A SURVEY OF DWARF IRREGULAR GALAXIES IN THE LOCAL SHEET
ROBIN LEIGH FINGERHUT
A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
GRADUATE PROGRAM IN PHYSICS AND ASTRONOMY YORK UNIVERSITY, TORONTO, ONTARIO
DECEMBER 2011 Library and Archives Bibliotheque et Canada Archives Canada
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The author retains copyright L'auteur conserv ownership and moral rights in this et des droits mor thesis. Neither the thesis nor la these ni des e: substantial extracts from it may be ne doivent etre ir printed or otherwise reproduced reproduits sans s without the author's permission. Abstract
This dissertation delineates and examines a sheet-like structure of galaxies embedded within the Local Supercluster. It is shown that this structure, coined "The Local Sheet", has a non-isotropic boundary at a mean distance of ~4 Mpc from the Milky Way, with the vast majority of its galaxies situated within 0.5 Mpc of the Supergalactic Plane. Its thickness is -10% of its length, which is over twice as thin as the Local Supercluster out to -10 Mpc.
Upon establishing the definition of Local Sheet membership, a set of diagnostics of the luminous mass and dynamics of the Sheet's most plentiful constituents, its dwarf irregular galaxies (dls), is presented. Luminous masses were gauged from photometry in the near-infrared (Ks), where more than 95% of a dl's light can be attributed to the stars which make up the bulk of a dl's mass. To this end, an extensive Ks imaging survey of over 70 Sheet dls was conducted. The spatial distributions of the dwarf properties were then investigated for environmental influences and compared with predictions for a dynamically equilibrated sheet-like structure. To establish a framework for these observations, the same investigations were pursued using a sample of 8 theoretical sheets extracted from an N-body simulation of structure formation in the standard ACDM cosmology.
Both the Local Sheet and the theoretical sheets are found to have vertical crossing times which exceed half the Universe's Age. This indicates that the Local Sheet's
- iv - galaxies are making their first traversal of the Sheet's vertical extent. Nevertheless, the vertical dynamics of the dark matter halos in the theoretical sheets exhibit similarities with the predictions for an equilibrated sheet for which the density declines exponentially with altitude. Further evidence of dynamical equilibrium is found in the vertical density profiles of both the Local Sheet and the theoretical sheets, which closely match an exponential model.
Tentative evidence is found for a correlation between a sheet's dynamical and structural parameters and its crossing time, in the sense that theoretical sheets with the shortest crossing times show the strongest agreement with the exponential density model for an equilibrated sheet. Based upon its comparatively long crossing time, the Local
Sheet appears to be in the process of evolving into an equilibrated system.
For a sheet to attain gravitational equilibrium, the exponential model requires that the sheet contains at least 1.5 times as much mass as is attached to its galaxies. The simulated sheets are estimated to possess at least this amount of mass in the form of dark matter below the approximate minimum mass of galaxy-sized halos. It is therefore concluded that the Local Sheet is pervaded by an intra-sheet population of dark matter which contains approximately as much mass as its galaxies.
- v - Acknowledgements
The completion of this dissertation was made possible by the unlimited patience and support of my supervisor, Dr. Marshall McCall. I have benefited immeasurably from his endless knowledge as well as from his demanding expectations. I am proud of all the work that we've done together.
I would also like to thank my supervisory committee for their very helpful feedback.
A special thank you goes to my entire examining committee for making the oral defense such a rewarding and memorable experience.
Thank you to my mom, Beverley Fingerhut, an incredible person who makes anything possible.
And finally, thank you to my husband, Dr. Robert Metcalfe, and my sons Jesse and
Joey Metcalfe. Every day, at least one of you makes me laugh, and that makes life amazing. Thank you for being such super-awesome people.
• VI - Table of Contents
Abstract iv
Acknowledgements vi
Table of Contents vii
L Introduction 1 1.1 Evidence for a Local Sheet of Galaxies 1 1.2 Scientific Justification for a Survey of dls in the Local Sheet 5 1.3 Features of this Study 8 1.4 Outline of this Dissertation 10
2. The Sample of Local Sheet dls 12 2.1 The Current Census of Nearby dls 12 2.1.1 Extinctions and cosmological corrections 15 2.1.2 Distances 16 2.2 The Local Sheet Reference Frame 22 2.3 Sample Definition 24 2.3.1 Extraction of the Local Sheet 24 2.3.2 Local Sheet candidates with uncertain distances 28 2.3.3 The Local Sheet survey region 29 2.4 The Local Sheet Sample 31 2.5 Sample Completeness 42
3. Theoretical Predictions of Sheets 48 3.1 The Current State of Cosmology 48 3.2 The ACDM Simulation and the Sheet Extraction Process 50 3.2.1 The overabundance of dwarf satellites predicted by the ACDM model 57 3.3 The Mass Functions of CDM Sheets 57 3.4 The Distribution of Mass in CDM Sheets 60 3.5 The Density Profiles of CDM Sheets 66 3.5.1 Construction of the vertical density profile 66 3.5.2 Vertical density models of equilibrated sheet-like systems 67 3.5.3 Fitting of the vertical density profile 70
- vii - 3.6 Peculiar Motions in CDM Sheets 81 3.6.1 The z-dependence of the vertical velocity dispersion 82 3.6.2 The relationship between the vertical and radial velocity dispersion 85 3.7 The Crossing Times of CDM Sheets 87 3.7.1 Are sheets evolving systems? 88 3.8 Surface Densities of CDM Sheets 95
4. The Near-Infrared Imaging Survey and Data Mining of Local Sheet dis 100
4.1 Ks Imaging Observations 100 4.1.1 CFHT observations 103 4.1.2 OAN-SPM observations 103 4.1.3 IRSF observations 104 4.1.4 CTIO observations 104 4.1.5 ESO observations 105 4.1.6 WIRCam observations 105
4.2 Ks Image Reduction 107 4.2.1 Image preprocessing 108 4.2.2 Sky subtraction 109 4.2.3 Photometric calibration 112
4.3 Ks Surface Photometry 113
4.4 Fitting of Surface Brightness Profiles in Ks 116 4.4.1 Radial range of the fits 117 4.4.2 Uncertainties in the fit parameters 118 4.4.3 Astrometry 119
4.5 Integrated Ks Magnitudes 132 4.5.1 Comparison with different facilities 133 4.5.2 Comparison with 2MASS photometric parameters 135 4.5.3 The dl Potential Plane 137 4.5.4 K, magnitudes for unobserved Sheet dis 138 4.6 Derived Masses of the Local Sheet dis 142 4.6.1 Stellar masses 142 4.6.2 Gas masses 143
5. Analysis of the Local Sheet 151 5.1 What We Can Learn From Our Own Backyard 151 5.2 The Distribution of Mass in the Local Sheet 154 5.2.1 Luminous mass 154 5.2.2 Gas fraction 160 5.2.3 (B-KJ o colour 164 5.3 The Density Profile of the Local Sheet 166 5.3.1 Effects of sample incompleteness 169 5.4 Peculiar Motions in the Local Sheet 171 5.4.1 The vertical velocity dispersion 176
- viii 5.5 The Crossing Time of the Local Sheet 177 5.6 The Surface Mass Density of the Local Sheet 179
6. Conclusions 185 6.1 Summary of Results 185 6.2 Future Work 191
7. References 193
- ix - 1. Introduction
Kirk: "No, I'm from Iowa. I only work in outer space."
1.1 Evidence for a Local Sheet of Galaxies
In the 1780s, the German-English musician William Herschel used a home-made
18.7-inch reflector to conduct the first systematic sweep of galaxies from his backyard in
Bath, England. In his On the Construction of the Heavens (Herschel 1785), he made one of the first documented observations of large-scale structure in the Universe by describing a "collection of many hundreds of nebulae which are to be seen in what I have called the nebulous stratum of Coma Berenices." Edwin Hubble, shortly after his determination of the 1st definitive extragalactic distance using a Cepheid in M31, observed that the Milky Way is found within a much smaller-scale "stratum" composed of, at least, 7 members. In his Realm of the Nebulae, Hubble named this system "The
Local Group" (Hubble 1936). The most recent census of the Local Group includes 35 galaxies spanning roughly 2 Mpc in extent (van den Bergh 2007).
In 1958, George O. Abell published The Distribution of Rich Clusters of Galaxies containing over 2700 galaxy clusters (Abell 1958). This was shortly followed by the
Catalogue of Galaxies and Clusters of Galaxies containing over 30,000 galaxies
(Zwicky, Herzog and Wild 1961). The distribution of galaxies in these surveys has confirmed that galaxy aggregation is found on a wide range of scales, from the Local
-1 - Group to superclusters composed of groups and rich clusters. More recently, in large redshift surveys such as the Las Campanas Redshift Survey of over 26,000 galaxies
(Shectman et al. 1996), a plexus of sheets, filaments, clusters and voids has been revealed to us, which Bond et al. (1996) have described as the "cosmic web".
In the 1950s, de Vaucouleurs showed that the spatial distribution of over 300 of the nearest galaxies suggests a "metagalactic cloud of galaxies" containing numerous galaxy aggregates, including the rich Virgo cluster at its nucleus and the Local Group as an
"outlying condensation" (de Vaucouleurs 1958). He described the structure as a "Milky
Way of galaxies" owing to its strong flattening, which is nearly perpendicular to the plane of the Milky Way (de Vaucouleurs 1953). De Vaucouleurs estimated the overall extent of this "local super-cluster" to be 20-30 Mpc with a width 1/5 of its extent. The midplane of the Local Supercluster has since been used to define the Supergalactic coordinate system, a convenient reference frame for investigating the distribution of galaxies and galaxy clusters within the Supercluster, and the location of the Supercluster relative to other superclusters.
In recent decades, as distance measurements have improved in accuracy for a rapidly increasing number of galaxies, the evidence has mounted that the Local Group may reside within a sub-filament of the Local Supercluster. Schmidt and Boiler (1992) investigated the three-dimensional distribution of nearly 300 galaxies with radial velocities relative to the Local Group within 500 km s"1. They concluded that the Local
Group is part of a filamentary structure with a larger concentration of galaxies than the
-2- surrounding Local Supercluster. McCall (1987) showed that the IC 342/Maffei group is part of a flattened system comprised of the galaxies within ~5 Mpc of the Milky Way.
More recently, Peebles et al. (2001) described The Local Group as "...part of an expanding filamentary structure, with the nearest big galaxies close to each other in the sky..." Karachentsev et al. (2003b) described the distribution of galaxies within ~5 Mpc of the Milky Way as "...rather inhomogeneous, showing concentration of the objects towards two opposite directions... which confirms the location of the Local Group in a filament, extending from Canes Venatici to Sculptor".
Karachentsev et al. (2004) have since released a set of galaxy distances to over 450 galaxies each having a distance estimate < -10 Mpc or a radial velocity < 550 km s"1. The authors refer to this volume of space as the "Local Volume". Based upon distances in the
Local Volume Catalogue, the vertical standard deviation of galaxies in the Local Volume
(i.e., the dispersion perpendicular to the Supergalactic Plane) corresponds to a thickness of 4.6 Mpc. However, at smaller spatial scales, the thickness of the galaxy distribution about the Supergalactic Plane hovers at only ~1.2 Mpc for all galaxies out to 4 Mpc from the Milky Way. In other words, the distribution of galaxies within 4 Mpc is nearly 4 times thinner than the Local Volume as a whole. Beyond 4 Mpc, the width of the galaxy distribution about the Supergalactic Plane increases rapidly to 2.0 Mpc for galaxies out to
5 Mpc and 2.8 Mpc for galaxies out to 6 Mpc. These statistics are illustrated in Figure
1.1.
-3- Figure 1.1: Mean Supergalactic Z-coordinate (SGZ) of Local Volume galaxies in running distance bins of 0.5 Mpc in width. The error bars, which are the standard deviation of the mean, increase significantly beyond 4 Mpc. This suggests the presence of a flattened distribution of galaxies within this distance, either aligned or tilted from the Supergalactic Plane. Distances are from the Local Volume Catalogue (Karachentsev et al. 2004).
•j - 1I 1? *1 1 1 1 1 J 1 1 1 r 1 1 1 1 1 1 1 1 I 1 t I f I t I 1 f 1 | • 1 • 1—•—1 1 « 0 1,r i * i I1—•—1 | „ T " 1 t 1 f 1 —i -L1 o 1 1—•—1 1
: 1 1 : I ° 1 a. T I 1 I 1 « S = - » i1 « T : 1 |
A -1 R RT tsq r i , t U : t :1 I 1 1 f 1 • 1 v : o :1 1 | 1 • -2- 1 -
I "i1 1 1 1 1 1 1 1 1 3 1i 1i 1i i i i i i i i i i i - i ! £ 0 2 4 6 8 Distance [Mpc]
Further evidence for a flattened distribution of nearby galaxies can be seen in the counts of galaxies as a function of vertical distance from the midplane of the Local
Supercluster. Figure 1.2 shows the number of Local Volume galaxies within 6 Mpc of the
Milky Way in 0.25-Mpc distance bins from the Supergalactic plane, revealing the large concentration of galaxies of all morphological types within 0.5 Mpc of the Supergalactic plane. Clearly, a thin sub-structure of galaxies exists on a scale intermediate to the Local
Group and the Local Supercluster. In keeping with the nomenclature established over a
-4- half-century ago by Hubble and de Vaucouleurs, this structure has been coined "The
Local Sheet" (McCall 2011).
Figure 1.2: Histogram of Local Volume galaxies within 6 Mpc in 0.25-Mpc bins along the Supergalactic z-axis (SGZ).
60 E galaxies _ to S galaxies dls o o 40
© jQ E 20
0 1 0 1 SGZ [Mpc]
1.2 Scientific Justification for a Survey of dls in the Local Sheet
The primary goal of this dissertation is to determine whether the Local Sheet is a dynamically equilibrated system, or whether it is expanding with the Hubble flow, unimpeded by its overdense midplane. To answer this question, the following investigations are pursued for both the Local Sheet as well for theoretical sheets extracted from N-body cosmological simulations:
-5- 1. Is a sheet's distribution of luminous mass consistent with the predictions of a
dynamically equilibrated sheet-like structure?
2. Are the dynamics of sheets consistent with the predictions of a dynamically
equilibrated sheet-like structure?
3. If sheets are dynamically equilibrated, in what ways would the global properties of
their galaxies be affected, and do the Local Sheet galaxies conform to these
predictions?
4. To what extent are the dynamics of sheets consistent with the potential field predicted
by their visible mass? Is additional mass required to justify the motion of a sheet's
mass tracers, and if so, how is this missing mass distributed throughout the sheet?
This dissertation addresses the above questions by focusing on dwarf irregular galaxies (dls), which are the most plentiful type of galaxy found throughout the Universe, dls typically have between 106 to 109 stars and are characterized by their predominantly disordered motion and lack of structural uniformity. They harbour star-forming environments, as evidenced by their gas content. However, pre-supposing an age, their relatively low metallicities indicate that chemical recycling has progressed at a slower rate, on average, compared to metal-rich galaxies. It has also long been suspected that dls are the building-blocks of more massive galaxies, dls are therefore most similar to first-generation galaxies, a feature which has been utilized to investigate the connection between the evolution of dls and the conditions of their environment (see §5.1).
-6- Our location within a structure composed mainly of dls makes our local dls especially useful for gaining insights into the evolution of structure on large scales. Owing to the proximity of these galaxies, measurements of their properties can be made with high accuracy. Thus, in surveying the dls of the Local Sheet, this study is essentially identifying the forest from the trees.
Fundamental to this study are reliable indicators of the mass components of a dl.
Large galaxy surveys, such as the Local Volume Catalogue of Karachentsev et al. (2004), often employ total B magnitudes as a global mass tracer of a galaxy. This is due to the availability of measurements of the apparent B magnitude for most galaxies.
Unfortunately, a B magnitude is only a reliable gauge of a galaxy's stellar mass if the galaxy's stellar mix is uniform, or if galaxies with similar proportions of stellar populations are being compared. Neither of these conditions apply to dls, which have stellar populations dominated by old stars yet still exhibit star formation activity in varying degrees. A dl which appears bright in the optical may be low in mass but in the possession of a recent population of O and B stars, which are enhancing the light in B.
Alternatively, a dl may be high in mass due to a large population of old stars, yet appear faint in the optical from a lack of hot, newborn stars. Thus, a dl's B magnitude is an ambiguous mass tracer.
The near-infrared (NIR), however, poses several advantages. Vaduvescu et al. (2005) showed that more than 95% of a dl's light in Ks (2.15 p.m) can be attributed to stars older than ~4 Gyr. Similarly, Cairos et al. (2003), Noeske et al. (2003) and Vaduvescu et
-7- al. (2007) have demonstrated for a large sample of dls as well as for blue compact dwarfs
(BCDs) that the surface brightness profile (SBP) of a dwarfs underlying old stellar population can be investigated near the profile's center owing to the lower contribution of the central starburst in the NIR compared to the optical. Thus, the NIR is well suited for tracing the old population dominating the stellar mass of a dl. Unfortunately, most dls are too faint to have been detected by the Two Micron All Sky Survey (2MASS)1 completed in 2001, and Kirby et al. (2008) show that the total magnitudes reported by 2MASS for low surface brightness galaxies can be underestimated by up to 2.5 mag. Furthermore, deeper NIR imaging surveys of dls (e.g., Hunter and Elmegreen 2006) have not focused exclusively on nearby objects, leaving the majority of dls within 5 Mpc of the Milky Way with unreliable gauges of mass. An extensive Ks imaging survey of nearby dls has therefore been conducted in support of this dissertation.
1.3 Features of this Study
To answer the questions posed above, the Ks imaging survey has been complemented by data mining for gas diagnostics, optical photometry, kinematical data and distance diagnostics for the Local Sheet dls. In drawing data from more than one source, it is important to adopt self-consistent processing methods and calibrations. Integrated properties, such as total magnitudes, require the adoption of an extrapolation method.
Distance-dependent properties, such as luminosity, require the adoption of an extinction-
1 This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.
-8- correction algorithm and a distance zero-point. Indirect measurements, such as mass, require the adoption of a mass-to-light calibration. Poor attention to these subjective aspects of data processing and mining can lead to underlying offsets in data sets drawn from multiple sources. In this study, a number of measures have been taken to ensure statistical self-consistency as well as the highest possible quality of data. In many cases, these measures are unique to this project, owing to the extensive expertise of the survey team in NIR imaging of dwarf galaxies. The unique features of the Ks survey and its associated analyses are described below.
• A robust algorithm has been developed for the reduction of Ks galaxy images
obtained from different facilities. Special attention is paid to the subtraction of the
high NIR sky level, which varies both temporally and spatially across detectors.
• Given that accurate and self-consistent distances are the cornerstone for establishing
the true 3D distribution of the Local Sheet's dls, great care has been taken to construct
a set of homogeneous distances based on the most accurate distance methods
available. Specifically, all distance diagnostics have been re-calibrated to a common
distance and extinction scale.
• As mentioned in the previous section, the masses of the Local Sheet dls presented in
this dissertation are derived from the galaxies' light in Ks. This bandpass best traces a
dl's old star population, which constitutes the bulk of a dl's mass. For each dl with
surface photometry in Ks (-55% of the sample), the luminosity is derived from a fit of
-9- a hyperbolic secant (sech) to the surface brightness profile. As discussed in §4.5, this
diagnostic has been proven to be a reliable tracer of a dl's total stellar mass.
• While several independent studies of nearby dls have been conducted for the purpose
of understanding the connection between their global properties and their
environment (see the discussion in §5.1), no such study has targeted galaxies confined
to a sheet-like structure. As such, this dissertation is the first to treat the Local Sheet
as a distinct environment.
• In addition to establishing a self-consistent set of mass diagnostics for the Local
Sheet's dls, this study identifies, for the first time, the theoretical predictions for the
spatial distribution of mass in simulated sheets, as conveyed by a sample of sheets
extracted from an N-body simulation of structure formation in the standard Big Bang
cosmology.
1.4 Outline of this Dissertation
In Chapter 2, the Local Sheet sample is introduced. The chapter begins with a detailed description of the methods used to obtain extinctions and distances for nearby galaxies, followed by a summary of the criteria for assigning Local Sheet membership, and finally a presentation of the Local Sheet dl sample and an analysis of the sample completeness.
Chapter 3 is an investigation of the theoretical predictions for the mass distribution and kinematics of sheet-like structures. The predictions are drawn from a sample of sheet-like distributions of cold dark matter halos extracted from an N-body simulation of
-10- the standard evolution in Big Bang cosmology. Specifically, the structural parameters of each sheet are measured from the sheet's density profile and compared with its internal kinematics so that a conclusion can be made regarding its dynamical stability.
Chapter 4 presents the observations, reductions and results of the Local Sheet dl imaging and mining survey. All the derived properties of the Sheet dls are tabulated in this chapter, including the galaxies' stellar and gaseous masses. Also provided are the Ks images and surface brightness profiles for the 72 dls observed in support of this dissertation.
In Chapter 5, the amassed properties of the Local Sheet dls are used to compare the
Sheet's mass distribution with its internal kinematics in the same manner as for the theoretical sheets of Chapter 3. In addition, the global properties of the Sheet dls are investigated for evidence of an environmental signature of their Sheet membership.
Finally, in Chapter 6, the dissertation objectives stated in §1.2 are revisited with the knowledge gained from this study.
-11- 2. The Sample of Local Sheet dls
Guinan: "No, it's a Samalian coral fish with its fin unfolded."
Data: "I believe what you are seeing is the effect of the fluid dynamic processes inherent in the large scale motion of rarefied gas. "
Guinan: "No, no. First it was a fish and now it's a Mentonian sailing ship."
Data: "Where?"
Guinan: "Right there. Don't you see the two swirls coming together to form the mast? "
Data: "I do not see it. It is interesting that people try to find meaningful patterns in things that are essentially random. I have noticed that the images they perceive sometimes suggest what they are thinking about at that particular moment. Besides, it is clearly a bunny rabbit."
2.1 The Current Census of Nearby dls
In order to extract the Local Sheet from the larger-scale distribution of galaxies in which it is embedded, we require an accurate 3D-map of the galaxies in the volume of space in and around the Local Sheet. The Local Volume Catalogue (hereafter LVC) by
Karachentsev et al. (2004) serves this purpose. The LVC is an all-sky catalogue of -450 galaxies with distance estimates within ~10 Mpc or heliocentric radial velocities within
550 km s"1. The objects included in the LVC were compiled from lists of galaxies from several sources, in which optical identification of the dwarf galaxies were primarily made from the photographic plates of the European Southern Observatory Sky Survey
(ESO/SERC; see Holmberg et al. 1974) or its northern counterpart, the Second Palomar
- 12- Sky Survey (POSS II; see Reid et al. 1991). The morphological types of the LVC objects were taken by Karachentsev et al. (2004) from the RC3 catalogue of de Vaucouleurs et al. (1991), in which objects were assigned a numerical index (T) along the Hubble sequence. In this scheme, known as the Revised Hubble system, ellipticals have either
T = -4 or T = -5, lenticulars have -3 < T < -1 and spirals have 0 < T < 9. In the original
Hubble sequence, objects assigned to the "irregular" family (denoted as "Irr") are those which do not "fall into a sequence of type forms characterized by rotational symmetry about dominating nuclei." (Hubble 1926). In the Revised Hubble system of de
Vaucouleurs et al. (1991), the "Irr" galaxies have been assigned either T = 9 or T = 10.
The classification T = 9 (Sm) applies to the irregular objects with evidence of very weak spiral structure. 52% of the objects in the LVC have T = 9 or T = 10. With the exception of the Large Magellanic Cloud (LMC), all such objects have absolute B magnitudes fainter than -18.
As stated in §1.3, one of the priorities of this dissertation has been to establish a self- consistent data set of dl properties, fundamental to which are distance estimates on the same distance and extinction scale. This objective must apply to the entire Local Volume of galaxies, as a dl's Local Sheet membership status can only be identified with confidence if accurate distances are known for its neighbours. The methods which have been employed to accomplish this objective for the 234 Local Volume dls are described
-13- below. The resulting distances, extinctions and K-corrections2 (for the 133 Local Sheet dls only) are listed in Table 2.2 and Table 2.4. The breakdown of the distance methods employed for the Local Sheet dls is depicted in Figure 2.1. For details on the distance methods used for more luminous galaxies (i.e., with absolute magnitudes brighter than -18 mag in B), the reader is referred to the survey of the 60 brightest galaxies in the
Local Volume by McCall (2011).
Figure 2.1: Breakdown of the distance methods employed for the Local Sheet sample of dls. The distance methods are abbreviated as follows: RGB = Tip of the Red Giant Branch; HUB = Hubble's Law; BBS = Brightest Blue Stars; PP = Potential Plane; MEM = Group Membership. Excluded from the diagram is the LMC, for which the Cepheid distance has been adopted (see notes to Table 2.2).
BBS (1 1%) HUB
PP (9%)
MEM (6%)
RGB
2 The K-correction accounts for the effects of the changing bandwidth and shape of a transmitted spectrum as a function of redshift. It is expressed as an offset, in magnitudes, to be subtracted from the apparent magnitude.
- 14- 2.1.1 Extinctions and cosmological corrections
All photometric quantities employed in this work, either measured as part of this dissertation or adopted from other sources, have been corrected for the obscuring effect of interstellar dust using the following treatment. First, each galaxy's Galactic reddening,
E(B-V), was obtained from the reddening maps ofSchlegel et al. (1998, hereafter SFD98) derived from 100-|am emission. Second, the reddenings were converted to broadband extinctions using the York Extinction Solver (YES)3.
Given E(B-V) in the direction of each target, YES computes the corresponding optical depth at l^m (ri) using an appropriate spectral energy distribution (SED) for the probe of extinction. For SFD98 reddenings, the appropriate SED to use is that of an elliptical galaxy, as ellipticals were used by these authors to calibrate the conversion from the dust's 100-|am intensity to reddening. Next, YES was used to convert ri into the target's broadband extinction in the desired filters. In this step, a SED representative of each target's morphological type, redshift and optical depth was adopted. From the same SED,
YES computes the cosmological K-correction for each filter. The advantage of computing extinctions in this manner is that source-dependent shifts in the effective wavelengths of broadband filters are avoided.
In this dissertation, the monochromatic reddening law of Fitzpatrick (1999) was adopted for all extinction analyses. This law was tuned to deliver a ratio of total to selective extinction for Vega of A^E(B-V) = 3.07, which is the mean value observed for
3 The York Extinction Solver (YES) is hosted by the Canadian Astronomy Data Centre, National Research Council of Canada, at http://www2.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/communitv/YorkExtinctionSolver/.
- 15- the diffuse component of the interstellar medium in the Milky Way. This form of the
Fitzpatrick law is therefore appropriate for objects whose radiation passes through long sightlines. Further details on the algorithms employed by YES can be found in
McCall (2004).
2.1.2 Distances
The Tip of the Red Giant Branch (TRGB)
Distance estimates for 137 (59%) of the Local Volume dls have been obtained via the
TRGB method, which has a typical uncertainty of -8%. This method makes use of the sharp cutoff that has been observed in the luminosity function of red giant branch (RGB) stars at an absolute magnitude in Cousins I (Mj) of ~ -4 mag. The astrophysical origin of this cutoff is the limiting core mass at which helium ignites, at which point a star's ascent up the giant branch is halted. The TRGB distance estimates for the Local Volume dls have been determined primarily by the authors of the LVC using Hubble Space
Telescope/WFPC2 images in V and I. These authors consistently adopt M/~-4.05 mag from Da Costa and Armandroff (1990). However, Rizzi et al. (2007) have established a metallicity dependence for Mi, in turn developing an algorithm for determining distances with a statistical uncertainty of only 1%. McCall (2011) has recalibrated the result of
Rizzi et al. (2007) to a common extinction scale (the same scale adopted in this dissertation) and to a distance scale anchored by the maser distance to NGC 4258 found by Herrnstein et al. (1999). With this calibration, which has been adopted in this dissertation, the absolute magnitude of the TRGB in Cousins I is given by
-16- Mj = 0.217±0.020 [(V- /)O,TRGB - 1.6] - (3.935±0.028) Equation 2.1
where (F-/)O,TRGB is the extinction-corrected Johnson-Cousins V-I colour of the
TRGB. The uncertainty in the zero-point is that due to random errors only and therefore excludes the 4% uncertainty in the distance to NGC 4258; as all distance methods have been anchored to this distance scale, the uncertainty in the distance to NGC 4258 represents a systematic error in distance moduli for all sample galaxies. The (F-/)TRGB colour and apparent /-band magnitude (/TRGB) of the TRGB were measured from each galaxy's published colour-magnitude diagram (CMD). In cases where the CMD was reported in HST filters, the measurements of the magnitude and colour of the TRGB were converted to the Johnson-Cousins filter system using the calibrations of Sirianni et al. (2005).
The uncertainty in a TRGB distance modulus is the quadrature sum of the measurement error in /TRGB and the standard error in M/ from Equation 2.1, which includes the measurement uncertainty in (F-/)TRGB as well as the errors in the slope and zero-point of the calibration of Mj. The mean uncertainty in the distance moduli found for the 137 dls in the LVC with TRGB distances is 0.13 mag, which corresponds to a mean uncertainty in distance of 6%.
The Potential Plane (PP) method
Twenty-two of the Local Volume dls which lack TRGB distances were observed as part of the near-infrared imaging survey conducted for this dissertation (Chapter 4). From the photometric parameters derived from their Ks surface photometry combined with their
- 17- HI fluxes and line widths, the distances to these dis could be estimated from the dl
Potential Plane (PP) formulated by McCall et al. (2011) in a study of -50 dis with TRGB distances. McCall et al. (2011) show that the potential, P, defined by the ratio of the baryonic mass in solar units to the scale length in parsecs, can be reliably estimated from the HI line width and central surface brightness as follows:
log P = (5.697±0.065) + (1.134±0.080)(log JF20-1.8) Equation 2.2 + (-0.198±0.018X/^,o-2.5 log q-20)
In the above equation, W20 is the width of the 21-cm line at 20% of the peak intensity and q and fij^o are the dl's axis ratio and central surface brightness in Ks (corrected for extinction), respectively. Knowing P in Mo pc"1, the distance modulus (u) follows from
( equating Equation 2.2 with P = ( YksLKs+Meas)/RQ, where Qfo = 0.883±0.199 is the adopted mass-to-light ratio in Ks in solar units (see §4.6.1), LKs are Mgas are the dl's luminosity and gas mass, respectively, in solar units, and Ro is the dl's scale length in parsecs. This leads to the following expression for the distance modulus:
/u/5 = log P + log ro - log(0.648/rc) Equation 2.3 0 4(m M o 5 - log [YKS i0" - ^°" ^ "12-5) + 10* K2\ F2I/XH]
In Equation 2.3, ro is the dl's scale length, in arcseconds, obtained from the hyperbolic secant (sech) fit to its surface brightness profile, mtcs,o is the dl's apparent magnitude in Ks after extinction and cosmological corrections have been applied, and MKs,Q = 3.32 is the
Sun's absolute magnitude in Ks computed from the Sun's absolute magnitude in V
(Holmberg, Flynn, and Portinari 2006) and the solar V-K colour (Flynn et al. 2006). The
-18- last term in Equation 2.3 defines the dl's gas mass, where
&21 = 2.356 x 10s M© K'1 km"1 s, Fix is the dl's flux at 21 cm in Jy km s"1 and XH = 0.735 is the fraction of a dl's gas mass in the form of hydrogen. The adopted value of Xh is that appropriate for the metallicities of the dwarfs in the Local Sheet sample (see McCall et al.
2011).
McCall et al. (2011) found that the cosmic scatter about the Potential Plane can be considered negligible in comparison to the uncertainty expected from observational quantities in Equation 2.2. Thus, the uncertainty in a PP distance modulus arises from the observational errors in mj&.o, Fix, ^20, and q as well as from the systematic uncertainty in The median uncertainty in the distance moduli for the 22 dls with PP distances is 0.50 mag (23% in distance).
The Brightest Blue Stars (BBS) method
For 26 of the Local Volume dls with neither TRGB nor PP distance estimates, distances could be estimated via the Brightest Blue Stars (BBS) method. This method utilizes the observed correlation between the mean luminosity of a galaxy's brightest blue stars and the apparent brightness of these stars relative to the galaxy's total apparent brightness (see, e.g. Karachentsev and Tikhonov 1994). The BBS calibration adopted in this dissertation is:
-19- In the above equation,
Galactic extinction, and Bo is the galaxy's total integrated magnitude in B. Equation 2.4 was produced using the BBS observations of Karachentsev and Tikhonov (1994) for 21
Local Volume galaxies with either TRGB or Cepheid distances. The distances and extinctions for the 21 galaxies were converted to the distance and extinction scale adopted in this dissertation. The scatter in Equation 2.4 is 0.46 mag, which is largely due to the standard deviation of the mean
The Membership (MEM) method
For 13 dls lacking Ks surface photometry or stellar photometry, reasonable estimates of their distances could be gauged from the mean distance of the galaxy group with which they are associated, as judged by their astrometry and kinematics. The group assignments for these galaxies were taken from the survey of nearby galaxy groups by
Karachentsev (2005). The mean distance of each galaxy group was computed after converting known distances of members to the distance and extinction scale adopted in this dissertation. The exception to the above treatment was Holmberg IX, which was assigned the distance to the bright spiral M81 owing to the evidence of a tidal interaction between these two galaxies (Chiboucas et al. 2009). The uncertainty in a MEM distance is the standard deviation in the distances of the member galaxies from which the group's mean distance was calculated. This amounts to a distance uncertainty of -0.6 Mpc, on average.
-20- The Hubble (HUB) method
For 35 dls in the LVC, the only available diagnostic of distance is the radial velocity, which yields a distance estimate via Hubble's Law (i.e., d- V^QIHQ). In this computation, vLg is the galaxy's radial velocity in the rest-frame of the Local Group (see Table 5.4).
The value of the Hubble constant (HQ) employed throughout this dissertation is
74.2 ± 3.6 km s"1 Mpc"1, which was determined from infrared photometry of extragalactic
Cepheid variables by Riess et al. (2009) using the maser host NGC 4258 to set the zero- point. Thus, Ho is on the same distance scale as the TRGB and PP distances calculated in this dissertation.
Karachentsev et al. (2009) found the local dispersion of velocities about the Hubble flow to be only ±25 km s"1. If the distance moduli for all 234 Local Volume dls were computed from Hubble's Law, their median uncertainty would be 0.2 mag, which takes into account the velocity dispersion as well as a typical error of 5 km s"1 in the velocity measurements, as adopted by Karachentsev et al. (2009). However, for the 132 Local
Volume dls with TRGB distances and positive vlg, the mean absolute difference between the TRGB distance modulus and the Hubble distance modulus was found to be
0.7±0.8 mag. The TRGB distance modulii have a typical uncertainty of <0.1 mag. Thus, there is, on average, a ~0.6±0.8 mag difference between the Hubble distances and the
TRGB distances which cannot be explained, suggesting that the true dispersion in the local Hubble flow is greater than that observed by Karachentsev et al. (2009). A conservative estimate of the uncertainty in a Hubble distance has therefore been adopted as the mean difference between the Hubble and TRGB distances after subtracting, in
-21 - quadrature, the error in the TRGB distance modulus. The error in a HUB distance estimate therefore amounts to 0.8 mag (37% in distance). Fortunately, Hubble distances were primarily restricted to galaxies beyond 5 Mpc. Only 3 dls within 5 Mpc had no other distance estimate available.
2.2 The Local Sheet Reference Frame
In the recent survey of our nearest and brightest galaxies by McCall (2011), it was found that the 14 nearest galaxies with absolute Ks magnitudes brighter than -22.5 mag are confined to a thin plane with a standard deviation in the plane's perpendicular direction of only 0.2 Mpc. These bright galaxies define the midplane of the Local Sheet, from which the Local Sheet coordinate frame can be established. By fitting these bright galaxies to a plane, McCall (2011) found:
• the polar axis of the Local Sheet has a Supergalactic longitude and latitude of
Lisp = 242° and BLSP = 82°, respectively;
• the Sheet is therefore tilted 8° from the Supergalactic plane; and
• the Sun is offset 0.12 Mpc perpendicularly from the midplane of the sheet.
