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Abstract the Mond External Field Effect on Dwarf ABSTRACT THE MOND EXTERNAL FIELD EFFECT ON DWARF SPHEROIDAL GALAXIES by Benjamin David Blankartz MOdified Newtonian Dynamics, or MOND, is an alternative to the dark matter paradigm which can be used to explain the missing mass problem. The External Field Effect(EFE) is a feature unique to MOND that has no counterpart in the dark matter paradigm. The EFE comes about when a gravitationally bound system is under the influence of an external gravitational field. We embed three Milky Way satellites, Fornax, Leo I and Crater II, as well as one Andromeda satellite, Andromeda XIX, in an external field produced by their host galaxy. We then model these systems kinetically. The velocity dispersion profiles are calculated statistically and then compared to observational data; acceleration profiles of each system are also presented. Fornax and Leo I show good agreement with observations in their velocity dispersion. We also examine structural similarities of the acceleration plots. THE MOND EXTERNAL FIELD EFFECT ON DWARF SPHEROIDAL GALAXIES Thesis Submitted to the Faculty of Miami University in partial fulfillment of the requirements for the degree of Master of Science by Benjamin David Blankartz Miami University Oxford, Ohio 2017 Adviser: Stephen Alexander, PhD Reader: Herbert Jaeger, PhD Reader: Jennifer Blue, PhD ©2017 Benjamin David Blankartz This thesis titled THE MOND EXTERNAL FIELD EFFECT ON DWARF SPHEROIDAL GALAXIES by Benjamin David Blankartz has been approved for publication by College of Arts and Science and Department of Physics Stephen Alexander, PhD Herbert Jaeger, PhD Jennifer Blue, PhD Contents 1 Introduction 1 1.1 Dark Matter . .1 1.2 ΛCDM ........................................2 1.3 MOND . .3 1.4 The MOND External Field Effect . .4 1.5 Dwarf Spheroidal Galaxies . .4 1.5.1 Previous Work . .5 1.6 Focus . .7 2 Method 8 2.1 Dwarf Spheroidal Model . .8 2.1.1 Code Conditions . .9 2.1.2 Hermite Individual Time Step Integrator . 10 2.1.3 Implementing MOND in Isolation . 12 2.2 Implementing the MOND External Field Effect . 13 2.2.1 The External Field Effect . 13 2.2.2 The External Field Effect in our Model . 13 2.2.3 The External Field Effect in Practice . 14 2.3 Data Analysis Technique . 18 2.3.1 Velocity Dispersion Profiles . 18 iii 3 Results and Discussion 19 3.1 Model Conditions . 19 3.2 Acceleration Plots and Velocity Dispersion Comparision . 20 3.3 Velocity Dispersion Profiles . 26 3.4 Possible Weaknesses of Work . 30 4 Conclusion 32 4.1 Future Work . 33 Bibliography 34 iv List of Tables 3.1 Initial data for four dwarf spheroidal galaxies. Rh is the half-light radius, L is the luminosity, D is the distance to the host galaxy, σobs is the observed bulk dispersion and ML is our best fit mass to light. 19 3.2 Mass data for host system. M is the unit solar mass. 26 v List of Figures 1.1 Eight Milky Way Dwarfs and their dispersion profiles. The red line is the isolated MOND dispersion while blue is the Newtonian velocity dispersion. The black circles are observational data published in Walker 2009 [14] . .6 2.1 Case 1 . 15 2.2 Case 2 . 16 2.3 Case 3 . 17 3.1 Fornax Acceleration and Dispersion Profiles . 21 3.2 Leo I Acceleration and Dispersion Profiles . 22 3.3 Crater II Acceleration and Dispersion Profiles . 23 3.4 Andromeda XIX Acceleration and Dispersion Profiles . 24 3.5 Fornax velocity dispersion profile. Black markers are observational data from Walker 2009[14]. 27 3.6 Leo I velocity dispersion profile. Black markers are observational data from Walker 2009[14]. 28 3.7 Crater II . 29 3.8 Andromeda XIX . 29 vi Dedication To my parents, Debbie and David. For all their work raising me and getting me to where I am today. vii Acknowledgements I would like to first and foremost thank my adviser, Dr. Stephen Alexander for the years of academic and professional guidance while I have attended Miami University. I would also like to acknowledge Matthew Walentosky and Tristan Clark for laying the ground work to our research into MOND. Joshua Schussler, Justin Messinger and Alex Staron for helping with the simulation work. I would also like to thank Dr. Jens Muller along with the entire Miami HPC group for all the support they gave me. Finally, I would like to thank my family, friends and professors for the uncountable amount of help given throughout my time as a graduate student. viii Chapter 1 Introduction 1.1 Dark Matter Midway through the 20th century, astronomers started using new technology to examine the kinematics of galaxies and stars within galaxies. This new access to the motion of stars within galaxies allowed for the testing of theories that described the dynamics of these systems. In 1933, astronomer Fritz Zwicky examined a clump of galaxies known as the Coma Cluster. He utilized the virial theorem, GM σ2 ≈ (1.1) R