit (collisional deexcitations negligible), every excitation to the ^1D level results in emission of a photon either in \lambda 5007 or \lambda 4959, with relative probabilities given by the ratio of the two transition probabilities, which is very close to 3 to 1. Every excitation of ^1S is followed by emission of a photon in either \lambda 4363 or \lambda 2321, with the relative probabilities again given by the transition probabilities. Each emission of a \lambda 4363 photon further results in the population of ^1D, which again is followed by emission of either a \lambda 4959 photon or a \lambda 5007 photon; but this contribution is small in comparison with the direct excitation of ^1D and can be neglected. Thus the ratio of emission-line strengths in the low-density limit is given simply by

\frac{j_{\lambda 4959} + j_{\lambda 5007}}{j_{\lambda 4363}} = \frac{\Upsilon(^3P, ^1D)}{\Upsilon(^3P, ^1S)} \left[ \frac{A(^1S, ^1D) + A(^1S, ^3P)}{A(^1S, ^1D)} \right] \frac{\bar{\nu}(^3P, ^1D)}{\nu(^1D, ^1S)} \exp(\Delta E/kT) \quad (5.1)

where

\bar{\nu}(^3P, ^1D) = \frac{A(^1D_2, ^3P_2)\nu(\lambda 5007) + A(^1D_2, ^3P_1)\nu(\lambda 4959)}{A(^1D_2, ^3P_2) + A(^1D_2, ^3P_1)}, \quad (5.2)

and \Delta E is the energy difference between the ^1D_2 and ^1S_0 levels.

Equation (5.1) is a good approximation up to n_e \approx 10^5 \text{ cm}^{-3}. However, at higher densities collisional deexcitation begins to play a role. The lower ^1D term has a considerably longer radiative lifetime than the ^1S term, so it is collisionally deexcited at lower electron densities than ^1S, thus weakening \lambda 4959 and \lambda 5007. In addition, under these conditions collisional excitation of ^1S from the excited ^1D level begins to strengthen \lambda 4363. The full statistical equilibrium Equations (3.27) can be worked out numerically for any n_e and T, but an analytic solution correct to the first order in n_e and to the first order in \exp(-\Delta E/kT) is that the right-hand side of (5.1) is divided by a factor

f = \frac{1 + \frac{C(^1D, ^3P)C(^1D, ^3P)}{C(^1S, ^3P)A(^1D, ^3P)} + \frac{C(^1D, ^3P)}{A(^1D, ^3P)}}{1 + \frac{C(^1S, ^3P) + C(^1S, ^1D)}{A(^1S, ^3P) + A(^1S, ^1D)}}, \quad (5.3)

where

C(i, j) = q(i, j)n_e = 8.629 \times 10^{-6} \frac{n_e}{T^{1/2}} \frac{\Upsilon(i, j)}{\omega_i} [\text{s}^{-1}].

Inserting numerical values of the collision strengths and transition probabilities from Chapter 3, this becomes

\frac{j_{\lambda 4959} + j_{\lambda 5007}}{j_{\lambda 4363}} = \frac{7.90 \exp(3.29 \times 10^4/T)}{1 + 4.5 \times 10^{-4} n_e / T^{1/2}}. \quad (5.4)

Here the representative values of the collision strengths from Table 3.6 have been used to calculate the numerical coefficients, but actually in \text{O}^{++}, there are several resonances and the resulting average collision strengths vary appreciably with temperature, so Equation (5.4) is not exact. However, in Figure 5.1 the intensity ratio is plotted (at a density of 1 \text{ cm}^{-3}) using the correct collision strengths at each T, but to within the thickness of the line it is the same as the result of Equation (5.4), so very little error results from the use of mean collision strengths.

An exactly similar treatment may be carried out for [N II], [Ne III], and [S III], and the resulting equations analogous to (5.4) are

[\text{N II}] \frac{j_{\lambda 6548} + j_{\lambda 6583}}{j_{\lambda 5755}} = \frac{8.23 \exp(2.50 \times 10^4/T)}{1 + 4.4 \times 10^{-3} n_e/T^{1/2}} \quad (5.5)

[\text{Ne III}] \frac{j_{\lambda 3869} + j_{\lambda 3968}}{j_{\lambda 3343}} = \frac{13.7 \exp(4.30 \times 10^4/T)}{1 + 3.8 \times 10^{-5} n_e/T^{1/2}} \quad (5.6)

[\text{S III}] \frac{j_{\lambda 9532} + j_{\lambda 9069}}{j_{\lambda 6312}} = \frac{5.44 \exp(2.28 \times 10^4/T)}{1 + 3.5 \times 10^{-4} n_e/T^{1/2}} \quad (5.7)

These equations form the basis for optical temperature determinations in gaseous nebulae. Since the nebulae are optically thin in forbidden-line radiation, the ratio of the integrals of the emission coefficients along a ray through the nebula is observed directly as the ratio of emergent intensities, so if the nebula is assumed to be isothermal and to have sufficiently low density that the low-density limit is applicable, the temperature is directly determined. Alternatively, the ratio of the fluxes

from the whole nebula may be measured in the case of smaller nebulae. No information need be known on the distance of the nebula, the amount of O^{++} present, and so on, as all these factors cancel out. If collisional deexcitation is not completely negligible, even a rough estimate of the electron density substituted into the correction term in the denominator provides a good value of T. The observed strengths of the lines must be corrected for interstellar extinction, but this correction is usually not too large because the temperature-sensitive lines are relatively close in wavelength.

The [O\ III] line-intensity ratio (\lambda 4959 + \lambda 5007)/\lambda 4363 is quite large and is therefore rather difficult to measure accurately. Although \lambda\lambda 4959, 5007 are strong lines in many gaseous nebulae, \lambda 4363 is relatively weak, and furthermore is close to Hg I \lambda 4358, which unfortunately is becoming stronger and stronger in the spectrum of the sky due to increasing light pollution. Large intensity ratios are difficult to measure accurately, and reasonably precise temperature measurements therefore require carefully calibrated photoelectric measurements with fairly high-resolution spectral analyzers. Most of the early work centered on the [O\ III] lines, partly because they occur in the blue spectral region in which detectors are most sensitive, and partly because [O\ III] is quite bright in typical high-surface brightness planetary nebulae. The [N\ II] lines are stronger in the outer parts of H II regions, where the ionization is lower and the O mostly emits [O\ II] lines. A great deal of more recent work has used all of these lines.

The ^2D^o and ^2P^o levels of [O\ II] and [S\ II] have different excitation potentials, and lines that originate from these two levels can also be used as temperature indicators. These have the advantage of lying in spectral regions that are relatively easy to observe, but the lines are widely separated in wavelength so the correction for interstellar extinction is larger. All of these temperature indicators are shown in Figure 5.1.

Let us first examine optical determinations of the temperatures in H II regions, some selected results of which are collected in Figure 5.2, which compares temperatures determined by several of the line ratios mentioned above. Note that in this and other figures, the observed intensity ratio has been corrected for interstellar extinction in the way outlined in Chapter 7, and the temperature has been computed using the methods described in this book.

It can be seen that all the temperatures of these H II regions are in the range 7,000–14,000 K. A good part of the dispersion in temperature is due to physical differences between the H II regions. As we shall see in Chapter 10, within a galaxy the abundances of the heavy elements tend to increase inward, resulting in the differences in temperatures shown in the figure.

Planetary nebulae have higher surface brightness than typical H II regions, and as a result there is a good deal more observational material available for planetaries, particularly [O\ III] determinations of the temperature. Most planetaries are so highly ionized that [N\ II] is relatively weak, but measurements of it are also available. A selection of the best observational material is collected in Figure 5.3, which shows that the temperatures in the hottest planetary nebulae are typically

somewhat higher than in H II regions. This is partly a consequence of higher effective temperatures of the central stars in planetary nebulae (to be discussed in Section 5.8), leading to a higher input of energy per photoionization, and partly a consequence of the higher electron densities in typical planetaries, resulting in collisional deexcitation and decreased efficiency of radiative cooling. Like the H II regions, some of the dispersion is due to physical differences among the nebulae. Some are halo objects with relatively low heavy-element abundances, as discussed in Chapter 10, and the lower cooling efficiency results in somewhat above-average temperatures.

From Figures 5.2 and 5.3, it is reasonable to adopt T \approx 10,000 K as an order-of-magnitude estimate for any nebula with near-normal abundances; with somewhat greater precision we may adopt representative values T \approx 9,000 K in the brighter parts of an H II region, and T \approx 11,000 K in a typical bright planetary nebula.