Figure 2.2 illustrates the orientation of the Local Sheet Cartesian coordinate frame relative to the Supergalactic Cartesian coordinate frame (SGX, SGY, SGZ). Throughout this dissertation, the Local Sheet axes are denoted simply by x, y and z. Equation 2.5 gives the transformation matrix from the Supergalactic Cartesian coordinate system to the
Local Sheet Cartesian system, where to is an arbitrary viewing angle for the xy-plane of the Local Sheet, measured counter-clockwise from the x-axis. Throughout this
-22- dissertation, co is set to -(LLSP-90°) = -151.74°, which represents a clockwise rotation of the xy-plane about the z-axis such that the x-axis coincides with the projection of the SGX axis in the xy-plane. In this manner, the Local Sheet frame is equivalent to the
Supergalactic frame with the exception of the 8° tilt of the x>>-plane from the
Supergalactic plane and the 0.12-Mpc offset of the origins of the two planes.
Figure 2.2: Orientation of the Local Sheet coordinate frame (x, y, z) relative to the Supergalactic coordinate frame (SGX, SGY, SGZ). The Supergalactic longitude (Lisp) and latitude (BLSP) of the Sheet's polar (z) axis are 241.74° and 82.05°, respectively, and the perpendicular offset of the Sheet's midplane (p) is 0.12 Mpc from the Sun in the -z direction (McCaI12011). The arbitrary viewing angle around the z-axis (a>) is set to -(Lis/^90°) = -151.74 (see text above). The SGX-SGY plane is defined by the plane of the Local Supercluster, which is tilted at an angle of nearly 90° from the Galactic plane. (The North Supergalactic Pole has a Galactic longitude and latitude of / = 47-37° and b = 6.32°, and the SGX-axis at L — 0° has a Galactic longitude and latitude of / = 137.37° and b = 0°, which is the intersection of the plane of the Local Supercluster and the Galactic plane.)
X Equation 2.5
X cosco SINLISP -cosco cosLISP -sinco cosBISP SGX 0 + sintu SINBISP cosLISP + sinco SINBISP sinLISP y - -sin -23- 2.3 Sample Definition Armed with the distances to the Local Volume dis and their 3D-distribution with respect to the Local Sheet midplane, it is now possible to identify the spatial extent of the Sheet dis. The procedure which was adopted to achieve this is as follows: 1. A group-finding algorithm was used to identify the dis that are participating in the same flattened system of galaxies as the Milky Way (i.e., the Local Sheet). 2. The extracted dis with uncertain distances were examined individually to ascertain the validity of their association with the Local Sheet. 3. The extracted dis were used to establish the volume of space containing the Local Sheet. A detailed description of the above steps is provided in the following sections. In Chapter 3, the same procedure is adopted for extracting theoretical sheets from a cosmological N-body simulation. 2.3.1 Extraction of the Local Sheet A natural way to extract the Sheet dis from the surrounding Local Volume is by way of the friends-of-friends (FOF) algorithm. The FOF algorithm assigns aggregate membership based on a specific linking length, such that every member of a group is within one linking length of another member. While this method may exclude spatially- isolated objects that are within a group's dynamical influence, it can be relied on to extract, at the very least, the skeleton of the Local Sheet. -24- The FOF algorithm was run in successive iterations on the Local Volume galaxies with linking lengths from 0.5 to 2 Mpc in 0.05 Mpc increments. Figure 2.3 shows, for each linking length, the number of galaxies in the aggregate containing the Milky Way. Also shown for each linking length is the ratio of the group's dispersion in the Local Sheet z-coordinate to its radial dispersion in the xy-plane relative to the Milky Way. This is a gauge of the group's relative thickness in the sense that the lower the ratio, the flatter the structure. It can be seen that with linking lengths less than 0.9 Mpc, only the Local Group galaxies are identified as friends of the Milky Way. Once the linking length increases beyond 0.9 Mpc, the number of friends makes a jump from -40 to -180 galaxies with a relative flattening of 18%. There is an additional jump in the number of friends when the linking length exceeds 1.1 Mpc, which is due primarily to the merging with the friends associated with the M83 group of galaxies. Despite the increase in the number of friends from 182 to 255 there is no appreciable increase in the relative flattening until the linking length exceeds 1.40 Mpc. Linking lengths of 1.35-1.40 Mpc yield the same 282 friends of the Milky Way with a relative flattening of 20%. A linking length of 1.45 Mpc causes the relative flattening to jump to 31% from the addition of 17 friends. Thus, 1.35 Mpc has been adopted as the linking length which best characterizes the Local Sheet. The 282 friends-of-friends of the Milky Way have a standard deviation of 0.6 Mpc in z, and -50% are dls. Figure 2.4 shows the locations of the FOF dls projected onto the xy-plane of the Local Sheet. -25- Figure 2.3: Number of Local Volume galaxies identified as friends-of-friends of the Milky Way as a function of the linking length. 5 400 c Rotio of z— to xy—dispersion * w— •<20% •21-30% 1 300 •31-40% .—• 0 1 •>40% H • m 200 1V 0 ^ 100 Q3 JD E 1 0 .5 1.0 1.5 2.0 FOF linking length [Mpc] Figure 2.4: xy-projection of the Local Volume dls within 3 1 Q r-•—•—1—•—i—1—<—•—1—i—1 v •—•—i—•—•—•— 5 M01 Grot CL S 0 h : ^ Cen A Group ~5: : . Local Group Maffei/1342 Group.Group • » 10 —. . . i . . . • ' » ?* -10 -5 0 5 10 x [Mpc] -26- Figure 2.4 reveals five overdense regions, four of which correspond to the four nearest galaxy groups. These are the Local Group (centred at x ~ 0.5 Mpc, y ~ -0.3 Mpc), the Maffei/1342 group (centred at x ~ 3.0 Mpc, y ~ 0.0 Mpc), the M81 group (centred at x ~ 2.5 Mpc, y ~ 2.0 Mpc), and the Cen A/M83 group (centred at x ~ -4.0 Mpc, y ~ 2.0 Mpc). In the recent study of nearby galaxy groups by Karachentsev (2005), it is shown that the centers of mass of all four groups are within 0.8 Mpc of the Supergalactic plane, and that the mean distance of each group member from its group centroid is <0.4 Mpc. Thus, the extents of the four nearest galaxy groups are all essentially within the thickness of the sheet-like structure of galaxies identified by the FOF algorithm. The fifth concentration of dls is centred at x~ 1.0 Mpc, y~ 3.5 Mpc, which corresponds to the central portion of the loose, low-density CVn I cloud. In their survey of -220 galaxies in the CVn I region, Karachentsev et al. (2003a) found two peaks in the 1 histogram of the galaxies' radial velocities. The first peak occurs at vLG = 200-350 km s" , where VLG is a galaxy's radial velocity with respect to the Local Group rest frame. This range of radial velocities includes M94, the brightest galaxy in the cloud, and spans the aforementioned dl concentration at x ~ 1.0 Mpc, y ~ 3.5 Mpc. Essentially all members of the CVn I group in this velocity range are confined to the z-range of the Milky Way's 1 friends-of-friends. In contrast, the second peak, which occurs at VLG = 500-650 km s" , primarily includes galaxies with uncertain distances, as can be seen in the region around x~2 Mpc, y ~ 6Mpc. Karachentsev et al. (2003a) suggest that this second peak "may correspond to a more distant galaxy group". In support of this, the z-coordinates of the -27- galaxies in this velocity range extend well beyond the range of the Milky Way's friends- of-friends. Furthermore, Karachentsev et al. (2003a) found a crossing time of-15 Gyr for the full extent of the cloud, while a more recent calculation using 9 galaxies in the proximity of M94 yielded only 6.9 Gyr (Karachentsev 2005). This suggests that the motion of the dls in the region around x ~ 2 Mpc, y ~ 6 Mpc is not characteristic of the CVn I group but rather of the Hubble flow. A sixth concentration of galaxies, known as the Sculptor group, is in the region x =-0.5 Mpc, =-3.5 Mpc. The concentration is not evident in Figure 2.4 because only 4 of the group's 11 known members are dls. Karachentsev (2005) described this group as a loose filament in which the giant spiral NGC253 and its companions comprise a semi-virialized core. All known members of this group were identified as friends-of- friends of the Milky Way. 2.3.2 Local Sheet candidates with uncertain distances Having extracted the Local Sheet candidates, it is prudent to examine the dls with uncertain distances. As described in the previous section, Figure 2.4 reveals a loose association of such dls centered at x - 2 Mpc and y~ 6 Mpc, which is beyond the densest region of the CVn I cloud. These dls share a velocity field with galaxies whose z-coordinates extend well beyond the Sheet's thickness. Furthermore, if the dl at x= 1.6 Mpc, y = 5J Mpc (PGC 41093) is merely ~0.5 Mpc more distant than its uncertain Hubble distance suggests, the 11 dls in this association are no longer friends-of- friends of the Milky Way. Thus, they have been denied Local Sheet membership based -28- on the likelihood that they are in a region of space where the galaxy distribution does not exhibit the characteristic flattening of the Local Sheet. The other friends-of-friends dls with uncertain distances are addressed individually below: E174-01 (JC = -5.8, y = 0.9) and E220-10 (x = -5.3, y = 1.0): The only distance estimates for these galaxies are their Hubble distances of 5.9 Mpc and 5.5 Mpc, respectively. Their astrometry and radial velocities place them in the range of the Cen A/M83 group, which qualifies both dls for Local Sheet membership. KK160 (JC = 1.3, y = 4.4): The astrometry and radial velocity of this galaxy places it within the densest region of the CVn I cloud. Given that essentially all the CVn I galaxies within the velocity range of this region are friends-of-friends of the Milky Way, KK160 has been included in the Local Sheet sample by right of its membership in this velocity field. 2.3.3 The Local Sheet survey region As a last step in defining the Local Sheet, a cylindrical volume of space has been established as the Local Sheet's minimum container, and Sheet membership has been extended to any galaxies which were not extracted by the FOF algorithm but which are found within this survey volume. The purpose of this is to enable the calculation of the Sheet's surface and volume densities (§5.6). -29- The survey volume is defined as the cylinder perpendicular to the Sheet's midplane with height equivalent to the z-range of the Milky Way's friends-of-friends. Similarly, the surface area of the cylinder is the minimum elliptical surface in the x>>-plane which contains all of the Milky Way's friends-of-friends. The axes, centroid and position angle of this ellipse were obtained with the aid of the IDL routine FIT ELLIPSE4, which establishes these parameters using a "number density" algorithm. The returned elliptical surface is overplotted in Figure 2.5 and its parameters are listed in Table 2.1. Table 2.1: Parameters of the Local Sheet survey region Parameter Value Height (h) of the cylindrical volume 3.2 Mpc Centroid of the elliptical boundary in the xy-plane (0.1 Mpc, 0.1 Mpc) Semimajor axis (a) of the elliptical boundary in the ^y-plane 6.3 Mpc Semiminor axis (b) of the elliptical boundary in the xy-plane 5.2 Mpc Position angle of the semimajor axis, measured ^ <.Q counter-clockwise from the +*-axis The survey volume defined above netted 133 dls, 7 of which were not identified by the FOF algorithm. Six of these new members are satellites of the late-type spiral NGC 672 at (x, y, z) = (4.5, 3.1, -0.2). Karachentsev (2005) associated this galaxy with a group comprised of the 6 dl satellites in addition to the late-type spiral IC 1727 at z = -0.3 Mpc. Thus, the entire NGC672 group is well contained within the z-range of the Local Sheet. Its members are all friends-of-friends of the 7th addition to the Local Sheet, 4 The FIT_ELLIPSE routine was developed by Fanning Software Consulting, Inc. (www.dfanning.comV The source code can be found at www.idlcovote.com/programs/fit ellipse.pro. -30- UGC685. This relatively isolated dl is less than 0.1 Mpc beyond the FOF linking length from the members of the IC 342/Maffei group, which is nevertheless what prevented the NGC 672 group from being extracted by the FOF algorithm. 2.4 The Local Sheet Sample Figure 2.5 shows the face-on view of the final collection of 133 dls. The dashed curve describes the elliptical boundary of the Sheet dls as defined in the previous section. The cylindrical volume perpendicular to the Sheet's midplane, which spans 3.2 Mpc in z and is bounded in the xy-plane by the ellipse, can be considered to define the membership of the Local Sheet. Thus, all dls within this volume and brighter than +17.5 mag in B (the approximate detection limit of the LVC) comprise the Local Sheet sample studied in this dissertation. An edge-on view of the Local Sheet dls is provided in Figure 2.6, and the edge-on histogram in Figure 2.7 reveals the decline of the dl number density with perpendicular distance from the Sheet's midplane. The basic properties of the Local Sheet dls are listed in Table 2.2. -31 - Figure 2.5: xy-projection of Local Sheet dis. The dashed ellipse contains the Local Sheet sample. Symbols are as in Figure 2.4 with the addition of the circumscribed triangles to denote the members of the Local Sheet which were not extracted by the FOF algorithm. 10 • # * 5 u Q_ ^ 0 it -5 -10 -10 -5 0 5 10 x [Mpc] Figure 2.6: xz-projection of Local Volume dis with x,y coordinates within the elliptical boundary of the Local Sheet. The solid circles are the Local Sheet dis. » i i—| -r i r r i » r "•! 'i » • I i i i 10 5 3 o OoO Q c N OS o o o °0 . O O &o -5 JS> o 0 ooo ^ -10 I .... I • » ' ' I -10 -5 0 5 10 x [Mpc] -32- Figure 2.7: The distribution of Local Sheet dis as a function of height above or below the Sheet's midplane. 40 I I F I T 'l I I I ™ 30 T> o Q) 20 _Q E 3 10 0L I i mt i t i i i i I a t tin. I 0.0 0.5 1.0 1.5 2.0 \z\ [Mpc] •33 • Table 2.2: Basic properties of the Local Sheet dis Galaxy <*J2000 8J2000 SGL SGB Bt Vo d d d * y z [h:m:s] [d:m:s] [°] H [mag] [km s"1] [Mpc] Method Reference [Mpc] [Mpc] [Mpc] ill _J2) *3}_ (5) (6) (V (8) (9) (10) (11) (12) (13) AM 1306-265 13:09:37 -27:08:27 143.55 -3.96 15.11 681 2.58±0.85 PP 44 -2.09 1.51 -0.11 AM 1321-304 13:24:36 -30:58:19 148.16 -1.90 16.50 487 4.24±0.38 RGB 19 -3.61 2.22 -0.06 Antlia 10:04:04 -27:19:52 139.58 -44.80 15.57 362 1.26±0.02 RGB 42 -0.74 0.47 -0.78 CGCG 269-049 12:15:47 +52:23:14 63.71 6.57 15.64 159 5.02±0.25 RGB 38 2.22 4.50 0.00 Cam A 04:25:20 +72:48:30 16.10 1.87 14.84 -46 3.74±0.45 RGB 26 3.59 1.03 -0.12 Cam B 04:53:07 +67:05:57 15.04 -4.21 16.71 77 3.19±0.48 RGB 26 3.05 0.78 -0.41 Cas 1 02:06:03 +68:59:59 6.26 8.48 15.29 35 3.34±0.62 MEM 28 3.31 0.41 0.35 DD0 210 20:46:52 -12:50:53 252.08 50.24 14.37 -141 0.91±0.02 RGB 31 -0.13 -0.47 0.90 Dwingeloo 2 02:54:08 +59:00:19 359.90 -0.81 20.50 94 4.03±1.07 PP 44 4.02 -0.03 -0.20 ESO 174-01 13:47:58 -53:20:51 170.08 -6.99 14.20 688 5.94±1.43 HUB ... -5.84 0.94 -0.34 ESO 222-10 14:35:03 -49:25:14 170.25 3.39 16.33 622 5.46±1.32 HUB ... -5.35 0.98 0.68 ESO 223-09 15:01:09 -48:17:26 171.55 7.63 13.82 588 6.24±0.29 RGB 35 -6.06 1.03 1.23 ESO 245-05 01:45:04 -43:35:53 255.14 -19.74 12.60 391 4.28±0.42 RGB 25 -1.11 -4.04 -0.77 ESO 269-58 13:10:33 -46:59:27 162.92 -8.83 12.20 400 3.62±0.16 RGB 35 -3.46 0.99 -0.33 ESO 321-14 12:13:50 -38:13:53 152.26 -17.62 15.36 610 3.16±0.11 RGB 42 -2.73 1.29 -0.82 ESO 324-24 13:27:37 -41:28:50 158.39 -4.42 12.91 516 3.61 ±0.42 RGB 19 -3.36 1.29 -0.10 ESO 325-11 13:45:01 -41:51:40 159.79 -1.46 14.02 545 3.32±0.39 RGB 19 -3.12 1.14 0.10 ESO 349-31 00:08:13 -34:34:42 260.18 0.40 15.70 227 3.09±0.26 RGB 32 -0.51 -3.02 0.55 ESO 379-07 11:54:43 -33:33:36 146.93 -20.99 16.61 641 4.97±0.42 RGB 19 -4.00 2.31 -1.70 ESO 381-18 12:44:42 -35:58:00 150.94 -11.19 15.73 624 5.13±0.35 RGB 35 -4.46 2.32 -0.88 ESO 381-20 12:46:01 -33:50:13 148.87 -10.50 14.24 589 5.28±0.13 RGB 35 -4.51 2.57 -0.87 ESO 384-16 13:57:01 -35:19:59 154.52 3.06 15.43 561 4.32±0.10 RGB 35 -3.88 1.89 0.38 ESO 443-09 12:54:54 -28:20:27 143.88 -7.41 17.38 645 5.83±0.34 RGB 35 -4.72 3.31 -0.73 ESO 444-84 13:37:20 -28:02:42 146.23 1.64 15.09 587 4.52±0.40 RGB 19 -3.75 2.52 0.19 HIDEEP J1336-3321 13:37:01 -33:21:47 151.24 -0.14 17.19 591 4.20±0.21 RGB 37 -3.68 2.02 0.10 HIPASS J1247-77 12:47:26 -77:34:17 193.54 -15.74 17.00 413 3.29±0.28 RGB 32 -3.13 -0.83 -0.47 HIPASS J1305-40 13:05:02 -40:04:58 155.90 -8.17 16.52 617 5.69±0.22 RGB 35 -5.19 2.20 -0.62 HIPASS J1321-31 13:21:08 -31:31:45 148.47 -2.77 17.10 571 4.90±0.27 RGB 37 -4.19 2.53 -0.15 HIPASS J1337-39 13:37:25 -39:53:48 157.46 -2.17 16.50 492 4.76±0.25 RGB 37 -4.40 1.81 0.01 HIPASS J1348-37 13:48:47 -37:58:29 156.39 0.56 16.90 581 5.61±0.50 RGB 35 -5.13 2.26 0.24 34- Galaxy OJ2000 §J2000 SGL SGB Bf vo d d d X y z [h:m:s] [d:m:s] l°] [°] [mag] [km s"1] [Mpc] Method Reference [Mpc] [Mpc] [Mpc] (1) (2) (3) (4) (5) («) (7) (8) (9) (10) (») (12) (13) HIPASS J1351-47 13:51:12 -46:58:13 164.96 -2.21 17.50 529 5.65+0.59 RGB 35 -5.46 1.45 0.08 Holmberg I 09:40:32 +71:10:56 38.77 1.33 13.64 139 3.80+0.06 RGB 42 2.95 2.36 -0.27 Holmberg II 08:19:05 +70:43:12 33.26 -2.36 11.14 142 3.30±0.14 RGB 42 2.73 1.77 -0.41 Holmberg IV 13:54:46 +53:54:03 64.15 21.40 13.76 144 4.34+0.89 PP 44 1.85 3.80 1.13 Holmberg IX 09:57:32 +69:02:45 41.26 0.69 14.53 46 3.55±0.15 MEM 42 2.66 2.32 -0.29 IC 10 00:20:17 +59:18:14 354.43 17.88 11.91 -348 0.80+0.02 RGB 40 0.77 -0.05 0.33 IC 1574 00:43:04 -22:14:49 274.23 -3.21 14.50 361 4.73+0.53 RGB 25 0.35 -4.71 0.41 IC 1613 01:04:48 +02:07:04 299.15 -1.78 9.88 -234 0.71+0.02 RGB 36 0.35 -0.62 0.15 IC 2574 10:28:23 +68:24:44 43.63 2.31 11.37 57 3.77+0.06 RGB 42 2.72 2.59 -0.22 IC 3104 12:18:46 -79:43:34 195.83 -17.06 13.66 429 2.24±0.17 RGB 20 -2.10 -0.65 -0.32 IC 3687 12:42:15 +38:30:12 78.42 7.27 13.79 354 4.42+0.45 RGB 19 0.90 4.33 0.09 IC 4182 13:05:50 +37:36:18 80.34 11.61 12.02 321 4.21+0.09 RGB 36 0.73 4.14 0.42 IC 4247 13:26:44 -30:21:45 147.71 -1.27 14.37 274 4.82+0.36 RGB 35 -4.08 2.56 -0.03 IC 4316 13:40:18 -28:53:32 147.25 1.99 14.55 674 4.00+0.43 RGB 19 -3.35 2.18 0.21 IC 4662 17:47:09 -64:38:30 199.19 8.61 11.97 302 2.31+0.19 RGB 32 -2.13 -0.69 0.70 IC 5152 22:02:42 -51:17:47 234.23 11.53 10.40 122 1.82+0.07 RGB 33 -1.01 -1.39 0.73 KDG 73 10:52:57 +69:32:58 44.03 4.75 17.09 116 3.56+0.24 RGB 18 2.55 2.47 -0.05 KKH 11 02:24:34 +56:00:43 355.32 1.12 17.17 310 4.66+1.08 PP 44 4.65 -0.39 -0.04 KKH 12 02:27:27 +57:29:16 356.81 1.45 17.80 70 2.17+0.49 PP 44 2.17 -0.12 0.05 KKH 18 03:03:06 +33:41:40 339.26 -15.93 16.70 216 4.18+0.44 RGB 24 3.68 -1.57 -1.08 KKH 34 05:59:40 +73:25:40 22.54 -0.41 17.10 110 3.96+0.29 RGB 24 3.64 1.49 -0.33 KKH 37 06:47:46 +80:07:26 26.14 5.84 16.40 -148 3.29+0.28 RGB 32 2.95 1.46 0.09 KKH 5 01:07:32 +51:26:26 347.21 10.31 17.10 61 4.10+0.34 RGB 24 3.98 -0.81 0.70 KKH 6 01:34:52 +52:05:30 348.94 6.39 17.00 53 3.72+0.31 RGB 32 3.65 -0.67 0.38 KKH 86 13:54:34 +04:14:35 116.34 15.47 16.88 287 2.61+0.09 RGB 42 -1.08 2.33 0.61 KKH 98 23:45:34 +38:43:04 332.35 23.17 16.70 -137 2.38+0.07 RGB 20 2.00 -0.90 1.05 KKR25 16:13:48 +54:22:16 56.09 40.37 16.45 -139 1.96+0.04 RGB 42 0.91 1.38 1.18 KKR 3 14:07:11 +35:03:37 84.56 23.55 17.90 62 1.97+0.04 RGB 42 0.21 1.88 0.67 LEDA 138451 01:42:17 +26:22:00 325.00 -3.23 16.62 357 5.48+0.97 MEM 43 4.47 -3.17 -0.09 LEDA 166062 01:44:43 +27:17:19 326.05 -3.44 18.08 423 5.48+0.97 MEM 43 4.52 -3.09 -0.13 LEDA 166063 01:46:42 +26:48:05 325.75 -4.02 18.08 366 5.48+0.97 MEM 43 4.50 -3.12 -0.18 LEDA 166064 01:55:20 +27:57:14 327.50 -5.41 16.71 207 5.28+0.15 RGB 33 4.40 -2.88 -0.31 Galaxy aj2ooo 5j2000 SGL SGB Bj- vo d d d x y z [h:m:s] [d:m:s] [°] [°] [mag] [km s"1] [Mpc] Method Reference [Mpc] [Mpc] [Mpc] (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) («> (12) (13) LEDA 166065 02:00:10 +28:49:53 328.71 -6.08 17.97 168 4.77±0.17 RGB 33 4.02 -2.52 -0.34 LEDA 166115 11:47:12 +43:40:18 70.24 -0.92 18.20 212 4.33±0.31 RGB 23 1.44 4.03 -0.54 LEDA 166142 12:43:58 +43:39:40 73.44 8.82 17.65 293 4.57±1.10 HUB 1.31 4.38 0.20 LMC 05:23:34 -69:45:22 215.79 -34.12 0.91 278 0.05±0.01 CEPH(a) 43 -0.03 -0.03 0.10 Leo A 09:59:26 +30:44:47 69.91 -25.80 12.92 24 0.79±0.07 RGB 11 0.22 0.62 -0.32 M81 Dwarf A 08:23:55 +71:01:56 33.52 -1.93 15.91 113 3.36±0.07 RGB 42 2.78 1.82 -0.40 NGC 1569 04:30:49 +64:50:53 11.91 -4.92 11.86 -104 2.66±0.06 RGB 39 2.57 0.51 -0.34 NGC 2366 07:28:55 +69:12:57 29.46 -4.86 11.68 80 3.13±0.06 RGB 42 2.69 1.48 -0.50 NGC 2915 09:26:12 -76:37:35 197.37 -26.06 13.25 468 3.59±0.42 RGB 24 -3.17 -1.14 -1.12 NGC 3077 10:03:19 +68:44:02 41.85 0.83 10.61 14 3.67±0.17 RGB 42 2.72 2.43 -0.30 NGC 3109 10:03:07 -26:09:35 137.96 -45.10 10.39 403 1.30±0.02 RGB 42 -0.74 0.50 -0.82 NGC 3738 11:35:49 +54:31:26 59.57 1.79 11.92 229 4.58±0.51 RGB 23 2.31 3.92 -0.37 NGC 3741 11:36:06 +45:17:01 67.96 -2.08 14.38 229 3.30±0.10 RGB 23 1.22 3.02 -0.45 NGC 4068 12:04:01 +52:35:18 62.95 4.93 13.19 210 4.11±0.35 RGB 32 1.87 3.65 -0.09 NGC 4163 12:12:09 +36:10:09 78.94 0.88 13.63 165 2.80±0.06 RGB 42 0.53 2.73 -0.21 NGC 4190 12:13:45 +36:38:03 78.61 1.33 13.90 228 3.18±1.28 BBS 8 0.62 3.10 -0.23 NGC 4214 12:15:39 +36:19:37 79.02 1.60 10.24 291 3.01 ±0.05 RGB 42 0.57 2.94 -0.19 NGC 4395 12:25:49 +33:32:49 82.31 2.73 10.84 319 4.30±0.59 RGB 23 0.57 4.25 -0.23 NGC 4449 12:28:11 +44:05:37 72.30 6.18 10.06 207 3.99±0.50 RGB 23 1.22 3.80 0.01 NGC 4789A 12:54:05 +27:08:59 90.13 6.90 14.17 374 1.53±0.20 PP 44 0.00 1.53 0.12 NGC 5204 13:29:37 +58:25:07 59.40 17.85 11.73 201 4.42±0.56 RGB 23 2.21 3.75 0.88 NGC 5264 13:41:37 -29:54:47 148.30 1.92 12.60 478 4.26±0.47 RGB 19 -3.62 2.25 0.23 NGC 5408 14:03:21 -41:22:40 160.59 1.90 12.20 506 4.90±0.39 RGB 19 -4.61 1.65 0.38 NGC 625 01:35:05 -41:26:10 257.27 -17.74 11.71 396 3.80±0.31 RGB 27 -0.86 -3.64 -0.54 NGC 6822 19:44:57 -14:47:21 229.08 57.10 9.83 -57 0.46±0.02 RGB 5 -0.14 -0.14 0.54 Pegasus Dwarf 23:28:36 +14:44:35 305.83 24.31 12.43 -183 0.87±0.02 RGB 31 0.49 -0.60 0.53 SMC 00:52:44 -72:49:43 224.23 -14.82 2.70 158 0.06±0.01 RGB 13 -0.04 -0.04 0.11 Sagittarius Dwarf 19:29:59 -17:40:41 221.27 55.52 14.12 -79 1.02±0.02 RGB 20 -0.38 -0.27 1.03 Sextans A 10:11:01 -04:41:34 109.01 -40.66 11.86 324 1.43±0.03 RGB 42 -0.42 0.90 -0.90 Sextans B 10:00:00 +05:19:56 95.46 -39.62 11.85 300 1.42±0.03 RGB 42 -0.17 0.97 -0.90 UGC3817 07:22:44 +45:06:31 32.03 -28.86 15.96 437 3.63±0.79 PP 44 2.57 1.45 -1.99 UGC 4459 08:34:07 +66:10:54 36.25 -6.04 14.55 20 3.52±0.05 RGB 42 2.78 2.00 -0.68 36- c c C c C C c c C C c c c c c c c c c c c c c c c c c c c c C c o o o a o o a o O O o o o o o o o o o o o a a a o o o o a a a o O o o o o o a o n o n o n o o o o o o o o o o o n o n o o o o o > > > > so so 00 00 00 00 00 00 00 00 00 00 00 -J -o -J -J -o -J -J Os OS ON ON v* Vl Vl 4*» to to »— N> >—» 00 -J o ON Vl CO co co tO tO o Os Vi Vl 4*. 4^ tO TO 00 00 Vl 4^ SO oo 4*. 4*» *0 Vi © 4*. to 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Vi b ON 00 O 00 to U> to 00 ON ON 00 NO -4 Vl u> to v» 0 to 00 Vl CO -0 to so 00 ON - < OJ uo u> 0 to to to u> to 42* to U) u> U> 4^ U) , • to 4^ CO CO CO UJ CO to CO to UI CO 42^ _l ON © 4^ 0 so 0 so O bs so O 42k 00 to 42» ON CO 42* ON SO to Vl CO bs 00 CO b SO U) s ON Os to -0 4^ 00 0 ON SO 00 0 so Vl t-o 4^ so 0 Vl 00 to CO -4 to 4^ 00 1 1 ( 1 1 1 1 1 1 1 1 O 0 0 0 , . 0 0 0 0 O p 0 O O p 0 0 0 0 O 0 0 O 0 0 p 0 0 p 0 0 b to to ON b 00 00 00 IA bs ON Vl Vl O CO 4*> CO CO 42k to Vl ON so 42* u> Vl 0 00 Vl u> 000 Vl 04^ -0 so4^ 0CO CO 4^ Vl U) Vl -p^4^ 00 CO 4^Vl -4 Galaxy OJ2000 &J2000 SGL SGB Bt vo d d d X y z [h:m:s] [d:m:s] H H [mag] [km s'1] [Mpc] Method Reference [Mpc] [Mpc] [Mpc] (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (») (12) (13) UGCA319 13:02:14 -17:14:15 133.54 -2.94 15.08 755 2.58±0.53 PP 44 -1.79 1.85 -0.12 UGCA 365 13:36:31 -29:14:06 147.30 1.09 15.59 573 5.07±0.26 RGB 35 -4.26 2.75 0.16 UGCA 438 23:26:28 -32:23:20 258.88 9.28 14.25 62 2.16±0.05 RGB 42 -0.38 -2.03 0.75 UGCA 86 03:59:48 +67:08:19 10.85 -1.17 14.70 67 2.42±0.21 RGB 32 2.36 0.44 -0.14 UGCA 92 04:32:05 +63:36:49 11.32 -6.01 15.22 -99 3.14±0.26 RGB 32 3.03 0.56 -0.48 UKS 1424-460 14:28:04 -46:18:06 166.95 3.84 16.50 390 3.50±0.33 RGB 19 -3.38 0.82 0.48 WLM 00:01:58 -15:27:39 277.81 8.09 11.03 -122 0.92±0.02 RGB 36 0.14 -0.88 0.35 (a) The quoted Cepheid distance to the LMC has been re-calibrated to the distance and extinction scale adopted in this dissertation (see McCall 2011). Notes. (1) Name of galaxy. (2-3) Right ascension and declination from the NASA/IP AC Extragalactic Database (NED)1. (4-5) Supergalactic longitude and latitude from NED. (6) Apparent B magnitude from published sources, not corrected for Galactic extinction or cosmological effects. Total asymptotic magnitudes from surface photometry have been adopted whenever possible (55% of the sample). 17% are from the RC3 (de Vaucouleurs et al. 1991). Uncertainties are typically between 0.1-0.3 mag. (7) Heliocentric radial velocity from NED. (8) Homogenized distance estimate and uncertainty as described in §2.1.2. (9) Adopted distance method: RGB = Tip of the Red Giant Branch, PP = Potential Plane, HUB = Hubble's Law, BBS = Brightest Blue Stars, MEM = Group membership. (10) Source of distance estimate from Table 2.3. (11-13) Cartesian coordinates in the Local Sheet frame of reference. 1 The NASA/IPAC Extragalactic Database (NED) is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. -38- Table 2.3: Distance references Num Reference Num Reference 1 Tully 1988 23 Karachentsev et al. 2003a 2 Karachentsev et al. 1996a 24 Karachentsev et al. 2003b 3 Karachentsev et al. 1996b 25 Karachentsev et al. 2003c 4 Sharinaetal. 1996 26 Karachentsev et al. 2003d 5 Gallartetal. 1996 27 Cannon et al. 2003 6 Georgiev et al. 1997a 28 Karachentsev et al. 2004 7 Georgiev et al. 1997b 29 Rekola et al. 2005 8 Tikhonov and Karachentsev 1998 30 Silva et al. 2005 9 Makarova et al. 1998 31 McConnachie et al. 2005 10 Makarova and Karachentsev 1998 32 Karachentsev et al. 2006 11 Tolstoy etal. 1998 33 Tully et al. 2006 12 Sharinaetal. 1999 34 Alonso-Garcia et al. 2006 13 Cioni et al. 2000 35 Karachentsev et al. 2007 14 Karachentsev et al. 2000 36 Rizzi et al. 2007 15 Crone et al. 2000 37 Grossi et al. 2007 16 Jerjen et al. 2001 38 Corbin et al. 17 Drozdovsky et al. 2001 39 Grocholski et al. 2008 18 Karachentsev et al. 2002a 40 Sanna et al. 2008 19 Karachentsev et al. 2002b 41 Tully etal. 2009 20 Karachentsev et al. 2002c 42 Dalcanton et al. 2009 21 Mai'z-Apellaniz et al. 2002 43 McCall 2011 22 Mendez et al. 2002 44 This dissertation Table 2.4: Extinctions and cosmological corrections for the Local Sheet dis Galaxy E(B-V) z Tl AB AKS AK,B Ak KS (1) (2) (3) (4) (5) (6) (7) (*) AM 1306-265 0.08 0.0023 0.087 0.33 0.03 0.0015 -0.0056 AM 1321-304 0.07 0.0016 0.078 0.30 0.03 0.0011 -0.0040 Antlia 0.08 0.0012 0.089 0.34 0.03 0.0008 -0.0030 CGCG 269-049 0.02 0.0005 0.028 0.11 0.01 0.0003 -0.0013 Cam A 0.22 -0.0002 0.251 0.95 0.08 -0.0001 0.0004 Cam B 0.22 0.0003 0.250 0.94 0.08 0.0002 -0.0006 Cas 1 1.02 0.0001 1.176 4.39 0.39 0.0001 -0.0003 DD0 210 0.05 -0.0005 0.057 0.22 0.02 -0.0003 0.0012 Dwingeloo 2 1.82(a) 0.0003 2.017 7.44 0.66 0.0002 -0.0008 ESO 174-01 0.50 0.0023 0.574 2.16 0.19 0.0015 -0.0057 ESO 222-10 0.27 0.0021 0.307 1.16 0.10 0.0014 -0.0051 ESO 223-09 0.26 0.0020 0.296 1.12 0.10 0.0013 -0.0049 ESO 245-05 0.02 0.0013 0.019 0.07 0.01 0.0009 -0.0032 ESO 269-58 0.11 0.0013 0.124 0.47 0.04 0.0009 -0.0033 ESO 321-14 0.09 0.0020 0.107 0.41 0.04 0.0014 -0.0050 -39- Galaxy E(B-V) z TL AK,B AK KS (1) (2) (3) (4) P) W (7) (8) ESO 324-24 0.11 0.0017 0.128 0.49 0.04 0.0012 -0.0043 ESO 325-11 0.09 0.0018 0.100 0.38 0.03 0.0012 -0.0045 ESO 349-31 0.01 0.0008 0.014 0.05 0.00 0.0005 -0.0019 ESO 379-07 0.07 0.0021 0.085 0.32 0.03 0.0014 -0.0053 ESO 381-18 0.06 0.0021 0.071 0.27 0.02 0.0014 -0.0052 ESO 381-20 0.07 0.0020 0.075 0.28 0.02 0.0013 -0.0049 ESO 384-16 0.07 0.0019 0.084 0.32 0.03 0.0012 -0.0046 ESO 443-09 0.06 0.0022 0.073 0.28 0.02 0.0014 -0.0053 ESO 444-84 0.07 0.0020 0.078 0.30 0.03 0.0013 -0.0049 HIDEEP J1336-3321 0.05 0.0020 0.055 0.21 0.02 0.0013 -0.0049 HIPASS J1247-77 0.77 0.0014 0.884 3.31 0.29 0.0009 -0.0034 HIPASS J1305-40 0.10 0.0021 0.116 0.44 0.04 0.0014 -0.0051 HIPASS J1321-31 0.06 0.0019 0.070 0.26 0.02 0.0013 -0.0047 HIPASS J1337-39 0.07 0.0016 0.085 0.32 0.03 0.0011 -0.0041 HIPASS J1348-37 0.08 0.0019 0.089 0.34 0.03 0.0013 -0.0048 HIPASS J1351-47 0.14 0.0018 0.164 0.62 0.05 0.0012 -0.0044 Holmberg I 0.05 0.0005 0.057 0.22 0.02 0.0003 -0.0012 Holmberg II 0.03 0.0005 0.036 0.14 0.01 0.0003 -0.0012 Holmberg IV 0.01 0.0005 0.017 0.06 0.01 0.0003 -0.0012 Holmberg IX 0.08 0.0002 0.089 0.34 0.03 0.0001 -0.0004 IC 10 0.77(b) -0.0012 0.835 3.13 0.27 -0.0007 0.0029 IC 1574 0.02 0.0012 0.017 0.07 0.01 0.0008 -0.0030 IC 1613 0.02 -0.0008 0.028 0.11 0.01 -0.0005 0.0019 IC 2574 0.04 0.0002 0.041 0.16 0.01 0.0001 -0.0005 IC 3104 0.41 0.0014 0.468 1.76 0.15 0.0010 -0.0036 IC 3687 0.02 0.0012 0.023 0.09 0.01 0.0008 -0.0029 IC 4182 0.01 0.0011 0.016 0.06 0.01 0.0007 -0.0026 IC 4247 0.06 0.0009 0.074 0.28 0.02 0.0006 -0.0023 IC 4316 0.05 0.0022 0.062 0.23 0.02 0.0015 -0.0056 IC 4662 0.07 0.0010 0.079 0.30 0.03 0.0007 -0.0025 IC 5152 0.03 0.0004 0.028 0.11 0.01 0.0003 -0.0010 KDG73 0.02 0.0004 0.021 0.08 0.01 0.0003 -0.0010 KKH 11 1.54(c) 0.0010 1.691 6.27 0.55 0.0007 -0.0026 KKH 12 1.54(c) 0.0002 1.691 6.27 0.55 0.0002 -0.0006 KKH 18 0.20 0.0007 0.226 0.85 0.07 0.0005 -0.0018 KKH 34 0.24 0.0004 0.273 1.03 0.09 0.0002 -0.0009 KKH 37 0.07 -0.0005 0.084 0.32 0.03 -0.0003 0.0012 KKH 5 0.28 0.0002 0.323 1.22 0.11 0.0001 -0.0005 KKH 6 0.35 0.0002 0.400 1.51 0.13 0.0001 -0.0004 KKH 86 0.03 0.0010 0.030 0.11 0.01 0.0006 -0.0024 KKH 98 0.12 -0.0005 0.140 0.53 0.05 -0.0003 0.0011 KKR25 0.01 -0.0005 0.010 0.04 0.00 -0.0003 0.0012 KKR3 0.01 0.0002 0.016 0.06 0.01 0.0001 -0.0005 LEDA 138451 0.09 0.0012 0.104 0.39 0.03 0.0008 -0.0030 LEDA 166062 0.07 0.0014 0.081 0.31 0.03 0.0009 -0.0035 LEDA 166063 0.08 0.0012 0.094 0.35 0.03 0.0008 -0.0030 -40- Galaxy E(B-V) z n AB AKS AK,B Akks (1) (2) (3) (4) (5) (6) (7) (8) LEDA 166064 0.07 0.0007 0.080 0.30 0.03 0.0004 -0.0017 LEDA 166065 0.05 0.0006 0.062 0.23 0.02 0.0004 -0.0014 LEDA 166115 0.02 0.0007 0.022 0.08 0.01 0.0005 -0.0018 LEDA 166142 0.03 0.0010 0.030 0.11 0.01 0.0006 -0.0024 LMC 0.92 0.0009 1.066 3.99 0.35 0.0006 -0.0023 Leo A 0.02 0.0001 0.024 0.09 0.01 0.0000 -0.0002 M81 Dwarf A 0.02 0.0004 0.023 0.09 0.01 0.0003 -0.0009 NGC 1569 0.69 -0.0003 0.797 2.99 0.26 -0.0002 0.0009 NGC 2366 0.04 0.0003 0.041 0.16 0.01 0.0002 -0.0007 NGC 2915 0.27 0.0016 0.313 1.18 0.10 0.0010 -0.0039 NGC 3077 0.07 0.0000 0.076 0.29 0.02 0.0000 -0.0001 NGC 3109 0.07 0.0013 0.076 0.29 0.02 0.0009 -0.0033 NGC 3738 0.01 0.0008 0.012 0.04 0.00 0.0005 -0.0019 NGC 3741 0.02 0.0008 0.028 0.11 0.01 0.0005 -0.0019 NGC 4068 0.02 0.0007 0.024 0.09 0.01 0.0005 -0.0017 NGC 4163 0.02 0.0006 0.023 0.09 0.01 0.0004 -0.0014 NGC 4190 0.03 0.0008 0.033 0.13 0.01 0.0005 -0.0019 NGC 4214 0.02 0.0010 0.025 0.09 0.01 0.0006 -0.0024 NGC 4395 0.02 0.0011 0.020 0.07 0.01 0.0007 -0.0026 NGC 4449 0.02 0.0007 0.022 0.08 0.01 0.0004 -0.0017 NGC 4789A 0.01 0.0012 0.011 0.04 0.00 0.0008 -0.0031 NGC 5204 0.01 0.0007 0.014 0.05 0.00 0.0004 -0.0017 NGC 5264 0.05 0.0016 0.058 0.22 0.02 0.0011 -0.0039 NGC 5408 0.07 0.0017 0.078 0.29 0.03 0.0011 -0.0042 NGC 625 0.02 0.0013 0.019 0.07 0.01 0.0009 -0.0033 NGC 6822 0.23 -0.0002 0.263 0.99 0.09 -0.0001 0.0005 Pegasus Dwarf 0.07 -0.0006 0.077 0.29 0.03 -0.0004 0.0015 SMC 0.42 0.0005 0.478 1.80 0.16 0.0003 -0.0013 Sagittarius Dwarf 0.12 -0.0003 0.141 0.53 0.05 -0.0002 0.0007 Sextans A 0.04 0.0011 0.051 0.19 0.02 0.0007 -0.0027 Sextans B 0.03 0.0010 0.035 0.13 0.01 0.0007 -0.0025 UGC3817 0.10 0.0015 0.116 0.44 0.04 0.0010 -0.0036 UGC 4459 0.04 0.0001 0.043 0.16 0.01 0.0000 -0.0002 UGC 4483 0.03 0.0005 0.039 0.15 0.01 0.0003 -0.0013 UGC 5423 0.08 0.0012 0.092 0.35 0.03 0.0008 -0.0029 UGC 5829 0.02 0.0021 0.027 0.10 0.01 0.0014 -0.0052 UGC 5918 0.01 0.0011 0.012 0.05 0.00 0.0008 -0.0028 UGC 6456 0.04 -0.0003 0.043 0.16 0.01 -0.0002 0.0009 UGC 6541 0.02 0.0008 0.021 0.08 0.01 0.0005 -0.0021 UGC 6817 0.03 0.0008 0.030 0.11 0.01 0.0005 -0.0020 UGC 685 0.06 0.0005 0.065 0.25 0.02 0.0003 -0.0013 UGC 7242 0.02 0.0002 0.021 0.08 0.01 0.0002 -0.0006 UGC 7298 0.02 0.0006 0.025 0.10 0.01 0.0004 -0.0014 UGC 7408 0.01 0.0015 0.013 0.05 0.00 0.0010 -0.0038 UGC 7490 0.02 0.0016 0.028 0.11 0.01 0.0010 -0.0038 UGC 7559 0.01 0.0007 0.016 0.06 0.01 0.0005 -0.0018 -41 Galaxy E(B-V) z *1 AB AKS AK,B Ks (1) (2) (3) (4) P) W (7) (8") UGC 7577 0.02 0.0007 0.023 0.09 0.01 0.0004 -0.0016 UGC 7605 0.01 0.0010 0.016 0.06 0.01 0.0007 -0.0026 UGC 8091 0.03 0.0007 0.029 0.11 0.01 0.0005 -0.0018 UGC 8201 0.02 0.0001 0.027 0.10 0.01 0.0001 -0.0003 UGC 8215 0.01 0.0007 0.012 0.05 0.00 0.0005 -0.0018 UGC 8308 0.01 0.0005 0.011 0.04 0.00 0.0004 -0.0014 UGC 8320 0.02 0.0006 0.017 0.07 0.01 0.0004 -0.0016 UGC 8331 0.01 0.0009 0.011 0.04 0.00 0.0006 -0.0022 UGC 8508 0.02 0.0002 0.017 0.07 0.01 0.0001 -0.0005 UGC 8638 0.01 0.0009 0.015 0.06 0.00 0.0006 -0.0023 UGC 8651 0.01 0.0007 0.007 0.03 0.00 0.0004 -0.0017 UGC 8760 0.02 0.0006 0.019 0.07 0.01 0.0004 -0.0016 UGC 8833 0.01 0.0008 0.013 0.05 0.00 0.0005 -0.0019 UGC 9128 0.02 0.0005 0.026 0.10 0.01 0.0003 -0.0013 UGC 9240 0.01 0.0005 0.014 0.05 0.00 0.0003 -0.0012 UGCA 105 0.31 0.0004 0.356 1.34 0.12 0.0002 -0.0009 UGC A 15 0.02 0.0010 0.019 0.07 0.01 0.0007 -0.0024 UGCA 276 0.02 0.0009 0.023 0.09 0.01 0.0006 -0.0023 UGCA 292 0.02 0.0010 0.018 0.07 0.01 0.0007 -0.0026 UGCA 319 0.08 0.0025 0.093 0.35 0.03 0.0017 -0.0062 UGCA 365 0.05 0.0019 0.060 0.23 0.02 0.0013 -0.0047 UGCA 438 0.01 0.0002 0.016 0.06 0.01 0.0001 -0.0005 UGCA 86 0.94 0.0002 1.082 4.05 0.36 0.0002 -0.0005 UGCA 92 0.78 -0.0003 0.903 3.38 0.30 -0.0002 0.0008 UKS 1424-460 0.13 0.0013 0.148 0.56 0.05 0.0009 -0.0032 WLM 0.04 -0.0004 0.043 0.16 0.01 -0.0003 0.0010 (a) E(B-V) is the value found for this galaxy's neighbour, Maffei 2, by Fingerhut et al. (2007). (b) E(B-V) is the value obtained from HII regions by Richer et al. (2001). (c) E(B-V) is the value found for this galaxy's neighbour, Maffei 1, by Fingerhut et al. (2007). Notes. (1) Name of galaxy. (2) Galactic reddening from Schlegel et al. (1998), unless otherwise noted for galaxies with Galactic latitudes within 5° of the plane of the Milky Way. (3) Redshift computed from vG (Table 2.2). (4) Optical depth of Galactic dust at 1 |im obtained from YES (see §2.1.1). (5-6) Galactic extinctions in B and Ks obtained from YES. (7-8) K-corrections in B and Ks obtained from YES. 2.5 Sample Completeness Since the sample of Local Sheet dls was drawn from the LVC, it has inherited the LVC detection limit of approximately +17.5 mag in B. This detection limit is reflected in the sample's luminosity function (LF) in B (Figure 2.8), which was produced using B -42- magnitudes from published sources (see Table 2.2) with total asymptotic magnitudes from surface photometry adopted whenever possible (55% of the sample). Figure 2.8 reveals that the number of dls per Mpc2 in the Local Sheet initially rises with decreasing luminosity following a power-law, then declines sharply at MB > -14 mag down to -8 mag, which corresponds to the LVC detection limit at a distance of 1 Mpc. Approximately 59% of dls in the Local Sheet sample have MB fainter than -14 mag. The observed LF contradicts the behaviour of the universal LF, which has been shown to be well described by the Schechter (1976) law (see, e.g., de Lapparent, Geller and Huchra 1989; Cole et al. 2001): cp(L) dL - (p* (L /L*)a exp(-L/L*) d(L/L*) Equation 2.6 Equation 2.6 is an expression of the number density of galaxies ((p) in a given luminosity range. The parameters a, L* and (p* represent, respectively, the exponent of the power- law component, the characteristic cut-off luminosity below which the power-law dominates, and a normalization factor in units of number density. The Schechter law approaches a power law at the faint end. Chiboucas et al. (2009) find that the LFs for dwarfs in nearby groups of galaxies, such as the M81 and CenA groups, have power-law exponents in the range of -1.3 -43- Given that a large majority of the Local Sheet dwarfs are found in groups such as those examined by Chiboucas et al. (2009), it is unlikely that the observed decline in the Sheet's LF at MB > -14 mag is a feature particular to sheets. In support of this, it is shown in §3.2.1 that the mass functions of theoretical sheets extracted from a cosmological simulation agree well with the mass function of the entire simulation, which does not decline at the low mass end. The decline in the Sheet's LF can, at least in part, be attributed to the exclusion of gas-poor dwarf galaxies from the Local Sheet sample. These are the low surface brightness, diffuse ellipticals classified as dwarf spheroidals (dSphs). However, Figure 2.8 shows that the LF for all dwarfs in the LVC exhibits the same decline as seen for dis in the Sheet sample at MB > -14 mag. The extent of the incompleteness of the Local Sheet sample can be evaluated by establishing the Sheet's LF at MB < -14 mag, where the sample appears to be complete. Integration under the LF after extrapolation to the LVC detection limit yields an estimate for the number of dwarfs missing from the Local Sheet sample. Overplotted in Figure 2.8 is the power-law fit to the LF for all known Sheet dwarfs in the LVC with MB<-14 mag. The power-law exponent is a = -1.4, which is consistent with the values found for wide-field surveys, as described above. Integration of Equation 2.6 over the regime -14 -44- with -12 Figure 2.8: Luminosity function of the Local Sheet dls (squares) in B, with photometry from Table 2.2. Galaxy counts have been normalized by the elliptical survey area (Table 2.1). The circles denote the count of all Local Sheet dwarfs from the LVC. The error bars represent the Poisson counting errors. The dashed line is the power-law fit to the circles in the region MB < -14 mag. o.o r J 1 I 1 1 ! 