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Structure of disk dominated galaxies MacArthur, Lauren Anne 2001

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STRUCTURE OF DISK DOMINATED GALAXIES By Lauren Anne MacArthur B. Sc. University of Guelph, 1999 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF MASTER OF SCIENCE in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS AND ASTRONOMY We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA October 2001 © Lauren Anne MacArthur, 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, B .C . , Canada V6T 1Z1 Date: 0<A W , ZJOQX Abstract A robust analysis of the structural properties of late-type spiral galaxies, based on the modeling of bulge and disk brightnesses, is presented. The resulting structural parameters provide important constraints to predictions of structure formation models for the bulges and disks of spiral galaxies. Two independent programs were developed which exploit ID profiles and 2D images. These algorithms use a non-linear least-squared fitting technique to reduce the profiles and images into bulge and disk components described by smooth mathematical functions. The reliability of the decomposition algorithms was thoroughly tested with ~ 250,000 simulated profiles and images. The simulations include testing for sensitivity to initial estimates and seeing and sky measurement errors, and cover the full range of late-type disk and bulge sizes, as well as the bulge shape. These tests provide us with a set of guidelines for the reliability of our decompositions. Among others, we find that an additional grid search must be applied in order to recover the bulge shape parameter reliably. Another important conclusion from these simulations is that the 2D decomposition technique does not provide a significant advantage over the ID algorithm for recovering axi-symmetric galaxian parameters to warrant the extra computational time and effort. This is likely only true for the small bulges considered here. Hence, our results for the actual data reductions are based solely on the ID decomposition technique. Decomposition results are presented for the largest multi-band data set of late-type galaxy images to date. We use a sub-sample which includes a total of 523 images in the B V R H passbands for 123 nearby late-type spiral galaxies with face-on and intermediate inclinations. The galaxies are divided into three major types according to the shape of their luminosity profiles in the optical and infrared passbands; type variations are ultimately linked to the presence of dust. For each galaxy brightness profile a total set of 1080 decompositions were reduced and ii analyzed according to the selection criteria provided by the simulations. We have used repeat observations to test the reliability of our decompositions. The typical parameter errors are ~20% for the bulge and ~5% for the disk components. We also point out that some types of disk profiles, dubbed "Type II," cannot be properly described by a simple two-component (bulge + disk) model. A new approach which includes, but may not be limited to, the effects of dust extinction is required. The final set of galaxy parameters is studied for parameter variations and correla-tions both in the context of profile type differences and wavelength dependence. The bulge shape parameter shows a clear range but has a mean value corresponding to an exponential profile. The size ratios of bulges and disks are also tightly coupled with (re/h) = 0.23 ± 0.10. A natural interpretation of this remarkable observation is that the bulge formed via viscous transport of material from the disk, in agreement with current models of secular evolution. The distribution of disk scale lengths shows a clear decreas-ing trend as a function of wavelength. This is interpreted as being due to dust extinction (and probably to some degree by stellar population effects). Detailed modeling of colour gradients in these galaxies will be addressed in a subsequent analysis. i n Table of Contents Abstract ii Table of Contents iv List of Tables vii List of Figures viii Acknowledgements xii 1 Introduction 1 1.1 Galaxy Formation and Evolution 1 1.2 Observational Constraints 8 1.2.1 Surface Brightness Profiles 11 1.3 Bulge-to-Disk Decompositions 17 1.3.1 Marking the Disk Method 18 1.3.2 Moments Method 19 1.3.3 Simultaneous Bulge-to-Disk Decompositions 19 1.4 Thesis Outline 20 2 Data and Basic Reductions 22 2.1 Sample Selection 22 2.2 Observations and Basic Reductions 25 2.2.1 Sky Background 26 iv 2.2.2 Seeing 27 2.3 Profile Extraction 28 2.4 Surface Brightness Corrections 36 2.5 Multiple Observations 38 3 Simulations of Bulge-to-Disk Decompositions 39 3.1 Description of the ID and 2D Algorithms and Model Components . . . . 40 3.2 Reliability of the Decomposition Results 49 3.2.1 Simulated Profiles and Images 53 3.2.2 Initial Estimates 55 3.2.3 Seeing Effects 64 3.2.4 Sky Uncertainty Effects 72 3.2.5 Sersic n Tests 79 3.3 Summary of Simulations 87 4 Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 91 4.1 Outline 91 4.2 Initial Estimates 93 4.2.1 Seeing and Sky Treatment 95 4.3 Data Filtering 96 4.4 Preferred Sky and Seeing 101 4.5 Decomposition Examples 102 4.6 Effect of r m a x 103 4.7 Distribution of the Sersic n parameter 108 4.7.1 Floating Sersic n 108 4.8 Error of a Single Measurement 110 4.9 Comparison with Other Authors 114 v 5 Discussion 117 5.1 Inclination Dependence 118 5.2 Bulge/Disk Parameters 118 5.2.1 Test of Secular Evolution 130 5.3 Future work 133 Bibliography 135 A Tables 144 v i List of Tables 2.1 Inclination correction coefficients 38 4.1 Mean rms deviations for repeat observations 113 4.2 Parameter comparison between us and de Jong (1996) 115 A. l Total Sample 145 A.2 Decomposition Results 152 vu List of Figures 1.1 Preliminary results of simulations from Scannapieco & Tissera 2002 of the distribution of the Sersic n parameter 11 1.2 Examples of Type I and Type II profiles 12 1.3 Example of a "transition" profile 13 1.4 Simulation of dust extinction effects on galaxy profiles (Evans 1994, Fig. 2). 16 2.1 Histogram of galaxy ellipticities 24 2.2 Histogram of galaxy heliocentric radial velocities 25 2.3 Histograms of sky measurements 27 2.4 Histograms of seeing F W H M measurements 29 2.5 Comparison of de Jong and Courteau & Holtzman profiles: U G C 463 and U G C 3080 31 2.6 Comparison of de Jong and Courteau & Holtzman profiles: U G C 3140. . 32 2.7 Comparison of major axis cuts (red circles) and azimuthally-averaged pro-files (blue squares) 34 2.8 Error curve from observations 35 3.1 Sersic n profiles for different values of n 44 3.2 Comparison between exact bn and several approximations 47 3.3 Results from fitting an incorrect Sersic n bulge: h 56 3.4 Results from fitting an incorrect Sersic n bulge: re 57 3.5 ID results from re initial estimates tests: re 59 3.6 ID results from re initial estimates tests: /xe 60 viii 3.7 2D results from re initial estimates tests: re 61 3.8 Results from fie initial estimates tests: re 62 3.9 Results from /ze initial estimates tests: fie 63 3.10 ID results from re initial estimates tests: n = 0.2 65 3.11 ID results from re initial estimates tests: n = 4.0 66 3.12 Are versus re from seeing tests: ID 68 3.13 A/x e versus re from seeing tests: ID 69 3.14 Ah versus re from seeing tests: ID. . 70 3.15 Afx0 versus re from seeing tests: ID 71 3.16 Are versus re from seeing tests: 2D 73 3.17 Effects of a 1% sky subtraction error 74 3.18 Results from sky subtraction tests: error on fitted h 75 3.19 Results from sky subtraction tests: error on fitted fi0 76 3.20 Results from sky subtraction tests: error on fitted re 77 3.21 Results from sky subtraction tests: error on fitted fie 78 3.22 Results from ID Sersic n tests: initial estimate n = 0.4 81 3.23 Results from ID Sersic n tests: initial estimate n = 1 82 3.24 Results from ID Sersic n tests: initial estimate n = 2 83 3.25 Results from ID Sersic n tests: initial estimate n = 4 84 3.26 Results from 2D Sersic n tests 85 3.27 Results from Sersic n tests: seeing errors 88 3.28 Results from Sersic n tests: sky errors 89 4.1 Examples of xlnner1 a n d xliobai' versus Sersic n distributions: well- and poorly-behaved solutions 99 ix 4.2 Examples of Xinnerf a n d xliobal* distributions: two different behaviors of X2gi for profiles with very well-behaved Xin 100 4.3 Comparison of different bulge fits for same profile (UGC 784 B-band). . . 101 4.4 Histograms of seeing F W H M and sky offsets 102 4.5 Decomposition examples: Type 1 104 4.6 Decomposition examples: Type II/Trans 105 4.7 Decomposition examples: truncated disk 106 4.8 Decomposition examples: bulgeless disk 107 4.9 Comparison of model parameters for two different baselines: 0.75 x r m a x versus r m a x 109 4.10 Histograms of Sersic n parameter: all fits and good fits only 110 4.11 Histograms of Sersic n parameter: n free parameter I l l 4.12 Comparison of parameters: floated n versus fixed n 112 4.13 Sersic n versus morphology 116 5.1 Sersic n versus ellipticity (1 — b/a) 119 5.2 Bulge and disk parameters versus ellipticity (1 — b/a) 120 5.3 Histograms of Sersic n bulge shape parameter 122 5.4 Histograms of Sersic bulge \ie parameter 123 5.5 Histograms of Sersic bulge rx (kpc) 124 5.6 Histograms of disk /x0 125 5.7 Histograms of disk hx (kpc) 126 5.8 Histograms of hB/hx , 127 5.9 Histograms of rx/hx 129 5.10 Histograms of rx/hx: n = l only 131 5.11 Distribution oire/h with Hubble types 132 x 5.12 S L O A N Profile 134 xi Acknowledgements First and foremost I would like to extend my greatest thanks to my advisor, Stephane Courteau. His tireless enthusiasm and encouragement provided me with courage and inspiration to explore my own ideas. I am forever grateful for the time and effort he gave selflessly to me over the last two years. A great deal of appreciation goes out to Adrick Broeils for providing his C version of the ID decomposition program, saving me a lot of programming headaches. Thanks also to Jon Holtzman and Stephane for providing me with an outstanding database, without which this project would not have been feasible. Jon, Stephane and Suresh Sivanandam devoted much effort to the development and testing of a 2D decomposition method for comparison with the ID method. Many thanks to Roelof de Jong for his inspirational and pioneering PhD thesis which provided much incentive for this project. A special thanks to Paul Hickson and Tony Noble for their careful readings and helpful suggestions. And finally, my most heartfelt appreciation and gratitude goes out to all my friends and family for their constant love and support. xii Chapter 1 Introduction 1.1 Galaxy Formation and Evolution How did the universe come to have its present form? How were its constituents formed? It has long been a puzzle how the early, homogeneous universe could have evolved into the clumpy universe we live in today. This puzzle has been compounded by a general lack of empirical information on the structure of the galaxies, both in the nearby and distant universe. However, all of this is changing. There have been dramatic new discoveries from microwave background measurements which have enabled scientists to detect a small, but finite, anisotropy indicative of tiny fluctuations in the density of the early universe. These asymmetries led to gravitational centers, seeding the accumulation of matter which would eventually account for all of the large scale structure we see in the universe today. There has also been great progress in understanding the matter and energy distribution in the universe. Information from the rotation curves of galaxies, supernovae, and meas-urements of the curvature of the universe have led us to a remarkably consistent picture. However, in this picture, only about 5% of the universe can be accounted for by ordinary baryonic matter— the protons, neutrons, atoms and molecules that make up our world. A further 25% is made up of exotic particles we have yet to identify, and the remaining 70% is a mysterious dark energy, also not understood! Clearly the formation and evolution of galaxies will be affected by the presence of 1 Chapter 1. Introduction 2 a substantial dark matter component. With the advent of new data, a greater under-standing of the matter/energy density of the universe, and the initial conditions, it is imperative that we better understand the structure of the galaxies we see today. This will allow an accurate comparison with older galaxies to expose evidence for evolution-ary changes. It can also be expected that a galaxy's appearance is a reflection of the underlying physics, and a better understanding of the galaxies structure will lead to an understanding of its formation and evolution. In this thesis, we develop techniques to characterize the distribution of structural parameters in spiral galaxies, with an eventual aim of being able to understand their formation and evolution. With ever-improving observational technologies, ambitious new surveys, and very large data bases becoming available (SDSS, 2MASS, 2dF) there is great potential for major advances in this field. In order to gain an appreciation for the outstanding questions regarding the formation of galaxies and their evolution into the structures we see today, we present here a brief historical account of the field. The foundation for theories of structure formation in the universe dates back to the late 1920s with the first studies of perturbations in a dust-filled Friedmann-Lemaitre universe (de Sitter 1930, Lemaitre 1931). The general picture involves the amplification, due to gravitational instability, of small inhomogeneities in an otherwise homogeneous distribution of matter. But a full treatment of linear perturbation theory (Peebles 1980, Bardeen 1980), has lead us back to the starting point: What are the initial amplitude and spectrum of the density fluctuations necessary to seed galaxy formation? By the late 70's, Alan Guth had begun developing his inflationary universe theory which provided theoretical predictions for the origin and spectrum of primordial density fluctuations. Inflation does not, however, offer a prediction for the amplitude of the fluctuations. An empirical measurement of the amplitude and spectrum of the primordial density fluctuations was provided by Smoot et al. (1992) based on microwave data from Chapter 1. Introduction 3 the Cosmic Background Explorer satellite (COBE). C O B E measured the temperature anisotropics in the cosmic background radiation to be (AT/T)rma « 1.1±0.2 x 10~5 which provided the necessary normalization to the structure formation models. Additionally, the spectrum measured by C O B E provides an excellent match to the scale-invariant spectrum of density perturbations predicted by inflation 1 . Equipped with a plausible inflationary theory and powerful computers, cosmologists in the 1990s were then in a much better position to study the evolution of the density fluctuations of the early universe. Cosmological N-body simulations, in which the matter distribution is described as a distribution of N particles that interact via gravity, provided a direct method of following the (non-)linear evolution of the density perturbations. However, given that the matter density of the universe is dominated by non-baryonic (dark) matter, whose physical nature and composition remains elusive, the simulations must adopt a basic framework providing a prescription for the dark matter particles. The two most extensively explored theories are referred to as the hot and cold dark matter models ( H D M and C D M , respectively). In H D M models, the dark matter particles are light, fast moving (relativistic), and interact only via the weak force. A primary motivation for the H D M models is the existence of a known candidate; the neutrino. C D M particles on the other hand are collisionless and have negligible random velocities. No known candidates for the C D M particle have been detected to date but a number of search efforts are under way (Primack 2000). Both the H D M and C D M theories have encountered a number of difficulties, but only the C D M models have stood up to most of the observational constraints available to date. Major weaknesses of the H D M models result from their lack of power on small scales. 1 While the inflationary theory still remains somewhat speculative, and a myriad of alternative theories have been suggested, the remarkable agreement between the COBE measurements and its predicted spectrum along with its solutions to the long standing flatness, horizon, and monopole problems, make the inflationary universe theory difficult to dismiss. Chapter 1. Introduction 4 Accordingly, structure on the scale of galaxies can only form after the collapse of cluster-sized objects, in a "top-down" scenario, by means of fragmentation. Simulations of H D M universes with reasonable values for z / o r m , the redshift at which galaxies begin to collapse, cannot be reconciled with the current measurements of the galaxy correlation function and observed clustering. Based on pure H D M models, galaxy formation should occur on timescales too close to the present time. Additionally, the C O B E measurements of anisotropy in the C B R rule out D M models in which free-streaming wipes out small scale power, which is a characteristic of H D M particles. In contrast, C D M models have the largest amplitudes on smallest scales which results in a hierarchical, "bottom-up", process of structure formation: large structures form out of smaller ones which formed first. C D M models have also faced their share of inconsistencies when comparing predictions with observations, but their overall success at reproducing many aspects of the large scale structure of the universe make them the preferred choice for cosmologists at the present time. In particular, C D M model predictions of the clustering of gas and galaxies as a function of redshift have been tested and verified observationally.2 Much attention has also been given to combined cold and hot dark matter (CHDM) universes, and comparison to the observed universe suggests a relatively small fraction of H D M particles (Primack 2000). Attempts have been made to extend the above models to the formation of individual galaxies, taking their results as initial conditions. Formation models begin with a "proto-galaxy" in the form of an isolated, uniformly rotating, sphere composed of primordial gas and dark matter. The mass and angular momentum distribution of these spheres are 2 As of this writing, the confrontation of CDM halo density profile shapes at the center of galaxies based on high resolution rotation curves remains one of the main challenges for CDM models (de Blok et al. 2001). Chapter 1. Introduction 5 taken from the above N-body simulations. The initial angular momentum of each proto-galactic sphere is acquired through tidal torquing from neighboring systems (Peebles 1969, Fall & Efstathiou 1980). The assumption of uniform rotation is based on an early observation that the angular momentum distribution of the galactic disk closely resem-bles that of a sphere in solid-body rotation (Mestel 1963). Another assumption adopted in these models is that angular momentum transport during the collapse is negligible. This assumption may hold for dissipationless, dark matter dominated systems, which is true of low surface brightness systems (LSBs) (Dalcanton et al. 1997), but in systems that are dominated by baryons, viscous dissipation must certainly play a major role in the evolution of the galaxy. Nevertheless, these simple models are able to reproduce many of the universal properties of low and high surface brightness disk galaxies; exponential disk profiles with a wide range of scale lengths, asymptotically flat rotation curves, and the Tully-Fisher relation between circular velocity and luminosity of spiral galaxies, but they do not simultaneously reproduce the observed local luminosity function. In models of disk galaxy formation which do incorporate the viscous dissipation of gas, an exponential distribution of the stars in the galactic disks naturally emerges as long as the character-istic timescale for star formation is equal to the viscous timescale (Lin &; Pringle 1987, Ferguson & Clarke 2001, and references therein, Slyz et al. 2001), other timescale com-binations would lead to truncated or power-law profiles. When gravitational instabilities are considered in models which incorporate viscous dissipation, galaxies often develop a bulge-like component whose properties also depend on the total angular momentum of the system (Saio & Yoshii 1990, Struck-Marcell 1991). However, the modeling of galaxies has been hampered by our limited knowledge of processes involving angular momentum transfer between the halo and disk, and star form-ation and feedback mechanisms in a multi-phase interstellar medium. Lacking detailed prescriptions for these fundamental astrophysical processes, it is difficult to develop a Chapter 1. Introduction 6 self-consistent model for the dynamical and chemical evolution of individual galaxies. Another class of models, referred to as semi-analytical (SAM) models, have attempted to incorporate analytical parameterizations of both well-constrained and elusive astrophys-ical processes into numerical simulations (Somerville &; Primack 1999, Cole et al. 2000). These require a number of adjustable parameters, which are fixed by reference to a set of local galaxy data. Based on the properties of present-day galaxies we can attempt to roll back the clock and predict how galactic structures may have appeared in the past. SAMs, however, have had limited success at reproducing observations that were not set by the tuning of parameters, and at making new hard predictions. Also of crucial importance to the study of the formation and evolution of disk galaxies is a scenario for the formation of the bulge. Such models are highly uncertain at present. Bulges are considered either very old and having formed before the disk, or the sum of an old and intermediate age population that would likely have formed via secular processes or minor mergers. We still know very little about stellar populations in disk galaxies owing mostly to contamination effects (bulge/disk, arm/interarm) of images, dust, low surface brightnesses, small velocity dispersion gradients and sizes. The classic view that the halo and the bulge are part of the same Galactic system predicts strong population differences between spheroids and disks. Measured colour gradients (Peletier et al. 1999) and M g 2 gradients (Halliday et al. 1996) suggest, on the contrary, that there is little or no colour or spectral change between the bulge and disk, at least for early-type spirals. It is of crucial importance for galaxy formation models to discriminate between competing bulge formation scenarios (in early and late-type spirals): (a) early monolithic collapse (Eggen, Lynden-Bell & Sandage 1962); (b) secular evolution from disk instabilities; (c) accretion from the halo after disk formation; (d) merger or tidal interactions (Courteau et al. 1996, Wyse et al. 1997, Noguchi 1998). What processes regulate these scenarios in producing the Hubble sequence and luminosity functions for near and distant galaxies Chapter 1. Introduction 7 is unclear at present. Our best models are ill-constrained due to the paucity of data for early and - especially - late-type bulges (Kauffmann 1996, Zhang & Wyse 2000). The processes that regulate these bulge formation scenarios are quite different and produce distinct signatures with respect to structural relationships between the bulge and disk components, star formation rates, and mixing between the bulge and disk parameters. For example, if bulges formed via monolithic collapse at high redshift, the bulges we see today would be much older than their corresponding disks. Additionally, the subsequently forming disk would not have a strong influence on the bulge component. Accordingly, no structural correlation would be expected between the bulge and disk components. Alternatively, in the secular evolution scenario, there is a strong coupling between the evolution of the bulge and disk components (Courteau 1996). In models of viscous transport, the efficiency of transporting disk material into the central regions will be enhanced by a bar or oval distortion, which can be triggered by a global dynamical instability in the disk induced by interaction with a satellite (Martinet 1995). Disk material will be heated vertically up to 1-2 kpc above the plane via resonant scattering of stellar orbits by the bar-forming instability. A "bulge-like" component with a nearly exponential profile will emerge due to relaxation induced by the bar. The properties of the disk's central regions are directly coupled to the relative time-scales of star formation and angular momentum transfer. Such a model is expected to produce correlated scale lengths and colours between the disk and its central regions. Gas redistribution by the bar can cause its own dissolution. Secular accumulation or satellite accretion of only 1-3% of the total stellar disk mass near the center is sufficient to induce dissolution of the bar into a lens or triaxial component and later into a spheroid (Kormendy 1982, Pfenniger & Norman 1990, Pfenniger 1993, Friedli et al. 1994, Martinet 1995, Norman, Sellwood, & Hasan 1996). It is estimated that about two-thirds of disk galaxies currently have a bar, especially as revealed in the infrared (Zaritzky, Rieke, & Chapter 1. Introduction 8 Rix 1993, Sellwood &; Wilkinson 1993, Martin 1995) and that most spirals have probably harbored a self-destructive bar at one time or another during their evolution (Friedli &; Benz 1993, Sellwood & Moore 1999), thus lending some support for bulge formation models of secular evolution. Further evidence for secular evolution in late-type spiral galaxies includes the continuation of spiral structure into center of galaxies and structural and photometric correlations between bulge and disk parameters (Courteau 1996). Secular evolution is a viable mechanism for producing the small, central accumula-tions of material in late-type disks. Bigger bulges, however, could not be formed this way without disrupting the disk. The energy required to heat up the central material is far greater than the total bar and disk's mechanical energy supply. Accretion of a small satellite to explain the bigger bulges of SO-Sa's provides an appealing alternative (Pfenniger 1993, Walker, Mihos & Hernquist 1996). It is clear that major developments in the theories of the formation and evolution of galaxies have been linked predominantly to major advances in observational technologies. There are a number of ambitious observational programs underway, in particular with the goal of probing to increasingly higher redshifts. However, given the tremendous difficulties in relating the unresolved high redshift objects to their present day analogues, it is imperative that we fully explore and understand the current state of these structures. This is a primary concern of the current study. 1.2 Observational Constraints Important constraints for any bulge and disk formation model come from stellar density distributions and kinematics. This thesis is concerned primarily with the former (velocity information for late-type spirals is scanty and inconclusive at present.) In modeling the stellar density distributions in spiral galaxies, one has often assumed an r 1 / 4 brightness Chapter 1. Introduction 9 law for the central regions (de Vaucouleurs 1948) and an exponential surface density for the outer regions of the disk (de Vaucouleurs 1959, Freeman 1970). However, departures from the standard de Vaucouleurs profile in the central light distribution of early and late-type systems have been demonstrated in a number of studies (de Vaucouleurs 1959, van Houten 1961, Frankston & Schild 1976, Kormendy & Bruzual 1978, Burstein 1979, Shaw & Gilmore 1989, Andredakis & Sanders 1994). For example, Kent, Dame & Fazio (1991) used the Space Shuttle Infrared Telescope to show that the Milky Way bulge is best described by an exponential luminosity profile with a scale length of 500 pc. Sersic (1968) proposed that the bulge is more appropriately characterized with a generalized law of the form E(r) = X0exp{-(r/r0)1/n} (1.1) where So is the central surface brightness (CSB), ro a scaling radius, and n is the shape parameter. Many recent studies have confirmed the inadequacy of the de Vaucouleurs profile (n = 4 in Eq. 1.1), for most bulges, but suggest instead a wide range of Sersic ras. For example, Caon, Capaccioli, &; D'Onofrio (1993) used their data on ellipsoids (E/SOs) and the low surface brightness (LSB) dwarf galaxies of Davies et al. (1988) to show that the parameter n correlates with absolute luminosity and half-light radius, such that bigger, brighter systems have larger values of n. This result was extended to brightest cluster galaxies by Graham et al. (1996). Andredakis, Peletier & Balcells (1995), de Jong (1996), Courteau, de Jong & Broeils (1996), and Carollo (1999) have used high-quality surface brightness (SB) profiles to show that central regions of disks are generally best described by an exponential luminosity profile. These authors concluded that most late-type spirals are best fitted by the double-exponential fit. This was interpreted by Courteau, de Jong &; Broeils (1996) as evidence for secular evolution of the bulge. Sa and Sab's are also generally best modeled with a Chapter 1. Introduction 10 n = 2 bulge. Central regions of earlier-type galaxies and a small fraction (~ 15%) of the later types are more appropriately fit by a de Vaucouleurs law. These results should serve to firmly establish the notion that central regions of late-type spirals are best described by nearly exponential profiles. A discussion of the shape of late-type bulges is also given in Graham (2001). Courteau, de Jong & Broeils (1996) also demonstrated a tight correlation between the bulge and disk exponential scale lengths, with hb/hd = 0.1 ± 0.05 (where h = r0 and n = 1 in Eq. 1.1). Taken at face value, a correlation of bulge-to-disk (B/D) scale lengths is best understood in a model where the disk forms first and the "bulge" that naturally emerges is closely related to the disk, such as expected via secular evolution. Self-consistent numerical simulations of secular evolution in disk galaxies evolve toward a double-exponential profile with a typical ratio between bulge and disk scale lengths near 0.1 (Friedli 1995, private communication), in excellent aggreement with observations. More recent simulations (Saiz et al. 2001, Scannapieco & Tissera 2002) show somewhat broader variations of the shape parameter n for collapsed objects at the present time. Fig. 1.1 shows preliminary results from the simulations of Scannapieco &; Tissera 2002. According to those simulations, the double-exponential structure of bulge and disk is not always the final relaxed state of an object, but whenever n = 1, the B / D scale length ratio takes its "nominal" value of 0.1. The analysis presented here is primarily concerned with the development of a set of observables that will serve as reliable constraints to predictions of structure formation models, such as those of Scannapieco & Tissera (2002). Specifically, we aim to character-ize and quantify the intrinsic structural properties of the bulge and disk and the extent of their variations. These characterizations are made through reliable modeling of bulge and disk parameters via surface brightness profile decompositions. Multi-waveband informa-tion also provides information about structural variations within galaxies due to dust and Chapter 1. Introduction 11 10 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I t l 0<z<2 I I I I I I I I I I I I I I I I I I ll I I I I I II I I I I I I I I I I I I I I I I II I I I I I I II I I I I I I I I I I I I I I I I I I I I I I I I I I I I I It z=0 11 I I I I I i i i i i i i i i i 4 0 2 n Figure 1.1: Preliminary results of simulations from Scannapieco & Tissera 2002 of the distribution of the Sersic n parameter. The left panel shows the predicted distribution over a redshift range 0 < z < 2 and the right panel shows the local distribution. stellar population effects. While some of these issues have been addressed before, there remains a number of significant measurement subtleties and interpretations that have not been thouroughly investigated. We shall describe the most commonly used methods and their shortcomings below, but first, a description of the nature of disk galaxy profiles is warranted. 