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Astrophysical plasmas near strongly magnetized compact objects Gill, Ramandeep 2012

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Astrophysical Plasmas Near Strongly Magnetized Compact Objects by Ramandeep Gill B.Sc. (Hons.), The University of British Columbia, 2007 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in THE FACULTY OF GRADUATE STUDIES (Astronomy) The University Of British Columbia (Vancouver) August 2012 c© Ramandeep Gill, 2012 Abstract The interaction of strong magnetic fields of compact objects with the surrounding plasma leads to novel and puzzling astrophysical phenomena. In this dissertation, we examine some of the properties of strongly magnetized plasmas as outlined in the following. A fully relativistic treatment of Bernstein waves in a uniform, magnetized, rel- ativistic electron-positron pair plasma has remained too formidable a task owing to the very complex nature of the problem. We perform contour integration of the dielectric response function and numerically compute the dispersion curves. If coupled to electromagnetic modes, these waves may be important for generating radiation in pulsar magnetospheres. The soft gamma-ray repeaters, classified as magnetars, unleash large amounts of magnetically stored energy in a spectacular event called the giant flare. What causes these flares to develop is an open question. We examine two trigger mech- anisms, one internal and the other external to the neutron star. In the internal mechanism, we propose that the strongly wound up poloidal magnetic field de- velops tangential discontinuities and dissipates its torsional energy in heating the crust. Alternatively, we argue that the shearing motion of the external magnetic field footpoints causes the materialization of a Sweet-Parker current layer in the magnetosphere. The thinning of this macroscopic layer powers the giant flare. The extreme environments of compact objects are conducive to the creation of exotic particles, that may not be discovered in laboratories. The light pseudoscalar particle, dubbed the axion, borne out of the Peccei-Quinn solution to the strong CP ii problem in quantum chromodynamics (QCD) is one such particle which remains elusive. We present a novel way of constraining its properties by examining the level of linear polarization in the radiation emerging from magnetic white dwarfs. On sub-meV mass scales, our study provides the strongest constraints on axion properties obtained astrophysically. The cooling theory of neutron stars is corroborated by comparison with obser- vations of thermally emitting isolated neutron stars. An important ingredient for such an analysis is the age of the object, which typically is highly uncertain. We conduct a population synthesis study of the nearby isolated thermal emitters and obtain their ages statistically. iii Preface The text of this dissertation includes reprints of previously pubished material as listed below. The co-author listed in these publications directed and supervised the research which forms the basis of this dissertation. The co-author suggested the important problems in the field of research and is responsible for designing the research program. The author of this dissertation is responsible for all the research conducted in its entirety and writing the manuscripts of the listed publications. • Chapter 2: R. Gill and J. S. Heyl, Dispersion relations for Bernstein waves in a relativistic pair plasma, Phys. Rev. E80, 036407 (2009). • Chapter 3: R. Gill and J. S. Heyl, Constraining the photon-axion coupling constant with magnetic white dwarfs, Phys. Rev. D84, 085001 (2011). • Chapter 4: R. Gill and J. S. Heyl, On the trigger mechanisms for soft gamma-ray repeater giant flares, MNRAS407, 1926 (2010). iv Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Strongly Magnetized Compact Objects . . . . . . . . . . . . . . . 1 1.1.1 Isolated White Dwarfs . . . . . . . . . . . . . . . . . . . 2 1.1.2 Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Bernstein Waves in Electron-Positron Pair Plasmas . . . . . . . . . 20 2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 What is a Plasma? . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Bernstein Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 v 2.4 Relativistic Dispersion Relation . . . . . . . . . . . . . . . . . . . 23 2.4.1 Numerical Approach . . . . . . . . . . . . . . . . . . . . 28 2.5 Properties of Dispersion Curves . . . . . . . . . . . . . . . . . . 30 2.5.1 Relativistic Effects . . . . . . . . . . . . . . . . . . . . . 35 2.5.2 Behavior for Small k̂⊥ . . . . . . . . . . . . . . . . . . . 37 2.5.3 Behavior for Large k̂⊥ . . . . . . . . . . . . . . . . . . . 37 2.5.4 Stationary Modes . . . . . . . . . . . . . . . . . . . . . . 39 2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 Axion Physics with Magnetic White Dwarfs . . . . . . . . . . . . . . 46 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 The Strong-CP Problem . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.1 Axion-Like Particles . . . . . . . . . . . . . . . . . . . . 48 3.3 Astrophysical and Laboratory Constraints on Properties of Axion and ALPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3.1 Constraints from Astrophysical Objects . . . . . . . . . . 50 3.3.2 Laboratory Constraints . . . . . . . . . . . . . . . . . . . 54 3.3.3 Magnetic White Dwarfs . . . . . . . . . . . . . . . . . . 57 3.3.4 Plan of this Study . . . . . . . . . . . . . . . . . . . . . . 58 3.4 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4.1 Off-Centered Non-Aligned Three Dipole Model . . . . . . 60 3.4.2 Fully Ionized Pure H Atmosphere . . . . . . . . . . . . . 62 3.4.3 ALP-Photon Mode Evolution in an Inhomogeneous Mag- netized Plasma . . . . . . . . . . . . . . . . . . . . . . . 65 3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.5.1 Constraints on gaγγ . . . . . . . . . . . . . . . . . . . . . 71 3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.6.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4 What Triggers the Soft Gamma-ray Repeater Giant Flares? . . . . . 78 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 vi 4.1.1 The Giant Flares . . . . . . . . . . . . . . . . . . . . . . 79 4.1.2 The Precursor . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 Magnetic Reconnection: An Overview . . . . . . . . . . . . . . . 82 4.2.1 Steady-State Reconnection . . . . . . . . . . . . . . . . . 82 4.2.2 Unsteady Reconnection: The Tearing Mode . . . . . . . . 88 4.2.3 Collisionless Plasma: Hall Reconnection . . . . . . . . . . 88 4.3 Internal Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4 External Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.4.1 Transition from Resistive to Collisionless Reconnection by Current Sheet Thinning . . . . . . . . . . . . . . . . . 96 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5 The Cooling Behavior of Nearby Isolated Neutron Stars:The Mag- nificent Seven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.2 The Magnificent Seven . . . . . . . . . . . . . . . . . . . . . . . 108 5.2.1 Spin-Down Ages: Poor Age Estimators . . . . . . . . . . 110 5.3 True Age Estimates of Isolated Neutron Stars . . . . . . . . . . . 118 5.3.1 Age Estimates from Population Synthesis . . . . . . . . . 118 5.3.2 RASS and Population Synthesis . . . . . . . . . . . . . . 119 5.3.3 Statistical Ages Vs The True Ages . . . . . . . . . . . . . 125 5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.1 Axion/ALP Physics with the Double Pulsar . . . . . . . . . . . . 129 6.2 High Energy Radiation Processes in Magnetar Bursts . . . . . . . 131 6.3 Mode Coupling of Bernstein Waves to Radiation in Electron-Positron Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 vii A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 A.1 Bernstein Waves in Fully Relativistic Pair Plasmas . . . . . . . . . 152 viii List of Tables Table 1.1 Spectral classification scheme of WDs . . . . . . . . . . . . . 3 Table 3.1 Model parameters used to construct the magnetic field of a white dwarf to study the properties of ALPs . . . . . . . . . . 59 Table 5.1 Properties of isolated NSs with SNR or massive star cluster associations . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Table 5.2 Properties of nearby thermally emitting isolated NSs . . . . . . 124 ix List of Figures Figure 1.1 Schematic of the internal structure of a neutron star. . . . . . . 7 Figure 1.2 The P − Ṗ diagram of pulsars. . . . . . . . . . . . . . . . . . 9 Figure 1.3 Illustration of a rotating neutron star . . . . . . . . . . . . . . 11 Figure 2.1 Contour of integration in the complex p̂-plane . . . . . . . . . 30 Figure 2.2 Dispersion curves for a relativistic plasma with η = 1, ω̂p = 3 32 Figure 2.3 Dispersion curves for a relativistic plasma with η = 5, ω̂p = 3. 33 Figure 2.4 Dispersion curves for a relativistic plasma with η = 20, ω̂p = 3 34 Figure 2.5 Dispersion curves for different values of η with ω̂p = 3 . . . . 36 Figure 2.6 Stationary modes at vanishing k̂⊥ as a function of non-dimensional reciprocal temperature parameter η . . . . . . . . . . . . . . . 40 Figure 2.7 Dispersion curves at two different plasma frequencies with η = 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Figure 2.8 Stationary modes as a function of the plasma frequency at dif- ferent values of the reciprocal temperature parameter η . . . . 43 Figure 3.1 Illustration of the coordinate system used to the obtain the ag- gregate magnetic field . . . . . . . . . . . . . . . . . . . . . . 61 Figure 3.2 Illustration of the coordinate system used to obtain the photon- ALP mode evolution, and a plot of the decline of the magnetic field strength with distance from the star . . . . . . . . . . . . 68 Figure 3.3 Polarization evolution of an unpolarized photon along a given line of sight . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 x Figure 3.4 The final state of polarization of radiation after traversal from the WD’s magnetosphere for different ma and gaγγ . . . . . . 72 Figure 3.5 Exclusion plot in the ma − gaγγ parameter space . . . . . . . 75 Figure 4.1 The Sweet-Parker model of steady-state reconnection . . . . . 83 Figure 4.2 The Petschek model of steady-state reconnection . . . . . . . 87 Figure 4.3 Illustration of the tearing mode instability . . . . . . . . . . . 89 Figure 4.4 A sketch of the different spatial scales involved in Hall recon- nection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Figure 4.5 Schematic of the different reconnecting current layers in the neutron star magnetosphere . . . . . . . . . . . . . . . . . . . 94 Figure 5.1 Comparison of the spin down ages and the SNR ages of some isolated NSs. The red line shows where the two ages are co- incident. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Figure 5.2 Change in the spin-down or characteristic age of an isolated NS due to various magnetic field decay mechanisms . . . . . . 116 Figure 5.3 RASS geometrical arrangement & exposure map . . . . . . . 120 Figure 5.4 Statistical cooling curve of the nearby thermally emitting iso- lated NSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 xi Acronyms ADMX Axion Dark Matter Experiment. AGB asymptotic giant branch. ALP axion-like particle. AXP anomalous x-ray pulsar. CAST Cern Axion Solar Telescope. DFSZ Dine-Fischler-Srednicki-Zhitnitsky. EDM electric dipole moment. EOS equation of state. GRB gamma-ray burst. ISM interstellar medium. KSVZ Kim-Shifman-Vainshtein-Zakharov. LMC large magellanic cloud. LOS line of sight. xii M7 magnificent seven. MHD magnetohydrodynamics. MWD magnetic white dwarf. NS neutron star. PSR pulsar. QCD quantum chromodynamics. QPO quasi-periodic oscillation. RASS ROSAT all-sky survey. SGR soft gamma-ray repeater. SHTCL super-hot turbulent current layer. SNR supernova remnant. TD01 Thompson and Duncan (2001). TD95 Thompson and Duncan (1995). WD white dwarf. xiii Dedication To my parents xiv Chapter 1 Introduction 1.1 Strongly Magnetized Compact Objects In astronomy, the term compact object encompasses three different kinds of de- generate objects - white dwarfs (WDs), neutron stars (NSs), and black holes (see for e.g. Shapiro & Teukolsky (1983) for the physics of compact objects). They have large masses but small sizes, and thus, are exceedingly dense in comparison to their progenitors. Stars currently on the main sequence are born with a predeter- mined fate as to what remnant they will leave based on their masses. The demar- cation is not so strict, but isolated stars with main sequence masses M < 8Ma (low mass) leave WDs as remnants; for 8M . M . 25M (intermediate mass) the end product is a NS and higher mass stars are fated to produce black holes. In this thesis, we are mainly concerned with the study of WDs and NSs and their environments because the physics of these objects is much more accessible due to the radiation that emanates from their surfaces and is also produced in their magnetospheres; the study of black holes, although interesting in its own right, is more accretion driven. a1 M = 1.99× 1033 g is the mass of the Sun. 1 1.1.1 Isolated White Dwarfs The evolution of WD and NS progenitors on the Hertzsprung and Russell (H-R) diagram is similar up to the core 4He ignition phase. In the case of low mass stars, 4He ignition commences at a temperature ∼ 108 K and produces 12C which only ignites for stars at the upper end of this mass regime. As the 4He reserves in the core are exhausted, the star evolves into an asymptotic giant branch (AGB) star, whose core resembles a proto-WD. The main source of luminosity at this point is the alternate burning of 1H and 4He in concentric shells. During the AGB phase, the star loses prodigious amounts of gas from its outer envelope at high rates of 10−4M yr−1. Eventually, the hot and electron degenerate 12C and 16O core, where 16O is produced by the capture of a helium nucleus by 12C at only slightly higher temperatures (Hansen et al., 2004), emerges from a cocoon of the stellar envelope. The shed envelope then may become photoionized by the newly born WD which has an effective temperature in the range Teff ≈ 103 − 105 K, and manifest as a planetary nebula. In general, low mass stars don’t reach temperatures and densities needed to burn 12C and 16O into heavier elements. However, observations of WDs in binary systems suggest that, in a narrow mass range, 12C is fused together to produce 20Ne followed by the production of 24Mg and more 16O. In this case, the remnant is no longer a C-O WD but a O-Ne-Mg WD instead. General Properties of White Dwarfs As per the foregoing discussion, WD progenitors have a broad mass range, how- ever, the remnants themselves are only found in a narrow one (see for e.g. Hansen et al. (2004); Kawaler et al. (1996) for a detailed discussion). Single WDs, that are not part of a binary system, have an average mass of 0.6M with a small dis- persion of 0.1M around this value (Hansen et al., 2004). This estimate doesn’t apply to WDs in binaries, for example the mass of Sirius B, whose binary com- panion is a main sequence star, is 1.034±0.026M (Holberg et al., 1998). A look at the H-R diagram of the nearby WDs residing in the Galactic disk reveals a tight 2 Table 1.1: Spectral classification scheme of WDs Spectral type Characteristics DA Strong neutral 1H absorption lines; no 4He. DB Strong neutral 4He lines; no 1H. DO Strong singly ionized 4He lines; None of the above. DC Star shows continuous spectrum with no strong lines. DZ Strong metal lines. No 1H or 4He. DQ Strong 12C lines. correlation between their intrinsic luminosity and effective surface temperatures. This observation leads to the finding that all WDs have sizes ∼ 1%R ≈ 7× 108 cm. Structurally, WDs are very simple objects. Although there is no direct way of ascertaining the internal structure of a WD, other than inferences derived from non-radial pulsations observed for pulsating WDs (see for e.g. Kawaler & Hansen 1989), the following inferences are purely based on theoretical modeling and its comparison with observations. The electron degenerate C-O core of the WD makes up almost all of its mass and volume. It is surrounded by a thin surface layer containing a mixture of 1H and 4He with some contamination from trace elements. There is an upper limit on the total mass of 1H and 4He of 10−4M and 10−2M, respectively, in the surface layer; Any excess will lead to further nuclear fusion. WDs are classified spectroscopically based on the elemental abundances in their atmospheres, which reflect the composition of their surfaces. The majority, almost ∼ 80%, of WDs have pure 1H surfaces, and they are classified as DA WDs (refer to Table 1.1). The remaining 20%, the DB WDs, predominantly show strong 4He lines and no 1H lines. WDs with peculiar features other than that found in the DA and DB types form an insignificantly small population. The pure hydrogen or helium surface layers are subject to contamination from accretion from the surrounding interstellar medium or by radiative levitation and thermal diffusion. These contaminants then add to the opacity and manifest as additional absorption lines depending on the effective temperature of the WD. For example, 3 stars that show singly ionized 40Ca lines, and lines of 56Fe and 16O are classified as DZ WDs. The spectral types listed in Table 1.1 are not entirely exclusive. In many instances, a spectrally classified DA WD will show some 4He lines and it is referred to as a DAB star; a DOZQ star is a DO star that also shows lines from carbon and other metals. The different spectral types of WDs are correlated with effective surface tem- peratures. The DA WDs, that are identified based on the strength of the Balmer lines, are found in a broad range of effective temperatures 5, 600 K . Teff . 80, 000 K. The DB WDs show a continuous progression for Teff . 30, 000 K. However, there is an abrupt cutoff in the number of observed DB WDs in the temperature range 30, 000 K . Teff . 45, 000 K. Within this range all WDs are found to be DA with no 4He in their atmospheres. At Teff & 45, 000 K, DO stars with singly ionized 4He are found up to effective temperature of ∼ 100, 000 K. The abrupt disappearance of helium-rich stars is perhaps the biggest mystery in the study of WD surface composition and evolution. Magnetic White Dwarfs The incidence of magnetism in WDs is somewhat of a novelty, rather than the norm. Only 3% of known WDs possess magnetic fields in the range 103 − 109 G, while most are non-magnetic (see for e.g. Wickramasinghe & Ferrario (2000) for a review). Kemp et al. (1970) discovered the first magnetic white dwarf (MWD), named Grw +70◦8247, by detecting strong circular polarization in its spectrum. The theory behind this observation was advanced by Kemp (1970) that electrons gyrating in the strong magnetic fields of WD in the presence of ions would emit highly polarized magneto-bremsstrahlung radiation. The two main techniques for finding MWDs are the detection of linear and quadratic Zeeman splittings in spectral lines and continuum circular polarization in strongly magnetized WDs (Hansen et al., 2004). In the former case, the lines have to be strong and readily recognizable such as the 1H and 4He lines in DA and DB WDs. In regards to the atmospheric composition, the MWDs almost exactly parallel the distribution 4 observed for non-magnetic WDs with the majority being hydrogen-rich and only ∼ 20% with helium-rich atmospheres. The number of MWDs has risen steadily to 170 objects which still is only a small (∼ 3%) percentage of the total observed population of WDs. There are 116 WDs within 20 pc of the Sun, of which 15 have been found to be magnetic, yield- ing an incidence of 13%± 4% (Kawka et al. (2007), and references therein). The magnetic field strength of these objects peaks around ∼ 16 MG where the field strengths in the range 105 − 109 G are easily measured by Zeeman spectropo- larimetry. The incidence of magnetism in these objects decreases with increasing temperature, contrary to the earlier belief that cooler WDs show higher incidence (Liebert & Sion, 1979). MWDs differ, most importantly, from their non-magnetic counterparts in terms of their masses. As stated earlier, the average mass of non- magnetic WDs is around 0.6M, whereas MWDs tend to be more massive with 〈M〉 ∼ 0.8M. This disparity can be understood by recognizing the role mag- netic fields play during the mass loss phase of the star; magnetic fields may inhibit excessive mass loss and produce massive remnants. The utility of detecting a cir- cular polarization in the spectra of these objects led to the discovery of rotation in MWDs (Angel & Landstreet, 1971). There appear to be two subclasses of MWDs, the ones that rotate so slowly that no periodic variations of spectra or po- larization components can be measured, and the others that have measured periods that range from tens of minutes to few days. The progenitors of MWDs have been argued to be the moderately magnetic Ap and Bp stars with fields in the range B ∼ 102 − 104 G (Wickramasinghe & Ferrario, 2000). If the fields acquired by the MWDs are indeed fossil then mag- netic flux conservation can easily accommodate their measured field strengths. 1.1.2 Neutron Stars Intermediate mass stars are massive enough to contract the 12C depleted 16O core and burn the latter into heavier elements. Thermonuclear burning of heavier ele- ments in an onion-like shell structure ensues until the core is entirely made of 56Fe. 5 Since 56Fe is the most tightly bound atomic nucleus resulting from the beta decay of symmetric nuclei, it cannot participate in an exothermic reaction. This upsets the hydrostatic equilibrium and forces further gravitational collapse. The contract- ing core that has reached a core temperature of Tc ∼ 1010 K can now commence the photodisintegration of heavier nuclei. The degenerate electrons are now cap- tured by heavy nuclei and protons producing neutrinos via inverse β-decay. The resulting loss in degeneracy pressure holding the imminent gravitational collapse throws the core in a free fall. When the density reaches ∼ 8 × 1014g cm−3, neu- trons become degenerate and the infalling core rebounds from a stiff inner core. An outward moving shock wave is launched which disrupts the outer envelope of the star in a spectacular type-II supernova explosion, giving birth to a NS. The existence of NSs in the form of extremely dense and compact stellar ob- jects was envisaged by Baade & Zwicky (1934). A little over thirty years later, it was realized that fast rotating NSs with strong dipole magnetic fields could be de- tected as a source of radiation energizing the surrounding nebula (Pacini, 1967). Then in 1968 came a remarkable serendipitous discovery of the first radio pul- sar (PSR) B1919+21 (Hewish et al., 1968). With an objective of finding sources similar to that of Hewish’s discovery, subsequent searches led to the discovery of two short-period pulsars, the Vela with a rotation period of 89 ms and the Crab with a smaller period of only 33 ms (Large et al., 1968; Staelin & Reifenstein, 1968). Furthermore, these searches were then extended into the optical and X-ray region of the spectrum that, ultimately, identified the Crab as an X-ray PSR. Over time, the population of X-ray PSRs grew in number as it now included the binary millisecond PSRs as well. General Properties of NSs The internal structure of the NS depends on the particular equation of state (EOS). In general, the inside of a NS is composed of a 1 km thick solid crystalline crust followed by a liquid core comprised mainly of neutrons. These can be further divided into four distinct zones based on density (see Fig. 1.1). In the following 6 Condensates of ~10 km~0.3 km ~0.6 km Outer crust: nuclei Inner crust: nuclei + neutron gas                              Outer core:                              Uniform nuclear matter Rod- and plate-like structures Quarks?π , Κ , Σ , . . .? Inner core: Figure 1.1: Schematic of the internal structure of a NS (adapted from Heisel- berg (2002)). discussion it is assumed that M = 1.4M and R = 10 km. • Outer Crust: It is comprised of a Coulomb lattice of nuclei, starting with 56Fe and becoming increasingly neutron rich as the density is increased, in a sea of degenerate electrons. The density in this layer ranges from 106 g cm−3 . ρ . ρdrip = 4× 1011 g cm−3. • Inner Crust: For ρ > ρdrip, massive nuclei become unstable and neutrons start dripping out of them, forming a degenerate fluid embedded in the de- generate sea of electrons. The inner crust extends to densities ρ . ρnuc = 2.8×1014 g cm−3 where the nuclei become non-spherical before dissolving fully into nuclear matter. • Outer Core: The range of densities found here are ρnuc . ρ . 2ρnuc where the composition is understood to be a fluid of neutrons, protons, and elec- 7 trons. The concentration of the charged components is only 5% of that of the neutrons and is set by the condition of equilibrium under β-decay. • Inner Core: The nature of the inner core is highly uncertain, where the matter density exceeds ρnuc and extends to a model dependent core den- sity ρc ∼ 1015 g cm−3. At such high densities, hyperons, pi and K-meson condensates, and a gas of free quarks may exist. The mass and size of a NS critically depends on the EOS as well as on its rotation period. There is a fairly good understanding of the EOS for ρ . ρnuc. Detailed models of the inside of a NS suggest that the maximum mass of a NS is about 3M (Lyne & Graham-Smith, 2006). However, observations of rapidly rotating NS have only revealed masses ranging from 1.06+0.11−0.10 M (van der Meer et al., 2007) up to 1.97± 0.04M (Demorest et al., 2010). Estimate of the radius can be obtained by simply equating the centrifugal force Ω2R to the gravitational force GM/R2 . The centrifugal force acts to expand the star due to its large rotational motion while the gravitational force acts in an opposing manner to collapse the star down to its central point of gravity. The equilibrium between the two yields a relation for the maximum radius depending on the rotational period of the star R = 1.5× 103 ( M M )1/3 P 2/3 km (1.1) From the shortest known period of 1.4 ms of PSR J1748-2446ad (Hessels et al., 2006), the above equation yields a maximum radius of 21 km. Typically, for calculations giving an order of magnitude estimate, a radius of 10 km and a mass of 1.4M is assumed for the NS. Rotation Powered Pulsars The population of more than 1700 rotation powered PSRs forms the bulk of the observed population of NSs in the Galaxy, as is evident from Fig. 1.2. The popu- lation of isolated rotation powered PSRs comprises of the normal radio PSRs and 8 ll l l l l l l l l l l l ll l l l l l l l l l l lll l ll l l ll l ll l ll l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l ll l lll ll l l l l l l l l l l l l ll l ll l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l llll l l l l l l l l l l l l l l l l l l l l l l lll l l l ll l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l lll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l ll l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l ll l l l ll l l l l l n n n n n n n n n n nn n n n H H H H H H H H H H H l n n H Period (s) L og (P er io d d er iv a ti ve ) (s s− 1 ) 1014 G 1013 G 1012 G 1011 G 1010 G 10 5 yr 10 6 yr 10 7 yr 10 8 yr Pu lsa r d ea th lin e 10−3 10−2 10−1 1 10 −20 −18 −16 −14 −12 −10 −8 Binary SGRS XDINS AXPS Figure 1.2: The P − Ṗ diagram of pulsars (PSRs): Gray dots are the normal radio PSRs, and the millisecond and other binary PSRs are denoted by open circles (Manchester et al. (2005); Data obtained from The open squares denote the X-ray dim isolated NSs (XDINSs), a.k.a. the mag- nificent seven (see Sec. 5.2 for discussion). The magnetars are rep- resented by the soft gamma-ray repeaters (SGRs; solid squares) and anomalous X-ray pulsars (AXPs; stars) (Data obtained from http: //∼pulsar/magnetar/main.html). The dot- ted lines show the polar magnetic field strengths and the dashed lines are the characteristic ages. The solid line is a model dependent death line beyond which all pulsar activity ceases. 