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Determining the Origin and Possible Mechanisms of QPOS in X-Ray Emissions of Neutron Stars and Black Holes Brent Thomson

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University of North Dakota UND Scholarly Commons Theses and Dissertations Theses, Dissertations, and Senior Projects January 2014 Determining The Origin And Possible Mechanisms Of QPOS In X-Ray Emissions Of Neutron Stars And Black Holes Brent Thomson Follow this and additional works at: https://commons.und.edu/theses Recommended Citation Thomson, Brent, "Determining The Origin And Possible Mechanisms Of QPOS In X-Ray Emissions Of Neutron Stars And Black Holes" (2014). Theses and Dissertations. 1721. https://commons.und.edu/theses/1721 This Dissertation is brought to you for free and open access by the Theses, Dissertations, and Senior Projects at UND Scholarly Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of UND Scholarly Commons. For more information, please contact [email protected]. DETERMINING THE ORIGIN AND POSSIBLE MECHANISMS OF QPOS IN X-RAY EMISSIONS OF NEUTRON STARS AND BLACK HOLES by Brent Wayne Thomson Bachelor of Science, University of North Dakota, 2003 Bachelor of Arts, University of North Dakota, 2003 A Dissertation Submitted to the Graduate Faculty of the University of North Dakota In partial fulfillment of the requirements for the degree of Doctor of Philosophy Grand Forks, North Dakota December 2014 Copyright 2014 Brent Thomson ii Title Determining the Origin and Possible Mechanisms of QPOs in x-ray emissions of neutron stars and black holes Department Physics and Astrophysics Degree Doctor of Philosophy In presenting this dissertation in partial fulfillment of the requirements for a graduate degree from the University of North Dakota, I agree that the library of this University shall make it freely available for inspection. I further agree that permission for extensive copying for scholarly purposes may be granted by the professor who supervised my dissertation work or, in his absence, by the Chairperson of the department or the dean of the Graduate School. It is understood that any copying or publication or other use of this dissertation or part thereof for financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to the University of North Dakota in any scholarly use which may be made of any material in my Dissertation. Brent Thomson December 10, 2014 iv TABLE OF CONTENTS LIST OF FIGURES…………………………………………………………………………….....vii ACKNOWLEDGMENTS………………………………………………………….........................x ABSTRACT…………………………………………………………………..…………......…….xi CHAPTER. I. INTRODUCTION……………………………………………………………….…….1 Ia. Detection Method of QPOs…………………………………………………...….11 Ib. General Methods in Asterseismology………………………………....................13 Ic. Optical Solar Oscillation ………………………………………………………...16 Id. Determining the Periodicity of X-Ray Light Curves……………….....................23 Ie. Quality Factor Q0…………………………………………………………………28 II. QPO MODELS…………………………………………….........................................30 IIa. Criteria for QPO Models………………………………………..……………….30 IIb. Relativistic Precession Models…………………………………….....................31 IIc. Relativistic Resonance Models…………………………………...……………..32 IId. Beat-Frequency Models………………………………………….……………...33 IIe. Beat Frequency Interaction……………………………………………………...35 IIf. Radii of Interest...…………………………………………..................................38 IIg. Boundary Layer of the Inner Disk………..…………………….….....................40 IIh. Coupling between the Vertical and Radial Epicylical Oscillations......................46 III. KERR GEOMETRY………………………...………………………………………..49 IIIa. Dimensionality of the Kerr Metric Terms ……………………….......................49 IIIb. The Kerr Metric...…………….………………………………...........................52 IIIc. Orbital Radii.…………………………………………………...........................62 IIId. Frequencies and Radii Relevant to the QPO Models…………………………..69 IIId. The Gravitational Potential of a Kerr Black Hole……………………………...73 IV. ALFVEN RADII OF ACCRETION DISKS………..……….…….............................80 IVa. Blandford-Znajek Mechanism………………………….…………....................83 V. GENERAL PRINCIPLES OF ACCRETION DISKS………………..........................96 Va. Angular Momentum Transport……………………………................................100 Vb. Hydrodynamics……………………………………………...............................105 Vc. Magnetohydrodynamics……………………………………….....…………….106 v Vd. Magnetorotational Instability (MRI)………………………………..................108 Ve. Vertical Pressure Balance……………………………………….......................117 Vf. Viscous Processes……………………………………………….......................118 Vg. Radiative Transport……………………………………………........................120 Vh. Conservation Equations……….………………………………...……………..121 Vi. Thick Disks and Tori…………………………………………..