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Order Number 8726612
The astrophysics of nebulae and active galactic nuclear emission-line regions: New methods and applications
Cota, Stephen Andrew, Ph.D.
The Ohio State University, 1987
U’lVI'I SOON.ZeebRd. Ann Arbor, MI 48106
The Astrophysics of Nebulae and Active Galactic Nuclear Emission-Line Regions:
New Methods and Applications
DISSERTATION
Presented in Partial Fulfillment of die Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
Stqihen A. Cota, B. S.
*****
The Ohio State University
1987
Reading Committee: Approved By
Dr. Gary J. Ferland
Dr. Jack A. Baldwin
Dr. Eugene R. C^niotti
Gary J. Ferland, Advisor Depariment of Astronomy To Sara
-1- ACKNOWLEDGEMENTS
It is with great pleasure that I thank my advisor. Dr. Gary J. Ferland. for his help and support in the completion of this dissertation. Thanks are due for both his oiginal research suggestions, which led to this dissertation, and for his almost daily guidance. I also want to warmly thank Dr. Eugene R. Capriotti for his constant help and advice. I owe special thanks to Dr. Jack A. Baldwin, Dr. Hagai Netzer, and Dr.
Bradley M. Peterson, for reading this dissertation and for their suggestions. Finally, thanks go to Dr.
David G. Lawrie, Dr. Gerald H. Newsom, and Dr. Bradley M. Peterson, for support and advice throughout the years.
Nearly 5 years of graduate school has only proved that the task of obtaining a Ph.D. ever was, and ever will be, beyond the limits set by my character. Without the faith and support of my family and
Mends, 1 never could have finished what 1 set out to do: 1 owe them all an enormous debt
-u- VITA
March 10,1960 Bom - Toledo, Ohio
1982 Phi Beta Kappa
1982 B. S., The Ohio State University, Columbus, Ohio
1983 Teaching Associate, Dq>artment of Astronomy, The Ohio State University, Columbus, Ohio
1983 - 198S Research Assistant, Departmmit of Astronomy, The Ohio State University, Columbus, Ohio
1984 Summer Research Assistant, National Astronomy Radio Observatory, Socorro, New Mexico
1986 Research Assistant, Department of Astronomy, The Ohio State University, at Lowell Observatory, Flagstaff, Arizona
1986 -1987 Presidential Fellow, The Ohio State University, Columbus, Ohio
1987 Research Assistant, Department of Astronomy, The Ohio State University, Columbus, Ohio
PUBLICATIONS
"The Double Broad-line Emitting Regions in NGC 5548 as Possible Evidence for a Supermassive Binary", B. M. Peterson, K. T. Kmrista, and S. A. Cota, Astrophysical Joiamal Letters , 312,1,1987.
"Spectrophotometry of the Seyfot 1 Galaxy Arakelian 120", B. M. Peterson, and S. A. Cota, Astronomical Journal , 94,7,1987.
"The Size of the Broad-Line Region in the Seyfert Galaxy NGC 4151", B. M. Peterson, and S. A. Cota, submitted to the AstropAyacof Journal, 1987.
"Hydrogen Emissivity in Realistic Nebulae: The Effects of Velocity Fields and Internal Dust", S. A. Cota, and G. J. Ferland, submitted to ihoAstrophysical Journal, 1987.
-u i- TABLE OF CONTENTS
DEDICATION i
ACKNOWLEDGEMENTS ü
VITA iü
LIST OF FIGURES iv
LIST OF TABLES v
CHAPTER PAGE
I. INTRODUCTION 1
n . THREE-BODY RECOMBINATION TO THE HIGHLY EXCITED STATES OF ATOMS AND IONS: AN IMPORTANT PROCESS FOR NOVAE SHELLS AND BROAD-LINE EMPTUNG REGIONS OF ACTIVE GALAXIES 5 2.1 Introduction 5 2.2 Calculations 8 (a) Formalism and Assumptions 8 0)) Computational Details 11 2.3 Results 14 (a) Numerical Results 14 (b) Comparison With Previous Work 15 (c) Evaluation of Approximations 17 2.4 Discussion 19 (a) Late-Fhase Novae Shells 19 0>) H n Regions and Planetary Nebulae 20 (c) Emission-Line Regions of Active Galaxies 21 (i) Conventional Density BLRs m iNLRs 22 (ii) High Density BLRs 23
m . A SIMPLE BUT ACCURATE TREATMENT OF THE HYDROGEN ATOM 33 3.1 Introduction 33 3.2 Factors Influencing the Accuracy of Calculations of the HI Recombination Spectrum 36 (a) /-Mixing 36 0>) Atomic Data 38 (c) Truncation of the Hydrogen Atom 40
-IV- 3.3 A Simple Model Hydrogen Atom 43 (a) Model Hydrogen Atmn 44 0)) Averaging of the Rates and Populations 44 (c) Avoaging of the Eneigies and Statistical Weights 47 (d) Temperature Dqtendence of Collision and Radiative Recombination Rates 49 (e) Choice of 6 . and Results SI 3.4 Comparison With Previous Authors 55 (a) Equations for a Photoionized Slab 55 0>) Comparison to Low Density Calculations with die n-M ethod 58 (c) Comparison to High Deisity Calculations 58 (d) Comparisons at Low Temperatures 60
IV. THE RECOMBINATION SPECTRUM OF HELIUM I: THE EFFECTS OF PHOTOIONIZATION OF HYDRCXÎEN BY HE I RESONANCE PHOTONS ON THE SINGLET EMISSIVmES 89 4.1 Introduction 89 4.2 Calculations 91 (a) Helium I Singlet Energies and Transition Rates 92 (b) Line Transfer 94 (c) Temperature Dqiaidence 98 (d) Collisional Population o f 2^f £rmn 2^S 100 4.3 Results 104 4.4 Discussion 107 (a) Photoionization Models 107 (i) AGN Model 109 (« ) H / / Regions 110 (h i ) Planetary Nebulae 111 (b) Predicted Singlet Spectrum 111 (c) NGC 7027 113
V. HYDROGEN EMISSIYITY IN REALISTIC NEBULAE: THE EFFECTS OF VELOCITY FIELDS AND INTERNAL DUST 131 5.1 Introduction 131 5.2 Calculations 133 (a) The Hydrogen Atom 133 (i) The High-Density Undt 134 (« ) Low^ensity Limit 134 (b) Velocity Structure and Optical Depths 136 (c) Dust to Gas Ratio 137 (d) Radiative Transfer 139
-V- (e) Results 141 5.3 Applications 143 (a) The Orion Nebula 143 Çâ) Giant H n Regions 145 (c) Planetary Nebulae 147 (d) Nova Envelopes 148 5.4 Discussion 149
VI. THE NATURE OF THE IONIZING CONTINUUM IN SGR A WEST 156 6.1 Introduction 156 6.2 Obsmred Properties of Sgr A West 159 (a) Photoionization Models 159 (b) Geometry of Sgr A West 160 ({) Distribution m d Kinematics of the Ionized Gas 161 (ü ) Identification and Nature cf the Central Object 164 (c) Extinction 166 (i) The Near-IR Extinction Curve 166 (ü ) The Far-IR Extinction Curve 168 (ü i) Adopted Extinction 169 (d) Fine Structure Line Spectrum 170 (i) Uncertainties in the line Data 170 ( a ) Adopted Spectrum 172 (a ;) Comparison with Extragalactic Objects 172 (e) The Ionizing Continuum Shape 173 (J) The Hard X-and gamma-Ray Contmua 175 (a ) The Soft X-Ray Continuum 177 (Jii) The IR Continuum 178 (f) The Ionizing Continuum Strength 180 (i) Estimates from [Ne II] 12 J8 pm Fluxes 180 (ii) Estimates from Brackett line Fluxes 182 (Hi) Estimates from Radio Flux Denrities 183 (iv) Other Estimates 184 (g) Density 185 (i) Evidence for lo w Densities 185 (ii) Evidence for High Densities 186 (h) Abundances 188 6.3 The Broad-Line Region, 1RS 16 191 (a) Geometry 191 (b) The //e / %2.058 pm Line 191 (c) Density 194 6.4 Photoionization Calculations 197 6.5 Results 199
-VI- (a) 35,000 K Stellar Atmo^hœ 200 (b) 30,000 K Stellar Atmosphœ 201 (c) AGN Model 201 (d) 40,000 K Stellar Atmosphere 202 6 .6 Discussion 203 (a) Variants on the AGN Models of Sgr A West 204 (i) Formation of [Ar / / ] 7 jjfli and [Ne //] 12.8 pm in AGN Models 204 (« ) Turbulent Heating 206 (fii) Depletion qf Coolants 206 (iv) Optical Depths in the Fine Structure Lines 207 (v) Radiative Excitation 210 (b) Uncertainties in the Calculations 211 (/) Uncertainties in the Stellar Atmospheres 211 (« ) Incomplete or Inaccurate Atomic Data for Argon 211 (c) PredictitHis 213 (d) The Nature of Sgr A West 214
Vn. CONCLUSION 239
APPENDIX 243
BIBLIOGRAPHY 246
-vu- UST OF TABLES
TABLE PAGE
1. Comparison o f N, vs. for Adams and Petrosian and the present work 2S
2. N, for Mh, (After Drake and Ulrich 1981) 61
3. Effective H p Recombination Coefficients and H I Line Ratios for Full and No I -Mixing; Case A 62
4. Effective H p Recombination Coefficients and H I Line Ratios for Full and No I -Mixing; Case B 62
5. Ctmparison of Collision Rates from the literature 63
6 . Real and Avemge Energies and Statistical Weights for Real and Fictitious Levels 64
7. Fits to the Radiative Recombination Coefficients for Level n 65
8 . Radiative Rates for m ^ 6 6 6
9. Avoaged Radiative Rates vs. rtc for Case A 67
10. Averaged Radiative Rates A vs. «g for Case B 67
11. Avaaged Radiative Rates A vs. for Case A 68
12. Avoaged Radiative Rates A vs. Re for Case B 6 8
13. Averaged Radiative RatesAy^ vs. Rg for Case A 69
14. Avraaged Radiative Rates A vs. Rg for Case B 69
15. Collision Strengths fbrm 26 (10,000 K) 70
16. Powtf-law Index p from Fit to Temperature Dqrendence of Collision Strengths for m 6 70
-vui- 17. Avoaged Collision Strengths (10,000 K) vs. «C for Case A 71
18. Power-Iaw Index p from Fit to Temperature Dependence of Averaged Collision Strengths vs. tig for Case A 71
19. Averaged Collision Strengths y (10,000 K) vs. rtg for Case B 72
20. Power-law Index P from Fit to Temperature Dependence of Averaged Collision Strengths vs. rig for Case B 72
21. Averaged Collision Strengths O*,, (10,000 K) vs. rig for Case A 73
22. Power-law Index p from Fit to Tonperatuie Dq%ndence of Avoaged Collision Strengths vs. rig for Case A 73
23. Avaaged Collision Strengths A , y (10,000 K) vs. rig for Case B 74
24. Power-law Index p from Fit to Temperature Dependence of Averaged Collision Strengths Q„,g' vs. rig for Case B 74
25. Avaaged Collision Strengths y (10,000 K) vs. rig for Case A 75
26. Powa'-law Index P from Fit to Temperature Dqiendence of Avaaged Collision Strengths vs. rig for Case A 75
27. Averaged Collision Strengths (10,000 K) vs. rig for Case A 76
28. Powa-law Index P fitom Fit to Temperature Dqtendence of Avaaged Collision Strengths A .y vs. tig for Case B 76
29. Ionization Strengths for r ^ 6 (10,(XX) K) 77
30. Powa-law Index y from Fit to Température Dqtendence of Ionization Strengths for n £ 6 77
-IX- 31. Ionization Strengths for Levels 7', 8 ', and 9' vs. tig for Case A (10,000 K) 78
32. Poww law Index 7 from Fit to Temperature Dq%ndence of Ionization Strengths for Levels T, 8 ', and 9 ' vs. tig for Case A 78
33. Imiization Strengths for Levels 7', 8 ', and 9 ' vs. tig for Case B (10.000 K) 79
34. Power-law Index 7 finom Fit to Temperature Dqwndence of Imtization Strengths for Levels T, 8 ', and 9 ' vs. tig for Case B 79
35. Case A Comparison of 6 ,exp(x,) Between Seaton (19596) and tiie 10-Level Hydrogmi Atom 80
36. Case B Comparison of 6 ,exp(x,) Between Seaton (19596 ) and the 10-Level Hydrogen Atom 80
37. Case A High Density Predictions ft»-10-Level Atom (6,000 K) 81
38. Case B High Density Predictions for 10-Level Atom (6,000 K) 81
39. Case A High Density Predictions for 10-Level Atom (10,000 K) 82
40. Case B High Density Predictions for 10-Level Atom (10,000 ^ 82
41. Case A High Density Predictions for 10-Level Atom (20,000 K) 83
42. Case B High Density Predictions for 10-Level Atom (20,000 K) 83
43. Case A High Density Predictions firom Adams and Petrosian (1974) 84
44. Case B High Density Predictions from Adams and Petrosian (1974) 84
45. Case A Low Temperature Predictions for 10-Level Atom (500 K) 85
46. Case B Low Temperature Predictions for 10-Level Atom (500 K) 85
47. Case A Low Temperature Predictions for 10-Level Atom (125 K) 8 6
48. Case B Low Tempoature Predictions fw 10-Level Atom (125 K) 86
-X- 49. Low Tempoature Predictions firom Martin (1987)
50. Indices for Case A Power-law Tempaature Fits to He I Singlet line Ratios (a) and Effective Recombination Coefficients (P) 115
51. Indices for Case B Power-law Tempoature Fits to He I Singlet line Ratios (a) and Effective Recombination Coefficients (P) 115
52. Collisional Transition and Radiative Recombination Rates Affecting Populations of Levels and 2*P 116
53. Population of Level 2^5 vs. Densi^ 116
54. Collisional Contributions to 2^P-2^S Emissivity; Case A 117
55. Collisional Contributions to 2^P—2*S Emissivity; Case B 117
56. Ratio of He I (4'D -2'f ) A4922 to He I (4^D -2^f ) X5876 118
57. Ratio of Case B to Case A He I Singlet Emissivities 119
58. Helium Singlet n -2^5 Intensities Relative to A4922 finom Photoionization Models 120
59. Helium Singlet Lines Relative to 24922 for NGC 7027 121
60. Case A Effective Recombination Coefficients and Ratio of B ra 24.05 ixm to Bry 22.17 pm 215
61. Case B Effective Recombination Coefficioits and Ratio of Bra 24.05 pm to Bry 22.17 pm 216
62. Power-Law Indices p for Fits to Published Total and Effective Recombination Coefficients 217
63. Atomic Line Data for 1RS 1 218
64. Adopted Fine Structure Line Spectrum for 1RS 1 221
65. Ctmtinuum Data for Sgr4*/IRS 16 222
6 6 . Schematic AGN Continuum (Afto* Mathews and Ferland 1987) 224
-XI- 67. Atomic Line Data fOT IRS 16 ("BLR") 225
6 8 . Comparison of Observed Line Ratios to Best-Fit Model line Ratios 226
69. Atomic Data fn Fine Structure Lines 227
-XU- USTOFHGURES
FIGURE PAGE
1. Dominant Recombination Processes for Electron Density and Temperature Regimes 26
2. Ao/aA vs.W,(500K) 27
3. Ao/aA vs. iV, (8,000 K) 28
4. A o/a^ vs. iV, (16,000 K) 29
5. Comparison of our Calculations of the Effect of 3-Body Recombination to those of Humm^ and Storey (1987) for HI and He II (3,000 K) 30
6 . Comparison of our Calculations of the Effect of 3-Body Recombination to those of Hummo’ and Storey (1987) and Summers (1979) for He n (20,000 K) 31
7 Critical Principal Quantum Number vs. Temperature for Aa=a.^ 32
8 . Comparison of H p Effective Recombination Coefficients, Predicted by our 10-Level Atom and by Hummer and Storey 88
9. ct^ossiw (2if _2ig) as a Function of and%«(X584) 122
10. j\5oi6(3‘P-2‘S)/j\4922(4*D-2*P)asaFunctionof'tx5g4andXc(X584) 123
11. yi3965(4‘P - 2 ‘S)//^396s(4^D-2‘P)asaFunctionofT3tsg4andXc(X584) 124
12. j3i36i4(5*P-2'S)/ju6i4(4^D-2‘P)asaFunctionofTx5g4andXc(X584) 125
13. jj344g(6*P-2*S)/jj344g(4’D-2*P)asaFunctionofXxsg4andXc(X584) 126
14. The Deviation o f j X72«i(3*5 -2*P )lj X3448(4*D -2*P ) firom its Case B Value as a Function of X;^g4 and %g(X584) 127
15. The Deviation o f ; wo 4g( 4 ^S-2 *P)/y^ 344g( 4 ‘D - 2 ‘P ) firom its Case B Value as a Function of Xx;g4 and%,(X584) 128
16. The Deviation of yx4438(5‘5-2*P )/j w 44g( 4 *D -2*P ) firom its Case B Value as a Function of Txs84andXe(X584) 129
-xm- 17. The Deviation of j x4169(6*S-2^F)//>3448(4‘D -2 ‘P ) from its Case B Value as a Function of and X,.(X584) 130
18. Deviations fipom Case Bin die High Density Limit for Two-Sided Escape 131
19. Case B Deviations for High Densities for One-Sided Escape 152
20. Case B Deviations for Very Low Densities for Two-Sided Escape 153
21. Case B Deviations for Very Low Densities fw One-Sided Escape 154
22. Effects of Internal Dust on Model Giant HU Regions 155
23. Two Possible Extincticm Curves for Sgr A West 228
24. Observed Continuum for Sgr A West, with Hypothesized AGN Continuum 229
25. Intensities of Obsaved Fine Structure lines Relative to Bry as a Function o fN h for 35,000 K Ionizing Stellar Spectrum 230
26. Intensities of Obsaved Fine Structure lines Relative to Bry as a Function ofNjf for 30,000 K Ionizing Stellar Spectrum 231
27. Intensities of Observed Fine Structure Lines Relative to Bry as a Function ofN u for an AGN Ionizing Spectrum 232
28. Intensities of Obsaved Fine Structure lines Relative to Bry as a Function of ^ for a 40,000 K Ionizing Stellar Spectrum 233
29. Intensities of Unobsaved Fine Structure Lines Relative to Bry as a Function ofNg for 35,000 K Ionizing Stellar Spectrum 234
30. Intensities of Unobserved Fine Structure Lines Relative to Bry as a Function of % for 30,000 K Ionizing Stellar Spectrum 235
31. Intaisities of Unobsaved Fine Structure Lines of Ions of N, Mg, and Ca Relative to Bry as a Function of for an AGN Ionizing Spectrum 236
32. Intaisities of Unobserved Fine Structure Lines of Ions of O, Ne, and Ar Relative to B ry as a Functiono f Njj for an AGN Ionizing Spectrum 237
-XIV- 33. Intensities of Observed Fine Structure Lines Relative to Rr y as a Function ofNh for a 40,000 K Ionizing Stellar Spectrum 238
34. Ionized Helium in the Orion Nebula 245
-XV- I. INTRODUCTION
The emission-line spectra of many astiophysical objects such as H n regions, planetary nebulae, nova and supernova ejecta, emission-line galactic nuclei such as active galactic nuclei (AGN), H n region, and starburst nuclei, and some components of the interstellar medium (ISM) all can be successfully modeled as dilute gases photoionized and heated by extonal sources of continuum radiation. The physics of nebulae photoionized by starlight has been studied since the early thirties (see references in Osterbrock
1974), and is well understood. Recent years have seen the extension of photoionization models to more extreme physical conditions and diverse objects, beginning in 1967 with Williams’ model for the Crab nebula, which is thought to be ionized by the synchrotron power-law spectrum emitted by the Crab pulsar. Photoionization models have most recently been reasonably successful in explaining the spectra of emission-line regions of more luminous AGN, whae the nonthermal continuum’s high density of X- ray photons and the ionized gas’ high electron-density lead to markedly different physics than obtains in classical nebulae such as H n regions and planetary nebulae (reviews in Davidson and Netzer 1979;
Ferland and Shields 1985).
Ongoing research in the physics of photoionized objects focuses on many problems, including extending models to still more extreme physical conditions by including the physical processes which are important under those conditions, on refining our undastanding of classical nebulae, and on developing a system of classification for photoionized objects which considers the shape of the ionizing spectrum as the fundamental paramet^. This dissotation includes contributions in each of these areas. We begin in chapter n by extending photoionization equilibrium calculations to regimes of low
temperature, high ionization, and/or high density where 3-body recombination to the excited states of atoms and ions becomes an important process. In brief, there are always highly excited states for any ion
whose populations are dominated by electron collisional transitions between states, 3-body recombinations, and collisional ionizations. For either low temperatures, high ionization, or high electron
densities, 3-body recombinations become the dominant collisional process for these states, leading to
very high excited-state populations. Radiative cascades from these states to the ground state result in
large enhancements in the total effective recombination rate. We derive the effects of this "3-body
recombination plus cascade" process over the attire range of physical conditions which are thought to
obtain in nebulae, and note at least 2 known classes of objects for which this process should be important
The first is the ejecta of novae, which ^pear to be characterized by very low temperatures while
retaining a relatively high density and level of ionization. The second is the broad-line emitting regions
(BLRs) of AGN, where high electron densities and very high levels of ionization obtain. Evidence from
variability studies suggests even higher datsity gas could be present than has previously been thought. At
these densities, 3-body recombination to excited states could be important even for low ionization
species.
Chapters lU-V deal with refinements of nebular theory. In chapter ID we consider deficiencies and
propose improvements in the existing treatment of the hydrogen recombination spectrum in
photoionization models. Although the theory of the H I recombination spectrum is well-developed and
highly accurate in some limiting cases, difficulties remain. For example, there are large uncertainties in
the published collisional rate coefficimits; In the density regime where collisional and radiative
transitions occur at comparable rates, these uncertainties are have a significant effect on the predicted spectnim. In many instances the spectrum can only be predicted in the context of photoionization models: In this event, there are difficulties which result from the limited number of levels for hydrogen which can be included in photoionization models. For example, the neglect of the /-sublevels through the simplifying assumption of /-mixing (wherein the /-sublevels are assumed to be relatively populated according to their statistical weights) introduces important errors in some regimes of temperature and resonance-line optical dq)ths. In other cases, such as those where 3-body recombination to the excited states is important, the limit of 10 - IS n-levels which a photoionization code can comfortably handle leads to considerable inaccuracy. We discuss these problems and derive a simple, 10-level model of the hydrogen atom, which formally retains the accuracy of a 1 0 0-level atom.
Chapter IV considas the helium singlet recombination spectrum. As is the case for hydrogen, the helium singlet spectrum has been modeled in two extremes. In case A, the gas is optically thin to scattering of resonance line photons, and excited states can decay to ground. In case B, the gas is optically thick to these photons, and excited states can only decay by means of higher series transitions.
These 2 cases have been modeled in great detail by previous authors, but it has also long been recognized that resonance photons can be destroyed through photoionizations of neutral hydrogen before they scatter often enough to degrade into higher series lines. We present the first detailed calculations which ftiUy account for this effect, and find that deviations firom case B should be quite large in high ionization objects like planetary nebulae or AGN. These effects will considerably influence helium abundance measurements which employ the singlets, and in principal will allow one to infer the s h ^ of the ionizing spectrum from the singlet spectrum. Cluster V considas the question of the tqiplicability of case B for hydrogen in real nebulae. In this case we ccHisider the consequences of destroying the resonance photons by absorption onto dust grains.
We also discuss how velocity gradients, turbul^ice, and the inhomogeneous distribution of matter can lower the optical depth scale. For realistic amounts of dust and line broadening, we find that deviations fipom case B reduce the flp emissivity by £ 15%. This introduces systematic errors into abundance measurements when they are made relative to //p. In applications where extremely accurate measurements are necessary, such as in measuring the primordial helium abundance, these small systematic effects are important.
Finally, in Chapter VI we consider the classification of the emission-line nucleus of our galaxy (Sgr A
West) in the context of the classification schemes used for extra-galactic emission-line nuclei, i.e. according to its ionizing spectrum. The optical and UV continuum, and the emission-line data which are commonly used to distinguish between photoionization by thermal and nonthermal spectra are unavailable for Sgr A West because of the 25 magnitudes of visual extinction along the line of sight. We demonstrate through comparison of the observed IR fine-structure line data for Sgr A West with our photoionization models that our galaxy’s nucleus is ionized by a igsectrum resembling a cool blackbody, and is therefore probably a starburst or H n region nucleus.
While all S of the following chapters relate to a common theme, each concerns a separate topic and could just as easily stand alone. Accordingly we have tried as much as possible to keep each chapter self-contained. Except where to do so would require an inordinate amount of space, we have repeated equations, arguments, and citations as necessary in each chapter, rather than referring back to previous chapters. n . THREE-BODY RECOMBINATION TO THE HIGHLY-EXCITED STATES OF ATOMS AND IONS:
AN IMPORTANT PROCESS FOR NOVAE SHELLS AND BROAD-LINE EMTITING REGIONS OF
ACTIVE GALAXIES.
2.1 Introduction
In computing the photoionization equilibrium of any ion, a large number of physical processes must commonly be consida:ed. These include photoionization firom the ground state, and radiative and dielectronic recombinations to all states. For metals, ionization by charge transfer with ionized hydrogen and helium, and recombination by charge transfer with neutral hydrogen and helium must also be
considered. In the low-density limit (LDL), all recombinations to excited states are assumed to undo-go
rapid radiative decay terminating at the ground state. Collisional ionization and Three-body
recombination are neglected, because for usual nebular tempaatures (of order 10,000 K) and densities
(Ne < 10^ cm ~^ it is straightforward to show that the rates for these processes are fast relative to
radiative de-excitation rates only for very high-lying states, cascades firom which have minimal influence
on the ground state.
There are three general regimes where the LDL fails to obtain. A well-known case occurs in the
line-emitting regions of active galactic nuclei (AGN), where X-ray heating and ionization is thought to
produce a warm, partially ionized zone at high resonance-line optical depth (e.g. Kwan and Krolik 1981).
Collisional ionization firom the excited states of hydrogen and helium becomes important because
resonance-line scattering effectively reduces the rate of radiative de-excitation, while the high
temperature and high electron firaction maintains the collisional ionization rate. Another well-known case where the LDL is inappropriate is the high density regime. At densities of order 10" cm~^ and higher, collisions dominate for states as low as n = 7. Three-body recombinations produce much larger populations for these states than they have in the LDL. At ordinary nebular temperatures, cascades from the LTE-states (throughout this chapter we will refer to high-lying, collisionally dominated states as
"LTE-states" since for them LTE effectively obtains; low-lying states which are populated by cascades and radiative recombination will be called "LDL-states") proceed at a rate comparable to the rate of radiative recombination, leading to significant enhancements in the populations of the LDL-states. The effect of this process on the Balmer decrement has been investigated by Adams and Petrosian (1974).
There is evidence that densities of this order obtain in some portions of AON line-emitting regions.
The physics of the last case where the LDL is iruq)prqpriate is identical to the physics of the high density case, but to the best of our knowledge this instance has to date gone unrecognized. At any density, some of the very high-lying, "Rydberg" states will be collisionally dominated and hence populated according to their LTE distribution. When the mean thermal energy kT of the gas is significantly lower than the ionization potential for these states, the Saha-Boltzmann equation indicates that three-body recombinations (recombinations resulting from the collision of 2 electrons and an ion) overwhelm coUisional ionizations, resulting in very high excited-state populations. The rate of cascades from the Rydberg states into the LDL-states then equals or exceeds the rate of direct radiative recombination to the LDL-states. In the hydrogenic approximation, the ionization potential of an ion of effective nuclear charge Z scales as Z*, so that when T/Z^ is small, three-body recombinations to the
Rydberg states are fast Thus, at low temperatures (of order 100 - 500 K) and modaate densities
(lO ^ - 10* cm~^), we find that singly and higher ionized species should have a greatly enhanced total
recombination rates. These conditions are expected to obtain for cold novae shells (Ferland et al. 1984). At ordinary nebular températures, the same i^ysics would apply to h i^ y ionized species, albeit at higher densities (10^ - lO’ cm~% Thus, in the case of very hard ionizing spectra (such as are characteristic of AGN) which produce very high ionization species at temperatures around 10,000 K, and for high densities (such as are typical of AGN broad-line emitting regions) three-body recombinations again strongly enhance the total recombination rate.
The situation is sununarized in figure 1, which is a result firom the present chapter. The figure divides the Ng - Tg plane into regimes of LDL-dominated recombination and LTE-dominated recombination for several values of Z. The rates of cascades fiom the LTE-states to the LDL-states, and of radiative recombinations to the LDL-states are equal along each curve, with cascades from the LTE-states dominating to the left and up (decreasing temperature; increasing density) and cascades from the LDL- states dominating to the right and down (increasing temperature; decreasing density). The figure clearly shows that coUisional processes can play a significant role in many low density regimes where they have not previously been expected.
