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TIME SERIES ANALYSIS OF THE CATACLYSMIC VARIABLE V1101 AQUILAE A Thesis by ALEXANDER C. SPAHN Submitted to the O ce of Graduate Studies of Texas A&M UniversityCommerce In partial ful llment of the requirements for the degree of MASTER OF SCIENCE August 2015 TIME SERIES ANALYSIS OF THE CATACLYSMIC VARIABLE V1101 AQUILAE A Thesis by ALEXANDER C. SPAHN Approved by: Advisor: Matt A. Wood Committee: William Newton Kurtis Williams Head of Department: Matt Wood Dean of the College: Brent Donham Dean of Graduate Studies: Arlene Horne iii Copyright c 2015 Alexander C. Spahn iv ABSTRACT TIME SERIES ANALYSIS OF THE CATACLYSMIC VARIABLE V1101 AQUILAE Alexander C. Spahn, MS Texas A&M UniversityCommerce, 2015 Advisor: Matt A. Wood, PhD This work reports on the application of various time series analysis techniques to a two month portion of the light curve of the cataclysmic variable V1101 Aquilae. The system is a Z Cam type dwarf nova with an orbital period of 4.089 hours and an active outburst cycle of 15.15 days due to a high mass transfer rate. The system's light curve also displays higher frequency variations, known as negative sumperhums, with a period of 3.891 hours and a period de cit of 5.1%. The amplitude of the negative superhumps varies as an inverse function of system brightness, with an amplitude of 0.70358 during outburst and 0.97718 during quiescence (relative ux units). These variations are believed to be caused by the contrast between the accretion disk and the bright spot. An OC diagram was constructed and reveals the system's evolution. In general, during the rise to outburst, the disk moment of inertia decreases as mass is lost from the disk, causing the precession period of the tilted disk to increase and with it the negative superhump period. The decline of outburst is associated with the opposite e ects. While no standstills were observed in this data, they are present in the AAVSO data and the results agree with the conditions for Z Cam stars. v ACKNOWLEDGMENTS I wish to extend my thanks to my advisor, Dr. Matt Wood, for the assistance that he has provided for this project; To Joe Patterson and the amateur astronomers who have provided us with the data used in this research; To my parents who have always believed in me and kept me motivated to do my best; To the National Science foundation for their generous STEM scholarship; To the Department of Physics and Astronomy and the Graduate School department for their nancial support and for o ering me the role as both a teaching assistant and a research assistant; To my high school chemistry teacher, Ms. Kim Froemming Nejedlo, who kindled my desire for knowledge by demonstrating the beautiful and thrilling world of science with an insurmountable level of passion and energy. I can only dream that I may one day be as in uential to my students as she was to hers. And nally, I extend my thanks to anyone with this passion; the passion for learning, the passion for science and knowledge of the world around us, and/or the passion to help others. Together, we can change the world. vi TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. STELLAR EVOLUTION TO CATACLYSMIC VARIABLES . . . . . . . . . 3 2.1. Stellar Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2. Stellar Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3. Stellar Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4. Binary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3. CATACLYSMIC VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1. Accretion Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2. Outbursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3. Disk Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4. Positive and Negative Superhumps . . . . . . . . . . . . . . . . . . . 51 4. ANALYSIS OF V1101 AQUILAE . . . . . . . . . . . . . . . . . . . . . . . . 57 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Appendix A. Permission Documentation . . . . . . . . . . . . . . . . . . . . . . . . 73 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 vii LIST OF TABLES TABLE 2.1. Results of a theoretical model of the Sun's interior. . . . . . . . . . . . . . . . . . . . . . . . . . . 9 viii LIST OF FIGURES FIGURE 1.1. An artist's representation of a cataclysmic variable. . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.1. A small volume of stellar material that is in hydrostatic equilibrium where the force of gravity is balanced by the outward force of the pressure gradient. . . . . . . . . . . 4 2.2. A spherically symmetric shell region of a star. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3. The cross section of a stellar interior depicting energy transport via radiation and convection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4. A pocket of stellar gas that rises in a region of convection. . . . . . . . . . . . . . . . . . . . 8 2.5. The luminosity, mass, temperature, and density pro les plotted as a function of radius for a theoretical model of the Sun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.6. The stellar evolution track model for a 1M star beginning at ZAMS and ending with the formation of a white dwarf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.7. The internal structure of a low mass star at several stages of its evolution. . . 15 2.8. Several examples of planetary nebulae. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.9. An image of Sirius A and its white dwarf companion Sirius B. . . . . . . . . . . . . . . . 18 2.10. The massradius relation for white dwarfs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.11. The three classi cations of binary star systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.12. A schematic of two stars in a binary system used to determine the gravitational potential experienced by a test mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.13. A cross section of the equipotential surfaces and the rst three Lagrangian points for a binary system and a surface representation of the potential. . . . . . . . . . . . 24 3.1. A distribution of cataclysmic variable orbital periods. . . . . . . . . . . . . . . . . . . . . . . . 30 ix 3.2. Evolution of the secondary near the minimum period of 75 min. . . . . . . . . . . 31 3.3. A topdown view of the stream trajectory of particles ejected through L1 at various velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4. A schematic of a CV depicting the mass transfer stream colliding with the accretion disk and causing a bright spot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5. The spreading of a ring of mass M with a Keplerian orbit at r = r0 due to viscous torques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.6. The total spectrum F for a steady state optically thick disk at di erent ratios rd=R1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.7. The vertical cross section of a concave accretion disk in the disk model. . . 40 3.8. The angular velocity (r) near the surface of the primary with 1. . . . . . . . . . . 41 3.9. A vertical cross section of an accretion disk's optically thick boundary layer. 42 3.10. A fouryear portion of U Gem's visual light curve. . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.11. A oneyear portion of SS Cyg's visual light curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.12. Schematics of three types of DN outburst pro les. . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.13. The bimodal distribution of SS Cyg's outburst durations. . . . . . . . . . . . . . . . . . . . 45 3.14. An analogy for magnetorotational instability in which two objects at di erent radii in Keplerian orbits are connected by a spring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.15. The growth of the magnetorotational instability displayed by a vertical eld line with slight deviations to larger and smaller radii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.16. A simpli ed Scurve relating the density of an accretion disk to its temperature T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.17. A DN disk's surface density plotted against its radius for several instances during an outburst cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 x 3.18. Luminosity, disk radius, disk mass, and angular momentum variations in DN. 52 3.19. An eightmonth portion of VW Hyi's visual light curve. . . . . . . . . . . . . . . . . . . . . . 53 3.20. Snapshots from a SPH elliptical accretion disk simulation for q = 0.25. . . . . . . 54 3.21. Luminosity, disk radius, disk mass, and angular momentum variations of a DN supercycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.22. A tilted accretion disk undergoing retrograde precession through one cycle. . 55 3.23. Superhump period excess and negative superhump period de cits plotted against the orbital period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.1. A threeandahalf year portion of Z Cam's visual light curve. . . . . . . . . . . . . . . . 57 4.2. The light curve of V1101 Aql in ux units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3. The original light curve of V1101 Aql and the light curve with the large amplitude signal removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.4. Discrete Fourier transforms of the original light curve of V1101 Aql and of the light curve with the large amplitude signal removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.5. An example of the aliasing of a sinusoidal signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.6. The average pulse shape of the negative superhumps for the entire set of data and for each of the high and low sets of data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.7. The OC phase diagram for the negative superhump period P = 3:89 hr and the amplitude of the tted sine curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.8. Observed negative superhump period de cit versus the orbital period for V1101 Aql and several other systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.9. A veyear portion of V1101 Aql's light curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 1 Chapter 1 INTRODUCTION Of the countless number of stars in the Universe, more than half are believed to exist within binary or multiple star systems. Cataclysmic variables (CVs for short) are particular types of binary systems which consist of a white dwarf primary star and a K  M type dwarf secondary star that interact in a semidetached binary system, with a binary separation roughly equivalent to the Sun's diameter, and orbit a common center of mass on a timescale of one to a few hours (Figure 1.1). Within these systems, mass transfers from the secondary star through the system's inner Lagrangian point and into the potential well of the primary. The infalling gas forms an accretion disk around the primary which often has a luminosity greater than that of the two stars combined (Hellier, 2001; Warner, 1995). Cataclysmic variables were rst observed in the mid1800's and have since been clas si ed into many classes and subclasses based upon the outburst pro les observed in the objects' light curves. This study focuses on the Z Cam subclass of the dwarf novae class of cataclysmic variables. Dwarf novae (DN) display various quasiperiodic outbursts in which the luminosity of the system increases by several magnitudes, lasting for anywhere from days to weeks. Z Cam stars have active outburst cycles and exhibit occasional standstill during which the system is stuck in outburst for a prolonged period of time. Figure 1.1. An artist's representation of a cataclysmic variable. Image is copyright Mark A. Garlick/spaceart.co.uk, used by permission. 2 The leading theory for the origin of such outbursts, rst developed in 1974 by Yoji Osaki, suggests that instabilities in the accretion disk are to blame. These instabilities are caused by the buildup of matter within the disk which raises the local temperature of the disk until it becomes ionized and highly viscous, leading to an outburst. The location of the instability when it forms governs the shape of the outburst. Standstills in Z Cam stars are believed to be caused by a mass transfer rate that is high enough to keep the disk in outburst for a signi cant amount of time. The study of cataclysmic variables is essentially the study of accretion disks. Accre tion disks are one of the most common astrophysical structures in the Universe, occurring around some neutron stars, black holes, quasars, active galaxies, and in protostellar and pro toplanetary systems. However, accretion disks are best observed in cataclysmic variables. The problem with other systems is that they are either too complex or too di cult to ob serve (typically both). For example, quasars are too distant and protoplanetary systems are often shrouded in dust. Cataclysmic variables provide astronomers with variations on a human timescale and are able to teach the basics of the mechanisms that govern the other more exotic systems. Furthermore, eclipsing cataclysmic variables may provide additional information as to the geometry of the systems. There are many known cataclysmic variables { more than can be observed with only the most e cient observatories in the World. As a result, these systems are most actively observed among amateur astronomers and provide them with opportunities to make worthwhile contributions to the scienti c community. Chapter 2 provides a relatively brief overview of stellar interiors and stellar evolution, laying the foundation for the discussion of interacting binary star systems. Chapter 3 presents a detailed discussion of cataclysmic variables and the outbursts observed within their light curves. In Chapter 4, information from the previous chapters is applied in the analysis of V1101 Aquilae's light curve in order to extract system parameters and to better understand its quasiperiodic nature. Information obtained from the data is then used to classify the system as a Z Cam dwarf nova. 3 Chapter 2 STELLAR EVOLUTION TO CATACLYSMIC VARIABLES 2.1. Stellar Structure Stars are remarkably dynamic objects and, while much of their interiors are not able to be directly observed, they can be modeled via several mathematical constructs known as the equations of stellar structure. First, consider a small cylindrical volume element of a stellar interior with mass dm and length dr as depicted in Figure 2.1. The force of gravity Fg acts toward the center of the star while an opposing force FP provided by pressure acts toward the surface of the star. Both the top and bottom surfaces of the volume element will experience pressure forces of FP,top and FP,bottom, respectively. The net force acting on the element is the sum of these forces and the application of Newton's second law F = ma gives FP,top FP,bottom + Fg = dm d2r dt2 : (2.1) The total pressure force dFP acting on the element is the di erence between the top and bottom forces and, given that F = PA, the total force dFP = AdP. With this, and the fact that the gravitational force is given by GMrdm=r2, Equation 2.1 can be written as AdP G Mrdm r2 = dm d2r dt2 (2.2) where Mr is the mass of the star within a radius r. Replacing dm with Adr, diving through by the volume, and noting that, for a static environment, the acceleration is zero, gives dP dr = G Mr r2 = g (2.3) where g = GMr=r2 is the acceleration due to gravity at a radius r and is the density of the gas. For spherically symmetric stars, Equation 2.3 is the equation of hydrostatic equilibrium. In short, this states the condition that a pressure gradient must exist in order to balance the force of gravity in a static star. 4 FP,top FP,bottom A Fg dm dr Figure 2.1. A small volume of stellar material that is in hydrostatic equilibrium where the force of gravity is balanced by the outward force of the pressure gradient. The volume is taken to be a cylinder of mass dm and length dr with end caps each of surface area A. Consider, now, a spherically symmetric shell region of a star, located a distance r from the center, that has a mass Mr, width dr, and density as depicted in Figure 2.2. The volume of a sphere is V = 4 3 r3 and so, for a thin shell (dr r), the volume is approximately dV = 4 r2dr: (2.4) From the familiar relation = Mr=V , the shell's mass may be expressed as dMr = (4 r2dr): (2.5) Simple rearranging gives the equation of mass continuity: dMr dr = 4 r2 : (2.6) Equation 2.6 prescribes how interior stellar mass varies with radius and is the second equation of stellar structure. The total mass of a spherical star can be thought of as the sum of all of the masses of each thin shell extending from the center to the star's radius R. 5 r dMr, dr Figure 2.2. A spherically symmetric shell with a mass dMr, a width dr, and a density . The inner surface of the shell is located at a radius r from the star's center. Up until this point, the origin of the gas pressure has been overlooked. It is necessary to derive an equation of state, which relates the pressure to properties of the gas, so that the origin may be understood. For a gas of volume V and temperature T consisting of N number of particles, the ideal gas law states that PV = NkT (2.7) where k is the Boltzmann constant. Equation 2.7 can be rewritten by de ning the particle number density n = N=V . The number density is related to the mass density by n = = m where m is the average mass of a gas particle within the volume. De ning one more quantity, the mean molecular weight = m=mH, allows for the equation of state to be given by Pgas = kT mH (2.8) where mH is the mass of a hydrogen atom. The radiation pressure due to photons is also signi cant and is often times more dominant than that supplied by the gas. The radiation pressure Prad with radiation constant a is expressed as Prad = 1 3 aT4: (2.9) 6 Taking the derivative of Equation 2.9 with respect to r gives an expression for the radiation pressure gradient: dPrad dr = 4 3 aT 3 dT dr : (2.10) The radiation pressure gradient may also be expressed in terms of the radiative ux Frad as dPrad dr = c Frad (2.11) where is the opacity and c is the speed of light. After equating the two expressions, using the de nition Frad = L=4 r2, where L is the luminosity at a radius r, and rearranging, an expression for the temperature gradient arises (Iben, 2013): dT dr = 3 L 16 acT 3r2 : (2.12) This is the equation of radiative energy transport. A transparent star with no opacity ( = 0) would absorb no energy and, therefore, would have no temperature gradient. The actual average temperature gradient between the Sun's center and the Sun's photosphere is roughly dT dr = 20 K km1 (Ryden & Peterson, 2010). Radiation is the most e cient process that carries energy away in the region from the center of the Sun out to about 0.7 R . However, from 0.7 to 1.0 R , convection dominates energy transportation (Figure 2.3). Convection typically occurs in regions of high opacity and causes hot pockets of gas to rise while cool pockets of gas sink further down. In order to construct a mathematical model of this type of energy transport, consider a pocket of gas of pressure Pp and density p that is perturbed some vertical distance dr as shown in Figure 2.4. The region surrounding the pocket has a pressure P and density . Before the pocket rises, it is in hydrostatic equilibrium, requiring that Pp = P. When the pocket rises by dr, the pressures and densities also change by some amount. If the pocket's density becomes greater than that of its surroundings, the pocket with begin to sink back down and no convection will occur. If, 7 convection radiation 0.25R 0.7R 1.0R core Figure 2.3. Solar energy transport via radiation and convection. Between 0.0 and 0.7 R , radiation is the dominant process of energy transportation. Convection dominates between 0.7 and 1.0 R . The core of the Sun, extending to 0.25 R , is included for scale. however, the pocket density is lower than that of its surroundings, it will continue to rise and convection will result. Therefore, the condition for the onset of convection is d p > d (2.13) where d p and d are the changes in the densities of the pocket and the surroundings, respectively, over a distance dr. In a convective stellar environment, the motion of these pockets is rapid and there is little time for them to gain or lose heat. Such a process is termed adiabatic (Rose, 1998) and gives PV = P = ln ln P = constant (2.14) where is the adiabatic index de ned by the ratio of the speci c heats for the constant pressure and constant volume cases. After taking the derivative of Equation 2.14, writing it in terms of the temperature gradient, and much rearranging, the expression becomes dT dr = 1 1 ! T P dP dr : (2.15) This is the equation of convective energy transport and it describes how the pocket's tem perature changes as the pocket rises due to convection. 8 surface center dr p, Pp , P + d , P + dP p + d p, Pp + dPp Figure 2.4. A pocket of stellar gas rises in a region of convection. Initially, the pocket has a density p and experiences a pressure Pp while the surroundings have a density and pressure P. After rising by a distance dr, the densities and pressures change by some amount (denoted by di erential forms). The nal equation of stellar structure discussed here is that of energy generation. A mass dm will contribute a luminosity dL = dm to the total luminosity, where is the rate of energy production. Recalling that dm = 4 r2 dr gives dL dr = 4 r2 : (2.16) Together, the equations of hydrostatic equilibrium (2.3), mass continuity (2.6), radiative energy transport (2.12), convective energy transport (2.15), and energy generation (2.16) make up the equations of stellar structure. These equations govern the structure of stellar interiors and solving them allows for models to be constructed. In general, as radius increases, the luminosity and mass increase while the temperature, density, and pressure decrease. For the Sun, the luminosity reaches maximum about a quarter of the way out to the surface and about 94% of the total mass lies within the inner half of the Sun. The luminosity, mass, temperature, and density pro les for a theoretical model of the Sun's interior are plotted as a function of radius in Figure 2.5 and the results are listed in Table 2.1 (Freedman & Kaufmann, 2008). 9 Table 2.1 Results of a theoretical model of the Sun's interior. Radius Luminosity Mass Temperature Density Pressure (R=R ) (L=L ) (M=M ) (106 K) (kg/m3) (P=Pc) 0:0 0:00 0:00 15:5 160 000 1:00 0:1 0:42 0:07 13:0 90 000 0:46 0:2 0:94 0:35 9:5 40 000 0:15 0:3 0:99 0:64 6:7 13 000 0:04 0:4 0:99 0:85 4:8 4 000 0:007 0:5 0:99 0:94 3:4 1 000 0:001 0:6 0:99 0:98 2:2 400 0:0003 0:7 0:99 0:99 1:2 80 4 105 0:8 0:99 1:00 0:7 20 5 106 0:9 0:99 1:00 0:3 2 3 107 1:0 0:99 1:00 0:006 0:0003 4 1013 Figure 2.5. The luminosity, mass, temperature, and density pro les plotted as a function of radius for a theoretical model of the Sun (Data from Freedman, 2008). 2.2. Stellar Nucleosynthesis Energy generation in stars is a result of nuclear fusion and the process of converting one element to another is known as nucleosynthesis. Most stars primarily convert hydrogen into helium and doing so releases energy. If the sun were made entirely of hydrogen atoms and all of them were converted into helium, the fusion processes would release roughly 1:2 1045 J of energy. This would allow the Sun to live for nearly 100 Gyr. However, only about 10% 10 of the Sun undergoes fusion and so the true lifetime of the Sun is closer to 10 Gyr (Ryden & Peterson, 2010). Before discussing the primary stellar nuclear reactions, a few de nitions are made. Electrons, positrons, photons, and neutrinos are depicted by e, e+, , and , respectively. Furthermore, nuclei are represented by AZ X, where A is the mass number, Z is the number of protons, and X is the element's symbol. Hydrogen is converted into helium in the cores of stars but it is highly unlikely that 4 H atoms will all collide to make 42 He. Rather than this direct interaction, there is a chain of protonproton reactions known as the PP chain that take place. The PP chain is the dominant form of hydrogen fusion for temperatures T < 1:8 107 K. The PPI chain is 11 H +11 H ! 21 H + e+ + 21 H +11 H ! 32 He + 32 He +32 He ! 42 He + 211 H: (69%) (2.17) Helium3 will react with another helium3 nuclei 69% of the time (Carroll & Ostlie, 2007). There is, however, a 31% chance that helium3 will interact with helium4 in the PPII chain: 32 He +42 He ! 74 Be + (31%) 74 Be + e ! 73 Li + (99:7%) 73 Li +11 H ! 242 He: (2.18) The beryliym7 nucleus will capture an electron 99.7% of the time while 0.3% of the time it will, instead, capture a proton to produce boron8 in the PPIII chain: 74 Be +11 H ! 85 B + (0:3%) 85 B ! 84 Be + e+ + 84 Be ! 242 He: (2.19) The net resultants of the entire PP chain are 411 H ! 42 He + 2e+ + 2 + 2 : (2.20) 11 Note that in these types of equations, charge and nucleon numbers are conserved. For temperatures T > 1:8 107 K, a separate process known as the CNO cycle becomes the dominant form of hydrogen fusion. It is given this name due to the fact that carbon, nitrogen, and oxygen are both reactants and products (i.e., they are regenerative, being both consumed and created). The chain of reactions in the CNO cycle are 12 6 C +11 H ! 13 7 N + 13 7 N ! 13 6 C + e+ + 13 6 C +11 H ! 14 7 N + 14 7 N +11 H ! 15 8 O + 15 8 O ! 15 7 N + e+ + 15 7 N +11 H ! 12 6 C +42 He (99:96%) ! 16 8 O + (0:04%) 16 8 O +11 H ! 17 9 F + 17 9 F ! 17 8 O + e+ + 17 8 O +11 H ! 14 7 N +42 He (2.21) and the net resultants of the CNO cycle are 411 H ! 42 He + 2e+ + 2 + 3 : (2.22) Over time, helium begins to accumulate within a star due to the PP chain and CNO cycle reactions. Eventually, the star will begin to run out of hydrogen and it will start to shrink. When the temperature rises to over 108 K, helium starts to fuse more rapidly than berylium8 is able to decay. As a result, berylium begins to build up and can fuse with additional helium nuclei to create carbon12. These reactions are quite rare, since berylium is very unstable and has a brief life, and so it takes a signi cant amount of time for carbon to be produced. 12 The transformation of three helium nuclei into carbon is known as the triple alpha process: 42 He +42 He *) 84 Be 84 Be +42 He ! 12 6 C + : (2.23) While the triple alpha process is occurring at these temperatures, carbon and oxygen may also capture alpha particles to produce even heavier nuclei in the following reactions: 12 6 C +42 He ! 16 8 O + 16 8 O +42 He ! 20 10Ne + : (2.24) More massive stars with higher temperatures may allow for carbon and oxygen burn ing. Carbon burning takes place at temperatures near 6 108 K: 12 6 C +12 6 C = 8>>>>>>>>>>>< >>>>>>>>>>>: 16 8 O + 242 He 20 10Ne +42 He 23 11Na + p+ 23 12Mg + n 24 12Mg + (2.25) while oxygen burning takes place at even higher temperatures near 109 K: 16 8 O +16 8 O = 8>>>>>>>>>>>< >>>>>>>>>>>: 24 12Mg + 242 He 28 14Si +42 He 31 15P + p+ 31 16S + n 32 16S + (2.26) While many di erent processes have been discussed here, only the PP chain is currently present in the Sun as it is not yet hot enough for the other reactions to take place. 13 2.3. Stellar Evolution The evolution of stars is inevitable and is primarily the result of changes in chemical composition due to the nuclear reactions discussed in the previous section. A model of the entire evolutionary track for a 1M star from the zeroage main sequence (ZAMS) phase to the formation of a white dwarf is given in Figure 2.6 and the internal structure at several log10(Te) log10(L=L ) ZAMS SGB RGB HB EAGB TPAGB PostAGB PN Formation Prewhite dwarf White dwarf H core exhaustion Core contraction H shell burning He core burning He core exhaustion He ash First dredgeup Second dredgeup First He shell ash superwinds Figure 2.6. The stellar evolution track model for a 1M star beginning at ZAMS and ending with the formation of a white dwarf. 