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Logarithmic Spiral Arm Pitch Angle of Spiral Galaxies University of Arkansas, Fayetteville ScholarWorks@UARK Theses and Dissertations 5-2015 Logarithmic Spiral Arm Pitch Angle of Spiral Galaxies: Measurement and Relationship to Galactic Structure and Nuclear Supermassive Black Hole Mass Benjamin Lee Davis University of Arkansas, Fayetteville Follow this and additional works at: http://scholarworks.uark.edu/etd Part of the Cosmology, Relativity, and Gravity Commons, and the External Galaxies Commons Recommended Citation Davis, Benjamin Lee, "Logarithmic Spiral Arm Pitch Angle of Spiral Galaxies: Measurement and Relationship to Galactic Structure and Nuclear Supermassive Black Hole Mass" (2015). Theses and Dissertations. 1041. http://scholarworks.uark.edu/etd/1041 This Dissertation is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of ScholarWorks@UARK. For more information, please contact [email protected], [email protected]. Logarithmic Spiral Arm Pitch Angle of Spiral Galaxies: Measurement and Relationship to Galactic Structure and Nuclear Supermassive Black Hole Mass Logarithmic Spiral Arm Pitch Angle of Spiral Galaxies: Measurement and Relationship to Galactic Structure and Nuclear Supermassive Black Hole Mass A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Space and Planetary Sciences by Benjamin Davis Pittsburg State University Bachelor of Science in Mathematics, 2008 Pittsburg State University Bachelor of Science in Physics, 2008 May 2015 University of Arkansas This dissertation is approved for recommendation to the Graduate Council. Dr. Julia Kennefick Dr. William Harter Thesis Director Committee Member Dr. Daniel Kennefick Dr. Claud Lacy Committee Member Committee Member Dr. Larry Roe Dr. Marc Seigar Committee Member Committee Member ABSTRACT In this dissertation, I explore the geometric structure of spiral galaxies and how the visible structure can provide information about the central mass of a galaxy, the density of its galactic disk, and the hidden mass of the supermassive black hole in its nucleus. In order to quantitatively measure the logarithmic spiral pitch angle (a measurement of tightness of the winding) of galactic spiral arms, I led an effort in our research group (the Arkansas Galaxy Evolution Survey) to modify existing two-dimensional fast Fourier transform software to increase its efficacy and accuracy. Using this software, I was able to lead an effort to calculate a black hole mass function (BHMF) for spiral galaxies in our local Universe. This work effectively provides us with a census of local black holes and establishes an endpoint on the evolutionary history of the BHMF for spiral galaxies. Furthermore, my work has indicated a novel fundamental relationship between the pitch angle of a galaxy’s spiral arms, the maximum density of neutral atomic hydrogen in its disk, and the stellar mass of its bulge. This result provides strong support for the density wave theory of spiral structure in disk galaxies and poses a critical question of the validity of rival theories for the genesis of spiral structure in disk galaxies. DEDICATION I dedicate this dissertation to my grandmother, Jacqueline Davis, and my aunt, Janet Green, who passed away during my time as a graduate student. EPIGRAPH I think music is about our internal life. It’s part of the way people touch each other. That’s very precious to me. And astronomy is, in a sense, the very opposite thing. Instead of looking inwards, you are looking out, to things beyond our grasp. —Brian May ACKNOWLEDGEMENTS Thank you to my mother, Sara Davis, for training me to be proficient at mathematics. Your excellent teaching allowed me to excel in an area that I initially struggled. Thank you to my father, Jeffrey Davis, for always discussing space and space exploration. Thank you for sharing your fascination and reverence of the heavens with me. Thank you to Dr. David Kuehn for inspiring me to study astronomy and for motivating me to become an astronomer. Thank you to my dear wife, Nicolette Davis, for being with me every step of my graduate career and my life in Arkansas. I love and cherish you. Thank you to Drs. Daniel and Julia Kennefick for allowing me to work with the Arkansas Galaxy Evolution Survey (AGES). Your kindness and guidance has made all of this work possible. Thank you to all my fellow students in AGES for your collaboration and friendship. Thank you to J. Adam Hughes for being a dear friend to Nicolette and I ever since we first visited the University of Arkansas. I will always cherish the adventures we have had together. I am much obliged for your Arkansan hospitality. Thank