1.2.1 Surface Brightness Profiles As discussed above, the traditional description of the radial surface brightness profile of spiral galaxies is the sum of two components: a de Vaucouleurs (n = 4 in Eq. 1.1) or exponential bulge and an exponential disk. Freeman (1970) pointed out that there are many cases where this description is not quite accurate. He distinguished two types of luminosity profiles: Type I for which the surface brightness at small radii always lies above the inward extrapolated disk brightness, and Type II for profiles whose surface Chapter 1. Introduction 12 111111111111111111111 i — ' — ' — ' — ' — i — i — i — 1 1 1 1 • i — i \ . . . . \ . . . . \ . . . . i i , . , , 11 0 6 10 16 20 25 30 36 40 45 60 0 6 10 16 20 26 30 36 40 Radius (") Radius (") Figure 1.2: Examples of Type I (left) and Type II (right) profiles. The solid black lines plotted on the B-band (blue circles) and H-band profiles (purple asterisks) are fits to the exponential disk profile. Note that in the Type I case the inner profile always lies above the surface brightness of the extrapolated inner disk, whereas in the Type II case, a portion of the profile lies beneath the inward disk extrapolation. brightness level falls below that of the extrapolated disk in an interval which encloses the bulge-disk transition. Fig. 1.2 shows examples of Type I and Type II profiles. The solid black lines plotted on the B-band (blue circles) and H-band (purple asterisks) profiles are fits to the exponential disks. In addition to Freeman's two profile types we observe a third "transition" type for galaxies whose profiles change from Type II at optical wavelengths, to Type I in the infrared. Freeman could not have made this observation since infrared photometry was not available in the 70s. A transition of this sort would be expected if the Type II dip is caused by dust extinction. A n example of a "transition" galaxy is shown in Fig. 1.3 and uses the same notation as Fig. 1.2. Many galaxies classified as Type II show a weakening of the inner profile dip at longer wavelengths and, in this sense there is no clear distinction between the Type II and Transition galaxies. Transition galaxies are likely just a case of lesser dust content, whereas Type II systems remain optically thick, even at H-band. Chapter 1. Introduction 13 CM o 0) VS IB O u a M E 20 c aa 0) o 10 b 3 EQ j i r r j T i T » j r f i r | | i i i i i i i _ ** Transition * H band - \ (UGC 10465) • R band -• V band • B band -** , , . , i , . . . i . , , . i , . . . i . . . i , . . . i . . . . i . . . T i . 0 & 10 IS 20 2S 30 35 40 Radius (") Figure 1.3: Example of a "transition" profile. The solid black lines plotted on the B-band (blue circles) and H-band (purple asterisks) are fits to the exponential disk profiles. Note how the galaxy profiles change from Type II at optical wavelengths to Type I in the infrared (see Fig. 1.2). Dust in Galaxies and Importance of Multiple Passband Information Dust extinction is the most likely explanation for Type II profiles (Evans 1994), although population effects at the bulge/disk transition could also be relevant (e.g., Prieto et al. 1992). Freeman (1970) argued that Type II profiles cannot be due to internal absorption as 8 out of 17 SO galaxies in his sample had Type II profiles, and SO galaxies were assumed to be dust-free. We now know, however, that SOs can harbour significant amounts of dust (e.g., Michard 1998). Over half of our sample is composed of Type II galaxies, none of which show bar-like morphology, but Type II profiles are also a common characteristic of barred galaxies (e.g., Weiner et al. 2001). The effects of recurring or dissolved bars could also be invoked as dust may have been generated and re-distributed by the action of a pre-existing bar. Models to constrain these scenarios are non-trivial and rely on assumptions about the mixing and distribution of stars and dust. Chapter 1. Introduction 14 Dust absorbs and scatters blue light and re-emits it at longer, far-IR wavelengths. The emission wavelength depends on the size and temperature of the dust grains. The overall effect of dust on the observed light distribution in a galaxy is strongly dependent on the dust geometry. In the simplest case of a homogeneous screen of dust, the observed galaxy is dimmed and reddened in the UV-to near-IR wavebands. However, for more complicated geometries where there is mixing between the stellar and dust components, the situation becomes much more complex as differential optical depths come into play. A comprehensive review of the current status of dust effects in galaxies is given by Calzetti (2001). A detailed picture of opacity and radiation transfer in galaxies is clearly not available and multi-wavelength datasets, scanty at present, are of crucial importance for such stud-ies. The near-IR wavebands, the H-band in particular, are largely free of dust extinction effects. Multi-wavelength datasets including optical and near-IR observations, such as the one used here, provide a possible indicator of the dust content and distribution in galaxies. However, great care must be taken in the interpretation of multi-wavelength information as dust extinction and scattering, stellar population mix, age, and metal-licity effects can all account for some of the differences seen in the light distributions of galaxies at different passbands. Studies using multi-wavelength data that solve for the intrinsic colour of stellar populations and reddening and extinction effects from dust simultaneously have measured typical central optical depths Ty(0) in the range 0.3-2.5 for types Sab-Sc (Peletier et al. 1995, Kuchinski et al. 1998, Xilouric et al. 1999), while lower values of T y ~ 0.15 have been observed for low surface brightness late-type galaxies (Matthews & Wood 2001). In the context of homogeneous stellar populations, galaxies with a non-negligible opacity, should display scale length variations at different wavelengths. Evans (1994) simulated the effect of dust absorption on galaxy profiles with exponential disks and Chapter 1. Introduction 15 de Vaucouleurs bulges. His simulations assume that the intrinsic disk scale length is the same at all wavelengths, which implies that there is no intrinsic colour gradient due to stellar populations or metallicity. The adopted geometry, based on the "triple exponential" model of Disney et al. (1989), is such that the stars and dust are well mixed, fall off exponentially in both the radial and z directions with the same radial scale lengths, but their vertical scale heights are allowed to be different. Accordingly, Evans defines a layering parameter ( = {3*/f3d which is the ratio of the stellar scale height (/3») to that of the dust (/3d). The B-band central face-on optical depth is defined as TB(0) = (3d/A, where A is the mean free path of a photon. The extinction law of Rieke & Lebofsky (1985) is used to convert the B-band optical depth to other wavebands. Fig. 1.4 shows examples of the effects of dust extinction on the B and H-band profiles of Sbc-like galaxies (with bulge-to-total luminosity (B /T) ratio of 0.15, note that Evans uses a de Vaucouleurs profile for the bulge which is not a good description of late-type galaxy bulges). In Figs. 1.4 (a) and (c) the central optical depth TB(0) is held fixed at 5 while the layering parameter is varied from £ = 0.0, corresponding to an infinitely thin layer of dust in the midplane of the disk, to ( = 1.0 where the dust and stars are uniformly mixed and have equal vertical scale heights. In Figs. 14(b) and (d) £ is held constant at 0.5 while TB(0) is varied from 0 (no absorption, i.e. optically thin) to 20 (optically thick). Note the Type II-like dips that develop in the profiles as a result of dust extinction and the observed variations from B to H , as we see in Transition galaxies. Evans (1994) pointed out that the intrinsic variation of disk scale lengths in galaxies at different wavelengths predicted by his dust models was smaller than the experimental error of the data sets available to him at the time. The data quality has improved and careful homogeneous studies like this one should significantly reduce the internal uncertainty. It is our goal to develop a new set of reliable galaxy structural parameters for the purpose of constraining the dust/stellar geometry in the bulge/disk transition Chapter 1. Introduction 16 Figure 1.4: Examples of B and H-band model profiles for Sbc-like galaxies (with B / T ratio 0.15) for different amounts and distributions of dust. The y-axes surface brightnesses are in mag arcsec - 2 and the x-axes scale lengths are in units of r/hd where hd is the exponential disk scale length. (Evans 1994, Fig. 2) Chapter 1. Introduction 17 region (Type I versus Type II) and the outer disk. We also plan to extend and improve upon these models in a future study. First, we need a data base. Another test for the opacity in disk galaxies, applicable in principle to our data set, is the measurement of a surface brightness dependence on inclination for a large sample of galaxies, also known as the "Holmberg test" (Holmberg 1958). Burstein et al. (1991) pointed out that inclination dependence tests such as these suffer from selection biases which can conspire to mimic the condition for transparent galaxies in the case of magnitude-limited samples, or the condition for opaque disks in apparent-diameter-limited catalogs. These biases may be overcome by combining samples with a variety of selection criteria. Such studies (Huizinga & van Albada 1992, Peletier & Willner 1992, Jones et al. 1996, Moriondo et al. 1998) have shown late-type spirals (mostly Sbc and Sc) to be opaque in their central regions but almost completely transparent in their outer disks, beyond about 2-3 scale lengths, and Giovanelli et al. (1995) observe a trend for more luminous galaxies to be more opaque than their fainter counterparts, though this latter result has been challenged by Willick et al. (1999). Bright disk galaxies with M j < — 21 have edge-on to face on magnitude corrections of A m / ~ 1 mag, corresponding to a face-on central optical depth of Ty(0) > 5 (Peletier & Willner 1992) (compare with the optical depths used by Evans in Fig. 1.4 above.) Our forthcoming study of colour gradients and dust content in spiral galaxies will address some of these issues. 1.3 Bulge-to-Disk Decompositions The literature is rich with measurements of structural parameters that describe the shape and scale of the bulge and/or disk components of spiral galaxies. The basic idea is quite simple: given an exponential disk profile and some profile for the bulge (e.g., the tradi-tional de Vaucouleurs r 1 / 4 law, pure exponential, Sersic law), compute the corresponding Chapter 1. Introduction 18 scale parameters for an individual galaxy. In practice, however, the determination of these parameters is fraught with difficulties. For example, sky subtraction errors and seeing effects can have a serious impact on measured surface brightnesses and must be properly treated in the decompositions. Exactly how the disk scale length is measured is also somewhat unclear, as no universal operational definition exists. Reported disk scale lengths often show discrepancies among different authors in excess of 20% (Knapen and van den Kruit 1991). Even harder to characterize are the small bulges. The different methods most commonly discussed in the literature are described below in order to highlight their respective strengths and weaknesses. 1.3.1 Marking the Disk Method The most simplistic method for determining a disk galaxy scale length is referred to as the "marking-the-disk method". Plotted on a logarithmic (magnitude) scale, the linear portion of a luminosity profile is "marked" and the selected range is fit using standard least squares techniques to determine its slope. While the resulting fits look acceptable to the human eye, this method has proved to be unreliable since the fitted slope (inverse scale length) is very sensitive to the adopted baseline. The inner fit boundary should extend beyond the bulge region or the disk parameters will be biased low (steeper slope, lower central surface intensity). The outer boundary is also crucial as it may reach into regions where the disk is truncated or where spiral features spoil the underlying disk profile. This could lead to both over- and under-estimates of the disk parameters. A truly objective and automated baseline selection may never be fully spelled out as it depends on such variables as bandpass, signal-to-noise, resolution, observing conditions, and galaxy properties (e.g., whether the disk is truncated, Type I, Type II, etc.) Chapter 1. Introduction 19 1.3.2 Moments Method Willick (1999) made an attempt at determining disk parameters meant to be objective and robust against irregularities in galaxy profiles and profiles which are not accurately exponential, while still recovering the exponential parameters for galaxies that are in fact truly exponential. The procedure involves computation of the statistical moments of the intensity distribution, where the nth moment is defined by /„= r rnI(r)dr (1.2) Jo Setting I(r) to be that of a pure exponential profile, Willick derived a transcendental equation for the scale parameter based on the 1 s t and 3 r d moments of the intensity distribution, the central surface brightness is then determined from the scale parameter and the 2 n d moment. Willick's computation includes the entire profile, alleviating the endpoint selection issues of the marking-the-disk method, but it still yields scale lengths that are biased low due to contamination from the bulge. This led Willick (1999) to claim a surface brightness dependence of the Tuny-Fisher relation (contrary to conventional wisdom as most modern Tully-Fisher residual analyses confirm the absence of surface brightness residuals (e.g., Courteau & Rix 1999).) Other published claims ("Freeman's law" (Freeman 1970), bimodality of galaxian central surface brightness (Tulfy &: Verheijen 1997)) have resulted, in part or in full, from erroneous disk scale length measurements. Reliable determination of structural parameters requires a full B / D profile separation. 1.3.3 Simultaneous Bulge-to-Disk Decompositions A major improvement in the determination of disk parameters was the development of ID and 2D simultaneous decompositions of the bulge and disk parameters (Kormendy 1977, Schombert & Bothun 1987, Andredakis & Sanders 1994, Byun & Freeman 1995, Chapter 1. Introduction 20 de Jong 1996). The results and short-comings of these studies will be discussed in § 4, but for now we highlight the pros and cons between the ID and 2D techniques. ID decompositions use major-axis cuts or azimuthally-averaged major-axis profiles from isophotal fits to the galaxy image. Only the mean radial surface brightness is pre-served. The decomposition of ID profiles thus depends on fewer data points than its 2D analog. The ID approach is also more robust to variations of the initial fit estimates. Use of isophotal fits to extract ID surface brightness profiles also provides information about boxiness/triaxiality of the dynamical system, otherwise unavailable. A disadvantage of the ID approach is its limitation to axisymmetric models. The ability to model and sub-tract out non-axisymmetric features, such as bars, rings, or spiral arms is a significant advantage (if needed) of the 2D approach. Computations for 2D decompositions can however be prohibitive due to the inclusion of millions of pixels at each operation. In general, parametric decompositions are also limited by the arbitrariness in the choice of the fitting function(s). Non-parametric approaches, to describe unambiguously the structure of galaxies, such as concentration and asymmetry indices, Fourier modes, colour gradients, axial ratios, magnitudes, will be tested and reported elsewhere. 1.4 Thesis Outline Ultimately, we wish to accurately describe the distribution of structural parameters in late-type spiral galaxies as a key to understanding their formation and evolution. As a first, and crucial step, this thesis provides the necessary data for this endeavor including: • an accurate characterization of the reliability of B / D decompositions resulting in robust measurements of bulge and disk structural parameters; • a clear picture of the distribution of the Sersic n shape parameter for late-type spiral galaxy bulges, needed for comparison with structure evolution models; Chapter 1. Introduction 21 • a characterization of the correlation between bulge and disk parameters, needed for discrimination between bulge formation scenarios; • a characterization of the wavelength variation of structural parameters, needed for the modeling of stellar populations and their dust content and distribution in late-type galaxies (especially for an understanding of Type II profiles). In the next chapter we describe the Courteau-Holtzman multi-band image collection of late-type spirals which is the backbone of our analysis. Prior to applying B/D decom-positions to those images and ID galaxy profiles, we first determine the limitations of our algorithms with a comprehensive set of simulations in § 3. We return to the actual data set in § 4 with a full investigation of galaxy B/D decompositions. Final results are presented in § 5, and we conclude with observations for future work in § 5.3. Chapter 2 Data and Basic Reductions Our multi-band analysis of galaxy luminosity profiles uses the compilation of 1063 digital images in the B , V , R, and H passbands of 324 nearby late-type spiral galaxies. The data were collected at the Lowell Observatory and Kitt Peak National Observatory (KPNO) by S. Courteau and J . Holtzman between 1992 and 1996. A full description of the sample selection, observations, and reductions are discussed in detail in Holtzman & Courteau (2002). A brief overview is given below with emphasis on the features that are important for this study. 2.1 Sample Selection The galaxies were selected from the Uppsala General Catalogue of Galaxies (UGC, Nilson 1973) with the following criteria: • Hubble type Sb-Sc • Zwicky magnitude m# < 15.5 • Blue Galactic extinction AB = 1 x E(B - V) < VT.h (Burstein & Heiles 1984) • Inclination bins covering face-on (i < 6°), intermediate (50° < i < 60°), and edge-on (i > 78°) projections • Blue major axis < 2'. 2. The diameter limit was constrained primarily by the field of view of the infrared cameras in use at K P N O (IRIM and COB) and Lowell Observatory (OSIRIS) between 1992-1996 22 Chapter 2. Data and Basic Reductions 23 and the requirement for blank areas in the field of view for accurate sky subtraction. Ad-ditionally, all peculiar and interacting galaxies (e.g., no visible tidal tails) were excluded to ensure that the sample consisted only of isolated disk dominated galaxies. For the purpose of selection only, crude inclination estimates were made based on the U G C red axial ratios and Holmberg's (1946) description of galaxies as oblate spheroids: cos2i = (r2 ~ el) (2.1) (1 - el) ' where r = b/a is the ratio of the (red) minor to the major axis (also referred to as the galaxy's intrinsic axial ratio) and e0 = c/a is the eccentricity of the spheroid (where a, b, and c are the spheroid's three axes) which is dependent on Hubble type. We used the values e0 = 0.18, 0.14, and 0.10 for Hubble types Sb, Sb-c, and Sc respectively (Haynes & Giovanelli 1984). More refined diameters and axial ratios can be obtained from digital images yielding more accurate measures of a galaxy's ellipticity. The conversion of ellipticity to inclination is a function of wavelength and relies critically on a dust extinction model. Our transformation assumes optically-thin disks. Fig. 2.1 shows a histogram of the measured ellipticities for all of the galaxies used in this study. A l l but two of the galaxies (UGC 2213 and 5153) have known radial velocities (from the Updated Zwicky Catalog (UZC) of Falco et al. (1999), available on the N A S A Ex-tragalactic Database) for transformation to physical scales. Fig. 2.2 shows a histogram of the distribution of heliocentric radial velocities (cz) for the galaxies used in this study. The survey has an effective depth (cz) ~ 5500 k m s - 1 . The vast majority of galaxies are too distant to worry about peculiar velocity effects on individual galaxies, such as caused by the Local Supercluster. A l l distances are corrected to the reference frame of the Local Standard of Rest (Courteau and van den Bergh 1999) and assume a Hubble constant H 0 = 70 km sec'1 Mpc - 1 . Chapter 2. Data and Basic Reductions 24 —i—1—i—1—i—1—r~ total Ii- 0.28 a - 0.16 N - 123 type I /j= 0.28 o= 0.16 N = 54 — - type II ix= 0 . 2 9 a= 0 . 1 6 N = 53 t r ons fi= 0 . 2 6 o= 0 . 1 4 N = 16 L , I . I "o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ( 1 - b / a ) Figure 2.1: Histogram of galaxy ellipticities (e = 1 — b/a) for the sub-sample of the Courteau-Holtzman database used for this study. The total catalog comprises 324 galaxies; more than half of them are of edge-on orientation and the rest are evenly divided between face-on and intermediate projections. For the purpose of bulge-to-disk (B/D) decompositions and the analysis of disk scale lengths, we consider only face-on and intermediate orientations. This leaves us with a sub-sample of 123 usable galaxies and 523 images. We shall return to our catalog of edge-on galaxies in an upcoming analysis of scale height variations and truncation radii in disk galaxies (see e.g., de Grijs & Peletier 1997). Table A . l in the Appendix lists the relevant catalog information for all galaxies used in this project. Chapter 2. Data and Basic Reductions 25 12 p 1 1 1 1 1 1 1 1 1 1 1 1 f 2000 4000 6000 8000 10 4 1.2x10* 1.4x10* cz (km s~1) Figure 2.2: Histogram of galaxy heliocentric radial velocities (cz) from N E D . 2.2 Observations and Basic Reductions A l l optical B V R images were obtained from 1992 to 1994 at Lowell Observatory with a TI 800x800 chip (scale = O'.'5/pix) on the Perkins 72" telescope. The infrared H-band images were acquired from 1993 to 1995 at the Kitt Peak National Observatory (KPNO) with the 2-meter and 4-meter telescopes equipped with either a HgCdTe (IRIM) or an InSb (COB) 256x256 array (scales are l'.'09/pix and O'.'5/pix respectively), and from 1995 to 1996 with the Lowell Perkins telescope using the OSIRIS imager (scale = l'.'493/pix). Typical exposure times were: 300s at R, 400s at V , 1400s at B , and on-target integration of 1200s at H. Landolt standards (Landolt 1992) were taken each night at Lowell, giving a photometric calibration good to better than 2%. Infrared observations were dithered, and flats and sky levels were also determined from multiple dithered Chapter 2. Data and Basic Reductions 26 observations. U K I R T standards (Guarnieri et al. 1991) taken each night, yielded H-band photometric calibrations good to ~3%. Stars and defects were edited from the images prior to further analysis. 2.2.1 Sky Background One of the most critical impediments in extracting accurate surface brightness distribu-tions from galaxy images is the measurement of the sky background level. The brightness of a moonless night sky comes from four major contributors, in decreasing order of impor-tance: (1) zodiacal light: sunlight scattered by the particulate matter in the solar system; (2) air glow: a fluctuating component most prominent at long wavelengths produced by photochemical processes in the upper atmosphere; (3) diffuse extragalactic light: back-ground from distant, faint, unresolved galaxies; and (4) faint and unresolved stars: from our own Galaxy. The sky brightness level in a given passband varies from one observing site to another, from one area of sky to another, from night to night, and can change on the order of minutes on a given night at infrared wavelengths. Hence, accurate flat fielding and wide fields of view are needed for each individual image in order to remove the sky component and determine accurate surface brightness levels for the galaxy image. Integration times must account for the shortest night sky fluctuation timescales. These vary from being quite stable in the optical, to significant variations on timescales of a few tens of seconds in the infrared. We measure mean sky levels (typical for a 5-6 day moon) for all of the calibrated images of B = 21.9, V = 21.2, R — 20.6, H = 14.1 mag arcsec - 2 with statistical errors equal to 0.8, 0.5, 0.5, and 1.2 mag arcsec - 2 respectively. Fig. 2.3 shows a histogram of all the sky measurements. Typical systematic sky errors per galaxy frame, computed from the deviations of the mean sky counts for 4 or 5 suitably located sky boxes around the Chapter 2. Data and Basic Reductions 27 25 n 20 15 E 3 10 i 1 i 1 i 1 i 1 i 1 i 1 i 1 i 1 i 1 r B - b o n d M= 21.9 a- 0.8 N - 148 V-bond H= 21.2 c = 0.5 N = 120 R-bond H= 20.6 o = 0.5 N = 167 H - b a n d M= u.1 <T= 1.2 N = 119 il i in J I I I I . I 23 22 21 20 19 18 17 16 15 14 13 S k y ( m a g a r c s e c - 2 ) Figure 2.3: Histograms of sky measurements in the four passbands for all profiles used in this project. galaxy, are ~0.5-1.0% in the optical and ~0.05% in the IR. We are able to trace profiles to ~26 mag arcsec - 2 in the optical and ~22 mag arcsec - 2 at H , about five to eight magnitudes fainter than the sky level. Accordingly, inaccuracies in the estimated sky background level can have serious effects on the measured light distribution of a galaxy, particularly for the fainter outer regions. 2.2.2 Seeing Astronomical images acquired through ground based observations are subject to blurring effects caused by turbulence in the Earth's atmosphere. The net result is a decrease in the angular resolution of the data, which is characterized by an increase of the seeing disk full-width at half-maximum F W H M (for a Gaussian seeing function). If we assume that Chapter 2. Data and Basic Reductions 28 the diffraction-limited image of a star free of atmospheric effects is a point source, the effect of the seeing on a given observation can be determined by measuring the angular size of a star on the image and comparing with the diffraction limit of the telescope. The typical seeing F W H M was 2'.'0 at Lowell and Kitt Peak. These measurements were computed as the mean of the F W H M s of all non-saturated stars measured auto-matically on each image frame; typically 10 to 30 measurements per frame were used for each F W H M estimate. The accuracy of the seeing estimate per frame is about 20% for the optical bands and 30% for the H band. Fig. 2.4 shows histograms of the seeing F W H M measurements for all the profiles used in this project. Contrary to theoretical expectations ( F W H M oc A - 1 / 5 ) , we do not see an improvement of the seeing at shorter wavelengths (this may also be site dependent). In cases where a seeing measurement could not be made, we adopted a nominal value of 2" . Seeing effects in our model galaxies (see § 3 were accounted for by convolving the theoretical ID and 2D bulge-disk surface brightness profiles and images with a two-dimensional radially symmetric Gaussian Point Spread Function (PSF), of the form J.(r) = < r - V 2 / 2 " 2 Itotal(x)I0(xr/o-2)e-x2^2xdx (2.2) Jo where Itotai(x) is the intrinsic surface brightness profile, a is the dispersion of the Gaussian PSF ( cr = F W H M / ( 2 V l n 4 ) ), and I0 is the zero-order modified Bessel function of the first kind (Pritchet & Kline 1981). 2.3 Profile Extraction For ID B / D decompositions it is necessary to extract radial luminosity profiles from the 2D digital images. A number of methods exist, each leading to differences in the resulting profile. We have explored and compared several methods to ensure consist-ency. Details and subtleties about surface brightness profile extraction can be found in Chapter 2. Data and Basic Reductions 29 B - b a n d med - 2.2 a= 0.6 N = 143 V-band med = 2.1 <7= 0.6 N = 116 R - b o n d med = 2.3 a= 0.5 N = 165 H - b a n d med = 2.2 o - 0.6 N - 118 rr i - n Irrnnnlh u 0 1 2 3 4 5 See ing FWHM (") Figure 2.4: Histograms of seeing F W H M measurements in B V R H for all profiles used in this project. Median values and standard deviations are shown; N corresponds to the total number of galaxy images for which a seeing measurement could be made. Courteau (1996). We first computed azimuthally-averaged surface brightness profiles for all the galax-ies using ellipse fitting with fixed center and extending nearly to the end of the frame (both in optical and IR). The center of the galaxy is selected as the intensity weighted centroid within a pre-defined box centered by eye on the galaxy. The ellipses, fitted along isophotes, can be allowed to twist and track the spiral structure or they can be forced to keep a constant position angle and ellipticity in a series of concentric annuli. The bulges of late-type spiral galaxies are assumed to be very mildly oblate and their flattening should be independent of viewing angle, thus yielding constant ellipticity isophotes. On the other hand, the ellipticity of the disk may change as a function of isophotal tilt. In Chapter 2. Data and Basic Reductions 30 this case it is desirable to allow the ellipticity to vary from the inner to outer regions. It is worth mentioning that either form of averaged profiles (constant or variable ellipticities) may not provide a realistic representation of the light distribution. In both cases, effects of deprojection of the bulge onto the disk can no longer be disentangled if the disk is tilted (Burstein 1979, Byun and Freeman 1995). Isophotes that map the spiral struc-ture (with occasional wild position angle excursions) yield lumpy SB profiles that favour either very bright or faint regions of the arm/inter-arm regions. Conversely, concentric isophotes are forced to cross over arm/inter-arm regions and thus yield a smoother SB profile. Neither one is "better" than the other in terms of physical representation; it is, however, crucial to use the same isophotal map when comparing SB variations with wavelength and measuring colour gradients. In Figs. 2.5 and 2.6, we show a comparison of our profiles with the 3 galaxies we have in common with de Jong (1996) (note the different red passbands for the U G C 463 and U G C 3140 profiles and the different IR passbands for the U G C 463 profiles). The seeing F W H M for each observation is indicated on the plots. Profiles obtained under poorer conditions show greater flattening in the bulge region, accounting for slight differences in the central regions of the profiles from the different observations. The Courteau-Holtzman profiles allow for variable position angle and ellipticity of each isophote while de Jong used concentric annuli for the extraction of radial profiles so some differences are expected, but the matching is excellent nonetheless. The zero-point difference between the respective profiles is essentially nil, attesting to the good quality of the individual calibrations. As expected, there are slight differences in the bumps and wiggles in the profiles extracted with the different methods but, for the most part, the underlying exponential slope is the same no matter how the profiles were extracted (see § 4.9 for a comparison of the structural parameters obtained in both cases). Major axis cuts provide another method of extracting a radial luminosity profile. Chapter 2. Data and Basic Reductions 31 dJ versus CH: UGC 463 T — c i — j — i — I I | I — i I | i — i i j r ~ i — ' — i | i i i — | — i — i — i — | — i — i — r 0 20 40 60 80 0 20 40 60 80 Figure 2.5: Comparison of de Jong (dJ) and Courteau & Holtzman (CH) surface bright-ness profiles for U G C 463 and U G C 3080. Note the different red and IR passbands in U G C 463. Chapter 2. Data and Basic Reductions 32 dJ ve r su s CH : UGC 3 1 4 0 T—i—I—[pi—I I | > I—r—|—i—i—r B-bond i i i I i i i I i—i—i—| i i i "Red"-bond o (JJ 24 |- FWHM - 2.0 " —i—i—i—L_i • • I • • 1 I * 20 40 60 Radius (") 80 Figure 2.6: Comparison of de Jong (dJ) and Courteau k, Holtzman (CH) surface bright-ness profiles for U G C 3140. Note the different red passbands. The advantage of this approach is that major axis cuts are mostly free from the effect of bulge projection onto the disk and thus should provide a more realistic measure of the bulge profile. This should be especially relevant for large bulged galaxies. Cuts, however, may not be representative of the galaxy as a whole due to the chance superposition of bright stars or spiral features. Major axis cuts also suffer from greater noise at large radii (since the relative number of averaged pixels at all radii decreases as r 2 ) . Thus, to take advantage of the higher fidelity of cut profiles in a galaxy's inner parts and the greater signal-to-noise of azimuthally-averaged SB profiles into the outer disk, the cuts and isophotal profiles could be merged into a single, idealized, surface brightness profile. The matching is done simply by replacing the surface brightness data in the isophotal profile from the center out with those from the major axis cuts up to a radius (usually Chapter 2. Data and Basic Reductions 33 between 1-2 disk scale lengths) where the cut and isophotal brightnesses are equal. The blending of the two profiles is in principle straightforward but in practice difficult owing to the noisy nature of luminosity profiles. In hopes of obtaining the above description of an idealized profile, major axis cuts were also extracted from the images. The cuts were made along the long axis of the most representative disk isophotes used in the computation of the disk position angle and ellipticity (Holtzman & Courteau 2002). The major axis cuts are 5 pixels wide and extend as far as the deepest isophote. Fig. 2.7 shows a comparison of the major axis cuts versus the azimuthally averaged profiles for three galaxies which span a range of ellipticities, where e = 1 — b/a, and major profile types. As expected, the averaging has a large effect on the shape of the outer profile. If the major axis cut crosses a prominent arm/inter-arm region of the galaxy, as in U G C 7852, an underlying exponential disk will be hard to recover. Also note how noisy the cuts get in the outer regions compared to the much smoother averaged profiles. The central regions of the profiles show a fairly good match, which is reassuring though not too surprising since the area difference between the isophotes and the slit width is small. We can conclude that the gain in combining the cuts and averaged profiles is not significant for the small bulges considered in this study; projection effects from the bulge onto the disk are negligible and our major axis profiles do not differ, in the central parts, from the azimuthally averaged profiles. However, it should be kept in mind that a similar study for larger bulges might require the use of profile cuts. For the rest of our analysis, we use profiles determined from non-concentric isophotes. Further details about profile fitting, ellipticity measurements, and magnitude calibration are found in Courteau (1996) and Holtzman & Courteau (2002). The surface brightness profiles are reliable (SB errors < 0.1 mag arcsec - 2 ) down to fairly deep levels: ~26 mag arcsec - 2 for optical bands and ~22 mag arcsec - 2 at H-band. Chapter 2. Data and Basic Reductions B-band R-band H-band 20 22 24 26 26 _ 1 1 1 1 1 1 l _ \ e = 0.09 -L U G C 1808^ Type I o ' . 