1 T" I I I i i i 1 1 r •0.5 O Q_ $ •1.0 • Cr> C) O • () •1.5 • O O •2.0 • •2.5 _i i i L _i i I i i i L _i I i I u -18 -16 -14 -12 -10 •8 MB [mag] While the predicted number of missing dwarfs may seem large, it should be noted that increasing numbers of dwarfs have been discovered in the Local Volume since the completion of the LVC. In the M81 group alone, Chiboucas et al. (2009) have discovered 22 new dwarfs and predict at least 70 dwarfs still to be discovered with Mr> < -6 mag. This would bring the total number of M81 dwarfs to -118. Based on the M81 study, a crude but empirical estimate of the minimum number of missing dwarfs in the Local -45- Sheet can be made by utilizing the dependence of the number of dwarfs in a group on the luminosity of the group's principal galaxies. In the census of nearby groups by Karachentsev (2005), which was constructed from the LVC, it can be seen that the brighter a group's principal galaxies, the more numerous their dwarf satellites. This relationship is illustrated below for the principal Local Sheet galaxies. Figure 2.9: The number of known dwarf satellites in the Local Sheet versus the luminosity of their parent galaxy in Ks. The Milky Way, M31, Maffei 1 and MafFei 2 are represented as solid circles. The number of detected satellites is likely lower than expected for these galaxies owing to the high extinction of Maffei 1 and Maffei 2 and the large sky coverage required for a complete census of the satellites of the Milky Way and M31. 1 1 1 r -i 1 1 1 1 1 1 1 1 r Messier 81 CO 25 Centaurus A CD o o -4->CD Messier 31 O CO 20 Milky Way essfer 83 o <5 15 c s o 10 NGC 253 Messier 94 0^342 o CD gtfoffei 2 N4826 Messier 82 Maffei 1 O ^IF#US # 0 i i f n i t i I i i i i i i i i 9.2 9.4 9.6 9.8 10.0 Log [L0] A rough approximation of the number of dwarf satellites in the Local Sheet can be made by simply scaling each principal's predicted number of satellites by the principal's luminosity relative to M81. Table 2.5 lists the principal galaxies in the Local Sheet -46- groups and their luminosity relative to the luminosity of M81, computed from the survey of the brightest galaxies in the Local Volume by McCall (2011). The third column of Table 2.5 contains the predicted number of dwarf satellites based on the principal galaxy's luminosity in Ks relative to the luminosity of M81. The last column lists the difference between the prediction in column 3 and the number of known dwarf satellites in the LVC, which is tabulated in column 4. The sum of the missing dwarfs is 1374. Of course, this does not include dwarfs in the Local Sheet which are not attached to the principals. It is therefore entirely conceivable that there are several hundred undetected faint dwarfs in the Local Sheet, as predicted by the Sheet's LF. Table 2.5: Predicted number of dwarf satellites in the Local Sheet based on the relative luminosities of the principal galaxies Principle LKJLKS,MS1 Number of Number of Number of Galaxy predicted dwarfs detected dwarfs undetected dwarfs (1) (2) (3) (4) (5) Milky Way 2.33 273 18 255 Centaurus A 1.76 206 24 182 Messier 31 1.74 204 21 183 NGC 253 1.03 121 8 113 Messier 81 1.00 117 25 92 Maffei 1 0.92 107 1 106 Messier 83 0.79 92 17 75 Maffei 2 0.67 78 4 74 Messier 82 0.62 72 1 71 NGC 4945 0.55 65 4 61 Circinus 0.55 64 0 64 IC 342 0.46 54 6 48 Messier 64 0.43 51 1 50 Messier 94 0.42 49 7 42 TOTAL 1553 137 1374 Notes. (1) Name of principle galaxy. (2) Luminosity in Ks of principle galaxy relative to M81 from McCall (2011). (3) Predicted number of dwarf satellites based on the luminosity of the principle galaxy relative to M81. (4) Number of known dwarf satellites in the LVC. (5) Difference between the predicted number of dwarf satellites and the number of known dwarf satellites. -47- 3. Theoretical Predictions of Sheets Spock: "Captain, I do not believe you realize the gravity of your situation." Kirk: "On the contrary, gravity is the foremost thing on my mind." 3.1 The Current State of Cosmology In the last two decades, we have experienced a convergence of the Big Bang model toward a universe in which ~23% of its density is comprised of cold dark matter (CDM) and ~73% is dark energy (denoted by QA), with the remaining matter density comprised of baryonic matter. This model is known as the ACDM model, so-called because of its non-zero value for QA. It has become commonly referred to as the standard Big Bang model as a result of the general agreement of its theoretical predictions with independent observations of cosmological probes, in particular the cosmic microwave background fluctuations observed by WMAP1 (see e.g. Larson et al. 2011). The ACDM model tells us that large scale structure in the universe is the result of hierarchical clustering. Specifically, small-scale fluctuations in the universe's initial density field are the first to collapse, forming bound objects known as dark matter halos. These are the progenitors of today's galaxies. Over time, the halos merge into clusters connected by filaments and sheets, producing the web-like distribution of matter that we 1 Launched by NASA in June 2001, the Wilkinson Microwave Anisotropy Probe (WMAP) completed a 7-year all-sky scan of the cosmic microwave background (CMB) radiation, the oldest light in the universe. -48- see today in large galaxy redshift surveys (see e.g. Peacock et al. 2001). The predictions of the ACDM model therefore have implications for the past, present and future of structures like the Local Sheet. In this chapter, an N-body simulation of the ACDM model is used to derive the structural and dynamical characteristics predicted for sheet-like distributions of dark matter halos. The purpose of this is to establish a framework in which to fit the results of the investigations of the Local Sheet in Chapter 5. Specifically, upon extracting a sample of sheets from a ACDM simulation, their vertical mass distributions are used to determine the manner in which a sheet's mass density depends on vertical distance from the sheet's midplane. Following that, a dynamical dependency on vertical distance is investigated using the vertical motions of the sheets' halos. The sheets' structural and dynamical parameters are then combined so that constraints can be placed on their current dynamical state. Finally, the surface density of each sheet is derived from the masses of its halos as well as from the motions of the halos under the assumption that the sheet is equilibrated. The ratio of these surface densities yields the amount of mass below the halo extraction limit which is required to equilibrate the sheet. This is compared to an estimate of the sheet's CDM particles which did not accrete into halos to determine whether the expected amount of intergalactic mass supports the assumption that the sheet is equilibrated. -49- 3.2 The ACDM Simulation and the Sheet Extraction Process This dissertation makes use of the dissipationless ACDM N-body simulation by Tikhonov and Klypin (2009). The choice of this simulation was based on two criteria: (1) that the simulation volume could be expected to contain at least one sheet-like distribution of halos with the same physical dimensions as the Local Sheet; and (2) that the mass coverage encompasses the range expected for dwarf galaxies (see §3.4). The simulation of Tikhonov and Klypin (2009) does not incorporate the accretion of gas or the subsequent formation of stars but simply tracks the motion and aggregation of the CDM particles up to the present epoch. The simulation parameters are those obtained from the WMAP first-year survey (Spergel et al. 2003), with Qm = 0.29 for the matter 2 = density parameter , 0.71 for the dark energy density parameter, Qc = 0.24 for the dark matter density parameter, 08 = 0.9 for the fluctuation amplitude on a scale of 8 h~x Mpc and h = //o/100 = 0.72, where HQ is Hubble's Constant. The simulation t O t / 1 contains 1.6 x 10 equal-mass dark matter particles of 5x10 h~ M© in a cube with sides 80 h'x Mpc in length. Halos were identified by Tikhonov and Klypin (2009) using a group-finding algorithm which extracted halos with masses greater than ~2 x 109 M© from a redshift-zero snapshot of the simulation. This minimum mass is comparable to estimates of the characteristic mass below which haloes start to fail accreting gas and therefore abstain from producing galaxies (see, e.g., Hoeft et al. 2006). In the context of this 2 The density parameter Q is defined as the ratio of the average density of the Universe to the critical value of that density. -50- simulation, halo masses were derived from a halo's virial radius (Ryn) and maximum circular velocity (Fc) via 1 2 MVir = G" Vc i?vir Equation 3.1 where Ryjr is the radius at which the halo's density falls below 200 times the halo's mean density and MVir (hereafter Mdark) is the mass within Rvir. For reference, the dark matter mass of the Large Magellanic Cloud is ~1.3xl010Mo, computed from = 10 Mdark Mtot Hc/Hm where Mtot = 1.6 x 10 M© is the LMC's total mass found from kinematical data by Alves (2004) and Qc/Qm = 0.83 from the first-year results of WMAP. Using this simulation, nearly 1400 groups of at least 10 dark matter halos were identified using a friends-of-friends algorithm (see §2.3.1) with a linking length of 1.0 Mpc. This is the maximum linking length with which sub-structure is identified; linking lengths greater than 1.0 Mpc link over 50% of all halos in the simulation into one cluster. The value adopted for the simulation is slightly less than the linking length of 1.3 Mpc which was used to extract the Local Sheet from the Local Volume Catalogue (LVC). As can be seen in Figure 2.3, a linking length of 1.0 Mpc only identifies the members of the Local Group as friends-of-friends of the Milky Way. Given the detection limit of the LVC, it is likely that there are undetected galaxies within 1 Mpc of the Local Group members which would link them to the rest of the Sheet with this shorter linking length if the LVC was 100% complete. Next, the coordinate frame of each group was rotated so that the z-axis had the least dispersion. The rotated z-axis therefore became a first approximation of the normal to the -51 - structure's midplane. A linear fit was then attempted on the projection of the halos on both the xz- and jz-planes; if the group contained a sheet-like or filament-like halo distribution, both projections revealed a well-defined line. Based on a visual inspection of the groups with obvious sheet-like distributions, a maximum cut-off for the rms of the linear fits was established at 1.0 Mpc. This first pass of the sheet extraction process led to the identification of 23 sheet candidates, all of which contained at least 100 halos. The coordinates of the halos in each of the sheet candidates were then transformed into a new coordinate frame with a z-axis normal to the sheet's midplane and an origin at the sheet's approximate geometric centroid, calculated from £(x„ y„ z,)]/£ In this way, the transformed z-coordinate measured the perpendicular distance of each halo from the sheet's midplane, analogous to the z-coordinate in the Local Sheet frame (see §2.2). Upon visual inspection of the transformed sheet candidates, it was found that the structures which appeared truly sheet-like (i.e., without significant warping) had a dispersion in z no greater than 50% of the dispersion in r, where r = (x2+y2)1/2 is a halo's distance in the xy- plane from the sheet's centroid. In other words, the sheet-like distributions were all at least twice as long as they were thick. Application of this criterion led to the rejection of 15 candidates, leaving 8 sheets. Halo memberships for each sheet candidate were then refined in a manner similar to that employed for the Local Sheet. Specifically, the boundaries of the sheet's xy-projection were established by fitting its friends-of-friend halos to a flat surface enclosed by an ellipse. The fit was obtained using the FIT_ELLIPSE IDL routine -52- developed by Fanning Software3. Sheet membership was then extended to all halos within the boundaries of the ellipse as well as within the vertical extent of the sheet, as defined by the z-range of its friends-of-friends halos. As a final step, the coordinate frame of each sheet was rotated once again to minimize the dispersion of its members about the z-axis, thereby maximizing the reliability of the z-axis as the normal to the sheet's midplane. This had the negligible effect of thinning each sheet's dispersion in z from a mean of 0.61±0.20Mpc for the 8 sheets to 0.58±0.19 Mpc. The final dispersions are consistent with the z-dispersion of 0.75 Mpc found for the Local Sheet (0.59 Mpc for the dls alone), taking into account that the Local Sheet's z-dispersion is enlarged by ~1% due to distance uncertainties. The basic properties of the 8 CDM sheets are listed in Table 3.1. Edge-on histograms and face-on projections are provided in Figure 3.1. Table 3.1: Basic properties of the CDM sheets 2 m SHEET Ntot A^DW log Mtof a b h Notes. (1) Sheet number. (2) Number of halos in the sheet. (3) Number of dwarf halos in the 9 sheet (i.e., with mass less than 6.9 x 10 M0; see §3.4). (4) Total mass of the sheet. (5-6) 3 E-mail: [email protected] -53- Semimajor and semiminor axis of the sheet's elliptical boundary in the .xy-plane. (7) Full z- range of the sheet's halos. (8) Standard deviation in z, where Z=z- Figure 3.1: Edge-on histograms (left) and face-on projections (right) of the CDM sheets extracted from the ACDM simulation of Tikhonov and Klypin (2009). In the face-on projections, the dotted line is the elliptical fit to the halos identified by the friends-of-friends algorithm. The hollow circles are the halos which lie within the sheet's z-range but are beyond the established boundaries of the sheet in the xy-plane. CDM SHEET 1 CDM SHEET 1 o CL -1 0 1 •10 -5 0 5 z [Mpc] x [MPC] CDM SHEET 2 CDM SHEET 2 o CL *ocPo« So p o° Oo o -5 0 5 z [Mpc] x [MPC] -54- CDM SHEET 3 CDM SHEET 3 10 *tTk v X. id . 0*°° "b^yi» o <9 $ o O flhiXa* -2-10 1 •10 -5 0 5 10 z [Mpc] z [MPC] CDM SHEET 4 CDM SHEET 4 1 M I I 10 6"# • 'V ' 5 ru r——t0 1 0 I o Qq$ao 0 6 a •mf* , ». o -2 •1 0 1 •10 -5 0 5 10 z [Mpc] a? [MPC] CDM SHEET 5 CDM SHEET 5 10 %, <*> o 3 0 ._L*k Loo, 7£bo ° >s <*<®Sfe °o%»' act o <». -10 -10 -5 0 5 10 z [Mpc] x [MPC] -55- CDM SHEET 6 CDM SHEET 6 10 37^7^CD 0 rtf 0 5 0 ° 1„ q,o0 ° o a. 0 o *?<,„ ° oo O oo I -5 0 -10 •® Hi -2-10 1 •10 -5 0 5 10 z [Mpc] x [MPC] CDM SHEET 7 CDM SHEET 7 M 7 WTF'TIRTTRMTRN'N'RR i" 10 -r " <* 5 o* o «g £ j~ 0 S» 1_I « -5 %«/ h . : -10 •<.. , ;rt»3 -2-10 1 -10 -5 0 5 10 z [Mpc] a? [MPC] CDM SHEET 8 CDM SHEET 8 jm >u' 8 o Q. JS ?» & •1 0 1 -5 0 5 z [Mpc] x [MPC] -56- 3.2.1 The overabundance of dwarf satellites predicted by the ACDM model The ACDM model is known to predict an overabundance of dwarf satellites in comparison to observations (see, e.g., Klypin et al. 1999). Despite recent discoveries of low luminosity dwarfs in nearby groups, this discrepancy between theory and observation remains (see, e.g., Simon and Geha 2007; Chiboucas et al. 2009). Several studies have shown that the observed abundance of dwarfs in the Local Group can be matched with the ACDM model by accounting for a reduction in the efficiency of galaxy formation as dwarf halos are accreted into galaxy groups (see, e.g., Kravstov, Gnedin and Klypin 2004). Specifically, tidal stripping would reduce a dwarfs gas retention, which could prevent star formation in dwarf halos with masses below ~109Mo- Since the halo extraction limit is above this limit, it can be assumed that the subsequent analyses of the distribution of dwarf halos in the simulated sheets are comparable to the analyses in Chapter 5 of the distribution of dwarf galaxies in the Local Sheet. 3.3 The Mass Functions of CDM Sheets The mass function of a sample of galaxies is the number density of galaxies as a function of galaxy mass. Figure 3.2 shows the mass functions for all halos in each of the theoretical sheets. For each sheet, the number of halos per Mpc"3 ( Overplotted on each figure is the mass function of the entire ACDM simulation from -57- which the sheets were extracted. The mass functions for the sheet halos all indicate that sheets contain more halos per Mpc'3 than the simulation as a whole, which is expected given that the universal halo counts per Mpc"3 are lowered by the inclusion of voids. For nearly every sheet, the slope of the mass function in the dwarf-mass range (i.e., masses less than 6.9 x 1010 M©; see §3.4) follows the slope of the simulation's universal mass function. The similarity observed between the slope of the universal dwarf mass function and that of the sheets may suggest that, as far as dwarf galaxies are concerned, the environmental influences that affect the relative abundances of dwarfs of different masses are no different in sheets than in the Universe has a whole. Figure 3.2: Mass functions of the CDM sheets. The solid line represents the mass function of the entire ACDM simulation. CDM SHEET 1 CDM SHEET 2 0.0 -0.5 TQ -1.0 O. 2 -1.5 •9- 1 <71 oS -2.0 O -2.0 -2.5 -3.0 10.0 10.5 11.0 11.5 12.0 10.0 10.5 11.0 11.5 12.0 log M«w [M0] log [M0] -58- CDM SHEET 3 CDM SHEET 5 CDM SHEET 6 g1 -2-0 10.0 10.5 11.0 11.5 12.0 10.0 10.5 11.0 11.5 12.0 log [M0] log [M0] -59- 3.4 The Distribution of Mass in CDM Sheets In the hierarchical formation scenario, high-mass halos are created from the merging of dwarf-sized halos. Consistent with this scenario is the apparent tendency of the most luminous galaxies in the Local Sheet to be found in regions where the Sheet's dwarfs are most abundant (see Figure 1.2). Thus, a sheet's full extent is best defined by its building blocks, the dwarfs, which represent a sheet's diffuse component. It is therefore useful to establish a limiting dwarf halo mass, so that the dwarf and non-dwarf mass ranges can be examined as two distinct halo populations, given the proposed difference in their formation histories. Dwarf galaxies are traditionally defined as galaxies having absolute magnitudes fainter than -15 or -16 mag in B (Hodge 1971; Tammann 1994) or fainter than -18 mag in V (Grebel 2001). In his Note on the Definition and Nomenclature of Dwarf Galaxies, Binggeli (1994) shows that dwarf irregulars are best defined by a two-parameter family for which absolute B magnitude (MB) and mean (or central) surface brightness (JU) follow a well-defined relation. His plot of the MB-H plane for galaxies of various morphological types suggests a continuity between spirals and dwarf irregulars in the range of -16 to -18 mag in MB, making it difficult to establish a clear line of demarcation between dwarfs and non-dwarfs. This difficulty is demonstrated by the statistics of the galaxy morphological types in the LVC: of the -50 galaxies with MB between -16 and -18 mag, 34% are classified as spirals while the remainder are classified as dwarfs. Thus, a cut-off value of dwarf mass based on a limiting value of MB is either going to exclude the -60- brightest dwarfs or include non-dwarfs. Unfortunately, since global mass is the only property available for the sheet halos, a balance between these overlaps must be struck. Binggeli (1994) points out that at MB > -16, dwarf galaxies in the MB-\X plane are distinct from both faint spirals and ellipticals. In support of this, only 5% of the LVC galaxies with MB >-16 have been classified as non-dwarfs. Thus, a brightness limit for dwarfs is adopted here as -16 mag in MB- From their study of the baryonic mass content of galaxies using data from large astronomical surveys, Read and Trentham (2005) found ranges for the distance- independent quantities M*/LB and MHI/LB of 0.9-1.2 and 0.55-0.99 MQ/LQ, respectively, for dwarf irregular galaxies. Here, M» is a galaxy's stellar mass and MHI is the mass of a galaxy's neutral hydrogen. Based on these limits, a galaxy with MB = -16 can have a maximum baryonic mass of 1.0 x 109 M0, assuming an absolute magnitude in B for the Sun of MB,O = +5.48 mag (Binney and Merrifield 1998) and MHi/Mgas = 0.735 (see §4.6.2). Read and Trentham (2005) conclude from their analysis that the density parameter of baryons in galaxies is Qb,gai= 0.0035, a result which was also found independently from the "The Cosmic Energy Inventory" by Fukugita and Peebles (2004). This means that a galaxy with 109 M0 of baryons has a CDM mass of 109 Mo £2c/£\gai, where Q.c is the dark matter density parameter. Adopting = 0.243 from the same set of WMAP parameters used in the simulation (i.e., the first-year results; see Spergel et al. 2003), the maximum halo mass for a dwarf galaxy with MB =-16 mag is -6.9 x 1010 M0. In Chapter 5, the same mass limit is applied to the Local Sheet dls in the analyses where comparisons are made between theory and observations. -61 - Figure 3.3 shows the distribution of the halo masses as a function of their sheet altitude (|z|) in each of the CDM sheets. This distributions are essentially scatter plots, with Spearman rank coefficients4 typically between 0 and -0.2 and reliabilities no better than 80%5. While this indicates that a linear correlation is unlikely, the fact that all sheets, with the exception of sheet #2, have negative correlation coefficients suggests some degree of mass segregation in the sense that the most massive halos are preferentially found closer to the midplane of their sheets. Restriction of the analysis to the non-dwarf halos significantly strengthens the negative dependency of mass on \z\, with coefficients ranging from -0.1 to as strong as -0.4 and reliabilities as high as 87%. In contrast, when only dwarf halos are considered, the negative correlation of mass with |z| is no longer seen in all sheets, with the correlation coefficients ranging from -0.1 to 0.2. All of this is revealed by closer inspection of Figure 3.3; the halos with masses which exceed the adopted dwarf mass limit are essentially confined to the innermost half of their sheet's vertical extent, where the number density of all halos is highest. In contrast, the dwarf halos are found throughout the sheets' full vertical extents. 4 The Spearman rank coefficient is a measure of the strength of a monotonic correlation between two variables. The absolute value of the coefficient is between 0 and 1, with 1 being indicative of a perfect correlation. A negative coefficient suggests the presence of a negative correlation. The absolute value of the coefficient is independent of the shape of the correlation; a coefficient of ±1 will result when the two variables are monotonically related, regardless of whether the correlation is linear or non-linear. 5 The significance of the Spearman rank coefficient quantifies the probability of the correlation being obtained by chance. The significance is a value between 0 and 1; a significance of 0.1 means that the same correlation will be found 9 times out of 10. The reliability of the correlation is therefore 1.0 minus the significance. -62- Figure 33: Masses of halos in CDM sheets as a function of sheet altitude, |z|. The grey line is the upper mass limit adopted for dwarf halos (see text below). CDM SHEET 1 CDM SHEET 2 Cr> o -o I I11 tsb 5 On en o FcP 0.5 1.0 1.0 1.5 Id [Mpc] Id [Mpc] CDM SHEET 3 CDM SHEET 4 I ' ' ' » | ' i > ' ' « l 13 - © r~0 2 O o cn cn 12 ° o O O Oo ° O u b 1 11 "O 11 % $09? ,n s o o cn Oi o o o o 10 0 a o On 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Id [Mpc] Id [Mpc] CDM SHEET 5 CDM SHEET 6 r • » i i 13 - 12 - en C7> o o o :° o c9 300 • CJ1 cr> o O o L* % O o 10 • 1 83 o O ° o 1 •. • , «-L . . 0.5 1.0 0.0 0.2 0.4 0.6 0.8 Id [Mpc] Izl [Mpc] •63 CDM SHEET 7 CDM SHEET 8 n> CJ °Q> o ® 0° o 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 \z\ [Mpc] Izl [Mpc] The mass segregation described above can be seen more clearly in Figure 3.4, which shows each sheet's standard deviation in |z|, denoted by <£2>1/2, as a function of halo mass, where C~ In 8 sheets, the dispersion in |z| increases with decreasing halo mass, then gradually flattens to a dispersion that characterizes the halos with masses less than ~2.5 x 1010 M©. This behaviour is emphasized in Figure 3.5, which combines the plots for the 8 sheets into a single averaged and normalized plot. Also plotted in Figure 3.4 and Figure 3.5 is the adopted limiting mass for dwarf halos, revealing that this value is a reasonable approximation of the mass at which a sheet's <£2>m flattens out. -64- Figure 3.4: Standard deviation of (z| as a function of halo mass for the 8 CDM sheets. The grey line is the upper mass limit adopted for dwarf halos (see text above). 0.8 o Q_ :> 0.6 A 0.4- V 0.2 0.0 12.5 12.0 11.5 11.0 10.5 10.0 9.5 log M [log M0] 2>m Figure 3.5: Same as Figure 3.4 except with the mean value of <^ aorm for the 8 CDM 2>1/2 2 m sheets along the ordinate, where <4" „orm is <£ > divided by the standard deviation in |z| for all of a sheet's halos. T 1 I I I , I . . , | I I I I | I. 1.1 :i • ' • | i | | • r ' ' i • 1.0 'r ^ \ > T D C C < j- \ i \ y z ri( O 4 *00 f / I/2 A \ ^^ : ^ 0.7 r J -E V \ f E v 0.6 \ / -E 0.5 I y\ -E 0.4 H ....1 V... 1 .... 1 . > . • i . • • i i . • i . I .= 12.5 12.0 11.5 11.0 10.5 10.0 9.5 log M [log M0] -65- 3.5 The Density Profiles of CDM Sheets 3.5.1 Construction of the vertical density profile Figure 3.6 presents the vertical density profiles of the CDM sheets. For each sheet, a density profile was produced for all its halos, its non-dwarf halos alone, and lastly its dwarf halos alone. The ordinate axis, S(z) = p(z)/p, is the ratio of the mass density at z to the total mass density of the sheet, computed from Equation 3.2 (2Az,)"']•>, S(zk)= 7=1 In the numerator of Equation 3.2, /«/ is the mass of each halo within two slabs on opposite sides of the sheet, parallel to the sheet's midplane, with midpoints ±z* and total thickness 2AZk. In the denominator, ntj is the mass of each halo within the full z-range of the sheet, denoted by h. The values of Az* were incremented geometrically; i.e. Azk+i = Az* (\+step). This was to insure sufficient sampling far from the sheet's midplane, where the halo density is lowest. The values of AZQ and step were optimized by identifying the minimum values which yielded the strongest correlation with |z| for the sheet with the greatest central concentration (sheet #3). The strength of the correlation was judged from the Pearson rank coefficient for the z-dependence of log 8, for which a coefficient of ±1.0 corresponds to a perfect linear correlation. In this manner, a peak at z = 0 would not be smoothed out, as the rank coefficient would increase with decreasing values of step until the correlation became hidden by the growing intrinsic scatter -66- produced by the decreasing sampling size. This process yielded Azo = 0.1 Mpc and step = 0.15. 