Another method that can be used to determine the temperature in a nebula is to compare the relative strength of a collisionally excited line, such as C III] \lambda 1909, with a recombination line of the next lower state of ionization, such as C II \lambda 4267, since both depend on the product of densities n(\text{C}^{++})n_e, which therefore cancels out of their ratios. Figure 5.4 shows a few of these types of ratios as a function of

T . This method has been applied to relatively few objects because the collisionally excited line usually occurs in the vacuum ultraviolet and it is difficult to obtain space-borne observations. There is the additional complication that a significant correction may be necessary if the spectrometers used for the different lines also have different entrance apertures since these are spatially extended objects. But the method has the advantage that the observed ratio is a very powerful function of the temperature, so that even modest signal-to-noise spectra can determine the temperature quite accurately.

5.3 Temperature Determinations from Optical Continuum Measurements

Although it might be thought that the temperature in a nebula could be measured from the relative strengths of the H lines, in fact their relative strengths are almost independent of temperature, as Table 4.4 shows. The physical reason for this behavior is that all the recombination cross sections to the various levels of H have approximately the same velocity dependence, so the relative numbers of atoms formed by captures to each level are nearly independent of T , and since the cascade matrices depend only

on transition probabilities, the relative strengths of the lines emitted are also nearly independent of T. These calculated relative line strengths are in good agreement with observational measurements.

However, the temperature in a nebula can be determined by measuring the relative strength of the recombination continuum with respect to a recombination line. Physically, the reason this ratio does depend on the temperature is that the emission in the continuum (per unit frequency interval) depends on the width of the free-electron velocity-distribution function—that is, on T.

The theory is straightforward, for we may simply use Table 4.4 to calculate the H-line emission, and Tables 4.7–4.9 and 4.12 to calculate the continuum emission, and thus find their ratio as a function of T. Figure 5.5 plots the calculated ratios for two choices of the continuum. The first is near H\beta\lambda 4861 and includes the H I recombination and two-photon continua as well as the He I recombination continuum. [A nebula with n(\text{He}^+) = 0.08n(\text{H}^+) and n(\text{He}^{++}) = 0 has been assumed, but any other abundances or ionization conditions determined from line observations of the nebula could be used.] The second choice is the Balmer discontinuity, j_\nu(\lambda 3646-)-j_\nu(\lambda 3646+), which eliminates everything except the H I recombination continuum due to recaptures into n = 2. [The He II recombination, of course, would also contribute if n(\text{He}^{++}) \neq 0 since recaptures to its n = 4 level produce light at this wavelength.] Note that the \lambda 4861 continuum has been calculated in the limit n_p \rightarrow 0 (no collisional deexcitation of H I 2^2S and hence maximum relative strength of the H I two-photon continuum) and also for the case n_p = 10^4 \text{ cm}^{-3}, n(\text{He}^+) = 10^3 \text{ cm}^{-3}, taking account of collisional deexcitation, while the Balmer discontinuity results are independent of density and n(\text{He}^+).

The continuum at \lambda 4861 is made up chiefly of the H I Paschen and higher-series continua, whose sum increases slowly with T, and the two-photon continuum, whose strength decreases slowly with T; the sum hence is roughly independent of T, and the ratio of this continuum to H\beta therefore increases with T. On the other hand, the strength of the Balmer continuum at the series limit decreases approximately as T^{-3/2}, and its ratio to H\beta therefore decreases slowly with T, as Figure 5.5 shows.

The observations of the continuum are difficult because it is weak and can be seriously affected by weak lines. High-resolution spectrophotometric measurements with high-sensitive detectors are necessary. To date the most accurate published data seem to be measurements of the Balmer continuum, which is considerably stronger than the Paschen continuum near H\beta. A difficulty in measuring the Balmer continuum, of course, is that the higher Balmer lines are crowded just below the limit, so the intensity must be measured at longer wavelengths and extrapolated to \lambda 3646+. Furthermore, continuous radiation emitted by the stars involved in the nebulae and scattered by interstellar dust may have a sizable Balmer discontinuity, which is difficult to disentangle from the true nebular recombination Balmer discontinuity. Some of the best published results for H II regions and planetary nebulae are collected in Figure 5.6, which shows that the temperatures measured by this method are generally somewhat smaller than the temperatures for the same objects measured from forbidden-line ratios. These discrepancies will be discussed again in the next section

following the discussion of temperature measurements from the radio-continuum observations.

5.4 Temperature Determinations from Radio-Continuum Measurements

Another completely independent temperature determination can be made from radio-continuum observations. The idea is quite straightforward—namely, that at sufficiently low frequencies any nebula becomes optically thick, and therefore, at these frequencies (assuming an isothermal nebula) the emergent intensity is the same as that from a blackbody—the Planck function B_\nu(T); or equivalently, the measured brightness temperature is the temperature within the nebula,

T_{b\nu} = T [1 - \exp(-\tau_\nu)] \rightarrow T \text{ as } \tau_\nu \rightarrow \infty \quad (5.8)

as in Equation (4.37). Note that if there is background non-thermal synchrotron radiation (beyond the nebula) with brightness temperature T_{bg\nu} and foreground radiation (between the nebula and the observer) with brightness temperature T_{fg\nu}, this equation becomes

T_{b\nu} = T_{fg\nu} + T[1 - \exp(-\tau_\nu)] + T_{bg\nu} \exp(-\tau_\nu) \rightarrow T_{fg\nu} + T \quad (5.9)

as \tau_\nu \rightarrow \infty

(assuming no foreground absorption).

The difficulty with applying this method is that at frequencies that are sufficiently low that the nebulae are optically thick (\nu \approx 3 \times 10^8 Hz or \lambda \approx 10^2 cm for many dense nebulae), even the largest radio telescopes have beam sizes that are comparable to or larger than the angular diameters of typical H II regions. Therefore, the nebula does not completely fill the beam, and a correction must be made for the projection of the nebula onto the antenna pattern.

The antenna pattern of a simple parabolic or spherical dish is circularly symmetric about the axis, where the sensitivity is at a maximum. The sensitivity decreases outward in all directions, and in any plane through the axis it has a form much like

a Gaussian function with angular width of order \lambda/d, where d is the diameter of the telescope. The product of the antenna pattern with the brightness-temperature distribution of the nebula then gives the mean brightness temperature, which is measured by the radio-frequency observations. The antenna pattern thus tends to broaden the nebula and to wipe out much of its fine structure. To determine the temperature of a nebula that is small compared with the width of the antenna pattern, it is therefore necessary to know its angular size accurately. But, of course, no nebula really has sharp outer edges, inside which it has infinite optical depth and outside which it has zero optical depth. In a real nebula, the optical depth decreases more or less continuously but with many fluctuations, from a maximum value somewhere near the center of the nebula to zero just outside the edge of the nebula, and what is really needed is the complete distribution of optical depth over the face of the nebula.

This can be obtained from high radio-frequency measurements of the nebula, for in the high-frequency region, the nebula is optically thin, and the measured brightness temperature gives the product T\tau_1,

T_{b1} = T[1 - \exp(-\tau_1)] \rightarrow T\tau_1 \text{ as } \tau_1 \rightarrow 0, \quad (5.10)

as in Equation (4.37). (In the remainder of this section, the subscript 1 is used to indicate a high frequency, and the subscript 2 is used to indicate a low frequency.) At high frequencies, the largest radio telescopes have considerably better angular resolution than at low frequencies because of the smaller values of \lambda/d, so that if the nebula is assumed to be isothermal, the high-frequency measurements can be used to prepare a map of the nebula, giving the product T\tau_1 at each point. Thus for any assumed T, the optical depth \tau_1/\tau_2 is determined at each point from the high-frequency measurements. The ratio of optical depths, \tau_1/\tau_2, is known from Equation (4.31), so \tau_2 can be calculated at each point, and then the expected brightness temperature T_{b2} can be calculated at each point:

T_{b2} = T[1 - \exp(-\tau_2)]. \quad (5.11)

Integrating the product of this quantity with the antenna pattern gives the expected mean brightness temperature at the low frequency. If the assumed T is not correct, this expected result will not agree with the observed mean brightness temperature, and another assumed temperature must be tried until agreement is reached. This, then, is a procedure for correcting the radio-frequency continuum measurements for the effects of finite beam size at low frequencies.

A few of the most accurate available radio-frequency measurements of temperature in H II regions are collected in Table 5.1. Many of the observations were obtained with single-dish telescopes, and so had limited spatial resolution. At 408 MHz, most of the nebulae listed have central optical depths \tau_2 \approx 1 to 10, while at 85 MHz, the optical depths are considerably larger. At lower frequencies the beam size of the antenna is larger and in addition the background (non-thermal) radiation is larger, so many of the nebulae are measured in absorption at this lower frequency. A few objects have been observed with the Very Large Array (VLA) at 330 MHz, with a spatial resolution

References for table: Shaver et al. 1983, MNRAS, 204, 53; Subrahmanyam, R., & Goss, W. M. 1996, MNRAS, 281, 239; and papers cited therein.

of roughly 1 arc minute. The radio continua give temperatures that are fairly consistent with other independent ways of measuring the mean temperature in a nebula.