14 di erent phases is shown in Figure 2.7. Once a star reaches the ZAMS and begins to fuse hydrogen into helium, the star's luminosity, radius, and temperature steadily increase. This is because PP chain reactions increase the mean molecular mass of the stellar gas. Hydrostatic equilibrium requires a constant central pressure and so the temperature and/or density must increase. As a result, the core contracts and the rate of energy generation increases which drives both the luminosity and radius up. For example, when the Sun reached the ZAMS 4.6 Gyr ago, it had a luminosity of 0.7 L and, when it runs out of hydrogen in about 6 Gyr, it will have a luminosity near 2.2 L (Ryden & Peterson, 2010). Eventually, after about 10 Gyr, the star will exhaust all of the hydrogen in its core and the PP chain reactions will end. At this point, the temperature will have increased enough to allow for a hydrogen burning shell around the isothermal helium core. Some of the energy generated by the thick shell does not reach the surface and so the temperature decreases and the evolutionary track (Figure 2.6) stars curve to the right. The prolonged hydrogen burning increases the core's mass until it is no longer able to support the rest of the star and reaches a collapse limit determined by the Sch onbergChandrasekhar limit: Mc M ! = 0:37 env c ! (2.27) whereMc and c are the core mass and mean molecular weight and env is the mean molecular weight of the envelope above the hydrogen burning shell. When the core mass reaches this limit, it will collapse and the star leaves the main sequence phase of evolution. The sudden collapse of the core releases gravitational energy and causes the envelope to expand while the temperature decreases. The evolutionary track moves to the right in this phase known as the subgiant branch (SGB). As the envelope of the star continues to expand, the temperature drops enough for a convective region to form near the surface. The bottom of the convective region may extend deep into the star and e ciently transport energy away from the core to the surface in a phase known as the rst dredgeup. This process increases the luminosity and the star moves 15 Main Sequence Star H burning core Envelope Red Giant Star Inert He core H burning shell Envelope Horizontal Branch Star He burning core H burning shell Envelope Asymptotic Giant Branch Star CO core He burning shell H burning shell Envelope Figure 2.7. The internal structure of a low mass star at several di erent stages of its evolution (not drawn to scale). As stellar nucleosynthesis increases the mean molecular mass of the star, the core contracts and the temperature increases enough for the onset of H and He shell burning around the core. upward on the evolutionary track. This phase is known as the red giant branch (RGB). When the temperature and density of the core become high enough, the triple alpha process reignites throughout the star and an incredibly rapid release of energy occurs for a few seconds in a process called the helium ash. A signi cant amount of the generated energy is absorbed by the star's envelope and never reaches the surface. As a result, the star may experience some mass loss. After the helium ash, the luminosity rapidly drops as the star begins to settle into a quiescent state, known as the horizontal branch (HB), in which helium burning ignites within the core and the temperature increases. At the end of the horizontal portion of the branch is the horizontal branch loop. At this point, the core begins to shrink due to the increased mean molecular weight and the envelope expands. 16 Shortly after the loop, the core will have converted all of its helium into carbon and oxygen and increased temperatures allow for the creation of a helium burning shell just outside of the CO core. The star is said to lie along the early asymptotic giant branch (E AGB) at this point. The primary di erence between a red giant star and an asymptotic giant branch (AGB) star is the fact that an AGB star has two shell burning regions (H, He) rather than the single H burning shell of a red giant star. In an EAGB star, the hydrogen shell is e ectively dormant and the helium shell dominates the generation of energy. The core, once again, begins to shrink, the envelope expands, the temperature decreases, and the star moves right on the evolutionary track. Convection within the envelope deepens, causing a second dredgeup of heavier elements and an increasing luminosity as the star moves upward on the evolutionary track to the thermalpulse asymptotic giant branch (TPAGB). The hydrogen shell reignites and dominates the energy production while the helium shell shrinks and alternates between phases of burning and dormancy causing helium shell ashes. The ashes push the hydrogen shell outward, causing it to temporarily shut o . Hydrogen burning eventually begins again and the process cycles. These ashes are easily seen in the evolutionary track as rapid changes in luminosity. The increasing radius of the star leads to a weaker surface gravity. This, coupled with pulsations caused by the helium shell ashes, leads to mass loss at rates as high as _M 104 M yr1 in what is known as a superwind (Winckel, 2003). Because of the hydrogen burning shell, the star's temperature will begin to rise without much change in the luminosity as the envelope loses mass. The star will move rightward on the evolutionary track and marks the postasymptotic giant branch (PostAGB), also referred to as the protoplanetary nebulae phase. The central star becomes incredibly hot and emits a signi cant amount of UV radiation which is absorbed by the expelled gas and causes the ionization of atoms. The emission of photons causes the gas within the expanding shell to glow and forms what is called a planetary nebula. Several examples of planetary nebulae are shown in Figure 2.8. Eventually, all of the envelope is expelled and, with little to no mass above the hydrogen 17 Figure 2.8. Several examples of planetary nebulae. From left to right: The Helix Nebula [Credit: NASA, NOAO, ESA, the Hubble Helix Nebula Team, M. Meixner (STScI), and T.A. Rector (NRAO).], the Cat's Eye Nebula [Credit: NASA, ESA, HEIC, and The Hub ble Heritage Team (STScI/AURA). Acknowledgment: R. Corradi (Isaac Newton Group of Telescopes, Spain) and Z. Tsvetanov (NASA).], HD 44179 (also known as the Red Rectangle Nebula) [Credit: NASA; ESA; Hans Van Winckel (Catholic University of Leuven, Belgium); and Martin Cohen (University of California, Berkeley).], and M29 [Credit: Bruce Balick (University of Washington), Vincent Icke (Leiden University, The Netherlands), Garrelt Mellema (Stockholm University), and NASA.]. and helium burning shells, the burning is extinguished and the luminosity rapidly decreases. Final pulses shed the shells and reveal the hot, dense CO core. After nearly 50,000 years, the planetary nebula will dissipate into the interstellar medium (Ryden & Peterson, 2010). The core will spend the rest of its life cooling down to become a white dwarf star. White dwarfs are stars that have roughly the mass of the Sun contained within a volume the size of the Earth. As a result, the density and pressure are immense. Figure 2.9 shows one of the most famous white dwarfs, Sirius B, a companion star to the main sequence star Sirius A. From the StefanBoltzmann law, the density of Sirius B is estimated to be 3.0 109 kg/m3. Integrating Equation 2.3 and solving for the central pressure gives Pc ' 2 3 G 2R2 = 3.8 1022 N m2. Furthermore, the central temperature is estimated, using Equation 2.12, to be 7.6 107 K (Carroll & Ostlie, 2007). To put these values into perspective, a teaspoon of white dwarf matter brought to Earth would weigh over 10 tons (Ryden, 2010). Typical gas and radiation pressure, as explored in Equations 2.8 and 2.9, are not su cient to support a white dwarf against such pressures. However, since these stars exist, there must be some other mechanism at work. This missing link arises in the form of degeneracy pressure. 18 Figure 2.9. An overexposed image of Sirius A (center) and its white dwarf companion Sirius B (bottom left). Sirius A is nearly 104 times as luminous as Sirius B. [Credit:NASA, H.E. Bond and E. Nelan (Space Telescope Science Institute, Baltimore, Md.); M. Barstow and M. Burleigh (University of Leicester, U.K.); and J.B. Holberg (University of Arizona).]. The Pauli exclusion principle states that no two fermions (particles that have half integer spin) may occupy the same quantum state or. More speci cally, no two electrons of an atom may have the same quantum numbers [principle (n), azimuthal (`), magnetic (m), and spin (s)]. As the temperature of a gas decreases, electrons move to lower energy states. However, as a result of the Pauli exclusion principle, they may not all move to the lowest state (the ground state). Instead, the electrons ll whichever unoccupied excited state has the lowest energy. When all of the lowest energy states are lled, the matter is considered to be degenerate. Any further addition of electrons would force the electrons into states of higher energy in order to make room and results in an outward compressionresisting pressure known as electron degeneracy pressure. An electron in a degenerate gas has a maximum energy F , known as the Fermi energy, of F = h2 2me (3 2n)2=3 (2.28) where me is the mass of an electron and n is the number of electrons in the volume. If the gas is fully ionized, the number of electrons per unit volume is n = #electrons nucleon ! #nucleons volume ! = Z A ! mH : (2.29) 19 Combining Equations 2.28 and 2.29, the Fermi energy becomes F = h2 2me " 3 2 Z A ! mH #2=3 : (2.30) If the Fermi energy is larger than the thermal energy of an electron (3/2kT), electrons will not transition into vacant states and the gas will become degenerate. This sets a condition for degeneracy dependent upon temperature and density: T 2=3 < h2 3mek " 3 2 mH Z A !#2=3 = 1261 Km2kg2=3 (2.31) for Z=A = 0:5. The level of degeneracy increases as T= 2=3 decreases. The Heisenberg uncertainty principle states that both the exact position and momen tum can not be simultaneously known. There is some uncertainty in both quantities such that x p h: (2.32) Since electrons are so tightly packed within the degenerate matter of a white dwarf, they must have a corresponding large uncertainty in their momentum, given in one dimension by px px h= x hn1=3 since the spacing between electrons in a degenerate gas is roughly n1=3. In three dimensional space, the momentum becomes p = p 3px p 3 h " Z A ! mH #1=3 : (2.33) Assuming that all electrons have the same momentum, the pressure may be expressed as P 1 3 npv (2.34) where v is the speed of the electrons in the gas. For nonrelativistic electrons, the speed is v = p=me. For relativistic electrons, the speed is v = c, where c is the speed of light. 20 Therefore, the pressure due to a completely degenerate gas of electrons is P = (3 2)2=3 5 h2 me " Z A ! mH #5=3 (2.35) for nonrelativistic electrons and P = (3 2)1=3 4 hc " Z A ! mH #4=3 (2.36) for the relativistic case. This pressure is what is responsible for maintaining hydro static equilibrium within a white dwarf. Equating the expressions for the central pressure and the degeneracy pressure (Equation 2.35), an expression for the radius arises: R = (18 2)2=3 10 h2 GmeM1=3 " Z A ! 1 mH #5=3 : (2.37) Close inspection of Equation 2.37 reveals the inverse relationship between mass and radius: R / M1=3: (2.38) This inverse relationship, due to the fact that electrons must be more tightly packed to generate a larger outward degeneracy pressure, suggests that white dwarfs with the highest masses actually have the smallest radii. There is, however, a limit on the maximum mass of a white dwarf that can be supported by electron degeneracy pressure. The mass limit, known as the Chandrasekhar limit MCh, is approximated by equating the central pressure and the relativistic degeneracy pressure, giving MCh 3 p 2 8 hc G !3=2" Z A ! 1 mH #2 = 1:44 M : (2.39) A sample set of data for various white dwarfs of varying mass and radii are plotted in Figure 2.10, revealing the massradius relation that cuts o near the Chandrasekhar limit. 21 Figure 2.10. The massradius relation for white dwarfs. As the mass of a white dwarf increases, the radius decreases. The dashed red line indicates the Chandrasekhar mass limit of 1.44 M . (Data from Burrows, 2015.) 2.4. Binary Systems If a pair of stars are close enough together, they may end up orbiting a common center of mass. These systems are called binaries and nearly half of all stars in the Universe are a part of binaries or multiplestar systems (Hilditch, 2001). There are three classi cations of binary systems: detached, semidetached, and contact. Around each star in a binary are imaginary surfaces called Roche lobes that de ne the limit of gravitationally bound matter. If both stars in a binary lie well within their respective Roche lobes, it is called a detached system. A binary in which one star lls its Roche lobe is called a semidetached system. The third classi cation, a contact binary, occurs when both stars ll and possibly over ow their Roche lobes. Sketches of the three types of binary systems are given in Figure 2.11. Consider two stars orbiting a common center of mass in a corotating twodimensional plane with masses M1 and M2 separated by a distance a, distances to the center of mass (at the origin) x1 and x2, and distances to a test mass (of mass m) r1 and r2, as depicted in Figure 2.12. Determining the total gravitational potential experienced by the test mass 22 Figure 2.11. The three classi cations of binary star systems. Left: a detached system in which neither star lls its Roche lobe. Middle: A semidetached system in which one star lls its Roche lobe. Right: A contact system in which both stars ll and over ow their Roche lobes, resulting in a common envelope. The loop represents the Roche lobes of the stars, the dark shaded regions indicate the initial sizes of the stars, and the light shaded regions indicate the sizes after expansion. will allow for an imaginary threedimensional surface to be constructed around the two stars, representing a region in which the test mass would have zero relative motion with respect to the coordinate system. This is analogous to surfaces of constant gravitational potential or regions where there is no force on the test mass. To begin, consider the gravitational potential energy U experience by the mass m. The gravitational attraction of the two stars will be balanced by the ctitious centrifugal force (F = m!2r) and the potential energy will have the form U = Ug + Uc where the gravitational potential Ug is Ug = G Mm r (2.40) and Uc is the potential energy due to the centrifugal force. Uc is determined by integrating the centrifugal force expression where Uc = 0 at r = 0: Uc = Z r 0 m!2rdr = 1 2 m!2r2: (2.41) Therefore, the total potential energy experienced by m is U = G M1m r1 + M2m r2 ! 1 2 m!2r2: (2.42) 23 M1 M2 m c x1 x2 r1 r2 r a Figure 2.12. Schematic of two stars in a binary system, each with an assumed circular orbit about the center of mass, used to determine the gravitational potential experienced by a test mass within a corotating frame. The gravitational potential is then determined by dividing the potential energy by m: = G M1 r1 + M2 r2 ! 1 2 !2r2 (2.43) where ! is the angular velocity of the binary, given by ! = 2 Porb = " G(M1 +M2) a3 #1=2 : (2.44) The equipotential surfaces, where d =dx = 0, may be found via Equations 2.43 and 2.44. Figure 2.13 shows a twodimensional cross section of the equipotential surfaces around a typical binary system, the rst three Lagrangian points, and a surface representation of the potential that shows two clear potential wells for the stars. The innermost surfaces that form a gure8 pattern are those which are referred to as the Roche lobes of the two stars. Lagrangian points (L) are locations where an object could maintain a stable position with respect to the two stars. The rst three Lagrangian points, as shown in Figure 2.13, lie along a line that passes through the two stars. Points 4 and 5 lie o to the sides, forming equilateral triangles with the stars. Points 4 and 5 are stable equilibrium positions. An object placed at either of these points, when perturbed, will stay in orbit around the respective point in the corotating frame of the stars. Points 13 are, however, unstable. These points lie at the 24 Figure 2.13. Top: A surface representation of the potential illustrating the two potential wells of the stars. Bottom: A cross section of the equipotential surfaces and the rst three Lagrangian points for a binary system with a mass ratio of 0.2 (van der Sluijs, 2006). three maxima seen on the surface given in Figure 2.13 (one between the two wells and one on each end). Any slight perturbation of an object placed at these points will cause it to fall into one of the potential wells and away from the equilibrium positions. Much like with water lling a hole, a star that is increasing in volume and lling its Roche lobe will begin to ll its potential well. When full, any additional expansion will cause it to over ow its potential well and matter will \spill" over the inner Lagrangian saddle point (L1) between the two stars and into the potential well of the companion star. The potential at any other location is higher (as seen as an increase in surface height in Figure 2.13), requiring more energy to overcome the barrier than that at L1. Thus, L1 is a point in space which allows for mass transfer from one star, when it lls its Roche lobe, to its companion in a semidetached binary system either by direct impact or via an accretion disk. It is possible that the star may expand more rapidly than the rate at which mass transfers through L1 and, in such a case, matter may be completely lost by both stars through the 25 other Lagrangian points or as stellar winds. The e ective radius RL gives the radius of the star when its volume is equal to that of the Roche lobe, de ned by Eggleton (1983) as RL a = 0:49q2=3 0:6q2=3 + ln(1 + q1=3) (2.45) where the mass ratio q is M2=M1 for the lowmass star and M1=M2 for the highmass star. Mass transfer may result from several di erent scenarios including standard stellar evolution, shrinking of the Roche lobe due to angular momentum loss, and/or the accretion of stellar winds. In the case of Roche lobe over ow, the rate _M at which mass is lost from the secondary star at the inner Lagrangian point L1 is _M = v (2.46) where , v, and are the stream's density, velocity, and crosssectional area at L1, respec tively. A simple approximation can be made assuming that the stars are of equal mass and that the transferred mass is purely hydrogen. The thermal velocity of hydrogen is estimated by the rootmeansquare speed given as v = p 3kT=mH and the mass transfer rate becomes _M dR r 3kT mH (2.47) where d is the amount that the secondary over lls its Roche lobe, R is the radius of secondary, k is the Boltzmann constant, T is the temperature of the gas, and mH is the mass of a hydrogen atom. A more detailed derivation (Pringle, 1985; Edwards & Pringle, 1987) applies Bernoulli's law to the mass ow to show that, for an envelope that is a polytrope of index 1.5, the mass transfer rate is _M = C M Porb R R !3 (2.48) where C is a dimensionless constant, Porb is the orbital period of the binary, R is the 26 amount that the secondary over lls its Roche lobe ( R = R RL), and M and R are the mass and radius of the secondary, respectively. Mass transfer and mass loss a ect the parameters of the binary system throughout its evolution, changing the orbital period Porb, separation distance a, angular frequency !, etc. For conservative mass transfer (M = M1 + M2 and _M 1 = _M 2), the total orbital angular momentum J of a system with circular orbits (e = 0) is J = (M1a21 +M2a22 )! = p GMa (2.49) where a1 = (M2=M)a, a2 = (M1=M)a, and = M1M2=M is the reduced mass. Di erentiat ing Equation 2.49 and dividing through by J, with _M 1 + _M 2 = 0 and J_ = 0, gives _a a = 2 _M 1 M1 M2 M1M2 (2.50) which describes how the binary separation is a ected by mass transfer. If M2 < M1, the separation increases and the Roche lobes of the stars will expand and may allow for stable mass transfer. If M2 > M1, the separation decreases and mass transfer will increase and is generally unstable. From Equation 2.44, the angular frequency ! / a3=2, and so !_ ! = 3 2 _a a : (2.51) As the binary separation increases, the angular frequency decreases. From Kepler's third law (GM = 4 2a3=P2) and by de ning initial values with the subscript i before any mass transfer, the nal to initial orbital period ratio is given by Porb Pi = M1iM2i M1M2 !3 : (2.52) Di erentiating Equation 2.52 with respect to time, noting that _M 1 = _M 2, yields _P orb Porb = 3 _M 1 M1 M2 M1M2 = 3 2 _a a : (2.53) 27 Equation 2.53 reveals that the orbital period is directly related to the separation distance. An increase in separation will cause the period to increase and vice versa. There are a di erent set of binary evolution expressions for nonconservative mass transfer but they are overlooked as they primarily deal with stellar winds or catastrophic mass losses and it is assumed that mass transfer is conservative in this study of a cataclysmic variable. 28 Chapter 3 CATACLYSMIC VARIABLES At this point, it is useful to de ne a speci c type of interacting binaries: a cataclysmic variable (CV). A CV is a semidetached binary system in which a white dwarf is the primary and the secondary is a larger, less massive main sequence star. CVs have orbital periods between 23 minutes and 5 days and are characterized by periodic outbursts that increase the system's brightness by a factor of 10106 before returning back to a quiescent state (Warner, 1995). CVs are classi ed according to the nature of their observed intensity variations: Classical novae (CN): Cataclysmic variables with only one nova outburst. These outbursts are accompanied by increases in intensity of anywhere between 6 and 19 magnitudes which last for hours to days before settling back to quiescence for decades to millions of years. These eruptions are the result of thermonuclear fusion of the material that is accreted onto the surface of a white dwarf. Recurrent Novae (RN): Novae that have multiple outbursts of 4 to 9 mag that repeat every 10 to 80 years (Mobberley, 2009). Dwarf novae (DN): Cataclysmic variables that display outbursts of 2 to 5 mag. These outbursts last for some 2 to 20 days and repeat every few days to tens of years. These eruptions are the result of the accretion disks accompanying these systems. Dwarf novae are further classi ed as Z Cam: Exhibit standstills 0:7 mag below maximum brightness, during which no outbursts occur, which last for days to years before returning to minimum. SU UMa: Have standard outbursts accompanied by occasional superoutbursts which are 0:7 mag brighter and ve times longer than standard DN outbursts. U Gem: All other dwarf novae which are neither of the previous two classi ca tions. These have standard DN outbursts at regular intervals of several weeks to months. Novalike (NL): All other cataclysmic variables that are neither novae or dwarf novae. Novalikes are further classi ed as 29 AM Her: Have strong magnetic elds that lock the system into synchronous rotation. Mass transfers via the eld lines and does not form an accretion disk. DQ Her: Have weaker magnetic elds than AM Her systems and an accretion disk with a substructure caused by the eld. UX UMa: Believed to be stuck in a permanent standstill. VY Scl: Similar to UX UMa systems, but occasionally drop by more than one mag and have several DNtype outbursts. SW Sex: Similar to DN but have steady state disks and display no outbursts. There is a fairly signi cant correlation between the type of many CVs and the orbital periods of the systems. A distribution of 591 CV orbital periods is given in Figure 3.1. A number of important conclusions are made (Warner, 1995) below. From the gure, it is clear that 1) there is a noticeable de ciency of CVs with 2:2 Porb 2:8 h in a region known as the period gap, 2) there is a distinct minimum near Porb ' 75 min, and 3) the number of systems with long orbital periods rapidly declines in what is known as the long period cuto . Furthermore, most nonmagnetic CVs lie above the period gap while most magnetic CVs lie below the gap (all with Porb 4:6 h). All U Gem and Z Cam stars (Porb 3:8 h) and nearly all NLs lie above the gap while almost all SU UMa stars lie below the gap. The sudden cuto at a minimum period of 75 min is a result of the lowmass secondaries responding to ongoing mass transfer. If the mass of the secondary is low enough, the star will become fully degenerate and follow the massradius relationship. As a result, a decrease in mass causes the star to expand and mass transfer causes the period to increase. Therefore, there must be a period minimum between when the nondegenerate star shrinks and when the degenerate star expands (Figure 3.2). The period gap may be explained by 30 Figure 3.1. The distribution of cataclysmic variable orbital periods consisting of 454 CVs from Ritter & Kolb (2003) in white and 137 CVs from the SDSS in gray. The light gray region depicts the 23 hour period gap. the sudden shuto of magnetic braking at 3 hrs, which is a theory that explains a loss of angular momentum due to gas being captured by the magnetic eld of the secondary and thrown out of the system. As the secondary loses its outer layers, the core's pressure decreases, there are less nuclear reactions, and the core shrinks. However, mass is transferring too quickly and the star is left with a radius that is too large for its mass. Magnetic braking shuts o , the secondary contracts, and mass transfer ceases. Gravitational radiation shrinks the orbit and mass transfer resumes when the orbital period is 2 hrs. As a result, the system becomes detached between 23 hrs and is too faint to be observed (Hellier, 2001). The long period cuto is due to the requirement that the secondary must be less massive than the primary, which is limited by the Chandrasekhar mass of 1.44 M . The size of the Roche lobes increase with orbital period and, in order to ll a larger Roche lobe, a star must be more massive. As a result, the mass limit leads to a period limit and the number of systems starts to rapidly decline above orbital periods of several hours. 31 Figure 3.2. Evolution of the secondary near the minimum period of 75 min. Orbital periods given in minutes are labeled along the evolutionary track. Note that minimum occurs between the nondegenerate and degenerate phases of the star (Ritter, 1986). 