9 the radio-quiet gamma-ray PSRs (see for e.g. Abdo et al. (2009); Ghosh (2007); Lyne & Graham-Smith (2006) for more details). Although the objects in both groups shine in other wavebands as well, they are primarily characterized by their pulsed radio and/or gamma-ray emission. The radio pulsations show high bright- ness temperatures and are beamed towards the observer as in a rotating lighthouse beam. The emission beam for gamma-rays is much broader, which also is taken as evidence for outer magnetospheric emission from open field lines over polar cap emission (Watters et al., 2009). The period of pulsations reveals the rate at which the NS is rotating, a property that is essential to the generation of radio emission. The normal radio PSRs have periods in the range∼ 0.1−10 s and they are slowly spinning down at the rate∼ 10−13−10−16 s s−1. Another class of rotation powered PSRs that have evolved differently from the normal radio PSRs are the millisec- ond PSRs, although many maybe found presently as isolated objects but they all at one point existed in a binary. They are characterized by very short spin periods in the range ∼ 1 − 10 ms and have spin-down rates of ∼ 10−19 − 10−20 s s−1. These objects represent a much evolved population in comparison to the normal radio PSRs and have been re-spun by accretion from their binary companions. In both groups, but especially in the case of the millisecond PSRs, the individual pulses show exceptional regularity and stability that is at par with atomic clocks. Some show variations of few tens of nanoseconds over a period of five or so years (Demorest et al., 2012). The high brightness temperatures detected in the beamed radio emission in- dicates the operation of a coherent emission mechanism as conjectured by Gold (1968), who in a prescient paper suggested rotating NSs to be the underlying source of the high intensity and very regular radio pulsations. He also noted the necessity of a corotating magnetosphere which would terminate at the velocity of light cylinder (r sin θ = c/Ω, where r is the radius at which any charge co-rotating with the NS at the angular frequency Ω will have a tangential velocity equal to the speed of light), and strong surface fields that would lead to charged particle accel- eration to relativistic speeds (see Fig. 1.3). Similar suggestions came from Pacini 10 Figure 1.3: Illustration of a rotating NS with a co-rotating magnetosphere. The closed magnetic field lines end at the velocity of light cylinder. Charged particles stream out along open field lines (adapted from Gol- dreich & Julian (1969)). 11 (1968) where he suggested the presence of a rotating NS that could power the Crab Nebula, as we show below. On the observational front, Radhakrishnan et al. (1969) presented evidence for the high degree of linear polarization of the radio pulses received from the Vela pulsar, the second fastest at the time after the Crab. Since the polarization profile traces the local magnetic field, a large change in the angle of the plane of polarization, in the case of Vela, led them to suggest that the observed radiation is generated close to the magnetic poles, and they are aligned at a considerable angle to the rotation axis. All these early works in conjunction with key revelations from observations bolstered the idea that radio PSRs are rota- tion powered NSs with strong magnetic fields. The pulsar emission mechanism is still an active area of research. Several early models explaining the global proper- ties of pulsar emission have been advanced by Cheng et al. (1986a,b); Goldreich & Julian (1969); Muslimov & Harding (2003); Ruderman & Sutherland (1975); Sturrock (1970) to cite a few. The typical polar magnetic field strength Bp of rotation powered NSs can be easily deduced from simple electrodynamics arguments and the observed proper- ties of these objects (see for e.g. Ghosh 2007). The rotational energy of a solid body is calculated from Erot = 1 2 IΩ2 (1.2) Since the pulsars are losing energy through radiation which is being generated at the expense of the rotational energy of the star, they ought to spin down. There- fore, the angular frequency is time variant, and power lost by this change is Ėrot = IΩΩ̇ = −4× 1032I45P−3Ṗ−14 erg s−1 (1.3) where we have neglected any time variance in the moment of inertia I45, expressed in units of 1045 g cm2. Also, the rotation period P = 2pi/Ω is expressed in s and period derivative Ṗ−14 in units of 10−14 s s−1. For an order of magnitude estimate we can assume the geometry of the magnetic field to be of a dipole. Then a 12 rotating dipole gives off electromagnetic radiation with power Ėdip = 2µ2Ω4 sin2 α 3c3 ≈ 1031B212R66P−4 sin2 α erg s−1 (1.4) where µ = BpR3/2 is the magnetic dipole moment, and α is the angle between the dipole and the rotation axis; for an orthogonal rotator sinα = 1. If the spin- down of the star is only caused by the emission of radiation, then equating Eq. 1.3 and 1.4 will yield the polar magnetic field, B12 ≈ 6 √ I45Ṗ−14P R36 (1.5) The above estimate gives the typical magnetic field strength of normal radio PSRs. Their millisecond counterparts typically have a much reduced field strength of 108 G, where the field has arguably been buried underneath the accreted matter (e.g. Stairs 2004 and references therein). Magnetars Over the last three decades, another class of isolated X-ray PSRs with character- istics very much at variance with the normal radio PSR population has emerged. The term magnetar, in particular, refers to an ultra-magnetized NS with a mag- netic field, as derived from the spin-down law (Eq. 1.5), much larger than that of normal radio PSRs, B & 1014 G, and whose emission is magnetically driven rather than rotationally (Duncan & Thompson, 1992). However, objects like SGR 0418+5729, with a dipolar magnetic field B < 7.5 × 1012 G similar in strength to some of the high-field normal radio PSRs, may manifest as transition objects bridging both the magnetar and normal radio PSR families together (Rea et al., 2010). Initially, the magnetar phenomenology was introduced to explain the be- havior of highly energetic soft gamma-ray repeaters (SGRs) exhibiting sporadic bursts of γ-rays. Later the group of the anomalous x-ray pulsars (AXPs) was inducted into the magnetar class (see e.g. Woods & Thompson 2006 for a com- 13 prehensive review). With the first soft γ-ray burst recorded on January 7, 1979 from the source SGR 1806-20, these Galactic sources wrongly earned their association with clas- sical gamma-ray bursts (GRBs). Soon after, on March 5, 1979 a high intensity γ-ray burst, a.k.a. a giant flare, from the source SGR 0526-66 was picked up by the Venera space probes. The highly energetic burst saturated the detectors of var- ious high energy space telescopes at the time. A sudden increase in flux over a period of ∼ 0.1 s was observed which was followed by a quasi-exponentially de- caying but coherently pulsating terminal pattern with a period of ∼ 8.0 s (Mazets et al., 1979; Terrell et al., 1980). Furthermore, detection of a faint pulsating signal recorded the next day from the same location as the March 5th event indicated that the source cannot possibly be any subtype of the classical GRB. Repeat occur- rences of bursts from the same sources negated the misidentified nature of SGRs, and in conjunction with other observed properties established them as a different class of NSs. Thus far, the total tally of SGRs, including the unconfirmed candi- dates, has reached to 11 objects b. The main characteristic of SGRs is the bursting phenomena which occurs on various different time scales. The short spin period as well as the quiescent steady X-ray spectrum of SGR 0526-66 led to the conclu- sion that the source must be a NS. Timing data collected for all the SGRs places them in a narrow range of spin periods, P ≈ 2.5 − 9 s, accompanied by large spin-down rates, Ṗ ≈ 10−13 − 10−10 s s−1. From the spin-down law, this corre- sponds to polar magnetic fields of B ∼ 1013 − 1015 G and characteristic ages of τ ∼ 103 − 104 yrs. Thus, SGRs are known to have the highest magnetic fields so far observed in the Universe, and the age estimates are suggestive of their youth as compared to the older population of normal radio PSRs. SGR like bursts have also been recorded from most of the AXPs, but the level of activity is not as pronounced (e.g. Woods & Thompson 2006). Fahlman & Gre- gory (1981) discovered the first isolated X-ray PSR associated with a supernova remnant (SNR), in this case SNR G109.1-1.0, from the data taken by the Einstein bSee∼pulsar/magnetar/main.html 14 space observatory. Currently, the tally of such objects has risen to 12 including the unconfirmed candidates. All AXPs save for one are Galactic objects with CXOU 010043.1-721134 residing in the Small Magellanic Cloud. These objects display spectral properties that are distinct from the well studied X-ray binaries. Hence, the name anomalous that identified with their inconsistent characteristics at the time of discovery. However, AXPs are not so different from their bursting cousins, the SGRs, and share many similarities with them. Such as, their spin periods likewise fall in the narrow range of ∼ 2− 12 s with high spin-down rates of∼ 10−13−10−11 s s−1. Similarly, the derived polar magnetic field strengths are in the range ∼ 1013 − 1014 G with characteristic ages slightly older than that of the SGRs, τ ∼ 103 − 105 yrs. Magnetar Phenomenology A characteristic property of both SGRs and AXPs is the persistence X-ray emis- sion with luminosities L ∼ 1035 − 1036 erg s−1. The origin of such an exces- sive amount of energy release is unaccountable by the spin-down mechanism that powers the normal radio PSRs. If the magnetars are simply radiating away their rotational energy then their luminosities must agree with that obtained from the spin-down law Eq. 1.3. For instance, the timing history of AXP 4U 0142-615 shows that its P ≈ 8.7 s and Ṗ ≈ 0.2× 10−11 s s−1, then its spin-down luminos- ity is Ėrot ≈ 3.6×1031 erg s−1. However, its radiated power inferred from distance estimates and measured flux at Earth amounts to almost ∼ 103Ėrot. Similar es- timates of the rotationally radiated power obtained for the rest of the SGRs and AXPs fall short of the measured values. This discrepancy hints at an alternative emission mechanism. The magnetar model has been proposed and developed in the works of Dun- can & Thompson (1992); Thompson (2008); Thompson & Duncan (1995, 1996, 2001); Thompson et al. (2002). Upon formation, a magnetar spins at an excep- tionally high angular frequency with spin period∼ 1 ms. Within the first 30 s after its collapse, vigorous convective action in its core coupled with a high spin rate 15 results in the formation of a dynamo that generates a strong magnetic field. Subse- quently, this leads to an immediate spin-down of the rotating NS, due to magnetic braking, and the energy released in the process powers the residual supernova ex- plosion, making it brighter. Furthermore, some of the energy goes into giving the remnant pulsar a large recoil or kick velocity ∼ 1000 km s−1. These magnetars continue to spin-down at rates still much higher than that observed for normal radio PSRs, in the process, powering the emission of soft γ-rays and X-rays as observed from SGRs and AXPs. According to the magnetar model, the X-ray emission from magnetars is mostly powered by magnetic field decay (Heyl & Kulkarni, 1998; Thompson & Duncan, 1995). The transition from rotation powered to magnetic powered emission occurs very early for magnetars at a characteristic age of (Thompson & Duncan, 1996) tmag = 15 yrs [ 1 BQED ( Bp 1015 G )]−4 (1.6) A similar calculation for the normal radio PSRs with a typical dipole field ∼ 1012 G yields tmag ∼ 1013 yrs. By this time the radio PSRs will have already gone past their death line after which these objects become undetectable. The core of a NS is primarily composed of a neutron superfluid with small fraction of protons and electrons (Lyne & Graham-Smith, 2006). The magnetic field of the star is strongly anchored to the crust via the charged entities. Ambipo- lar diffusion limited by pressure gradient forces ensues in strongly magnetized neutron stars that leading to heating of the stellar crust and producing thermal emission from their surfaces (Goldreich & Reisenegger, 1992). It is this emission that is most noticeable in the spectra of SGRs and AXPs. van Riper (1988) pro- vides an estimate of the surface X-ray luminosity caused by magnetic dissipation, Lx = 1.2× 1035 ( Tc 6× 108 K )2.2 R26 erg s −1 (1.7) 16 where R6 is the size of the NS in units of 106 cm, and Teff = 1.3× 106 ( Tc 108 K ) 5 9 K (1.8) From spectral fits, the typical effective surface temperature of magnetars is Teff ∼ 106 K, which yields a core temperature of Tc ∼ 6 × 107 K. Then, from Eq. 1.7, the X-ray surface luminosity is on the order of Lx ∼ 1035 erg s−1, which agrees well with the observed luminosities of SGRs and AXPs. Furthermore, as the core magnetic field diffuses through the crystalline surface there is a tendency for the crust to yield and crack, ultimately injecting a large amount of energy∼ 1041 erg into the stellar magnetosphere. Thompson & Duncan (1996) note that the crustal cracks produced by diffusive action of the magnetic field are responsible for the observed short bursts from SGRs. In addition, they argue that large scale magnetic reconnection events are responsible for creating an expanding photon-electron-positron fireball on time scales ∼ 0.15 s. Such events manifest as the ultra-luminous giant flares with magnetic energy released in the form of hard gamma rays of E & 1 MeV. Some of the plasma, following the initial fireball, remains trapped in the magnetosphere and is detected as the pulsating tail of the flare. Based on the energetics of the flare, strong magnetic fields, B & 1014 G, are required to trap the escaping plasma fireball. 1.2 Thesis Outline The objects studied in this thesis fall in the upper end of the magnetic field spec- trum in their respective categories. For example, we look at WDs PG 1015+14, PG 1031+234 and SDSS J234605+385337 that have magnetic field strengths, de- termined from Zeeman spectropolarimetry, of 108 G and ∼ 109 G, respectively. We study NSs that are part of the Magnificent Seven (M7; see Sec. 5.2) group and have empirically deduced magnetic field strengths of ∼ 1013 G. At the extreme end of the spectrum, we look at one of the most magnetized objects in the Galaxy, the magnetars, that have fields in excess of the quantum critical field strength of 17 4.4× 1013 G, which introduce novel and rich physical effects. The two main ingredients that lie at the heart of the astrophysical phenom- ena investigated in this thesis are strong magnetic fields embedded in electron- ion/positron plasmas. The emphasis on strong here implies magnetic field strengths much in excess of that generated terrestriallyc and that found in majority of stars in the Universe. Compact objects prove their utility in providing extreme condi- tions that allow one to test fundamental physics principles as in the case of the axion discussed in Chapter 3. In addition, they offer puzzling new revelations that not only provide deep insight into their makeup but also help one to break new ground in advancing the existing physical theories. This is indeed true in the case of the magnetars and the M7 objects that are shining new light at the physics of NSs. The common theme of magnetized plasmas and compact objects is explored in the next three chapters. We start with the exploration of plasma waves in mag- netized relativistic electron-positron pair plasmas in Chapter 2. Here, we are par- ticularly interested in understanding the behavior of Bernstein waves. We discuss the properties of this wave mode by calculating its dispersion relation in a fully relativistic framework. For the very first time, we show how the dispersion solu- tion of Bernstein waves in pair plasmas changes with an increase in the thermal energy of the plasma. The importance of this study in relation to the pulsar ra- diation mechanism via mode coupling to electromagnetic waves is discussed in Chapter 6. The extreme environments of compact objects facilitate in the investigation of some of the very important questions in fundamental physics. One such avenue that can be studied with strongly magnetized WDs is the properties of the invisible axion. In the past, several astrophysical sources have been used to constrain the mass of the axion and its coupling strength to photons. We study the creation of axions in the tenuous plasmas surrounding magnetic WDs and their interaction cThe strongest magnetic field strength ever attained in a laboratory is . 106 G for only a few ms. Fields vastly exceeding BQED are assumed to exist in heavy ion collision but are limited in space to a region of a few femtometers in radius and a duration of a few yoctoseconds. 18 with photons emanating from the surface in Chapter 3. In Chapter 4, we transition from mWDs to the extremely magnetized NSs. The magnetars display a variety of bursting phenomena, especially the soft gamma- ray repeaters (SGRs), and although the source of their high luminosities has been hypothesized to be the decay of their strong magnetic fields, many open questions remain. The SGRs unleash up to ∼ 1045 erg of energy in a matter of few seconds in the form of hard photons and energetic charged particles in their signature burst - the giant flare. We explore several trigger mechanisms that lead to the buildup and impulsive release of large amounts of magnetic energy. In the case of the radio pulsars, the study of the interior of NSs is complicated by the predominance of non-thermal emission from their magnetospheres. Since there is no thermonuclear burning occurring inside the isolated NSs, these hot cinders also cool via the emission of thermal radiation from their surfaces. The nearby M7 objects are pure thermal emitters or coolers. Corroboration of NS cooling theories with observations of such objects can lead to important insights into the complex structure of NSs beyond nuclear densities. In Chapter 5, we present a statistical approach to study the cooling behavior of M7 objects. 19 Chapter 2 Bernstein Waves in Electron-Positron Pair Plasmas 2.1 Motivation The goal of this thesis is to investigate the properties of plasmas pervading the strongly magnetized environment of compact objects. The interest in studying electrostatic plasma waves in e+e− pair plasmas is sparked from the still open problem of understanding the emission processes in normal radio PSRs. More importantly, we are interested here in providing a plausible solution to the prob- lem of drifting subpulses in radio PSR emission, which has been a long stand- ing puzzle. Subpulses are narrow intensity enhancements observed in individual pulses. They tend to drift longitudinally within the main pulse over several rota- tion cycles with remarkably persistent widths and heights (see (Rankin, 1986) for a review). A good understanding of this phenomena is critical for cracking the pulsar emission mechanism. Bernstein waves (Bernstein, 1958), by virtue of their transverse propagation to the magnetic field lines, can be used to explain subpulse drifts. These electrostatic waves can circle around the magnetic pole in the polar cap region and excite electromagnetic waves by mode coupling. Mode coupling of Bernstein waves in pair plasmas has not been considered yet. Astrophysically, 20 this holds a lot of promise as pair plasmas are found in abundance near highly energetic compact sources. 2.2 What is a Plasma? A plasma is a collection of oppositely charged particles that are equal in number so that the net charge over macroscopic distances is zero, and the plasma is said to be quasi-neutral. For example, ionization of neutral hydrogen gas leads to the formation of a gas containing protons and electrons with equal number densities. There are three parameters that characterize a plasma. First, is the length scale over which the quasi-neutral aspect of a plasma becomes apparent. The electric charge q of each particle is shielded by the neighboring charges and the electric potential is described by the Debye potential φD = q r e−r/λD (2.1) where λD is the Debye length which depends on the temperature T and the number density n0 of the plasma constituents, in this case assuming that the electron and ion number densities and temperatures are equal (ne ' ni = n0 and Te ' Ti = T ) λD = ( kBT 4pin0e2 )1/2 (2.2) A fully ionized plasma is considered quasi-neutral if the characteristic length scale of the system exceeds the Debye length, such that L λD  1 (2.3) The number of oppositely charged particles inside the Debye sphere, when the plasma is at equilibrium, must meet the following condition to satisfy Eq. 2.1 4pi 3 n0λ 3 D  1 (2.4) 21 This condition is referred to as the plasma approximation, and the associated pa- rameter Λ = n0λ3D  1 is called the plasma parameter. In the case of an electron- ion plasma with equal temperatures, the ions are considered immobile due to the large mass ratio. When the quasi-neutrality condition is upset the electrons oscil- late about the stationary ions at the electron plasma frequency ωpe = ( 4pinee 2 me )1/2 (2.5) In the presence of neutral species, the charged particles, say the electrons, will un- dergo collisions with them on a typical time scale of τn. If this time scale is much shorter than the time over which electrons undergo space-charge oscillations, the charged particles establish thermal equilibrium with the neutrals quickly. This de- stroys the plasma nature of the medium and behaves as an ideal gas. Therefore, the collision time must be much longer such that ωpeτn  1 (2.6) Detailed exposition on the properties of plasmas and plasma waves can be found in many pedagogical works, for example Baumjohann & Treumann (1996); Krall & Trivelpiece (1973); Stix (1992). 2.3 Bernstein Waves Relativistic electron-positron pair plasmas are found in many astrophysical objects such as neutron star magnetospheres (Ruderman & Sutherland, 1975), relativistic jets and accretion disks associated with black holes in the centers of active galactic nuclei (Begelman et al., 1984; Takahara & Kusunose, 1985). To study the proper- ties of the pair plasma in these objects it is imperative to employ a fully relativistic approach. In the study of magnetized pair plasmas one finds that a number of os- cillation modes exist (see Gedalin et al. (1998); Melrose (1997)). The focus of this study is to investigate the behavior of Bernstein modes in a uniform, magne- 22 tized, relativistic e+e− pair plasma. Bernstein waves are electrostatic undulations that are always localized near the electron cyclotron harmonics in an electron-ion plasma (Bernstein, 1958). These waves can propagate undamped only very close to the plane perpendicular to the static magnetic field. Such waves are of great interest since, in an electron-ion plasma, they strongly interact with the electrons and are excellent candidates for plasma heating and driving currents as compared with the electromagnetic (O)rdinary and the e(X)traordinary modes (Decker & Ram, 2006). The non-relativistic treatment of Bernstein waves in an electron-ion plasma is well understood (Baumjohann & Treumann, 1996; Dougherty, 1975; Krall & Trivelpiece, 1973; Swanson, 2003). It is the fully relativistic case that is marred by difficulties such that a closed form analytic solution is hard to formulate or maybe even impossible. Many workers have expounded on the fully relativistic treat- ment of electron Bernstein waves (Georgiou, 1996), the ultrarelativistic case (Buti, 1963), and have successfully obtained approximate dispersion relations (Saveliev, 2005, 2007). The Bernstein mode in a weakly relativistic pair plasma has been in- vestigated by Keston et al. (2003) where they have found closed curve dispersion relations that are remarkably distinct from the classical case. All of these studies either discuss the fully relativistic case to the point where no closed form analytic solution is found, or simplify the analysis by either treating the limiting case only or employ various approximations that may not yield an entirely correct result. 2.4 Relativistic Dispersion Relation The evolution of the distribution function, f(r,p, t), of plasma particles in phase space is governed by the Vlasov equation, which in the momentum representation is given as ∂fs ∂t + v · ∇rfs + qs(E+ v ×B) · ∇pfs = 0 (2.7) where s indicates different species constituting the plasma. To investigate the behavior of small amplitude waves with oscillation periods much smaller than 23 particle collision times, we make the following assumptions fs(r,p, t) = f0s(p) + f1s(r,p, t) (2.8) B = B0 +B1e i(k·r−ωt) (2.9) E = E1e i(k·r−ωt) (2.10) where the subscripts 0 and 1 indicate equilibrium and perturbed functions, re- spectively. Furthermore, we restrict the equilibrium distribution function to only depend on the momentum of the particles to account for the anisotropy introduced by the ambient static magnetic field. As a result, we write the Vlasov equation in its linearized form d dt f1s = −qs(E1 + v ×B1)ei(k·r−ωt) · ∇pf0s (2.11) The main idea here is to calculate the perturbation of the distribution function by integrating along the unperturbed orbits from some time in the past, say t0, to the present time t. Next, we use Ohm’s law to write the current density induced by the perturbed distribution J = ∑ s qs ms ∫ psf1s(r,p, t)d 3p = ∑ s σ s · E1 (2.12) This enables us to write the effective dielectric permittivity tensor in terms of the conductivity tensor σ (Swanson, 2003) (ω,k) = 0 ( I − σ iωs0 ) (2.13) In the fully relativistic approximation, the energy and linear momentum of parti- cles in the rest frame of the plasma are given as E = γmc2 = √ p2c2 +m2c4 (2.14) 24 p = γmv where the Lorentz factor is given in terms of the momentum as γ = ( 1 + p2 m2c2 ) 1 2 (2.15) The complete details of the rest of the calculation can be found in the Appendix A.1 and in various monographs and textbooks on plasma waves (see Baumjohann & Treumann (1996); Swanson (2003)), and we only provide the salient points of the derivation in what follows. We adopt B0 = B0ẑ for the equilibrium mag- netic field and restrict both the wave vector and the perturbed electric field to be k = k⊥x̂ and E1 = E1x̂ as dictated by the purely electrostatic mode where the perturbed magnetic field B1vanishes. After some mathematical manipulations we arrive at the relativistic dielectric tensor, (ω,k) =  xx xy 0−xy yy 0 0 0 zz  (2.16) where the different components have been summed over both species, e+ and e−, of the pair plasma, xx = 0 [ 1 + 4piq2m2 k2⊥m0 ∫ { piγa sin piγa Jγa(ξ)J−γa(ξ) −1 } γ p⊥ ∂f0(p) ∂p⊥ p2 sin θdpdθ ] yy = 0 [ 1 + 4piq2 ωm0 ∫ p⊥ ωc ∂f0(p) ∂p⊥ { pi sin piγa ×J ′γa(ξ)J ′−γa(ξ) + a γξ2 } p2 sin θdpdθ ] zz = 0 [ 1 + 4piq2 ωm0ωc ∫ p‖ ∂f0(p) ∂p‖ 25 ×piJγa(ξ)J−γa(ξ) sin piγa p2 sin θdpdθ ] xy = i 4piq2m2 mk2⊥ ∫ γ p⊥ ∂f0(p) ∂p⊥ p2 sin θdpdθ (2.17) In the above, ‖ and ⊥ subscripts denote components parallel and perpendicular to the equilibrium magnetic field, Jγa(ξ) is the Bessel function of non-integer order, J ′γa(ξ) is the derivative of the Bessel function with respect to ξ, where ξ = k⊥p⊥ qB , and a = ω ωc with ωc denoting the non-relativistic cyclotron frequency. Finally, one finds the dispersion relation, ω = ω(k), by setting the dielectric response function to zero, (ω,k) = k · (ω,k) · k = 0 (2.18) which in our case simply picks out the xx component. To keep the treatment fully relativistic we adopt the Maxwell-Boltzmann-Jütner distribution function (Schlickeiser, 1998; Swanson, 2003), f0(p) = (4pim 3c3)−1 η K2(η) e−ηγ (2.19) where η ≡ mc 2 kBT (2.20) is the ratio of the rest mass energy of the particles to that of their thermal energy, and K2 is the modified Bessel function of the second kind and of order two. Also, the equilibrium plasma distribution has been normalized to unity 1 = n0 = ∫ f0(p)d 3p (2.21) Taking the derivative of f0(p) with respect to p⊥ and p‖ yields ∂f0 ∂p? = − η 2 4pim5c5K2(η) p? γ e−ηγ (2.22) 26 where ? can be replaced by either ‖ or ⊥ components. At this point, we can carry out the integration over the polar angle and by defining β = k⊥p qB we can write xx as the following xx = 0 [ 1− ω 2 pη 2 k2⊥m3c5K2(η) ∫ ∞ 0 p2e−ηγ (2.23) × ∫ pi 0 { piγa sinpiγa Jγa(β sin θ)J−γa(β sin θ)− 1 } sin θdθdp ] where the non-relativistic plasma frequency is defined as ω2p = n0e 2 m0 (2.24) Next we use the following Bessel function identity (Swanson, 2003)∫ pi 0 sin θJa(b sin θ)J−a(b sin θ)dθ (2.25) = 2 sinpia pia 2F3 ( 1 2 , 1; 3 2 , 1− a, 1 + a;−b2 ) to express the integral over the polar angle in terms of a hypergeometric function, and redefine all constants and variables to make xx dimensionless p̂ = p mc k̂⊥ = k⊥c ωc ω̂ = ω ωc (2.26) ω̂p = ωp ωc β = β̂ = k̂⊥p̂ a = ω̂ Then, we can write xx as the following where the integral now is just over the dimensionless momentum p̂, xx = 0 [ 1− 2ω̂ 2 pη k̂2⊥ { η K2(η) ∫ ∞ 0 p̂2e−ηγ (2.27) ×2F3 ( 1 2 , 1; 3 2 , 1− γω̂, 1 + γω̂;−β̂2 ) dp̂− 1 }] 27 In writing this equation we have made use of the following integral identity∫ ∞ 0 p̂2e−ηγdp̂ = K2(η) η (2.28) The integrand in (2.27) consists of a hypergeometric function which is singular for 1− γω̂ = −n for integer n = 0, 1, 2, . . . ,. This singular behavior, as we shall see, is associated to the phenomenon of cyclotron resonance where plasma particles are in resonance with the wave at the cyclotron harmonics. Also, this very res- onance poses a real challenge for any numerical computation of the integral and has to be dealt with using advanced numerical techniques. Since our interest lies in finding the oscillation frequency ω̂ as a function of the wavenumber k̂⊥, it is clear from (2.27) that this operation is explicitly nonlinear. Therefore, one is left with an exercise of root finding for a given ω̂. Alternatively, one could simplify the analysis by making some approximation. However, by adopting such method- ology one risks losing the subtleties of the solution and may obtain something that is not entirely correct, as we show in the weakly relativistic case. We remain op- timistic and decide to compute the dispersion relation using a brute force method, that is by simply integrating (2.27). 