Exactly the same method can be used to measure the temperatures in planetary nebulae, but as they are very small in comparison with the antenna beam size at the frequencies at which they are optically thick, the correction for this effect is quite important. Nearly all the planetary nebulae are too small for mapping at even the shortest radio-frequency wavelengths with single-dish radio telescopes, but it is possible to use the surface brightness in a hydrogen recombination line such as H\beta, since it is also proportional to the proton emission measure, the integral

I(H\beta) \propto \int n_p n_e ds = E_p, \quad (5.12)

to get the relative values of \tau_2 at each point in the nebula. Even the optical measurements have finite angular resolution because of the broadening effects of seeing. Then, for any assumed optical depth of the nebula at one point and at one frequency, the optical depths at all other points and at all frequencies can be calculated. For any assumed temperature, the expected flux at each frequency can thus be calculated and compared with the radio measurements, which must be available for at least two (and preferably more) frequencies, one in the optically thin region and one in the optically thick region. The two parameters T and the central optical depth must be varied to get the best fit between calculations and measurements. The uncertainties are largely due to the lack of accurate optical isophotes, from which the distribution of brightness temperature over the nebula must be calculated.

With the Very Large Array (VLA) radio interferometer it is possible to achieve angular resolution as small as 0.05'' at high frequency (\lambda = 1.3 cm), ideal for these measurements. Mean temperatures determined in this way for planetary nebulae include T = 8,300 K for NGC 6543, T = 18,500 K for IC 418, and T = 14,000 K

for NGC 7027. The high-resolution radio images, obtained at these high frequencies, provide excellent information on the spatial structure of planetary nebulae.

5.5 Temperature Determinations from Radio and UV Absorption Lines

The H I 21-cm line occurs as an electron spin-flip transition, with a transition probability of A = 2.85 \times 10^{-15} \text{ s}^{-1}. The calculation of A is exactly analogous to atomic transition probabilities but here it involves coupling of I, the nuclear spin of the proton, with J, the angular momentum of the electron. The latter is made up of spin (non-zero) and orbital (zero) angular momenta to form F, the total spin of nucleus plus electron. It is an allowed magnetic dipole transition in these variables, but has a very small transition probability because of the \nu^3 dependence of A.

Observations of 21-cm absorption, together with \text{L}\alpha absorption, can measure the excitation temperature T_{ex} of the hyperfine structure levels that produce the 21 cm line. T_{ex}, often referred to as the spin temperature in radio-astronomy papers, is defined in terms of the level populations as

\frac{n_u}{n_l} = \frac{\omega_u}{\omega_l} \exp(-\chi_{ul}/kT_{ex})

where \chi_{ul} is the excitation energy of the line and the n 's and \omega 's are the populations and statistical weights of the lower and upper levels.

T_{ex} is equal to the gas kinetic temperature if the level populations are determined mainly by collisions. This is usually the case when the electron density is above the critical density of the transition, and the radiation density at the frequencies which can populate and depopulate the upper level are low. Collisions with electrons and hydrogen atoms are the most important. At temperatures of 10^3 \text{ K} and 10^4 \text{ K}, the downward collisional rate coefficients are q_{10} = 2.4 \times 10^{-10} and 5.2 \times 10^{-10} \text{ cm}^3 \text{ s}^{-1} for collisions with hydrogen atoms, and 6.0 \times 10^{-9} and 7.9 \times 10^{-9} \text{ cm}^3 \text{ s}^{-1} for collisions with electrons. The corresponding critical electron density is 3.6 \times 10^{-7} \text{ cm}^{-3}, so the level populations are collisionally dominated for most conditions. Scattering by \text{L}\alpha photons can alter the level populations, and this can be important under some conditions. However, the assumption that T_{ex}(21 \text{ cm}) = T_e is often a good approximation.

The line optical depth \tau of any transition is given by

\tau = \kappa L = a(n_l - n_u \omega_l / \omega_u) L \quad (5.13)

where \kappa is the line opacity, L is the path length, and a is the atomic absorption cross section (\text{cm}^2). The first term in parentheses represents direct absorption of radiation from the lower level, while the second term corrects for stimulated emission.

Using the definition of T_{ex}, the opacity \kappa can be written as

\kappa = a n_l [1 - \exp(-\chi_{ul}/kT_{ex})] [\text{cm}^{-1}]

There are two limiting forms to this equation. If \chi_{ul} \gg kT_{ex}, the limit that holds for L\alpha at T \sim 10^4 K, the exponential can be ignored and the opacity is given by

\kappa_{UV} = a_{UV} n_I.

The opposite limit occurs for H I 21 cm. Here \chi_{ul} \ll kT_{ex} and the exponent can be expanded as \exp(-\chi_{ul}/kT_{ex}) \approx 1 - \chi_{ul}/kT_{ex}. Then, the opacity is given by

\kappa_{radio} = a_{radio} n_I \frac{\chi_{ul}}{kT_{ex}}

i.e., in this case it is proportional to the ratio n/T_{ex} and not n itself. So, 21-cm absorption is strongly weighted to colder regions.

The ratio of the optical depths of the H I 21 cm and L\alpha lines can be combined to measure the excitation temperature.

\frac{\tau_{radio}}{\tau_{UV}} = \frac{n_{I,radio} \kappa_{radio} L}{n_{I,UV} \kappa_{UV} L} = \frac{n_{I,radio} a_{radio} \chi_{ul}}{n_{I,UV} a_{UV} kT_{ex}}.

For the particular case of 21 cm and L\alpha this becomes

\frac{\tau_{21cm}}{\tau_{L\alpha}} = \frac{1}{4} \frac{a_{21cm}}{a_{L\alpha}} \frac{\chi_{ul}}{kT_{ex}} = \frac{1}{4} \frac{A_{21cm} \lambda_{21cm}^3}{A_{L\alpha} \lambda_{L\alpha}^3} \frac{\chi_{ul}}{kT_{ex}} = 3.84 \times 10^{-7} T_{ex}^{-1}.

The factor of \frac{1}{4} enters if the hyperfine structure levels that produce 21 cm absorption have populations that are proportional to their statistical weight, since both levels absorb L\alpha while only the lower level absorbs 21 cm.

5.6 Electron Densities from Emission Lines

The average electron density in a nebula may be measured by observing the effects of collisional deexcitation. This can be done by comparing the intensities of two lines of the same ion, emitted by different levels with nearly the same excitation energy so that the relative excitation rates to the two levels depend only on the ratio of collision strengths. If the two levels have different radiative transition probabilities or different collisional deexcitation rates, the relative populations of the two levels will depend on the density, and the ratio of intensities of the lines they emit will likewise depend on the density. The best examples of lines that may be used to measure the electron density are [O II] \lambda 3729/\lambda 3726, and [S II] \lambda 6716/\lambda 6731, with energy-level diagrams shown in Figure 5.7.

The relative populations of the various levels and the resulting relative line-emission coefficients may be found by setting up the equilibrium equations for the populations of each level as described in Section 3.5. However, direct physical reasoning easily shows the effects involved. Consider the example of [O II] in the low-density limit n_e \rightarrow 0, in which every collisional excitation is followed by emission

of a photon. Since the relative excitation rates of the ^2D_{5/2} and ^2D_{3/2} levels are proportional to their statistical weights [see Equation (3.22)], the ratio of strengths of the two lines is j_{\lambda 3729}/j_{\lambda 3726} = 1.5. On the other hand, in the high-density limit, n_e \rightarrow \infty, collisional excitations and deexcitations dominate and set up a Boltzmann populations ratio. Thus, the relative populations of the two levels ^2D_{5/2} and ^2D_{3/2} are in the ratio of their statistical weights, and therefore the relative strengths of the two lines are in the ratio

\frac{j_{\lambda 3729}}{j_{\lambda 3726}} = \frac{n(^2D^o_{5/2}) A_{\lambda 3729}}{n(^2D^o_{3/2}) A_{\lambda 3726}} = \frac{3}{2} \frac{3.6 \times 10^{-5}}{1.6 \times 10^{-4}} = 0.34

The transition between the high- and low-density limits occurs in the neighborhood of the critical densities [see Equation (3.30)], which are n_e \approx 3 \times 10^3 \text{ cm}^{-3} for ^2D^o_{5/2} and n_e \approx 1.6 \times 10^4 \text{ cm}^{-3} for ^2D^o_{3/2}. The full solution of the equilibrium equations, which also takes into account all transitions, including excitation to the ^2P^o levels with subsequent cascading downward, gives the detailed variation of intensity ratio with the electron density that is plotted in Figure 5.8. Note from the collisional transition rates that the main dependence of this ratio is on n_e/T^{1/2}. There is also a very slight

temperature dependence (as a consequence of the cascading from {}^2P^o) that cannot be seen on this graph.

An exactly similar treatment holds for [S II]; the calculated ratio j_{\lambda 6716}/j_{\lambda 6731} is also shown in Figure 5.8. Other pairs of lines from ions with the same type of structure, which may also be used for measuring electron densities, are [N I], [Cl III], [Ar IV], and [K V] in the optical region, as well as [Ne IV] \lambda\lambda 2422, 2424 in the vacuum ultraviolet.