In this chapter we have examined the effects on the ionization equilibrium of including three-body recombination to excited states over electron temperatures from 1 K to 10^ K, and at densities from .1 to
10'^ cm ~ \ Section 2.2 describes the calculations used to study these coUisional effects. Section 2.3 discusses the results, and Section 2.4 discusses some astrophysical objects in which these effects should be important. The effects of three-body recombination on the recombination spectrum of H I are discussed in chapt^ 3. 2.2 Calculations
(a) Fonnalism and Assumptions
For most photoionization problems, the rate of radiative de-excitation greatly exceeds the rate of photoionization for all excited states. Thus it is correct to assume that photoionizations occur from the ground state only. Ignoring charge exchange for simplicity, the ionization balance between stages of effective nuclear charge Z and Z + 1 may be written;
JV(Z)«bi(Z) =N.Af(Z + l)af(Z) (2.1) where are the number densities of ions Z and Z + 1, and aj^(Z) is the total effective rate coefficient for recombination to the ground state of ion Z. In the LDL, af^(Z) is, for gases optically thin at the ionization limit of ion Z (case A), simply af{Z)=aDR{Z)+aAiZ) (2.2) where agg(Z) is the total dielectronic recombination coefficient for ion Z (tabulated for selected ions in Burgess, 1964; and Nussbaumer and Storey, 1984), and «^(Z) is the sum of the rate coefficients for radiative recombination to each level of ion Z. In case B the gas is optically thick at the ionization limit of ion Z and we assume that radiative recombinations to the ground state are identically balanced locally by photoionizations from their resultant photons. In this case, % , the sum of the radiative recombination coefficients for all levels except ground, replaces in equation 2.2. Case B would not ordinarily occur for ions of elements otb« than hydrogen and helium because of their low abundance. For hydrogenic species, q* can be calculated as in Ferland (1980), who follows the method of Seaton (1959a), but with the gaunt factors of Kaizas and Latter (1961). Power-law tmnperature fits to case A radiative recombination coefficients for more complex ions are available in Aldiovandi and Pequignot (1973) arid Gould (1978). We will assume case A hereafter for concreteness and because this case is the one applicable to the majority of ions. The ratio of the coUisional excitation rate to the radiative de-excitation rate is: (2.3) where is the rate coefficient for coUisional excitation firom level n to n+ l, andA,u, is the radiative de-excitation rate firom level n tom . The ratio ^(n) increases with n : Thus, there are always high enough lying states for which collisions dominate and which are driven to their LTE populations. For a given density, the division between LTE-states and LDL-states occurs at n^, the critical principal quantum numba chosen firom: K C ( n ,- l ) = l . (2.4) It turns out that is never less than 5, ev«r ficn the highest density we considered (10^^ and we therefore assume hereaft^ that a hydrogenic structure reasonably represents an arbitrary ion. Also note that for levels above n^, equation 2.4 is a sufficient condition for / -mixing to obtain. Le. the I-sublevels are populated according to their statistical weights. For levels below n^, f-mixing need not necessarily obtain, but as we do not need to know the populations of these levels explicitly, we can assume / -mixing 10 hereafter without any loss of generality. Far above «g it is clear that LTE obtains, while far below it, the LDL obtains. For levels near iig neither LTE nor the LDL are exact and coUisional and radiative processes should be simultaneously included in the rate equations. We make the simplifying assumption, however, that LTE obtains exactly for levels and above, while the LDL obtains exactly for all levels below. This approximation is very similar to that used by Adams and Petrosian (1974) who investigated the Balmer decrement at densities up to = 10*^ cm~^ at Tt = l( fK . The justification for this ^proximation lies in the fact that Ç(n) rises sharply with n , being of order unity for only one or two values of n . Further, the only collision rates which are appreciable are for collisions between adjacent levels, so aU but the lowest few of the LTE levels are wholely decoupled collisionaUy firom the LDL levels. AU in aU, only a few levels are not correctly treated, and as we wiU be summing over the contributions of cascades firom aU excited states to the ground population, the net contribution of these levels near itg wiU be smaU. FinaUy, a comparison of published coUision rates for hydrogen (for example, compare Johnson, 1972 and Vriens and Smeet, 1980; or see section 2.3a or chapter m ) shows that most coUision rates are not known to better than a factor of two, even for the lowest lying states. Clearly then, no level of computational sophistication wiU give results of higher accuracy than this, and we feel justified in using a simple approximation. For ease of use with existing photoionization codes we cast our results in a correction of the form Aa(Nf,TfZ) which can be added to the published case A or case B radiative recombination coefficient of an arbitrary ion of unscreened nuclear charge Z to give a^(N g, J , , Z): a fiN ,J ,; Z ) = aiDR{T,:Z) + aj,(J',X) + itaiN „T „Z) . (2.5) 11 (b) Computational Details We began by using equation 2.4 to calculate tic. We computed for neutral species from the semi-empirical approximation of Johnson (1972) based on the Bethe approximation, and for charged species we used a simple Coulomb-Born approximation given by Bates, Kingston and McWhirter (1962). Having found for a given electron density, we calculated Aa(mg, T„ Z): The rate at which any LDL- state is populated by direct radiative recombination and by cascade from out of the LTE-states is: N,N(Z4.1)o.(Z)+ %N.(Z)A_(Z) (2.6) m =n, where o,(Z) is the rate coefficient for direct radiative recombinations to the ground state of a hydrogenic ion, calculated again aft^ Ferland (1980). The population of the m ’th LTE-level, is given by the Saha-Boltzmaim equation for a hydrogenic ion: W .(Z)=N,N (Z+1)P,(Z) (2.7) where Æ is the Rydberg constant. The rate of radiative transitions, from level m to n is ^proximately given for the hydrogenic case by: where a is the fine structure constant, is the Bohr radius, u = -{tilm Ÿ and where gi(.n,u) is the 12 bound-bound Gaunt factor: &(«,«)= 1 -0.1728(I-«)(l+«r^#i-®^-0.0496(l+-|-«+«2)(l+«)-^3„^ (2.10) This expression for the Gaunt factor is Burgess’ (1958) correction to the approximation of Menzel and Pekeris (1935), and is most accurate when m»n. Applied to states of interest in this problem, it introduces an error of about 1.5% to 2.5% for resonance transitions of the low-lying states and for cases when m = n +1. It is even more approximate for transitions between the very low-lying states, but these states do not concern us much here. Compared to the other approximations and the imcertainty in the collision rates the use of this expression introduces an acceptable level of error. While the LDL-states are populated by transitions from out of the continuum and the LTE-states, they may only return their population to the continuum by first populating the ground state. In equilibrium the total rate at which all the LDL-states, including the ground state, are populated by transitions firom the continuum and the LTE-states must equal the photoionization rate. It follows that: af(Z)=i;a»(Z)+S Z P.(Z)A_(Z) + aoj,(Z) (2.11) 11 = 1 » = 1 m = n . and from equation 2.5: Aa(n„r,.Z)=2 2MZ)A™.(Z)-fiMZ) (2.12) »=1 IB =51, m=B. One can easily calculate Aa firom the formulae provided in equations 2.8 - 2.10. The summation to infinity for Om is approximated as a sum to m = 1000. The summation to infinity for the first term of 13 equation 2 .1 2 is approximated by a sum over the first 500 levels and a continuous approximation for higher levels: M 500 2 P«(ZMm,(Z) = 2 (2.13) iRsn. Note that the many approximations and uncertainties in the data (discussed below) make this level of sophistication really unnecessary. It can be readily verified from equations 2.8 and 2.9 and firom the standard hydrogenic formula for a , that Aa scales with Z as: A a(r„ Z) = Z A a (r„ 1) (2.14) = (2.15) allowing the application of the Z = 1 case to all ions. Similarly, it is easy to verify firom equations 2.4, 2.9, and die expression of Bates, Kingston, and McWhirter, that scales with Z as: n,(N., Z) = n,Qf„ 1). (2.16) whore N, sN ,IZ ’' (2.17) 14 2.3 Results (a) Numerical Results We calculated Aa for JV, from .1 to 10'® cm”® and for T, from 1 K to lO’ IT. The results of these calculations have already been presented in part in figure 1. They are given in anotha- form in figures 2- 4, which show curves of Ao/a^ against N, for several values of Z at SCO, 8,000, and 16,000 K. Note that as r , increases, a higher and higher N, is required to produce a given ratio of Aa to a^ . The effects of three-body recombination are small at high temperature, so it seemed unnecessary to present very high temperature results here. For example, above T, ” 100,000 K, Aa was found to always be less than S% of a^i, even for N, " 10'® cm”®. The physical interpretation of our results is straightforward: For any given N„ when the mean thermal energy AT, is small compared to the ionization potentials of the Rydberg states, three-body recombinations occur so fast that the LTE-states become much more highly populated than is the case for the LDL:  typical departure coefficient (the ratio of die true population to its LTE value; see equation 3.1) for the LDL is around .5 at 10,000 K, whereas for LTE it is, of course, unity. When these large populations obtain, the rate of cascades from the LTE-levels exceeds the rate of direct radiative recombination to the LDL-states. The ionization potential scales like Z®, so for any given T,, the same situation obtains as Z increases. 15 (b) Comparison with Previous Work These results predict that in many regimes of N, and T„ three-body recombination is the dominant process. Therefore it is important to evaluate how accurate the approximations used in deriving it are. Adams and Petrosian (1974) computed the HoJHft ratio at h i^ densities and temperatures from 6,000 - 20,000 K, using a formalism only slightly more complicated than our own (they included An = 1 collision rates explicitly for levels near n^). The only portion of their results which is directly comparable to our results in this chapter is their curve of N, vs. n^. Their results and ours are given in table 1. Over the range tabulated, factors of 1.3 - 2 differences are noticeable. It appears that agreement worsens for lower densities, but it is not clear if the trend would continue if more complete data w ee available. In practice, disagreement at low densities for 10,000 K would not have any serious consequences since the three^body recombination plus cascade process is negligible here. The reason for the disagreement between Adams and Petrosian and ourselves can only be differences in collision rates. Adams and Petrosian used different collision data than we did, and in particular, used rough scaling arguments for rates involving levels above n = 9. Except when is small this usually translates into a much smaller uncertainty in Aa (as a rule of thumb, we found that roughly a facttv of 10 uncertainty in the collision rate led to a factor of 2 uncertainty in Aa). Hummer and Storey (1987) have computed total effective case B recombination coefficients for temperatures from 1,000 K to 20,000 K, explicitly taking account of all coUisional processes for H I and He II. They used the collision data of Percival and Richards (1978). Their results extend to densities as high as 10*® cm~^ for H I and 10*® cm“® for He n . We computed their value for Aa by subtracting off our own computation of Og from their tabulated total effective case B recombination coefficients, and compared this to our value for Aa. Figure 5 compares the two at 3,000 K for both H I and He II. \ \ .. Hummer and Storey’s results appear as dashed lines, ours as solid. The dotted line will be discussed shortly. Also shown on figure 5 are the total radiative recombination coefficients and a/SH ell). Considering first H I, we see that the agreement is clearly not very good. At low densities, it is especially bad, but this is probably due to the fact that our value for O/ is somewhat different firom theirs: In this regime even a percent difference in is large compared to Aa. Regardless of the explanation, the discrepancy here is irrelevant since Aa is negligible next to a^. At IV, = 10^° cm"^, the disagreement is more like 20%. While this is larger than we would have liked to have seen, it is probably as good as can be expected, given the uncatainties in the collision data. We suspect that the primary reason for the disagreement is just the differences in collision data used. For He II the agreement is extremely poor, with our values of Aa larger by factors of anywhere firom 1.5 to 3, and worsening when Aa is large relative to a^(He II). This result appears to be most likely due to differences of about a factor of 10 in the collision rates adopted by Hummer and Storey, and those used by us. We have illusüated this by recomputing Aa using vey different collision rates. We used those of Johnson (1972), which are rgrpropriate for neutrals, and scaled them for Z = 2. This result tg)pears as the dotted line. It appears firom the improved agreement with Hummer and Storey that the collision data for He n which they used are much like our coUisirm data for neutrals. We also compared our results for He n to Aa computed firom Summers (1979), who tabulated total effective case A recombination coefficients for hydrogenic ions for temperatures of 20,000 K and higher. The result of this comparison is given as figure 6 , which shows Hummer and Storey and our own data as before, with Summers’ data appearing as the dot-dashed line. The principal points of this comparison are; the substantial, but unimportant, disagreement at low densities; substantial disagreement between our 17 results and those of Hummer and Storey; and very good agreement at high densities with Summers. While the divergences we have found are disheartening, it must be noted that they reflect a fundamental problem which is not unique to the present case: Collision rates really are very imcertain for all ions. Had we approached the problem of three-body recombination in a more formally correct manner, by including all processes and solving simultaneously for all levels, we would have found uncertainties of the same order. Despite the large imcertainties, we are gratified to find agreement with all authors on the basic behavior of the three-body recombination plus cascade process over the range of physical conditions where we overlap. (c) Evaluation of Approximations Since three-body recombination gives a large enhancement to the total effective recombination rate because of the enhanced populations of the LTE states, it is important that these populations are well approximated for arbitrary ions. We have already noted that the populations of these states should not deviate firom LTE except near n^, and that these deviations are small when compared to the sum over all collisionally-populated states. The populations comptited &om the assumption of LTE then depend upon how well a hydrogenic approximation rqtroduces the «lergies of these high lying Rydberg states. From figure 7 which shows the curves Aa= for several values of Z in tire plane, it is clear that at low T, or high Z, where Aa is largest, is also large and the hydrogenic tqtproximation is very good. At higher 7,, can become as small as S, and the hydrogenic approximation is not as good. Note, howeva, that the hydrogenic ^proximation is usually used to derive the radiative recombination to excited states of heavy ions. The current treatment is thus consistent with and no worse than the prevailing treatment of radiative recombination. 18 Another problem relating to the hydrogenic a^Hoximation is that equation 2.12 involves radiative transitions to the ground and very lowest-lying states which are definitely not hydrogenic for arbitrary ions. The hydrogenic ^roxim ation should still be reasonable if the difference between the hydrogenic and true radiative rates is small in comparison to the total sum, which involves many terms that really are hydrogenic in nature. We assumed that this would usually be true, without tabulating the results for arbitrary ions. One otha possible shortcoming is the fact that we sum over an infinite number of levels whereas the highest levels of real ions are not bound because of the perturbations of neighboring ions. The highest bound level is given by (Mihalas 1978): n i x =4% 10*-Z-iV7^2j w (2.18) We verified, howeve, for representative electron densities and temperatures within our range, that the decline in radiative rates makes contributions of levels near unimportant compared to contributions fiom levels near In fact, at 1,000 K there is negligible contribution to ground state population from LTE-states above n " ICX) at any density, and this declines with increasing electron temperature, so that at Tf ~ 10,000 K there is no contribution &om states above n ~ 30. Computing Aa firom the method given in the preceding sections is much too slow for use in photoionization codes. Apparently because of the discrete nature of thejproblem, and its dependaice on 2 variables, we had great difficulty in obtaining smooth analytic fits to Aa and resorted to interpolation on the computed values. The necessary tables are too extatsive to rqnoduce here, but are available upon request firom the author, along with a FORTRAN subroutine to interpolate on them. 19 2.4 Discussion In this concluding section, we consider some classes of objects where we expect the "three-body recombination plus cascade" jvocess to be important The basis for the discussion are figures 1 and 2-4, which have been discussed in sections 2.1 and 2.3, respectively. (a) Late-Phase Novae Shells From figure 1 it is clear that at temperatures of order 100 - 500 K and electron densities of order 10^ - 10* cm~^, recombination to modaately (singly and higher) ionized species can be strongly «ihanced by three-body recombination. It is hard, however, to find instances where gas can be photoionized to even this moderate degree without also being heated to much high» tanperatures than 500 K. An instance where low temperatures coexist with moderate levels of ionization, is for the class of objects for which DQ Herculis is the prototype. The 1934 nova DQ Herculis is surrounded by a nebula of ejected gas. For this nebula, ~ 500 K has been measured firom the sharp emission Balmer jump (Williams et ai. 1978; Gallagher et ai. 1980; Williams 1982), and predicted by photoionization models using the observed continuum (Ferland et al. 1984). Temperatures as low as 200 - 500 K are a consequence of the very high abundances of the metals, which cool the gas efficiently through IR fine- structure line emission. The coolest models require the lowest densities, because at high» densities the fine-structure lines are collisionally de-excited and radiate less efficiently. A model with Ng, the total hydrog»! density, of about lO^ cm~^ produces an av»age temp»ature of 600 K. These models also piedietSutmg[Wl/73 57p«j,[0/ff3S S and52psi,and[0/y]2S jiw lines. Tbs IN HI] sad \G U!] lines may have been observed: DQ Her has be»i detected in the 60 and 100 \un bands by IRAS 20 (Dinerstein 198Q, suggesting either dust or line emission. In a future paper we plan to present the results of detailed photoionization calculations for DQ Herculis including the effects of three-body recombination, which can be compared to models which neglect it. For the moment we engage in the somewhat hazardous enterprise of speculating upon the effect of including three-body recmnbination in a successful model of DQ Her. Consulting hgure 2 we find that for 500 K and lO^ cm~^, the enhancements to the radiative recombination rate should be 6 %, 18%, and 25% for recombination to neutral, singly, and doubly ionized species. We expect that enhancements of this low order will produce some decrease in the average abundances of N m and O m , but probably will not alter the role of these fine-stmcture lines as coolants. The model would probably equilibrate at a somewhat higher temperature. If we insist upon matching an observed tempaature of 500 K, then the best-fitting model would require a lower doisity than was previously necessary. These effects are fairly small for the probable density and temperature of this object, but we note that if densities w oe only a factor of 10 higher or a few hundred degrees lower, much larger three-body recombination effects would occur. If such conditions were thought to obtain (they may in fact, given the uncerËûnty in the tneasurem^ts), dien the inclusion of three-body recombination might lead to the prediction of other, Iow a ionization lines than the N m and 0 m lines, as the principal coolants. (b) H n Regions and Planetary Nebulae. At more typical nebular temperatures (8,000 - 16,000 K), figures 1, or 3 and 4 indicate that three- body recombination is unimportant except for highly ionized species at extremely high electron densities. This fact makes it seem unlikely that three-body recombination is an important process for H EE regions and planetary nebulae. 21 H n regions have O IE temperatures typically around 8,000 - 9,000 K, but densities on the wder of lO^ cm~^ (Osteibrock 1974). Figure 3 shows that even for recombinations to ions with Z = 5, the three- body recombination rate is barely a few percent of the radiative rate. Planetary nebulae have measured densities typically of order 10^ cmbut as high as in some instances. High ionization species are also not unusual because of the the hot stellar ionizing spectra: For example, the spectrum of Kskx et al. (1976) for NGC 7027 showed strong lines &om Ne V, Ar V, and Fe VI, among others. Shields’ (1978) model for NGC 7027 predicted considerable abundances of similarly high ionization species for an ionizing stellar spectrum of T« = 166,000%. The problem is, however, that these hot central stars produce thomal equilibrium at higher temperatures. From [O m] temperatures it sqrpears that 10,000 - 1S,(XX) K is a typical temperature for a planetary nebula (Osterforock 1974). From figures 3 and 4 it would seem that only for recombinations to ions with effective nuclear charges of Z ^ S would the three-body recombination rate be ev«i of order 10% of the radiative rate, and then only for the highest densities. While it is possible that in some rare cases three-body recombination could play a role for these objects, we consider it unlikely. (c) Emission-line Regions of Active Galaxies. Historically, the mnission-line regions of AGN such as Seyfert 1 galaxies or QSOs, which exhibit broad emission lines have been divided into the low-density narrow line-emitting regions Q'Q.Rs) and the higher density broad line-emitting regions (BLRs). More recent evidmice suggests a range of densities are present in the NLR, and that pertiaps it blends smoothly into the BLR (e.g. DeRobertis and Ostobrock 1986). Th»e is also evidence (discussed below) that densities in the BLR can be much high» than were previously thought to obtain. We consider the implicatitms of the present work for both 22 broad and narrow AGN emission-line regions, first for cmventional densities and fiien for higher densities. (i) Conventional Density BlUs mANLRs. Because of the strength of narrow forbidden-line emission, it has classically been thought that densities around 10*-10^ cm~^ obtain in the NLRs of broad-line objects. Recent measurements of high 7([0 ///] A4363 )//([O III] A5007 + X4959) ratios for selected Low-Ionization Nuclear Emissicm Regions (LINERs) imply that densities as high as 10^ cm~^ obtain in at least some regions of these objects (Filipenko 1985; Carswell et al. 1984). LINERs are narrow-line objects whose mission-line regitms are possibly ionized by ntmthermal AGN-like spectrum, but it is not yet clear to what degree LINERs are similar to AGN such as Seyfert 1 and 2 galaxies (for more extensive discussion of this point, see section 62). The BLRs of Seyfats and QSOs show no forbidden lines, but datsities of around 10^'^ are implied by the observed strength of the C ///] X 1909 intercombination line, which is collisionally de-excited at higher densities. Both NLRs and BLRs are ionized by hard, nonthermal continua, which simultaneously produce strong high and low ionization lines (from O I to Fe X and Fe XI; Osteibrock and Mathews 1986). The high-ionization lines of both the BLRs and NLRs are formed in the ionized zone, for which pholoionization models predict tmperatures of order 16,000 K. From figure 4 it appears that for BLR densities of 10^^ cm~\ even recombination to neutral species will be affected at the 10% level, while the recombination rate to Fe X and Fe XI will be almost doubled. Thus, three-body recombination will be an important process for BLRs. For very dense (i.e. 10^- lO ’’ cnt~^ NLRs, a 10% effect is expected for the highest stages of ionization, and if the lower density estimates are typical of most of the NLR there is probably no appreciable affect 23 The low-ionization lines for both regions form beyond the ionized zone, in the X-ray ionized and heated partially ionized zone. Here an electron fraction as high as 50% (our calculation using the continuum of Mathews and Ferland 1987) to 80% (Kwan and Krolik 1981) wiU obtain, along with a temperature of about 8,000 K. The degree of ionization is much more modoate, however. For recombination to neutral and singly ionized species (the dominant species in this zone) the enhancements in the BLR due to three-body recombination are a more modest 20% and 40% respectively. The NLR is affected at the 5% level if it is very dense, and not at all otherwise. (Ü) High Density BLRs. Recently there has been evidence that BLRs contain at least some gas at higher densities (lO" cm~^ than indicated by /(C ///] X 1909). This evidence includes the tendency of current photoionization models to overpredict the HaJH^ ratio (reviewed in Ferland and Shields 1985) and variability studies of Seyfert 1 galaxies, which suggest that their BLRs are much smaller than previously believed (For NGC 4151 see Antonucci and Cohen 1983; Ulrich et al. 1984; Gaskell and Sparke 1986; Petorson and Cota 1987. For Akn 120 see Peterson et al. 1985; Gaskell and Sparke 1986). A density high» than the canonical value should low» the laedicted Hct/H P ratio (e.g. Adams and Petrosian 1974; Drake and Ulrich 1980; Hubbard and Puett» 1985). A high density would also keep the ionization paramet» F, the ratio of ionizing photon density to gas density at the ionized face of a BLR cloud: (where r is the distance firom an ionizing continuum source of monochromatic luminosity Lv, and 24 Vo= 329x lO" Hz is the ionization limit of HI), in the neighborhood of 1(T\ as has been required by the most successful photoionization models (e.g. Kwan and Krolik 1981), in spite of the low value of r implied by variability timescales. Against this argument stands criticism of the continuum used in the Kwan and Krolik model and the possibility that a single value of F is not appropriate to all AGN. Netzer (1987) argues that Kwan and Kiolik’s continuum is harder in the X-ray than has been observed, which accounts for the good agreement of their predicted emission-line spectrum with those observed. Since more realistic continua have difficulty accounting for the observed spectrum of AGN, it is not clear that a strong constraint really exists on the ionization parameter. Further, while equation 2.19, combined with variability studies implies a high density for Arakelian 120 (whose spectrum has never been modeled), Ferland and Mushotzky (1982) found F " 0.1 fiom their models of NGC 4151. This high value of the ionization parameter could be met by a BLR of the size implied by variability studies, with a density (Hily a little higher than the canonical value. Needless to say, our present results show that if BLR densities as high as 10" cm~^ do obtain, three- body recombinations would be important even for recombination to neutral species: Correctly accounting for three-body recombination would then be crucial to the undersfânding of BLRs. 25 Table 1 Cmiparison of N, mcm~^' «c N,{AP) N ,iC ) 10 1.93(9) 3.56(9) 9 8.33(9) 1.04(10) 8 4.64(10) 3.50(10) 7 2.24(11) 1.43(11) 6 1.34(12) 7.65(11) 5 9.64(12) 6.06(12) 4 7.74(13) 9.40(13) 3 7.40(15) 26 2 3 5 8 10 N, 10® 1 0 1 0 ® 10® 'e Figure 1. Dominant Recombination Processes for Electron Density and Temperature Regimes. This figure divides the N, • plane into regimes where radiative recombinations plus cascades dominate the pq)uWions of the lower-lying states, and where three-body recombinations plus cascades dominate their populations, fat the indicated values of the effective nuclear charge Z. The processes make equal contribution to the low-lying state populations along each curve, with three-body recombination dominating to the left and up (decreasing temperature; increasing density) and radiative recombination dominating to the right and down (increasing temperature; decreasing density). 27 Z - 5 100.00 T .» 500K 10.00 à a 1.00 0.10 0.01 Figure 2. Ao/a^ vs. (500 K). The contribution of three-body recombination to the total recombination coefficient (Aa), relative to the total hydrogenic recombination coefficient a^. is plotted against electron daisi^ for a constant temperature of 500 K and the indicated values of effective nuclear chargez. 28 T. « 8,000 K 10.00 0.10 0.01 .2 ,«3 ,_ 4 ,.5 ,«6 ,_7 .8 i/%9 ,,^10 N, Figure 3. Ao/a^ w. N, (8,000 K). As figure 2, but for a constant temperature of 8,000 K. 29 Z>8 10.00 T@ > 16.000 K 1.00 0.10 0.01 ,2 ,-3 I...® ,10 ,0,11 F io'= N. Figure 4. Aot/a^i vs. N , (16,000 K). As figure 2, but for a constant temperature of 16,000 K. 30 : I I i I— r “ T— I— I— I— 1— j— r = 3,000 K H el -10 — Goto I 1 0 ••••Cota I — Hummer 8 Storey - Il 1 0 a» (H el) A a a^(Hl) 10"'® t - 1 0 " '* fc- r l4 1 0 N, Figure 5. Comparison o f our Calculations o f the Effect o f 3-Body Recombination to those of Hummer and Storey (1987) for H I and H ell (3,000K). Our value of Aa and that derived firom Hummer and Storey (1987) are shown in absolute units (cm Vscc) plotted against electron density for H I and He II at a constant temperature of 3,000 K, Our value appears as the solid line, and Hummer and Storey’s appears as the dashed line. The dotted line is our calculation of Aa for He n computed using the collision data of Johnson (1972) which is appropriate to neutrals only: This is meant to show the sensitivity of the value of Aa to die collision data used. 31 g * (HeB) -12 1 0 T ^ - 2 0 ,0 0 0 K Goto ——— Hummer 8t Storey Summers -13 10 ,—14 10 .-IS 10 10 10 10 10® 10® to lo'® io'® Me* Figure 6. Comparison of our Calculations of the Effect o f 3-Body Recombination to those of Hummer and Storey (1987) and Summers (1979) for H e ll (20,000K). Our value of Aa and those derived from Hummer and Storey (1987) and Summers (1979) are shown in absolute units (cm^fsec) vs. electron density for H I and He H at a constant temperature of 20,(X)0 K. Our value appears as the solid line. Hummer and Storey’s appears as the dashed line, and Summers’ appears as the dot-dashed line. 32 160 ISO 140 Aa = a 130 120 80 70 60 50 40 30 20 2 3 .4 3 Figure 7: Critical Principal Quantum Number vs. Temperature for Aa = q*. This figure is intended to show the regimes of electron temperature for which the hydrogenic approximation is best The curves correspond to A a= a^ for the indicated values of Z: To the left and down (decreasing temperature, decreasing »«) three-body recombinations dominate the photoionization balance. The low temperature regime, where Aa = corresponds to large n ^, is whae the hydrogenic approximation is best m . A SIMPLE BUT ACCURATE TREATMENT OF THE HYDROGEN ATOM 3.1 Introduction Until the I970’s, die tqipioach to predicting the relative intensities of the hydrogen recombination lines had been primarily to perform very detailed calculations for isothermal, homogeneous slabs of a given Lyman series optical depth (e.g., Seaton 19596 ; Pengelly 1964; Brocklehurst 1971). In most cases the optical dqiths were taken to be either zero (case A) or infinite (case B). The justification for f ly in g such simple models to real nebulae was that for temperatures and densities estimated for H n regions (We = lOi^ cm~h Tg = 9,0(X) K) and planetary nebulae (W, = 10* -10* cm~^; T, = 15,(XX)%), the low- density limit (LDL) tgiplies. In the LDL, the only processes which are important for the determination of the populations of the lower levels, where the commonly obsaved optical lines are formed, are radiative recombinations to excited states and cascades to ground. Collisional processes strongly affect higher lying states, but these states have little effect upon the low-lying states for classical nebular parameters. When collisions can be neglected, the optical and IR recombination spectrum is highly insensitive to the physical conditions, and a detailed knowledge of the temperature and density structure of the nebula is unnecessary. At the higher densities of about 10^° (estimated from the obsaved strength of the C ///] À1909  intercombination fine) charactmstic of the broad-line emitting regions (BLRs) of active galactic nuclei (AON), collisional processes become important for the low-lying levels and the H I spectrum becomes more sensitive to the physical conditions. The so-called "Lyo/wp problem" of QSO research in the late 70 s conclusively demonstrated that accurate, detailed photoionization calculations 33 34 were needed to predict the hydrogen recombination spectrum under more genoal physical conditions: The LyaJH^ problem refers to the fact that the observed I(Lya)II(ffp) ratio for the BLRs of these objects is much smaller than standard recombination theory predicted. To explain this, photoionization models were constructed which produce theLy a emission in the ionized zone, and which produce excess Balmer line emission beyond the ionization fiont in a X-ray heated, partially ionized zone (PLZ) where L y a is thermalized (e.g. Kwan and Krolik 1981; for a recent review of the LyaJH^ problem, see Ferland and Shields 1985). Without considering the question here of whether or not the LyoJH^ problem has been fully resolved, this example clearly demonstrates that there are real objects for which the observed hydrogen spectnun is formed in two more zones of diqtarate physical crmditions. The spectrum can in such cases only be readily predicted by means of photoionization models. It is the purpose of this chuter to consider the accuracy with which existing photoionization models can predict the hydrogen spectrum, and to propose a model hydrogen atom which can significantly improve this accuracy in some physical regimes without any increase in computation time. We consider three foctors which have bearing on the accuracy of photoionization calculations: First, in low density cases the j-sublevels need to be consid^ed separately to accurately predict the spectrum. Photoionization models at present cannot expediently solve for the populatimis of as many states as this would imply, so they must assume for some values of the principal quantum numbern that its / -sublevels are mixed, i.e. they are relatively populated according to their statistical weights. We can offer no alternative to this simplification, but we show by comparison of published results that in some important temperature regimes f-mixing coincidentally gives good agreement with more detailed calculations. Second, we compare published collision rates to show that these are uncertain to about a factor of 2. This is a source of considerable uncertainty in any density regime whae collisions must be considered. We 35 can offer no solution for this problem either, except to wait for the literature to converge on a single value. Finally, there are many instances where large numbers of n-levels should be included in the calculation. For example, in the preceding c h ^ t^ we have seen that for high-densities and normal nebular tempaatures or for low-temperature and normal nebular densities, 3-body recombinations to excited states will produce significant changes in the populations of the lower-lying states. In the high density regime perhaps 10 -15 levels would be required to account for this accurately, while in the low- temperature regime the number is more like 100. Another instance whoe many levels are required is in low-density situations, where radiative recombinations to excited states above n = 10 produce cascades which influence the lower levels at the 5% level; this is a significant amount of error in this instance because the precision to which the radiative rates are known makes flight accuracy possible. Again because of practical limits on computation time, it is not possible for photoionization models to solve for so many level populations as a rule. We demonstrate below, however, that a 10-level atom can be constructed which acts as if it had 100-levels. As a quick aside, we note that there are difficulties concmfing the radiative transfer, such as to what extent the simple and widely used escape probability method is a good approximation (for discussion of this point see, e.g. Canfield and Puetta 1980; Collin-Souffirin et a l. 1982; Hubbard and Puetta 1985; Faland and Shields 1985), or to what extent case B actually obtains (see chapter V). We will not be further concerned with problems of radiative transfer in this chapter. 36 3,2 Factors Influencing the Accuracy of Calculations of the H I Recombination Spectrum. (a) I-Mixing One of the most widely used simplifications for predicting the hydrogen spectrum in photoionization modeling is to assume that the /-sublevels of a given n-level are "/-mixed.” As will be discussed at length in section 3.3, it is always possible to simplify the population equilibrium equations by averaging any number of levels together to form fictitious levels in a manner which formally satisfies the equilibrium equations. This is done by taking the average of the transition rates of the individual levels, weighted by their relative populations, to form rates for the fictitious levels. Obviously, the trick is to know what the relative populations of the component levels are before solving the equations for the complete set of level populations. For an n-level which is /-mixed, it is assumed that /-changing collisions occur much faster dian radiative de-excitations, driving the / -sublevels to their LTE populations relative to one another. For degenoate / -sublevels, this means that they are relatively populated according to their statistical weights. Using the simplifying assumption of / -mixing is also called the n -method, and using partial / -mixing is also called the nl -method. Full /-mixing (all /-levels mixed) only obtains for vay high densities because of the very high radiative de-excitation rates for the low levels. Except in the case of zero-density, however, there are always states which lie high enough for /-changing collision rates to exceed the radiative de-excitation rates. The lowest level, Ufa,, for which /-mixing obtains is that level for which the rate of /-changing collisions equals the rate of radiative de-excitati(m (actually, this level and levels of comparable n are neither fully /-mixed nor fully unmixed, but in most cases it introduces negligible error to ignore the 37 intermediate cases). The collisional rate is proportional to the density, therefore n^, is a function of density. In table 2 we reproduce the results of Drake and Ulrich (1980), who give versus N, for the lowest 7 levels of hydrogen. Earlier authors who discuss this point are Pengelly and Seaton (1964) and Brocklehurst (1971). In most photoionization models, it is simply not possible to avoid assuming /-mixing even for very low levels for which it is clearly false, because of the tremendous increase in computation time resulting firom including so many levels. In our model below we will use f -mixing for all levels except n = 2. As table 2 makes clear, this degree of /-mixing (which we will refer to as "nearly-fuU" / -mixing hereafter) is never formally true for any real nebula except those as dense as the BLRs of AGN. Thus, in the majority of cases, full /-mixing is an approximation to die real, partially /-mixed case. How reasonable an t^proximation it is depends upon the application. Nearly-fuU /-mixing is most seriously wrong for the LDL, where collisions are negligible except for vay high levels. Tables 3 and 4 compare for cases A and B respectively, the effective recombination coefticient for H p, and the ratios of selected hydrogen lines for no /-mixing and full /-mixing. The n- mediod results are computed from the results of Seaton (1959b) at 5,000, 10,000, and 20,000 K, and taken from Faland (1980) for 500,1,000, and 2,500 K. The nl -method results come from Martin (1987). Tables 3 and 4 bring out several interesting points. In the vicinity of 10,000 K, full and partial /- mixing happen to agree to within a few percent for case B for the Balmer lines, but for case A th^e are unaccqttably large diffaences. At low temperatures (T, < 1,000 ÜT) there are also unacceptably large differences even for case B. The M-method results agree so well with the case B n/-method results at 10,000 K because for this temperature the relative populations of the /-sublevels for case B and for /-ntixing are very similar. In 38 full /-mixing, the assumption that the /-sublevels are populated according to their statistical weights means that they all have populations of about the same order. This is also true in case B for 10,000 K for two reasons. First, the p-states have populations of the same order as the other /-states, because the p - states cannot decay to ground. This transition is always the fastest transition for a given value of n , with the result that when it occurs (as in case A) it leads to very low p-state equilibrium populations (see e.g. Pengelly’s plots of the departure coefficients vs. /). Second, the recombination rate coefficients to the /-sublevels at 10,000 K relatively populate the sublevels in a manner very similar to /-mixing. As the temperature decreases, the relative rates of population through radiative recombination change (see the tables of Burgess 1964), which is why die n - and nl -methods fail to agree at low Tg. Note also that while the Balmer lines agree well for case B and 10,000 K, the Paschen a Une does not Assuming / -mixing, the /(P a)//(//p ) ratio must be constant with tMnperatuie, because both lines arise out of the same upper state. If we do not assume /-mixing, however, the ratio varies with temperature as the different sublevels of the upper state acquire different relative populations. All oth^ things being equal, it is possible to get good and //y predictitms at any density from the n-method in the usual nebular range of 5,000 - 20,000 K, for case B only. This is extremely fortunate, since this is the most important appUcation. (b) Atomic Data. For hydrogoi, of course, the radiative rates and the energies of the levels are, in principle, known exactly. In practice, the riata for the iqrper levels are subject to a smaU amount of error. To the best of our knowledge, exact radiative rates are only available for transitions begiiming firom levels of n = 2 0 or less (Green, Rush, and Chandler 1957) for and fircm n =20 or loss (%%so, Smid:, and Glenncn 1966) for the /-mixed case (A^g). To go to higher levels one must either calculate the rates quantum 39 mechanically, or use the ^proximate gaunt factors given by Burgess (1958; originally given by Menzel and Pekeris 1935, albeit with an error which Burgess corrected; see equation 2.10), which differ by as much as 2.5% when compared to the exact values for highest values of n' and n which are tabulated. Errors of this order would only be noticeable in the LDL where the larger errors in the collision data are not noticeable, and even there they should have a negligible effect on the emissivities of lower-lying lines which are of the most interest The biggest obstacle to predicting an accurate hydrogen spectrum is that the crucial collisional rates are all vay much in doubt Table 5 presents collision rates at 10,000 K drawn from a number of sources in the literature. It is immediately evident that these rates vary substantially (about a factor of 2) from author to author. Furtho', this is at a single temperature and says nothing about possible uncertainties regarding the temperature dependence. We have already seen in ch^ta II that when 3-body recombination is important, uncertainties of a factor of 2 in the collision rates introduce at least 2 0 % uncertainties in the total recombination rate. If the larga discrepancies found there for He II are due to iimate uncertainties of a factor of 10 in the He n collision rates and not an error on the part of either Hummer and Storey (1987) or ourselves, then factors of 2 and 3 uncertainties occur in the resultant level populations. The imcertainty in the collision rates is, of course, most important in those density and tempoature regimes where collisional and radiative transitions occur with similar rates. At higher densities where collisions dominate LTE is established and the level populations are known to the same precision as the energy levels. At lower densities, where collisions affect the higher levels, the imcertainty in the collision rates will affect the determination of quantities like r/„, but the lower levels are insensitive to «to, unless it is less than about 10 (W, " 10’cm~^; Brocklehurst 1971). 