2.4.1 Numerical Approach As the Lorentz factor is a function of momentum, the integrand remains singular over the domain of integration. A workaround for avoiding the singular points on the real axis is by analytically continuing the momentum to the complex domain. We can easily shift the integration contour below the real axis by writing p → p − iδ where δ is reasonably small. Ideally, one would like to keep the contour on the real axis but go below the singular points to avoid divergence from the singularities. A similar result can be achieved by closing the contour of integration in the lower half of the complex plane as shown in Fig. 2.1. Here we are interested in finding the principal value of the integral in Eq. 2.27, 28 which is given by the following: P.V. ∫ ∞ 0 dp̂ f(p̂) = ipi ∑ Residue R0 − (I2 + I3 + I4) (2.29) where I2 = ∫ ∞−iδ ∞ dp̂ f(p̂) ≈ 0, I3 = ∫ −iδ ∞−iδ dp̂ f(p̂), I4 = ∫ 0 −iδ dp̂ f(p̂), (2.30) andR0 is the residue from the poles on the Re(p̂) axis. In the above, for I2 we note that the integrand vanishes sufficiently rapidly for large p̂ due to the exponential. When ω̂ and k̂⊥ are kept real, one finds that R0 is also real, which makes the first term on the right in Eq. 2.29 purely imaginary. However, the second term has both real and imaginary components, and the negative sign is indicative of the clockwise sense of the contour. Since the left-hand side is real then so must be the right, which suggests that the imaginary components cancel each other and yield the following result: P.V. ∫ ∞ 0 dp̂ f(p̂) = −Re(I3 + I4) (2.31) Although it appears that with the given prescription one can have an arbitrarily large δ, we find that the integral does start to lose its accuracy as δ approaches unity. Therefore, we set δ = 0.1 for all numerical computations. Moreover, since the hypergeometric function is complex-valued, analytical continuation of p̂ will render xx to be complex. Nevertheless, we are only interested in the zeros of Re(xx) for a given ω̂ and k̂⊥. This reasoning is valid as the function has a behavior like 1 x for x →± 0 near the poles. Thus, if one were to integrate along the Re(p̂) axis then the contribution from the singularities must be negligibly small. Since 29 Re p̂ Im p̂ δ Figure 2.1: Contour of integration in the complex p̂-plane used to avoid sin- gular points of the integrand. Only the general case is shown here to guide the reader. the imaginary part results from the evaluation of the integrand at the poles, due to the Cauchy principle, it too must be negligibly small. Ultimately one is interested in only the principal value of the integral which is hard to compute directly given the complexity of the integrand. To compute the integral numerically over the hypergeometric function we used Mathematica (V.6) for it is capable of calculating generalized hypergeometric functions. The poles of the integrand were dealt with by employing a globally adaptive integration routine available in Mathematica. To speed up the integration over the singular points we used the Double Exponential Quadrature singularity handler built into Mathematica’s integration routine. 2.5 Properties of Dispersion Curves In the non-relativistic case of the electron Bernstein modes in an electron-ion plasma, one finds that there are no wave modes below the first harmonic. The dispersion curves above the hybrid frequency, that is given by ω̂2H = 1 + ω̂ 2 p, (2.32) 30 are all bell shaped with local maxima corresponding to stationary modes. Further- more, band gaps are present above the hybrid frequency between each dispersion curve. It is not at all surprising to say that the picture is remarkably different in the relativistic pair plasma scenario. We plot the dispersion curves for a relativistic pair plasma in Fig. 2.2, 2.3, 2.4 for different values of η = 1, 5, 20, respectively. We assume a plasma frequency of ω̂p = 3 for these plots. The dispersion relation for a moderately relativistic pair plasma, shown in Fig. 2.2, clearly has two wave modes. This is further accompanied by the existence of two stationary modes as the tangent to the dispersion curve vanishes at two distinct oscillation frequencies for k̂⊥ = 0. Also, above the higher stationary mode there are two wavenumber solutions for a given ω̂. This behavior persists in the case of a mildly relativistic pair plasma, shown in Fig. 2.3. However, one sees some drastic changes in the shape of the dispersion curves as the particles lose their energy. We readily notice the appearance of the curve near the cyclotron fundamental frequency. This marks the onset of the cyclotron resonance where the plasma particles oscillate at the same frequency as the perturbing electrostatic wave. Although not very significant at this point, the higher harmonic resonances also start to emerge. Furthermore, the dispersion relation now extends to higher wavenumbers and the turnover from the lower wave mode into the upper mode is not as sharp as it was in the previous case. We also notice a shift in the stationary modes, and we discuss this point in a later section. Keston et al. (2003) reported the dispersion relation for a weakly relativis- tic pair plasma. They found island shaped curves occurring mostly between the cyclotron harmonics. We find a similar solution for the weakly relativistic case, shown in Fig. 2.4, but with a few exceptions. Firstly, the dispersion curves are no longer closed but extend to higher wavenumbers as they approach cyclotron harmonics. This behavior is very similar to what we observe in the case of a non-relativistic electron-ion plasma. However, in the pair plasma case the solu- tion does not extend to an infinitely large wavenumber, and there are not infinitely many wave modes as we increase ω̂. There appears to be a cutoff in frequency, 31 k̂⊥ ω̂ 0 1 2 3 4 1 2 3 4 Figure 2.2: Dispersion curves for a relativistic plasma with η = 1, ω̂p = 3. Hatted variables are expressed in terms of the non-relativistic cy- clotron frequency ωc. See text for more detail. 32 k̂⊥ ω̂ 0 1 2 3 4 5 6 7 1 2 3 4 5 Figure 2.3: Dispersion curves for a relativistic plasma with η = 5, ω̂p = 3. Hatted variables are expressed in terms of the non-relativistic cy- clotron frequency ωc. See text for more detail. 33 k̂⊥ ω̂ 0 5 10 15 20 25 30 1 2 3 4 5 Figure 2.4: Dispersion curves for a relativistic plasma with η = 20, ω̂p = 3. Hatted variables are expressed in terms of the non-relativistic cy- clotron frequency ωc. See text for more detail. 34 the point where the overturn takes place, beyond which there does not exist any solution. Secondly, there exists a wave mode below the fundamental cyclotron harmonic which was absent in the solution provided in Keston et al. (2003). In fact, we find that a solution below the cyclotron fundamental exists for all cases, regardless of η, for ω̂p = 3 as we show below. This, again, is in contrast with the non-relativistic electron-ion plasma where there is no wave mode below the first harmonic. In comparison to the moderately relativistic case, the weakly relativistic case is richer in its behavior as well as much more structured. The former only has two frequency modes for a given wavelength, namely a high and a low mode, and the latter has many. Moreover, for a moderately relativistic pair plasma none of the wave modes found in between the two stationary points (discussed below) extend to infinitely small wavelengths. In fact, there is a limiting wavelength above which these modes exist. As reported in Keston et al. (2003), one can find similar, although much less severe, imperfections in the graphics produced by the contouring algorithm. The culprit here is the non linearity of the equation from which the solutions are obtained. 2.5.1 Relativistic Effects As the plasma particles become strongly relativistic, with decreasing η, the dis- persion curves undergo drastic changes. We now plot all of the dispersion curves shown earlier onto a single plot and analyze the progression from the weakly rel- ativistic case to the strongly relativistic one (see Fig. 2.5). The Bernstein waves are strongly absorbed near the cyclotron harmonics in the weakly relativistic limit where η  1. This effect is the strongest at the first cyclotron harmonic at which point the phase velocity of the wave vph → 0 and the wave loses all of its energy in heating up the pair plasma. At higher harmonics the same phenomenon is re- peated, however with decreased efficiency. Upon increasing the thermal energy of the plasma particles, the cyclotron resonances become much less pronounced and start to disappear completely. In a hot magnetized pair plasma no resonant 35 k̂⊥ ω̂ 0 5 10 15 1 2 3 4 5 Figure 2.5: Dispersion curves for different values of η with ω̂p = 3. (Solid) The weakly relativistic case with η = 20, (Dots) the mildly relativistic case with η = 5, and the two strongly relativistic cases: (a) (Dash) with η = 1 (b) (Dot-dash) with η = 0.5 36 interaction between the Bernstein wave and the plasma occurs, and the solution occupies only a small region of the ω̂ − k̂⊥ space. 2.5.2 Behavior for Small k̂⊥ The hypergeometric function in the integrand of (2.27) can also be written in the form of an infinite power series (Swanson, 2003) 2F3(a1, a2; b1, b2, b3;x) = Γ(b1)Γ(b2)Γ(b3) Γ(a1)Γ(a2) (2.33) × ∞∑ m=0 Γ(a1 +m)Γ(a2 +m) Γ(b1 +m)Γ(b2 +m)Γ(b3 +m) xm m! The series expansion of Eq. 2.27 gives a power series in k̂⊥ of the following form xx = 0[1 + α1(ω̂) + α2(ω̂)k̂ 2 ⊥ + α3(ω̂)k̂ 4 ⊥ + . . .] (2.34) Taking the limit k̂⊥ → 0 only leaves the first two terms in the series above, and yields the dispersion solution for large wavelengths xx = 0 [ 1 + 2ω̂2pη 2 3K2(η) ∫ ∞ 0 p̂4e−ηγ (1− γ2ω̂2)dp̂ ] (2.35) We numerically solve the above equation for vanishing xx and find the normalized frequency ω̂ as a function of the temperature (see Fig. 2.6). 2.5.3 Behavior for Large k̂⊥ In the weakly relativistic case, one finds wave-particle resonances occurring at cyclotron harmonics. These resonances extend to large values of k̂⊥ but not to in- finity. Intuitively, one may argue, by looking at the rest of the dispersion solution, that such a behavior is expected as the dispersion curves start and end at k̂⊥ = 0. The plot remains connected over the whole domain, as is evident in the strongly relativistic case, and the resonances only extend to some maximum value of the 37 wavenumber k̂Max⊥ . This hypothesis can be ascertained by computing the integral in (2.27) for k̂⊥  1, however for large values of k̂⊥ the calculation becomes very computationally expensive. Ideally, one would like to find the asymptotic behavior of the hypergeomet- ric function in (2.27) to simplify the problem. The asymptotic behavior of the hypergeometric function can be gleaned by asymptotically expanding the Bessel function in the integral representation of the hypergeometric function given in (2.26). The Bessel function expansion for large arguments is given as (S. & M., 2007) J±ν(z) = √ 2 piz cos ( z ∓ pi 2 ν − pi 4 ) +O (z−3/2) (2.36) Plugging this back into (2.26) and carrying out the integral over the polar angle yields, 2 pib ∫ pi 0 dθ cos ( b sin θ − pi 2 a− pi 4 ) cos ( b sin θ + pi 2 a− pi 4 ) = cos(api) +H0(2b) b (2.37) where H0 is the Struve function of order 0. For our specific case, b = k̂⊥p̂ and a = γω̂, and upon substitution of this result into the dielectric response function we find xx = 0 [ 1− 2ω̂ 2 pη k̂2⊥ { η 2k̂⊥K2(η) ∫ ∞ 0 p̂e−ηγ × ( piγω̂ sin piγω̂ ) [cospiγω̂ +H0(2k̂⊥p̂)]dp̂− 1 }] (2.38) This integral again has a similar singularity for γω̂ = n for integer n. We can further simplify this equation by noting the leading order behavior of the Struve function in the limit b →∞, which goes like H0(b) ∝ 1√b . Then, in this limit the dominant term in (2.38) is the cosine term. Although this step is not justifiable given the limits of integration where the integrand is evaluated for p̂  1, this 38 does not modify the overall behavior of the dielectric response function in the large k̂⊥ limit. Next, we define I(η, ω̂) = piω̂ ∫ ∞ 0 dp̂ γp̂e−ηγ cot(piγω̂) (2.39) and solve for xx = 0. After some rearrangement of terms, we arrive at a cubic equation which we then solve for k̂⊥ k̂3⊥ + 2ω̂ 2 pηk̂⊥ − ω̂2pη 2 K2(η) I(η, ω̂) = 0 (2.40) To do the integral in (2.39) we again employ the same contour integration scheme as was done earlier (see Fig. 2.1). For η = 20 and ω̂ = 1, which is the strongest resonance of all occurring at higher cyclotron harmonics, we find a maximum wavenumber k̂Max⊥ ∼ 72. We find values of the same order for resonances at higher harmonics as well. This limit shows the maximum value of k⊥ for which a solution exists using the simplified dielectric tensor (appropriate for large values of k⊥). The actual limits to the wavenumber in the realistic dispersion relation are typically lower. This exercise demonstrates that the resonances do not extend to infinitely small wavelengths, that there is some cutoff at k̂⊥ ∼ k̂Max⊥ , and the dispersion curves remain connected over the whole domain. More importantly, one must not forget that this estimate is particularly inaccurate for the strongly relativistic case where the solution exists for only modest values of k̂⊥. 2.5.4 Stationary Modes There are two stationary modes (vg = dω/dk⊥ = 0) present for vanishing k̂⊥ in all the cases shown above. We plot the evolution of both stationary points as a function of η in Fig. 2.6. For ωp = 3ωc the upper stationary mode remains above the cyclotron fundamental and the lower stationary mode remains below it for all η. Contrastingly, in the electron-ion case a solution exists for vanishing k̂⊥ at all cyclotron harmonics, except at the fundamental. It appears that the upper station- 39 ηω̂ 0 0 5 10 15 20 1 2 3 4 Figure 2.6: Stationary modes at vanishing k̂⊥ as a function of non- dimensional reciprocal temperature parameter η. The upper curve cor- responds to the upper stationary mode and the lower curve to the lower stationary mode. For this plot we assume ω̂p = 3. ary mode turns into the hybrid resonance, as given in Eq. 2.32, in an electron-ion plasma. This picture is slightly modified as we lower the plasma frequency so that it equals the cyclotron frequency, ω̂p = 1, while remaining in the weakly relativistic 40 limit with η = 20 (see Fig. 2.7). We still find those two stationary points and the strong resonance at the cyclotron fundamental, however, the rest of the plot has disappeared, and, interestingly, been replaced by a single closed curve. By comparing the present case to the one treated previously, with ω̂p = 3, we find that upon decreasing the plasma frequency the lower branch of the dispersion curve in the first harmonic band separates from the upper branch. Also the upper branch in the first harmonic band connects to the lower branch in the second harmonic band. This is the first incidence where a closed curve solution, like the ones found by the authors of (Keston et al., 2003), has appeared in our analysis. Furthermore, the number of stationary modes are now double of what was observed previously. We also provide a plot of the position of stationary points in frequency as a function of the plasma frequency for different values of η = 1, 5, 20 (see Fig. 2.8). As the slope of the curves varies with a change in η, the upper hybrid frequency, if one was to associate the upper stationary mode with it, necessarily depends on the temperature of the plasma, as shown in Fig. 2.6. Interestingly, in the case of a hot pair plasma (η ≤ 1) we find that the upper stationary mode extends below the cyclotron fundamental for ω̂p < 1. Thus, an underdense (ωp < ωc) strongly relativistic pair plasma does not have any Bernstein wave modes above the cyclotron fundamental. Another consequence of this situation is that the two wave modes exist for only small wavenumbers and therefore for extremely large wavelengths. We see that for a given plasma temperature, the upper stationary mode depends linearly on the plasma frequency for ω̂p > 1. On the other hand, the lower mode remains constant. 2.6 Discussion In this article, we investigate the behavior of Bernstein waves in a uniform, mag- netized, relativistic electron-positron pair plasma and provide dispersion curves for different values of the non-dimensional reciprocal temperature. The disper- sion solutions in all cases are found to be remarkably different than the Bernstein 41 k̂⊥ ω̂ 0 5 10 15 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 Figure 2.7: Dispersion curves at two different plasma frequencies with η = 20. (a) For ω̂p = 3, and (b) (bold) for ω̂p = 1. 42 ω̂p ω̂ 0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 Figure 2.8: Stationary modes as a function of the plasma frequency at differ- ent values of the reciprocal temperature parameter η. (a) (Solid) For η = 20, (b) (Dash) for η = 10, and (c) (Dot) for η = 1. 43 modes found in the non-relativistic electron-ion case. For a moderately relativis- tic pair plasma we find two Bernstein wave modes accompanied by two stationary modes for vanishing wavenumber. We do not find closed curve solutions (Keston et al., 2003) in all cases but one where the plasma frequency equals the cyclotron frequency in the weakly relativistic limit. As stated earlier, the Bernstein waves in a non-relativistic electron-ion plasma propagate undamped in the direction orthogonal to the equilibrium magnetic field. This might not be true for such waves in a relativistic pair plasma. These waves were found to be very weakly damped in a weakly relativistic pair plasma (Laing & Diver, 2005). In this article, we only report the real component of ω̂ as the imaginary component is quite non-trivial to calculate for the following reason. With ω̂ complex Eq. 2.31 does not hold since the principal value of the integral is no longer purely real. As a result, one must calculate the residue R0, which itself is a difficult task as the integrand is Eq. 2.27 cannot be easily transformed into the following form where the singularity arises due to a simple pole of order m: f(p̂) = g(p̂) (p̂− p̂0)m (2.41) and where g(p̂) is analytic as p̂→ p̂0. Nevertheless, we do expect very mild damp- ing of the waves (ω̂i  ω̂r) at least for the weakly relativistic case as discussed in (Laing & Diver, 2005), where they find Im(ω̂) to be the largest on the upper half of the curve that advances toward the cyclotron harmonics, and relatively much smaller on the lower half. It remains to be seen if damping is at all observed in the moderately relativistic case. The results presented in this article have important implications for all astro- nomical objects where a magnetized hot pair plasma is present. Especially, in the case of radio pulsars this aspect of the e+e− plasma, in the polar cap and outer magnetospheric region where the observed radio emission maybe generated, has never been explored. In particular, the existence of drifting subpulses in pulsar pulse profiles can potentially be explained by the mode coupling of Bernstein 44 waves with electromagnetic modes in the pulsar pair plasma. Subpulses are nar- row intensity enhancements that appear in individual pulses and tend to drift lon- gitudinally within the main pulse over several rotation cycles (see Rankin (1986) for a review). Explaining the drifting phenomenon is a three part process. This study forms the first part where we have shown the behaviour of Bernstein modes in relativistic pair plasmas with a uniform equilibrium magnetic field. To mode couple these waves, the equilibrium magnetic field has to be non-uniform (see for e.g. Stark et al. (2007)). Thus, the investigation of Bernstein modes in inhomo- geneous plasmas is needed first. Finally, if these waves do indeed exist in such plasmas, the problem of mode coupling can then be investigated. 45 Chapter 3 Axion Physics with Magnetic White Dwarfs 3.1 Introduction The Standard model of particle physics identifies the source particles that exist in nature and describes their interactions via the electromagnetic, weak, and strong forces (Griffiths, 1987). At the fundamental level, the source particles can be divided into two groups: quarks and leptons. Neglecting gravity, all processes occurring in nature can be stripped down to interactions of these two groups that are mediated by the force carriers - photons, intermediate vector bosons, gluons. The leptons, as they are devoid of any color degrees of freedom, only participate in electromagnetic and weak interactions by exchanging photons and W±, Z0 vector bosons. The quarks, on the other hand, interact strongly (by the exchange of gluons which carry color), weakly, and electrodynamically. All these particle interactions can be described using the lagrangian formalism, L = LQED + Lweak + Lstrong (3.1) 46 whereL is the total lagrangian density. After the construction of any quantum field theory that describes nature, one of the important tasks is to check for symmetries or violations thereof and their manifestation in experiments. One such symmetry that is inextricably linked to the matter-antimatter imbalance in our Universe is the discrete symmetry CP (Sakharov, 1967). It is a combination of the two discrete symmetry operators C (charge conjugation) under whose operation a positively charged particle becomes negatively charged, and the P (Parity) operator which swaps the handedness of the particle from left to right or vice versa. The viola- tion of CP symmetry was first realized in K0-meson decays that are described by the theory of weak interactions (Christenson et al., 1964). Therefore, weak in- teractions necessarily violate CP symmetry, however, no such violation has been found experimentally for strong interactions. The violation of CP symmetry is directly linked to the violation of the time reversal discrete symmetry T according to the CPT theorem (Branco et al., 1999), which states that nature is invariant under the joint action of the three discrete operators. Thus, if T is violated, then so is CP . One of the strong indicators of T violation in nature is the existence of an electric dipole moment (EDM) of baryons (Crewther et al. (1979); also see Dar (2000) for a review). The magnitude of the EDM, for which only upper limits exist, such as the upper limit for the neutron EDM is ‖dn‖ < 2.9 × 10−26e cm (Baker et al., 2006), can be used, as we discuss below, to constrain the severity with which CP is violated. 3.2 The Strong-CP Problem Quantum chromodynamics (QCD) has emerged as a phenomenologically accu- rate theory that describes strong interactions among the six quark flavors that are bound into two families of hadrons, namely mesons (quark and anti-quark pair) and baryons (quark triplet). From experiments, we understand that strong inter- actions enjoy C, P , T discrete symmetries of nature. Therefore, QCD must also obey such symmetries, both separately and any combinations formed thereof (Kim & Carosi, 2010). However, CP symmetry may be broken in QCD due to the pres- 47 ence of the following term in the QCD Lagrangian (Callan et al., 1976; Jackiw & Rebbi, 1976; ’t Hooft, 1976) Lint = ( θg2 32pi2 ) tr Gµνa G̃aµν (3.2) where 0 ≤ θ ≤ 2pi is a periodic parameter, g is the QCD coupling constant, Gµν is the color field strength tensor, and G̃µν is its dual. The value of the θ- parameter is not set theoretically, but it can be measured from the EDM of the neutron (dn), for which many theoretical estimates exist but we only quote one, ‖dn‖ ∼ 2.7× 10−16θ̄e cm (Baluni, 1979). Here θ̄ = θ + arg det mq, where mq is the quark mass matrix. The latest experimental estimate of ‖dn‖ < 2.9 × 10−26e cm (Baker et al., 2006) constrains ‖θ̄‖ . 10−11 (Kim & Carosi, 2010). This inexplicably small value of θ̄ gave rise to the strong CP problem. One of the solutions, also the most favored, to this problem was envisioned by Peccei & Quinn (Peccei & Quinn, 1977), whereby the θ̄ parameter is driven precisely to zero under a global chiral symmetry, later named U(1)PQ. The pseudo-Nambu- Goldstone boson that results upon the spontaneous breakdown of this symmetry was dubbed the axion (Weinberg, 1978; Wilczek, 1978). Not unlike the Higgs boson, the axion has proven to be extremely difficult to observe as it couples only weakly to ordinary matter and radiation. 3.2.1 Axion-Like Particles Due to non-perturbative effects, the QCD axion has a non-vanishing mass, hence, the identification with a pseudo-Nambu-Goldstone boson, which depends on the U(1)PQ symmetry breaking scale (also referred to as the axion decay constant) fa, ma ' 6 µeV ( 1012 GeV fa ) (3.3) The initial estimate of the PQ-symmetry breaking scale placed the axion decay constant close to the electroweak scale fa ' fEW = 250 GeV, where fEW is 48 the typical energy of electroweak processes and, in particular, corresponds to the vacuum expectation value of the Higgs field. Laboratory experiments conducted immediately after the axion was postulated ruled out the posited energy scale. Constraints from astrophysical considerations, as discussed below, ruled out even higher energy scales (fa & 108 GeV), reinforcing the idea that the axion must couple weakly to matter and radiation, and, therefore, must be invisible. This notion further led to the realization, which is also supported by several extensions of the SM, that axion-like particles (ALPs) should exist in nature (see for e.g. Massó (2008) for a review). ALPs are the generalized version of the QCD axion, that are light pseudo-scalar spin zero bosons with two-photon couplings at the one-loop level given by the triangle interaction (e.g. De Angelis et al. (2011) and references therein). They don’t obey the QCD axion mass-decay-constant relation as given in Eq. 3.3, but instead their mass ma and coupling strength to photons gaγγ are treated as independent parameters, where gaγγ = −gγ α pifa (3.4) with gγ = −0.97 and 0.36 in the Kim-Shifman-Vainshtein-Zakharov (KSVZ) and Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) QCD-axion models, respectively; α = 1/137 is the fine structure constant. It is to be noted that this study is mainly trying to constrain the properties of ALPs in general and not only the QCD axion. 3.3 Astrophysical and Laboratory Constraints on Properties of Axion and ALPs Despite several attempts to experimentally observe axions and ALPs, they remain elusive to this day. Nevertheless, the experimental efforts have not gone in vain, but have been able to place serious constraints on the coupling strength of ALPs to photons gaγγ < 10−10 GeV−1 (see Eq. 3.13). Stringent constraints have been placed on the mass of the axion 10−6 . ma . 10−2 eV with the lower limit 49 arising from cosmology (Preskill et al., 1983) and the upper limita from the neu- trino flux recorded for SN 1987A, which placed strong limits on the cooling flux through other channels namely, right-handed neutrinos or axions (Raffelt, 2004). If gaγγ > 10−10 GeV−1, the production of ALPs through the Primakoff process (Primakoff 1951; γ + γ∗ → a; here γ∗ is a virtual photon provided by the elec- tromagnetic field) will significantly alter the core He burning timescales of post main sequence stars, a possibility excluded by the ratio of horizontal branch stars in globular clusters (Raffelt, 1996). Several reviews on the properties of axions have been forthcoming in the past decade, for example see (Asztalos et al., 2006; Kim & Carosi, 2010; Raffelt, 1999), to which we point the reader for a more de- tailed and comprehensive exposition. In the following, we briefly describe the theory behind obtaining these constraints on ma and gaγγ from astrophysical and laboratory measurements. 3.3.1 Constraints from Astrophysical Objects Like neutrinos, ALPs are weakly interacting particles as established by the in- visible axion models (Sikivie, 1983). Thus, their production in the cores of non- degenerate stars, like the Sun, and uninhibited streaming leads to an energy deficit. The star responds to an increased rate of energy loss by shrinking its radius and escalating its fuel consumption to sustain hydrostatic equilibrium. As a result, the surface photon luminosity increases. In principle, one can estimate the ALP rate of emission for a solar like star by including the various emission processes by which ALPs can be produced in the core, viz. Compton process (γ + e→ a+ e), Bremsstrahlung emission (e + Ze → Ze + e + a), and Primakoff process, to re- produce the observed stellar radius and photon luminosity. However, the effect of energy loss due to such processes can be well compensated in the standard solar aDue to large uncertainties in the axion mass derived for the Dine-Fischler-Srednicki- Zhitnitskii (DFSZ) model (Dine et al., 1981; Zhitnitskii, 1980) from SN 1987A observations (0.004 . ma . 0.012 eV), a more relaxed upper limit is ma . 0.01 eV (Raffelt, 2004) 50 model by adjusting the pre-solar helium abundance Y b, for which no definitive es- timate exists. In the solar case, only the knowledge of the age of the Sun, obtained from radiochemical dating, provides constraints on axion properties because it is model independent. Thus, solar evolution models can accommodate an axion lu- minosity of La ≤ 0.04L for gaγγ . 