3.5.2 Vertical density models of equilibrated sheet-like systems It is now possible to investigate whether the observed density profiles of the CDM sheets match the profile of an equilibrated sheet-like system of particles. An analogue presents itself in the vertical distribution of light emitted by a disk galaxy viewed edge- on. In his study of edge-on disk galaxies, van der Kruit (1988) suggested that the vertical density profile of a self-gravitating system of plane parallel layers of particles might be described by a family of functions ranging from a pure exponential function (Equation 3.3) to the square of the hyperbolic secant (sech) function6 (Equation 3.4): p(z) = po exp(-z/z0) Equation 3.3 2 p(z) =po sech (z/z0) Equation 3.4 In the above equations, p(z) is the mass density at z and po and zo are the central density and scale length of the fit, respectively. The viability of the sech model arises from the assumption that the system is virialized as well as locally isothermal (i.e., its kinetic energy is independent of z). For a steady-state system of plane parallel layers, Poisson's equation is: a2o Equation 3.5 -r = 4nGp(z) oz 6 sech x is equal to (cosh x)"1, which is 2 (e*+e"x)"'. -67- where O is the gravitational potential. Liouville's Theorem tells us that if the total energies of the individual particles can be regarded as conserved, the density of particles in position-momentum space is constant. If the system is assumed to be stable with time (i.e., equilibrated), Liouville's Theorem leads to the expression Equation 3.6 where az is the dispersion of the particles' velocity components which are perpendicular to the layers (vz). By imposing the isothermal approximation, the second term on the right of Equation 3.6 is zero. Differentiation of Equation 3.6 and substitution into Equation 3.5 leads to the solution for p(z), which is the sech2 function given by Equation 3.4, where 112 ZO = oz (2nGpo)~ Equation 3.7 At z » ZQ, the sech function approaches the exponential function with central density 4po and scale length zo/2. This model was successfully used to fit the vertical light distributions in a sample of edge-on galaxies (see, e.g., van der Kruit and Searle 1981). However, deviations from the model were seen at small z, where the light is most contaminated by dust lanes as well as most enhanced by newborn stars. The case for the exponential model is based on near-infrared observations of disk galaxies, for which the contamination of light by dust and newborn stars is significantly reduced. Such studies have revealed an exponential peak in the vertical distribution of light at small z (see, e.g., Wainscoat, Freeman and Hyland 1989). Moreover, star counts -68- in our Galaxy have been shown to favour an exponential dependence on z (see, e.g., Pritchet 1983). However, a pure exponential model is not isothermal, as az varies with z according to Equation 3.9, which requires that az rise steeply at z < zo (see Fig 3. of van der Kruit 1988). This conflicts with observations of our own Galaxy, which suggest that the z-dependence of oz is more moderate, if not completely isothermal (see, e.g., Fuchs and Wielen 1987; Bahcall 1984). The characterization of the vertical density profile of a disk galaxy by a family of functions ranging from the sech to the exponential is, according to van der Kruit (1988), substantiated by the fact that the isothermal approximation is expected to break down near the system's midplane owing to the decrease in the velocity dispersion of successively younger generations of stars, which could be responsible for the peak in the vertical density profile. As a bridge between a non-isothermal and isothermal system, van der Kruit proposes the sech function as an intermediate solution; namely, p(z) =p0 sech (z/z0) Equation 3.8 At z » zo, Equation 3.8 approaches the exponential function with central density 2po and scale length ZQ. The vertical velocity dispersion as a function of z is given for all three models by van der Kruit (1988): 2 sech model: oz = oz$ Equation 3.9 sech model: -69- 2 2 exp model: a = az Here, "J ID sech model: 112 exp model: AZ>O ~ (UGIZQ) Equation 3.14 Equation 3.10 and Equation 3.11 tells us that the rise in az with z is more gradual for an equilibrated sheet with a sech vertical density profile than for the pure exponential case (see Fig. 3 in van der Kruit 1987). 3.5.3 Fitting of the vertical density profile Given that the non-dwarf halos in CDM sheets are generally confined to low z, it may be expected that they play a similar role as a disk galaxy's young stars in defining the shape of a sheet's vertical density profile in the core, while the sheet's diffuse component, the dwarf halos, play a role analogous to that of the disk galaxy's older stellar population in determining the sheet's vertical extent. This motivates the separate fitting of the non-dwarf and dwarf density profiles of the CDM sheets. Fits of the exponential, sech and sech2 functions are superimposed on the observed density profiles in Figure 3.6. The coefficients and rms of each fit are listed in Table 3.2. A comparison of the fits yields the following observations: -70- As expected from the mass segregation seen in §3.4, the values of <50 (the overdensity in the midplane) from the fits to the non-dwarf profiles are, in most cases, in general agreement with the global values, while the values of for the dwarf profiles are, on average, 48% lower than the global values. This difference is not significantly dependent on the fitting function; the values of So derived from the sech2, sech and exp fits to the dwarf profiles are, on average, 46%, 47% and 51% of the global values, respectively. Also not surprisingly, given the more diffuse distribution of the dwarf halos, the fits to the non-dwarf profiles yield the lowest values of zo, while the dwarf profiles produce values of ZQ which are, on average, ~5 times larger than the scale lengths of the non-dwarf profiles. As a result of the thinning effect of the non-dwarf halos on the global profile, the global values of z<> are, on average, 27% lower than the values which characterize the dwarf profiles. Thus, the scale lengths of the dwarf profiles most accurately model the sheet's full vertical extent. The mean rms scatters for the exponential fits to the global, non-dwarf and dwarf profiles are 0.51±0.16, 0.53±0.20 and 0.31±0.07, respectively. For the sech profiles, the corresponding means are 0.52±0.18, 0.53±0.22 and 0.37±0.10. For the sech2 profiles, the mean rms scatter of the global, non-dwarf and dwarf profiles are 0.53±0.19, 0.53±0.23 and 0.38±0.11. Thus, as evidenced by the central peak seen in nearly all of the profiles, the exponential function provides the best fit for all three halo mass regimes, with the lowest rms scatter associated with the dwarfs. Figure 3.6: Vertical overdensity profiles of the CDM sheets in running bins of geometrically-incremented size. Overplotted is the exponential fit (solid line), the sech fit (dotted line) and the sech2 fit (dashed line). CDM SHEET 1 4.0 dwarf halos 3.0 i.iTrr*. p'.w all halos 4.0 3.0 2.0 5.0 non—dwarf halos 4.0 2.0 v 0.0 0.0 0.5 1.0 1.5 2.0 \z\ [Mpc] -72- Figure 3.6 (continued) CDM SHEET 2 dwarf halos 2.0 - H ' •—• « 1—•- h all halos •T"4# non—dwarf halos o.o 1.0 Izl [Mpc] -73- Figure 3.6 (continued) CDM SHEET 3 4.0 dwarf halos 3.0 2.0 K> 12.0 all halos 10.0 8.0 "O 6.0 4.0 2.0 18:8 non-dwarf halos 10.0 5.0 0.0 •• 0.0 0.5 1.0 1.5 2.0 Izl [Mpc] -74- Figure 3.6 (continued) CDM SHEET 4 3.0 dwarf halos 2.0 •O * "*"1 **i"!*" **' all halos 3.0 ^<0 2.0 •> non—dwarf halos 3.0 2.0 *o 0.0 0.0 0.5 1.0 1.5 2.0 Izl [Mpc] -75- Figure 3.6 (continued) CDM SHEET 5 4.0 dwarf halos 3.0 2.0 •o all halos 8.0 6.0 4.0 2.0 10.0 non-dwarf halos 8.0 6.0 4.0 2.0 0.0 0.0 0.5 1.0 1.5 2.0 Izl [Mpc] -76- Figure 3.6 (continued) CDM SHEET 6 4.0 dwarf halos a I halos non—dwarf halos 1.0 \z\ [Mpc] -77- Figure 3.6 (continued) CDM SHEET 7 4.0 dwarf halos 3.0 ^ 2.0 K> all halos 6.0 ^ 4.0 "O 2.0 1 8.0 non—dwarf halos 6.0 *o 4.0 2.0 0.0 0.0 0.5 1.0 1.5 2.0 Izl [Mpc] -78- Figure 3.6 (continued) CDM SHEET 8 5.0 dwarf halos 4.0 i- 3.0 •o 2.0 12.0 all halos 10.0 8.0 6.0 4.0 2.0 non—dwarf halos 10.0 5.0 0.0 0.0 0.5 1.0 1.5 2.0 \z\ [Mpc] -79- Table 3.2: Parameters of fits to the vertical overdensity profiles of the CDM sheets SHEET <*o Zo [MpcJ RMS (1) (2) (3) (4) global non-dwarfs dwarfs global non-dwarfs dwarfs global non-dwarfs dwarfs sech2 fit 1 4.47±0.38 4.84±0.49 2.26±0.34 0.32±0.04 0.31±0.04 0.62±0.12 0.41 0.48 0.48 2 1.97±0.33 5.50±0.62 1.44±0.21 0.94±0.21 0.21±0.03 1.47±0.34 0.58 0.60 0.43 3 11.51±0.89 12.75±0.86 2.71±0.19 0.12±0.02 0.11±0.01 0.80±0.07 0.83 0.81 0.30 4 2.41±0.40 2.46±0.47 2.02±0.14 0.53±0.11 0.52±0.13 0.67±0.06 0.53 0.61 0.21 5 8.94±0.63 9.80±0.71 2.57±0.29 0.08±0.02 0.07±0.03 0.53±0.08 0.58 0.58 0.38 6 5.40±0.39 6.23±0.44 2.46±0.28 0.09±0.02 0.06±0.04 0.30±0.04 0.33 0.24 0.27 7 6.81±0.79 7.64±0.91 2.35±0.33 0.13±0.02 0.11±0.02 0.59±0.11 0.73 0.76 0.46 8 11.76±0.28 13.01±0.24 3.21±0.38 0.10±0.01 0.10±0.00 0.48±0.07 0.26 0.18 0.49 sech fit 1 4.51±0.41 4.87±0.54 2.40±0.34 0.21±0.03 0.20±0.03 0.37±0.07 0.43 0.51 0.46 2 1.98±0.35 5,47±0.63 1.46±0.22 0.62±0.15 0.14±0.02 0.97±0.24 0.58 0.61 0.43 3 11.49±0.84 12.73±0.83 2.76±0.19 0.08±0.01 0.07±0.01 0.52±0.05 0.79 0.78 0.29 4 2.47±0.43 2.53±0.50 2.06±0.15 0.34±0.08 0.33±0.09 0.43±0.04 0.53 0.62 0.21 5 8.93±0.62 9.79±0.71 2.67±0.30 0.05±0.01 0.04±0.02 0.33±0.05 0.57 0.58 0.37 6 5.40±0.38 6.23±0.44 2.52±0.29 0.05±0.01 0.04±0.02 0.19±0.03 0.32 0.24 0.27 7 6.78±0.76 7.62±0.89 2.47±0.33 0.09±0.02 0.08±0.02 0.36±0.07 0.70 0.74 0.43 8 11.76±0.27 13.01±0.24 3.39±0.38 0.06±0.00 0.06±0.00 0.29±0.04 0.26 0.18 0.47 exponential fit 1 4.76±0.54 5.10±0.71 2.85±0.30 0.32±0.06 0.32±0.07 0.53±0.09 0.54 0.64 0.33 2 2.08±0.47 5.44±0.68 1.65±0.28 1.06±0.42 0.24±0.05 1.59±0.58 0.66 0.66 0.43 3 11.42±0.73 12.68±0.74 3.13±0.24 0.13±0.02 0.11±0.02 0.78±0.09 0.69 0.69 0.31 4 2.90±0.48 3.01±0.56 2.37±0.17 0.46±0.12 0.44±0.13 0.63±0.08 0.51 0.58 0.20 5 8.90±0.59 9.78±0.70 3.18±0.24 0.08±0.02 0.06±0.02 0.44±0.05 0.54 0.57 0.25 6 5.39±0.36 6.23±0.44 2.73±0.29 0.08±0.02 0.04±0.03 0.28±0.05 0.30 0.23 0.26 7 6.72±0.66 7.54±0.81 2.84±0.29 0.15±0.03 0.13±0.03 0.55±0.09 0.62 0.68 0.33 8 11.76±0.26 13.01±0.24 3.86±0.35 0.09±0.01 0.08±0.00 0.43±0.06 0.24 0.18 0.38 Notes. (1) Sheet number. (2) Central overdensity of the fit. (3) Scale length of the fit. (4) Root-mean-square deviation of the fit. -80- 3.6 Peculiar Motions in CDM Sheets Included in the output of the ACDM simulation is the co-moving velocity vector (V*, Vy, v-) of each halo. This vector, known as the peculiar motion, represents the three- dimensional motion that would be observed from a frame that is expanding with the Universe. When the velocity vectors of sheet halos are transformed into the coordinate frame of their sheet, it is possible to compute The velocity dispersion of a system of galaxies is best gauged by its isolated dwarfs, as the motions of these probes are less likely to have been perturbed by other sources of mass. The motion of isolated dwarfs therefore reflect their primordial dynamics. In addition, the small sizes of dwarfs allow them to be regarded as test particles of their local potential. The identification of isolated dwarfs in the CDM sheets is based on the simple prescription of Tikhonov and Klypin (2009) in their comparison of ACDM simulations with the Local Volume. Specifically, a dwarf is considered to be isolated if it is (1) beyond 1.0 Mpc of any non-dwarf galaxy; and (2) lacking companions brighter than itself within 200 kpc. Based on these criteria, 30%-55% of the dwarf halos in the CDM sheets were identified as isolated members and were therefore selected for inclusion in the -81- computation of the velocity dispersion. The resulting values of az for each sheet are listed in Table 3.3. In §3.7, these measurements are used to evaluate the crossing time of each sheet. Table 3.3: Velocity dispersions of the CDM sheets 1 1 SHEET trz [km s" ] a, [km s" ] oMz (1) (2) (3) (4) (5) 1 63 88.94±11.21 104.86±13.21 1.18±0.21 2 52 126.76±17.58 155.91±21.62 1.23±0.24 3 92 83.93±8.75 105.87± 11.04 1.26±0.19 4 52 82.33±11.42 78.15±10.84 0.95±0.19 5 67 115.61±14.12 104.11±12.72 0.90±0.16 6 46 64.39±9.49 49.51±7.30 0.77±0.16 7 55 77.06±10.39 88.82± 11.98 1.15±0.22 8 42 60.69±9.36 61.78±9.53 1.02±0.22 Notes. (1) Sheet number. (2) Number of isolated dwarf halos used to compute the velocity dispersion. (3) Observed vertical velocity dispersion of the isolated dwarf halos. The quoted r uncertainty is the standard error of the mean (i.e., cz/VA Uo). (4) Observed line-of-sight velocity dispersion of the isolated dwarf halos (see §3.6.2). The quoted uncertainty is the standard error of the mean. (5) Ratio of the line-of-sight velocity dispersion to the vertical velocity dispersion. 3.6.1 The z-dependence of the vertical velocity dispersion In §3.5, it was found that the vertical density profiles of the CDM sheets are best described by an exponential decline with z, primarily due to the presence of sharply peaked cores. If the CDM sheets are equilibrated systems, then the exponential shape of their vertical density profiles implies that the sheets are non-isothermal. More specifically, the system's kinetic energy per unit mass, and therefore the halos' vertical velocity dispersion ( 3.2 reveals that for the majority of the sheets, the rms scatter in the exponential fits is -82- only marginally less than the scatter in the sech and sech2 fits. It is therefore worthwhile to investigate whether oz is constant across a sheet's vertical extent, or alternatively, whether oz varies with z according to the predictions of the sech or exponential models, given by Equation 3.10 and Equation 3.11, respectively. Figure 3.7 presents the profiles of the vertical velocity dispersion for each CDM sheet. The profiles have been smoothed in the same manner as the vertical density profiles constructed in §3.5. Overplotted are the profiles predicted by the sech , sech and exponential models, which were produced by fitting the observed profiles to Equation 3.9, Equation 3.10 and Equation 3.11 with 2 a free parameter. For the sech model, o^o corresponds to the weighted mean of az(z). Figure 3.7: Vertical velocity dispersion profiles of isolated dwarfs in the CDM sheets in running bins of geometrically-incremented size. The solid, dotted and dashed lines are the profiles predicted by the sech2, sech and exponential models, respectively. The solid grey line is the observed vertical velocity dispersion. The error bars are the standard error of the mean az. CDM SHEET 1 CDM SHEET 2 200.0 250.0 200.0 150.0 150.0 b" 100.0 100.0 50.0 50.0 0.0 0.0 0.0 0.5 0 1.5 2.0 0.0 0.5 2.0 Izl [Mpc] -83- CDM SHEET 3 CDM SHEET 4 160,0 200.0 140.0 150.0 - 120.0 < * o 100.0 [J» t> 100.0 : rt"1 * • I I I I I I • I I • • > > I > > • • 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 Izl [Mpc] \z\ [Mpc] CDM SHEET 5 CDM SHEET 6 200.0 200.0 180.0 160.0 150.0 < i 140.0 b 100.0 120.0 < i oo 100.0 i/ 50.0 i'' 80.0 60.0 0.0 . . . *i . . * ...... 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Izl [Mpc] \z\ [Mpc] CDM SHEET 7 CDM SHEET 8 200,0 120.0 100.0 150.0 80.0 llll 100.0 60.0 40.0 20.0 f \ 0.0 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 Izl [Mpc] Izl [Mpc] -84- The z-dependence of az does not appear smooth for any of the sheets. It is possible that this is due in part to the small numbers of isolated dwarfs in each smoothing bin, as indicated by the large errors in the values of az. Despite the scatter, it can be recognized that the majority of data points in sheets #1 through #5 are increasing with z, which is evidence of non-isothermality. In particular, sheets #4 and #5 exhibit the sharp minimum predicted by the exponential model, while for sheet #2 the rise in az is much more gradual and appears to plateau beyond ~1.4 Mpc. Plateaus are also seen in the outermost regions of sheets #1, #3 and #7. The shapes of sheets #6 and #8 aren't smooth enough to deduce a trend in their z-dependence, either because a trend doesn't exist or because the trend is hidden by the uncertainty in az. In support of the latter, these two sheets are the least populous of the 8, as can be seen in Table 3.1. The implications of the shapes of the profiles are discussed further in §3.7.1. 3.6.2 The relationship between the vertical and radial velocity dispersion Listed in Table 3.3 is the total line-of-sight velocity dispersion of each sheet, < > z m or = <(vr- vr j > , where vr is the radial component of the co-moving velocity vector of a sheet's halo after subtraction of the three-dimensional motion of the halo nearest the sheet's origin. In other words, vr is the radial motion that would be observed from the rest frame of a galaxy near a sheet's centroid, which is akin to the motion that we observe from the rest frame of the Local Group. Since o> is the only observable diagnostic of the velocity dispersion of the Local Sheet, it is of value to investigate whether or is coupled with the total vertical velocity dispersion (ov), so that an estimate of the latter can be obtained from observable motions from within a real-life sheet. Of particular interest to -85- this study is the nature of the relationship between or and az for dwarf galaxies, as these constitute the sample used to investigate the Local Sheet in Chapter 5. For the family of vertical density functions considered in the previous section, van der Kruit (1988) shows that ar is linearly correlated with oz. The slope of the correlation depends on the shape of the vertical density profile as well as on the observation point. Since the 8 CDM sheets are being viewed from their approximate centroids, variations in aJ Figure 3.8: Dependence of a, on az for the isolated dwarf halos in the 8 CDM sheets. The solid line is the linear fit. 180 140 ^ 100 i i b 60 20 20 60 100 140 180 1 uz [km s" ] -86- 3.7 The Crossing Times of CDM Sheets In the above sections, the vertical scale lengths (zo) and the vertical velocity dispersions (ov) were computed for the 8 CDM sheets. The combination of these structural and dynamical gauges leads to an estimate of a sheet's vertical crossing time, tc = 2W/(Tz , where w is half the sheet's vertical extent. Given that a sheet's dwarf halos represent its most extended component, w can be approximated as the exponential scale length (ze) of the vertical density profile of a sheet's dwarfs (i.e., the vertical length at which the density drops to Me). Since oz is calculated for isolated dwarfs, tc represents the typical time it takes for a dwarf to complete a vertical traversal of its sheet. The crossing time is therefore an indicator of whether a sheet's dwarfs possess sufficient motion in the z-direction to have obtained a state of dynamical equilibrium. The values of tc obtained for the CDM sheets are listed in Table 3.4. For each sheet, crossing times were computed using scale lengths from the sech2, sech and exponential fits to the dwarf density profiles. Values of ZQ were converted to ze via ze = 1,085zo for the sech model, ze = 1.657zo for the sech model and ze = zo for the exponential model. Table 3.4 reveals that the values of tc from the 3 vertical density models are consistent within errors. The crossing times of the CDM sheets range from ~7 to -25 Gyr, with a mean value of 15.3±5.0 Gyr. Thus, the vertical crossing times are all over half the current estimate of the age of the Universe, assuming to = 13.75±0.13 Gyr from the most recent results of the WMAP mission (Larson et al. 2011). This can be interpreted as saying that dwarf halos in sheets do not possess enough vertical motion to have made more than one traversal of their sheet's vertical extent. In comparison, the crossing times for galaxies in -87- virialized groups in the Local Volume have been found by Karachentsev (2005) to range from 2 to 4 Gyr. The median crossing time of these groups is 2.3 Gyr, or roughly one- sixth the value of to. Conversely, the Sculptor and CVnl filaments, which Karachentsev (2005) described as having dynamics "...rather in the free Hubble expansion than in a state of dynamical equilibrium...", were found to have crossing times of 6.6 and 6.9 Gyr, respectively. Given that all 8 theoretical sheets have vertical crossing times greater than ~7 Mpc, they could not have evolved into dynamically equilibrated systems by the present epoch. This means that the current vertical distribution of a sheet's halos must still retain information about the initial density fluctuations of the universe. Table 3.4: Crossing times of the CDM sheets SHEET tc [Gyr] (1) (2) sech2 sech exp 1 14.86±3.43 13.52±3.11 11.62±2.39 2 24.63±6.60 24.92±7.15 24.46±9.61 3 20.14±2.76 19.94±2.77 18.18±2.90 4 17.22±2.90 16.88±2.92 14.98±2.80 5 9.78±1.85 9.22±1.77 7.53±1.28 6 10.03±2.09 9.63±2.06 8.59±2.01 7 16.29±3.68 15.23±3.42 13.92±2.93 8 16.67±3.60 15.33±3.30 13.70±2.84 2 Notes. (1) Sheet number. (2) Crossing times computed with ze from the sech , sech and exponential density models, respectively. 3.7.1 Are sheets evolving systems? It was found above that the crossing times of CDM sheets are no less than half the age of the Universe, which precludes the possibility that sheets have evolved into steady- -88- state systems. However, their vertical density profiles (§3.5) and vertical velocity dispersion profiles (§3.6.1) contain features that may be evidence of dynamical equilibrium, in spite of their long crossing times. Specifically, the vertical density profiles of all 8 sheets were found to be well fit by an exponential function, a result which has also been found in near-infrared observations of the cores of disk galaxies (see §3.5.2). Moreover, it was found that the erz-profiles of 2 of the sheets exhibit sharp minima at their cores, which is expected for an equilibrated, exponential sheet. A possible explanation for these observations is that present-day sheets are in the process of evolving into steady- state systems. The above hypothesis can be explored by examining whether the sharpness of the central minimum in a sheet's §3.6.1 with increasing -89- The above observations are illustrated in Figure 3.9, in which az>o/ Figure 3.9 shows that the mechanism behind the reduction of the vertical velocity dispersion at small z may indeed be correlated with crossing time. The Spearman rank coefficient for a correlation between azsJoz$A and tc is 0.89, which is indicative of a high degree of monotonicity between these variables, and the probability that the correlation is being observed by chance is less than 1%. However, if the sheet with the longest crossing time (sheet #2) is excluded from the statistical test, the rank coefficient drops to 0.71 and the probability of a chance correlation increases to 7%. While the rank coefficient is still high relative to values which are characteristic of scatter plots (i.e., <0.1), the robustness of a correlation between -90- Figure 3.9: Dependence of ov,n..P/qyin.t on the sheet crossing time (fc), where o^o and o^n,, are derived from the exponential fit to a sheet's 0.78 [ 0.76 0.74 o it b 0.72 0.70 0.68 5 10 15 20 25 30 tc [Gyr] Another probe of a sheet's dynamical model, albeit less direct, is the shape of its vertical density profile. In particular, if sheets are evolving toward or away from the exponential model, the relative strength of the exponential peak in a sheet's vertical density profile may also correlate with crossing time. The relative strength of the peak can be quantified by the ratio of the central overdensity derived from the exponential fit to the vertical density profile for dwarfs (<5o,exp) to the central overdensity derived from the sech fit (<$o,iso)- This ratio is therefore a measure of the strength of a sheet's -91 - exponential peak relative to the flat core that would be expected if the sheet were isothermal. This ratio is plotted against the sheet's crossing time in Figure 3.10. Figure 3.10: Dependence of <$o,eXp/<$o,iso on the sheet crossing time (/c), where tc is derived from the scale length of the exponential fit to the vertical density profile of a sheet's dwarfs. The dashed line represents the age of the Universe. 'I I ' ' . |,."J. 7' T I " I I" 1.5 P T 1.4 J 1.3 - H®>. Y ' 1.1 I 1.0 5 10 15 20 25 30 *c [Gyr] The measurements of ^o.exp/^o.iso are not accurate enough to make a definitive statement about their dependence on crossing time. However, it is observed that the sheet with the shortest crossing time has <5o,exp/<5o,iso among the highest of the observed values, while the two sheets with the longest crossing times have <5o,exp/ -92- If sheets which have the closest agreement with the exponential model have comparatively shorter crossing times, then a tentative hypothesis presents itself; namely, that sheets are evolving into equilibrated, exponential systems. This implies that the mechanism that is responsible for a sheet's exponential peak must be a continuing process as opposed to an evolutionary stage of the past. Given the mass segregation observed in §3.4, it is possible that a sheet's halos are being accreted by the population of high mass halos in the sheet's innermost slab. This is, in fact, a large-scale analogy to the proposal by van der Kruit (1988) that the exponential peaks occasionally seen in disk galaxies at small z, and the corresponding minima in a:, are due to the newer generations of stars with lower velocity dispersions that preferentially populate the disks' midplanes. In the hierarchical galaxy formation scenario, a sheet's high-mass halos represent the newer generation of galaxies with reduced random motion. As in disk galaxies, these newly-formed members of the sheets are essentially confined to low z. A sheet with a crossing time less than the age of the Universe has therefore had enough time for its dwarfs to respond to the draw of the high-mass halo population, the visible evidence of which is the peak in the vertical density profile of the sheet's dwarfs at small z. It is therefore proposed that sheets are evolving into exponential systems as a result of mergers leading to the development of high-mass halos in the midplane. The above hypothesis can be explored by investigating the vertical motions of a sheet's isolated dwarf halos for a signature of infall. If a sheet's dwarfs are indeed responding to a pull toward the sheet's midplane, then the dwarfs with z < 0 Mpc should have positive vertical motions, while the dwarfs with z > 0 Mpc should have negative -93- vertical motions. Figure 3.11 shows the vertical motions (vr) of each sheet's dwarfs as a function of z, where vz is the vertical motion that would be observed from the rest frame of a galaxy near the sheet's centroid (see §3.6.2). Inflow signatures are evidenced in all 8 sheets by a drop in the mean vz at z > 0 Mpc. The signature appears strongest in the 5 sheets with increasing Figure 3.11: Vertical motions of isolated dwarf halos in the theoretical sheets as a function of z. The dashed lines at z < 0 Mpc and z > 0 Mpc represent the mean vz of the halos below and above the sheet's midplane, respectively. CDM SHEET 1 CDM SHEET 2 8b © 0 z [MpcJ z [Mpc] CDM SHEET 3 CDM SHEET 4 300 200 100 0 > o -100 00°cP Q 0 ^ -200 o -300 --Q o z [Mpc] -94- CDM SHEET 5 CDM SHEET 6 -100 -100 go ©b Q c#>^P -300 0 -1 0 z [Mpc] z [Mpc] CDM SHEET 7 CDM SHEET 8 300 200 O 100 <&>° {L&8 ? 0 O o O -100 o°o °J° & o -100 [*> % O -200 -200 -300 -300 -2-10 1 2 -2-10 1 z [Mpc] z [Mpc] 3.8 Surface Densities of CDM Sheets The surface density, I, of a sheet is its mass divided by its face-on surface area. For a particular vertical density model, the predicted surface density, 2flt, is the integral of the vertical density function over all z. The sech and exponential models are considered here as limiting cases for the derivation of For both models, Zflt = 2poZo, where po and zo are the central densities and scale lengths from the respective fits. -95- Dynamical estimates of a sheet's surface density, Z,dyn, are derivable for isothermal and exponential sheets using equations Equation 3.12 and Equation 3.14, respectively. Thus, for either an isothermal or an exponential sheet, •y t Z&yn-0zfl (nGzo)" Equation 3.15 Owing to the small numbers of isolated dwarf halos in the CDM sheets, it is not possible to measure #5 in Figure 3.7 agree well enough with the exponential model that Table 3.5 lists Zflt,dw and 2dyn for the 8 CDM sheets. Also provided is 2dW, which is the observed surface density of a sheet's dwarfs. Comparison of 2dW with 2® shows that the surface densities obtained from the fits of the dwarf density profiles to the sech2 and exponential models are consistent, within their uncertainties, with each other as well as with the observed surface density. Each model is the more reliable gauge of the observed surface brightness in 4 out of 8 sheets, so neither model is universally superior in this context. A comparison of the observed surface density of all of a sheet's halos (Z"tot) with Ziyn reveals that if the sheets are assumed to be equilibrated, isothermal systems, then the observed vertical velocity dispersions require 2.5 to 7 times more mass than the total halo masses. Alternatively, if the sheets are assumed to be equilibrated, non-isothermal systems with dynamics described by the exponential-sheet model, then the sheets require -96- only 1.5 to 4 times more mass. The possible sources of this missing mass are (1) the CDM particles which did not accrete into halos more massive than the halo extraction limit of the simulation (i.e. 2 x 109 M©); and (2) the particles that are beyond their halo's jRvir (the radius at which the halo's density falls below 200 times the halo's mean density; see §3.2). The ratio EiJE&yn *s impervious to the overabundance of dwarf satellites predicted by the ACDM model (see §3.2.1), as both numerator and denominator would decrease by the same factor if a correction for the overabundance were applied. A direct measurement of the unextracted CDM mass cannot be made from the simulation products that were acquired for this study. However, this measurement is constrained by the simulation's global statistics. The simulation begins with 1.6 x 108 equal-mass particles. The total number density in the simulation cube with sides 1 l 80 h~ Mpc in length is 116.6 Mpc", where h = 0.72 as adopted in the simulation. The total number of CDM particles in the extracted halos is only 7.7 x 107. Thus, only 48% of the simulation ends up in the form of galaxy-sized halos, while the rest is unextracted. Lemson and Kauffmann (1999) showed that the shape of the halo mass function depends on the local overdensity in the sense that the denser the environment, the greater the number of high-mass halos relative to low-mass halos. The CDM sheets are, on average, 3.9±1.7 times denser than the simulation's mean density. It can therefore be assumed, in principle, that the sheet environment is overabundant in high-mass halos compared to the simulation mean. Correspondingly, it can be assumed that 48% is a lower limit on the percentage of CDM particles that are extracted in sheets. It can also be reasonably assumed that the upper limit is not significantly larger than 48%, given that the -97- aforementioned dependence of the halo mass function on environmental density is essentially negligible for halo masses less than 1013 M©. This is the maximum halo mass observed in all sheets except #5, which contains one halo with mass 1.5 x 1013 M©. It is therefore expected that the CDM sheets contain approximately twice as much mass as has been extracted in the form of galaxy-sized halos. This is consistent with the amount of mass predicted for an equilibrated, exponential sheet. Given that the expected amount of unextracted mass is an approximate upper limit, the larger mass requirements of the isothermal model do not appear to be met. As early as 1933, Fritz Zwicky observed that the motions of galaxies in the Coma cluster require the presence of -90% more mass than the total mass derived from the luminosity of the cluster's galaxies (Zwicky, 1937). Since then, numerous studies have corroborated this result, despite the discovery of previously undetected sources of mass, such as the low surface brightness galaxies first detected by van den Bergh (1959) and the warm-hot intergalactic medium theorized by Cen and Ostriker (1999) and observed by Nicastro et al. (2003). In the theoretical analysis of this chapter, the motions of isolated galaxies in sheets were used to estimate the amount of dark matter required to equilibrate a sheet's galaxy-sized halos. Thus, if sheets are indeed evolving into equilibrated systems, as is suggested by their vertical density profiles and their vertical velocity dispersion profiles, then their intergalactic dark matter is not confined to galaxy groups within the sheet, but rather constitutes an intra-sheet population of dark matter -98- particles or dark matter aggregates which are below the minimum mass of galaxy-sized halos adopted by Tikhonov and Klypin (2009). Table 3.5: Observed and predicted surface mass densities of the CDM sheets SHEET iogrdw log 10g2"dyn -^"tot^dyn 2 2 2 [M© Mpe" ] [Mo Mpc' ] ]M0 Mpc ] (1) (2) (3) (4) (5) sech2 exp sech2 exp sech2 exp 1 10.69 10.64±0.15 10.67±0.11 11.97±0.14 11.90± 0.07 0.284±0.090 0.333±0.054 2 10.51 10.58±0.16 10.67±0.22 11.91±0.16 11.68± 0.16 0.140±0.051 0.240±0.088 3 10.51 10.46±0.07 10.51 ±0.08 11.82±0.10 11.59± 0.05 0.388±0.088 0.650±0.079 H- O 00 OP UJ 4 10.57 10.57±0.07 10.62±0.08 T—» 0.383±0.112 5 10.68 10.69±0.12 10.71±0.08 12.27±0.12 11.95± 0.05 0.242±0.069 0.499±0.059 6 10.56 10.56±0.12 10.58±0.12 12.00±0.14 0.153±0.050 & 7 10.63 10.55±0.15 © © 11.87±0.14 0.331±0.108 1 0 8 10.59 10.53±0.12 d 11.76±0.15 0.390±0.134 Notes. (1) Sheet number. (2) Surface density of a sheet's extracted dwarf halos (i.e. the total dwarf mass divided by nab, with a and b from Table 3.1). (3) Surface density derived from integration of the sech2 and exponential fits to the vertical density model for dwarfs. (4) The dynamical surface density; i.e. the surface density required for an isothermal or exponential sheet with vertical velocity dispersion az and scale length Za, computed via Equation 3.15. (5) Ratio of the surface density of a sheet's extracted halos (including dwarf and non-dwarf) to the dynamical surface density obtained from the isothermal and exponential models. -99- 4. The Near-Infrared Imaging Survey and Data Mining of Local Sheet dls Malcolm: "Does that sound modulated enough for you?" Tucker: "Modulated? " Malcolm: "The radio. Or is it just the galaxy giggling at us again?" Tucker: "It can giggle all it wants, but the galaxy's not getting any of our bourbon." 