where M is the total mass, and R is a radius, and σ is velocity dispersion to make an estimate of the mass of the system by utilizing observations of the velocity. This led him to discover that the galaxies were moving too fast for the system to be gravitationally bound. Furthermore, he estimated that the mass required for the system to remain bound would be on the order of 400 times the visible matter; he coined the term ‘dunkle Materie’. This was one of the first published indications that there may be a discrepancy between the theory and observation which was attributed to dark matter[1]. Almost thirty years after Zwicky’s proposal, astronomer Vera Rubin was examining the rotation curves of spiral galaxies. Rotation curves are the relationship between the radial velocity of an 1 object and its distance from the center of mass of the gravitating body. Assuming that a system is gravitationally bound and stable, the tangential velocity of the bodies should fall off as the radial distance increases. Rubin’s observations showed that the velocities did not fall off, but instead remained abnormally high. This was further proof for scientists that the gravitaional mass in galaxies were higher than the observationed mass would suggested. Since these discoveries, astronomers and physicists have been trying to unravel the mysteries of this missing mass known as unseen dark matter [2]. Beyond the initial observational problems, several other key discoveries point to a missing mass problem. Dwarf Spheroidal galaxies have been shown to have a large mass discrepancy[3]. Galaxy clusters also exhibit mass discrepancies, such as the in-falling of the Milky Way and Andromeda[4] as well as the X-Ray red shift of gas within individual galaxies in a cluster[5]. On cosmological scales, two main telling issues stand out: low baryonic mass density and large scale structural formation. Mass density estimates for the universe are significantly lower than those derived from big bang nucleosynthesis[6]. Furthermore, formation rates of large scale structures due to gravitational attraction is much higher than should be possible with only observable matter[7]. These last two observations point towards two facts: dark matter must be non-baryonic and must be cold or not moving fast. 1.2 ΛCDM The current theory of dark matter is known as Lambda Cold Dark Matter(ΛCDM), which is the culmination of years of observational, computational and theoretical research[8]. The Λ reflects the addition of General Relativity and by extension Dark Energy, to the theory of dark matter[9]. Dark Energy is a large amount of extra energy in the universe identified from observations of distant supernovae. The ’Cold’ in Cold Dark Matter refers to the fact that for dark matter to clump, the matter cannot be moving quickly. These theories combined give us our current estimates of the amount of baryonic matter, dark matter, and dark energy that exist in the universe. 2 1.3 MOND Initial theories that aimed to correct for the ’missing mass problem’ proposed adding additional mass to the systems although this would not be the only option. Some people theorized that instead of adding additional mass to the system, another method for solving the missing mass problem would be to modify how gravity functions. Mordehai Milgrom proposed his theory of MOdified Newtonian Dynamics in 1983 [10]. The modification done by this theory affects Newtonian dynamics at low accelerations, thus keeping the rotation curves of galaxies flat. The modification that he proposes appears in the form1, jgi j gn = µ gi ; (1.2) ao where ao is a characteristic acceleration, µ is the matching function and gn is the Newtonian acceleration. The term gi is what the actual acceleration would be after the MOND correction. The matching function can be several types of functions that behave with the limits: 8 >µ = 1; x >> 1 µ¹xº = >< (1.3) > >µ = 0; x << 1 : : Several other matching functions could be used [11]; our research will utilize a simple function. The choice of matching function does little to change the results obtained. Finally, m a = 1:2 × 10−10 ; (1.4) o s2 is the characteristic acceleration that establishes the transition between Newtownian dynamics and Modified Newtonian dynamics. This has been established by fitting functions to rotational velocity curves although there are other method to establish this value. 1Bold face variable are to be treated as vector notation. 3 1.4 The MOND External Field Effect When Milgrom first proposed his idea, an immediate issue stood out: why do we not experience MOND in our every day life? His solution to this is the external field effect. Planets, stars, star clusters and galaxies do not exist in isolation, they are influenced by the gravitational field of a nearby object which will change the dynamics of the system.
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