1 , 1 , . o, 10 20 30 40 24 26 1 1 1 1 1 1 1 1 1 1 * lu 0.70 • UGC 7852 V \ " Type II • S o N . i 0 10 20 30 40 50 I 1 I 1 I 26 18 J 20 0 10 20 30 40 50 60 0 10 20 30 40 50 60 18 1 1 1 1 1 1 1 1 1 1 V € = 0.45 J ^UGC 271 Trans . i . i . i . l i t 0 10 20 30 40 50 0 10 20 30 40 50 Radius (") 10 20 30 Figure 2.7: Comparison of major axis cuts and azimuthally-averaged profiles. Chapter 2. Data and Basic Reductions 35 These levels are typical for integrations of ~ 10-15 mins on medium-sized telescopes. Fig. 2.8 shows the variation of surface brightness errors for our profiles in all four bands. The error range is comparable to that derived by Courteau (1996; see his Fig. 9) for a different, but similar, data set collected with Lick Observatory's 1-meter telescope. Surface Brightness (mag arcsec-2) Figure 2.8: Surface brightness errors from all SB profiles for the four different band-passes: B (blue circles), V (green squares), R (red triangles), and H (purple asterisks). The error model (black curve), fitted after the R-band data, is used to model realistic noise in artificial profiles (see § 3.2.1). Note that the error curve has been shifted by 3 mag arcsec - 2 to match the H-band data. For the computation of homogeneous structural parameters and colour gradients, we overlay the R-band isophotal maps onto all other images (BVH) of a given galaxy. Even though dust effects can still play a role at 7000 A, the R-band has a stable sky and the deepest profiles. H-band profiles may be more sensitive to older stellar populations, and thus smoother than optical profiles, but they are intrinsically too shallow and the most Chapter 2. Data and Basic Reductions 36 affected by night sky variations. Thus, systematic errors are likely smallest at R, making it the best choice for the skeleton isophotal maps. Prior to use in the decomposition routine, all profiles were truncated beyond radii where the surface brightness errors systematically exceed 0.12 mag arcsec - 2 . The final profiles are all re-sampled at 0.5 mag arcsec - 2 . 2.4 Surface Brightness Corrections The observed surface brightness of a galaxy can change when viewed at different inch-nation angles, depending on the opacity of the galaxy in the observed waveband. For example, for a purely transparent disk, the surface brightness at any radius increases with inclination. Conversely, the surface brightness of an opaque disk is independent of viewing angle, as the line-of-sight penetrates just as deeply into the disk from any direc-tion. In order to establish a standard for comparative purposes, we attempt to correct the observed surface brightness to its face-on value. Exactly how this correction should be done is the subject of great debate and controversy (see Bur stein et al. 1995 for a review), but the most commonly used expression is the so-called geometric correction of the form /4 = ML + CA2.51og(a/6) (2-3) where fix0 is the face-on value, a and b are the semi-major and semi-minor axes of the galaxy respectively, and Cx is a wavelength dependent coefficient characterizing the transparency of a galaxy for a given wavelength. In Eq. 2.3, Cx must satisfy the condition Cx < 1. In the outskirts of the galaxy, where the interstellar medium is practically transparent, Cx = 1, while in the inner regions, where a higher concentration of dust is present, Cx < 1. However, a generic prescription for the radial dependence of the transparency of late-type galaxies is not yet available, and may prove unachievable if Chapter 2. Data and Basic Reductions 37 the dust content differs from galaxy to galaxy. The transparency coefficient is often determined from isophotal radii (a, b) measured in the outer disk. This may be an adequate correction for the extrapolated disk central surface brightnesses, computed from the outer disk slope, but less appropriate for bulge effective surface brightnesses which are more sensitive to the inner distribution of fight. Cx depends not only on the opacity of the disk, but also on the relative thickness of the stellar and the absorbing layers. In a perfectly exponential disk, without bulge, bar or other non-axisymmetric structure, preferentially stronger extinction towards the center will produce a more slowly rising surface brightness profile inward than that of the expo-nential. This phenomenon is observed in Type II and Transition galaxies (Evans 1994, § 1.2.1). The dust extinction in the central parts of galaxies is counterweighted by the presence of any bulge or bar enhancement. Thus, comparison of Transition/Type II sur-face brightness profiles at different wavelengths should enable us to quantify the amount of dust extinction in the inner regions of galaxies. The global transparency coefficient for CH should be very close to unity as expected for a transparent model. We adopt CH = 1. Other estimates of global values for Cx by Burstein et al. (1995) give CB = 0.42, CR = 0.50, and C1 = 0.64. TuUy & Verheijen (1997) find nearly identical values for the R, I, and H bands but their B-band estimate, CB = 0.23, differs substantially. Valentijn (1990) also found CB ~0.2 from his statistical analysis of Sb-Sc photometric properties. This value is surprisingly low, especially given our ability to see galaxies through the disk of other galaxies (White, Keel & Conselice 2000). We adopt the values of Burstein et al. (1995), but caution about possibly large uncertainties in the B-band and use an interpolated value for Cv = 0.47. Table 2.1 lists our adopted values for Cx. We also correct for Galactic foreground extinction using the reddening values, A \ , of Schlegel et al. (1998) and assuming an Rv = 3.1 extinction curve (e.g., Cardelli et al. Chapter 2. Data and Basic Reductions 38 A B V R H Cx 0.42 0.47 0.50 1.00 Table 2.1: Inclination correction coefficients. The B R H values are from Burstein et al. (1995); the V-band was interpolated. 1989), Me.<w = - Ax (2.4) Finally, we correct for the (1 + z)3 cosmological redshift dimming (Weinberg 1972) which translates to magnitude units as Mc.z = Hob, - 7.5log(l + z) (2.5) where the redshifts are corrected to the Local Group standard of rest and assuming H o=70 km/s/Mpc. The final correction to observed surface brightnesses is, = ~^x + CA2.5log(a/6) - 7.5log(l + z). (2.6) 2.5 M u l t i p l e Observations We take advantage of our many multiple observations to assess the level of systematic errors in our reductions. Multiple observations for a given galaxy and photometric band may have been taken in different seeing and sky conditions. These are not prioritized as we trust our corrections for atmospheric effects and we should not bias our data base against non-ideal conditions. We return to the issue of systematic errors computed from repeat measurement in § 4.8. Chapter 3 Simulations of Bulge-to-Disk Decompositions In order to measure galaxy structural parameters, we have developed independent pro-grams to decompose their ID and 2D light distributions into bulge and disk components. These bulge-to-disk (B/D) decomposition programs allow for a generalized Sersic bulge, an exponential disk, and, for the 2D decompositions, a central bar. It is crucial to test the reliability of our decompositions with simulated images (2D) and profiles (ID), es-pecially in light of significant cross-talk between some of the fit parameters. There are several issues involved with accurate decompositions, particularly with the measurement of bulge parameters, including; the sensitivity of final results to starting guesses, effects of statistical and systematic errors in sky brightness and seeing estimates, variable fit base-line, etc. We explore these issues in great detail below using both ID and 2D analyses to determine the robustness of our codes and the reliability of our final solutions. Because azimuthally-averaged profiles or major axis cuts contain much fewer data points than full 2D images, it is possible to compute ID simulations on significantly shorter timescales than 2D models. Full-blown 2D simulations and decompositions as we describe in this and the following chapter can sometimes involve many hours, if not days, of computer operations even on the fastest workstations available today. Thus our most extensive tests rely on ID simulations which are shown to be fully consistent with 2D simulations when considering axisymmetric features. We first describe below our ID and 2D B / D decomposition programs and then apply them in parallel to a series of specific tests performed with a mock catalog of simulated 39 Chapter 3. Simulations of Bulge-to-Disk Decompositions 40 galaxies. We conclude with a discussion of the parameter space of reliable solutions. 3.1 Description of the ID and 2D Algorithms and Model Components ID algorithm. The C-program used here to perform one dimensional B / D decomposi-tions of major-axis galaxy luminosity profiles (see § 2) was initially developed by Broeils &; Courteau (1997) and subsequently improved by the author. This program reduces ID luminosity profiles into bulge and disk components simultaneously using a non-linear least-squares fit of the logarithmic intensities (i.e., magnitude units). The non-linear fit uses the methods of steepest descent and Taylor expansion to locate the minimum of the surface brightness %2 distribution (see Levenberg-Marquardt tech-nique, § 15.5 in Press et al. 1992 or Chapter 11 in Bevington &: Robinson 1992 for details), profile data point are included in the fitting procedure as a combination of read-out and photon noise, averaged over each isophote. Systematic errors such as uncertainties in the sky background and determination of the image mean PSF are accounted for separately in a series of experiments designed to calibrate their effects. Seeing is accounted for by convolving the profile fitting functions with a Gaussian seeing disk ( F W H M ) specified by the user (see Eq. 2.2). Similar decomposition methods have been devised and tested, most notably, in Schombert & Bothun (1987) and Andredakis &; Sanders (1994). We will return to these works below. 2D algorithm. The 2D decomposition program is based on the same Levenberg-Marquardt technique as above but uses the full 2D image in intensity units instead of a radial surface brightness profile. While computationally more intensive than its ID analogue, the 2D decomposition yields, in principle, more physically meaningful results since all the pixel information averaged over to create ID profiles is now fully accounted for in the 2D analysis. Byun & Freeman (1995) and de Jong (1996) have discussed the Chapter 3. Simulations of Bulge-to-Disk Decompositions 41 merits of the 2D approach, such as the greater ability to recover true parameters (based on simulations), the potential to model non-axisymmetric features such as bars, rings and spiral arms, and being able to bypass isophotal profile fitting to make ID profiles. The need for the implementation and testing of a robust 2D B / D decomposition package is thus obvious, but we shall verify that ID decompositions still hold their own for reliability and predictive power provided the right profiles are used. Note that both ID and 2D decompositions are not impervious to dust extinction effects; a proper recovery of the true stellar density profile would require a full 3D radiation transfer treatment, but such an analysis is beyond the scope of this work. However, the ultimate goal of studies like this one is to eventually achieve some knowledge about the contribution of dust and stellar populations to the integrated light of galaxies. Methodology. A fundamental aspect of the profile decompositions is the choice of fit-ting functions. While the broad exponential nature of disk profiles has been established observationally (Freeman 1970, see also Courteau & Rix 1999) and emerges naturally in analytical models of disk galaxy formation (e.g., Dalcanton et al. 1997 , Ferguson &; Clarke 2001, and references therein), the true shape and range of bulge profiles in spiral galaxies remain somewhat more elusive. Observations suggest a range of bulge shapes from early-to-late-type spirals (Graham 2001), and preliminary simulation results (Scan-napieco & Tissera 2002, in preparation) indicate a range of bulge shapes (see Fig. 1.1). We address this issue here in great detail, but are restricted to late-type spirals. The disk light is modeled by the usual exponential function, 7d(r) = 7oexp(-0 (3.1) or, in magnitudes, fid(r) = p0 + 2.5\og(e){£) (3.2) Chapter 3. Simulations of Bulge-to-Disk Decompositions 42 where /zo = — 2.5 log To and h are the disk central surface brightness (CSB) and scale length respectively. In the 2D decompositions, the computation of the radius at each pixel requires two additional parameters: the position angle (PA) of the disk major axis on the sky and the disk ellipticity (e = 1 — b/a where a and b are the major and minor axes of the disk respectively). Hence, r is determined as r = sj[{x cos (PA) + y sin (PA))6/a] 2 + (-x sin (PA) + y cos (PA)) 2 . (3.3) To test for the shape of the bulge luminosity profiles we adopt the formulation of Sersic (1968) who showed that exponential and de Vaucouleurs intensity profiles are special cases of a general power law: I(r) = I 0 e x p j - ( ^ ) 1 / n j . (3.4) or, in magnitudes, M r ) = /xo + 2 . 5 l o g ( e ) | ( ^ ) 1 / n | . (3.5) where fi0 (lo) is the central surface brightness (intensity), r0 is a scaling radius, and the exponent 1/n is a shape parameter that describes the amount of curvature in the profile. For n = 1 or 4 one recovers a pure exponential (Eq. 3.2) or the de Vaucouleurs r 1 / 4 profile respectively. The distribution of profile shapes for different values of the Sersic n parameter is shown in Fig. 3.1. The top panel shows profiles with fie = 21 mag arcsec - 2 and re = 3'.'5 for values of n in the range 0.2 < n < 4. The bottom panel shows the same profiles except for a constant central surface brightness of //o = 18 mag arcsec - 2 . For n < 1 the profiles are shallow at small radii (< r e ) and fall off rapidly as r increases. Conversely, profiles with n > 1 are very steep at small radii (<C re), but level off as r increases. (The transition at n = 1 is, of course, the usual exponential profile, a straight line in log space.) Given the large differences in the profile shapes above and below n = 1, one might Chapter 3. Simulations of Bulge-to-Disk Decompositions 43 expect different physical mechanisms (formation/transport/dynamics/interactions) to be at work in systems whose light profiles are well represented by n > 1 versus n < 1 Sersic profiles. Additionally, for the small bulges of late type galaxies, poor seeing conditions could conceivably smear the image such that an intrinsically n > 1 bulge would appear more like an n < 1. (Here is yet another reason to correct data properly for seeing effects). The Sersic "law" can also be parameterized in terms of effective quantities as 7b(r) = / e e x p | - 6 n ( f ) ^ - ! } (3-6) where re, the effective radius, encloses half the total extrapolated luminosity. Ie is the intensity at this radius and bn is chosen to ensure that f°° Ib 2irr dr = 2 f ' Ib 2irr dr. (3.7) Jo Jo In magnitudes Eq. 3.6 translates to pb(r) = p,e + 2.5log(e)6 n | l /n where /xe is the effective surface brightness. It is trivial to convert from Eq. 3.5 to Eq. 3.8 by noting that (3.8) re = {bn)nr0 fie = fio + 2.5log(e)6„ (3.9) (3.10) It has become customary to express the disk parameters in terms of scale length and CSB (h and u-o), while the bulge parameters are expressed in terms of effective parameters (r e and fie). We also adopt this formalism, so it is implied that parameters with subscript e always represent bulge parameters. Chapter 3. Simulations of Bulge-to-Disk Decompositions 44 0 5 10 15 20 Radius (") Figure 3.1: Sersic n profiles for different values of n. The top panel shows profiles with He = 21 mag arcsec - 2 and r e = 3'.'5 for values of n in the range 0.2 < n < 4. The table lists the relative light contributions of the different profiles normalized to the n = 1 case. The bottom panel shows the same profiles except for a constant central surface brightness of fio — 18 mag arcsec - . Chapter 3. Simulations of Bulge-to-Disk Decompositions 45 Eq. 3.7 implies that r(2n) = 2-y(2n,bn) (3.11) where T(a) is the gamma function and 7(0, x) is the incomplete gamma function. Eq. 3.11 cannot be solved analytically for bn. Various numerical approximations have been dis-cussed in recent years (Caon et al. 1993, Graham & Prieto 1999, Khosroshahi et al. 2000, Mollenhoff & Heidt 2001). One often encounters the approximation bn « 2n — 0.32, valid for supposedly all values of n (sic). Khosroshahi et al. (2000) even contend that this approximation is accurate to one part in 105, with a range of validity on n unspecified. However, since the gamma function diverges near the origin, most utilized approxima-tions are inaccurate for values of the Sersic exponent n < 1. The difference between the numerical solution for bn (Eq. 3.11) and the two most commonly adopted approximations can yield brightness differences greater than 0.1 mag arcsec - 2 for n < 1. Fig. 3.2 shows the offset between our exact numerical solution of bn and commonly used approximations (a similar figure is also found in Graham 2001). The numerical solution and analytical solution agree well for n > 1 but differ signif-icantly for smaller values of n. However, since spiral bulges exhibit Sersic n's as small as 0.1, we have tested for a best-fitting parameterization which would be valid for all n. The exact value of bn was computed to a numerical precision of one part in 107 for all n < 10. Due to the rapidly changing curvature in the gamma function, and thus bn, at small n, a fit accurate to better than 0.25% for all n < 10 could most easily be achieved if the curve was subdivided into three different segments. For computational simplicity, we adopted a polynomial function of the form m bn = Y,ai*ni (3.12) i=0 for each region, where m is the order of the polynomial and the a; are the coefficients of the fit. The polynomial fits and their ranges of applicability are as follows: Chapter 3. Simulations of Bulge-to-Disk Decompositions 46 PI : (n < 0.5) a 0 = 0.01945 ax = -0.8902 a 2 = 10.95 a 3 = -19.67 a 4 = 13.43 P2 : (0.5 <n< 3.0) a 0 = -0.2820 a a = 1.9265 a 2 = 0.04561 a 3 = -0.02901 a 4 = 0.00363 P3 : (n > 3.0) a 0 = -0.32769 oj = 1.9992 a 2 = 0.00003. Note that solution P3 is close to the standard parameterization adopted by numerous authors. It should not be used for n < 3. The difference between our approximation to bn (Eq. 3.13) is also displayed in Fig. 3.2 as the solid curve, broken into the three sections by colour; P I is blue, P2 is cyan, and P3 is green. The wiggles in P I are the result of the polynomial nature of the fit and limited numerical precision where the gamma function approaches infinity. Also of potential relevance to the study of galaxy structure is the relative light frac-tions contributed by the bulge and disk. This is expressed in terms of a bulge-to-disk luminosity ratio, B/D, derived by integrating the bulge and disk luminosity profiles to infinity. For the Sersic profile the total extrapolated luminosity is given by (3.13) (3.14) and for an exponential disk (3.15) giving a bulge-to-disk light ratio of (3.16) Chapter 3. Simulations of Bulge-to-Disk Decompositions 47 0.06 0.04 r i c 0.02 h 1 — i — i — i — i — i — i — i — i — i — i — i — i — r -b„-(2n-0.332) bn-(1.9992n-0.3271) bn-(Pl,P2,P3) n Figure 3.2: Difference between the exact numerical value for bn and several commonly adopted approximations. The dashed (red) and dashed-dotted (magenta) lines are the two most commonly used approximations found in the literature. The solid Une shows our adopted approximation which consists of a combination of three different polynomials (see text) for the ranges 0.05 < n < 0.55 (blue), 0.55 < n < 3.0 (cyan), and n > 3.0 (green). A weakness of B/D ratios for systematic comparisons of galaxy light profiles is the dependence on a particular model and the strong covariance between some of its para-meters. For instance, consider the table in Fig. 3.1 which lists the relative light fractions contributed by the profiles with different ns, normalized to n — 1. The integrated bulge light increases steadily as a function of n, for given values of r e and /xe. Thus, the adopted n value in a bulge-to-disk decomposition has a strong influence on the computed B/D ratio. Additionally, since larger n values contribute light out to large r, the combination of a high n with a bright [ie could take away light from the disk and artificially boost the B/D ratio. It is obviously critical to devise a robust goodness-of-fit statistic to alleviate such degeneracies in the fits. Chapter 3. Simulations of Bulge-to-Disk Decompositions 48 We model the total galaxy luminosity profile as a sum of bulge + disk components: hotai{r) = h(r) + Id(r) (3.17) Seeing is simulated in the decompositions by convolving the models with a Gaussian PSF of the form of Eq. 2.2. Similar analyses have also considered additional terms for a Gaussian bar (de Jong 1996), a lens or ring (Prieto et al. 2001), spiral arms, and disks with inner and/or outer truncations (Baggett et al. 1998). We restrict our choice of fitting functions to a Sersic bulge and an (infinite) exponential disk for a number of reasons. We find no prominent bars in our sample and our disk profiles are fairly linear (in magnitude space). Elliptical averaging (for ID profiles) smoothes out spiral arm features (to a different extent depend-ing on whether the position angle was fixed or allowed to vary in the profile extraction) and trying to remove spiral arms from those profiles would require a serious investment of time and effort that is not warranted by our analysis at this stage. The 2D technique would, in principle, allow for the removal of spiral arms by an iterative technique, but this has not been attempted here. There is no theoretical basis for an inner truncation of the disk and, moreover, some studies show evidence that the spiral structure of disks continues all the way to the center of the galaxy (Courteau 1996, Elmegreen et al. 2001). Non-linear least-squares modeling requires that a merit function or likelihood be de-fined in order to characterize the agreement between data and model for a given set of parameter values. The "best-fit" parameters are those which minimize the adopted merit function, defined here as the usual chi-square statistic, %2 •> described in intensity units as 1 N XDOF = J J — ^ E where N is the number of data points used, M is the number free parameters (i.e., N -M = DOF = Degrees of Freedom), and <7; is the intensity error at each pixel (2D) or I,(ri\h, I0, r e, Ie, n) - Igai{ri) (3.18) Chapter 3. Simulations of Bulge-to-Disk Decompositions 49 surface brightness level (ID). From here on the DOF subscript will be omitted and the X2 variable is to be taken as %2 per degree of freedom (unless otherwise specified). As noted by Graham (2001), the global x2 °f intensities will be dominated by the contribution from the disk, virtually irrespective of the fitted bulge. The domination of the disk to the %2 statistic might be even worse in galaxies with prominent features (e.g., spiral arms, bar, rings, lenses) which are not accounted for in our pure exponential disk models. Cases are found where B / D fits with significantly different bulge exponent n values for a given profile have nearly the same global %2 value (see Fig. 4.3). In order to refine our search for a best fit model and to determine a relative goodness-of-fit parameter for our bulge models in general, a separate, inner, %2 statistic is computed out to twice the radius where the bulge and disk contribute equally to the total luminosity of the galaxy (rb=a = 2r(Ib = Id))- We represent this statistic as Xin • For cases where the bulges are so small that they never truly dominate the light profile (i.e., Tbd is undefined), we compute the X;„ out to the radius at which DOF = 1. This definition was also adopted by Graham (2001), but, as we shall discuss in § 4.9 he did not use it to search for an optimal solution. 3.2 Reliability of the Decomposition Results This section describes extensive testing of our bulge-to-disk decomposition programs. Nearly half a million artificial surface brightness profiles and images with a wide range of parameters using various bulge profile shapes and exponential disks were generated and made to resemble genuine data with realistic noise and seeing effects. Real galaxies are clearly more complicated than the sum of two idealized smooth mathematical functions, but these tests provide a reasonable test-bed for an understanding of the reliability of B / D decomposition algorithms. Chapter 3. Simulations of Bulge-to-Disk Decompositions 50 The mock catalog of surface brightness profiles and images will be used to address the following questions: • How reliable and meaningful are the bulge-to-disk decompositions and fitted para-meters? • How important are initial estimates? Are our fits achieving the lowest minimum in the x2 distribution? • How does seeing influence the decompositions, and can seeing effects be properly accounted for? • How do sky subtraction errors affect the decompositions, and can they be properly accounted for? • Are bulges in late-type disk galaxies resolved well enough that the Sersic n para-meter can be fit reliably as a free parameter? • What is(are) the principal contributor(s) to the errors on the determined para-meters? Tests of profile fitting routines on artificial data are not new. In a pioneering study, Schombert k Bothun (1987) performed a double-blind experiment in which one of the authors created artificial luminosity profiles and the other fitted the data. The model galaxies were composed of a de Vaucouleurs bulge and an exponential disk. Photon noise, at a level matching typical blue C C D performance at a given magnitude, and a systematic error in the range of 0.5-3.0% of the sky background were also added to the profiles. Schombert Sz Bothun found that the simultaneous fitting of disk and bulge using a standard non-linear least-squares fitting routine was able to reproduce the input parameters to within 10-20% in cases where the reduced \ 2 (using logarithmic intensities) Chapter 3. Simulations of Bulge-to-Disk Decompositions 51 is less than a threshold value of 2. For x2 > 2 the B/D ratios were deemed ambiguous and not to be trusted. In fact, the basic concept of the B/D ratio loses any meaning if all galaxy profiles can not be decomposed nearly perfectly as the sum of a given bulge and disk. For example, we know that this condition is not met with Type II profiles whose inner disk features are significantly different (see § 1.2.1). In this respect, the notion of B/D ratios should really be replaced, Schombert & Bothun surmised (correctly), with one of a light concentration. They also make claim that a sky background uncertainty of up to 3% does not have an effect on the derived parameters. This conclusion is, however, not borne out in their analysis as the comparisons between their model and fit parameters (see their Fig. 3 or Table II) do not indicate what sky errors were used and what tests were actually performed. Contrary to this claim, we find (see § 3.2.4) that even a 1% error in the sky background can have a significant effect on the shape of the outer disk profile, which consequently alters the final fitted parameters for both the bulge and disk. Moreover, all of their B / D decompositions for real galaxy profiles had %2 > 2.5 which suggests that real galaxies do not reduce perfectly to an idealized de Vaucouleurs bulge profile and an exponential disk. The authors do not consider other fitting functions but recognize that bulges may not be adequately described by the de Vaucouleurs r 1 / 4 function. The model inadequacies noted by Shombert & Bothun were addressed by Andredakis & Sanders (1994) who examined differences between the r 1 / 4 and pure exponential func-tional forms for the bulge profile. They showed that late-type spirals are better modeled by the sum of two exponential profiles, for the bulge and disk. Their simulations also showed that intrinsically r 1 / 4 profiles fit with an exponential function have much larger X 2 ' s than intrinsically exponential profiles fit with r 1 / 4 functions. In light of these results, if profile decompositions have xlxp < xii/*, this is a good indication that the true shape of the bulge more closely resembles the exponential than an r 1 / 4 function. In practice, Chapter 3. Simulations of Bulge-to-Disk Decompositions 52 however, deviations in the disk profiles of real galaxies from a pure exponential will dominate the global %2 and wipe out this effect. 2D B / D decomposition techniques, which exploit the full galaxy image, have also been developed and tested in a context similar to our decompositions. Byun & Freeman (1995) and de Jong (1996) were among the first to propose and make use of robust 2D decompositions. The main advantage of 2D technique over ID is the ability to model or remove non-axisymmetric features, such as bars and spiral arms, in addition to the bulge and disk. Following the results presented by Andredakis & Sanders (1994) discussed above, de Jong performed extensive tests of his 2D decomposition routine, modeling only pure exponential bulges along with the usual exponential disks, de Jong used the artificial galaxies to determine how well the routine reproduced the model values for different disk and bulge parameters. He also explored the effect of errors in the measured observables including the seeing F W H M , sky background level, minor over major axis ratio, 6/a, and position angle, PA. These observables are fixed parameters in the fitting routine, so to test the effect of measurement error on the determined parameters he re-decomposed the artificial galaxies using erroneous values for the observables. His main conclusions are as follows: errors in po are predominantly caused by sky subtraction errors and can be as large as 0.1 mag arcsec - 2 ; errors in h can reach 10% and are dominated by sky background and ellipticity measurement errors; bulge parameter errors, of order 20%, are controlled by their size and brightness relative to the disk; bright bulges are most affected by seeing errors, and fainter bulges can also be effected by sky background errors. Our simulations also corroborate the conclusions above, though we explore in some-what greater detail the robustness of the fitting procedure and accuracies of the derived parameters with various values for the fit initial estimates, seeing F W H M , sky value, and, additionally, we model the bulge with a generalized Sersic profile. These simula-tions were initiated by Broeils & Courteau (1997) but are extended here in much greater Chapter 3. Simulations of Bulge-to-Disk Decompositions 53 detail, especially with respect to the determination of the bulge shape parameter ra. Our tests are performed using both the one and two-dimensional decomposition algorithms. In § 3.2.1 to § 3.3 we develop a complete decomposition formalism and derive general conclusions for the applicability and reliability of ID and 2D decompositions. 