……………….122 Vj. Papaloizou-Pringle Modes……………………………………….......................125 Vk. Equation of State……………………………………………………………….125 VI. ACCRETION RATE………..…………………………………................................131 VIa. Magnetic Influence on the Disk………………………….................................141 VIb. Accretion onto a Kerr Black Hole……………………………….....................143 VIc. Luminosity of an Accretion Disk………………………………......................146 VId. Conservation of Rest-Mass……..……………………………….....................149 VIe. Conservation of Angular Momentum………………………………………....149 VIf. Energy Conservation…………………………………………………………..150 VIg. Thin Disks, Slim Disks, and ADAFs…………………………….....................152 VII. DISKOSEISMOLOGY………………..………………………..………..................156 VIIa. Vibration modes……………………………………………...…....................158 VIIb. Basic Oscillation properties…………………………………….....................161 VIIc. Properties of the Nodes…………………………………………....................162 VIId. Variations from Spherical Symmetry……………………………..................165 VIIe. Oscillations and Instabilities in rotating fields……………..……...................168 VIIf. Rayleigh Instability for differential rotation………………………………….170 VIIg. Magnetorotational (Balbus-Hawley) Instability…………………...................171 VIIh. Global Instabilities…………………………………………………………....177 VIIi. Non-radial Stellar Pulsation…………………………………...…...................178 VIIj. Oscillation motion for g-modes…………………………………………….....179 VIIk. Standing Wave Characteristics……………………………….........................183 VIIl. Classification and Coupling of Radial and Vertical oscillations………………………........................186 VIIm. Trapped Oscillations…………………………………………………………194 VIII. RADIAL PULSATION (HELIOSEISMOLOGY)……………................................198 VIIIa. Helioseismology in a Cylindrical Reference Frame………….…..................199 VIIIb. Adiabatic Index Relation for a Disk………………………………………...203 IX. DISKOSEISMOLOGICAL APPROACH………………...………………………...208 IXa. Physical context of the Radial Potential Energy Expression……….………….213 IXb. Context of the Orbital Frequencies………………………….............................219 IXc. Physical Context of the Fractional Displacements……………………………223 IXd. Thermodynamics and Internal Energy Transport……………………………..225 vi IXe. Determining the Adiabatic Index in the Corona of a star…………………….234 X. PULSATION ACTIVITY.…………..……………………………..……................239 Xa. The Fractional Displacement Solutions per the Simple Harmonic Oscillation Equation………………………………………………………….243 Xb. The Radial Displacement Solutions per the Elliptic Equation………...............254 XI. CONCLUSIONS…………………………………………………………………....260 REFERENCES…………………………………………………………………………………..274 vii LIST OF FIGURES Figure Page 1. X-ray light curves of GRS 1915+105 on day 152 with finer time resolution…………………....12 2. A detailed view of the kilohertz QPO in Sco X-1………………………………………………..25 3. Anatomy of the Boundary/Cusp Layer model in an Accretion Disk……………………………..45 4. The Length of the Event Horizon versus the ISCO………………………………………………78 5. Comparison of the Magnetic Fields of the Black Hole and Disk………………………………...90 6. Alfven Radius for n = 2 versus ISCO for 6 M BH……………………………………………….90 7. Alfven Radius for n = 3 versus ISCO for 6 M BH……………………………………………….91 8. Alfven Radius for n = 4 versus ISCO for 6 M BH……………………………………………….91 9. Alfven Radius for n = 2 versus ISCO for 8 M BH……………………………………………….92 10. Alfven Radius for n = 3 versus ISCO for 8 M BH……………………………………………….92 11. Alfven Radius for n = 4 versus ISCO for 8 M BH……………………………………………….93 12. Alfven Radius for n = 2 versus ISCO for 10 M BH…..………………………………………….93 13. Alfven Radius for n = 3 versus ISCO for 10 M BH…..………………………………………….94 14. Alfven Radius for n = 4 versus ISCO for 10 M BH…..………………………………………….94 15. Alpha coefficient per adiabatic index…………………………………………………………...137 16. Radius of Influence per adiabatic index…………………………………………………………138 17. Density coefficient per adiabatic index………………………………………………………….139 18. Diagram of the Radial Potential Energy Curve with the roots shown for each case of q………215 19. Lengths of Event Horizon versus ISCO (Marginally Stable Circular Orbit) versus spin……….241 20. First solution of the radial fractional displacement versus spin for fixed r……………...………243 21. First solution of the vertical fractional displacement versus spin for fixed r……………………244 viii 22. First Radial Displacement for fixed index = 5/3………………………………………………..245 23. First Radial Displacement for fixed index = 3/2………………………………………………..246 24. First Radial Displacement for fixed index = 4/3………………………………………………..246 25. First solution of radial fractional displacement versus spin for fixed r = ISCO…………..…….248 26. First solution of radial fractional displacement versus spin for fixed r = ISCO……………...…249 27. First Vertical Displacement for fixed index = 5/3……………………………………………....250
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