3.1. Accretion Disks The main component of a CV is the accretion disk that is created as matter transfers from the secondary to the primary. The outbursts observed in these systems are believed to be caused by sudden increases in the rate at which matter ows through these disks. Matter that is ejected through L1 travels at a speed vk which is approximately equal to the speed of sound in the gas cs ' 10(T=104 K)1=2 km s1 and gives vk 10 km s1. In addition, there is a component of the speed v? ' 1 2a! that is perpendicular to this motion as a result of the L1 point orbiting with the system. Here, by use of Kepler's third law, a is a = 3:53 1010M1=3 1 (1 + q)1=3P2=3 cm: (3.1) Applying Equations 3.1 and 2.44 give v? 100 km s1. This Coriolis force causes the stream to swing into an orbit around the white dwarf primary instead of directly impacting the star. The stream will sweep past a closest approach location, loop around the white dwarf, and intersect its earlier path as depicted in Figure 3.3, which displays the trajectory of the mass transfer stream at various ejection velocities for a system with a mass ratio q = 0:67. The 32 Figure 3.3. A topdown view of the stream trajectory of particles ejected through L1 at various velocities for a system with a mass ratio q =0.67. The stream collides with itself and will eventually form a circular ring around the white dwarf (Flannery, 1975). distance of closest approach rmin is obtained from trajectory computations (Lubow & Shu, 1975) to within 1% accuracy as rmin a = 0:0488q0:464 (3.2) for 0.05 < q < 1. The interactions caused by the stream colliding with itself at supersonic speeds shock the gas and, thus, dissipate energy. Although energy is radiated away, angular momentum must be conserved and so the stream will settle into the lowestenergy circular orbit and create a ring of orbiting gas around the primary. A particle in a circular orbit a distance r from the primary has a Keplarian velocity v of v = GM1 r ! : (3.3) Applying Equation 3.3 and conservation of angular momentum, an accurate (1%) value for 33 the radius of the orbiting ring rr is obtained (Hessman & Hopp, 1990) to be rr a = 0:0859q0:426 (3.4) for 0.05 < q < 1. The ring rotates di erentially as di erent annuli orbit at di erent speeds and causes a viscous shear ow that heats the gas. As energy is radiated away, the outer annuli must move to larger orbits in order to conserve angular momentum. As a result, the ring spreads out into a thin disk known as an accretion disk. rr is the minimum radius of the disk. The outer radius rd may extend 8090% of the way out to L1, but is limited by tidal interactions with the secondary (Carroll & Ostlie, 2007). A rough estimate is then rd ' 2rr: (3.5) Once an accretion disk is established, the mass transfer stream will impact the outer edge of the disk at supersonic speeds and create an area of shockheated material known as a bright spot (Figure 3.4). The bright spot gets its name from the fact that it has the ability to radiate as much or more energy than all other components (including both stars and the accretion disk) combined (Warner, 1995). It is possible that the stream misses the disk's edge and ows over its surface. In such a case, the stream will follow a single particle trajectory until striking the disk at a radius closer to the primary (Lubow & Shu, 1975). Figure 3.4. A schematic of a CV depicting the mass transfer stream colliding with the accretion disk and causing a bright spot. If the disk is tilted with respect to the orbital plane, the stream may collide with disk at a smaller radius (dashed line). 34 A steady mass transfer stream, from in nity, striking at the location of the bright spot on the disk's outer edge will release energy Ls at a rate Ls ' GM1 rd _M : (3.6) Note that this is simply an upper limit since the stream originates at a nite distance. Material in the disk will be owing at a rate of _M d and the disk's luminosity Ld is Ld = 1 2 GM1 R1 _M d (3.7) where R1 is the radius of the white dwarf. Since rd=R1 30, the luminosity of the bright spot is greater than that of the disk and it is expected that _M d < _M . The other half of the luminosity is at the boundary layer between the disk and the surface of the primary. Accretion disks can be approximated as twodimensional ows since the material is typically con ned closely to the orbital plane (z = 0). The material in the disk, with a surface density = 2 R dz, will move with a Keplerian angular velocity given by = GM1 r3 !1=2 ; (3.8) a circular velocity given by Equation 3.3, and will have a radial drift velocity vr. An annulus of the disk, with inner radius r and outer radius r + r, will have a mass 2 r r and angular momentum 2 r r r2 . The mass conservation equation is obtained by nding the rate of change of the mass in the limit r ! 0 and has a value @ @t + 1 r @ @r (r vr) = 0: (3.9) The conservation equation for angular momentum is similarly obtained and has a value @ @t ( r2 ) + 1 r @ @r (r vrr2 ) = 1 r @ @r r3 d dr ! (3.10) where is the coe cient of e ective kinematic viscosity of the gas (Warner, 1995). Com 35 bining Equations 3.9 and 3.10, eliminating vr, and assuming circular Keplerian orbits ( / r3=2) gives a nonlinear di usion equation for : @ @t = 3 r @ @r " r1=2 @ @r r1=2 # : (3.11) This equation reveals that matter di uses inward toward the white dwarf while angular momentum di uses outward toward the disk's edge. Choosing to be constant, Equation 3.11 can be solved (Pringle, 1981; Frank, King, & Raine, 2002) to display how a ring of material is spread out into a disk structure (Figure 3.5). The gure gives as a function of x = r=r0 for values of = 12 tr2 0 and shows that changes on a viscous timescale t of t r2 r vr (3.12) where the radial drift velocity is implied to be vr r : (3.13) For viscous timescales 1, most of the initial mass M has accreted onto the white dwarf and the angular momentum has been shifted to very large radii. Figure 3.5. The spreading of a ring of mass M with a Keplerian orbit at r = r0 due to viscous torques. is constant and is a function of x = r=r0 and the dimensionless time variable = 12 tr2 0 (Frank et al., 2002). 36 If the accretion rate _M 2 is much slower than the viscous timescale, a stable disk will settle into a steadystate structure with @=@t = 0 and, from integration of Equation 3.9, the mass ow rate within the disk _M d is then expressed as _M d = 2 r(vr) : (3.14) If the disk extends all the way down to the surface of the primary (r = R1), the primary must rotate more slowly than the material in the disk at that location, i.e., 1 < (R1). There must be a thin boundary layer in which the disk material is decelerated to equal the primary's rotational velocity and there is no torque on the disk (d =dr = 0 or 1 = ). This, along with Equations 3.10 and 3.14, set an inner boundary condition = _M d 3 " 1 R1 r 1=2 # (3.15) which leads to an expression for the energy generation by viscous shearing D(r): D(r) = r d dr !2 = 3G _M 1 _M d 4 r3 " 1 R1 r 1=2 # 3 4 2 _M d (3.16) where the last expression is for the limit r R1. It is by integration of Equation 3.16 from r = R1 to 1 over the area 2 rdr that the disk luminosity Ld of Equation 3.7 came to be. The energy of Equation 3.16 is radiated at a rate of 2 T4 e away from the two surfaces of the disk, where the e ective temperature is Te = T1 r R1 3=4 " 1 R1 r 1=2 #1=4 T1 r R1 3=4 (3.17) and T1 = " 3G _M 1 _M d 8 R3 1 #1=4 : (3.18) where the second expression of Equation 3.17 is for the limit r R1 and its derivative yields the maximum disk temperature of 0:488T1 at r = (49=36)R1 (Pringle, 1981). 37 The motion of gas is governed by the Euler equation @v @t + v rv = rP + f: (3.19) Since there is little to no ow in the zdirection, hydrostatic equilibrium must hold and the zcomponent of Equation 3.19, with the neglection of any velocity terms, becomes @P @z = @ @z " GM1 (r2 + z2)1=2 # = zGM1 r3 (3.20) for a thin disk (z r). This may be solved for to get = cez2=2H2 (3.21) where c is the central plane (z = 0) density of the disk and H is the disk's scale height. The scale height is estimated by noting that P = c2s , where cs is the speed of sound in the gas and P is the sum of the gas and radiation pressures, and setting @P=@z P=H and z H: H = cs r3 GM1 1=2 = cs : (3.22) H r for a thin disk and so cs . This reveals that the local Keplerian velocity is highly supersonic and that density rapidly falls o with height above the central plane. If the disk is optically thick, Equation 3.20 is solved along with the equation for radiative transfer to get the radiative ux F through the disk's faces: F = 16 T3 3 R @T @z : (3.23) If the disk is optically thin and isothermal, the emitted ux is then F = T4 e = R T4 c (3.24) where R is the Rosseland mean opacity and Tc is the midplane temperature. Since an annulus of the disk subtends a solid angle 2 rdr cos i=d2, an observer a distance d away from 38 the system will view the disk's ux distribution F as F = 2 cos i d2 Z rd R1 I rdr / 1=3 Z 1 0 x5=3 ex 1 dx (3.25) where i is the inclination of the disk, is the frequency, and I is the intensity, which may be approximated as a blackbody distribution B : I = B = 2h 3 c2 eh =kT 1 1 (3.26) where h is the Planck constant and c is the speed of light. The second term in Equation 3.25 is valid in the frequency range kT(rd)=h kT1=h where x = h =kT. As a result, if T1 T(rd), the steady state disk spectrum is characterized by F / 1=3 or F / 7=3 (LyndenBell, 1969). For frequencies kT(rd)=h, the spectrum takes the RayleighJeans form and F / 2. Frequencies kT1=h yield a Wien spectrum 2h 3c2eh =kT . The full spectrum, given by Equations 3.25 and 3.26, is shown in Figure 3.6 where the dependency in the three regimes is rather clear and resembles a stretchedout blackbody spectrum. Shakura and Sunyaev (1973) hypothesized that viscosity is enhanced by the presence of turbulence, in the form of subsonic eddie currents, within the disk's gas. Their model, Figure 3.6. The total spectrum F for a steady state optically thick disk at di erent ratios rd=R1. The frequency is normalized to kTout=h where Tout = T(rd) and there are three primary regimes de ned by kT(rd)=h and kT1=h (Frank et al., 2002). 39 known as the disk model, estimates the viscosity to be = csH = H2 (3.27) where is a free parameter with a value that is roughly between 0 and 1. Although the presence of a free parameter is typically detrimental, most of the observables only weakly depend on and so the theory still gives much insight into the physics of accretion disks. With an opacity given by Kramers' law R = 5 1024 T7=2 c cm2g1, the ShakuraSunyaev solution for many of the disk parameters in terms of is = 5:2 4=5 _M 7=10 d;16 M1=4 1 r3=4 10 f14=5 g cm2 H = 1:7 108 1=10 _M 3=20 d;16 M3=8 1 r9=8 10 f3=5 cm = 3:1 108 7=10 _M 11=20 d;16 M5=8 1 r15=8 10 f11=5 g cm3 Tc = 1:4 104 1=5 _M 3=10 d;16 M1=4 1 r3=4 10 f6=5 K = 190 4=5 _M 1=5 d;16f4=5 = 1:8 1014 4=5 _M 3=10 d;16 M1=4 1 r3=4 10 f6=5 cm2s1 vr = 2:7 104 4=5 _M 3=10 d;16 M1=4 1 r1=4 10 f14=5 cm s1 9>>>>>>>>>>>>>>>>>>= >>>>>>>>>>>>>>>>>>; (3.28) where r10 = r=(1010cm), M1 = M=M , _M d;16 = _Md=(1016g s1), and f = [1 (R1=r)1=2]1=4 (Shakura & Sunyaev, 1973). The solution has many implications including con rmation that the disk is optically thick, uniform in the vertical direction, may extend out to a large radii on the order of the Roche lobe, has a negligible mass, does not self gravitate, etc. (Frank et al., 2002). From Equation 3.28, the ratio of the disk's height to its radius, for r R1, gives H / r9=8: (3.29) This implies that the disk's faces are concavely shaped and may be heated/illuminated due to radiation from the central regions and boundary layer (Figure 3.7). Just beyond the surface of the white dwarf, r = R1, there exists a boundary layer of radial extent b in which the angular velocity of the disk material must decelerate to match 40 White Dwarf Disk Disk Figure 3.7. The vertical cross section of a concave accretion disk in the disk model. that of the star's surface 1. An approximate form of (r) is plotted in Figure 3.8. The energy released within the boundary layer LBL can be solved for by use of conservation of energy and angular momentum (Kley, 1991) and has a value LBL = Ld " 1 1 (R1) #2 (3.30) which reduces to Ld for 1 (R1). As a result, the boundary layer may emit as much radiation as the entire accretion disk. A vertical cross section of the boundary layer geometry is given in Figure 3.9. For an optically thick disk, the energy radiated by the boundary later must traverse a region of width H on both faces of the disk. This radiation is roughly that of a blackbody of area 2 R1H 2 and must equal Ld as given by Equation 3.7. Therefore, the e ective temperature of the boundary layer TBL for an accreting white dwarf is TBL 1 105 _M 7=32 d;16 M11=32 1 r25=32 9 K (3.31) (Frank et al., 2002). Due to these high temperatures, the boundary layer typically emits in the soft Xray and EUV regions while the outer portions of the disk may be cool enough to be IR radiators. If, however, the boundary layer is optically thin, radiation from the shock front where the disk material meets the primary's surface escapes with a temperature Tsh of Tsh = 3 16 mH k GM1 R1 = 1:85 108M1r1 9 K (3.32) (Warner, 1995) and so an optically thin boundary layer may emit in hard Xrays. 41 Figure 3.8. The angular velocity (r) near the surface of the primary with 1. The outer edge of the boundary layer at R1+b is given by the dotted line and the dashed line represents (r) if there were no boundary layer. (Frank et al., 2002). With a general understanding of what happens at the inner portion of the disk where material encounters the primary, it is useful to now shift to the outer portions of the disk where angular momentum is disposed of. Since the primary absorbs angular momentum at a rate ( _M p GMR1) that is much less than that which is supplied by the secondary ( _M p GMr), there must be a signi cant torque acting on the edge of the disk that soaks up angular momentum. The disk may extend out to radii that approach that of the primary's Roche lobe and the orbits of the particles will become distorted from circular motion due to the presence of the secondary, forming a bulge in the disk which takes an elliptical shape. The disk material orbits more quickly than the secondary, but the gravitational attraction of the secondary holds the bulge back, slightly ahead of a line that connects the two stars in the binary. The pulling of the bulge material slows it down and, thus, reduces its angular momentum. The balance of angular momentum can be expressed as _M p GMr = _M p GMR1 + Gtidal (3.33) where Gtidal is the tidal torque. Because of the loss of angular momentum at the disk's edge, 42 White Dwarf H Disk H b Boundary Layer Figure 3.9. The vertical cross section of an accretion disk depicting an optically thick bound ary layer (not to scale). Gtidal will halt any further expansion of the disk and therefore de nes the disk radius. The maximum radius of the disk rtidal set by the tidal limit is approximated by rtidal = 0:6a 1 + q ' 0:9RL: (3.34) A more detailed discussion of elliptical disks will follow in the discussion of superoutbursts. 3.2. Outbursts Accretion disks occur in many di erent types of astrophysical systems beyond those of dwarf novae and cataclysmic variables, however those of dwarf novae are the most useful to astronomers. Accretion disks are key components of quasars and protoplanetary systems, but quasars are too distant and protoplanetary systems are too heavily shrouded in dust for the study of their accretion disks. On the other hand, dwarf novae occur everywhere in the Universe and astronomers are able to make clear observations of them to obtain their light curves. The most prominent features of a dwarf novae's light curve are the semiperiodic outbursts that appear as rapid increases of several magnitudes in brightness. The rst ever dwarf novae, U Geminorum, was discovered in 1855, but the mechanisms of their outbursts were not well understood until 1974 when Brian Warner used the data of Z Chamaeleontis to 43 reveal that they are caused by sudden increases in the luminosity of accretion disks (Warner, 1995). A fouryear portion of U Gem's visual light curve displaying several outbursts is given in Figure 3.10. The outbursts of this system are semiregular and repeat every 100 20 days. Each outburst increases the system's luminosity by a factor of 100 in one day and by 5 mag, from 14 to 9 mag, in total. No two outbursts, even those in the same star, are exactly alike. There are, in fact, a few di erent shapes that DN outbursts may take. There may be variations in the rises, declines, and total duration. Take, for example, the light curve of SS Cyg (Figure 3.11). This system appears to alternate between short and long duration outbursts. All but the second outburst have rapid rises of 2 days and slow declines of 8 days. The second outburst is more symmetric due to the longer rise. The longer outbursts appear to have a plateaulike structure at maximum brightness that lasts for 10 days. Schematics of the three most common types of DN outburst pro les are given in Figure 3.12. Furthermore, one of the more distinct properties of DN outbursts is the bimodality of their duration. Figure 3.13 gives a clear separation into two groups of outburst durations from SS Cyg (Bath & Paradijs, 1983). Bimodality is observed in many DN but is most dramatic in SU UMa systems. Figure 3.10. A fouryear portion of U Gem's visual light curve displaying semiperiodic outbursts of 5 mag that repeat every 100 20 days [Data compiled by the American Association of Variable Star Observers (AAVSO)]. 44 Figure 3.11. A oneyear portion of SS Cyg's visual light curve displaying various outburst shapes (Data compiled by the AAVSO). Osaki (1974) proposed a cause for outbursts based upon an accretion disk's instability (DI) and it was quickly agreed upon by many that the passage of matter through the accretion disk is the source of the outbursts. Consider a system for which the mass transfer from the secondary _M is greater than the rate at which matter is transported through the accretion disk _M d. In this case, matter would begin to build up in the disk and lead to an instability which increases the viscosity, angular momentum, and the radius of the disk. The spreading of the disk leads to enhanced accretion onto the white dwarf and increases the system's luminosity (seen as an outburst) while depleting the disk of matter. The disk will eventually return to a lowviscosity quiescent state where it will gain its mass back before returning to a new outburst phase. Figure 3.12. Schematics of three types of DN outburst pro les (Hellier 2001). 45 Figure 3.13. The bimodal distribution of SS Cyg's outburst durations (Bath 1983). 3.3. Disk Instability The study of DI begins with a theoretical, and widely accepted, explanation for the source of viscosity within accretion disks: magnetorotational instability (MRI). This the ory, also known as the BalbusHawley instability, suggests that magnetic instabilities drive turbulence within the disk. To understand how this works, consider a magnetic eld line that connects two pockets of ionized matter at di erent radii (di erent annuli). Matter at a smaller radii orbits more quickly and so acts to stretch the eld line and accelerate the pocket at a larger radii. The acceleration of the outer pocket gives it angular momentum and so it moves to a larger radii and slows down the inner pocket, causing it to lose angular momentum and shift to a smaller radius. As a result, the eld lines are further stretched and the magnetic eld is strengthened, leading to turbulence. A simple analogy for this e ect is to consider the magnetic eld as a spring connecting two objects in orbit (Figure 3.14). The inner object is moving faster than the outer one and so the inner object is pulled back while the outer one is pulled forward. This causes angular momentum to be transported outward. In order to better understand the transport of disk material in this manner, consider a ver tical magnetic eld line that is very slightly perturbed to larger and smaller radii in several locations. Figure 3.15 shows the sequence, from left to right, of the growth of the 46 Figure 3.14. An analogy for magnetorotational instability. Two objects at di erent radii in Keplerian orbits are connected by a spring. The inner object orbits more quickly and gets pulled back while the outer object is pulled forward. The inner object falls inward while the outer object moves to larger radii. magnetorotational instability. Here, the line's deviations are enhanced until the line recon nects and pockets of ionized material are transported to di erent radii. A hot disk of ionized matter will posses many free electric charges that may couple with the magnetic eld and induce MRI and high viscosity. On the other hand, a cold disk will posses many neutral charges since the electrons will combine with various nuclei and MRI will no longer apply. As a result, DN outbursts are caused by the disk ipping between hot and cold states. Cold (a few thousand Kelvin) disk material is unionized and has a low opacity to radiation. As the disk material heats up to 5000 10,000 K, the material Figure 3.15. The growth of the magnetorotational instability displayed by a vertical eld line (perpendicular to the plane of the disk) with slight deviations to larger and smaller radii. The deviations are enhanced and the magnetic eld strengthens until it reconnects and transports material to di erent radii (Hellier, 2001). 47 becomes partially ionized and the opacity increases. The opacity becomes extremely sensitive to temperature (Faulkner, Lin, & Papaloizou, 1983): R = 1:0 1036 1=3T10 cm2g1 / T10: (3.35) and leads to instability. Consider a slight increase in temperature at a location within the disk. The higher temperature increases the motion of the particles and, therefore, the viscosity. Due to a more viscous material, the disk spreads out to larger radii and the situation is considered to be stable. However, once the temperature is high enough for partial ionization to occur, any increase in temperature greatly increases the opacity. The high opacity traps the heat generated by viscous interactions and so the temperature rapidly increases. Once fully ionized, the opacity loses its sensitivity to temperature (Faulkner et al., 1983) and the disk becomes stable at a higher temperature than before: R = 1:5 1020 T2:5 cm2g1 / T2:5: (3.36) In the now highly viscous state, material is transported through the disk more rapidly than it is supplied by mass transfer and so the hot stable state is only brie y sustained before the disk returns to its original quiescent state. The transitions detailed above may be visualized by a simpli ed cycle known as an Scurve (Figure 3.16). Consider a stable disk at quiescence that begins at point A on the Scurve. If the mass transfer from the secondary supplies the disk with matter more quickly than it is able to move through the disk, matter will build up in the disk and increase its density . The more massive disk will have an increased viscosity and increase in temperature until it reaches point B where ionization sets in and the critical density max is de ned to be (Cannizzo, Shafter, & Wheeler, 1988) max = 11:4r1:05 10 M0:35 1 0:86 C g cm2 (3.37) where C is determined for the cold disk. At this point, any increase in density causes a 48 T A B C D min max Ionized Partially Ionized Unionized 20,000 K 7,000 K 3,000 K Figure 3.16. The simpli ed Scurve relating an accretion disk's density to its temperature T. A disk follows the cycle from A ! B ! C ! D ! A as described in the text. runaway heating e ect that is much quicker than the time it takes matter to move through the disk. The result is a temperature increase without any associated change in the surface density. Once the matter is ionized (point C), the disk reaches a new stability at a higher temperature with a greater luminosity. During this period of stability, viscosity is high enough that the infall of matter is more rapid than that which is supplied by the secondary. As a result, the disk is drained of material and decreases in temperature until point D where the critical density min is de ned to be (Cannizzo et al., 1988) min = 8:25r1:05 10 M0:35 1 0:8 C g cm2 (3.38) where H is determined for the hot disk. From here, partial ionization sets in again and the opacity returns, allowing for the temperature to rapidly decline until the ions recombine and the disk is returned to its colder quiescent state at point A. Outbursts begin as a heating transition wave formed by an instability in an annulus of the disk that spreads into adjacent annuli. How an outburst evolves is entirely dependent 49 upon the radius and distribution of matter where the outburst begins. A stable disk will have a density between the two critical values (Figure 3.17). If the mass transfer is rapid, matter will not e ciently di use inward and will build up at large radii. As a result, the instability would form in the outer disk ( rst panel on the right of Figure 3.17). The heating wave will progress inward and enhance the density in the annuli it passes through (second panel on the right). If, however, the mass transfer is slow, the matter will di use inward and build up at small radii. As a result, the critical density will be reached in the inner disk ( rst panel on the left) and the resulting heating wave will propagate outward. As the heating wave moves into adjacent annuli, the viscosity within them increases and so the inward di usion of matter increases. Therefore, the inner regions of the disk increase in density and the accretion rate increases (second panel on the left). At the end of the outburst phase, the result is generally the same: the density pro le is reversed so that it is now greater in the inner regions of the disk and the accretion rate is enhanced (Hellier, 2001). The increased accretion rate drains the disk of matter until it reaches min near the outer edge of the disk where min is the largest. At this point, the region leaves the outburst phase and a cooling wave propagates inward, lowering the density, pulling matter back outward, and returning the disk to a steady quiescent state that is similar to that before outburst. It is the nature of these waves that de ne the various outburst shapes displayed in Figure 3.12. The short rise times are due to outsidein outbursts. The heating wave propa gates rapidly due to a number of factors. The viscosity leads to more matter owing inward rather than out. The outer annuli also have a higher density and so the spreading of the matter has a signi cant e ect on the inner annuli. Furthermore, the density is increased by the infalling matter since annuli at smaller radii are smaller in size. As a result, the wave overtakes adjacent annuli quickly and the enhanced accretion is sudden. Slow rise times are due to insideout heating waves. The same arguments made above act to decrease the wave's outward velocity. Accretion is more steadily enhanced in this case as an increasing amount 50 of the disk acts to transport matter inward. The time scale for outburst rise r is given by r = 0:14P1:15(h) d mag1 (3.39) where P(h) is the orbital period in hours (Warner, 1995). Figure 3.17. A DN disk's surface density plotted against its radius for several instances during an outburst cycle. The disk's pro le begins at the top. High mass transfer rates cause an outsidein heating wave and low mass transfer rates cause an insideout heating wave. Both routes leave the disk with a greater density in the inner regions and an enhanced accretion rate which drains the disk and forms an inwardmoving cooling wave that returns the disk to quiescence. (Hellier, 2001). 51 Short outbursts occur when the heating wave is unable to reach the edge of the disk. The cold region at the disk's edge then acts to pull matter from the hot region and quickly causes a cooling wave that ends the outburst phase. Plateaus, on the other hand, occur when the entirety of the disk partakes in sustained outburst. The minor decline in magnitude during a plateau is a result of the motion from points C to D on the Scurve. When short outbursts occur, they leave the outer regions of the disk with more material and so the next outburst will likely engulf the entire disk. Because of this, it is often observed that long and short outbursts alternate. The width of an outburst T0:5 is given by (Warner, 1995) as T0:5 = 0:90P0:80(h) d: (3.40) The decline of an outburst are all similar since the cooling waves always originate in the disk's outer region and propagate inward. The time scale for outburst decline is given by d = 0:53P0:84(h) d mag1: (3.41) The overall e ects of an outburst on luminosity, disk radius, disk mass, and angular momen tum are simulated (Ichikawa & Osaki, 1992) and displayed in Figure 3.18. 