3.2.1 Simulated Profiles and Images Our tests use a large set of artificial surface brightness profiles and images which span a wide range of the bulge, disk, and seeing parameters. The mathematical forms of the bulge and disk components are those discussed in § 3.1. Noise was added to the model profiles and images from a Gaussian distribution with standard deviation representative of the observed profiles in our sample, at a given surface brightness level. The nomi-nal surface brightness error curve from which representative brightness level errors are sampled is shown in Fig. 2.8 for all four passbands in our data set. The adopted error model for our simulations (black curve in Fig. 2.8) is based on the R-band data (after Courteau 1996, see his Fig. 9). The curve is a close enough match to all optical data, and can be shifted by R-H = 3 mag arcsec - 2 to match the slightly noisier H-band surface brightness error distribution. Any constraints from these simulations must be shifted by —3 mag arcsec - 2 when applying them to H-band data. One hundred SB profiles and fourty images were created and co-added for each bulge, disk, and seeing combination, where the only difference between each profile or image is the resulting noise. Most of the simulated profiles and images had the same disk parameters, /xo = 20 mag arcsec - 2 , h = 12" , which are representative of a typical galaxy in our sample. (A few different values of h Chapter 3. Simulations of Bulge-to-Disk Decompositions 54 were also tested and will be discussed below.) For exponential bulges (n = 1), the bulge parameters were selected from He = 16, 17,..., 22 mag arcsec - 2 , re = 0.1, 0.2,..., 3'.'0 corresponding to B/D ratios ranging from 0 to 5 and B/T from 0 to 0.8 (see Eq. 3.16). Finally, all profiles/images were convolved with a seeing disk of F W H M = 1.0, 1.5,...,3'.'0. These models span the full range of the parameters typically found in late-type bulges (e.g., de Jong 1996, Broeils & Courteau 1997) and the seeing values match those typical of our observing sites (Lowell, K P N O , see § 2). In the interest of time, a smaller sampling of the parameter space was explored for the full range of Sersic n values n = 0.2, 0.4,..., 4.0 for the set of combinations with re = 0.8, 1.5, and 2'.'5, \ie = 18, 20, and 22 mag arcsec - 2 and for seeing F W H M s of 1.5, 2.0 and 2'.'5. Here the corresponding B/D ratios range from about 0 to 1, and B/T from 0 to 0.5. We also simulated the full range of re, F W H M , and pe = 18, 20, 22, and 24 mag arcsec - 2 for n = 0.2 and n = 4.0 for a test regarding the initial estimates (see § 3.2.2). A total of about 223,000 artificial surface brightness profiles and 16,200 images were created and fitted. The mock profiles were all sampled at O'.'5/pixel to match the data. The ID and 2D decompositions of the artificial profiles and images would be deemed satisfactory if they met the following rather liberal criteria: • solution found within 100 iterations Chapter 3. Simulations of Bulge-to-Disk Decompositions 55 • 0 < re{fit) < 500" • 0 < He(fit) < 30 mag arcsec - 2 • 0.05 < n ( / i t ) < 16 • 0 < hifit) < 100" • 0 < fio(fit) < 30 mag arcsec - 2 Furthermore, more than 66% of the simulated profiles and images for each given set of parameters had to pass these criteria to be included in the plots. Otherwise, parameters for solutions which did not meet these criteria were set to ±9.99 and appear as outliers (nearly vertical streaks) in Figs. 3.5 to 3.19. The 2D tests were not developed as fully due to prohibitive computing times, but the results from both techniques corroborate each other. Thus we have relied more heavily on the ID modeling approach for greater efficiency. 3.2.2 Initial Estimates Non-linear least-squares algorithms require initial estimates as input parameters, and we must verify whether our solutions are sensitive to our initial guesses. Different initial es-timates may yield different solutions with comparable x2 values especially if the topology of the x2 distribution is non-trivial or shallow (e.g., Schombert &; Bothun 1987, de Jong 1996). Our simulations confirm the robustness of our algorithmss to a wide range of ini-tial disk parameter estimates. The results are slightly more sensitive for large values of re, but in general the disk parameters were perfectly recovered independent of the initial guesses. One must still caution that if the bulge is fit with the wrong n, the fitted disk parameters will differ from their intrinsic values, even if the initial estimates were good. Fig. 3.3 shows the relative fit error for h, Ah, where Chapter 3. Simulations of Bulge-to-Disk Decompositions 56 Ah EE M m e a n ) - h m o d e l ( 3 i g ) •^model versus the fitted n value (held as a fixed parameter in the decomposition) for bulges with n — 0.2, 1.0, and 4.0. (i.e., Ah = 0 indicates a perfect recovery of the model parameter h). model n = 0.2 model n = 1.0 model n = 4.0 fixed n Figure 3.3: Effect of fitting an incorrect Sersic n bulge on the disk scale length h. Each panel plots the average relative fitted h errors (Ah = (/i^(mean) - ^ modelV^model) with solid symbols and connected by solid lines as a function of the model n for a bulge with re =2'.'5 and u.e = 20 mag arcsec - 2 . Circles (red), triangles (green), and squares (blue) correspond to seeing values of 1.5, 2.0, and 2'.'5 respectively. The three panels are for model ra values of 0.2, 1.0, and 4.0 from left to right. In the decompositions the correct inital values were given for the bulge and disk parameters and the seeing F W H M was assumed to be known. Different seeing values are represented by three point types: circles (red), triangles (green), and squares (blue) for seeing values of 1.5, 2.0, and 2'.'5 respectively. We see that the fitted disk parameters can have errors as large as 10% when the bulge is modeled with an incorrect shape parameter. The reason for this is obvious if one considers the different shapes for the different values of ra shown above in Fig. 3.1, where the different shapes contribute differently to the Chapter 3. Simulations of Bulge-to-Disk Decompositions 57 outer profile. Even more sensitive to the fitted n, Fig. 3.4 shows the relative fit error on re, A r e , where A r e == r «*»(™an) ~ Eggjgjj (3.20) .model fixed n Figure 3.4: Same as in Fig. 3.3 except for re where Are = (r e^ t(mean) -re,model)/ re,model)- Note the different scale on the y-axis. We now turn to investigate the importance of the bulge initial estimates. Given that the fit baseline for the bulge is much smaller than that of the disk and the apparent size of these bulges is comparable to the seeing disk, the bulge model is likely to be much more sensitive to the input parameters. The sensitivity to the initial estimate of re was investigated first. For these decom-positions the seeing is assumed to be known (the effects of incorrect seeing estimates are explored in §3.2.3) and we use as input the correct initial estimates for the disk para-meters as well as u-e. Offsets of ± 25% and ± 50% from the model values were used as initial estimates for these tests. The results are summarized in Fig. 3.5. Each row of panels corresponds to a different seeing value, increasing from top to bottom, and the columns show under- to over-estimated initial estimates for re from left to right respec-tively. The seven curves in each panel are for different values of fie'. 16 (dark purple), Chapter 3. Simulations of Bulge-to-Disk Decompositions 58 17 (blue), 18 (red), 19 (green), 20 (magenta), 21 (cyan), and 22 (orange) mag arcsec - 2 . The y-axis plots the average (of the successful fits, according to the criteria defined in § 3.2.1, of the 100 simulated profiles for each set of parameters) relative fit errors with Are given by Eq. 3.20. As expected, the initial estimates for re become more critical as the bulge becomes smaller and seeing conditions worsen. (A Une extending to the plot Umits indicates that that particular fit did not meet our passing fit criteria as defined in § 3.2.1.) For bulges with re > 0.3 * F W H M the sensitivity of the algorithm to the initial input parameter for re is almost negUgible. Below this Umit initial estimates become more important and errors on re can exceed 50%. Fig. 3.6 shows the effect of re initial estimates on the fitted pe. The plot is the same as Fig. 3.5 except that we show the average absolute error in the fitted pe, A/z e , on the y-axis where Afie = /i e i f i t(mean) - nCtiaodei (mag arcsec - 2), (3-21) Again we see that the algorithm is insensitive to the re initial estimate for the larger bulges, but for bulges with re < 0.3 * F W H M , errors of up to Afie ± 0.3 mag arcsec - 2 can occur. The corresponding results for the 2D decomposition algorithm are shown in Fig. 3.7. The plots are similar to those in Fig. 3.5 but in the interest of time, a restricted parameter space was considered. Here the re initial estimates used were ± 35% of the model re and only two values of fie are shown; u,e — 18 (red circles), and 22 (blue triangles) mag arcsec - 2 . The soUd symbols connected by soUd Unes indicate the average (of the 40 image decompositions for each parameter and initial estimate combination) relative error on re, Are (Eq. 3.20). Also displayed are the 25th and 75th percentiles (i.e., 50% of the successful fits Ue between these two boundaries) indicated by the open symbols Chapter 3. Simulations of Bulge-to-Disk Decompositions 59 -50% -25% A r . l n l l = 0% +25% +50% 0.6 -0.26 -0 \-. -0.25 -0.6 • i i i i i i i i i i U oh -0.25 i i i i i i i i i i i i i 11 II i II 0.5 -0.26 -0 F--0.25 n 1111111111 rrj 11111111111 0.25 h-r r -0.5 H 0.5 0.25 O h -0.26 -0.6 0.5 I-0.25 -0 H -0.26 -0.5 f ¥ 11111111111 i • • • • i r 11111111111 if I I I | M I I | I I I • I • • 1 • I • * » " 1 1 " 1 1 1 F W H M - 1.0" i 1111 i 111111 F W H M - 1.6" I I I | I I I I | I I I F W H M - 2 .0" I I | I I I I | I I iTl I I | I II I | I I iTl I I | I I I I | I I iTl II | I I I I | II iTl I I | I I I I | I I H F W H M - 2 .6" I I I | I I I I | I I I X F W H M - 3.0" -i l i ' i I ' ' i i I I I I I I I I I I I I I I | I I I I | I I I | I I I I | I I l" I I I | I I I I | I I I • • • i • ' • • * • ' ' 0.5 1 0.5 1 0.6 1 0.6 1 0.5 1 model r (") Figure 3.5: Effect on the fitted re value as a result of using a wrong value for the re initial estimate in the ID decomposition. The columns show different offset values for the re initial estimate used in the fit, where A r e j n i t = (r e_ u s ea — r-e^ odeO/r-e-modei- Each row is for a different value of the model F W H M , increasing from top to bottom. Each panel shows the average relative error on r e , A r e (Eq. 3.20), versus the model re. The seven curves are for different values of (ie: 16 (dark purple), 17 (blue), 18 (red), 19 (green), 20 (magenta), 21 (cyan), and 22 (orange). Chapter 3. Simulations of Bulge-to-Disk Decompositions -50% i i i I i i i i i r t i -25% Ar., n I. = 0% +25% +50% •J lt i i i | i i i i | i i i l l i i | i i i i | i i I I I i i | i i i i | i i i I i i 0.2 0 -0.2 0.2 0 -0.2 0.2 0 -0.2 0.2 0 -0.2 T FWHM - 1.0" ~t~ 11111111 I I I 1111111111 l " " l 111111111II " l 1111111111 \"\ 111II111111 FWHM - 1.6" 1111111111ITI1111111111ITI1111111111ITI1111111111 ITI 11111111111 — - - FWHM - 2.0" 11111111111"\ 11111111111~~11111111111 iTi 11111111111 111111111111 FWHM - 2 . 5 " 1111111111iTi1111111111iTi1111111111iTi1111111111iTii'111111111 0.2 h o -0.2 H FWHM - 3.0" I • • • • I I • • • • I I . . . . I . . . T . . . t 0.6 1 0.5 1 0.5 1 0.5 1 0.5 1 model r (") Figure 3.6: Same as Fig 3.5 exept for Ape. Chapter 3. Simulations of Bulge-to-Disk Decompositions 61 and are connected dashed-dotted lines. The results match those from the ID tests (see Fig. 3.5). -35% A W = 0% +35% 0.2 0.1 0 -0.1 -0.2 v < 0.2 0.1 0 -0.1 -0.2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 • M . = 18 -* A*. = 22 - A - » A - -fa* « — i , , , , i , | , , > , | , , i i | , ft i i 1 i i i i l i i i i l i i i i l i— - A \ ' A m f _ * _ / - A FWHM = 1.5" -. ^ 11 J . / • A A - V A - \ • " v ^ « ^ A A -- A -\ 1 \ -\ 5- * ~— \ = ^ r ^r^ if • i " • ^ / W H M = 2.5" . . i /. . . . i . . . . i . . . . i . . . i y . . . i . . . . i . . . . i . 0.5 1 1.5 2 0.6 1 1.6 2 0.5 1 1.5 2 model r e (") Figure 3.7: Effect on he fitted re value as a result of using a wrong value for the re initial estimate in the 2D decomposition (compare with Fig. 3.5.) The solid symbols connected by solid lines indicate the average (of the 40 image decompositions for each parameter and initial estimate combination) relative error on r e , Are (Eq. 3.20). Circles (red) and triangles (blue) are for fie values of 18 and 22 mag arcsec - 2 respectively. The open symbols show the width of the distribution of A r e within the 25th and 75th percentiles and are connected dashed-dotted lines. An analogous test for /xe is shown in Fig. 3.9. The initial estimates for \ie were ± 25% and ± 50% of the model values. Figs. 3.8 and 3.9 and the previous suite of tests (Figs. 3.5 to 3.7) show that the bulge parameter initial estimates are not important in both the ID and 2D decompositions, as long as re > 0.3 * F W H M . This result is unchanged if re and \ie are varied simultaneously. Chapter 3. Simulations of Bulge-to-Disk Decompositions -50% -25% AM^ta,. = 0% +25% +50% + FTHM - 1.0* f I 1 1 1 1 I 1 1 1 1111 111 FWHM - 8.0* 1 I I | I I I I | I I I I | I I I I | I I FWHM - 1.6" 11111111111 rn 1111111111 rn • 1 1 1 1 1 1 1 1 1 1 1 1111111111 1111 111 w 111111111 iTj 11111111111 FWHM - 2.6" + I I I | I I I I | I I I -0.6 r-FWHM - 3.0* f • • I • • i i I • • • 1 • * • 1 • 1 • 1 1 1 1 • 0.6 1 0.6 1 0.6 1 0.6 1 0.6 1 model r (") Figure 3.8: Similar to Fig. 3.5 but for A/z e_init. Chapter 3. Simulations of Bulge-to-Disk Decompositions 0.2 H o - ° - 2 " I - f F W H M - 1 .0" j i i | i i i i | i i I " I i i | i i i i | i i I " I i i | i i i i | i i i T I I i | i i i i | i i i T I I i | i i i i | i i i 0.2 h 0 -0.2 h 0.2 H -0.2 h 0.2 H 0 -0.2 h 0.2 0 -0.2 -50% -25% 11 1 1 11 " '± AM._i„,i = 0% +25% 11 | I I 11 | I I I"I I I | I I I 11 11 I I 11 11 I I I 11 I \\~-• I 1 1 i i I i i i i i 1 1 | i i i i | ' • • • • i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 FWHM - 1 .6" FWHM - 2.0" II|IIII|IIITIII|IIII|IIITIII|IIII|IIITIII|MII|IIIT FWHM - 2.6" j M | I I I I | I I I ' j I I | I I I I | I I I "l i I | I I I I | I I I J i I | I I I I | I I l"j I I | I I I I | I I I FWHM - 3.0" • • ' I • I 1 ' 1 1 I I | I I I I | I I I j-' . i • • • • i +50% 11111 I 4 | i i i i | i i i" •.tar 111111 11 11 I • • ' 1 • ' • 0.6 1 0.6 1 0.6 1 0.6 1 0.6 1 m o d e l r ( " ) Figure 3.9: Similar to Fig. 3.6 but for A/ie Jnit. Chapter 3. Simulations of Bulge-to-Disk Decompositions 64 The above conclusions hold for the n = 1 case. In order to define a reliable parameter space for the entire range of bulge shapes we also performed the above tests for n = 0.2 and n — 4.0. These extreme cases should tell us how representative the n = 1 case is. The results are shown in Figs. 3.10 and 3.11. In the n — 0.2 case we see that the parameter recovery is largely independent of the seeing F W H M , which is to be expected for profiles which are flat towards the center. Given the correct initial estimates the parameters are recovered perfectly, but if the initial estimates are not exactly equal to the intrinsic values, profiles with re < l'.'O cannot be trusted. For n = 4.0 profiles, the parameter recovery is very dependent on the seeing F W H M such that profiles with re < 0.7* F W H M are not to be trusted. Additionally, even given the correct initial estimates, the parameters are not well recovered for profiles with re < l'.'O. In order to interpolate between the values of n and define a set of constraints for the full range we have adopted the following range of parameter space for which the solutions are not effected by the choice of initial estimates: re £ (0.3) 1 / n * F W H M and > -0.7571 + 1.15 for n < 1.0 0.2ra + 0.2 for n > 1.0 3.2.3 Seeing Effects In § 3.2.2 we found that our B / D decompositions are insensitive to the choice of initial parameters for exponential bulges with effective radius in the range re > 0.3 * F W H M . However, this assumes perfect knowledge of the seeing measurement (and sky background, Chapter 3. Simulations of Bulge-to-Disk Decompositions 65 -50% -25% Ar._,B,. = 0% +25% +50% 0.6 0.25 0 -0.25 -0.5 0.6 0.26 0 -0.25 -0.6 0.5 -0.25 -0.5 0.6 0.25 0 -0.25 -0.6 0.5 0.26 0 -0.25 -0.6 : • ' ' i • ' ' ' i_ j . • • • i • • • • i_ - FWHM - 1.0" -i i i i 1 i i i i 1 C l y d I 1 1 I 1 I I • 1 1 • FWHM - 1.6" • M 1 I | 1 l 1 1 | r i i 1 | 1 r t 1 1 M 1 1 | 1 1 1 1 |_ 1 1 1 1 j f 4 i t I t I r t | 1 1 I 1 | • FWHM - 2.0" • ;-- FWHM - 2.6" • -1111111111 I I I I | I I I I | 1 1 1 1 | 1 1 1 1 | 1 111111 [ i I I I 11 1111 • m 1 1 1 1 1 1 1 1 1 1 • V/" : _j—i—i—i—i—i—i—i—i—1_ - FWHM - 3.0" -—i—i—i—i—1—i—i—i—i L _ • • • • 1 • • • • * 1 2 1 2 1 2 1 2 1 2 model r (") ( n = 0.2 ) Figure 3.10: Effect on the fitted re value in ID decomposition from an incorrect ini-tial estimate for r e initial estimate with bulge shape parameter n — 0.2. The col-umn plots show different offset values for the re initial estimate used in the fit, where Ar-eJnit = (re_used — remodel)/^e_modei- Each row is for a different value of the model F W H M , increasing from top to bottom. Each panel shows the average relative error on r e , A r e (Eq. 3.20), versus the model re. The seven curves are for different values of fie: 18 (dark purple), 20 (blue), 22 (red), and 24 (green). Chapter 3. Simulations of Bulge-to-Disk Decompositions 66 -50% -25% Ar . , B l . = 0% +25% +50% 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 model r (") ( n = 4.0 ) Figure 3.11: Same as Fig. 3.10 except for bulge shape parameter n = 4.0. Chapter 3. Simulations of Bulge-to-Disk Decompositions 67 see §3.2.4). Reality is, however, not so simple; the uncertainty in the seeing F W H M measurement should be accounted for in our simulations to model its effects on the model parameters (r e , f i e , h, and f i 0 ) . Accordingly, we not only fitted each model using the nominal seeing value, but we also varied the seeing F W H M by ± 15% and ± 35%, as dictated by normal seeing measurement errors (1 cr[~ 20% for optical, ~ 35% for infrared]; see § 2.2.2). The results for an exponential bulge profile with 0'.'8 < re < 3'.'0, 16 < fie < 22 mag arcsec - 2 , and l'.'O < seeing(FWHM) < 3'.'0 are shown in Figs. 3.12 to 3.15. These figures show the average fit error of a particular parameter versus the model re. Each row of panels corresponds to a different seeing value, increasing from top to bottom, and the column plots show under- to over-estimated seeing values from left to right respectively. When the correct seeing is used as input, the bulge and disk parameters are recovered perfectly for the full range of bulge parameters and for all values of the seeing F W H M tested. Even moderate seeing conditions and uncertainties can severely affect bulge para-meters depending on the size of the bulge relative to the F W H M . In Fig. 3.12 we see that if the seeing is under(over)-estimated re is systematically over(under)-estimated, worsening for smaller and fainter bulges and larger seeing values. Similar trends are seen in Fig. 3.13 for the errors on the fitted f i e . These figures demonstrate that the fit errors will be larger by about a factor 2 if the seeing F W H M is over-estimated than if it is under-estimated. As a rough rule of thumb, for re ~ F W H M and a seeing measure-ment uncertainty at the 35% (15%) level, the bulge re can be trusted to within 10-25% (0-10%), and fie to within ± 0.1-0.4 (0-0.2) mag arcsec - 2 , the lower end of the range applying to the brightest bulges and increasing towards the upper end for the fainter bulges. For r e ~ F W H M + 1 the errors improve to within 0-15% (0-10%) for re, and ± 0.0-0.2 (0.0-0.05) mag arcsec - 2 for f i e . Chapter 3. Simulations of Bulge-to-Disk Decompositions 68 - 3 5 % - 1 5 % AF.HU = 0 % + 1 5 % + 3 5 % 0.5 h II 1 1 1 1 I 1 1 1 ' I 11 ' 1 1 1 I 1 1 1 ' I -0.5 0.5 -0.5 0.5 3 0 -0.5 0.5 -0.5 0.5 | I I I I | I I I I | | I I I I | I I I I | I I I I | I I I I | I I I I | I I I I | | I I I I | I I I I | I I | I I I I | O h | I I I I | I I I I | -0.5 h * I I I I I I 1 1 I I J i i i i I i i i i L i i i i i i i i i 111 i i i i I i i i i 111 i i i i I i i i i I FWHM - 1 . 0 " I I I I | I I I I | FWHM - 1 . 6 " _ I l I I | I I I I | l " l l | U M | | I I I I | I I I I | | i i i i | i i i i | V\ i i i i I i i i i I FWHM - 8 . 0 " _ I I I I | I I I I | | I I I I | I I I I | FWHM - 8 . 5 | I I I I | I I I I | I\ I I I I | I I I I | FWHM = 3 . 0 " i i i i I i i i i I | v*r\ i | i i i i | , . i , , , . \ Y\ . . . . . i 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 model r (") Figure 3.12: Effect on the fitted re value of an incorrect seeing value in the ID decom-position. The column plots are based on different values for the fractional seeing error used in the fit, where A F W H M = ( F W H M u s e d - F W H M m o d e i ) / F W H M m o d e i . Each row is for a different value of the model F W H M , increasing from top to bottom. Each panel shows the average relative error on re, A r e (Eq. 3.20), versus the model re. The seven curves are for different values of /xe: 16 (dark purple), 17 (blue), 18 (red), 19 (green), 20 (magenta), 21 (cyan), and 22 (orange). Chapter 3. Simulations of Bulge-to-Disk Decompositions - 3 5 % -15% Ap.™ = 0% +15% +35% 1 2 31 2 31 2 31 2 31 2 3 model r (") Figure 3.13: Same as in Fig. 3.12 except for Ape. Chapter 3. Simulations of Bulge-to-Disk Decompositions 70 -35% -15% A „ H H = 0% +15% +35% J I_ 1 1 1 1 1 1 1 1 1 1 1 1 , , , , , , , , , ! | , i i , | i . , , | 1 . . , i | i i i i | _ FWHM - 1.0" _ 1 1 1 1 1 1 1 1 1 1 1 _ FWHM - 1.5" _ 1 • • • • 1 • • • • 1 1 1 1 1 1 I I 1 1 I I I I 1 1 1 j 1 1 1 1 j ' I M 1 1 1 1 1 1 1 1 | 1 T 1 t | I t I I | | 1 1 1 1 | 1 1 1 1 | _ FWHM - 3.0" _ • 1 I I 1 1 1 1 I I I 1 i • • * • 1 1 1 1 • i _ FWHM - 2.6" _ l • • • . l i • • • l 1 1 1 1 1 | T T 1 1 | f i t f r f i i t t l I I I I I I I I 1 1 t 1 1 | 1 T 1 1 f 1 . . . . i . . . . r i . . . . i , , , . i _ FWHM = 3.0" _ 1 . . . . 1 . . . . 1 "1 . . . . i . . . . i 1 _ 0.05 h -0.05 0.05 -0.05 0.05 -0.06 0.05 -0.05 0.05 -0.06 1 2 31 2 31 2 31 2 31 2 3 model r e (") Figure 3.14: Same as in Fig. 3.12 except for Ah. Chapter 3. Simulations of Bulge-to-Disk Decompositions 71 -35% 0.1 0.05 0 -0.05 -0.1 0.1 I 0 g -0.05 o SH 0.1 C O 0.05 c d - ° - 0 5 0.1 J 3 0.05 < ° -0.05 -0.1 0.1 0.05 0 -0.05 -0.1 FWHM - 1.0" [| i i i i | i i i i |T| i i i i | i i M |T| i i i i | i i i i |T| i i i i | i i i i |T| i i i i | i i i i | 111111 II 11 ['| 111111111\"\ 1111111111"] 111111111 \"\ 111111111 [ I • • • • I * 1 • • I -15% A p , H M = 0% +15% +35% 1 1 1 1 I 1 1 1 1 111 1 ''' I 1 1 1 1 I T X FWHM - 1.5" X \\ I I I I | I I I I [] I I I I | I I I I |"| I 11 I | I I I I |~~ FWHM - 2.5" -| i i i i | i i i i |T| i i i i | i i M |T| i i i i | i i i i | I • • • 1 I • • • • I X FWHM - 2 .0" X X FWHM = 3.0" X I ' • • • * ' • ' • I 1 2 31 2 31 2 31 2 31 2 3 model r (") Figure 3.15: Same as in Fig. 3.13 except for A/io-Chapter 3. Simulations of Bulge-to-Disk Decompositions 72 Figs. 3.14 and 3.15 show the same results but for the disk parameters where Ah = /ifit(mean) - h m o d e i •^model (3.22) and A/x 0 = /i0,fit(mean) - / x 0 ,modei (mag arcsec 2 ) . (3.23) Note the different scales for the y-axes in Figs. 3.14 and 3.15. As one would expect, for the worst case of a F W H M of 3'.'0 and a seeing over estimate of 35%. In all other cases, the disk parameters are virtually unaffected by seeing, as the size of the disk is much larger than the seeing profile. The results for the corresponding test with the 2D decomposition algorithm are shown in Fig. 3.16 (only the plot for Are is shown). Again, for the sake of time, the parameter space explored was restricted to just a few different values of the bulge and seeing para-meters. The ID and 2D results match almost exactly. These tests show that it would be unwise to combine multiple observations of one galaxy taken under various seeing conditions into one surface brightness profile. Perhaps the most important conclusion is that, B / D decompositions must include seeing errors in order to sample the true range of bulge parameters. 3.2.4 Sky Uncertainty Effects In a similar spirit as the previous discussion, we now test for the effects of an improper sky subtraction. The tremendous sensitivity of B / D decomposition and scale length determinations to sky errors has been highlighted several times before (Courteau 1992, de Jong 1996). Here we aim to provide a firm quantitative assessment of such errors. We re-model the same simulated profiles that were used in the previous section but using sky values that are ± 1 % of the nominal sky value, as dictated by actual data. We use there is no appreciable effect due to seeing on the disk parameters (less than 1%) except Chapter 3. Simulations of Bulge-to-Disk Decompositions 73 -35% -15% FWHM = 0% +15% +35% < 0.4 0.2 0 0.2 * -0.4 0.4 0.2 0 -0.2 -0.4 1 ' 1 ' 1 : J _ 1 1 1 ' _ 1 ' 1 ' 1 1 ' 1 1 •_ • M. = i s : = 22 • 1 i 1 i 1 i i i i i ~ 1 FWHM = 1.6" J 1 i 1 i 1 i i i a i [ i i i i i i 1 1 1 1 ' i i • ' • J i 1 i 1 i i 1 i 1 i i . i i i . i . i ! FWHM - 2.5" j l . l . l ' I . I . I ' • /"*'• i / ? i . i 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 model r e (") Figure 3.16: Effect on the fitted re value from an incorrect seeing value in the 2D decom-position (compare with Fig. 3.12.) The solid symbols connected by solid lines indicate the average (of the 40 image decompositions for each parameter and initial estimate com-bination) relative error on re, Ar€ (Eq. 3.20). Circles (red) and triangles (blue) are for u-e values of 18 and 22 mag arcsec - 2 respectively. a typical optical sky brightness of 21 mag arcsec - 2 , though our results do not depend sensitively on this exact value (it is the relative difference between the profile luminosity and that of the sky that matters). Fig. 3.17 illustrates the effects of a 1% sky subtraction error on a simulated profile with re = 1'.'5, u.e = 18 mag arcsec - 2 , h = 12" , fi0 = 20 mag arcsec - 2 , n = 1, F W H M = 2'.'0, and fi,ky = 21 mag arcsec - 2 . The idealized, over- and under-subtracted profiles are shown with circles (blue), squares (red), and triangles (green) respectively. Since the bulge brightness is typically significantly higher than that of the sky, at least in the optical, one might expect the bulge parameters to be somewhat insensitive to sky subtraction errors. However, the outer disk, which is measured out to ~26 mag arcsec - 2 (optical), is clearly sensitive to sky subtraction errors (see Fig 3.17), and a modified disk Chapter 3. Simulations of Bulge-to-Disk Decompositions 74 Radius (") Figure 3.17: Effects of a 1% sky subtraction error on a simulated profile with re = 1'.'5, He = 18 mag arcsec - 2 , h = 12" , Ho = 20 mag arcsec - 2 n = 1, and F W H M = 2'.'0. The luminosity profiles with proper, over-, and under-subtraction of the sky background are shown with circles (blue), squares (red), and triangles (green) respectively. ultimately affects bulge structure due to their coupling. The effects of a 1% sky error on the ID decompositions are shown in Figs. 3.18 to 3.21. Quantitatively, if the sky is over- or under-estimated by 1%, the error on the disk scale length Ah will be of order 5-15% and the disk CSB A/x 0 will be ± 0.1-0.25 mag arcsec - 2 . These dispersions hold for the full range of bulge brightnesses except the two faintest bulges which are one and two orders of magnitude fainter than the sky (the orange fine, which represents a bulge with fie — 22 mag arcsec - 2 extends off scale in most of the panels in Figs. 3.18 to 3.20). The parameter dispersions are controlled by the relative sizes of the bulge and disk such that the disk parameter errors increase slightly from smaller to larger bulges. This is simply because a larger bulge weakens the importance of the disk in the central parts, thus giving more weight to the sky-sensitive outer disk. The errors on the bulge parameters are negligible for bulges with (fi»ky — He) > 1 Chapter 3. Simulations of Bulge-to-Disk Decompositions 75 A«ky = A«kjf = 0% A , k , = +1% 0.16 0.1 0.06 h 0 -• -0.05 --0.1 --0.16 -0.,. ' " ' l ^ ^ 0.1 0.06 h 0 ---0.05 --0.1 --0.16 '-•| i i i i | i 0.16 ^ 0.1 r - i 0.06 h <J -0.05 --0.1 --0.16 -0.16 0.1 0.05 0 h -0.05 --0.1 --0.15 1 0.15 0.1 0.06 h 0 -• -0.06 --0.1 --0.16 -1 1 111 ' - U J A J : H I ' 1 1 ' I 1 1 1 1 IXI 1 1 1 1 I 1 1 1 1 IJ FWHM - 1.0" i | I I I I | I I I I | : ' | i i i i | i i i i | ; FWHM - 1.6" jF"" | I I I I | I I I I | " . | I I I I | I I I I FWHM - 2.0" I I I I | I I I I FWHM = 2.5" j; | I I I I | lv> i - < J | | I I I I | I I I I | J | FWHM = 3.0" I i i i i I i i i i I I I I I | I I I I |" I I I | i I I I | ; i . . . . i' 1 2 3 1 2 3 1 2 3 model r e (") Figure 3.18: Effects of inaccurate sky determination on the disk scale length. The sky level was taken to be 21 mag arcsec - 2 . The left and right column plots give results for an under/over-estimate of the sky by 1% ( A s k y = ( sky u s e d - s k y m o d e l ) / s k y m o d e i with sky in intensity units). The middle column uses the correct sky level. The rows show the results for different values of the seeing F W H M , increasing from top to bottom. The axes and curves are as in Fig. 3.14. Chapter 3. Simulations of Bulge-to-Disk Decompositions 76 0 . 2 5 0 - 0 . 2 5 0 . 2 5 0 - 0 . 2 5 0 . 2 6 O ^ 0 <l - 0 . 2 5 0 . 2 5 0 - 0 . 2 5 0 . 2 5 0 - 0 . 2 5 • "v\y\}- 1 I 1 I I 1 , 1 I 1 1 , i . i . | i i , , | " FWHM - 1 . 0 " " 1 1 1 1 1 1 1 1 1 1 1 1 i i i 1 1 i i i i 1 i i i i i P i i i i i " FWHM - 1 . 5 " " 1 i i i i l • < • • i I • • • • I i • • • I " FWHM - 2 . 0 " "j •' V ^ ~ v ! _ A • 1J 1 1 1 Ii f ti 1 il 1 ||j " FWHM - 2 . 5 " " 1 i i i 1 I I i i 1 I i i i i | KJf'HJ 1—1—1—1—1—1—1—1—1—1—1_ " FWHM = 3 . 0 " " " i i i i i i i i i i r i . . . i . . . , i" 1 2 3 1 2 3 1 2 3 m o d e l r e ( " ) Figure 3.19: Same as in Fig. 3.20 but for p0. Chapter 3. Simulations of Bulge-to-Disk Decompositions 77 = - « V . = 0 % A r t y = +1% SH < -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 ll ' I ' 1 1 1 I -0.5 k I •- IV I I I I • • I I I I I O J I I I I I | I 1 1 LA 1 1 1 1 1 I J i—_J i i I i i i i L FWHM - 1.0" | I I I I | I I I I | FWHM = 1.5" I ' ' ' ' I ' ' ' ' I FWHM - 2.0" I I I I I I I I I FWHM = 2.5" | I I I I | I 1 I I | FWHM = 3.0" J i i i i L_i i i i L i i i i I i i i i I I i i i i | i i i i | I rTi i | i i i i | J i * • * I i_i • • I 1 2 3 1 2 3 1 2 3 m o d e l r ( " ) Figure 3.20: Same as in Fig. 3.18 but for r e . Note the different scale on the y-axis. Chapter 3. Simulations of Bulge-to-Disk Decompositions 78 \ k , = -1% A * , = 0% A . k y = +1% 0.25 0 -0.25 0.25 0 -0.25 0.25 0 -0.26 0.25 0 -0.25 0.26 0 -0.26 1 . 1 1 I , 1 1 . 1 , 1 I • • I 1 I • • • 1 " FWHM - 1.0" " l i t i i l i t t i l I j j i m i i i i . n O r f M ~ I I I I I I I I I I I " FWHM - 1.5" " ^ — ( -" FWHM - 2.0" " " FWHM = 2.5" " 1 , , , , 1 , , , , 1' _ FWHM = 3.0" -"1 , , , , 1 , , , , 1" 1 2 3 1 2 3 1 2 3 model r e (") Figure 3.21: Same as in Fig. 3.19 but for /xe. Chapter 3. Simulations of Bulge-to-Disk Decompositions 79 mag arcsec - 2 for the entire range of re and seeing F W H M s , but increase up to A r e > 15% and |A/z e | > 0.1 mag arcsec - 2 (increasing as the bulge gets smaller and as seeing conditions degrade) for bulges with (fi,ky — Me) < 1 mag arcsec - 2 . In other words, if the bulge effective surface brightness is less than one magnitude greater than the sky brightness, the bulge parameters will be strongly affected by sky subtraction errors. This effect is too often neglected in major studies of bulge/disk structure. The bulge and disk parameters are most affected for the case of an under-subtracted sky. This is largely due to a magnitude threshold of 26.5 mag arcsec - 2 in our decompo-sition algorithm. The data are too noisy below this value (cf. Fig. 2.8) and are excluded from the fits. This provides some protection against over-subtracted skies in the meas-urement of the disk scale length. Similar tests were performed with our 2D decomposition algorithm and corroborate, once again, the results above. The plots are not shown. 