2.4× 10−9 GeV−1, which corresponds to an axion mass of ma . 17 eV (Raffelt, 1986) for the hadronic axion (Kim-Shifman- Vainshtein-Zakharov (KSVZ) model (Kim, 1979; Shifman et al., 1980)) where the axions are only produced via the Primakoff process. A better constraint is ob- tained from solar helioseismology, with gaγγ . 1.0× 10−9 GeV−1, by comparing model solar sound speed profiles with observations (Schlattl et al., 1999). The most stringent astrophysical constraint on the photon-ALP coupling strength has come from the study of horizontal branch (HB) stars (see Raffelt (1990a) for a comprehensive discussion on astrophysical constraints). A HB star is character- ized by a helium burning core, where the nuclear burning proceeds via the triple α process fusing three 4He nuclei into 12C, and a hydrogen burning shell. The luminosity of these objects remains approximately constant until the helium is ex- hausted in their cores and they begin their ascension onto the AGB as depicted in the Hertzsprung-Russell diagram. The important constraint on the photon- ALP coupling strength is derived by measuring the helium burning lifetime (tHe) of low-mass stars in globular clusters. Globular clusters are roughly spherical agglomerations of typically ∼ 106 stars that are approximately coeval and ho- mogeneous in their chemical makeup (Binney & Merrifield, 1998). They are also among the oldest structures in the Galaxy with estimated ages ∼ 11 Gyr (Krauss & Chaboyer, 2003). Therefore, their brightness is dominated by low mass (. 0.8M) stars, while the massive coevals have already evolved past their post main-sequence (MS) stages. The helium burning lifetime of HB stars can be es- timated by the ratio R of the number of HB stars to that of the red giant branch (RGB) stars (Raffelt, 1990b), which also is equivalent to the ratio of HB to RGB lifetimes. As mentioned earlier, the primary effect of the production of any ALPs bThe formal definition of helium abundance is Y ≡ Total mass of heliumTotal mass of gas 51 is the drainage of energy followed by an increase in the nuclear burning rate. As a result, tHe is shortened by an amount allowed by the uncertainty in R, which is ∼ 10%. This effectively provides an upper limit on the energy loss rate due to ALP production, such that La . 0.1L3α where La and L3α are the luminosities due to ALP production and the triple α process. The constraint on La is very gen- eral and applies to any non-standard energy loss process. For pseudoscalars with ma . 0.7 eV, the upper limit on La yields a constraint on the coupling strength gaγγ . 0.6× 10−10 GeV−1 (3.5) altering the HB lifetime from 1.2 × 108 yr to 0.7 × 108 yr (Raffelt & Dearborn, 1987) (see Fig 3.5). A strong constraint on the mass of the axion came from the neutrino signal observed from the SN 1987A. On 23 February 1987, approximately three hours prior to optical brightening, neutrinos were detected from a source∼ 50 kpc away in the large magellanic cloud (LMC), a satellite galaxy of the Milky Way. A total of 11 events in the time span of 12.6 s were registered at the Kamiokande II neutrino observatory (Hirata et al., 1987) and, likewise, the Irvine-Michigan- Brookhaven detector registered 8 neutrino events over 5.6 s (Bionta et al., 1987). It is the duration of the observations that provides the main constraint on axion parameters. If axions were produced via nucleon-nucleon bremsstrahlung (N + N → N + N + a), their rate of emission could not have been more than that of the neutrinos. This yields an upper limit on the axion emission rate εa . 1019erg g−1s−1 for a nucleon core average density of ρ = 3 × 1014g cm−3 and an average temperature during the initial burst of T = 30 MeV (Raffelt, 2000). At this rate of axion production, the observed neutrino signal would be halved in duration, which yields a strong upper limit on the mass of the axion with ma . 0.01 eV (3.6) Axions can be produced both thermally and non-thermally in the early Uni- 52 verse, but it is only the non-thermal population that can be non-relativistic in the current epoch. The mass of the axion, derived from current algebra techniques, is inversely proportional to the axion decay constant fa as shown in Eq. 3.3 (see for e.g. Kuster et al. 2008 for an extended review of axion cosmology). Any relic thermal population of axions, which was not wiped out during the inflationary epoch, has a temperature Ta = 0.905K ( 106.75 ND )1/3 (3.7) where ND is the number of spin degrees of freedom at the time when the ax- ions decoupled from the thermalizing medium. This corresponds to an average momentum 〈pa〉 = 2.1 × 10−4 eV  ma, thus, making the thermal axions rel- ativistic. On the other hand, axions that were produced non-thermally are much colder and potential cold dark matter candidates. For example, the velocity dis- persion of a subpopulation of axions that were produced by vacuum realignment or string decay is estimated to be σ c ' 3× 10−17 ( 10−5 eV ma )5/6 R0 R (3.8) where R0/R is the ratio of the scale factor today to that at time t for an expanding Universe. For R0 = R, this velocity dispersion corresponds to a temperature of ' 0.5× 10−34 K (10−5/ma)2/3. Such axions are indeed non-relativistic and have an energy density that increases with the decay constant Ωa = ρa ρc ∼ ( fa 1012 GeV )7/6( 0.7 h )2 (3.9) where ρa is the axion energy density, ρc = 3H20/8piG is the critical energy den- sity of the Universe in the current epoch, h relates to the Hubble constant H0 = 100h km s−1Mpc−1. Requiring Ωa < ΩCDM = 0.22 yields a constraint on the 53 mass of the axion, ma > 10 −6 eV (3.10) 3.3.2 Laboratory Constraints A number of attempts have been made to detect ALPs in the laboratory. Their detection relies on the Primakoff process, whereby a photon polarized parallel to a strong magnetic field would oscillate into an ALP or vice versa. This technique was employed in a light shining through a wall or photon regeneration experiment (van Bibber et al., 1987) by the Rochester-Brookhaven-Fermilab (RBF) collabo- ration (Cameron et al., 1993). Their experiment consisted of passing a λ = 514 nm laser beam through a B = 3.7 T magnetic field in an optical cavity of length L = 4.4 m with 200 reflections to build up a high signal to noise ratio. The laser photons would oscillate into ALPs with a probability of Pγ→a ∝ (gaγγBL) 2 4 1− cos(qL) (qL)2 (3.11) where q ≡ kγ−ka is the difference in the photon and ALP momenta. These ALPs would then freely pass through an absorbing wall and emerge on the other side. The interaction of these ALPs with virtual photons from another magnetic field of the same strength would cause them to reconvert into real detectable photons. The total probability of conversion is then P 2γ→a. No photons were ever detected but the experiment yield the constraints gaγγ < 6.7× 10−7 GeV−1 for ma < 10−3 eV with 95% confidence. ALPs can also be detected by the magneto-optical effects of the vacuum. The conversion of a linearly polarized photon, oriented at some angle to the magnetic field, into an ALP acts as a dichroic filter and results in the rotation of the plane of polarization. Higher order QED processes both with and without any ALP involvement, for example light-by-light scattering, lead to the birefringence of the vacuum when subjected to a magnetic field. Early attempts at this were made by the RBF group but the experiment lacked enough sensitivity to detect extremely 54 small ellipticities caused by photon to ALP conversion (Cameron et al., 1991). The Sun is a potential ALP emitter. The Primakoff generation of ALPs dom- inates over the Compton and Bremsstrahlung processes. Thermal photons in the core of the Sun couple to the electromagnetic fields and produce ALPs with a thermal distribution having an average temperature of 4.2 keV and an integrated flux of ΦA = g210 3.67× 1011 cm−2s−1; Here g10 ≡ gaγγ1010 GeV (Zioutas et al., 2005). The free streaming ALPs from the Sun can be reconverted back into soft X- rays via the reverse Primakoff process in an external magnetic field with the same probability as given in Eq. 3.11. Notice that the maximum conversion probability is obtained when the ALP and photon fields are coherent, such that qL → 0. In vacuum kγ = ω, and this condition is always met for a massless boson ka = kγ . In the case where ma 6= 0, the photon and ALP fields are never degenerate in vacuum and this problem can be circumvented by introducing a buffer gas which gives the photon an effective mass, mγ = ωp where ωp is the plasma frequency of the buffer gas. An experiment exploiting this idea based on the axion helioscope concept of Sikivie (1983) was launched by the Brookhaven National Laboratory in 1992 (Lazarus et al., 1992). Another helioscope was developed by the Uni- versity of Tokyo employing a 4 T superconducting magnet in 1997 (Moriyama et al., 1998). A collaboration between CERN and the University of Tokyo led to a third-generation axion helioscope (Cern Axion Solar Telescope (CAST)) utilizing an LHC test magnet with B = 9.0 T. The CAST phase-I results surpassed the previous bounds on the coupling strength obtained from HB stars with gaγγ < 8.8× 10−11 GeV−1 for ma . 0.02 eV (3.12) with 95% confidence (Andriamonje et al., 2007). Alternatively, cosmic axions that fell into the gravitational well of the Milky Way can be detected with the use of an axion haloscope (Sikivie, 1983). It em- ploys the same idea as mentioned above where cosmic axions are resonantly con- verted into microwave photons in high-Qc microwave cavities. The cavity is also cQuality factor Q = E/δE where E is the average axion energy and δE is the spread in 55 equipped with metallic or dielectric rods that are used to tune its resonant fre- quency to search for axions in some given mass range. Two such experiments are currently underway - the Axion Dark Matter Experiment (ADMX) at Lawrence Livermore National Laboratory (Asztalos et al., 2004, 2010) and the Cosmic Ax- ion Research with Rydberg Atoms in Cavities at Kyoto (CARRACK) (Tada et al., 2006). The ADMX experiment uses a 8.5 T superconducting magnetic and has scanned for axions in the mass ranges 1.86 − 3.36 µeV excluding KSVZ axions at the 90% confidence level (see Fig. 3.5). Still, there is no denying the fact that none of the laboratory experiments con- ducted thus far have been able to secure a positive detection of this mysterious particle. The detection of very weakly coupled particles demands extremely sen- sitive laboratory experiments. So far, as discussed above, only a handful of exper- iments, have been able to surpass the astrophysically derived limits on gaγγ in the axion and ALP mass range of interest. Yet, the sensitivity envelope needs to be pushed even further by a few orders of magnitude to be able to draw any definitive conclusions about the existence of the axions or ALPs. Plans are afoot to modify the existing experiments and devise new ones to improve upon current limits on gaγγ (see Section 3.6.1). Unlike the laboratory experiments, the odds are in favor for detecting ALPs in astrophysical systems. This optimism stems from the fact that the ALP to photon conversion probability scales with large magnetic field strengths and longer co- herence lengths (Chelouche et al., 2009), such that Pa→γ ∝ g2B2L2, where L is the length over which both the photon and ALP fields are in phase. Thus, there is a very good chance of finding ALPs in strongly magnetized compact objects, namely magnetic white dwarfs (mWDs) and neutron stars (NSs). The possibility in the latter case has been expounded by many (see for example (Chelouche et al., 2009; Lai & Heyl, 2006; Pshirkov & Popov, 2009; Raffelt & Stodolsky, 1988); also see (Jimenez et al., 2011) where constraints on gaγγ are derived from the dimming of radiation by photon-ALP conversion in astrophysical sources), how- energy due to virial distribution 56 ever, the case of the mWDs has not been investigated in great detail and warrants further study. 3.3.3 Magnetic White Dwarfs After the discovery of the first mWD by Kemp (Kemp et al., 1970), the number of white dwarfs with magnetic fields ranging from a few kG to 103 MG has grown to about 170. The size of this subpopulation is only 3% of the total population of known WDs comprising of 5447 objectsd. The main channel for identifying mag- netism in WDs is through Zeeman spectropolarimetry, which not only allows one to discern the strength of the field but also the direction of the field lines, and also through cyclotron spectroscopy (see for e.g. (Wickramasinghe & Ferrario, 2000) for a review on isolated and binary mWDs). Nevertheless, reconstruction of the field topology has proven to be very difficult, mainly due to its highly non-dipolar structure. Over the last decade Zeeman tomography of mWDs has enjoyed some success in elucidating the underlying field structure. This technique is based on calculating a database of model spectra, where different field geometries compris- ing of single/multiple dipole, and higher multipoles, that may also be off-centered and misaligned with the rotational axis, are considered. Then a least-squares fit using the pre-calculated synthetic spectra is performed through a highly optimized algorithm on the phase-resolved Zeeman spectra to obtain the complex field struc- tures (Euchner et al., 2002). The generality of the models not only allows greater flexibility but also renders a closer fit to the actual field geometry of the source for a given rotational phase. The presence of even a small degree of circular polarization in the spectrum of a WD is a strong indicator of the object possessing a magnetic field upwards of 106 G (Kemp, 1970). The degree of circular polarization typically reaches up to∼ 5%, and sometimes beyond that in a few selective objects, near absorption features and also in the continuum. Continuum circular polarization stems from the magnetic dG.P. McCook and E.M. Sion, web version of the Villanova White Dwarf Catalog, 57 circular dichroism of the atmosphere, where the left and right circularly polarized waves propagating through a magnetized medium encounter unequal opacities (Angel, 1977). A relatively higher degree of circular polarization also appears near the red and blue shifted wings of the Zeeman split absorption lines (σ+ and σ− components, (Wickramasinghe & Ferrario, 2000)). On the other hand, most observations of mWDs indicate that the linear po- larization component never exceeds that of the circular one, and the spectrum remains dominantly circularly polarized until field strengths ≥ 108 G are reached (Angel, 1977). In a magneto-active plasma, the plane of linear polarization un- dergoes many Faraday rotations, an effect that arises due to the magnetic birefrin- gence of the medium, so that on average the degree of linear polarization of the emergent radiation is much reduced (Sazonov & Chernomordik, 1975). The very fact that linear polarization is of the order of a few percent (∼ 5%) in the continuum spectra of most mWDs can be exploited to draw meaningful con- clusions on the extent of ALP interaction with photons traversing the magnetized plasma of mWDs. We explain how this can be implemented in the next section. 3.3.4 Plan of this Study The purpose of this study is to conduct a survey of the ma− gaγγ parameter space by modelling photon-ALP oscillations in the magnetosphere of a mWD. To this end, we model the field structure of a strongly magnetized WD PG 1015+014, for which high resolution optical spectropolarimetric observations are available (Eu- chner et al., 2006). In the same article, the authors also conduct a phase-resolved Zeeman tomographic analysis and derive a best-fit model of the magnetic field topology. Despite fitting the spectrum with a range of field geometries, they were only able to pin down the field geometry for a single rotational phase by fitting it with a superposition of three off-centered and non-aligned dipoles of unequal surface field strengths (see Table 3.1 for model parameters). To model the effect of photon-ALP oscillation in the magnetosphere on the emergent polarization, we propagate an unpolarized photon of a given energy from the photosphere through 58 Table 3.1: Top: Magnetic field geometry adopted from the spectropolarimet- ric analysis by (Euchner et al., 2006) of the the mWD PG 1015+014. The model comprises of three off-centered and non-aligned dipoles D1, D2, D3 with unequal surface field strengths Bs, polar (θB) and az- imuthal (φB) angles of the magnetic field axes. The center positions of the dipoles relative to the center of the star are given by (ax, ay, az). Bottom: Collection of parameters used in the study. Model Parameters D1 D2 D3 Bs (MG) -40 92 -38 θB( ◦) 44 63 63 φB( ◦) 339 276 134 ax(R?) 0.04 -0.012 0.27 ay(R?) 0.35 -0.136 0.080 az(R?) 0.33 -0.28 0.21 R? 7× 108 cm θk 23 ◦ Ye 1 BQ 4.413× 1013 G T 104 K g? 10 8 cm s−2 ρ0 10 −10 g cm−3 ρ∞ 10−20 g cm−3 the encompassing magnetosphere, that has been populated by a diffuse, cold ion- ized H gas. The emergent intensity and polarization is then averaged over the whole surface of the star. Finally, we compare the degree of polarization from our model simulation to what is observed in mWDs with field strengths in excess of a few 106 G, for example PG 1015+014, and draw conclusions on the strength of the coupling constant for a given ALP mass. 59 3.4 Model Equations The interaction of the ALP field with an external electromagnetic field is given by the following Lagrangian density (Raffelt & Stodolsky, 1988), in natural units where ~ = c = 1, L = −1 4 FµνF µν + 1 2 (∂µa∂ µa−m2aa2)− 14gaγγFµνF̃ µνa + α 2 90m4e [ (FµνF µν)2 + 7 4 ( FµνF̃ µν )2] (3.13) where the first term describes the external electromagnetic field, with Fµν as the antisymmetric electromagnetic field strength tensor and F̃ µν = 1 2 εµνρσFρσ as its dual. The second term is simply the Klein-Gordon equation for the ALP field a wherema represents its mass. The next term is the interaction Lagrangian density, which upon simplification, using the given definitions, yields Lint = gaγγaE · B, where gaγγ is the photon-ALP coupling strength, E is the polarization vector of the photon field, and B is the external magnetic field. Quantum corrections to the classical electromagnetic field, due to the constant creation and annihilation of electron-positron pairs in vacuum, are given in the last term of (3.13) by the Euler-Heisenberg effective Lagrangian, causing the vacuum to be birefringent. 3.4.1 Off-Centered Non-Aligned Three Dipole Model We start by writing down the aggregate magnetic fieldB0 in the coordinate system Σ0 where the ẑ-axis coincides with the rotational axis of the star B0 = 3∑ i=1 RTi B ′ i (3.14) The three off-centered dipole fields B′i are first rotated before they are added to- gether by operating on each of them with a rotation matrix RT = RTz,iR T y,i, where the superscript T indicates the transpose, and the rotation matrices are given as 60 x̂ŷ ẑ ai φB,i θB,i x̂′ ŷ′ ẑ′ r′i r Σ0 Σi Figure 3.1: This illustrates the coordinate system used to obtain the aggre- gate magnetic field. Here Σ0 represents the coordinate system cen- tered on the star with the ẑ-axis aligned with the rotational axis. Σi represents the coordinate system in which the different dipole mag- netic field equations are written. This system is misaligned with Σ0 by a polar angle θB,i and an azimuthal angle φB,i, and then it is displaced from the center by the vector ai. For clarity we have chosen the vector ai to lie along the ẑ′-axis. 61 follows Rz,i =  cosφB,i sinφB,i 0− sinφB,i cosφB,i 0 0 0 1  (3.15) Ry,i =  cos θB,i 0 − sin θB,i0 1 0 sin θB,i 0 cos θB,i  (3.16) Here the polar and azimuthal angles θB,i and φB,i, respectively, are defined with respect to the axis of rotation. In cartesian coordinates, the dipole fields are ex- pressed as B′i = Bs,iR 3 ? 2r′5i (3x′iz ′ ix̂ ′ + 3y′iz ′ iŷ ′ + {3z′i2 − r′i2}ẑ′) (3.17) where the fields are shifted from the coordinate center, such that r′i = r−ai. In the above equation, Bs,i is the surface field strength of the ith dipole field component, R? ' 7 × 108 cm is the radius of the WD, and r′i is the magnitude of the radial vector in the coordinate system Σi. The profile of the aggregate field B0 as a function of distance is shown in Fig. 3.2. 3.4.2 Fully Ionized Pure H Atmosphere The presence of a magnetic field necessarily introduces anisotropy in the plasma dielectric tensor ε p . In the case of a nonuniform field, none of the dielectric tensor components vanish, as compared to the homogeneous case. Below we write all the dielectric components, which one can easily derive from Maxwell’s equations, for completeness. ε p =  ε11 ε12 ε13ε21 ε22 ε23 ε31 ε32 ε33  (3.18) 62 ε11 = 1− ∑ s=e,p ω̂2p,s [ 1− ω̂2c,sB̂20x 1− ω̂2c,s ] ≈ 1− ω̂2p,e [ 1− ω̂2c,e(1 + ω̂2c,p)B̂20x (1− ω̂2c,e)(1− ω̂2c,p) ] (3.19) ε12 ≈ ω̂c,eω̂ 2 p,e (1− ω̂2c,e) (iB̂0z + ω̂c,eB̂0xB̂0y) (3.20) ε13 ≈ − ω̂c,eω̂ 2 p,e (1− ω̂2c,e) (iB̂0y − ω̂c,eB̂0xB̂0z) (3.21) ε21 ≈ − ω̂c,eω̂ 2 p,e (1− ω̂2c,e) (iB̂0z − ω̂c,eB̂0xB̂0y) (3.22) ε22 ≈ 1− ω̂2p,e [ 1− ω̂2c,e(1 + ω̂2c,p)B̂20y (1− ω̂2c,e)(1− ω̂2c,p) ] (3.23) ε23 ≈ ω̂c,eω̂ 2 p,e (1− ω̂2c,e) (iB̂0x + ω̂c,eB̂0yB̂0z) (3.24) ε31 ≈ ω̂c,eω̂ 2 p,e (1− ω̂2c,e) (iB̂0y + ω̂c,eB̂0xB̂0z) (3.25) ε32 ≈ − ω̂c,eω̂ 2 p,e (1− ω̂2c,e) (iB̂0x − ω̂c,eB̂0yB̂0z) (3.26) ε33 ≈ 1− ω̂2p,e [ 1− ω̂2c,e(1 + ω̂2c,p)B̂20z (1− ω̂2c,e)(1− ω̂2c,p) ] (3.27) In the above set of equations, ω̂c,s = qsB0/ωmsc is the normalized cyclotron frequency for species s = (e, p), where e and p signify electrons and protons; ω̂p,s = √ 4pins/msω2 is the normalized plasma frequency, where np = ne = Yeρ/mp are the electron and proton number densities, Ye is the electron fraction, and ρ is the proton mass density of the plasma; the normalized magnetic field components are defined as B̂0,i=x,y,z = B0,i/B0. 63 Vacuum Corrections Due to the polarizability of the vacuum in strong magnetic fields, the plasma di- electric tensor ε p , and the inverse permeability tensor µ−1 are modified (Lai & Ho, 2003b; Meszaros & Ventura, 1979), such that ε p+v = ε̃ = ε p + ∆ε v and µ−1 p+v = µ̃−1 = I + ∆µ−1 v , where ∆ε v = (av − 1)I + qvB̂0B̂0 (3.28) ∆µ−1 v = (av − 1)I +mvB̂0B̂0 (3.29) av = 1− 2δv qv = 7δv mv = −4δv δv = α45pi ( B0 BQ )2 and BQ = 4.413× 1013 G is the quantum critical field for which the separation in energy between Landau levels of the electron exceeds its rest mass. Plasma Density Profile That many mWDs are surrounded by hot coronae has been suggested by many to explain the polarized flux of those WDs that show comparable degree of linear and circular polarization (Ingham et al., 1976; Sazonov & Chernomordik, 1975; Zheleznyakov & Serber, 1994). The thermal electrons in the hot tenuous plasma with temperature T ∼ 106−8 K radiate at the cyclotron frequency that falls in the optical wavelength for field strengths of B ∼ 108 G. This radiation appears to be polarized both linearly and circularly, depending on the orientation of the line of sight to the magnetic field, and traverses the corona without any absorption. Furthermore, slightly polarized radiation emanating from the photosphere, with very low degree of linear polarization due to Faraday rotation, gets added to that generated in the corona, as a result increasing the amount of flux that is polar- ized linearly. Several hot isolated WDs, with effective temperatures in excess of ' 25, 000 K, emitting X-rays were detected by ROSAT (Fleming et al., 1996), however all cases were linked to subphotospheric thermal emission (Musielak et al., 2003). Although the non-detection of any coronal emission may indicate 64 the absence of a hot tenuous corona, it is not at all unreasonable to suggest the presence of a tenuous cold plasma of fully ionized H. In this study, we envisage that the mWDs are encompassed by cold isothermal electron-proton coronae with the following barometric density profile, ρ(r) = ρ0 exp ( −r −R? Hρ ) + ρ∞ (3.30) where ρ0 is the density near the surface of the star, ρ∞ is the density that remains far away from the star as the strength of the magnetic field becomes significantly weaker than that at the surface, and Hρ = 2kBT/mpg? ' 1.65 × 104 cm is the density scale height with an effective temperature T ' 104 K and surface gravity log g?(cm/s2) = 8. There is no clear agreement on the surface plasma density with 10−11 . ρ0 . 10−6 g cm−3. Here, we assume that the plasma is sufficiently tenuous with ρ0 = 10−10 g cm−3 and ρ∞ = 10−20 g cm−3. 3.4.3 ALP-Photon Mode Evolution in an Inhomogeneous Magnetized Plasma We are interested in knowing the evolution of the ALP field and the polarization vector as the radiation propagates out from the surface of the star, traversing the region with an inhomogeneous plasma density and magnetic field. Here we follow the discussion given in (Lai & Heyl, 2006; Raffelt & Stodolsky, 1988), and derive the photon field mode evolution from the EM wave equation ∇× (µ̃−1 · ∇ × E) = ω 2 c2 ε̃ · E (3.31) Next, we assume the ansatz E = Ẽ exp(ikz) where the wave is propagating along the rotational axis of the star, which in this case is also the line of sight direction, and the wavenumber k = ω/c. Plugging this ansatz into the wave equation, and 65 ignoring second order derivatives, we find d dz ( Ẽx Ẽy ) = ( χ11 χ12 χ21 χ22 )( Ẽx Ẽy ) (3.32) where the matrix elements are given below χ11 = Υ −1 3 [ k2ε̃11 −Υ4 − ( 1− Υ 2 1 Υ3Υ5 )−1 Υ1Υ −1 5 × ( k2ε̃21 −Υ2 − Υ1 Υ3 {k2ε̃11 −Υ4} )] (3.33) χ12 = Υ −1 3 [ k2ε̃12 −Υ2 − ( 1− Υ 2 1 Υ3Υ5 )−1 Υ1Υ −1 5 × ( k2ε̃22 −Υ6 − Υ1 Υ3 {k2ε̃12 −Υ2} )] (3.34) χ21 = ( 1− Υ 2 1 Υ3Υ5 )−1 Υ−15 × ( k2ε̃21 −Υ2 − Υ1 Υ3 {k2ε̃11 −Υ4} ) (3.35) χ22 = ( 1− Υ 2 1 Υ3Υ5 )−1 Υ−15 × ( k2ε̃22 −Υ6 − Υ1 Υ3 {k2ε̃12 −Υ2} ) (3.36) Υ1 = d dz (mvB̂xB̂y) + i2kmvB̂xB̂y (3.37) Υ2 = ik d dz (mvB̂xB̂y)− k2mvB̂xB̂y (3.38) Υ3 = − d dz (av +mvB̂ 2 y) + i2k(av +mvB̂ 2 y) (3.39) Υ4 = −ik d dz (av +mvB̂ 2 y) + k 2(av +mvB̂ 2 y) (3.40) Υ5 = − d dz (av +mvB̂ 2 x) + i2k(av +mvB̂ 2 x) (3.41) 66 Υ6 = −ik d dz (av +mvB̂ 2 x) + k 2(av +mvB̂ 2 x) (3.42) Line of Sight Geometry The Zeeman tomography analysis of mWD PG 1015+014 indicates that the line of sight (LOS) is inclined at an angle θk = 23◦ to the rotational axis of the star. Following (Dupays & Roncadelli, 2006), we modify the matrix Eq. 3.32 to obtain the mode evolution of the photon-ALP system in a coordinate system oriented along the LOS (see Fig. 3.2). Again, we assume the ansatz a ∝ exp (ik′s− iωt) i d ds  aEx′ Ey′  =  ∆a − k ′ ∆Mx′ ∆My′ ∆Mx′ iχ ′ 11 − k′ iχ′12 ∆My′ iχ ′ 21 iχ ′ 22 − k′   aEx′ Ey′  (3.43) where ∆a = m2a/2ω, ∆Mx′ = −gaγγBx/2, ∆My′ = −gaγγBy/2. Notice that Eq. 3.32 applies to a system for which the LOS vector coincides with the rotational axis of the star. For a different LOS vector, such as shown in Fig. 3.2, we perform a rotation of the plasma dielectric tensor around the ŷ-axis by an angle θk, ε̃′ = RTy (θk)ε̃Ry(θk) where Ry is given in Eq. 3.16. The total degree of polarization can be found by integrating Eq. 3.43 from a given point on the surface outwards to a distance beyond which the amplitude of photon-ALP oscillations and plasma effects become negligible, and then by averaging the Stokes parameters (Rybicki & Lightman, 2004) over the whole ob- servable hemisphere. I = ‖Ex′‖2 + ‖Ey′‖2 (3.44) Q = ‖Ex′‖2 − ‖Ey′‖2 (3.45) U = Ex′E ∗ y′ + Ey′E ∗ x′ (3.46) V = −i(Ex′E∗y′ − Ey′E∗x′) (3.47) 67 s/R? B 0 (G ) 10−3 10−2 10−1 1 10 102 10 102 103 104 105 106 107 108 Figure 3.2: This figure illustrates the coordinate system used to obtain the photon-ALP mode evolution along a given LOS (top), and the decline of the magnetic field strength with distance s from the surface (bot- tom). Here the LOS vector is represented by s that is tilted at an angle θk to the rotation axis, and is in parallel to z′ which is not to be con- fused with ẑ′ in Fig. 3.1. Several different points on the star’s surface with spherical coordinates (R?, θR, φR) are chosen and then averaged to determine the final polarization of the photon leaving the magneto- sphere. 68 The mode amplitudes are in general complex, and in the above set of equa- tions ∗ gives the complex conjugate. We sample the Stokes vector from differ- ent points on the surface, with coordinates (R?, θR, φR), that are spread around the LOS vector k̂ with ∆φR = 30◦ and ∆θR = 10◦, where 10◦ ≤ θR ≤ 80◦ and 0◦ ≤ φR ≤ 330◦. Because the sampling in the azimuthal angle is sparse for larger polar angles, we take a weighted average, as shown below for one of the Stokes parameters, to determine the average degree of polarization of the whole hemisphere 〈I〉 = ∑ θR,φR I(θR, φR) sin θR∑ θR sin θR (3.48) 3.5 Results In the following, we look at how an unpolarized photon emitted from the pho- tosphere of a mWD gets polarized as it traverses through the magnetosphere. Photon-ALP interaction and the intervening plasma make the medium birefrin- gent, consequently, altering the state of polarization of the unpolarized photon. We obtain the degree of polarization from the averaged Stokes parameters PL = √〈Q〉2 + 〈U〉2 〈I〉 (3.49) PC = 〈V 〉 〈I〉 (3.50) where PL and PC represent linear and circular polarization. In Fig. 3.3, we present the evolution of the Stokes vector with distance s from the surface of the star for the case of radiation with Eγ = 3 eV, and ALP pa- rameters ma = 10−5 eV, gaγγ = 10−9 GeV−1. The oscillations in the solution arise due to the mixing of the ALP and photon eigenstates, an effect analogous to neutrino oscillations due to the MSW effect (Mikheev & Smirnov, 1985; Wolfen- stein, 1978). However, notice that the interaction is non-resonant because a 50% drop in intensity would be observed if the ALP and photon modes were to achieve 69 I0.95 1.00 Q −0.05 0.00 0.05 U −0.05 0.00 0.05 s/R? V 10 −3 10 −2 10 −1 1 10 10 2 −0.02 0.00 Figure 3.3: Polarization evolution of an unpolarized photon along a given LOS as it starts at the photosphere and propagates through the magne- tosphere. In this case,Eγ = 3 eV,ma = 10−5 eV, gaγγ = 10−9 GeV−1. The magnetic field geometry assumed is that of mWD PG 1015+014. 