3.4. Positive and Negative Superhumps Consider, now, the light curve of VW Hyi (Figure 3.19). While there are numerous typical DN outbursts, there are also several superoutbursts which are roughly 0.51.0 mag brighter and 510 times longer. Accompanying each supoeroutburst is a superhump; a humpshaped photometric modulation which appears near maximum intensity. Initially observed by Nicholas Vogt (1974), these superhumps have a period that is about 3% longer than the orbital period. Vogt correctly predicted that superoutbursts were the result of 52 Figure 3.18. Luminosity, disk radius, disk mass, and angular momentum variations in DN, based upon model simulations (Ichikawa & Osaki, 1992). an accretion disk becoming elliptical during outburst. An elliptical disk will precess in the prograde direction with a period P+ prec given by 1 P+ prec = 1 Porb 1 P+ (3.42) where Porb is the orbital period and P+ is the superhump period. Snapshots from a smoothed particle hydrodynamics (SPH) simulation of such a disk, for a system with q = 0:25, are given in Figure 3.20. Disks that extend to large radii are subjected to tidal forces due to the presence of the secondary. Material in the inner regions of the disk remain in nearly Keplerian orbits while the orbits of those closer to the disk's edge become increasingly elliptical. A consequence of this is that the orbits of adjacent particles are no longer parallel and will intersect, causing the dissipation of energy due to the collision of particles. Therefore, as the 53 Figure 3.19. An eightmonth portion of VW Hyi's visual light curve displaying both normal DN outbursts and superoutbursts (Data compiled by the AAVSO). secondary passes by the disk's bulge, the tidal stresses will lead to an increased luminosity (superoutbursts) where the orbits intersect at the outer regions of the disk. A 3:1 resonance, for which disk material orbits three times for every orbital period, is thought to be the driving force of ellipticity in accretion disks. The resonant orbits will precess at a rate dependent upon the mass ratio of the system: Porb=P+ prec = 0:233q(1 + q)1=2 (Patterson, 1998). This may also be expressed as a superhump period excess + in terms of Porb: + = P+ Porb Porb : (3.43) The superhump period excess is plotted against the orbital period in Figure 3.23. The evolution of a system leading to a superoutburst is known as a supercycle. The general outline of events in such a case was rst proposed by Osaki (1996) and is summarized in Figure 3.21 and the preceding discussion. Mass transfer from the secondary supplies the disk with more mass than what is removed by regular DN outbursts and so the disk will gradually gain mass and spread to larger radii throughout the outbursts. At some point, when r 0:46a, the disk will expand enough to where it is tidally in uenced by the secondary. The disk will then become elliptical and begin to precess. Frames 1 and 6 in Figure 3.20 show such an elliptical disk as superhump maximum where convergent ows are strongest. 54 Figure 3.20. Snapshots from a SPH elliptical accretion disk simulation for q = 0:25. The 100,000 particles are colorcoded by their luminosity (Wood et al., 2011). In frame 3, the mass transfer stream impacts the disk edge much deeper into the potential well, releasing more energy and causing what are known as late superhumps. The tidal in uence of the secondary further drains the disk of angular momentum and enhances the inward ow of matter through the disk so that the disk is sustained in the hot, highly viscous state. The accretion rate will begin to exceed the rate of mass transfer and the disk will shrink. The disk will drain and remain elliptical until r 0:35a, at which point most of the acquired mass has been accreted. Because this involves much more of the disk's mass, superoutbursts have a much longer duration than regular DN outbursts. While many systems display superoutbursts with superhump periods a few percent longer than the orbital period, some may display humps that are a few percent less than the 55 Figure 3.21. Luminosity, disk radius, disk mass, and angular momentum variations of a supercycle, based upon model simulations (Ichikawa, Hirose, & Osaki, 1993). orbital period. These humps are known as negative superhumps (or infrahumps). Negative superhumps are the result of a tilted accretion disk relative to the orbital plane. Disks that are tilted will precess in what is called nodal precession, where nodes are the points at which the disk's edge passes through the orbital plane. Nodal precession is opposite to the direction of the orbital motion (retrograde) as seen in Figure 3.22. Figure 3.22. A tilted accretion disk undergoing retrograde precession through one cycle. In this gure, the disk precesses in the clockwise direction, whereas the orbital motion is counterclockwise (Hellier, 2001). 56 The period P prec of the retrograde precession of the tilted disk is given by 1 P = 1 Porb + 1 P prec (3.44) where P is the negative superhump period. Furthermore, the fractional period o set of the negative superhumps, or period de cit, is = P Porb Porb : (3.45) The negative superhump period excess is plotted against the orbital period in Figure 3.23. The physical source of negative superhumps is the varying radial location of the bright spot within the white dwarf's potential well (Wood, Thomas, & Sampson, 2009). In a standard system with an nontilted disk, the mass transfer stream will strike the edge of the disk. However, if the disk is tilted, the stream will pass the edge of the disk and collide with one of its two faces. Negative superhump maxima occur when the stream collides deepest into the potential well (when the bright spot is located at a minimum radius). Having fallen further, the stream will collide with more energy and the luminosity will increase. Figure 3.23. Superhump period excess (circles) and negative superhump period de cits (squares) plotted against the orbital period with linear ts included (Hellier, 2001). 57 Chapter 4 ANALYSIS OF V1101 AQUILAE V1101 Aql is a cataclysmic variable star that is often considered to be a Z Camtype dwarf nova, though this classi cation is not yet con rmed. V1101 Aql was rst classi ed as an irregular variable (Kholopov, 1987), but observations by Meinunger (1965), Vogt & Bateson (1982), Downes & Shara (1993), and Masetti & Della Valle (1998) have all suggested the Z Cam nature of the star. Downes, Hoard, Szkody, & Wachter (1995) also suggests that V1101 Aql could be a Herbig Ae/Be star. Z Cam stars, such as Z Cam in Figure 4.1, have orbital periods between 3.05 and 8.40 hours, active outburst cycles between 10 and 30 days, and occasional standstill in which the disk becomes stuck in outburst due to an irradiated secondary (Meyer & MeyerHofmeister, 1983). As will be shown in the following discussion, many parameters and characteristics of V1101 Aql match those of typical Z Cam stars. Observations for this study were made by several amateur astronomers from the Center for Backyard Astrophysics between July 15, 2013 (BJD 2456488) and September 17, 2013 (BJD 2456553) and were compiled by Joe Patterson of Colombia University before being sent for data reduction. The total combined meanzero light curve for the 2 months of data is plotted in ux units in Figure 4.2. The data displays four clear DN outbursts and Figure 4.1. A threeandahalf year portion of Z Cam's visual light curve displaying two standstills (Data compiled by the AAVSO). 58 Figure 4.2. The light curve of V1101 Aql in ux units. A portion of the plot is enlarged in order to display several of the system's negative superhumps. the enlarged section in the top panel reveals a higher frequency signal the system's negative superhumps. In order to study the nature of the negative superhumps, the large amplitude signal of the DN outbursts must rst be removed. There are clear gaps in the data corre sponding to times when the system was not being observed. The data was split if the gaps were greater than 0.2 days and each chunk of data was placed within its own le giving 67 total les. Each chunk of data was then set to mean zero ux by nding the average 1 2(max + min) values of the data points of each chunk and subtracting the average from them. At this point, many of the chunks still had a signi cant overall slope to them (such as those during the rise/decline times of outbursts). To remove them, a polynomial was t to appropriate chunks and then subtracted o from the original data. For most chunks, a rst degree polynomial of the linear form y = mx+b was su cient. The 67 modi ed chunks were then recombined and the nal result of these processes is displayed in the bottom panel of Figure 4.3 where the large amplitude signal has successfully been removed. 59 Figure 4.3. Top: The original light curve. Bottom: The same light curve, but with the large amplitude signal removed and all chunks shifted to have a meanzero ux. It is clear that the data of V1101 Aql displays oscillatory characteristics. Often, a single signal is composed of multiple sinusoidal waves that are added together. A signal y(t) consisting of N sine waves can be expressed as y(t) = XN j yj sin(2 jt + j) (4.1) where yj is the amplitude, j is the frequency, and j is the phase of the jth component. A signal in the time domain, such as that of Eqn 4.1, may be transformed into the frequency domain Y ( ) via a Fourier transform: Y ( ) = Z 1 1 y(t)e2 i tdt: (4.2) The Fourier transform Y ( ) gives a direct measure of the frequencies (and hence the periods), allowing for a convenient way to view a signal in the frequency domain. 60 A discrete Fourier transform (DFT) replaces the in nite integral with a nite sum: Y ( k) = NX1 n=0 yn exp2 ikn=N: (4.3) DFTs were performed on both the original light curve (top panel of Figure 4.4) and the light curve with the large amplitude signal removed (bottom panel of Figure 4.4) in order to extract the frequencies of the DN outbursts and negative superhumps, respectively. The dominant peak at 0.066 cyc/day in the top panel corresponds to the DN outburst frequency and gives a period PDNO = 15:15 days. The other peaks near 1 cyc/day are a result of gaps in the data sampling and are not real. The dominant peak at 6.168 cyc/day in the bottom panel corresponds to the negative superhump frequency and gives a period PNSH = 3:891 hrs. The two peaks at 5.168 and 7.168 cyc/day are the result of an e ect known as aliasing. Figure 4.4. Top: The DFT of the original light curve with a single dominant peak at 0.066 c/d (P ' 15.15 days) which identi es the period of the DN outbursts. Bottom: The DFT of the light curve after removing the large amplitude signal. The dominant peak at 6.168 c/d (P ' 3.89 hr) identi es the negative superhump period. The two peaks at 5.168 and 7.168 c/d are one day aliases. The orbital period has been detected at 5.870 c/d (4.09 hr). 61 The spectral components of a signal that consists of a number N of data points sampled at an interval t can be recovered by a Fourier transform if the frequency is below the Nyquist frequency Nyquist = 1=2 t. Consider a sine function f(t) = sin(2 t) with = 2 Hz and t = 0:4 s (top panel of Figure 4.5). In this case, > Nyquist = 1=2(0:4) = 1:25 Hz. The DFT of such a signal will result in the power spectrum given in the middle panel of Figure 4.5. There are peaks at 0.5 Hz and 2.0 Hz, where the latter corresponds to a negative frequency = 0:5 Hz. This frequency derives from the fact that the Fourier transform above and below the Nyquist frequency are related by YN=2n = Y N=2+n. A peak at 2.0 Hz, 2:0 1:25 = 0:75 Hz above the Nyquist frequency, will generate a peak 0.75 Hz below the Nyquist frequency as well (at 0.5 Hz). This folding back of the frequency is known as aliasing. To visualize this e ect, a 0.5 Hz sinusoid is overlaid upon the existing sine function t y(t) f(t) = sin(4 t) (Hz) jF(t)j2 Nyquist = 1:25 0:5 2:0 t f(t); g(t) f(t) = sin(4 t) g(t) = sin(2 t) Figure 4.5. Top: A sine function f(t) = sin(2 t) with = 2 Hz and t = 0:4 s. Middle: The power spectrum of the above sine function. Note the existence of an additional peak at 0.5 Hz opposite to the 2.0 Hz peak about the Nyquist frequency. Bottom: The original signal overlaid with a 0.5 Hz sinusoid. The two frequencies are indistinguishable. 62 in the bottom panel of Figure 4.5. Both sinusoids pass through all of the sampled data points and their frequencies become indistinguishable. The two peaks at 5.168 and 7.168 cyc/day are 1day aliases of the primary negative superhump frequency of 6.168 cyc/day. These aliases appear as a result of the fact that frequencies of 1 day also tend to t the data well and so we see a partial signal at the corresponding frequencies in the power spectrum. A second pair of weaker 1day aliases are present at 4.870 and 6.870 cyc/day and help to reveal the system's orbital period of Porb = 4:089 hrs (5.870 cyc/day). Following the Fourier analysis, the data was split into two separate les of high and low states. Referring back to Figure 4.3, the high state data are the points in the bottom panel that correspond to relative ux values in the upper panel greater than zero. The remaining points constitute the low state data. The average pulse shapes of the negative superhumps, as shown in Figure 4.6, were then generated with the frequency input of 6.168 cyc/day for the entire data set (top panel) and for each of the high and low states (bottom panel). The average high state pulse shape was o set by +0:5 for clarity so that the data points of the two data sets did not overlap. It is immediately apparent that the amplitude of the high state data is less than that of the low state data. In order to extract further information from the system, a single sine curve of the form y(t) = Asin[2 (t T0)=P] (4.4) was t to the light curve with the large amplitudes removed and to the high and low state data sets using the LevenbergMarquardt method for nonlinear least squares curve tting (Press et al., 2007). For example, consider a set of m data points (ti; yi) for which a function ^y(ti; p) is t, where t is an independent variable and p is a vector of parameters of the tted curve. The goal is to minimize the sum of the squares of the deviations between the data and the tted function and is given by the chisquared error criterion as 2(p) = Xm i=1 = " y(ti) ^y(ti; p) wi #2 (4.5) 63 where wi is a measure of the error of y(ti). For nonlinear tting, this process is iterative and each iteration nds some parameter perturbation that reduces the value of 2. The results of the t to the full set of V1101 Aql data are P = 0:16214 2:7800 106 d T0 = BJD 2456489:9814 0:0006 A = 0:83344 0:0095649: This period is consistent with that which was determined from the DFT. The important results from the ts to the high and low data are the amplitudes of 0.70358 0.014024 and 0.97718 0.011770, respectively. This further proves that the amplitude is changing and the it is highest during the quiescent phases. Figure 4.6. The average pulse shape of the negative superhumps for the entire set of data (top) and for each of the high and low sets of data (bottom), where the high data is o set by +0:5 for clarity. The dashed lines represent sine ts to the data. 64 The period of the full data set was then used to construct an OC diagram. The program used to create the diagram steps through the data with a dt determined by time boundaries and shows ts over the data. An OC diagram compares the observed time of an event (O), such as the midpoint of a pulsation cycle peak, to the calculated time of the event (C) as determined by a tted curve with a constant period. For each event, the calculated time is subtracted from the observed time and the di erence is plotted versus time. The resulting plot has three possible outcomes as it may either be linear and horizontal, linear and sloped, or curved. A horizontal line in an OC diagram implies that the t perfectly matched what is observed and that there is no change in the period. A straight but sloped line implies that, while constant, the period of the t is di erent from that which is observed. A positive slope or a negative slope imply that the actual period is longer or shorter than that of the t, respectively. The discrepancies will accumulate and lead to larger OC values over time (Batten, 1973). Curves present in the diagrams indicate that the period, or phase, is variable and changing with time. Deviations to a smaller OC value means that the period is decreasing and the events are occurring earlier than predicted. Deviations to a larger OC value means that the period is increasing and the observed times are delayed. These diagrams give a signi cant amount of information about the system's time evolution. The middle panel of Figure 4.7 shows the OC diagram for the light curve with the large amplitude signals removed for the period determined by the sine t mentioned above. Recall that the negative superhump light source is the bright spot sweeping across the face of an accretion disk, which precesses in a retrograde direction. Variations in the OC diagram indicate a variable disk precession rate. The precession rate may vary as a result of changes in the moment of inertia in the disk. A disk with a small radial extent will have a long precession period and a correspondingly large negative superhump period. A disk that does not precess (i.e., a disk with an in nite precession period) will have a negative superhump period equal to the orbital period of the system. On the other hand, a disk with a large radius will have small precession period and negative superhump periods. 65 Consider four regions, de ned by the BJD 2456000, in the OC diagram of Figure 4.7: region I (BJD 518  524), region II (BJD 524  534), region III (BJD 534  540), and region IV (BJD 544  548). Regions I and III have a generally positive trend, indicating that the actual period is longer than the mean period, the precession rate is slower, and that the disk's mass is weighted to smaller radii. Regions II and IV have a generally negative trend, indicating that the actual period is shorter, the precession rate is faster, and that the disk's mass is weighted to larger radii. Furthermore, there is slight curvature visible throughout the regions. For example, in region I, while the values are increasing, they are increasing at a decreasing rate. As a result, the retrograde precession rate is increasing throughout this region and nearly matches that of the t for BJD 522  524 where the dots lie on a nearly horizontal line. In region III, however, the values are increasing at an increasing rate, indicating that the precession rate is decreasing and deviating further from the t. Figure 4.7. The OC phase diagram for the negative superhump period P = 3:89 hr. 12 cycles were t to each point. The bottom panel shows the amplitude of the tted sine curve, which is highest near quiescence and lowest at maximum light during outburst. 66 The bottom panel of Figure 4.7 shows the negative superhump amplitude variations over time. It is immediately apparent that the amplitude plot mirrors the large amplitude variations in the original light curve. In general, minimum amplitude occurs during outburst maximum and maximum amplitude occurs slightly before minimum light in quiescence. These amplitude variations, which were also apparent in the phase diagrams and sine t results, are believed to be caused by contrast between the bright spot and the accretion disk. During outburst, the disk is bright and so it will be closer in brightness to the bright spot. As a result, the contrast and negative superhump amplitude will decrease. During phases of quiescence, the disk is not as bright and so the contrast and amplitude will increase. The negative superhump period de cit was calculated using Equation 3.45 to be 5.1%. Wood, Thomas, and Simpson (2009) list 22 systems with known negative super hump period de cits, giving the orbital period and published period de cits for each system. This data is used to generate a plot of period de cit versus orbital period, as given in Figure 4.8, where V1101 Aql is represented by a red diamond, other stars are represented by blue circles, and a linear t to the smoothed particle hydrodynamics model results of Wood et al. (2009) is added as the solid line. While the scatter is large, the results of this study place V1101 Aql within a reasonable position among the other stars and helps to con rm the validity of the results. Figure 4.8. Observed negative superhump period de cit versus the orbital period for V1101 Aql (red diamond) and several other systems (blue circles). A linear t is included. 67 The data presented in this study have revealed an orbital period of 4.089 hours and a DN outburst period of 15.15 days, both of which are consistent with those of typical Z Cam stars (3:05 h Porb 8:40 h and 10 d PDNO 30 d). While the data in this study only show 4 DN outbursts and no standstills, data on the object from the AAVSO extend back some 20 years and gives better insight into the time evolution of the system. Figure 4.9 shows a veyear portion of V1101 Aql's AAVSO light curve. In this portion of the data, there appears to be a standstill from, roughly, BJD 2455800  2456200 where the relative ux amplitude is much smaller than the rest of the data and the average ux of the region is about 1 mag below maximum light { a requirement of Z Cam standstills. As a result of this detection, all three primary criteria for Z Cams are ful lled by V1101 Aql. Figure 4.9. A veyear portion of V1101 Aql's light curve displaying a single standstill from BJD 2455800  2456200 (Data compiled by the AAVSO). 68 REFERENCES Bath, G., & Paradijs, J. (1983). Outburst period{energy relations in cataclysmic novae. Nature, 305, 3336. doi: 10.1038/305033a0. Batten, A. (1973). Binary and multiple systems of stars (Vol. 51, p. 85). Burrows, A. (2014). White dwarfs (degenerate dwarfs). Retrieved July 4, 2015, from http://www.astro.princeton.edu/ burrows/classes/403/white.dwarfs.pdf. Carroll, B., & Ostlie, D. (2007). An introduction to modern astrophysics (2nd ed.). San Francisco: Pearson AddisonWesley. Cannizzo, J., Shafter, A., & Wheeler, J. (1988). On the outburst recurrence time for the accretion disk limit cycle mechanism in dwarf novae. The Astrophysical Journal, 333, 227235. doi:10.1086/166739. Downes, R., & Shara, M. (1993). A catalog and atlas of cataclysmic variables. Publications of the Astronomical Society of the Paci c, 105 (684), 127245. doi:10.1086/133139. Downes, R., Hoard, D., Szkody, P., & Wachter, S. (1995). Spectroscopy of Poorly Studied Cataclysmic Variables. The Astronomical Journal, 110, 18241824. Edwards, D., & Pringle, J. (1987). Numerical calculations of mass transfer ow in semi detached binary systems. Monthly Notices of the Royal Astronomical Society, 383394. Eggleton, P. (1983). Approximations to the radii of Roche lobes. The Astrophysical Journal, 268, 368369. Faulkner, J., Lin, D., & Papaloizou, J. (1983). On the evolution of accretion disc ow in cataclysmic variables  I. The prospect of a limit cycle in dwarf nova systems. Monthly Notices of the Royal Astronomical Society, 205, 359375. Flannery, B. (1975). The Location of the Hot Spot in Cataclysmic Variable Stars as Deter mined from Particle Trajectories. Monthly Notices of the Royal Astronomical Society, 170, 325331. Frank, J., & King, A. (2002). Accretion power in astrophysics (3rd ed.). Cambridge: Cam bridge University Press. 69 Freedman, R., & Kaufmann, W. (2008). Universe (8th ed.). New York, NY: W.H. Freeman and Co. Hellier, C. (2001). Cataclysmic variable stars: How and why they vary. London: Springer Praxis. Hessman, F., & Hopp, U. (1990). The massive, nearly faceon cataclysmic variable GD 552. Astronomy and Astrophysics, 228, 387398. Hilditch, R. (2001). An introduction to close binary stars. Cambridge: Cambridge University Press. Iben, I. (2013). Stellar evolution physics (Vol. 1). Cambridge: Cambridge University Press. Ichikawa, S., & Osaki, Y. (1992). Time evolution of the accretion disk radius in a dwarf nova. Publications of the Astronomical Society of Japan, 44, 1526. Ichikawa, S., Hirose, M., & Osaki, Y. (1993). Superoutburst and superhump phenomena in SU Ursae Majoris stars  enhanced masstransfer episode or pure disk phenomenon? Publications of the Astronomical Society of Japan, 45 (2), 243253. Kholopov, P. (1987). General catalogue of variable stars, IV edition, vol. 3. Moscow. Kley, W. (1991). On the in uence of the viscosity on the structure of the boundary layer of accretion disks. Astronomy and Astrophysics, 247, 95107. Lubow, S., & Shu, F. (1975). Gas dynamics of semidetached binaries. The Astrophysical Journal, 198, 383405. doi: 10.1086/153614. LyndenBell, D. (1969). Galactic nuclei as collapsed old quasars. Nature, 223, 690694. doi:10.1038/223690a0. Masetti, N., & Della Valle, M. (1998). A possible orbital period for the dwarf nova V1101 Aql. Astronomy and Astrophysics, 331, 187192. Meinunger, L. (1965). Mitt. Verand. Sterne 3, 110. Meyer, F., & MeyerHofmeister, E. (1983). A model for the standstill of the Z Camelopardalis variables. Astronomy and Astrophysics, 121 (1), 2934. 70 Mobberley, M. (2009). Cataclysmic cosmic events and how to observe them. New York: Springer. doi: 10.1007/9780387799469. Osaki, Y. (1974). An accretion model for the outbursts of U Geminorum stars. Publications of the Astronomical Society of Japan, 429436. Osaki, Y. (1996). DwarfNova Outbursts. Publications of the Astronomical Society of the Paci c, 108 (719), 3960. Patterson, J. (1998). Late evolution of cataclysmic variables. Publications of the Astronomical Society of the Paci c, 110 (752), 11321147. doi:10.1086/316233. Press, W., Teukolsky, S., Vetterling, W., & Flannery, B. (2007). Numerical recipes: the art of scienti c computing (3rd ed.). Cambridge: Cambridge University Press. Pringle, J. (1981). Accretion discs in astrophysics. Annual Review of Astronomy and Astro physics, 19, 137160. Pringle, J. (1985). Interacting binary stars. Cambridge: Cambridge University Press. Ritter, H. (1986). Secular Evolution of Cataclysmic Binaries. The Evolution of Galactic XRay Binaries, 167, 271293. doi: 10.1007/9789400945944 24. Ritter, H., & Kolb, U. (2003). Catalogue of cataclysmic binaries, lowmass Xray binaries and related objects (Seventh edition). Astronomy and Astrophysics, 404, 301303. doi: 10.1051/00046361:20030330. Rose, W. (1998). Advanced stellar astrophysics. Cambridge: Cambridge University Press. Ryden, B., & Peterson, B. (2010). Foundations of astrophysics. San Francisco: Addison Wesley. Shakura, N., & Sunyaev, R. (1973). Black Holes in Binary Systems. Observational Appear ances. Astronomy and Astrophysics, 24, 337355. van der Sluijs, M. (2006). Formation and evolution of compact binaries. Utrecht University. Vogt, N. (1974). Photometric study of the dwarf nova VW Hydri. Astronomy and Astro physics, 36, 369378. 71 Vogt, N., & Bateson, F. (1982). An atlas of southern and equatorial dwarf novae. Astronomy and Astrophysics Supplement Series, 48, 383407. Warner, B. (1995). Cataclysmic variable stars. Cambridge: Cambridge University Press. Winckel, H. (2003). PostAGB Stars. Annual Review of Astronomy and Astrophysics, 391. Wood, M., Thomas, D., & Simpson, J. (2009). SPH simulations of negative (nodal) su perhumps: A parametric study. Monthly Notices of the Royal Astronomical Society, 21102121. Wood, M., Still, M., Howell, S., Cannizzo, J., & Smale, A. (2011). V344 Lyrae: A touchstone cataclysmic variable in the Kepler eld. The Astrophysical Journal, 741 (2), 105105. doi:10.1088/0004637X/741/2/105. 72 APPENDICES 73 APPENDIX A PERMISSION DOCUMENTATION 74 PERMISSION DOCUMENTATION Figure 1.1 is copyright Mark A. Garlick/spaceart.co.uk and used by permission. Be low is a transcript of the conversation via email with Mark in which permission to use the image is given. 75 VITA Prior to entering the physics master's degree program at Texas A&M University Commerce, Alexander C. Spahn earned bachelor's degrees in 2013 for both astrophysics and mathematical sciences from the Florida Institute of Technology in Melbourne, FL. Mr. Spahn aspires to continue his education to earn a PhD, either in the eld or in atmospheric sciences, before entering the workforce as a university professor. Mr. Spahn may be reached by mail at 2304 D Yellowstone Park Ct., Maryland Heights, MO 63043, or by email at aspahn2009@gmail.com.
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Title  Time Series Analysis of the Cataclysmic Variable V1101 Aquilae 
Author  Spahn, Alexander C. 