3.2.5 Sersic n Tests Initial Estimates A number of recent studies have described the variation of bulge shapes as a function of Hubble types (Andredakis et al. 1995, Moriondo et al. 1998, Khosroshahya et al. 2000, Mollenhoff & Heidt 2001), going so far as suggesting that precise values of n (i.e., ±0.1) could be determined (Graham 2001). However, to our knowledge, no study to date has tested the reliability of such decompositions to recover the Sersic n parameter reliably. In order to test the sensitivity of the decomposition to the full range of bulge profile shapes we use mock luminosity profiles with values of the Sersic n parameter ranging from n = 0.1 to n = 4.0. The suite of profiles used all combinations of re = 0.8, 1.5, and 2'.'5, u-e = 18, 20, and 22 mag arcsec - 2 and seeing FWHMs of 1.5, 2.0, and 2'.'5. The profile fits used initial estimates of n = 0.4, 1, 2, and 4, and correct initial estimates for r e , /xe, and the disk parameters. The seeing F W H M Chapter 3. Simulations of Bulge-to-Disk Decompositions 80 was fixed to the correct model value. The results are presented in Figs. 3.22-3.25, where we plot the average relative fit error on n (for the 100 profiles with the same simulated parameters) where An = n f i t ( m e a n ) ~ n m o d e l (3.24) ^•model versus the model n (the dashed Une at A n = 0 indicates a perfect recovery of the model n parameter.) Each panel represents one particular combination of pe and re, and the panels are arranged such that the B/D ratio decreases from top to bottom and right to left. The open symbols indicate the width of the distribution of Are within the 25th and 75 t h percentiles (i.e., 50% of the successful fits Ue between these two boundaries) and are connected with dotted and dashed-dotted Unes respectively. The different seeing F W H M values are represented by three point types: circles (red), triangles (green), and squares (blue) for seeing values of 1.5, 2.0, and 2'.'5 respectively. These figures clearly demonstrate that the bulges of even nearby late-type spirals are smaU and not sampled at high enough spatial resolution to yield a stable, robust solution for n as a floating parameter. Given the correct value of n as an initial estimate (along with the correct initial estimates for the other four parameters), the algorithm normally finds the correct value of the model n, but any departure from the model value by even a small amount, yields significantly erroneous solutions for n, or the fit may simply fail (as indicated by the vertical Unes in the figures). For most of the parameter combinations, an offset of ~50% in the initial estimate of n wiU yield a ~50% error in its determined value. Fig. 3.26 shows the corresponding results for a similar set of tests using the 2D algorithm. The 2D algorithm does a sUghtly better job at recovering the model n value given incorrect initial estimates, but, the recovery rate is stiU poor and the n value based on a floating initial estimate is highly questionable. Chapter 3. Simulations of Bulge-to-Disk Decompositions 81 Figure 3.22: Difference between modeled and recovered values of n are shown for a range of artificial profiles from n = 0.1 — 4. The Sersic exponent n is a free fit parameter and the initial estimate is set here to n = 0.4. Each panel shows the average relative fitted n errors (An = (njj t(mean) - ^ m o del) / n model) w i t h solid symbols and connected by solid lines versus the model n for the 9 combinations of re = 0.8, 1.5, 2'.'5, and pe = 18, 20, 22 mag arcsec - 2 . The open symbols show the width of the distribution of A n within the 25 t h and 75th percentiles and are connected with dotted and dashed-dotted fines respectively. Circles (red), triangles (green), and squares (blue) correspond to seeing values of 1.5, 2.0, and 2'.'5 respectively. The panels are ordered such that the B/D ratio decreases from top to bottom and right to left panels. Chapter 3. Simulations of Bulge-to-Disk Decompositions 82 Sersic n tests: init est n = 1.0 r i i i i l i i i i l i i i i l i i i i l T i t i i l i i i i t i t t i l i i i i l T t i i i l i i i i l i i i i l i i i i l * 1 2 3 4 1 2 3 4 1 2 3 4 model n Figure 3.23: Same as in Fig. 3.22 but with initial estimate n = 1. Chapter 3. Simulations of Bulge-to-Disk Decompositions 83 Sersic n tests: init est = 2.0 1 2 3 4 1 2 3 4 1 2 3 4 model n Figure 3.24: Same as in Fig. 3.22 but with initial estimate n = 2. Chapter 3. Simulations of Bulge-to-Disk Decompositions 84 Sersic n tests: init est = 4.0 Figure 3.25: Same as in Fig. 3.22 but with initial estimate n = 4. Chapter 3. Simulations of Bulge-to-Disk Decompositions 85 model Sersic n Figure 3.26: Residual difference for the 2D decompositions between modeled and re-covered values of n for a range of artificial images from n — 0.4 — 4. The Sersic exponent n is kept as a free fit parameter and the initial estimate is varied from n = 0.4 — 4. Chapter 3. Simulations of Bulge-to-Disk Decompositions 86 From these tests we conclude that the small bulges of late-type spiral galaxies are not sampled at high enough resolution to fit the Sersic ra as a free parameter in addition to the other bulge and disk parameters. The bulge profiles are defined by a very small distribution of independent intensity measurements and failure to recover ra as a free parameter should be interpreted as a limitation of the data information content. We suggest that the best-fit ra should be found by solving for a range of ra, holding it as a fixed parameter, and using the xln a s defined in § 3.1 to determine the best fit. Seeing Errors We know from § 3.2.3 that seeing errors can seriously affect the deter-mined bulge re and \ie. We must also consider the effect of seeing measurement errors on the shape of the bulge profile. To this end we re-modeled the profiles used in § 3.2.3 for the same five values of A F W H M , but this time letting ra float as a free parameter. The simulated profiles all have model n — 1 and were given TJ. = 1 as the initial estimate. We know from the previous discussion that the model ra will be recovered if (and only if) it was given the correct initial estimate. Accordingly, if ra = 1 in not recovered in any of these decompositions, we can conclude that the seeing measurement error had an effect on the shape of the bulge. The results are presented in Fig. 3.27. This figure is identical to Fig. 3.12 except that each panel shows the average error in ra, Ara (Eq. 3.24) on the y-axis. Indeed, if the seeing value is accounted for properly the correct ra is found. However, if an incorrect F W H M is used in the fit, the ra parameter is not recovered for all re < ( F W H M + 0.5) with A F W H M = 35%, and for all re > (FWHM) with A F W H M = 15%. It is difficult to quantify the error on ra here as the results from the previous section clearly demonstrated that if the initial estimate is not exactly equal to the bulge ra, the fitted ra can be very different from the true value. However, given that ra = 1 was not recovered for the simulations with seeing errors, we can conclude that the seeing did in fact change the apparent shape of the bulge and, hence, seeing errors must be accounted Chapter 3. Simulations of Bulge-to-Disk Decompositions 87 for when determining the best fit n for the bulge. Sky Errors We now turn our attention to the effects of sky subtraction errors on the bulge shape parameter n. The usual galaxy models are re-analysed but using an over- and under-estimate of the sky background determination of ±1.0%. The results are shown in Fig. 3.28. Again, since n is not always recovered when the sky is not properly accounted for, we can conclude that sky subtraction errors can affect the shape of the bulge. The recovery of n under the same 1% over- or under-estimate of the sky background worsens for smaller and fainter bulges, and for larger seeing F W H M values. The effect is negligible for bulges with re > ( F W H M — 0.5) and (ptky — Me) ~ 4 mag arcsec - 2 , but outside of these ranges, sky subtraction errors must be accounted for when determining the best fit n for the bulge. 3.3 Summary of Simulations The tests perfomed in this chapter allow us to define a set of guidelines for the reliability of our 1D/2D decompositions as follows: • Initial estimates for bulge and disk parameters are unimportant provided that re > (0.3) 1 / n * F W H M and r > -0.75n + 1.15 for n< 1.0 0.2n + 0.2 for n > 1.0 Seeing errors must be accounted for in all bulge parameter studies. Chapter 3. Simulations of Bulge-to-Disk Decompositions 88 Armw " -35% ^ T m u - -16% A^, - OX 1^, - + 15% A„,HM " +35% 0.5 -0.5 0.5 -0.5 0.5 <j 0 ^ -0.5 0.5 -0.5 0.5 -0.5 h I 1 ' 1 1 I 1 1 1 1 111 1 1 1 ' I 1 1 1 1 111 1 1 1 1 I 1 1 1 1 I 11 1 1 1 1 I 1 1 1 1 11 I ' ' ' 1 I 1 1 ' 1 I | I I I I | I I I I | I I I I | I I I I | | I I I I | I I I I | I I I I | I I I I | I i i i i I | I I I I | I I I I | | I I I I | I I I I | | I I I I | I I I I | | I I I I | I I I I | i • . . . i FWHM - 1 .0" | I I I I | I I I I FWHM - 1 .6" | I I I I | I I I I FWHM - 2 . 0 " | I I I I | I I I I | FWHM - 2 . 6 " I I I I | I I I I | FWHM - 3 . 0 " • i • ~ • i i"N 1 2 31 2 31 2 31 2 31 2 3 model r (") Figure 3.27: Effects of seeing errors on the Sersic n parameter, n is a free parameter for these tests (the model n = 1 for all profiles here). Each panel plots A n (Eq. 3.24) as a function of the model re. The curves and panels are as in Fig. 3.12. Chapter 3. Simulations of Bulge-to-Disk Decompositions 89 A r t y = - 1 % = 0 % Atfqr = + 1% < 0 . 5 - 0 . 5 0 . 5 - 0 . 6 0 . 5 - 0 . 5 0 . 5 - 0 . 5 0 . 5 - 0 . 5 h J T 1 1 1 , 1 1 1 1 | 1 i i i i | i i i i | FWHM - 1 . 0 " 1 1 */\y \> ¥ * M I i i I I 1 1 1 I i 1 i FWHM - 1 . 5 " 1 1 1 1 1 1 1 1 1 1 1 ^ V\\ i t r t Y l 1 1 1 1 1 1 1 1 1 1 1 /K/ N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i i FWHM - 2 . 0 " lYi t - / ^ ^ ' T M ^ l T FWHM = 2 . 5 " i . . . . i . . . . i FWHM = 3 . 0 " i . . . . i . . . . i 1 2 3 1 2 3 1 2 3 m o d e l r ( " ) Figure 3.28: Effects of sky subtraction errors on the Sersic n.n is a free parameter for these tests (the model n = 1 for all profiles here). Each panel plots A n (Eq. 3.24) as a function of the model re. The curves and panels are as in Fig. 3.20. Chapter 3. Simulations of Bulge-to-Disk Decompositions 90 • Sky subtraction errors dominate disk parameter errors and must also be taken into account for bulges whose effective surface brightnesses are less than one magnitude brighter than the sky brightness (i.e., for (fi,ky — He) ~ 1)-• The sampling and resolution of bulges for even nearby late-type spirals is not high enough to constrain the Sersic n exponent, n cannot be fitted as a free parameter. It must be held fixed and the full range of values should be sampled in steps and the relative % 2 n can be compared to determine the best fit. • The 2D decomposition technique does not provide a significant improvement over the ID method at recovering axisymmetric structural parameters of late-type spiral galaxies to warrant the extra computational time and effort. Armed with these basic guidelines we can now turn our attention to actual data, focussing on the ID technique. Chapter 4 Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 4.1 Outline We now apply our tools for bulge-to-disk decompositions to real images of late-type spiral galaxies in order to determine their structural properties and the variation of galaxian parameters as a function of wavelength. The Courteau-Holtzman data set used for this analysis is described in § 2. This sub-sample includes 123 late-type spiral galaxies with face-on and intermediate inclinations for a total of 523 images. Edge-on systems that are part of the larger data set sample will be used later for an analysis of stellar and dust scale heights and truncation radii in spiral disks. Most galaxies have at least one set of B V R H images, and we use multiple observations for 54 galaxies to estimate systematic errors. This multi-band sample is the largest of its kind to date for this sort of analysis. Other B / D decomposition analyses have used larger samples (e.g., Baggett et al. 1998) but lacked the crucial multi-wavelength information. Reliable measurements of disk and bulge scale parameters, with errors less than 10%, are paramount for meaningful comparisons with any given bulge/disk formation scenario. In a general sense, we seek to develop a stable and versatile prescription to characterize structural evolution of the bulges and disks of galaxies. However, just as any morpholo-gical description of galaxies (e.g., Hubble types) depends on the wavelength of detection, intrinsic structural parameters are also expected to vary with wavelength due to stellar population and dust extinction effects. Thus, a lack of multi-waveband information could 91 Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 92 potentially impede any attempt to achieve an accurate description of galaxian structure parameters for any one galaxy. Physical differences among galaxies of bulge shapes and sizes are also expected de-pending on how they were formed. Formation by accretion processes can account for steeply rising de Vaucouleurs light profiles in the central parts of galaxies, while secular evolution would yield exponential distributions, with (ra < 1) or without (n > 1) a core, of the central light (see § 1.) Our analysis should provide new constraints on these pro-cesses and effects. It is fair to say that, as of this writing, the formation of small bulges is largely attributed to secular processes and redistribution of disk material (Courteau et al. 1996, Zhang &; Wyse 2000). The present study should confirm this picture but also raise it to a new level of precision with more robust analysis tools and a unique data base. Prior to this study, some of the best analyses of B / D parameters in disk galaxies have relied on de Jong's 1996 multi-band thesis sample of 86 face-on spirals. De Jong's ID and 2D B / D decompositions established significant parametric variations at different wavelengths (most notably the decreasing disk scale length at longer wavelengths). His B / D fits used a fixed Sersic ra parameter as he realized, as have we (see § 3.2.5), the intrinsic limitations of the data modeling (i.e., over-determination of the parameter space.) De Jong used fixed values for ra = 1, 2, and 4 and concluded that late-type bulges are best described by an exponential, ra = 1, profile. A clear correlation of bulge/disk scale lengths, hb/hd = 0.1 ± 0.05, was found by de Jong (1996), and Courteau et al. (1996) with a different data set, who also adopted ra — 1 for bulge parameterization of late-type disk galaxies. Evidence for this correlation has been challenged by Graham (2001) who re-modeled de Jong's thesis galaxies with a ID B / D decomposition technique and using an uncon-strained ra. His results support a wide spectrum of Sersic shape parameters for any Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 93 given Hubble type, in agreement with recent N-body simulations (Scannapieco Sz Tissera 2002). However, aware of the inadequacy of basic B / D decompositions to fit for the bulge shape parameter due to poor data resolution and strong covariances with other bulge parameters (§ 3.2.5, de Jong 1996, Broeils & Courteau 1997), we felt that it was important to revisit this issue. Our approach involves B / D decompositions with fixed n values that sample the full parameter space of spiral bulges, from n = 0.1, 0.2, ...,4.0. Final solutions are filtered out on basis of relative x2 a n d other exclusion criteria determined in the previous chapter. This section is thus concerned with the derivation of robust B / D parameters for each galaxy profile. We shall compare our results with Graham's (and others) and test for any B / D parameter correlations in § 5. Note that this chapter focuses exclusively on results from ID B / D decompositions. The fact that we do not model non-axisymmetric shapes (bars, rings, oval distortions) lessens the need for more computationally intensive 2D B / D decompositions. We shall return to the comparison of ID and 2D parameters in the published version of this chapter. 4.2 Initial Estimates In order to determine the range of best fitted bulge and disk parameters, we need to assist the minimization program in finding the lowest possible model-data \ 2 - From analysis of our mock images and profiles, we found that any reasonable initial estimates for the disk parameters (e.g., marking the disk, fit by eye) yields a robust solution. Flexibility in the choice of bulge initial parameters is, however, only afforded outside a certain range of bulge sizes relative to the seeing disk. Moreover, observed galaxy profiles show significantly more variety than the idealized profiles from which these conclusions were Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 94 drawn (e.g., we did not model Type II galaxies). Accordingly, we explore at least three different sets of initial bulge parameter estimates to protect against local minima in the parameter space. Initial estimates for the disk parameters h and fi0 are based on the "marking the disk" technique (§ 1.3.1). The fit baseline extends from 20% to 100% of the radius at which the surface brightness error has systematically reached values greater than 0.12 mag arcsec - 2 The inner boundary is chosen to exclude the major contribution of a putative bulge or Type II dip. Initial bulge effective parameters were determined from: • Subtraction of the disk fit (based on the "marking the disk" technique) from the original profile leaving only the bulge light. re is then computed non-parametrically from the data by summing up the light up to the radius which encloses half the total light of the bulge. Thus fie = H(re)-• re — 0.15h, and fie = fio • re = (i n/log(e)) * 0.15A; fie = (bn - bn=1) + fi0. The second set of initial estimates was motivated by the bulge/disk correlation found by Courteau et al. (1996) 2 . Given that we are fitting for a range of Sersic ra's, we added the third set of initial estimates which attempt to scale re and fie more appropriately to the different values of n. No specific set of initial estimates worked best in all cases, though the 3 r d method may be the least attractive. It failed to provide reliable solutions in most cases, but in a few cases also provided the only viable solution. 1We tested various choices for the minimum and maximum fit ranges for our galaxy profiles including: full profile fit, fit from 20% and 40% to the end of the profile, a boxcar fit of fixed length which was shifted along the length of the profile and fit at 8 different locations, and Willick's moments method. The discrepancies between the different fits are large (~10% on average and up to ~100% for the worst cases), as expected, though the 0.2rmax to rmax baseline resulted in the most stable fits. 2Note that Courteau et al. (1996) quote the ratio in terms of the exponential scale length for the bulge and disk. We have scaled their value to the effective radius for the bulge according to Eq. 3.9. Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 95 4.2.1 Seeing and Sky Treatment "Bulges" of late-type spirals are small and their luminosity profile can be severely affected by atmospheric blur. In principle, if the blurring from the atmosphere (seeing) can be measured accurately, it can also be corrected by Fourier transform deconvolution. In practice, however, deconvolution amplifies noise in the profile, and the seeing F W H M is subject to measurement errors. In § 3 we used extensive simulations with a wide range of input parameters and various values of n to derive a space of recoverable parameters under specific seeing conditions, accounting for the typical measurement errors of our data. Seeing is accounted for by convolving the model profiles with a Gaussian PSF whose dispersion is measured from field stars. Typically, 5 to 40 average-luminosity (non-saturated) stars were used in each image PSF measurement. The typical seeing is 2'.'0 ±0.7 (see Fig. 2.4) with standard excursions of ~20% (optical) and ~35% (IR). In order to account for seeing measurement errors, each profile is modeled with three different values of the seeing F W H M : the nominal measured value and ±15% of that value. A mean seeing uncertainty of ±15% was used rather than the individual errors per measurement as these fluctuate greatly due, in large part, to the different number of stars in each measurement. Sky subtraction errors, another critical source of uncertainty in B / D decompositions, were also examined carefully (§ 3.2.4). The measured sky brightnesses in each passbands are shown in Fig. 2.3. Typical rms sky errors are ~0.5-1.0% in the optical and ~0.05% in the IR. The sky brightness measurement error is accounted for in B / D decompositions by using three different sky levels: the nominal measured value and ±0.5% of that value. Note that our profiles are already sky subtracted (Holtzman & Courteau 2002); our sky uncertainty test simply consists of subtracting/adding the extra 0.5% of the nominal sky value. Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 96 Accounting for seeing and sky statistical measurement errors is a most crucial part of the decomposition that is often ignored. We will see below that our solutions favour sky and seeing estimates that actually differ from our nominal estimates, thus yielding a slightly different set of structural parameters. Whether these offsets are physical or shear numerical artifacts remains to be clarified (§ 4.4). Note also that these tests would be extremely time-consuming for 2D decompositions. De Jong (1996) only included sky errors in his ID profile decompositions to determine parameter uncertainties. So did Graham (2001). The effects of seeing errors have also been addressed by Mollenhoff and Heidt (2001) but no details of their treatment were given. 4.3 Data Filtering Each optical (BVR) profile is thus reduced 3 times for each different combination of r e , pe initial estimates, times 3 seeing F W H M values, times 3 sky values, and times 40 different fixed values of n, for a total of 1080 decompositions. Sky measurement errors in the H-band are small so we did not test for statistical sky variations, yielding a smaller set of 360 solutions for each H-band profile. The 1080 (360) decompositions for each profile are first vetted on the basis of struc-tural criteria determined from our simulations (§3) . A decomposition is deemed accept-able if it meets the following criteria: re £ (0.3)1/" * F W H M and re > { -0.75n + 1.15 for n < 1.0 0.2n + 0.2 for n > 1.0 (fiaky — Me) < — 2 (optical B V H passbands only) • B/D < 5 • < 15 kpc ; re < 50 kpc Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 97 • re/h > 1 The first two constraints are derived from our simulations and effectively eliminate the faint and/or small bulges whose sizes are comparable to, or smaller than, the seeing disk. The remaining constraints are based on physical considerations and help eliminate solutions with small \ 2 values but unrealistic parameters for late-type galaxies. Note, however, that these physical constraints are rather generous and do not contribute any subjective bias. The successful decompositions are then ranked on the basis of two indicators: (a) a global x2, X2gii computed for all data from r — 0 to rmax; and (b) an inner x2, X2ni which includes only the central regions of the galaxy from r = 0 to twice the radius, rb=d, where the intensities of the fitted bulge and disk are equal. xln was adopted to increase the sensitivity of the goodness-of-fit indicator to the bulge area 3 . The radius rj,^ is clearly a function of the bulge shape and may change from a small to large n (see e.g., Fig. 3.1). Thus we use a %2 per degree of freedom to remove any dependence of the normal %2 to a changing re. Because of the presence of spiral arms (bright foreground stars removed) and other non-axisymmetric features which we do not attempt to model, the reduced X2 is always, in principle, greater than unity. However, some of our solutions may have X2 values less than unity which is indicative of an over-determined system (correlated parameters). We preserve only the better half of the decompositions based on their xgi- The reduced set is then ranked according x\n v a l u e s a n d the bottom half of the distribution is discarded. This process is iterated at least twice, or until we reach 50 or fewer solutions. Solutions with x2gi greater than 50 in this final subset are discarded. Ideally, the minima for the distributions of xgi a n d Xin values should agree to a 3Note that the algorithm minimizes the x2,! only. The x2„ is calculated and used as a discriminator only after the algorithm has converged. Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 98 common value of n, but differences may exist. We search the final < 50 solutions for a common x2 minimum, starting at the minima of each %2 distribution. If the n values corresponding to the two x2 minima do not agree, the solutions for the next smallest X 2 values are compared, and this process is iterated until a match is found, up to three times. If this process did not converge, i.e., there is no true minimum in the (x2au Xin) " n space), a final solution is chosen corresponding to the minimum value of (x2j/min(x2j) + X?n/min(x2 n))- This grid search was deemed necessary due to the cross-talk between the fit parameters. Figs. 4.1 and 4.2 show example distributions of the (x2ai, Xin) v e r s u s n where Xgiobai1 = X2gi/™™{x2gijiit) and x 2 n n e p ' = Xin/™™(xL,filt), w h e r e ™™(x2ut)is t h e minimum X 2 value from the set of (< 50) filtered solutions. These minima do not necessarily correspond to the lowest x2 oi the respective distributions from all 1080 points, as the initial absolute minima may have been filtered out (i.e., poor combination of xgi-, Xin f ° r a g i y e n solution). Thus, the normalized %2s may be less than one, as seen in the leftmost plot of Fig. 4.2. In these figures the left panels show the x2 distributions for all 1080 decompositions, while the right panels display only the < 50 solutions remaining after the iterative filtering scheme described above. Fig. 4.1 highlights the sensitivity of our technique for two V -band observations of UGC 929 taken under different seeing/sky conditions. The left figures show a fairly well-behaved solution favouring n = 0.6 and the figures on the right plot show a rather messy solution favouring n = 0.8. The seeing conditions were worse and the sky was much brighter for the observation shown on the right which could explain the noisy distributions of both the xgi a n d Xin-Fig. 4.2 shows two different behaviors of x2gi f ° r profiles with very well-behaved xln-The plot on the left for our UGC 784 B-band profile illustrates the need for an additional, more discriminating statistic for the bulge region. Decompositions based solely on the X2gi goodness-of-fit indicator may result in fits, like the one shown on the right side of Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 99 0 1 2 3 4 0.4 0.6 0.8 1 1.2 0 1 2 3 4 O.S 0.7 0.8 0.9 Sersic n Sersic n Figure 4.1: Examples of % 2 n n e r / (open triangles) and xgiobai1 (filled squares) versus Sersic n distributions for the 1080 decompositions of two different V-band observations of the same galaxy (UGC 929). In the two sets of plots, the left panel displays all 1080 points and the right panel shows only the (< 50) points remaining after iterative filtering. The set of plots on the left shows a reasonably well-behaved solution favouring n = 0.6 while the set on the right shows a rather noisy solution favouring n = 0.8. Fig. 4.3 (dashed-dotted blue line), with unrealistic bulges. Our treatment which includes both the Xin a n d X^i statistics yields solutions that conform with expected trends (even though there is no a priori restriction on the physical size of the bulge). Our algorithm protects against bulges that can be as bright or brighter than the disk in the outskirts of the galaxy. This is in agreement with deep photometry of edge-on galaxies, which fails to reveal signatures of overwhelming bulges at faint surface brightness levels (Zheng et al. 1999). Visual inspection of the final decompositions is the final step of this filtering procedure. The criteria for this user examination include information from multiple exposures and multi-band reductions for a given galaxy. Profiles and/or solutions with the following pathologies were eliminated from the final sample : • disk profile too short for proper fitting Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 100 0 1 2 3 4 0.6 0.8 1 1.2 1.4 0 1 2 3 4 0.3 0.4 0.0 0.6 0.7 Sersic n Sersic n Figure 4.2: Examples of xLner' a n d X^iobai1 distributions for a solution with a well-behaved Xinner'i D u t a n a t X^iobai1 distribution ( U G C 784 B-band) (left) and for a very well-behaved solution in both %2 distributions ( U G C 929 B-band) (right). • no obvious, extended, underlying exponential structure for the disk (occurs pre-dominantly in Type II profiles) • unphysically large fitted bulge • unrealistic disk fit for Type II profiles. The fit is tilted below the true disk to account for the Type II dip near the bulge-disk transition region leading to scale lengths that are biased high (e.g., see Fig. 4.6 for U G C 12527 for a series of "bad" fits which were eliminated from the final sample) • large deviations between multiple observations of a given galaxy. Not surprisingly, most of the eliminated profiles are Type II. We caution that even those Type II profile decompositions that survived the full sorting process do not provide ideal descriptions of the complex surface brightness features of Type II profiles. It is clear that these Type II profiles can not be properly modeled with only a Sersic bulge and exponential disk; a new approach that includes the effects of dust extinction is required. Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 101 R a d i u s ( k p c ) 0 5 10 16 20 1 I  || p 1 Jj M UGC 784 1 M 1  | | ' • II II 1 d i s k Mo * 21.0 h = 1 1 1 8.8 sr b a n d , . b u l g e M, = 22.0 r . = 0 9 o S 22 '_ \ n = 0.6 arc: FWHM - 1 . 8 " M * 24 Sky = 22.2 £ 26 l • • • l , , , , l , , , , l 0.4 r r •1 1 1 | 1 1 1 1 | 1 1 1 1 | 1 1 1 1 | 0.2 i .o i 0 -0.2 V""" " ^ " ^ hiatal 16.0 j -0.4 r i , . , 1 , . , , 1 . , , , 1 . . , ". 1 R a d i u s ( k p c ) s 10 15 0.4 0.2 0 -0.2 -0.4 UGC 784 B b a n d , d isk Ho - 21.1 h - 7.8 bulge / i . = 28.4 r. " 1 2 8 . 8 n - 3.9 W XL»r • 15.0 ' >4*a " 10.5 20 30 R a d i u s ( " ) 20 30 R a d i u s ( " ) Figure 4.3: Comparison of different bulge fits for the same profile (UGC 784 B-band). The plot on the right has a bulge fit (dashed-dotted blue line) which is likely unphysical. Its x\ii however, is lower than that of the decomposition on the left plot, whose bulge fit looks more realistic. Without adopting the x | n statistic, the plot on the right is favoured. Using the xjn m addition to the x\i a s a discriminator, the plot on the left is favoured. (See lef plot of Fig. 4.2 for the corresponding x2 v s - n distributions.) 4.4 Preferred Sky and Seeing Fig. 4.4 shows histograms of the preferred seeing F W H M and sky offsets for all decom-positions in all four bands. The H-band sky error is negligible and was not modeled. Typically, a lower sky brightness level is preferred by our algorithm. In most cases, this can be explained by an over-estimated sky level, but this may also be due in part to profiles with genuine truncated outer disks as in Fig. 4.7. The seeing F W H M is typically under-estimated and our program prefers a slightly larger value on average. We offer no explanation for these trends at present, but remind ourselves that solutions with variable sky/seeing estimates were retained in the final solution set. Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 102 80 60 i m E 3 40 B -band V - b a n d R - b a n d H - b a n d 20 h -0.5 0 . 0 0 . 5 - 1 5 15 Jsky (%) FWHM (%) Figure 4.4: Histograms of seeing F W H M and sky offsets preferred in our analysis for all profiles surviving the final cut, separated into the four different bands. Note that no sky variation was allowed for the H-band profiles as the error in the H-band sky measurememt is negligible. 4.5 Decomposi t ion Examples Not all galaxies were successfully decomposed into bulge/disk models. A large fraction of Type II profiles could not be modeled adequately by our procedure, a somber fact we return to in § 5. Out of 523 images/profiles, a total of 376 passed our acceptance criteria. There is no room for a full display of our catalog of all final (pass/reject) decompositions, but a few examples are shown in Figs. 4.5-4.8. In these figures, the solid black circles in the upper panel are the data points, the black dots show the sky error envelope (from the measured sky error), the dashed green and blue dashed-dotted lines show the disk and bulge fits respectively, and the red solid line is the total (bulge+disk) fit. The fits are all seeing-convolved using the best selected Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 103 seeing values. The bottom panel shows the fit residuals where A/x(r) represents the data minus the model. Fig. 4.5 shows an example of the quintessential Type I profile at all wavelengths; all solutions are very well-matched. Note, though, that the discrete and noisy nature of these data often prevents \ 2 values close to 1. Fig. 4.6 shows a Type II-Transition galaxy whose Type II signature significantly weakens from the optical to the infrared. Fig. 4.7 shows a Type I profile with a truncated disk. Such decompositions will presumably favour an under-subtracted sky in attempt to align the inner and outer parts of the disk. Here is an example where our procedure with an over/under-estimation of the sky may not be adequate since the disk truncation is real. We conclude this display of final B / D decompositions with the example of a nearly bulgeless system in Fig. 4.8. 