70 maximal mixing and undergo level crossing. Eventually, as the photons travel far- ther away from the surface, the decline in the magnetic field strength reduces the probability of conversion, hence the diminishing of intensity variation. We find that the change in polarization is primarily brought about by the ALP interaction with the photon. In the event this interaction is made negligible, no significant po- larization or change in intensity of the emergent radiation is found. The origin of circular polarization in mWDs, as alluded to earlier, is understood in terms of the difference in opacities for the two modes of radiation, making the plasma dichroic, in the presence of a magnetic field. Linear polarization, on the other hand, was explained by the cyclotron radiation that emanates from the tenuous corona com- posed of an ionized plasma. In this study, since the treatment of radiative transfer effects is very simplistic only an upper limit can be placed on how strongly the ALP couples to photons, as shown in the next section. 3.5.1 Constraints on gaγγ ALP production in the mWD magnetosphere can enhance the degree of linear polarization of the observed optical radiation. The goal here is to not determine the precise value of the photon-ALP coupling strength but only constrain it from above. To this end, we look at the amount of linear polarization that is produced for a givenma and gaγγ . The underlying assumption here is that all of the observed linear polarization is generated due to photon-ALP interaction, and not by the plasma, which effectively yields the absolute upper limit on gaγγ . In Fig. 3.4, we plot the emergent intensity and state of polarization for different ALP masses and for photons in the UV - optical waveband with energies between 2 − 5 eV. The ma and gaγγ in Fig. 3.4 were chosen specifically so that PL & 0.05 for all photon energies. In Fig. 3.5, we use the same parameters to draw an exclusion region in the ma − gaγγ parameter space, along with regions excluded by lab experiments and astrophysical considerations. The shaded region in red excludes all ma - gaγγ val- ues for the case of mWD PG 1015+014, that is for a typical surface field strength 71 ma = 10 −4 eV gaγγ = 2× 10−7 GeV−1 0.0 0.5 1.0 ma = 10 −5 eV gaγγ = 5× 10−9 GeV−1 0.0 0.5 1.0 Eγ (eV) ma = 10 −6 eV gaγγ = 2× 10−10 GeV−1 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.0 0.5 1.0 〈I〉 PL PC Figure 3.4: The final state of polarization of radiation after traversal from the WD’s magnetosphere for different ma and gaγγ . Here 〈I〉 is the average Stokes intensity, and PL and PC are the degrees of linear and circular polarizations. These results apply to the case of mWD PG 1015+014. 72 B ' 108 G and degree of linear polarization PL ∼ 5%. We find that for the range of masses that are of relevance, in particular, to the ALP models, the constraints on gaγγ from this study are superseded by that from horizontal branch (HB) stars. Still, the limiting linear polarization criterion used in this study is able to probe smaller gaγγ values in comparison to works that only look at radiation dimming (for e.g. see (Jimenez et al., 2011)). The constraints can be further improved by looking at mWDs with higher magnetic field strengths. The highest field strength that has ever been discovered in a mWD is B ' 1000 MG in two such objects namely, PG 1031+234 and SDSS J234605+385337 (Jordan, 2009). Both objects show linear polarization as low as ∼ 1% for some rotational phases (Piirola & Reiz, 1992; Vanlandingham et al., 2005). Based on these two facts and assuming that the magnetic field geometry of these two mWDs is at least as complex as that found in PG 1015+014, we produce two exclusion regions shown in Fig. 3.5 with colors blue and green. The former corresponds to a surface field strengthB = 1000 MG with the same level of linear polarization as before, and the latter studies the case with PL ' 1%. For these two cases, we have only looked at ma ≤ 10−5 eV since higher mass values don’t constrain gaγγ better than limits derived from HB stars and CAST (Phase-I). On the other hand, we have extended our treatment to smaller particle masses. It is worth mentioning that the change in gaγγ is not linear with the change in magnetic field strength, as evident from the comparison between the red and blue regions in Fig. 3.5. For higher field strengths one observes higher degree of polarization of the emerging radiation. Naively, one would expect the plane of polarization to rotate by an amount that is O(g2aγγB2l2), which is valid strictly in the absence of plasma when the photon and ALP are treated as massless particles (Raffelt & Stodolsky, 1988; van Bibber et al., 1987). Therefore, for a fixed de- gree of polarization an increase in B should also decrease gaγγ by the same factor, when l, the length over which the magnetic field remains homogeneous, is kept constant. However, as shown by (Heyl et al., 2003) in the case of NSs, an increase in magnetic field strength also increases the level of polarization by effectively 73 shifting the polarization-limiting radius, the distance beyond which the two polar- ization modes couple and which depends weakly on the magnetic field strength Rpl ∝ B2/5, farther away from the star. The farther the polarization-limiting ra- dius, the more coherently the polarization states from different LOSs add, yielding a higher degree of polarization. 3.6 Discussion This study looks at how the production of ALPs in mWD magnetospheres can alter the state of polarization of the observed radiation. We find that unpolar- ized photons of photospheric origin become linearly polarized upon their traver- sal through the inhomogeneous magnetic field of a mWD. We have modeled the magnetospheric plasma, as fully ionized pure H with a barometric profile. Since the majority of mWDs are strongly circularly polarized and only show a relatively small degree of linear polarization, at most 5%, we have used this observation to constrain the coupling strength gaγγ of ALPs to photons. We find that for the case where the plasma component only contributes negligibly to the state of polariza- tion, the coupling strength gaγγ increases with the mass of the ALP ma. The level of linear and circular polarization observed in mWDs is sensitive to the properties of the magnetospheric plasma. The limits on gaγγ can be improved by modelling all the radiative transfer effects in WD atmospheres and fitting the model spectra to real observations. Magnetic fields stronger than that of mWDs exist in NSs. Going back to the argument of how astrophysical objects, compared to laboratory experiments, ben- efit from longer coherence lengths (see Section I), in comparing mWDs with NSs, one finds that the latter are ∼ 104 times more efficient in converting photons to ALPs and vice-versa. A number of studies have expounded on the subject of propagation of polarized radiation through the NS magnetosphere, where they have considered IR/Optical radiation (Shannon & Heyl, 2006), and thermal X- rays (Lai & Ho, 2003a,b) produced at the surface of the NS. Unfortunately, no X-ray polarimetry observations have been conducted partly due to the very low 74 ma (eV) g a γ γ (G eV − 1 ) Horizontal Branch Stars A D M X R B F KS VZ DF SZ C os m ol o gy S N 19 87 A CAST (Phase-I) mWD 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 10−15 10−14 10−13 10−12 10−11 10−10 10−9 Figure 3.5: Exclusion plot in the ma − gaγγ parameter space. The red re- gion corresponds to the case of mWD PG 1015+014 with B ' 108 G and limiting linear polarization PL ∼ 5%. The blue and green re- gions correspond to the case with B ' 109 G, but with PL ∼ 5% and PL ∼ 1% respectively. The mass of the axion is constrained to 10−6 . ma . 10−2 eV from cosmology and SN 1987A measure- ments. The photon-ALP coupling constant is capped from above with gaγγ < 10 −10 GeV−1 by the number of horizontal branch stars in glob- ular clusters. KSVZ and DFSZ are two different theoretical models that predict how gaγγ scales with ma. Other exclusion regions are from lab experiments by the CAST experiment (Andriamonje et al., 2007), ADMX group (Asztalos et al., 2004, 2010) and by Rochester- Brookhaven-Fermilab collaboration (De Panfilis et al., 1987; Wuensch et al., 1989). 75 flux in X-rays from these objects, and also because none of the high energy tele- scopes are equipped with a polarimeter. X-ray polarimetry has been neglected for the last 30 years but it is hoped that some of the future space missions will fill this void in X-ray astronomy (Muleri et al., 2010). In any case, as discussed by (Chelouche et al., 2009; Lai & Heyl, 2006; Raffelt & Stodolsky, 1988), NSs are excellent laboratories for the detection of any light, weakly coupled pseudoscalar particle. 3.6.1 Outlook The ADMXe project, that employs a microwave cavity to search for cold dark matter axions, will begin its phase II of testing in the year 2012. With the new up- grades the ADMX project will be able to exclude gaγγ up to the DFSZ line in the same mass range as before. Although outside of the range of axion masses probed in this study, the modified CAST experiment has been able to exclude axions with gaγγ & 2.2×10−10 GeV−1 forma . 0.4 eV, becoming the first experiment to ever cross the KSVZ line (Arik et al., 2009; Aune et al., 2011). The currently running CAST experiment in its phase-II will be able to exclude axions with ma . 1.15 eV with unprecedented sensitivity in this mass range. An improved version of the light shining through wall (LSW) experiment ((van Bibber et al., 1987), also see (Redondo & Ringwald, 2011) for a recent review on such experiments) using Fabry-Perot optical cavities to resonantly enhance photon-axion conversion has been proposed (Hoogeveen & Ziegenhagen, 1991; Mueller et al., 2009; Sikivie et al., 2007). The projected limit in sensitivity to gaγγ & 2.0 × 10−11 GeV−1 typically for axion masses ma . 10−4 eV achieved using 12 Tevatron supercon- ducting dipoles appears quite promising. Further improvements in experiment design and optimization techniques yielding increased sensitivity to even smaller coupling strengths have also been suggested by many workers in the field, for ex- ample the use of the dipole magnets, each providing a field strength of 5 T, from the Hadron Electron Ring Accelerator (HERA) at DESY in Hamburg in a 20+20 e 76 configuration can potentially exclude gaγγ & 10−11 GeV−1 for ma < 10−4 eV (Arias et al., 2010; Ringwald, 2003). Another proposed line of investigation to search for ALPs is the use of resonant microwave cavities which are much similar in design to the optical LSW experiments discussed above (Caspers et al., 2009; Hoogeveen, 1992; Jaeckel & Ringwald, 2008). This method has already been em- ployed to search for hidden sector photons (Povey et al., 2010) and can prove to be a powerful tool in the case of ALPs. Finally, the simplistic model assumed for the mWD atmosphere only yields an absolute upper bound on gaγγ . A much tighter constraint can be obtained by adopting a more realistic atmospheric model and solving the equations of radiative transfer with the photon-ALP oscillations included. Such an analysis is outside the scope of this study, but it is hoped that the novel method discussed in this work will prove to be extremely useful in better constraining the properties of any ALP. 77 Chapter 4 What Triggers the Soft Gamma-ray Repeater Giant Flares? 4.1 Introduction The soft gamma-ray repeaters (SGRs) showcase flux variability on many dif- ferent time scales. The quiescent state, with persistent X-ray emission (LX ∼ 1035 erg s−1), punctuated by numerous sporadic short bursts of gamma-rays, with peak luminosities up to∼ 1042 erg s−1 and typical duration in the range∼ 0.01−1 s, mark the defining characteristics of SGRs (see Mereghetti 2008 and Woods & Thompson 2006 for a review). Out of the seven confirmed SGR sources, SGR 0525-66, SGR 1806-20, SGR 1900+14, SGR 1627-41, SGR 1150-5418, SGR 0418+5729, SGR 0501+4516 (with the last three added recently to the SGR fam- ily; see Kaneko et al. 2010; Kumar et al. 2010; van der Horst et al. 2010), the first three have been known to emit giant flares. A rare phenomenon compared to the commonly occurring short bursts, the giant flares unleash a stupendous amount of energy (∼ 1044 erg) in gamma-rays over a time scale of ∼ 0.2− 0.5 s in a fast ris- ing initial peak. The initial high energy burst is followed by a long (∼ 200− 400 s), exponentially decaying pulsating tail of hard X-ray emission, the period of which coincides with that of the rotation of the neutron star (NS). Additionally, 78 intermediate strength but rare outbursts lasting for few tens of seconds have been observed in the case of SGR 1900+14. 4.1.1 The Giant Flares The first extremely energetic giant flare from a recurrent gamma-ray source, SGR 0525-66, was detected on March 5, 1979 by the gamma-ray burst detector aboard the Venera 11 & 12 space probes and the nine interplanetary spacecraft of the burst sensor network (Helfand & Long, 1979; Mazets et al., 1979). The position of the source was found to be coincident with the supernova remnant N49 at a distance of ∼ 55 kpc in the Large Magellanic Cloud. The flare consisted of a sharp rise (∼ 15 ms) to the peak gamma-ray luminosity, Lγ ∼ 1044 erg s−1, subsequently followed by an exponentially decaying tail with Lγ ∼ 1042 erg s−1. Remarkably, the initial burst only lasted for ∼ 0.1 s compared to the longer lasting (∼ 100 s) tail that pulsated with a period of ∼ 8 s (Terrell et al., 1980). The total emitted energy during the initial peak and the decaying tail amounted to an astonishing ∼ 1044 erg. An even more energetic flare was detected from SGR 1900+14 on August 27, 1998 by a multitude of space telescopes in the direction of a Galactic supernova remnant G42.8+0.6 (Hurley et al., 1999a), making it the second exceptionally energetic event detected in the past century from a recurrent gamma-ray source. The burst had properties similar to that of the March 5 event, with a short (< 4 ms) rise time to the main peak that lasted for ∼ 1 s and then decayed into a pulsating tail with a period identical to the rotation period of the NS of 5.16 s. The flare had a much harder energy spectrum compared to the March 5 event (Feroci et al., 1999), with peak luminosity in excess of ∼ 4× 1044 erg s−1, assuming a distance of ∼ 10 kpc to the source. The total energy unleashed in this outburst amounted to ∼ 1044 erg in hard X-rays and gamma-rays (Mazets et al., 1999). Finally, on December 27, 2004 the most energetic outburst ever to be detected came from SGR 1806-20 (Hurley et al., 2005), a Galactic source that was found to have a possible association with a compact stellar cluster at a distance of∼ 15 kpc 79 (Corbel & Eikenberry, 2004). The initial spike had a much shorter rise time (≤ 1 ms) to the peak luminosity of∼ 2×1047 erg s−1 which persisted for a mere∼ 0.2 s. Like other giant flares, a hard X-ray tail, of duration∼ 380 s, followed the main spike pulsating at a period of 7.56 s. The total energies emitted during the initial spike and the harmonic tail are ∼ 4× 1046 erg and ∼ 1044 erg, respectively. 4.1.2 The Precursor Hurley et al. (2005) reported the detection of a ∼ 1 s long precursor that was observed 142 s before the main flare of December 27. A similar event, lasting a mere 0.05 s (Ibrahim et al., 2001), was observed only 0.4 s prior to the August 27 giant flare (Feroci et al., 2001; Hurley et al., 1999a), albeit at softer energies (15 − 50 keV, Mazets et al. 1999); a non detection at harder energies (40 − 700 keV) was reported in Feroci et al. (1999). Such a precursor was not detected at all for the March 5 flare for which the detectors at the time had no sensitivity below ∼ 50 keV, which suggests that a softer precursor, if there indeed was one, may have gone unnoticed. Unlike the August 27 precursor, which was short and weak and for which no spectrum could be obtained (Ibrahim et al., 2001), the relatively longer lasting December 27 precursor had a thermal blackbody spectrum with kT ≈ 10.4 keV (Boggs et al., 2007). In comparison to the common short SGR bursts, that typically last for∼ 0.1 s and have sharply peaked pulse morphologies, the December 27 precursor was not only longer in duration but also had a nearly flat light curve. Nevertheless, the burst packed an energy∼ 3.8×1041 erg which is comparable to that of the short SGR bursts. The possible causal connection of the precursors to the giant flares in both cases indicates that they may have acted as a final trigger (Boggs et al., 2007; Hurley et al., 1999a). A strong case for the causal connection of the precursor to the giant flare in both events can be established on statistical grounds. For example, SGR 1900+14 emitted a total of 50 bursts during its reactivation between May 26 and August 27, 1998 following a long dormant phase lasting almost 6 years (Hurley et al., 1999b). Here we are only interested in the burst history immediately prior to the Aug 27 event as this time period is in- 80 dicative of the heightened activity that concluded with the giant flare. From these burst statistics, the rate of short bursts of typical duration∼0.1 s is∼ 6×10−6 s−1, which then yields a null hypothesis probability of ∼ 2.4× 10−6 for the August 27 precursor. Additionally, we find a null hypothesis probability of ∼ 8.6 × 10−4 in the case of the December 27 precursor, assuming similar burst rates. Although the magnetar model (particularly the phenomenological models developed in Thomp- son & Duncan 1995, hereafter TD95, and Thompson & Duncan 2001, hereafter TD01), as we discuss below, offers plausible explanations for the occurrence of short bursts and giant flares, the connection between the precursor and the main flare has remained unknown. In the event the precursor indeed acted as a trigger to the main flare, it is of fundamental significance that the association between the two events is understood. As magnetars, SGRs are endowed with extremely large magnetic fields with B ∼ 102BQED, where BQED = 4.4× 1013 G is the quantum critical field, and all the energetic phenomena discussed above are ascribed to such high fields (TD95). In the TD95 model, the short bursts result due to sudden cracking of the crust as it fails to withstand the building stresses caused by the motion of the magnetic footpoints. The slippage of the crust, as a result, injects Alfvén waves into the external magnetic field lines, that subsequently damp to higher wavenumbers, and ultimately dissipate into a trapped thermal pair plasma. Such a mechanism may not be invoked for the giant flares due to energy requirements. Alternatively, a large-scale interchange instability (Moffatt, 1985), driven by the diffusion of the internal magnetic field, in combination with a magnetic reconnection event can power the giant flares. The plausibility of these mechanisms is well supported by the observed energetics of the bursts and the associated time scales. Nevertheless, a clear description of the reconnection process, which indubitably serves as one of the most efficient mechanisms to convert magnetic energy into heat and particle acceleration, has not been forthcoming. Furthermore, an alternative mechanism, motivated by the coronal heating problem in the solar case, can be formulated to give a reasonable explanation for the association of the precursor and the main 81 flare. 4.2 Magnetic Reconnection: An Overview Magnetic reconnection is a topological restructuring of the magnetic field whereby field lines undergo a change in connectivity due to their diffusive motion in a plasma. This results in the sudden release of magnetic energy that manifests in the form of electrodynamical bursts, that are characterized by the emission of hard electromagnetic radiation and acceleration of charged particles to high energies. When two oppositely directed but equal in magnitude magnetic field lines inter- sect at a point, they give rise to an X-type neutral point where B = 0 (see Fig. 4.1). The formation of a neutral point is a general phenomenon when multiple sources of magnetic field are present. In a plasma, the collapse of the X-type neu- tral point gives rise to the formation of a current sheet, across which the magnetic field changes direction and/or magnitude. Mathematically, distinct current sheet geometries can be constructed from various different initial field configurations. Several pedagogical texts (for e.g. (Priest & Forbes, 2000; Somov, 2006)) have been devoted to the study of current sheet formation, and the reader is advised to consult those. Detailed description of current sheet formation is outside the scope of this work. Here, we are only interested in describing the different reconnec- tion methods that are relevant to the problem at hand, and assume that a current layer already exists which facilitates the diffusion of magnetic field lines, their break-up, and reconnection. 4.2.1 Steady-State Reconnection In steady-state reconnection, the time scale for the change in the global configura- tion of the diffusion region is much smaller than the typical time scales at which irregularities in the plasma and the magnetic field propagate across some typical length scale L0 associated to the system. In this case, the rate at which field lines 82 Figure 4.1: The Sweet-Parker model of steady-state reconnection. Outside the current sheet, the magnetic field lines are frozen in the plasma and are advected into the diffusion layer by the fluid flow of the plasma. Field lines only diffuse, break, and reconnect inside the current layer due to its finite conductivity. The plasma is expelled through a width of size 2δ and the ideal MHD condition is restored. The aspect ratio of the current sheet is what sets the rate of reconnection (adapted from Zweibel & Yamada (2009)). break and reconnect is prescribed by Faraday’s law ∇× E = −1 c ∂B ∂t (4.1) where E and B are the electric and magnetic fields. Up to some gauge condition, the magnetic field can be expressed in terms of a vector potential A, such that B = ∇ × A. Then, from Eq. 4.1 one finds that the rate of reconnection is determined by the electric field at the neutral point N − 1 c ∂A ∂t |N= E (4.2) 83 The first model of steady-state reconnection was put forth by Sweet (1958) and Parker (1957, 1963), where they envisioned the merging of two oppositely di- rected, but equal in magnitude, magnetic field lines in a narrow diffusion region. The hydromagnetic equation (or the induction equation) that governs the flow of field lines into the current layer and the rate of reconnection is the following ∂B ∂t = ∇× (v ×B) + ηB∇2B (4.3) where v is the velocity of the plasma flowing into the diffusion layer, ηB = c2/4piσ is the magnetic diffusivity, σ is the electrical conductivity. This equa- tion dictates that the change in magnetic field in a plasma is brought about by the motion of the plasma element, in which the field lines are “frozen”, and by the diffusive motion of the field lines themselves, for example inside a current layer. The ratio of the two terms in the right hand side of Eq. 4.3 provide a good measure of the behavior of the magnetic field in a plasma. This quantity is defined to be the magnetic Reynolds number Rm = ∇× (v ×B) ηB∇2B ∼ L0V0 ηB (4.4) where the fluid flow velocity of the plasma has a typical value of V0, and L0 is the typical length scale of the system. Thus, for magnetic field lines to diffuse through the plasma and undergo reconnectionRm  1 is a necessary condition, otherwise the field lines are effectively tied to the plasma and remain frozen (Rm  1). These are, of course, limiting cases. Some diffusion of magnetic field still occurs even if Rm ∼ 1. However, it is generally difficult to describe the behavior of the field and the plasma in the middle regime. In the Sweet-Parker model of steady-state reconnection (see Fig. 4.1), the length of the current sheet is set by the global length scale of the field, L. As it will be discussed later, this very assumption makes the reconnection process occur at a slower rate as compared to some of the faster mechanisms, such as the one advanced by Petschek (1964). Maxwell’s equations along with Ohm’s law 84 and some fluid equations allow one to make some order of magnitude estimates that yield important insight into the reconnection process. In the two-dimensional steady-state process, the continuity equation ∇ · (ρv) = 0 (4.5) gives a relation between the size of the current sheet and the inflow (vi) and out- flow (vo) velocities, such that δ = ( vi vo ) L (4.6) where δ is the thickness of the diffusion region. The homogeneous electric field E is directed perpendicular to the plane of the current sheet and is given by Ohm’s law E = −v c ×B+ j σ (4.7) Notice that just outside the current sheet j = 0 and at the neutral line B = 0. From Ampère’s law, where we have ignored the displacement current term as it is negligible in steady-state reconnection, ∇×B = 4pi c j (4.8) the current at the neutral line is simply j = cBi/4piδ, and Bi is the upstream magnetic field. Then, matching the Ohm’s law equation inside and outside the current layer yields the inflow velocity vi = ηB δ (4.9) To calculate the outflow velocity, we look at the equation of motion of plasma inside the current layer where the particles are accelerated due to the j×B Lorentz 85 force and pressure gradient. ρ(v · ∇)v = −∇p+ 1 c j×B (4.10) Ignoring any changes in the pressure along the length of the current sheet, and noting that magnetic flux is conserved, such that BoL = Biδ where Bo is the downstream magnetic field, one can obtain an estimate of the outflow velocity which is defined to be the Alfvén velocity upstream of the current sheet. The j×B Lorentz force accelerates the plasma to the Alfvén velocity. vo = Bi√ 4piρ ≡ vAi (4.11) By combining Eq. 4.4,4.6,4.9, and 4.11 and writing all quantities in terms of the magnetic Reynolds number, we find Mi = vi vAi = δ L = Bo Bi = R−1/2m (4.12) where Mi is the Alfvén Mach number which gives the upstream plasma velocity in terms of the upstream Alfvén velocity. Although it is not a rate, higher Mi cor- responds to a faster reconnection rate. Evidently, for Rm  1, the reconnection proceeds rather slowly and occurs in an extremely thin current sheet. For exam- ple, in the solar corona the magnetic Reynolds number is quite high Rm ≈ 109, where the reconnection rate is∼ 10−5 that of the Alfvén speed. The characteristic Alfvén crossing time τA τA = L0 vA (4.13) of the solar corona of length L0 ∼ 107 cm is roughly 10 ms, and solar flares release magnetic energy on time scales of 102 s. Based on the above results, the Sweet-Parker mechanism can only operate on time scales of order 103 sec, much longer than needed to explain the bursty release of energy in solar flares. Any model of fast steady-state reconnection must operate at rates Mi > R −1/2 m to be 86 Figure 4.2: The Petschek model of steady-state reconnection. It is based on the Sweet-Parker mechanism but the half-length of the Sweet-Parker current layer L∗ is much smaller than the global length scale of the field, L. Shortening the length changes the aspect ratio and increases the rate of reconnection. In addition, reconnection is made faster by two pairs of slow-mode shock waves emanating downstream of the current layer (adapted from Zweibel & Yamada (2009)) considered in any astrophysical flaring phenomenon where Rm  1, typically. Petschek (1964) proposed a new mechanism where the reconnection rate scales as Mi ∝ (lnRm)−1. The roadblock to faster reconnection in the Sweet-Parker case was essentially the smallness of the width of the current layer, as evident from Eq. 4.12, through which large amounts of plasma fluid needs to be expelled. Thus, affecting the reconnection rate drastically. In Petschek’s model, the length of the current layer is progressively made smaller, yielding faster reconnection rates, until a maximum reconnection rate is established with Mi = pi/8 lnRm corresponding to a length that is only 8 lnRm/pi √ Rm of that in the Sweet-Parker case. The novelty is his model comes from the introduction of two pairs of slow- mode shock waves that emanate from both ends of the current sheet, accelerating 87 the plasma column to Alfvén speeds and converting most of the magnetic energy to kinetic energy of the particles and heat. 4.2.2 Unsteady Reconnection: The Tearing Mode In the absence of any plasma flows (v = 0), Eq. 4.3 becomes ∂B ∂t = ηB∇2B (4.14) the diffusion equation, where the plasma is assumed to have finite electric con- ductivity, such that magnetic field lines can diffuse on time scales given by the diffusion time τd = l2 ηB (4.15) In current sheets with τd  τA, such that any perturbations in the field can grow much faster than they are diffused away, the onset of resistive instabilities leads to the formation of magnetic islands, i.e. plasmoids with an O-type neutral point at their centers and X-type neutral points in between neighboring plasmoids. One such instability is the tearing mode which operates on a time scale τ ' √τdτA and has a wavelength exceeding the size of the current sheet, kl < 1 (Furth et al., 1963). The growth rate of any small perturbations in the current sheet is τtm = (kl) 2/5R3/5m (4.16) which reaches its maximum value when kl ≈ R−1/4m and the tearing mode pro- ceeds at a much faster rate (Mi ≈ R1/2m ) in comparison to the Sweet-Parker mech- anism. 