From the observational point of view, it is unfortunate that the [O II] \lambda\lambda 3726, 3729 are so close in wavelength; a spectrograph, spectrometer, or interferometer with good wavelength resolution must be used to separate the lines. However, a fair amount of data is available on both H II regions and planetary nebulae.

Table 5.2 compares some densities measured in H II regions. It can be seen that typical densities in several H II regions are of order n_e \approx 10^2 \text{ cm}^{-3}. (NGC 1976 M is a position in the outer part of the Orion Nebula.) Several H II regions have dense condensations in them, though—for instance, the central part of the Orion Nebula, near the Trapezium (NGC 1976 A), with \lambda 3729/\lambda 3726 = 0.50, corresponding to n_e \approx 3.0 \times 10^3 \text{ cm}^{-3}. In fact, observations of the [O II] ratio at many points in NGC 1976, of which only A and M are listed in Table 5.2, show that the mean electron density is highest near the center of the nebula and decreases relatively smoothly outward in

all directions. The three-dimensional structure of the nebula thus presumably must have a density maximum, and the intensity ratio observed at the center results from emission all along the line of sight, so the actual central density must be higher than 4.5 \times 10^3 cm ^-3 . A model can be constructed that approximately reproduces all the measured [O II] ratios in NGC 1976; this model has n_e \approx 1.7 \times 10^4 cm ^-3 at the center and decreases to n_e \approx 10^2 cm ^-3 in the outer parts. Furthermore, measurements of the [S II] ratio at many points in the inner bright core of NGC 1976 (about 8' diameter) show good agreement between the electron densities determined from the [S II] lines and the [O II] lines. Similarly, in M 8 the [O II] measurements show that the density falls off outward from the Hourglass, a small dense condensation in which n_e \approx 2 \times 10^3 cm ^-3 .

Some information on electron densities in planetary nebulae derived from [O II] and [S II] is shown in Figure 5.9, in which the densities derived from these two ions are mostly within a factor of two of each other. In most planetaries the degree of ionization is high, and most of the [O II] and [S II] lines that arise in fairly low stages of ionization are emitted either in the outermost parts of the nebula or in the densest parts, where recombination depresses the ionization the most. Thus the densities derived from these ions may not be representative of the entire nebula. The higher stages of ionization, [Ar IV], [K V], and so on, are more representative, but their lines are weaker and more difficult to measure. An example is NGC 7662, for which the [Ar IV] lines give n_e = 1.0 \times 10^4 cm ^-3 , while the [Ne IV] pair at \lambda\lambda 2422, 2424 give 9.6 \times 10^3 cm ^-3 , both at an assumed T = 10^4 K.

The electron densities derived from these line ratios may be used in Equations (5.4) and (5.5) to correct the observations of the temperature-sensitive lines of [O III] and [N II] for the slight collisional deexcitation effect; and actually these corrections have already been taken into account in Tables 5.1 and 5.2. Though the electron density derived from [O II] line measurements may not exactly apply in the [O III] emitting region, the density effect is small enough that an approximate correction should be satisfactory.

In the densest planetaries known, collisional deexcitation of [O III] ¹ D ₂ is strong enough that the (\lambda 4959 + \lambda 5007)/\lambda 4363 ratio is significantly affected. The best example is IC 4997, with \lambda 3729/\lambda 3726 = 0.34, corresponding to n_e poorly deter-

mined in the high-density limit but certainly greater than 10^5 \text{ cm}^{-3}. The measured (\lambda 4959 + \lambda 5007)/\lambda 4363 \approx 22, which would correspond to T \approx 4 \times 10^4 \text{ K} if there were no collisional deexcitation. This temperature is far too large to be understood from the known heating and cooling mechanisms in photoionization equilibrium, and the ratio is undoubtedly strongly affected by collisional deexcitation. If it is assumed that T \approx 12,000 \text{ K}, the [O III] ratio gives n_e \approx 10^6 \text{ cm}^{-3}; higher assumed temperatures correspond to somewhat lower electron densities, and vice versa.

Transitions with higher critical densities can be used as diagnostics in denser environments. The ion C III, whose energy-level diagram is shown in Figure 5.10, has two observed emission lines, [C III] 3s^2 \ ^1S_0 - 3s \ 3p \ ^3P_2^o \ \lambda 1907, a highly forbidden magnetic quadrupole transition, and C III 3s^2 \ ^1S_0 - 3s \ 3p \ ^3P_1^o \ \lambda 1909, an intercombination or “semi-forbidden” electric-dipole transition. The two lines therefore have a ratio of intensities fixed in the low-density limit by collision strengths alone. It is approximately I(^1S_0 - ^3P_2^o)/I(^1S_0 - ^3P_1^o) \approx 5/3 by Equation (3.22), but more nearly exactly = 1.53, because the downward radiative transition ^1S_0 - ^3P_0^o is completely forbidden; hence collisional excitation of ^3P_0^o is always followed by a further collisional process, either deexcitation to ^1S_0, or excitation to ^3P_1^o or ^3P_2^o. In the high-density limit, on the other hand, the ratio is fixed by the ratio of statistical weights and transition probabilities, and \approx 9 \times 10^{-5}. The detailed form of its variation with electron density is shown in Figure 5.11.

The \lambda 1907/\lambda 1909 ratio has mainly been measured for bright planetary nebulae. They mostly have ratios in the range 0.8 to 1.4, corresponding to n_e = 3 \times 10^4 to 3 \times 10^3 \text{ cm}^{-3}. The large, low-density nebulae like NGC 650/1, 3587, 6720, 6853, and 7293 are too faint to have yet been measured in these lines. But the small, high-surface-brightness planetary IC 4997 has \lambda 1907/\lambda 1909 \approx 0.03, showing that its mean electron density n_e \approx 10^6 \text{ cm}^{-3}, agreeing quite well with the [O III] results.

5.7 Electron Temperatures and Densities from Infrared Emission Lines

Sensitive infrared detectors, together with airborne or orbiting telescopes that are above most of the infrared absorption in the Earth's atmosphere, have made it possible to measure "fine-structure" lines such as [O III] ^3P_0 - ^3P_1 \lambda 88 \mu\text{m} and ^3P_1 - ^3P_2 \lambda 52 \mu\text{m} (see Figure 3.1 and Table 3.12). These far-infrared lines have much smaller excitation potentials than the optical lines such as ^3P_2 - ^1D_2 \lambda 5007. Thus a ratio like j_{\lambda 5007}/j_{\lambda 88 \mu\text{m}} depends strongly on temperature but, since the ^3P_2 level has a much lower critical electron density than ^1D_2 does, the ratio depends on density also. On the other hand, the ratio j_{\lambda 52 \mu\text{m}}/j_{\lambda 88 \mu\text{m}} hardly depends on temperature at all (since both excitation potentials are so low in comparison with typical nebular temperatures), but does depend strongly on density (since the two upper levels have different critical densities). Hence by measuring two [O III] ratios, we can determine the average values of the two parameters, T and n_e. Figure 5.12 shows calculated curves of the values of the two [O III] intensity ratios for various values of temperature and electron density. Observed values of the line ratios are entered on the diagram for several planetary nebulae, from which the average T and n_e can be immediately read off. They agree reasonably well with values determined independently from optical lines alone. Including the infrared lines makes determinations of temperature and density possible for many more ions than the optical lines. The chief difficulty, given an airborne telescope and spectrograph, is to be certain that exactly the same area is measured in both spectral regions. For this reason the measurements available to date are chiefly for entire planetary nebulae.

5.8 Electron Temperatures and Densities from Radio Recombination Lines

Information can be obtained on the temperature and density in gaseous nebulae from measurements of the radio recombination lines. Practically all the observational results refer to H II regions, which have considerably larger fluxes than planetary nebulae and hence can be much more readily observed with radio telescopes. The populations of the high levels of H depend on T and n_e, as explained in Section 4.4, and the strengths of the lines emitted by these levels relative to the continuum and to one another therefore depend on n_e, T, and the optical depth, which is conventionally expressed in terms of the emission measure E defined in Equation (4.32). Comparison of measured and calculated relative strengths thus can be used to calculate mean values of n_e, T, and E.

To calculate the expected strengths, we must solve the equation of radiative transfer, since the maser effect is often important, as was shown in Section 4.4. Furthermore, the continuum radiation is not weak in comparison with the line radiation and therefore must be included in the equation of transfer. The observations are generally reported in terms of brightness temperature. We will use T_C for the measured temperature in the continuum near the line and T_L + T_C for the measured brightness temperature at the peak of the line (see Figure 5.13), so that T_L is the excess brightness temperature due to the line.