4.6 Effect of r m a x Of potential importance to the fit results is the maximum radius used in the decompo-sitions. For all of the above fits we have used the entire profile out to radii where the surface brightness errors systematically reached values greater than 0.12 mag arcsec - 2 . Although this may not be the ideal criterion for r m a x , and a different selection could po-tentially yield different results, we have explored other alternatives and this one seemed most reasonable and stable. To test the sensitivity of our parameter determinations to the chosen value of r m a x we re-decomposed the profiles following exactly the same proced-ure for initial estimate, seeing and sky values, and filtering as described above, except that the fit baseline extended to only 0.75 x r m a x . A comparison of the results from the two techniques, prior to any eyeball filtering is shown in Fig. 4.9. (The few points lying on the x and y axes indicate that no suitable fit was found and all parameters were set to zero.) The agreement between parameters is quite good for the most part. A slight systematic trend appears for the disk scale lengths such that the shorter baseline fits Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 104 R a d i u s (kpc) 0 2 4 6 8 0 20 40 R a d i u s ( " ) R a d i u s (kpc) o 2 4 6 8 10 R a d i u s ( " ) R a d i u s (kpc) 0 2 4 6 6 T — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — r 0 20 40 R a d i u s ( " ) R a d i u s (kpc) 0 20 40 60 R a d i u s ( " ) Figure 4.5: Decomposition results for a Type I galaxy (UGC 9908). In the upper panels, the data points and measured sky error envelopes are shown with solid black circles and dots respectively. The dashed-dotted blue and dashed green lines show the bulge and disk fits respectively, and the red solid line is the total (bulge+disk) fit. The fits are all seeing-convolved using the best selected seeing values. The bottom panel shows the fit residuals where A/x(r) = data(r) — fit(r). Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles R a d i u s (kpc) 10 1 5 UGC 12527 B b a n d . . d i s k * 9 h = 11.3 b u l g e M. - 22 I R . " 23 n - 0.4 20 40 R a d i u s ( " ) R a d i u s (kpc) 10 16 R a d i u s ( " ) R a d i u s (kpc) 10 16 R a d i u s (kpc) 6 10 « 18 s 0.4 02 0 -0.2 -0.4 • 1 1 1 1 1 1 1 : UGC 12527 1 1 1 1 1 1 d i s k ii, - 17.1 h 1 1 - 12.3 A H b a n d , w b u l g e /x. = 16.7 r , n - 2.2 - 1.0 FWHM - 0 . 9 " \ \ 1 1 1 1 1' 1 1 t 1 1 1 S k y 1 I 1 1 1 1 1 1 1 • 13.9 1 1 1 1 Am " 8.0 ~. ' 1 1 . . . . 1 *v . . ' 3.3 : 20 30 R a d i u s ( " ) Figure 4.6: Decomposition results for a Type II/Transition galaxy (UGC 12527). Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles R a d i u s ( k p c ) 1 0 R a d i u s ( k p c ) 10 16 2 0 4 0 R a d i u s ( " ) R a d i u s ( k p c ) 0 5 1 0 1 5 2 0 2 5 1 8 1 1 n i i j , i . n i , i ;|| i UGC 1937 ' ' ' 1 1 disk Mo - 19.6 h i i i 1 i - 16.6 M R b a n d „ bulge M. " 20.0 r. - 2.6 o 8 2 0 O fa • ri F W H M - 1.1 - 1.4" ? 2 2 - \ sky = 20.8 \ 2 6 H 1*1 1 1 1 1 1 i i i 1 i i i 1 i 0 . 4 -J 1 1 | 1 1 1 1 i i i 1 i i 0 . 2 1 o - 0 . 2 A <~-- 1.9 -. - 4.2 j - 0 . 4 •1 . . i . . . -i o 2 0 V g CD 2 2 c C Ul) 2 4 2 6 0 4 0 . 2 0 - 0 . 2 - 0 . 4 . 1 1 1 ' ' 1 ' ' : UGC 1937 disk Mo = i , • , 20.3 h -i | i i 1 8 4 • \ V b a n d M bulge Ma • 20.6 r. - 2.9 n = F W H M « 1.1 1 4 " \ \ V \ 1 1 1 * 1 1 1 I I 1 I Sky = i i 1 21.3 J 1 1 1 1 1 r. • •• • 1 1 | 1 1 1 1 3.1 -. " i . i I % global _ I .V. I 5 . 8 j 0 2 0 4 0 R a d i u s ( " ) R a d i u s ( k p c ) 6 0 0 6 1 0 1 5 2 0 UGC 1937 H b a n d disk Mo - 169 h - 14 2 bulge M. = 16.6 r, « 2.6 4 0 R a d i u s ( " ) 2 0 4 0 R a d i u s ( " ) Figure 4.7: Decomposition results for a galaxy with a truncated disk (UGC 1937) Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles R a d i u s (kpc) R a d i u s (kpc) 2 4 1 ' 1 1 1 UGC 10757 1 disk M o - 21.5 h = i 14.3 ^ ^ > J 3 b a n d ^ . bulge M. " 24.1 r, = 2.0 n « 0.1 F W H M - 3.1 » Sky = 22.3 • 1 i i i 1 i i i 1 J < i i | i 1 1 1 -1 1 j 0.8 -: _ 3.1 j r i . . i I i ~ 0 20 40 60 R a d i u s ( " ) R a d i u s (kpc) 2 4 _ 1 UGC 10757 I I | i i i | . disk M o - 20.3 h - 13.0 20 JT W R b a n d „ bulge M. • 22.9 r. = 3.0 0 OJ n - 0.4 n u 22 fi F W H M - 1.9" M fi 'Slfcs, Sky = 20.9 E 5 2 4 26 - | ' i i 1 i I I 1 l l I 1 I 0.4 -J i i i | i 1 1 | 1 1 1 | 1 ^ 0.2 i- Xi»- " 1.2 1 4 n <1 0 -0.2 -0.4 r l i I i 1 0 20 40 60 R a d i u s ( " ) R a d i u s (kpc) 2 E . 1 , , I I , I 1 1 UGC 10757 i 1 i i i disk M o " i | i i 18.4 h 1 1 | 1 - 10.4 - ^ ^ ^ H b a n d a r bulge M. 21.3 r . - 2.8 n = 0.1 F W H M - 2.7 " • Sky = 14.2 • , I I • i • i • l l 1 1 1 1 1 1 1 L | 1 1 l 1 | l l l | 1 1 1 1 1 1 - 0.2 -. • i . . . i . . . 1 , , , ^$Ii«lMl 1 . . = 1.3 j , i . ,• 20 30 R a d i u s ( " ) Figure 4.8: Decomposition results for a galaxy with a bulgeless disk (UGC 10757) Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 108 have bigger disk scale lengths. This can be attributed to the large number of profiles with outer truncated disks; a more extended baseline at large radii would include the disk truncation and yield a steeper profile fit. 4.7 Distribution of the Sersic n parameter Fig. 4.10 shows histograms of the Sersic n parameter from our decompositions for all of the fits (left) and for the good fits only (right), after user examination as described above. The distribution of n has a definite range, implying that not all late-type bulges are adequately described by an exponential profile, but the mean value is very close to one, especially for the final, user filtered, distribution. Implications from this result for galaxy structure evolution models will be discussed in § 5.2.1. 4.7.1 Floating Sersic n In § 3.2.5 we showed that data limitations prevented stable fitting of the Sersic n as a free parameter in our B / D decompositions. To illustrate the effect a floating n can have on fitted parameters we re-decomposed all of our galaxy profiles leaving n as a free parameter. The results are shown in Fig. 4.11 for three different initial guesses for n (0.2, 1.0, and 4.0). The histograms of the resulting distributions of n reveal a strong bias towards the chosen initial estimate. A l l 3 distributions show a large peak at n = 0.1, indicative of poor bulge fits. The histogram for the n = 1.0 initial estimate looks some-what similar to our own constrained solution (Fig. 4.10), but this is somewhat fortuitous given the closely-exponential nature of spiral bulges. Note also the non-Gaussian tail in Fig. 4.10 not reproduced in Fig. 4.11 for the n = 1 initial estimate case. A closer look at the differences between the two methods is shown in Fig. 4.12. The scatter is quite significant, particularly for the bulge parameters (all plotted in the right column). Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 109 Figure 4.9: Comparison of model parameters for two different fit baselines: 0.75 x rmax versus rmax. The colours and point types are as follows: Type I (blue), Type II (red), Trans (green), B-band (triangles), V-band (squares), R-band (pentagons), H-band (as-terisks). Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 110 Histograms of Best Sersic n (oil 3 sky and seeing: all fits) 40 i -o £ *30 20 nJ i ' * • • i * ' 1 1 1 * * 1 1 1 1 1 • total IV 0.81 II- 0.51 N - 523 t y p e I H - 1.10 o - 0.50 N - 240 - - type II H - 0.73 o - 0.42 N - 195 trans M- 0.76 o - 0.53 N — 88 0 0.5 1 1.5 2 2.5 3 3.5 4 Sersic_n Histograms of Best Sersic n (all 3 sky and seeing: good fits only) 1 • 1 1 I 1 • • • I 1 1 1 1 I 1 • 1 total (i- 0.94 o- 0.48 N - 376 t y p e I Ii- 1.06 o- 0.47 N - 219 - - type II ( i - 0.70 o - 0.34 N - 105 trons H - 0.91 o - 0.57 N - 52 30 h u i . . . . i • • • - . i . n T , . . i . -0 0.5 1 1.5 2 2.5 3 3.5 4 Sersic_n Figure 4.10: Histograms of Sersic n parameter for "final" solutions (left), and the reduced set of solutions after further visual examination (right). Note the huge discrepancies in \in a n d xii between the two methods. Thus, while the final destributions for the n = 1 initial estimate case and our constrained n procedure look similar, extraordinary differences may exist between individual decompositions. The sampling approach we have used gives us confidence about the stability of our results. Fig. 4.12 shows that our extra efforts are justified. 4.8 Error of a Single Measurement An important test of the robustness of any decomposition technique is to see if parameters are reproduced for different observations. Our sample includes 54 profiles for which we have multiple (2-4) observations, allowing for a direct measure of the reliability of our decomposition results. Table 4.1 lists the mean and mean standard deviation of the five Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 111 Histograms of Sersic n: n FREE PARAM (observed sky and seeing only: init est n - 0.2) I 1 1 1 1 I 1 1 1 ' I 1 1 1 1 I 1 1 1 1 I 1 1 ' total n- 0.42 <r- 0.35 N - 343 type I Ii- 0.46 o- 0.37 N - 170 - - • type II H - 0.38 o- 0.34 N . 108 trons H« 0.36 u- 0.29 N = 65 I — . . — I .... I .... I 0.5 1 1.5 2 2.5 3 3.5 4 Sersic_n Histograms of Sersic n: n FREE PARAM (observed sky ond seeing only: init est n - 1.0) 40 £ 30 J l ' 1 1 I total Ii- 0.89 c- 0.43 N . 407 type I M- 1.02 c- 0.32 N - 196 - - • type II H - 0.75 <T- 0.47 N . 136 trans )L- 0.77 0- 049 N - 75 E 0 0.5 1 1.5 2 2.5 3 3.5 4 Sersicn Histograms of Sersic n: n FREE PARAM (observed sky and seeing only: init est n-4.0) I 1 1 1 1 I 1 1 1 1 I 1 E 4 total Ii- 1.66 a- 1.78 N - 18 type I It- 1.86 a - 2.02 N - 7 type II ix- 1.95 o- 1.73 N - 6 t r a n s (i= 1.02 a- 1.68 N - 5 • I. ' ilk i ' 0 0.5 1 1.5 2 2.5 3 3.5 Sersicn Figure 4.11: Histograms of Sersic n parameter fitting n as a free parameter in the de-compositions. Results using three different values for the initial estimate of n are shown: n — 0.2 (top left), n = 1.0 (top right), n — 4.0 (bottom). Note the different y-axis scales in each of the plots. The selection criteria for the fits is as described in the text. Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 112 Figure 4.12: Comparison of parameters: floated n versus fixed ra. The colours and point types are as follows: Type I (blue), Type II (red), Trans (green), B-band (triangles), V-band (squares), R-band (pentagons), H-band (asterisks). Note the different axis scales for Xin-Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 113 model parameters from repeat observations with, n-l 1/21 N J N (4.1) where x is the fit parameter, n is the number of observations for a given profile, and N is the number of profiles with repeat observations. n crn r*7 ( m a g / D * ) ( k p c ) (mag / o ' ) h ( k p c ) N I B 1.00 0.13 22.06 0.19 1.13 0.07 21.08 0.03 5.35 0.18 7 V 0.98 0.07 20.31 0.27 0.81 0.09 20.10 0.03 4.06 0.08 7 R 0.93 0.18 20.16 0.21 0.83 0.18 19.78 0.05 3.93 0.10 14 H 1.17 0.24 18.29 0.40 1.26 0.33 17.76 0.18 3.62 0.27 8 Total 1.01 0.16 20.14 0.26 0.98 0.17 19.65 0.07 4.16 0.15 36 II B 0.72 0.17 22.94 0.28 1.05 0.36 21.55 0.06 3.60 0.13 2 R 0.61 0.19 20.10 0.34 0.41 0.05 19.90 0.02 3.01 0.02 4 H 0.65 0.28 17.83 0.06 0.78 0.04 17.72 0.11 5.20 0.42 2 Total 0.65 0.21 20.24 0.26 0.66 0.13 19.76 0.05 3.71 0.15 8 Tr V 0.65 0.14 21.56 0.27 0.78 0.11 20.84 0.05 5.00 0.14 2 R 0.65 0.11 20.31 0.33 0.71 0.37 19.72 0.03 4.51 0.06 5 H 1.12 0.31 18.39 0.20 0.62 0.07 17.61 0.03 2.57 0.09 3 Total 0.79 0.18 19.98 0.28 0.69 0.06 19.31 0.04 4.03 0.09 10 Table 4.1: Table of mean values and mean rms deviations for repeat observations. There are not enough statistics to see any clear trends in Table 4.1 for the different passbands and profile types. The average errors from repeat observations for Type I profiles are ± 16% for n, ± 0.3 mag arcsec - 2 for /xe, ± 18% for re, ± 0.1 mag arcsec - 2 for /Xo, and ± 4% for h. Clearly the disk parameters are much more stable than those of the bulge. Based on image simulations and J H K images of 40 bright spirals, Mollenhoff and Heidt (2001) estimate that recovery errors from their 2D B / D decompositions on all Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 114 standard fit parameters (Id, h, Ie,re, and n) are less than 15%. They too have tested for wrong estimates of the sky level, seeing width, etc (although no clear description of their technique was offered). Their error figure is slightly more optimistic than ours (judging from Table 4.1) but the techniques differ in our use of a constrained n which they keep free, and perhaps also in the treatment of sky and seeing measurements. 4.0 Comparison with Other Authors As mentioned in § 2.3 we have three galaxies in our sample in common with de Jong's 1996 thesis sample. Our respective profiles are plotted together in Figs. 2.5 and 2.6 which reveal excellent calibration (zero-point) and overall shape agreement between Courteau-Holtzman and de Jong. However, our decompositions have arrived at somewhat different parameters. Table 4.2 lists the B and H-band results from de Jong's and our own ana-lyses. For U G C 463 and U G C 3080 our disk scale lengths are systematically longer that de Jong's. We have confirmed that possible sky under/over-estimates are not responsible for this difference. A closer look at Fig. 2.5 reveals Type II-like dips in the profiles for these two galaxies. The fact that de Jong's disk scale lengths are shorter than our own suggests that his algorithm gives more weight to the outer part of the disk. U G C 3140 has a more clearly defined exponential disk profile, and our results are in slightly better agreement. We cannot directly compare our parameters with Graham (2001) for the same three galaxies as he does not provide individual decomposition results. Recall that Graham (2001) re-decomposed de Jong's (1996) thesis sample, but fitting n as a free parameter. We can, however, broadly compare our respective distributions of n with morphological type. In Fig. 4.13 we plot our best fit Sersic n versus the galaxy's morphological type index, to be compared with Graham's Fig. 10. The general features are similar, but we Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 115 Author U G C band po h pe re n de Jong us 463 B 20.76 13.5 20.60 1.3 1 463 B 20.93 14.2 20.49 1.1 0.4 de Jong us 463 H 16.80 12.0 16.73 1.9 1 463 H 17.12 14.1 16.58 1.6 0.6 de Jong us 3080 B 21.99 17.2 19.88 0.2 1 3080 B 22.34 24.3 23.09 2.0 0.5 de Jong us 3080 H 18.21 15.1 18.79 1.7 1 3080 H 18.34 18.3 19.29 2.8 0.9 de Jong us us 3140 B 20.90 13.1 20.46 2.0 1 3140 B 20.88 12.9 21.11 2.8 1.1 3140 B 21.02 13.6 21.64 3.6 1.4 de Jong us 3140 H 16.96 11.3 15.99 2.1 1 3140 H 17.32 11.8 17.40 4.1 1.9 Table 4.2: Parameter comparison between us and de Jong (1996). Note that the surface brightnesses shown here have not been corrected for Galactic or internal extinction in order to compare with de Jong. Repeat observations are also listed for U G C 3140 B. may have a wider range of n for the later type galaxies where we have a larger number of galaxies in our sample. Chapter 4. Bulge-to-Disk Decompositions of Galaxy Surface Brightness Profiles 116 H-band r j . i - r — « t B-band 2 h T r-M o r p h o l o g y Figure 4.13: Sersic n versus morphological type index. Circles (blue), triangles (red), and squares (green) indicate Type I, Type II, and transition galaxies respectively. Chapter 5 Discussion Simulations of galaxy profiles and images (§ 3) and careful B / D decompositions (§ 4) have led to a final set of structural parameters for late-type spiral galaxies (Table A.2 in the Appendix). These long-sought data can now be examined for intrinsic structural variations and sensitivity to dust and stellar population effects. The latter effects, we reiterate, will be discussed in a subsequent analysis. The outline of this chapter is as follows. First, we confirm in § 5.1 that are results are not affected by any inclination dependences not already accounted for in the data reduction. We then discuss in § 5.2 B / D parameter variations both in the context of profile type differences and wavelength dependence. B / D decompositions for Type I surface brightness profiles were shown to be fairly dependable (§ 4.8), but the Type II profiles are not as reliably modeled by our 2-component technique. Additional model ingredients, which account for the effects of extinction as observed from the B to H bandpasses, are required for more accurate decompositions. New model ingredients would include, though not necessarily be limited to, central optical depths and dust-to-stars layering parameters. In light of existing limitations in our modeling of Type II profiles, our final conclusions will be based mostly on properties from Type I profiles. These will enable us to re-assess the viability of secular evolution models for bulge formation. We conclude with plans for future work including large-scale structural studies of galaxies from photometric surveys such as S L O A N . 117 Chapter 5. Discussion 118 5.1 Inclination Dependence Before we can proceed with our analysis of structural parameters, we must first test for projection effects. Any detected trends that affect our analysis would necessarily require corrective measures. To this end, we plot the distributions of Sersic n, f i e , and re, as well as disk f i 0 , and h as a function of ellipticity, (e = 1 — b/a) in Figs. 5.1 and 5.2. Other than a possible effect for f i e , which we return to below, no trends with ellipticity can be seen. Furthermore, Types I, II and Transition1 are not confined to any particular inclination range showing that the Type II phenomenon is not an accentuated feature due to line-of-sight extinction (e.g., Type II galaxies are not preferentially inclined with the plane of the sky). A l l three Types are evenly distributed (see Fig. 2.1). Our surface brightnesses have already been corrected for internal extinction (§ 2.4) with a mean correction coefficient which is derived from galaxies outer disks. Thus, detection of a trend with ellipticity in the disk fiQ would have been surprising. However, as noted above, the distribution of the corrected fie shows a trend with ellipticity. This may be understood as an inadequacy of our inclination correction for small radii. The dust concentration in the inner parts of galaxies is greater than in the outer disk region, thus requiring a radius-dependent correction, unavailable at present. As a result, our computed fies are upper limits (i.e., our effective brightnesses are too high), and this effect is stronger for shorter wavelengths. Note that our main results do not rely on f i e . 5.2 Bulge/Disk Parameters Figs. 5.3-5.7 show the distribution of fitted parameters at B V R H wavelengths for all galaxy profile types (Type I, II, and Transition). The number of Type II and Transition 1 Recall that Transition profiles show Type II features (central dip) at optical passbands (BVR) but a straight exponential disk akin to Type I's at infrared passbands). Chapter 5. Discussion 2 h I. ' I 1 B-bond-- - j J 1 * mm * J i i _ a L <u 0 • R-band— 1 M A * «> A 0 _T_J • i •> I L 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 - b / a ) Figure 5.1: Plot Sersic n versus ellipticity (1 — b/a). The colours and point types are as follows: Type I (triangles), Type II (squares), Transition (circles); B-band (blue), V-band (green), R-band (red), H-band (purple). fits included in these figures (e.g., only 4 decompositions for Transition galaxies in the B-band) is drastically reduced from our original selection. This is because many de-compositions failed our criteria and were excluded, which explains the paucity of good decompositions in the lower sections of these figures. Fig. 5.3 for the Sersic n parameter is a broken down version of Fig. 4.10 (good fits). It shows a value of n which stabilizes near, or slightly above, unity. Both Type I and II profiles show a systematic increase of n at longer wavelengths. Note that Transition profiles at H-band have a distribution of n which broadly matches that of Type I's at the same wavelength. Thus, we advocate that the more natural, intrinsic distribution of the Sersic ra parameter for late-type spirals has a mean near 1.0 (and large deviation of o~n ~ 0.6). Thus, by all accounts, bulges of late-type spirals are well-approximated, on average, by a pure exponential (luminosity/mass) density distribution. Any stricter Chapter 5. Discussion 19 20 21 22 o 23 OJ u 24 o CP 18 o E 19 o 20 21 22 23 I 1 I B-bond-| 19' 7 2 0 hv; > H 2 1 r 22 -23 -"I ' I ' V-bond 0 0.2 0.4 0.6 0 0.2 0.4 0.6 (1-b/a) o o • 4 19 20 21 22 23 24 25 18 19 20 1 1 ' i 1 i '_ • B-band -' ?' 1 jj. * -A. -_ . •« .( 1 . E • 18 * # •* 21 h' 22 h •• 2 3 ^ 24 0.2 0.4 0.6 H 19 - . - * • . * * 20 - * -m\ A 4 I 2 1 k / i 22 - •. 23 » 24 -T — 1 — I r V-band~ • 4 * * 15 1 6 h 17 18 19 t • 20 2 1 b 22 1 1 I 1 . H-band-0.2 0.4 0.6 (1-b/a) 12 10 8 - 12 h • B-band -- 10 6 .*. *»• - • • » 4 :<-\ ,> J i L i 1 r . V-band H - 8 - 6 h-""* - 2 b J i I i L - 1 1 1 i 1 i A • H—bond • • vr -_ •. " K " . " i 0 0.2 0.4 0.6 0 0.2 0.4 0.6 (1-b/a) 0.2 0.4 0.6 0.2 0.4 0.6 ( 1-b/a) Figure 5.2: Plot of bulge and disk parameters versus ellipticity (1 — b/a). The colours and point types are as in Fig. 5.1. Chapter 5. Discussion 121 definition (e.g., smaller mean with errors better than 2-<r) is not supported by the data. Figs. 5.4 and 5.5, for the distribution of Sersic re and /xe, do not reveal much new information2. The standard deviations are large in both cases, attesting to the large covariances between these two variables. Bulge effective-brightness differences can be explained as the sum of bandpass zero-point offsets and intrinsic colour effects in the bulge, e.g., B — H~<±,B — R~ 1.6,5 — V ~ 0.9. Typically, bulges have effective radii less than 2 kpc as seen in Fig. 5.5. The re estimates for Type II profiles are much smaller and seemingly better determined than Type Is but is an artifact of our limited 2-component modeling. Visual examination of Type II profile fits shows that the model disk is typically shallower (with respect to the true disk) as it tries to account for the fainter (extincted) bulge/disk transition feature. This has the effect of pushing the bulge re to lower values (since some of the bulge light is being provided by the disk's inward extension), resulting in sharply peaked profiles for the inner bulge. Fig. 5.6 shows the distributions of corrected disk central surface brightnesses. If disks have less extinction than bulges, the colour differences between the extrapolated disk CSB should be smaller than those for bulge effective surface brightnesses (since there is more dust and old stars in the bulge). This is exactly what we observe for fi0 with e.g., B — H ~ 3.6, B — R ~ 1.4, B — V ~ 0.8 (contrast with values above for pe). The B-band CSB for Type I, (fiB) — 21.34±0.72 mag arcsec - 2 , can be compared with Freeman's "classic" result, (pB) = 21.65 ± 0 . 3 mag arcsec - 2 , derived from photographic photometry for 28 out of 36 spirals galaxies (Freeman 1970)3. The two values agree 2Distances are computed as in § 2.1, and surface brightnesses are corrected as in § 2.4. 3 A tribute to Freeman and a celebration of his famous 1970 paper, one of the most cited in the entire astronomical literature, is presented by van der Kruit (2001). If the central M/L is approximately constant among galaxies, Freeman's "law" would translate directly into a constant central surface density of matter associated with the luminous material, thus calling for an unusually fine-tuned process of galaxy formation. Freeman's "law" has been shown to be no more than a selection (tip-of-the-iceberg) artifact since his measurements were only representative of the brighter end of the galaxy luminosity function (e.g., de Jong 1996, Dalcanton et al. 1998). Chapter 5. Discussion 122 10 8 6 4 2 0 8 6 4 2 0 1 I 1 I 1 I 1 Type II M = 0.64 o= 0.33 N = 24 DO in ii 1 1 1 1 • i 1 1 1 1 • Trans /*= 0.50 a= 0.37 N = 4 1 i i | i | i M= 0.69 . - a - 0.30 -- N = 24 - p [ L -1 I 1 I 1 I 1 M= 0.89 <j= 0.41 N = 12 4 0 1 T—r-r—1—•—|—r M= 0.70 a= 0.31 N - 33 L J L 1 I 1 I 1 I 1 M - 0.79 a= 0.40 N = 19 4 0 1 I ' M M - 1.24 a - 0.57 N = 42 nn u i l L 4 0 1 I 1 I 1 /*= 0.75 a - 0.44 N - 23 1 I 1 I 1 I 1 M= 1.01 o= 0.49 N = 16 rY n H Figure 5.3: Histograms of Sersic n bulge shape parameter. Chapter 5. Discussion 123 MS Figure 5.4: Histograms of bulge /xe. Chapter 5. Discussion 124 10 8 h - i 1 1 1 1-Type I M - 1.18 o= 1.04 Ni m 60 hType j M= 0.71 o= 0.41 N - 24 4 h T 1 1 1 1— I- Trons M = 0.40 <7= 0.14 N = 4 - i 1 1 r T M= 0.68 o= 0.27 N = 12 . . i i i_ 1 — 1 — r M - 0.69 <7= 0.31 N = 19 i — 1 — r M= 0.75 •7= 0.41 N = 16 J i , L (kpc) ^ (kpc) r* (kpc) Figure 5.5: Histograms of bulge rx (kpc). Chapter 5. Discussion 1 1 i M - 20.48J _ o= 0.71 — N 1 = 42 -A ll. " 22 1 1 1 — M= 20.72" <»= 0.47 N = 24 H= 20.57 a= 0.56 N = 12 20 18 22 1 1 1 - 19.95 - 0 = 0.59 • = 19 1  1 I 20 18 20 Mo Mo Figure 5.6: Histograms of disk central surface brightness. Chapter 5. Discussion 126 1 1 1 1 1 1 Type II • i 1 i i i i M = 4.89 - o = 1.83 " N = 24 ll 1 li l 1 i n . - i i i i | i i i i | i i i i T r a n s M = 4 - 0 5 - o = 1.07 " N = 4 -i i i i i I i i i i I I I I I I I I I I I I / x - 5 .20 o = 2.49 " N = 42 • 1111111111111 M = 4.67 a = 1.79 N - 24 i l l i l l i i i i i i i i | i i i i | i i i i M - 4.51 o= 1.58 N = 12 i i i i I 1 1 1 1 I 1 1 1 1 M = 4.34 a = 1.73 N = 19 ill i I i i i i i i i i | i i i i | i i i i M = 3.28 a = 1.32 • N = 16 I I i • I i i • 5 10 15 0 h B (kpc) 5 10 15 0 5 10 15 0 5 10 15 h v (kpc) h R (kpc) h H (kpc) Figure 5.7: Histograms of disk hx (kpc). Chapter 5. Discussion 20 b 15 10 5 T Type I 0 10 M - 1.07 o= 0.09 N = 56 i i I i 1 1 1 1 I 1 1 1 1 I 1 Type II M = 1.06 o= 0.07 N = 20 I 11 • 1 1 I 1 1 1 1 I 1 M - 1.15 £7= 0.17 I N = 85 1 1 I 1 1 1 1 I 1 M= 1.1* o= 0.09 IN = 25 I I I I I I I M - 1.29 ff- 0.21 N = 58 -I I I L i I i i i i M= 1.33 a= 0.18 N = 14 11111 111 • T 1 1—I 1 1 1 1—I 1 1 -M= 1.09 o= 0.10 N = 3 0.5 _L 1 1.5 h B / h R 1 1 I 1 1 1 1 I 1 1 1 ' M= 1-33 o= 0.14 N = 3 0.5 1 1.5 2 h B / h H Figure 5.8: Histograms of hB/hx. Chapter 5. Discussion 128 by virtue of their proper calibration, and the fact that we studied similar galaxies. No other conclusion should be drawn from this agreement. The R-band CSB can also be compared with the value from Courteau (1996) derived from his r-band study of 349 northern late-type spirals. The two studies find (U-Q) = 19.99 ± 0.63 mag arcsec - 2 , and (U-Q) = 20.08 ± 0.55 mag arcsec - 2 . Again, this good agreement attest to the matching calibration of each sample. The distribution of disk scale lengths, in Fig. 5.7, shows a clear decreasing trend as a function of wavelength. This effect is statistically significant and seen for all profile types. A natural explanation for this effect is the higher concentration of older stars in the bulge relative to the younger, bluer disk. The infrared bands are most sensitive to light emission from old stars and far less so to extinction by dust. Thus, disk surface brightness profiles ought to be steeper (shorter fio and h) at longer wavelengths. This effect is even more clearly seen in Fig. 5.8 where we normalize Z?-band disk scale lengths to values at other wavelengths. Again, this variation is most significant and we plan to study it with dust/stellar distribution models in a subsequent paper. Fig. 5.9 provides the basis for a renewed discussion of the suggestion by Courteau et al. (1996) of structural coupling between the bulge and disk of late-type galaxies, with ( ^ b u i g e / ^ d i s k ) ~ 0.10 ± 0.05 (or (re/h) = 0.15 ± 0.08) for all spiral galaxies (early and late-types)4. Shown in Fig. 5.9 are distributions of the ratio of the bulge effective radius, r e , to the disk scale length, h, at all wavelengths. Note that the large dispersions in the rx and hx nearly cancel to yield significant re/h correlations. Focusing only on the redder Type I/Transition profiles, we have (re/h) = 0.30 ± 0.18. Thus, we find a somewhat weaker correlation than that reported by Courteau et al. (1996). This could be due in part to the fact that our effective radii are computed for a wide range of Sersic n, whereas 4Recall that the study of Courteau et al. (1996) combines the r-band decompositions of Broeils & Courteau (1997) and the K-band decompositions de Jong's (1996) thesis study. Chapter 5. Discussion T—i—i—i—|—i—i—i—r Type II M - 0.15 o= 0.10 N - 24 /j.- 0.23 a= 0.11 N = 42 III, • ! I 1 1 1 1 /*= 0.15 o= 0.05 N = 24 _i • i i_ l — i — i — | — i — i — i — r ix= 0.16 o= 0.08 • N = 12 0.5 Jll I 1 1 1 1 M- 0.24 . o= 0.11 N - 66 Ul i • • 1 1 M- 0.17 a= 0.07 N - 33 i I , I 1 1 1 1 M= 0.16 o= 0.04 " N - 19 1 0 0.5 r « / h R T 1 1 1 1 1 1 1—I M= 0.22 ff= 0.08 N = 16 0.5 r £ / h H Figure 5.9: Histograms oirx/hx. Chapter 5. Discussion 130 Courteau et al. (1996) and de Jong (1996) enforced n = 1. Given that Type I profiles are statistically equivalent to a double-exponential brightness distribution, we are compelled to recompute (re/h) for n = 1 fits only. These new fits are shown in Fig. 5.10. We find, for the R and H-bands, (re/h) = 0.23 ± 0.10. The significance of this detection does improve for the n = 1 case. The new re/h ratio has a higher amplitude and smaller relative error than the previous claim by Courteau et al. 1996. This result makes clear the notion of coupling between bulge and disk in late-type spirals. 5.2.1 Test of Secular Evo lut ion This work has served to confirm two important statements which must be addressed by theoretical models of structure formation: • The underlying surface brightness distribution of late-type spiral galaxies is best described, on average, by a double-exponential model of bulge and disk, such as found in Type I profile galaxies5. Type II profiles are most likely an artifact due to dust extinction. • Bulges and disks of late-type spirals are tightly coupled as shown by the size ratio (rjh) = 0.23 ± 0.10. The later result is also in agreement with Graham's (2001) re-analysis of de Jong's thesis sample which finds (re/h) — 0.2 (no quoted error) at the K-band. This result applies both to early and late-type systems, also in agreement with de Jong's (1996) earlier study. To illustrate this point, we plot in Fig. 5.11 the variation of the bulge-to-disk ratio as a function of Hubble type for the cases of unconstrained Sersic n parameter (left 5This result describes the large-scale appearance of bulges. Preliminary results indicate that a sig-nificant fraction of bulges have power-law profiles in their very inner parts (r < 500pc; e.g., Phillips et al. 1996; Balcells 2001) Chapter 5. Discussion 131 - i — I — I — I — | — i — i — i — r Type II it= 0.15 o= 0.08 N = 33 ~l 1 1 1—| 1 1 1 1-Trons n= 0.11 o= 0.05 ' N a 18 J i i_ 0 .5 r - B / h B i—r p P 0.21 0 = 0.09 N = 47 n i l .11.10. n • • • • - i — i — i — i — I — i — i — i — r M= 0.15 0 = 0.07 N - 37 _i_L - i — i — i — i — i — i — i — i — r -H= 0.14 0 = 0.04 N - 22 0 .5 r v / h v _L 0 .5 r « / h R i 1 1 1 1 M - 0.24 o= 0.10 N = 53 t l I f l 1 I 1 1 1 1 M= 0.18 0 = 0.12 N = 43 I T. M l , ' i i II 11 il I II Mi | | i_ T — i — | — i — i — i — r M= 0.20 0 = 0.08 • N = 20 " 1 " L . i l • 1 0 0 .5 r J / h H Figure 5.10: Histograms of rx/hx for n = 1 only. Chapter 5. Discussion 132 43 0.8 0.6 0.4 o.z 0 0.8 0.6 0.4 0.2 0 1 1 1 i 1 1 — • _ H-band ] • . 1 • i ! i ! ; i i i i i i • • • : i : 1 1 1 1 i R-band • i i 0.8 0.6 e 0.4 O M 0.2 0 C "-a * X i 0.6 \ v U 0.4 0.2 0 1 1 1 • • • : . i i : : ' ! 1 i i i i i H-band • ! ;| i i • « j * , : • ! ! i i i i i R-band -i • J ! ! Morphology Morphology Figure 5.11: Distribution of rejh with Hubble types. The left side is for fits with un-constrained Sersic n parameter and the right side is for n = 1. A very mild trend for increasing B / D size ratio can be seen from later to earlier types. side) and n = 1 (right side). A very mild trend of increasing B / D size ratio can be seen from later to earlier types, though this relation shows considerable scatter. More data in the T=2 bin would greatly help secure this assertion. Fig. 5.11 should be compared with Graham's (2001) Fig. 20 which shows a stronger trend for earlier types. As we await our own analysis of earlier type spirals we can at least contend with certainty that all un-barred late-type spirals have constant B / D size ratios. A natural interpretation of this remarkable observation is that the bulge formed via secular evol-ution of the disk. This scenario is possible if disks are bar-unstable and if significant angular momentum transport is feasible. The secular evolution of collisionless stellar disks, especially through three-dimensional N-body simulations, have seen significant de-velopments in the early 90s (e.g., Pfenniger 1993; see Wyse et al. 1997 for a broad review). Because disks are bar-unstable, in particular to deformations out of the plane of the disk, stars initially in the inner disk end up in the bulge, which provides a natural explanation Chapter 5. Discussion 133 for the continuity observed in the properties of the stellar populations in disks and in bulges (Courteau 1997, Wyse et al. 1997). Gas flows must also be invoked to explain the higher spatial densities of bulges compared to the inner disk. The hierarchical hydrodynamical simulations of disk-like objects by Saiz et al. (2001) and Scannapieco & Tissera (2001) attempt to constrain the evolution of the bulge and disk as a function of time. The double-exponential characterization of disks naturally emerges after a relaxation time of a few Gyr (see also Pfenniger & Friedli 1991). However, none of these simulations have the required resolution to constrain re/h adequately at present and we await further developments from these groups. 