4.2.3 Collisionless Plasma: Hall Reconnection In a collisionless plasma, the mean free path of the constituent particles is larger than the global length scale of the system, λmfp  L0. The particle dynamics in 88 Figure 4.3: The tearing mode is a non-steady process caused by a resistive instability that leads to the formation of magnetic islands in the current sheet (adapted from Birn & Priest (2007)). Figure 4.4: A sketch of the different spatial scales involved in Hall recon- nection. For Hall reconnection to become the dominant process the size of the Sweet-Parker layer δ must be smaller than the ion dissipa- tion region δi = c/ωpi. As the field is advected into the current layer both the electrons and ions are coupled to the field lines. The ions decouple first when reaching the ion inertial length scale δi while the electrons are still frozen to the field lines. At length scales smaller than the electron inertial length, the electrons decouple from the field lines (adapted from Zweibel & Yamada (2009)). such a system are governed by the generalized Ohm’s law (me ne2 )[∂j ∂t +∇ · ( vj+ jv − jj ne )] = E+ v ×B c − j×B nec + ∇ · Pe ne − ηj (4.17) The terms on the left hand side in the above equation describe electron inertia 89 where the three terms (vj, jv, and jj) form a dyadic product. The inductive and convective electric fields are given by E and v×B terms, respectively. The j×B term gives rise to the Hall effect, the term with the electron pressure tensor Pe de- scribes electron gyro-viscosity and gives rise to kinetic Alfvén waves, and finally the last term describes Ohmic dissipation due to electron-ion collisions (Priest & Forbes, 2000). All the terms that are absent in the MHD version of Ohm’s law (see Eq. 4.7) define a particular spatial and time scale, estimated from their corre- sponding gradients, which gives important insight into when these effects become dominant; In the ideal MHD case, Ohm’s law has no gradients and, thus, is scale invariant. Upon examining one of the electron inertial terms (since they are all of the same order) inside the diffusion region and comparing its magnitude to the convective electric field outside the current layer, we find with j ∼ cB0/4piL0, ∂/∂t ∼ v0/L0 v0B0 c ∼ (me ne2 ) cB0v0 4piL20 ⇒ L0 ∼ ( mec 2 4pine2 )1/2 = c ωpe = δe (4.18) where ωpe is the electron plasma frequency, and c/ωpe defines the electron inertial- length or skin-depth. For the Hall term, we find with B0/v0 = B0/vAM , where M is the reconnection rate (defined in Eq. 4.12), ρ = nmi, and ne = ni = n (quasi-neutrality condition), v0B0 c ∼ cB 2 0 4piL0nec ⇒ L0 ∼ 1 M ( mic 2 4pine2 )1/2 = c Mωpi = δi M (4.19) Here δi is the ion skin-depth. According to the forgoing analysis, it is clear that the electron inertia term and the Hall effect don’t affect the particle dynamics until the characteristic length scale of the diffusion region is comparable to that of electron and ion skin depths, respectively. 90 4.3 Internal Trigger In the magnetar model, the magnetic field in the interior of SGRs is considered to be strongly wound up which then generates a strong toroidal field component, possibly even larger than the poloidal component (TD01). The relative strengths of the poloidal and toriodal magnetic field components have been quantified by constructing relativistic models of NSs and testing the stability of axisymmetric fields by Lander & Jones (2009) and Ciolfi et al. (2009). Both studies arrive at the conclusion that the amplitude of the two field components may be comparable but the total magnetic energy is dominated by the poloidal component as the toroidal component is non-vanishing only in the interior, with EB,tor/EB ≤ 0.1. However, in another study Braithwaite (2009) arrived at a somewhat different conclusion where he found a significant enhancement in the toroidal component to sustain a stable magnetic field configuration, with 0.20 ≤ EB,tor/EB ≤ 0.95. In the interior, the twisted flux bundle, composed of several flux tubes, can be envisioned to stretch from one magnetic pole to the other along the symmetry axis of the dipole field that is external to the star. It has been shown by Parker (1983a,b) that any tightly wound flux bundle is unstable to dynamical nonequilibrium, and will dissipate its torsional energy as heat due to internal neutral point reconnection. Although Parker had provided such a solution to the long standing problem of coronal heating in the solar case, with a few exceptions, the same applies to the case of magnetars as the arguments are very general. In the case of the Sun, flux tubes are stochastically shuffled and wrapped around each other due to convective motions in the photosphere. Unlike in the Sun, where the flux tube footpoints are free to move in the photospheric layer, the footpoints are pinned to the rigid crust in NSs. Nevertheless, for exceptionally high magnetic fields (B > 1015 G) the crust responds plastically (TD01), and any moderate footpoint motion can still occur. It is understood that this is only true to the point where the crustal stresses are below some threshold, which depends on the composition. Thus, as the imposed strain exceeds some critical value, the crust will yield abruptly (Horowitz & Kadau, 2009), but may not fracture (Jones, 2003). 91 Parker’s solution is at best qualitative, however, it serves as a reasonably good starting point in the context of the present case. As we have noted earlier, the precursor may be causally connected to the main flare, and so can be argued to act as a trigger in the following manner. Immediately after the precursor the internal field evolves toward a new state of equilibrium. Since the crust has yielded to the built up stresses, and may deform plastically under magnetic pressure, some of the footpoints can now move liberally. Understandably, the turbulent dynamics, due to the high Reynolds number (Peralta et al., 2006), of the internal fluid in response to the burst translates into chaotic motion of the footpoints. As a result, the flux tubes are wrapped around each other in a random fashion. Current sheets then inevitably form leading to reconnection followed by violent relaxation of the twisted flux bundle. The heat flux resulting from the dissipation of the torsional energy of the flux bundle is given by Parker (1983a), P = ( B2v2τ 4piL ) (4.20) where B is the strength of the internal magnetic field, v is the footpoint displace- ment velocity, and L is the length scale of the flux tubes. Here τ is the time scale over which accumulation of energy by the random shuffling and wrapping of flux tubes occurs until some critical moment, after which neutral point reconnection becomes explosive. Having knowledge of the burst energetics, equation (4.20) can be solved for τ τ ∼ 142 B−215 L−16 E46T−10.125 ( v 8.4× 103 cm s−1 )−2 s (4.21) where we have used the event of December 27 as an example, with internal field strength measured in units of 1015 G, flux tube length scales in 106 cm, the total energy of the flare in 1046 erg, and the time scale of the initial spike in 0.125 s (RHESSI PD time resolution). It is clear from equation (4.21) that the preflare quiescent time scales linearly with the total energy emitted in the initial spike but 92 is inversely proportional to the internal magnetic field strength: τ ∝ EspikeB−2in . Following TD95, we have assumed that almost all of the energy of the flare was emitted in the initial transient phase during which the lightcurve rose to its maxi- mum. Additionally, we find that the footpoints are displaced at a rate of few tens of meters per second, which is a reasonable estimate considering the fact that it is insignificant in comparison to the core Alfvén velocity VA ∼ 107 cm s−1. A noteworthy point is that in regards to the burst energetics there is nothing special about the precursor when compared to the common SGR bursts, other than that it occurs at the most opportune time when the internal field undergoes a substantial reconfiguration. The mechanism outlined above is activated after every SGR burst after which significant footpoint motion ensues. However, whether the entanglement of flux tubes is sufficient to reach a critical state such that an explosive release of energy can occur depends on the evolution of the internal field configuration. Alternatively, the twisted flux bundle can become unstable to a resistive insta- bility, such as the tearing mode. The resistivity here is provided by the turbulent motion of the highly conductive fluid which is in a state of nonequilibrium im- mediately after the precursor. The growth time of the tearing mode instability is given by the geometric mean of the Alfvén time, say in the core, and the resistive time scale τ = (tAtR) 1/2 (4.22) = ( 4piσL3 VAc2 )1/2 (4.23) ∼ 142 L3/26 ( VA 107 cm s−1 )−1/2 ( σ 1013 s−1 )1/2 s (4.24) where σ is not the electrical conductivity, but corresponds to the diffusivity of the turbulent fluid. In this case, the scaling for the preflare quiescent time becomes τ ∝ E1/2spikeB−3/2in (4.25) 93 V NS Sweet-Parker Layer Hall Reconnection Layer SHTCL Plasmoid Figure 4.5: This figure displays the setup of the different reconnecting cur- rent layers. The macroscopic Sweet-Parker layer with length L ∼ 105 cm and width δ ∼ 0.01 cm is the largest of the three. This layer is then thinned down vertically as strong magnetic flux is convected into the dissipation region. The Hall reconnection layer, represented by the orange region, develops when δ becomes comparable to the ion- inertial length di. The system makes a transition from the slow to the impulsive reconnection and powers the main flare. The tiny region embedded inside the Sweet-Parker layer is the super-hot turbulent cur- rent layer, which aids in creating sufficient anomalous resistivity to facilitate the formation of the Sweet-Parker layer. The strongly accel- erated plasma downstream of the reconnection layer is trapped inside magnetic flux lines and forms a plasmoid moving at some speed V . This plasmoid is then finally ejected during the initial spike when the external field undergoes a sudden relaxation (After Lyutikov 2006). where we have assumed that the twisted flux bundle occupied the entire internal region of the NS. 94 4.4 External Trigger The notion that the giant flares are a purely magnetospheric phenomenon appears very promising and requires further development. A magnetospheric reconnection model has become the favourite of many for two main reasons. First, it can easily explain the millisecond rise times of the explosive giant flares in terms of the Alfvén time of the inner magnetosphere, which for exceptionally low values of the plasma beta parameter is very small; τA ∼ R?/c ∼ 30µs. Second, the SGR giant flares have much in common with the extensively studied solar flares, for which reconnection models explaining nonthermal particle creation, plasma bulk motions, and gas heating have been developed over the last few decades (Lyutikov, 2002). The most powerful solar flares release an equally impressive amounts of energy ∼ 1032 erg, which is mainly divided into heating the plasma and radiation in multiple wavebands, for example γ-rays, X-rays, and radio. The impulsive rise in the soft X-ray emission to peak luminosity occurs over a timespan of few hundreds of seconds, which is then followed by a gradual decay lasting several hours (Priest & Forbes, 2002). In the magnetar model, because of the shearing of the magnetic footpoints caused by the unwinding of the internal field, a twist can be injected into the external magnetic field (TD95,TD01). Depending on how the crust responds to the stresses, either plastically or rigidly, the gradual or sudden (in the event of a crustal fracture) transport of current from the interior creates a non-potential region in the magnetosphere where a reconnecting current layer can develop (Mikic & Linker, 1994; Thompson et al., 2002). Lyutikov (2003, 2006) has explained the impulsive nature of the giant flares in terms of the tearing mode instability, which has a magnetospheric growth time of τtear ∼ 10 ms. Impulsiveness of the underlying magnetic reconnection mechanism explain- ing the origin of giant flares is a primary requirement. The tearing mode insta- bility is quite befitting in that regard; however, it has not been shown to bear any dependence on the precursor, which as we argue, triggered the main hyperflare. Hall reconnection, which is another impulsive reconnection mechanism, has been completely ignored on the basis that it is unable to operate in a mass symmetric 95 electron-positron pair plasma. Nevertheless, a mild baryon contamination may be enough to render it operational. Therefore, what is needed here is the syn- ergy of two distinct mechanisms — a slow reconnection process, like the Sweet- Parker solution, that only dissipates magnetic energy at a much longer time scale (Parker, 1957; Sweet, 1958), and a fast process that is explosive, like Hall recon- nection (Bhattacharjee et al., 1999). To put this in the context of the December 27 event, we envision that immediately after the precursor a macroscopic current layer developed as a result of the sheared field lines. Then began the slow dissi- pation of magnetic field energy by Sweet-Parker reconnection, which continued throughout the quiescent state that followed the precursor. Finally, the transition to Hall reconnection resulted in the explosive release of energy (see figure 4.5). We describe this process in more detail in the following section. 4.4.1 Transition from Resistive to Collisionless Reconnection by Current Sheet Thinning The steady state reconnection process of Sweet and Parker is severely limited by its sensitivity to the size of the macroscopic dissipation region, such that the plasma inflow velocity is regulated by the aspect ratio of the current layer vi = δ L vA (4.26) where δ is the width and L is the length of the dissipation region, with δ  L generally. The downstream plasma flow speed coincides with the Alfvén veloc- ity, which in the magnetar case approaches the speed of light. The Sweet-Parker mechanism is a resistive reconnection process where the resistivity is either colli- sional or anomalous. It is understood that the electron-positron pair plasma per- vading the inner magnetosphere is collisionless. Nevertheless, if enough ions are present in the dissipation region, as we show below, then a source of anomalous resistivity can be established. We argue that the energy released during the precur- sor was enough to heat the crust to a point where a baryon layer was evaporated 96 into the magnetosphere. TD95 provide an upper limit to the mass of the baryon layer ablated during a burst by comparing the thermal energy of the burst to that of the potential energy of the mass layer ∆M ∼ EthR? GM? (4.27) ∼ 1017 ( Eth 1038 erg )( R? 106 cm )( M? 1.4M ) g (4.28) where we have assumed a more conservative estimate of Eth. Then, assuming that ∆M amount of baryonic mass, in the form of protons, was injected into the magnetospheric volume of ∼ R3? yielding a baryon number density of nb ∼ 6× 1022 ( Eth 1038 erg )( R? 106 cm )−2( M? 1.4M ) cm−3 (4.29) Even with the large amount of baryons, the magnetospheric plasma is still colli- sionless. The Spitzer resistivity for a quasi-neutral electron-ion plasma is only a function of the electron temperature ∝ T−3/2e , which for electron temperatures as high as ∼ 108 K yields a negligible resistivity. Super-Hot Turbulent Current Layer For plasma temperatures higher than T > 3 × 107 K, the reconnecting current layer turns into a super-hot turbulent current layer (SHTCL), for which the theory has been well developed by and documented in (Somov 2006, pp. 129-151). The anomalous resistivity in the current layer arises due to wave-particle interactions, where the ions interact with field fluctuations in the waves. As a result, the re- sistivity and other transport coefficients of the plasma are altered. The electrons are the current carriers and participate mainly in the heat conductive cooling of the SHTCL. The current layer is assumed to have been penetrated by a relatively weak transverse magnetic field component (transverse to the electric field in the current layer), where B⊥  B0 with B0 as the strength of the external dipole 97 field. In the two temperature model, where the electrons and ions are allowed to have dissimilar temperatures, the effective anomalous resistivity is generally a combination of two terms. One resulting from the ion-acoustic turbulence and the other from the ion-cyclotron turbulence. ηeff = ηia + ηic (4.30) In addition, each turbulent instability has two separate regimes – marginal and saturated. The former applies when the wave-particle interactions are described by quasilinear equations, and the latter becomes important in the case of strong electric fields when the nonlinear contributions can no longer be ignored (see for e.g. Somov 1992 pp. 115-217 for a detailed description). For an equal temper- ature plasma (Te ∼ Ti), the saturated ion-cyclotron turbulent instability makes the dominant contribution to the effective resistivity. Thus, we ignore any other terms corresponding to the ion-acoustic instability. The effective resistivity in the present case is given as (Somov, 2006), depending on the dimensionless tempera- ture parameter θ ≡ Te/Ti, ηeff = 2m 1/2 e pi1/4 ec1/2m 1/4 p [ (1 + θ−1)1/2 N1/4(θ)Uk(θ) ] (B⊥E0)1/2 B 1/2 0 n 3/4 b (4.31) where N(θ) = 1.75 + f(θ)√ 8(1 + θ−1)3/2 (4.32) f(θ) = 1 4 ( mp me )1/2 for 1 ≤ θ ≤ 8.1 (4.33) Uk(θ) ∼ O(1) for θ ∼ 1 (4.34) E0 = αB0 (4.35) α ≡ v0/c is the effective reconnection rate determined by the inflow fluid velocity v0 into the current layer, and the rest of the variables in equation (4.31) retain 98 their usual meaning. Equation (4.35) conveys the frozen-in field condition. Next, we write the magnetic diffusivity of the plasma due to the effective anomalous resistivity ηdiff = ηeffc 2 4pi (4.36) ' 6× 1023(αB⊥)1/2n−3/40 (4.37) To calculate the inflow plasma velocity, we assume that the SHTCL is embed- ded in a macroscopic Sweet-Parker current layer. The primary role of the SHTCL is to provide enough resistivity in a collisionless plasma so that the magnetic field lines can diffuse through it and ultimately undergo reconnection. From equation (4.26) we know that for a Sweet-Parker current layer the inflow fluid velocity is regulated by the aspect ratio of the current layer. The outflow velocity is limited by the speed of light. By expressing the width of the Sweet-Parker current layer in terms of the magnetic diffusivity, we find that the inflow velocity has to be on the order of v0 ∼ 103 cm s−1 (so that α 1), with the width of the layer given as δ ∼ √ ηdiffL c (4.38) ∼ 0.01 cm ( v0 103cm s−1 )1/4( B⊥ 1011 G )1/4 (4.39) × ( nb 6× 1022 cm−3 )−3/8( L 105 cm )1/2 where the transverse magnetic field is B⊥ ∼ 10−3B0, and L is the length of the current layer. The size of the SHTCL can now be obtained from the following a = c e √ me 2pinb [√ 1 + θ−1 N(θ) 1 Uk(θ) ] (4.40) ∼ 2.5× 10−6 ( nb 6× 1022 cm−3 )−1/2 cm (4.41) 99 b = B0 h0 √ 2v0 B⊥ [ pimpnb N(θ) ]1/4 (4.42) ∼ 80 cm ( R? 106 cm )( v0 103 cm s−1 )1/2 (4.43) × ( B⊥ 1011 G )−1/2( nb 6× 1022 cm−3 )1/4 where a and b are, respectively, the half-width and the half-length of the SHTCL, and h0 ∼ B0/R? is the magnetic field gradient in the vicinity of the current layer. Current Sheet Thinning The main flare is triggered when the transition is made from the steady state, slow reconnection process to an impulsive one. In the present case, Sweet-Parker reconnection makes a transition to Hall reconnection when the width of the current layer δ drops below the ion-inertial length di, where di = c ωp,i = c e √ mp 4pinb (4.44) ∼ 10−4 ( nb 6× 1022 cm−3 )−1/2 cm (4.45) and ωp,i is the non-relativistic ion plasma frequency. Cassak et al. (2005) show that for a given set of plasma parameters, the solution is bistable such that the slow Sweet-Parker solution can operate over long time scales, during which the system can accumulate energy, while the faster Hall solution starts to dominate as the resistivity is reduced below some critical value. Lowering the resistivity would naturally reduce the width of the current layer to the point where the sys- tem can access the Hall mechanism. Alternatively, as Cassak et al. (2006) argue, the same result can be achieved by thinning down the current layer by convect- ing in stronger magnetic fields into the dissipation region during Sweet-Parker 100 reconnection. The critical field strength needed to thin the current layer is Bc ∼ √ 4pimpnb ( ηdiff d2i L ) (4.46) ∼ 4× 1014 G ( nb 6× 1022 cm−3 )3/4 (4.47) × ( v0 103 cm s−1 )1/2( B⊥ 1011 G )1/2( L 105 cm ) Due to flux pile up outside the current layer, it can be argued that the system is able to achieve such high field strengths. The time scale for thinning down the current sheet until its width is comparable to the ion-inertial length is given as τthin ∼ 2Ws √ L ηdiffc ( Bc B0 ) (4.48) ∼ 130 s ( Ws 105 cm )( L 105 cm )( B0 1014 G )−1/2 (4.49) × ( nb 6× 1022 cm−3 )3/4 where Ws is the magnetic shear length, that is the length scale over which the field lines are severely sheared. What we find here is that the thinning down time τthin of the current layer from the Sweet-Parker width to the ion-inertial length, where Hall reconnection dominates, is on the order of the preflare quiescent time of ∼ 142 s for the December 27 event. The scaling relation of the thinning down time in terms of the initial spike energy and the external magnetic field strength can be deduced to be the following τthin ∝ E2/3spikeB−11/60 (4.50) Again, we emphasize here that this same mechanism may operate after every SGR burst which is energetic enough to inject the requisite baryon number density, 101 as calculated in equation (4.29), to facilitate the development of a Sweet-Parker current layer. However, this mechanism will fail if the twist injected into the magnetosphere by the unwinding of the internal field is not sufficient to create a tangential discontinuity at the first place. In that instance no current sheet will form. Giant Flare Submillisecond Rise Times In Hall reconnection, a multiscale dissipation region develops with characteristic spatial scales on the order of the ion and electron inertial lengths (Shay et al., 2001). Within a distance di of the neutral X-line, the ions decouple from the electrons and are accelerated away at Alfvénic speeds in the direction perpendic- ular to that of the inflow. The electrons continue their motion towards the neutral line as they are frozen-in, and only decouple from the magnetic field when they are a distance de, the electron-inertial length, away from the neutral line. Within the ion-inertial region, the dynamics of the electrons are significantly influenced by the nonlinear whistler waves. Subsequently, the electrons too are accelerated away in an outflowing jet at Alfvénic speeds. The time scale associated to Hall reconnection then is in good accord with the rise times of giant flares (Schwartz et al., 2005), that is τHall ∼ R? 0.1c ∼ 0.3 ms (4.51) 4.5 Discussion In this paper, we present an internal and an external trigger mechanism for the SGR giant flares, where we strongly emphasize the causal connection of the pre- cursor to the main flare. The quiescent state that follows the precursor has been argued, in our model, to be the time required for the particular instabilities to de- velop, along with the accumulation of energy just before the flare. The internal mechanism is based on the hypothesis that poloidal field component in the interior of the NS is strongly wound up. The solution is motivated by Parker’s reasoning 102 that such a twisted field would inevitably develop tangential discontinuities and dissipate its torsional energy as heat. The time scale for the accumulation of en- ergy that is to be released in the main flare is on the order of the duration of the preflare quiescent state. The external trigger mechanism makes use of the fact that a Sweet-Parker re- connection layer may develop between significantly sheared field lines if a source of resistivity is established. Such a source may be embedded inside the macro- scopic Sweet-Parker layer in the form of a super-hot turbulent current layer. To make the reconnection process impulsive, we invoke the non-steady Hall recon- nection which is switched on as the width of the Sweet-Parker layer is thinned down to the ion-inertial length. Again, the time scale over which the layer is thinned down roughly coincides with that of the preflare quiescent state duration. We have shown detailed calculations of the time scales for the December 27 event in particular. However, a similar analysis can also be carried out for the Au- gust 27 event. For the internal mechanism, we find the time scale to be comparable to the observed preflare quiescent time with τ ∼ 0.4B−215 L−16 E44T−11 ( v 5.6×104 cm s−1 )−2 s, where we have assumed the same internal magnetic field strength and length of flux tubes. For the external mechanism, assuming ∆M ∼ 1015 g, since the pre- cursor was short and weak, Ws ∼ 2 × 104, and L ∼ 5 × 104, we find τthin ∼ 0.4 s, δ ∼ 0.04 cm, a ∼ 2.5× 10−5 cm, b ∼ 25 cm, and di ∼ 10−3 cm. A significant nonthermal component, with an average power-law index of Γ ∼ 2 as in E−Γ, was observed during the decaying phase of the flare in both the August 27 and December 27 events (Boggs et al., 2007; Feroci et al., 1999). In the magnetar model, the nonthermal emission originates much farther out from the star, almost at the light cylinder (TD01). At this distance, inverse Compton cooling by X-ray photons has been invoked to explain the nonthermal spectrum. Nonthermal particle generation is one of readily identified features of magnetic reconnection, especially in the case of Hall reconnection where outflow velocities approach the Alfvén speed of the medium. Such acceleration of high energy parti- cles due to meandering-like orbits in the presence of strong electric fields has also 103 been seen in particle-in-cell simulations (Zenitani & Hoshino, 2001). Therefore, the Hall reconnection process that gives rise to the main flare can easily explain the origin of nonthermal particles. Israel et al. (2005) and Strohmayer & Watts (2005) reported the detection of quasi-periodic oscillations (QPOs) in the burst spectra of the December 27 and August 27 events, respectively. QPOs in the December 27 event were detected at 92.5 Hz, and 18 and 30 Hz at, respectively, 170 s and∼ 200−300 s after the initial spike. Watts & Strohmayer (2006) confirmed the detection of the first two QPOs and reported the presence of two additional QPOs at 26 and 626.5 Hz. Similarly, in the August 27 event, QPOs at 84 Hz, 53.5 Hz, 155.1 Hz, and 28 Hz (with lower significance) were detected at about a minute after the onset of the flare. Torsional oscillation of the NS crust appeared to be the natural explanation for the QPOs. However, Levin (2006) argues that purely crustal oscillations rapidly lose their en- ergy to an Alfvén continuum in the core by resonant absorption. He demonstrates that steady low frequency oscillations can be associated with MHD continuum turning points in the core (Levin, 2007), while others have reproduced the QPOs in toy models governed by global MHD-elastic modes of the NS (Glampedakis et al., 2006; Sotani et al., 2007). Both trigger mechanisms presented in this paper can be linked to the initiation of such an oscillatory behavior, whether in the core or as a global mode of the star, by the realization that during the giant flare the global magnetic field of the star undergoes sudden magnetic relaxation (TD95). In the internal trigger, the sudden loss of helicity can be argued to be sufficient to launch Alfvén waves in the interior. On the other hand, although the external trigger, is not directly tied to the crust, however, a sudden relaxation of the internal toroidal field in the sense of the Flowers & Ruderman (1977) instability can be realized. The loss of magnetic energy in the form of a plasmoid, which has been know to form during an eruptive flare (Magara & Shibata, 1997), serves to relax both the sheared external dipole field and the twisted internal toroidal field. Since both fields thread through the entire star, a sudden relaxation during the initial spike can easily excite global 104 elastic modes in the NS. The ejection of a plasmoid also naturally explains the origin of the radio afterglow observed for both the August 27 and the December 27 events (Frail et al. 1999; Gaensler et al. 2005; also see Lyutikov 2006 for afterglow geometry and parameters). In our calculations, we have shown how the preflare quiescent time scales with the total energy released in the initial spike. In the internal trigger mecha- nism, we find that Parker’s solution yields a linear dependence. Although, as we have remarked earlier, this mechanism is simple and elegant but based on quali- tative arguments. Nevertheless, it does reproduce the observed result, that is the longer the preflare quiescent time the energetic the flare, if the internal magnetic field strengths for both NSs are assumed to be similar: τAug/τDec ∼ EAug/EDec. On the other hand, for the scaling to reconcile with observations in the case of the tearing mode instability and the external trigger, either EAug/EDec  10−3 or B1900+14/B1806−20  10−1. Based on the measured P and Ṗ values for SGR 1900+14 and SGR 1806-20, which suggest that B1806−20 ∼ 3B1900+14, and with the revised distance estimate of D ∼ 12− 15 kpc for SGR 1900+14 (Vrba et al., 2000), neither condition may be satisfied. However, one should not ignore the fact that the external mechanism also depends on the size of the current layer and the field line shearing lengthscale. Therefore, the scaling relation may not be as simple as that argued in equation (4.50). In any case, with only two events it is pre- mature to observe any trends regarding the preflare quiescent times and the burst energies. Future giant flares from SGRs will certainly improve our understanding of such correlations. 