40 (c) Truncation of the Hydrogen Atom All photoionization calculations must truncate the hydrogen atom at srnne n -level if the calculation is to remain expedient. The amount of error this introduces dqtends upon the application. Calculations of the hydrogen spectrum in the LDL have historically been extrapolated to an infinite number of levels (e.g. Seaton 19596; Pengelly 1964; Brockldiurst 1971; and most recently Hununer and Storey 1987; and Martin 1987). From the 10-level n/-method calculations of Searle (1958) and our own R-method calculations we find at lO.CKX) K that 10 n-levels will correctly predict the HP emissivity to 5%, and 20 n -levels will predict it to about 2%, providing the total recombinafion coefficient is piesorved (i.e., the sum of the recombination coefficients for all the levels which are included equals the recombination coefficient for an infinite number of levels; in practice the recombination coefficient to the top level is adjusted to make this so). While initially, detailed calculations were necessary to leam how accurate less detailed calculations would be, the primary motivation for continuing with infinite level calculations is to exploit the high precision possible in radiative rates. Clearly calculations of such detail cannot be done in photoionization modeling, at least not as a rule. Even neglecting all coUisicmal processes on the high-lying states and solving recursively for the level populations is too time-consuming when it must be dCHie repeatedly as the model iterates on a temperature structure at each of the 100 or so points in the nebula. Thus, photoionization models have, to date, been forced to accept the 5% or so errors introduced by simple calculations. While this error is small, it can be important in abundance studies (see chapter V). In any situation where collisions strongly affect the Iowa levels, of course, this level of error is trivial. Infinite level LDL calculations assuming zero-density have recently been done by Martin (1987), who extaided fiiem to the low-tempaature {T, = 100 K - 500K) regime. As we have sear in chapter n, in 41 this régime 3-body recombinations to levels higher than 30 and cascades firom there become important for the populations of the lower levels at densities of 10^ an~^ and higher: We doive this estimate by noting for SCO K, the density and critical quantum number rig at which the rate of cascades firom the coUisionally populated states equals the rate of direct radiative recombinations to the lower states on figures 1 and 7. For these values we see a factor of 2 enhanconent in the total recombination coefficient, but in fact appreciable enhancements are seen at densities as low as l(f cm~^ (see figure 2). At these low densities, the important cascades originate in much higho states, so 3-body recombinations to states as high as about n = 100 - 200 are important. While chapt^ n is concerned with the total recombination rate, similar changes in the recombination spectrum would be expected. This process would be entirely missed in any photoionization calculation which truncated the atom at 10 or evoi 20-levels. As it is, howew, even including 1 0 0 n -levels in the calculation will not produce a fully accurate spectrum in this temperature regime since any photoionization calculation will have to assume /-mixing, which we have seen to be a poor approximation at low temperatures. Nonetheless, including the additional n-levels at least gives some insight into the probable consequences of 3-body recombination at low temperatures. In the high daisity regime (N, = 10^° cm~^ and high^) where direct collisional transitions between the low-lying states become important, all previous calculations have used a very limited numb^ of levels. For example, Krolik and McKee (1978) have used 12 n-levels without assuming /-mixing for densities up to 10" cm~^, while Adams and Petrosian (1974) and Drake and Ulrich (1981) used 20 n- levels for densities as high as 10" cm^ assuming respectively full and partial /-mixing. All diese calculations were for single slabs of homogeneous conditions. In {diotoionization models Kwan and Krolik have used a 6 n -level, fully /-mixed model. For the h i^ density (N, > 10" cm~^) regime at mdinary nebular temperatures ( 8 ,(XX) - 16,000 K), 42 our results in chapter n indicate that recombinations to and cascades firom highly excited levels are an important process for populating the lower levels (see figures 3 and 4). The excited levels ûom which appreciable cascades occur are not readily discernible from the figures givat in chapter U, but firom our tabulated results we find that cascades from levels as high as n = IS - 20 result in a significant rate of population of the lower states. The 20-level calculations of previous authors are therefore almost certainly complete. The 12-level calculation of Krolik and McKee is probably just adequate for the highCT density regimes. Kwan and Krolik’s calculation is also correct because they only considered ~ 1 0 ^° whoe collisional population of the high-n states enhances the total recombination rate by only about 8 %; givai that die calculations of chapter n are as imcertain as any calculation involving collisions, this is an entirely negligible change. Had they tried to extend their model to higher densities, howev», their simplified model of the hydrogen atom would have beoi incomplete. If temperatures dropped to around 1,000 K in their model, 3-body recombination to states as high as 20 would again have been important, and their model incomplete. The high daisity regime is thus one where the truncation of the atom could be a serious omission. At 10^° the truncation can be serious in, for example, the PIZ of a BLR, where tanpoatures are less than 10,000 K. If the recent evidence for higher BLR densities than 10^** cm~^ is borne out (see section 2.4c) then the truncation of the atom could be seious at even 20,000 K. In this regime it would be extremely useful to be able to include the additional levels in a photoionization model, since I -mixing would actually be formally correct at these densities. 43 3.3 A Simple Model Hydrogen Atom As discussed above, there are a number of T, - N, regimes where we need to include a vesy large number of n-levels to accurately predict the spectrum of HI. These regimes include very low temperatures and moderate densities (7^ = 500K or less, N, = lO^ cm~^ or more), nebular temperatures and high densities (15,000 K and N , = lO " cm~% and any regime of temperature and density for which the LDL applies, when accuracies exceeding 5% are sought If the assumption of f-mixing carmot be done away with, the inclusion of the additional n-levels will not lead to an accurate spectrum at 500 K, nor to accurate case A results at any temperature, but will at least give a sense of the sort of changes 3- body recombination to excited states will produce on the spectnun. The practical difficulty is to find a way to include a large number of n-levels in a photoionization model without increasing computation time. We hae present a model hydrogen atom which will do this. The basic idea of our model is to include up to ICX) levels by averaging the majority of them together into 3 "fictitious" levels, retaining only 7 real levels. The fictitious levels will mimic the bdiavior of their component levels, but as the total number of levels whose populations we must solve for is only 1 0 , there will be no increase in the computation time. In section (a) we outline the model and its 10 levels. In sections (b) and (c) we show how the populations, rates, statistical weights, and enagies of the fictitious levels are defined. In section (d) we discuss our treatment of the temperature dependence of the collisional rates, and in section (e) we compute the averaged atomic data for the fictitious levels under various assumptions about the level populations. 44 (a) Model Hydrogen Atom We modeled hydrogen with 100 n-levels. We assumed all levels wore /-mixed excq>t level n=2, where we treated 2p and 2s separately. As discussed above, this assumption is not formally correct at densities below 1 0 ^® cm~^. Levels with n ^ 6 were considered explicitly, but the remaining 94 / -levels were weighted by their populations and averaged together to form 3 fictitious levels, 7’, 8 ’, and 9’ (hereafter primed quantum numbers refer to the fictitious levels and unprimed quantum numbers refer to real atomic levels). Level T is the average of levels 7 - 10, level 8 ’ the average of levels 11 - 20, and 9* the average of levels 21 - 100. Using these averaged levels it is possible to obtain the 10 level populations under a wide range of circumstances by eq)licitly including all rates, collisional and radiative, in a simultaneous solution. Since the weighting factors (i.e., the level populations) vary with temperature, density, and resonance line optical depth (i.e., case A or case B) clearly one set of averages is not applicable to aU conditions. We chose to optimize the fictitious atom for the range 5,000 K - 20,000 K, for / -mixing (high densities), and for case B. This is the most widely tqtplicable case. In practice this meant tabulating values at the mid-range (10,000 K) and fitting simple powa-laws to the quantities which vary strongly with temperature. (b) Averaging of the Rates and Populations It is easy to show from examination of the equilibrium equations that they will be exactly satisfied by any fictitious level whose population is the sum of the populations of its component levels, and whose transition rates are averages of the transition rates of its real, component levels weighted by their relative populations. We write these avaages here in terms of the departure coefficients b, of a component level 45 n , defined as the ratio of the true pqpulation to its LTE-pqpulation: wha^e can be expressed variously as: Xn = = 1.4388 ). (3.2) kTn^ Tn^ T and whae R is the Rydberg constant, v(cm~*) is the fiequency in wavenumbers, and all other atomic constants have their usual meaning. For convenience we will now ejqplicitly show the averaging equations. For a fictitious upper state m' which is the average of real levels mo to m i and fictitious Iowa state n' which is the average of real levels Hq to to Ri, the radiative de-excitation rate the collisional de-excitation rate and the collisional excitation rate ae: 2 2 b„mhxp(x„)q„^ w ür:------<3-4) 2 b„mhxp(x„) 46 QriM = ------7. ------(35) £ 6 „n*exp(xJ The population of a fictitious level n' is: N., = 4.1416* 1(T»‘T-*2 2 6.m^exp(*J (3.6) nsn» Essentially the same results hold for collisional ionization rates q^>^, the 3-body recombination rates and the radiative recombination rates a^\ Z i»«exp(*,)^,^- 9 ^ . = ^ ------(3.7) Ê *»exp(*,) *=*# 4»,t'= Z * a (3.8) »=»« a»' = S “« (3.9) M=a# « Note that in equations 3.8 and 3.9, the continuum is effectively a single level, and so averaging over it is unnecessary. 47 (c) A v^ging of the Energies and Statistical Weights We defined the energy h \ ^ and the statistical weight In TE, detailed balancing obtains. Each process is balanced by its invose, and so for any real levels m and n the following equation must apply: =^m?m« (3-10) Also in TE, N„fN^ is equal to the relative Boltzmann population: where exp(.ï„) (3.12) Hence in TE, the collision rate coefficients are related by: <0 - = — 9 m.»exp(-j^) (3.13) By the usual arguments, equation 3.13 must hold undo’ all circumstances if it is to hold under TE. 48 In TE, equation 3.10 must also hold for the fictitious levels, with the result that: Z m„exp(x„) = — 9 mV (3.14) Z M»exp(x„) Equation 3.14 will be of the same form as equation 3.13 if: Z ®« (3.15) m=m« and Z w«exp(x„) exp(x„0 H (3.16) Z Ml—fR# with expressions of the same form for n '. For collisional ionization and 3-body recombination it is easy to show that the definitions of equations 3.1S and 3.16 also presave the correct relationship between these coefficients, which follow again through considerations of TE: = 4.141& 10r‘« ^ cxpOv) (3.17) 49 The results of all these averages will be presented below, after we have discussed appropriate choices for the dq)arture coefficients. Only the averages of the statistical weights and the averaged energies do not depend on the departure coefficients. Table 6 presents the results of these averages, done at 10,000 K. (d) Temperature Dqiendence of CollisicHi and Radiative Recombination Rates There are two ways in which we must craisida^ the treatment of the temperature dq>endence. The departure coefficients and Boltzmann factors depend on temperature and densiQr, and so introduce these dependences even into averages over temperature independent quantities like A ^ . This will be dealt with in the section following. Here we deal with the innate temperature dependence of the collision rates and the radiative recranbination coefficients for the real levels. We can express the collision rates in terms of collision strengths. The collision strength, between real upper level m and real lower level n is defined via the collisional de-excitation rate coefficient by: “'m Collision strengths are most useful for ions, where they are nearly constant with temperature. This is not the case for a neutral atom. For example, collision strengths calculated from expressions given in Johnson (1972) vary more strongly than do the rate coefficients themselves. To account for this, we fit the collision strength with a simple power-law of index p. cmstrained to be e:mct at 10,000 K. These fits worited well over the desired range of 5,000 - 20,000 K with typical errors of a few percent and in the 50 worst cases no more than 10%. The fits become much worse, however, outside this temperature range and should not be used there. With these fits to £2^, the collisional de-excitation rate may be written: 8.629% 10-* Q^ilOfiOOK) r / ««.» =------(3-19) whoe we have absorbed the factor of from equation 3.18 into P; and where r^ is the temperature in units of 10^^. Given the definitions for the fictitious rates in equations 3.4, 3.5, 3.15, and 3.16, it is straightforward to show that equation 3.18 will also hold for the fictitious levels. Because the fits to the fictitious levels vary slightly with different assumed departure coefficients in the averaging, and because we desired to keep the collision data for both real and fictitious levels together, we defer presentation of these fits until the next section. For collisional ionization and 3-body recombination an analogous ’ionization strength’ can be defined: 4.1416xir’£2,^ ^ ^ ^ ----- exp(-%„) (3.20) Again, we fit the temperature dq)endence of this ionization strength with a simple power-law of index y. The result is: 4.1416% i(rn.,(io,oooAr)r,;y , ^ = ------— ------exp(-%„) (3.21) wh«e the factor of T ~ ^ has been absorbed into y. Again, because the values of y for n' = 7', 8 ', and 9 ' 51 vary depaiding upon the assumed values of the departure coefficients, we defer presenting them until the next section. For the radiative recombination coefficients we calculated their values for the real levels and their averages for the fictitious levels at tempaatures over the range 1,250 K to 1,256,000 K, by the method of Seaton (1959a), but using the gaunt factors of Karzas and Latter (1980). For levels 2s and 2p we took the coefficients from the tables of Burgess (1964). We fit the recombination coefficients at each value of n or ttf with a function of the form: logio(a„r) = ao(rt) + ho(nXlogio(r) - Co(n))* (3.22) The values of ao(n), ^nd Co(n) are given in table 7. The fits have been constrained to be exact at 10,000 K, and deviate about 1 - 5% at othor tempaatures. Note that Og, and have different dependences, as discussed in section 3.2a above. (e) Choice of 6 ^ and Results While the equations presented in the preceding two sections are formally correct under any circumstances, they are only of utility if the relative populations of levels 7 - 100 do not change much as physical conditions change. Clearly all the averages vary to some extent with and T, dqiending on the former through the departure coefficients, and on the latter through both the departure coefficients and the Boltzmann factors. The departure coefficients also depend upon optical depth, being very différait in case A than case B. By avaaging togetha levels which are reasonably close in energy (e.g. levels with n ^7) we 52 minimize the variations due to the Boltzmann factors. These variations are most serious for level 7’, which has the greatest difference in energy across it. While the absolute values of the departure coefficients change considerably with temperature, density, and optical depth, averaging ova: levels closely spaced in energy exploits the fact that their relative values over a fictitious level are more nearly constant (e.g., notice in the tables of Seaton 1959b that the departure coefficients for the high-n states are very slowly varying for both cases A and B). To investigate the manner in which the averages over the fictitious levels change with density, we performed the ava^ges for both case A and case B in 5 different cases, corresponding to 5 different densities. These averages were all done at 10,000 K. Over a wide range of electron densities and temperatures the populations of the highly excited levels should be given by their LTE-populations. The level where b, = 1, referred to as n^, is found by equating radiative rates and collisional rates for the level immediately below it W.-Ç(n,-l) = l (3.23) where (3.24) m=l Below tic we assumed that no collisional processes occurred, so that the low-density limit (LDL) tqpplies. In the LDL, b. is given by: 2 m%exp(%^)A,« + a. 4.1416* 10-^*r-^2/j2g^p(x„) 53 where ^ -= 2 ^ » « + P(tM„.i (3.26) m=2 where P(x) is the escape probability of arescmance photon, equal to unity for case A and zero for case B. The 5 cases investigated are given by ric= 7,11,21,101, and <». Using the collision data of Johnson (1972) to calculate -1 ) at 10,000 K, we found that these values of «g correspond respectively to densities of 1.4* 10", 1.4* 10’, 2.6* 10®, 1.3, and of course, 0 cm~K The results of the averaging may be found in tables 8-34, along with adopted data for the 7 real levels, and power-law indices for fits to all collision rates. All collision rates were calculated from the expressions of Johnson. Since these expressions are just for transitions between f-mixed levels, we assumed for collisional transitions and collisitxial ionizations involving levels 2s and 2p that the strengths for the component levels summed to the total strength for level 2, and that they were in the ratio 1:3, the ratio of the statistical weights of 2s and 2p . For the radiative rates we used the gaunt factors discussed in section 3.2b above. Radiative rates from levels with n < 20 to 2 j o rlp were calculated from the rates of Capriotti (1964), or else computed from the tables of Green, Rush, and Chandler (1964). Transitions to these same lower levels from levels with n >20 were again assumed to sum to the total rates A, ; and to be in the ratio of the statistical weights of 2s and 2p . The collision strength for transitions between levels 2s and 2p , and the rate for the 2-photon 2 s-ls decay process both came from Osterbrock (1974). Since protons are primarily responsible for the 2s-2p collisional transition, we related the strength to the electron density by assuming a helium abundance by number of 0.1. Because Ostobrock cmly quotes the collision rate for 2s-2p a t 2 temperatures, we made no attempt to fit a power-law to this rate. 54 Note that the maximum deviation in the final, averaged rates between cases of Mg= 7 (all the fictitious levels in LTE) and = <» (all levels in LDL) was at most 20%, and most often 10%. Similar deviations exist from case A to case B. To investigate the dependence of the avoages on changes in temperature we computed average radiative transitions at temperatures from 1,0(X) K to 1,000,000 K assuming LTE (6 , = 1), and compared the values to the value at 10,000 K. These results are not presented here, but we found that the temperature dependence of the Boltzmann factors introduced an error of order 5% fw level T and smaller errors for the 2 higher fictitious levels over the range of 5,000 -10,000 K. At very much lower temperatures (e.g. 1,000 K) the deviation is as bad as 30% - 40%, while at higher temperatures the ^ro ach of the Boltzmann factor towards unity leads to a constant deviation of less than 4%. By contrast, when we computed avaages assuming the LDL with case B, the temperature variations were less severe for the Iowa* fictitious levels than the higher ones. Avoages for level 9’ changed by as much as 20% at 1,000 K and at high tanperatures, but over the range of 5,000 K - 20,000 K the avaages w ae constant to within 5%. Avaages for the other fictitious levels changed by no more than 5% with temperature. 55 3.4 Comparison With Previous Authors As a check on the accuracy of our model hydrogen atom, we compared its predictions with the single slab calculations of Seaton (19596), Adams and Petrosian (1974), Hummer and Storey (1987), and Martin (1987). (a) Equations for a Photoionized Slab. For a 10-level atom, the equilibrium equations of the n *th level are: 10 « -I Nn(A^+N, 2 (3.27) 10 fi-1 10 m=w+l m=l m=a+l where is the photoionization rate from level n ; and whmre refers to de-excitation if m > n and excitation if m < n ; and similarly for q ^ . In terms of departure coefficients, equation 3.27 is: 10 M—1 n\exp{x„)(A^+N, % + & E f». (3-28) m=R+l m=l 10 »-l m=n+l 111=1 10 . jV 2 56 Of course, there are 10 such equations for a 10-level atom. We used the rates given in the preceding section. We have already noted for the several values of for which averages were done that the differences were small. We chose to use the averages appropriate to case B and the LDL, because this would optimize our results in the low-density cases, while in the h i^ density case, the uncertainties in the collision rates exceed the differences in the averaging. This left only the photoionization rates to be specified: Usually the photoionization rate is negligible compared to the radiative de-excitation rates, so this term is kept only for the ground state. The rate coefficient is givai in genaal by: “ L *1 = f - — T— a vexpHv)dv (3.29) Vo■' Aiar^hv where Vq = 3.29x 10*^ Hz, Ly is Ac monochromatic luminosity of the ionizing source, r is the distance separating the ionizing source fiom the gas, a y is the photoionization cross-section, and Xy is the optical depth to photoionization. In what follows we will use a simple approximation to this rate. We assume the gas is optically thin, s o X y = 0. hi this case, equation 3.29 can be written: “ L d>i = Ï (3.31) “ I 57 and where the ionization parameter, F, is given by: We now assume <0v> is equal to its value at threshold, which is 63xlQT^*cm\ and that Ng = 12Nff. The first assumption is most correct for spectra with sharp exponential cutoffs, and the second is correct if the gas is fully ionized. Given these assumptions: While this is highly simplified, the exact rate is only important if we desire to know the neutral fiaction (we will be using as our free paramétra'), or in determining if densities are high enough for excitations from n = 1 to be fast relative to the photoionizations. In all cases we will use F = .01, for which it may be verified that <&i is much larger than the rate of collisions from n = ltOR=2forall temperatures less than or equal to 20,000 K and for all densities which we will consider. The system of equations can be solved immediately for the 10 level populations by any standard matrix inversion routine, as in Bevington (1969). The predictions which most interest us are the effective recombination coefficients and Hy. These are given by: a ^ ( / / a ) = 1.638% 10 ^"(J^^ 6 3 exp(xg) (3.34) a ^ m = 5.5789% 10-"*f;^%4exp(%4) (3.35) a^(/7y) = 2.6196%10-%-3/26gexp(%;) (3.36) 58 (b) Comparison to Low Density Calculations with the n-Method. We expect that if our formalism, our fits to the radiative recombination coefficiaits, and our averages are correct, we should be able to exactly reproduce the results of more detailed n-method LDL calculations. Seaton (19596) presents die most complete calculations of this nature. Tables 35 and 36 give, for cases A and B respectively, the departure coefficients obtained solving the above system of equations, and those of Seaton for temperatures of 2,500 K, 5,000 K, 10,000 K, and 20,000 K. It will be noted that the agreement is exceptional. For n =4 at 10,000 K the agreement is about .3%, and at 20,000 K about .8%. At lower temperatures, as expected, the variations of the Boltzmann factors in the averages and the errors in the recombination coefficient fits produce a maximum error of 3.7% at 2,500 K. It will also be noted that the case A results are just as good. This confirms that while the case A and case B departure coefficients for the high levels are very different, they both vary in about the same way. (c) Comparison to High Density Calculations. From densities of 10® - 10'° cw“®, we compared our results with those of Hununer and Storey (1987), who computed the case B spectrum in the r/ -method for an infinite number of levels, allowing fully for I -changing and n -changing collisions. At densities of 10® cm~^ and higher, we compared our results to those of Adams and Petrosian, who used the n -method and a limited number of levels for cases A and B. Both authors neglect excitations from n -1 , which is equivalent to assuming a high photoionization rate. In order to compare our results to theirs, we have made similar assumptions. Figure 8 shows the deviation of the caseB Ap effective recombination coefficient &om its zero- density value as given by both our calculations and those of Hummer and Storey at 10,000 K. This 59 quantity reflects collisional effects such as three-body recombination to the higha- levels, and eliminates the one percent or so difference between the n- and nl-method results for a^(^P ). The agreement is good, although it must be noted that we have made several changes in our method to make this so: For example, used the collision data of Vriens and Smeet (1980) taùiex than Johnson’s because this is closer to what was used by Hummer and Storey. We also neglected collisional excitations firom n = 2 to conform with Hummer and Storey, even though they themselves stress that these are important in case B at densities as low as 10^ cm~^ for typical nebular tanperatures (the precise value depends actually on the total optical depth in Lya). To show the importance of neglecting this, we also show the case where these collisions are not neglected for 1 ^ ,0 = 10® in figure 8. Table 37 - 42 give for cases A and B at 6,000,10,000, and 20,000 K, the departure coefficients and the effective recombination coefficients which we obtained for Hct, Hfi, and Hy, as well as the J/o/i/p and HylH^ ratios in the high density limit (10® cm~® and above). The latter two quantities, plus a^ff (H P) may be compared to the results of Adams and Petrosian (1974), given in tables 43 and 44. It will be noted that at 10’° c/n~® our results for a ,j(//p ) agree well with Adams and Petrosian for either cases A or B. This is as expected, because collisional transitions are very minor here and the radiative data we used should agree with theirs. As densities rise we find good agreement in the high density lim it In intermediate cases (densities around 10^^-10^® cm~^) we can disagree by factors of 1.5 - 2, although usually we agree better. We interpret this as due to a difference in collision rates, especially those which involve direct collisions to J/p. Unfortunately we were unable to check this directly because the reference for Adams and Petrosian’s collision data (Adams 1973) was unavailable to us. 60 Comparison at Low Températures The final case we consider is the low-temperature case. Martin (1987) has calculated the zero-density recombinafion spectrum for the low temperatures which are applicable to late-phase novae shells (see chaptCT D). We chose for comparison, his results for cases A and B at 125 K and 500 K. Our results are given in tables 45 - 48. and his are given in table 49. We presait a ^ (tf P) at densities from 10 cm~^ to 10^ cm~^. Only the low density case is directly comparable to Martin’s results. As expected, our value is 80% larger at 500 K and 66% largar at 125 K. This is predominantly the result of using the n -method: Comparison to Ferland (1980), who computed ci^(H p) in the n-method at 500 K (repeated here in table 4) shows that our calculation is about 12% off firom his more detailed calculaticms. This is due to the poorness of the temperature fits and the variation of Boltzmann factors in this regime. Our main puipose in presenting our low-tempoature results in q)ite of the fiact that the n -method is so inappropriate here, is that they demonstrate that at densities as low as 10^- 3-body recombinations to the highly-excited states will have an important effect on the /f P emissivity. This result is also probably on firmer footing than might at first be thought: Note that for the uppermost fictitious level, the n -method is not inappropriate, especially for densities above l( f cm~^, so that we can be reasonably certain that we have correctly estimated the total rate at which then =4 level is populated by cascades from this level (subject, of course, to the uncatainties in the collision data). The exact effect on the //p emissivity depends upon exactly how the / -sublevels of n = 4 get pq>ulated by these cascades, but we suspect that our estimate will be not be found to be too 6 r off when more detailed calculations becmne available. 61 Table 2 TV, (in cm "®) for Hte (AAa Drake and Ulrich 1981). T=l%10*f T = 2%l(y»% nim N, N, 2p 1.6(12) 2.0(12) 3 2.5(9) 3.1(9) 4 5.7(8) 7.4(8) 5 1.4(8) 1.9(8) 6 3.7(7) 5.0(7) 7 1.1(7) 1.5(7) 62 T ables Effective Recombination Coefficients and H I Line Ratios for Full (jifa, = 1) and No l -Mixing (n,* = <»); Case A. Te «ta (.cm~^lsec) HaJH^ W8///P PaJH^ 5,000 K 1 2.45(-14) 1.84 .607 .400 .277 oo 3.739(-14) 3.09 .459 252 .557 10,000 K 1 1.36(-14) 1.91 .589 .378 .277 oo 2.019(-14) 2.90 .473 .264 .463 20,000 K 1 7.18(-15) 1.99 .569 .356 277 oo 1.028(-14) 2.67 .484 .272 .391 Table 4 Effective H p Recombination Coefficients and H I Line Ratios for Full (Wfa, = 1) and No / -Mixing (n^, = e o ) ; Case B. Te «fa. (cm~Vsec) HaJH^ Hy/H^ /75///P /•o/Z/p 500 K 1 3.21(-13) 2.50 OO 2.606M 3) 4.22 1,000 K 1 1.95(13) 2.52 oo 1.690C-13) 3.79 2,500 K 1 9.79(14) 2.57 oo 9.040(-14) 3.33 5,000 K 1 5.356(-14) 2.63 .600 281 277 oo 5.37(-14) 3.07 .458 251 .419 10,000 K 1 3.017(-14) 2.71 .506 298 277 oo 3.03(-14) 2.88 .468 .259 .343 20,000 K 1 1.56(-14) 2.79 .491 282 277 oo 1.610(-14) 2.76 .475 264 .286 63 Table 5 Comparison of Collision Rates (in cm ^Isec ) from the Literature. (Transitions from level m to n ; c denotes transitions to the continuum) m - n DU VAL Johnson VS Aggarwal 2s -1 1.14(-8) 8.67(-9) 1.12(8) 4.76(-9) U 2 (-8 ) 2 p-\ 7.40(-9) 5.02(-9) 7.47(-9) 3.17(-9) 7.03(-9) 3 - 1 7.17(-10) 6.18(10) 7.18(10) 4.24(-10) 1.67(-9) 4 - 1 1.48(-10) 1.40(10) 1.48(10) 1.05(-10) 3-2s 5.62(-8) 6.47(-8) 6.33(-8) 1.97(-8) 3-2p 1.96(-7) 1.94(-7) 1.90(-7) 5.91(8) 4-2s 6.09(-9) 6.03(-9) 7.24(-9) 2.83(-9) 4-2p 2.28(-8) 1.81(-8) 2.17(-8) 8.50(-9) 4 - 3 1.28(-6) 2.56(-6) 1.29(-6) 5.14(-7) 5 - 4 1.22(-5) 4.55(-6) 1.99(-6) 6 - 5 3.82(-5) 1.27(-5) 5.72(-6) 1 - c 4.88(-16) 5.80(16) 4.67(-16) 1.45(15) 2 - c 1.75(-9) 1.53(-9) 1.65(9) 3.03(-9) 3 - c 6.58(-8) 6.51(-8) 6.08(-8) 1.18(-7) 4 - c 3.72(-7) 4.08(-7) 3.35(-7) 6.80(-7) 5 - c 1.09(-6) 1.31(-6) 9.49(-7) 2.0U-6) 6 - c 2.35(-6) 2.90(-6) 1.96(-6) 4.32(-6) References: (Aggarwal) Aggarwal (1983); (DU) Drake and Ulrich (1980); (Johnson) Johnson (1972); (VAL) Vemazza, Avrett, and Loeser (1981); (VS) Vriens and Smeet (1980) 64 Table 6 Real and Average Energies and Statistical Weights for Real and Fictitious Levels. n v(cm~^) to- I 109,737.35 2 2s 27.434.34 2 2p 27,434.35 6 3 12,193.04 18 4 6,858.58 32 5 4389.49 50 6 3,048.26 72 T 1304.85 588 8 ’ 443.77 4,970 9 ' 26.22 670,960 65 Table 7 Fits to the Radiative Recombination Coefficients for Level n logio(a»r) = 0 .( 0 ) + 6.(0) {log io (r) - c.(0))^ (a. in units ofcm^lsec) n fl-(0) 6.(0) c.(0) 1 -8.3281 -.10809 6.1135 2s -8.9776 -.06256 7.2302 2p -9.1508 -.12667 4.9748 3 -92150 -.12898 4.9933 4 -9.4583 -.12236 4.7567 5 -9.6516 -.11481 4.5614 6 -9.8112 -.10747 4.3913 7’ -9.4942 -.093811 4.0573 8’ -9.5858 -.070572 3.2727 9’ -9.5028 -.045593 1.4592 66 Table 8 Radiative Rates for m <,6. (in units of 10^sec~^) m —• > n 2s 2p 3 4 5 6 1 8.230(-7) 62.57 5.574 1.279 .4123 .1644 2s .7478 .1812 .05935 .02380 2p 3.660 .6603 .1936 .07345 3 .8993 .2201 .07780 4 .2700 .07714 5 .1025 67 Table 9 Avaaged Radiative Rates A vs. for Case A. (in units of sec"' ) « e —- > n 7 11 21 101 OO 1 3.253 2.883 2.991 3.007 3.007 2s .4734 .4197 .4354 .4376 .4377 2p 1.392 1.230 1.277 1.284 1.284 3 1.398 1.232 1.280 1.287 1.287 4 1.219 1.067 1.111 1.118 1.118 5 1.222 1.058 1.105 1.112 1.112 6 1.504 1.271 1.336 1.346 1.346 Table 10 Averaged Radiative Rates A vs. n, for Case B. ______(in units of 10^ sec~^)______n 7 11 21 101 CO 1 3.253 2.995 3.068 3.078 3.079 2s .4734 .4359 .4465 .4480 .4481 2p 1.392 1.279 1.311 1.315 1.315 3 1.398 1.282 1.314 1.319 1.319 4 1.219 1.113 1.143 1.147 1.147 5 1.222 1.107 1.139 1.144 1.144 6 1.504 1.341 1.385 1.392 1.392 68 Table 11 Averaged Radiative Rates A vs. «« for Case A. (in units of 10^ sec~*) fig — > n 1 11 21 101 OO 1 17.36 17.36 15.20 15.94 15.96 2s 2.539 2.539 2.224 2.332 2.335 2p 7.144 7.144 6.252 6.558 6.564 3 6.867 6.867 6.003 6.299 6.305 4 5.462 5.462 4.766 5.004 5.009 5 4.674 4.674 4.068 4.274 4.278 6 4.233 4233 3.671 3.862 3.886 T 18.04 18.04 15.28 16.18 16.20 Table 12 Averaged Radiative Rates A g,_, vs. n , for Case B. ______(in units o f 10^sec~^) ______ R 7 11 21 101 OO 1 17.36 17.36 15.87 16.37 16.38 2s 2.539 2.539 2.322 2.395 2.396 2P 7.144 7.144 6.528 6.733 6.738 3 6.867 6.867 6.271 6.469 6.474 4 5.462 5.462 4.982 5.141 5.145 5 4.674 4.674 4.255 4.394 4.397 6 4.233 4.233 3.845 3.973 3.976 r 18.04 18.04 16.12 16.73 16.74 69 Table 13 Averaged Radiative Rates A , „ vs. rtc for Case A. (in units of sec ~^) Rg •—> R 7 11 21 101 OO 1 42.89 42.89 42.89 34.69 36.51 2s 5.958 5.958 5.958 4.822 5.075 2p 17.87 17.87 17.87 14.47 15.23 3 16.57 16.57 16.57 13.41 14.11 4 12.77 12.77 12.77 10.33 10.88 5 10.46 10.46 10.46 8.454 8.898 6 8.913 8.913 8.913 7.201 7.580 T 27.16 27.16 27.16 21.89 23.05 8’ 55.43 55.43 55.43 43.76 46.17 Table 14 Averaged Radiative Rates A vs. for Case B. (in units o f sec Rg — > R 7 11 21 101 OO 1 42.89 42.89 42.89 37.33 38.63 2s 5.958 5.958 5.958 5.188 5.368 2p 17.87 17.87 17.87 15.56 16.10 3 16.57 16.57 16.57 14.43 14.93 4 12.77 12.77 12.77 11.12 11.51 5 10.46 10.46 10.46 9.098 9.415 6 8.913 8.913 8.913 7.752 8.022 T 27.16 27.16 27.16 23.59 24.42 8’ 55.43 55.43 55.43 47.49 49.22 70 Table 15 Collision Strengths form ^6 (10,000 K). tn > n 2s 2p 3 4 5 6 1 .3030 .5150 .1497 .05490 .02639 .01476 2s 10,880. 13.19 2.685 1.027 .5123 2p 39.57 8.055 3.081 1.537 3 477.6 98.48 38.38 4 2636. 517.0 5 10,610 Table 16 Power-law Index P firom Fit to Tempmture Dependence ______of Collision Strengths for OT ^ 6 . ______ffl ™ > n 2s 2p 3 4 5 6 1 -.2753 -.2682 -.2650 -.2635 2s 0.0649 0.0792 0.0861 0.0897 2p 0.0649 0.0792 0.0861 0.0897 3 0.3271 0.2739 0.2637 4 0.4423 0.3480 5 0.4840 71 Table 17 Averaged Collision Strengths y (10,000 K) vs. Me for Case A. ttg — > n 7 11 21 101 OO 1 .02331 .02062 .02141 .02152 .02152 2s .7316 .6436 .6691 .6728 .6729 2p 2.195 1.931 2.007 2.018 2.019 3 45.51 39.62 41.32 41.56 41.57 4 423.5 362.0 379.4 382.0 382.1 5 3,498. 2,894. 3,061. 3,086. 3,087. 6 46,890 36,540 39,280 39.700 39,710 Table 18 Power-law Index p from Fit to Tamperature Dqœndence of Averaged Collision Strengths VS.«efor Case A. «c — > n 7 11 21 101 1 -.3067 -.2223 -.2413 -.2440 -.2444 2s 0.0459 0.1351 0.1146 0.1117 0.1116 2p 0.0459 0.1351 0.1146 0.1117 0.1116 3 0.2084 0.3039 0.2825 0.2794 0.2793 4 0.2618 0.3693 0.3452 0.3415 0.3417 5 0.2922 0.4203 0.3916 0.3876 0.3873 6 0.3790 0.5450 0.5083 0.5029 0.5028 72 Table 19 Averaged Collision Strengths (10,000 K) versus «e for Case B. Mg — > n 7 11 21 101 OO 1 .02331 .02144 .02196 .02204 .02204 2s .7316 .6702 .6874 .6898 .6900 2p 2.195 2.010 2.062 2.070 2.070 3 45.51 41.40 42.54 42.71 42.71 4 423.5 380.5 392.3 394.0 394.1 5 3.498. 3,075. 3,188. 3,205. 3,206. 6 46,890 39,600 41,470 41,760 41,770 Table 20 Power-law Index p firom Fit to Temperature Dqtendence of Averaged Collision Strengths y ______vs. Mg for CaseB. ______n 7 11 21 101 OO 1 -.3068 -2242 -2429 -2453 -.2455 2s 0.0459 0.1330 0.1132 0.1104 0.1104 2p 0.0459 0.1330 0.1132 0.1104 0.1104 3 0.2093 0.3019 0.2807 0.2778 02779 4 0.2980 0.3669 0.3435 0.3402 0.3400 5 0.2922 0.4176 0.3898 0.3858 0.3860 6 0.3789 0.5412 0.5062 0.5009 0.5007 73 Table 21 Averaged CoUisicm Strengths A , , ' (10,000 K) vs. He for Case A. Mg — > n 7 11 21 101 OO 1 .01029 .01029 .009006 .009445 .009455 2s .2948 .2948 2515 .2703 .2705 2p .8843 .8843 .7725 .8108 .8116 3 15.65 15.65 13.63 14.32 14.33 4 110.4 110.4 95.62 100.6 100.7 5 546.8 546.8 470.1 495.8 496.4 6 2,212. 2,212. 1,882. 1,991. 1,993. T 1.387(6) 1.387(6) 1.067(6) 1.159(6) 1.161(6) Table 22 Power-law Index P from Fit to Temperature Dependence of Averaged Collision Strengths y vs. Mg for Case A. Mg — > n 7 11 21 101 OO 1 -.2896 -.2896 -2079 -2322 -.2324 2s 0.0666 0.0666 0.1495 0.1250 0.1248 2p 0.0666 0.0666 0.1495 0.1250 0.1248 3 0.2262 02262 0.3113 0.2865 0.2862 4 02723 0.2723 0.3605 0.3348 0.3343 5 0.2830 0.2830 0.3756 0.3486 0.3486 6 02828 02828 0.3815 0.3533 0.3526 7 0.3640 0.3640 0.5256 0.4834 0.4828 74 Table 23 Avmged Collision Strengths (10,000 K) vs. Mg for CaseB. »g ™ > n 7 11 21 101 OO 1 .01029 .01029 .009403 .009698 .009705 2s .2948 .2948 .2691 .2776 .2778 2p .8843 .8843 .8072 .8328 .8334 3 15.65 15.65 14.25 14.72 14.73 4 110.4 110.4 100.2 103.5 103.6 5 546.8 546.8 493.8 511.0 511.5 6 2,212. 2.212. 1,983. 2,057. 2,059. T 1.387(6) 1.387(6) 1.162(6) 1.225(6) 1.227(6) Table 24 Power-law Index p £rom Fit to Temperature Dependence of Averaged Collision Strengths vs. Mg for CaseB. B g — > n 7 11 21 101 OO 1 -.2895 -.2895 -2098 -2332 -.2332 2s 0.0666 0.0666 0.1481 0.1242 0.1239 2p 0.0666 0.0666 0.1481 0.1242 0.1239 3 0.2262 0.2262 0.3097 0.2854 0.2850 4 0.2723 0.2723 0.3585 0.3337 0.3335 5 0.2830 02830 0.3736 0.3474 0.3474 6 0J2828 0.2828 0.3794 0.3522 0.3515 7 0.3640 0.3640 0.5322 0.4809 0.4807 75 Table 25 Averaged Collision Strengths (10,000 K) vs.tte for Case A. Kg — > n 7 11 21 101 1 .003435 .003435 .003435 .002781 .002926 2s .09481 .09481 .09481 .07672 .08074 2p .2844 .2844 2844 .2302 .2422 3 4.742 4.742 4.742 3.834 4.0351 4 30.54 30.54 30.54 24.66 25.96 5 132.8 132.8 132.8 107.1 112.7 6 447.2 447.2 447.2 359.9 379.0 7’ 26,800 26,800 26,800 21,360 22,520 8’ 1.163(8) 1.163(8) 1.163(8) 8.439(7) 8.964(7) Table 26 Powa-law Index p firom Fit to Temperature Dqiendence of Averaged Collision Strengths VS.R« for Case A. Rç —- > R 7 11 21 101 OO 1 -.2773 -.2773 -2773 -.1693 -.1761 2s 0.0803 0.0803 0.0803 0.1887 0.1817 2p 0.0803 0.0803 0.0803 0.1887 0.1817 3 0.2389 02389 0.2389 0.3478 0.3411 4 0.2828 02828 0.2828 0.3924 0.3855 5 0.2912 0.2912 0.2912 0.4013 0.3944 6 0.2876 02876 0.2876 0.3989 0.3918 7 02610 02610 0.2610 0.3772 0.3701 8 0.3137 0.3137 0.3137 0.4831 0.4744 76 Table 27 Averaged Collision Strengths (10,000 K) vs. Mg for Case A. « g — > n 7 11 21 101 OO 1 .003435 .003435 .003435 .002991 .003095 2s .09481 .09481 .09481 .08254 .08541 2p .2844 .2844 .2844 .2476 .2562 3 4.742 4.742 4.742 4.126 4.270 4 30.54 30.54 30.54 26.55 27.48 5 132.8 132.8 132.8 115.3 119.4 6 447.2 447.2 447.2 388.0 401.6 T 26,800 26,800 26,800 23,100 23,930 8’ 1.163(8) 1.163(8) 1.163(8) 9.431(7) 9.820(7) Table 28 Power-law Index p from Fit to Temperature Dqiendence of Averaged Collision Strengths vs. tie for Case B. itg ™ > It 7 11 21 101 OO 1 -.2772 -.2773 -2112 -.1702 -.1746 2s 0.0803 0.0803 0.0803 0.1876 0.1830 2p 0.0803 0.0803 0.0803 0.1876 0.1830 3 0.2389 0.2389 0.2389 0.3466 0.3423 4 0.2828 0.2828 0.2828 0.3914 0.3866 5 0.2912 0.2912 0.2912 0.4004 0.3955 6 0.2876 0.2876 0.2876 0.3979 0.3931 7 0.2610 0.2610 0.2610 0.3761 0.3711 8 0.3137 0.3137 0.3137 0.4801 0.4744 77 Table 29 Ionization Strengths for n ^ 6 (10,000 K) n 0 » 1 1.623 2s 41.43 2p 124.3 3 1429 4 6,945 5 21460 6 52,950 Table 30 Power-law Index y from Fit to Temperature Dependence ______of Ionization Strengths forw $ 6 ______« £2, 1 0.7641 2s 0.8178 2p 0.8178 3 0.7382 4 0.6442 5 0.5671 6 0.5064 78 Table 31 Ionization Strengths for Levels 7', S', and S' vs. for Case A (10,000 K). « C > n' 7 11 21 101 OO T 1.273(6) 1.353(6) 1.327(6) 1.323(6) 1.323(6) 8’ 5.362(7) 5.362(7) 5.640(7) 5.525(7) 5.523(7) 9 ’ 2.557(11) 2.557(11) 2.557(11) 2.646(11) 2.605(11) Table 32 Power-law Index y from Fit to Temperature Dependence of Ionization Str^gths for Levels 7% 8% and 9* vs. üç for Case A. n' 7 11 21 101 OO 7 0.3875 0.3306 0.3444 0.3464 0.3463 8 0.2521 0.2521 0.2168 0.2291 0.2289 9 0.0517 0.0517 0.0517 0.0467 0.0365 79 Table 33 Ionization Strengths for Levels T, 8 ', and 9 ' vs. tic ______for Case B (10,000 K).