5.3 Future work Much of this study has focused on the development of rigorous B / D decomposition tech-niques based on a new, comprehensive, multi-band survey of late-type spiral galaxies. Having established the validity of our galaxy B / D decompositions, we can now pursue our study of dust extinction and stellar populations in late-type spirals via colour gradi-ents and the wavelength-dependence of various structural parameters with this sample. Future tests with 2D decomposition algorithms will also focus on their ability to recover non-axisymmetric features, something we could not address here for lack of time. Galaxy-structure studies of this scope will benefit greatly from larger data bases. Many of the important correlations in this work have, after all, fairly large statistical scatter. Ultimately, we aim to apply our techniques to survey data bases such as those from the SDSS. We have obtained images from the SDSS early data release and compared it with ours to assess their value for future work. The S L O A N optical system delivers 1'.'8 images, comparable to our data. S L O A N pixels are 0'.'40 Fig. 5.12 shows the decomposition of a S L O A N image for U G C 929 using our 2D Chapter 5. Discussion 134 V 20 in o L to M CO £22 •5,84 '£ cn o o e -26 3 in 1 1 1 1 1 . 1 . 1 . 1 1 . 1 I UGC929 g ,V-band - D U S T • S L O A N • S L O A N F i t • B u l g e F i t -• D i s k F i t ->• i . . . . . . . II . . , 1 , . .* , 20 30 R a d i u s (") Figure 5.12: Comparison of S L O A N profile with our data for U G C 929. The decompo-sition used our 2D algorithm on the SLOAN image. algorithm. The 2D bulge and disk fits are shown as dotted lines 6. For comparison, we show the ID profile from Holtzman h Courteau (2002) and a major-axis cut extracted along the major axis of the S L O A N image for U G C 929. The depth of the S L O A N data is superb, yielding reliable surface brightnesses even below 26 mag arcsec - 2 . The ID and 2D B / D fit parameters match to within 4% for the disk and 20% for the bulge. 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Fan, L. Fang, J . J . Hester, Z. Jiang, Y . L i , W. Lin, W. Sun, W. Tsay, R. A . Windhorst, H . Wu, X . Xia , W. X u , S. Xue, H . Yan, Z. Zheng, X . Zhou, J . Zhu, Z. Zou, and P. Lu. Deep Intermediate-Band Surface Photometry of N G C 5907. Astron. J., 117:2757-2780, June 1999. Appendix A Tables We present tabular information for the full sample of galaxies used in this study and the bulge/disk decomposition results for a subset of the sample. Table A . l lists relevant catalog information for each galaxy. The entries are arranged as follows: Column (1): Galaxy number from the Uppsala General Catalogue of Galaxies (UGC, Nilson 1973); Columns (2) and (3): Galactic longitude and latitude, I and b, taken from NED; Column (4)'- Heliocentric redshift, Vheiio, in kilometers per second, taken from NED. Redshifts could not be found for U G C 2213 and U G C 5153. These two galaxies were removed for analyses involving calibrated surface brightnesses; Column (5): Redshift corrected to the reference frame of the Local Group Standard of Rest, VLG (Courteau & van den Bergh 1999), assuming a Hubble constant H o=70 km sec"1 Mpc" 1; Column (6): Blue Galactic foreground extinction using the reddening values of Schle-gel et al. (1998). For conversion to other wavebands, we assume an R^ = 3.1 extinction curve (e.g., Cardelli et al. 1989); Column (7): Galaxy morphological classification from the Third Reference Catalog of Bright Galaxies (RC3); Columns (8): Profile type as defined in § 1.2.1: Type I, Type II , or Transition (Tr); Columns (9)-(12): Number of observations per passband for each galaxy. 144 Appendix A. Tables 145 o CQ O i P H oo IH P H CD r^H - — • I" C O V "tf »© CO 6 0 a. ^  i fc ^ ;cn cn co < C N T-H T-IT—I C N T- I .—I T-H C N T-H O T-I r H 1—II—I O T-I O T-H O T-I © T—I T — I C N • - H I - 1 >—I | _ H | | _ ! | _ | | _ ! | _ | • — 1 | _ , r — I | E H T3 C O CO < m m < < U T J U T j T j H3 PQ CO CO C O ^ P Q ^ c o c o c o c o c o ^ j ^ ^ P Q <; < . ^ C O S^ l •*£ r.-I C O co <^  co ^ cn ^ co ^ o CN CO CN © © CO O i © T t f O CN t-H © ' © 2 o o o o c o CD co o S * ^ "5 N 2 w « l O ^ O i t - CN CO i d CN CN CN CO l O l O l f l ^ io <D io q C N CO CN CO "tf l O l O W ( D O O l O o CO CN "tf CO O i o o o o o o C D CN O i o c o o o o o o o o CO O i co CD CN O i CO CO o T-H O i CO T-H O i "tf T-H r~ CD oo oo O i t— O i CN oo T-H "tf O i O i C N 1—I co c o T-H t— "tf CN CO o o CO CN CN T-H CO T-H © C N C N c o c o T-H c o CO © O © o © ' o © ' © ' © ci © ' ci © © ' ci o ci ci ci ci o O o o © o o o o o o o o o o o o o o o oo l O CD T-H O i o oo co •*tf o c o t-- OO I O T-H "tf T-H co CO O i "tf oo co O i T-H T-H o CO CO T—1 o CN o T-H T-H oo CD i O "tf "tf CO CD t ~ O i C O l O CO i O t>- l O "tf T-H l O CD c o CN "tf O i c o T-H co i O "tf t — c o c o T-H T-H T-H O i T-H o t— "tf c o o T-H o o CD CN o O i o C N CN CN CO co CD CN CD CD O i O i O i o © oo O i oo o O i co "tf CO i O l O i O "tf l O "tf CO O i C N "tf c o t ~ "tf "tf T-H l O T-H CO oo co "tf T-H t- 00 T-H o C D oo CD o 1—1 CN cp O i oo T t f \o T t f T-H o 00 T-H © c o i O oo T-H O i cp cp cp T—i OO oi oi oo © ' oi ci CO CN oi T-H oo CO CO 00 c o c o 1 T-H I t o 1 1 "tf 1 c o 1 T-H 1 CN 1 co CN 1 CD 1 1 CN 1 CN . 1 c o 1 ( 1 c o 1 "<tf 1 c o 1 T-H co "tf o o CO T t f oo C D oo c o oo O i CN CD *p CO O i O i © CO cp O i "tf T-H o "tf cp O i ip ip T-H oi CD oo © CD i o o © T-H © CN od "tf CN oi CN i - H T-H T-H T-H CN CN CN CN c o c o "tf CO c o c o "tf i O "tf "tf "tf O i CD "tf c s CO "tf oo c o t - O i c o o O i T-H CN t - T-H O i co O i c o T-H co CD T-H CN oo oo CN C N t— OO O i CN CN c o CN c o "tf "tf t - t - t - t - O i O i o o o CN c o "tf i O o o o o o o o o o o o T-H T-H T-H T-H T-H T-H T-H T-H T-H o o o o o o o o o o o o o o o o o o o o Appendix A. Tables 146 CN fl o 'I rt OJ co ll-O o > S PQ os OH oo to - f l - — • r> co — CJ o bO i t * CS fl u _ liz; E> O T — I r H C S O O r H O T—I r H T—I T—I T—I T — I T — I T — I CS T — I T — I T — I C S T — I r H C O C O C S r H C O r H C S r H C S r H T — I T — I T — I CS O T—IT—I O CS T — I T — I T — I CS T — I T — I T — I T — I T — I O CO T — I T — I T — I T — I O T — I T — I O T — I T — I T — I T — I T — I CS T — I T — I T — I CS T—IT—I CO CS CS T—I CO T—I CS O CS T—I M M H ^ M H ^ H H H H H M H H H ^ H M r H H H M H rO -O CO CO o CO PQ PQ < < CO CO u OO OO <J CO PQ < CO H3 - d o o CO CO o _ to , . — . . O 2.>i u PQ ^ ^ P Q S m < o l OT •<J CO CO o •< cn CO -O PQ <! CO rt X . o <o PQ T3 cn CO •H ^ - P Q CO co 1 -H CO Oi oo T-H CS oo o CO •HH Oi CS Oi r H o CO oo Oi Oi co I O oo oo oo T-H T—1 CD CS CD CS o CS Oi CS T}H CS t— o co CO r H CO t~ CS CS co oo co CO t- CS CO CO i p T-H CS co CO CO i p co CO I p co cs CO o o o o o o o o o o o o o o CO o o o o o o o o o o o o o o o o o o o o o o o o o o —H o o o o o o o o CS CD co CS LO Oi oo Oi HO 1— CS t~ CS oo CD T-H Oi oo co CD CO co t~ CO CS CO CS l O l O CN " 0 " 0 l O CD 0 0 o CO Oi CO oo t~ o CO Oi Oi " 0 I O •HH CD CS CS T-H CO CS CO i~H i — . " 0 oo oo " 0 CO CO oo oo CS 0 0 o *o oo t- Oi CS " 0 r H co t— " 0 t- CS oo I O T-H T—1 t- OO CS oo co o T-H CD CO r H co CO Oi co T-H t- Oi T-H oo T-H o T-H t—— o I O ' ' m l O " 0 CO oo co Oi T-H " 0 CO Oi CS Oi Oi u 0 •HH CO CS CN r H CO CS co T-H "* oo oo " 0 CO CS oo o oo CS CO T-H CO oo co o CS o b- Oi "0 oo Ol i p "0 T-H CS o co oq Oi CS Oi o T-H o i p o Oi T-H CS CO o CO CS CD CS 1 CO 1 CO 1 co CS 1 CS 1 lO CS 1 oi T-H 1 CO T-H 1 CO co 1 o lO 1 o 1 T-H co 1 r H co 1 oi CS 1 CS 1 o CS 1 t>^  T-H r H CS 1 oo T-H 1 o CS CS od T-H t-^  CS co CS co co T-H CO T-H co o r H oo co CS o co CO CO T—1 oo CS oo r H co o IO r H CS T-H "0 co CO CO CO T-H Oi o co IO CO "0 o T-H Oi Oi CO oo Oi oo CN "0 CD Oi CS CD oo «*< CO oo "0 oo co oo "0 05 OS Oi CO Oi CS Oi r H Oi Oi lO Oi o UO CS CO O i o o b - o c s o o c o c o o o e o c o c s c s M O M O ) N C J i O H W O i i T | t o C O O O O i O r H r H r H C S C S C O r H T—I T—I CS CN O O O O O oo o o o t- oo o o o i O CO Oi •HH I O CO Oi t-T-H C S CS CS C S CO r H CO T-H U 0 N O O N r)t l O N 0O c s c s c s c s c s c s c s c o c o c o c o c o c o c o c o c o c o c o c o c o o o o o o o o o o o o o o o o o o o o o Appendix A. Tables 147 H 3 fl ^ 0 ^ 1 * -IH <u O > § P Q oT P H oo co IIH PH ||^3? O CO II" CO *—• o „ d •a fl o Js fc ^ O T - H O C S C N O O O O C N O T - H O T — I C S C N I - H T - I C S T - H I - H O T - H I — I C O r — l O O O O O O O O i — l O O O i — I O i—I i-H i—I i—I T-I CN O i—I O H a H CO CO co <3 •< ^ CO CO o CO CO CO CO CO CO m CO o CO t o o , -D CO CO CO PQ < •< CO CO H3 T3 o o CO CO -Us U T J ^ CO CO CO oo CO T-H "tf O co lO CO t- T-H cs co CN CO CO Oi Oi 0 0 "tf t- o cs cs oo lO co oo "tf CO T-H o o T—1 CO oo o CN CN CO Oi T-H Oi CO T-H CD o CN T-H T-H CN T-H CN T-H T-H CN CN T-H T-H T-H T-H T-H T-H T-H o T-H o o T-H T—1 o T-H o o o O o O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o iO o t~ t- CO Oi t>- Oi Oi Oi co Oi Oi CO 1 l-\ co CN T-H oo tO Oi o iO CO o to oo Oi Oi Oi co Oi o Oi lO u J t- CO t~ o lO CO CO T-H oo ^HH iO "tf co CO co co co T-H T-H co CN fc CO Oi CO T-H r- T-H CO co CO "tf CN CO T-H co co •*tf Oi oo CN T-H oo 0 0 m o co Oi "tf CD co t- T-H CN co CO o oo oo co co oo o oo CN CN "tf o lO co "tf o oo oo CN oo oo oo Oi T-H CO Oi co lO oo lO oo Oi CO T-H T-H cs T-H T-H T-H CO io Oi CO oo o CD co t— CN Oi +^1 m "tf co "tf co co "tf "tf T—H T-H "tf CN fc "tf Oi CD T-H t~- T-H CD CO CD "tf cs co v i T-H CO co CO co "tf co lO Oi o o T-H o lO co co T-H T-H oo co o co "tf o CN cp CN oo i>- cp © oo Oi "tf cp cp cp oo Oi CN oq oq co T-H i p t- t— Oi CN CN o CN c i eo o i T-H T-H "tf o i co oo co CD id CD oo T-H id co o i o CN CN CO c s CO CO CN co "tf co CO "tf "tf m lO lO CD lO oo oo oo oo "tf CN CO CN "tf CN oo o c s o i>- CN o CO IO i-H CO o CS t~ t~ cs IO ,—N Oi CN cs lO i p CN i-H oq CO Oi "tf t- 1-H Oi CN "tf CO Oi Oi co o CO "tf lO Oi CN "tf o i 1—1 i-H "tf t>^  co "tf CN o i o i c s "tf co id CO c i co t-^  cs Oi i-H oo co o oo o i-H T-H co CN cs lO T-H CN i-H co o lO Oi r-H co Oi CN o T-H T-H CN i-H CN cs CN T-H CN CN i-H CN CN cs CN cs c s ! — t CN cs CN CN co c s "tf Oi t- Oi CO "tf oo oo i-H co "tf "tf CD oo CO co c s t-- CN CN I>- o o oo o co CN i-H t~- 0 0 I>- t— oo Id "tf Oi o CN i-H i-H lO co lO CN 0 0 1-H Oi i-H oo T-H T-H CN to t~ Oi o i-H co CD CD oo T-H • v t f o co CD oo Oi "tf iO cs ' ^ co "tf "tf "tf "tf "tf "tf "tf ITS tD tO co CO t~ t- t~ t- oo oo Oi o o o o o o o o o o o o o o o o o o o o o o o o o Appendix A. Tables 148 C N Pi O 1 3 tH OJ . to CQ oi P H o o g o !IH rt C5 £ i 2 «© o o „ * •a CN OJ r - . CO CN T - H HH T - H T - H T - H CN r H r H CN CN 00 T - H r H CN CO T -H CN T - H t-H CO CN T - H CD C N r H C N T - H C N C N T - H T - H T t l r H r H T - H C N T - H - ^ O T - H C N C N C N C N r H C N r H C N CN r H CN T - H T - H CO T - H T - H CN T - H T - H T - H CN CO CN C O T - H OO CN CO T - H T - H CN r H CO HH T - H CN T - H T - H T - H T - H T - H CO T - H T - H CN CN r H CN H ^ r H r H r H CN r H T - H CO T - H T - H ^ H ^ H ^ - ^ ^ " i — l " " ! — ! * — ! " " ^ ! — H r — I " C J rO u r O CJ CO CO c— T3 CJ CO r Q CO co pg ^ co <! CO r H O CJ 0 CO OT CO CO OT CJ CO P H -m T3 T ) C J o co co rt o CO CJ <3 co CO E H CJ rO CJ rd CO CO '—- u CO OJ CM PQ x CO - — ' ^< rt wa •HH T - H xO O l o o " 0 CN •* 01 O o XO l O r - C O CN o C O HH T - H T - H CN C O t - T - H T - H CN c o o o XO O J o o o o HH o T - H " 0 HH T - H " 0 o r H C O OO xO CN o O CN T - H T - H c o o o CN CN T - H T - H T - H CN T - H CN CN CN o o •HH c o -HH c o o o o O o o o o o o o O O o o o o o O o o O o o O o o o o o o OO CO O l HH N ^ X N CO CO r H XO o o o O o o o o CO c o o o o o o o O o o o o o o o r H t - •HH l O O l xO O l o t ~ o r H CN CN c o O l HH CO O l CO c o CN c o CO XO xo •HH CO r ~ CN •HH o t~- CO o HH o CN CO O l •HH CO r H •HH O l r H CO t - t-- CO CN XO XO l O CO N C f t M H / N i o r J H O O ' * C O O i a i c o C N O O C O i O O l N N O O i n O l N H O M i O t - -HH oo T - H O C O O O T — I L O O i C N C O O i O l CN o e o - H H o o - H H c o c o o o i a c N c o o o c o c o - ^ - ^ c o o i o o • H H e O r H r H O C N t - - H H H H H O i H C D N N i O I M i O L O C O OO O l CN t D H CN H N •HH CO OO O l C O •HH HH lO lO O l 00 CN CO o H io N O CN t--CN O l •HH C O 00 00 r H C O XO XO XO C O CN CN HH C O CO CN CN C O O l t— xO C O CN O i C O O l r H O T - H T-H C O CN c o T - H Ol •>* . T - H T - H I>-oo C O •HH o o c o CO T - H c o c o Ol C O ! > • ] > -XO XO CN 00 "HH c o HH o c o CO* c o CO o o a i c o o i 00 o o o o c o c o CO CN CN CN a i T - H CO T - H t - to T - H Ol C O Ol 00 T - H r -T - H C O • * oo •HH oq T - H CN •* T - H 00 T - H l O i>-o r H o i t— l O C O T—1 c o •* •HH c o xO o c o o T - H a i oo c o VO T - H C O CN O T - H oo C O x6 CO t - o o C O O l •HH -HH Ol Ol Ol Ol Ol Ol o o o o C O C O C O - ^ C O C O i O O C O O l C O t - -O C O C O i O O l O O C O O l C N C N O O t O 00 T J H T—I C O xo o o o CO "HH o o HH o •<H XO XO XO o o o o c o ai o o o O O CN CO -HH o CN C O CN T - H b - o o C O oo o i c o T - H T - H i T - H 1 CN 1 00 i •HH o T - H c o Oi oq a i CN oo C O CN o i HH •HH •HH xo CN xO XO xo C O oo Ol xO xo xo C O Appendix A. Tables M 3 SH O <L> C O O T - H =tt= PQ oT oo OQ T ^ H K e n cj o • I-l o bO , o I H I - H I H C N O C 3 I H I H I H C N C O I - H C ^ C N O I H I - H C S I - H I H I H I - H I - H C S C N T - H T - H C O T-H T-H T-H T-H T-H T-H T-H T-H T-H C S C N T"H T " H T " H T - H ^tf T - H C S T - H T - H T - H T - H C S C N C S T - H T - H C S T - H T - H T - H T - H T - H T - H T - H C S T - H C O "tf T - H C N "tf T - H CN t-H >J t-H J-l - d o »H C J ^ P Q ^ ^ c o ^ ^ P Q ^ c n P Q P Q c d < CO CO < CO H H 1 H F - H r—1 o CO CO tH tH o ^— << Q PQ CO < < CO CO- Q - d - d co co CO PQ CO H IT PQ co ••d ' d " d co co co T ^ rt ^TPQ •< co CO ^ oo I O CN oo "tf i-H oo CN i-H t— o CN cs "tf O i co O i I>- t - co l O i-H to oo CN oo CO CS i-H "tf oo t - CD "tf o CN CO r-- oo CD "tf i-H cs CO T H co CO CN CN CO CO i O cs eo i-H "tf cs CN co O i o i p co co l O CN i-H o o o O o o o O c i o c i o c i c i i-H o T H c i c i o c i c i O o o o o o o o o o o o o o o o o o o o © o o o ^ ~ ^ t~- CO O i "tf cs l O CO CO "tf "tf T H oo "tf O i CD co CD i-H i O CD oo oo oo t - o oo i-H l O CO ' j o O i t - co o "tf l O c s i-H kO 1 i O o oo O i co co l O oo co t— t - i-H l O O i co c s l O t - oo CD CO i-H "tf "tf l O o o co l O i-H O i co "tf co CO CD oo oo CN t - O i cs c— CN l O O i IO —44 t v . CN t - o i-H o co t— o o t-H co co co o co O i O i co r - CO CD i O t - O i CO co oo t - t - - t o i-H oo c s O i O i cs VI ' cs t - CO "tf co CO co co "tf CN t~- O i "tf O i CO cs "tf t - f - l O CD i-H "tf co i O l O l O CO co "tf oo CO O i cs O i i-H l O ir- O i o CN o "tf cs c s t~ oo t>- CO t -^ v T H "tf O i t>- CO O i CO o O i C i es oq "tf O i "tf O i CN co oq O i CN CN CO CO o co co o i "tf "tf oo "tf CD o o i o i o i "tf CO "tf 1—i o i CN oo CN c i co CN 1 T H l co i co i CN 1 CN 1 i-H 1 co 1 "tf 1 "tf 1 "tf 1 co 1 CN 1 co 1 co 1 CO 1 co 1 cs 1 t o 1 co 1 l O 1 CO 1 co O i oo "tf cs "tf cs i-H co oo "tf o CO "vt< i o CD O i CO cs O i r—i "tf oo CO CN O i t— i O c q CO i-H r H cp oq CO i-H CO o oq !>• i-H oq CN co CN cs c d T H o o i c s c s t-^ i d i d "tf cs c d CO o i co i-H co o i c d CO 00 O i 00 t - O i O i O i 00 oo oo O i O i O i O i O i T H O i i-H O i O i l O oo c s l O cs CN "tf co o "tf o i-H CO co i o oo co t - t - "tf "tf CO ^ — ^ CO c— i-H "tf "vt< CO co o cs t - £— oo o cs t— oo CN CO i-H CD CO "tf T H t— O i o o i-H i-H i-H CN cs CN cs CN co co co co " 0 CO t— oo oo " — ' T H H CN CN CN CS CN CN cs c s cs c s CN CN cs c s c s CN CN cs CN cs CN Appendix A. Tables 150 Table A.2 gives relevant photometric information and ID B / D decomposition results for the final set of Type I galaxy profiles only. The entries are arranged as follows: Column (1): (UGC number) _ (observation number) _(passband) for each profile; Column (2): Ellipticity, e = (1 — b/a). The final estimates of ellipticity (and position angle) correspond to an average of those values for the five contours surrounding the best isophotal fit in the outer disk, as determined by eye. This estimate is clearly sensitive to the presence of spiral arms. The typical inclination error is ~ 3° independent of ellipticity; Column (3): Sky brightness in mag arcsec - 2 , measured from 4 sky boxes located between the chip/array edges and a fair distance away from the galaxy. Typical rms sky errors, computed from the deviations of the mean sky counts for 4 or 5 suitably located sky boxes around the galaxy, are ~ 0.5 — 1.0% in the optical and 0.05% in the IR. The subscripts indicate the sky offset preferred by our selection process as described in § 4.4, where "+" and "—" indicate 0.5% over- and under-subtracted skies respectively. No subscript indicates that the measured sky was preferred; Column (4): Seeing F W H M values, computed as the mean of the F W H M s of all non-saturated stars measured automatically on each image frame; typically 10 to 30 measurements per frame were used for each F W H M estimate. The accuracy of the seeing estimate per frame is roughly 20% for the optical bands and 30% for the H-band; The upper and lower boundaries in the remaining columns correspond to the max-imum and minimum values of the < 50 [out of 1080 (optical) or 360 (H-band) total] solutions remaining after our filtering process as described in § 4.3; Column (5): Best fit Sersic n bulge shape parameter. Column (6): Bulge effective surface brightness, fie, in mag arcsec - 2 corrected for Galactic foreground extinction, cosmological redshift dimming, and inclination as de-scribed in § 2.4. Note, however, that the inclination correction adopted is probably not Appendix A. Tables 151 appropriate for the bulge region; Column (7): Bulge effective radius, re, in arcseconds; Column (8): Bulge effective radius, re, in kiloparsecs. Converted to a physical scale using the Local standard of rest velocity, Via (see Table 1, Column (7))\ Column (9): Disk central surface brightness, UQ in mag arcsec - 2 , corrected for Galac-tic foreground extinction, cosmological redshift dimming and inclination as described in §2.4; Column (10): Disk scale length h, in arcseconds; Column (11): Disk scale length h, in kiloparsecs. Converted to a physical scale using the Local standard of rest velocity, VLG (see Table 1, Column (7)); Column (12): Bulge-to-disk luminosity ratio, B/D, calculated using Eq. 3.16 in § 3.1. Appendix A. Tables 152 CN CJ « o CN © CN O co o o CN o CN o co q H rH O CN O H O O rH o q o o rH O T|< q o o o o q q d d rH O . - i 6 d d d d d d d d d d d H d d d d d d d d d d d d d d d d d d d d d d d d d d + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 4- 1 + 1 + 1 + 1 co T-H CO T-H co CN Oi O CO •HH CN CN oi oq CO T-H T-H o t - - CO CO LO CO CO •* TJH cd CN CN CN T-H T-H T-H o a . • ^ (8 ^ CJ , — ^ PH CO r * •3 ^ OJ c3 o n PH • ^ "bC CO* c3 LO 2 fa H«i OO to 60 CO 05 v o o CN rH O co o o N O O rH CN CN o co rH O O io O O o o rH O rH O o o o o o o O rH rH O o o o o o o O O o o o o o o o o o o o o o o O 01 O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O o o o o o o o o o o o o o o o o o o o o o o o o + 1 + 1 -1-1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 t - co co co CN co CN t— T-H T-H CN CN co co T-H T-H CN O o o o o CN CN r H CN o o O o o O O O O CJ o o O O CO o o o o O o r - H o O O o o O O O O O o o o H O co © t o o i f o q i o H N O N © c© O H W H TJ< © I A O e s q q <N H O q q O H co q co q q q H O H O ridriOHdoddodddoodonddosoododd + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + ( ^ C O b - L O - H H C O - ^ L O C O C O O O H H C N O O O i r H C O O i C O L O O LO HH CO Oi OO o CN OO CO OO CO CN H H « H CO CN CO CN oo t - t— ©CO CO CO OCT) O N OCN H H i r tO W O COO O H O O O H H O COO t- O H O CN O H O o o o o o o o o o o O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o + 1+1 CO CN rH CN CN r-i CN CN o o o o + 1+1 lO CO CO Oi o t-^  CN r H o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o + I CN "* T-H CN + I L O OO o CN + I CN CN o CN + I + I + I + I + I + I + I + I + I + I + I + I + I + I C O - H H | > - - H H L O C O b - L O t - C N O T - H T - H C N O i C O r H C N C O C O L O O i C O T — I T J H ' C J H O T - H Oi 00 T-H T-H T-H CN o CN Oi Oi o CN O i O i t— CN T-H T-H CN CN CN o CN + I oo L O oi *n © c o q t>q * r q O H CN CN O C N O H q q q q H O H O H O H O O H O H O H T T O coq H O q q q q d d H O r i d d d d d d d d d d d d N d d d d d d d d d d d d d d d d d d d d d + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I N C N T H H T ^ O T H H T - H T - H C N t ~ O T - H O C O L O L O L O C a C O C O T j H - < r f H CN CN CN CN T-H LO O T-H O O O O T-H O O O o o o o H O in q q CN q q © t-; qco qco H O coq ^ q coq c o q qco H CN O N H O coq c o q H O q q d o T ^ d e i d d d d d d o d d d d d a o o d d d d d d d d d + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I C N O O O b - O i O O O C N O i O O O O i r H C O t - C O C O O i O i C O C O CN CN CO C N C O C O C O - H H C N C O C O C N CO H H O OiO t- © O O CO CN O U 5 Mil) in co O H O W ^ » * O CN O CO CN «NO COCO O I A O I H csco c o o O I O O CN CO H H I A O r - o COO " f l *H O H H H O H O C 0 OOi H O H O H H H O N H O C N O H H O * q H O t- ;0 H H H O o o o o d d d d d d d d d d d d d d d d d d d d d d d d d + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I H H L O t - T - H C O L O T - H C O C i O C N ^ T - H C N e O C O C O O O t ^ O O N C O ' H H O H H O i O O C 3 i T - H r t i L O O C N C O O O O T - H b - e O L O ^ O i O C O C O ( ^ T - H O ^ O T - H O o i o i c N T - H r H O O o i o i c O C N r H c ^ CN CN CN rH CN CN CN rH T-H CN CN CN CN CN T-H T-H T-H CN CN CN CN CN l O O W O l / J O C O H H H H H O H O O ^ O C O H O H q H H H q H H H H H H H q W q H q ^ q H H H © d d d d d d d d d d d o d o d o d o d o o d d o d o d o d d d d d o d d d o d d d d d o + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I O H H L O c O C N O i O ^ L O b - C X j T - H O ^ L O L O C O N O l c O L O L O T-H T-H T-H T-H rH rH CN rH CN C i d r—H T"H d C^i ^O OJ OJ ^Zi CO OJ C i I I 1 + + + I I I + + + 0 0 O 0 0 C N H ^ ^ L 0 H ^ C X > T - H C N C 0 0 0 C ^ < ^ L 0 L 0 ' * C 0 C 0 rH CN rH T—H T—H CN T-H T—H CN CN T—H CN rH CN CN CN CN rH T—H T—H CO Oi I I I + + + I I + + + I I I C O L O O i C O L O ^ C O C O L O C O O - " N H t ~ L O O i L O ' H H O O t ~ - O H H C O L O O a i C N H H L O L O O O O i L O C O O O L O L O C O L O C O r H C N O O CN r H T-H co CN T—H o o co T-H T-H o r H CN r H o co CN o CN T-H o CN CN CN T-H CN CN CN CN T-H CN CN CN CN CN CN CN r H CN CN CN CN CN LO LO LO LO b - t - b - b - t— LO LO LO LO t - b - t - CO CO oo 00 co O i O i O i O i LO LO LO LO LO t - b - t— b - O i O i O i O i r H r H CN CN CN CN CN CN CN •<* o o o O o o CO CO o o o o O O O o o o CO CO O O > Pi m PQ j > rt rt( w cq 1 > rt rt CQ 1 > rt I m CQ rt 1 °S > | rt i 1 co CN •HH CO O i LO co CO o LO r H r H LO o O i CN T-H O i co b -LO co co co co CO CN CN co co co o CO CO CO o o O t ~ o | o o o -HH o 1 o O | O 1 O 1 o | o 1 -HH | o I o 1 o 1 O i O I o o o O CO CO CO CO CO CN CN CN CN co co co co 00 oo LO LO LO LO CO CO CO CO CN CN co oo CO "HH t— b -Appendix A. Tables 153 <u 3 C c3 -9 CO <L> rt fl o CO o o o Q CN < CN o a. • ^ to OJ rt ' a, c o JA ^ • bO CD to CO cf l o (H PH S 0 o O rH o o o o O H o o o o O r H o o o o O CO o o o o o o o o o o o o o o o o O r H O r H O CN o o O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 T-H c o CN c o "tf CN CN CN CN CO T -H CN CN CN c o m CO C N m r - H T-H o o O o o O O O O O o O O O o O o O O O i-H o o O c i c i c i c i c i c i c i o c i c i c i O c i O O c i c i c i O o o o N O o o O TH O CO r i O o o r-t O o o O CN r H O O r H O r H o o O r H O r H o o r H r H O r H O r H O O o o o o O O o o o o o o o o o o o o o o O O o o o o o o o o o o O O o o o o o o O O O O o o + 1 + 1 + 1 + 1 + 1 4- 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 p t— Jt— c o O O l i n i p T -H o q C O T -H i>- C O i p CO c o CO 0 0 O J C N "tf "tf "tf "tf CO CO CO CO CO "tf c d c d CO C N C N C N i d i d i d -tf "tf O H o O H O CN O S co o O r - l CO o o o O IA CN © O CN O CN O r H p CO p CN O r H r H r H O CN O r H O O p p d 6 6 6 o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 4- 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 r -H c o t — o q c o O CN O i p c o CN c o CO O O t — T-H T -H CO t — CO o i o i c o c o c o c i O O O CO CN i - H c i o i o o c o o i o i o o t-^ O CN CO o o o O r H O CO CO o O r H r H O O CN O O r H O O r H O r H O CO O CO O CN r H O o ia O CO O CN o o O O o o o o o o o o o o o o o o o o O r H o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O o o o o o o o o o o o o o o o o o o o o o o o o + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 05 l O t — t - co co "tf T-H CO o O i O CO C O o O J T-H o CN oo T-H co T-H l O O S o> l O "tf l O l O T-H CN C O CN p C O l>- O J CN c o co l O i - H c i O o i o i c i o i o i T—i O o i c i o i od O O o i o r-H CN CN CN T-H T-H CN T-H T—H CN CN T-H T-H CN T-H T -H CN C N T-H T-H CN p 1A r H p p p O r H p o r H O O r H o p O r H p CO O r H p p p p p p o o o o p p r H O O CN O r H O r H O r H d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d O + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 O J t - -tf CO "tf CO CO i p "tf i O i p i p "tf "tf i p i p CN c p T-H i p o O O o O O O O O r - i c i c i c i c i O O c i O T-H T - l c i O - P H O O H O CO O H N O O " * H H OC0 O01 O CN H O O H H O O H H O 0 0 H O OC0 O N O N O C N O H d d d d d d d d d d d d d d d d O H d o d o d d d d d d d o d o d o d d d d d d d d + I 0 0 " t f 0 J C N i 0 0 J 0 0 0 0 C 0 T - H i 0 i 0 C D C 0 i O C 0 t - - T - H O t — t ^ - 0 0 CN T-H O O * O f f l O C 0 « O ^ « W « l f l ( 0 O W H H ^ O W © ^ « H M N O O H O H O O - f O H e N H O H e N O O O H H O O O O H O H e N O 0 H o o + 1 + 1 01 t — "tf c o i-H i—J CN CN O O O O O O O O O O O O O O O H o o o o o o o o o o o o o o o o + I i-H c o c i CN + I i p c i CN + I i-H CO o CN + I i O o q o i + i o o q i-H CN + 1 + 1 c o "tf c i CN + I oo CN o i + i r-H C O CN + I OO "tf c i CN + I CN CO c i CN + I "tf + I o CN + I CO i p o i + I CO c p i-H CN o o o o + 1 + 1 CN t — O J o q c i CN H U ) S i f O H O H od oo + 1 + 1 I O r-H o CN O ) N H i—I i—I CN p c R H O O H W O O O M O M H M O C N ^ O S H O N O H p p N p H N p H p H p p e s H H H H ^ ^ d d d d d d d o d o d o d o d o d o d o d d d d d o d o d o d o d o d o d d d d d o d o + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + . I + 1 + 1 i p CN i p ci o q ci c p c i c o ci c o o o q ci c i "tf o c p CN c o o o q ci c o c i o q o o o q c i c o c i o q ci + + CN CO oo o I I CO CO "tf CN CO OJ CO CN + + I O OJ "tf i O o CN O CN T-H T-H CN CN CN CO T-H I—I T-H I-H C N O T-H T-H T-H I + CN O CN "tf i-l c i I i co O J o q o + CO OJ CO CO OJ o i O i-H CO CN CO i O CN OJ oo •vtf CO CN CN "tf i-H o o CN CO t— "tf OJ CN O oo l O o CN CD CN CN "tf "tf CO t— OJ OJ CN c o T-H o CN c o i-H o CN CO i-H o CN i-H CN o CN O CN "tf i-H CN CN r-H CN o CN c o i-H i—H CN o CN o CN CN CN r-H CN o CN c o i-H CN CN CN 0 0 CN "tf CN CN CN CN CN CN CN CN CN CN o oo o c o O oo o oo "tf oo "tf oo "tf 0 0 "tf oo OJ l O c o OJ m c o OJ m c o m oo o m CO o m 0 0 o m oo o CO "*tf "tf o O o O O O o o o o o o o o o o o o o o o o a PQ > > rt rt PQ rt( rt W PQ > rt( W PQ i > rt j PQ j > rt j W j PQ i i CN oo 1 co t co r- t - co oo CO CN m co i-H o OJ -tf co co "tf OJ CN oo i-H i-H oo "tf "tf oo CO l O CO m o OJ OJ. oo m CD m o OJ OJ o | i-H p i-H o o I o p CN o 1 o o | "tf o 1 o 1 o | o i o 1 o i co | o 1 "tf OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ oo oo oo oo co CN CN CN CN CN co co oo oo CN CN CN CN CN CN CN o o o o co t - OJ OJ OJ OJ OJ o o o o m m m m co CO CO oo oo oo oo OJ Appendix A. Tables 154 o a. • bD Ol cfl ~—^  a o ,—^ PH 0 0 p CO rH CO o q q m CN q CN q rH O rH O q q q co q q q q q co l» O Tf O co q O CN O CN CO o CN q q co H q 6 6 d d d d d d •-i d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 r f CO CN CN CO co CO CO CO CO O CN CN CO LO Oi i-H r« r H Oi oo CO r f CN oi oi od od oo CO l>" i-H O O od r f r f r f CO CN oi t-H t-H t-H i-H t-H i-H t-H i-H t-H t-H t-H i-H t-H O CN CO CO rH O O rH O O o o o rH O rH O O CO rH O O rH O CO C - O X O CO rH O CN O CO H< O CO O O O CN O o o o o o o o o rH O o q q q q q q q q q q q q q O O o o o q q q q q o o o o q q q H o o 6 6 d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 CM o oo LO CO CO CO r f i-H t- co O oo CN CO r f Oi t- - H H CO CO Ol co r H oo CO co co CO CO CN i-H oo oo oq co CO Oi Oi CN Oi CO Oi oi t-- i-H o o o oi oi oi oi CO o oi oi CO o O O oi t-^  !>" t-H t-H t-H CN CN CN CN t-H i-H i-H i-H t-H CN t-H i-H t-H CN CN CN t-H r H r H O H rH rH rH O q q q q q q O rH q q q q O rH o o o q q q q q rH O rH O O rH q IA co q CN O O f q q 6 d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 -(- 1 + 1 oo oo O O CO co co co co CN i-H i-H CN Oi oo Oi CN CN b- CO o o c i o o c i o o c i o o O i—l i—l i—i O c i o t-H i—H i—H c i • ^ bD c o ce ^—' a L O O O O O O O O O H O o o o o o o o o O O O H H O O H O H H O H H O O O H H O H O O l O o o q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q o o o o o q q q o o o o o o o d d d d o d d d d d d d d d d d o o d d d o d d d d + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I C 0 r f 0 0 C N C O r H i - H i - H < T q C 0 C O C ? 