Subject  Physics; Astrophysics 
Abstract  TIME SERIES ANALYSIS OF THE CATACLYSMIC VARIABLE V1101 AQUILAE A Thesis by ALEXANDER C. SPAHN Submitted to the O ce of Graduate Studies of Texas A&M UniversityCommerce In partial ful llment of the requirements for the degree of MASTER OF SCIENCE August 2015 TIME SERIES ANALYSIS OF THE CATACLYSMIC VARIABLE V1101 AQUILAE A Thesis by ALEXANDER C. SPAHN Approved by: Advisor: Matt A. Wood Committee: William Newton Kurtis Williams Head of Department: Matt Wood Dean of the College: Brent Donham Dean of Graduate Studies: Arlene Horne iii Copyright c 2015 Alexander C. Spahn iv ABSTRACT TIME SERIES ANALYSIS OF THE CATACLYSMIC VARIABLE V1101 AQUILAE Alexander C. Spahn, MS Texas A&M UniversityCommerce, 2015 Advisor: Matt A. Wood, PhD This work reports on the application of various time series analysis techniques to a two month portion of the light curve of the cataclysmic variable V1101 Aquilae. The system is a Z Cam type dwarf nova with an orbital period of 4.089 hours and an active outburst cycle of 15.15 days due to a high mass transfer rate. The system's light curve also displays higher frequency variations, known as negative sumperhums, with a period of 3.891 hours and a period de cit of 5.1%. The amplitude of the negative superhumps varies as an inverse function of system brightness, with an amplitude of 0.70358 during outburst and 0.97718 during quiescence (relative ux units). These variations are believed to be caused by the contrast between the accretion disk and the bright spot. An OC diagram was constructed and reveals the system's evolution. In general, during the rise to outburst, the disk moment of inertia decreases as mass is lost from the disk, causing the precession period of the tilted disk to increase and with it the negative superhump period. The decline of outburst is associated with the opposite e ects. While no standstills were observed in this data, they are present in the AAVSO data and the results agree with the conditions for Z Cam stars. v ACKNOWLEDGMENTS I wish to extend my thanks to my advisor, Dr. Matt Wood, for the assistance that he has provided for this project; To Joe Patterson and the amateur astronomers who have provided us with the data used in this research; To my parents who have always believed in me and kept me motivated to do my best; To the National Science foundation for their generous STEM scholarship; To the Department of Physics and Astronomy and the Graduate School department for their nancial support and for o ering me the role as both a teaching assistant and a research assistant; To my high school chemistry teacher, Ms. Kim Froemming Nejedlo, who kindled my desire for knowledge by demonstrating the beautiful and thrilling world of science with an insurmountable level of passion and energy. I can only dream that I may one day be as in uential to my students as she was to hers. And nally, I extend my thanks to anyone with this passion; the passion for learning, the passion for science and knowledge of the world around us, and/or the passion to help others. Together, we can change the world. vi TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. STELLAR EVOLUTION TO CATACLYSMIC VARIABLES . . . . . . . . . 3 2.1. Stellar Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2. Stellar Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3. Stellar Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4. Binary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3. CATACLYSMIC VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1. Accretion Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2. Outbursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3. Disk Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4. Positive and Negative Superhumps . . . . . . . . . . . . . . . . . . . 51 4. ANALYSIS OF V1101 AQUILAE . . . . . . . . . . . . . . . . . . . . . . . . 57 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Appendix A. Permission Documentation . . . . . . . . . . . . . . . . . . . . . . . . 73 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 vii LIST OF TABLES TABLE 2.1. Results of a theoretical model of the Sun's interior. . . . . . . . . . . . . . . . . . . . . . . . . . . 9 viii LIST OF FIGURES FIGURE 1.1. An artist's representation of a cataclysmic variable. . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.1. A small volume of stellar material that is in hydrostatic equilibrium where the force of gravity is balanced by the outward force of the pressure gradient. . . . . . . . . . . 4 2.2. A spherically symmetric shell region of a star. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3. The cross section of a stellar interior depicting energy transport via radiation and convection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4. A pocket of stellar gas that rises in a region of convection. . . . . . . . . . . . . . . . . . . . 8 2.5. The luminosity, mass, temperature, and density pro les plotted as a function of radius for a theoretical model of the Sun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.6. The stellar evolution track model for a 1M star beginning at ZAMS and ending with the formation of a white dwarf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.7. The internal structure of a low mass star at several stages of its evolution. . . 15 2.8. Several examples of planetary nebulae. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.9. An image of Sirius A and its white dwarf companion Sirius B. . . . . . . . . . . . . . . . 18 2.10. The massradius relation for white dwarfs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.11. The three classi cations of binary star systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.12. A schematic of two stars in a binary system used to determine the gravitational potential experienced by a test mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.13. A cross section of the equipotential surfaces and the rst three Lagrangian points for a binary system and a surface representation of the potential. . . . . . . . . . . . 24 3.1. A distribution of cataclysmic variable orbital periods. . . . . . . . . . . . . . . . . . . . . . . . 30 ix 3.2. Evolution of the secondary near the minimum period of 75 min. . . . . . . . . . . 31 3.3. A topdown view of the stream trajectory of particles ejected through L1 at various velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4. A schematic of a CV depicting the mass transfer stream colliding with the accretion disk and causing a bright spot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5. The spreading of a ring of mass M with a Keplerian orbit at r = r0 due to viscous torques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.6. The total spectrum F for a steady state optically thick disk at di erent ratios rd=R1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.7. The vertical cross section of a concave accretion disk in the disk model. . . 40 3.8. The angular velocity (r) near the surface of the primary with 1. . . . . . . . . . . 41 3.9. A vertical cross section of an accretion disk's optically thick boundary layer. 42 3.10. A fouryear portion of U Gem's visual light curve. . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.11. A oneyear portion of SS Cyg's visual light curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.12. Schematics of three types of DN outburst pro les. . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.13. The bimodal distribution of SS Cyg's outburst durations. . . . . . . . . . . . . . . . . . . . 45 3.14. An analogy for magnetorotational instability in which two objects at di erent radii in Keplerian orbits are connected by a spring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.15. The growth of the magnetorotational instability displayed by a vertical eld line with slight deviations to larger and smaller radii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.16. A simpli ed Scurve relating the density of an accretion disk to its temperature T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.17. A DN disk's surface density plotted against its radius for several instances during an outburst cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 x 3.18. Luminosity, disk radius, disk mass, and angular momentum variations in DN. 52 3.19. An eightmonth portion of VW Hyi's visual light curve. . . . . . . . . . . . . . . . . . . . . . 53 3.20. Snapshots from a SPH elliptical accretion disk simulation for q = 0.25. . . . . . . 54 3.21. Luminosity, disk radius, disk mass, and angular momentum variations of a DN supercycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.22. A tilted accretion disk undergoing retrograde precession through one cycle. . 55 3.23. Superhump period excess and negative superhump period de cits plotted against the orbital period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.1. A threeandahalf year portion of Z Cam's visual light curve. . . . . . . . . . . . . . . . 57 4.2. The light curve of V1101 Aql in ux units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3. The original light curve of V1101 Aql and the light curve with the large amplitude signal removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.4. Discrete Fourier transforms of the original light curve of V1101 Aql and of the light curve with the large amplitude signal removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.5. An example of the aliasing of a sinusoidal signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.6. The average pulse shape of the negative superhumps for the entire set of data and for each of the high and low sets of data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.7. The OC phase diagram for the negative superhump period P = 3:89 hr and the amplitude of the tted sine curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.8. Observed negative superhump period de cit versus the orbital period for V1101 Aql and several other systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.9. A veyear portion of V1101 Aql's light curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 1 Chapter 1 INTRODUCTION Of the countless number of stars in the Universe, more than half are believed to exist within binary or multiple star systems. Cataclysmic variables (CVs for short) are particular types of binary systems which consist of a white dwarf primary star and a K  M type dwarf secondary star that interact in a semidetached binary system, with a binary separation roughly equivalent to the Sun's diameter, and orbit a common center of mass on a timescale of one to a few hours (Figure 1.1). Within these systems, mass transfers from the secondary star through the system's inner Lagrangian point and into the potential well of the primary. The infalling gas forms an accretion disk around the primary which often has a luminosity greater than that of the two stars combined (Hellier, 2001; Warner, 1995). Cataclysmic variables were rst observed in the mid1800's and have since been clas si ed into many classes and subclasses based upon the outburst pro les observed in the objects' light curves. This study focuses on the Z Cam subclass of the dwarf novae class of cataclysmic variables. Dwarf novae (DN) display various quasiperiodic outbursts in which the luminosity of the system increases by several magnitudes, lasting for anywhere from days to weeks. Z Cam stars have active outburst cycles and exhibit occasional standstill during which the system is stuck in outburst for a prolonged period of time. Figure 1.1. An artist's representation of a cataclysmic variable. Image is copyright Mark A. Garlick/spaceart.co.uk, used by permission. 2 The leading theory for the origin of such outbursts, rst developed in 1974 by Yoji Osaki, suggests that instabilities in the accretion disk are to blame. These instabilities are caused by the buildup of matter within the disk which raises the local temperature of the disk until it becomes ionized and highly viscous, leading to an outburst. The location of the instability when it forms governs the shape of the outburst. Standstills in Z Cam stars are believed to be caused by a mass transfer rate that is high enough to keep the disk in outburst for a signi cant amount of time. The study of cataclysmic variables is essentially the study of accretion disks. Accre tion disks are one of the most common astrophysical structures in the Universe, occurring around some neutron stars, black holes, quasars, active galaxies, and in protostellar and pro toplanetary systems. However, accretion disks are best observed in cataclysmic variables. The problem with other systems is that they are either too complex or too di cult to ob serve (typically both). For example, quasars are too distant and protoplanetary systems are often shrouded in dust. Cataclysmic variables provide astronomers with variations on a human timescale and are able to teach the basics of the mechanisms that govern the other more exotic systems. Furthermore, eclipsing cataclysmic variables may provide additional information as to the geometry of the systems. There are many known cataclysmic variables { more than can be observed with only the most e cient observatories in the World. As a result, these systems are most actively observed among amateur astronomers and provide them with opportunities to make worthwhile contributions to the scienti c community. Chapter 2 provides a relatively brief overview of stellar interiors and stellar evolution, laying the foundation for the discussion of interacting binary star systems. Chapter 3 presents a detailed discussion of cataclysmic variables and the outbursts observed within their light curves. In Chapter 4, information from the previous chapters is applied in the analysis of V1101 Aquilae's light curve in order to extract system parameters and to better understand its quasiperiodic nature. Information obtained from the data is then used to classify the system as a Z Cam dwarf nova. 3 Chapter 2 STELLAR EVOLUTION TO CATACLYSMIC VARIABLES 2.1. Stellar Structure Stars are remarkably dynamic objects and, while much of their interiors are not able to be directly observed, they can be modeled via several mathematical constructs known as the equations of stellar structure. First, consider a small cylindrical volume element of a stellar interior with mass dm and length dr as depicted in Figure 2.1. The force of gravity Fg acts toward the center of the star while an opposing force FP provided by pressure acts toward the surface of the star. Both the top and bottom surfaces of the volume element will experience pressure forces of FP,top and FP,bottom, respectively. The net force acting on the element is the sum of these forces and the application of Newton's second law F = ma gives FP,top FP,bottom + Fg = dm d2r dt2 : (2.1) The total pressure force dFP acting on the element is the di erence between the top and bottom forces and, given that F = PA, the total force dFP = AdP. With this, and the fact that the gravitational force is given by GMrdm=r2, Equation 2.1 can be written as AdP G Mrdm r2 = dm d2r dt2 (2.2) where Mr is the mass of the star within a radius r. Replacing dm with Adr, diving through by the volume, and noting that, for a static environment, the acceleration is zero, gives dP dr = G Mr r2 = g (2.3) where g = GMr=r2 is the acceleration due to gravity at a radius r and is the density of the gas. For spherically symmetric stars, Equation 2.3 is the equation of hydrostatic equilibrium. In short, this states the condition that a pressure gradient must exist in order to balance the force of gravity in a static star. 4 FP,top FP,bottom A Fg dm dr Figure 2.1. A small volume of stellar material that is in hydrostatic equilibrium where the force of gravity is balanced by the outward force of the pressure gradient. The volume is taken to be a cylinder of mass dm and length dr with end caps each of surface area A. Consider, now, a spherically symmetric shell region of a star, located a distance r from the center, that has a mass Mr, width dr, and density as depicted in Figure 2.2. The volume of a sphere is V = 4 3 r3 and so, for a thin shell (dr r), the volume is approximately dV = 4 r2dr: (2.4) From the familiar relation = Mr=V , the shell's mass may be expressed as dMr = (4 r2dr): (2.5) Simple rearranging gives the equation of mass continuity: dMr dr = 4 r2 : (2.6) Equation 2.6 prescribes how interior stellar mass varies with radius and is the second equation of stellar structure. The total mass of a spherical star can be thought of as the sum of all of the masses of each thin shell extending from the center to the star's radius R. 5 r dMr, dr Figure 2.2. A spherically symmetric shell with a mass dMr, a width dr, and a density . The inner surface of the shell is located at a radius r from the star's center. Up until this point, the origin of the gas pressure has been overlooked. It is necessary to derive an equation of state, which relates the pressure to properties of the gas, so that the origin may be understood. For a gas of volume V and temperature T consisting of N number of particles, the ideal gas law states that PV = NkT (2.7) where k is the Boltzmann constant. Equation 2.7 can be rewritten by de ning the particle number density n = N=V . The number density is related to the mass density by n = = m where m is the average mass of a gas particle within the volume. De ning one more quantity, the mean molecular weight = m=mH, allows for the equation of state to be given by Pgas = kT mH (2.8) where mH is the mass of a hydrogen atom. The radiation pressure due to photons is also signi cant and is often times more dominant than that supplied by the gas. The radiation pressure Prad with radiation constant a is expressed as Prad = 1 3 aT4: (2.9) 6 Taking the derivative of Equation 2.9 with respect to r gives an expression for the radiation pressure gradient: dPrad dr = 4 3 aT 3 dT dr : (2.10) The radiation pressure gradient may also be expressed in terms of the radiative ux Frad as dPrad dr = c Frad (2.11) where is the opacity and c is the speed of light. After equating the two expressions, using the de nition Frad = L=4 r2, where L is the luminosity at a radius r, and rearranging, an expression for the temperature gradient arises (Iben, 2013): dT dr = 3 L 16 acT 3r2 : (2.12) This is the equation of radiative energy transport. A transparent star with no opacity ( = 0) would absorb no energy and, therefore, would have no temperature gradient. The actual average temperature gradient between the Sun's center and the Sun's photosphere is roughly dT dr = 20 K km1 (Ryden & Peterson, 2010). Radiation is the most e cient process that carries energy away in the region from the center of the Sun out to about 0.7 R . However, from 0.7 to 1.0 R , convection dominates energy transportation (Figure 2.3). Convection typically occurs in regions of high opacity and causes hot pockets of gas to rise while cool pockets of gas sink further down. In order to construct a mathematical model of this type of energy transport, consider a pocket of gas of pressure Pp and density p that is perturbed some vertical distance dr as shown in Figure 2.4. The region surrounding the pocket has a pressure P and density . Before the pocket rises, it is in hydrostatic equilibrium, requiring that Pp = P. When the pocket rises by dr, the pressures and densities also change by some amount. If the pocket's density becomes greater than that of its surroundings, the pocket with begin to sink back down and no convection will occur. If, 7 convection radiation 0.25R 0.7R 1.0R core Figure 2.3. Solar energy transport via radiation and convection. Between 0.0 and 0.7 R , radiation is the dominant process of energy transportation. Convection dominates between 0.7 and 1.0 R . The core of the Sun, extending to 0.25 R , is included for scale. however, the pocket density is lower than that of its surroundings, it will continue to rise and convection will result. Therefore, the condition for the onset of convection is d p > d (2.13) where d p and d are the changes in the densities of the pocket and the surroundings, respectively, over a distance dr. In a convective stellar environment, the motion of these pockets is rapid and there is little time for them to gain or lose heat. Such a process is termed adiabatic (Rose, 1998) and gives PV = P = ln ln P = constant (2.14) where is the adiabatic index de ned by the ratio of the speci c heats for the constant pressure and constant volume cases. After taking the derivative of Equation 2.14, writing it in terms of the temperature gradient, and much rearranging, the expression becomes dT dr = 1 1 ! T P dP dr : (2.15) This is the equation of convective energy transport and it describes how the pocket's tem perature changes as the pocket rises due to convection. 8 surface center dr p, Pp , P + d , P + dP p + d p, Pp + dPp Figure 2.4. A pocket of stellar gas rises in a region of convection. Initially, the pocket has a density p and experiences a pressure Pp while the surroundings have a density and pressure P. After rising by a distance dr, the densities and pressures change by some amount (denoted by di erential forms). The nal equation of stellar structure discussed here is that of energy generation. A mass dm will contribute a luminosity dL = dm to the total luminosity, where is the rate of energy production. Recalling that dm = 4 r2 dr gives dL dr = 4 r2 : (2.16) Together, the equations of hydrostatic equilibrium (2.3), mass continuity (2.6), radiative energy transport (2.12), convective energy transport (2.15), and energy generation (2.16) make up the equations of stellar structure. These equations govern the structure of stellar interiors and solving them allows for models to be constructed. In general, as radius increases, the luminosity and mass increase while the temperature, density, and pressure decrease. For the Sun, the luminosity reaches maximum about a quarter of the way out to the surface and about 94% of the total mass lies within the inner half of the Sun. The luminosity, mass, temperature, and density pro les for a theoretical model of the Sun's interior are plotted as a function of radius in Figure 2.5 and the results are listed in Table 2.1 (Freedman & Kaufmann, 2008). 9 Table 2.1 Results of a theoretical model of the Sun's interior. Radius Luminosity Mass Temperature Density Pressure (R=R ) (L=L ) (M=M ) (106 K) (kg/m3) (P=Pc) 0:0 0:00 0:00 15:5 160 000 1:00 0:1 0:42 0:07 13:0 90 000 0:46 0:2 0:94 0:35 9:5 40 000 0:15 0:3 0:99 0:64 6:7 13 000 0:04 0:4 0:99 0:85 4:8 4 000 0:007 0:5 0:99 0:94 3:4 1 000 0:001 0:6 0:99 0:98 2:2 400 0:0003 0:7 0:99 0:99 1:2 80 4 105 0:8 0:99 1:00 0:7 20 5 106 0:9 0:99 1:00 0:3 2 3 107 1:0 0:99 1:00 0:006 0:0003 4 1013 Figure 2.5. The luminosity, mass, temperature, and density pro les plotted as a function of radius for a theoretical model of the Sun (Data from Freedman, 2008). 2.2. Stellar Nucleosynthesis Energy generation in stars is a result of nuclear fusion and the process of converting one element to another is known as nucleosynthesis. Most stars primarily convert hydrogen into helium and doing so releases energy. If the sun were made entirely of hydrogen atoms and all of them were converted into helium, the fusion processes would release roughly 1:2 1045 J of energy. This would allow the Sun to live for nearly 100 Gyr. However, only about 10% 10 of the Sun undergoes fusion and so the true lifetime of the Sun is closer to 10 Gyr (Ryden & Peterson, 2010). Before discussing the primary stellar nuclear reactions, a few de nitions are made. Electrons, positrons, photons, and neutrinos are depicted by e, e+, , and , respectively. Furthermore, nuclei are represented by AZ X, where A is the mass number, Z is the number of protons, and X is the element's symbol. Hydrogen is converted into helium in the cores of stars but it is highly unlikely that 4 H atoms will all collide to make 42 He. Rather than this direct interaction, there is a chain of protonproton reactions known as the PP chain that take place. The PP chain is the dominant form of hydrogen fusion for temperatures T < 1:8 107 K. The PPI chain is 11 H +11 H ! 21 H + e+ + 21 H +11 H ! 32 He + 32 He +32 He ! 42 He + 211 H: (69%) (2.17) Helium3 will react with another helium3 nuclei 69% of the time (Carroll & Ostlie, 2007). There is, however, a 31% chance that helium3 will interact with helium4 in the PPII chain: 32 He +42 He ! 74 Be + (31%) 74 Be + e ! 73 Li + (99:7%) 73 Li +11 H ! 242 He: (2.18) The beryliym7 nucleus will capture an electron 99.7% of the time while 0.3% of the time it will, instead, capture a proton to produce boron8 in the PPIII chain: 74 Be +11 H ! 85 B + (0:3%) 85 B ! 84 Be + e+ + 84 Be ! 242 He: (2.19) The net resultants of the entire PP chain are 411 H ! 42 He + 2e+ + 2 + 2 : (2.20) 11 Note that in these types of equations, charge and nucleon numbers are conserved. For temperatures T > 1:8 107 K, a separate process known as the CNO cycle becomes the dominant form of hydrogen fusion. It is given this name due to the fact that carbon, nitrogen, and oxygen are both reactants and products (i.e., they are regenerative, being both consumed and created). The chain of reactions in the CNO cycle are 12 6 C +11 H ! 13 7 N + 13 7 N ! 13 6 C + e+ + 13 6 C +11 H ! 14 7 N + 14 7 N +11 H ! 15 8 O + 15 8 O ! 15 7 N + e+ + 15 7 N +11 H ! 12 6 C +42 He (99:96%) ! 16 8 O + (0:04%) 16 8 O +11 H ! 17 9 F + 17 9 F ! 17 8 O + e+ + 17 8 O +11 H ! 14 7 N +42 He (2.21) and the net resultants of the CNO cycle are 411 H ! 42 He + 2e+ + 2 + 3 : (2.22) Over time, helium begins to accumulate within a star due to the PP chain and CNO cycle reactions. Eventually, the star will begin to run out of hydrogen and it will start to shrink. When the temperature rises to over 108 K, helium starts to fuse more rapidly than berylium8 is able to decay. As a result, berylium begins to build up and can fuse with additional helium nuclei to create carbon12. These reactions are quite rare, since berylium is very unstable and has a brief life, and so it takes a signi cant amount of time for carbon to be produced. 12 The transformation of three helium nuclei into carbon is known as the triple alpha process: 42 He +42 He *) 84 Be 84 Be +42 He ! 12 6 C + : (2.23) While the triple alpha process is occurring at these temperatures, carbon and oxygen may also capture alpha particles to produce even heavier nuclei in the following reactions: 12 6 C +42 He ! 16 8 O + 16 8 O +42 He ! 20 10Ne + : (2.24) More massive stars with higher temperatures may allow for carbon and oxygen burn ing. Carbon burning takes place at temperatures near 6 108 K: 12 6 C +12 6 C = 8>>>>>>>>>>>< >>>>>>>>>>>: 16 8 O + 242 He 20 10Ne +42 He 23 11Na + p+ 23 12Mg + n 24 12Mg + (2.25) while oxygen burning takes place at even higher temperatures near 109 K: 16 8 O +16 8 O = 8>>>>>>>>>>>< >>>>>>>>>>>: 24 12Mg + 242 He 28 14Si +42 He 31 15P + p+ 31 16S + n 32 16S + (2.26) While many di erent processes have been discussed here, only the PP chain is currently present in the Sun as it is not yet hot enough for the other reactions to take place. 13 2.3. Stellar Evolution The evolution of stars is inevitable and is primarily the result of changes in chemical composition due to the nuclear reactions discussed in the previous section. A model of the entire evolutionary track for a 1M star from the zeroage main sequence (ZAMS) phase to the formation of a white dwarf is given in Figure 2.6 and the internal structure at several log10(Te) log10(L=L ) ZAMS SGB RGB HB EAGB TPAGB PostAGB PN Formation Prewhite dwarf White dwarf H core exhaustion Core contraction H shell burning He core burning He core exhaustion He ash First dredgeup Second dredgeup First He shell ash superwinds Figure 2.6. The stellar evolution track model for a 1M star beginning at ZAMS and ending with the formation of a white dwarf. 