______ 11 21 101 T 1.273(6) 1.328(6) 1.311(6) 1.309(6) 1.308(6) 8’ 5.362(7) 5.362(7) 5.550(7) 5.475(7) 5.473(7) 9’ 2.557(11) 2.557(11) 2.557(11) 2.615(11) 2.588(11) Table 34 Power-law Index y fiom Fit to Temperature Dependence of Ionization Strengths for Levels T, 8% and 9' vs. tiç for Case B. n ' 7 11 21 101 OO 7 0.3875 0.3345 0.3472 0.3489 0.3489 8 0.2521 0.2521 0.2184 0.2298 0.2299 9 0.0517 0.0517 0.0517 0.0347 0.0359 80 Table 35 Case A Comparison of b„exp(x^) Between Seaton (19596) and the 10-Level Hydrogen Atom. Te = 2^00 «■ 5.000 f 10,000 20,000 f n Seaton Cola Seaton Cota Seaton Cota Seaton Cota 3 .0770 .0795 .131 .133 .213 .214 .332 .332 4 .0939 .0974 .155 .157 .244 .245 .364 .366 5 .1113 .116 .179 .182 .273 .274 .394 .399 6 .129 .134 .201 .205 J299 .301 .421 .427 Table 36 B Comparison of 6aexp(*„) Between Seaton (19596) and the 10-Level Hydrogen Atom. T, = 2,500 g 5,000 ÜC 10,000 AT 20,000 iS: n Seaton Cota Seaton Cota Seaton Cota Seaton Cota 3 .257 .266 .422 .429 .668 .672 1.013 1.02 4 .218 .226 350 .356 .540 .542 .792 .798 5 .220 l i s 346 .351 .519 .521 .739 .748 6 .231 .241 .355 .361 .520 .524 .725 .735 81 T ables? Case A High Density Predictions for 10-Level Atom (6,000 K). (a,flT in cm^/sec; = 6.exp(x.)) logQf,) P3 p4 Ps a ^ ( « 7 ) HaJH^ i/y /H P 9 .171 6.03(-14) .203 2.43C-14) .236 1.33(-14) 1.84 .612 10 .187 6.62(-14) .225 2.70C-14) .266 1.50C-14) 1.82 .622 11 .220 7.78(-14) .271 3.25(-14) .336 1.90(-14) 1.77 .653 12 .285 1.01(-13) .375 4.50C-14) .538 3.03C-14) 1.66 .755 13 .441 1.56(-13) .729 8.75(-14) 1.30 7.31(-14) 1.32 .936 14 1.03 3.63(-13) 2.21 2.65C-13) 2.41 1.36(-13) 1.01 .574 15 3.87 1.37(-12) 4.27 5.13(-13) 2.78 1.57(-13) L98 .343 Table 38 Case B High Density Predictions for 10-Level Atom (6,000 K). {titffVicmrlsec', P „ = b.exp(%.)) log{N,) Pa a ^ ( i / a ) p4 a.ff(«P) p5 a*ff(^7) H o J H ^ # 7 /# P 9 .643 2.27(-13) .457 5.49(-14) .447 2.52(-14) 3.07 .515 10 1.22 4.31(-13) .544 6.53(-14) .509 2.87(-14) 4.89 .492 11 2.07 7.33(-13) .701 8.42(-14) .643 3.63(-14) 6.45 .483 12 2.73 9.66(-13) .923 1.11(13) .928 5J23(-14) 6.45 .528 13 3.94 1.39(12) 1.54 1.85(13) 1.76 9.92(-14) 5.58 .601 14 6.82 2.41(-12) 3.28 3.94(-13) 2.06 1.47(-13) 4.54 .418 15 12.4 4.39(-12) 4.76 5.72(-13) 2.82 1.59(13) 5.68 .312 82 Table 39 Case A High Di snsity Predictions for 10-Level Atom (10,000 K). ([a^r in cinVsec ; P. = 6,exp(%,)) iog(N^) Pa a ^ ( H a ) P4 a^C H p) Ps OeffiHy) HaJH^ Hy/H^ 9 .231 3.80(-14) 267 1.49C-14) .302 7.92(15) 1.89 .597 10 .245 4.03(-14) .286 1.59(-14) .329 8.61(15) 1.87 .605 11 .275 4.5K-14) .327 1.82(-14) .393 1.03(14) 1.83 .632 12 .329 5.41(-14) .420 2.34(-14) .579 1.52(-14) 1.71 .725 13 .468 7.70(-14) .747 4.17(-14) 1.18 3.08(-14) 1.37 .827 14 .966 1.59(-13) 1.75 9.74(-14) 1.73 4.54(-14) 1.21 .522 15 2.83 4.66C-13) 2.50 1.39(13) 1.86 4.87(-14) 2.48 .392 Table 40 Case B High Density Predictions for 10-Level Atom (10,0(X) K). (a,ff in cm^/sec; p , =b„exp(x„)) log(K ) Pa a .ff(//a ) P4 Ps HaJH^ i/y /tfp 9 1.01 1.66(13) .622 3.47(-14) .580 1.52(-14) 3.53 .490 10 1.35 2.23(-13) .700 3.9CK-14) .790 1.66(14) 4.22 .477 11 1.56 2.56(-13) .790 4.40(-14) .732 1.92(-14) 4.30 .487 12 1.79 2.95(-13) .945 5.27(-14) .951 2.49(-14) 4.14 .529 13 2.30 3.78(-13) 1.38 7.70(14) 1.47 3.86(-14) 3.63 .561 14 3.36 5.52(13) 2.22 1.24(-13) 1.81 4.74(-14) 3.30 .429 15 4.95 8.13C-13) 2.61 1.46(-13) 1.87 4.90(-14) 4.13 .377 83 Table 41 Case A High Di ïnsity Predictions for 10-Level Atom (20,000 K). in cm^lsec ; = b,exp(%,)) Iog(N,) Pa P , (W(«P) Ps # o / # P H ylH ^ 9 .345 2.0U-14) .383 7.56C-15) .420 3.89C-15) 1.97 .577 10 .357 2.07(-14) .399 7.87C-15) .442 4.09C-15) 1.95 .583 11 .381 2.21(-14) .434 8.56C-15) .499 4.62(-15) 1.92 .604 12 .427 2.48(-14) .517 1.02C-14) .667 6.18C-15) 1.80 .678 13 .552 3.21(-14) .810 1.60(-14) 1.09 1.01(-14) 1.49 .705 14 .980 5.69(-14) 1.37 2.71C-14) 1.33 1.23(-14) 1.56 .507 15 1.91 l.lK-13) 1.60 3.16(-I4) 1.37 1.26(-14) 2.60 .449 Table 42 Case B High D oisi^ Predictions for 10-Level Atom (20,000 IQ. (fx^ in cm^/sec ; g . = b,exp(jc«)) log(N,) Pa P4 a ^ (H B ) Ps arfr(//Y ) i / a / i / p Hy/Hfi 9 1.21 7.01(-14) .853 1.68(-14) .786 7.28(-15) 3.09 .485 10 1.24 7.23(-14) .875 1.73(-14) .810 7.50(-15) 3.10 .486 11 1.30 7.55(-14) .922 1.82(-14) .868 8.04(-15) 3.08 .495 12 1.39 8.08(-14) 1.02 2.00(-14) 1.01 9.33(-15) 2.99 .522 13 1.59 9.22(-14) 1.26 2.49(-14) 1.25 1.16(-14) 2.75 .524 14 1.95 1.14(-13) 1.55 3.05(-14) 1.35 1.25(-14) 2.75 .460 15 2.31 1.34(-13) 1.63 3.2K-14) 1.37 1.27(-14) 3.10 .443 Table 43 Case A High Density Predictions from Adams and Petrosian (1974). (g«ff in cm^lsec) T, = 6,000 f 10,000 f 20,000% logiN,) a.ff(i/P ) HoJHfi //0 ///P HylHfi « ^^(//p ) # o / # p 9 1.77 .602 1.87 .600 1.97 .568 10 3.07(-14) 1.74 .602 1.67(14) 1.84 .600 8.01(15) 1.95 .568 11 4,13(-14) 1.66 .705 2.06(-14) 1.74 .684 9.22(-15) 1.85 .663 12 6.50(-14) 1.46 1.07 3.29(-14) 1.51 .947 U 6(-14) 1.59 .842 13 1.98(-13) .905 .726 8.58(-14) 1.00 .632 2.32(-14) 1.18 .542 14 3.35(-13) 1.23 .347 1.45(13) 1.60 .379 3.02(-14) 2.01 .426 15 6.50(-l3) .284 1.45(-13) .353 3.02(14) .426 16 .284 .353 .426 max 7.84 4.70 Table 44 Case B High Density Predictions from Adams and Petrosian (1974). T, = 6,(M)0% 10,000% 20,000% Iog(N^) a ,ff(« P ) HoJHW HaJH^ W o /# p Hy/Hp 9 2.57 .516 2.68 .516 2.78 .500 10 6.72(-14) 2.54 .516 3.67(44) 2.65 .516 1.70(44) 2.77 .489 11 8.91(44) 2.47 .579 4.52(44) 2.57 .517 1.83(44) 2.69 .537 12 1.35(43) 2.24 .747 6.41(44) 2.31 .653 2.33(44) 2.47 .579 13 3.35(43) 1.74 .411 1.20(43) 1.87 .458 3.08(44) 2.27 .458 14 6.18(43) 2.36 .321 1.59(43) 2.80 .368 3.30(44) 3.0 :432 15 6.50(43) .284 1.59(43) .358 3.30(44) .432 16 .284 .358 .432 max • 7.84 4.70 3.20 Table 45 Case A Low Temperature Predictions for 10-Level Atom (500 IQ. (a,ff in cmvsec);'^„ = ^ e x p (* J) logiN,) P s P4 a . f f ( f f p ) 0 5 HaJHfi Hy/Hfi 1 2.20(-2) 3.23(-13) 2.88(-2) 1.44(-13) 3.66(-2) 8.58(-14) 1.67 .669 2 2.20C-2) 3.24(13) 2.88(-2) 1.44(-13) 3.66(-2) 8.59(14) 1.67 .669 2.5 2.20(-2) 3.24(-13) 2.88(-2) 1.44(-13) 3.67(2) 8.59(14) 1.67 .669 3 2.21(2) 3.25(13) 2.89(-2) 1.44(-13) 3.68(2) 8.62(14) 1.67 .669 3.5 2.23(-2) 3.28(13) 2.92(-2) 1.45(-13) 3.71(2) 8.69(14) 1.67 .669 4 2.29(-2) 3.36(-13) 2.99(-2) 1.49(13) 3.80(.2) 8.91(14) 1.67 .669 4.5 2.44(-2) 3.58(-13) 3.18(-2) 1.59(-13) 4.04(-2) 9.48(14) 1.67 .669 5 2.73(-2) 4.01(13) 3.56(2) 1.77(13) 4.52(-2) 1.06(-13) 1.68 .668 5.5 3.07(-2) 4.51(13) 3.99(-2) 1.99(13) 5.06(2) 1.19(13) 1.68 .668 6 3.29(-2) 4.84(-13) 4.27(-2) 2.13(13) 5.42(-2) 1.27(13) 1.68 .667 Table 46 Case B Low Temperature Predictions for 10-Level Atom (500 K). (a,ff incmVsecy, g .= b„exp(*,)) log(N^) 03 04 a ,ff(//0 ) 05 / /0 / // 3 1 8.07(-2) 1.19(12) 7.18(2) 3.58(-13) 7.65(2) 1.79(13) 2.45 .560 2 8.07(-2) 1.19(12) 7.18(-2) 3.59(13) 7.65(2) 1.79(13) 2.45 .560 2.5 8.08(-2) 1.19(12) 7.19(2) 3.59(13) 7.66(-2) 1.79(-13) 2.45 .560 3 8.11(-2) 1.19(12) 7.22(-2) 3.60(13) 7.68(2) 1.80(13) 2.45 .560 3.5 8.19(-2) 1.20(12) 7.29(-2) 3.64(13) 7.76(-2) 1.82(-13) 2.45 .560 4 8.43(-2) 1.24(12) 7.49(-2) 3.74(-13) 7.98(-2) 1.87(-13) 2.45 .560 4.5 9.02(-2) 1.33(12) 8.01(2) 4.00(-13) 8.53(-2) 2.00(-13) 2.46 .560 5 1.01(1) 1.48(12) 8.93(-2) 4.46(-13) 9.50(-2) 2.23(-13) 2.46 .559 5.5 1.11(1) 1.64(-12) 9.88(-2) 4.93(-13) 1.05(-1) 2.46(-13) 2.46 .559 6 1.18(-1) 1.73(12) 1.04(-1) 521(-13) 1.11(1) 2.60(-13) 2.46 .559 Table 47 Case A Low Température Predictions for 10-Level Atom (125 K). (.Utff iacmVsec); P« =b.exp(%.)) logiN,) & P4 a«flr(WP) Ps OL.ffiHy) H a/H ^ Hy/H^ 1 5.83C-3) 9.59(-13) 7.87(-3) 4.39(-13) 1.04(-2) 2.73(13) 1.62 .697 1 7.0G(-3) 8.23(-13) 9.43(-3) 3.76(13) 1.25(-2) 2.33(-13) 1.62 .695 2 7.01(3) 8.24(13) 9.44(3) 3.77(13) 1.25(-2) 2.34(13) 1.62 .695 2.5 7.03(-3) 8.27(-13) 9.47(-3) 3.78(-13) 1.25(2) 2.34(-13) 1.62 .695 3 7.10(-3) 8.35(13) 9.55(3) 3.81(13) 1.26(2) 2.36(-13) 1.62 .694 3.5 7.3K-3) 8.60(-13) 9.83(-3) 3.92(13) 1.30(-2) 2.43(-13) 1.62 .693 4 7.96(-3) 9.37(-13) 1.07(2) 4.26(-13) 1.40(-2) 2.63(13) 1.63 .691 4.5 9.73(-3) 1.14(-12) 1.29(2) 5.16(-13) 1.69(2) 3.16(-13) 1.64 .686 5 1.35(-2) 1.59(-12) 1.77(-2) 7.08(-13) 2.29(-2) 4.3(K-13) 1.66 .680 5.5 1.85(2) 2.18(12) 2.42(2) 9.65(13) 3.11(2) 5.82(13) 1.67 .676 6 2.23(-2) 2.62(-12) 2.90(-2) 1.16(12) 3.71(2) 6.96(13) 1.68 .674 Table 48 Case B Low Temperature Predictions for 10-Level Atom (125 K). ioL,ifiacmVsec); p .= l»,exp(*,)) log(N,) Pa a ,ff(//a ) P4 a.ff(/fP ) Ps dM H y) HoUH^ H y m 1 2.80(-2) 3.29(-12) 2.53(-2) 1.01(-12) 2.77(-2) 5.19(13) 2.41 .575 2 2.80(-2) 3.30(-12) 2.54(-2) 1.01(12) 2.77(-2) 5.19(-13) 2.41 .574 2.5 2.81(-2) 3.31(12) 2.54(-2) 1.02(12) 2.78(-2) 5.21(-13) 2.41 .574 3 2.84(-2) 3.34(-12) 2.57(2) 1.03(12) 2.81(-2) 5.26(13) 2.41 .574 3.5 2.94(-2) 3.46(-12) 2.66(2) 1.06(12) 2.90(-2) 5.43(13) 2.41 .574 4 3.22(-2) 3.79(-12) 2.90(2) 1.16(12) 3.16(-2) 5.92(-13) 2.42 .572 4.5 3.97(-2) 4.67(-12) 3.56(2) 1.42(-12) 3.85(-2) 7.22(-13) 2.43 .570 5 5.43(-2) 6.39(-12) 4.84(-2) 1.93(-12) 5.22(-2) 9.78(-13) 2.45 .566 5.5 7.16(-2) 8.42(12) 6.36(-2) 2.54(12) 6.82(-2) 1.28(12) 2.46 .564 6 8.31(-2) 9.77(-12) 7.37(-2) 2.95(-12) 7.89(-2) 1.48(-12) 2.46 .563 cR Table 49 (a,ff iacm^/sec) Low Temperahire Predictions fiom Martin (1987) (Zero-Density) T, = 500% 125% Case a ,ff(//P ) Ha/H^ Hy/H^ a^(jH a) a,ff(//p) //ot///p HyfH^ A 8.185( 13) 1.985(13) 4.54 .408 4.488(-13) 2.640(-12) 5.84 .388 B 8.418(-13) 2.606(-13) 4.22 .417 5.773(13) 2.688(12) 5.27 .401 23 8 8 10 Hummer 8 Storey This work (q.Z|ll « 0) This work (q, included) 2 | l l 1.0 0.1 0.01 J5 T ,9 II 13 IS Ne Figure 8. Comparison cfH^ Effective Recombination Coefficients, Predicted by our 10-Level Atom and by H um m er a nd Storey. Our effective recombination coefficients for and those of Hummer and Storey (1987) are expressed as the deviation of the total effective recombination coefficient for (a) including coUisional effects, fiom its purely radiative value (ag), and are plotted against election density. Our results, using the collision data of Vriens and Smeet (1980) and neglecting collisions fiom out of the n = 2 level are shown as the solid ctirve. Our results, including collisions fiom out of n = 2, are shown od utw uvrilÂÀ* vuA vv. The dashed car/e are the Ksalts of Hummer and Storey. IV. THE RECOMBINATION SPECIRUM OF HELIUM I: THE EFFECTS OF PHOTOIONIZATION OF HYDROGEN BY HE I RESONANCE PHOTONS ON THE SINGLET EMISSIYITIES. 4.1 Introduction The recombination spectrum of the helium I singlets has been modeled in considerable detail in two limiting cases of the low-density limit (LDL), wherein collisions are negligible for the low-lying states. Brocklehurst (1972) and Robbins and Robinson (1971), have calculated emissivities and line ratios for the helium singlets in the case A limit, which assumes that the gas is optically thin in the n 'f - l'S "Lyman" series. Brocklehurst has also calculated the singlet spectrum in the case B limit, which assumes that the gas is infinitely optically thick to scattering in the resonance lines so that resonance photons are degraded into higher series photons plus L ya X584. Both authors included the effects of elastic collisions on the higher-lying states, and Brocklehurst also included inelastic collisions for these levels. Both the methods and the approximations to the physical conditions which these authors used are directly analogous to those used in calculating the hydrogen spectrum, some of which were discussed in the preceding chapter. Real nebulae have finite Lyman series optical depths, of course, with the higher Lyman series members optically thin. Calculations allowing for finite optical depths, however, show that caseB provides almost the exact emissivities for lines originating firom low-lying states for as low as l(f (see section 4.3 below). Any radiation-bounded nebula should attain He I Lya optical depths of this order near its Stromgren radius, and since volume averages of the line entissivities are heavily weighted towards outer radii, caseB should be a good approximation to the average optical recombination 89 90 spectrum, all other things being equal. There are, howevo', a number of other effects which can prevent the case B limit from obtaining for either H I or He I. The destruction of Lyman {Notons on dust mixed with the gas, and the lowering of the optical depth scale by velocity gradients and tuibulent line- broadening, will be discussed for hydrogai in chapter V. For He I these same effects may occur. More important, however, is the destruction of helium singlet resonance photons through photoionization of H I, before they can scatter often enough to attain case B (Osterbrock 1974). The problem of finite optical depths has been approached previously by Robbins and Bemat (1974), who considered XS84 optical depths of 2,000 and below. They did not, however, include destruction of the scattered resonance photons through photoionization of H I. Thompson and Tdcunaga (1980) and Geballe et al. (1984) have done approximate calculations of the strength of the He I 2^P-2^S X2.058 pm forbidden line relative to hydrogen Bry allowing for neutral H photoionizations, but neglecting cascades from higher levels. In this chapter we calculate the emissivities of the helium singlets for XS84 optical depths between zero and the case B limit, allowing frilly for cascades from excited levels and for the destruction of resonance photons on neutral hydrogen. We predict substantial deviations from the caseB He I spectrum for many classes of objects. Because the lines most sensitive to the optical depth in the resonance lines, i.e. those miginating from an upper nf-state, are unblended for helium, it should be possible to very accurately measure this deviation. We also show that the deviation is simply related to the H I to He I abundance ratio, which in turn is related to the shape of the ionizing spectrum. We begin in section 4.2 by describing our calculations. In section 4.3 we display our results, and in section 4.4 we predict the singlet spectra of some tj'pica! H H Regions, planetary nebulae, and active galactic nuclei (AGN), and finally we compare our results to the well-studied planetary nebula NGC 7027. 91 4.2 Calculations We calculated the He I singlet spectrum in the LDL, i.e. we assumed that n-changing collisions were slow relative to radiative transitions so that only radiative recombinations and cascades need to be considered. The methods used are very similar to those used to obtain the H I level populations in chapta m , but we did not consider the L -sublevels separately there. When the L -sublevels are included in the LDL, the population of any level nL is given by: ------(4.1) where •AfiL= £ £ (4.2) it=a r,'=i±i where P(x) is the escape probability of the resonance photon, as in chapter m . In strict case A, P(t) = 1 for all resonance lines, and in strict case B, P(x) = 0 for all resonance lines. Since there are no upward transitions, it is possible to solve recursively for all the level populations of the atom, beginning with the highest level Our goal was to find the emissivities of the He I singlet lines, which can be expressed in tm ns of the effective recombination coefficient: (4.3) 92 Calculations in the LDL depend upon only a very few free parameter. These are temperature, density, the optical depth in the resonance lines, and the ratio of any continuous opacities at the resonance line frequencies to the local resonance line opacity. Our principal interest was in studying the effects of varying the latter two parameters. We therefore performed our calculations at a single tempoature, using simple scaling laws to approximately (5% or better) account for the temperature variation. This will be described in the section following. We also calculated the spectrum for a single density: The density enters through the assumption of partial or full Z.-mixing. As discussed at length in the preceding chapter, at any density, there are high-lying levels for which elastic, L-changing collisions are so fast that the L-sublevels are populated according to their statistical weights. In our case we chose to consider L -sublevels as unmixed for levels n =20 and below, and L-mixed above there. From Brocklehutst’s calculations it appears that this assumption corresponds to a density of l(f cm~^. Robbins and Robinson’s results suggest this assumption corresponds to a density of less than K f cm~^. The diffeences between these authors are probably due to the use of different collision rates. Regardless of the precise density for which our calculatirais are formally valid, BrocMehurst’s calculations show that the intensities of lines originating from levels n ^ 6 differ by only 2% - 3% between densities of 10^ cm~^ and lowar density cases. The other uncertainties, which we will deal with below, are cartainly as great as this, so we consider our results to be applicable across the entire low-density regime. (a) Helium I Singlet Energies and Transition Rates. We included 320 n levels in our He I atom, with separate 1 -sublevels 155 through n = 20. We used the energy levels given in Moore (1971) for n ^ 20 and Z, ^ 3, and assumed hydrogen aiergies for the 93 remainder. This seemed appropriate since the energy of 20'P is 0.1% different from the analogous hydrogen level. We used the radiative transition probabilities in Wiese, Smitli, and Glennon (1966) where given, otherwise those in Green, Johnson, and Kolchin (1965). These refoences are complete to n = 8 . For L' > 2 and L > 2 (throughout this chuter, primed quantum numbers refer to states of higher principal quantum number), we used hydrogen transition rates for all r . These came from Capriotti (1964) and Green, Rush, and Chandler (1965). For L ^ 2 or L' ^ 1 the hydrogen and helium rates do not converge to the same values as n increases, at least not over the range of n for which the tables provide values. We therefore scaled published hydrogen rates by the ratio of helium to hydrogen rates from the lower n -levels as suggested by examination of the published rates: = ' (4.4) forn <7;and (4.5) for n > 7 and n' = n+1; and (4.6) fatn > 7an d > n+1. 94 For transitions from the L-mixed levels with n > 20 to the L-sublevels of states with n ^ 20, we computed the transition rates fiom the hydrogenic, fully L -mixed rates by: (4.7) This result was further scaled by equations 4.4 to 4.6 in cases w h ae L ^ 2. For the radiative recombination rates we used those given by Burgess and Seaton (1960) for n<:10 and L £ 1, and 0.25 times the hydrogen rates for L >1. For n > 10 we scaled the hydrogen recombination rate coefticiMits by the ratio For n > 20 we used the total hydrogen recombination coefficients ot,, scaled by a factor of .247, to approximately account for the summation over the S and F states. (b) Line Transfer For cases intomediate between A and B it was necessary to allow for finite optical depths in the resonance lines. The resonance line transfer was treated with the escape probability formalism (Capriotti 1965), using an escape probability appropriate to a Voigt profile firom Slat» et al. (1982): p(T) = [1+1[0.6 + (4.8) where a is the damping width: b ' - 1 L '= » -1 (Z Z A^v.isr B = 1 L = 0 a =■ (4.9) 4 tcAv 95 and Av, the line-width, is: Av = - ^ = - ^ (4.10) C À where the velocity width, AV is: AV = 6.405% lO*COT/sec [<4 + ( . w T * , (4.11) 6.405 km I sec and where is the turbulent velocity. The optical depth in the rescHiance line, T, can be expressed as a constant ratio to the optical dq)th of Txss4 ‘ Therefore, qtecification of saved to specify all optical depths in the problem. Absorption of each of the n^P-1^5 resonance photons in neutral hydrogen photoionizations was included by Netzer, Elitzur, and Ferland (1985)’s formalism for the escape of a photon in the presence of a continuous absorbing medium. The total effective escape probability for an arbitrary line is: p^(T) = b3T4+X,p(x) (4.12) where if equation 4.12 exceeds unity, it is set equal to unity. We chose the constant to be 6 = 5.6(-^) °^ to match the numerical results of Hummer and Kunasz (1980). The ratio of the continuous to the total opacity, X^, given in terms of the photoionization opacity of neutral hydrogen (k^) and the scattering opacity in the line (k,), is: Xc = Ktl(}Cc + %,) = (4.13) 96 The ratio of the line scattering opacity to the total opacity, is: X, s K,/(Kg + K,) = 1 - Xj = 1 (4.14) The ^proximate expressions hold for x ^/K; small, which is the principal ^plication. Although we included the full expressions in our calculations, we will use the approximate expressions haeafter in the text for simplicity. We chose Xc(A584) = k;.(XS84)/K|(X584) as the free parameter through which to specify the continuous opacity. This is the most general treatment of the problem: Although we assumed that the continuous opacity implicit in Xg was due to photoionizations of neutral hydrogen alone, other continuous opacity sources such as dust ntight be important If these opacities have a wavelength dependence not too different from the H I photoionization opacity over the wavelength region spanned by the n^P -l^S series, then results parameterized in terms of Xc(X584) will apply equally to these opacity sources. Given Xg(%584), we obtain for an arbitrary line: (4.15) where KX.-I.MS) =«,(»« (4.16) 97 and k,(X584) =NiiHe I ) ^ ^ = 3.75% /) (4.17) In the above equations, we have used V interchangeably with f , Thus we have written (0„x' for the statistical weight, although it is in general equal to (27' +1). This is permissible because we are dealing with singlet states, where /'=L '. The continuous opacities are proportional to Nh* and are given by equation 2.4 of Osterbrock (1974). A t 2584, is 1.87% IQ-^cm^ -JV„.. Clearly, in equation 4.13, the dependences on level populations and temperature cancel out, leaving only quantities which are independent of the physical conditions, and %c(2584). The ratio X^(2584) is the only quantity directly coupled to the physical conditions, and it can be expressed as a function of the ratio of helium to hydrogen ionization fractions. Ratioing iq.(2584) = 1.87%l(T^^cm^ VVg» to equation 4.17, gives: X,(2584)=4.96% ir^( ^ ) [ t 4+=4.96% l(T'(-^)Tl[t 4 + (418) N}jf* 6.405 Ajjg 6.405 where is the abundance of helium, and where n = (— ) (4.19) %«• where %»• is the fraction of neutral hydrogen, and is the fraction of neutral helium. The ratio of 98 these fractions is, in turn, a function of the ionizing spectral shtgte, dqtending on the ratio of He I ionizing photons to H I ionizing photons. In practice we require detailed photoionization models to accurately predict this ratio: In any radiation-bounded nebula, if changes across the nebula as the rising optical depths change the spectral s h ^ , so that any simple relationship between t], the singlet spectrum of the entire nebula, and the incident spectral shape is without predictive power. In section 4.4 we predict values o ft ) for several different assumed ionizing spectra through the use of photoionization models. (c) Tanperature Dependence All our calculations were done for 10,000 K, but can be easily extended to other temperatures through simple power-law fits to Brocklehurst’s (1972) results at temperatures of 5,000,10,000, and 20,000 K. We made fits to both the line ratios and effective recombination coefficients quoted by Brocklehurst for the first 6 members of the b*P-2^S, «*S-2*P, and n^D-7}P series for both case A and B. For the n}P-7}S series, for example, these fits are of the form: with analogous relationships for the other series. The power-law indices a and P are given in tables 50 and 51 for cases A and B respectively. Blank entries in the tables indicate that Brocklehurst quoted no value for the quantity, or in the case of the 4^D-2^f line, that the ratio was, of course, constant for all temperatures. 99 The fits are by definition exact at 10,000 K. The goodness of these fits at otha temperatures varied with the series: For the n^P-2^S series, the fits to the ratios to A4922 were better than a percent everywhere; the fits to the effective recombination coefficients had an m or of 3% at 5,000 K and 1% at 20,000K. For the n^S-2^P series, the fits to the ratios had an error of .5% at 5,000K and 5% at 20,000 K; the fits to the effective recombination coefficients were much better than .5% everywhere. For the n^D-2^P series, the fits to the ratios had an error of about 1% evoywhae; the fits to the effective recombination cofficients were in error by 4% at 5,000 K and 1.5% at 20,000 K. Since all these fits are for 3 points only, however, the reliability of these error estimates for intermediate temperatures is minimal. As is apparent from the tables, the temperature dependence is nearly the same for both cases A and B for the n*P-2*S, and especially the n*D-2^P series, but it is different for the n^S-2^P series. This implies that for the n'P-2 'S and n^D-2^P soies, temperature and optical depth effects vary independently, and the results at 10,000 K can easily be extended to any temperature by means of equations 4.20 and 4.21 with either case A or case B fits for any value of Tus 4 tmd %g(X584). Loss of accuracy should occur, however, if the caseB fits are applied to the n^S-2^P soies for either high %g(X584) or low unless the temperature is near to 10,000 K: The case B fits differ from the case A fits by 10% for 3*S-2*P, 6.7% for 4‘S-2‘P , 5% for 5^S-2*P, and 3.6% for 6^S-2‘P . Since the errors in the case B fits at case B are much better than .5%, the inapplicability of the case B fits in general is the largest source of oror for these fits. The reason for this coupling of temperature and optical depth effects in the n ^5-2^P series, and their independence fcff the other series, follows from the way in which rising optical depths lead to an Mihancement of each series’ emissivities. The reason that the n *P -2^5 series intensities are enhanced as 100 the case B limit is approached is that the n states have a higher population simply because the rate of transitions to the ground state is decreasing. This decrease is produced by the decreasing value of the escape probability, which is only weakly dependent on temperature. The n^S states, on the other hand, have no transitions directly to ground, and the enhancement of the n^S-2^P series as caseB is t^proached occurs because the increased rate of cascades firom the n^P states leads to hi^er n^S populations. The cascade process dq»nds in turn on temperature through the recombination coefficients to the upper levels. Thus, as cascades increase the temperature dqtendence of these lines change. Of course, the n^P states have contributions to their populations firom cascades too, but these are less important The n^D states also are populated in part by cascades, but the variation of this series’ emissivities firom cases A to B is almost negligible. The remedy for the difficulty with the n *5-2*P series would be either to expand our calculations into a third dimension, tabulating emissivities for the n^S-2^P series vs. Xe(A584), and T«, or to specify the values of txsg 4 and %g(%584) at which to switch between the case B and case A temperature fits. We feel either approach to be unnecessary and recommend the use of the case B fits for most circumstances: For most real objects, conditions will be nearer to case B than case A. (d) CoUisional Population of 2^P from 2%. Although most coUisional effects should be negUgible for the densities we are considering, we do need to consider briefly the rate of coUisional population of the 2*P level from 2^S, which can be t^preciable at densities as low as 10^ cm~^. These densities are oth^wise typical of the LDL, i.e., other coUisitHis are unimportant. This coUisional process wiU affect the 2^P-2^S emissivi^ only and in a simple fashion, without changing any results which we obtain for lines originating fiom higher levels. 101 The 2^S level is highly metastable, with a radiative transition rate of A(2^S) = 157* 10^ sec"* (Osterbrock 1974). Comparison of this rate to the rate coefficients for coUisional transitions to levels of comparable energies (Cox and Daltabuit 1971 give rates for coUisions to 3^D, 4^0, and 1*S ; Osterbrock 1974 gives rates for excitations to 2*5 and 2*P), shows that coUisional depopulation is the dominant process de-populating the 2^5 level at densities around 3x 10^ cnT^. Previous authors (e.g. GebaUe et al. 1984) have estimated factors of 2 enhancement in the m issivity of the 2*P-2*5 22.058 pm line as the result of coUisional excitation from the 2^5 level. We demonstrate hae that whUe this result is a good approximation, the enhancement at 10,000 K is more nearly a factor of 1.5. It is straightforward to show firom the population balance equation for level 2*P, that the emissivity of the 22.058 pm line, including the effects of population and depopulation by coUisional transitions with the Q?S level, is given by: 4rt7f**"* + Af,N(2%)^ (2^5-2*P)P (2*P-2*5) Av [l+lV,g(2*P-2%)/A(2*P)] where g(2%-2*P) is the rate coefficient for coUisional excitation from 7?S to 2*P, and where f (2*P-2^5) is the rate coefficient for coUisional de-excitation. The teanching ratio, P(2*P-2*S), is equal to: P(2*P-2*5) 5 (4.23) A(2*P) where 102 i4 (2 ‘P ) =A (2‘P - 2 'S )+A (2^P-1^S)P^ (4.24) and where A(2^P-2^S)=1.976xl0^sec~^ and i4(2^P-1*S)= 1.799x10^ sec"^ In pure case A, P(2‘P-2^S) = .0011, while in pure case B, P(2‘P-2*S) = 1.00. For Tf = 5,000,10,000, and 20,000 K , table 52 gives the coUisional excitation coefficients for the 2^5-2^P transition from Ostabrodc (1974), and the de-excitation coefficients for this transition, computed firom: « (2‘P-2"S) = (2"S-2‘P)exp(-^^^^) (4.25) Using the derived values we see that even for 20,000 K and the low value of A (2'P) ^tpropriate to caseB , Ngq(2^P-2^S)/A(?}P) is negligible for aU densities below about 3x10''*o n a n d the denominator of equation 4.22 is unity. To get the rate o f excitation fixtm out o f level 2^S we need to know that level’s population. This is easily found: Since none of the triplet levels can readily decay to l^S, aU recombinations to triplet states must result in a population of level 2^5. The rate of coUisional transitions into level 2^S is also negUgible for these densities. Thus the rate of population of level 2^S is simply N,Nng*a.B{He^;n}L). The rate of depopulation is the sum of the coUisional transition rates to levels 2^P, , 4^Z), 2 'P , 1*5, and 2^5, plus the rate of the radiative transition to level l^S. Both the recombination coefficient and the sum of the coUisional rate coefficients are also given in table 52. 103 From these arguments, it follows that the population of level 2 5 is given by: Computed values of N(2^S)/Nut* are given in table 53, for cases where radiative decays dominate (Ng = 10c m ~ \ are equal to (N, =3xl(Pcm~% and are negligible beside (Ng = l(fcm~^ and Ng = 10^ cm~^) coUisional processes. Note that, in the latta limit, the population of level 2^5 tends to a constant with regard to Ng. The emissivity of X2.058 can now be written: 4nj 3LZOS8 whore _ N(2^f )g(2^P-2»P)P(2»P-2»5) kb = «W.0S8 is the ratio of the rate of coUisional population of level 2^P to the rate of its popidation by radiative recombination and cascade. This term is given for cases A and B in tables 54 and 55 respectively, and it can be seen that for densities above lO^ cm~^ coUisional excitation from 2% plays an important role. The coUisional contributions for cases A and B are nearly the same. This was expected: To first order (i.e., neglecting the changes in the contribution of cascades from higher lying states onto 2^F), scales as P (2^f-2^5), just as does the coUisional term, so that Jce should be almost constant 104 4.3 Results We computed the helium singlet spectrum for values of ranging £rom 0 (pure case A) to 10^ (effectively pure case B for the lower series members), and for values o f %g(X584) firom 0 to 10"\ I Except for the 2*P-2*S we have quoted all results as ratios relative to 4'Z)-2*P %4922. Where we talk of "intensity ratios" below, we mean the ratio relative to 4922. This can be related to the 4^D-2^P XS876 intensity through the ratios of table 56, which have been computed firom the results of Brocklehurst (1972). Figure 9 shows the variation of the 2‘P-2*S A2.058 pm effective recombination coefficient with x«g4, for values of Xg(A584) o f KT^, 1(T^, 1(T \ and 10"*. Figures 10 to 13 show the intensity ratios of the higher members of the « *P-2*S series fora = 3 - 6 fbr%g(X584) = 10"*, 10"^, and 10"^. The results for Xg(XS84) = 10"* are omitted firom figures 10 to 13 because they are indistinguishable firom the case A results for these high^ sories lines. The case of Xg(X584) = 10"^ is omitted firom figure 13 because for 6*P-2*5 X3448  this case too is indistinguishable firom case A. All these figures are for a temperature of 10,000 K, but as discussed above can be easily extended to other temperatures through simple relationships. The results for the n *5-2*P series for n = 3 - 6 are given in figures 14 to 17. In this case, we have plotted against txs 84 for several values of Xg(X584), the deviation of the intensity ratios firom their case B values. This makes the smaller amount of variation in these ratios much more noticeable. As for the n *P-2*5 series, values of Xg(%684) which are not given produce ratios indistinguishable firom the case A ratios. The results for the other optical series, n *J9 -2*P, are not given in gn^hical form because these lines scarcely vary. The situation for all 3 series is sununarized in table 57, which specifies for each series, the 105 total amount of variation fiom the case A limit to the case B limit through the ratio of case B to case A emissivities. Note that since A4922 barely changes from case A to B (ff„f sfJem a = 1*028), figures 10 through 17 are equivalent to the variation of the emissivities of the n ^P-2*5 series lines. Note that for %g(X584) of zero, case B emissivities and ratios are effectively attained for all lines by 1x584 = 10*. Such A584 optical depths are usually attained in model radiation-bounded nebulae at their mean radii (see section following) barring large velocity gradients or turbulent velocities. Therefore, in general, observed deviations from case B are the result of absorption of resonance photons in hydrogen photoionizations, and not of finite optical depths. It is possible to read off the case A and B limits for the n ^F-l^S series on figures 9 -13 to compare them directly with previous authors. Figures 14 - 17 give ratios relative to their case B values, but we have provided the case B value on the figure. Comparison of line intensity ratios for n < 10 with Robbins and Robinson (1971) and Brocklehurst (1972) gave agreement for the n*P-2'S and n^S-2^F series of about 5% - 9%. The n ^P-2*S series agreed much bett^ with Brocklehurst for case B. We found 1-2% agreement for the and n^D-2^P intensity ratios at caseB. Our emissivity for X4922 was lowo^ than Brocklehurst’s by 2.8% at both cases A and B. These differaices are larger than we would have liked to have seat, particularly for the higher members of the n ^S-2^P series which change by only 10% from case A to case B. The errors seem to be inh erit in the problem since comparisons between Robbins and Robinson, and Brocklehurst show similar differences. They are probably attributable to the differences in the adopted atomic data, the mode of extrtp>lation to high n , and the fact that Brockldiurst included collisions explicitly. We are not convinced dnd oar absolute values fcr the singlet cnnssivitiss and intensity ratios ere less raliable then those of our predecessors, but regardless of whether or not they are, our results for the deviations firom 106 case B should be more accurate than 10%. These results can easily be applied to Brocklehurst’s results, for example, by simply scaling our case B value to his. 107 4.4 Discussion (a) Photoionization Models As we have shown above, the singlet spectrum depends upon the ratio of continuous to total opacities, which can be related to i) =Xji*lxn,*. In turn, Ti dq>ends inversely upon the local ratio of H I ionizing photons to He I ionizing photons. The higher the ratio of 1.8 Ryd to 1 Ryd photons, the greater the value of T), and the greater the deviations &om case B. As optical dq>ths rise, the local shape of the ionizing spectrum changes, and therefore r\ varies ccmtinuously across die nebula. This introduces large variations in the local singlet spectrum, which must be averaged over to produce the observed spectrum. This calculation requires photoimizadon models. The volume-averaged anissivity of an arbitrary recombination line is given by: ^ = - J4w*dr wh 0 reJVj,lVi^<‘,a n d o t^ » ^ = alldqwndonr. The detailed c^ture-cascade calculations performed for section 4.3 are too time consuming to perform over the large grid of points necessary to produce an accurate average with equation 4.29, and so we sought some simple fits to our results from the previous section which could be added to existing photoionization codes. Since they exhibit the most dramatic variation from case A to case B, we were only ooncsmed to tnodel the n ^P-2^S lines. 108 We assume that the bulk of a line’s luminosity comes from a region of large enough that case B would fully obtain for Xg(X584) of zero. This is usually a reasonable assumption because volume averages tend to be heavily weighted towards outer radii, where the optical depths should also be high. Under this assumpticm, we need only fit the Xe(XS84) dqtendence of the n^P-2^S lines. For the lower m emboa o f the n ^P-2^S series we find: 9 52x lOr^* r7‘™ whore we have inducted the temperature dependence discussed in section 4.2 for convenience. Note that these fits are not valid over the whole range of %g(XS84): Equations 4.30 - 4.33 are good for %c(X584) ^ 10"^ but equation 4.34 is only good for Xc(W84) £ 10~^ For higher values of X, (1584) the real effective recombination coefficients tend towards the constant, case A value, whereas these simple equatkHis would indicate that the zsctnnbinatioa coefficients continue to decline K) zcso. 109 We used the above fits in several photoionization models. The code used for all the models was most recently described in Ferland and Osteibrock (1987). We present the singlet spectrum results below, after a brief discussion of the models. (i ) AGN Model. The emission-line regions of active galactic nuclei (AGN) such as Seyfert 1 and 2 galaxies are thought to be characterized by densities firom as low as 10* cm~^ to as high as 10^° cm~^ (Osterbrock and Mathews 1987). Historically the high density, broad-line emitting regions (BLRs) have been thought to be distinct &om the the low density regions, narrow-line emitting regions (NLRs), although it is no longer clear if this is so (see discussion in chapter U). BLR densities are much too high for the LDL calculations presented here to be applicable, but we e^^ect oiu: results to be tgtplicable at least to NLRs. Also, there are a numbo' of narrow-line AGN such as LINERs fw which densities of the order considered here may apply. We sought to produce photoionization models of typical, low-density active galactic nucleus (AGN) emission-line regions. As we discuss at somewhat greater length in section 6.2a below, a photoionization model of a nebula can be fully specified by specification of the shape and strength of the ionizing spectrum at the inner radius of the nebula, the elemental abundances of the ionized gas, and the density of the ionized gas. We will assume all our photoionizaiion models to be radiation bounded in this section, so that it is not necessary to specify the gas column density. In the case of an AGN, where spatial information is hard to obtain, the strength of the ionizing spectrum at the mni^on-line region is most conveniently described by the ionization parameter, T: 110 where r is the (usually unknown) distance from the ionizing continuum source, jg is the rate of production of ionizing photons (i.e., those beyond the hydrogen Lyman limit), and Na is the numbo* density of neutral plus ionized hydrogen. We made 2 models, more or less appropriate to a LINER and to the NLR of a Seyfot 2 galaxy. We used solar abundances (see section 6.4 below for the adopted values). We adopted the mean AGN spectrum of Mathews and Ferland (1987) as the photoionizing spectrum in both cases (for discussion of the particulars of this continuum, and whether or not it is appropriate to a LINER, see section 6.2). For the LINER we chose a density of Ng = 10^ cm~^ and F = 1(T^^. Hiese parameters are of the same order as those which Ferland and Netzer (1983) found to well represent a mean LINER spectrum. Two differences are that Ferland and Netzor used a simple power-law continuum and defined their ionization parameter in terms of rather than N^. For the present application these differences were not considered to be important For the Sy 2 galaxy we used the same parameters, but an ionization parameter of F = 10~^ which is also similar to the value found by Ferland and Netzer. We stopped the models when the temperature had fallen to 4,000 K and line emission had ceased. Both models were plane parallel. The resulting models both include X-ray heated and ionized zones beyond the H^IH^ ionization front (m) H U Regions. We modeled HII regions using stellar spectra from Kurucz (1979), with r. = 35,000,40,000, and 50,000X. These were all tog(g)=4 stars, except the last, which was /og(g) = 4.5. We adopted a density of 10^ cm~^ fm the ionized gas in each case. We specified the ionizing flux, however, through values of Q and r (see equation 4.35) rather than F. From Osterbrock (1974) we took log(Q) = 48.84,49.23, and 49.77 as typical of the respective stars. We used abundances I l l appropriate to H U regions (He = .10; C = 3.2* 10”*; N = 5.0* 10”*; O = 4.0* 1(T*; Ne = 8.0* 1(T®; Mg = 4.2* 10”^; A1 = 2.7* 10”’ ; Si = 4.3* 10”®; S = 2.0* 10”^; Ar = 3.7* 10"^; Ca = 2.3* 10”’; Fe = 5.0* 1(T«). We chose an inner radius of 5.25* 10*^ cm for all models (effectively zero). We also stopped these models at a temperature of 4,000 K. In this case, of course, the models all stopped just past the ionization front, because there was no X-ray flux. (lii) Planetary Nebulae. We modeled planetary nebulae using simple blackbodies of 1(X),000 K and ISO,(XX) K. We used a density o f 6* 10* cm~^. We specified the incident flux through r = 5.25* 10^^ cm again, and log(Q) = 47.2 and 48.114 for the 100,(XX) K and 150,(X)0 K stars respectively. These values for Q came in the latto" case firom radio observations of NGC 7027 (Terzian, Balick, and Bignell 1974), which we assumed to be a ionized by a star with T* > 150,(XX) K (Shields 1978). In the former case we used a crude extrapolation based on the position of planetary nebulae stars with T* = 100,000 K relative to 150,000 K stars on the HR-diagram given in Osterbrock (1974). Our justificatimi for using such crude estimates of the parameters was again that our purpose here is to get a general, not definitive, idea of the behavior of the singlets for diffeent objects. We used abundances appropriate to planetary nebulae (He = .10; C = 7.8*10"*; N = 1.8* 1(T^ O = 4.4* 1(T*; Ne = 1.1*10^; Mg = 1.6* KT’; A1 = 2.7* i r ’; Si = 1.0* 10^®; S = 1.0* 10"^; Ar = 2.7* 10"*; Ca = 2.3* 1(T’; Fe = 5.0* 1(T’). We again stopped our models just beyond the ionization firont, at a temperature of 4,(K)0 K. (b) Predicted Singlet Spectrum Our results for the 7 models discussed above are givea in table 58. hi the table we have quoted for the n ‘P-2*S series for n ^ 5, the ratio of the predicted effective recombination coefficient allowing for inclusion of photoionizations of H I by resonance photons, to the caseB effective recombination 112 coefficients. As discussed in section 4.3 this is less than 3% different from the equivalent quantity for the intotsily ratios. Also included in the calculation were all other photoionization opacities which could be ^preciable at 584Â. The results are largely as expected: As the spectrum hardens, the deviations from case B grow (e.g., compare the sequence of H E region models). The change in ionization parameter between the 2 AGN models produces almost no change in the singlet spectrum, also as expected. Thus, the deviations of the He I singlet lines should provide a useful measure of the ionizing spectral shape. Future predictions of the singlet spectrum for a large grid of photoionization models would be necessary to fully exploit this information. The deviations from case B can be used to derive an average %c(X584) from equations 4.30 - 4.34. The derived value varies slightly for each line since each depends somewhat differently on %g(A584). For the first 3 lines of the series, the average value of %g(X584) is almost identical, but for S^F-2^5 it is usually significantly larger. We quote a single value of Xc(X584) for the first 3 lines of the n ^P-2^S series, and a separate value for S^F-2'S in the last 2 columns of table 58. As a good first tqyproximation, however, it can always be said that a single value of %g(X584) charactmizes the spectrum. It will be noticed that even for the hardest spectrum (the planetary nebula with T, = 150,000 if) which produces the maximum deviation frmm case B, the deviations for 2^F-2*5 are equivalent to about %c(X584) = 2.2x lOr^, much lower than the maximum values for which our figures present results. The calculations for the higher values of %g(X584) are still worthwhile, however, because they are appropriate and oftmi attained for optically thin gases. Finally, it will be noticed that even the softest spectrum we have used (the HU region with T, = 35,000 AT) has 10% • 25% deviations from its case B values. 