0 r f i — l O C O r H i — I CO CO t— r H O O O O O O O O O O O O r H C N C N C N O O O O r H O o o o o o o o o o o o ' o o o o o o o ' o o o o O H O H q Q o q q o H O o q o q o s o r i o o q d o d o d o d o d o d o d d d o d o d o 6 6 d d d d d d d d d o d o d d d d 6 6 6 6 66 + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I t - I C O O i C N C N C O L O H ^ H ^ H ^ c O O i b - - * C O C O C O - H H C N t - H C O C O L O r f C O O O C N C N C N C N C N C N r f r f r f C O C O C O C O C O C N C N oco N N coo O N o o O H O C N H H O H O N O H N O O H H O coq c o q M W qco N O N O O N H O 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 O N H 6 66 O H 66 + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I O i C O C O O C O O r - l O r H C O C O C O C O C O b - C N L O C ^ r f i - H C O r f CN CN CN CN CN i—i i—i i—' i—' i—H i—* 0 CN CN CN CN CO CO LO L O t>" rH O N *J> CO W O I A H N O N t - O O H C 0 O C N H H CN O 1 0 O C O N H O H O O H CO "3 O N CO CO H O O C 0 C O O O H H H C O O H O C O O H O O H H H H H O H H O N O O H H O N O N O H N O H N O N O O N H O 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 6 H 66 66 66 66 + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I C » b - C 0 L 0 0 0 0 i 0 i O C N 0 i C 0 O t — l O i L O L O C O - H H i — I t — O C O C O t — O O C O t — I b - O i O O i L O r f H c O - H H C N O O O i O i O O ^ O O O C N C N r H r H C N C N C N C N C N r H r H i - H T H C N i - H i — I r H C N C N C N C N r H r H q H H H C o O H O N O N O H O H H N q H q N q H H H H H q N q H H N H q N N O N O O N H O 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + I + I + I + I rH O O + 1+1 C N C O + I CO + CO + I CO CO + 1+1 b- CO + 1+1 Oi rH + I O + I o + 1+1 •HH O + I CN + 1+1 O L O + I CO O O O O O O O O 0 1 1 1 1 1 1 1 1 1 + 1 1 1 + 1 1 1 1 + 1 Oi SB •HH •HH co r H LO Oi LO CN LO CO co Oi Oi co O CN t— 0 0 0 CO r H is "—' i-H r H t-H CN r H r H r H CN t-H t-H r H t H t-H t-H CN CN t-H CN i-H r H CN 0 + + + 1 1 1 1 1 1 + 1 + + r f >> • r 0 0 LO Oi •H CN HH CN LO Oi co rf* CO t- CO CN CN Oi O CN i-H 60 co CN t- CO CO t-H r H r H CO t— O r f CN r H Oi t-H co 0 r f O r H CO CO fi — ' r H 0 r f r H O r H r H O O O O co CN t-H 0 r f CN CN r H t-H r f co fi CN CN t-H CN CN CN CN CN CN CN CN t-H CN CN CN i-H CN CN CN CN r H r H C3 CO CO CO CO co 0 0 CO CO OO 0 0 co OO r H t-H r H r H r~ t~ t— b- b-^—v •* • H •HH t-H t-H co CO co co co CO co O O O O 0 0 0 0 0 0 0 0 OO 0 0 10 CN •HH r H r H CN CN CN CN CN CN CN r H t-H r H r H r f r f r f r f r f r f rH O O O O O O O O O O 0 O O O O O O O O O O O fi > °1 W PQ PQ PQ > > (t\ P4 w PQ j > | w PQ 1 PQ 1 > | Pf* | W j W | r ^ H 8 1 0 0 t— 1 CN O Oi 0 r j* Oi CO t~ CN co CO Oi CN LO t-H O Oi CO r f r H Oi Oi LO co CN rJH Oi co Oi co Oi 0 0 CO Oi CO LO 0 0 Oi Oi 0 0 O CO C+H O r H 0 O co 0 O O O | 0 0 0 0 1 10 O | O | O | CO | 0 i 0 1 O | 0 1 O 1 LO I rH °, ^—' 1 b-t- t— co co 0 0 0 0 0 0 CO 0 0 0 0 0 0 co co co co CN CN CN CN CN CN P H 0 co co co r H r H LO LO LO LO LO LO LO 0 0 0 0 CO CO CO CO CO co 0 Oi Oi Oi CN CN CN CN CN CN CN CN CN co co CO co O O O O O 0 D i-H t-H r H CN CN CN CN CN CN CN CN CN CN CN CN CN co co CO co co co Appendix A. Tables 155 5" CM o to Ol «3 — ' O , — ^ P H 00 5- -O O O H H H H O H H O N H O N O N O H O - f N O O O O O O O O O O O O O O N H T f O C S H O N p p p p p p p p p p p o p p p p p p p p p p p p p p p p p p q p q p p p p p p p p 0 6 0 6 0 0 6 0 6 6 6 6 6 0 0 6 0 0 6 0 0 0 0 0 0 6 0 6 6 0 + I 0 0 H C S 0 0 O > N ( D N ( D t D 0 O N C M ( » 5 W C S ' * ' * N N i C I C X ) O 1 - H 1 — < O O i — l r — I T—I 1 — I T — I C N O O O O O O O O O T — I O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O O O H O 1 N Q N O O H H H O N O C O O C O O O O O H O O H C ' - O - f O O O O C O 0 0 O O 0 0 0 0 0 0 0 0 0 0 0 0 O O O O O O O O O O O O 0 0 O O O O O O 0 0 O O rH O 0 0 + 1 + 1 4- 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 0 O 0 oq O T—1 T-H oq Oi i n C O O 1—I t - O O l p CN c o -tf CN CN CN CO "tf "tf "tf cd CO CO CO i n i n i n ed CO "tf i n CD 'Ctf O CN O CO CN CO in 0 CO CO CO CO x 0 CO 0 rH iii CN CO O Ui 0 X 0 0 rH rH rH rH m 0 0 -f O 0 CN O X 0 0 X O O 0 0 0 0 0 0 0 0 0 0 O O 0 0 0 0 0 0 O O 0 0 0 0 O rH O O O O 0 0 0 0 rH O rH O CN O 0 0 + 1 + 1 4- 1 4- 1 4- 1 + 1 + 1 + I 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 cp i n CN Oi CO 01 O l oq CO 0 oq O i n t— c o i n CO 0 "tf CO 1—1 CO CN CN CN CO CO CO CN CN CO r-H 00 i n i n r-1 t>^  06 oi oi 06 i-H rH CN O CO x 0 CO CO 0 e» Hi CO rH O CO 0 CO iii X t~ O CN O CN O CO rH CN rH O CN O O CO CO CO CO 0 e» 0 O Tf O O 0 0 0 0 0 0 0 0 0 0 0 0 rH O O O O O O O O O O O 0 0 O O O O O O 0 0 rH O rH O CN 0 O rH 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I •-tf O "vt4 O ^ t— - t f co C N 00 m co o o 4- 1 c o "tf 4- I 4- I 4- I 00 m 00 C O - t f 1—I 4- 1 00 4 - 1 4 - 1 4 -1—1 m c o -tf "tf o C N CN O CN O CN O CN O CN O CN O l O l O l t — r-H CN O CN O CN 00 4 - 1 4 - 1 O l i-H C O C O i - i O CN CN 4- I m 00 4- I 4- I 4- I 4- I o 1—1 C N m N N C D CX) O C N i-H CN CN p p OH HO HO H H O CN H H NO NO HO H H OH OO O H OO OO OO OH t>CN 0)0 NO O CO 0 6 0 0 0 0 0 0 0 0 0 0 0 6 6 6 6 6 6 0 6 0 6 6 0 6 6 6 6 0 6 0 0 6 6 + I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I o o o i o o c o i - H c y 5 C N i - H i - H i - H C N c o i n ^ " t f i n i n c o i — t e o c o o O O O O i - i i - i i - i i - i i - i i - i r - H O O O N N O W T f N T f O W C O O t ^ C O N C O O W O C O H W l A O N H H p M p H H H H O p N p c O - f q c O O p t - ; 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 CN6 N 6 COO 6 6 4 - I + I + I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I c ^ i n c N c o c o c o o c o i n c o T - H c o i n T - H - t f o i o o o t ^ c o o t — i n i n i n c M c d - t f - t f c o c d c d - t f i — i i - H C M T - H T - H i - H C M i n c o t — c N H O N O N O -f cx» in m O CO Tf O Tf o Tf O CN Tf t - CD CO X o -f O CN CO X X X H 0> O Hi CN CD CO o Hi O O -f H O O H H O H p H H p CO H H CN p CN O H O H CN p p O H p CO O o N O H O O CN CO H co p CO p O CO • 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 ft) , — V 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 6 0 CO C N m m CO CO i-H O O l CO CO O O O l CO CO co m CO O i O CN cS — ' O "tf oq t— CN co CN -tf CO "tf CN i n CN cp CO O l i n co co O l CO CN cd CN r - H o i - H c i c i oi oi oi i - H c i o c i oi oi CN CN i—l O O CN CN CN C N CN CN CN i-H r-H i-H i-H CN CN CN 1—t CN i-H i-H C N CN CN i-H H O O H H O H H H H O CO H H N O N O H O H N H O O N O N H H CN O H H O CN CN H CN O N O O CN o o o o o o o o o o o o o o o o O o o o O O o o O O O O o o O O O O o o o o O O o o O O i n 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 O l oo i - H i n "tf CO CO co O l oo O CN t - r-H C N "tf m m t— -tf O O o _ 1 4- 1 4- + 1 1 1 4- 1 4- 1 1 4- 4- 4- 4- 4-S O l i—i oq i n cp -tf CO oq oq CN cp i n i O !>• "tf O l CN O CN FW " — ' T - H CN l - i i - H CN CN CN i-H i—i i-H CN i—i CN i - H i—i i - i i—l i—l CN CN CN i—l 4- 4- 4- 1 i 4- 4- 4- 4- 4- 1 4- 1 >> • r m m i—i t— O -tf oo CO o i-H eo m t— i>- m t— m m i n O J*t 60 co -tf CO oq O Oi CO oq -tf co co co t>- CN co O l t>- "tf o T-H co cp CO (8 s V — ' CN i—l o CN i—i <~> c i o c i c i -tf l—i c i c i cd c i c i c i i - l CN c i co M CN CN CN CN CN CN CN CN CN CN i - H C N CN C N i-H C N CN C N CN CN CN t-H t— t - t— co CO CO co co CO C O CO t— t~ t - co CO CO m m m m —^s O l O l O l t - t — t - I — oo oo CO oo C N C N CN t - t - i>- t -o CN C O co co o o o o o o o o CN CN CN C N m m m co co eo co i - H o o o o o o o o o o o o O O O o o o o o o o TJ C 03 > P4 P Q pq| > > °1 rt w P Q rt P ^ W pq j > rt 1 pq j P Q i rt j w 1 r^H 3 "tf 1 co CN co r-H \ CO o CO m O l co CN i - H o CN CN i-H o •* co co "tf • 1 * l O l O l O l co CO O l co co m -tf "tf "tf co "tf i - H r-H r-H "tf co "tf o Cf5 O rO i - H p p o o | o 1 o p p o o •-tf O | O | o | O | T - H 1 T-H 1 i-H j o 1 o i o i 00 1 M Cj V — ' o o o o o o o o o o o m m m m O l O l O l i-H T-H i-H i - H P H O *— "tf -tf "tf -tf -tf "tf "tf -tf •* "tf "tf "tf t - t - t— t— t -o o o o i - H i - H i - H i-H i-H r-H i-H i - H CN CN CN CN co co co "tf "tf -tf "tf C O co co co co co CO co co co co CO co co co CO co co CO CO co co Appendix A. Tables 156 5" CN -a : o a. • ^ 60 Ol c3 ^—' O C M r * co s- -• ^ 60 <0 cS ^—' B LO 2 a is M CO r f M S O IH P H rO o O O D o o O r H CN O CO o CN O CO f CN r H r H r H r H O CN O CO o r H O o o r H O CO r H O r H r H O CO CN r H CO CO o r H O r H O o o o o o o o o o o CN CN o o o o o o o o r H O o o o o o o o o O r H o o o o O r H o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 CN c o b- CO c o b- o CO c o b- Oi T -H CN LO c o CN b- CO b- r f b- fc-rH T -H T—1 T -H T -H LO CN o o o T -H o o o o CN o T -H c o o o CS o c i c i c i c i c i c i c i c i c i c i c i c i c i o c i o c i c i c i o o TH O r H CN CN O CN O O CN r H O O CO CN r H •* o CN r H o o o o o o o o o o o o O CO O u i r 1 O i r H O CN r H o o o o o o o o O r H O CN O r H o o o o o o CN O o o o o o o o o o o o o o o o o r H O o o o o + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 CO OS CO CO o T -H o CO Oi LO b- CN T -H o OS OS b- LO CO CO LO LO c o c o c o c o c d c d CO CN c d c d b- r f T—H T-H c i c i r f r f r f LO r f r f to © es t - o O i o O CO r H CO O r H O i CN t - o CO CN CN O r H O CN O r H O U i CN r H O H t - r H r H 0 1 r H u l o u l CN 0 1 o o o o o o o o o O r H O CN O io o o o o o o CO o o o o o o o o o o o o o O r H O CN CN O o o o o + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 T-H c o c o CN o CN LO CN LO Oi CO LO LO CO b- o LO CO CN b- b-r f u i rf* r f T —i r - i CO c i r f CO CN oo CN T -H c i o r - i c i O c o c i c i CO o CN I f o o CN O O O i CO o O r H CO u l X o CO CN • 0 1 o CO o CN O CN CN CN U i o o CO CD CO to r H CO o o o o r H O o o O O H o TH O o o •<f CO O CN r H O o o o o •* o o o o o o o r H O O r H o o O r H r H U 5 CO o r H O r H O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O O O o o o o O O o o o o + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 b- T-H o s o o CN CN1 r f CN b- o 1—1 LO Ol LO Ol c o T-H o c o LO O CN CO CO CN CN CN c o CO r f LO LO r f OS CN o o q rH t - t - T-H OO t - r -T-H o c d o o T-i c i c o T—i T-i T-H o o c i o i CO t— T-H o o i T-H O o CN CN T-H T-H CN CN CN CN CN CN CN CN CN T-H T-H r-H CN CN T-H CN CN CN o o O r H r H O CN O o o X CN O CN CN O CO o CO o T f O in o o o o o r H O O 1 " r H r H CN CN O CO t- o r H O CN O o o o o o o o o o o O r H o o o o o o o o r H O o o o o o o O O O O o o O O O o o o o o o o + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 o o Oi Ol Ol o s o s CN b- T-H T-H c o Ol T-H CN CO o q LO CO O CN CN o c i o c i o r-H c o CN o CN CN CN c i O O c i c i T—i T-i CN i — i r - i T-H O r H O CN TT O CO O o o r H CO O CO CO CN O i o 91 O CO o o o r H O 1 " r H CN I A o <- CO CN r* io o f t- o CN r H r 1 o o o o o o o o o o o r H r H o o o o o o o o CN O r H O O O o o r H O o o o o o O r H r H O O O o o + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 CN CO CO CO CO CO LO o q r f r f o q CO LO CO c o CO c o LO OS a s LO CO CO CO CO CN L6 o o CN CO CO CO T—H T-H CN CO o i CO co- r f CN CN CN O CN X CN t- o X o 1 " o I A X I A O CO X o o X o o o r 1 o CO o X t- O i O 01 CN O co O i O O i r* o x o CO o r H O o o r H O r H O o o r H CN o o r H O CO o CO o CO o CO o r H O CN q CO r H q q CN O r H O q co CO o o q r H O ci 6 6 6 6 6 6 ci ci 6 6 6 6 ci ci 6 6 6 6 6 6 ci d d 6 d d d d d d d 6 d d d d d d d d d d d + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 b- b- Ol LO c o o o o CO r f LO t— CO o c o CO 00 o o O CN LO Ol OO Ol Ol o q Oi b- CN o q CN o t - H r f Ol CO t— o CO o q CN CN LO O Ol c i OS t— b-" o c i CO CN CN CN o CN c i o i o d o i CN T-i o i CN T-H c i CN T-H r-H T-H CN CN CN CN CN CN CN CN CN T-H r-H T-H CN CN T-H CN CN CN H O H O H O H O O O H CN O O H H C N O C O O T * 0 CO O H O C N O CN H O t D C N O H H qco coq H O H O d o d o d o 6 6 6 6 6 6 66 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 + 1 + 1 + 1 + 1 + 1 + 1 + 1 - 1 - 1 - 1 - 1 - 1 - 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 C O O O O l O i C O L O O C O C O C O r f O b - C O C q L O O r - n r f C O L ^ O O O O O T - H T - i o T - H T - H T — i i — J O T — ! T — i c N T - i r - I T — I O O O I I + + + I + I I I I + I I + I I + 1 + C 0 0 1 C X ) C N C O C O L O C O O O O C O C O r f C N r f O S C O T - H C O r f b -T - i T - i c q C O O C N r - i c N r - i C N T - i T - i + + + I + + o OS O CO OS CO c o O o o CO t - r f o b- CN T-H CN O Ol o c o Ol c o r-H CO CN OS CO CN o t-- OS o LO i - H o q CN b- CO LO O CO b- CN CO V —' CN o r f CO o i o i CN O t — i CN o CN r - i O c o r f T-i CD CO T-i O o CN CN T-H t - H rH t - H CN CN CN CN CN CN CN CN T-H T-H CN CN T-H CN CN CN LO LO LO LO CO CO LO LO CO CO CO LO r f r f r f r f LO LO LO t - H i - H i - H y v t - H T-H T-H t - H c o c o CN CN OS Ol Oi c o CO CO CO CO CN CN CN LO LO LO CN t - H T-H T-H l—1 o o CN CN CN CN CN o CN CN CN o o o o o o O O o o o o O O o O O O O O O o PQ rt w W PQ rt PQ rt PQ PQ rt PQ PQ rt m W PQ j rt W PQ j rt j rt i LO 1 r f I T-H i CO t - H o o Oi CN r f r-H c o LO r f LO LO o Ol CN Ol CO b-CO CO CN CN b- t - Oi o o Ol LO Ol b - c o c o c o CO c o b- CO o o c o b-T-H o O o t - H o o o O o o 1 o o | o I o | CN T-H o | o 1 CN j o i o i o I -—' t— b- t - t— o o c o c o o o r f r f r f o o CO c o c o OO c o c o CO b- b- b-CN CN CN CN t - t— t— t— t— b- t - o CN CN CN CN T-H T-H T-H LO LO LO CN CN CN CN t - t— Ol Ol CO CO CO o o i - H i - H T-H r-H r f r f r f c o c o CO r f r f r f r f r f r f r f r f LO LO LO LO c o CO CO CO CO CO CO b- b- b-Appendix A. Tables 157 CN O a. • ^ c8 s — ' O ,—.. PH OO • ^ «e s—-m s is 6H CO DO CO a) ' CN «3 o tH P H o O o O O H O H O U 3 N O O O O O O O O H O O O O O O H © H H O O N T 1 < © H C N C O C O H N O N O O O C O q o o q o o o o o o o q o q o q o q q q q q q q q q o o q q q o q o o q q q q q o q o q 6 0 0 6 6 0 0 6 0 6 0 6 6 0 0 6 6 0 0 6 6 0 0 0 6 0 0 + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I - t f ^ t — i n i - H i - H i - H C > 3 C O C O C O ^ C O i n " t f O O a i O i — I t — O l O O O O O i - H O O O O O O O O O O C N t ^ C N e O C O C O O i - J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 q n o v a n H H q o q q c o c O H O q o H q s o q r i s q o o o q O H H q q 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I o i o o o i o ^ r v - t - ^ c O " t f - t f e o e > i ' o - t f c o c o c N , c N O c o O O O O O O C O C N C N C N C N C N C N C N C N C ^ © < O H X O t - N c o © H O H © H O H C o o N O c o © © H © c O H ^ © T r H N © i r t i A ^ H C O H N q c o © x 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 0 0 0 0 0 0 6 0 0 0 + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I O O C O O I O I C N C N C O C N C O O O I " — " t f O i — l i n - c t f C O C O t ^ C O C O m OO OO CD CO i-H i — l O O l O l O l O l O O i — I O O 0 1 0 1 0 0 " t f m ©CO T P T P COCO X t- H H H H H H O C N CCO COO "d* O OCO ©00 H N OOO OOO OOl N IA C N b- T T CO OCO O H o o 0 0 0 0 0 0 0 0 o © o o o o o o 0 0 0 0 O O O O O O O O O O O O H © O O © O O © © H 0 0 O O O O O O 0  O O 0 0 O O O O 0  O O O O  O 0 0 O O O O  O O O  O O   O O O + 1 + 1 + 1 + 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 O l CN CO O -tf -tf CO i-H 00 CO CO O CO 10 CO CD t-H "tf O i CN -tf 0 "tf oq oq i-H r-H i-H i-H "tf O l O l CN t-H O 10 CO t - O C3 i O oq CN i-H 0 06 r-H 1—i T-H 1—i O o i o i 00 00 06 0 a i o i o i o i c d i-H 0 CN CN CN i-H CN CN CN CN CN i-H i-H i-H i-H i-H CN i-H i-H i-H r-H i-H CN CN q H CO CO CO H X Hi q q q q H q q q H O 0 0 0 q q H q H H q q H O O 0 q q q q q O O q q q co 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 + 1 + 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 r-H t-H i p i p 10 i O "tf -tf 10 10 -tf t~ CO i p i n "tf -tf "tf "tf CN 00 i-H CN 1—5 CN O O O O O O O O O O 0 O O O O O T-H i-H O CN ce co C - CO CO 0 CN H CN H CN O H H -c 0 CN O CN O O CN 0 -f CN O O U) *p O 0 CO CO CO CN CN O H H H O X O O 0 0 0 0 H H O O O O O O O O 0 0 O O O O O O 0 0 O O 0 0 O O 0 0 0 0 O O O O O O O O + 1 + 1 + 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 4- 1 c o CN CN r-H t-H 00 O l i-H 00 t— CO O i-H i p CO "tf i O i-H CN Oi CN "tf CO i d CN CN 1—i i—i i—l CN i - l CN CN CN -tf c d c d CO c d CO c d -tf O CN x cn CN m CO iii •0" CO rr CO co -r CN O 0 0 GN O CO 0 0 CO 0 0 X Hi 0 m CO 0 0 ->r TJ* CN H CO t- CO H H 0 X O H H H CO H CO CO H O H O H H H H CO 0 H O H O O H 0 CO O O O CN H O O H H H H H O H H H O CN O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 4- I 4- I 4- I 4- I 4- I Oi O O CO -tf CN O t - t -CN CN CN CN CN 4- I -tf !>• CN CN 4-CN O CN CN 4 - 1 4 - 1 i O Oi O CN CN CN 4- I 00 o 4- I c o CO 4- I CO O 4- I Oi + I c o 00 CN i-H O CN CN Oi 00 00 4- I CN t -O CN 4- I 4- I 4- I 4- I 0 CO CN in 10 Oi Oi 01 oi 06 00 4 - 1 4 - 1 t -CN CO 4- I -tf CN CN O H H H CN H CN CN H O H O H CN H H CN O CN O N O O N O N O H O N H O O H H H H H H H H H O N 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 o o 4 - 1 4 - 1 4 - 1 4 - 1 + 1 + 1 + 1 + 1 + 1 4 - 1 4 - 1 4 - 1 + 1 4 - 1 4 - 1 4 - 1 4 - 1 4 - 1 4 - 1 4 - 1 4 - 1 4 - 1 o i o c o o i O i o o o i o o i c o ^ o i T - H o o - t f c o e o e o c o o c N i o O C N I - I C N ' O O O O O O O O ' T H + + + + + + + 1 1 + + + 1 + + 1 1 + - t f o i ^ t - c o c D O c o t ^ - o i - v t f o o o i - t f c o t — t ~ - t — oo H oo oo O i - l i - H C N i - i t - l c N O i - l i - l t - l c N r - l c r q C N + + + + T - H i - H C O C O C O C O C O l O ! - j C N t - - - " * t f » O i O - t f - t f C N i - l o ' - t f C N C N C N C N i - H C N C N C N i - H C N C N C N C N C N C N C N t - H t - H i - H + + + I + c o c o c o H^H —ctf " tf c o c o "tf "tf c o c o "tf "tf o o o o o o o -tf o -tf "tf ITi o OS CO CO oo o m cp co co co -tf co -tf co o Oi Oi o CN oo co m -tf 1—1 CN o CN o CN -tf t-H "tf r-H CO 1—t CN CN i — l CN 1-1 CN i — l CN o ' CN •-tf i-H CN CN I - l CN co "tf CO -tf CO -tf co "tf co "tf CO "tf "tf m cp -tf l O cp m cp -tf m cp -tf m cp "tf m cp m o -tf m o -tf o o ' o o o o ' o o o ' o o o ' o o P3 > w PQ PQ PQ PQ > rt( rt W w W PQ j > > rt 1 rt 1 w 1 PQ 1 > i CO 1 m "tf 1 c o CO "tf m CO i n t - c o CO r-H O l CO t— m o i-H c o t -c o c o c o CN CO c o c o co CO CD CD o CN O l t - t— t - t— oo oo m m o o p CN o o o o o o O CN CN o o 1 o p o 1 o 1 CN 1 o i o | CN 1 CN CN 1 CN t -tr- t— t— t - t - fc-t— oo oo oo 00 00 c o c o c o c o c o c o c o CO c o c o ee CO CD CD CD CO c o o o o o o o c o c o CD CO CD CO "tf "tf "tf "tf "tf "tf "tf -tf -tf O l O l O l O l O l O l O l O l r— Oi Oi Oi O l O l O l O l Oi a i O l O l O l O l O l O l O l O l O l Appendix A. Tables 158 CN o a. • ^ bO O i OH JA co • 60 tO B tO x fa C O f t - 1 Ul >-i i O H C3 O O t - 0 0 - f - H T | ' 0 0 5 - H - H O > 0 « l A - H O O O - H O i - l O - H O N O O O O O O O ' » ' O G O - H - H - H O N O - H O o o - H O o o o o - H O o o o o - I N o o q q q q q q q q q q q q q q q q q q q q q q q q q q 0 0 6 0 0 0 6 6 0 0 6 6 6 0 6 0 0 6 6 0 0 6 0 0 0 6 6 0 6 0 6 0 0 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 > o > i O " t f " t f i o o e o o i C N c o " t f i o t O i H i H C N o o c o " t f c o c N i o C N C N r H C N C N c o c o i o o o o o o o o o i - j i - j i - j r H r H i - j 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 oco © co-H - H O 10 o O - H rn CN CN i—1 o - H - H O 1-H-H o o - H O o q q - H --jo O N CO - H . O q q co © - H - H 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 + I + I + I + I + I + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 w n CV O) CO CV <x> ^  ^ 10 n >^ CV ^ ^ n ^ c^t q N H q a ^ ^ o n o H n n ^ e n o H N O H H q o H o q o o O O H O O O O O H O O O O O O O O O O O O O O O O O O O O O O O O H O O O O O O O O O + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I c C i O t O C O t O C » O O C N C « " t f O O i C O e O O O t O " t f t O O u O C N C N T t f CO "tf CN o oi ai to to 00 t— t— 10 o o 00 o O T J I O O 00 b- O O O C O O O C O N CO COCO O - H - H O - H - H O O - H O - H O - H O C O O O C O C O O C O O N - H C O O —H LA O - H C O O - H O O O C O O O - H O - H - H C N O O O O O O O O O O O O O O O O O - H O - H O O O O - H O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O + I co CO o CN + I + I + I + I 00 CN 00 t -T-H tr— O r-H + I + I + I 00 CO CN 0 1 CO O + I CO CO 00 t— 00 00 r-H r-H CN CN CN CN + I co 0 CN + 1+1 co to "tf CN + I •«tf CN + 1+1 m m t - r H r H C N O CN O CN + I co CN CN + I CN "tf O C N + I "tf "tf O CN + I r H IO o CN + 1+1 r H O t O O O i O CN o t - c ^ o c o - H N O N O o b - o t r t T t * c o o N q - H q - H q - H q c i q - H q - H H d d r * d d d d d r * d d d d d d d d d d d d d d d d o d d d d d d d d d d d d d + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I l 0 0 i O 0 0 0 i t ^ ^ O ^ 0 0 l T - t 0 1 > - 0 0 i - H r H " t f " t f p t - . O l > - ; O i-i O CN r H C O C N C O O O O O O O r H "tf CN O O C O O COCO C O O N O O C O - H N O C O O C O O N O N O N O C O O N O - H C 4 —H O O © L A CN O - H © N O —H —H drimdddddriddr*dT*T*T*dddddddddddddddd + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I O i r H t— O O C N O » 0 " t f C N C O C O O r H C O t O t — 1—j O t O t>- CO CO I D l O d ^ i n d c C N H H H H ^ H H H N ' r f CN r-i CN r H 0 CO LA O t - 00 CO C - CN O 0 0 -H t ~ T T X O O O CN OJ -H CN CO O co N CO LA X LA O N O 0 CO T T O IS- T T C~ N O CN q LA N ; O CO -H N O C - O O CN O CN -H CN O CN q -H q —H q CN q -H —H -H 0 q -H q q -H 0 -H -H q q -H O O O CN 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + .1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 to "tf t— •n r H co t— CN 00 "tf "tf r H "tf r H CN to to t - 00 "tf 10 0 CN co to co CO m r H to CO CO t - oq 00 O CN to O O i CN m m r H r H 06 r>^  06 00 co CN i—i r H 0 o i to to CO CN r H c d c i O o i o i o i CN r H r H r H r H CN CN CN CN CN r H r H r H CN CN CN CN CN CN r H r H r H 0 T r LO O CN -H CN -H LO O O CN O N -H N -H O -H -H -H -H -H O —H O CN O -H O N O O —H O —H -H O O -H -H O O N 0 0 O O O O O O O O O O 0 0 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 O to to m O i to to m oq m "tf CN to t— r H O i 00 r H m CN CN 0 O 0' 0 c i O c i c i r H 0 0 i—l 0 r H + + 1 1 1 1 1 1 1 1 1 + + + + 1 + 00 co co t— CN O i r H O i t - to O i 0 00 r H to O i co r H CN O i co O i CN CN r H CN CN r H CN r H r H r H r H CN r H CN r H r H CN CN CN r H r H r H + + I „ v "tf CO to "tf 00 "tf m 00 r H t— "tf CN m 0 co to 00 m 00 to "tf co 0 CN co to O i "tf m O i IO m CN CN 00 "tf in r H "tf O i in O i r H to v — ' r H "tf "tf ed ed CN 1—i c i CN i—l i—l "tf c d CN i—l i—l CN c i i—l 0 c i c i CN r H r H r H r H CN CN CN CN CN CN r H r H CN CN CN CN CN CN CN CN CN iO m m m IO to to to co co CO O i O i "tf "tf "tf "tf CV O 0 0 0 O O 0 O to to to to to "tf "tf "tf CN CN O O O O "tf "tf "tf "tf "tf m m in O 0 O 0 0 CN CN CN r H r H co co co co O c i c i c i O CD 0 c i c i c i O 0 0 c i c i O O O 0 0 0 0 C4 W K a PQ j > >, w 1 pq 1 > 1 rt j pq 1 rt 1 > | >; rt 1 rt 1 to 10 0 t - CN co "tf m co CN r H r H CN CN r H 0 r H O i 0 CN O i 1—1 m "tf 00 0 "tf co CO CO 00 00 00 to 00 00 00 O i 00 t— CO to co 0 CN m r H r H 0 0 0 0 0 O CN r H 0 | O 0 1 0 1 0 | 0 | 0 1 0 1 0 1 co co co co CO to to to to to to to to to to to to to 00 00 00 00 CO CO co co co O i O i O i co co CO co CO CN CN CN 00 00 00 00 00 00 O i O i O i O i O i co co co "tf "tf "tf "tf "tf m to lO m 10 00 00 00 00 O i O i O i O i O i 0 0 0 0 0 O 0 O 0 O O 0 0 0 0 0 0 Appendix A. Tables 159 CM u ~S PM JA o a. • ^ 60 O i ce B o ,—^ CH 00 JA ^  • ^ c3 • f r~! tn o -g CL, O O O rH O O d d + I O O O O O O O O O O O O O O H O O C N O O H O O O O O O O O O O O O H o o o o q o o o q q q q q q q q q q q q q o q o q o o o q q q q q o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d + 1 + 1 . -r f i-H CN H O O H O O O O q q q q q q q q _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ d d d d d d d d d o d d d d d d d d d  + I + I + I + I + I + I + I  I + I + I + I + I + I + I + I + I + I + I + I + I + I tO r H CM CO r H r H r H f r  CM r f CN b— CO HH OO r f r H CM CM CM r H C N O O O O O O O O O O O O O O O O O O O " o o o o o o o o o o o o o o o o o O O O O O H H N O O t H i s q q K O O O O H O O o o o O H O q H q o w q q d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I i - H c o b - o i t - O i o o c o c N t - c o r f i - H C N i - H C N t o c o c o o i o o t - -• H H CO CO r f r f r f r f r f r f r f r f r f C O C O - H H C O C O t O t O C O c o c s q q q ^ H q q c N q q O H C N q q H q H r H C N H r H d d d d d d T * d d r * d d d d d d d d d d d d d d d d d d d + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I O C O i - H b - i - H r f i - H r H O S i - H C N t O C O O r H C O O l i - H O i C ^ O l r f ° t O C O O O C « C » C N i - i i - i O i - i o d o 6 c N C N C N r H r H i—I i-H i—I i-H i-H r H r H r H r H r H O O i O i H C S O i O OOO O C 0 CN T f O O O O H O O C 0 H O OCN CO H r* © O Ui COO I 0 O O C 0 H O H O H CN OCN HCO o o o o o o o o o o o o o o o o o o o o o o o o q q q q q q q q q q q q q q q q q q q q d o d d d d d d d o d o d o d o d o d o d o d d d o d o d d d o d o d o d o d o d o d o + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I b - r f r f C N r H O O t O r H O O r H C N C N r H i - H O t O t — t O I O r H CO r f O o C O t - r H C O b - O t O C O O O O O r f t O O i O l O O C N O r H t O C O O i O i b - CM CM O CM CN CM O CM oo oq o CM o CN o CN O i O i CO CO O CN O i r H r H CM O CN O i CN o o H O O H H O C N H H O H O H O o o H O H H C N O CN q o r * q q c o © H q H q q q q q q q q q d o d o d o d o d o d o d o d o d o d o d o d d d o d o d o d d d d d o d o d o d o d o + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I b - b - b - r H - f C O C O t ^ O b - r f r H C N b - C O C O b - t ^ b - b - O O C O O O O O O O r H O O O O O O O O o o C N O O C N C N H T T C O c o o C N H C N O O H c o n COCN H r ; © q q H O r - © c o q C O H O H H H H H q c o d d d d d d d d d d d o d d d o d o d o d o d o d d O H d o d d d d d d d o d o d o O H + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I c o c o c o c o o c o t - t o r H O i c o c o i - H t o c o o o O T O i o i o a i O C O 1 A O O O O CO © U5 CD O O C N H O C N O ^ * CD t— C O H r* CO O O O H H O t— O LO O H Tj* O r * •«* © U> 00 O O o o H O O C N H O C N H r* © H H C N O O H c o o H H C N O H O q c o C N O C N O c o n O H H O q q q q d d d d d o d d d d d d d d d d d d d d d d d o o d d d d d d d d d d d d d d o d d O H + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I + I O r H r H r f C O C O C N O O b - O O C O O i O O O O b - C O b - O i C O O O b -OS b - O C O C O t 0 0 1 C O C N C O C O t - - r H C N C O C O t 0 0 1 oo r H 00 r H CO r H r H CN o CN co CM CN CM r H CM O i r H r H CM CN CN r H CM r H CN r H CN r H oo i-H r H CN O i i-H r H CM o CN O i r H r f CM O H H O O CN H O CN H CN O H O H O O H CO o CN H CN O H O O CN CN O CO o CN O CN H O H CN O O H O ui e tO O O + 1 O o o + 1 oq O O + 1 CM o o + 1 O o o + 1 co O O + 1 co o o + 1 co o o + 1 co o o + 1 b -o o + 1 t -O O + 1 O o o + 1 O o o + 1 t o O O + 1 t o O O + 1 oq o o + 1 r H O O + 1 oq o o + 1 O i o o + 1 b -O O + 1 o o + 1 O i o o + 1 CO r - i o r H r H r H o O O O O i—i i—i O i—i o r H o O O o O CM HM rf"" + b -+ + o 1 1 CO + CO + CO + t o 1 O O i O i + CO CO + oo i CO 1 co 1 CO 1 CO CO + O FW "—' r H r H CN i-H r H CM CN CM CN >> JA *o 60 CrT 1 CO CO 1 r H co O i co 1 CN r H + co o + r f t o + O i t o O i r H tO CO + co oq + CM c q + O i b -r H o + b -CO t o CO o CO i CN i-H + oo r H + r H c q + O i CO + c q 1 co CM CO o CN o CM CO r H r-H CN o CM CN CN r - i CN i—i CM c d r H r - i CM r - i CN o CN o CN o CM co r H co r H i—i CN O CN 1—i CN o CM o CM CM CM 10 '«? i -O r f O CO r f O CO r f O CO tO O i CM tO O i CM r H O i CO r H O i c q r H O i co r H O i CO r f » -r f r f r f r f t -r f r H t -r f r f t -r f r f r f r f b -r f b -O r H b -o r H O i oo r H O i co r H O i co r H co r f r H O o o o o O O o o O O O O O O O o o c i c i o O (-\ rt *\ *] rt PQ j >^ rt, a >t >; rt { rt j w 1 m 1 rt 1 PQ i > ) rt PR r f to b - 00 t - O i oo t - CO CN r H r H CM O to co CO tO CO to r f oo CO r f O i r H i-H r f r f r f CN r f »- r f t - r f O i O CN CN CO co 00 to o o o o O O O O tO O o O C | O I O r H O 1 o 1 o 1 o I o | o , co co CO r f r f r H r H r H i-H CO CO CO CO CO CO CO CM CM to to tO toCO oo oo CO CO O i O i O i O i CO co CO co co co CO CO CO CO oo oo CO CO co co O O o O o O CN CM CM CN CN CN CN tO to to to to o o o Appendix A. Tables 160 5" C N o 6 0 O l - ,—-PH CO JA ^—' • ^ tO 2 33 is 6-JA C O bo CO a) - ' 53 _> rH O rH O O CN O CN rH LA O LA IA CO •0" rH "91 rH rH CO CN O N CN O N O CO TH •<(< O H O H O H O O LA LA N CO O H o o O O O LA O O CO o o o o o o o o o o o o O O O O O H O O O O o o o o O O O CN O CN O O + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 CD O Ol t O CO O b- tq CN CO b- Ol b- CO O t o oo t o Ol CN oq •HH CO CO O ci r - i oi oo fr- od od b- b- b- od b- lO r f r f CO r - i r - i t -T-H H T—1 rH rH CN O CN O N O CO CO rH O X t- LA ee CS CO CN rH Tf CO o H CO O H O H CO CO o f O Tl> O IA O H O) X O CN o o O O H o O X O S o o o o o o o o o o o o H O O H O CN o o o o o o o o o o o •* H xr o o + 1 + 1 O rH + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 cq 01 + 1 +^H t o oq oq CO CO t o o O HH Ol CN CN Ol CN cq "* oq CO CO vi • r - i co co CO r - i CN CO CN CN CN CO O cd O od CO r - i CN b-CN CN CN CN 1—1 r-1 rH i—1 rH rH rH rH rH rH CN CN rH rH CN CN i H CN O 01 o o X o N •01 H o N H xr O LA CN CO CO CO CO H CO X o O CN CO CO O H O H O H O H H CO O LA o o O O o o O LA O CN O N o o H O H O H O q o q q CN q q co q LA r H H o o o o q q q q q N O CO q q 6 6 d d d H O r H d d d d d d d d d d d d d d d d d d d d d d 6 d d d d d d d o d d d d d + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 b- CO CO rH CN rH CO Ol co co co O co rH b- tO Ol b- b- CN Ol rH Ol Ol cq CO O rH cq t o oq r f CO CO oq O CN oq Ol "** rH CN b- O oi oi co CN CN CN r - i ci od r - i ci ci od oi od r - i O ci od cd CN r - i rH rH CN CN CM CN CN CN rH CN CN CN rH rH rH CN CN CN rH CN CN CN O O 1- o 9 9 © CN o co q CN LA CN H q q H CN H H H CO CO O CO o q co r; q q q q O o q q q q q q o o d d d d d H d H d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 CO tO tO O O tO O tO b- b- b- CO rH c o tO "* t o tO O b- •HH CD ci rf ' rf ' r f r - i CN r - i r - i CN CN co r f rf ' CO ci ci ci tO •* O O O H O H r H O N O H O H C N r H r H O f O H O O H t - C O H O H C 0 N O O O O O O O O O O H C 0 O O o q q o q N O i o O N o q o o q o o o q o q o q o q N O H o q q q q d o d o d o d o d o d o d o d o d o d o d o d d d o dco d o d o d o d o d d d o d o d o + I d + I + I + I + I + I + I + I + I C O C N t O r H C N C O r H C N b - r H C O - H H H H + I N r H r H C N r H r H C O O O O r H r H C O O r H r H r H r H r H C N b - - t H C O O O O O r H C O O CO O O r H o o o o o o o o o O O O O O O H COO O N 4 1 O r 1 H M O r 1 XCO C N O O C N COrH CN rH OlA O Ci O V ^ q O C N OrH O C N C N O O O O H O H d o H O dco O C N O H d d d d d d d o d d d o H O d d O H d d d o d d d d d o O H O H d d + I r f rH CO CN rH HH rH CN CO CN CN CN rJH CN t o b— Ol CN CO rJH Ol Ol oi od o r H O l C O C O C N C O C N C N - J H - J H i o c O t o CN XCO N O COO OCN OCO OCN LAN 0 1 X CO X O H 01CN TfCN OLA OlH O O X H O Oi TflA O *f OCO X O q q q q H C O qco O H q r ; C O H H O L A H H O q q H H O H O H O H O C N O H q o H O O H O H H O d o r ^ d o d c i o o d d d o d d d d d d d d d d d d d d d + I + I + I + I + I + I + I + I H - I + I + I + I + I + I + I + I + I + I + I + I + I + I O b - t O C O C O C ^ t O C O t O b - r H b - t O C O C O b - C O C O ' ^ C O O l ' H H 0 0 1 C N C O C O r H l O O O C 0 0 1 r H b - C O b - t O O e O C » r H C N r H C O r—i ci CN r H C5 CN r—i Ol CO r—i rH ci CO CO 00 CN r—i CO CO CN r—i r-i C N C N C N C N C N C N C N r H r H C N C N C N r H r H r H C N C N C N r H C N C N C N H q q q H c c q N q H q ^ H H N H t O H H q H q H H O H q H H H O H r H q q o O H O H O H w o d d H O d o d o d o d o d d d d d o d o d o d o d d d d d o d o d d d o d o d d d d d o + l + l + l + l + l - | - l - ) - l - l - I H - l - l - l - r - l + l + l + I H - l + l + l + l + l - ( - l - ( - l + l C O r H b - b - C O c y 5 0 0 b - O S 0 1 r H C O r J H C O O b - b - b - . C O t O b - t O O r H r H r H r H r H r H O r H O r H r H r H r H i — i o O O O r H r H O + I + Ol CO Ol Ol CO Ol 00 CO CN t o t o CO r f CN CN CO t o t O t O b- co t O o r H r H r H r H CN CN o CN 1 o o 1 b-Ol r H r H 1 Ol cq 1 CO o + r f O t o CN 1 Ol oq CN CO + CO CN i t O r H 1 oo oq CO r H CO cq co b-+ r f r H + o r H + co b-r H CO i o r f I b-cq + Oi CN o CN ci CN CN CN r - i CN r - i CN CN CN r H CN ci CN cd r H CN CN r - i CN o CN r f r H CO i— i CO r H CN CN r - i CN O CN CO r H CN CN r H CN CN CN b-eo r f b-co r f r f r H r f r f r H r f r f r H r f OO CN r f CO CN r f co CN r f CO CN r f OO Ol t o OO Ol t o oo Ol t o 00 Ol t O CO Oi t o CO Ol t o Ol t o o Ol t o o Ol t O o Ol t O o O Oi r H o Ol r H t O t o r f o ci O O O O O ci ci o o o o o o o o ci o o o O rt rt >^ rt 05 > rt w cq 1 > rt W W pq i >; rt { W i pq i > 1 cq I 1 r f b-o 1 b-t O o r f CN O co CN O CN CN O CO 00 O b-co o CO co o t O o t o b-t O o 1 CO t o o t o t o o t O Oi CN Ol r f CO co Ol r H o CO O 1 Ol t O o 1 co t o o 1 r H b-t O 1 r H CO o 1 o oo o I .060 11765. 11765. 12162. 12162. 12162. 12164. 12164. 12164. 12164. 12200. 12200. 12200. 12200. 12200. 12200. 12224. 12224. 12224. 12224. 12303 12303. 12378 Appendix A. Tables CN r * d o a . • ^ bO OJ cS • fi o i O o D o o o o o o o o o o o o o o o o o o o o o o o o o o + I + I + I + I + I + I + I + I + I + I + I + I + I O r H C N O O C O C O ' v t f i O - ^ i O i O t O O J O O O O O O O O O O O O O o o o ' o o o o o o o o ' o ' o p q H O q q O H H CO q O H O H q es q q O H O N O N 6 6 0 0 6 6 6 6 0 6 0 0 0 0 0 0 6 6 0 6 6 0 6 0 0 0 + I - t f O O - t f t O C O l O t O l O C O C N C N C O r H O O N O O O O N N b- O O H N H T J * O CO H H H CO O tD l""! q d o o o o ' d o d o o d H o d o d o d d d o o d o d o 4- I 4- I 00 4- I C D 10 ec o 4- I OJ 00 CN 4- I OJ i d CN 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I -tf CO O CO O CN CO CN CN CN r-H O CN CN O CN 4- I iO O CN 4- I tr-OJ O O O O O O O O O O O O O O O O O O O O O O O O O O 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I 4 - I OJ tr- 1 — I r H C O O O O O C N - t f O J C N - t f i O t o c o c o c o c o c o c o c o t o t — tr— lO J3, 0 CN OJ rH tr-^  rH 1—1 CN 1—1 CN i—l CN 0 CN 0 CN CJ CN OJ 1—1 OJ rH oi rH 1—t O H O H O O 0 0 0 0 O O q q q q q H q q q q q H q H '"tT OH 00 6 6 4- 1 -tf O 6 6 4- 1 "tf O 6 6 4- 1 10 O 6 6 4- 1 O 6 6 4- 1 CN 6 6 4- 1 CN 6 6 4- 1 O 6 6 4- 1 rH 6 6 4- 1 co 6 6 4- 1 CJ 6 6 4- 1 rH 6 6 4- 1 CN 6 6 4- 1 rH O N O N O H O H H H O H O H H H O N 0 0 H O O N H CO t—^ O O 4- 1 O O O 4- 1 O O O 4- 1 rH O O 4- 1 C— O O 4- 1 CO O O 4- 1 -tf O O 4- 1 oq O O 4- 1 CN O O 4- 1 iO 0 0 4- 1 OJ O O 4- 1 rH O O 4- 1 CO O O 4- 1 rH H r-i rH CN ed CO CN cd CO CN CO cd 00 O H O CO O T|< O H O 01 O H 0J N O O 0> -1" 0 0 10 CO 0 0 OJ N O O 0 0 H O O N O H IO H O O N O H O 0 CO O H N IA H H a . • bC CS to^  0 0 4- 1 t -OJ O O 4- 1 10 O O O 4- 1 tr-eo O O 4- 1 to O 0 0 4- 1 to co 0 0 4- 1 rH -tf O O 4- 1 rH O O  4- 1 CO rH O O 4- 1 iO "tf O O 4- 1 CO CO O O 4- 1 rH IO 0 0 4- 1 00 to O O 4- 1 tr-ee O CN O CN rH i—l CN i—l CN r-1 CN O CN O CN O CN OJ rH oi rH oi rH t>5 rH H H H O H H H O H O H O H O H O O H H O H O O H H H e 10 O O 4- 1 to O O 4- 1 to O O 4- 1 to O O 4- 1 OJ O O 4- 1 oq O O 4- 1 oq O O 4- 1 OJ O O 4- 1 CO O O 4- 1 OJ O O 4- 1 CO O O 4- 1 OJ O O 4- 1 O 0 0 4- 1 "tf O O O O 0 0 O O O O O i—l 1-1 HM 1 t -1 1 CO 1 iO 1 CN 1 oq 1 1 CN 1 "tf 1 O 1 OJ 1 OJ 1 0 FW "—' rH r-1 1—1 1—1 CN CN i—l i—l CN i—l 1—1 i—l 1—1 >•> r* tc co 4-tr-CN to OJ 0 rH 4--tf CN 4-to oq 4-t -oq 4-OJ rH 4-00 -tf 4-OJ "tf 4-rH OJ 4-co CN 4-co CN tr-CO J, rH CN 0 CN -tf rH CN CN 0 CN 0 CN i—l CN O CN O CN CO CN O CN O CN ed rH to 1? - O CNT iO iO -tf lO lO -tf IO -tf CN 00 "tf CN 00 -tf CN CO -tf CN 00 -tf CN 00 -tf CN 00 "tf CN 00 -tf CN 00 -tf CN 00 "tf CN 00 -tf 1-t O O 0 o" 0 0 0 cz? c5 0 0 0 ci tf w PQ j PQ 1 PQ j >^ >^ > rt l rt j rt j OJ 00 rH rH CN O OJ 0 OJ tr- 00 O 10 10 IO "tf •«t f lO co -tf •*tf ee co O 0 0 00 O O O O 0 0 0 O 0 | -tf 00 00 00 "tf "tf "tf "tf -tf -tf -tf -tf -tf "tf t— tr- tr- co co co 00 co CO co co co CO 00 co co 00 00 00 00 CO 00 00 00 00 00 CN CN CN CN CN CN CN CN CN CN CN CN CN 

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