14 di erent phases is shown in Figure 2.7. Once a star reaches the ZAMS and begins to fuse hydrogen into helium, the star's luminosity, radius, and temperature steadily increase. This is because PP chain reactions increase the mean molecular mass of the stellar gas. Hydrostatic equilibrium requires a constant central pressure and so the temperature and/or density must increase. As a result, the core contracts and the rate of energy generation increases which drives both the luminosity and radius up. For example, when the Sun reached the ZAMS 4.6 Gyr ago, it had a luminosity of 0.7 L and, when it runs out of hydrogen in about 6 Gyr, it will have a luminosity near 2.2 L (Ryden & Peterson, 2010). Eventually, after about 10 Gyr, the star will exhaust all of the hydrogen in its core and the PP chain reactions will end. At this point, the temperature will have increased enough to allow for a hydrogen burning shell around the isothermal helium core. Some of the energy generated by the thick shell does not reach the surface and so the temperature decreases and the evolutionary track (Figure 2.6) stars curve to the right. The prolonged hydrogen burning increases the core's mass until it is no longer able to support the rest of the star and reaches a collapse limit determined by the Sch onbergChandrasekhar limit: Mc M ! = 0:37 env c ! (2.27) whereMc and c are the core mass and mean molecular weight and env is the mean molecular weight of the envelope above the hydrogen burning shell. When the core mass reaches this limit, it will collapse and the star leaves the main sequence phase of evolution. The sudden collapse of the core releases gravitational energy and causes the envelope to expand while the temperature decreases. The evolutionary track moves to the right in this phase known as the subgiant branch (SGB). As the envelope of the star continues to expand, the temperature drops enough for a convective region to form near the surface. The bottom of the convective region may extend deep into the star and e ciently transport energy away from the core to the surface in a phase known as the rst dredgeup. This process increases the luminosity and the star moves 15 Main Sequence Star H burning core Envelope Red Giant Star Inert He core H burning shell Envelope Horizontal Branch Star He burning core H burning shell Envelope Asymptotic Giant Branch Star CO core He burning shell H burning shell Envelope Figure 2.7. The internal structure of a low mass star at several di erent stages of its evolution (not drawn to scale). As stellar nucleosynthesis increases the mean molecular mass of the star, the core contracts and the temperature increases enough for the onset of H and He shell burning around the core. upward on the evolutionary track. This phase is known as the red giant branch (RGB). When the temperature and density of the core become high enough, the triple alpha process reignites throughout the star and an incredibly rapid release of energy occurs for a few seconds in a process called the helium ash. A signi cant amount of the generated energy is absorbed by the star's envelope and never reaches the surface. As a result, the star may experience some mass loss. After the helium ash, the luminosity rapidly drops as the star begins to settle into a quiescent state, known as the horizontal branch (HB), in which helium burning ignites within the core and the temperature increases. At the end of the horizontal portion of the branch is the horizontal branch loop. At this point, the core begins to shrink due to the increased mean molecular weight and the envelope expands. 16 Shortly after the loop, the core will have converted all of its helium into carbon and oxygen and increased temperatures allow for the creation of a helium burning shell just outside of the CO core. The star is said to lie along the early asymptotic giant branch (E AGB) at this point. The primary di erence between a red giant star and an asymptotic giant branch (AGB) star is the fact that an AGB star has two shell burning regions (H, He) rather than the single H burning shell of a red giant star. In an EAGB star, the hydrogen shell is e ectively dormant and the helium shell dominates the generation of energy. The core, once again, begins to shrink, the envelope expands, the temperature decreases, and the star moves right on the evolutionary track. Convection within the envelope deepens, causing a second dredgeup of heavier elements and an increasing luminosity as the star moves upward on the evolutionary track to the thermalpulse asymptotic giant branch (TPAGB). The hydrogen shell reignites and dominates the energy production while the helium shell shrinks and alternates between phases of burning and dormancy causing helium shell ashes. The ashes push the hydrogen shell outward, causing it to temporarily shut o . Hydrogen burning eventually begins again and the process cycles. These ashes are easily seen in the evolutionary track as rapid changes in luminosity. The increasing radius of the star leads to a weaker surface gravity. This, coupled with pulsations caused by the helium shell ashes, leads to mass loss at rates as high as _M 104 M yr1 in what is known as a superwind (Winckel, 2003). Because of the hydrogen burning shell, the star's temperature will begin to rise without much change in the luminosity as the envelope loses mass. The star will move rightward on the evolutionary track and marks the postasymptotic giant branch (PostAGB), also referred to as the protoplanetary nebulae phase. The central star becomes incredibly hot and emits a signi cant amount of UV radiation which is absorbed by the expelled gas and causes the ionization of atoms. The emission of photons causes the gas within the expanding shell to glow and forms what is called a planetary nebula. Several examples of planetary nebulae are shown in Figure 2.8. Eventually, all of the envelope is expelled and, with little to no mass above the hydrogen 17 Figure 2.8. Several examples of planetary nebulae. From left to right: The Helix Nebula [Credit: NASA, NOAO, ESA, the Hubble Helix Nebula Team, M. Meixner (STScI), and T.A. Rector (NRAO).], the Cat's Eye Nebula [Credit: NASA, ESA, HEIC, and The Hub ble Heritage Team (STScI/AURA). Acknowledgment: R. Corradi (Isaac Newton Group of Telescopes, Spain) and Z. Tsvetanov (NASA).], HD 44179 (also known as the Red Rectangle Nebula) [Credit: NASA; ESA; Hans Van Winckel (Catholic University of Leuven, Belgium); and Martin Cohen (University of California, Berkeley).], and M29 [Credit: Bruce Balick (University of Washington), Vincent Icke (Leiden University, The Netherlands), Garrelt Mellema (Stockholm University), and NASA.]. and helium burning shells, the burning is extinguished and the luminosity rapidly decreases. Final pulses shed the shells and reveal the hot, dense CO core. After nearly 50,000 years, the planetary nebula will dissipate into the interstellar medium (Ryden & Peterson, 2010). The core will spend the rest of its life cooling down to become a white dwarf star. White dwarfs are stars that have roughly the mass of the Sun contained within a volume the size of the Earth. As a result, the density and pressure are immense. Figure 2.9 shows one of the most famous white dwarfs, Sirius B, a companion star to the main sequence star Sirius A. From the StefanBoltzmann law, the density of Sirius B is estimated to be 3.0 109 kg/m3. Integrating Equation 2.3 and solving for the central pressure gives Pc ' 2 3 G 2R2 = 3.8 1022 N m2. Furthermore, the central temperature is estimated, using Equation 2.12, to be 7.6 107 K (Carroll & Ostlie, 2007). To put these values into perspective, a teaspoon of white dwarf matter brought to Earth would weigh over 10 tons (Ryden, 2010). Typical gas and radiation pressure, as explored in Equations 2.8 and 2.9, are not su cient to support a white dwarf against such pressures. However, since these stars exist, there must be some other mechanism at work. This missing link arises in the form of degeneracy pressure. 18 Figure 2.9. An overexposed image of Sirius A (center) and its white dwarf companion Sirius B (bottom left). Sirius A is nearly 104 times as luminous as Sirius B. [Credit:NASA, H.E. Bond and E. Nelan (Space Telescope Science Institute, Baltimore, Md.); M. Barstow and M. Burleigh (University of Leicester, U.K.); and J.B. Holberg (University of Arizona).]. The Pauli exclusion principle states that no two fermions (particles that have half integer spin) may occupy the same quantum state or. More speci cally, no two electrons of an atom may have the same quantum numbers [principle (n), azimuthal (`), magnetic (m), and spin (s)]. As the temperature of a gas decreases, electrons move to lower energy states. However, as a result of the Pauli exclusion principle, they may not all move to the lowest state (the ground state). Instead, the electrons ll whichever unoccupied excited state has the lowest energy. When all of the lowest energy states are lled, the matter is considered to be degenerate. Any further addition of electrons would force the electrons into states of higher energy in order to make room and results in an outward compressionresisting pressure known as electron degeneracy pressure. An electron in a degenerate gas has a maximum energy F , known as the Fermi energy, of F = h2 2me (3 2n)2=3 (2.28) where me is the mass of an electron and n is the number of electrons in the volume. If the gas is fully ionized, the number of electrons per unit volume is n = #electrons nucleon ! #nucleons volume ! = Z A ! mH : (2.29) 19 Combining Equations 2.28 and 2.29, the Fermi energy becomes F = h2 2me " 3 2 Z A ! mH #2=3 : (2.30) If the Fermi energy is larger than the thermal energy of an electron (3/2kT), electrons will not transition into vacant states and the gas will become degenerate. This sets a condition for degeneracy dependent upon temperature and density: T 2=3 < h2 3mek " 3 2 mH Z A !#2=3 = 1261 Km2kg2=3 (2.31) for Z=A = 0:5. The level of degeneracy increases as T= 2=3 decreases. The Heisenberg uncertainty principle states that both the exact position and momen tum can not be simultaneously known. There is some uncertainty in both quantities such that x p h: (2.32) Since electrons are so tightly packed within the degenerate matter of a white dwarf, they must have a corresponding large uncertainty in their momentum, given in one dimension by px px h= x hn1=3 since the spacing between electrons in a degenerate gas is roughly n1=3. In three dimensional space, the momentum becomes p = p 3px p 3 h " Z A ! mH #1=3 : (2.33) Assuming that all electrons have the same momentum, the pressure may be expressed as P 1 3 npv (2.34) where v is the speed of the electrons in the gas. For nonrelativistic electrons, the speed is v = p=me. For relativistic electrons, the speed is v = c, where c is the speed of light. 20 Therefore, the pressure due to a completely degenerate gas of electrons is P = (3 2)2=3 5 h2 me " Z A ! mH #5=3 (2.35) for nonrelativistic electrons and P = (3 2)1=3 4 hc " Z A ! mH #4=3 (2.36) for the relativistic case. This pressure is what is responsible for maintaining hydro static equilibrium within a white dwarf. Equating the expressions for the central pressure and the degeneracy pressure (Equation 2.35), an expression for the radius arises: R = (18 2)2=3 10 h2 GmeM1=3 " Z A ! 1 mH #5=3 : (2.37) Close inspection of Equation 2.37 reveals the inverse relationship between mass and radius: R / M1=3: (2.38) This inverse relationship, due to the fact that electrons must be more tightly packed to generate a larger outward degeneracy pressure, suggests that white dwarfs with the highest masses actually have the smallest radii. There is, however, a limit on the maximum mass of a white dwarf that can be supported by electron degeneracy pressure. The mass limit, known as the Chandrasekhar limit MCh, is approximated by equating the central pressure and the relativistic degeneracy pressure, giving MCh 3 p 2 8 hc G !3=2" Z A ! 1 mH #2 = 1:44 M : (2.39) A sample set of data for various white dwarfs of varying mass and radii are plotted in Figure 2.10, revealing the massradius relation that cuts o near the Chandrasekhar limit. 21 Figure 2.10. The massradius relation for white dwarfs. As the mass of a white dwarf increases, the radius decreases. The dashed red line indicates the Chandrasekhar mass limit of 1.44 M . (Data from Burrows, 2015.) 2.4. Binary Systems If a pair of stars are close enough together, they may end up orbiting a common center of mass. These systems are called binaries and nearly half of all stars in the Universe are a part of binaries or multiplestar systems (Hilditch, 2001). There are three classi cations of binary systems: detached, semidetached, and contact. Around each star in a binary are imaginary surfaces called Roche lobes that de ne the limit of gravitationally bound matter. If both stars in a binary lie well within their respective Roche lobes, it is called a detached system. A binary in which one star lls its Roche lobe is called a semidetached system. The third classi cation, a contact binary, occurs when both stars ll and possibly over ow their Roche lobes. Sketches of the three types of binary systems are given in Figure 2.11. Consider two stars orbiting a common center of mass in a corotating twodimensional plane with masses M1 and M2 separated by a distance a, distances to the center of mass (at the origin) x1 and x2, and distances to a test mass (of mass m) r1 and r2, as depicted in Figure 2.12. Determining the total gravitational potential experienced by the test mass 22 Figure 2.11. The three classi cations of binary star systems. Left: a detached system in which neither star lls its Roche lobe. Middle: A semidetached system in which one star lls its Roche lobe. Right: A contact system in which both stars ll and over ow their Roche lobes, resulting in a common envelope. The loop represents the Roche lobes of the stars, the dark shaded regions indicate the initial sizes of the stars, and the light shaded regions indicate the sizes after expansion. will allow for an imaginary threedimensional surface to be constructed around the two stars, representing a region in which the test mass would have zero relative motion with respect to the coordinate system. This is analogous to surfaces of constant gravitational potential or regions where there is no force on the test mass. To begin, consider the gravitational potential energy U experience by the mass m. The gravitational attraction of the two stars will be balanced by the ctitious centrifugal force (F = m!2r) and the potential energy will have the form U = Ug + Uc where the gravitational potential Ug is Ug = G Mm r (2.40) and Uc is the potential energy due to the centrifugal force. Uc is determined by integrating the centrifugal force expression where Uc = 0 at r = 0: Uc = Z r 0 m!2rdr = 1 2 m!2r2: (2.41) Therefore, the total potential energy experienced by m is U = G M1m r1 + M2m r2 ! 1 2 m!2r2: (2.42) 23 M1 M2 m c x1 x2 r1 r2 r a Figure 2.12. Schematic of two stars in a binary system, each with an assumed circular orbit about the center of mass, used to determine the gravitational potential experienced by a test mass within a corotating frame. The gravitational potential is then determined by dividing the potential energy by m: = G M1 r1 + M2 r2 ! 1 2 !2r2 (2.43) where ! is the angular velocity of the binary, given by ! = 2 Porb = " G(M1 +M2) a3 #1=2 : (2.44) The equipotential surfaces, where d =dx = 0, may be found via Equations 2.43 and 2.44. Figure 2.13 shows a twodimensional cross section of the equipotential surfaces around a typical binary system, the rst three Lagrangian points, and a surface representation of the potential that shows two clear potential wells for the stars. The innermost surfaces that form a gure8 pattern are those which are referred to as the Roche lobes of the two stars. Lagrangian points (L) are locations where an object could maintain a stable position with respect to the two stars. The rst three Lagrangian points, as shown in Figure 2.13, lie along a line that passes through the two stars. Points 4 and 5 lie o to the sides, forming equilateral triangles with the stars. Points 4 and 5 are stable equilibrium positions. An object placed at either of these points, when perturbed, will stay in orbit around the respective point in the corotating frame of the stars. Points 13 are, however, unstable. These points lie at the 24 Figure 2.13. Top: A surface representation of the potential illustrating the two potential wells of the stars. Bottom: A cross section of the equipotential surfaces and the rst three Lagrangian points for a binary system with a mass ratio of 0.2 (van der Sluijs, 2006). three maxima seen on the surface given in Figure 2.13 (one between the two wells and one on each end). Any slight perturbation of an object placed at these points will cause it to fall into one of the potential wells and away from the equilibrium positions. Much like with water lling a hole, a star that is increasing in volume and lling its Roche lobe will begin to ll its potential well. When full, any additional expansion will cause it to over ow its potential well and matter will \spill" over the inner Lagrangian saddle point (L1) between the two stars and into the potential well of the companion star. The potential at any other location is higher (as seen as an increase in surface height in Figure 2.13), requiring more energy to overcome the barrier than that at L1. Thus, L1 is a point in space which allows for mass transfer from one star, when it lls its Roche lobe, to its companion in a semidetached binary system either by direct impact or via an accretion disk. It is possible that the star may expand more rapidly than the rate at which mass transfers through L1 and, in such a case, matter may be completely lost by both stars through the 25 other Lagrangian points or as stellar winds. The e ective radius RL gives the radius of the star when its volume is equal to that of the Roche lobe, de ned by Eggleton (1983) as RL a = 0:49q2=3 0:6q2=3 + ln(1 + q1=3) (2.45) where the mass ratio q is M2=M1 for the lowmass star and M1=M2 for the highmass star. Mass transfer may result from several di erent scenarios including standard stellar evolution, shrinking of the Roche lobe due to angular momentum loss, and/or the accretion of stellar winds. In the case of Roche lobe over ow, the rate _M at which mass is lost from the secondary star at the inner Lagrangian point L1 is _M = v (2.46) where , v, and are the stream's density, velocity, and crosssectional area at L1, respec tively. A simple approximation can be made assuming that the stars are of equal mass and that the transferred mass is purely hydrogen. The thermal velocity of hydrogen is estimated by the rootmeansquare speed given as v = p 3kT=mH and the mass transfer rate becomes _M dR r 3kT mH (2.47) where d is the amount that the secondary over lls its Roche lobe, R is the radius of secondary, k is the Boltzmann constant, T is the temperature of the gas, and mH is the mass of a hydrogen atom. A more detailed derivation (Pringle, 1985; Edwards & Pringle, 1987) applies Bernoulli's law to the mass ow to show that, for an envelope that is a polytrope of index 1.5, the mass transfer rate is _M = C M Porb R R !3 (2.48) where C is a dimensionless constant, Porb is the orbital period of the binary, R is the 26 amount that the secondary over lls its Roche lobe ( R = R RL), and M and R are the mass and radius of the secondary, respectively. Mass transfer and mass loss a ect the parameters of the binary system throughout its evolution, changing the orbital period Porb, separation distance a, angular frequency !, etc. For conservative mass transfer (M = M1 + M2 and _M 1 = _M 2), the total orbital angular momentum J of a system with circular orbits (e = 0) is J = (M1a21 +M2a22 )! = p GMa (2.49) where a1 = (M2=M)a, a2 = (M1=M)a, and = M1M2=M is the reduced mass. Di erentiat ing Equation 2.49 and dividing through by J, with _M 1 + _M 2 = 0 and J_ = 0, gives _a a = 2 _M 1 M1 M2 M1M2 (2.50) which describes how the binary separation is a ected by mass transfer. If M2 < M1, the separation increases and the Roche lobes of the stars will expand and may allow for stable mass transfer. If M2 > M1, the separation decreases and mass transfer will increase and is generally unstable. From Equation 2.44, the angular frequency ! / a3=2, and so !_ ! = 3 2 _a a : (2.51) As the binary separation increases, the angular frequency decreases. From Kepler's third law (GM = 4 2a3=P2) and by de ning initial values with the subscript i before any mass transfer, the nal to initial orbital period ratio is given by Porb Pi = M1iM2i M1M2 !3 : (2.52) Di erentiating Equation 2.52 with respect to time, noting that _M 1 = _M 2, yields _P orb Porb = 3 _M 1 M1 M2 M1M2 = 3 2 _a a : (2.53) 27 Equation 2.53 reveals that the orbital period is directly related to the separation distance. An increase in separation will cause the period to increase and vice versa. There are a di erent set of binary evolution expressions for nonconservative mass transfer but they are overlooked as they primarily deal with stellar winds or catastrophic mass losses and it is assumed that mass transfer is conservative in this study of a cataclysmic variable. 28 Chapter 3 CATACLYSMIC VARIABLES At this point, it is useful to de ne a speci c type of interacting binaries: a cataclysmic variable (CV). A CV is a semidetached binary system in which a white dwarf is the primary and the secondary is a larger, less massive main sequence star. CVs have orbital periods between 23 minutes and 5 days and are characterized by periodic outbursts that increase the system's brightness by a factor of 10106 before returning back to a quiescent state (Warner, 1995). CVs are classi ed according to the nature of their observed intensity variations: Classical novae (CN): Cataclysmic variables with only one nova outburst. These outbursts are accompanied by increases in intensity of anywhere between 6 and 19 magnitudes which last for hours to days before settling back to quiescence for decades to millions of years. These eruptions are the result of thermonuclear fusion of the material that is accreted onto the surface of a white dwarf. Recurrent Novae (RN): Novae that have multiple outbursts of 4 to 9 mag that repeat every 10 to 80 years (Mobberley, 2009). Dwarf novae (DN): Cataclysmic variables that display outbursts of 2 to 5 mag. These outbursts last for some 2 to 20 days and repeat every few days to tens of years. These eruptions are the result of the accretion disks accompanying these systems. Dwarf novae are further classi ed as Z Cam: Exhibit standstills 0:7 mag below maximum brightness, during which no outbursts occur, which last for days to years before returning to minimum. SU UMa: Have standard outbursts accompanied by occasional superoutbursts which are 0:7 mag brighter and ve times longer than standard DN outbursts. U Gem: All other dwarf novae which are neither of the previous two classi ca tions. These have standard DN outbursts at regular intervals of several weeks to months. Novalike (NL): All other cataclysmic variables that are neither novae or dwarf novae. Novalikes are further classi ed as 29 AM Her: Have strong magnetic elds that lock the system into synchronous rotation. Mass transfers via the eld lines and does not form an accretion disk. DQ Her: Have weaker magnetic elds than AM Her systems and an accretion disk with a substructure caused by the eld. UX UMa: Believed to be stuck in a permanent standstill. VY Scl: Similar to UX UMa systems, but occasionally drop by more than one mag and have several DNtype outbursts. SW Sex: Similar to DN but have steady state disks and display no outbursts. There is a fairly signi cant correlation between the type of many CVs and the orbital periods of the systems. A distribution of 591 CV orbital periods is given in Figure 3.1. A number of important conclusions are made (Warner, 1995) below. From the gure, it is clear that 1) there is a noticeable de ciency of CVs with 2:2 Porb 2:8 h in a region known as the period gap, 2) there is a distinct minimum near Porb ' 75 min, and 3) the number of systems with long orbital periods rapidly declines in what is known as the long period cuto . Furthermore, most nonmagnetic CVs lie above the period gap while most magnetic CVs lie below the gap (all with Porb 4:6 h). All U Gem and Z Cam stars (Porb 3:8 h) and nearly all NLs lie above the gap while almost all SU UMa stars lie below the gap. The sudden cuto at a minimum period of 75 min is a result of the lowmass secondaries responding to ongoing mass transfer. If the mass of the secondary is low enough, the star will become fully degenerate and follow the massradius relationship. As a result, a decrease in mass causes the star to expand and mass transfer causes the period to increase. Therefore, there must be a period minimum between when the nondegenerate star shrinks and when the degenerate star expands (Figure 3.2). The period gap may be explained by 30 Figure 3.1. The distribution of cataclysmic variable orbital periods consisting of 454 CVs from Ritter & Kolb (2003) in white and 137 CVs from the SDSS in gray. The light gray region depicts the 23 hour period gap. the sudden shuto of magnetic braking at 3 hrs, which is a theory that explains a loss of angular momentum due to gas being captured by the magnetic eld of the secondary and thrown out of the system. As the secondary loses its outer layers, the core's pressure decreases, there are less nuclear reactions, and the core shrinks. However, mass is transferring too quickly and the star is left with a radius that is too large for its mass. Magnetic braking shuts o , the secondary contracts, and mass transfer ceases. Gravitational radiation shrinks the orbit and mass transfer resumes when the orbital period is 2 hrs. As a result, the system becomes detached between 23 hrs and is too faint to be observed (Hellier, 2001). The long period cuto is due to the requirement that the secondary must be less massive than the primary, which is limited by the Chandrasekhar mass of 1.44 M . The size of the Roche lobes increase with orbital period and, in order to ll a larger Roche lobe, a star must be more massive. As a result, the mass limit leads to a period limit and the number of systems starts to rapidly decline above orbital periods of several hours. 31 Figure 3.2. Evolution of the secondary near the minimum period of 75 min. Orbital periods given in minutes are labeled along the evolutionary track. Note that minimum occurs between the nondegenerate and degenerate phases of the star (Ritter, 1986). 