you to Dr. Marc Seigar for inviting me to Little Rock in 2010, and teaching me the technique and providing me with the software necessary to perform 2-D FFT pitch angle measurements. Without your guidance, my graduate student career would have been entirely different. Thank you to Dr. Joel Berrier for mentoring me and being a dear friend to Nicolette and I. Thank you to Dr. Robert Mueller for allowing me to play with the University Symphony Orchestra during my time at the University of Arkansas. The chance you have afforded me to practice, perform, become involved with the music department, and make lasting friends has served to be an invaluable part of my graduate student life. Finally, thank you to Ryan Cockerham for being a dear friend to Nicolette and I for the entirety of our marriage. I hold dear all the memories of your company, making music together, and our synergistic creative ideas. TABLE OF CONTENTS 1 Introduction . .1 1.1 Logarithmic Spiral Pitch Angle . .2 1.2 Relation Between Pitch Angle and Supermassive Black Hole Mass in Disk Galaxies4 1.3 Black Hole Mass Function . .6 1.4 Spiral Arm Pitch Angles as a Probe of Disk Galaxy Mass Profiles . .7 1.5 Spiral Density Waves and Pitch Angle . .7 1.6 Outline . .8 Bibliography . 10 2 Measurement of Galactic Logarithmic Spiral Arm Pitch Angle Using Two-Dimensional Fast Fourier Transform Decomposition . 13 2.1 Abstract . 13 2.2 Introduction . 13 2.3 Observations and Data Analysis Techniques . 16 2.3.1 Two-Dimensional Fast Fourier Transformations of Galaxy Images . 17 2.3.2 Image Preprocessing . 19 2.3.2.1 Deprojection . 19 2.3.2.2 Image Cropping . 21 2.3.2.3 Star Subtraction . 22 2.3.3 Image Measurement . 23 2.4 Pitch Angle as a Function of Inner Radii . 24 2.5 Error Determination . 28 2.5.1 Inclination Angle and Galactic Center Position Errors . 33 2.5.2 Bulges and Bars . 37 2.5.3 Problems with Underlying Presumptions . 37 2.5.3.1 The Effect of Wavelength on Pitch Angle . 39 2.5.3.2 Variable Pitch Angle with Galactic Radius . 45 2.5.3.3 Flocculence . 46 2.6 Image Analysis . 53 2.6.1 Symmetrical Component Significance . 55 2.6.2 Two-Dimensional Inverse Fast Fourier Transform . 59 2.7 Discussion and Future Work . 61 2.7.1 Comparison to Other Methods . 61 2.7.1.1 NGC 7083 . 67 2.7.1.2 NGC 4995 . 68 2.7.1.3 NGC 1365 . 68 2.7.1.4 NGC 3513 . 70 2.7.2 Pitch Angle - SMBH Relation . 70 2.7.3 Evolution of Pitch Angle with Redshift . 71 2.7.4 Continuing Efforts . 72 2.8 Acknowledgements . 73 Bibliography . 74 3 The Black Hole Mass Function Derived from Local Spiral Galaxies . 77 3.1 Abstract . 77 3.2 Introduction . 78 3.3 Methodology . 84 3.3.1 How We Measure Pitch Angle . 84 3.3.2 The M–P Relation . 85 3.4 Data . 86 3.5 Pitch Angle Distribution . 98 3.6 Black Hole Mass Distribution . 103 3.7 Black Hole Mass Function for Local Spiral Galaxies . 108 3.8 Discussion . 116 3.9 Conclusions . 121 3.10 Acknowledgements . 126 3.11 Appendix . 126 3.11.1 The Golden Spiral . 126 3.11.2 The Milky Way . 129 Bibliography . 130 4 A Fundamental Plane of Spiral Structure in Disk Galaxies . 134 4.1 Abstract . 134 4.2 Introduction . 134 4.3 Data and Analysis . 137 4.4 Results . 141 4.5 Discussion . 144 4.6 Appendix . 147 Bibliography . 159 5 Conclusions . 161 Bibliography . 164 Vitæ............................................... 168 LIST OF FIGURES Figure 1.1: Graphical demonstration of the definition of logarithmic spiral arm pitch angle. There are four elements in the plot above: blue - a logarithmic spiral with φ = 25◦, teal - a circle with r = 2:08 intersecting the logarithmic spiral at (0;2:08), red - a line tangent to the circle at (0;2:08), and green - a line tangent to the logarithmic spiral at (0;2:08). The angle subtended between the line tangent to circle and the line tangent to the logarithmic spiral is 25◦, which is equivalent to the pitch angle of the logarithmic spiral. .3 Figure 1.2: Black hole mass vs. pitch angle for all spiral galaxies with directly mea- sured black hole masses available (34). The best linear fit to this data is illus- trated and gives the following relation log(MBH =M ) = (8:21 0:16)-(0:062 0:009)P. The fit has a reduced χ2 = 4:68 with a scatter of 0:38± dex. Black hole± masses measured using stellar and gas dynamics techniques are labeled with black ’s (10 points), reverberation mapping masses with red triangles (12 points),× and maser measurements with blue squares (12 points). This figure is reproduced from Berrier et al. (2013). .5 Figure 2.1: Fig. 2.1a (top) - B-band (inverted color) image of NGC 5054. NGC 5054 is measured to have a position angle (PA) of 160◦ and an inclination angle (α) of 53:84◦.
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