4.1 Ks Imaging Observations Between May 2004 and July 2006, a Ks (2.15 fi.m) imaging survey of 72 Local Sheet dls was conducted over 8 observing runs'. 68 of the survey objects were detected sufficiently well for surface photometry to be obtained. The details of the full survey have been published by Fingerhut et al. (2010) and are described in the paragraphs below. Ks imaging for an additional 25 Sheet dls were obtained between 2002 and 2008 as part of another project (Vaduvescu et al. 2005; Vaduvescu et al. 2008; McCall et al. 2011). In total, 93 out of 133 Sheet dls have been imaged in Ks. The observing runs for the 72 Sheet dls presented in Fingerhut et al. (2010) were conducted at the 3.6m Canada-France-Hawaii Telescope (CFHT), the 2.1m telescope of the Observatorio Astronomico Nacional at San Pedro Martir (OAN-SPM), the Blanco 4m telescope at the Cerro Tololo Inter-American Observatory (CTIO), the 3.6m New 1 The Principal Investigator on all runs was R. Fingerhut. However, due to travel restrictions, the images were acquired by the Co-Investigators specified in the facility-specific sections 4.1.1 to 4.1.6. All image reductions were performed by R. Fingerhut. -100- Technology Telescope (NTT) operated by the European Southern Observatory (ESO) at La Silla, and the 1.4m Infrared Survey Facility telescope (IRSF) hosted by the South African Astronomical Observatory (SAAO). A few galaxies were observed at multiple facilities to check the pre-processing method used for each site, the quality of the sky subtraction, and the zero-points computed for each image. Obtaining deep images of dls in the NIR poses several challenges which require great care to overcome in both the observing and reduction processes. Not only is the sky background high in the NIR, it also varies both spatially across the chip as well as temporally. Thus, the total on-target integration times must be split into sub-exposures, and the sky background must be repeatedly sampled close in time to each sub-exposure so that the sky pattern in each sub-exposure can be adequately subtracted. The low surface brightnesses that are typical of dls make it especially important to ensure that the low galaxy signal is not corrupted by inadequacies in the background determination. In their detailed analysis of NIR imaging strategies at CFHT for faint extended sources, Vaduvescu and McCall (2004) found that the mean brightness of the background signal in K' (2.12 (im) varied on average by 0.5% per minute. Thus, to measure the sky pattern in each frame with a precision of at least 1%, it is necessary to chop between on- target and off-target positions with no more than 90 seconds between each pointing. In consideration of these findings, the general procedure for observing a dl in the Local Sheet Ks imaging survey was as follows: -101 - 1. Sets of 1 to 6 on-target exposures were taken for 30-75 seconds per set, dithering the pointings for each exposure by a few pixels to aid in the removal of detector artifacts. The total on-target integration time varied from ~10-150 minutes, depending on the target brightness and the detector sensitivity. 2. Between each set of on-target exposures, an equal-length set of off-target (sky) exposures was obtained. The sets were dithered by at least 20" so that stars and distant galaxies could be removed. These star-subtracted sky images could then be used to create an interpolated image of the sky pattern at the time of each set of on-target exposures. The procedure for creating the interpolated sky image is detailed in §4.2.2. In cases where the galaxy was less than half the size of the detector field (in B), observing times were minimized by alternating the galaxy's position between halves or quadrants of the detector field. In this fashion, a reliable image of the galaxy's underlying sky pattern at each pointing could be obtained from the half or quadrant of the field that contained the galaxy in the previous pointing. A log of the Ks observations is provided in Table 4.1. All the observing configurations and typical exposure times at each facility are described in the sections below in chronological order. The sky was primarily photometric throughout the observing sequences for all of the detected galaxies except at CTIO and ESO, where the detected galaxies were observed through thin clouds. Nonphotometric conditions were -102- accounted for by deriving the photometric zero-point for each galaxy image from 2MASS stars on the galaxy image itself (see §4.2.3). 4.1.1 CFHT observations On the nights of Feb 23 and Mar 6-8 in 2004,21 survey targets were observed by Co- Investigators M. McCall2 and O. Vaduvescu3 with the 3.6m CFHT atop Mauna Kea in Hawaii. The telescope was equipped with the CFHT-IR detector installed at the f 8 Cassegrain focus. The detector employed a HgCdTe 1024x1024 pixel array with a scale of 0".211 pixel"1, yielding a 3'.6 x 3'.6 field of view. All observations were made with the K' filter. For each galaxy, single-frame exposure times were 75 seconds, and the total on- target integration time was 10 minutes. 4.1.2 OAN-SPM observations In March and April of 2005,9 survey targets were observed by Co-Investigator M. Richer4 with the 2.1m telescope at OAN-SPM in Baja California, Mexico. The CAMILA NIR camera contained a NICMOS3 256x256 pixel array and was installed at thefl\2>.5 Cassegrain focus (see Cruz-Gonzalez et al. 1994). The pixel scale was 0".85 pixel"1, resulting in a 3 '.6 x 3 '.6 field of view. All observations were made with the K' filter. Individual on-target exposures times were typically 60 seconds for a total of 40 minutes 2 Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada; [email protected] 3 Isaac Newton Group of Telescopes, Apartado 321, E-38700 Santa Cruz de La Palma, Spain; [email protected] 4 Institute de Astronomfa, Universidad Aut6noma de Mexico, P.O. Box 439027, San Diego, CA 92143, USA; [email protected] - 103- per galaxy. Of the 9 targets observed, ESO 443-09 and KKR 25 could not be detected sufficiently well for photometry to be obtained. 4.1.3 IRSF observations In June and July of 2005 and July of 2006, 18 survey targets were observed by S. Nishiyama5 and M. Cluver6, respectively, with the 1.4m IRSF, a joint Japanese/South African facility located at the SAAO near Sutherland, South Africa. The IRSF employed the Simultaneous-3color InfraRed Imager for Unbiased Survey (SIRIUS), which consisted of three 1024x1024 HgCdTe arrays for simultaneous observing in the J(1.25|j.m), H (1.63nm), and Ks filters. The field of view was 7'.7 x 7'.7 with a pixel scale of 0".45 pixel"1. On-target integration times were typically between 40-100 minutes of 30-second exposures. Only UKS 1424-460 could not be detected sufficiently well for analysis. 4.1.4 CTIO observations In July of 2006, 2 nights were granted with the Blanco 4m telescope at CTIO in Chile, where images were acquired by Co-Investigator O. Vaduvescu. The telescope was equipped with the Infrared Side-Port Imager (ISPI) mounted at the fl% Cassegrain focus. The camera used a Hawaii 2048x2048 pixel array with a scale of 0".3 pixel"'and a 10'.25 x 10'.25 field of view. Each target was observed using sequences of either 3 co- added 20-second exposures or 6 co-added 10-second exposures, with total on-target 5 Department of Astronomy, Kyoto University Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan; [email protected] 6 IP AC, California Institute of Technology, MC 220-6, Pasadena, CA 91106, USA; [email protected], [email protected] -104- integration times of approximately 30 minutes. All observations were made with the Ks filter. The first night was clouded out, and due to lengthy cloudy periods on the second night, it was only possible to obtain complete observing sequences for 3 survey targets. 4.1.5 ESO observations On the night of July 2, 2006, 4 survey targets were observed by O. Vaduvescu and R. Rekola7 with the 3.6m NTT at ESO at La Silla, Chile. The SofI (Son of ISAAC) infrared spectrograph and imaging camera was installed at the Nasmyth A focus. SofI consists of a Hawaii HgCdTe 1024x1024 pixel array with a scale of 0".288 pixel"1 and a 4'.92 x 4'.92 field of view. Each target was observed with sequences of 6 co-added 10- second exposures for a total on-target integration time of approximately 45 minutes. 4.1.6 WIRCam observations During the 2005B and 2006A semesters, Ks images of 17 targets were obtained via Queued Service Observing (QSO) with the Wide-Field IR Camera (WIRCam) installed on the 3.6m CFHT. WIRCam contains four 2048x2048-pixel HAWAII2-RG detectors spanning a 20' x 20' field of view with a sampling of 0".3 pixel"1. Each target was observed for at least one 16-minute sequence consisting of 20-second exposures with the target alternately positioned within each of the four 10' x 10' arrays. Thus, sky images could be produced using the three galaxy-free fields. 7 Tuorla Observatory, Department of Physics and Astronomy, University of Turku, FI-21500 Piikki' o, Finland; [email protected] -105- Table 4.1: Log of K, Observations of Local Sheet dls Galaxy Instrument Date (UT) tint [s] Image Size ['] Seeing (1) (2) (3) (4) (5) (6) AM 1306-265 SPM-CAMILA 2005 MAR 21 2340 2.8x2.8 2.9 Antlia SPM-CAMILA 2005 APR 24 2400 2.8x2.8 4.2 Cam B CFHT-IR 2004 MAR 07 600 3.2x3.2 1.9 Cas 1 CFHT-WIRCAM 2006 JUL 11 1140 8.0x8.0 3.3 DD0 210 CFHT-WIRCAM 2006 MAY 19 1020 8.0 x 8.0 3.1 Dwingeloo 2 CFHT-WIRCAM 2005 NOV 18 959 8.0 x 8.0 2.1 ESO 269-58 IRSF-SIRIUS 2006 JUN 11 4320 6.9 x 6.9 1.6 ESO 321-14 IRSF-SIRIUS 2005 MAY 16 2070 3.0x3.0 2.1 ESO 324-24 IRSF-SIRIUS 2005 JUN 02 4140 3.0x3.0 2.1 ESO 325-11 CTIO-ISPI 2006 JUL 15 1020 9.7x9.7 2.6 ESO 349-31 NTT-SOFI 2006 JUL 02 2640 4.6 x 4.6 3.0 ESO 379-07 IRSF-SIRIUS 2005 MAY 16 6210 3.0x3.0 2.0 ESO 381-18 IRSF-SIRIUS 2005 MAY 16 2520 3.0x3.0 2.3 ESO 381-20 CTIO-ISPI 2006 JUL 15 1800 9.7x9.7 2.6 ESO 384-16 IRSF-SIRIUS 2005 JUN 06 1530 3.0x3.0 2.2 ESO 443-09 SPM-CAMILA 2005 MAR 23 2820 ESO 444-84 CFHT-IR 2004 MAR 08 600 3.2x3.2 1.5 HIPASS J1247-77 NTT-SOFI 2006 JUL 02 2580 2.2 x 2.2 1.5 HIPASS J1247-77 IRSF-SIRIUS 2006 JUN 08 6750 3.0x3.0 2.0 HIPASS J1305-40 NTT-SOFI 2006 JUL 02 2580 2.2 x 2.2 1.5 HIPASS J1337-39 NTT-SOFI 2006 JUL 02 2580 2.2 x 2.2 1.5 HIPASS J1348-37 IRSF-SIRIUS 2006 JUN 09 6750 3.0x3.0 2.1 HIPASS J1351-47 IRSF-SIRIUS 2006 JUN 07 6750 3.0x3.0 2.1 Holmberg I CFHT-WIRCAM 2005 DEC 10 840 8.0 x 8.0 2.9 IC 1574 IRSF-SIRIUS 2006 JUN 10 4320 6.9 x 6.9 2.9 IC 3104 IRSF-SIRIUS 2006 JUN 10 4320 6.9 x 6.9 1.5 IC 3687 CFHT-IR 2004 MAR 08 600 3.2x3.2 1.2 IC 4182 CFHT-WIRCAM 2006 JUL 09 960 8.0 x 8.0 2.5 IC 4247 SPM-CAMILA 2005 APR 20 2220 2.8x2.8 2.4 IC 4316 SPM-CAMILA 2005 APR 21 2160 2.8x2.8 2.4 IC 4662 IRSF-SIRIUS 2006 JUN 06 4320 6.9 x 6.9 1.6 IC 5152 IRSF-SIRIUS 2006 JUN 08 4320 6.9 x 6.9 1.2 KKH 11 CFHT-IR 2004 MAR 08 600 3.2x3.2 1.2 KKH 12 CFHT-WIRCAM 2005 DEC 09 980 8.0x8.0 2.7 KKH 6 CFHT-WIRCAM 2006 JUL 11 960 6.0 x 6.0 4.8 KKH 86 IRSF-SIRIUS 2006 JUN 08 6750 3.0x3.0 2.1 KKH 98 CFHT-WIRCAM 2006 JAN 17 750 8.0 x 8.0 2.7 KKR25 SPM-CAMILA 2005 APR 21 4800 ... KKR3 CFHT-WIRCAM 2006 JUL 09 1200 8.0 x 8.0 2.8 LEDA 166065 CFHT-WIRCAM 2006 JAN 20 740 8.0 x 8.0 2.9 M81 Dwarf A CFHT-WIRCAM 2005 DEC 13 956 8.0 x 8.0 3.0 Pegasus Dwarf CFHT-WIRCAM 2006 JAN 17 965 8.0 x 8.0 2.7 Sagittarius Dwarf CFHT-WIRCAM 2006 MAY 11 960 8.0x8.0 2.9 Sextans A CTIO-ISPI 2006 JUL 14 1680 9.7x9.7 2.5 - 106- Galaxy Instrument Date (UT) tmi [s] Image Size ['] Seeing ("J (1) (2) (3) <4> (5) (6) Sextans B IRSF-SIRIUS 2005 MAY 20 990 6.9 x 6.9 2.3 UGC3817 CFHT-IR 2004 MAR 08 600 3.2x3.2 1.1 UGC 4459 CFHT-WIRCAM 2005 DEC 13 955 8.0 x 8.0 3.0 UGC 4483 CFHT-IR 2004 FEB 24 525 3.2x3.2 1.6 UGC 5423 CFHT-IR 2004 MAR 07 525 3.2x3.2 0.9 UGC 5829 CFHT-IR 2004 MAR 08 600 3.2x3.2 1.3 UGC 5918 CFHT-IR 2004 MAR 07 1200 3.2x3.2 1.1 UGC 6817 CFHT-IR 2004 MAR 08 600 3.2x3.2 1.5 UGC 7298 SPM-CAMILA 2005 APR 25 2400 2.8x2.8 3.1 UGC 7408 CFHT-IR 2004 MAR 08 600 3.2x3.2 0.6 UGC 7490 CFHT-IR 2004 MAR 08 600 3.2x3.2 1.2 UGC 7559 CFHT-IR 2004 MAR 08 600 3.2x3.2 1.9 UGC 7577 CFHT-IR 2004 MAR 07 600 3.2x3.2 1.3 UGC 7605 CFHT-WIRCAM 2006 JUL 13 1200 8.0x8.0 2.8 UGC 8201 CFHT-IR 2004 MAR 08 600 3.2x3.2 1.2 UGC 8320 CFHT-IR 2004 MAR 07 600 3.2x3.2 1.3 UGC 8331 CFHT-IR 2004 MAR 08 600 3.2x3.2 1.4 UGC 8638 SPM-CAMILA 2005 APR 25 2400 2.8x2.8 2.1 UGC 8651 CFHT-IR 2004 MAR 08 600 3.2x3.2 0.7 UGC 8833 SPM-CAMILA 2005 APR 20 2400 2.8x2.8 0.9 UGC 9240 CFHT-IR 2004 MAR 08 600 3.2x3.2 1.3 UGC A 15 IRSF-SIRIUS 2006 JUN 09 4500 3.0x3.0 2.3 UGCA 292 CFHT-WIRCAM 2006 JUL 12 960 8.0 x 8.0 2.2 UGCA319 CFHT-IR 2004 MAR 07 600 3.2x3.2 1.0 UGCA 365 CFHT-IR 2004 MAR 07 600 3.2x3.2 1.5 UGCA 438 IRSF-SIRIUS 2006 JUN 07 4500 3.0x3.0 2.4 UGCA 86 CFHT-WIRCAM 2006 JAN 18 689 8.0x8.0 1.6 UKS 1424-460 IRSF-SIRIUS 2006 JUN 06 4320 ... Notes. (1) Name of galaxy. (2) Facility and instrument with which Ks or K' or image was obtained. (3) Date (UT) on which image was obtained. (4) Total on-target integration time. (5) Dimensions (width by height) of image displayed in Figure 4.2. (6) FWHM of the seeing disk. 4.2 Ks Image Reduction Given the diversity of the facilities used in this imaging survey, great care has been taken throughout the reduction process to insure that all photometric data are self- consistent. The image reductions for all of the survey targets have been performed using methods specially designed for isolating faint galaxies from the high NIR sky level, -107- which varies significantly from site to site. The methods employed follow the recommendations of Vaduvescu and McCall (2004) and were automated with the aid of the Image Reduction and Analysis Facility (IRAF)8. The details of each stage of the reduction process are provided in the following paragraphs. The reduced galaxy images are presented in Figure 4.2. 4.2.1 Image preprocessing The first stage of the reduction process was to correct each sky and target exposure for bad pixels. Bad pixel maps were built for each run, excluding WIRCam, by comparing pairs of flat-field images taken with two different exposure times. The maps were then applied to each exposure using the IRAF fixpix task. For the WIRCam run in the 2005B semester, bad pixels were corrected using the QSO bad pixel mask that had been constructed closest in time to the dl observations. For the WIRCam run in the 2006A semester, the transmitted images had been cleaned of detector imprints and flat- fielded by the WIRCam QSO pre-processing pipeline (not available for the 2005B semester). Next, each exposure was divided by a flat field image produced on the night of the observation. The flat-fielded images retained the bias and dark current, but these instrumental signatures were readily removable during the sky-subtraction process, as indicated in §4.2.2. All flat fields were produced from a sequence of equal exposures of 8 IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation. - 108- the darkening (or brightening) twilight sky. After removal of the dark current from the twilight images, a flat field image was generated which consisted of the relative sensitivity of each pixel, quantified by the slope of the linear fit to the pixel value versus the image mean over the course of the twilight sequence. This algorithm was implemented with the MAKEFLAT IDL routine written by Olivier Lai.9 For the WIRCam run in 2005, flat fielding was implemented using the QSO flat field that had been constructed closest in time to the dl observations. For the WIRCam run in 2006, flat fielding was applied by the WIRCam QSO pre-processing pipeline, as explained above. 4.2.2 Sky subtraction As explained in §4.1, the low surface brightnesses typical of dls necessitate that the subtraction of the sky pattern from each galaxy exposure be accomplished using a sky image which closely represents the sky pattern at the time of each galaxy exposure. This has been accomplished with an IRAF script which takes a set of exposures in the form of a sky-galaxy-sky sequence and creates an interpolated sky image from the sky exposures taken before and after each galaxy exposure. This process is illustrated in Figure 4.1. The specific steps are as follows: 1. Each pair of dithered sky exposures was leveled to the average of their statistical modes, then subtracted from each other to reveal their stars and extended sources. The objects on each of the subtracted images were then masked using IRAF's objmask task, which identifies pixels that are above a user-specified sigma from a 9 CFHT Corporation, 65-1238 Mamalahoa Highway, Kamuela, HI 96743 USA, Email: [email protected] - 109- spatially-varying mean background. Care was taken to select the masking parameters which best reflected the seeing, signal-to-noise ratio and the degree of crowding. 2. For each sky image in a pair, the mask of stars and extended sources was used to replace the stars and extended sources with the underlying sky signal obtained from the other dithered, leveled sky exposure, which resulted in star-free images of the sky pattern immediately before and after the galaxy exposure. 3. The two sky images were then averaged into an image which approximated the underlying sky pattern of the galaxy exposure. Having taken sky frames with the same exposure time as the galaxy exposure, the mean sky image contained the same instrumental signatures as the galaxy image, which meant that the bias and dark current were removed from each galaxy image when the sky was subtracted. 4. Lastly, a final galaxy image was produced by aligning and combining the individual sky-subtracted galaxy images with the aid of IRAF's imalign and imcombine tasks. -110- Figure 4.1: The image reduction process akp^flaMleMedsiy I^K>1l6nB09qf mwQs Image corrected for bad taken vttHn9Qsee of sky,, mask of panto and ImM to the (poeffiw) slam and mean ofmod^sfcy,) and tawBtodtotfwiDMnof extended source* in n»M%) modeftfcy^md wodetsfcy;) •Iquffl sky, with of sftxjiaetais, and sJquKwtofs, 4 bfjjf! target imaoe taken between sky, and •k^W^ehtedand gal^sfcy^uMracted find! average of corrected far bad ptete 0**i—0®l» - Ill - Despite the effort that was made to sample the sky frequently enough and long enough to overcome rapid variability of the NIR background, it was impossible to obtain a perfect match to a galaxy's underlying sky pattern. As a result, background residuals can occasionally be seen in the sky-subtracted images presented in Figure 4.2. The residuals have been accounted for by estimating the uncertainty in a galaxy's outermost isophotes owing to background residuals, and incorporating this estimate into the uncertainties in the photometric parameters. This is described in more detail in §4.4.2. Many of the galaxy images also display some shallow negative residuals, which are a result of imperfect removal of stars from the sky images. As explained in §4.3, these negative stellar residuals were masked prior to producing a galaxy's surface brightness profile. 4.2.3 Photometric calibration Using IRAF's ccxymatch task, the stars in each galaxy field were matched with stars in the 2MASS Point Source Catalogue (PSC) with 2MASS photometric quality flags of either A or B (i.e., with photometric measurement uncertainties <0.15 mag). The number of matched stars was typically between 3 and 20. The photometric zero-point for each galaxy field was then computed from the average difference between the instrumental and 2MASS magnitudes, for which the rms was typically -0.1 mag. The colour term in the magnitude transformation equation could be dropped owing to the fact that Vaduvescu et al. (2005) found a remarkably constant J-Ks colour of 0.8-1.0 mag among 34 dls, which overlaps the mean J-Ks colour of +0.8±0.2 mag found for the -2000 2MASS stars in our survey fields. -112- Five galaxy fields (DD0120, DD0122, DD0169, DD0181, and U7559) contained fewer than three 2MASS stars. Since all of these fields were observed at CFHT on the photometric night of March 8 2004, their photometric zero-points could be estimated by interpolating, to the desired airmass and atmospheric extinction, the zero-points measured for the other galaxies observed on the same night. The zero-points were generally steady throughout each run, varying by only 0.1 mag with the CFHT-IR, 0.1 mag with the IRSF, 0.2 mag with SPM-CAMILA, 0.1 mag with the CTIO-ISPI, 0.4 mag with the NTT-SOFI and 0.2 mag with WIRCam. 4.3 Ks Surface Photometry In the NIR, dls generally display regular elliptical morphologies without significant perturbation from starburst emission (essentially by definition). The validity of adopting an elliptical isophotal model for dls has been demonstrated by Vaduvescu et al. (2005). In their NIR imaging survey of 34 dls in the Local Volume, they found that the total flux of each dl agrees closely with the flux obtained by integrating the fit to the galaxy's surface brightness profile (SBP), which was generated from elliptical isophotes of increasing size. Furthermore, their study of the dls' resolved components revealed that more than 95% of a dl's light in KS comes from stars older than ~4 Gyr. Thus, deviations from an elliptical isophotal model due to recent star formation are not significant. The isophotes of a dl can therefore be characterized by a global centroid (xo, yo), ellipticity (e) and position angle {(p), defined here as the angle of the semimajor axis eastward from north. -113- The elliptical parameters and the corresponding SBPs for the survey dls have been measured using the STSDAS IRAF ellipse task. Ellipse fits a series of elliptical isophotes along the semimajor axis of a 2D galaxy image. Prior to applying the task, many of the galaxy images were binned so that each galaxy's shape and centroid could be reliably identified. Typical binning dimensions were 4x4 pixels (0".8x0".8) for the CFHT-IR images, 4x4 pixels (l".8xl".8) for the IRSF images, 8x8 pixels (2".4x2".4) for the WIRCam images, and no binning for the SPM-CAMILA, CTIO-ISPI and NTT-SOFI images. To avoid contamination from resolved sources (either in the galaxy itself or in the foreground) and from negative stellar residuals arising from imperfect sky subtraction (see §4.2.2), IRAF's objmask task was used to produce a mask of all resolved stars, small extended sources and negative stellar residuals in the galaxy field. The ellipse task ignores the masked pixels when computing the mean intensity of each isophote. The steps involved in the measurement of a dl's elliptical parameters were as follows: 1. Initial estimates of the dl's centroid, e and (p were obtained by approximating, through visual inspection, the galaxy's elliptical extent. 2. The estimated parameters were used with the ellipse task to produce a first approximation of the galaxy's SBP, with the centroid allowed to vary freely with radius but e and 3. The variation in the centroid with radius was examined in order to estimate the coordinates which best centred the outermost isophotes, where the old stellar -114- population dominates the NIR light and therefore most clearly reveals the geometry of the entire galaxy. 4. The centroid, e and second stage, ellipse was run iteratively with a range of coordinates within ~5" of the estimate of the centroid from the outermost isophotes. In this manner, it was possible to identify the centroid for which elliptical isophotes could be identified down to intensity levels comparable to the noise of the sky background. 5. Once the final centroid of the light profile was established, the centroid was held fixed while e and (p were estimated by following the same iterative fitting process. The parameters which were found to best characterize the elliptical profile of each galaxy are presented in Table 4.2. Uncertainties in the profile centroid, ellipticity and position angle were computed by ellipse from the errors in the coefficients of each elliptical fit. These uncertainties were typically 0".7, 2% and 8° for the centroid, ellipticity and position angle, respectively. SBPs were produced by running ellipse with all four parameters fixed. In all runs of the ellipse task, the semimajor axis was sampled at a geometric rate of o,+i = l.la„ where a, is the previous position sampled. This is the minimum sampling rate recommended in the ellipse documentation for boosting the signal-to-noise ratio of the outermost isophotes while insuring that the innermost isophotes can be properly fit. The SBPs are displayed in Figure 4.2 with the accompanying galaxy images. The error bar at each surface brightness level is the square -115- root of the quadrature sum of the uncertainty computed by the ellipse task and the uncertainty arising from the sky subtraction (see §4.4.2). 4.4 Fitting of Surface Brightness Profiles in Ks Most of the SBPs exhibit a central plateau and an exponential component extending down to a surface brightness of 23-24 mag arcsec'2. This is consistent with the findings of Vaduvescu et al. (2005), who found that the shapes of the near-infrared SBPs of dls most closely follow a hyperbolic secant (sech) function. In magnitude units, this fitting function is given by: H = yUo - 2.5 log [(sech (r/r0)] Equation 4.1 where N is the surface brightness at radius r, /UQ is the central surface brightness, and ro is the scale length of the sech (i.e., the radius at which the intensity falls to 65% of the central value). At small radii, Equation 4.1 flattens out, converging to /XQ. At large radii, Equation 4.1 approaches an exponential function with scale length ro. After calibrating the SBPs generated by the ellipse task using the measured zero- points for each frame, the fit coefficients //o and ro were obtained for each galaxy using the STSDAS IRAF nfitld task with the USERPAR/FUNCTION parameter set to Equation 4.1. The sech coefficients for each dl are listed in Table 4.2. The sech fits are superimposed on the SBPs in Figure 4.2. The average rms about the sech fit is 0.11 mag, with the largest rms of 0.23 mag associated with the IRSF observation of HIPASS J1247-77. -116- Of the 68 detected dls in this survey, 18 have been identified for which their SBPs exhibit a central peak, thereby deviating from the flattening of the sech function at small radii. This is likely due to starburst activity within the galaxy core (see Vaduvescu, Richer and McCall 2006). In such cases, a better fit to the SBPs was attempted using either an exponential function or the sum of a sech and Gaussian function. For all 18 galaxies, the sum of a sech and Gaussian function resulted in a lower rms about the fit than with a sech or exponential function alone. In Figure 4.2, the plots of the SBPs for these galaxies include the fits to both the sech and Gaussian components of the sum of the two functions. 4.4.1 Radial range of the fits The isophotes included in the fit to each SBP begin at the smallest semimajor axis for which an isophote could be identified by ellipse. This innermost isophote was usually within 5" of the galaxy centroid. The outermost isophote included in the fit was the faintest isophote that could be recovered down to the background noise level of the image. In most of the SBPs, the semimajor axis corresponding to this limiting isophote was easily identifiable. However, in cases where the amplitude of the background residuals was particularly large, a definitive identification of the limiting isophote could not be made. A representative example is the SBP obtained from the NTT-SOFI image for HIP ASS J1247-77, one of the faintest dls in the Sheet survey. The sensitivity of the fitting parameters to the limiting semimajor axis was evaluated by fitting the SBP out to a range of limiting radii. When the SBP of HIPASS J1247-77 was fit out to semimajor axes -117- ranging from 50"-90", neither /uq nor the integrated sech magnitude, m$ (see §4.5), varied by more than 0.1 mag, which is within the uncertainties in these quantities. Several of the SBPs in Figure 4.2 exhibit bumps and dips, particularly in the galaxy core and at the outermost isophotes. The bumps and dips reflect background residuals and imperfect masking of resolved stars prior to generating the SBPs with the ellipse task. As can be seen in the SBP of HIP ASS J1247-77, the low surface brightness of this galaxy relative to the sky brightness makes its isophotes particularly susceptible to imperfections in the mask. To evaluate the extent to which this effects its fitting parameters, the fit to its SBP was compared to the fit after deleting the isophotes in the dips at -15", -40" and -60". The exclusion of the dips induced decreases of only 0.13 mag arcsec"2, 0".8 and 0.04 mag in pio, r<> and ms, respectively. As HIPASS J1247-77 is representative of the lowest surface brightness objects in the Sheet survey, such bumps and dips in the SBPs do not appear to be a significant source of uncertainty in the photometric parameters. 4.4.2 Uncertainties in the fit parameters The formal uncertainties in fxo and ro, as computed by nfitld, were typically found to be 0.02 mag arcsec" and 0".3, respectively. An additional and important source of uncertainty in these parameters arises from the sky subtraction process. If the sky image does not accurately match the underlying background pattern of the galaxy image, the galaxy image will be contaminated by positive or negative background residuals. If there is an excess of the latter, the profile can appear steeper at large radii, resulting in a -118- truncated value for ro- In the former case, the profile can appear flatter, resulting in a larger ro and correspondingly brighter integrated magnitude. This source of uncertainty due to sky subtraction has been accounted for using a method similar to that discussed in Cairos et al. (2003). Specifically, for each galaxy image, a constant offset ±ASky was added, which represents the upper and lower limit of 1 the background residuals. Asicy was computed from the median absolute deviation of the background estimated from a star- and galaxy-free region of each image. The resulting deviated galaxy images were used to regenerate the SBP and determine the extremes for each fit parameter. The differences between the extremes and the mean values were generally symmetric, which allowed for the adoption of half the difference between the extremes as the uncertainty due to sky subtraction. The uncertainties in no and ro that are listed in Table 4.2 were computed from the quadrature sum of the formal uncertainty computed by nfitld and the uncertainty in the sky subtraction. The uncertainty in no also includes, in quadrature, the uncertainty in the photometric zero-point measured for the galaxy frame. 4.4.3 Astrometry Table 4.2 lists the right ascension and declination of the centre of each galaxy's light profile found with the ellipse task. The coordinates are in the J2000.0 International Celestial Reference System (ICRS). The plate solution for each galaxy image was 1 The median absolute deviation (MAD) is the median of the absolute deviations from the data's median (i.e. MAD = median flx, - median(x)|)). The MAD is a more robust measure of dispersion than the standard deviation in that it is less effected by outliers. - 119- computed using the stars on the image that were matched with 2MASS stars during the calibration process (see §4.2.3). For the 5 galaxy fields with fewer than 3 2MASS stars, other stars in these fields were matched with stars in the U.S. Naval Observatory All-Sky Catalogue version B1.0 (USNO-B1; Monet et al. 2003). The plate solutions were then used to transform the centroid of each light profile from pixel to celestial coordinates. The accuracy in the astrometry is limited by the uncertainties in the catalogue coordinates of the stars and the accuracy of the ellipse centering algorithm and plate solution. According to the 2MASS PSC documentation2, 2MASS star coordinates are accurate to < 0".l over most of the magnitude range. Typical uncertainties in the adopted galaxy centroids, as reported by IRAF's ellipse task, are 0".7. The average uncertainty in the plate solutions, as reported by IRAF's ccmap task, is 0".2. Thus, the formal uncertainty in the galaxy coordinates listed in Table 4.2 is ~1 2 http://www.ipac.caltech.edu/2mass/releases/allsky/doc/sec6\_l d.html - 120- Table 4.2: Ks photometric parameters of the Local Sheet dis Galaxy <*J2000 5j2000 e Pnooo Po r0 Pc rc ms mT [h:m:s] [d:m:s] [°] [mag arcsec" ["1 [mag arcsec" ["] [mag] [mag] (1) (2) (3) (4) (5) 2J (7) (9) (10) (11) (6) (8) AM 1306-265 13:09:37 -27:08:28 0.51 -36 19.56+0.23 24.91+0.57 ... 10.70+0.27 10.80+0.43 Antlia* 10:04:04 -27:20:01 0.56 -67 21.69+0.20 35.22+3.58 ... 12.20+0.37 12.73+0.49 Cam B 04:53:07 +67:05:55 0.56 +55 21.95+0.23 26.25+1.08 21.81 3.26 13.09+0.28 13.36+0.31 Cas 1 02:06:04 +69:00:13 0.58 +89 19.48+0.09 52.80+4.86 ... 9.16+0.24 9.34+0.21 DD0 210 20:46:52 -12:50:54 0.18 -68 20.90+0.16 30.49+0.81 22.14 4.09 11.05+0.22 11.29+0.27 Dwingeloo 2 02:54:09 +59:00:16 0.76 +56 19.80+0.12 63.19+1.91 ... 