3.1. Accretion Disks The main component of a CV is the accretion disk that is created as matter transfers from the secondary to the primary. The outbursts observed in these systems are believed to be caused by sudden increases in the rate at which matter ows through these disks. Matter that is ejected through L1 travels at a speed vk which is approximately equal to the speed of sound in the gas cs ' 10(T=104 K)1=2 km s1 and gives vk 10 km s1. In addition, there is a component of the speed v? ' 1 2a! that is perpendicular to this motion as a result of the L1 point orbiting with the system. Here, by use of Kepler's third law, a is a = 3:53 1010M1=3 1 (1 + q)1=3P2=3 cm: (3.1) Applying Equations 3.1 and 2.44 give v? 100 km s1. This Coriolis force causes the stream to swing into an orbit around the white dwarf primary instead of directly impacting the star. The stream will sweep past a closest approach location, loop around the white dwarf, and intersect its earlier path as depicted in Figure 3.3, which displays the trajectory of the mass transfer stream at various ejection velocities for a system with a mass ratio q = 0:67. The 32 Figure 3.3. A topdown view of the stream trajectory of particles ejected through L1 at various velocities for a system with a mass ratio q =0.67. The stream collides with itself and will eventually form a circular ring around the white dwarf (Flannery, 1975). distance of closest approach rmin is obtained from trajectory computations (Lubow & Shu, 1975) to within 1% accuracy as rmin a = 0:0488q0:464 (3.2) for 0.05 < q < 1. The interactions caused by the stream colliding with itself at supersonic speeds shock the gas and, thus, dissipate energy. Although energy is radiated away, angular momentum must be conserved and so the stream will settle into the lowestenergy circular orbit and create a ring of orbiting gas around the primary. A particle in a circular orbit a distance r from the primary has a Keplarian velocity v of v = GM1 r ! : (3.3) Applying Equation 3.3 and conservation of angular momentum, an accurate (1%) value for 33 the radius of the orbiting ring rr is obtained (Hessman & Hopp, 1990) to be rr a = 0:0859q0:426 (3.4) for 0.05 < q < 1. The ring rotates di erentially as di erent annuli orbit at di erent speeds and causes a viscous shear ow that heats the gas. As energy is radiated away, the outer annuli must move to larger orbits in order to conserve angular momentum. As a result, the ring spreads out into a thin disk known as an accretion disk. rr is the minimum radius of the disk. The outer radius rd may extend 8090% of the way out to L1, but is limited by tidal interactions with the secondary (Carroll & Ostlie, 2007). A rough estimate is then rd ' 2rr: (3.5) Once an accretion disk is established, the mass transfer stream will impact the outer edge of the disk at supersonic speeds and create an area of shockheated material known as a bright spot (Figure 3.4). The bright spot gets its name from the fact that it has the ability to radiate as much or more energy than all other components (including both stars and the accretion disk) combined (Warner, 1995). It is possible that the stream misses the disk's edge and ows over its surface. In such a case, the stream will follow a single particle trajectory until striking the disk at a radius closer to the primary (Lubow & Shu, 1975). Figure 3.4. A schematic of a CV depicting the mass transfer stream colliding with the accretion disk and causing a bright spot. If the disk is tilted with respect to the orbital plane, the stream may collide with disk at a smaller radius (dashed line). 34 A steady mass transfer stream, from in nity, striking at the location of the bright spot on the disk's outer edge will release energy Ls at a rate Ls ' GM1 rd _M : (3.6) Note that this is simply an upper limit since the stream originates at a nite distance. Material in the disk will be owing at a rate of _M d and the disk's luminosity Ld is Ld = 1 2 GM1 R1 _M d (3.7) where R1 is the radius of the white dwarf. Since rd=R1 30, the luminosity of the bright spot is greater than that of the disk and it is expected that _M d < _M . The other half of the luminosity is at the boundary layer between the disk and the surface of the primary. Accretion disks can be approximated as twodimensional ows since the material is typically con ned closely to the orbital plane (z = 0). The material in the disk, with a surface density = 2 R dz, will move with a Keplerian angular velocity given by = GM1 r3 !1=2 ; (3.8) a circular velocity given by Equation 3.3, and will have a radial drift velocity vr. An annulus of the disk, with inner radius r and outer radius r + r, will have a mass 2 r r and angular momentum 2 r r r2 . The mass conservation equation is obtained by nding the rate of change of the mass in the limit r ! 0 and has a value @ @t + 1 r @ @r (r vr) = 0: (3.9) The conservation equation for angular momentum is similarly obtained and has a value @ @t ( r2 ) + 1 r @ @r (r vrr2 ) = 1 r @ @r r3 d dr ! (3.10) where is the coe cient of e ective kinematic viscosity of the gas (Warner, 1995). Com 35 bining Equations 3.9 and 3.10, eliminating vr, and assuming circular Keplerian orbits ( / r3=2) gives a nonlinear di usion equation for : @ @t = 3 r @ @r " r1=2 @ @r r1=2 # : (3.11) This equation reveals that matter di uses inward toward the white dwarf while angular momentum di uses outward toward the disk's edge. Choosing to be constant, Equation 3.11 can be solved (Pringle, 1981; Frank, King, & Raine, 2002) to display how a ring of material is spread out into a disk structure (Figure 3.5). The gure gives as a function of x = r=r0 for values of = 12 tr2 0 and shows that changes on a viscous timescale t of t r2 r vr (3.12) where the radial drift velocity is implied to be vr r : (3.13) For viscous timescales 1, most of the initial mass M has accreted onto the white dwarf and the angular momentum has been shifted to very large radii. Figure 3.5. The spreading of a ring of mass M with a Keplerian orbit at r = r0 due to viscous torques. is constant and is a function of x = r=r0 and the dimensionless time variable = 12 tr2 0 (Frank et al., 2002). 36 If the accretion rate _M 2 is much slower than the viscous timescale, a stable disk will settle into a steadystate structure with @=@t = 0 and, from integration of Equation 3.9, the mass ow rate within the disk _M d is then expressed as _M d = 2 r(vr) : (3.14) If the disk extends all the way down to the surface of the primary (r = R1), the primary must rotate more slowly than the material in the disk at that location, i.e., 1 < (R1). There must be a thin boundary layer in which the disk material is decelerated to equal the primary's rotational velocity and there is no torque on the disk (d =dr = 0 or 1 = ). This, along with Equations 3.10 and 3.14, set an inner boundary condition = _M d 3 " 1 R1 r 1=2 # (3.15) which leads to an expression for the energy generation by viscous shearing D(r): D(r) = r d dr !2 = 3G _M 1 _M d 4 r3 " 1 R1 r 1=2 # 3 4 2 _M d (3.16) where the last expression is for the limit r R1. It is by integration of Equation 3.16 from r = R1 to 1 over the area 2 rdr that the disk luminosity Ld of Equation 3.7 came to be. The energy of Equation 3.16 is radiated at a rate of 2 T4 e away from the two surfaces of the disk, where the e ective temperature is Te = T1 r R1 3=4 " 1 R1 r 1=2 #1=4 T1 r R1 3=4 (3.17) and T1 = " 3G _M 1 _M d 8 R3 1 #1=4 : (3.18) where the second expression of Equation 3.17 is for the limit r R1 and its derivative yields the maximum disk temperature of 0:488T1 at r = (49=36)R1 (Pringle, 1981). 37 The motion of gas is governed by the Euler equation @v @t + v rv = rP + f: (3.19) Since there is little to no ow in the zdirection, hydrostatic equilibrium must hold and the zcomponent of Equation 3.19, with the neglection of any velocity terms, becomes @P @z = @ @z " GM1 (r2 + z2)1=2 # = zGM1 r3 (3.20) for a thin disk (z r). This may be solved for to get = cez2=2H2 (3.21) where c is the central plane (z = 0) density of the disk and H is the disk's scale height. The scale height is estimated by noting that P = c2s , where cs is the speed of sound in the gas and P is the sum of the gas and radiation pressures, and setting @P=@z P=H and z H: H = cs r3 GM1 1=2 = cs : (3.22) H r for a thin disk and so cs . This reveals that the local Keplerian velocity is highly supersonic and that density rapidly falls o with height above the central plane. If the disk is optically thick, Equation 3.20 is solved along with the equation for radiative transfer to get the radiative ux F through the disk's faces: F = 16 T3 3 R @T @z : (3.23) If the disk is optically thin and isothermal, the emitted ux is then F = T4 e = R T4 c (3.24) where R is the Rosseland mean opacity and Tc is the midplane temperature. Since an annulus of the disk subtends a solid angle 2 rdr cos i=d2, an observer a distance d away from 38 the system will view the disk's ux distribution F as F = 2 cos i d2 Z rd R1 I rdr / 1=3 Z 1 0 x5=3 ex 1 dx (3.25) where i is the inclination of the disk, is the frequency, and I is the intensity, which may be approximated as a blackbody distribution B : I = B = 2h 3 c2 eh =kT 1 1 (3.26) where h is the Planck constant and c is the speed of light. The second term in Equation 3.25 is valid in the frequency range kT(rd)=h kT1=h where x = h =kT. As a result, if T1 T(rd), the steady state disk spectrum is characterized by F / 1=3 or F / 7=3 (LyndenBell, 1969). For frequencies kT(rd)=h, the spectrum takes the RayleighJeans form and F / 2. Frequencies kT1=h yield a Wien spectrum 2h 3c2eh =kT . The full spectrum, given by Equations 3.25 and 3.26, is shown in Figure 3.6 where the dependency in the three regimes is rather clear and resembles a stretchedout blackbody spectrum. Shakura and Sunyaev (1973) hypothesized that viscosity is enhanced by the presence of turbulence, in the form of subsonic eddie currents, within the disk's gas. Their model, Figure 3.6. The total spectrum F for a steady state optically thick disk at di erent ratios rd=R1. The frequency is normalized to kTout=h where Tout = T(rd) and there are three primary regimes de ned by kT(rd)=h and kT1=h (Frank et al., 2002). 39 known as the disk model, estimates the viscosity to be = csH = H2 (3.27) where is a free parameter with a value that is roughly between 0 and 1. Although the presence of a free parameter is typically detrimental, most of the observables only weakly depend on and so the theory still gives much insight into the physics of accretion disks. With an opacity given by Kramers' law R = 5 1024 T7=2 c cm2g1, the ShakuraSunyaev solution for many of the disk parameters in terms of is = 5:2 4=5 _M 7=10 d;16 M1=4 1 r3=4 10 f14=5 g cm2 H = 1:7 108 1=10 _M 3=20 d;16 M3=8 1 r9=8 10 f3=5 cm = 3:1 108 7=10 _M 11=20 d;16 M5=8 1 r15=8 10 f11=5 g cm3 Tc = 1:4 104 1=5 _M 3=10 d;16 M1=4 1 r3=4 10 f6=5 K = 190 4=5 _M 1=5 d;16f4=5 = 1:8 1014 4=5 _M 3=10 d;16 M1=4 1 r3=4 10 f6=5 cm2s1 vr = 2:7 104 4=5 _M 3=10 d;16 M1=4 1 r1=4 10 f14=5 cm s1 9>>>>>>>>>>>>>>>>>>= >>>>>>>>>>>>>>>>>>; (3.28) where r10 = r=(1010cm), M1 = M=M , _M d;16 = _Md=(1016g s1), and f = [1 (R1=r)1=2]1=4 (Shakura & Sunyaev, 1973). The solution has many implications including con rmation that the disk is optically thick, uniform in the vertical direction, may extend out to a large radii on the order of the Roche lobe, has a negligible mass, does not self gravitate, etc. (Frank et al., 2002). From Equation 3.28, the ratio of the disk's height to its radius, for r R1, gives H / r9=8: (3.29) This implies that the disk's faces are concavely shaped and may be heated/illuminated due to radiation from the central regions and boundary layer (Figure 3.7). Just beyond the surface of the white dwarf, r = R1, there exists a boundary layer of radial extent b in which the angular velocity of the disk material must decelerate to match 40 White Dwarf Disk Disk Figure 3.7. The vertical cross section of a concave accretion disk in the disk model. that of the star's surface 1. An approximate form of (r) is plotted in Figure 3.8. The energy released within the boundary layer LBL can be solved for by use of conservation of energy and angular momentum (Kley, 1991) and has a value LBL = Ld " 1 1 (R1) #2 (3.30) which reduces to Ld for 1 (R1). As a result, the boundary layer may emit as much radiation as the entire accretion disk. A vertical cross section of the boundary layer geometry is given in Figure 3.9. For an optically thick disk, the energy radiated by the boundary later must traverse a region of width H on both faces of the disk. This radiation is roughly that of a blackbody of area 2 R1H 2 and must equal Ld as given by Equation 3.7. Therefore, the e ective temperature of the boundary layer TBL for an accreting white dwarf is TBL 1 105 _M 7=32 d;16 M11=32 1 r25=32 9 K (3.31) (Frank et al., 2002). Due to these high temperatures, the boundary layer typically emits in the soft Xray and EUV regions while the outer portions of the disk may be cool enough to be IR radiators. If, however, the boundary layer is optically thin, radiation from the shock front where the disk material meets the primary's surface escapes with a temperature Tsh of Tsh = 3 16 mH k GM1 R1 = 1:85 108M1r1 9 K (3.32) (Warner, 1995) and so an optically thin boundary layer may emit in hard Xrays. 41 Figure 3.8. The angular velocity (r) near the surface of the primary with 1. The outer edge of the boundary layer at R1+b is given by the dotted line and the dashed line represents (r) if there were no boundary layer. (Frank et al., 2002). With a general understanding of what happens at the inner portion of the disk where material encounters the primary, it is useful to now shift to the outer portions of the disk where angular momentum is disposed of. Since the primary absorbs angular momentum at a rate ( _M p GMR1) that is much less than that which is supplied by the secondary ( _M p GMr), there must be a signi cant torque acting on the edge of the disk that soaks up angular momentum. The disk may extend out to radii that approach that of the primary's Roche lobe and the orbits of the particles will become distorted from circular motion due to the presence of the secondary, forming a bulge in the disk which takes an elliptical shape. The disk material orbits more quickly than the secondary, but the gravitational attraction of the secondary holds the bulge back, slightly ahead of a line that connects the two stars in the binary. The pulling of the bulge material slows it down and, thus, reduces its angular momentum. The balance of angular momentum can be expressed as _M p GMr = _M p GMR1 + Gtidal (3.33) where Gtidal is the tidal torque. Because of the loss of angular momentum at the disk's edge, 42 White Dwarf H Disk H b Boundary Layer Figure 3.9. The vertical cross section of an accretion disk depicting an optically thick bound ary layer (not to scale). Gtidal will halt any further expansion of the disk and therefore de nes the disk radius. The maximum radius of the disk rtidal set by the tidal limit is approximated by rtidal = 0:6a 1 + q ' 0:9RL: (3.34) A more detailed discussion of elliptical disks will follow in the discussion of superoutbursts. 3.2. Outbursts Accretion disks occur in many di erent types of astrophysical systems beyond those of dwarf novae and cataclysmic variables, however those of dwarf novae are the most useful to astronomers. Accretion disks are key components of quasars and protoplanetary systems, but quasars are too distant and protoplanetary systems are too heavily shrouded in dust for the study of their accretion disks. On the other hand, dwarf novae occur everywhere in the Universe and astronomers are able to make clear observations of them to obtain their light curves. The most prominent features of a dwarf novae's light curve are the semiperiodic outbursts that appear as rapid increases of several magnitudes in brightness. The rst ever dwarf novae, U Geminorum, was discovered in 1855, but the mechanisms of their outbursts were not well understood until 1974 when Brian Warner used the data of Z Chamaeleontis to 43 reveal that they are caused by sudden increases in the luminosity of accretion disks (Warner, 1995). A fouryear portion of U Gem's visual light curve displaying several outbursts is given in Figure 3.10. The outbursts of this system are semiregular and repeat every 100 20 days. Each outburst increases the system's luminosity by a factor of 100 in one day and by 5 mag, from 14 to 9 mag, in total. No two outbursts, even those in the same star, are exactly alike. There are, in fact, a few di erent shapes that DN outbursts may take. There may be variations in the rises, declines, and total duration. Take, for example, the light curve of SS Cyg (Figure 3.11). This system appears to alternate between short and long duration outbursts. All but the second outburst have rapid rises of 2 days and slow declines of 8 days. The second outburst is more symmetric due to the longer rise. The longer outbursts appear to have a plateaulike structure at maximum brightness that lasts for 10 days. Schematics of the three most common types of DN outburst pro les are given in Figure 3.12. Furthermore, one of the more distinct properties of DN outbursts is the bimodality of their duration. Figure 3.13 gives a clear separation into two groups of outburst durations from SS Cyg (Bath & Paradijs, 1983). Bimodality is observed in many DN but is most dramatic in SU UMa systems. Figure 3.10. A fouryear portion of U Gem's visual light curve displaying semiperiodic outbursts of 5 mag that repeat every 100 20 days [Data compiled by the American Association of Variable Star Observers (AAVSO)]. 44 Figure 3.11. A oneyear portion of SS Cyg's visual light curve displaying various outburst shapes (Data compiled by the AAVSO). Osaki (1974) proposed a cause for outbursts based upon an accretion disk's instability (DI) and it was quickly agreed upon by many that the passage of matter through the accretion disk is the source of the outbursts. Consider a system for which the mass transfer from the secondary _M is greater than the rate at which matter is transported through the accretion disk _M d. In this case, matter would begin to build up in the disk and lead to an instability which increases the viscosity, angular momentum, and the radius of the disk. The spreading of the disk leads to enhanced accretion onto the white dwarf and increases the system's luminosity (seen as an outburst) while depleting the disk of matter. The disk will eventually return to a lowviscosity quiescent state where it will gain its mass back before returning to a new outburst phase. Figure 3.12. Schematics of three types of DN outburst pro les (Hellier 2001). 45 Figure 3.13. The bimodal distribution of SS Cyg's outburst durations (Bath 1983). 3.3. Disk Instability The study of DI begins with a theoretical, and widely accepted, explanation for the source of viscosity within accretion disks: magnetorotational instability (MRI). This the ory, also known as the BalbusHawley instability, suggests that magnetic instabilities drive turbulence within the disk. To understand how this works, consider a magnetic eld line that connects two pockets of ionized matter at di erent radii (di erent annuli). Matter at a smaller radii orbits more quickly and so acts to stretch the eld line and accelerate the pocket at a larger radii. The acceleration of the outer pocket gives it angular momentum and so it moves to a larger radii and slows down the inner pocket, causing it to lose angular momentum and shift to a smaller radius. As a result, the eld lines are further stretched and the magnetic eld is strengthened, leading to turbulence. A simple analogy for this e ect is to consider the magnetic eld as a spring connecting two objects in orbit (Figure 3.14). The inner object is moving faster than the outer one and so the inner object is pulled back while the outer one is pulled forward. This causes angular momentum to be transported outward. In order to better understand the transport of disk material in this manner, consider a ver tical magnetic eld line that is very slightly perturbed to larger and smaller radii in several locations. Figure 3.15 shows the sequence, from left to right, of the growth of the 46 Figure 3.14. An analogy for magnetorotational instability. Two objects at di erent radii in Keplerian orbits are connected by a spring. The inner object orbits more quickly and gets pulled back while the outer object is pulled forward. The inner object falls inward while the outer object moves to larger radii. magnetorotational instability. Here, the line's deviations are enhanced until the line recon nects and pockets of ionized material are transported to di erent radii. A hot disk of ionized matter will posses many free electric charges that may couple with the magnetic eld and induce MRI and high viscosity. On the other hand, a cold disk will posses many neutral charges since the electrons will combine with various nuclei and MRI will no longer apply. As a result, DN outbursts are caused by the disk ipping between hot and cold states. Cold (a few thousand Kelvin) disk material is unionized and has a low opacity to radiation. As the disk material heats up to 5000 10,000 K, the material Figure 3.15. The growth of the magnetorotational instability displayed by a vertical eld line (perpendicular to the plane of the disk) with slight deviations to larger and smaller radii. The deviations are enhanced and the magnetic eld strengthens until it reconnects and transports material to di erent radii (Hellier, 2001). 47 becomes partially ionized and the opacity increases. The opacity becomes extremely sensitive to temperature (Faulkner, Lin, & Papaloizou, 1983): R = 1:0 1036 1=3T10 cm2g1 / T10: (3.35) and leads to instability. Consider a slight increase in temperature at a location within the disk. The higher temperature increases the motion of the particles and, therefore, the viscosity. Due to a more viscous material, the disk spreads out to larger radii and the situation is considered to be stable. However, once the temperature is high enough for partial ionization to occur, any increase in temperature greatly increases the opacity. The high opacity traps the heat generated by viscous interactions and so the temperature rapidly increases. Once fully ionized, the opacity loses its sensitivity to temperature (Faulkner et al., 1983) and the disk becomes stable at a higher temperature than before: R = 1:5 1020 T2:5 cm2g1 / T2:5: (3.36) In the now highly viscous state, material is transported through the disk more rapidly than it is supplied by mass transfer and so the hot stable state is only brie y sustained before the disk returns to its original quiescent state. The transitions detailed above may be visualized by a simpli ed cycle known as an Scurve (Figure 3.16). Consider a stable disk at quiescence that begins at point A on the Scurve. If the mass transfer from the secondary supplies the disk with matter more quickly than it is able to move through the disk, matter will build up in the disk and increase its density . The more massive disk will have an increased viscosity and increase in temperature until it reaches point B where ionization sets in and the critical density max is de ned to be (Cannizzo, Shafter, & Wheeler, 1988) max = 11:4r1:05 10 M0:35 1 0:86 C g cm2 (3.37) where C is determined for the cold disk. At this point, any increase in density causes a 48 T A B C D min max Ionized Partially Ionized Unionized 20,000 K 7,000 K 3,000 K Figure 3.16. The simpli ed Scurve relating an accretion disk's density to its temperature T. A disk follows the cycle from A ! B ! C ! D ! A as described in the text. runaway heating e ect that is much quicker than the time it takes matter to move through the disk. The result is a temperature increase without any associated change in the surface density. Once the matter is ionized (point C), the disk reaches a new stability at a higher temperature with a greater luminosity. During this period of stability, viscosity is high enough that the infall of matter is more rapid than that which is supplied by the secondary. As a result, the disk is drained of material and decreases in temperature until point D where the critical density min is de ned to be (Cannizzo et al., 1988) min = 8:25r1:05 10 M0:35 1 0:8 C g cm2 (3.38) where H is determined for the hot disk. From here, partial ionization sets in again and the opacity returns, allowing for the temperature to rapidly decline until the ions recombine and the disk is returned to its colder quiescent state at point A. Outbursts begin as a heating transition wave formed by an instability in an annulus of the disk that spreads into adjacent annuli. How an outburst evolves is entirely dependent 49 upon the radius and distribution of matter where the outburst begins. A stable disk will have a density between the two critical values (Figure 3.17). If the mass transfer is rapid, matter will not e ciently di use inward and will build up at large radii. As a result, the instability would form in the outer disk ( rst panel on the right of Figure 3.17). The heating wave will progress inward and enhance the density in the annuli it passes through (second panel on the right). If, however, the mass transfer is slow, the matter will di use inward and build up at small radii. As a result, the critical density will be reached in the inner disk ( rst panel on the left) and the resulting heating wave will propagate outward. As the heating wave moves into adjacent annuli, the viscosity within them increases and so the inward di usion of matter increases. Therefore, the inner regions of the disk increase in density and the accretion rate increases (second panel on the left). At the end of the outburst phase, the result is generally the same: the density pro le is reversed so that it is now greater in the inner regions of the disk and the accretion rate is enhanced (Hellier, 2001). The increased accretion rate drains the disk of matter until it reaches min near the outer edge of the disk where min is the largest. At this point, the region leaves the outburst phase and a cooling wave propagates inward, lowering the density, pulling matter back outward, and returning the disk to a steady quiescent state that is similar to that before outburst. It is the nature of these waves that de ne the various outburst shapes displayed in Figure 3.12. The short rise times are due to outsidein outbursts. The heating wave propa gates rapidly due to a number of factors. The viscosity leads to more matter owing inward rather than out. The outer annuli also have a higher density and so the spreading of the matter has a signi cant e ect on the inner annuli. Furthermore, the density is increased by the infalling matter since annuli at smaller radii are smaller in size. As a result, the wave overtakes adjacent annuli quickly and the enhanced accretion is sudden. Slow rise times are due to insideout heating waves. The same arguments made above act to decrease the wave's outward velocity. Accretion is more steadily enhanced in this case as an increasing amount 50 of the disk acts to transport matter inward. The time scale for outburst rise r is given by r = 0:14P1:15(h) d mag1 (3.39) where P(h) is the orbital period in hours (Warner, 1995). Figure 3.17. A DN disk's surface density plotted against its radius for several instances during an outburst cycle. The disk's pro le begins at the top. High mass transfer rates cause an outsidein heating wave and low mass transfer rates cause an insideout heating wave. Both routes leave the disk with a greater density in the inner regions and an enhanced accretion rate which drains the disk and forms an inwardmoving cooling wave that returns the disk to quiescence. (Hellier, 2001). 51 Short outbursts occur when the heating wave is unable to reach the edge of the disk. The cold region at the disk's edge then acts to pull matter from the hot region and quickly causes a cooling wave that ends the outburst phase. Plateaus, on the other hand, occur when the entirety of the disk partakes in sustained outburst. The minor decline in magnitude during a plateau is a result of the motion from points C to D on the Scurve. When short outbursts occur, they leave the outer regions of the disk with more material and so the next outburst will likely engulf the entire disk. Because of this, it is often observed that long and short outbursts alternate. The width of an outburst T0:5 is given by (Warner, 1995) as T0:5 = 0:90P0:80(h) d: (3.40) The decline of an outburst are all similar since the cooling waves always originate in the disk's outer region and propagate inward. The time scale for outburst decline is given by d = 0:53P0:84(h) d mag1: (3.41) The overall e ects of an outburst on luminosity, disk radius, disk mass, and angular momen tum are simulated (Ichikawa & Osaki, 1992) and displayed in Figure 3.18. 