We will consider an idealized homogeneous isothermal nebula. The optical depth in the continuum, which we will write \tau_C, is given by Equation (4.32). The optical depth in the center of the line is

\tau_{cL} = \tau_L + \tau_C (5.14)

where \tau_L is the contribution from the line alone,

d\tau_L = \kappa_L ds

and

\kappa_L = n_n k_{0L}. \quad (5.15)

Here we consider a n, \Delta n line between an upper level m = n + \Delta n and a lower level n; the central line-absorption cross section, corrected for stimulated emission as in Equation (4.43), is

\begin{aligned} \kappa_{0L} &= \frac{\omega_m}{\omega_n} \frac{\lambda^2}{8\pi^{3/2} \Delta \nu_D} A_{m,n} \left[ 1 - \frac{b_m}{b_n} \exp(-h\nu/kT) \right] \\ &= \frac{\omega_m}{\omega_n} \frac{\lambda^2 (\ln 2)^{1/2}}{4\pi^{3/2} \Delta \nu_L} A_{m,n} \left[ 1 - \frac{b_m}{b_n} \exp(-h\nu/kT) \right] \end{aligned} \quad (5.16)

In this equation a Doppler profile has been assumed, with \Delta \nu_D, the half width at e^{-1} of maximum intensity and \Delta \nu_L the full-width at half-maximum intensity, the conventional quantity used in radio astronomy. Combining (5.16) with

n_m = b_n n^2 \left( \frac{h^2}{2\pi m k T} \right)^{3/2} \exp(X_n/kT) n_p n_e \text{ [cm}^{-3}] \quad (5.17)

and using \exp(X_n/kT) \approx 1 to a good approximation for all observed radio-frequency recombination lines, expressing A_{m,n} in terms of the corresponding f -value, f_{nm}, and expanding the stimulated-emission correction as in Equation (4.43) gives, for the

special case of local thermodynamic equilibrium (b_m = b_n = 1, which we shall denote by an asterisk throughout this section),

\begin{aligned}\tau_L^* &= 1.53 \times 10^{-9} \frac{n^2 f_{nm} \nu}{\Delta \nu_L T^{2.5}} E_p \\ &= 1.01 \times 10^7 \frac{\Delta n f_{nm}}{n \Delta \nu_L T^{2.5}} E_p.\end{aligned}\quad (5.18)

The proton-emission measure, defined in Equation (5.12), is expressed in \text{cm}^{-6} \text{pc} in both forms of Equation (5.18), and

\nu = \frac{\nu_0}{n^2} - \frac{\nu_0}{m^2} \approx \frac{2\nu_0 \Delta n}{n^3}.

In the true nebular case,

\begin{aligned}\tau_L &= \tau_L^* b_n \frac{\left[1 - \frac{b_m}{b_n} \exp(-h\nu/kT)\right]}{\left[1 - \exp(-h\nu/kT)\right]} \\ &= \tau_L^* b_m \left(1 - \frac{kT}{h\nu} \frac{d \ln b_n}{dn} \Delta n\right)\end{aligned}\quad (5.19)

by the power-series expansion, while the continuum optical depth is the same as in thermodynamic equilibrium, because the free electrons have a Maxwellian distribution.

Now we will use these expressions and the formal solution of the equation of transfer to calculate the ratio of brightness temperatures r = T_L/T_C in the special case of thermodynamic equilibrium,

\begin{aligned}r^* &= \frac{T_L + T_C}{T_C} - 1 = \frac{T[1 - \exp(-\tau_{CL})]}{T[1 - \exp(-\tau_C)]} - 1 \\ &= \frac{1 - \exp[-(\tau_L^* + \tau_C)]}{1 - \exp(-\tau_C)} - 1.\end{aligned}\quad (5.20)

If \tau_L^* \ll 1 (this is a good approximation in all lines observed to date), and in addition, \tau_C \ll 1 (this is generally but not always a good approximation),

r^* \approx \frac{\tau_L^*}{\tau_C}.

Under the assumption of local thermodynamic equilibrium, the observed ratio of brightness temperatures in line and continuum thus gives (in the limit of small optical

depth) the ratio of optical depths, which, in turn, from Equations (4.32) and (5.18), measures T. Note that the continuum emission measure E_c, given by

E_c = \int n_+ n_e ds, \quad (5.21)

involves all positive ions, but the proton emission measure E_p involves only H^+ ions, so their ratio depends weakly on the helium abundance, which, however, is reasonably well known. This scheme was used in the early days of radio recombination-line observations to determine the temperatures in H II regions, but it is not correct because in a nebula the deviations from thermodynamic equilibrium are significant, as is shown by the fact that measurements of different lines in the same nebula, when reduced in this way, give different temperatures.

To calculate the brightness-temperature ratio r = T_L / T_C in the true nebular case, we note that the brightness temperature in the continuum is still given by

T_C = T[1 - \exp(-\tau_C)].

However, both the line-emission and line-absorption coefficients differ from their thermodynamic equilibrium values. The line-emission coefficient depends on the population in the upper level, so

j_L = j_L^* b_m,

while the line-absorption coefficient, as shown in Equation (5.19), is

\kappa_L = \kappa_L^* b_m \beta,

where

\beta = 1 - \frac{kT}{h\nu} \frac{d \ln bn}{dn} \Delta n. \quad (5.22)

The equation of transfer, in intensity units is

\frac{dI_\nu}{d\tau_{CL}} = -I_\nu + \frac{j_L + j_C}{\kappa_L + \kappa_C} = -I_\nu + S_\nu, \quad (5.23)

where

\begin{aligned} S_\nu &= \frac{j_L^* b_m + j_C}{\kappa_L^* b_m \beta + \kappa_C} \\ &= \frac{\kappa_L^* b_m + \kappa_C}{\kappa_L^* b_m \beta + \kappa_C} B_\nu(T) \end{aligned} \quad (5.24)

from Kirchhoff's law, so that the brightness temperature at the center of the line is

T_L + T_C = \left[ \frac{\kappa_L^* b_m + \kappa_C}{\kappa_L^* b_m \beta + \kappa_C} \right] T \left\{ 1 - \exp \left[ - (b_m \beta \tau_L^* + \tau_C) \right] \right\} \quad (5.25)

Hence finally,

r = \frac{T_L}{T_C} = \left[ \frac{\kappa_L^* b_m + \kappa_C}{\kappa_L^* b_m \beta + \kappa_C} \right] \left[ \frac{1 - \exp(-b_m \beta \tau_L^* + \tau_C)}{1 - \exp(-\tau_C)} \right] - 1, \quad (5.26)

which depends only on one optical depth, say \tau_C, the ratio of optical depths, \tau_L^*/\tau_C = \kappa_L^*/\kappa_C given by Equations (4.42) and (5.14), and the b_n factors, which, in turn, depend on n_e and T.

Thus, when the deviations from thermodynamic equilibrium are taken into account, r depends not only on T, but also on n_e and \tau_C (or equivalently, E). Therefore, observations of several different lines in the same nebula are necessary to determine T, n_e, and E from measurements of radio-frequency recombination lines. The procedure is to make the best possible match between all measured lines in a given nebula, and the theoretical calculations for a given T, n_e, and E, using the b_n(T, n_e) calculations described in Chapter 4. There are observational problems connected with the fact that the radio recombination lines, coming as they do from levels with large n and thus large atomic radii, suffer significant impact broadening even at the low densities of nebulae. This makes the wings of the line difficult to define observationally except with very good signal-to-noise ratio data. Otherwise significant contributions from the wings may easily be overlooked. Another problem is that measurements are made at different frequencies and with different radio telescopes, so the antenna beam patterns are not identical for all lines. Model calculations show that over a wide range of nebular conditions, measurements of lines with \Delta n = 1 at frequencies near 10 GHz (such as 109\alpha at 5.009 GHz) are only slightly affected by maser effects and by deviations from thermodynamic equilibrium, and therefore are especially suitable for determining nebular temperatures. Data on some of the best observed nebulae are collected in Figure 5.14, which compares the recombination line measurements with temperatures determined from collisionally excited lines. The temperatures determined from the recombination lines tend to be lower than those determined from collisionally excited lines, as was also found with the Balmer jump and radio continuum.

The range in temperatures is largely due to physical differences among the nebulae. Radio techniques have the advantage that nebulae can be detected at large distances within the galactic plane, where interstellar extinction prevents optical spectroscopy. The measured temperatures show a clear increase with increasing distance from the center, which is consistent with the decrease in heavy-element abundance outward from the center that we shall discuss in Chapter 10.

Average electron densities can also be found from the radio recombination line measurements. The best procedure is to compare lines of two different frequencies,

such as 85\alpha and 109\alpha, or 66\alpha and 85\alpha. Naturally it is important to match the antenna beam widths as closely as possible. Very high n lines cannot be used, because impact (Stark) broadening becomes important, making the wings difficult to define and measure accurately. The mean electron density derived in this way for the Orion Nebula, NGC 1976, n_e = 2.4 \times 10^3 \text{ cm}^{-3}, is comparable with an emission-weighted average of the [O II] determinations. Likewise, radio recombination line measurements for a few of the highest surface brightness planetary nebulae give mean electron densities ranging from n_e = 8.5 \times 10^3 \text{ cm}^{-3} in NGC 6543 to 1.6 \times 10^5 \text{ cm}^{-3} in NGC 7027.