113 (c) NGC 7027 Kaler, Aller, Czyzak, and Epps (1976) (KACE hereafter) have measured line fluxes for n'D-2'f, n*S-2^P, and «*P-2^S, for n = 3 to 6 for the planetary nebulae NGC 7027. Their results, ratioed to A4922, are repeated in table 39, along with a measurement of the 2'P-2*S X2.058 pm/BrYX2.17 iim line ratio, due to Treffeis et a l. (1976). As discussed above, it should be possible to roughly characterize the measurements of KACE by a single value of Xg(X584). Consulting figure 10, it is immediately clear that the quoted 3*F-2^S intensity is quite close to its case A value, implying either Xxsg 4 < 10 or a value o f %g(XS84) k 10~^. We did not find so great a deviation firom case B for any object considered above. The other 3 lines, furthermore, are much closer to their case B values. The line ratios fall into a regime of TwM = 5 0 0 -1000 and %g(A384) - 10~^ to 0, in figures 11,12, and 13. We did not find such a large divergence between the average %c(A384) derived from different lines in the n ^P-2^3 series for any model above. Nor did we find such a low value of %c(A384) for our model planetary nebula of T* = 150,000fiT, which had parametos much like those used by Shields (1978) in his model of NGC 7027. For the other series, the n^D-2}P sales ratios are leastmably close to their recombination values, which of course show almost no dependence on either %c(2384), or T,. The n ^S-2‘P saies is the most puzzling, in that the ratios here a e actually larger than their case B values at 10,000 K. They si& roughly equal to the caseB values at 20,000 K, but the KACE data provide no other evidence for temperatures much différait than 11,000 - 13,500 K. We can offa no explanation for this result All in aU, the agreement between the KACE results and our theory is not good, and the reason for this is unclear. Most troublesome forthen -2^5 series is the extreme weakness of 3*P-2*5. Pahaps it is because 3*P-2*5 is difficult to measure as it is near to [O III] X5007, or perhaps die weaker, higher 114 series lines are in error because of their lower signal-to-noise. Alternatively, thae may be an important process which affects the emissivity of the 3^P-2^S line which has not been included in our calculations. In this context, we also note that Diaz et al. (1987) found for the giant H n region NGC 604, a /xsoi6/fM876 ratio of .16 - 20, whereas the case B value should be around .77. This implies a value of Xc(A584) of 2.5x 10"® to 1.8% lOr®, which is high for the 35,000 K - 40,000 K spectrum which Diaz et al. claim ionizes NGC 604, although not too high for a 50,000 K star. This suggests a similar problem for X5016 for NGC 604 as we found for NGC 7027, but as no other n'P-2^5 lines w^e quoted in Diaz et al. we cannot be sure. 115 Table 50 Indices for Case A Power-law Temperature Fits to He I Singlet Line Ratios (a) and Effective Recombination Coefficients (3) n^S-2^P n^D-2^P n a 3 a 3 a 3 2 .166 -.864 3 .231 -.7995 .557 -.462 -.0991 -1.119 4 .261 -.770 .559 -.460 -1.019 5 .265 -.765 .557 -.462 .0430 -.976 6 .260 -.770 .550 -.470 .0621 -.957 Table 51 Indices for Case B Power-law Temperature Fits to He I Singlet Line Ratios (a) and Effective Recombination Coefficients (3) n^P -2*S n*S-2*P n^D-2^P n a a P a 3 2 3 224 -.790 .414 -.601 -.0994 -1.114 4 254 -.761 .460 -.554 -1.015 5 260 -.755 .483 -.532 .0425 -.972 6 258 -.757 .494 -.521 .0616 -.953 116 Table 52 Collisional Transition and Radiative Recombination Rates Affecting Populations of Levels 2^5 and 2*P. ______T, qÇ?S-2^P) qC2^P-2^S) aj(//eV^L) 5.000 1.05(-9) 8.99(9) 3.26(-13) 2.03(-8) 10.000 4.70(-9) 7.94(-9) 2.10(-13) 4.27(-8) 20,000 1.00(-8) 7.50(-9) 1.29C-13) 6.82(-8) Table 53 Population of Level 2^S vs. Density. N(2^S)fNae* Te Ne = 10 COT' 3% l(f COT-^ 10® COT 10* COT"* 5.000 2.56rl(T* 5.2(kl0r® 1.51% lOr® 1.61% 1(T® 10.000 1.65% lOr» 2.47% 10-^ 4.78% 10-^ 4.92% 10*^ 20,000 1.01% 10r« 1.17% lOr* 1.86% 1(T* 1.89% lOr® 117 Table 54 ICE , Te Ne = 10 3*l0®cm"^ 10* ctn^ 10* cm“^ 5,000 7.92* lOr” .04% 8% 22% 23% 10,000 4.35*10"” 2% 29% 57% 58% 20,000 2.39*10"” .5% 54% 86% 87% Table 55 Collisional Contributions to 2^P-2*5 Emissivity; CaseB. Y c e r . Ng = 10 cm"* 3*10* cm"* lO*cm"* 10* cm"* 5,000 7.66*10"*'* .04% 7% 21% 22% 10,000 4.21* lOr*'* 2% 28% 53% 55% 20,000 2.31* KT*'* .4% 51% 81% 82% 118 Table 56 Ratio of Hel (4‘Z)-2‘P) M922 to He I X5876. T. Case A CaseB 5.000 K .353 .362 10.000 K .350 .359 20.000 K .345 .356 119 Table 57 Ratio of Case B to Case A He I Singlet Emissivities n n^P -2 ‘S n^S -2 ‘P n ‘£>-2*F \(Â) ^ xlt X4922 X(Â) f xU X4922 X(Â) ^ xf^ M922 2 20581 968.9 3 5016 43.92 7281 1.916 6678 1.027 4 3965 29.26 5048 1.441 4922 1.027 5 3614 25.26 4438 1.291 4388 1.029 6 3448 21.57 4169 1.186 4144 1.029 Table 58 Hélium Singlet n ^P-2^S Intensities Relative to X4922 firom Photoionization Models, given for X2.058 \m in cm ^Jsec )• AGN Spectra Planetary Nebulae H n Regions n X(Â) r=i0"^® r=io~^ 100,000 K 150,000 K 35,000 K 40,000 K 50,000 K 2 20581 .174 .178 .136 .0804 .752 .353 .153 3 5016 .452 .450 .387 .273 .891 .651 .412 4 3965 .280 .280 .233 .157 .789 .463 .253 5 3614 .392 .392 ,327 .216 .881 .612 .355 X,(X584) n = 2 -4 9.2(-4) 9.2(4) 1.2(-3) 2.2(-3) 6.8C-5) 3.6(4) l.K-3) %c(X584) n = 5 l.K -3) l.K -3) 1.5(-3) 2.6(-3) 1.0(-4) 4 .7(4) 1.3(-3) o 121 Table 59 Helium Singlet Lines Relative to M922 for NGC 7027. (2 ^P-2 ^S is relative VsBryU-M \m). n^P-2‘S n ‘S-2*P -2 ‘P n X(Â) ^ M922 X(Â) M k) Ix/^iA92l 2 20581 .22 3 5016 .109 7281 .69 6678 3.17 4 3965 .70 5048 .24 4922 1.00 5 3614 .28 4438 .063 4388 .57 6 3448 .19 4169 .059 4169 .36 7 4024 .21 8 3936 .16 122 X 2 . 0 S 8 | i m a.„(2’P-2'S) . - 1 4 10" S 4. a> iia r 2 X,(X5@ 4)= 10 X584 Figure 9. (2*P-2^S) as a Function o f t>sg 4 and %g(XS84). The effective recombination coefficient {cm}! sec) for the 2'P -2 'S transition is plotted against the optical depth in He I Lya XS84  (tu 84 ), for the indicated values of the ratio of continuous to line opacity for XS84 (Xg(X584)). The results are for 10,000 K, but can be simply scaled to any temperature by (r.)l_ = „ W .0 5 8 (10,000 K)tl-.864 123 1X5016 (3 'P -2 'S ) .-4 1X4922 (4'0-2'P) 1.00 ,-3 <0 5 - 2 ' m X^(X584)> 10 •< 0.10 X584 Figure 10. J %3oiô(3^f -2^^ )/j ) as a Function o fr ^ s i and X^(X584). The ratio of the 3}p-2^S intensity to the fiducial 4^D-2Y intensity is plotted against Tx 5S4 for the indicated values of Xe(XS84). Hie results are for 10,000 K, but can be simply scaled to any temperature by JxsoiefJmn O') ^JxsoiefJum ^4^^ • Because7 ^9 2 2 varies by at most 3% for any combination of 1^584 and Xe(A584) this figure also effectively gives the variation of the 3^P-2^S effective recombination coefficient 124 1X3965 (4‘P-2'S) 1.00 - 1X4922 (4'0-2'P) 10" N Si -3 ■V lO X, (X584) »10"® T X584 Figurell. as aFunctionqfXxsuOttàXcO^^). This figure is similar to figure 10, except the plotted ratio is the 4*P-2*S intensity relative to 14922. The temperature scaling is given by Jw 96 s/Jm922 = Jwsss/J>4922 * • 125 .-4 1X3614 (S'P-2'S) 1X4922 (4'0-2'P) r 3 X„ (X584)»10“^ 0.01 T X584 Figure Î2. 7\36i4(5^^“2*S)/iw6i4(4*D-2'F) osaFunction ofXj^sM andXcQ^M). This figure is similar to figure 10, except the plotted ratio is the 5'P-2'S intensity relative to X4922. The tempaature scaling is given by y U 6i4/y ^4922 (^) == V weiVJ wzz ( ^ • 126 . JX3448(6'P-2*S) CM % 0.10 % CD V » X. (X584) » 10'* n 0.01 T X584 Figure 13. jX3^ (. 6 ^P -2^S )/jX3C48(4^^ -2*F ) as a Function ofxxsn and %e(XS84). This figure is similar to figure 10, except the plotted ratio is the 6^P-2^5 intensity relative to X4922. The temperature scaling is given by 7 x34480\4922 (^) ~ • 127 1.00 -3 _ iX728l(3'S-2'P) ,-4 X728I “ 1X4922(4'0-2'P) • 0.10 0.01 XS84 Figure 14. The Deviation o f j x i 2 s i O ^ S - 2 ^ P y j f r o m its Case B Value, as a Function of o«d%c(X584). The deviation of the îaîio of the 3*S-2*P intensity to the fiducial 4^7)-2^P intensity from the case B intensity ratio, is plotted against for the indicated values of Xg(X584). The results are for 10,000 K, but can be scaled to any temperature by jximUMm (T)=jKimfJMm ^4 Because y M922 varies by at most 3% for any combination of T)js4 and J^(XS84) this figure also effectively gives the variation of the 3^5 -2^P effective recombination coefficient 128 1.00 -3 >.161 X6048 .-4 /< 0.10 0.01 ,2 A 10 10 X584 Figure 15. The Deviation o /j u448(4^Z)-2‘P)yrofft its Case B Value, as a Function of Xjjg 4 and %g(X584). This figure is similar to figure 14, excq)t the plotted ratio is the 4‘S-2^P intensity relative to X4922. The temperature scaling is given by j Mow/Jwgzz (%') = J wo 4s//wzz ( 4 - 129 1.00 1X4438 (5 'S -2* P) X4438 jX4922 (4 '0 -2 'P ) r» •9 /< 0.10 I , " 0G35 0.01 10.2 10.4 X584 Figure 16. Tks Deviation o f /W438(5'^ )(/ -2 ^f ) from its Case B Value, as a Function of txs84 and %g(X584). This figure is similar to figure 14, except the plotted ratio is the intensity relative to A4922. The temperature scaling is given hy JiM’aUum =7 M43g/7vt922 ^4 130 1.00 jX4l69(6'S-2'P) X4I6B 1X4922 (4 '0 -2 'P ) 0.10 .-4 > .0330 0.01 ,2 .4 10 10 X584 Figure 17. The Deviation o f jMi69(6^S-2*P)/Jy44g(4*i>-2‘/*) jîioüî its Case S Value, as a Function of Xx5g4 and %c(X584). This figure is similar to figure 14, exc^t the plotted ratio is the 6'S-2*P intensif relative to X4922. The temperature scaling is given by j m&fJ U9n (r)=V vues^/v m i ( 4 ™ • V. HYDROGEN EMISSrvriY IN REALISTIC NEBULAE: THE EFFECTS OF VELOCITY FIELDS AND INTERNAL DUST 5.1 Introduction The accurate and reliable measurement of the primordial helium abundance Y s He IH will offer a definitive test of big-bang nucleosynthesis (Boesgaard and Steigman 1985); unfortunately the accuracy required, much better than 5%, presents a challenge to conventional methods of determining abundances from ratios of emission lines (see the reviews by Kunth 1986, Pagel, Terlevich, and Melnick 1986, Shields 1987, and Pagel 1987). The refinement of conventional nebular theory to the precision required to test the big-bang is an important goal. Although this chapter does not address measurements of 7 , it does discuss two effects which affect abundance determinations at this level of precision. Hydrogen and helium lines ate commonly assumed to be produced by recombination under "case B" conditions. The deduced abundance then has little depmidence on the physical conditions (see Osterbrock 1974) and great precision is possible. The presence of either collisional or radiative transfer effects would complicate the situation. Davidson and Kinman (1985) discuss the possibility that collisional excitation of Balmer lines would raise the hydrogen emissivity and hence cause Y to be underestimated. Ferland (1986) discussed the possibility that collisional excitation of He I lines is important; this effect would cause 7 to be overestimated. The role of both collision processes is uncertain; Pagel (1987) and Shields (1987) both conclude that collisional effects amount to a few percent at most. A possibly related issue is the slope of, and scatte* in, the dY/dZ relation illustrated, for instance, by Peimbert (1986) and Pagel, Te’levich, and Melnick (1986). At a given value of OlH or 131 132 NIH the scatter in Y often amounts to +10%, possibly suggesting the presence of a third parameter. This chapter addresses the question whether hydrogen lines are actually formed under caseB conditions in real nebulae. For case B to apply, all Lyman lines, including those produced by high-n levels, must scatter often enough to be degraded into L y a and Balmer lines. Hiis will not occur if dust destroys the photon or if the photon escapes the cloud (see Capiiotti 1966, Cox and Mathews 1969). Case B //p emissivities may not be valid when very accurate results are needed; Lyman line escape or absorption by dust will lower the emissivity of ffp and cause abundances of oth^ elements to be overestimated. 133 52 Calculations We are inta-ested in computing the emissivity for conditions ^Jpropriate to H E regions, planetary nebulae, and nova envelopes. Lyman line leakage and destruction by dust can occur under some circumstances. Here we discuss our treatment of the hydrogen atom, Lyman line transport, and outline our assumptions concerning the possible velocity structure and dust content of the line-forming regions. Our approach is basically similar to that followed by Capriotti (1966) and Cox and Mathews (1969); these studies were primarily interested in changes in the Balmer decrement, and they did not report emissivities. The latter is the point of the present chapter. Many details of our calculations are analogous to those done for hydrogen and He I in the preceding chapters and are not presented here. (a) The Hydrogen Atom Hydrogen lines are formed under conditions which Ue between two extreme limiting circumstances; in case A all Lyman photons escape while in case B all high-n Lyman photons scatter often enough to be degraded into L ya and Balmer lines (see Osterbrock 1974). Most analyses assume case B even though this assumption formally requires infinite optical depth in all Lyman lines. A second pair of limiting assumptions concerns the density; in the low-density limit /-mixing collisions can be neglected (see Pengelly 1964), while in the high-drasity limit (W 5:5x 10* c/n“*, Drake and Ulrich 1980; Mathews, Blumenthal, and Grandi 1980) full / -mixing can be assumed for n ^ 4 and the formalism developed by Seaton (19596) applies. The low-density limit should be valid for H n regions and planetary nebulae, while the high-density limit applies to nova shells. Although the case B emissivity of H P is quite close for these two assumptions concerning the density, the case A values differ by substantial amounts. 134 (i) The high-density limit. We treat the hydrogen atom as in Seaton (1959a). This assumes that n-changing collisions, along with collisional ionization and thiee-body recombination, are negligible. The calculation also assumes that /-changing collisions are fast enough for die /-sublevels to be populated according to their statistical weights. Following Seaton, for n > 30, we use the closed form for the transition probabilities given by Maizel and Pekeris (1938) and corrected by Burgess (1958). For n ^ 30 we use the gaunt factors of Baker and Menzel (1935). Radiative recombination coefficients are computed as in Seaton (19596), but with gaunt factors from Karzas and Latta (1961). We treat principal quantum numbers up to 200 exactly, and add the remaining portion of the recombination coefficient to the 200* level. Tests show that the predicted emissivi^ is in exact (four significant figure) agreemait with Seaton’s (19596) results for both case A and case B at temperatures of 5000 AT, 10*Ar,and2xl0^AT. Results of these calculations are shown in figures 18 and 19, and will be discussed furth^ below. (ii) Low-density limit. In this limit both / -changing and n -changing collisions are assumed to be slow relative to radiative processes (see Pengelly 1964). The differences between case A and caseB emissivities are not as extreme as in the high-density case because only np levels are effective in producing decays to ground; as a result only these levels are underpopulated by ground state transitions in case A. (When full /-mixing is assumed, all levels within principal quantum number n are affected by the inclusion of the decays firom np to ground.) We computed the hydrogai capture-cascade problem for the low-density limit with the following assumptions. Complete / -mixing was assumed fra n ^ 21 and the formalism described above was used for these high- n states, with the excqrtion that n-levels up to 320 were included. This is not formally 135 correct at densities below -10^ cm~^ (Bioclddiurst 1971), but the emissivities predicted below are in good agreement with more complete calculations. Below n =21 all /-sublevels were considered. Radiative rates were taken from Capriotti (1964) or computed 6om Green, Rush, and Chandler (1957). Recombination coefficients were taken from Burgess (1964). Transitions from the 300 /-mixed levels above n = 20 to the 210 / -states below were assumed to occur in the ratio of the statistical weights of the lower level, as suggested by examination of tables of transition rates. Full results are shown in figures 20 and 21. Hummer and Storey (1987) have recomputed hydrogen line emissivities and decr^ents treating the / -sublevels exactly and with a complete treatmoit of collisions. Our predicted 7/p emissivity for case B is ~ 0.4% lower than theirs for a density of N, = 10* cm~^. Comparison with Pengelly (1964)’s results shows that our predicted f/p emissivity was smaller than his by 2.7% for case A and 0.3% ftn* case B. Our results were small» than those of Martin (1988) by 1.8% and 0.5% for the two cases. These relative deviations are expected to be computed to rath» more precision than the absolute emissivity since small systematic errors cancel in the difference. This precision was considered acceptable since the goal of this chapt» is not to recompute die absolute caseB »nissivity to great precision, but rath» to study deviations from case B due to two effects: partial Lyman line leakage and destruction of trt^ped Lyman lines by dust. The main source of uncertainty in the following calculations is probably our use of the escape probability formalism in the treatment of the transport of the - 300 Lyman lines in the model hydrogen atom. 136 (b) Velocity Structure and Optical Depths The velocity structure of the H* zone, where is fbrnied, must be known if optical depth effects are to be determined. Emission-lines from giant extragalactic H U regions typically have widths which correspond to thermal broadening together with an additional turbulence of -2 5 km/sec (see, for example. Smith and Weedman 1970; Melnick 1977; Talevich and Melnick 1981; Melnick et al. 1987). The origin of this supasonic motion is not clear, but it may be a form of micro-turbulence or provide an indication of the depth of the potential well at the line-forming region. Galactic H II regions generally have thermal line-widths, although in the well-studied case of the Orion Nebula a component of sub sonic microturbulence is present (Castaneda and O’Dell 1987). Planetary nebulae have expansion velocities of order 25 km/sec (see, for example, Osterbrock 1974), while in nova shells velocities of order 500 - 20(X) km/sec are encountered, hi both cases a velocity gradient is present When the line-width is due to random motions their effect would be to reduce the optical depth scale through the velocity term in the Voigt function. In cases of expansion with a velocity gradient the Sobolev or large-velocity-gradient approximation may be valid. Intermediate cases are discussed by Hummer and Rybicki (1982). All of these velocity fields diminish the optical depth scale, increase the escape probability to some extent, and make Lyman Une escape more likely. The geometry of the onission-fine regimis could also affect the line transfer undra some circumstances. If the gas is distributed in smaU clumps which statisticaUy average into a uniform structure on large scales, as is assumed in most photoionization calculations and emission-line analyses, then inhomogeneities should not affect the transfix problem. Real H n regions and planetary nebulae are clumpy even on large scales, so we are not guaranteed that a photon created inside the H* zone must traverse the full optical depth of the Stromgren sphere to escape. Depending on the geometry, the large- 137 scale clumps could allow Lyman lines to escape more readily than we assume so that the actual deviations from case B may be larger than we predict A second uncotainty is the role played by neutral gas outside the Strmngren sphae. The Lya optical depth through this material is very large, but the gas is expected to be very cold and hence have a very small line-width. Depending on the geometry, it seems likely that Lyman lines could interact with this material. Depending on circumstances, we consider two cases; the first in which resonance lines e sc t^ without interacting with a large optical depth in the direction away from the central star; and a second, in which the optical depth in the outward direction is effectively infinite. (c) Dust to Gas Ratio Dust is known to exist within ionized zones of H II regions (see Mathis et al. 1981) and planetary nebulae (see, for example, the review by Barlow 1983). Dust acts to diminish the intensity of both by attenuating X4861 as it escapes (a correction is usually applied for this effect, see for example, Mathis 1983), and by destroying Lyman line photons before they scatter often enough to be degraded into Balmer lines (Capriotti 1966, Cox and Mathews 1969); these papers first considered the effects we discuss hee, but did not rqxtit changes in emissivity, the point of the present chapt^. This second correction to deduced abundances for diminished Balmer line emissivity has never been ^plied, to the best of our knowledge. Although the detailed propmies of dust are often not known, it is known that the scattering cross section per hydrogen atom in galactic H n regions is generally in the neighborhood of o««f(V ) f N^, = 5x 10r“ cm® (5.1) 138 for visible light (O’Dell, Hubbard, and Peimbert 1966, Perinotto and Patriarchi 1980, Mathis et al. 1981). In this equation is the total hydrogen density. There is nearly a half-order of magnitude range in this value among galactic H n regions (O’Dell, Hubbard, and Peimbert 1966). If the albedo is a = 0.5 (a typical value, see Savage and Mathis 1979), then this is also the absorption cross section per atom. Mathis (1986) argues that typical H n regions have dust properties similar to the general ISM, and studies of correlations between gas column densities and interstellar reddening (Savage and Mathis) find absorption cross sections of this magnitude in the general interstellar medium. This value of the absorption cross section is for visible wavelengths; in the genaal interstellar medium the total cross section at = 1000  is known to be roughly five times larger (Savage and Mathis). Assuming that the .wavelength dqtendence of dust in nebulae is the same as the ISM, then the reddening curve given by Savage and Mathis, an assumed albedo of -0.5, die absorption cross section per H atom given by equation 5.1, and extrapolating the Savage and Mathis extinction curve to 950  (the wavelength of typical Lyman lines), we find c ^ ( 950 k ) I N ^ ,~ 3 x i r " cm ' . (5.2) In the following discussion we take this as the "standard" absorption cross section per H atom. Dust in the Orion Nebula is known to be peculiar, perhaps because of the very large electron density. Although the scattaing cross section p « atom is fairly typical of the genial interstellar medium in the visible (Perinotto and Patriarchi 1980), the absorption cross section appears to have a much flatter wavelength dependence than the general ISM; Bohlin and Savage (1981) measured extinction of the stars near the cellier ofiheûjdoûNcûula, and they find^(X-V)/£l {£ - V )= 1.88 at 1250 Â. Ws assume that the extinction continues to rise with the same slope as that between 1390  and 1250  to = 950 Â, 139 sothat£(9S0Â-V) I E {B -V )~ 3 . Then the ratio of cross sections between X.9S0 and the visible i s - 2 and the absorption cross section per atom for the ionized gas within the Orion Nebula is given by CTofc,(950 k ) /N u ,,~ 1.0% 10r“ COT* . (5.3) In the following discussion we will consido^ various values of the absorption cross secticm given by 950  ) /N ^, = / 3.0% 10-zi COT* . (5.4) and defined in terms of the variable / , which equals 1 for the general ISM, 1/3 for the Orion Nebulae, and 3 for high values of the ISM. For planetary nebulae it is possible to estimate the effects of dust on certain ultraviolet emission-lines (see, for example, Barlow 1983). Harrington et al. (1982) find an absorption cross section per atom of 0 ’afc,( 1549Â)/Nft,,=2rlO"“ cOT* in the planetary nebula NGC7662, in agreement with Bohlin, Marionni, and Stecher (1975)’s results for NGC 7027. The wavelength dependence of the absorption curve is unknown for dust in planetary nebulae, but this result seems roughly consistent with that for Orion quoted above ( f = 1/3). ((^ Radiative Transfer Lyman line transfer is treated with the escape probability fbrmal&m (Capriotti 1965). We use the form of the escape probability function m = 140 given by Slater et al. (1982). A temperature of l( fK was assumed in our evaluation of the damping constant a for each hydrogen line. In the following discussion the optical depth scale will be parametrized by the optical depth in Lya. The expression given above was used for the case where Lyman photons do not interact with neutral material outside the line-forming region; for the case of one sided escape the line was assumed not to escape in the outer direction and half this value was used for the total escape probability. In the following discussion we will neglect the effects of dust scattering on the line transport; this is justifiable if die grains are forward scattering and the scattering process coherent Overlapping of lines originating in high-n levels is a possible concern. Levels above n =30 overlap in frequency for typical velocity fields. In this case the very high-n Lyman lines form a psuedo- continuum with an opacity roughly equal to that of the highest unblended line. In the following calculations we evaluated the opacity by averaging over the overlapping lines assuming a Doppler profile. For the optical depths of interest here (10^ -10^) level 30 is optically thin and line ovalapping is of secondary importance. For situations where gas is mixed with another absorbing medium, such as dust, the interplay between escape and destruction of the resonance line photon is parameterized by the ratio of continuous to line opacities siq. /(K; + Xg ) (Hummer and Kunasz 1980, Netzer, Elilzur, and Ferland 1985). Netzer et a l. show that the form of the escape probability in this situation is given by P^('i) = ft^c + P('t) (5.6) where P(x ) is givai by equation 5.5. Hummer and Kunasz’s nummcal results suggest that b = 5.6 for line optical dq>ths x = 10^ and = iCT^. The coefficient ô is a very weak function of optical deptli according to Hummo' and Kunasz’s results. We treated this as a power-law in the optical depth. 141 We can express Xg as a function of the hydrogen neutral fraction if the gas and dust are well-mixed (which may not be the case for the inner-most regions of Orion, see Mathis et al. 1981). Using the absorption cross section for dust given by equation 5.4, we find for Ly a X,(Lya) = 5.1x10^ f |r4+ [-^1 I (5.7) where the turbulent velocity is given in km/sec and is the temperature in units of lO^liT. Our results will be presented in terms of Xc (L y a ), although the ratio of cross sections was evaluated individually for each Lyman line assuming a fiat dust opacity. (e) Results Results of these calculations are shown in figures 18 - 21. All calculations are for a temperature of VfK, although the calculated deviations should have only a very slight temperature dependence. Figures 18 and 19 are for the high-density case where full / -mixing occurs. Figures 20 and 21 are for the low-dotsity limit The optical depth in Lya is given on the lower axis, and the four curves in each figure are for %g(Lya) = 0, 10”^, 10 ^, and 10"\ The quantity plotted is the deviatitm from case B emissivity fCBc Hfl. Hie emissiviQr of is diminished because some fraction of the Lyman lines are not converted into Ly a and Balmer lines for all cases except those which are dust-free and Xi^ya large. Optical depths greatmr than X iy ^ -10* are needed to approach caseB to better than 1% in the dust-free cases. The preseiree of dust causes the emissivity to diminish for all optical depths. Photoionization calculations which examine the importance of this euiect are presenied below. The case of greatest interest is one in which expansion velocities are small and large; some H n regions 142 and planetary nebulae are examples of objects in this category. We have fitted our predictions of the deviation from caseB for Xiya= 10® and X ^Lyo) < 10"® as a simple power law for the low-density limit: . = 1 .- 11.008X ^ ( Z , y . (5.8) For the high-density limit (where full I -mixing is assumed and the effects are larger) the results are well approximated by = 1 -13.403 X,(jLya)®«®® . (5.9) As a check on the temperature dependence of these effects, we also computed the case B deviations for the case of an qrtical depth of %,a= 10® and T ,= 2 x l(fK . This higher temperature is more characteristic of very metal deficient nebulae. The results are very well fitted by ■ = 1. -15.447X c iL y a f^ ^ . (5.10) ^KJ(H\i)caseB All of these fits are virtually exact. 143 5.3 Applications (a) The Orion Nebula The Orion Nebula is a benchmark in any study of H E regions (see Osterbrock and Rather 1959; Peimbert and Torres-Peimbat 1977; Torres-Peimbert, Peimbert, and Daltabuit 1980); for instance, it helps define the dYldZ correlation sometimes used to detamine the primordial helium abundance (see Peimbert 1986). Here we examine the conditions under which hydrogen lines are formed in this object. Osterbrock and Rather’s (1959) work established the presence of a large density gradient across the nebula (ranging from N, > 10'* cm~^ to < 300 c m ~ \ Peimbert and Torres-Peimbert (1977) and Torres- Peimbert, Peimbert, and Daltabuit (1980) firrtha^ established an ionization gradient. Apparently both densi^ and ionization decrease with increasing projected distance firom the central stars. Peimbert and Torres-Peimbert (1977) also note that the H e*IH* ratio ^jpeared to decrease by about 30% across the nebula. Collisional excitation and optical depth effects may be responsible for part of this, and the correction for the possible presence of He" is jsobably also important We estimate how close hydrogen emissivities will be to case B for intomediate regions of the nebula. Dust is likely to be mixed with gas in regions outside the dust cavity discussed by Schiffer and Mathis (1974), Mathis et a l. (1981), and Patriarchi and Perinotto (1985). Taking - 2' for the inner edge of the dusty zone, we can estimate the deviations firom case B once the hydrogen neutral fraction is known. The following estimate oiH * IH ” is approximate, but further sophistication seems inappropriate in light of otha uncertainties due to the velocity, density, and ionization structure, and the dust-to-gas ratio in this object 144 Bohlin and Savage (1981) quote a spectral class of 0 6 for 6^ Ori C, and Ostobrock (1974) gives Teff=4xl(fK for the effective temperature and Q(H)= sec~^ for the number of ionizing photons emitted per second for a star of this class. These parameters are consistent with the models presented by Simpson et a l. (198Q. Neglecting photoelectric and dust absorption, the photon flux at a distance of 2 '—9x10^’’ cm from the central star is <|>= 1.7x 10*^ If the electron density is N ,~ 2 x 10^ cm~^ OPeimbMt and Torres-Peimbert 1977) and the electron temperature is <4 = 0.85 (Mathis 1985), then the hydrogen ionization balance can be written as K | 1 = < ^ = 7800 LU (5.11) rJ 2000 cm where < or > = 2.5x10“** cmis the hydrogen photoionization cross section at the mean energy of 131 Ryd, appropriate for a blackbody with T = A xV f K , and a is the case B hydrogen recombination coefhcienL Assuming = 0.85, a component of turbulence of - 8 km/sec (Castafieda and O’Dell 1987), thatN^, =N*, and equation 5.7 w ith/ = 1/3 we find r %c(Z.ya) = 1.5x10“ —2 -1 (5.12) 2' 2000 cm“\ The small velocity gradients in the Orion Nebula (Balick, Gull, and Smith 1980) make it unlikely that Lyman lines see an optical depth any smaller than that for the full Stromgren sphere, which we take to be tiya ~ l(f. Our calculations (figure 21) suggest that theffp emissivity should be = 4.9% below case B for this region. Observations suggest that both the level of ionization and the dust-to-gas ratio (both of which detmmine X^Lya)) vary across the ùcc of the nebula. It seems plausible that changes in 145 hydrogen emissivity, caused by changes in XdLya), could contribute to the variations ia the He*IH* latio found by Peimbert and Torres-Peimbert (1977). This issue is discussed further in the tqqiaidix. This result may apply to other HII regirms with dust charactoistics and ionization similar to the Orion Nebula. Dust in Orion is peculiar, however, and if other nebulae have the same level of ionization but an average or high dust opacity ( / ~ 1 - 3), then X dLya) < KT® and the HP emissivity will be reduced by as much as -15.8% below case B. Errors as large as this would have significant effects on such correlations as the riT /dZ relationship for H II regions since the range of dP is only of this order (Peimbert 1986; Pagel 1987). (b) Giant HII regions A successful model of the giant H II region NGC4861 ("The HP Nebula") was produced by Dinerstein and Shields (1986). Their single-component model was recmnputed using the photoionization code most recently described by Poland and Osterbrock (1987) in ordar to detomine the physical conditions throughout the HP forming region. The parameters given by Dinerstein and Shields (n = 100 cm~\ a filling factor of 0.017, Q (H®) = 6r 10*^ sec"*), their abundances, spherical symmetry, and the 7« = 45,000 K stellar atmo^here firom Shields and Searle (1978), were assumed. Our predictions for the ionization and thermal structure were in good agreement with Dinerstein and Shields’. The calculation predicts that HP is aihanced by collisions by 1.0%, H a by 2.9%, and He I k5876 by 1.9% (rate coefficients for collisional excitation are firom Aggarwal 1983 and Herrington and Kingston 1987). The effects of inteiual dust, which is present in sane of these objects (b&lnick 1979), csuM change the hydrogen emissivity. The effects of dust on the ionization structure were included in the 146 photoionization calculation as in Martin and Poland (1980). The firequency dependence of the absorption cross section was taken to be that apprcq>riate for graphite dust with a radius of 0.05 \un . This is simplistic, but gives results not too different firom Mathis (1986). We predicted the case B emissivity by numerically integrating equation 5.8 o v a the model ionization structure. This assumes a Lya optical depth of 10^, so tiiat escape of high-n Lyman lines is negligible and deviations firom case B are only caused by dust absorption. We assumed cases of thermal broaderting alone, as well as thermal plus a turbulence of 25 ktn/sec (Smith and Weedman 1970, Melnick 1979, Talevich and Melnick 1981; Melnick et al. 1987); the emissivity would reduced much further if velocity gradients as large as 400 km/sec are found across the line-forming region, as suggested by Clayton and Meabum (1987). Conditions in the model ranged firom %c(Lya) = 0 near the ionization firont to ~3xlO~^ at the edge nearest the iortizing star, for the Orion dust opacity. The dust absorption per H atom was varied, by varying the factor / defined in equation 5.4, and the results are shown in figure 22. Both assumptions about die velocity structure of the nebula are shown. Dust is probably not as abundant in extrmiely metal deficiait objects as it is in the ISM or Orion; these results show that measuremoits of the primordial helium abundance (i.e.. Page! 1987; Shields 1987) are probably not affected. Actually, for small values of / the slightly reduced onissiviiy is almost exactly compensated by collisional excitation of /f p, and case B is an excellent assumption. However, significant effects occur when / > 1. If the dust absorption per H atom is correlated with the metalicity, as seems likely, then figure 22 suggests that the slope of the d ï !dZ relation could be affected by the reduction of the H p missivity in metal-rich objects. Figure 22 cannot be compared directly with obsawations of dïldZ , of course, since the temperature of the ionizing star was not also varied with metalicity (see, for instance. Shields and Searle 1978). 147 (c) Planetary Nebulae Evidence now suggests that dust is mixed with ionized gas in at least a few well studied planetary nebulae (see, for example, Harrington et al. 1982 and Barlow 1983). A formalism very similar to that adopted in this chapter was used to estimate the fractional destruction of certain ultraviolet emission lines (Barlow 1983). The ultraviolet gas-to-dust ratios quoted in Barlow’s review correspond to dust-to-gas ratios roughly similar to Orion, and Seaton (1979) finds fairly normal extinction properties in NGC 7027. We assume the dust properties outlined above, even though dust in planetary nebulae is Ukely to be quite diffluent from dust in H n regions. We have recomputed the simple model of NGC 7662 presented by Harrington et al. (1982, their Model 1). This was chosen over the more realistic Model 2 because of its simplicity. We assume a micro-turbulence of S km/sec and calculate the deviation from case B expected for various assumptions concerning the grain properties and velocity field by numerically integrating the {xedictions shown in figures 20 and 21. This nebula is matter-botmded, so Lyman lines can escs^ in the outward direction. If the expansion velocity is large then photons emitted towards the star will not interact with gas on the 6 r side; for smaller velocities Lyman photons can only e r c ^ in the outer directimt. We find Hfl emissivities, for the situation whore photons freely escape afto* being onitted in the direction toward the central star, which are 93.9% and 97.5% of case B for "high ISM” (f = 3) and Orion grain opacities (f= 1/3) respectively. For circumstances whoe Ly a cm only escape in the outo* direction (in the direction pointing away firom the star) the values are increased to 94.2% and 98.1% for the two cross sections. If, instead of neglecting the expansion velocity ( - 2 5 km/sec) of the nebula entirely, we assume that its effect is to broaden the lines and reduce the opacity in a manner similar to turbulence (that is, through the koadening torn in the Voigt function) then the anissivities are decreased to 91.5% and 148 96.7% of case B for the two assumptions concerning grain opacities, and considering only outward escape of Lya. Emissivities are reduced to 90.8% and 96.0% when two-sided escape is allowed. The actual effects of the gross velocity field, which lies between the range of validity of the large velocity gradient assumption and the assumption of strictly thmnal broadening (Hummer and Rybicki 1982), is also probably between these two cases. (d) Nova Envelopes Envelopes of classical novae expand with velocities l(f km/sec, and have densities high oiough for complete /-mixing to occur. Grain formation occurs in some objects (see, for example, Gallagher and Starrfield 1978), but here we focus on the effects of the velocity gradient observed in VISOO Cygni (Ferland 1980). Assuming that velocity and radius are proportional, the line optical dqtth seen by a typical photon in the H* zone is given by dx = a N ^ dr = r (5.13) where a is the cross section for Ly ct absorption, and Vexp are the thermal and expansion velocities, and the calculations for two-sided escape are used (figure 18). Ferland and Shields (1978) present a model of the envelope of VISCX) Cygni; assuming their parameters for day 121 ('■««r = 1.0* 10“cm,Q(H) = 9.5x 10^’ s~ \ U = 0.87) we findiV® - 3x 1(P cm"^ and dx^y „ ~ 250. The calculations for the case of ftill /-mixing show that the HP emissivity is reduced by -15% . The number is extremely uncertain because the velocity and ionization structure, and even the ionization mechanism of nova shells (see Ferland, Lambot, and Woodman 1986) are imcertain. 149 5.4 Discussion We have shown that the presence of dust or velocity gradiaits can cause the emissivity of HP to fall below case B predictions by < 15%. This consideration is important if abundances are to be measured to an accuracy much better than 5%. Such accuracy is possible (and needed) in the measurement of helium abundances in metal deficient H n regions to test current models of nucleosynthesis in the early Universe. Dust destruction of Lyman lines can also introduce spurious correlations between abundances since the emissivity is affected by the dust-to-gas ratio, which may scale with the metalicity, as well as the mean H * /H ° ratio of the gas, which may be correlated with the temperature of the ionizing star. Changes in the emissivity of H P could introduce t(pparent changes in abundances of otha" elements at the 2% to 15% level. These effects may influence the slqte of the dY idZ correlation, for instance, or cause scatter in Y when the helium abundance is actually constant Exttanely metal-deficient objects, such as those used to measure the primordial helium abimdance, should have hydrogen emissivities close to case B if they are also dust-fiee and die line-widths are not great. Although the simple models presented above suggest deviations firom case B emissivity by as much as 15%, actual nebulae are vastly more complicated than our theoretical constructs, and the line- fmmation process can be affected by flows, inhomogeneities, and patchy covering. Lyman line escape may be easier or more difficult than we assume. It is possible to determine how close the hydrogen spectrum of actual objects is to case B through simple infirared measurements, however. Hydrogen lines firom different series overiap in this wavelength range; for instance the 14 - 6 and 5 - 4 transitions occur at wavelmigths of 4.02 |un and 4.05 pm respectively, lines originating in different series are affected differently by departures from caseB, sometimes by percentages comparable to changes in the HP emissivity. For instance, the relative intensity of the two lines mentioned here differ by 23% in the 150 case A and case B limits. Precise measurements of such hydrogen decrements should offer clues to details of the line formation process and the validity of the case B assumption. 