3.4. Positive and Negative Superhumps Consider, now, the light curve of VW Hyi (Figure 3.19). While there are numerous typical DN outbursts, there are also several superoutbursts which are roughly 0.51.0 mag brighter and 510 times longer. Accompanying each supoeroutburst is a superhump; a humpshaped photometric modulation which appears near maximum intensity. Initially observed by Nicholas Vogt (1974), these superhumps have a period that is about 3% longer than the orbital period. Vogt correctly predicted that superoutbursts were the result of 52 Figure 3.18. Luminosity, disk radius, disk mass, and angular momentum variations in DN, based upon model simulations (Ichikawa & Osaki, 1992). an accretion disk becoming elliptical during outburst. An elliptical disk will precess in the prograde direction with a period P+ prec given by 1 P+ prec = 1 Porb 1 P+ (3.42) where Porb is the orbital period and P+ is the superhump period. Snapshots from a smoothed particle hydrodynamics (SPH) simulation of such a disk, for a system with q = 0:25, are given in Figure 3.20. Disks that extend to large radii are subjected to tidal forces due to the presence of the secondary. Material in the inner regions of the disk remain in nearly Keplerian orbits while the orbits of those closer to the disk's edge become increasingly elliptical. A consequence of this is that the orbits of adjacent particles are no longer parallel and will intersect, causing the dissipation of energy due to the collision of particles. Therefore, as the 53 Figure 3.19. An eightmonth portion of VW Hyi's visual light curve displaying both normal DN outbursts and superoutbursts (Data compiled by the AAVSO). secondary passes by the disk's bulge, the tidal stresses will lead to an increased luminosity (superoutbursts) where the orbits intersect at the outer regions of the disk. A 3:1 resonance, for which disk material orbits three times for every orbital period, is thought to be the driving force of ellipticity in accretion disks. The resonant orbits will precess at a rate dependent upon the mass ratio of the system: Porb=P+ prec = 0:233q(1 + q)1=2 (Patterson, 1998). This may also be expressed as a superhump period excess + in terms of Porb: + = P+ Porb Porb : (3.43) The superhump period excess is plotted against the orbital period in Figure 3.23. The evolution of a system leading to a superoutburst is known as a supercycle. The general outline of events in such a case was rst proposed by Osaki (1996) and is summarized in Figure 3.21 and the preceding discussion. Mass transfer from the secondary supplies the disk with more mass than what is removed by regular DN outbursts and so the disk will gradually gain mass and spread to larger radii throughout the outbursts. At some point, when r 0:46a, the disk will expand enough to where it is tidally in uenced by the secondary. The disk will then become elliptical and begin to precess. Frames 1 and 6 in Figure 3.20 show such an elliptical disk as superhump maximum where convergent ows are strongest. 54 Figure 3.20. Snapshots from a SPH elliptical accretion disk simulation for q = 0:25. The 100,000 particles are colorcoded by their luminosity (Wood et al., 2011). In frame 3, the mass transfer stream impacts the disk edge much deeper into the potential well, releasing more energy and causing what are known as late superhumps. The tidal in uence of the secondary further drains the disk of angular momentum and enhances the inward ow of matter through the disk so that the disk is sustained in the hot, highly viscous state. The accretion rate will begin to exceed the rate of mass transfer and the disk will shrink. The disk will drain and remain elliptical until r 0:35a, at which point most of the acquired mass has been accreted. Because this involves much more of the disk's mass, superoutbursts have a much longer duration than regular DN outbursts. While many systems display superoutbursts with superhump periods a few percent longer than the orbital period, some may display humps that are a few percent less than the 55 Figure 3.21. Luminosity, disk radius, disk mass, and angular momentum variations of a supercycle, based upon model simulations (Ichikawa, Hirose, & Osaki, 1993). orbital period. These humps are known as negative superhumps (or infrahumps). Negative superhumps are the result of a tilted accretion disk relative to the orbital plane. Disks that are tilted will precess in what is called nodal precession, where nodes are the points at which the disk's edge passes through the orbital plane. Nodal precession is opposite to the direction of the orbital motion (retrograde) as seen in Figure 3.22. Figure 3.22. A tilted accretion disk undergoing retrograde precession through one cycle. In this gure, the disk precesses in the clockwise direction, whereas the orbital motion is counterclockwise (Hellier, 2001). 56 The period P prec of the retrograde precession of the tilted disk is given by 1 P = 1 Porb + 1 P prec (3.44) where P is the negative superhump period. Furthermore, the fractional period o set of the negative superhumps, or period de cit, is = P Porb Porb : (3.45) The negative superhump period excess is plotted against the orbital period in Figure 3.23. The physical source of negative superhumps is the varying radial location of the bright spot within the white dwarf's potential well (Wood, Thomas, & Sampson, 2009). In a standard system with an nontilted disk, the mass transfer stream will strike the edge of the disk. However, if the disk is tilted, the stream will pass the edge of the disk and collide with one of its two faces. Negative superhump maxima occur when the stream collides deepest into the potential well (when the bright spot is located at a minimum radius). Having fallen further, the stream will collide with more energy and the luminosity will increase. Figure 3.23. Superhump period excess (circles) and negative superhump period de cits (squares) plotted against the orbital period with linear ts included (Hellier, 2001). 57 Chapter 4 ANALYSIS OF V1101 AQUILAE V1101 Aql is a cataclysmic variable star that is often considered to be a Z Camtype dwarf nova, though this classi cation is not yet con rmed. V1101 Aql was rst classi ed as an irregular variable (Kholopov, 1987), but observations by Meinunger (1965), Vogt & Bateson (1982), Downes & Shara (1993), and Masetti & Della Valle (1998) have all suggested the Z Cam nature of the star. Downes, Hoard, Szkody, & Wachter (1995) also suggests that V1101 Aql could be a Herbig Ae/Be star. Z Cam stars, such as Z Cam in Figure 4.1, have orbital periods between 3.05 and 8.40 hours, active outburst cycles between 10 and 30 days, and occasional standstill in which the disk becomes stuck in outburst due to an irradiated secondary (Meyer & MeyerHofmeister, 1983). As will be shown in the following discussion, many parameters and characteristics of V1101 Aql match those of typical Z Cam stars. Observations for this study were made by several amateur astronomers from the Center for Backyard Astrophysics between July 15, 2013 (BJD 2456488) and September 17, 2013 (BJD 2456553) and were compiled by Joe Patterson of Colombia University before being sent for data reduction. The total combined meanzero light curve for the 2 months of data is plotted in ux units in Figure 4.2. The data displays four clear DN outbursts and Figure 4.1. A threeandahalf year portion of Z Cam's visual light curve displaying two standstills (Data compiled by the AAVSO). 58 Figure 4.2. The light curve of V1101 Aql in ux units. A portion of the plot is enlarged in order to display several of the system's negative superhumps. the enlarged section in the top panel reveals a higher frequency signal the system's negative superhumps. In order to study the nature of the negative superhumps, the large amplitude signal of the DN outbursts must rst be removed. There are clear gaps in the data corre sponding to times when the system was not being observed. The data was split if the gaps were greater than 0.2 days and each chunk of data was placed within its own le giving 67 total les. Each chunk of data was then set to mean zero ux by nding the average 1 2(max + min) values of the data points of each chunk and subtracting the average from them. At this point, many of the chunks still had a signi cant overall slope to them (such as those during the rise/decline times of outbursts). To remove them, a polynomial was t to appropriate chunks and then subtracted o from the original data. For most chunks, a rst degree polynomial of the linear form y = mx+b was su cient. The 67 modi ed chunks were then recombined and the nal result of these processes is displayed in the bottom panel of Figure 4.3 where the large amplitude signal has successfully been removed. 59 Figure 4.3. Top: The original light curve. Bottom: The same light curve, but with the large amplitude signal removed and all chunks shifted to have a meanzero ux. It is clear that the data of V1101 Aql displays oscillatory characteristics. Often, a single signal is composed of multiple sinusoidal waves that are added together. A signal y(t) consisting of N sine waves can be expressed as y(t) = XN j yj sin(2 jt + j) (4.1) where yj is the amplitude, j is the frequency, and j is the phase of the jth component. A signal in the time domain, such as that of Eqn 4.1, may be transformed into the frequency domain Y ( ) via a Fourier transform: Y ( ) = Z 1 1 y(t)e2 i tdt: (4.2) The Fourier transform Y ( ) gives a direct measure of the frequencies (and hence the periods), allowing for a convenient way to view a signal in the frequency domain. 60 A discrete Fourier transform (DFT) replaces the in nite integral with a nite sum: Y ( k) = NX1 n=0 yn exp2 ikn=N: (4.3) DFTs were performed on both the original light curve (top panel of Figure 4.4) and the light curve with the large amplitude signal removed (bottom panel of Figure 4.4) in order to extract the frequencies of the DN outbursts and negative superhumps, respectively. The dominant peak at 0.066 cyc/day in the top panel corresponds to the DN outburst frequency and gives a period PDNO = 15:15 days. The other peaks near 1 cyc/day are a result of gaps in the data sampling and are not real. The dominant peak at 6.168 cyc/day in the bottom panel corresponds to the negative superhump frequency and gives a period PNSH = 3:891 hrs. The two peaks at 5.168 and 7.168 cyc/day are the result of an e ect known as aliasing. Figure 4.4. Top: The DFT of the original light curve with a single dominant peak at 0.066 c/d (P ' 15.15 days) which identi es the period of the DN outbursts. Bottom: The DFT of the light curve after removing the large amplitude signal. The dominant peak at 6.168 c/d (P ' 3.89 hr) identi es the negative superhump period. The two peaks at 5.168 and 7.168 c/d are one day aliases. The orbital period has been detected at 5.870 c/d (4.09 hr). 61 The spectral components of a signal that consists of a number N of data points sampled at an interval t can be recovered by a Fourier transform if the frequency is below the Nyquist frequency Nyquist = 1=2 t. Consider a sine function f(t) = sin(2 t) with = 2 Hz and t = 0:4 s (top panel of Figure 4.5). In this case, > Nyquist = 1=2(0:4) = 1:25 Hz. The DFT of such a signal will result in the power spectrum given in the middle panel of Figure 4.5. There are peaks at 0.5 Hz and 2.0 Hz, where the latter corresponds to a negative frequency = 0:5 Hz. This frequency derives from the fact that the Fourier transform above and below the Nyquist frequency are related by YN=2n = Y N=2+n. A peak at 2.0 Hz, 2:0 1:25 = 0:75 Hz above the Nyquist frequency, will generate a peak 0.75 Hz below the Nyquist frequency as well (at 0.5 Hz). This folding back of the frequency is known as aliasing. To visualize this e ect, a 0.5 Hz sinusoid is overlaid upon the existing sine function t y(t) f(t) = sin(4 t) (Hz) jF(t)j2 Nyquist = 1:25 0:5 2:0 t f(t); g(t) f(t) = sin(4 t) g(t) = sin(2 t) Figure 4.5. Top: A sine function f(t) = sin(2 t) with = 2 Hz and t = 0:4 s. Middle: The power spectrum of the above sine function. Note the existence of an additional peak at 0.5 Hz opposite to the 2.0 Hz peak about the Nyquist frequency. Bottom: The original signal overlaid with a 0.5 Hz sinusoid. The two frequencies are indistinguishable. 62 in the bottom panel of Figure 4.5. Both sinusoids pass through all of the sampled data points and their frequencies become indistinguishable. The two peaks at 5.168 and 7.168 cyc/day are 1day aliases of the primary negative superhump frequency of 6.168 cyc/day. These aliases appear as a result of the fact that frequencies of 1 day also tend to t the data well and so we see a partial signal at the corresponding frequencies in the power spectrum. A second pair of weaker 1day aliases are present at 4.870 and 6.870 cyc/day and help to reveal the system's orbital period of Porb = 4:089 hrs (5.870 cyc/day). Following the Fourier analysis, the data was split into two separate les of high and low states. Referring back to Figure 4.3, the high state data are the points in the bottom panel that correspond to relative ux values in the upper panel greater than zero. The remaining points constitute the low state data. The average pulse shapes of the negative superhumps, as shown in Figure 4.6, were then generated with the frequency input of 6.168 cyc/day for the entire data set (top panel) and for each of the high and low states (bottom panel). The average high state pulse shape was o set by +0:5 for clarity so that the data points of the two data sets did not overlap. It is immediately apparent that the amplitude of the high state data is less than that of the low state data. In order to extract further information from the system, a single sine curve of the form y(t) = Asin[2 (t T0)=P] (4.4) was t to the light curve with the large amplitudes removed and to the high and low state data sets using the LevenbergMarquardt method for nonlinear least squares curve tting (Press et al., 2007). For example, consider a set of m data points (ti; yi) for which a function ^y(ti; p) is t, where t is an independent variable and p is a vector of parameters of the tted curve. The goal is to minimize the sum of the squares of the deviations between the data and the tted function and is given by the chisquared error criterion as 2(p) = Xm i=1 = " y(ti) ^y(ti; p) wi #2 (4.5) 63 where wi is a measure of the error of y(ti). For nonlinear tting, this process is iterative and each iteration nds some parameter perturbation that reduces the value of 2. The results of the t to the full set of V1101 Aql data are P = 0:16214 2:7800 106 d T0 = BJD 2456489:9814 0:0006 A = 0:83344 0:0095649: This period is consistent with that which was determined from the DFT. The important results from the ts to the high and low data are the amplitudes of 0.70358 0.014024 and 0.97718 0.011770, respectively. This further proves that the amplitude is changing and the it is highest during the quiescent phases. Figure 4.6. The average pulse shape of the negative superhumps for the entire set of data (top) and for each of the high and low sets of data (bottom), where the high data is o set by +0:5 for clarity. The dashed lines represent sine ts to the data. 64 The period of the full data set was then used to construct an OC diagram. The program used to create the diagram steps through the data with a dt determined by time boundaries and shows ts over the data. An OC diagram compares the observed time of an event (O), such as the midpoint of a pulsation cycle peak, to the calculated time of the event (C) as determined by a tted curve with a constant period. For each event, the calculated time is subtracted from the observed time and the di erence is plotted versus time. The resulting plot has three possible outcomes as it may either be linear and horizontal, linear and sloped, or curved. A horizontal line in an OC diagram implies that the t perfectly matched what is observed and that there is no change in the period. A straight but sloped line implies that, while constant, the period of the t is di erent from that which is observed. A positive slope or a negative slope imply that the actual period is longer or shorter than that of the t, respectively. The discrepancies will accumulate and lead to larger OC values over time (Batten, 1973). Curves present in the diagrams indicate that the period, or phase, is variable and changing with time. Deviations to a smaller OC value means that the period is decreasing and the events are occurring earlier than predicted. Deviations to a larger OC value means that the period is increasing and the observed times are delayed. These diagrams give a signi cant amount of information about the system's time evolution. The middle panel of Figure 4.7 shows the OC diagram for the light curve with the large amplitude signals removed for the period determined by the sine t mentioned above. Recall that the negative superhump light source is the bright spot sweeping across the face of an accretion disk, which precesses in a retrograde direction. Variations in the OC diagram indicate a variable disk precession rate. The precession rate may vary as a result of changes in the moment of inertia in the disk. A disk with a small radial extent will have a long precession period and a correspondingly large negative superhump period. A disk that does not precess (i.e., a disk with an in nite precession period) will have a negative superhump period equal to the orbital period of the system. On the other hand, a disk with a large radius will have small precession period and negative superhump periods. 65 Consider four regions, de ned by the BJD 2456000, in the OC diagram of Figure 4.7: region I (BJD 518  524), region II (BJD 524  534), region III (BJD 534  540), and region IV (BJD 544  548). Regions I and III have a generally positive trend, indicating that the actual period is longer than the mean period, the precession rate is slower, and that the disk's mass is weighted to smaller radii. Regions II and IV have a generally negative trend, indicating that the actual period is shorter, the precession rate is faster, and that the disk's mass is weighted to larger radii. Furthermore, there is slight curvature visible throughout the regions. For example, in region I, while the values are increasing, they are increasing at a decreasing rate. As a result, the retrograde precession rate is increasing throughout this region and nearly matches that of the t for BJD 522  524 where the dots lie on a nearly horizontal line. In region III, however, the values are increasing at an increasing rate, indicating that the precession rate is decreasing and deviating further from the t. Figure 4.7. The OC phase diagram for the negative superhump period P = 3:89 hr. 12 cycles were t to each point. The bottom panel shows the amplitude of the tted sine curve, which is highest near quiescence and lowest at maximum light during outburst. 66 The bottom panel of Figure 4.7 shows the negative superhump amplitude variations over time. It is immediately apparent that the amplitude plot mirrors the large amplitude variations in the original light curve. In general, minimum amplitude occurs during outburst maximum and maximum amplitude occurs slightly before minimum light in quiescence. These amplitude variations, which were also apparent in the phase diagrams and sine t results, are believed to be caused by contrast between the bright spot and the accretion disk. During outburst, the disk is bright and so it will be closer in brightness to the bright spot. As a result, the contrast and negative superhump amplitude will decrease. During phases of quiescence, the disk is not as bright and so the contrast and amplitude will increase. The negative superhump period de cit was calculated using Equation 3.45 to be 5.1%. Wood, Thomas, and Simpson (2009) list 22 systems with known negative super hump period de cits, giving the orbital period and published period de cits for each system. This data is used to generate a plot of period de cit versus orbital period, as given in Figure 4.8, where V1101 Aql is represented by a red diamond, other stars are represented by blue circles, and a linear t to the smoothed particle hydrodynamics model results of Wood et al. (2009) is added as the solid line. While the scatter is large, the results of this study place V1101 Aql within a reasonable position among the other stars and helps to con rm the validity of the results. Figure 4.8. Observed negative superhump period de cit versus the orbital period for V1101 Aql (red diamond) and several other systems (blue circles). A linear t is included. 67 The data presented in this study have revealed an orbital period of 4.089 hours and a DN outburst period of 15.15 days, both of which are consistent with those of typical Z Cam stars (3:05 h Porb 8:40 h and 10 d PDNO 30 d). While the data in this study only show 4 DN outbursts and no standstills, data on the object from the AAVSO extend back some 20 years and gives better insight into the time evolution of the system. Figure 4.9 shows a veyear portion of V1101 Aql's AAVSO light curve. In this portion of the data, there appears to be a standstill from, roughly, BJD 2455800  2456200 where the relative ux amplitude is much smaller than the rest of the data and the average ux of the region is about 1 mag below maximum light { a requirement of Z Cam standstills. As a result of this detection, all three primary criteria for Z Cams are ful lled by V1101 Aql. Figure 4.9. A veyear portion of V1101 Aql's light curve displaying a single standstill from BJD 2455800  2456200 (Data compiled by the AAVSO). 68 REFERENCES Bath, G., & Paradijs, J. (1983). Outburst period{energy relations in cataclysmic novae. Nature, 305, 3336. doi: 10.1038/305033a0. Batten, A. (1973). Binary and multiple systems of stars (Vol. 51, p. 85). Burrows, A. (2014). White dwarfs (degenerate dwarfs). Retrieved July 4, 2015, from http://www.astro.princeton.edu/ burrows/classes/403/white.dwarfs.pdf. Carroll, B., & Ostlie, D. (2007). An introduction to modern astrophysics (2nd ed.). San Francisco: Pearson AddisonWesley. Cannizzo, J., Shafter, A., & Wheeler, J. (1988). On the outburst recurrence time for the accretion disk limit cycle mechanism in dwarf novae. The Astrophysical Journal, 333, 227235. doi:10.1086/166739. Downes, R., & Shara, M. (1993). A catalog and atlas of cataclysmic variables. Publications of the Astronomical Society of the Paci c, 105 (684), 127245. doi:10.1086/133139. Downes, R., Hoard, D., Szkody, P., & Wachter, S. (1995). Spectroscopy of Poorly Studied Cataclysmic Variables. The Astronomical Journal, 110, 18241824. Edwards, D., & Pringle, J. (1987). Numerical calculations of mass transfer ow in semi detached binary systems. Monthly Notices of the Royal Astronomical Society, 383394. Eggleton, P. (1983). Approximations to the radii of Roche lobes. The Astrophysical Journal, 268, 368369. Faulkner, J., Lin, D., & Papaloizou, J. (1983). On the evolution of accretion disc ow in cataclysmic variables  I. The prospect of a limit cycle in dwarf nova systems. Monthly Notices of the Royal Astronomical Society, 205, 359375. Flannery, B. (1975). The Location of the Hot Spot in Cataclysmic Variable Stars as Deter mined from Particle Trajectories. Monthly Notices of the Royal Astronomical Society, 170, 325331. Frank, J., & King, A. (2002). Accretion power in astrophysics (3rd ed.). Cambridge: Cam bridge University Press. 69 Freedman, R., & Kaufmann, W. (2008). Universe (8th ed.). New York, NY: W.H. Freeman and Co. Hellier, C. (2001). Cataclysmic variable stars: How and why they vary. London: Springer Praxis. Hessman, F., & Hopp, U. (1990). The massive, nearly faceon cataclysmic variable GD 552. Astronomy and Astrophysics, 228, 387398. Hilditch, R. (2001). An introduction to close binary stars. Cambridge: Cambridge University Press. Iben, I. (2013). Stellar evolution physics (Vol. 1). Cambridge: Cambridge University Press. Ichikawa, S., & Osaki, Y. (1992). Time evolution of the accretion disk radius in a dwarf nova. Publications of the Astronomical Society of Japan, 44, 1526. Ichikawa, S., Hirose, M., & Osaki, Y. (1993). Superoutburst and superhump phenomena in SU Ursae Majoris stars  enhanced masstransfer episode or pure disk phenomenon? Publications of the Astronomical Society of Japan, 45 (2), 243253. Kholopov, P. (1987). General catalogue of variable stars, IV edition, vol. 3. Moscow. Kley, W. (1991). On the in uence of the viscosity on the structure of the boundary layer of accretion disks. Astronomy and Astrophysics, 247, 95107. Lubow, S., & Shu, F. (1975). Gas dynamics of semidetached binaries. The Astrophysical Journal, 198, 383405. doi: 10.1086/153614. LyndenBell, D. (1969). Galactic nuclei as collapsed old quasars. Nature, 223, 690694. doi:10.1038/223690a0. Masetti, N., & Della Valle, M. (1998). A possible orbital period for the dwarf nova V1101 Aql. Astronomy and Astrophysics, 331, 187192. Meinunger, L. (1965). Mitt. Verand. Sterne 3, 110. Meyer, F., & MeyerHofmeister, E. (1983). A model for the standstill of the Z Camelopardalis variables. Astronomy and Astrophysics, 121 (1), 2934. 70 Mobberley, M. (2009). Cataclysmic cosmic events and how to observe them. New York: Springer. doi: 10.1007/9780387799469. Osaki, Y. (1974). An accretion model for the outbursts of U Geminorum stars. Publications of the Astronomical Society of Japan, 429436. Osaki, Y. (1996). DwarfNova Outbursts. Publications of the Astronomical Society of the Paci c, 108 (719), 3960. Patterson, J. (1998). Late evolution of cataclysmic variables. Publications of the Astronomical Society of the Paci c, 110 (752), 11321147. doi:10.1086/316233. Press, W., Teukolsky, S., Vetterling, W., & Flannery, B. (2007). Numerical recipes: the art of scienti c computing (3rd ed.). Cambridge: Cambridge University Press. Pringle, J. (1981). Accretion discs in astrophysics. Annual Review of Astronomy and Astro physics, 19, 137160. Pringle, J. (1985). Interacting binary stars. Cambridge: Cambridge University Press. Ritter, H. (1986). Secular Evolution of Cataclysmic Binaries. The Evolution of Galactic XRay Binaries, 167, 271293. doi: 10.1007/9789400945944 24. Ritter, H., & Kolb, U. (2003). Catalogue of cataclysmic binaries, lowmass Xray binaries and related objects (Seventh edition). Astronomy and Astrophysics, 404, 301303. doi: 10.1051/00046361:20030330. Rose, W. (1998). Advanced stellar astrophysics. Cambridge: Cambridge University Press. Ryden, B., & Peterson, B. (2010). Foundations of astrophysics. San Francisco: Addison Wesley. Shakura, N., & Sunyaev, R. (1973). Black Holes in Binary Systems. Observational Appear ances. Astronomy and Astrophysics, 24, 337355. van der Sluijs, M. (2006). Formation and evolution of compact binaries. Utrecht University. Vogt, N. (1974). Photometric study of the dwarf nova VW Hydri. Astronomy and Astro physics, 36, 369378. 71 Vogt, N., & Bateson, F. (1982). An atlas of southern and equatorial dwarf novae. Astronomy and Astrophysics Supplement Series, 48, 383407. Warner, B. (1995). Cataclysmic variable stars. Cambridge: Cambridge University Press. Winckel, H. (2003). PostAGB Stars. Annual Review of Astronomy and Astrophysics, 391. Wood, M., Thomas, D., & Simpson, J. (2009). SPH simulations of negative (nodal) su perhumps: A parametric study. Monthly Notices of the Royal Astronomical Society, 21102121. Wood, M., Still, M., Howell, S., Cannizzo, J., & Smale, A. (2011). V344 Lyrae: A touchstone cataclysmic variable in the Kepler eld. The Astrophysical Journal, 741 (2), 105105. doi:10.1088/0004637X/741/2/105. 72 APPENDICES 73 APPENDIX A PERMISSION DOCUMENTATION 74 PERMISSION DOCUMENTATION Figure 1.1 is copyright Mark A. Garlick/spaceart.co.uk and used by permission. Be low is a transcript of the conversation via email with Mark in which permission to use the image is given. 75 VITA Prior to entering the physics master's degree program at Texas A&M University Commerce, Alexander C. Spahn earned bachelor's degrees in 2013 for both astrophysics and mathematical sciences from the Florida Institute of Technology in Melbourne, FL. Mr. Spahn aspires to continue his education to earn a PhD, either in the eld or in atmospheric sciences, before entering the workforce as a university professor. Mr. Spahn may be reached by mail at 2304 D Yellowstone Park Ct., Maryland Heights, MO 63043, or by email at aspahn2009@gmail.com. 
Date  2015 
Faculty Advisor  Wood, Matt A 
Committee Members 
Wood, Matt A Newton, William Williams, Kurtis 
University Affiliation  Texas A&M UniversityCommerce 
Department  MSPhysics 
Degree Awarded  M.S. 
Pages  85 
Type  Text 
Format  
Language  eng 
Rights  All rights reserved. 



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