5.9 Filling and Covering Factors

Direct photographs of nearly all planetaries and H II regions show chaotic structure to some degree. In some cases nebulae appear to have large hollow central regions. As we shall see in the next chapter, these can be understood as resulting from a highly ionized, low-density, high-velocity “wind,” flowing out from the central star or stars. In other

cases large departures from spherical symmetry are present, with significant regions being relatively free of matter. These density condensations, low-density hollows, etc., are an important feature of the structure of gaseous nebulae.

They can be detected quantitatively if the densities derived from [O II] line ratios in a large, well resolved nebula of known distance, such as NGC 1976, are used to predict the expected high-frequency radio continuum brightness temperature, by combining Equations (4.32) and (4.37) and assuming that the nebula is spherically symmetric. They give, in the limit of small optical depth, which is a good approximation for high-frequency observations,

T_{bv} = 8.24 \times 10^{-2} T^{-0.35} \nu^{-2.1} E_c \quad (5.27)

with the continuum emission measure E_c given by Equation (5.21), with units in \text{cm}^{-6} \text{ pc}. Note that the predicted brightness temperature depends only very weakly on the nebular temperature.

The measured values of T_{bv} are invariably smaller than those predicted in this way, typically by a factor of order ten. This can only be understood if the nebula is thinner along the line of sight than perpendicular to it (which is actually the case for the Orion nebula, as we shall see), or in terms of density fluctuations. The line-ratio density measurements are heavily weighted toward the regions of strongest emission—that is, of highest density. These measured densities thus deviate greatly from the average density along a typical path or ray through the nebula. Fluctuations in density must be taken into account in describing the structure of the nebula.

The simplest, though extreme, way to do so is to idealize the nebula as containing gas in small clumps or condensations, with electron density n_e within the condensations, but with zero electron density between them. The “filling factor” \epsilon is then the fraction of the total volume occupied by the condensations. The space between the condensations, in this simple picture, is a vacuum, which makes no contribution to the emission, mass, opacity, etc., of the nebula. The filling factor may be assumed to be constant throughout a nebula, for simplicity, or it can be allowed to vary with position, to match either an image or the spectrum of the real nebula. For NGC 1976, assuming a constant filling factor, the comparison of density and radio-continuum measurements gives \epsilon = 0.03, and values ranging from 0.01 to 0.5 or so have been determined for other H II regions and planetary nebulae.

The covering factor, \Omega/4\pi, is the fraction of 4\pi sr that is covered by gas, as viewed from the location of the central star. The covering factor takes into account regions where the gas is either nonexistent or has insufficient column density to fully absorb the ionizing continuum, and so appear fainter.

Note that under the filling-factor and covering-factor description of nebulae the intensity of an emission is given by

I_l = \int j_l ds = \int \epsilon n_i n_e \epsilon_l(T) ds; \quad (5.28)

the luminosity in the same line, integrated over the volume of the nebula, is

L_l = \frac{\Omega}{4\pi} \int \varepsilon n_i n_e \varepsilon_l(T) dT \quad (5.29)

the number of recombinations is

Q(H^0) = \frac{\Omega}{4\pi} \frac{4\pi}{3} r_1^3 \varepsilon n_p n_e \alpha_B(H^0) \quad (5.30)

replacing Equation (2.19) or

Q(H^0) = \frac{\Omega}{4\pi} \int_0^{r_1} \varepsilon n_p n_e \alpha_B(H^0) dV, \quad (5.31)

replacing its analogue in Section 5.10, and the total mass of H in the nebula is

M_H = m_H \frac{\Omega}{4\pi} \int_0^{r_1} \varepsilon n_p dV \quad (5.32)

Likewise the radial optical depth becomes

\tau_v(r) = \int_0^r n(H^0, r') a_v dr' \quad (5.33)

replacing Equation (2.12). Similar generalizations can be made in other equations, always on the basis that n stands for the density in the condensations, which are assumed to fill a fraction \varepsilon of the total volume, with vacuum (or hot, low-density invisible gas) between them.

5.10 Ionizing Radiation from Stars

Observations of gaseous nebulae may be used to find the number of ionizing photons emitted by a star and thus to determine a long base-line color index for it between the Lyman ultraviolet region and an ordinary optical region, from which the effective temperature of the star can be derived. The idea of the method is quite straightforward. If the nebula around the star is optically thick in the Lyman continuum, it will absorb all the ionizing photons emitted by the star. Thus the total number of ionizations in the nebula per unit time is just equal to the total number of ionizing photons emitted per unit time, and since the nebula is in equilibrium, these ionizations are just balanced by the total number of recaptures per unit time, so

\int_{\nu_0}^{\infty} \frac{L_\nu}{h\nu} d\nu = Q(H^0) = \int_0^{r_1} n_p n_e \alpha_B(H^0, T) dV,

where L_\nu is the luminosity of the star per unit frequency interval. In this and the following equations we have set the covering and filling factors to unity, but these

could easily be included following the discussion in the previous section. Note that by using the recombination coefficient \alpha_B, we have included the ionization processes due to diffuse ionizing photons emitted in recaptures within the nebula—see Equation (2.19). The luminosity of the entire nebula in a particular emission line, say H\beta, also depends on recombinations throughout its volume:

\begin{aligned} L(H\beta) &= \int_0^{r_1} 4\pi j_{H\beta} dV \\ &= h\nu_{H\beta} \int_0^{r_1} n_p n_e \alpha_{H\beta}^{eff}(H^0, T) dV \end{aligned}

Thus, dividing

\begin{aligned} \frac{\frac{L(H\beta)}{h\nu_{H\beta}}}{\int_{\nu_0}^{\infty} \frac{L_\nu}{h\nu} d\nu} &= \frac{\int_0^{r_1} n_p n_e \alpha_{H\beta}^{eff}(H^0, T) dV}{\int_0^{r_1} n_p n_e \alpha_B(H^0, T) dV} \\ &\approx \frac{\alpha_{H\beta}^{eff}(H^0, T)}{\alpha_B(H^0, T)} \end{aligned} \quad (5.34)

gives the result that the number of photons emitted by the nebula in a specific recombination line such as H\beta is directly proportional to the number of photons emitted by the star with \nu \geq \nu_0. Note that the proportionality between the number of ionizing photons absorbed and the number of line photons emitted does not depend on any assumption about constant density, and that replacing the ratio of integrals by the ratio of recombination coefficients is a good approximation because \alpha_{H\beta}^{eff}/\alpha_B depends only weakly on T. Note further that any other emission line could have been used instead of H\beta, or alternatively the radio-frequency continuum emission at any frequency at which the nebula is optically thin could have been used, except that then the ratio of nebular photons emitted to ionizing photons would involve the ratio of the number of protons to the total number of positive ions, which depends weakly on the He abundance. The number of ionizing photons may be compared with the luminosity of the star at a particular frequency \nu_f in the observable region,

\begin{aligned} \frac{L_{\nu_f}}{\int_{\nu_0}^{\infty} \frac{L_\nu}{h\nu} d\nu} &= \frac{L_{\nu_f}}{\frac{L(H\beta)}{h\nu_{H\beta}}} \frac{\frac{L(H\beta)}{h\nu_{H\beta}}}{\int_{\nu_0}^{\infty} \frac{L_\nu}{h\nu} d\nu} \\ &= h\nu_{H\beta} \frac{\alpha_{H\beta}^{eff}(H^0, T)}{\alpha_B(H^0, T)} \frac{\pi F_{\nu_f}}{\pi F_{H\beta}} \end{aligned} \quad (5.35)

where the ratio of luminosities has been expressed in terms of the ratio of the observed fluxes at the earth from the star at \nu_f and from the nebula at H\beta. This ratio is independent of the distance, and is in addition independent of the interstellar extinction if the nebula and the star are observed at the same effective wavelength by choosing \nu_f = \nu_{H\beta}.