151 1.00 FULL /-MIXING 2-SIDED ESCAPE PROBABILITIES CASE A ( d ) «a, X 0.10 — CD • a (c) (b) (a) 0.01 (Ly a) Figure 18. Deviations from CaseB in the High Density Limit and Two-Sided Escape. The plotted quantily is the deviation in the emissivity from case B predictions is plotted against the Lya optical depth. These calculations are for the case of full {-mixing and two-sided escape and should represent high density gas. The curve marked (a) is for the case of no dust, while those marked (b), (c), and (d) are for the case of a ratio of continuum to line opacity of Xc(Ly a) = 10~®, 10"^, and 10”® respectively. 152 1.00 FULL /-MIXING I-SIDED ESCAPE PROBABILITIES CASE A (d) «EL X 0.10 ffi • o (b) 0.01 T(Ly a) Figure 19. Case B Deviations for High Densities and One-Sided Escape. The calculations ate similar to those in figure 18 eq)ect that the escape probabilities have been reduced by a factor of two to represent the effects of a very large optical depth in one direction. 153 LOO PARTIAL ^-MIXING 2 - SIDED ESCAPE PROBABILITIES CASE A (d) % X 0.10 m • a (C) (o) (b) 0.01 2 T {Ly â ) Figure 20. Case B Deviations for Very Low Densities and Two-Sided Escape. The lowest 20 principal quantum numbers are treated with all angular momentum sub-levels considered, but with no /-changing or n -changing collisions. Thus, this calculation is appropriate for densities found in planetary nebulae or H II regions. This case is for two-sided escape. 154 1.00 PARTIAL /-MIXING I-SIDED ESCAPE PROBABILITIES CASE A (d ) « a . 0.10 ( 0 ) ( b ) 0.01 T (Ly a ) Figure 21. Case B Deviations for Very Low Densities and One-Sided Escape. The calculation is similar to that in figure 20 expect that the escape probabilities have been reduced by a factor of two to represent the ejects of a very l^ge optical depth in one direction. 155 turb = 25 km s ID O turb ' 0 km s 0.01 0.1 10 f Figure 22. Effects o f Internal Dust on Model Giant H II Regions. The Dinerstein and Shields model of NGC 4861 was recomputed with various values of the dust absorption per H atom assumed. Dust opacity was included in calculations of both the ionization structure and the //p emissivity. The ratio of the assumed absorption cross section to that for the general interstellar medium is given as the factor / . The deviations from case B are shown for two cases; line broadening by thermal motions alone, and line broadening by both thermal and turbulent - 25 km/sec) motions. VI. THE NATURE OF THE IONIZING CONTINUUM IN SGR A WEST. 6.1 Introduction The classification of emission-line galactic nuclei as regards to the presence of Seyfert-Uke or starbuist activity is expected to provide important clues to the nature of such activity. A number of criteria distinguish luminous active galactic nuclei (AGN) such as Seyfert 1 nuclei and QSOs, whose line-emitting regions are excited by a nonthermal continuum, firom other emission-line nuclei such as H n region nuclei (French 1980) or more luminous starburst galaxies (Weedman et al. 1981) where young O and B stars provide the ionizing flux. These criteria include emission line FWHM > 250 km/sec (Feldman et al. 1982), the UBV colors (Weedman 1973) or JHK colors (Balzano and Weedman 1981) of a nonthermal continuum source, or emission-line spectra which reflect the wide range of ionization produced in a gas excited by a nonthmnal continuum (Baldwin, Phillips, and Terlevich 1981; Stauffer 1982; Balzano 1983; Veilleux and Osteibrock 1987). For low-luminosity ( l ( f erg/sec) emission-line nuclei the ionizing continuum is seldom strong enough to be directly observed, while emission-line widths can be comparable for disparate classes of objects (see review in Keel 1985). These objects must be classified through their emission-line spectra alone, often resulting in considerable ambiguity as to the nature of or even existence of, the ionizing spectrum. For example, the range of ionization and the relative strengths of the emissim lines of the Low-Ionization Nuclear Emission Regions (UNERs) (Heckman 1980) suggest either shock heating (Heckman; Baldwin, Phillips, and Talevich 1981), photoionization by a weak, nonthermal continuum (Ferland and Netzer 1983; Halpem and Steiner 1983), or possibly even photoionization by a hot 156 157 (80,000 K) bladcbody (Pequignot 1984; Carswell et al. 1984). It is not clear if every LINER’S emission lines are excited by the same mechanism. In&ared observations of fine structure lines from Sgr A West, the ionized region at die crater of our galaxy, demonstrate that it is also a low luminosity (L = 10^ erg/sec) narrow emission-line nucleus (for comprehensive reviews see Brown and Liszt 1984; Oort 1984; and Townes et al. 1983; also see AIP Conference Proceedings 155: The Galactic Center). Identifying Sgr A West as an example of either a Seyfert galaxy or a starburst or H n region nucleus would have important implications for the study of galactic nuclei in general because of the unparalleled spatial resolution available for Sgr A West Observations of broad wings (FWHM k 700 km/sec) on H I and He I recombination lines (Hall, Kleinmann, and Scoville 1982; Geballe et al. 1984) suggest that Sgr A West is an AGN, but a demonstration of the nature of the ionizing continuum is the preferred discriminant because the physical basis for differences in line-widths in extra-galactic objects is unclear. Neither the ionizing continuum nor the optical emission-lines generally used to classify emission-fine nuclei can be observed through the roughly 25 magnitudes of visual extinction towards Sgr A West. The modest amount of IR spectrophotomeuy available for known AGN and starburst nuclei (Aitken and Roche 1985 and references therein) suggests that Sgr A West is similar to the latter. The X-ray and IR continua are also most consistent with the continua of known starburst nuclei. Comparison of the fine structure fine data with optically thin, homogeneous models (Lacy et al. 1980) has shown that a thermal spectrum with T* = 31,000 - 35,000 K can reproduce the low-level of ionization which Sgr A West exhibits. This is the same result obtained from IR spectrophotometry for M82 (Willner et al. 1977), one of the best known starburst nuclei (Riekeet at. 1980). It is not obvious that the line data exclude a nonthermal ionizing spectrum however, because such a spectrum’s strong X-ray flux is predicted to produce a partially 158 ionized zone beyond the hydrogen ionization front, whore strong, low-ionization lines can be formed. There are no published studies of the formation of the IR fine-structure lines under these conditions, and only a few studies for cases of ionization by thermal spectra (e.g. Laçasse et al. 1980; Mathis 1985). In the present work we consider the observational data which pertain to the question of Sgr A West’s classification, and use detailed photoionization models to predict the fine structure line-spectrum resulting from typical nonthermal and thamal spectra. We demonstrate that all available evidence suggests a starburst model for Sgr A West. 159 6.2 Observed Properties of Sgr A West. In this section we discuss those observed properties of Sgr  West which save to constrain photoionization models of it We begin with a brief digression on the free parameters of photoionization models. (a) Photoionization Models. The principal free parameters for a photoionization model of a constant density gas cloud are: (i) the total hydrogen density, Nu, of the gas; (ii) the spectral s h ^ of the ionizing continuum; (iii) the strength of the ionizing continuum, specified in this chapter through Q , the rate of production of quanta beyond the Lyman limit; (iv) the distance r , of the ionizing continuum source from the ionized gas; and (v), the elemental abundances of the ionized gas. The thermal structure is not among die free parameters, but is found by balancing heating and cooling rates. In constant pressure photoionization models the density distribution is also solved for self-consistendy, but the free parameters stay the same, with Ng then specifying the pressure at the inner face of the cloud. In a plane-parallel model where curvature need not be described, the third and fourth parameters may be replaced by a single parameter, called the ionization parameter T. It is the ratio of ionizing photon density to gas density, given by: wnere 160 e =ï^rfv (6a) where Ly is the monochromatic luminosity of the ionizing source per # z , and Vq = 3^9x10^^/fz is the frequency of the Lyman limit Even in sphaical models F is an uniquely useful parameter because it characterizes so many properties of the model. In the units which will be most useful in this chapta, F is: F = ^ ^ 0 5 o (6.3) where Tpe is r in parsecs, is in units of 10^ cm and is also roughly equal to 1.2 times the electron density in most cases, and |2 sois Q in units of l(f° photons/sec. In addition to these S parameters, there may be others as mandated by the geometry. These include: (i) column density Wg for matter-bounded or optically thin models; and (ii) volume filling factor, e. Finally, for very complex geometries where optically thick inner regions such as perhaps BLRs shadow the outer regions, the distribution of such regions and their fiee parameters must also be specified. Qo) Geometry of Sgr A West The distribution of the ionized gas is of primary importance to photoionization models of Sgr A West. Since much of our information on the spatial distribution of the gas comes from its kinematics, however, in section (i) we will review the motions of die gas together with the qratial infc^ation provided by radio and IR m ^s. These data also provide evidence that the ionizing radiation in Sgr A West comes from the vicinity of a possible supermassive object, which we will consider further in section (ii). 161 (f)Distribution aadKinematics o f the Ionized Gas. Sgr A West is a roughly 3 pc diameter region of ionized gas, seen in the radio through its thermal free-free emission. It is one of 2 components to the radio source Sgr A. The other, a nonthermal radio source called Sgr A East, qipears to be a supernova remnant (Ekers, Goss, and Schwarz 1975; Pauls et al. 1976; Ekas et al. 1983), located on Sgr A West’s far side (Yusef-Zadeh et al. 198Q. The projected distance between the 2 components of Sgr A is 2.5 pc. The high resolution VLA 6 cm maps of Lo and Claussen (1983; 1987; also presented in Lo 1987), and the 2 cm map of Morris and Yusef-Zadeh (1987) provide the best information on the spatial distribution of ionized gas in Sgr A West We refer to features of the map by the names given in Serabyn and Lacy (1985) for ease of discussion. The central features of the map are a q>iral-like structure consisting of 3 - 4 distinct arms, and a compact, nonthermal source, Sgr A* more or less at its coiter. One high surËtce brightness arm, called the "bar", crosses Sgr A* running east-west About 30 arcsec east of Sgr A *. it turns north and begins to decline in surface brightness. This easton extension is called the "eastern arm". West of Sgr A* the bar crosses a long, low surface brightness, semicircle which extends from 45 arcsec south to 30 arcsec north of the compact object This semicircle is called the "western arc”. At its northern extait it coincides with the "northern arm", which extends south to join the bar about 8 arcsec east of Sgr A*. The northern arm grows in siuface brightness as it nears the bar and SgrA*. The bar, eastmt, and northon arms are also readily apparent in high resolution 10 pm mt^s (Becklin and Neugebauer 1975; Rieke, Telesco, and Harpar 1978), and in RraA4.05pm images (Forrester al. 1987). The IR mt^s also show discrete sources which are less evident or in some cases not evident at all on the radio maps. 1RS 1 and 1RS 16 are discrete 10 pm sources, the lattm: being the most nearly coïncidait with SgrA* (but evidently not exactly coincident; see below). The 2.2 pm maps (e.g. Becklin and Neugebauer 1975) show discrete, primarily stellar sources. 162 Velocity mapping of the spiral pattern via the [Ne //] 12.8 pm fine structure line has clarified the kinematics of the pattern and is the basis for dividing the spiral into the discrete "arms" described above (Lacy er al. 1979, 1980; Serabyn and Lacy 1985; Lacye/ al. 1987). Reviewing the conclusions of Serabyn and Lacy, the western arc is a single feature. It appears to be the ionized inner edge of the circumnuclear ring of dust and neutral hydrogen. The opposite side of the ring is not visible on eitha- the 6 cm or 2 cm maps, but has been seen on a 1.3 cm map (Ishiguro et al. 1986, cited in Geballe 1987). The presence of the circumnuclear ring is inferred from a number of other obsavations: The dust of the ring has been observed in thermally reradiated light at 30, 50, and 100 im (e.g. Rieke, Telesco, and Harpar 1978; Becklin, Gatley, and Werner 1982), while the gas has been observed in the fine structure lines of [O /] 63 pm (Genzel et al. 1984; Genzel et al. 1985), and in [C //] 158 pm and molecular rotational lines of CO and OH (Genzel et al. 1985). The circumnuclear ring extends out to about 10 pc. The velocity curve of the western arc is consistent with a circular orbit in the plane of the galaxy, at a radius of 1.7 pc from the dynamical galactic cotter, coincident with SgrA* to within 2". Adopting the circular orbit hypothesis, the velocity of the arc at this radius is 110 km/sec. For a virialized ring of gas (GM Ir = V% with negligible nongravitational forces: implying an enclosed mass of Mg=4.7 ± 1.0. The northern arm is also all one feature, ratho than a number of sqtarate clouds. From the velocity msssursmsnts. it does not appear that the northern arm and the western arc are physically connected, despite tppearances to that effect The velocity curve of the northern arm cannot be adequately 163 accounted for by a circular orbit, but can be fit by a noncircular orbit which approaches to within .5 pc of the dynamical center. About Afg=3 are enclosed within this orbit, implying that about Mg = 1.7 of the total mass seen by the circumnuclear ring resides in nuclear stars. The bar and eastern arm ^tpear to consist of discrete clouds at many velocities. In general higher velocities and greater velocity dispersions are seen nearer the center. Broad [Ne //] 12.8 pm lines are also observed (FWHM about 100 km/sec, but in one case up to 260 km/sec). There is also a trend for clouds west of center to be blueshifted and east of center to be redshifted. H and He IR recombination lines with extremely broad wings are seen in 4 arcsec apertures centred on 1RS 16. For He I 2.058 pm (2'P-2^S) Hall, Kleinmann, and Scoville (1982) reported a FWHM of 1500 km/sec. Geballe er a /. (1984) reported FWHM of about half this for BraX4.05pm , and He / 2.058 pm. The large recombination-line widths imply an enclosed mass of M g = 4 - 7 within as little as .1 pc of Sgr A *. Most of the available fine structure line data come firom beams centered on 1RS 1, the peak 10 pm continuum source which lies about 1 arcsec ntnth and .3 seconds (4.5 arcsec) east of 1RS 16 within the northan arm. The projected separation betwe«i 1RS 1 and 1RS 16 is .225 pc (assuming 10 Iqpc distance to Sgr A West, 1 arcsec = .05 parsec), but as the [Ne II] 12.8 pm velocities are consistent with others on the arm, a separation of .5 pc is more likely. Nongravitational forces might be responsible for the observed gas motions, especially nearer to 1RS 16 where conditions arc likely to be more extreme. For example, Geballe et aJ. (1984) suggest that the broad Brackett line-widths reflect outflow which results in shocked molecular hydrogen in the cool circumnuclear ring, as suggested by the observations of Gatley et al. (1984). Allen (1987) suggested that these widths might be due to nothing more exotic than mass loss firom Wolf-Rayet stars. In this context it 164 should be noted that Lacy et al. (1987) find evidence that the nonhem arm is connected with at least one filament in the bar, and that the northern arm undergoes an abrupt change of velocity and flow direction at 1RS 1. Thus, estimates of the mass of the central object may be suspect (ii) Identification and Nature o f the Central Object. We now review the evidence that a central object is responsible for the photoionization of Sgr A West and that this object is, or is near, a supeimassive object The existence of a supermassive object is an important question, given the prévalait belief that such objects are the prime movers in known AGN (e.g. Osterbrock and Mathews 1986). Thore is also speculation that such an object might be the natural result of a starburst (Weedman 1983). As we have seen, assuming that the gas is in firee-fall, the gas-streamer orbits and the line-width data provide evidence that the dynamical center is in the vicinity of Sgr A "/IRS 16. The K-band surface brightness distribution, presumably due to unresolved population n stars, also peaks in this vicinity. It is thought that all of the 3 - 4x 10^ Mq are concentrated in a single object, presumably a black hole, because the velocity curve suggests that the enclosed mass tends towards a constant (Crawford et al. 1985; Lugten et al. 1986). An star cluster with this much mass within .5 pc is thought unlikely: An ordinary cluster is not predicted to be this centrally concentrated, and if it were, it would have a higher central 2.2 pm surface brightness than observed (Crawford et al. 1985). Further evidence for a single supeimassive central object would be a cusp in the surface brightness distribution the stellar cluster centered on it (assuming the cluster is relaxed), rather than a turnover as is appropriate to an isodiermal King model. In actuality, a turnover must be present in any physical model, so a black hole’s signature would be a voy small value forrgg„, the radius at which the turnover occurs. Allen, Hyland, and Jones (1983) found < .5 arcsec, strongly suggesting a black hole. Rieke and 165 Lebofsky (1987) found that the galactic center lies at a local minimum in the interstellar extinction, which together with the confusion caused by the discrete 2.2 |jm sources present in the galactic center leads to the artificial tgtpearance of a cusp. They find fa,» =20 arcsec from the K-band surface brightness. Allen (1987) notes that patchy extinction would make this result an upper limit, and that the blighter population I stars, excluded from Rieke and Lebof&y's analysis, do show a cusp. The intapretation of the western arc as die ionized inno' 6ce of the circumnuclear ring suggests an ionizing source or sources lying interior to this 1.7 pc orbit The dust tempmature also declines in the ring and for highly luminous 10 pm sources within the central parsec, implying excitation from within the .5 pc orbit of the northern arm rather than locally (review in Gatley 1987). Geballe et al. (1984) noted a decline from 1RS 16 to 1RS 1 in the X2.058 ]un/Bra ratio, implying a decline in He* abundance. This suggests that the BLR at 1RS 16 is closer to the ionizing source than the gas of the northern arm. In the radio, SgrA* is the most obvious candidate for the central ionizing object The identification of the nearby source 1RS 16 as SgrA**s IR counterpart has beat frustrated, howeva, by the failure of ever more precise positional measurements to demonstrate coincidence. Although Hairy, DePoy, and Becklin (1984) concluded that 1RS 16 C, one of die 4 discrete sources which make up 1RS 16, was coincident with the radio source, lata high angular resolution work (Forrest, Pipher, and Stein 1986) found that the nearest source is actually 1RS 16 NW. wlfich further does not coincide with SgrA* to within an error of ±.5". Currently, it seems that 1RS 16 SW is the neara 1RS 16 source to SgrA*, but the displacement remains (Allen and Sanders 1986; Forrest et al. 1987). The dynamical evidence is also ambiguous. The BLR is definitely associated with 1RS 16, while if îîicîe is a supennassive object in Sgr A V/cst, the prcpa motion limits set by Backer and Sramek (1987) f a SgrA * require that it be the supermassive object The peak in the population I clusta is a i 1RS 16, 166 while near IR spectroscopy of the components of 1RS 16 has shown than to he very blue (Storey and Allen 1983) and lacking in CO absoiption (Allen 1987), so that none of them can be readily identified with the peak of the population n clusta. With evidence for 2 ^patently different objects as the dynamical center, the exact identification of the central object ranains uncertain. This will not affect our results in this chuter (we will refer to the ionizing object as 1RS 16 for convenience haeafter, even though this may not be the actual photoionizing source), but remains an important question that bears on the nature of Sgr A West (c) Extinction (t ) The Near IR Extinction Curve . Becklinet al. (1978) measured extinctions from 1.25 to 4.8 pm . Assuming a standard reddening law (van de Hulst's curve 15, as in Johnson 1968) they derived Ay = 30 from the near-lR cola excesses for the 10 pm sources 1RS 7,11, and 12, which they assumed to be late type stars. At 2.2 pm they found an extinction of 2.7 magnitudes. Their complete results may be seen in figure 23, curve a . The uncertainty in the inuinsic colors of the stars led them to quote a 4 magnitude uncatainty inAy. By the same method, Henry, DePoy, and Becklin (1984) used the .98 - 2.2 pm colors of 1RS 12 to get Ay - 38 from the curve of Savage and Mathis (1979), and Axo. 9 g = 14.5. It should be noted that the extinction curves f a stars in H n Regions and field stars differ, with the former exhibiting a larger differential extinction firom the near-IR to the visual (Osterbrock 1974). This is not a problem for the n ^ IR extinction measures, but it would underestimateAy. As Ay is only a secondary quantity in the present context, quoted for its familiarity, this is of little concern. Baiiy, Joyce, and Scoville(1979) measured die SralA.ljZ pm to Sry% 2.ll pm raüc an 8" beam for several of the non-stellar 10 pm sources including 1RS 1 and 16. Effective recombination 167 coefficients a,g for the Brackett lines are computed firom the results of Pengelly (1964) and presMited in table 60 - 61, for both cases A and B, and table 62 gives power-law indices p fi)r fits to of the form; <%(r) = «^(10,000K ) i t ^ (6.5) where is the temperature in units of 10,000 K. These fits are exact at 10,000 K and good to about 5% at 5,000 and 20,000 K. Also given in this latter table are useful fits of the same form to % for recombination to hydrogen, % for recombination to He I, and a ^ (H p). Using the fits to the Brackett line ratios for case B gives: £(2.17 m -4.05 \un) = -2.5 log - 1.178 + .3403 log(t 4) (6 .6 ) f 4.1 whae ( 4 is the temperature in units of 10,000 K. Assuming t4 = 1, Bally, Joyce, and Scoville found E(2.17-4.05) = 1.71 and 1.48 for 1RS 16 and 1RS 1 respectively. Note that if densities were higher, full /-mixing for these levels would obtain and the theoretical ratio would be closer to 2.3, as in Seaton (19596). The density at which this becomes a consideration is roughly 10^ cm~^ (Drake and Ulrich 1980). This density is much higher than the density thought to obtain for 1RS 1 (see section g), although it may be appropriate to 1RS 16 (see section 6.3). For the Brackett color excess, the standard reddening law gives: E ( 2.17 \un - 4.05 pm ) = .053 Ay (6.7) which gives Ay= 32 and 28 for Bally, Joyce, and Scoville’s respective excesses, consistent with the 168 reddaiing derived from the late type stars. Bally, Joyce, and Scoville quote an error of at least ± S magnitudes because of the marginal detection ofBry. For 1RS 1, the detection is less than 2 a. Willner and Hpher (1983) also measured B ra and B ry fluxes for several objects near the galactic center with a 3.9" beam. They found excesses of ordo-1.15 for the sources 1RS 2,4, S, and 20, and an excess of 2.53 for 1RS 6. They found similar excesses when ratioing either Brackett line to the 5 GHz data of Brown, Johnston, and Lo (1981). An excess of 1.15 implies Ay = 21.7 from equation 6.7 above, and t u .17 = 18. Assuming a lA, extinction law gives Tj^i? = 2.5, in better agreement with Becklinet al. (1978) at this wavelength, but not at visual wavelengths. The 4.05/2.2 pm flux ratios which Willner and Pipher measured for 1RS 1 and 16 are 1.6 and 1, respectively. These ratios are lower than predicted, which is exactly opposite the effect reddening would have. Presumably this is due to the rather large uncertainties for the B ry flux for these objects. The uncertainties in the Bry fluxes of 1RS 2, 4, 5, and 20 are smalls, so this doesn't account for the divergence of their extinction estimates from those of other authors. A possibility is that reddening is not constant for every object in Sgr A West. (ii) The Far IR Extinction. To measure the extinction due to the 10 pm silicate absorption feature, Becklinet al. (1978) assumed a 350 K biackbody emission for the circumstellar dust ^ ell of 1RS 7. This allowed them to derive relative extinctions due to the silicate absorption feature, but as the intrinsic 8-13 pm luminosity of 1RS 7 in this region relative to its intrinsic 1.25 - 3.5 pm luminosity was not known, they could only guess at how to relate the 10 pm curve to the near IR curve. To do this, they extrapolated &om 3.5 pm with a lA. law. Anotha drawback to their results is that dust spectra, such as seen in the Orion nebula, are only approximately biackbody. 169 Willner and Pipher (1983) also present 10 |Xm absoiption estimates for objects other than 1RS 1 and 16. After Gillettef al. (1975), they assumed an emitting hot dust spectra, either optically thin graybody emission (Type I) like that of the Trapezium (e.g. Forrest, Gillett, and Stein 1975) or silicate emission (Type D). They also assumed a wavelength dependence for interstellar cold dust absoiption, which is daived firom the assumption that the dust absoiption is proportional to its emissivity. They then fit the spectra of observed objects in the galactic center with these assumptions and doived qptical depths at 9.7 fim of about 2.8 and 4 2 for the respective assumed emissivity laws. In this case they have an absolute measure of the optical dqith at 10 pm. At 20 pm and above, Becklin el al. (1978) assumed t^3o= -25 Xxg.j from measurements of moon rocks and meteorites. McCarthy et al. (1980) derived optical depths averaged over all of Sgr A firom their own large beam X19 pm data, assuming emissivity and extinction laws for dust. They found Xug.? around .8 for a 1/X emissivity law, 1.1 for a biackbody emissivity, and 2.4 for a silicate emissivity law. Herter el a l. (1984) adopted Xnn/XyiA = 3.1 on the basis of the opacity curve of Forrest, McCarthy, and Houck (1979) calculated firom meteorite compositions and properties. 0») Adopted Extinction. Figure 23 presents two extremes in the extinction estimates. Curve a is simply the absolute extinctions quoted by Becklin et al. (1978), with interpolation from near-IR to far-IR by means of a 1/X law. Curve b uses the absolute extinction quoted by Willner and Pipher (1983) at 2.17 pm, and interpolates into the far-IR using a standard extinction law. At 9.7 pm curve b uses Willner and Pipher’s absolute extinction, but the rest of the far-IR curve is constructed with the relative extinctions of Becklin et al. (1978) in this wavelength regime. Beyond the regime shown in the figure, we extended the curves to 18.7 pm by the extremes of the measuranents of McCarthy et al. (1980) 170 (Xxi8.7 = 2.4 for curvea and Txis.? = .8 for curve 6 ) and extended both these results to 33.4 pm by use of Herter et a l ’s (1984) Xxi 8 .7/x«3.4 = 3.1. We favor curve a in the near-IR because most estimates we have reviewed are reasonably close to it. Above 4.8 pm we favor somewhat the type I fits ofW illna and Pipher because they are based on the observed spectrum of the Trapezium, for which it can be hoped that it is similar to the galactic center. Because of the imcertainties in the extinction measures we have corrected the line data by both curves in the section following, in order to measure the sensitivity of these data to the assumed reddenings. (d) Fine Structure Line Spectrum (i) Uncertainties in the Line Data. Table 63 lists all the atomic line fluxes for 1RS 1 which we were able to find in the literature. Also listed in the table are the beam widths of the measurements, and (for circular beams only) the fluxes reduced to a common beam of 4 arcsec and de-ieddened by both curves a and b . All beams are centered on 1RS 1. In most cases where 2 or more measurements of the same line are available, they are reasonably consistait when reduced to a common beam. A puzzling exception is the Brackett lines. We can offer no leascmable explanation for why different authors present such divergent measurements for these lines. Ideally we would have liked to have had the line data for tte entire nucleus, as we would observe it if Sgr A West were at extra-galactic distances. This would most readily allow comparisons with extra- galactic objects. Data of this sort are unavailable, howeva. Therefore we sought to get a spectrum for 1RS 1 alone. In principal 1RS 1, defined here as encompassing the high surface brightness part of the northem arm with an angular diameter of about 4 - 8 arcsec, should be of constant excitation, making it easy to compare to models. Unfortunately the obsovations come from a variety of beamwidths, many 171 much larger than this. These beams include apparently empty space as well as light &om areas of potentially different density and excitation such as the bar and the ciicunmuclear ring. In order to get a dux for 1RS 1 for an arbitrary line measured with a beam of arbitrary size, we multiplied by the ratio of the [iVe //] 12.8 |im dux in that beam (interpolating between the available ]Ne //] 12.8 pm dux measurements of Lacy, Townes, and Hollenbach 1982) to the X12.8 pm dux in a 4 arcsec beam. This accurately reproduces the spectrum of 1RS 1 only if the ratio of the line’s dux to the \Ne //] 12.8 pm dux is everywhere constant (ie., they are formed in the same regions). For thermal ionizing spectra this is often a reasonable assumption, but it need not be for nonthermal spectra. In AGN models the [Ne //] 12.8 pm line is formed primarily in the H* zone, while lines such as [C //] 158 pm, [O / ] 63 pm and [St //] 35 pm are formed primarily in warm, mostly neutral gas. The duxes of the neutral zone lines increase with increasing column of neutral gas, so their measured duxes must be regarded only as upper limits on their dux at 1RS 1 unless measured with a beam of tmder 8 arcsec. The X-ray dux of nonthermal spectra also creates large abundances of singly ionized species in die neutral zone such as A ril and Ne n, again requiring that large beam measurements be considered uppor limits, hi this case the limit may be less stringent, because these species might be emitting strongly in the zone. Another danger in this method of obtaining the spectrum of 1RS 1, is that for thermal qiecua with T* < 40,000 K the number of photons capable of ionizing He ° is limited, causing the He* zone to be smalls than the H* zone (see 6gure2.5 of Osterbrock 1974). Many observed lines are formed in the He* zone (e.g., [Ar ///] 9pm, [Ar V] 13.1 pm, and [S IV} 10.5pm) and if a stellar specttum with r . g 40,000 K ionized Sgr A West, the duxes of these lines relative to lines fmmed in the zone would be undeiesthnated when measured in large beams. 172 AU these effects tend to make the spectrum look as if it is lower ionizatitxi than it might really be. They are not easily corrected for, since the corrections to be adopted would depend upon the ionizing spectrum, which is exactly the model parameter we are seeking to deduce from die line spectrum, as well as details of the geometry which are not well known. Also note that the greatest uncotainty in the extinction occurs at about 10\ m , where the high ionizaticm lines [Ar ///] 9 pm and [5 /V] 10.5 pm are located. This effect makes the ionization level uncertain, since these lines might be either over- or underestimated by our reddening cmrections. (Ü) Adopted Spectrum. Table 64 provides an adopted set of line ratios for each of the 2 extinction curves. AU the fine-structure lines, ratioed toBry, are first presented foUowed by some ratios which are independent of abundance effects and of the large uncertainties in the Brackett-line fluxes. For the most part, the adopted ratios are unweighted avaages of the line data in table 63, with grossly different values excluded. Qearly, most of the line fluxes are uncertain to at least a factor of 2 because of uncertainties in extinction and measurements. (fii) Comparison With Extragalactic Objects. A limited number of IR line data are available for extra-galactic objects. They are reviewed in Aitken and Roche (1985). Prominent [Ne //] 12.8 pm emission and unidentified dust emission features at 3.%, 8.65, and 11.25 pm are seen for 11 out of 13 starburst nuclei, whUe for 18 out of 21 active galaxies this spectral regime is virtuaUy featureless. The prominence of die fine structure lines in Sgr A West seems to indicate a connection with starburst or HU region nuclei. Willnergf al. (1977) obtained 2 - 8pm spectrophotometry of the nucleus of M82, one of the proto^ical starburst galaxies. The spectrum of this nucleus is also vay similar to Sgr A West, with 173 with strong [Ne //] 12.8 ym and with a vay low /([>lr ///] 9 juw )//([Ar II] 7 ym ) ratio. Willner et al. quoted an ionizing stellar temperature of 30,000 K based on their derived Ar into Ar n abundance ratio. Peimbert and Spinrad (1970) found a similar temperature for M 82 firom optical emission lines. This is very similar to the result quoted by Lacy et al. (1980) for Sgr A West from consideration of the same line ratio. The reason the Ar ID to Ar n abundance ratio requires such a low-temperature star for these objects is that for any reasonable values of the physical parameters, Ar III wiU predominate in the H'*' zone. As discussed above, stars with T« g 40,000 K do not produce enough photons to ionize He I throughout the H* zone. Since the ionization limit for Ar D is at about 2 Ryd, the Ar Dl-ionizing photons will be used up in the H en zone as well, resulting in a zone beyond the He*IHe^ ionization firont where Ar D can exist As we have already mentioned, another case where a separate Ar D zone forms is for nonthermal ionizing spectra. In this case the Ar D zone is an X-ray heated, partiaUy-icxiized zone, which lies beyond the ionization firont Whether or not this zone can produce sufficient [Ar II] 7 ym emission to match that observed is the question addressed by the photoionization models presented below. (e) The Ionizing Continuum Shape. Figure 24 displays much of the available continuum data for Sgr A West These data are listed in table 65. No attempt has been made to be complete, the object being to get a crude idea of what the continuum looks like, rather than present a definitive spectrum for i t Many of the data are estimated firom figures presented in the literature. These data are daioted "est", because we doubt that this procedure is very accurate. All the IR data are unconected for extinction in üie table, but have been corrected using extinction curve a of figure 23 before use in figure 24. The high enegy data employ the 174 corrections given in the original papers in both the table and the figure, since it is usually not possible to quote luminosities in this regime without specifying the extinction. The data come from a great many authors, using instruments of widely differing angular resolution. The symbols on figure 24 roughly bin the objects by the size beam used in obtaining the data, with open circles denoting resolutions poorer than or equal to 1", closed circles denoting resolutions between 1" and IS", and crosses denoting observations of 1RS 16 alone with a few arcsec resolution. Table 65 gives the exact beamwidths used. All beams are roughly centered on 1RS 16. All are circular, excqpt for the case of Soifer, Russell, and Merrill (1976), which is the difference between a 17" and 8.5" aperture centered on 1RS 7. As Table 65 indicates, most of the low-enagy data were given in the litoature as flux densities at the earth, while most of the X-ray and y-ray data were given in terms of a luminosity and a photon powar law index over some band. We converted the integrated quantities into a single flux density each by simply dividing by the lower bound value of v for the band. This is correct for a powar-law decreasing faster than and a wide enough band. We note as an aside that the 511 keV positron aruiihilation line and the positronium continuum have been detected as well (see review in Ramaty and Lingenfelter 1987). These data are not considered in table 65 because of the lack of such data for other emission-line nuclei, but clearly they are important clues to Sgr A’s nature. Superimposed on the figure is the mean AGN continuum adopted in the present work to rqrresent one hypothesis for the intrinsic spectrum of 1RS 16. This mean continuiim is summarized in table 66. It comes firom Mathews and Ferland (1987) who sought to derive the mean continuum of luminous AGN like Seyfert Î nuclei or QSOs. As was alluded to in the introduction, the continuum of a low-luminosity AGN may be substantially différait firom this. We will consider this point, and also the uncertainties in 175 their derivation of the mean continuum, in more detail below as we compare it to the obsoved continuum of Sgr A West. Here we briefly note the properties of their AGN continuum in die spectral regimes unobservable for Sgr A West. It is a flat, power-law throu^ the optical and UV, which is the well- established spectral s h r^ of a luminous AGN. It also has a "blue bump". i.e. a peak in the XUV, inferred from energy balance considerations (Netzer 1985) and the equivalent width of He 1 1640  (MacAlpine 1981; MacAlpine et al. 1985). It is ^propriate to a radio quiet AGN, with the power- law index from the optical (2500 Â) to the X-ray (2 keV), equal to 1.46 (see Zamorani et a l. 1981). In figure 24 the mean continuum has been scaled to provide about the numb^ of photons required to ionize the gas, as discussed in the section (f). Our other hypothesized ionizing (i.e. starburst) crmtinua are not plotted on die figure because we know of no refermice for a "mean" IR, X-ray, and y-ray continuum for these objects. (i) The Hard X- and y -Ray Continua. For the very high energy points (ES 10 keV) detectors with resolutions of better than one degree are not available, and usually the resolution is much worse. These high energy points are obviously suspect, since they could result from large-scale diffuse emission or from discrete objects far outside of Sgr A. Hiey could also come from the supanova remnant Sgr A East rathor than Sgr A West The situation with regards to resolution can be helped somewhat by variability, although this then makes the shape of the dmved spectrum uncmain. For example, HEAO-3 with 30° resolution, measured a change in the 100 keV - 500 keV luminosity from 7x10^ to 2x10^^ erg/sec o \ c t a period of 6 months (refermiced in Matteson 1982), implying that the bulk of this flux comes from a region < 2 ■pc. This does not prove that this emitting region is within Sgr A West, but it does rule out a large-scale diffuse origin for the flux. 176 The very high energy end (> 200 keV) is where the continuum of Sgr A West is most consistent with the mean AGN spectrum in both luminosity and slope, but it must be noted that this continuum is also voy uncertain for known AGN. Observations at energies approaching an MeV are available for only a few AGN (references in Mathews and Ferland). The hard AGN continua are all consistent with a continuation of the better observed hard X-ray powa-law. The change in slope at 100 keV for the AGN spectrum is set by the requirement that the slope become steeper than unity at some point (otherwise the total energy integral diverges) and from considerations of the y-ray background, which would be exceeded without a change in power-law index in this general vicinity (Rothschild er al. 1983). The exact location of the break and the spectral index of the continuum beyrnid it are thaefore rather imcortain. Similar high oiergy data for starburst and H n region nuclei are unavailable. Many galactic objects, of course, are y-ray sources (for a review of their prqterties see Bignami and Hermsen 1983). For Ktample, both the Crab Nebula and its compact source have high enargy spectral slopes which are consistent with that obswred for Sgr A W est The typical luminosity of a galactic y-ray source is about l(f^ -10®® erg/sec, however, so from 10 -100 such objects would be necessary for the galactic center’s 10^’ -10®* erg/sec luminosity over this spectral region. Cyg X-3 has a 100 MeV luminosity perhaps exceeding 10®^ ag/sec (Bignami and Hermsen), so only 1-10 such objects would be neeeded to explain the high enargy luminosity of Sgr A West. Large numbers of supernova remnants (Weedman er al. 1981), or collapsed objects like high mass X-ray binaries (Watson, Stanga, and Griffiths 1984) are often invoked to explain the soft X-rays from starburst galaxies. In tiie piesait case, the lack of radio objects with a remnant morphology or large-scale nonthermal radio emission within Sgr A West seem to preclude the presence of supemovae remnants within Sgr A West For a starburst picture. 