105 Chapter 5 The Cooling Behavior of Nearby Isolated Neutron Stars:The Magnificent Seven 5.1 Introduction The observed thermal states of isolated neutron stars have become the primary source to glean useful and interesting information about the internal structure of neutron stars (NSs). Reconciliation of theoretical cooling curves with observa- tions of nearby isolated cooling NSs is a challenging task (see for e.g. Page et al. (2006); Yakovlev & Pethick (2004) for a comprehensive review). On the theo- retical front, the problem arises from incomplete knowledge of the composition and equation of state (EOS) of matter in the NS core at supernuclear densities (ρ > 1014 g cm−3). Many possibilities can be realized: Depending on the com- position the EOS can be either soft or stiff, where the stiffness characterizes the compressibility of matter, and strongly depends on the internal degrees of free- dom of the system (Schaab et al., 1996). For example, for a polytropic EOS, P = KρΓ, a larger adiabatic index Γ yields stiffer EOS. Such an EOS generally produces larger maximum masses and radii of NSs than its softer counterpart. 106 Softer EOSs can be obtained by the introduction of phase transitions in the the- ory where the core of the NS may be composed of boson condensates, quark or hyperonic matter. The composition of matter at densities ρ . 2ρ0 ∼ 5.6 × 1014 g cm−3 is rea- sonably well understood. Matter at such densities constitutes the outer and inner crust, and the outer core of the NS. Starting with a lattice structure of atomic nuclei and a relativistic, degenerate gas of electrons, in the first few hundred meters that forms the outer crust of the stars, the nuclei tend to become more and more neu- tron rich with increasing density. The neutron drip density, ρND ≈ 4×1011g cm−3, demarcates the outer crust from the inner, which is mainly composed of neutron- rich atomic nuclei in a sea of free neutrons and electrons at low densities and a fully degenerate fluid of neutron, protons, and electrons at high densities. The main uncertainty lies in accurately predicting the composition of matter beyond ρ ∼ 2ρ0 that makes up the inner core and occupies ∼ 80% of the star’s volume. In their review article, Yakovlev & Pethick (2004) outline several possibilities for the composition of the inner core, namely nucleon matter, hyperonic matter, pion and kaon condensates, and quark matter. Important constraints on the various EOSs have come from the mass-radius relation of the observed NS population. For example, the discovery of a 1.97±0.04 M NS has ruled out many EOSs that predicted the existence of hyperonic matter and kaon condensates at high densities (Demorest et al., 2010). The cooling behavior of NSs depends very sensitively on the composition of the inner core. The rate of cooling from the proto-neutron star stage up to 104−105 years is governed by which neutrino process is dominant in the core (see for e.g. Yakovlev et al. (2001) for a detailed description of all the possible neutrino emis- sion processes), although neutrino emission does occur in the stellar crust as well, for example due to Cooper pairing. In the case of nucleonic matter or hyperonic matter, the most efficient of the neutrino emission processes is the direct Urca cy- cle (also referred to as the enhanced cooling mechanism). For it to operate in the absence of hyperons, the proton fraction in the core must exceed a critical thresh- 107 old of∼ 11−15%, however (Lattimer et al., 1991). Otherwise, the modified Urca cycle (the standard cooling mechanism), which is six orders of magnitude slower than the direct cycle, with its two branches - the proton branch and the neutron branch, and the slowest of all processes - the nucleon-nucleon bremsstrahlung, fa- cilitates the cooling. Alternatively, in the presence of meson condensates or quarks other fast neutrino cooling processes, similar to the nucleon direct and modified Urca cycles, become available. For a given EOS, the cooling behavior of low-mass stars is considerably differ- ent from that of their high-mass coevals. The cooling in low-mass NSs proceeds mainly via the modified Urca process in their outer cores. Thus, low-mass NSs are much hotter than the high-mass ones that allow the enhanced cooling mech- anism to operate in their inner cores in the absence of superfluidity (Levenfish & Haensel, 2007). At temperatures less than that of the density dependent neutron and proton critical temperatures, T < Tcn(ρ) and Tcp(ρ), superfluidity in neutrons and su- perconductivity in protons appears in the dense core. As a result, all nucleon neutrino emission channels are blocked and cooling proceeds via neutrino emis- sion by the formation of Cooper pairs (see for e.g. (Yakovlev et al., 1999) for a detailed discussion); Such pairing is also possible in quark matter. Several mi- croscopic models of baryon superfluidity suggest the occurrence of singlet-state neutron superfluidity in the inner crust for ρ . ρ0 and triplet-state neutron super- fluidity for ρ > ρ0 in the core. Due to the small percentage of protons in the core, only singlet-state proton superfluidity can be established. 5.2 The Magnificent Seven Nearby isolated cooling neutron stars are particularly important for confronting cooling models with observations. However, detection of thermal radiation from their surfaces is complicated by the non-thermal component that originates in the magnetosphere and/or the presence of polar hot spots that may dominate the thermal spectrum of the star. Nearby isolated NSs were discovered by Einstein, 108 ROSAT, and ASCA space telescopes (Becker & Pavlov, 2002). These were further observed with the extremely sensitive and high resolution high-energy space tele- scopes Chandra and XMM-Newton. Among all the discovered objects, the most interesting are the seven radio-quiet thermally emitting isolated NSs (a.k.a. the magnificent seven (M7)) discovered in the ROSAT all-sky survey (RASS) (see for e.g. (Haberl, 2007) for a review). These objects radiate predominantly in X-rays with high X-ray to optical flux ratios, fX/fopt > 104−5. Their soft X-ray spectra is reasonably well fit by an absorbed blackbody-like spectrum with kT . 100 eV and a hydrogen column density nH ∼ 1020 cm−2, indicating small distances d ∼ few × 100 pc. That the thermal emission is coming from majority of the stellar surface is confirmed by the small pulse fractions . 20% of the X-ray light curves. Initially, it was speculated that these objects may be a revived population that is accreting from the interstellar medium (ISM) and, as a result, powering their X-ray luminosities (e.g. Ostriker et al. 1970). Later it was discovered that three sources among the magnificent seven had high proper motions making ac- cretion from ISM highly unlikely (Haberl, 2005). Spin periods ranging from 3 - 12 s have been measured for all but one (RX J185635-3754) of the M7 objects (see for e.g. (Mereghetti, 2011), Table 1 and 2, and references therein). This in conjunction with the measured spin-down rates, Ṗ ∼ 10−14 − 10−13 s s−1, yields an estimate of the polar magnetic field strengths Bp ∼ 1013 G and the characteristic spin-down ages tsd ∼ 106 years. Measure- ments of the high proper motions of three of the M7 objects and their association thus established to the Sco OB2 complex comprising the Gould Belt yield slightly smaller kinematic ages. This discrepancy strongly suggests that the spin-down ages are overestimates and the M7 objects in reality are much younger (see dis- cussion below). Broad absorption features have been observed in the spectra of these thermal sources suggesting proton or heavy ion cyclotron resonances in the atmosphere caused by magnetic fields whose strengths are consistent with that es- timated from timing (van Kerkwijk & Kaplan, 2007). No radio emission has been detected from any of these objects. Haberl (2005) argues that the absence of radio 109 emission is possibly linked to narrow widths of the radio beam originating much higher in NS’s magnetosphere. 5.2.1 Spin-Down Ages: Poor Age Estimators According to the standard magnetic dipole model of pulsars (Shapiro & Teukol- sky, 1983), a pulsar with a magnetic field of strength Bp at the pole and initial angular frequency Ωi spins down over time by emitting magnetic dipole radiation. The time evolution of the angular frequency obtained from classical electrody- namics is Ω(t) = Ωi ( 1 + 2Ω2i ‖Ω̇0‖ Ω30 t )−1/2 (5.1) where quantities with subscript 0 are the measured values. For Ω(t) = Ω0, and with the assumption that Ωi  Ω0 one finds the spin-down age of the pulsar tsd = Ω0 2‖Ω̇0‖ (5.2) The above assumption that newly born NSs have much smaller spin periods than their older counterparts can be easily confirmed by studying some of the younger rotation powered PSRs. The prime examples are the Crab and Vela PSRs with rotation periods of 33 ms and 89 ms and spin-down ages of 1.3 × 103 yr and 1.1 × 104 yr, respectively (e.g. Lyne & Graham-Smith 2006). Also, such short initial periods (Pi ∼ 1 ms) are demanded by the magnetar model, where the generation of magnetar-like fields in NSs occurs under the operation of an efficient dynamo within the first few tens of seconds after their birth (Duncan & Thompson, 1992). The angular frequency evolution given in Eq. 5.1 implicitly assumes that none of the other physical characteristics of the pulsar vary over time. This may not be the case and, in general, the spin-down law (Lyne & Graham-Smith, 2006) written as the following can be allowed to include variation of Bp, the moment of inertia 110 I , and the angle between the rotation axis and the magnetic dipole axis α. dΩ dt = −κ(t)Ω(t)n (5.3) where κ is usually assumed to be a constant and n is the braking index which assumes a value of 3 for magnetic dipole braking. Any change in κ with time naturally yields ages of pulsars that are in conflict with their spin-down ages; Generally, the spin-down age should only be taken as a rough estimate to aid in quick calculations. 111 l ll l l l llll ll l n n n H H H H l n H SNR Age (yrs) S p in -d ow n A ge (y rs ) 103 104 105 106 103 104 105 106 107 Young pulsar CCO Magnetar Figure 5.1: Comparison of the spin down ages and the SNR ages of some isolated NSs. The red line shows where the two ages are coincident. 112 Table 5.1: Properties of isolated NSs with SNR or massive star cluster associations Object P (s) Ṗ (10−15 s s−1) tsd(10 3 yrs) tsnr(103 yrs) SNR Young Pulsars 0531+21 0.033 421 1.3 0.93 Crab Nebulaa 1509-58 0.150 1540 1.5 1.5 MSH 15-52a 0833-45 0.089 125 11 18− 31 Vela XYZa 1853+01 0.267 208 20 > 1.3 W44a 0540-69 0.050 479 1.7 ∼ 0.60 SNR 0540-693b 1610-50 0.232 493 7.4 ∼ 3.0 Kes 32c 1338-62 0.193 253 12 ∼ 32.5 G308.8-0.1d 1757-24 0.125 128 15 & 70 G5.4-1.2e 1800-21 0.134 134 16 15− 28 W30f 1706-44 0.102 93 17 ∼ 5.6 G343.1-2.3g 1930+22 0.144 58 40 ∼ 486 G57.1+1.7h 2334+61 0.495 192 41 ∼ 10 G114.3+0.3i 1758-23 0.416 113 59 33 W28j CCOs RX J0822.0-4300 0.122 < 8 > 220 3.7± 0.4 Pupppis Ak 1E 1207.4-5209 0.424 < 25 > 27000 ∼ 7 G296.5+10.0l CXOU J185238.6+004020 0.105 8700 > 8000 5.4− 7.5 Kes 79m Magnetars CXOU J164710.2-455216 10.6 920 200 (4± 1)× 103 Westerlund 1?n 1E 1841-045 11.77 41000 4.8 ∼ 2 Kes 73o 1E 2259+586 6.98 480 230 15 CTB 109p SGR 0526-66 8.05 65000 3.4 ∼ 5 N49q a(Braun et al., 1989); b(Seward et al., 1984); c(Caraveo, 1993; Vink, 2004); d(Caswell et al., 1992); e(Blazek et al., 2006); f(Finley & Oegelman, 1994); g(McAdam et al., 1993); h(Kovalenko, 1989); i(Furst et al., 1993); j(Velázquez et al., 2002); k(Gotthelf & Halpern, 2009); l(Gotthelf & Halpern, 2007); m(Halpern et al., 2007); n(Muno et al., 2006)? Westerlund 1 is a massive star cluster; o(Vasisht & Gotthelf, 1997); p(Sofue et al., 1983); q(Klose et al., 2004) 113 An independent age estimate is provided by the age of the associated super- nova remnant (SNR) or massive star cluster for younger objects. Establishing such an association for older NSs may prove to be difficult since SNRs fade away in ∼ 60 kyr, and in the same time, due to natal kicks (∼ 500 km s−1), NSs may move significantly far away from their birth sites (Frail et al., 1994). In Table 5.1, we list the spin-down ages and the estimated SNR ages for young pulsars (tsd < 105 yrs), central compact objectsr (CCOs), and magnetars (SGRs and AXPs) along with their timing properties. This data is plotted in Fig. 5.1 and it is clear that for most NSs the spin-down age is a poor age estimator. The objects that have spin-down ages smaller than the SNR ages can be explained by having a braking index less than the canonical value, n < 3. An excellent example supporting this notion is the Vela pulsar which has a very small braking index n = 1.4 ± 0.2 estimated from an impressive 25-year long observation (Lyne et al., 1996), albeit under the assumption that κ is still a constant. This yields a spin-down age of 25.6 kyr, making it appear more than twice as old as its age inferred from the standard magnetic braking scenario. This result is well supported by the estimated age of the Vela SNR (tSNR ∼ 18 − 31 kyr) (Aschenbach et al., 1995). The disparity in the braking index from the spin-down law can be attributed to a time varying κ, such that the measured breaking index is given as nobs = Ω̈Ω Ω̇2 = n+ κ̇Ω κΩ̇ (5.4) On the other hand, for objects that appear to be younger than their spin-down ages, it has been argued that the magnetic moments decrease in strength over time (See Fig. 5.2; Lyne et al. 1975). There are three main mechanisms by which magnetic fields can decay in iso- lated NS (Goldreich & Reisenegger, 1992), namely Ohmic dissipation, ambipolar rCentral compact objects are a group of young radio-quiet but X-ray bright NSs that were found near the centers of their respective SNRs. They are characterized by thermal X-ray emission well modeled by a two component blackbody model with Tbb ' 2−7×106 K and smaller emitting areas Rbb ∼ 0.3− 5 km (Becker, 2009) 114 diffusion, and Hall drift. The time scale over which the field decays substantially due to these processes, as given below (Heyl & Kulkarni, 1998) tohm ∼ 2× 1011 ( L5 T8 )2( ρ ρnuc )3 yr (5.5) tsamb ∼ 3× 109 ( L5T8 B12 )2 yr (5.6) tirramb ∼ 5× 1015 T 68B 2 12 yr + tsamb (5.7) tHall ∼ 5× 108L 2 5T 2 8 B12 ( ρ ρnuc ) yr (5.8) determines the dominating process at different stages in the evolution of an iso- lated NS. In the above, the length scale of the magnetic field L5 is given in units of 105 cm, its strength B12 in units of 1012 G, and the core temperature T8 in units of 108 K. Out of the given three processes, Ohmic dissipation and ambipolar diffusion are responsible for the decay of magnetic energy; the Hall mechanism conserves magnetic flux. Ohmic dissipation arises in conducting fluids and leads to the diffusion of magnetic field lines with respect to the charged species of the partially ionized plasma. It operates both in the inner core and the crust at a rate that depends on the density and is independent of the magnetic field strength. In ambipolar diffusion, the magnetic field lines move through the plasma along with the charged particles with respect to the neutral species. This process is decomposed into an irrotational (curl-free) and solenoidal (divergence-free) part where both processes operate on very different time scales. Since the Hall drift mechanism does not dissipate magnetic energy, it may act to transport magnetic flux from the inner crust to the outer crust. Ohmic dissipation is much faster in the outer crust and can observe enhanced rates due to the Hall cascade effect, whereby magnetic energy is transported to smaller scales that is much more conducive to Ohmic decay. An important consequence of field decay is that it leads to an overestimation of the real age of the NS. Following the analysis of (Colpi et al., 2000), the decay 115 t (yrs) C h ar ac te ri st ic A ge τ (y rs ) 1 10 102 103 104 105 106 107 10 102 103 104 105 106 107 108 Ambipolar diffusion I-mode Ambipolar diffusion S-mode Crustal Hall cascade Figure 5.2: Change in the spin-down or characteristic age of an isolated NS due to various magnetic field decay mechanisms, namely the ambipo- lar (I)rrotational (a = 0.01, α = 5/4) and (S)olenoidal (a = 0.15, α = 5/4) modes , and the Hall cascade (a = 10, α = 1) (also see Eq. 5.6, 5.7, 5.8). The solid black line denotes t = τ , meaning no field decay. This assumes an initial field strength B0 = 1016 G and period P0 = 1 ms. 116 in magnetic field strength due to the aforementioned processes can be written as dB dt = −aB1+α (5.9) where α and a are constants corresponding to the three field decay modes; note this does not include Ohmic dissipation since it is independent of the magnetic field. This gives the evolution equation for the field strength, where the initial field is B0 B(t) = B0 [ 1 + aα ( B0 1013 G )α( t 106 yr )]−1/α G (5.10) Combining the above with the spin-down law given in Eq. 5.1, also written as PṖ = bB2 with b ≈ 3 when B is expressed in units of 1013 G, P in s, and Ṗ in s per 106 yr, yields the period evolution equation P (t) = ( P 20 + 2b a(2− α) ( B0 1013 G )2−α × [ 1− ( 1 + aα ( B0 1013 G )α( t 106 yr ))(α−2)/α])1/2 s(5.11) where P0 is the initial period. We can write Eq. 5.2 in terms of the period and the magnetic field strength, such τ(t) = P (t)2 2bB(t)2 (5.12) This relation is plotted in Fig. 5.2 for the different field decay processes. The effect of field decay at late times is apparent from the divergence of the character- istic age from the real age. 117 5.3 True Age Estimates of Isolated Neutron Stars The M7 objects don’t have any SNR or massive star cluster associations. There- fore, ages for these objects have been derived from their P and Ṗ measurements. In addition, since these are nearby objects (d . 500 pc), and due to their large proper motions, kinematics ages became a possibility and have been estimated for only three of the group members (see Table 5.2). In this case, one finds that the spin-down ages are larger by 3 − 10 times the kinematic ages. Accurate age estimates are extremely important in determining the cooling behavior of isolated NSs. Overestimated ages used to fit model cooling curves can obscure the deter- mination of the true thermal state of these objects. 5.3.1 Age Estimates from Population Synthesis In the following, we estimate the true ages of the M7 members by a method that is motivated by another method devised by Schmidt (1968) and then applied by Huchra & Sargent (1973) to calculate the luminosity function of field galaxies. The original idea is implemented as follows. An apparent magnitude limited sam- ple is first obtained and it is assumed that the objects in the sample, field galaxies for instance, are distributed uniformly in Euclidean space such that the luminos- ity function is independent of the distance. Then, to each object an accessible volume V (M) is assigned (Avni & Bahcall, 1980). This volume depends on the absolute magnitude of the object and gives a measure of the volume surveyed for a given object with an absolute magnitude M . Basically, V (M) is the max- imum volume in which the object would be detected had its apparent magnitude been equal to the limiting magnitude of the survey. The whole sample is then divided into bins of size dM with objects having absolute magnitude in the range [M − dM/2,M + dM/2]. The luminosity function for this bin is estimated by adding the inverse of the accessible volumes for each object in that bin Φ(M)( Mpc−3 mag−1) = ∑ i 1 Vi(M) (5.13) 118 This method provides a non-parametric way of estimating the luminosity func- tion and it exactly reproduces the true luminosity function within statistical errors (Hartwick & Schade, 1990). Furthermore, this is a very general method which re- lies on only one underlying assumption that the objects are distributed according to their intrinsic brightnesses. 5.3.2 RASS and Population Synthesis In the following, we develop a slight variant of the Schmidt (1968) estimator to calculate the true ages of the members of the M7 family. The method we develop cannot be non-parametric as the distribution of NSs, unlike that of galaxies over large scales, is not uniform. Since the progenitors of NSs mainly reside in the arms of a spiral galaxy, and for a natal kick velocity of, say ∼ 500 km s−1, the NSs only travel a distance of ∼ 0.05 kpc from their birth sites within ∼ 105 years. This is small compared to the scale height of the thin disk ∼ 0.3 kpc (Binney & Merrifield, 1998). As a result, the Schmidt (1968) estimator cannot be used here. Instead, we look at the NS progenitor population and calculate the normalization for each M7 member by counting the number of massive OB stars that are found in the accessible volume Vmax for that object. The population synthesis method is given in our earlier study (Gill & Heyl, 2007), and essentially requires the assumption of the luminosity function of massive OB stars in the galaxy (Bahcall & Soneira, 1980), and the distribution of HI, which we model as a smooth exponential disk both radially and vertically (Foster & Routledge, 2003). The ROSAT All-Sky Survey ROSAT, short for ROentgen SATellite, was an X-ray space observatory launched on June 1, 1990. Equipped with a Position-Sensitive Proportional Counter (PSPC) with a 2◦ field of view, ROSAT produced an unprecedented all sky survey in a period of 6 months. The RASS covered the whole sky in the ∼ 0.12 − 2.4 keV (∼ 100Å − 6Å) energy range at a typical limiting count rate of 0.015 cts s−1 (see Hünsch et al. (1999) for more details). The survey was conducted between 119 July 1990 and January 1991 with additional observations carried out in February and August of 1991. ROSAT had an orbital period of ∼ 96 min with 40 min nighttime per orbit and it scanned the whole sky in great circles perpendicular to the line connecting the Earth and the Sun (see figure 4.1). Over a period of 6 months, while the Earth finished half of its revolution about the Sun, ROSAT accumulated X-ray data over the whole sky with 92% coverage for a count rate of 0.1 cts s−1, and has yield the most complete and sensitive survey of the X-ray sky (Voges et al., 1999). As the symmetry axis of ROSAT pointed towards the ecliptic poles, regions on the sky falling at the ecliptic registered much less exposure time (typically ∼ 400 s) than that at the ecliptic poles (∼ 40,000 s); see Fig. 5.3 for orbital geometry and exposure map. The exposure map displays scans with fairly large gradients of exposure time with numerous regions where the telescope did not register any time at all. The two isolated dark spots at the top and bottom of the exposure map mark the position of the ecliptic poles. With a rotation rate of Figure 5.3: Left: Geometrical arrangement of RASS; Right: RASS expo- sure map; x-axis represents Right Ascension (RA) (deg) with 0◦ at the bottom-right corner; y-axis represents Declination (DEC) (deg) with 0◦ in the middle. Two dark spots receiving the maximum exposure time represent the ecliptic poles. (Image Credit: ROSAT mission and Max-Planck-Institut für extraterrestrische Physik) 1◦ per day, any source at the ecliptic had a maximum detection frequency of 2 per survey. Conversely, sources at the ecliptic poles were detected with the highest 120 frequency. Besides the aforementioned timing issues, many other problems were experienced during the survey, an account of which is given in Belloni et al. (1994) with an elaborate discussion on the source detection process. Method All M7 objects were discovered in the RASS, which provides a perfect flux- limited sample for our study. Next, we calculate the weights for each object in the sample by simulating the RASS and finding the total number of massive OB stars in the volume Vmax, such that the estimated age is given in terms of the typical age of their progenitors tOB. τi ∼ tOB j≤i∑ j=1 Nj N j,OB (5.14) where Ni is the number of NSs in a small absolute magnitude bin of size dM centered at Mi. Since the sample is of marginal size, Nj = 1 in this case. Then, 1/Nj,OB gives the number of massive OB stars per ith object in the sample. The ages of NS progenitors are highly uncertain and are usually obtained by estimating the main sequence turn-off ages of the massive star cluster to which the NS may be associated (see for e.g. (Smartt, 2009) for a review). Typical ages of ∼ 3−15 Myr have been estimated for the progenitors of NSs and magnetars. Figer et al. (2005) report the age of the cluster of massive stars, containing three Wolf- Rayet stars and a post main-sequence OB supergiant, associated to the magnetar SGR 1806 − 20 to be roughly 3.0 − 4.5 Myr. Also, Muno et al. (2006) report an age of 4 ± 1 Myr for the cluster Westerlund 1 which seems to be the birth site of another magnetar CXOU J164710.2− 455216. In yet another study, Davies et al. (2009) find the age of the cluster associated to the magnetar SGR 1900 + 14 to be 14 ± 1 Myr. Since the spin-down ages of magnetars are much smaller (∼ 103−4) than that of the clusters, the notion that the cluster age reflects the age of the progenitor, under the assumption of coevality of its members, is a valid one. In the case of SGR 1806− 20 and CXOU J164710.2− 455216, both groups find that 121 the progenitor must be a massive star with M > 40M, except in the last study where the progenitor of SGR 1900 + 14 is claimed to be a lower mass star with initial MS mass of 17±2M. Notwithstanding this last result, it has been claimed that magnetars may be the progeny of only sufficiently massive stars (M & 25M) (Gaensler et al., 2005) that would, otherwise, have resulted in the formation of a black hole. Although the members of the M7 family are endowed with fields an order of magnitude higher than the normal radio pulsars, they are not magnetars and can be argued to be the descendants of progenitors not much more massive than that of the normal radio pulsars. In that case, it is expected that the progenitor age tOB will be considerably longer in comparison to that of magnetar progenitors. From Eq. 5.14 the ages of the sample objects are proportional to tOB, thus sig- nificant uncertainty in the progenitor age will yield erroneous ages. To circumvent this problem, we normalize tOB such that the estimated age of RX J1856.5− 3754 matches its kinematic age tkin = 0.46±0.05 Myr, which has been measured using its high proper motion (Tetzlaff et al., 2011). This yields a slightly larger age for the progenitor tOB ∼ 55 Myr. Over the last few years, two new candidates have been added to the M7 group. The first object, 1RXS J141256.0 + 792204 dubbed Calvera (Rutledge et al., 2008), was actually cataloged in the RASS Bright Source Catalog (Vo- ges et al., 1999) for having a high X-ray to optical flux ratio FX/FV > 8700 (see Table 5.2 for properties). However, its large height above the Galactic plane (z ≈ 5.1 kpc), requiring a space velocity vz & 5100 km s−1, presents a chal- lenge for its interpretation as an isolated cooling NS like the M7 members (see Rutledge et al. (2008) for a detailed discussion). Also, recent X-ray observa- tions of Calvera done with XMM-Newton space telescope found unambiguous evidence for pulsations with period P = 59.2 ms (Zane et al., 2011). The authors of this study argued that Calvera is most probably a CCO or a slightly recycled pulsar. The uncertainty in its nature (see Halpern (2011)) doesn’t warrant its in- clusion into our sample of radio-quiet isolated NSs. The second object 2XMM J104608.7 − 594306 (Pires et al., 2009), discovered serendipitously in an XMM- 122 Newton pointed observation of the Carina Nebula hosting the binary system Eta Carinae, appears to be a promising candidate (see Table 5.2 for properties). This object was not detected in the RASS due to its larger distance (≈ 2.3 kpc, based on its association to the Carina nebula) and higher neutral hydrogen absorption column density (NH = 3.5± 1.1× 1021 cm−2). Therefore, the accessible volume Vmax is the ROSAT surveyed volume plus the additional volume probed by the XMM-Newton’s pointed observation. In Table 5.2, we provide all the relevant data on the sample objects including the spectral fit parameters that were used to simulate the RASS to obtain Vmax. We take the calculated ages and plot them against the effective temperatures ob- served at infinity in Fig. 5.4. The errorbars on the ages correspond to the maxi- mum of the difference in ages obtained due to uncertainties in the Tbb, NH (these two parameters are covariant), and the distance. For comparison, we also plot some cooling curves from (Yakovlev & Pethick, 2004), where the non-superfluid (No SF) model for a 1.3M cannot explain the data. Other model curves are shown for NS cooling behavior if proton superfluidity in the core is taken into ac- count ( see (Yakovlev & Pethick, 2004) for more details on the 1P and 2P models). 