It is often more convenient to make the stellar measurements with a fairly wide filter of the type ordinarily used for photometry (for instance, the V filter of the UBV system), and we can then write a similar equation in terms of

L_V = \int_0^\infty s_\nu(V) L_\nu d\nu

and

\pi F_V = \int_0^\infty s_\nu(V) \pi F_\nu d\nu,

where s_\nu(V) is the sensitivity function of the telescope-filter-photocell combination, known from independent measurements. For measurements of stars in bright nebulae, it is advantageous to use a narrower-band filter that isolates a region in the continuum between the brightest nebular emission lines, to minimize the correction for the "sky" background. In principle, any observable frequency \nu_f can be used, and likewise any observable recombination line, for instance H\alpha, might be measured instead of H\beta. The method of using the nebular observations to measure the stellar ultraviolet radiation was first proposed by Zanstra, who assumed that the flux from a star could be approximately represented by the Planck function B_\nu(T_*), so that

\frac{L_{\nu_f}}{\int_{\nu_0}^\infty \frac{L_\nu}{h\nu} d\nu} = \frac{B_{\nu_f}(T_*)}{\int_{\nu_0}^\infty \frac{B_\nu(T_*)}{h\nu} d\nu}

and the measurements thus determine T_*, the so-called Zanstra temperature of a star that ionizes a nebula. However, modern theoretical work on stellar atmospheres shows that there are important deviations between the emergent fluxes from stars and Planck functions, particularly in the regions where there are large changes in opacity with frequency, such as at the Lyman limit itself and at the various limits, due to other ions at shorter wavelengths, so that it is not a very good approximation to set F_\nu = B_\nu(T_*). As illustrations, Figures 5.15 and 5.16 show calculated models for stars with T_* = 40,000 K, \log g = 4, approximately an O6 main-sequence star, and T_* = 100,000 K, \log g = 6, a fairly typical planetary-nebula star. Thus the ratios

\frac{L_{\nu_f}}{\int_{\nu_0}^\infty \frac{L_\nu}{h\nu} d\nu} = y(T_*) = \frac{\pi F_\nu(T_*, g)}{\int_{\nu_0}^\infty \frac{\pi F_\nu(T_*, g)}{h\nu} d\nu} \quad (5.36)

should be determined from the best available sequences of model stellar atmospheres, and it can be seen that there is a one-parameter relationship y = y(T_*) for a fixed value of g or along a fixed line in the T_*, \log g plane.

We shall first use these relationships to examine the effective temperatures of population I O stars in H II regions, and then generalize these equations and use them to describe the higher-temperature planetary-nebula central stars.

Many H II regions are observed, but a fairly large fraction of them contain several O stars that contribute to the ionization and thus complicate the determination of the effective temperature of individual stars. The best cases for measurement are clearly nebulas with only a single involved hot star. Furthermore, the basic assumption of the method is that the nebula completely absorbs the stellar ionizing radiation and is a true Strömgren sphere (radiation-bounded rather than density-bounded). It is difficult to be certain that this assumption is fulfilled in any particular nebula, though well-defined ionization fronts at the outer edge of a nebula suggest that it is and thus indicate that it is a good candidate to be measured. However, as we shall see in Chapter 7, absorption of ionizing photons by dust can still cause serious errors in the results.

Today a large body of optical, infrared, and radio-frequency measurements are available. The infrared and radio measurements have the advantage that they are

relatively unaffected by interstellar extinction, although they have the defect that this extinction enters the ratio of optical stellar flux to radio-frequency nebular flux in full force. From these measurements of about 25 nebulae, the best available model stellar atmospheres were used to derive the temperature scales for the main-sequence stars shown in Table 2.3.

In many planetary nebulae the number of ionizing photons emitted by the star beyond the \text{He}^+ limit can also be measured from \lambda 4686, the strongest optical He II recombination line. This line is in general not present or only very weakly present, and has only been observed in some giant extragalactic H II regions. This indicates that the flux of \text{He}^+ -ionizing photons is small in all main-sequence O stars, confirming the calculated models in this respect. Many of the planetary-nebula central stars, however, are considerably hotter and emit an appreciable number of photons with h\nu > 4h\nu_0 = 54.4 eV. Thus, from the He II observations and from Equation (2.29), we have the relation analogous to that in Equation (5.35),

\frac{L_{\nu_f}}{\int_{4\nu_0}^{\infty} \frac{L_{\nu}}{h\nu} d\nu} = h\nu_{\lambda 4686} \frac{\alpha_{\lambda 4686}^{eff}(\text{He}^+, T) \pi F_{\nu_f}}{\alpha_B(\text{He}^+, T) \pi F_{\lambda 4686}} \quad (5.37)

Hence from the measured H I and He II line fluxes of the nebula, together with the measured stellar flux at some observable frequency, two independent determinations of T_* can be made by the Zanstra method. In some nebulae these two determinations agree, but in other nebulae they disagree badly. For instance, in NGC 7662, the H I measurements indicates T_* = 70,000 K, while the He II measurement indicates that T_* = 113,000 K, which, in fact corresponds to over 100 times more \text{He}^+ -ionizing photons than does the lower temperature. The discrepancy may be understood as resulting from the fact that the nebula is not optically thick to the H-ionizing radiation as Equation (5.35) assumes. If the nebula is density-bounded rather than ionization-bounded, then we must replace Equation (5.35) with

\frac{L_{\nu_f}}{\int_{\nu_0}^{\infty} \frac{L_{\nu}}{h\nu} d\nu} = \eta_{\text{H}} h\nu_{\text{H}\beta} \frac{\alpha_{\text{H}\beta}^{\text{eff}}(\text{H}^0, T) \pi F_{\nu_f}}{\alpha_{\text{B}}(\text{H}^0, T) \pi F_{\text{H}\beta}} \quad (5.38)

where \eta_{\text{H}} represents the fraction of the H-ionizing photons that are absorbed in the nebula. Likewise it is possible to imagine that all the \text{He}^+ -ionizing photons are not absorbed within the nebula—that is, that even the \text{He}^{++} zone is density-bounded rather than ionization-bounded. However, this does not seem to occur in most of the observed planetaries because nearly all observed planetaries have He I lines in their observed spectra, indicating the existence of an outer \text{He}^+ zone, which, as the discussion of Chapter 2 shows, is certainly optically thick to \text{He}^+ ionizing radiation. In a similar way, if [O I] lines are observed in a nebula, they indicate the presence of \text{O}^0, and therefore also of \text{H}^0, which has the same ionization potential as \text{O}^0, and thus indicates that the nebula is optically thick to H-ionizing radiation and that \eta = 1 (assuming spherical symmetry).

One further item of information can be obtained from measurements of the flux in a He I recombination line, such as \lambda 4471 or \lambda 5876 —namely, the number of photons emitted that can ionize \text{He}^0. This condition is

\frac{L_{\nu_f}}{\int_{\nu_2}^{\infty} \frac{L_{\nu}}{h\nu} d\nu} = \eta_{\text{He}} h\nu_{\lambda 5876} \frac{\alpha_{\lambda 5876}^{\text{eff}}(\text{He}^0, T) \pi F_{\nu_f}}{\alpha_{\text{B}}(\text{He}^0, T) \pi F_{\lambda 5876}}. \quad (5.39)

If the nebula is known to be optically thick to the He-ionizing radiation, either because the \text{He}^+ zone is observed to be smaller than the \text{H}^+ zone, or because the apparent abundance ratio n(\text{He}^+)/n_{\text{p}} \leq 0.1 (presumably indicating that the \text{He}^+ zone is smaller than the \text{H}^+ zone, even though this was not directly observed), then \eta_{\text{He}} = 1.

It should be noted that although the integrals giving the numbers of photons that can ionize \text{He}^+, \text{He}^0, and \text{H}^0 in Equations (5.26), (5.28), and (5.27), respectively, overlap, the equations are nevertheless essentially correct, because as indicated in Chapter 2, nearly every recombination of a \text{He}^{++} ion leads to emission of a photon that can ionize \text{He}^0 or \text{H}^0, and nearly every recombination of a \text{He}^+ ion leads to emission of a photon that can ionize \text{H}^0.

The observational data on the fluxes in H\beta, \lambda 4686, and \lambda 4471 are fairly complete and fairly accurate for planetary nebulae. The measurements of the stellar continuum fluxes are less accurate, because the stars are faint and must be observed on the bright background of the nebula. The best measurements are those that avoid the wavelengths of the nebular emission lines.

Figure 5.17 compares Zanstra temperatures obtained from straightforward application of Equations (5.35) and (5.37). The He II Zanstra temperature often exceeds the H I temperature. This is generally ascribed to the nebulae being partially transparent to hydrogen-ionizing radiation, in which case the He II temperature is the more accurate value.

Another, related method of determining the temperature of the central star is to measure directly all the cooling radiation. If the fluxes in all the emission lines plus the continuum, from the ultraviolet to the infrared, are added together, this gives directly the energy radiated by the nebula. Comparing this quantity with the flux in a hydrogen recombination gives the energy radiated per recombination (through the ratio \alpha_B/\alpha_{H\beta}^{eff})—that is, the energy input to the nebula per photoionization.

\frac{\text{total cooling}}{L(H\beta)} = \frac{\text{total heating}}{L(H\beta)} = \frac{\alpha_B}{h\nu_{H\beta}\alpha_{H\beta}^{eff}} \frac{\int_{\nu_0}^{\infty} \frac{L_\nu}{h\nu} h(\nu - \nu_0) d\nu}{\int_{\nu_0}^{\infty} \frac{L_\nu}{h\nu} d\nu} \quad (5.40)

This can be viewed as the next higher moment of the radiation field measured by the Zanstra method, and determines the effective temperature of the star, either in the blackbody approximation, or from a series of model atmospheres. This method of temperature determination is called the energy balance or Stoy's method and, as observations into the ultraviolet and infrared spectral regions become more straightforward, it will become increasingly useful. Note that if the continuum flux is not measured, it can be estimated as the energy lost (in the Balmer, Paschen, etc., continua) by the recombining electrons as outlined in Section 3.3.