9.70+0.16 10.22+0.23 ESO 269-58 13:10:33 -46:59:32 0.37 +71 19.14+0.22 36.22+3.74 18.46 18.24 9.20+0.10 9.04+0.43 ESO 321-14 12:13:49 -38:13:46 0.73 +33 21.21+0.16 34.09+1.40 21.55 6.00 12.31+0.22 12.70+0.25 ESO 324-24 13:27:39 -41:28:57 0.07 +90 20.47+0.17 35.99+0.57 10.11+0.20 10.35+0.21 ESO 325-11(a) 13:45:01 -41:51:34 0.65 -45 20.96+0.13 48.10+1.39 ... 11.04+0.18 11.32+0.25 ESO 325-11(a) 13:45:01 -41:51:34 0.65 -45 20.96+0.13 48.10+1.40 11.04+0.18 11.32+0.25 ESO 349-31(a) 00:08:14 -34:34:40 0.42 +18 21.51+0.17 21.64+0.84 12.77+0.24 13.02+0.32 ESO 349-31(a)* 00:08:14 -34:34:41 0.42 +15 21.55+0.05 18.80+0.70 13.12+0.23 13.37+0.27 ESO 379-07 11:54:43 -33:33:36 0.15 +90 22.12+0.20 18.41+1.38 21.16 1.92 13.32+0.32 13.60+0.48 ESO 381-18 12:44:43 -35:58:01 0.3 +85 20.45+0.10 9.06+0.65 13.40+0.21 13.52+0.39 ESO 381-20* 12:46:01 -33:50:17 0.68 -49 21.01+0.07 34.81+0.62 11.89+0.11 12.11+0.14 ESO 384-16 13:57:01 -35:19:58 0.08 0 19.47+0.16 10.24+0.25 20.40 4.46 11.86+0.17 12.00+0.18 ESO 444-84* 13:37:20 -28:02:37 0.11 -1 20.62+0.05 14.55+0.29 12.28+0.11 12.60+0.13 HIPASS J1247-77(a)* 12:47:32 -77:34:54 0.15 +45 21.48+0.12 8.57+0.28 14.34+0.27 14.39+0.33 HIPASS J1247-77(a)* 12:47:32 -77:34:54 0.18 +43 21.55+0.06 7.40+0.40 ...... 14.76+0.12 15.13+0.23 HIPASS J1305-40(a) 13:05:02 -40:04:58 0.32 +78 21.31+0.03 14.50+0.42 13.27+0.23 13.47+0.29 HIPASS J1305-40(a)» 13:05:02 -40:04:58 0.32 +72 21.35+0.12 10.50+0.40 14.02+0.22 14.23+0.34 HIPASS J1337-39* 13:37:25 -39:53:52 0.83 +37 21.65+0.09 9.36+0.74 16.07+0.28 16.61+0.52 HIPASS J1348-37* 13:48:32 -37:57:42 0.47 +69 20.55+0.36 3.47+0.40 15.88+0.26 16.70+0.62 HIPASS J1351-47* 13:51:21 -46:59:55 0.95 +59 21.40+0.14 29.49+1.31 21.99 3.41 14.65+0.16 15.79+0.41 Holmberg I* 09:40:27 +71:11:02 0.31 -37 22.15+0.09 63.18+1.22 10.90+0.14 11.20+0.19 IC 1574 00:43:04 -22:14:52 0.85 0 20.58+0.11 42.83+0.79 11.82+0.13 12.48+0.19 IC 3104 12:18:46 -79:43:40 0.55 +36 19.00+0.14 43.08+0.91 9.04+0.16 9.26+0.20 IC 3687 12:42:15 +38:30:15 0.4 0 20.72+0.06 33.50+1.35 21.23 12.78 11.00+0.14 11.12+0.23 IC 4182 13:05:49 +37:36:16 0.21 0 20.42+0.48 49.85+1.13 21.29 17.36 9.54+0.49 9.72+0.50 - 121- Galaxy <*J2000 5j2000 e 9nooo Mo n Pc fc ms mT [h:m:s] [d:m:s] [°] [mag arcsec" ["] [mag arcsec" ["1 [mag] [mag] (1) (2) (3) (4) (5) 2] (7) 2] (9) (10) (11) (6) (8) IC 4247 13:26:44 -30:21:47 0.65 -28 18.82+0.03 17.19+0.29 11.13+0.08 11.25+0.22 IC 4316 13:40:18 -28:53:33 0.6 +63 20.10+0.22 34.03+1.21 20.13 13.88 10.78+0.23 10.96+0.32 IC 4662 17:47:08 -64:38:32 0.27 -69 17.42+0.14 21.60+0.44 8.44+0.19 8.53+0.31 IC 5152 22:02:42 -51:17:46 0.34 -74 18.09±0.18 37.98+1.57 18.11 14.42 8.00+0.18 7.97+0.29 KKH 11 02:24:35 +56:00:38 0.45 -40 19.95+0.14 19.34+0.25 ... 11.51+0.16 11.74+0.17 KKH 12 02:27:28 +57:29:18 0.55 -15 20.17+0.14 23.50+0.69 ...... 11.53+0.17 11.79+0.24 KKH 6* 01:34:52 +52:05:35 0.1 +15 20.66+0.17 9.00+0.13 ...... 13.35+0.20 13.73+0.21 KKH 86* 13:54:33 +04:14:45 0.39 -3 21.57+0.08 14.27+0.47 ... 13.68+0.13 14.09+0.17 KKH 98* 23:45:34 +38:42:55 0.41 -5 21.44+0.15 15.95+0.27 13.35+0.20 13.82+0.24 KKR 3* 14:07:11 +35:03:35 0.05 0 22.57+0.18 14.67+0.50 22.80 2.75 14.14+0.23 14.55+0.29 LEDA 166065 02:00:10 +28:49:51 0.69 -31 21.69+0.09 19.71+0.52 22.63 2.97 13.84+0.12 14.24+0.26 M81 Dwarf A* 08:23:56 +71:01:50 0.42 -7 22.81+0.16 34.05+2.58 13.08+0.28 13.63+0.19 Pegasus Dwarf* 23:28:37 +14:44:32 0.45 -65 20.96+0.09 78.81+0.57 9.48+0.12 9.87+0.12 Sagittarius Dwarf* 19:29:59 -17:40:51 0.16 +90 21.95+0.38 36.91+1.29 11.65+0.44 12.13+0.45 Sextans A(a) 10:11:03 -04:41:01 0.05 0 21.10+0.14 52.28+4.24 9.91+0.30 10.18+0.25 Sextans A(a)* 10:11:03 -04:41:01 0.05 0 20.90+0.09 30.10+0.50 10.91+0.14 11.25+0.16 Sextans B 10:00:00 +05:19:47 0.13 +90 20.58+0.17 37.05+2.17 10.23+0.29 10.38+0.37 UGC3817* 07:22:44 +45:06:29 0.08 0 21.00+0.07 23.12+0.45 11.62+0.16 11.89+0.17 UGC 4459 08:34:08 +66:10:47 0.14 +90 22.16+0.11 27.10+0.53 ... 12.51+0.16 12.77+0.18 UGC 4483* 08:37:04 +69:46:28 0.45 -4 20.71+0.17 17.90±0.43 12.44+0.21 12.70+0.24 UGC 5423 10:05:31 +70:21:53 0.51 -38 19.61+0.19 14.82+0.79 11.88+0.24 11.99+0.39 UGC 5829 10:42:44 +34:27:04 0.36 +80 20.65+0.18 32.62+1.95 10.91+0.25 11.17+0.24 UGC 5918 10:49:37 +65:31:48 0.5 -87 21.51+0.23 38.90+1.93 22.04 11.89 11.66+0.25 11.86+0.32 UGC 6817 11:50:53 +38:52:49 0.29 +70 21.31+0.42 30.71+1.52 21.59 6.17 11.60+0.45 11.86+0.45 UGC 7298* 12:16:19 +52:13:01 0.59 -39 21.96+0.18 29.41+4.44 21.93 5.39 12.93+0.43 13.29+0.65 UGC 7408 12:21:13 +45:52:43 0.5 -77 20.59+0.13 38.06+1.73 21.05 12.58 10.79+0.16 10.77+0.31 UGC 7490 12:24:25 +70:20:05 0.22 +87 20.03+0.15 26.41+2.63 20.14 7.60 10.54+0.21 10.55+0.35 UGC 7559 12:27:05 +37:08:30 0.48 -46 20.87+0.13 34.49+0.77 ... 11.24+0.22 11.53+0.23 UGC 7577 12:27:42 +43:29:41 0.41 -68 20.24+0.25 42.45+0.65 ... 10.02+0.26 10.23+0.27 UGC 7605 12:28:38 +35:43:07 0.33 -25 20.75+0.19 19.20+0.49 ... 12.11+0.23 12.40+0.31 UGC 8201 13:06:24 +67:42:27 0.45 -89 21.10+0.09 42.28+1.81 ... 10.96+0.20 11.14+0.24 UGC 8320 13:14:27 +45:55:26 0.62 -33 20.36+0.11 36.13+0.96 10.97+0.17 11.11+0.26 UGC 8331 13:15:30 +47:29:58 0.31 +45 20.20+0.13 16.10+0.95 11.92+0.22 12.17+0.22 - 122- Galaxy * Uncertainty flag; indicates that the faintest detectable isophote is within 2.5 mag arcsec"2 of the peak surface brightness (see §4.5.1). * For galaxies observed at multiple facilities, the IRSF measurements are listed first. Notes. (1) Name of galaxy. (2-5) Centroid, ellipticity (1 - b/a) and position angle (eastward from north) of the ellipse fit For a discussion of the uncertainties in these quantities, see §4.3. (6-7) Central surface brightness and scale length of the sech fit to the SBP in Ks. For a discussion of the uncertainties in these quantities, see §4.4.2. (8-9) Central surface brightness and scale length of the Gaussian component of the profile fit (see §4.4). (10) Sech magnitude in Ks. The uncertainty is the square root of the quadrature sum of the scatter about the sech fit, the uncertainty in the frame zero point, and the uncertainty due to sky subtraction (see §4.4.2). (11) Total magnitude in The uncertainty is the square root of the quadrature sum of the measurement uncertainty in the total flux, the uncertainty in the frame zero point, the uncertainty due to sky subtraction, and the uncertainty in the asymptotic component of the total magnitude. - 123- Figure 4.2. Left panels: SBPs in Ks. In each panel, the solid line represents the sech fit to the profile. In cases where a better fit was obtained using the sum of a sech and a Gaussian function, the solid and dashed lines represent the sech and Gaussian components alone, respectively. Right panels: Ks images, with the adopted galaxy centroid at the center of each frame. (North is up, east is to the left; the image size is given in Table 4.1). Note: in the CTIO-ISPI image of ESO 381-20, the extended signal that appears to the west of the galaxy is a defect caused by the saturated star to the east of the galaxy. Ait i sm-jss <0AJ(. fet'M) AWTUA (QAK-SPX) c i i 'ft' 'nfio wnnlwwjor nxts (") nmtnugor axis f) CAMS (CTHT) CA31 (WWCAM) >•*' ft. i Z waljiKjor {") Mtnimtjor oxiit {") DD0210 (WiKCAM) mva T- i W" 'A1"' '«•" "Witr-tir Or nnlnajor «xt> (") xnisM^or *xu (") E2B9-B8 (OtSD E321-14 (IRST) *4 -- *»..i • -f,?* •»- v * V •j —T« IB »—Us—*6 * wmliiwlnr »xl» <") *xt* (*) 8324-a* (tasr) E326-U (cno) X- jr J? 1 "ifr • i6"'rtr »cmiOTA)or axis (") veixUirittjar «xii {") - 124- Figure 4.2 (continued) C3S6-U (tssr> E348-31 (ESO) M-H $ 5 M ,*V . l>" • • "m ft Mminujor «xb (") V ' Mmtnu}«r «xt» (*) t'A E340-31 (tRSF) E379-0T <|R5F) r I t J. 4. «#mifa«i»r «xn C) nralmajor ud* {*) ttmt-iB tmsn casi-ao (cno) M- C-•tg A' • Kb'' T » Msdmliv ttxbt {") nnimajnr oil (") E384-18 (fflST) *»( E444-M (CFHT) f •A ife axis (•) «? s a>mlm*Jor »xt» (•) HW*ssiiS4T-rr (£so) HIJ»AS8J1sm-7» («sr) r- 5%> t Jf f #*: * wmlmajor axis (') •rmlmcjor axis <*) mPASsJiwiodwo) HJPASSJlJOMOflBSTI •t'tit'.Mi it w Mmlnuior tub (*) wrolm«Jor »xl# (") - 125 - Figure 4.2 (continued) HtPASSJ 1337-30 (ISO) H»>*sms>4»-3T (msr) r- Jf V**F % M :U Minimajor oxi* (") wttntmajor axis (•) HIPA5SJ1351-47 (mST) HOI (VIRCAH} ' to' » **mltn*Jor »xl* (*) •MlOHMlMr *xt» <"> X15"H ORSF) CJJ04 (IRST) r* jr * >1 nmlmajw axis ("I *emim«jar axu (") 13M? (CTHT) MtBg (W1KCAM) r* t "A "&1 Sfe lis1—1*r Hmtmnjor axl* (*) MAtaoltr »xl* {") UatT (OAM-SPU) 14318 ft—~it •»••••«• "ft' "W'TW •emimajor oxii (*) v. * •emimajor axis (") t«eax (jxsft Kitsa (WSF) NnttMiw «xi« (") MmlmifM' *xn <") - 126- Figure 4.2 (continued) KKHU (crwr) KXHta (VJRCAM) r- jr £u:Af &>xV m it - A 'ft &Ss. •ulmtjor udi (H) wmlm»|or axis (") KKHC (WmCAiO KKHaa (asr? Mmlmajor «xi> (") MmiaMjOt1 «xl» {") KKK8B (W1RCAM) KXA3 (WlkCAM) A"""y w •«niina>or axis {") nemimajor «xi» {*) LEDAIMOW (WttCAM) MM DWAfirAOMRCAM> A'"ft1 •walawlw *xl* (") •amimajer «xt» (*) PEGDKS (WHtCAM) SAGDIC FMACAM) wemlroajor udi (") M*nij»«jor axis (") SEX* (cnoy 88XA <-tr 3. -a "« <*' 1 'ft TS iM li •*mUn«jor axl» (") wmim«Jor txlt {") - 127- Figure 4.2 (continued) SEXB (iwr) U38J7 (CTHT) «'* - r P r •i r 4 % as Ol R seroimajor axis (") •euilrnajor axis (") WWSKWHtCAAO U44B3 (CTHT) r ;v»,i .-CWf• r~ * '6'1 'fr ifc- A' •MttMjtr aid* (*) HMUmtlw axfc» (") uw3 nmlnijiir ndi (*) semiro*|or txis f) CW18 (CFHTJ irninarwrj r* jr •A"1 'A"' "b Mmlm4«r aid* f) walB*|w aula (") imw> (OAK-SHI) ro«qe ccrar> u r 5 sr HI 1s >»»• « a ta.fr T4r •tmlonjor udj (*) Mmteajor axis (*) •w f u?«o cmm i». U7SS8 (CTHT) ao r* |#i Jf f MM a. »] •A' • 'ft' Mnlnwiw ani* (*) •amtmajor axl* (*) - 128- Figure 4.2 (continued) fcl U7«t» <*mc«K) r „ * *" n - aambxu^or axi* ("> (CFHT) vraaofcnrn J" W • J. ° 1 * » •arotmajor txto (") •wnltaajor axt* {"> U863C (OAK-SPM) #? V? f a> $ n Momzn*)or axis (*) somimojar axil C") IWH (CFHT) LT8833 (OAN-SPM) & wmltnajor axia (") •*mlma}or Ml* (*) l»MB(CFHT> UAI5 Jf »I * «* nnliru|or axis (") StfVV semltnajor axic (") UA2BZ (VtDOUt) a*1 Mmtm«jor «xl» (") Mmimator *xi» {*> - 129- Figure 4.2 (continued) UA3SS ccrar) C !jr Kmlfflijiir axis (") mmixnajor mart* {*) UM3» (E50) UA430 flRST) ^ y PL.WJ* r* jr v\H •»mim»tor tub (") •trainiiiM1 *xi* CM> UAM (WTHCAM) f jr wntiau^or axht (") - 130- Figure 4.3. Comparison of SBPs for dls observed at multiple facilities (see §4.5.1). The solid line is the SBP obtained with the IRSF. The dotted line is the SBP obtained with the comparison facility (NTT-SOFI or CTIO-ISPI). E325-11 E349-31 $ 1 ft' ' ' '!« *-a» HIPASS J1247—77 SUB »'''A''' w (Mnlmajor ud* (") mmnrimajar uli (") KK182 SEXA 81.5 r 1 * 23.5 Tar wnrimnlw udi (") Mmlmajor udi (") UA438 iNHRtklittAjCMr rrlff (*) -131- 4.5 Integrated Ks Magnitudes Two types of integrated magnitudes have been computed to quantify the total amount of light emitted in KS. One is the sech magnitude, MS, which is computed from the integral of a dl's sech fit out to infinity; i.e.: ms = f*o- 2-5 log [11.511 ro2 (1-e)] Equation 4.2 where e is the ellipticity obtained with the ellipse task (see §4.3). In principle, ms is the best measure of the brightness of the old stellar population of a dl because it is least effected by deviations arising from recent star formation. The uncertainty in ms is computed from the quadrature sum of the scatter about the sech fit, the uncertainty in the zero-point measured for the galaxy frame, and the uncertainty due to sky subtraction (see §4.4.2). The measurements of ms and its associated uncertainty are given in Table 4.2. The average uncertainty in ms is 0.2 mag, with most of the uncertainty coming from the error in the photometric zeropoint. Also provided is the total magnitude, mj, which is computed from the flux within the outermost elliptical isophote generated by the ellipse task and extrapolated to infinity using the fit of an exponential to the outermost isophotes. Given that resolved stars are masked so that their light does not contribute to the isophotal intensities computed by ellipse, mr and should closely agree, except in those cases where starburst activity in a galaxy's core or elsewhere causes its light profile to deviate from the sech function. For the 46 profiles which have been fit to a sech function alone, the average difference between ms and mr is only 0.28±0.17 mag, which is comparable to the average - 132- measurement uncertainty in ms. The differences between ms and my do not appear to be systematic with respect to either no, ro or ms- The sech fits can therefore be considered a reliable gauge of the light profiles of the dls in this survey. 4.5.1 Comparison with different facilities As described in §4.1 and §4.2, all of the galaxies in this survey have been observed in a similar manner and images have been processed in a similar fashion to ensure that the data set is self-consistent. To confirm this, a comparison is provided of a few dls observed with multiple facilities. These objects are UGCA 15, ESO 325-11, ESO 349-31, Sextans A, UGCA 438, HIPASS J1305-40 and HIPASS J1247-77. The targets UGCA 15, ESO 349-31, HIPASS J1305-40, and HIPASS J1247-77 are among the lowest surface- brightness objects in the dl survey, having central surface brightnesses fainter than 21.5 mag arcsec'2. Figure 4.3 overplots the SBPs of the above dls for each site at which the dl was observed. For each dl, the isophotes obtained from the different sites differ by a mean value of only 0.15±0.03 mag arcsec"2 at radii where surface brightnesses are within 0.5 mag arcsec"2 of the central surface brightness of the deeper profile (i.e., within 1 sech scale length). This difference is comparable to the uncertainty in the central surface brightness, which confirms the reliability of the frame zero-points and the integrity of the subtraction of the sky background level. The mean difference between isophotes from different sites increases slightly to 0.18±0.05 mag arcsec"2 for surface brightnesses within 1.4 mag arcsec"2 of the central surface brightness (i.e., within 2 sech scale lengths). -133- Within 2.5 mag arcsec"2 of the central surface brightness (3 sech scale lengths), the mean difference between isophotes increases to 0.20±0.05 mag arcsec"2. This is within the uncertainty associated with the estimation of the sky background for these galaxies (see §4.4.2), which is the primary cause of discrepancies in the outer isophotes of the SBPs obtained from different sites. In the case of all 7 galaxies, the deepest profiles were obtained with the IRSF, where the amplitude of the sky residuals was lowest. The lower this amplitude, the fainter the outermost isophote that can be distinguished from the background residuals. With the exception of the two faint dls UGCA 15 and HIP ASS J1247-77, the IRSF profiles reach surface brightnesses that are at least 2.5 mag arcsec"2 below the peak. In comparison, the CTIO-ISPI and NTT-SOFI profiles do not reach 2 mag arcsec"2 below the peak for 5 of the 7 galaxies. In all 7 cases, the non-photometric conditions during the CTIO-ISPI and NTT-SOFI runs led to smaller scale lengths than measured from the IRSF profiles. This can be seen most dramatically in the SBPs for UGCA 15, HIPASS J1305-40 and Sextans A, where the high background residuals in the CTIO-ISPI and NTT-SOFI images have resulted in an oversubtraction of the sky. For ESO 325-11, ESO 349-31 and UGCA 438, the effect on ms is small because profiles were sampled out to large radii. However, for the other 4 galaxies, the measurements of ms from different sites is 0.5-1.0 mag fainter than measured with the IRSF. This demonstrates the importance of isophotes far-removed from the flat core in properly defining the slope of a dl's SBP outside the core. - 134- Based on the analysis above, integrated magnitudes should be treated as faint limits in cases where the faintest detectable isophote is within 2.5 mag arcsec'2 (i.e., 3 sech scale lengths) of the central surface brightness. The 23 profiles which do not meet this sensitivity criterion are indicated in Table 4.2. For the 7 dls for which images were obtained at multiple sites, the photometric parameters measured from the IRSF images were adopted for all further analyses. 4.5.2 Comparison with 2MASS photometric parameters Four dls in the Sheet survey (ESO 269-58, IC3104, IC4662, and IC 5152) have KS photometry published in the 2MASS Extended Source Catalogue. Below, the 2MASS photometric parameters are compared with the same quantities measured from the Sheet survey. Included in this comparison is the faint spiral PGC 47885, which was observed with the NTT-SOFI as part of the larger KS survey by Fingerhut et al. (2010). The 2MASS quantity km5is the integrated magnitude within a circular aperture of 5" in radius. This quantity has been measured for the five galaxies listed above using the images obtained from this survey. The measured values, which are listed in Table 4.3 are reasonably consistent with the values published by 2MASS. Again, this confirms the reliability of the sky subtraction and photometric zero-points employed in the Sheet survey. The largest difference of 0.4 mag observed for IC 3104 is most likely due to the galaxy centroid adopted by 2MASS, which differs by over 30" from the value obtained in the Sheet survey. Also, the 2MASS light profile for this galaxy spans less than 10% of the galaxy's detectable extent in our deeper IRSF image, and less than half of its sech - 135- scale length. With such limited sampling, it is unlikely that the 2MASS centroid accurately coincides with the centre of the galaxy. The 2MASS extrapolated magnitude k_m_ext is the magnitude within the isophote at 20 mag arcsec"2 corrected via extrapolation to radius rext, which is the deduced extent of the 2MASS profile. For each of the five Sheet galaxies observed by 2MASS, the SBPs produced here were used to compute integrated magnitudes out to the 2MASS value of r ext. The resulting magnitudes are significantly brighter than the 2MASS values, despite the close match in k_m_5. This indicates that the scale lengths of the profile fits obtained by 2MASS are smaller than the true profiles, and explains the significant discrepancy between the values of kmext reported by 2MASS and the measurements of mj from the Sheet survey. Table 4.3 is sorted in order of descending %r_ext, the ratio in percent of the 2MASS measurement of r ext to the limiting radius of the deeper profiles in the Sheet survey. The ratios show that the discrepancy between k m ext and mj can be significant when the 2MASS integration aperture does not extend far enough beyond a galaxy's core. -136- Table 4.3. Comparison with 2MASS Photometric Parameters Galaxy k_m_5 [mag] rext ["] k m ext [mag] Am [mag] %#• ext (1) (2) (3) (4) (5) (6) 2MASS This 2MASS This 2MASS This 2MASS This study study study study IC 3104 14.25 14.57 14 148 13.61 9.31 4.35 0.05 9 ESO 269-58 13.36 13.49 55 171 10.63 9.05 1.59 0.01 32 IC 5152 12.80 12.52 67 199 9.28 7.98 1.31 0.01 34 PGC 47885 12.47 12.40 22 52 11.74 11.66 0.08 0.00 43 IC 4662 12.82 12.76 68 123 9.51 8.54 0.98 0.01 55 Notes. (1) Name of galaxy. (2) Integrated Ks magnitude within a 5" circular aperture as measured from the 2MASS and Sheet survey images. (3) Extent of the galaxy's light profile in K, as deduced from the 2MASS and Sheet survey surface photometry. (4) Integrated Ks magnitude from the 2MASS and Sheet survey fit to the SBP out to r_ext. (5) Difference between k_m_ext and the total magnitude (mT) in Ks as deduced from the Sheet survey surface photometry. (6) Ratio in percent of the 2MASS measurement of rext to the Sheet survey measurement. 4.5.3 The dl Potential Plane Twenty-eight dls in the Local Sheet survey have measurements of TRGB distances as well as HI line widths. McCall et al. (2011) have combined the KS photometry for these objects with KS data for about -20 other dls in the Local Volume for the purpose of establishing the dl Potential Plane (PP) as a reliable distance indicator. The physical basis for the existence of the dl PP is a correlation between a galaxy's baryonic potential and the kinetic energy of its gas. As described in §2.1.2, McCall et al. (2011) employ the sech magnitude in KS as their gauge of the stellar component of the baryonic mass of a dl. McCall et al. (2011) have determined that the uncertainty in a PP distance modulus is only 0.38 mag. This is comparable to the scatter of -0.35 mag characteristic of the Tully Fisher relation for spirals (see, for example, Sakai et al. 2000). The strong correlation observed among the PP photometric parameters for the subset of Local Sheet dls -137- demonstrates that the sech magnitudes derived in this survey can be considered reliable gauges of the total stellar masses of the dls. 4.5.4 Ks magnitudes for unobserved Sheet dls Listed in Table 4.5 are the absolute Ks magnitudes (MKs) for all 133 dls in the Local Sheet sample. Also listed in this table are the sources of the Ks measurements from which MKS was derived. The sources are primarily from either the imaging survey presented in this dissertation or the related surveys by co-authors Vaduvescu et al. (2005, 2008) and McCall et al. (2011). Of the 37 Sheet dls which were not observed in the above surveys, 8 were detected by 2MASS. For these objects, the published 2MASS value of k_m_ext has been adopted. It was shown in §4.5.2 that kjnjext is unreliable if the extent of galaxy contained within the 2MASS integration aperture (r_ext) is not sufficiently large. Such galaxies can be flagged byexamination of the concentration oftheir light. The2MASS parameter kreff measures the radius at which the light in Ks drops to 50% of the peak value. Galaxies for which most of the light in Ks is contained within rext will have lower values of k_r_eff/r_ext than galaxies for which a significant fraction of their light is beyond r ext. Of the 12 dls in the Local Sheet which were observed by 2MASS, k_r_eff/r_ext ranges from 0.25 to 0.65. Of the 5 galaxies observed by 2MASS as well as by Fingerhut et al. (2010) (see Table 4.3), k_r_eff/r_ext exceeds ~0.4 for the 4 galaxies for which k m ext is over ~1 mag fainter than the values measured from the deeper profiles of Fingerhut et al. (2010). By comparison, the late-type spiral PGC 47885, which has -138- k_m_ext in agreement with the value of Fingerhut et al. (2010), has k_r_eff/r_ext = 0.25. Based on this, it can be concluded that a 2MASS measurement of k m ext can be considered reliable if kj-_eff/r_ext is approximately 0.25 or less. This applies to just 2 dls (ESO 174-01 and NGC 4449) of the 8 for which only 2MASS measurements of k_m_ext are available, as the remaining 6 dls have k_r_eff/r_ext greater than 0.39. These 6 dls have been flagged as uncertain in Table 4.5 as well as in the analyses of Chapter 5. For the 29 Sheet dls which lack published KS magnitudes, it was possible to estimate the KS magnitude from photometry in another band using colour-colour diagrams for Local Volume dls with published KS magnitudes from one of the 4 surveys listed above. The useful correlations are (F-Ks)o vs (V-I)o, (B-Ks)o vs (B-I)Q, (B-KS)Q vs {B-V)Q and (B-KS)O vs (B-mm)o- In the latter relation, mm is a galaxy's HI flux in magnitude units, which was computed via the RC3 calibration mm = -2.5 log FHI + 17.40, where FHI is the HI flux in units of Jy km s'1. The dls which were used to formulate each colour-colour correlation were required to have total asymptotic magnitudes derived from surface photometry in all bands of the correlation. All magnitudes were converted to the same extinction scale adopted in this dissertation. The colour-colour correlations are plotted in Figure 4.4. The coefficients and rms for each linear least-squares fit are listed in Table 4.4 along with the number of dls from which each fit was constructed and the number of dls with KS magnitudes derived from the fit. -139- Figure 4.4. Colour-colour diagrams found for dis with published K, photometry (hollow circles). The solid line is the linear least-squares fit. The filled circles represent the dis with Ks magnitudes derived from a known abscissa. . " 1 ' I ' 1 " M 1 ' ' | " 7 0 P c O § 3 O P P ^ « 5 oo 1 2 -i a 1 o 0 ... . i. J———i—. i . 1 . .L 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 (V-I), ——1—'—r '—r-i— <— ' 6 ' ' ' ' : o r O °o ^ O P : O 3 ^ A ° £ h o u* f : ^9 °o° § j Pi ? : &0 ° : o 1 d i o 0 0, —....J—*.,-. • -0.5 0.0 0.5 1.0 1.5 2.0 2.5 -10 -5 0 5 10 (B-V)o (B—7ngi)a Table 4.4. Colour-colour correlations found for Local Volume dis with published Ks photometry, in order of preference based on the RMS. Colour-colour #dls Zero-point Slope RMS # JT, mags derived correlation in fit from correlation (1) (2) (3) (4) (5) (6) (B-Ks)o vs (B~I)o 50 1.67+/-0.22 0.85+/-0.19 0.53 15 (V-Ks)0 vs (V-I)0 65 1.28+/-0.20 1.43+/-0.30 0.57 7 (B-Ks)q vs (B-V)0 55 2.31+/-0.12 0.67+/-0.20 0.58 5 (B-Ks)o vs (B-mH,)0 67 2.62+/-0.10 -0.02+/-0.05 0.78 2 - 140- The scatters about the three broad-band correlations in Table 4.4 are comparable, with the (B-Ks)o vs (B-I)o relation exhibiting the strongest correlation by a small margin. Ks magnitudes were preferentially estimated from (B-I)o when photometry in I was available. For dls which lacked surface photometry in B but which possessed surface photometry in V, Ks was estimated from (V-I)o. The (B-Ks)o vs (B-V)o correlation was used for the dls which lacked photometry in I but which had published total magnitudes in B and V. Not surprisingly, the weakest of the colour-colour correlations is the relationship between (B-Ks)o and (B-mm)o- Since the stellar population of a dl is typically dominated by stars older than 4 Gyr (Vaduvescu et al. 2005), a dl's B-Ks colour is an indicator of recent starburst activity, in the sense that the bluer the dl, the more young stars it has. The fraction of young stars in a dl is not directly correlated with its global amount of remaining HI gas, since the rate and magnitude of star formation activity also depend on local conditions. Despite this, the direction of the correlation for the Local Volume dls is in the expected sense; the dls that are relatively gas-poor (low (B-mni)o) have relatively higher values of (B-Ks)o, reflecting a relative lack of young stars. Regardless, a measurement of Ks from the fit to (B-Ks)o vs (B-mni)o is only a marginal improvement to deriving Ks from the mean value of (B-Ks)o, which has a standard deviation of 0.83 mag. Fortunately, the (B-Ks)0 vs (B-mm)o correlation was only called upon to estimate Ks for two dls which lacked photometry in both V and I (ESO 222-10 and UKS 1424-460). -141 - 4.6 Derived Masses of the Local Sheet dls 4.6.1 Stellar masses The total Ks magnitudes and distances amassed for the Local Sheet dls now make possible a derivation of reliable estimates of the total mass of each dl's stars. A dl's stellar mass, M*, is computed from the appropriate stellar mass-to-light ratio, % for light in Ks 5 (i.e. rK = M./Lk). From their study of over 10 galaxies from the Sloan Digital Sky Survey (SDSS), Kauffinann et al. (2003) observed that, in the dwarf regime, Tcorrelates with a galaxy's luminosity in the SDSS bandpasses g, r, i and z. Fortunately, the range of log ^in the dwarf regime appears to decrease with increasing wavelength, falling steadily from -0.7 dex in the ultraviolet g-band to ~0.3 dex in the near-infrared z-band. This suggests that %c may be even better constrained. Portinari et al. (2004) confirm this by showing, using population synthesis models, that 1* is nearly constant for (V-K)o colours that are typical of dwarfs; i.e., 1.9-2.5 mag. This range encompasses the mean (V-Ks)O of the Local Sheet sample, which is +2.2±0.7 mag. For a Salpeter IMF1, Portinari et al. (2004) find IFR - 0.6-0.9 for typical dwarf colours (see Figure 28 of Portinari et al. 2004). McCall et al. (2011) find a consistent result (%: = 0.883±0.19) by minimizing the dispersion of -50 dls about the Potential Plane, which is an observational expression of the Virial Theorem (i.e., potential versus HI line width and surface brightness). Thus, 1 The initial mass function (IMF) describes the distribution of the initial masses of stars in a star-forming event. Several independent studies have found that the IMF may be universal (see, e.g. Kroupa et al. 2001). The IMF is typically represented by the empirical Salpeter law, which refers to a power law with an exponent of -2.35 (Salpeter 1955). - 142- %c = 0.883 has been adopted throughout this dissertation as it was derived from a sample of dls using the same observable quantities derived in this study. The adopted absolute Ks magnitudes (MKS) of the Local Sheet dls and the derived stellar masses are listed in Table 4.5. In computing LKs from MKs in units of solar luminosities, the absolute Ks magnitude of the Sun was adopted to be +3.32 mag. This value is based on a solar (V-Ks)o colour of +1.505 mag (Holmberg, Flynn and Portinari 2006) and an apparent solar F-band magnitude of +4.82 mag, as adopted by Flynn et al. (2006). In Table 4.5, stellar masses are flagged as uncertain if their apparent Ks magnitude comes from an unreliable 2MASS estimate of kjn ext (§4.5.4) or if their faintest detectable isophote is within 2.5 mag arcsec'2 of the peak surface brightness (see §4.5.1). Flagged galaxies comprise 21% of the sample. 4.6.2 Gas masses The mass of a galaxy's neutral hydrogen (HI) gas is one of its fundamental evolutionary diagnostics. As the progenitor of the galaxy's stars, its presence reflects an environment in which star formation is continuing or still possible. Conversely, its absence in a galaxy is an indicator of the galaxy's infertility and subsequent aging stellar population. A galaxy's HI content can be gauged from the flux of its 21-cm emission line, which results from the hyperfine transition of neutral hydrogen. If a galaxy's distance is known, its total HI mass can be inferred from the following analytic expression (Roberts 1975, Equation 1): -143- MHI = 2.36 x 105 £F HI Equation 4.3 where d is the distance to the galaxy in Mpc and FHI is the integrated flux of the galaxy's 1 21-cm line in Jy km s" . Division of Equation 4.3 by X= MHi/Mgas, the mass fraction of hydrogen (accounting for helium), yields the galaxy's total atomic gas mass (Mgas). Mass in the form of molecular hydrogen (H2) is ignored. While H2 is expected in galaxies with continuing star formation, its tracer, carbon monoxide (CO), has been detected in only a few dls. Consequently, the proportion of hydrogen in molecular from is believed to be small. For consistency with the Potential Plane analysis by McCall et al. (2011), from which distances to several Sheet dls were computed, the value of X was adopted to be 0.735. This value is based on measurements of the rate of change of the helium and metal fractions with the oxygen abundance in dwarfs (Izotov and Thuan 2010). Since the 21-cm line was first detected in the Milky Way in 1951, it has been observed in hundreds of thousands of galaxies. Most of these observations come from surveys that employ single-dish instruments, which have the advantage of higher sensitivity and greater coverage of the u-v plane compared to higher-resolution multi-dish arrays. 