177 then, the high eneigy photons would probably have to be attributed to Sgr A East or other objects distributed around the nucleus. Tateyama, Strauss, and Abraham (1986) have used radio frequency spectral index data to identify close to 20 possible supemovae remnants within 1 degree of Sgr A West. At 4.1’ resolution these objects also have size limits consistent with such an identification, but evidently lack the X-ray emission which would be expected for remnants. (ii) The Soft X-ray Continuum. With the exception of the .5 keV point, all the soft X-ray data are of resolutions of 1" or poor^, so that diey too might not be associated with Sgr A. The .5 keV data are of 1’ resolution and definitely firom Sgr A West, but not necessarily from 1RS 16. There is a significant departure from the mean AGN ctmtinuum in that the soft X-ray fluxes, particularly at .5 keV (firom Watsonet al. 1981), are measured to be much too weak. To reconcile the AGN continuum with the data, we would have to assume that the data are much more strongly absorbed than assumed in the original papers. Watson et al. (1981) adopt a column densify of Mg = 6 x 10^ cm~^. With a standard gas-to-dust ratio expressed as: Mg»= 1.6œlO^‘cm“^Ajf (6.8) this implies Ay = 37, roughly consistent with other extinction measures givmt above. Watson et a l. note that the column density could range firom 2 x l( P - l(f^cm ~ \ The rollover from .SkeV to SOkeV is very suggestive of the need to correct for additional absorption, although it is likely given the many uncertainties that much of the specific shape is merely a coincidence. The cross-section at 25  (.5 keV) is 1.26xl(T^cm^ firom figure 2 of Cmddace er a/. (1974), which assumes interstellar medium abundances for tte heavy elements. To raise the flux at .5 keV the firctor of 1,8(X) required to meet the AGN-like continuum, would require an additional coliurm of 6x10^ cm for a total column of 178 1.2% 1(P cm~^, not far outside the limits quoted. Equation 6.8 would imply Ay = 75 for the extinction to the soft X-ray source if the additional column has the standard amount of dust associated with it. This suggests that the additional column, if present, might be so hot as to be inimical to dust A more reasonable intapretation of the X-ray data on the AGN hypothesis is probably that the continua of low-luminosity AGN are diffoent from those of higher luminosity AGN: Willner et al. (1985) note that the soft X-ray data are inconsistent with a continuation into the X-ray of the optical-UV power-law inferred for LINERs firom the photoionization models of Ferland and Netzer (1983) or Halpem and Steiner (1983). The radio to X-ray luminosity ratio for the galactic center is smnewhat lower than those for the starburst galaxies M 82 and NGC 253 (Fabbiano and Trinchieri 1984), but the agreement is better than with an AGN. At a .5 • 4.5 keV luminosi^ of 1.5% 10^^exgfsec the galactic center is less luminous than the 3% 10^ - 4% l(f ^ eig/sec typical of siqtemova ronnants over a similar band in the LMC, but not inconsistent with the 10^^ - 1 0 ^ erg/sec typical of galactic remnants (Long and Helfand 1979; Seward and Mitchell 1981). The lack of any other indication of supernova remnants in Sgr A West, however, argues that that the soft X-ray luminosity is just due to hot, diffuse gas. (iii) The IR Continuum. The IR continuum is probably not important to a photoimization model of Sgr A West: It is usually of importance when Compton cooling is important, which occurs at higher than nebular temperatures. We consider hoe, however, whether the IR continuum is more consistent with a starburst or AGN nucleus. Below .5 keV, most of the measurements are of arcminute resolution. All are definitely associated with Sgr A West Most of the large beam IR data are apparently consistent with the mean AGN continuum shown in figure 24, but it is important to note that there is no doubt that these data for Sgr A 179 West are due to dust distributed throughout the nucleus, because of the dependence of IR luminosity on beamsize and because of the overall spectrum (see review in Gatley 1987). Ferland and Mathews’ continuum, on the other hand, assumes the IR continuum of an AGN is an extension of the optical-UV power-law. In this interpretation, the IR emission should come from a central point source, not the entire nucleus. 1RS 16, the supposed continuum object, is far too underluminous at 2 |im to be consistent with this interpretation. Whether or not this is a critical point against the hypothesis of an AGN depends on whether the IR continuum for low-luminosi^ AGN is the same as for high luminosity objects, and on whether or not the nature of the IR continuum in luminous AGN is really well established. Both are points of continuing controversy (e.g. Rieke 1985). Evidence that the IR emission is a continuation of the optical-UV power-law in luminous AGN is provided by the observed correlations between between 2-10 keV flux and 10 and 3.5 \xm fluxes (Elvis et al. 1978; Malkan 1985). The shape of near-IR spectra for Seyfert I ’s are goierally flat and unsuggestive of dust mission (e.g. Rieke 1985; Edelson and Malkan 1986). On the other hand, there are clearly some Seyfert I ’s which show evidence for dust-emission, such as the dust emission-band features at 3.28,8.65, and 11.25 \m (summarized in Aitken and Roche 1985) or far-IR dust spectra (Ward et at. 1987). The question could be answered deflnitively if IR variability for AGN could be detected on tunescales too short to correspond to dusty nuclear clouds. Detections of IR variability have been claimed, but not on timescales short enough to rule out the possibility that they are the response of nuclear dust to continuum variations (reviews in Rieke and Lebofsky 1979; Rieke 1985). Willner et al. (1985) could find no evidence of the optical-UV pown-law extending into the near-IR for LINERs. Similarly, Lavirsncs st cl. (1985) foîind the 1 - 5 spectra of LINERs to be dominated by a typical stellar population. The less luminous Seyfert 2 nuclei generally have IR spectra which 180 appear more like dust than power-laws (Elieke 1985). These results suggest that regardless of the nature of the IR continuum for luminous AGN, in at least some low-luminosity AGN, it is due to dust and stars. Considering the controversy concerning AGN IR spectra, the IR continuum of Sgr A West probably is consistent with an AGN. Strong»' evidence would have been a correlation between 2 keV and the near-IR, which does not exist A dusty nucleus is definitely typical of starburst galaxies (Rieke and Lebofsky 1978). Starburst galaxies have the near-IR JHK colors of starlight (Balzano and Weedman 1981). Lawrence et al. (1985) found them to be flatter from 1 - 5 [tm than LINERs. No lepwts of the IR dust bands are available for Sgr A West, however, despite the correlation between starburst galaxies and fliese features noted by Aitken and Roche (1985). (f) The Ionizing Continuum Strength. (i) Estimates from [Ne II] I2B ]m Fluxes. There are a number of estimates of Q so available in the literature. In the optical, Q is usually estimated flirough a recombination line such as HB. For Sgr A West, the most extensive line flux measurements are for [Ne ff] 12.8 pm. For [Ne //] 12.8 |im 3^2) luminosity in the line is equal to: Z-MZS = AVmzsK (6.9) wh»e the integral is over volume, k the rate of collisional excitation from the state, and wh»e Ng is the critical daisity for collisional de-excitation, given in terms of the radiative de-excitation coefificimit A 21. and die statistical weight of the upper state of ( 2 /2 + 1)> by: 181 where e is the volume filling factor, SÎ21 k the collision strength, and the final equality uses the data of Mendoza (1983). For an isothomal nebula this is equal to = (6.11) where is the abundance of singly ionized neon relative to singly ionized hydrogen, and EM, denotes the volume proton emission measure: EMy = lN,Np£dV (6 .12) The luminosity in a line of wavelength Vq, is given in units most iqtpropriate to the line data in this chapter, by: F L^= 1.197X IC^erg/sec[ „ (6.13) JLU ww Cfi% where x,, is the reddening correction, and is the distance from the earth to the galactic center in units of lO^pc. Finally, the emission measure in in terms of the measured flux in the Iine,Exiz8 i" units of 10r” iy cTO"2.is: EM, = 3.306* 10* (-J^)ex p (^i^)r 1 + N /53.9)[^^] expdxizg) d } (6.14) 182 The volume emission measure can be related to 6% , (he rate at which photons must be supplied to keep a volume ionized: 12 50 = 2.939% Kf -EK-ag = 7.642% 1(T* EM^ (6.15) Rom an argument of this form, Lacy et cd. (1980) estimated that 6so = are needed to keep 1RS 1 ionized, and that |25o=2 are required to ionize all the gas within die central 3 parsecs assuming all neon is in the form of N eH Rodriguez and Chaisson (1979) argue for i4/v« = 4%10^, 4 times the solar abundance. If they are correct this reduces S so to 0.5 (Lacy, Townes, and Hollenbach 1982). (ii) Estimates from Brackett Line Fluxes. To avoid questions concerning the ionized neon abundance, we can use a hydrogen recombination line, eith^ S r a A4.05 \im at B ry 72.11 pm . For an isothermal nebula, the volume proton emission measure derived from a recombination line from upper state m to lower state n with luminosity L y, is: For the 2 Brackett lines we have: £My = 3.965* 10«ri-“ 5 [^^^]exp(Tx4.os)d| (6.17) EM, = 1.340* lO^rl”’ [^^]exp(T^i7)d4* (6.18) 183 Using all the Bra measurements of Geballe et ed. (1984) we obtain (3 so = 84. (iii) Estimates from Radio Flux Densities. The flux density, S y for a resolved, thermally-emitting source of low optical depth is given by: S v = B v 0 \ , (6.19) where 8 is the beamwidth, B y is the Planck function, given at radio frequencies by: B y = - ^ ^ (6.20) and Ty is the free-free optical depth, given for an isothermal nebula by: %v= (6.21) where gjy, the free-free Gaunt factor is given in this regime by: ql/2 7-3/2 g .= — ln(4.9xlO^-— ) (6.22) " 7C V and EMc is the column emission measure: EM^-^jN,N^eds (6.23) If gy is given in flux units (f.u. = 1 Jy = lOr^ erg/seclcm^lHz), and 0 in arcseconds, EMg is given in 184 an*pc by: 2.53Q*10’e-2t4® = ------— (6.24) 8ff The column emission measure is related to j2 50 by: Qso= 1.796% (6.25) where (6.26) Paulset a l. (1976) measured the 10.7 GHz continuum flux for all of Sgr A (East and West). If all the flux w ae thermal, Ga, = 6.8 are absorbed in Sgr A. In fact the 10.7 GHz continuum of Sgr A East is nonthomal in origin and at any rate. Sgr A E?st is not photoionized by the source at the galactic center, so this must be a rath» high upper limit on Qso- Paulset al. estimate that about one half of the 10.7 GHz flux comes &om Sgr A West. (iv) Other Estimates. Becklin, Gatley, and Wmter (1982) find 1 - 3x10’’ Lg of q>tical and UV radiation are required to power the dust observed in the far-IR. The adopted AGN-like continuum with Qso = 1 supplies f-km =6.6% 10^^ »g/sec, consistent with this constraint. Any stellar spectrum from 30,000 K to 45,000 K which is scaled to this same value of Ôso also meets the constraint. Finally we note that, a value for j2 so of 2 provides the best fit to the observed hard X-ray and y-ray 185 points for the schematic AGN continuum. On the basis of the available estimates it seems that something of order G 5o= .5 - 3 isthe best value for the rate at which photons are absorbed. Since the IR and radio maps indicate that the covering factor is clearly not unity, the source must be more luminous than this. Adopting a covering factor of order .5 to .1 would suggest G 50 = 2-10. We use a covering factor of orda .5 and G so= 2 hereafter, because the low brightness of the eastern edge of the ciicumnuclear ring suggests that the ionizing source is heavily shadowed by nearer gas (Serabyn and Lacy 1985). (g) Density (i) Evidence for Low Densities. There have been several estimates of N, from radio obsovations. From continuum observations, Brown, Johnston, and Lo (1981) and Brown and Johnston (1983) found 1^4 = 2 in the ionized gas near 1RS 1. Bregman and Schwarz(1982) found N^=.2. Lo and Qaussen (1983) again found # 4 = 2 for the high brightness temperature clumps, and .37 for the low brightness tempaature clumps. Ekessetal. (1983) estimated # 4 = .4. Densities of order # 4 = .2 - .5 have been deduced from the radio recombination line observations of Bregman and Schwarz (1982). From the ratio of the [S ///] 33.4 ]un and 18 \m line intensities, Heiteset at. (1984) deduced a value of # 4 = .18. If one adopts extinction curve a , which gives the largest X18.7 optical depth, this ratio is consistent with a density as high as # 4 = 1 and might be high enough to be collisionally saturated. Genzel et al. (1984) failed to detect the [O ///] 88 pm line, and fipom their adopted upper limit on its flux, found that the /([O ///] 52 pm )//([0 ///] 88 pm) ratio gave a lower limit of #4 = .8 on the density. Hertereroi. found # 4 = .! when tliey reanalyzed the /([O ///] 52 pm)// i[0 ///] 88 pm) ratio of Genzel et al. (1984) with updated collision strengths and a 186 higher, 3a limit on X 88 |im. Using the data compiled in table 63 with our somewhat different extinction estimates we find that the ratio is collisionally saturated. (it) Evidence fat High Densities. We are particularly interested in whether densities could be much higher than N 4 = .1 to 1, since we shall see that rather high densities are indicated by all our models. The existence of high density gas would require density inhomogeneities because of the low densities obtained by the radio estimates. The radio estimates come from the column emission measure, given in equation 6.24. If the filling factor is not unity, these measurements are, of course, an avoage over an inhomogeneous region which might include both higher and lowar density gas. In this context it is worth noting that the lowest density estimates come from the poorest angular resolution observations, e.g. 3x20 arcsec (a x 5) resolution for Bregman and Schwarz. Clumping on scales of around .05pc is demonstrated by the high resolution maps, especially the map of Morris and Yusef-Zaddi (1987) and the Lo and Claussen (1983) map. These m ^ s clearly demonstrate that the poor resolution observations average togetha clumps of low (JV 4 = .1) and high ( 1^4 = 1) density gas. Thoe might be still further clumping at scales below even their resolution. Initially the [Ne II] 12.8 pm data of Lacy et al. (1979, 1980) suggested this was the case, because they identified discrete clouds through their velocities. The data of Serabyn and Lacy (1985) showed that most of the arms were smooth features, but for the bar at least, the data still suggest discrete clouds. The line observations correspond to beamwidths of 25" for the S m lines, and 44" FWHM for the [O III] 88 pm line. These beams also average ov«- the known density inhomogeneities. The postulation of density inhomogaieities to account for evidence of higher density gas is not without precedent: In the narrow line-mitting regions of AGN and in LlNERs a range of densities have been inferred by the observed correlation betwe^i linewidth and critical density (see Fillipenko 1985; 187 Wilson and Heckman 1985). In some cases the fact that the /([S //] X6716)//([S //] X6731) ratio indicates low densities, while high densities are needed to understand the high 7([0 ///] X4363)//([0 ///] X5007) ratio demonstrates a range in density from 200 - 300 cmto 10^ - 10^ cm~^ (Carswell et al. 1984; Fillipenko 1985). In these objects it tgtpears that densities decrease as distance fiom the continuum source increases, while the ionization parameter stays roughly constant. With current data it is not possible to look for a similar stratification for Sgr A W est We note however, that the apparent absence of broad components of forbidden lines near 1RS 16, if verified, suggests that at least some density stratification may be present in Sgr A West If dmisities do exceed 1^ 4 =.1 to 10, then the highly clumped medium is characterized by the filling factor, e, which is the ratio of filled to total volume. It can be found firom any estimate of the emission measure if the density is assumed or known independently. Assuming a spherical cloud of the same angular size as the beam used to observe it gives: 3EM, 1.676* IQ-^EJlfv £ =------= ------—----- (6.27) 4n(ci-|)X^ ^4 whae 8 is given in arcsec and EMy is in ctn^pc^. From equation 6.14, adopting F M2.8 =.62* 10~^^ W/cm ( 4 = .5 (frmn the radio recmnbination-line measurements of Rodriguez and Chaisson 1979), 0 = 4 arcsec, aoAAjf,* = .8 gives: or, from equation 6.17, adopting f 3^4 05 = .077* 10^^^ WIcm}, *4 = .5, and 8 = 4.2 arcsec: 188 107 e = ÿ T r2 exp(XM.os) (6.29) Note that if the cloud is very much deeper than the .2 pc assumed in equations 6.28 and 6.29, these would be significant ov^estimates of e. This does not seem likely. We can obtain very rough limits on Ar, the total column of neutral and ionized hydrogar from examination of the high surface brighmess ridges in the Lo and Claussen (1983) radio map. It appears that the luminous region at 1RS 1 has a projected thidoiess of about .25 pc. If the cloud is at .5 pc from the caitral source as is thought, then it is foreshortened by a factor of 2 and the real luminous surface is about .5 pc across. If the ionization parameter is vay low, the ionized zone will not extend very deep into the cloud (i.e., it is sheedike). and the luminous surface we see must be primarily the part of the arm that faces the ionizing source. If the arm is totally ionized, then roughly half the luminous surface we see would be the inner imiized face, and the other half gas deeper within the cloud. In the first case, if the arm is roughly cylindrical at 1RS 1, Ar would be about .5 pc. fri the second case, for a cylindrical arm, it would be about J25 pc. These estimates are within a factor of 2 of the result obtained using 8 = 4.2 arcsec. Finally, we note that density estimates from the ciicumnuclear disk are high», which may or may not have any relevance to the question of densities in the ionized zone. In order to undastand the ratio for the disk, Genzel et al. (1985) postulated densities of N 4 = 10. This would suggest a filling factor of roughly for the disk, since the mean density is roughly 10^ cm~^. (h) Abundances There are only a few data concerning abundances. Rodriguez and Chaisson (1979) suggested that McOii was îOüguly 4 solar abmuance. The argument for &is is that the value of EM, found from the [Ne II] 12.8 |im data of Wollman et al. (1976) assuming solar abundances, and £My implied by the 189 S GHz continuum agree to within a factor of 2. Yet Wollman et tü. *s data had FWHM ~ 400 bn/sec while radio recombination lines have FWHM ~ 200 km/sec typically. Further, the radio data were of poorer resolution and included a larger volume than did the neon data. These facts together suggested that the neon lines arose in only a portion of the total volume from which the ionized hydrogen continuum comes, requiring neon to be about 4 times more abundant dian solar. Additionally, Rodriguez and Chaisson’s fit to their radio recombinatitHi line data suggested an unusually low tempoature of roughly 5,000 K, consistent with an enhanced abundance of coolants. Recent data show more typically FWHM "50 km/sec for 2,12.8 (e.g. Serabyn and Lacy 1985), presumably because the higher spatial resolution used has resolved velocity components, while current measurements of hydrogen radio recombination line widths are about 2-3 times this, the reverse of the situation quoted in Rodriguez and Chaisson. Velocity mapping in the radio recombination lines finds for the most part excellent agreement with the X12.8 pm m ^ in g (Bregman and Schwarz 1982; van Gorkomet a l. 1984) so the evidoice that the 2.12.8 \un emitting gas and the 5 GHz emitting gas must be sq)arate is not strong. Further, the total flux for Sgr A in 2,12.8 pm quoted by Lacy et al. (1980; or see table 63), which was daived fiom the continuum map of Rieke, Telesco, and ifetper (1978) by assuming the ratio of 2,12.8 pm fiux to the 8-13 pm flux was constant, gives an emission measure which agrees better with Pauls er a /.’s (1976) 10.7 GHz data assuming solar abundances than with the proposed enhancement Finally, we show below that plausible models can be constructed without recourse to neon abundance enhancements. A factor of 3 overabundance of argon has beat suggested by Willner et a l. (1979) and a factor of 2 by Lestar et al. (1981a). Willner et al. ’s result came firom their measurement of the [Ar II] 7 pm fiux, Lacy et al.'s limits on the [Ar III] 9 pm fiux, and the thermal radio fiux. Lester et al. measured the 2.7 190 and \9 flux, and additionally the hydrogen P fa lA S \im (6-->S) and £raX4.0S \m lines. As we show below, all our models require an argon enhancement to reproduce the observed strengths of XI pm relative to Bry. This is either further evidoice for an overabundance of argon or for possible inaccuracies in the collision strengths for XI \an. Finally, it is well known that thae is a galactic tempraatuie gradient for H H regions such that dervied firom radio recombination lines increases with distance firom the galactic center (see Wink, Wilson, and Sieging 1983). This is interprétable as due to an increase in the abundance of coolants, and T, can be empirically related to the 0/H ratio. Rodriguez and Chaisson found a temperature firom radio recombination lines of about 5,000 K, implying an overabundance of metals. Wink, Wilson, and Sieging find that the temperature of Sgr A West is consistent with a solar O/H. 191 6.3 The Broad Line Region, IRS 16. Line data collected from the liteiatuie for 1RS 16 are pres^ted in table 67. Note that these data are for composite broad and narrow profiles. Broad helium accounts for 40% of the X2.0S8 pm fiux at 1RS 16, broad hydrogen accounts for 20% of the Bra%4.05 pm flux there, and no broad component can be identified for Bry X2.17 pm (Geballe et of. 1984). The data are not de-reddened in the table. Where we use the X2.0S8 and B ra fluxes below, they are both de-reddened and multiplied by the respective factors of 40% and 20%. Recall from section 6.2b above that the recombination line mdths are of order FWHM = 700 -1500 km/sec. (a) Geometry. Much of the geomeuy relevant to the BLR was discussed in section 6.2b. Geballe (1987) indicates that the BLR is spatially resolved and .15 pc in size. The narrow line components and the narrower [/Vie //] 12.8 pm emission seen along the bar at the position of 1RS 16 presumably comes firom gas farther out along the line of sight. The lack of broad componaits to the fine structure lines suggests a higher density for the BLR. Evidence for broader neon lines (FWZI = 800 km/sec) than have previously been seen has been cited recently by Lacy et al. (1987), however. This is still a factor of 2 below the widths of the recombination lines. (b) The He I X2.058 pm Line. We have discussed the helium singlet lines at length in cluster IV. H ae we briefly review some of ths argmncRts presented there. We show th^ the iooizing spectrum of Sgr A West is somewhat constrained by the/(//c I A2.058 pm )//(fir a X4.05 pm) ratio. 192 The He I 2.058 \m line results firom the 2^P - 2^5 transition. The spontaneous transition rate for this line is lOr^ the rate of a 2^P - 1'5 resonance transition which produces the X584  photon. Thus, in order for the X2.058 |Jm line to be strong, the scattering optical depth at 584  must be high. On the other hand, if there is a large ratio of neutral hydrogen to neutral helium, the X584 photons are destroyed in photoionizations of neutral hydrogen before they can scatter often enough to degrade into A2.058 pm and higher series photons. From a measured ^.058/B ra ratio it is possible to infer the effective recombination coefficient for X2.0S8, which can be used to constrain the ratio of neutral hydrogen to neutral heUum. This in turn constrains the ionizing spectrum. For the broad components of Geballe et al. ’s (1984) measured fluxes (see table 67) and diKaential extinctions of 2.2 and .725 (for curves a and b respectively) we find that the X2.058/A4.05 ratio is between 0.61 and 0.14. The ratio of A2.058 pm to Br a 4.05 pm is given by: where denotes the effective recombination coefficients. The A2.058 pm effective recombination coefficients are given in chtpter IV. It was shown there they scale horn their value at 10,000 K as in equation 6.5 above, with the power-law index P equal to .864. This, along with the case B fit to a^ffiBra) gives: = 0.0939(-^ '^ (6 .31) 7 * 4.05 10 " 0.10 where is the singly ionized faction of helium, %«isthe ionized fraction of hydrogen, and is the 193 abundance of helium. In cases of moderate ionization, is about unity, which we will assume to be the case. Then, for normal helium abundance and * 4 = 1, we find that the measured value of « ^ “ *(10,000AT) is between 6 .SxlO~^* and 1.49xlO~^* cm^/sec. These values are rather near the case B limit, although there may be as much a factor of 1.5 • 1.6 enhancement of the A2.0S8 emissivity by coUisional excitation out of 2^5. The value of at 10,000 K deviates firom its case B value when either is small, or when %c(X584), the ratio of continuous to line opacities, is large. Usually, will be large enough that only Xe(X584) affects the singlet emissivities. This latter parameter is in turn proportional to the ratio of neutral hydrogen firaction to neutral helium fraction, %««, which increases as the ratio of He I ionizing photons to H I ionizing photons increases. From the results of ch^to^ IV (table 58} we see that the current, nearly case B value of is suggestive of a soft ionizing spectrum, such as that of a star with r . ^ 40,000 AT. On the other hand, the uncertainties are quite large, and additionally the emissivity of B ra may not have its pure case B recombination value: If, by analogy with AGN BLRs, the density in the BLR of Sgr A West is around 10^° B ra would likely deviate firom its recombination value. Also, as discussed in chapter V, dust destruction of HI Lyman photons drives the Balm^ lines firom their case B values, and ought do likewise for the Brackett lines (owing to deviations between our dq)arture coefficients for the high f-states of level n = 5 and those of Pengelly 1964 and Martin 1987, we did not feel confident in calculating the effects of dust opacity on the B ra emissivity in that chtq)ta). Because of these concerns, we feel Aat the I(H eI X2.058 pun)//(BraM.05 pm) ratio is consistent with a soft ionizing spectrum, but does not require it. 194 Finally, as an aside we note that an ionizing spectnun as soft as could be inferred firom the A2.0S8/A4.05 ratio would produce a He^lUe^ ionization front at shorts' columns than the ionization finont. Therefore the BLR must not be optically thick at either 1 Ryd or 1.8 Ryd, or else such a firent would form and the observed high A2.0S8A4.0S ratio could again not be explained. Also consistent with this picture is the fact that the ratio does decline at 1RS 1, which given the result found for the ratio at 1RS 16, can most readily be explained by the combination of a soft spectrum and a radiation bounded nebula at 1RS 1. (c) Density. Because the linewidths for the BLR of Sgr A West approach those of BIRs in AGN, it is interesting to see if the density in the current example might be as high as the canonical 10^ ^ cm~^ inferred from the strength of the C ///] X1909 intercombination line observed in QSOs. The apparent low ratios of broad forbidden-line emission to B ra or Bry from 1RS 16 suggests a higher density for it than the rest of Sgr A West As more data become available, howev^, it is possible that broad components to the fine-structure lines will be seen, as evidently is the case for [Ne //] 12.8 \un (Lacy et al. 1987). The only firm constraint on density aside firom the linewidths, comes firom the emission measure. From equation 6.17 the mean density of the BLR as measured by the Br a line is: (632) (04“) Using Geballe et a i *s (1984) fiux for B ra in a 4 2 arcsec beam (see table 67)» and Î4 = 1, we find a not 195 too stringent lower limit o{  very loose limit on the density comes from the strength of the He I recombinatimi line. For this line to be present, either the gas must be optically thick to radiation at 4 Ryd, or else there must be no He U ionizing photons in the spectrum, or else for a hard ionizing spectrum and an optically thin gas, the ionization parameter must be small. The He m to He n ionization balance is given at any point by: (6.33) a a iH e y where L ..IM = (634) , L y ^IKyd f — dv where / = A^/A^, and where the final equality in equation 6.33 results from assuming zero optical depth and from assuming that the average of the photoionization cross-section is equal to the threshold value of a v(H’e'*) = 1.58jc 1(T** (Osterbrock 1974). To get an abundance ratio He II > He in for a value of Çy*- of .35 (sppropiiate to the AGN spectrum of Mathews and Feiland 1987), F must be of order 2r 1(T^. Returning to equation 6.3 with Ô 50 = 2 and = .1, we obtain N 4 =8,900. While this argument demonstrates consistency with an AGN picture, it is so obviously model dependent that it is hardly cmnpelling. Evoi assuming a spectrum with copious He II ionizing photons. 196 the simple lequirement that the gas be optically thick at 4 Ryd is met at a much lower densi^: The Stromgren column of the He m zone is: where the final equality used the fit to the H I recombination coefficient givoi in table 62 and the fact that for a hydiogenic ion of effective nuclear charge Z, ag(Z, T) = Z qg(l, TIZ^). For III) = .01 pc (1/10’tA the BLR radius), Nu could be as low as JV4 = 5 for even a vey high F = 1. 197 6.4 Photoionization Calculations. We have constructed photoionization models using the code most recoitly described in Ferland and Osterbrock (1987). This code includes all ionization stages of the elements He, C, N, O, Ne, Mg, Al, Si, S, Ar, Ca, and Fe, and predicts all the fine stmcture lines which could be of interest for these ions. The code includes all physical processes (e g. X-ray photoionization, charge transfer, dielectronic recombination) for these ions for which rates are available. We constructed models for a number of photoionizing spectra. Those to be presented here include the mean AGN continuum discussed previously, and main sequence stellar atmo^heres at tmperatures of 30,000 K, 35,000 K, and 40,000 K taken firmn Kurucz (1979). Because the 40,000 K model clearly did not produce a viable model, hotter spectra were not tried. We also tried the continuum given by Terlevich and Melnick (1985) for a "warmer", a hypothetical nucleus photoionized by a collection of highly evolved, very hot Wolf-Rayet stars. The results for this case are not presented because they are not substantially different than those for the AGN. This is hardly surprising since the motivation for Terlevich and Melnick’s work was to demonstrate that low-luminosity AGN-like continua could be produced by a collection of stars. All the models used solar abundances (H=1.0; He=.10; 04.7x10"^; N=9.8*10"®; 0=8.3x10"^; Ne=1.0xlCr^; Mg=4J2xlO"®; Al=2.7xl(T*; Si=4.3xl(T®; S=1.7xl(T^ Ar=3.8xl(T*; Ca=2.3xlO"^; and Fe=3.3x 10~^), an ionizing photon flux ofG;o = 2, a distance finm the ionizing source at 1RS 16 to 1RS 1 of Tpc = .5, and a total column of .25 pc. Only the density was varied. This also resulted, of course, in the variation of ionization parameter and filling factor through equations 6.3 and 6.28 or 6.29 reflectively. The fixed parametas, and especially are not known exactly, of course, and so other models could have been constructed by varying <|> = —^ = 8.36r 10“ the ionizing photon flux at 1RS 1. 4jtr* r4 198 Because we felt that i|> was known at least to within the uncertainties in the fine-structure line data, the atomic data base, the extinction, and the abundances, we felt a detailed fit to it was unnecessary. Furthermore, to first order, the effect on die predicted spectrum of a small change in i|> can be t^roximated by an inversely proportional change in Nu because of equation 6.3. The abundances are also not known to be solar, but to vary them would have introduced far too many firee parameters, leaving us with no information on the ionizing spectrum. We instead chose to hope that solar abundances wne not too far firom the real case, and that a reasonable fit to the observed spectrum would result without tinkering with the abundances. Figures 25 - 28 show how the fluxes of the fine structure lines which have been obsoved to date, change relative to B ry as the density is varied for each assumed ionizing spectrum. We will discuss these diagrams in the section following. 199 6.5 Results In the discussion following we compare the predicted and observed spectra over the entire range of density given in figures 25 - 28. For quicker considaation, we read off a single "best fitting" density for each of the assumed ionizing spectra and tabulated the corresponding qtectrum in table 68 . Our criterion for a best fit was an agreement between models and observations to within a factor of two or three for as many lines as possible. These wide error bars are ^ ro p riate to the large imcertainties present in the data, the abundances, and the atomic data base: It may be that even wider error bars would be suitable. The uncertainty due to the extinction may be estimated by comparing the observed values fm the line strengths for extremes in the published extinction. This information is available in either table 64 or 68 . No model which we investigated produced an ideal fit, and in all but one case the best fit model was not even an acceptable fit for some lines. The best fitting model is a 35,000 K star, which fit the observed spectrum well for a density of 10^^ cm~^. This ionizing spectrum is probably a little hotter than what would actually fit best, but it seems certain that a star as cool as 30,000 K produces a worse fit, as do hotter stars and the AGN spectrum. We made no effort to find a more exact stellar tempaature because of the numerous uncertainties in the data on Sgr A West, and because of uncertainties in stellar atmospbae thewy which we will discuss in section 6 .6 . In the remainder of this section, we consider each of the models in detail, starting with 35,000 K which establishes the standards by which the otiier models are judged. 200 (a) 35,000 K Stellar Atmosphere From a comparison of table 64 to figure 25, we find that a 35,000 K star produces the best fit to the observed spectrum at a density of roughly 10^^ cm~^ (T=7.Q9x 10"^, somewhat higher than has been previously thought to obtain. At this density, the predicted fluxes in [Ne II] 12.8 |im , [S ///] 18.7 \un , [i4r ///] 9|im, and [O III] 52pm relative to fir y are all in good agreement with the obsovations. All the neutral lines are predicted to be lowra' than observed, which is consistent with the possibility that the measurements have included areas of differing excitation in their beams. Despite the goodness of fit, there are still discrepancies. First, the predicted [Ar //] 7 pm is a factor of 3 • 8 Iowa than observed. This is probably not an objection to the 35,000 K model, since none of the ionizing spectra which we tried predict X7 to be as strong as observed. This is a likely reason to believe that either the argon abundance is high, or the atomic data for argon are in error. The abundance insensitive /([Ar ///] 9 pm)//([Ar //] 7 pm) ratio is predicted to be a little higha than observed, but is within the uncertainties. If densities w ae only a little higha than 10^^ cm~^ this ratio could easily be matched. The objection to higher densities is that the observed /([S ///] 18.7 pm)//([S III] 34 pm) ratio probably implies densities lower than 10* cm~^. This constraint holds for all ionizing spectra, since this ratio is almost totally insensitive to all parametos but density. For densities as low as it requires, however, all the ionizing spectra we have tried predicted much stronger high ionizaticxi lines than obsaved, because of the high value of the ionization parameter which such densities imply. Unless the ionizing source is an orda of magnitude less luminous than 6 so=2, or unless 1RS 1 is 3 times as distant from 1RS 16 as it is believed to be, the low density estimate from the S QI line ratio cannot be reconciled with the observed line spectrum for any assumed ionizing spectrum, unless the ionizing spectrum is 201 cooler than 30,000 K. We consider the latter possibility unlikely (see below). Therefore we are not greatly concerned about the failure to match the/([S ///] 18.7 pm )// ([S III] 34 pm) ratio in the instance of the 35,000 K star. Finally, the /([S IV] 10.5 pm)//(Fry) ratio is a little high here too, which p a h ^ s argues for a somewhat softer spectrum. A soft» spectrum would probably also result in a lowor best fit density. (b) 30,000 K Stellar Atmosphere. Comparing table 64 to figure 26 we find that this model probably cannot be reconciled with the line data. The biggest problem this model faces at the best fit density of 10^ cm~^ (see table 68) is that it predicts /([A r///]9 p m )//([A r//]7 p m ) ratio to be a factor of 10 higher than observed. Higher densities would lowe* this ratio, but then [O III] 52 pm quickly becomes very much weaker than observed. (c) AGN Model Comparing table 64 to figure 27 it is clear that the AGN model is a very poor fit to the observed spectrum, in that it predicts that the low ionizaticm lines will be much weaka than obiMrved. There are the fewest discrepancies between the model and the observations for densities of Abut 10^^ cm~^ (see table 68), but even at this density, relative to Bry, [Ar //] 7 pm is predicted to be a factor of 15 - 30 too weak, and [5 ///] 18.7 pm and [Ne II] 12.8 pm are predicted to be factors of 3 - 10 too weak. Even the lines which are nearly within a factor of 2 - 3 of their observed values only barely agree; [SIV] 10.5 pm is just at the tçpsr limit on its strength set by the observations, [4r ///] 9 pm isa factor of about 2 too strong, and [0 III] 52 pm is about a factor of 2 - 3 too weak. All the neutral lines are weaker than 202 observed, which indicates consistency for the present continuum. The weakness of some many lines relative to Bry might be interpreted as due to uncertainties in the hydrogen line’s measured flux or extinction, but even then the /([Ar ///] 9 |im)//([Ar //] 7 |Jm) ratio remains a crucial constraint: The predicted /([A r///]9 |x m )//([A r//]7 (jin ) ratio is weaker than observed for all densities less than about 10^^ cm~^. At densities this high, all lines (including the A rm and Ar n lines) but for [Ne //] 12.8 ]vn are predicted to be grossly below their observed values relative toBry. Most importantly, at these densities, die/([O ///]S 2pin)//([0 /] 63 pm) ratio would be far too small. (d) 40,(X)0 K Stellar Atmo^here. Comparing table 64 to figure 28 we find that this model is not in good agreement with the line data either. It has many of the same problems as the AGN model, such as that it carmot match the /([A r///]9 |im )//([A r//]7 p m ) and /([0 ///]5 2 p » i)//([0 /]6 3 |w i> simultaneously (the best fit densi^ of 10^ cm~^ given in table 68 was chosen to match all lines but die argon lines). Like all models considered here, an acceptable spectrum is only predicted for densities high enough to violate the observed I([S III] 18.7 pm)//([S III] 34 p m ) ratio. Worse than for other models, [Ar //] 7 pm is too weak everywhere by at least a factor of 20. The [Ne II] 12.8 pm strength is clos^ to the observed value at low densities than it is for an AGN model. 203 6.6 Discussion To summarize the preceding sections we have found that the observed X- and y^ray continua appear to be like that of a starburst of H n region nucleus rather than an AGN, unless the extinction has been substantially underestimated. The fine-structure Une data, principally the strength of the [Ne //] 12.8 pm line and the /([Ar ///] 9 pm)//([Ar //] 7 pm) ratio, have also been found to be like those observed for staibiust and H n region nuclei, and unlike those obsmved for AGN. We have found from the large observed I [He I 2.058 pm)// (Bra) ratio for the BLR that the ionizing spectrum is somewhat more consistent with a cool, stellar ionizing spectrum, but this is not a strong constraint From comparison of a grid of models to the fine structure line data we find that a stellar atmosphere of 35,000 K firom Kurucz (1979) provides the best fit, while an AGN ionizing spectrum as in Mathews and Ferland (1987) cannot reproduce the spectrum to the same accuracy. The models also suggest that eith« argon is overabundant or that the atomic data for the [Ar //] 7 pm line are inaccurate. The conclusion must be that the galactic center is ionized by a stellar spectrum with T* ~ 35,000 K. Note that a simple blackbody of the same spproximate tempaatuie would also be allowed. The most likely interpretation of this ionizing spectrum is that it arises finm young stars, making our nucleus a starburst or H E region nucleus. This is not the only interpretation, but it is the one which we will implicitly adopt throughout the following discussion. In this section we will first discuss some variations on the AGN models presented above and discuss some physical processes which our models did not include, but which might be plausibly thought to change the results; these variant models and processes fail to produce viable AGN models, which adds furthw support to the conclusion tiiat young stars ionize il'te galactic center. The topics discussed include turbulent heating, depletion of coolants, large optical depths in the forbidden lines, and radiative 204 excitation of the forbidden lines. After discussing these issues, we will discuss some questions concerning the accuracy of our results, such as how well the crucial /([Ar ///] 9 |xm)//([Ar //] 7 |jm) ratio is predicted, and how firm our knowledge of the ionizing stellar temperature is. We then present predictions for the strengths of lines for which fluxes or flux-limits have not yet been reported in the literature, and conclude by discussing the implications of our results. (a) Variants on the AGN Models of Sgr A West Ignoring for the moment the other lines of evidence which have suggested that a stellar spectrum ionizes Sgr A West, the Mlure of the AGN models to reproduce the fine-structure spectrum is surprising, since it has been shown by previous authors that such ionizing spectra can produce strong low-ionization optical lines for a low F of 1(F^^ (Ferland and Netzer 1983; Halpem and Steiner 1983). Even if Sgr A West is not in fact itmized by an AGN-like spectrum, we expected that it should be possible to make a more plausible model using this continuum than we in fact obtained. We therefore ctxisidaed in detail why this failed to be the case and tried several possible complications which might account for the discrqtancy. (i) Formation of [Ar //] 7pm and [Ne II] 12.8 pm in AGN models At a densiQr of l ( f ^ cm~^, the biggest, but by no means single, discrq>ancy between the AGN model and the obsavations is the /([A r///]9 p m )//([A r//]7 p m ) ratio. Another problem is that [V e//] 12.8 pm is too weak. A cursory compariscm of the rates of ionization from and recombination to A ril and Ne n demonstrates that for no value of the ionization parameta* likely to obtain in Sgr A West will they predominate in the ionized zone in any model: Even a 30,000 K star produces too many photons at their ionization limits for 205 Ar n and Ne n to remain abundant. These species must therefore be found in another zone. For stars with T* < 40,000 JÜT, as we have discussed already, the rate of production of photons above 1.8 Ryd is too low to keep helium ionized throughout the H* zone. All the high energy photons are absorbed in He I photoionizations in the inner zones of the nebula, leaving a zone behind the He*!He^ ionization front where Ar n and Ne n can exist. For AGN models the He* and H* zones are coextensive, but Ar n and Ne n can be formed beyond the ionization front, in the "partially ionized zone" (PI^. The PIZ is a standard prediction of AGN models, and is a consequence of a strong X-ray flux (for discussion of these sorts of models see Kwan and Krolik 1981; Ferland and Shields 1985). The mean cross-section for the absorption of the X-rays is low. This allows them to penetrate beyond the H*IH^ ionization front, where they ionize and heat the predominantly neutral gas. Each of these ionizations produces on the order of several hundreds of secondary ionizations, as the so-called "suprathermal" electrons (electrons with several keV kinetic energies in a gas with average thermal energy of a few eV) liberated by X-ray photoionizations collisionally ionize other species. Further ionizations result from thermal coUisional ionizations, and photoionization by low-energy photons fiom excited states of hydrogen. These states are highly populated because of the high optical depths to Lya scattering. The low ionization lines in LINERs are produced in the PIZ, and the wide range of ionization charactaistic of all AGN is a consequence of the existence of fliis zone. ArH and N ell are among the ionized species which are abundant in the PIZ, and so we expected that their fine structure lines would be strongly enhanced by emission fiom the PIZ. They are «ihanced, but not to the extent observed for Sgr A W est One reason fw this is that the PIZ quickly becomes so cool (500 - 1,(XX) K) that even the smaU energy difference corre^nding to 7 and 12.8 pm is large compared to the mean thermal energy of the gas. CoUisional excitations of these lines are then too slow to produce much line emission. 