5.11 Abundances of the Elements in Nebulae

It is clear that abundances of the observed ions in nebulae can be derived from measurements of the relative strengths of their emission lines. All the individual nebular lines are optically thin, so that no curve-of-growth effects of the kind that complicate stellar atmosphere abundance determinations occur. Many light elements are observable in the optical spectra of nebulae, including H, He, N, O, and Ne, although unfortunately C is not. However, it can be observed in the satellite ultraviolet spectral region. Collisionally excited lines can be quite bright, but their strengths depend strongly on temperature, which complicates the determination of relative abundances. Furthermore, all stages of ionization of an element are generally not observable in the optical spectral region; for instance, though [O II] and [O III] have strong lines in diffuse nebulae, O IV and O V do not. However, there is an [O IV] line in the far infrared, and O IV] and O V lines in the satellite ultraviolet. Opening up these new wavelength regions has greatly aided abundance determinations.

In general, as we have seen in Chapter 4, the observed intensity I_l of an emission line is given by the integral

I_l = \int j_l ds = \int n_i n_e \varepsilon_l(T) ds \quad (5.41)

taken along the line of sight through the nebula, where n_i and n_e are the density of the ion responsible for the emission and the electron density, respectively.

For the recombination lines, the emission coefficients have been discussed in Chapter 4, and we have, for instance,

\begin{aligned} I_{H\beta} &= \frac{1}{4\pi} \int n_p n_e h\nu_{H\beta} \alpha_{H\beta}^{eff}(H^0, T) ds, \\ I_{\lambda 5876} &= \frac{1}{4\pi} \int n(\text{He}^+) n_e h\nu_{\lambda 5876} \alpha_{\lambda 5876}^{eff}(\text{He}^0, T) ds, \\ I_{\lambda 4686} &= \frac{1}{4\pi} \int n(\text{He}^{++}) n_e h\nu_{\lambda 4686} \alpha_{\lambda 4686}^{eff}(\text{He}^+, T) ds. \end{aligned}

For all the recombination lines, \varepsilon_l \propto T^{-m} can be fitted over a limited range of temperature, with m \approx 1. For instance, for H\beta, m = 0.90, while for \text{He I } \lambda 5876, m = 1.13. Thus the recombination-emission coefficients are not particularly temperature

sensitive and the abundances derived from them do not depend strongly on the assumed T . But their intensities scale with the abundances, making recombination lines of elements heavier than helium faint and hard to observe.

Less abundant ions, such as C II, O IV, and O V, have weak permitted emission lines as observed in planetary nebulae, and these lines have often been interpreted as resulting from recombination, and have been used to derive abundances of the parent ions. However, some of these lines may be excited by resonance-fluorescence, and their emission coefficients therefore depend not only on temperature and density but on the local radiation field as well, so they cannot be used to derive abundances in any straightforward way. Other lines, such as C II 3^2D-4^2F \lambda 4267, cannot be excited by resonance-fluorescence, and are suitable for abundance determinations. Abundances determined from collisionally excited and recombination lines should agree, but as we shall see, there are several nebulae where they do not, and this is a current area of active research.

It is also possible to measure relative abundances of He ⁺ in H II regions from relative strengths of the radio recombination lines of H I and He I. At the very high n of interest in the radio region, both H and He are nearly identical one-electron systems except for their masses, so that the relative strengths of their lines (separated by the isotope effect) are directly proportional to their relative abundances as long as the lines are optically thin, and the nebula is a complete H ⁺ , He ⁺ region, with no H ⁰ , He ⁰ zone.

For abundance determinations of elements other than H and He, strong collisionally excited lines are available, and for these lines, in contrast to the recombination lines, the emission coefficient depends more sensitively on the temperature,

\begin{aligned} I_\nu &= \frac{1}{4\pi} \int n_i n_e h\nu q_{1,2}(T)b \, ds \\ &= \frac{1}{4\pi} \int n_i n_e h\nu \frac{8.63 \times 10^{-6}}{T^{1/2}} \frac{\Upsilon(1, 2)}{\omega_1} \exp(-\chi/kT)b \, ds \end{aligned}

in the low-density limit, where b is the fraction of excitations to level 2 that are followed by emission of a photon in the line observed.

The temperature must be determined from observational data of the kind discussed in the beginning sections of this chapter. From the measured relative strengths of the lines and the known emission coefficients, the abundances can be determined on the basis of a model of the structure of the nebula. The simplest model treats the nebula as homogeneous with constant T and n _e and thus might be called a one-layer model. From each observed relative line strength, the abundance of the ion that emits it can be determined. In some cases two successive stages of ionization of the same element are observed, such as O ⁺ and O ⁺⁺ , and their relative abundances can be used to construct an empirical ionization curve giving n(\text{A}^{+m+1})/n(\text{A}^{+m}) as a function of ionization potential. Thus finally, the relative abundance of every element with at least one observed line can be determined. Discrepancies (for instance, in n _e and T ) determined from different line ratios indicate that this model is too simplified to

give highly accurate results, though the abundances determined from it are generally thought to be correct to within a factor of order two or three.

Previous sections have described several methods of determining the temperature, and the discussion has shown that temperatures determined from hydrogen recombination lines, the Balmer jump, and free-free brightness temperatures are lower than the temperature determined from forbidden-line ratios. The probable explanation of this discrepancy is that the temperature is not constant throughout the nebula as has been tacitly assumed, but rather varies from point to point due to variations in the local heating and cooling rates. Under this interpretation, a more complicated comparison between observation and theory is necessary. An ideal method would be to know the entire temperature structure of the nebula, to calculate from it the expected forbidden-line ratios and radio-frequency continuum brightness temperatures, and then to compare them with observation; this is a model approach that will be discussed in Section 5.12.

However, the general type of effects that are expected can easily be understood. The forbidden-line ratios determine the temperature in the region in which these lines themselves are emitted—that is, the [O III] ratio measures a mean temperature in the O^{++} zone and the [N II] ratio measures the mean temperature weighted in a different way. The emission coefficient for the forbidden lines increases strongly with increasing temperature, and therefore the mean they measure is strongly weighted toward high-temperature regions. On the other hand, the recombination and free-free emission coefficients decrease with increasing temperature, and therefore the mean it measures is weighted toward low-temperature regions. We thus expect a discrepancy in the sense that the forbidden lines indicate a higher temperature than do the Balmer jump or radio-frequency measurements, as is in fact confirmed by observation. It is even possible to get some information about the range in variation of the temperature along a line through the nebula from comparison of these various temperatures, but as the result depends on the ionization distribution also, we shall not consider this method in detail.

A somewhat more sophisticated scheme takes into account the spatial variations of temperature along the line of sight and uses the observations themselves to get as much information as possible on these variations. The emission coefficient is expanded in a power series,

\varepsilon_l(T) = \varepsilon_l(T_0) + (T - T_0) \left( \frac{d\varepsilon_l}{dT} \right)_0 + \frac{1}{2} (T - T_0)^2 \left( \frac{d^2\varepsilon_l}{dT^2} \right)_0, \quad (5.42)

correct to the second order. It is clear that for recombination lines with

\varepsilon_l(T) = CT^{-m}

or for collisionally excited lines with

\varepsilon_l(T) = \frac{D \exp(-\chi/kT)}{T^{1/2}},

(in the low-density limit) the necessary derivatives can be worked out analytically. Then, integrating along the line of sight,

\begin{aligned} \int n_i n_e \varepsilon_i(T) ds &= \varepsilon_i(T_0) \int n_i n_e ds \\ &+ \frac{1}{2} \left( \frac{d^2 \varepsilon_i}{dT^2} \right)_0 \int n_i n_e (T - T_0)^2 ds \end{aligned} \quad (5.43)

where T_0 is chosen so that

T_0 = \frac{\int n_i n_e T ds}{\int n_i n_e ds}. \quad (5.44)

If all ions had the same space distribution n_i(s), then from two line ratios, such as [O III] (\lambda 4959 + \lambda 5007)/ \lambda 4363 and [N II] (\lambda 6548 + \lambda 6583)/ \lambda 5755, both T_0 and

t^2 = \frac{\int n_i n_e (T - T_0)^2 ds}{T_0^2 \int n_i n_e ds}

could be determined instead of the one constant T_0 from one line ratio, as in the single-layer model. Then T_0 and t^2 could be used to determine the abundances of all the ions with measured lines. The difficulty with this method is that all ions do not have the same distribution; for instance, O^{++} is more strongly concentrated to the source of ionizing radiation than N^+, so other more or less arbitrary assumptions must be made.