85% of the sample galaxies have FHI obtained from single-dish instruments, which were favoured in cases where a galaxy has multiple measurements of FHI. Adopted values of FHI for the Local Sheet dls are tabulated in Table 4.5 along with the source of each measurement, the derived gas mass, and the ratio of the gas mass to the stellar mass (Mgas/M»). -144- Table 4.5: Absolute Ks magnitudes, HI fluxes and derived masses of the Local Sheet dis Galaxy Mgs Ks log M» log FHI FHI log Mpj Mj^M. [mag] Reference [Mo] [Jy km s"1] Reference [Mo] (1) (2) (3) (4) (5) (6) (7) (8) AM 1306-265 -16.39±0.76 1 7.83±0.32 0.833±0.042 10 7.16±0.21 0.22±0.19 AM 1321-304 -14.59±0.29 3 7.11±0.15 0.230±0.042 17 6.99±0.06 0.76±0.29 Antlia* -13.32±0.37 1 6.60±0.18 0.491±0.042 17 6.19±0.03 0.39±0.16 CGCG 269-049 -14.98±0.54 6 7.26±0.24 0.719±0.012 26 7.62±0.03 2.30±1.27 Cam A -16.68±0.64 8 7.95±0.27 1.656±0.042 17 8.31±0.08 2.30±1.51 Cam B -14.51±0.43 1 7.08±0.20 0.699±0.042 17 7.21±0.10 1.37±0.70 Cas 1 -18.85±0.47 1 8.81±0.21 1.699±0.042 17 8.25±0.12 0.28±0.16 DD0 210 -13.78±0.23 1 6.78±0.13 1.107±0.048 13 6.53±0.04 0.56±0.18 Dwingeloo 2 -18.99±0.60 1 8.87±0.26 1.489±0.007 7 8.20±0.17 0.22±0.15 ESO 174-01 -19.46±0.53 5 9.05±0.23 1.741±0.047 20 8.79±0.16 0.55±0.35 ESO 222-10 -16.11±0.94 9 7.72±0.39 0.845±0.124 20 7.82±0.18 1.28±1.26 ESO 223-09 -19.72±0.25 3 9.16±0.14 1.983±0.054 8 9.08±0.05 0.83±0.28 ESO 245-05 -18.01±0.31 3 8.47±0.16 1.941±0.042 8 8.71±0.07 1.71±0.68 ESO 269-58 -18.64±0.14 1 8.73±0.11 0.857±0.042 1 7.48±0.04 0.06±0.02 ESO 321-14 -15.22±0.23 1 7.36±0.13 0.806±0.109 20 7.31±0.08 0.89±0.32 ESO 324-24 -17.71±0.32 1 8.36±0.16 1.717±0.042 8 8.34±0.08 0.95±0.39 ESO 325-11 -16.60±0.31 1 7.91±0.16 1.405±0.063 8 7.95±0.09 1.10±0.46 ESO 349-31 -14.68±0.30 1 7.14±0.16 0.763±0.042 20 7.25±0.06 1.27±0.49 ESO 379-07 -15.18±0.37 1 7.34±0.18 0.716±0.142 20 7.61±0.12 1.86±0.91 ESO 381-18 -15.17±0.26 1 7.34±0.14 0.519±0.042 24 7.44±0.05 1.27±0.45 ESO 381-20* -16.75±0.13 1 7.97±0.11 1.504±0.050 8 8.45±0.04 3.05±0.82 ESO 384-16 -16.34±0.18 1 7.81±0.12 0.185±0.042 3 6.96±0.03 0.14±0.04 ESO 443-09 -13.99±0.55 6.87±0.24 0.000±0.042 17 7.04±0.05 1.47±0.83 ESO 444-84* -16.01±0.22 1 7.68±0.13 1.324±0.066 20 8.14±0.07 2.90±1.01 HIDEEP J1336-3321 -13.06±0.54 6.50±0.24 0.000±0.087 25 6.75±0.07 1.80±1.03 HIPASS J1247-77* -13.53±0.33 1 6.69±0.16 0.672±0.129 20 7.21±0.11 3.35±1.52 HIPASS J1305-40 -15.54±0.24 1 7.49±0.14 0.326±0.042 16 7.34±0.04 0.71±0.24 HIPASS J1321-31 -12.62±0.25 1 6.32±0.14 0.771±0.118 20 7.66±0.09 21.73±8.46 HIPASS J1337-39* -12.35±0.30 1 6.21±0.16 0.820±0.118 20 7.68±0.09 29.48± 12.31 - 145- Galaxy Mgj Ks log M* IogFra ^HI logM^ [mag] Reference [Me] [Jy km s"1] Reference [Mo] (1) (2) (3) (4) (5) (6) (7) (8) HIPASS J1348-37* -12.89±0.32 1 6.43±0.16 0.633±0.042 1 7.64±0.07 16.22±6.52 HIPASS J1351-47* -14.15±0.28 1 6.93±0.15 0.643±0.042 1 7.65±0.07 5.23±1.99 Holmberg I* -17.02±0.15 1 8.08±0.11 1.603±0.022 32 8.27±0.02 1.54±0.41 Holmberg II -20.21±0.24 3 9.36±0.14 2.340±0.022 32 8.88±0.03 0.33±0.11 Holmberg IV -17.17±0.50 4 8.14±0.22 1.515±0.042 15 8.29±0.14 1.42±0.85 Holmberg IX -15.95±0.54 6 7.65±0.24 1.380±0.042 21 7.99±0.04 2.16±1.19 IC 10 -18.65±0.23 3 8.73±0.13 2.970±0.042 15 8.28±0.03 0.35±0.11 IC 1574 -16.55±0.27 1 7.89±0.15 0.732±0.121 20 7.59±0.11 0.49±0.21 IC 1613 -17.24±0.58 8 8.17±0.25 2.468±0.042 15 7.68±0.03 0.33±0.19 IC 2574** -17.18±0.11 5 8.14±0.11 2.588±0.022 32 9.25±0.02 12.66±3.18 IC 3104 -17.86±0.22 1 8.42±0.13 1.013±0.105 20 7.22±0.09 0.06±0.02 IC 3687 -17.23±0.26 1 8.17±0.14 1.336±0.008 13 8.13±0.07 0.93±0.34 IC 4182 -18.58±0.49 1 8.71±0.22 1.846±0.021 11 8.60±0.02 0.78±0.40 IC 4247 -17.31±0.18 1 8.19±0.12 0.531±0.064 25 7.40±0.07 0.16±0.05 IC 4316 -17.24±0.33 1 8.17±0.16 0.322±0.041 25 7.03±0.07 0.07±0.03 IC 4662 -18.41±0.26 1 8.63±0.14 2.114±0.040 20 8.35±0.06 0.51±0.19 IC 5152 -18.31±0.20 1 8.60±0.13 1.988±0.042 20 8.01±0.04 0.26±0.08 KDG73 -13.57±0.55 6.70±0.24 0.000±0.042 17 6.61±0.05 0.81±0.46 KKH 11 -17.38±0.53 1 8.23*0.23 1.358±0.042 18 8.20±0.15 0.94±0.60 KKJH 12 -15.71±0.52 1 7.56±0.23 1.212±0.042 17 7.39±0.15 0.68±0.43 KKH 18 -14.94±0.61 7 7.25±0.26 0.477±0.042 17 7.22±0.07 0.94±0.60 KKH 34 -14.38±0.59 7 7.02±0.26 0.380±0.042 17 7.08±0.06 1.14±0.69 KKH 37 -14.86±0.60 7 7.22±0.26 0.204±0.042 17 6.74±0.06 0.33±0.21 KKH 5 -14.57±0.60 7 7.10±0.26 0.241±0.042 18 6.97±0.06 0.74±0.45 KKH 6* -14.63±0.27 1 7.13±0.14 0.613±0.042 17 7.26±0.06 1.36±0.49 KKH 86* -13.41±0.15 1 6.64±0.11 -0.301±0.042 17 6.04±0.04 0.25±0.07 KKH 98* -13.58±0.21 1 6.70±0.13 0.613±0.042 17 6.87±0.04 1.46±0.45 KKR25 -13.11±0.57 7 6.52±0.25 0.176±0.042 17 6.27±0.03 0.56±0.32 KKR3* -12.33±0.23 1 6.21±0.13 0.415±0.042 17 6.51±0.03 2.00±0.64 LEDA 138451 -15.32±0.65 6 7.40±0.28 0.093±0.014 27 7.08±0.11 0.47±0.33 - 146- Galaxy mKs K, log M> log FHI ^HI log [mag] Reference [Mo] [Jy km s"1] Reference [Mo] (1) (2) (3) (4) (5) (6) (7) (8) LEDA 166062 -13.00±0.65 6 6.47±0.28 0.447±0.006 27 7.43*0.11 9.06±6.29 LEDA 166063 -12.87±0.65 6 6.42±0.28 -0.143±0.018 27 6.84±0.11 2.63±1.82 LEDA 166064 -14.61±0.53 6 7.12±0.23 -0.155±0.042 17 6.80±0.04 0.48±0.26 LEDA 166065 -14.57±0.14 1 7.10±0.11 -0.022±0.042 16 6.84±0.04 0.55±0.15 LEDA 166115 -12.67±0.55 6 6.34±0.24 -0.155±0.042 17 6.62±0.06 1.91±1.09 LEDA 166142 -13.69±0.75 6 6.75±0.31 -0.155±0.042 17 6.67±0.16 0.83±0.67 LMC -23.42±0.74 8 10.64±0.31 5.874±0.009 5 8.72±0.14 0.01±0.01 Leo A -14.19±0.61 8 6.95±0.26 1.684±0.042 17 6.99±0.07 1.10±0.69 M81 Dwarf A* -14.56±0.29 1 7.09±0.15 0.613±0.022 32 7.17±0.02 1.19±0.42 NGC 1569 -18.93±0.23 4 8.85±0.13 1.924±0.022 32 8.28±0.02 0.27±0.08 NGC 2366** -16.87±0.14 5 8.02±0.11 2.367±0.022 32 8.86±0.02 6.97±1.82 NGC 2915 -18.41±0.34 3 8.64±0.17 2.161±0.042 23 8.78±0.08 1.38±0.59 NGC 3077 -20.43±0.24 3 9.44±0.14 2.408±0.022 32 9.04±0.03 0.40±0.13 NGC 3109** -16.31±0.08 5 7.80±0.10 3.060±0.037 20 8.79±0.03 9.93±2.44 NGC 3738 -18.63±0.33 4 8.72±0.16 1.342±0.017 13 8.17±0.07 0.28±0.11 NGC 3741 -15.37±0.23 4 7.42±0.14 1.718±0.042 28 8.26±0.04 6.92±2.23 NGC 4068 -17.60±0.56 6 8.31±0.25 1.551±0.010 13 8.28±0.06 0.94±0.54 NGC 4163 -16.28±0.23 4 7.79±0.13 0.982±0.014 13 7.38±0.02 0.40±0.12 NGC 4190 -16.78±0.90 4 7.98±0.37 1.365±0.021 13 7.88±0.26 0.78±0.82 NGC 4214 -19.29±0.23 3 8.99±0.13 2.301±0.022 32 8.76±0.02 0.60±0.18 NGC 4395** -18.20±0.30 5 8.55±0.16 2.519±0.003 14 9.29±0.09 5.50±2.27 NGC 4449 -20.76±0.28 5 9.57±0.15 2.420±0.022 32 9.13±0.08 0.36±0.14 NGC 4789A -13.75±0.36 4 6.77±0.17 1.914±0.022 32 7.79±0.08 10.33±4.61 NGC 5204** -18.72±0.28 5 8.76±0.15 2.057±0.034 13 8.85±0.08 1.23±0.49 NGC 5264 -18.68±0.33 4 8.75±0.16 1.107±0.081 20 7.87±0.09 0.13±0.06 NGC 5408 -17.89±0.28 2 8.43±0.15 1.816±0.044 8 8.70±0.06 1.88±0.70 NGC 625 -19.32±0.60 7 9.00±0.26 1.512±0.042 8 8.18±0.06 0.15±0.09 NGC 6822 -16.32±0.25 3 7.80±0.14 3.355±0.048 9 8.19±0.05 2.44±0.83 Pegasus Dwarf* -15.25±0.13 1 7.37±0.11 1.449±0.042 19 6.83±0.03 0.29±0.08 SMC -20.41±0.67 8 9.44±0.29 5.676±0.009 5 8.78±0.10 0.22±0.15 - 147- m m CO 00 00 t- m oo c> m m Tf o 2 * O CN r- r- m fN OiriVKSNW^ -t~~ -*3- t— -—1 «\0'HTj'rvV0Tl-t^Mirt^-t^*0NTt'V1(Nlm oo (S —< — ^j- — a O ON O VO CN r» N rt « vi m o vo vo m vo VO CO TF O ON TF ON CO M CN M •—» CN "T VO N M "T ft js "3 * ON M N OS 00 vo — r- * Uncertainty flag; indicates that the faintest detectable isophote is within 2.5 mag arcsec"2 of the peak surface brightness (see §4.5.1). ** 2MASS flag; indicates that was derived from an uncertain 2MASS measurement of k_m_ext (see §4.5.4). Notes. (1) Name of galaxy. (2) Absolute Ks magnitude and its standard error, computed from the quadrature sum of the uncertainty in ms (Table 4.2) and the uncertainty in the distance modulus (Table 2.2); in cases where the uncertainty in nts wasn't published, the mean value of 0.22 mag was adopted. (3) Source of Ks from Table 4.6. (4) Stellar mass derived from MK„ along with its standard error (see §4.6.1). (5) HI flux and its measurement error; in cases where the measurement error was not published, the mean value of 11% was adopted. (6) Source of FHI from Table 4.7. (7) Gas mass derived from Fm and its standard error (see §4.6.2). (8) The ratio of gas to stellar mass and its standard error. - 149- Table 4.6: References tor Ks photometry Num Reference 1 This dissertation (also published in Fingerhut et al. 2010) 2 McCalletal. 2011 3 Vaduvescu et al. 2008 4 Vaduvescu et al. 2005 5 2MASS 6 (B-I)o (see §4.5.4) 7 (V-I)o (see §4.5.4) 8 (B-V)o (see §4.5.4) 9 (B-mHi)o (see §4.5.4) Table 4.7: References for HI fluxes Num Reference Num Reference 1 Banks et al. 1999 18 Karachentsev et al. 2001 2 Barnes and de Blok 2004 19 Kniazev et al. 2009 3 Beaulieu et al. 2006 20 Koribalski et al. 2004 4 Begum et al. 2008 21 Lang et al. 2003 5 Briins et al. 2005 22 Longmore et al. 1982 6 Bureau and Carignan 2002 23 Meureretal. 1996 7 Burton et al. 1996 24 Meyer et al. 2004 8 Cote et al. 1997 25 Michin et al. 2003 9 de Blok and Walter 2006 26 Pustilnik and Martin 2007 10 Doyle et al. 2005 27 Saintonge et al. 2008 11 Fisher and Tully 1981 28 Schneider et al. 1992 12 Giovanelli and Haynes 1993 29 Springob et al. 2005 13 Huchtmeier and Richer 1986 30 Theureau et al. 1998 14 Huchtmeier and Seiradakis 1985 31 Tifift and Cocke 1988 15 Huchtmeier et al. 1981 32 Walter et al. 2008 16 Huchtmeier et al. 2000 33 Warren et al. 2006 17 Huchtmeier et al. 2003 - 150- 5. Analysis of the Local Sheet Riker: "They were just sucked into space." Data: "Blown, sir." Riker: "Sorry, Data." Data: "Common mistake, sir." 5.1 What We Can Learn From Our Own Backyard What deductions about the Local Sheet can be made from the appearance of its most plentiful members, the dwarf irregular galaxies? Moreover, are these deductions consistent with the theoretical predictions of sheets that emerged in Chapter 3? In that chapter, it was found that the vertical crossing times of 8 sheets extracted from an N-body ACDM simulation are over half the age of the Universe. This suggests that present-day sheets have not enough time to evolve into a state of dynamical equilibrium. However, it was also observed that the vertical density profiles and the vertical dynamics of the sheets with the shortest crossing times are in close agreement with the predictions of the exponential model for an equilibrated sheet. This may indicate that sheets are presently evolving into steady-state, exponential sheets. With the masses of Local Sheet dls now in hand, it is possible to test the validity of this hypothesis on a real-life sheet by constructing the Local Sheet's vertical density profile and by computing its scale length and crossing time. It is also possible to employ the motions of dls to constrain the amount and distribution of unseen matter required to equilibrate the Sheet. -151 - The dl properties amassed in this dissertation also make possible an investigation of whether the evolution of galaxies in a present-day sheet has been affected by the relatively crowded locale. Several independent studies have confirmed a connection between the density of a galaxy's environment and global properties such as its morphological type, luminosity, star formation rate, colour, gas fraction and surface brightness. For example, early-type galaxies (i.e., ellipticals and lenticulars) tend to be found in denser environments (Dressier 1980), as do dwarf spheroidals compared to dwarf irregulars (van den Bergh 1994). Bouchard et al. (2009) have furthered our understanding of the link between dwarf galaxies and their environment by showing that dwarfs in high-density environments of virialized groups display relatively lower star formation rates as well as lower gas fractions. On the theoretical side, Lemson and Kauffmann (1999) have demonstrated through CDM simulations that the only property of dark matter halos - the progenitors of today's galaxies - that displays an environmental dependence is the initial halo mass function, in the sense that high-mass halos are overabundant in overdense regions and underabundant in underdense regions. They find no correlation between an environment's density and the formation times, spins, concentrations and shapes of its halos. Thus, they propose that observed environmental signatures can be attributed to the environmental dependence of the halo mass function. In particular, the lower star formation rates, lower gas fractions and aging stellar populations observed in dwarfs in high-density environments can be explained by the tidal stripping of their surrounding dark matter - and hence their gas - 152- supply - as they are accreted into groups and clusters by the larger galaxies that are in higher abundance in these overdense systems. With a treasury of data on the Local Sheet dls now in our possession, we are in a position to test for the presence of a sheet-like signature of environmental trends in our own extragalactic backyard. In the following sections, the above hypotheses are addressed by first constructing the vertical distributions of various global properties of the Local Sheet dls to expose any connection with the environment's geometry. Following this, the vertical density profile and structural parameters of the Local Sheet are presented. Concluding this chapter is a derivation of the Sheet's crossing time and a comparison of the Sheet's observed surface mass density with the density derived from its dynamics. For convenience, Table 5.1 lists the ranges and medians of the observed values for all global properties considered in this chapter. Included in these tables is the statistics for the baryonic mass (Mbar), which is the sum of M» and Mgas. Also provided is the statistics for the dark matter mass (Mdark), the computation of which is discussed in §5.6. All statistics in Table 5.1 exclude the LMC, as the stellar mass of this galaxy exceeds the sample mean by over 10 -153- Table 5.1: Range and median of global properties for the Local Sheet dis, excluding the LMC and dis with uncertain masses. Property Units Range Median isolated non-isolated all dis isolated non-isolated dis dis dis dis stellar mass (M.) log MG 6.52-9.16 6.32-9.57 7.79 7.95 7.72 ±8.80 ±8.55 ±8.84 gas mass (Mg„) log M0 6.27-9.08 6.53-9.13 7.77 7.79 7.67 ±8.43 ±8.42 ±8.44 baryonic mass (MB,,) log M0 6.71-9.42 6.81-9.71 8.08 8.21 8.01 ±8.94 ±8.77 ±8.98 dark matter mass (M^) log M© 8.52-11.24 8.62-11.52 9.89 10.02 9.82 ±10.75 ±10.58 ±10.79 gas fraction (MGAS/M.) 0.15-10.37 0.06-21.83 0.90 0.95 0.86 ±2.68 ±2.43 ±2.77 absolute B mag (MB) mag (-17.44M-10.05) (-18.26M-10.06) -13.69 -13.68 -13.69 ±1.91 ±1.74 ±1.97 absolute K, mag (MKs) mag (-19.72H-13.il) (-20.76K-12.62) -16.18 -16.68 -16.11 ±1.90 ±1.77 ±1.97 (B-Ks)o colour mag (-0.06>3.45 (0.56)-4.11 2.61 2.54 2.61 ±0.68 ±0.70 ±0.68 5.2 The Distribution of Mass in the Local Sheet 5.2.1 Luminous mass In §3.4, the ACDM simulation revealed that the mass distribution of dwarf galaxies as a function of sheet altitude (|z|) appears essentially uniform. High-mass and low-mass dwarfs appear equally likely to form in a sheet's overdense midplane as in the underdense environment of its outermost vertical extent. The vertical distributions of the stellar masses (M.), gas masses (Mgas) and baryonic masses (Mbar) of Local Sheet dis are shown in Figure 5.1, Figure 5.2 and Figure 5.3. All 3 figures reveal that, within ~1 Mpc of the Sheet's midplane (where -92% of the Sheet's dis are found), the high-mass dis and their low-mass counterparts are fairly evenly distributed across the Sheet's vertical extent. -154- Table 5.2 lists the Spearman rank coefficients and their associated significance for the probability of the stellar, gaseous or baryonic masses of Sheet dls having a |z|-dependency. The statistics are provided for correlations among all Sheet dls as well as for isolated dls alone and non-isolated dls alone. The latter subsample is considered so that sheet-induced and group-induced effects can be differentiated. For consistency with the analysis of theoretical sheets in Chapter 3, the definition of an isolated dl is one that is (1) beyond 1 Mpc of any non-dwarf galaxy; and (2) lacking in companions brighter than itself within 200 kpc. 24 out of the 133 dls in the Local Sheet sample meet these criteria. Also listed in Table 5.2 is the D statistic of the Kolmogorov-Smirnov (KS) test1 for agreement between the Q vs \z\ distributions of the isolated and non-isolated dls, where Q is the global property being investigated. For the number of dls in the two datasets, the critical value of the D statistic is Da — 0.29 at the 95% significance level . Thus, the hypothesis that the isolated and non-isolated dls exhibit similar distributions can be rejected with 95% confidence if the D statistic is greater than Da. When the dls with uncertain mass estimates are excluded (see Table 4.5), Da increases to 0.31. 1 The Kolmogorov-Smirnov (KS) test, in two dimensions, measures the agreement between two two- dimensional distributions. The test output (the D statistic) represents the maximum difference between the fraction of data in one distribution and the fraction of data in the second distribution in one quadrant of the 2D-plane encompassing both datasets. E.g., a D statistic of 0.20 indicates that one distribution has, at most, 20% more data per quadrant than the second distribution. 2 For N, > 15, the critical value of the KS D statistic (Da) at the 99% significance level can be approximated as 1.22 [W+Ay/iVtiVj"2, where N, is the number of objects in the /* dataset (Press et al. 1995). -155- Table 5.2: Probability of correlations among global properties of Local Sheet dis Correlation Spearman rank" Probability by chance3 KS test1 all dis isolated non-isolated all dis isolated non-isolated dis dis dis dis including dis with uncertain mass estimates M. vs |z| -0.05 -0.18 -0.02 0.54 0.35 0.87 0.38 Mgas VS |z| -0.02 -0.12 0.03 0.82 0.53 0.76 0.27 Mbar VS |Z| -0.06 -0.15 -0.02 0.49 0.43 0.86 0.27 Mga/M. VS |z| 0.10 -0.03 0.14 0.27 0.86 0.16 0.20 (B-Ks)0 vs \z\ 0.06 0.41 -0.05 0.51 0.03 0.59 0.03 Mgas/M. vs MB 0.25 0.32 0.24 <0.01 0.09 0.01 ... Mgas/M. VS Mbar -0.16 -0.11 -0.16 0.07 0.56 0.10 (B-Ks)0 vs Mbar -0.08 -0.07 -0.09 0.35 0.74 0.39 (B-Kj)o VS Mgaj/M. -0.56 -0.29 -0.61 <0.01 0.14 <0.01 •. • excluding dis with uncertain mass estimates M. vs |z| -0.05 -0.08 -0.06 0.59 0.70 0.60 0.32 M^ VS |z| 0.05 0.01 0.08 0.63 0.98 0.51 0.34 Mbar VS \z\ -0.06 -0.03 -0.05 0.55 0.90 0.63 0.40 Mgas/M« VS |z| 0.17 -0.06 0.25 0.07 0.77 0.03 0.29 (B-Ks)0 vs |z| 0.23 0.38 -0.08 0.24 0.06 0.47 0.16 Mgaj/M. VS Mg 0.38 0.40 0.30 <0.01 0.05 <0.01 •. • Mgas/M. VS Mbar -0.30 -0.13 -0.33 <0.01 0.53 <0.01 (B-Ks)0 VS Mbar 0.03 0.20 -0.02 0.76 0.35 0.89 ... (B-Ks)o vs Mgas/M. -0.48 -0.37 -0.50 <0.01 0.07 <0.01 •«• (a) The Spearman rank coefficient and its significance are defined in §3.4. Table 5.2 reveals that neither M», Mgas nor Mbar have a statistically significant dependency on a dl's vertical distance from the Sheet's midplane, regardless of whether the Spearman rank is restricted to isolated or non-isolated dis. Table 5.2 also shows that the correlation strengths do not improve if the dis with uncertain masses are excluded. The strongest correlation is seen in the stellar masses of isolated dis, in the sense that M» decreases with |z|. However, the significance of the Spearman rank coefficient indicates that this correlation has only a 65% chance of being real. -156- Figure 5.1: Stellar masses of Sheet dis vs |z|. The dis with reliable mass estimates, as judged by the uncertainty flag in Table 4.5, are denoted by circles, while dis with uncertain mass estimates are indicated by squares. The filled symbols are isolated dis. The dotted line is the running median for all dis in bins of 0.5 Mpc. The solid line is the running median for isolated dis alone. The grey line marks the M. below which galaxies can be expected to be missing (see text below). The LMC, which has M. = 4.4 x 1010 Mo and |z| = 0.10 Mpc, is beyond the mass range of the plot. 10 o o © 9 JO ° • o O OO O ° °.o °o^ra" CD o • 1 » ' • ' * » 0.0 0.5 1.0 1.5 2.0 \z\ [Mpc] Figure 5.2: Same as Figure 5.1 but with gaseous mass on the ordinate. © o<7> cn O 0.0 0.5 1.0 1.5 2.0 \z\ [Mpc] - 157- Figure 5.3: Same as Figure 5.1 but with baryonic mass (M*+Mg„) on the ordinate. I IF I I | !I I I !I | 1I 1I I •I | II I I I ^ 9i I ,• I 81 | 0 i c % n Q> ocLr"V,~J • :QD "• % cn 7 £<*> o % O °°aBB • • 6 : : 0.0 0.5 1.0 1.5 2.0 \z\ [Mpc] In §2.5, it was shown that there could be several hundreds of undetected dwarfs within the survey region of the Local Sheet sample. It was also shown that the large majority of the undetected dwarfs are expected to be in groups. The effect of this incompleteness on the above observations can be evaluated as follows. For a galaxy at the LVC detection limit of B\\M = +17.5 mag, the brightest that such a galaxy could appear in KS can be estimated as A^.iim = BUM-[<(B-KS)O> + GBKS + where <(B-KS)O> = +2.60 mag is the median extinction-corrected (B-KS)o colour of the Local Sheet dls, -158- Milky Way. For a Sheet galaxy at the LVC detection limit of Z?ijm = +17.5 mag and located at the maximum possible distance from the Milky Way in the xy-plane (rmax), M» iim(c() can be expressed as M»jim(rmax, \z\) as follows, where rmax and z are in Mpc: 2 2 log M.Jim = logr* - 0.4(7^,|im - = Here, = 0.88 is the adopted mass-to-light ratio in Ks (see §4.6.1) and MKS.O 3.32 is the adopted absolute magnitude of the Sun in Ks (see §2.1.2). Within the survey region's = elliptical boundaries in the xy-plane, rmax 7.09 Mpc, which was computed from fmax — V[(a cos (p + xo)2+(a sin Here, a = 6.3 Mpc is the semimajor axis of the elliptical survey region in the xy-plane, >o = 0.1 Mpc are the Local Sheet coordinates of the ellipse's centroid, as given in Table 2.1. Equation 5.2 is overplotted in Figure 5.1 and delineates the region of the plots where the undetected dls would be found. When the dls with M» < M» Hm are excluded from the Spearman test for monotonicity between M» and z, the rank coefficient for the isolated dls drops to 0.04 with a reliability of 22%, while the coefficient for the non-isolated dls increases in strength to -0.31 with a reliability of 99%. The inclusion of the hundreds of undetected dls which are suspected to be in the Sheet's groups could reduce the strength of this correlation to an insignificant level. However, Figure 5.1 displays an apparent tendency of dls with M* > 109 M© to be -159- found within 0.5 Mpc of the Sheet's midplane, which would not be negated by the addition of the undetected dis. While a definitive correlation with \z\ is not presently seen for the full range of dwarf masses, the same cannot be said for all Sheet galaxies. In his survey of the brightest nearby galaxies, McCall (2011) shows that nearly all galaxies in the Sheet with Mks < -22.5 mag are found within 0.3 Mpc of the Sheet's midplane. More specifically, the mean |z| for the Sheet's 14 most luminous galaxies is 0.17 Mpc, while the mean |z| for the dis is 0.45 Mpc. The same behaviour was found for the theoretical sheets; in §3.4, it was shown that the halos more massive than dwarfs are primarily found in the innermost half of a sheet's vertical extent, where the number density of its dwarf galaxies is highest. 5.2.2 Gas fraction As discussed above, it has been observed that gas-poor dis are more likely to be found in high-density environments. The distribution of dl gas fractions as a function of |z| is shown in Figure 5.4, where gas fraction is defined here as the ratio of the gaseous to stellar mass (i.e., Mgas/M*). Unlike the scatter plots of Figure 5.1 and Figure 5.2, there is a |z|-dependent signature in the vertical distribution of the gas fractions. Figure 5.5 reveals that the range of gas fractions decreases with z, while Table 5.2 indicates that the dis with the lowest gas fractions are more likely to be found at small z. It is unlikely that the positive trend suggested by the rank coefficient is due to sample incompleteness. Figure 5.6 and Figure 5.7 show that dis with faint absolute B magnitudes (or, more fundamentally, low baryonic masses) tend to have higher gas fractions. Table -160- 5.2 reports a strong correlation between M»/Mgas and MB with a probability of less than 1% that the correlation is being observed by chance. Thus, the Sheet's undetected dis are more likely to have values of Mgas/M» above the Sheet's median value. The observed correlation is therefore more likely to be strengthened if the Sheet sample were complete down to the limiting magnitude. Table 5.2 also indicates that the strength of the correlation is due to the non-isolated dis, for which the significance of the correlation is 97% when the galaxies with uncertain masses are excluded. In comparison, the rank coefficient for the isolated dis is within ±0.1 and is less than 33% significant. However, the KS test statistics suggest that the z-distributions of the gas fractions for the isolated and non-isolated dis are in better agreement than the z-distributions of the other 3 mass quantifiers (i.e., M», Mgas and Mbar). Moreover, the hypothesis that the z-distributions of gas fractions for isolated and non-isolated dis are in agreement cannot be rejected at the 95% confidence level. It is therefore possible that the z-dependence of the gas fractions for the isolated dis would reveal itself if the number of isolated dis in the Local Sheet were larger. -161- Figure 5.4: Gas masses of Sheet dis vs the absolute value of their Local Sheet z-coordinate. The dis with reliable mass estimates, as judged by the uncertainty flag in Table 4.5, are denoted by circles, while dis with uncertain mass estimates are indicated by squares. The filled symbols are isolated dis. The dotted line is the running median for all dis in bins of 0.5 Mpc. The solid line is the running median for isolated dis alone. - • an (ft O CD cn O 0.0 0.5 1.0 1.5 2.0 [Mpc] Figure 5.5: Standard deviation of gas fractions of Sheet dis vs the absolute value of their Local Sheet z-coordinate. The dotted line is the standard deviation for all dis in running bins of 0.5 Mpc. The solid line is the running standard deviation for isolated dis alone. 5 ET I | I ' I ' I T 4 H § 3r > _ CD 2 "O T3 O ~o 1 E- c o co 0L ,.JL .1 I I » I I I « \ . I • I T 0.0 0.5 1.0 1.5 2.0 Izl [Mpc] -162- Figure 5.6: Ratio of gaseous to stellar mass versus absolute B magnitude for Sheet dis, excluding those galaxies with masses flagged as uncertain. Isolated dis are defined by filled circles. 1 i -M Mi, i 1 1 1.5 T-"T » i 7 —r r— "T r-"f "T "T" "T" "r i 1.0 $ c* o O o 0.5 ° O <$*° Figure 5.7: Same as Figure 5.6 but with baryonic mass (Mblr) on the abscissa. 1.5 7 , "J ., | , , , | -R T R IT 1.0 r * 0.5 : ° Q« S Oo^ log(Mb0() [M0] -163- 5.2.3 (B-KJ o colour The (B-KS)O colour of a dl is a gauge of the relative abundance of newborn, hot stars; the bluer the galaxy, the more abundant its newcomers compared to its aged stars. As was observed with Mgas/M*, Figure 5.8 reveals a z-dependent signature in the vertical distribution of (B-KS)Q. Table 5.2 reports a strong and significant correlation for the isolated dis, in the sense that the bluest isolated dis are found closest to the Sheet's midplane. The z-dependence of (B-KS)o for the non-isolated dis is weak and statistically insignificant. The detection of non-isolated dis at high-z which are bluer than the median suggests that the correlation observed for the isolated dis is not due to selection effects. Figure 5.9 reveals that the (B-KS)O colours are found to be strongly correlated with gas fraction in the sense that the higher the gas fraction, the bluer the dl's colour. This implies that the gas fractions of the isolated dis should exhibit a negative z-dependence in correspondence with the z-dependence observed in the (B-KS)o colours. The gas fraction trend is not observed, which suggests that the colour trend is due to an external cause. A possible culprit is the population of intergalactic dark matter predicted for the simulated sheets in Chapter 3; i.e., a population of dark matter halos which lack the mass to host galaxies (see §3.8). If these halos are concentrated along a sheet's midplane, then the flow of the isolated dis through the small-z environment may be causing tidal triggering, which could manifest itself as a rash of newborn O and B stars in the dis that are nearest to their sheet's midplane. The fact that the colour trend is not observed among the non isolated dis in the Local Sheet may indicate that the virialized galaxy groups do not currently possess a concentration of intergalactic dark matter at small z. - 164- Figure 5.8: (M,)o colours of Sheet dis vs the absolute value of their Local Sheet z-coordinate. The dis with reliable mass estimates, as judged by the uncertainty flag in Table 4.5, are denoted by circles, while dis with uncertain mass estimates are indicated by squares. The filled symbols are isolated dis. The dotted line is the running median for all dis in bins of 0.5 Mpc. The solid line is the running median for isolated dis alone. 0 :o •s ° 1 • ° ° 1=1 o i 3 <8 4 5 » 1 ' • 0.0 0.5 1.0 1.5 2.0 \z\ [Mpc] Figure 5.9: (B-Ks)0 versus gas fractions, excluding those galaxies with masses flagged as uncertain. Isolated dis are defined by filled circles. 0 1 2 ?CQ 3 4 5 -2-10 1 2 log(Mg0S/M„) - 165- Figure 5.10: (M,)o versus baryonic mass (Mb,,), excluding those galaxies with masses flagged as uncertain. Isolated dls are defined by filled circles. " I" T 111 T"r 1T " T T V 0 1 o O o 0°0 to <© •ood* • <>- o °- Ik ° %cgl° o O O ° log(Mbor) [M0] 5.3 The Density Profile of the Local Sheet The fundamental dynamical nature of the Local Sheet is embedded in its density profile. From the density profile, one can gauge the Sheet's thickness, which, in combination with the vertical velocity dispersion of its constituents, can be used to determine the Sheet's crossing time. As was shown for the theoretical sheets in Chapter 3, these quantities can provide constraints on whether a sheet is presently a dynamically equilibrated system. In the previous chapter, it was shown that the structural parameters of a dl can be determined from the surface brightness profile of the NIR light from its most numerous kinds of stars. Analogously, a density profile for the Local Sheet can be constructed from -166- its most numerous contingent, the dl population. Of course, such a profile is not representative of the total Sheet density, since by far the dominant reservoirs of mass in the Local Sheet are its minority population of spirals and ellipticals. However, the distributions and motions of the dis are tied to the gravity of all Sheet constituents. Thus, the dis offer a means of probing the state of equilibrium of the Local Sheet as well as the total amount of mass within it. In §2.4, it was shown that the number density of Local Sheet dis declines with altitude. In Figure 5.11, it can be seen that this trend translates into an exponential decline in the overdensity S(z) = p(z)/p. Here, p is the sum of stellar and gaseous mass densities (M»+Mgas) of all known dis in the Sheet and p(z) is the mass density of dis within running cylindrical slices parallel to the Sheet's midplane. As in §3.5, p(z) was computed from Equation 3.2 with Azo = 0.1 Mpc, h = 3.2 Mpc (from Table 2.1), and step = 0.15 for the sampling rate along the z-axis (i.e., Az,+/ = Az, (1+step)). Excluded from p(z) and p is the unknown mass contribution from the dark matter halos of the dis. However, if the relative proportion of dark matter mass in dis does not vary significantly with z, then <5(z) should reflect the trend for dark matter halos. For consistency with the dwarf density profiles of the theoretical sheets (§3.5), the maximum baryonic mass of a dl has been capped at 1.0 x 109 Mo (see §3.4). This limit excludes only 14 out of 133 dis (i.e., 11% of the Local Sheet sample). Figure 5.11 reveals that the mass attributed to the Sheet dis is concentrated within -0.5 Mpc of the Sheet's midplane, beyond which the mass from dis becomes underdense -167- (i.e., <5 < 1). As for the theoretical sheets, the density profile was fit to a sech2 (Equation 3.4), a sech function (Equation 3.8), and an exponential function (Equation 3.3). The fits are plotted in Figure 5.11 and their parameters are listed in Table 5.3. The structural parameters of the Local Sheet are found to be comparable to those of the simulated sheets (see §3.5). The best empirical fit to the dl overdensity profile is obtained with the exponential function. Table 5.3: Parameters of fits to the vertical overdensity profile of the Local Sheet Sample 8a Zo [Mpc] RMS (1) (2) (3) (4) \ 4 * sech sech exg sech sech exp sech sech exp Local Sheet 3.23±0.38 3.33±0.24 3.64±0.38 0.38±0.05 0.24±0.05 0.38±0.04 0.40 0.37 0.27 MCI 3.22±0.30 3.32±0.29 3.66±0.24 0.40±0.05 0.25±0.04 0.39±0.04 0.35 0.32 0.24 MC 2 2.93±0.33 3.02±0.32 3.35±0.24 0.41±0.06 0.26±0.04 0.40±0.05 0.40 0.36 0.25 Notes. (1) Galaxy sample used to produce the vertical overdensity profile (see text below for details of the MC samples). (2) Central overdensity of the fit. (3) Scale length of the fit. (4) root-mean-square deviation of the fit. -168- Figure 5.11: Vertical overdensity profile of the Local Sheet dls in running bins of geometrically-incremented size. Overplotted is the exponential fit (solid line), the sech fit (dotted line) and the sech2 fit (dashed line). 