206 (lï) Turbulent Heating. Since the PIZ produced by our models was too cool to produce sufficient [Ar II] 7 |im and [Ne II] 12.8 ym emission, we considered plausible ways in which the PIZ might remain hotter. We added turbulent heating to AGN models with Nh ~ 10^'^ cm~^ in order to maintain a high temperature in the PIZ. Non-radiative heating could, in fact, be a process actually imdmway in Sgr A West, as suggested by the molecular hydrogen observations (Gatley 1984), and by Lacy et al. (1987) who present evidence that the gas stream called the northern arm undergoes a sudden change in direction at 1RS 1. For rates of the same order as the rate of heating by photoionization (about 1(T'^ to ICT^^ erg cm~^ sec“*) the /([Ar ///] 9 ]un)II ([Ar //] 7 ym ) ratio could be brou^t into agreement with the observations. When this htqtpened, howev^, the neutral gas was found to emit far too strongly in otha lines formed in the PIZ, such as [0 I] 63 ym, [Si II] 35pm. and [CII] 158 pm. For turbulent heating rates of this order, our models predicted that these lines should be as strong as sevoal hundred times Rry, in gross contradiction of the observed upper limits on these ratios. This result also constrains the rate of turbulent heating for models ionized by stellar spectra, as these models would produce the same strong neutral line emission in a heated zone beyond their ionization fironts. (ÎÜ) Depletion o f Coolants. In models without turbident heating, of the cooling rate for the very low temperature regions of the PIZ is attributable to [Si II] 35 ym line emissitm. If silictm were imderabimdant, the PIZ might again remain hot Models using planetary nebulae abimdances (for which silicon is 5 times less abimdant; the abimdance set used was given in chapter IV, section 4.4) did not produce substantially better agreement for/([A r //] 7 ym ) relative to either/(Fry) or/([Ar ///] 9 ym). 207 (fv) Optical Depths In the Fine Structure Lines. The high densities predicted by all models, along with the independent constraint on the total column of material raises the possibility diat the column daisilies are large enough for some fine structure lines to become optically diick. We expected this to have important implications for models. For example, lines formed in the same region where they are thick might be self-absorbed and therefore less intense. If important cooling lines are thick, the gas temperature would rise until other, opticaUy-thin lines become strong enough to cool the gas. This effect, if it in fact occurs, might be particularly important in the PIZ of an AGN model, where optical depths would be large, but the gas still warm. The optical depth in a line of an arbitrary ion with upper state population and Iowa state N{, is given by: whae the velocity width, àV, is given by: (6.37) and a = - 2.245% 10"*^ cm^Isec (6.38) Values for a, along with otha data pertaining to the fine structure lines which we considaed in o a 208 models, are given in table 69. All the data come from Mendoza’s (1983) compilation. As a first estimate of the optical depth, we neglected stimulated emission. The correction for stimulated emission (the tm n NJ(ùu) is actually rather important for tk se lines in the current situation, because at densities where the column is large enough for optical dq>ths to be large many of the fine structure level populations are also collisionally dominated: This follows from consideration of their critical densities against coUisional de-excitation, which for many of the lines are as low as l(f cm~^. Because of the small energy differences between the levels, for tempaatures of a few thousand degrees and higher the excited states and the ground state have comparable populations, making the rate of stimulated emission comparable to the rate of radiative excitation. In fact, it was expected that some lines would mase. Note that this high excited state population also means that middle states of triplets could become nearly as optically thick as the ground states. Neglecting the stimulated mission for the moment, the optical depth can be estimated for an ion denoted by the subscript i , by: where A,- is the elemental abundance associated with the ion, and where X, is the column averaged fraction of the ion: Jx,dr (6.40) jd r To 209 where r i is the outer radius, tq is the inner radius, and Ar = r i - tq . To estimate X,- for a given line we assumed 2 limits. In the low density limit, 1RS 1 is entirely ionized and can be approximated as being in a single ionization state. Then X; is unity and equation 6.39 indicates that for a fixed column, the optical depth rises linearly with N,. In the limit of densities high enough to produce an ionization front within 1RS 1, most ions are only abundant in theff* zone. In this case, X,- is equal to the Stromgren column, r, ovex 25 pc. The Stromgien column is given by: r, = = .04 (6.41) where the last equality makes use of equation 6.3. In this case the optical depth is inversely proportional to N,. The optical depth is at a maximum when r, = Ar = .25 pc, which occurs for our adopted parameters at iV 4 = 1.13, t^ = 1, and no turbulent broadening. Note, however, that for lines fcvmed in neutral gas, the Stromgren column argument does not apply; their optical depths continue to rise linearly with density. It is straightforward to see by these arguments and the data in table 69 that essentially all lines should become optically thick for Sgr A West for densities of order ^ 4 = 1, except for those with abundances under about 1(T\ Whether or not this results in any appreciable effect on the observed ipecirum is another matter. For a thermal ionizing spectrum, the material beyond the front, which would be optically thick, produces no appreciable emission. Furth^, it is much colder dian ionized material and is unlikely to interact substantially with line photons emitted in the ionized zone because of the difference in linewidths. For a nontheimal ionizing spectrum the material in the PIZ should be much warmer, so that this mataial can emit appreciably and also affect photons firom the ionized zone. The question is 210 best answered with models. Our models above did not include the fine stnicture line optical depths, but afterwards we constructed models which did. These models achieved large optical depths in the forbidden lines, and some lines even mased. The emitted spectrum and the tempoature in the PIZ failed to change, however, by even 1%. Evidently, the optical depths rose to rgtpreciable values at depths where line formation had for the most part ceased. Because of this, we have chosen not to present the results of these models here. We still think it possible, however, that qttically thick fine-structure lines may play a role in some objects. (v ) Radiative Excitation. Because 1RS 1 is a strong 10 pm source, the possibility exists that intonal dust emission could excite the IR lines near 10 pm. Whether or not this is an important process is easily estimated by comparing the photoexcitation rate to the coUisional excitation rate. The Einstein B value for abscnption is givra in c.g.s. units by: The rate of radiative excitation is where /y is the mean intensity, related to the observed flux density by: wo = —^expC tw o) = 1.354% 10"" erg Isec Icm toer-^exp(Xj,i(0 (6.43) where t^io is the estincticn ccasction, is to be given in Jenskys, and where 0 is given in arcsec. Using the 10 pm flux measurement of 40 Jy in a 2.3" beam centered on 1RS 1 (Becklin and Nragebaua 211 1975), gives /xio=2.4xl(T*° and = for extinction curves a and b respectively. For [Ar 111^9\m and [5/V] 10.5pm we find maximum respective radiative excitation rates of 1.4x10”* sec”^ and 1.1x10”^ se c T h e se rates are impressive, but an order of magnitude below the coUisional excitation rates for these lines at densities as low as 10^ cm~^. Unless we are prepared to conclude that densities are much lower than any prevailing estimates, we conclude that radiative excitation of the fine-structure-Unes is probably unimportant (b) Uncertainties in the Calculations. (z) Uncertainties in the Stellar Atmospheres. While the atmospheres we used were highly detailed, it should be noted that there are considerable differences between various authcus concerning the UV spectra of early-type stars (discussed in AUer 1985, p. 167). Mathis (1985) discusses uncertainties in the ionization equilibria of nebular neon and argon which are introduced by the diffidences in the Kurucz and Mihalas (1972) atmo^heres. For the present it should be noted that the best-fitting temperature of 35,000 K may be subject to revision, but the general conclusion that Sgr A West is ionized by a soft spectrum which has insufficient photons at 1.8 Ryd and above to maintain an He II zone throughout the nebula stands. Despite diffidences in the details of stellar models, a stellar or blackbody qtectrum must be chosen over a hard, AGN-like spectrum. (ii) Incomplete or Inaccurate Atomic Data for Argon. Inctunpleteness and inaccuracies in the atomic dat?i base are always a concern for photoionization models, particularly in cases where a single line ratio is the only diagnostic available. There are several constraints on the ionizing spectnun for Sgr A West, but the/([i4r ///] 9 pm)//([Ar //] 7 pm) ratio does weigh particularly heavily on the conclusions. In the case of M 82 this ratio has been used alone to constrain the ionizing spectrum. 212 There has been some evidence prior to this investigation that either the Ar m to Ar n abundance ratio, or the /([Ar ///] 9 pm)//([Ar //] 7 |im) line ratio, both, are not being correctly predicted: The /([A r III] 9 pm)//([Ar //] 7 pm) line ratio has been observed to be low, perii^s anomalously so, in many H H regions: Herter, Heifer, and Pipho- (1983) predicted this ratio and also the /([S IV] 10.5 pm)//([S ///] 18.7 pm) for H II regions for which they believed they independently knew the temperature of the ionizing stars. The predictions and the observations diverged by as much as 1 -2 orders of magnitude. Laçasse et al. (1980) noted a similar effect for W 31RS 1. In both cases, the authors invc&ed dust with a substantial increase in opacity around 2 Ryd ov^ its value at 1 Ryd to selectively soften the ionizing spectrum. Although the dust opacity in this regime is poorly known, Martin and Faland (1980) argue that the more Ukely effect of dust is to harden the spectrum. In addition, Mathis (1985) suspected that the temperatures Hater, Heifer, and Pipha had derived for the ionizing stars were too high, and that the Ar M to Ar n and SIV to S m line ratios were consistent with the real stellar temperatures. Anotha possibility is that the atomic data are inaccurate, or that an important physical process is being ovalooked. It is difficult to evaluate the accuracy of the available atomic data. Comparison between authors provides some measure of consistency, but not necessarily accuracy. Two instances for which accuracy is definitely a concern are for photoionization cross-sections and collision strengths. The photoionization cross-sections given by Chapman and Henry (1971) are consistently larger than those of Redman and Manson (1979), often by almost an order of magnitude. Our code used unpublished cross-sections of Mendoza which are midway between the two, but this illustrates the uncertainties in these parameters. Mendoza (1986) has noted that recent calculations of collision straigths for fine structure lines which take resonances more fully into tccount, have shown significant increases over previous calculations. 213 Completeness is an easier question to answer. There are rates available for most processes which could affect the Ar m to Ar n balance. One exception is that there is no dielectmnic recombination rate published which is valid at nebular temperatures: Shull and van Steenbuig (1982) provide a rate valid only in the high temperature approximation (Burgess 1965). The dielectronic recombination rate for Ar m recombining to A ril at nebular tmnperatures dqpends on the existence of autoitmizing states from out of the ground term, which may or may not exist. Thus, the rate may be negligible, or it may be quite fast. In AGN models not only the Ar Œ/Ar II balance, but also the Ar H/Ar I balance is important to the /([Ar Iff] 9 iim)/I([Ar //] 7 pm) line ratio. This is because the bulk of the [Ar //] 7 pm emission is formed in the PIZ where Ar m is essentially nonexistent Ibere are no published rates that we are aware of for Ar* recombining to Ar** through charge transfer with neutral H, but if the rate coefficimit for this process were as high as 10~^ cm^lsec there would be no Ar n in the PIZ under any circumstances likely to obtain in Sgr A West (c) Predictions Figures 29 through 33 provide predictions of intensities relative to Bry, for fine structure lines which have not yet been reported in the literature. While we feel the current set of data has probably provided sufficient constraints to choose the ionizing continuum with certainty, we provide these predictions so that future observations can confirm or falsify our results, and resolve any nagging questions concerning the prediction of the Arm and A ril lines. Particularly useful would be observations of the /([.Afe///] 15.6pm)//([N«//] 12.8p»»)»atio, which would strongly discriminate between ionizing spectra. 214 (d) The N ature o f Sgr A W est While we have shown that a blackbody or stellar spectrum is probably respmsible for the ionization of Sgr A West, a number of questions remain. For example, the linewidths of the H I and He I recombination lines would lead, in the absence of otha- data, to an AGN classification for Sgr A West. Hie evidence for a supermassive object also suggests such a classification, although this is not really compelling since we lack any hard evidence for the presence of black holes in AGN or for their absence firom other types of objects. We also note that although the evidence suggests a central ionizing source, there is no evidence of star firvmation currently underway in the region where this source must be localized. Neitha* are Ihore any objects which are readily identifiable as supemovae or stellar remnants within Sgr A West. Although throughout this chapter we have discussed our results as if only a stellar spectrum could produce the ionization, it must be emphasized that what we really have shown is that a soft, blackbody like spectrum is responsible for the ionization. Such a spectrum need not actually be stellar. If Sgr A West is not a starburst or H n region galaxy after ail, it may be that what we really have shown is that some low-luminosity AGN have ionizing spectra more like blackbodies. This is not a new idea: We have already cited arguments for this possibility for LINERs, and the blue bump is often intapreted as a blackbody component supoimposed on a power-law spectrum, perii^s arising firom an accretion disk (see e.g. Ferland and Shields 1985). Both the blue bump and die proposed thermal componait of ONER spectra correspond to much hotta temperatures dian would characterize the ionizing spectrum of Sgr A West, but it may be fiiat "bumps" of otho’ temperatures might be ^tpropriate to otha AGN. 215 Table 60 Case A Elective Recombination Coefficients and Ratio of Bra X4.05 ym to BryX2.17 \un. (In units o f 10"*^ cm Vsec ) T atff(Bra4.0S fw i) a ^ (S ry 2 .1 7 pun) JM.osfJx2.n 1,250 23,26 2.811 4.24 2,500 10.82 1.516 3.82 5,000 4.778 0.7608 3.36 10,000 2.011 0.3541 3.04 20,000 0.8131 0.1553 2.80 40,000 0.3214 0.06498 2.64 80,000 0.1251 0.02652 2.52 216 Table 61 Case B Effective Recombination Coefficients and Ratio of E ra A4.0S \un to BryA2.17 \\m. (In units of 10"^^cm^lsec ) T a,ff(Bra4,05 im ) a^ff(Bry2.n iim) j>A.OslivLYI U50 23.58 2.905 4.34 2400 11.03 1.578 3.74 5,000 4.913 0.8007 3.28 10,000 2.095 0.3787 2.96 20,000 0.8624 0.1697 2.72 40,000 0.3481 0.07280 2.56 80,000 0.1387 0.03046 2.43 217 Table 62 Power-Law Indices p for Fits to Published Total and Effective Recombination Coefficients. case A caseB a ^iB ra ) -1.2775 -1.2550 a^(B ry) -1.1462 -1.1190 «a (if) - -1.3333 OLs(He) - -1.2024 aw f(//P) -1.4379 -1.3645 218 Table 63 Atomic Line Data for 1RS 1 (in units of 1(T*’ WIcm^) line e (a ) Flux^‘Hb) Ref. RrY 2,17 Jim 2” .0014 .059 .024 SA 3.6" .0038 ±.0006 .067 .027 Geb 3.8" .0036 ±.0009 .059 .024 WSL 3.9" .0093 ±.005 .15 .059 WP 8"x8" .023 (<2a) - - BJS 32" .11 ±0.01 .078 .032 N Bra4.05]}m 3.9" .015 ±.014 .047 .047 WP 4.2" .077 ±15% .22 22 Geb 8"x8" .26 ± .0 2 - - BJS 10" 1.8 1.4 1.4 Le81a P f a 7.45 pro 14"x28" .77 -- LeSla He 1 2.058 \un 3.6" .0023 ±.0005 .071 .018 Geb [C/7] 158 pm 55" 1.26 .028 .028 Lu r 1.4 .028 .028 Gen85 [0 7] 63 pm 25" 6.9 .50 .50 H86 30" 4.2 3A .23 Gen84 44" 1 7 ± 5 .55 .54 Gen84 V 11 .23 .23 Le81b [0 7] 146 pm V .67 .013 .013 Gen85 [0 777] 88 pm 44" .7 ±.2 .022 .022 Gen84 r < 1.7 <.034 <.034 Wa80 4’x4.4’ .09 ± .03 - - D [0 777152 pm 44" 5 .16 .16 Wa85 1’ 4.3 .091 .090 Wa80 219 Table 63 (C ont) line e Fiux^o) Flux^‘Ha) Flux^^Hb) Réf. [Ne //] 12.8 HOT 3.6" 1.0 2.7 1.8 L79 4" .62 1.41 .92 L80 5" .7 ± .13 1.2 .76 Wi78 8" 1.2 ± .3 1.0 .65 Wo76 8" 1.7 1.41 .92 L80 13" < 2 <.8 < .5 AJ 16" 5 1.41 .92 L80 22" 8 1.41 .92 L80 25" 10.9 ± 2.5 1.66 1.08 AJP 25" 7.5 ± .7 5 1.1 .75 A 6"x31" 7.0 ± .3 - . - Wo76 7"xl4"EW 1.3 - - Wo77 ail Sgr A 50 1.41 .92 L80 [S i//] 35 MOT 25" 2.3 ± il .34 .21 H86 [S u n 18.7 MOT 20" 1.7 ±.2 1.76 .39 H83 25" <1.5 <1.1 < .2 5 A 25" 1.8 ± . l 1.32 .30 H84 2.7’ < 1 9 - - M [S lin 34 MOT 25" 2.6 ± .4 .39 .23 H84 [S IV] 10.5 MOT 7" <0.01 < .1 <.05 L79 8" <0.01 < .l < .0 4 L80 25" < 2 < 4 .0 < 1.4 A [Ar II] 7 MOT 28” 6.4 ±1.2 .71 .64 W179 14"x28"NS 4.6 - - LeSla [Ar ///] 9 MOT 7" 0.03 .3 .09 L79 8" 0.03 ±.015 .2 .07 L80 10" <0.05 <.3 <.09 Le81a [Ar V] 13.1 MOT 3.6" <0.02 < .0 4 < .04 L79 4" <0.04 < .0 6 < .06 L79 References: (AI) Aitken and Jones (1972); (AJP) Aitken, Jones, and Penman (1974); (A) Aitkenera/. (1976); (BJS) Bally, Joyce, and Scovilie (1979); (D) Dain ei ai. (i57S); (Geb) Geballe eî al. (1984); (Gen84) Gwizel et al. 0984); (GenSS) Genzel et al. (1985); (H83) Herter e/ al. (1983); (H84) Herter et al. (1984); (H86) Herter et al. (198Q; (L79) Lacy et al. (1979); (L80) Lacy et al. (1980); (LeSla) 220 Lester et al. (1981a); (LeSlb) Lest^ et al. (1981b); (Lu) Lugten et al. (198Q; (M) McCarthy et al. (1980); (N) Neugebauer et al. (1978); (SA) Storey and Allen (1983); (Wa80) Watscm et al. (1980); (Wa85) Watson et al. (1985; referenced in Genzel et al. 1985); (Wi78) Willner (1978); (Wi79) Willneref al. (1979); (Wi79) Willner «r al. (1979); (WP) Willnw and Pipher (1983); (Wo76) Wollman et al. (1976); 0Vo77) Wollman « al. (1977); (WSL) Wollman, Smith, and Larson (1982). 221 Table 64 Adopted Fine Structure Line Spectrum for 1RS 1 Ratioed to B r y = .066% 10"*’ Wlcm ^ for Extinction Curve a , and toB rY = .027% 10"*’ Wlcm^fca Extinction Curve 6 . Line Ratios Curve a Curve 6 [C //] 158 jun/Bry 1.0 .42 [0 /] 63 pm/Bry 5.8 14 [0 / ] 146 ymlBry 20 .48 [0 ///] 88 pm/Bry .33 .81 [0///]52n»i/Bry 3.8 4.6 [Nein 12.8pm/Bry 19.7 31.5 [Si / / ] 35 pmIBry 5.2 7.8 [S ///] 18.7 pm/Bry 23 13 [S///]34pm/Bry 5.9 8.5 [S m 10.5pm /B ry <1.5 <1.7 [Ar //] 7 pm/Bry 11 24 [Ar///]9pm/Bry 3.7 3.0 [Ar V] 13.1 vmIBry <.45 <1.1 [S IV] 10.5 \m /[S u n 18.7 \un .065 .13 [S ///] 18.7 nm/[S ///] 34 pm 3.9 1.5 [A r///]9jJin/[A r//]7|im .34 .13 [0 ///] 88 vm![0 ///] 52 pm .087 .18 [0/]146pm /[0/]63pm .034 .034 222 Table 65 Continuum Data for Sgr A */IRS 16 Wavelength/ Flux/ Energy Band Luminosity 8 Réf. 1 mm 25 Jy (est,) r G77 1 0 0 (un 1,800 Jy (est.) 30" BGW 100 |im 8,000 Jy (est) 1 ' G77 8 8 Mm 8,000 Jy (est.) r G78 8 6 MM 77,500 Jy (est) 4 ’x4.4’ D 63 MM 4,300 Jy (est.) 30' Gen 63 MM 9,000 Jy (est) 1 ’ G78 50 MM 3,200 Jy (est.) 30" BGW 50 MM 10,000 Jy (est) 1’ G77 30 MM 5,000 Jy (est.) 30’ BGW 30 MM 7,500 Jy (est) 1 " G77 27.8 MM 14.9 Wlcm^lVJn 30" MC 18.9 MM 16.1 W/cm^/\im 30" MC 3.7 MM 13.7 Jy (est) 14.7" S 3 mm .1 Jy (est) .5" SF 2.5 MM 1.3*10"“ W/cm*/MM (est) 14.7" S 2 .2 MM .27 Jy 2.3" BN 1.7 MM .0 2 Jy (est.) .5" SF iMM 10 r» Jy (est) .5" SF 0.98 MM 44.6 MJy 1.17" HDB 0.5-4.5keV 1.5* 10® erg/sec 1’W 1-10 keV <2 * 10“ ag /sec 2.5’-15’ M 10-100 keV 3* 10® erg/sec 5" M 10-100 keV 6 * 10® eg /sec 1.6 " M 10-100 keV 4*10®CTg/sec 1.6 " M 10-100 keV 7*10®org/sec 1.6 " M 50keV 2xl(T^phlsec/cm^lkeV (est.) 35" R 50-500 kéV 10® erg/sec 8 " C 223 Table 65 (Cont.) Wavelength/ Flux/ Energy Band Luminosity e Réf. 100keV SxlCr*ph/seclcm^lkeV (est.) 35" R 100 keV 2xl(r*ph /sec ion ^/keV (est.) 15" P 100-500 keV 7% lO” erg/sec 30" M 100-500 keV 2 * 10^ erg/sec 30" M 200 keV 4xl0~^phlsecIcm^lkeV (est) 15" P 500keV-10MeV 2 k 10“ erg/sec 43" M IM eV < Sx lOr^ ph /sec /cm ^/kéV (est) 35" R IM eV ixl(f*ph/sec/cm^/keV (est.) 35" R >100M eV 8 lx 10“ erg/sec 1" M References: (BGW) Becklin, Gatley, and W ema (1982); (BN) Becklin and NeugebauCT (1975); (Q Coe et al. (1981); (D) Dain, et al. (1978); (G77) Gatley et al. (1977); (G78) Gatley et al. (1978); (G) Genzel et al. (1985); (HDB) Heniy, Depoy, and Becklin (1984); (M) Matteson (1982); (MC) McCarthy et al. (1980); (P) Paciesas, et al. (1982); (R) RiegW, et al. (1985); (S) Soifer, Russell, and Merrill (1976); (SF) Stein and Forrest (1986); (W) Watson, et o/. (1981). 224 Table 6 6 Schematic AGN Continuum (After Mathews and Ferland 1987). (2 sohi units of i f photons/sec. Fy X 8.8417% = flux in Janskys at earth. F y (erg/sec/Hz) Vi(Ryd) V2(Ryd) .0000912 .18382 (1 mm-4960 Â) .18382 1.7426 (4970Â-525Â) 1.7426 4.1323 (525Â-220Â) 1.7426’“ 4.1323 26.84 (220 -34 ) 7.52»itf“( - j^ r ’eso 26.84 7353 (34Â-100keV) i.4&io'’( ^ r ‘4’e» 7353 7353000 (lOOkeV-lOOMeV) 225 Tabled? Atomic Line Data for 1RS 16 ("BLR") (in units of /cm*) line e flux Ref. Bry2.VJ pm 3.6" .0029 ±.0006 Geb 3.8" .0017 ±.0002 HKS 3.8" .0019 ±.0005 WSL 3.9" .009 ±.005 WP 8 "x8 " .018 ±.007 BJS B ra 4.05 pm 3.9" .009 ±.005 WP 4.2" .092 ±15% Geb 8 "x8 " .261 ± .0 2 BJS He 7 2.058 pm 3.6" .0024 ±.0004 Geb 3.8" .0022 ±.0003 HKS 3.8" >.00019 WSL 4.2" .0031 ± .0005 Geb References: (BJS) Bally, Joyce, and Scovilie (1979); (Geb) Geballe et al. (1984); (HKS) Hall, Kleinmann, and Scovilie (1982); (WP) Willner and Pipher (1983); (WSL) WoUman, Smith, and Larson (1982). 226 Table 6 8 Comparison of Observed Line Ratios to Best-Fit Model Line Ratios (The observed range corresponds to the 2 adopted reddening curves) Line Ratios Observed 35,000 K 30,000 K AGN 40,000 K Best Fit Density {cm~^ - 1CJ4.5 10^ l ( f ^ 10* [C i n 158 pm/Br Y 1.0-.42 .039 .12 .064 .0 0 2 [0 /] 63 pm/Bry 5 .8 -1 4 .035 < .0 0 1 1.3 .065 [0 7] 146 pm/BrY .2 0 -.4 8 < .0 0 1 < .0 0 1 .074 < .0 0 1 [0 /// ] 8 8 pm/BrY .3 3 -.8 1 .2 0 .65 .15 .18 [ 0 / // ] 52 pm /Br Y 3 .8 -4 .6 1.85 2 .0 1 .6 1.6 [Ne II] 12.8 \lm/Bry 19.7-31.5 21 25 6 .2 19 [Si II] 35 ]lmlBry 5 .2 -7 .8 .51 .115 2.4 .38 [S III] 18.7pm/BrY 23-13 11.5 40 3.6 3.8 [S/7/]34pm/BrY 5 .9 -8 .5 1.4 31 .45 .38 [S IV] 10.5 \imtBry <1.5 -1 .7 .98 .1 1.35 .70 [Ar II] 7 \m /B ry 1 1 -2 4 2.1 .52 .72 .38 [Ar 777] 9 \unlBry 3 .7 -3 .0 1.95 5.6 5 5.1 [Ar V] 13.1 \m lB ry <.45-<1.1 < .0 0 1 < .0 0 1 .0 0 2 < .0 0 1 [S IV] 10.5 pm /[S 777] 18.7 pm .065 - .13 .085 .0025 .38 .18 [S 777] 18.7 pm/[S 777] 34 pm 3 .9 -1 .5 8 .2 1.29 8 10 [Ar 777] 9 pm/[Ar 77] 7 pm .3 4 -.1 3 .93 10.8 6.9 13.4 [0 III] 8 8 m / [ 0 III] 52 pm .087 - .18 .33 .13 .11 227 Table 69 Atomic Data for Fine Structure Lines (For triplets, transition from middle state Line ®« (0, Ou! A|ii a [C /7 ] \51.1\m^Pvx-Pi^ 4 2 125 2.29(-6) 8.07(13) [iV/7] 205.3 Mm (3Pi-3Po) 3 1 .401 2.08(-6) 2.02(43) IN m 121.8 pm CPx-^Pi) 5 3 1.13 7.46C-6) 9.06(43) [NlinSlSvrnCP^rr^PvH 4 2 .701 4.47(-5) 8.08(43) [0/163.2nm(3Pi-®P2) 3 5 8.92(5) 1.52(42) [ 0 7] 145.6 pm C P trP i) 1 3 1.74(-5) 1 2 1 (4 2 ) [ 0 777] 88.3 pm 3 1 .542 2.62(-5) 1 2 2 (4 2 ) [0 777] 51.8 pm 5 3 129 9.76(-5) 1.52(42) [0 7V] 25.9 pfw (^P 3/2~^7* 1/2) 4 2 2.33 5.20(-4) 8.11(43) [We 77] 12.8 pm ^Pia~^Pzrù 2 4 .368 8.55(-3) 8.04(-13) [We 777] 15.6 pm 2) 3 5 .527 5.97(-3) 1.53(42) [We 717] 36.0 pm 0-^7»1 ) 1 3 .185 1.15(3) 121(42) [We V] 24.2 pm (®Pi-®Po) 3 1 244 1.19(3) 1.14(42) [We V] 14.3 pm ^P jr^P i) 5 3 .578 4.43(-3) 1.42(42) [Mg m 4.49 pm & i a - P v ù 2 4 .300 1.99(1) 8.08(43) [Mg V'[ 5.61 pm CPi- P ^ 3 5 .400 1.27(-1) 1.51(42) [Mg V] 13.5 pm i^Pa-^Pi) 1 3 .156 2.17(-2) 120(42) [Si 77] 34.8 pm ( ¥ 3/2- ^ ? ,a) 4 2 2.17(-4) 8 2 1 (4 3 ) [S 777] 33.6 pm (3Pi-3Po) 3 1 2.59 4.72(-4) 121(42) [S777] 18.7 pm (^Fz-^Pi) 5 3 5.81 2.07(-3) 1.51(42) [S7F] 10.5 pm (^f 1,2) 4 2 6.42 7.73(-3) 8.02(43) [At 77] 6.99 pm (2Pi^-2p 3,2) 2 4 .635 5.27(-2) 8.08(43) [i4r 777] 8.99 pm 3 5 2.24 3.08(-2) 1.51(42) [Arlin 2l.im & (rP i) 1 3 1.18 5.17(-3) 120(42) [Ar V] 13.1 pm (^Fi-^Po) 3 1 257 7.99(-3) 121(42) [Ar V] 7.90 pm (^Pz-^Pi) 5 3 1.04 2.72(-2) 1.51(42) [Ca 7V] 3.21 pm (*P vr-^Pvù 2 4 1.06 5.45(-l) 8.09(43) [Ca V] 4.16 pm CPi~^P^ 3 5 .760 3.10(-1) 1.50(42) [Ca V] 11.5 pm & ff-^P i) 1 3 2 0 2 3.54(-2) 1.21(42) 228 8 Extinction for SgrAWost 7 6 5 4 ,( 0 ) 3 2 \__ X ( jjLm) Figure 23. Two Possible Extincüon Curves for Sgr A Wes:. Ths Sguis shows two extremes in the extinction estimates. Curve a uses the absolute extinctions quoted by Becklinei ai. (1978), with interpolation from near-lR to fai-IR by means of a l/k law. Curveb uses the absolute extinction quoted by Willner and Pipher (1983) at 2.17 pm, and inteipolates into the far-IR using a stendard extinction law. At 9.7 pm curve b uses Willner and Pipher’s absolute extinction, but the rest of the far-IR curve is constructed with the relative extinctions of Becklin et al. (1978) in this wavelength regime. Beyond the regime shown in the figure, we extended the curves to 18.7 pm by the extremes of the measurements of McCarthy et al. (1980) (twg .7 = 2.4 for curve a and = .8 for curve 6) and extended both these results to 33.4 pm by use of Hateret al. ’s (1984) % g. 7/tM3.4 = 3.1. Note that because of the uncertainties in the reddening we have used tx. nnd A% interchangeably, without using the 1.086 conversion factor. 229 I S «t^in^SmScviSH 8 OBSERVED CONTINUUM FOR >18 SGR A WEST 10' >18 10' >* s It^ 10>18 >12 10' ,12 ,14 ,18 .18 >20 10' 10 10' 10' 10' 10“ viWz) Figure 24. Observed Contùiuumfor Sgr A West, with Hypothesized AGN Continuum. Continuum Data fiom the Literature (summarized in table 65) are plotted as open circles (for resolutions equal to or exceeding 1'), closed circles (for resolutions from IS" to 1'), or crosses (for 1RS 16 alone, the proposed continuum source). The units on the ordinate axis areJy -Hz. The hypothesized AGN continuum from Mathews and Ferland (1987)is shown as the set of solid lines. 230 r 4 10 10" 10 10" 35.000 K Observed Lines 10 [Nel] 12 I - [ArE3 7 X CO [SIS] 35 10 [01363 \ \ \ '[SHÎ 18.7 '^ W \ ' ' w-2 10 X\ \ V T—[s m ]3 4 [Om]52 L,[s m l 10.5 L -lO B ] 88 ■ICZ] 158 ± 6 10 10" 10 10 NH Figure 25. Intensities of Observed Fine Structure Lines Relative to B ry as a Function o f Nh for 35,000 K Ionising Stellar Spectrum. Predictions of the ratios to Bry of the fine structure lines for which fluxes or uppo^ limits have been reported in the literature (see table 63), are plotted against the total hydrogen density Nh for a 35,000 K ionizing stellar atmosphere. The top axis gives the ionization parameter F, which is inversely proportional to the density. All stellar atmospheres used are fiom Kurucz (1979). 231 ri -2 r 3 10 10 10 10" 30,000 K Observed Lines 10 [NeZ] 12.8 J [ArUlT X w œ ri 10 [sm]i8.f ^[SIE ]35 • [A rS]9 10 [OS] 52 [S m ]3 4 [Om]88 [CE] 58 ‘— [S JE 110.5 «0^ 10^ 10* 1 0 ' NH Figure 26. Intensities c f Observed Fine Structure Lines Relative to B ry as a Function c f Nu for 30,000 K Ionizing Stellar Spectrum. The same quantities are shown as for figure 25, but for a model using a 30,000 K ionizing stellar spectrum. 232 10 10' 10' —r —r A6N Obssrved Lines 10 CNel] 12.8 - **•«. I - — -tA rM 7 s . ^ CO I] 63 X \ \^ C S iI]3 5 ^CA rII39 m \ 10 ------ lOX3 146 CCS3I58 10"® h 'csm:34 CArZ] I COE] 88 —I COE] 52 ^CSBCl 10.5 Figi^e 27. Intensities of Observed Fine Structure Lines Relative to Bry as a Function offor an AON Ionizing Spectrum, The same quantities are shown as for figure 25, but for a model using an AGN ionizing spectrum from Mathews and Ferland (1987). 233 2 10 10" 1 0 "^ 10 —r —r n ------40.000 K Observed Lines 10 - [N el3l2.8 _ I X v> tArlE37 m [Arm]9 % \ \ ^..ISiI3 35 .“I 10 \ \ \ v COI3 63 [ s m ] i8 10 «*«» W\ tS B l 34 CCÏÏ1 158 J Icom] "52 [0M3 88 [S3Z] 10.5 _L 10^ 10" 10" 10 ' NH Figure 28. Intensities of Observed Fine Structure Lines Relative to B ry as a Function of Nh for a 40,000 K Ionizing Stellar Spectrum. Hie same quantities are shown as for figure 25, but for a model using a 40,000 K ionizing stellar spectrum. 234 z 10 10 10 10 35.000 K Predicted Lines 10 K m " IQ -' 10 v \ 10 10 10" 10 N. Figure 29. Intensities of Unobserved Fine Structure Lines Relative to Br"^ as a Function of Nh for 35,000 K Ionizing Stellar Spectrum. Predictions of the ratios to Bry of the fine structure lines for which no published data are yet available, plotted against the total hydrogen density Nh for a 35,000 K ionizing stellar atmosphere. 235 3 0 , 0 0 0 K Predicted Lines k_ QQ l-l r 2 - CNem ] 15.6 [Nm]57—j CN13 2 0 5 - 3 ,4 6 NH Figure 30. Intensities o f Unobserved Fine Structure Lines Relative to B ry as a Function of Nu for 30,000 K Ionizing Stellar Spectrum. The same quantities are shown as for figure 29, but for a model using a 30,000 K Ionizing stellar spectrum. 236 .-2 r 4 AGN Predicted Lines for ions of N, Mg, Co 10 X w 00 ^[MgIZ]4.5 CC03T] 11.5 -J CCoZl 4.2 CNÏÏ1 122 -J 0MglE39.6 - .4 .9 .6 .7 10 !û N. Figure 31. Intensities o f Unobserved Fine Structure lines o f Ions ofN, Mg, and Ca, Relative to Bry as a Function ofNg for an AGN Ionizing Spectrum. The same quantities are shown as fw figure 29, but for a model using an AGN ionizing spectrum from Mathews and Ferland (1987). Only ions of N, Mg, and Ca are shown on this figure to limit the number of overlapping curves. See figure 32 for other ions. 2 3 7 10" 10 10 AGN Predicted Lines for ions of 0, No, Ar •A. »—1 I 10 10 [Ar Z ] 7.9 - J 10" 10= 10" 1 0 ' N, Figure 32. IntensiHes c f Unobserved Fine Structure lines of Ions c f O, Ne, and Ar, Relative to Bry as a Function cfN u for an AGN /onizing Spectrum. The same quantities are shown as fcff figure 29, but for a model using an AGN ionizing spectrum from Mathews and Ferland (1987). Only ions of O, Ne, and Ar are shown on this figure to limit the number of ov^l^ping curves. See figure 31 for other ions. 238 - 2 40 .0 0 0 K Predict«d Lines X m C N Il 122 10’ 10= 10= K) NH Figure 33. Intensities o f Observed Fine Structure Lines Relative to Bry as a Function of for a 40,000 K loTÛzing Stellar Spectrum. The same quantities are shown as for figure 29, but for a model using a 40,000 K ionizing stellar spectrum. m œNCLusioNS In this final section we briefly review the principal conclusions of each chuter. In chapter n we showed that three-body recombination to excited states is an important process for the ionization balance of all species. This process is important not only for high density regimes (We > 10^^ cm~^) where it had been expected previously for ordinary nebular tempaatuies (" 10,000 K), but also for low density regimes (W, = 10*-10* cm~^) at very low temperatures (500 K). In addition for recombination to very highly itmized species the process can be important at ordinary nebular temperatures and densities in excess of 10* cm~^. We compared our results to those of previous authors (Adams and Petrosian 1974; Humma and Storey 1987) and found that our general conclusions are borne out by the limited cases they treated, but that imcertainties as great as a factor of 2 in the published collision data results in comparable uncertainties in the total recombination rate when three-body recombination is appreciable. We expect these results to be most important for our undasfânding of cold nova shells and especially the broad line-emitting regions (BLRs) of active galactic nuclei (AGN), and would hope that they are included in future photoionization models of these objects. In chapter m we considered the treatment of the hydrogen atom in the context of photoionization models. We discussed the inaccuracy introduced into the calculated hydrogen spectrum when it is assumed that angular momentum changing collisions are fast relative to radiative de-excitations (i.e., when I-mixing is assumed); at 10,000 K and case B this assumption poduces a good approximation to the true hydrogen spectrum, but for case A at all temperatures and for case B at tempoatures under about 239 240 2^00 K /-mixing cannot be used. We also showed by comparison of published collision rates that the collision data are not known to better than a factor of 2. Finally we considered die effects of truncating the hydrogen atom at very low n-levels. This procedure neglects the effects of three-body and radiative recombinations to excited states which are often important. We proposed a genoal method by which an arbitrary number of n-levels can be reduced to 10 or so real plus fictitious levels which formally retain the accuracy of a more complete atom. We also presented a specific example where this was done (we reduced a 100-level atom to 10 levels), and showed by comparison to previous authors that our 10-level atom indeed retains the accuracy of a 100-level atom. We expect both the method and the specific example given to allow researches using photoionization models to make more accurate calculations of the hydrogen spectrum without changing eithe the structure of their photoionization codes or the computation time which their codes currently require. In chapter IV we considered the helium singlet recombination spectrum. It has been suggested previously (Osterbrock 1974) that case B for the helium singlets might not fiilly obtain because resonance photons would be destroyed in photoionizations of neutral hydrogen before they could scatter often enough to degrade into higher series photons. We calculated the effects of this process on the singlet emissivities and showed how the departures from case B increased with the ratio of neutral hydrogen to neutral helium. This ratio is in turn a function of the spectral shape of the ionizing star. Results are presented for all important optical and IR lines which are sensitive to the assumption of case A or B. Most of the lines show considerable sensitivity to the ionizing spectral shtpe, and we expect that our results should prove indispensible in analyzing observations of the singlet recombination spectrum in the future: For example, if these calculations are crxnbined with a grid of detailed, realistic photoirmization models of planetary nebulae and H n regions, it should be possible to use the singlet spectrum to measure 241 the temperature of the photoionizing star. In chapter V we continued our investigations of the effects of deviations from case B, focusing on the hydrogen recombination spectrum. In dus case, absoiption of resonance photons onto dust produced the most significant departures fnmi case B, although effects such as large scale velocity fields were also considered. We found that the onissivity decreases fipom 5% and 15% as the dust-to-gas ratio increases. This indicates that helium abundance measurements for dusty H n regions are systematically overestimated by this amount This effect is an important source of oror for measurements of the primordial helium abundance, and for measurements of the correlation between helium and metal abundances; the former constrains models of die big bang, while the latter constrains theories of post-big bang galactic chemical evolution. Both triplications require that helium abundances be determined to accuracies exceeding 5%. In the future, helium abundance detmninations for these purposes must be corrected for the effects o f dust, requiring the calculations o f this ctuqpter. Finally, in chtgita^ VI we considoed the nature of the ionizing continuum for Sgr A West, the ionized gas spiral at the center of our galaxy. By comparing the observed IR fine structure line spectrum to a grid of models we found that a soft ionizing spectrum like that of a 35,000 K star or group of stars provided the best fit An AGN-like spectrum was found to be unable to rqiroduce the observed high intensities of the low-i(Hiization lines. We concluded that if Sgr A West were to be classified according to the schemes used for extragalactic emission-line nuclei it would be considered an H II region or starburst nucleus. More generally, it can be concluded that a soft ionizing spectrum is necessary for Sgr A West; the spectrum need not necessarily be stellar. This result is important for our understanding of both our own galaxy’s nucleus and extragalactic nuclei. For Sgr A West we have identified the nature of the spectrum of the object or objects which maintain its ionization; For extragalactic nuclei we have demonstrated that 242 a neaiby example of an H n region nucleus exists, one for which we can achieve unparalleled spatial resolution. Our calculations of the fine structure line intensities are also the first that we know of for AGN; although our calculations are not completely general because we coupled the ionization parameter to the density through independent constraints on the distance from the ionizing source to the ionized gas, they provide a first look at the formatitm of these lines under conditions charactoistic of the narrow line- emitting regions of those nuclei. APPENDIX In this appendix we examine some consequences of collisions and radiative transfer on the helium abundance of the Orion Nebula. We use the data presented by Peimbert and Torres Peimbert (1977; FTP). They found that the abundance of singly itmized helium appeared to decrease with increasing distance from the central stars, and deduced an empirical ionization correction factor, in part based on the requirement that the helium abundance not vary with position. In this appendix we show that part of the change I was caused by collisional excitation of helium lines and the suppression of hydrogen lines, as discussed in this paper. Figure 34 shows the abundance of singly ionized helium in Orion. The top panel shows the apparent He*IH* ratio as a function of electron density as taken from PTP. The reddening-corrected f(A5876)//(//p) ratio was converted into an abundance of ionized helium using the tables in Osterbrock (1974) and assuming a temperature of 8500 X; these values are plotted as the +'s. The apparent ionic abundances were then corrected for collisional effects assuming the density given by PTP but a constant electron temperature of 8500K. This tonperature is suggested by radio data (see Mathis 1985) and consistent with optical observations (McCall 1979). The rate coefficient for excitation of X 5876 given by Berrington and Kingston (1987) was used. (The collision strength for the 2^5 - 3^2? collision is very close to the value proposed by Feldman and MacAlpine 1978 and is ~ 37% smaller than the previous value given in Berrington et al. 1985). The correction factor is given by ICKSmcou 7.7 /(X 5876),„ ( 1. + 3 3 9 0 /N,) ' 243 244 This correction is important because of the very high density in the Orion Nebula. These collision- corrected icHiic abundances are the lower of the values given for each density in the t(^ panel and are plotted as filled circles. The mean isN(,He*)/N{H'*') = 0.072 ± 0.009. We next correct the ionic helium abundance for the diminished hydrogen emissivity using the values of %c(Lya) given by eqn 5.12. The lower panel plots the collision-corrected data versus distance for the central stars as +'s. These were corrected for the effects of deviations fimm case B assuming x^ya = 10^ and the deviations given by equation 5.7. The f/^-emissivity corrected data were plotted as filled circles. The mean helium abundance is N{He*)/N(H*) = 0.Q67±0.007. The emissivity corrections appled here may be too large; it seems likely that the dust to gas ratio decreases in inner regions of the nebula (see Mathis et al. 1981), and absorption of radiation by dust and gas will increase the hydrogen neutral ffactim in outa regions. These considerations introduce uncertainties in %c(Ly a) at the factor of 2 -3 level, both in the sense of lowering it, but this additional uncatainty is no greater than that already present in the derivation of equation 5.12 A correction for neutral helium must be made to obtain the total helium abundance. We have nothing new to add to the extensive discussion on this subject (see FTP; Stasinska 1980; Mathis 1982; Mathis 1985). The analysis presented hae is only offered as an example of some of the systematic errors which can affect spatially resolved observations. A reliable helium abundance for the Orion Nebula must await measurements of density indicators which co-exist with He*; some examples are [Ne JV ], [Ar IV1, or the [0 ///] fine-structure lines (the latter are discussed by Simpson et al. 1986). Secondly, the hydrogen neutral Section, which determines XdLyai), must be determined by detailed modeling of the ionization structure of the Nebula, including the effects of dust absmption and scattering. Lastly, the question of the dust to gas ratio across the face of the nebula must be resolved. 245 0.10 1 1 ' "" 1 " "1. T----- r - T- 1 + — + + a • a He+ + 0.08 - H+ ► • • i + • + 0.06 a — 1 1 .. 1 1 1 1 1 - 1 : 3.0 3.2 3.4 3.6 3.8 log Ng (cm'®) i 1 1 i — + + + • - 0.08 — o + H e ^ A 4- + H+ + -# • + • - e a w + + + a 0.06 — a a +- © 1 1 1 1 1' 2' 3* 4' Figure 34. Ionized Helium in the Orion Nebula. The upper panel shows the apparent helium abundance, as measured by the reddening corrected intensity ratio A5876///p>, versus the electron density. The deduced density and the intensity data were taken form Peimbert and Torres-Peimbert (1977). The data plotted as + marks are the original data, while the filled circles have been corrected for collisional excitation using rate coefficients from Barrington et al. (1985). In the lower panel the collision- corrected ionic abundances are plotted as a function of the ^stance from the central stars, as + marks. The filled circles show the ratio N(He*) f N(,H*). 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