123 Table 5.2: Properties of nearby thermally emitting isolated NSs with estimated statistical ages τ . The uncer- tainties in the age estimates are due to uncertainties in the inferred temperature, HI column density, and distance. Object P Ṗ D Tbb NH Fx NOB τ (s) (10−14 s s−1) (pc) (eV) (1020 cm−2) (Myr) RBS 1223a 10.31 11.20 ≥ 525b 118± 13 0.5− 2.1 4.5 368+32−28 0.062± 0.005 2XMM J104608.7?c - - 2000 117± 14 35± 11 0.097 1702+668−477 0.076± 0.007 RX J1605.3 + 3249d - - 325− 390e 86− 98 0.6− 1.5 1.15 181+414−23 0.20± 0.15 RBS 1774f 9.43 4.1± 1.8 390− 430g 92+19−15 4.6± 0.2 8.7 588+138−75 0.24± 0.15 RX J0806.4− 4123h 11.37 5.5± 3.0 240± 25i 78± 7 2.5± 0.9 2.9 211+26−27 0.35± 0.15 RX J0720.4− 3125j 8.39 7.01 330+170−80 k 79± 4 1.3± 0.3 11.5 278+42−35 0.43± 0.15 RX J0420.0− 5022l 3.45 2.8± 0.3 350m 57+25−47 1.7 0.69 713+99−89 0.47± 0.15 RX J1856.5− 3754n 7.06 2.97± 0.07 161+18−14o 57± 1 1.4± 0.1 14.6 604+256−115 0.50± 0.15 Fx(10 −12 erg cm−2 s−1) - The absorbed X-ray flux in the ROSAT energy band (0.12− 2.4 keV). ? 2XMM J104608.7− 594306 a(Schwope et al., 1999); b(Posselt et al., 2007); c(Pires et al., 2009); d(Motch et al., 1999); e(Posselt et al., 2007); f(Zampieri et al., 2001); g(Posselt et al., 2007); h(Haberl et al., 1998); i(Motch et al., 2008); j(Haberl et al., 1997); k(Kaplan et al., 2007); l(Haberl et al., 1999); m(Posselt et al., 2007); n(Walter et al., 1996); o(Kaplan et al., 2007) 124 5.3.3 Statistical Ages Vs The True Ages The ages of isolated NSs have been estimated using different methods, namely from the spin-down law, cooling models, and kinematics. The method we pro- pose in this study to estimate the true ages of these objects has only been applied, in its original form, to calculate the luminosity function of field galaxies and white dwarfs. The method itself is purely a statistical one, for which the underlying as- sumption is that the objects in the sample follow a Poisson distribution (see for e.g. (Felten, 1976)). The number of objects in the sample is a random variable and the accessible volume that each object occupies is also a random variable. The important question to ask here is how good of an estimate of the true age is the statistical age. What is the inherent statistical error associated to this method of predicting ages? Consider the youngest object, RBS 1223, which is also the hottest among the eight isolated NSs. The kinematic age of RBS 1223 has been found based on its association to possible OB associations and young star clusters to be ∼ 0.5 Myr (Tetzlaff et al., 2010), although this age should only be regarded as the upper limit due to large uncertainties. The statistical age of RBS 1223 that we find in our study, given that the sample only contains eight such objects, is much smaller ∼ 0.062± 0.005 Myr, with uncertainties corresponding to system- atic errors. The statistical error is of course much larger than the systematic one. Since the discovery of an object in a given volume follows Poisson statistics, the error scales as 1/ √ N where N is the sample size with ages ti ≤ tN . Therefore, the error in the age of the first object is±t1 where t1 is its statistical age. Likewise, the error in the age of the eighth object is ±t8/ √ 8. Evidently, one needs a much larger sample to reduce the statistical errors to that comparable to the systematic errors. 5.4 Discussion The ages of isolated NSs are primarily important for constraining their inner struc- ture. In addition, they can be useful for accurately determining the birthplace of 125 l l l l l l ll Log t (yr) L o g T (K ) R B S 12 23 J 1 04 60 8 .7 -5 9 43 0 6 R X J 1 60 5. 3 + 3 24 9 R B S 17 74 R X J 08 06 .4 -4 1 23 R X J 07 20 .4 -3 12 5 R X J 04 20 .0 -5 02 2 R X J 18 56 .5 -3 7 54 3.0 3.5 4.0 4.5 5.0 5.5 6.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0 No SF 1.3M 1P 1.36M 1P 1.35M 2P 1.49M 2P 1.35M Figure 5.4: Statistical cooling curve of the nearby thermally emitting iso- lated NSs with systematic uncertainties only; the data is listed in Ta- ble 5.2. We plot sample theoretical cooling curves from Yakovlev & Pethick (2004) for comparison. 126 the object, if its proper motion is known. Similarly, if the SNR-NS association has been made, then the age of the object can reveal its space velocity and the kinematics of the SNR. The knowledge of ages of such objects is also useful for population synthesis models which rely on the spatial and velocity distributions, and the birthrates of NSs. In this study, we propose a statistical method to esti- mate the true ages of the ROSAT discovered sample comprising the M7. We then use the age estimates along with the derived spectral temperatures to compare the data with some cooling models. The strength of this technique, as discussed earlier, lies in obtaining a larger sample of coolers. With only eight objects, the statistical error is indubitably much larger. The statistics can be improved by locating more of these objects in the disk of the Galaxy. In a recent population synthesis study, (Posselt et al., 2010) find that young isolated NSs, that are both hot and bright, with ROSAT count rates below 0.1 cts s−1 (the ROSAT bright source catalog had a limiting count rate of 0.05 cts s−1) should be located in OB associations beyond the Gould belt. They also remark on the possibility of finding new isolated cooling NSs by conduct- ing yet another careful search of such objects in the RASS and using the recently published XMM-Newton Slew Survey (Esquej et al., 2006). However, they note that ROSAT observations are incapable at locating the isolated sources with suf- ficient spatial accuracy, such that many optical counterparts can be found in its large positional error circle. On the other hand, although XMM-Newton is much more sensitive, albeit with strong inhomogeneities, and can probe deeper into the Galactic plane, the slew survey only covers 15% of the sky currently. Searching the RASS for new isolated NSs appears to be a promising avenue, however, it is unlikely that any new sources will be identified. What is needed at the moment is another all-sky survey that is able to surpass ROSAT in both sensitivity and posi- tional accuracy. In that regard, the upcoming eROSITA mission (Cappelluti et al., 2011) shows a lot of promise, and its planned launch in 2013 makes it very timely. The X-ray instrument eROSITA will be part of the Russian Spectrum-Roentgen- Gamma (SRG) satellite, equipped with seven Wolter-I telescope modules with an 127 advanced version of the XMM-Newton pnCCD camera at its prime focus. The telescope will operate with an energy range of 0.5−10 keV, a field of view (FOV) of 1.03◦, an angular resolution of 28′′ averaged over the FOV, and a limiting flux of ∼ 10−14 erg cm−3 in the 0.5 − 2 keV energy range and ∼ 3 × 10−13erg cm−3 in the 2 − 10 keV energy range. The all-sky survey will reach sensitivities that are ∼ 30 times that of the RASS where the entire sky will be scanned over a period of four years. The greater sensitivity offered by eROSITA to distant or highly absorbed isolated NSs promises an increase in their numbers by an order of magnitude (Posselt et al., 2010). Currently, with only 8 objects and large statistical uncertainties, it is hard to constrain any theoretical cooling models. However, it is hoped that as more and more isolated thermal emitters are discovered the reduction in such uncertainties will lead to strong constraints on not only the EOS of NSs but also the dominant mode of field decay by comparison of their characteristic and statistical ages. 128 Chapter 6 Future Work The study of strongly magnetized plasmas near compact objects is absolutely cru- cial to the understanding of the generation of high energy radiation. It is com- plicated by the inherent complexity of the system, poorly understood particle dy- namics and radiation transfer mechanisms. The following projects will examine some key features of magnetized plasmas and their influence in generating high energy emission in compact objects, particularly neutron stars. The secondary goal of this research is to exploit the extreme conditions that exist near neutron stars to uncover the properties of the hypothetical axion and axion-like particles (ALPs). 6.1 Axion/ALP Physics with the Double Pulsar The discovery of the axion, like the Higgs boson, will have far reaching con- sequences. It will help answer the long standing question of why nature is not CP symmetric at the fundamental level. Among the four forces of nature, only the weak force has been known to violate CP. Experiments strongly suggest that strong interactions are CP-invariant. However, the QCD lagrangian necessarily violates CP resulting in a mismatch between theory and experiments. One way to avoid this problem is by introducing axions into the Standard Model of ele- 129 mentary particles (see Chapter 3). From astrophysical considerations, we find that the axion and ALP must be a light (ma . 10−2 eV) and very weakly interacting particle. As a consequence, it could very well be the mysterious cold dark matter particle that pervades the whole Universe. If found, its implications in cosmology will be extensive. Efforts to discover the axion/ALP in laboratory experiments have not been very fruitful. So far, they have only yielded upper limits on its coupling strength to photons (gaγγ) in a small mass range. To survey the remaining broad ma− gaγγ parameter space, much higher sensitivity is required. On the other hand, astro- physical sources, such as the double pulsar, isolated neutron stars, and magnetic white dwarfs, are ideal for such an experiment. They have the added benefit of being endowed with extremely high magnetic fields and larger system sizes; Both are inaccessible in terrestrial experiments and needed to probe weaker coupling strengths. The double pulsar system J0737-3039 offers a unique testbed for testing fun- damental physics. It is the only known system that comprises of two radio pulsar; Pulsar A is a recycled millisecond pulsar with a spin period of PA = 22.7 ms, and pulsar B is a slowly spinning young pulsar with PB = 2.77 s (see Kramer & Stairs (2008) for a review). The nearly edge-on orbital inclination (i ≈ 88.69◦) is what makes this system so remarkable and perfectly suited to study the axion/ALP. As the pulsars spin, the radio beam of pulsar A passes through the magnetosphere of pulsar B. This effect has been observed in the radio signal as an eclipse of pulsar A when it is at conjunction. In addition, both pulsars have also been ob- served by XMM-Newton and Chandra to emit pulsed x-rays. The non-thermal x-ray radiation is most likely emitted as a broad fan-beam by pulsar A. Due to the edge-on orbital inclination, it too must propagate through pulsar B’s magneto- sphere. Photon-ALP oscillations in pulsar B’s magnetosphere will cause a drop in intensity and change in polarization in the x-ray and radio emission of pulsar A at conjunction. The magnitude of this change directly correlates to gaγγ for a given ma. 130 This project entails the modeling of pulsar B’s rotating magnetosphere (Lyu- tikov & Thompson, 2005) and the integration of photon modes as they propagate through it. The calculations are very similar to our earlier work (Gill & Heyl, 2011) with added effects due to the strong magnetic field (B ≈ 1.2 × 1012 G) and strong gravity. A similar study (Dupays et al., 2005) has looked at the inten- sity modulation due to the same effect, albeit using the incorrect ALP parameters. Moreover, their work ignored any plasma and vacuum effects which are absolutely important when photon mode evolution in strong fields is considered. Thus, a re- evaluation of the radiation transfer effects is warranted. Finally, no polarization data exists for x-ray sources due to the lack of x-ray polarimeters in space. Al- though, at the moment only radio observations (see Yuen et al. (2012)) can be used to probe ALP parameters, predictions can still be made of polarization changes in the x-ray emission from the double pulsar. Future polarimeter equipped X-ray missions observing strong magnetic field NSs will allow for better constraints. 6.2 High Energy Radiation Processes in Magnetar Bursts Magnetars are strongly magnetized neutron stars (NSs) with fields in excess of 1014 G. Unlike the rotationally powered radio and x-ray pulsars, magnetars are powered by the dissipation of their ultra-strong magnetic fields. In comparison to the much commonly observed radio pulsars, active magnetars constitute only ∼ 10−20% of the total neutron star population in our Galaxy (Gill & Heyl, 2007). The magnetar class consists of two distinct NS sub-populations: the soft gamma- ray repeaters (SGRs) and the anomalous x-ray pulsars (AXPs) (see Mereghetti (2008) for a review). Both show flux variability on many different timescales and have been observed to burst repeatedly. The focus of this study, however, is on the extremely luminous bursts given off by SGRs - the giant flares (see Chapter 4). According to the TD95 model (Thompson & Duncan, 1995), magnetar bursts occur due to the sudden release of magnetic energy. In Gill & Heyl (2010) (see Fig. 4.5), we examined several mechanisms that trigger giant flares and how mag- 131 netic reconnection can explain the impulsive nature of these bursts and the resul- tant particle acceleration. Ultimately, a stupendous amount of energy E ∼ 1044 erg is injected into the magnetosphere which inevitably leads to the formation of an optically thick photon-electron-positron plasma that is trapped between field lines in the form of a plasmoid. It is not clear how this hot cloud of plasma cools in the next 200−400 s, the time over which the intensity of the giant flares returns to quiescence after the onset. Also, detailed understanding of what radiative pro- cesses are dominant in shaping the emergent spectrum and what are the different sources of opacity is lacking. Several studies have already investigated in detail the radiation transport properties and the emerging spectrum and polarization of the atmospheres of strongly magnetized neutron stars (e.g. (Ho & Lai, 2003; Ho et al., 2003)). None have treated the time dependent evolution of the photon-pair plasma that gives rise to the super-Eddington luminosity in giant flares. In the atmosphere codes, generally the spectrum and polarization is obtained by solving the equation of radiative transfer in a magnetized plasma using the nor- mal mode approach. However, this procedure is time independent and does not trace the electron-positron distributions in a self-consistent manner as the mag- netic trap shrinks in size. A time dependent coupled partial differential equation needs to be solved to determine the evolution of photon and particle distributions. The initial conditions for this work will be drawn from our earlier work on mag- netar bursts (Gill & Heyl, 2010). A brief description of the processes involved in giant flares that are to be simulated in my code is as follows. High energy blackbody photons emanating from the heated neutron star surface are injected into the magnetic trap. At the same time power-law electrons, accelerated by the current layer formed in magnetic reconnection, are injected at high Lorentz num- bers. This leads to the comptonization of the blackbody photons and production of primary electron-positron pairs at the pair-production threshold. This distribution of pairs not only cools by giving off synchrotron and bremsstrahlung radiation, but also undergoes pair annihilation, producing more photons in the system. The process then repeats in a self-consistent way and the plasma quickly evolves to 132 an optically thick state. Allowance is made for the trapped radiation to escape. This depends critically on the assumed particle scattering model, such as resonant cyclotron scattering. Understanding the escape of radiation from the trap also provides deep insights into how the plasma fireball is cooled radiatively. The evolution of the photon and pair distributions as a function of energy and time are given by three coupled partial differential equations (Coppi, 1992). An implicit finite difference scheme, that also conserves the total energy of the system, is employed to calculate the photon and particle spectra at any given time. To understand the emission properties of hard x-rays and gamma-rays in giant flares, Fermi observations are desperately needed. 6.3 Mode Coupling of Bernstein Waves to Radiation in Electron-Positron Plasmas In our earlier work (Gill & Heyl (2009); see Fig. 2.5), we have examined the nature of B-waves in homogeneous, relativistic pair plasmas. This work was the first to investigate the dispersion relation of Bernstein waves in a fully relativistic framework. However, astrophysical plasmas, particularly in NS magnetospheres where the field itself is non-uniform, are generally inhomogeneous. Therefore, the study of Bernstein waves in inhomogeneous plasmas is needed. Again, this work will be original and lay out the foundations of the kinetic theory of Bern- stein waves in relativistic inhomogeneous pair plasmas. The phenomena of sub- pulse drifting can then be explained by coupling these modes to the electromag- netic modes. So far, mode coupling of Bernstein waves has only been studied for electron-ion plasmas where it has proven to be an essential tool for core plasma heating and current drive. However, no work has been done towards coupling Bernstein modes to radiation in pair plasmas. For mode coupling to occur the plasma needs to have a density gradient. So, the first project will be the extension of my previous work on the fully relativistic treatment of Bernstein waves in homogeneous pair plasmas (Gill & Heyl, 2009). The proposed study will draw upon the numerical sophistication gained in my ear- 133 lier work to calculate the dispersion curves in inhomogeneous pair plasmas. The second project will be the study of mode coupling of B-waves to electromagnetic modes in pair plasmas. 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Notice that there cannot be any equilibrium electric field E0, as the plasma particles would arrange themselves in such a way so that the field is shorted out. fs(r,v, t) = f0s(v) + f1s(r,v, t) (A.2) B = B0 +B1 exp i(k·r−ωt) (A.3) E = E1 exp i(k·r−ωt) (A.4) 152 Substitution of the above quantities then yield the modified Vlasov equation, where we have ignored terms of second order in the perturbed quantities, ∂f1s ∂t + v · ∇rf1s + qs ms (E1e i(k·r−ωt) + v × {B0 +B1ei(k·r−ωt)}) · ∇vf1s = − qs ms (E1 + v ×B1)ei(k·r−ωt) · ∇vf0s (A.5) The left hand side of the above equation is simply the time derivative of the per- turbed distribution function df1s dt = − qs ms (E1 + v ×B1)ei(k·r−ωt) · ∇vf0s (A.6) Integrating the above equation over the unperturbed particle trajectories, also known as the method of characteristics, parameterized by the time t′, and using Faraday’s law (∇× E1 = −∂B1/∂t) to replace B1 with E1, we find f1s(r,v, t) = − qs ms ∫ t −∞ [( E1 + v ′ × { k× E1 ω }) ei(k·r ′−ωt′) · ∇v′f0s ] dt′ (A.7) Here the cross product rule is useful, A × (B × C) = B(A · C) − (A · B)C, which yields f1s(r,v, t) = − qs ms ∫ t −∞ [ E1 · ( I { 1− v ′ · k ω } + v′k ω ) ei(k·r ′−ωt′) · ∇v′f0s ] dt′ (A.8) In the above equation, the term v′k forms a dyadic product and I is the identity matrix. The current induced by the perturbed distribution is defined as J = ∑ s qs ∫ vf1s(r,v, t)d 3v = ∑ s σs · E1 (A.9) where summation is done of different particle species s, electrons and positrons in this case. The following expressions will be kept general and the sum over different species will be done at the very end. The electric conductivity can now 153 be written as the following σ = −q 2 m ∫ v (∫ t −∞ ( I { 1− v ′ · k ω } + v′k ω ) ei(k·r ′−ωt′) · ∇v′f0dt′ ) d3v (A.10) Expressing the dielectric tensor in terms of the conductivity tensor gives ε = ε0 ( I− σ iωε0 ) (A.11) = ε0 [ I+ q2ei(k·r−ωt) iωmε0 ∫ (∫ ∞ 0 Te−i(k·̃r−ωτ)dτ ) d3v ] (A.12) where r̃ = r− r′ and T = v ({ 1− v ′ · k ω } I+ v′k ω ) · ∇v′f0 (A.13) Next, we work out the tensor components of T by writing v and k in component form. The equation of motion for particles on their unperturbed trajectories is given by dv′ dt′ = q m (v′ ×B0) = ωc(v′ × ẑ) (A.14) From the boundary condition, r′(t′ = t) = r,v′(t′ = t) = v, we find v′x = vx cosωcτ − vy sinωcτ = v⊥ cos(ωcτ + φ) (A.15) v′y = vx sinωcτ + vy cosωcτ = v⊥ sin(ωcτ + φ) (A.16) v′z = vz = v‖ (A.17) The velocity equations can be integrated to find the 0th-order particle trajectories x′ = x+ v⊥ ωc (sinφ− sin{ωcτ + φ}) (A.18) y′ = y − v⊥ ωc (cosφ− cos{ωcτ + φ}) (A.19) z′ = z − v‖τ (A.20) 154 Since v′⊥ and v ′ ‖ are constants of the motion and they coincide with the unprimed speeds at t′ = t, such that v′⊥ = v⊥ and v ′ ‖ = v‖. The equilibrium distribution is uniform in space and time, and only depends on the velocity of the particles f0 = f0(v⊥, v‖), such that ∇v′f0 = v ′ x v⊥ ∂f0 ∂v⊥ x̂+ v′y v⊥ ∂f0 ∂v⊥ ŷ + ∂f0 ∂v‖ ẑ (A.21) The exponential factor can now be expressed as e−i(k·̃r−ωτ) = exp [ −i ( kx v⊥ ωc (sin{ωcτ + φ} − sinφ) +ky v⊥ ωc (cosφ− cos{ωcτ + φ}) + (kzv‖ − ω)τ )] (A.22) Since we are investigating the Bernstein mode, with waves propagating perpen- dicular to the equilibrium magnetic field B0 and the electric field parallel to the wave vector, then kz = k‖ = 0 and k×E1 = 0 for an electrostatic mode. Without any loss of generality we can let k⊥ = kx and ky = 0, Tij = vi ({ 1− k⊥v⊥ cos(ωcτ + φ) ω } ∂f0 ∂v′j + v′j ω ∂f0 ∂v⊥ k⊥ cos(ωcτ + φ) ) (A.23) The matrix elements for T are given below Txx = v⊥ ∂f0 ∂v⊥ cosφ cos(ωcτ + φ) (A.24) Tyy = v⊥ ∂f0 ∂v⊥ sinφ sin(ωcτ + φ) (A.25) Tzz = v‖ [ ∂f0 ∂v‖ − k⊥ ω cos(ωcτ + φ) { v⊥ ∂f0 ∂v‖ − v‖ ∂f0 ∂v⊥ }] (A.26) Txy = v⊥ ∂f0 ∂v⊥ cosφ sin(ωcτ + φ) (A.27) Tyx = v⊥ ∂f0 ∂v⊥ sinφ cos(ωcτ + φ) (A.28) 155 Txz = v⊥ cosφ [ ∂f0 ∂v‖ − k⊥ ω cos(ωcτ + φ) { v⊥ ∂f0 ∂v‖ − v‖ ∂f0 ∂v⊥ }] (A.29) Tzx = v‖ ∂f0 ∂v⊥ cos(ωcτ + φ) (A.30) Tyz = v⊥ sinφ [ ∂f0 ∂v‖ − k⊥ ω cos(ωcτ + φ) { v⊥ ∂f0 ∂v‖ − v‖ ∂f0 ∂v⊥ }] (A.31) Tzy = v‖ ∂f0 ∂v⊥ sin(ωcτ + φ) (A.32) Also, if the distribution function is isotropic then, v⊥ ∂f0 ∂v‖ = v‖ ∂f0 ∂v⊥ (A.33) The integral over the unperturbed particle trajectories can now be carried out and we show the steps involved in the calculation for the Txx component above.∫ ∞ 0 Txxe −i(k·̃r−ωτ)dτ = v⊥ ∂fo ∂v⊥ cosφ ∫ ∞ 0 cos(ωcτ + φ) × exp [ i k⊥v⊥ ωc {sinφ− sin(ωcτ + φ)}+ iωτ ] (A.34) The cosine term can be included into the exponential by noticing that i k⊥v⊥ d dτ exp [ i k⊥v⊥ ωc {sinφ− sin(ωcτ + φ)} ] = cos(ωcτ + φ) exp [ i k⊥v⊥ ωc {sinφ− sin(ωcτ + φ)} ] (A.35) We can write the exponentials using Bessel functions by the following identity eib sin θ = ∞∑ n=−∞ Jn(b)e inθ (A.36) 156 Let b = k⊥v⊥/ωc and assuming Im(ω) > 0 then∫ ∞ 0 Txxe −i(k·̃r−ωτ)dτ = v⊥ ∂f0 ∂v⊥ cosφ ∞∑ n,m=−∞ m b Jn(b)Jm(b)e i(n−m)φ × ∫ ∞ 0 ei(ω−mωc)τdτ (A.37) = i ωc k⊥ ∂f0 ∂v⊥ cosφ ∞∑ n,m=−∞ mei(n−m)φ (ω −mωc)Jn(b)Jm(b) Next, we carry out the integral over d3v = v⊥dv⊥dφdv‖. In order to do the integral over φ we use another Bessel identity ∞∑ n=−∞ Jn(b) ∫ 2pi 0 cosφei(n−m)φdφ = 2pi mJm(b) b (A.38) This yields∫ ∞ 0 Txxe −i(k·̃r−ωτ)dτ = i ω2c k2⊥v⊥ ∂f0 ∂v⊥ ∞∑ m=−∞ 2pi m2 (ω −mωc)J 2 m(b) (A.39) The infinite sum is numerically expensive to calculate and can be avoided by using the Newberger sum rule ∞∑ n=−∞ Jn(z)Jn−m(z) a− n = (−1)mpi sin pia Jm−a(z)Ja(z)m > 0 (A.40) Then for a = ω/ωc we get∫ ∞ 0 Txxe −i(k·̃r−ωτ)dτ = i 2piωc k2⊥v⊥ ∂f0 ∂v⊥ [ pia2 sin pia Ja(b)J−a(b)− a ] (A.41) 157 The dielectric tensor components can now be written in terms of integrals over v⊥ and v‖, expressed in Fourier space ε(ω, k): εxx = ε0 [ 1 + ∑ s 2piq2 k2⊥msε0 ∫ { pias sin pias Jas(bs)J−as(bs)− 1 } ×∂f0s ∂v⊥ dv‖dv⊥ ] (A.42) εyy = ε0 [ 1 + ∑ s 2piq2 ωmsε0 ∫ { pi sin pias J ′as(bs)J ′ −as(bs) + as b2s } × v 2 ⊥ ωcs ∂f0s ∂v⊥ dv‖dv⊥ ] (A.43) εzz = ε0 [ 1 + ∑ s 2piq2 ωmsε0ωcs ∫ piJas(bs)J−as(bs) sin pias ×v⊥v‖∂f0s ∂v‖ dv‖dv⊥ ] (A.44) εxy = ε0 [∑ s i 2piq2 msε0k2⊥ ∫ { pibs sin pias Jas(bs)J ′ −as(bs) + 1 } ×∂f0s ∂v⊥ dv‖dv⊥ ] (A.45) εyx = ε0 [ −i ∑ s 2piq2 msε0k2⊥ ∫ { pibs sin pias Jas(bs)J ′ −as(bs) + 1 } ×∂f0s ∂v⊥ dv‖dv⊥ ] (A.46) εxz = ε0 [∑ s 2piq2 k⊥ωmsε0 ∫ { pias sin pias Jas(bs)J−as(bs)− 1 } ×v⊥∂f0s ∂v‖ dv‖dv⊥ ] (A.47) εzx = ε0 [∑ s 2piq2 k⊥ωmsε0 ∫ { pias sin pias Jas(bs)J−as(bs)− 1 } 158 ×v‖∂f0s ∂v⊥ dv‖dv⊥ ] (A.48) εyz = ε0 [ − ∑ s i 2piq2 k⊥ωmε0 ∫ { pib sin pia Ja(b)J ′ −a(b) + 1 } ×v⊥∂f0s ∂v‖ dv‖dv⊥ ] (A.49) εzy = ε0 [∑ s i 2piq2 k⊥ωmε0 ∫ { pib sin pia Ja(b)J ′ −a(b) + 1 } ×v⊥∂f0s ∂v‖ dv‖dv⊥ ] (A.50) Double integration over the parallel and perpendicular velocities can be avoided by using spherical coordinates such that the volume element reads v⊥dv⊥dv‖dφ = v2dv sin θdθdφ (A.51) For a relativistic distribution function it is convenient to have all velocity depen- dent quantities expressed in terms of the momentum. εxx = ε0 [ 1 + 4piq2m2 k2⊥mε0 ∫ { piγa sinpiγa Jγa(ξ)J−γa(ξ)− 1 } × γ p⊥ ∂f0(p) ∂p⊥ p2 sin θdpdθ ] (A.52) εyy = ε0 [ 1 + 4piq2 ωmε0 ∫ { pi sin piγa J ′γa(ξ)J ′ −γa(ξ) + a γb2 } ×p⊥ ωc ∂f0(p) ∂p⊥ p2 sin θdpdθ ] (A.53) εzz = ε0 [ 1 + 4piq2 ωmε0ωc ∫ p‖ ∂f0(p) ∂p‖ piJγa(ξ)J−γa(ξ) sin piγa p2 sin θdpdθ ] (A.54) εxy = ε0 [ i 4piq2m2 mε0k2⊥ ∫ γ p⊥ ∂f0(p) ∂p⊥ p2 sin θdpdθ ] (A.55) εxz = ε0 [ 4piq2m k⊥ωmε0 { piγa sin piγa Jγa(ξ)J−γa(ξ)− 1 } 159 ×∂f0(p) ∂p‖ p2 sin θdpdθ ] (A.56) εyz = ε0 [ −i 4piq 2m k⊥ωmε0 ∫ ∂f0(p) ∂p‖ p2 sin θdpdθ ] (A.57) The dielectric components obey the Onsager symmetry relations, such that εxy = −εyx (A.58) εxz = εzx (A.59) εyz = −εzy (A.60) Summation over electrons and positrons in a pair plasma leaves the dielectric tensor components unchanged in form but doubles them in magnitude (Jν(x) and J ′ν(x) are even (odd) for even (odd) ν)∑ s εij,s = 2εij − ε0 (A.61) To calculate the dispersion relation, we need to find the dielectric response func- tion which is defined by the following and picks out the εxx component in this case, (ω,k) = k · (ω,k) · k = 0 (A.62) The distribution of particles in momentum space is described by the relativistic Maxwellian f0(p) = (4pim 3c3)−1 η K2(η) e−ηγ (A.63) where the temperature dependence is encoded into the parameter η = mc2/kBT . The derivative of the distribution function with respect to momentum gives ∂f0 ∂p? = − η 2 4pim5c5K2(η) p? γ e−ηγ (A.64) 160 where the p? → p⊥ or p‖. Integral over the polar angle θ can be carried out first in closed form. For example by letting β = k⊥p qB , we get εxx = ε0 [ 1− ω 2 p k2⊥ η2 m3c5K2(η) ∫ ∞ 0 p2e−ηγ (A.65) × ∫ pi 0 { piγa sin piγa Jγa(β sin θ)J−γa(β sin θ)− 1 } sin θdθdp ] Here the following Bessel function integral identity is useful∫ pi 0 sin θJa(b sin θ)J−a(b sin θ)dθ = 2 sinpia pia 2F3 ( 1 2 , 1; 3 2 , 1− a, 1 + a;−b2 ) (A.66) By plugging this into the integral and using the following substitutions to make the integral dimensionless p̂ = p/mc, k̂⊥ = k⊥c/ωc, ω̂ = ω/ωc, ω̂p = ωp/ωc, β = β̂ = k̂⊥p̂, a = ω̂ we find εxx = ε0 [ 1− 2ω̂ 2 pη k̂2⊥n0 { η K2(η) ∫ ∞ 0 p̂2e−ηγ ×2F3 ( 1 2 , 1; 3 2 , 1− γω̂, 1 + γω̂;−β̂2 ) dp̂− 1 }] (A.67) εyy = ε0 [ 1− ω̂ 2 p n0 η2 ω̂K2(η) ∫ ∞ 0 p̂ γ e−ηγ { 2ω̂ k̂2⊥p̂2 + 4 3ω̂ 2F3 ( 1 2 , 2; 5 2 ,−γω̂, 2 + γω̂;−β̂2 ) + 4 3ω̂ 2F3 ( 1 2 , 2; 5 2 , γω̂, 2− γω̂;−β̂2 ) − 8k̂ 2 ⊥p̂ 2 15γ2ω̂3 2F3 ( 3 2 , 2; 7 2 , 2− γω̂, 2 + γω̂;−β̂2 )} dp̂ ] (A.68) εzz = ε0 [ 1− 2ω̂ 2 p 3ω̂2n0 η2 K2(η) ∫ ∞ 0 p̂2e−ηγ ×2F3 ( 1 2 , 1; 5 2 , 1− γω̂, 1 + γω̂;−β̂2 ) dp̂ ] (A.69) 161 εxy = ε0 [ −i 2ω̂ 2 p k̂2⊥n0ηK2(η) e−η{2 + η(2 + η)} ] (A.70) 162


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