4 3 ^ 2 1 0 0.0 0.5 1.0 1.5 2.0 \z\ [Mpc] 5.3.1 Effects of sample incompleteness In §2.5, the completeness of the Local Sheet sample was evaluated. It was concluded that the sample is missing over 750 dwarfs in the range -14 45 dwarfs at MB = -13 mag, 179 dwarfs at MB = -11 mag and 532 dwarfs at MB = -9 mag. -169- As in §3.4, these magnitudes were converted into baryonic masses using M*/LB = 1.2 and MHIILB = 0.99 MQ/LQ from Read and Trentham (2005) and MHi/Mgas = 0.735 (see §4.6.2). It was shown in §5.2.1 that the baryonic masses of Sheet dls do not exhibit a significant dependence on their Local Sheet z-coordinate. The vertical distribution of the undetected dwarfs is therefore unknown. The distribution scenario first considered (MC 1) was that which would lead to the maximum thinning of the vertical density profile of the Local Sheet's dwarfs; namely, the extreme case where all of the undetected dls are contained within |z| = 0.5 Mpc. After 10 iterations, this simulation converged on the fit coefficients listed in Table 5.3. The second scenario considered (MC 2) was that where the undetected dwarfs are distributed randomly throughout the Sheet's full ~4Mpc vertical extent. Again, convergence of the fit coefficients was rapid (after 20 iterations), and the results for this simulation are given in Table 5.3. Given that the most likely vertical distribution of the missing dwarfs is somewhere in between the two simulated scenarios, the values of So and z0 resulting from the simulations are approximations of the upper and lower limits for these quantities. For all 3 models, the upper and lower limits are consistent, within errors, with the values obtained from the fit to the Local Sheet's detected dls alone. The effect of sample incompleteness on the Sheet's vertical density profile is therefore negligible. Table 5.1 shows that the minimum dark matter mass found for the Local Sheet dls (4.2 x 108 Mo) is -0.5 dex lower than the minimum halo extraction limit for the CDM sheets studied in Chapter 3. However, only 21% of the Local Sheet dls have Mdark below -170- the halo extraction limit, and all such dls have MB > -14 mag. Thus, having found above that the low-mass dwarfs appear to have a negligible effect on a sheet's density profile, it can be assumed that the central densities and scale lengths of the CDM sheets would not be significantly altered if the halo extraction limit matched the minimum halo mass found for the Local Sheet dls. 5.4 Peculiar Motions in the Local Sheet The motion of galaxies is the net result of the cosmological expansion and the local gravitational potential. The dispersion of the latter component, the peculiar motion, is a gauge of the strength of the local potential field; a strong field induces a larger range of peculiar motions, and vice versa. As explained in §3.6, the dispersion of peculiar motions ( Furthermore, the widespread distribution of the dls within the Sheet allows them to trace out the Sheet's potential over a larger spatial extent. -171 - The only visible component of a galaxy's motion through space is its radial heliocentric motion (v©), which is usually measured from the Doppler shift seen in the 21-cm emission line emitted by the galaxy's neutral hydrogen gas. A measurement of v© contains the Milky Way's motion relative to the other members of the Local Group, which must be removed from v© in order to obtain the motion that would be observed from within a rest frame common to all members of the Sheet. The chosen rest frame is the barycentre of the Local Group, the transformation to which is obtained using the standard equation VLG = v© + va [(cos b cos ba cos (/ - la) + sin b sin ba] Equation 5.4 where VLG is a galaxy's motion relative to the Local Group barycentre, / and b are the galaxy's Galactic longitude and latitude (respectively), and va is the Sun's motion toward the solar apex given by Galactic longitude la and Galactic latitude ba. The solar apex vector adopted in this dissertation is from Karachentsev and Makarov (1996), who found va = 316km/s, la = 93°, and ba = -4°. This is the same apex vector employed by the NASA/IPAC Extragalactic Database (NED). A galaxy's radial motion also contains, in addition to its peculiar motion, a component of motion due to the expansion of the Universe. According to Hubble's Law, this motion can be removed by simply subtracting Hod; i.e. vpee = VLG - Hod, where vpec is the radial component of the galaxy's peculiar motion, Ho = 74.2 km s"1 Mpc"1 is Hubble's Constant (see §2.1.2) and d is the galaxy's distance. The radial peculiar motions of the Sheet dls are tabulated below. - 172- Table 5.4: Kinematical properties of the Local Sheet dis 1 1 Galaxy Isolated / [°] M°] VLG [km s" ] Vp« [km s" ] (1) (2) (3) (4) (5) (6) AM 1306-265 307.90 35.56 457 266±64 AM 1321-304 311.26 31.36 264 -50±32 Antlia 263.10 22.31 66 -26±4 CGCG 269-049 135.34 63.87 241 -130±26 Cam A 137.25 16.20 164 -112±35 Cam B 143.38 14.42 266 29±37 Cas 1 129.56 7.09 283 35±47 DD0 210 34.05 -31.34 9 -58±3 Dwingeloo 2 138.16 -0.19 316 17±80 ESO 174-01 311.41 8.59 440 0±108 ESO 222-10 319.70 10.05 405 0±100 ESO 223-09 * 324.12 9.17 389 -74±32 ESO 245-05 * 273.08 -70.29 305 -12±35 ESO 269-58 306.32 15.76 140 -128±17 ESO 321-14 294.85 24.05 333 99±15 ESO 324-24 310.17 20.88 273 5±34 ESO 325-11 313.50 19.91 312 65±31 ESO 349-31 351.48 -78.12 235 6±22 ESO 379-07 289.59 27.84 363 -4±36 ESO 381-18 301.40 26.89 366 -13±32 ESO 381-20 301.64 29.02 336 -55±22 ESO 384-16 317.73 25.65 349 28±18 ESO 443-09 303.86 34.52 409 -22±33 ESO 444-84 315.12 33.74 380 44±34 HIDEEP J1336-3321 313.76 28.55 370 59±22 HIPASS J1247-77 302.71 -14.70 153 -90±23 HIPASS J1305-40 305.75 22.71 363 -57±27 HIPASS J1321-31 310.31 30.92 344 -18±27 HIPASS J1337-39 312.43 22.13 258 -95±25 HIPASS J1348-37 315.22 23.52 358 -58±43 HIPASS J1351-47 313.41 14.68 291 -127±48 Holmberg I 140.73 38.66 290 8±15 Holmberg II 144.28 32.69 296 51±16 Holmberg IV 103.70 60.80 275 -46±68 Holmberg IX 141.98 41.06 187 -75±17 IC 10 118.96 -3.33 -63 -123±3 IC 1574 101.20 -84.76 411 60±43 IC 1613 129.74 -60.58 -90 -143±3 IC 2574 140.21 43.61 196 -83±14 IC 3104 301.41 -16.95 170 4±15 IC 3687 131.95 78.46 381 53±37 IC 4182 107.70 79.09 357 44±18 IC 4247 311.90 31.89 54 -303±31 IC 4316 315.66 32.77 467 170±36 IC 4662 328.55 -17.85 139 -32±16 IC 5152 343.92 -50.19 72 -61±8 -173- 1 1 Galaxy Isolated /[°] b[°) vLG [km s' ] Vp« [km s" ] (1) (2) (3) (4) (5) (6) KDG73 136.88 44.23 263 0±22 KKH 11 * 135.74 -4.53 542 196±82 KKH 12 ... 135.58 -3.01 302 141±37 KKH 18 * 152.03 -21.63 374 64±36 KKH 34 140.42 22.35 298 5±26 KKH 37 ... 133.98 26.54 55 -188±23 KKH 5 • 125.49 -11.35 326 21 ±29 KKH 6 ... 129.68 -10.21 305 29±27 KKH 86 * 339.04 62.60 208 14±12 KKH 98 • 109.09 -22.38 151 -24±10 KKR 25 * 83.88 44.41 67 -77±7 KKR 3 * 63.71 71.99 126 -19±8 LEDA 138451 136.88 -35.12 555 148±75 LEDA 166062 137.25 -34.10 622 215±75 LEDA 166063 137.91 -34.46 562 155±75 LEDA 166064 139.75 -32.82 400 8±23 LEDA 166065 140.63 -31.67 360 6±22 LEDA 166115 156.85 68.98 241 -79±28 LEDA 166142 127.67 73.40 345 7±83 LMC 280.47 -32.89 27 24±0 Leo A 196.90 52.42 -39 -98±6 M81 Dwarf A 143.82 33.01 268 18±14 NGC 1569 143.68 11.24 87 -109±10 NGC 2366 146.42 28.54 234 2±12 NGC 2915 291.97 -18.36 191 -74±33 NGC 3077 141.90 41.66 154 -118±18 NGC 3109 262.10 23.07 109 13±5 NGC 3738 * 144.55 59.31 310 -29±41 NGC 3741 * 157.57 66.45 262 17±15 NGC 4068 * 138.92 63.04 289 -14±30 NGC 4163 163.20 77.70 166 -41±11 NGC 4190 160.62 77.59 232 -3±95 NGC 4214 160.25 78.07 294 71±12 NGC 4395 162.09 81.53 313 -5±46 NGC 4449 136.85 72.40 254 -41±40 NGC 4789A 35.16 89.41 353 240±17 NGC 5204 113.50 58.01 338 10±44 NGC 5264 315.72 31.71 269 -46±38 NGC 5408 317.15 19.50 285 -78±34 NGC 625 273.68 -73.12 325 43±27 NGC 6822 25.35 -18.39 63 29±2 Pegasus Dwarf 94.78 -43.55 60 -4±3 SMC 302.80 -44.30 -22 -27±0 Sagittarius Dwarf 21.06 -16.28 20 -54±3 Sextans A 246.15 39.88 94 -11±5 Sextans B 233.20 43.78 109 4±5 UGC3817 172.93 24.11 478 208±60 UGC 4459 149.30 34.95 150 -110±13 - 174- Galaxy Isolated /[°] b[°] VLG [km s'1] Vp« [km s"1] (1) (2) (3) (4) (5) (6) UGC 4483 144.97 34.38 303 42±15 UGC 5423 * 140.03 40.81 495 34±135 UGC 5829 • 190.07 61.53 591 309±55 UGC 5918 * 140.90 47.12 467 73±105 UGC 6456 ... 127.84 37.33 89 -231±20 UGC 6541 * 151.90 63.28 303 26±38 UGC 6817 * 166.20 72.75 247 52±12 UGC 685 * 128.43 -46.02 351 14±18 UGC 7242 ... 128.87 50.60 213 -166±37 UGC 7298 135.21 64.06 255 -45±26 UGC 7408 * 138.75 70.38 515 246±67 UGC 7490 126.24 46.62 630 283±97 UGC 7559 ... 148.60 78.74 231 -119±42 UGC 7577 • 137.76 72.94 239 50±11 UGC 7605 ... 151.00 80.14 316 -3±43 UGC 8091 * 310.74 76.98 136 -7±8 UGC 8201 ... 120.75 49.36 195 -136±34 UGC 8215 ... 114.58 70.03 297 -24±31 UGC 8308 ... 111.63 70.32 242 -55±37 UGC 8320 ... 110.77 70.66 270 -40±39 UGC 8331 ... 111.49 69.09 346 60±59 UGC 8508 * 111.14 61.31 186 2±12 UGC 8638 * 23.27 78.99 273 -31±30 UGC 8651 ... 89.73 73.12 272 42±13 UGC 8760 ... 77.79 73.45 257 24±13 UGC 8833 ... 69.71 73.96 285 61±12 UGC 9128 * 25.57 70.46 172 11±9 UGC 9240 * 82.01 64.48 263 61±11 UGCA 105 ... 148.52 13.66 279 49±25 UGC A 15 ... 119.39 -83.88 347 109±19 UGCA 276 ... 161.10 78.06 286 58±14 UGCA 292 148.28 83.72 305 34±15 UGCA 319 * 306.62 45.56 555 363±41 UGCA 365 ... 314.61 32.61 362 -13±27 UGCA 438 ... 11.87 -70.86 98 -61±8 UGCA 86 ... 139.76 10.65 275 95±18 UGCA 92 144.71 10.52 89 -144±22 UKS 1424-460 ... 319.83 13.38 175 -84±27 WLM 75.86 -73.62 -15 -84±3 Notes. (1) Name of galaxy. (2) Isolated galaxy indicator; an indicates that this galaxy is isolated according to the criteria outlined in §5.2.1. (3-4) Galactic longitude and latitude from NED. (5) Radial velocity relative to the rest frame of the Local Group, computed via Equation 5.4 with v© from Table 2.2. (6) Radial component of the peculiar motion (i.e. Vpcc = vLG - Hod) and its standard error, which includes the uncertainty in the distance, the 5% uncertainty in H0 and a 2% uncertainty in vG as in Karachentsev et al (2003b). - 175- 5.4.1 The vertical velocity dispersion As stated above, the z-component of the dispersion in the peculiar motion of the Local Sheet is not a directly observable quantity. Fortunately, it was shown in §3.6.2 that az is linearly correlated with the line-of-sight component of a sheet's peculiar motion, ar. The slope of the correlation depends on a sheet's viewing angle as well as on its vertical density profile. For a viewer stationed at a sheet's approximate centroid, = or (1.06±0.18) az Equation 5.5 From their study of field galaxies in the vicinity of the Local Group, Karachentsev et 1 al. (2003b) found ar = 25-30 km s' after correcting for a 10% uncertainty in the galaxies' distance estimates and a 2% uncertainty in the measured radial velocities. In corroboration with this result, Karachentsev (2005) found that the centroids of the 9 nearest galaxy groups have A measurement of ar for the Local Sheet is computed here by restricting the mass tracers to the Sheet's isolated dis which have TRGB distances. Reliable distances insure accurate removal of the Hubble flow from the galaxies' observed radial motions. Based on this subset of 21 dis, the radial dispersion is 39.8 km s"1. The component of this quantity due to uncertainties in the distance measurements can be removed by quadrature -176- subtraction of Atr2 = (HQ2 N~l) Z(Ad2), where Ad, is the uncertainty in the distance to the z'th galaxy and the summation is over the N= 21 dls. This quantity evaluates to 1 1 Aa = 19.6 km s" , which in turn yields ar = 34.6 ±7.6 km s" , where the quoted uncertainty is the standard error of the mean. Employing Equation 5.5, the z-component of the 1 Sheet's peculiar motion evaluates to oz = 32.7±9.0 km s" , where the uncertainty is the standard error in Equation 5.5. 5.5 The Crossing Time of the Local Sheet In §5.3 and §5.4.1, the vertical scale length (zo) and vertical velocity dispersion ( (w). The Sheet's crossing time is therefore an indicator of whether its galaxies possess sufficient motion in the z-direction to have virialized the Sheet. As in the computation of tc for the CDM sheets (§3.7), the crossing time of the Local Sheet was computed by approximating w as ze, where ze is the vertical length at which a 2 sheet's density drops to Me. For an exponential, sech and sech density profile, ze equals zo, 1.657zo and 1.085zo, respectively. The possible values of the Local Sheet's crossing time are therefore 22.5±6.7 Gyr, 23.8±7.3 Gyr and 24.7±7.6 Gyr for the exponential, sech and sech2 models, respectively. As for the CDM sheets, the most reliable estimate of the Sheet's crossing time is the value computed with zo from the exponential fit to the vertical density profile, given that - 177- the exponential fit yielded the lowest relative uncertainty for zo in comparison to the relative uncertainties associated with the sech and sech fits. Although the uncertainty in this estimate of /c is considerable (30%), its minimum value is over half the current estimate of the age of the Universe, as was found for the CDM sheets. This suggests that the Local Sheet does not possess enough vertical peculiar motion to have achieved a state of dynamical equilibrium in the time elapsed since the Big Bang. In §3.7.1, it was proposed that sheets are evolving into equilibrated, exponential systems. The observations which support this are (1) the trend with crossing time observed for azs/^z,tut> which is a gauge of the depression of a sheet's vertical velocity dispersion at small z; and (2) the trend with crossing time observed for Sheet. However, the observables required to test the 2nd observation are at hand. For the Local Sheet, CDM sheets; namely, that sheets with long crossing times have weaker exponential peaks in their vertical density profiles. While the mechanism responsible for the exponential peak is clearly at work in the Local Sheet, as judged by its vertical density profile, the vertical motions of the Sheet's dwarfs are not large enough for the exponential peak to -178- have attained the strength seen in the CDM sheets with crossing times less than the Universe's age. 5.6 The Surface Mass Density of the Local Sheet With the central overdensity (<5o) and scale length (zo) of the Local Sheet's dwarfs in hand, it is now possible to estimate their surface density, E, which is the total mass of the sheet divided by its surface area in the xy-plane. As stated in §3.8, integration of the sech and exponential fits to the vertical density profiles over all z yields 2f,t = 2poz0. The sech model is not considered here, as the sech2 and exponential models lead to the limiting values of 2flt. The derivation of po from do requires the total mass density (p), since S(z) =p(z)/p. The total mass of the Sheet's dwarfs (Mm,dW) which was employed in the computation of po is listed in Table 5.5 along with its two components; namely, the total baryonic mass of the Sheet's dls (Mbar.dw) and the total dark matter mass (Mdark,dw). The dark matter = masses of the Sheet's dls were computed by adopting £2b,gai 0.0035 and Qc = 0.243, which are the same density parameters that were employed for the CDM sheets. To ensure that the derived surface density of the Local Sheet's dwarfs is directly comparable with the surface densities found for the CDM sheets, the mass summation was restricted to galaxies with Mdark within the mass range of the dwarf halos in the CDM 9 sheets. The lower limit on Mda* is therefore 2 x 10 M0 (the halo extraction limit) and the upper limit is 6.9 x 1010 M© (the adopted maximum mass for dwarf halos established in §3.4). These mass limits encompass 70% of the Local Sheet sample. -179- Also provided in Table 5.5 are the total masses predicted by the completeness analysis of §5.3.1. This amounts to an additional 45 galaxies with baryonic mass 6.3 x 10 MQ. The undetected galaxies with MB > -13 mag are not considered here as their dark matter masses are below the halo extraction limit of the CDM simulation. The surface mass densities of the various sources of mass in the Local Sheet are tabulated in Table 5.6, followed by the surface mass densities which incorporate the 45 undetected galaxies predicted by the completeness analysis. The second column in Table 5.6 is the surface mass density of baryons in the Sheet's dwarf galaxies (2bar,dw)- The surface density in the third column, Zi^dw, includes the baryonic mass from the Sheet's dwarfs as well as the mass from their dark matter halos, which was computed with Qc and Qb.gai as above. The third column, 2fitidw> is the surface mass density derived from the sech and exponential fits to the vertical density profile for the Sheet's dwarfs. Included in 2flt,dw is the surface mass density of the galaxies' dark matter halos. The surface density predicted by the exponential model is consistent with 2"m dw within errors, which indicates that the exponential fit to the Sheet's vertical density profile yields the most reliable estimate of the total mass of the Sheet's dwarfs. The fifth column of Table 5.6,is the surface mass density of baryonic and dark matter associated with both the dwarf and non-dwarf galaxies in the Local Sheet. This quantity was estimated by dividing RM,DW by = 0.18±0.06. The latter quantity is the mean ratio of the surface mass density of dwarf halos to the surface density of all halos found for the theoretical sheets (see Table 3.1). -180- The sixth column of Table 5.6 contains the dynamical surface densities (2dyn) predicted by the isothermal and exponential model based on the Sheet's vertical velocity dispersion (Equation 3.12 and Equation 3.14). The latter equation requires a measurement of the vertical velocity dispersion at z - 0. While this quantity cannot be observed from our observation point within the Local Sheet, it can be approximated from the mean ratio of ov o to found for the two CDM sheets (#2 and #3) with similar crossing times as the Local Sheet, given the trend with crossing time that was observed for this ratio in §3.7.1. Figure 3.7 shows that the cz-profiles for these two sheets are flat enough that (Xj fiat ~ az, where az is the observed vertical velocity dispersion. Therefore, oz$loz for the Local Sheet can be approximated as 0.740±0.004, which is the weighted mean of (fzfiloz,flat for CDM sheets #2 and #3. This yields Sheet. Following 2dyn in Table 5.6 is ZmtoJZiyn, which is the surface density of the total baryonic and dark matter in all of the Local Sheet's galaxies relative to the dynamical surface densities predicted by the isothermal and exponential model. The fractional amount of missing mass required to equilibrate the Local Sheet is therefore 1 - •Tm.tot/'^dyn- Unlike in the theoretical sheets, for which all motion is due solely to cold dark matter, the missing mass of the Local Sheet contains some fraction of baryonic matter, the most likely candidate of which is the warm-hot intergalactic medium theorized by Cen and Ostriker (1999) and observed by Nicastro et al. (2003). The mass density of the undetected intergalactic baryonic matter (2bar,igm) can be evaluated from -181- •^bar.igm [•^bar.dw C^dw/^tot) ] (^b/^b,gai— 1) Equation 5.6 Here, Zbar.dw is the baryonic surface density of dwarfs as listed in Table 5.6, and Zdw/^tot =0.18±0.06 as above. Thus, the first factor in Equation 5.6 is the baryonic surface density of both dwarf and non-dwarf galaxies in the Local Sheet. Multiplication of this factor by CV^b,gai =13.0, which is the ratio of baryonic matter to baryonic matter trapped in galaxies (see §3.2 and §3.2.1), yields the Sheet's total baryonic surface density. Thus, 2bar,igm is the difference between the Sheet's total baryonic surface density and the surface density of the baryons which are trapped in the Sheet's galaxies. The surface density of intergalactic dark matter required to equilibrate the sheet is then •^dark.igm — ^dyn — ^ni.tot " -^bar.igm Equation 5.7 Equation 5.7 lists -Tdark,ign/£dyn for the isothermal and exponential models. The ratio is -80% for the former and -40% for the latter. It can therefore be concluded that if the Local Sheet is an equilibrated, isothermal system, it requires -5 times more mass in the form of intergalactic dark matter. Alternatively, if the Local Sheet is an equilibrated system with a vertical density profile characterized by an exponential decline, then only -1.5 times more mass is required. Both conclusions agree with the observations of the theoretical sheets; namely, that the isothermal model requires 2.5 to 7 times more mass for dynamical equilibrium, while the exponential model requires 1.5 to 4 times more mass. In §3.8, it was shown that the mass requirements for the exponential model can be met by the simulation's unextracted dark matter, which was estimated to amount to approximately twice as much mass as the extracted halos. It can additionally be -182- concluded here, as in §3.8, that the dark matter required for equilibration must pervade the Local Sheet as opposed to being confined to the Sheet's galaxy groups, since 2dyn was measured from the motion of the Sheet's isolated dwarfs. Table 5.5: Mass and volume density parameters of the Local Sheet Sample log M^>ar,dw log IVXdirk,dw log l0g/>m,dw 2 2 2 2 [M© Mpc" ] {M© Mpc ] [M0 Mpc ] [M© Mpc ] (1) (2) (3) (4) (5) Local Sheet 10.38±0.02 12.19±0.02 12.19±0.04 9.63±0.02 MC 1 & 2 10.40±0.02 12.21±0.02 12.22±0.02 9.66±0.02 Notes. (1) Galaxy sample used to produce the vertical overdensity profile (see §5.3.1 for details of the MC samples). (2) Total baryonic mass of the dwarfs (dls only for the Local Sheet sample), after restricting the dwarf mass range to the same mass range of the dwarf halos in the CDM sheets (see text above). The quoted error is from the uncertainties in the individual baryonic mass estimates. (3) Total mass of the dark matter dwarf halos, employing nb,(*i = 0.0035 and fic = 0.243 as in §3.2. (4) Total mass of the dwarfs, including baryonic and dark matter. (5) Volume mass density of the dwarfs (i.e., Mm>dw divided by nabh, with a, b and h from Table 2.1). - 183- Table 5.6: Surface mass density of the Local Sheet Sample log iTb.r,dw log log ^m,tot^dyn ^d*rlMgm^dyn 2 2 2 2 2 2 2 [Me Mpc" ] [Me Mpc" ] [Me Mpc' ] [Me Mpc ] [MeMpc ] [M©Mpc" ] [Me Mpc ] ID (2) (3) (4) J5) (6) (7) (8) sech exg sech2 exp sech exp sech exp Local Sheet 8.32±0.02 10.14±0.02 10.02±0.08 10.07±0.06 10.89±0.03 11.32±0.25 11.06±0.25 0.38±0.20 0.68±0.35 0.82 0.46 MCI 8.35±0.02 10.17±0.02 10.05±0.07 10.09±0.06 10.92±0.03 11.30±0.25 11.05±0.24 0.42±0.19 0.75±0.33 0.77 0.38 MC 2 8.35±0.02 10.17±0.02 10.02±0.08 10.06±0.07 10.92±0.03 11.29±0.25 11.04±0.25 0.43±0.20 0.77±0.35 0.76 0.35 Notes. (1) Galaxy sample used to produce the vertical overdensity profile (see §5.3.1 for details of the MC samples). (2) Baryonic surface density of the Local Sheet dwarfs (i.e., Mbir,dw from Table 5.5 divided by nab, with a and b from Table 2.1). (3) Surface density of the Sheet's dwarfs including baryonic and dark matter, computed from Mm>dw as in Table 5.5). (4) Surface density of baryonic and dark matter derived from integration of the sech2 and exponential fits to the vertical density profile for the Local Sheet's dls. (5) Total surface density of all galaxies in the Local Sheet including baryonic and dark matter (i.e., derived from the mean ZAJZtot found for the CDM sheets; see text above). (6) Surface density required for an isothermal or exponential sheet with vertical velocity dispersion - 184- 6. Conclusions Q: "Humans. I'd have thought by now you would have scampered back to your own little star system." 6.1 Summary of Results Unlike the Universe as whole, for which -74% is composed of something we do not currently understand, the nature of sheets of galaxies is gradually becoming comprehensible using the physics at hand. In the previous chapter, observations of the mass and motion of dis in the Local Sheet were used to quantify the Sheet's structural and dynamical properties. The results agreed with the predictions of theoretical sheets derived from the ACDM simulation in Chapter 3. A consistent picture of sheets has emerged as systems comprised of two galaxy populations with distinct evolutionary histories, embedded within an intra-sheet medium containing roughly as much mass as the total mass of the sheet's galaxies and predominantly in the form of dark matter halos which could not host galaxies. In addition, there is preliminary evidence that sheets are evolving into steady-state systems with vertical density profiles characterized by an exponential decline, possibly due to the population of high-mass galaxies that appears to delineate a sheet's midplane. The specific observations that have led to these results are summarized below. -185- The distribution of luminous mass in sheets • In §2.3.1, a friends-of-friends algorithm was used to extract a structure of ~300 galaxies on an intermediate scale between the Local Group and the Local Supercluster. The structure, coined "the Local Sheet", has an exponential scale height of -0.4 Mpc and spans -12.5 Mpc along its longest length. Thus, the Sheet's thickness is only -6% of its length. The midplane of the Local Sheet is tilted at an angle of ~8° from the Supergalactic plane. -45% of its known members are dwarf irregular galaxies (dls). An analysis of the Local Sheet's completeness suggested that there are -750 low-mass dwarf galaxies still to be detected within the Sheet's boundaries. In §5.3.1, it was shown that the inclusion of a simulated population of the missing dwarfs would have a negligible effect on the Sheet's global structural parameters. • A search for sheets in a ACDM simulation yielded 8 sheet-like distributions of CDM halos from an 80-Mpc cube (§3.2). The thicknesses of the CDM sheets ranged from 17% to 40% of their lengths, which spanned from -8 to -14 Mpc. The similarity of these structural parameters with those of the Local Sheet suggest that the Local Sheet is a real-life example of this family of theoretical structures. • A deep near-infrared (Ks) imaging survey was conducted for the purpose of amassing reliable mass estimates for the Local Sheet dls. The galaxy images were obtained at five different facilities between 2004 and 2006. The image reductions and surface photometry were performed using methods specifically designed for isolating faint -186- galaxies from the high and varying near-infrared sky level. The majority of the dls have surface brightness profiles which could be fit to a hyperbolic-secant (sech) function, while the remaining profiles could be fit to the sum of a sech and a Gaussian function. The fits were employed to measure the central surface brightnesses, scale lengths, and integrated magnitudes of the dls, which in turn led to estimates of their stellar masses. The details of the full survey have been published by Fingerhut et al. (2010). An investigation of the vertical distribution of galaxy masses in the theoretical sheets (§3.4) as well as in the Local Sheet (§5.2.1) revealed that a sheet's dwarfs have vertical distances from their sheet's midplane which are 40% longer than the vertical distances of a sheet's non-dwarfs. This segregation of mass, which was observed in the theoretical sheets as well as in the Local Sheet itself, suggests that sheets are composed of two distinct galaxy populations. The baryonic masses of a sheet's dwarfs, its more widely-dispersed component, do not appear to correlate with vertical distance from a sheet's midplane. However, the number density of dwarfs peaks at the midplane. All of these observations can be explained by the hierarchical model of galaxy formation, which predicts that the most massive galaxies are formed from the cannibalization of their dwarf neighbours, making the cannibals more likely to be found in environments that are rich with their prey. The vertical density profiles of dwarf galaxies in the theoretical sheets (§3.5) and the Local Sheet (§5.3) appear similarly exponential. The exponential peak observed at small z may be a consequence of the accretion of dwarfs by the high-mass galaxy population concentrated along a sheet's midplane. Preliminary observations of the diagnostics of the sheets' kinematics were found to support this conclusion, as discussed below. The dynamics of sheets • In §3.6.1, the vertical profiles of the z-component of the peculiar velocity dispersions (a.) derived from the motion of isolated dwarfs were examined for the theoretical sheets. Owing to the small number of mass tracers in each sheet, the profiles were not smooth enough to definitively deduce the nature of the z-dependency of az. However, the five most populated sheets exhibit a minimum in their dispersion profiles at z = 0, followed by a gradual plateau, which approximates the dispersion profile predicted by the exponential sheet model. Four of the sheets have dispersion profiles which can be characterized as isothermal at large z. • The crossing times of the theoretical sheets were found to exceed one-half of the Universe's age (to), with two of the sheets having crossing times of nearly double the value of to (§3.7). This precludes the possibility that the sheets have evolved into dynamically equilibrated systems since their formation, since any sheet-induced infall is generally occurring for the first time. Thus, the near-isothermality observed in the majority of the dispersion profiles at large z must have been inherited from the sheets' host environment. -188- In §3.7.1, it was investigated whether a sheet's dynamics depend on its crossing time. Among the theoretical sheets for which the approximate shape of their vertical It is proposed that this evolution is being driven by the accretion of a sheet's dwarfs toward the high-mass galaxies at a sheet's midplane. In support of this, an inflow signature is found in the vertical motions of the isolated halos in all of the theoretical sheets. The most reliable estimate of the crossing time of the Local Sheet, which was computed from AZ for isolated dwarfs and ZQ from the exponential fit to the vertical density profile, is -20 Gyr (§5.5). While the uncertainty in this estimate is large (31%), even the lowest possible value of the Sheet's crossing time is greater than to. Thus, as was found for the theoretical sheets, the Local Sheet does not possess enough vertical motion to have become dynamically equilibrated since its formation. While the vertical dynamics of the Local Sheet cannot be observed from our location within its midplane, a comparison of the best value of the Sheet's crossing time with the crossing times of the theoretical sheets suggests that the kinematics of the Local Sheet should resemble the theoretical sheets which are furthest from the exponential model; namely, with only a moderate depression in the vertical velocity dispersion at small z. In support of this, the relative strength of the exponential peak in the Sheet's vertical density profile is consistent with the relative strengths observed for the theoretical sheets with the longest crossing times. The vertical distribution of the global properties of the Local Sheet's dls • In §5.2.3, a z-dependence was observed among the colours of the Sheet's dls, in the sense that within the Sheet's innermost 0.5-Mpc layer (-25% of its full vertical extent), the majority of isolated dls are bluer than the median. This colour trend is not seen among the dls associated with galaxy groups. It is proposed that the bluer colours found near the midplane may be a reflection of an enhanced likelihood of recent star formation in the Sheet's small-z environment outside of its virialized groups. This could be the result of tidal triggering caused by the motion of the isolated dls through the densest layer of the population of intergalactic dark matter halos predicted for the simulated sheets in §3.8. Missing matter in sheets • The dynamics of the isolated dwarfs in both the Local Sheet and the theoretical CDM sheets were found to require ~2.5 to 7 times more mass if the sheets are equilibrated, isothermal systems but only ~1.5 to 4 times more mass if the sheets are assumed to be equilibrated, non-isothermal systems with vertical density profiles characterized by an exponential decline (§3.8). The mass requirement of the exponential model is consistent with the expected amount of cold dark matter below the halo extraction limit of the ACDM simulation. Thus, theory predicts that there is enough dark matter in an exponential sheet for dynamical equilibrium to be achieved. -190- • Since the dynamical surface densities of the Local Sheet and the CDM sheets were computed from the vertical velocity dispersions of the sheets' isolated dwarfs, it can be concluded that the missing non-baryonic mass is not restricted to a sheet's virialized groups, but rather comprises an intra-sheet population of dark matter halos which lack the mass to host galaxies. 6.2 Future Work The observations presented in this dissertation have led to the proposal that sheets are evolving into steady-state systems. Given the agreement between the structural parameters of the Local Sheet and the theoretical sheets, a definitive confirmation of this proposal is, in principle, at hand, upon removal of the following limitations of the present study: 1. The small number of theoretical sheets that could be extracted from the ACDM simulation decreased the statistical significance of a possible correlation between the sheets' dynamics and their crossing times. 2. The small number of isolated dwarf halos within each sheet induced large uncertainties in measurements of the vertical velocity dispersion. Thus, for 3 of the sheets, agreement with a dynamical model could not be reliably ascertained. Moreover, the central value of the vertical velocity dispersion for 4 of the sheets could not be reliably estimated, which precluded a meaningful estimate of the surface mass density required to equilibrate these sheet. -191- 3. The lack of information regarding the mass and spatial distribution of the CDM particles below the halo extraction limit precluded a definitive statement about whether there is enough mass available to equilibrate the sheets. Fortunately, all of the above limitations can be addressed using a simulation with (1) larger spatial coverage, to increase the number of sheets; (2) a lower halo extraction limit, to increase the number of probes of the potential field; and (3) access to all particles in the simulation, so that the total amount of mass in each sheet can be ascertained. The algorithms which were developed in support of this dissertation could be operated on a simulation which meets the above criteria in order to refine the theoretical analyses in Chapter 3 and thereby corroborate or refute the aforementioned proposal regarding the dynamics of sheets. Moreover, an additional test of the proposal could be made by comparing snapshots of the simulation at different epochs, this being the most direct probe of the dynamical and structural evolution of sheets. 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