Black Hole Binaries in Our Galaxy: Understanding Their Population and Rapid X-Ray Variability by Kavitha Arur, M.Phys, M.Sc a Di

Home , Anna Watts, Daryl Haggard, Rudy Wijnands

Black hole binaries in our Galaxy: Understanding their population and rapid X-ray variability

Kavitha Arur, M.Phys, M.Sc

A Dissertation

Physics

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulﬁllment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

Approved

Thomas J. Maccarone Chairperson of the Committee

Benjamin Owen

Simone Scaringi

James F. Steiner

Beth Thacker

Mark Sheridan Dean of the Graduate School

May 2020 c 2020, Kavitha Arur Texas Tech University, Kavitha Arur, May 2020

ACKNOWLEDGMENTS

First I would like to thank my advisor, Prof. Tom Maccarone for the many years of support and encouragement, especially when it came to voicing my ideas. Thank you for teaching me so much, and for always believing in me. I could not have asked for a better mentor! I would also like to thank my committee members for participating in my defense and approving this dissertation.

This work would not have been possible without the extensive HEASARC data archives, the Astrophysics Database System (ADS), arxiv.org, endless cups of tea and Stackoverﬂow who have helped me and many other PhD students get their research done. I am also incredibly grateful to the wider X-ray astronomy community, who have been incredibly encouraging throughout my PhD. I also want to thank Adam, Phil and Jack for all their help during the job application process.

Coming to TTU has been one of the best decisions I have made, and this is largely because of the wonderful people I have met. A huge thank you to Chris Britt, Paul Sell, Denija Crnojevic, Lennart van Haaften, Michael Holcomb, Rob Coyne, Ra Inta, Liliana Rivera Sandoval, Dario Carbone, Robert Morehead and Anna McLeod for always cheering me on, and always being willing to help me out. An extra spe- cial thank you to the most amazing group of ladies (Denija, Bonnie, Beth, Kris Coyne, Kristen Holcomb, Jess, Pam, Lili, Anna, Nipuni) who kept me going even when things were hard. Thanks for all the laughs, hugs, advice and chocolate. You guys are the best! I also want to thank Manuel, Deven, JP, Matteo, Arvind, Binod, and everyone else in this amazing TTU group - thank you for making Lubbock truly feel like home during these PhD years.

I also have to mention the various D&D groups (both in Lubbock and in the UK) that I am fortunate to have been a part of - thank you for all the amazing memories and for keeping me entertained and sane!

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Huge thanks to my parents, grandparents and other family members for their endless support. Thanks also to Lavanya, Abishek, Samiksha, Sindhuja and Sarah for keeping in touch and cheering me on from thousands of miles away. I would also like to express my gratitude to my fellow TANASTRO members, who have always encouraged me to study the stars.

And ﬁnally, thank you Paul for your endless love and support. I am so glad that we are a team, and that we could do this together.

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STATEMENT OF WORK

Chapters 2,3 and 4 of this dissertation have been published in the Monthly No- tices of the Royal Astronomical Society as articles co-authored by Kavitha Arur and Thomas Maccarone.

Thomas Maccarone and Kavitha Arur worked together to deﬁne the scope of the project and on the interpretation of the results. Kavitha Arur performed the data analysis and wrote up the results, with comments from Thomas Maccarone.

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TABLE OF CONTENTS

Acknowledgments ...... ii Statement of work ...... iv Abstract ...... ix List of Figures ...... xi 1. Introduction ...... 1 1.1 Black Hole X-ray Binaries ...... 1 1.2 Population of BHXBs ...... 3 1.3 Properties of BHXBs ...... 5 1.3.1 Spectral Properties ...... 5 1.3.2 Timing Properties ...... 10 1.4 Quasi-Periodic Oscillations ...... 11 1.4.1 QPO Models ...... 12 1.5 Non-linear Variability ...... 14 1.6 Higher Order Statistics ...... 15 1.6.1 Bispectrum ...... 16 1.6.2 Biphase ...... 16 1.6.3 Bicoherence ...... 17 1.6.4 Calculating the bicoherence ...... 19 1.7 RXTE ...... 20 1.8 Outline of this dissertation ...... 21 2. Selection Effects on the Orbital Period Distribution of Low Mass Black Hole X-ray Binaries ...... 22 2.1 Abstract ...... 22 2.2 Introduction ...... 22 2.3 Data set and correction of selection effects ...... 25 2.3.1 Galaxy model and data set ...... 25 2.3.2 Extinction of Optical Counterpart ...... 26 2.3.3 Detection of the X-ray outburst ...... 27 2.3.4 Probability of outburst detection ...... 27

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2.3.5 Implied Orbital Period Distribution using Observational Data ...... 31 2.4 Monte Carlo Simulation ...... 33 2.4.1 Generation of simulated sources ...... 33 2.4.2 Optical counterpart detection ...... 35 2.4.3 Probability of an X-ray outburst and its detection . . . . . 36 2.4.4 Implied Orbital Period Distribution using Simulated Data 36 2.4.5 Effect of mass transfer on the companion ...... 36 2.5 Discussion ...... 38 2.6 Conclusions ...... 41 3. Nonlinear Variability of Quasi-Periodic Oscillations in GX 339-4 ... 42 3.1 Abstract ...... 42 3.2 Introduction ...... 42 3.3 Bicoherence ...... 44 3.4 Data Overview ...... 45 3.4.1 QPO Classiﬁcation ...... 46 3.5 Results ...... 46 3.5.1 Bicoherence Patterns ...... 46 3.5.1.1 The ‘hypotenuse’ pattern ...... 47 3.5.1.2 The ‘web’ pattern ...... 47 3.5.2 Type B QPOs ...... 48 3.5.3 Type A QPOs ...... 49 3.5.4 Evolution during state transition ...... 51 3.5.5 Reconstructing the QPO waveforms ...... 54 3.6 Discussion ...... 56 3.6.1 Physical Interpretation ...... 56 3.6.1.1 Is this a reasonable value for τ?...... 59 3.7 Conclusions ...... 60 4. A Likely Inclination Dependence in the Non-linear Variability of Quasi Periodic Oscillations from Black Hole Binaries ...... 61 4.1 Abstract ...... 61 4.2 Introduction ...... 61

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4.3 Statistical Methods ...... 63 4.3.1 Bicoherence ...... 63 4.3.2 Biphase ...... 64 4.4 Observations and Data Reduction ...... 65 4.4.1 Data Sample ...... 65 4.4.2 Data Reduction and Analysis ...... 66 4.5 Results ...... 67 4.5.1 Bicoherence Patterns ...... 67 4.5.1.1 Hypotenuse ...... 67 4.5.1.2 Cross ...... 67 4.5.1.3 Web ...... 68 4.5.2 Evolution of type C QPOs ...... 69 4.5.2.1 Low inclination sources ...... 69 4.5.2.2 High inclination sources ...... 71 4.5.2.3 Hardening phase of the outburst ...... 71 4.5.2.4 Intermediate inclination sources ...... 73 4.5.2.5 Quantifying the inclination dependence ...... 75 4.5.3 Statistical Signiﬁcance ...... 75 4.5.4 Type B QPOs ...... 77 4.6 Discussion ...... 77 4.6.1 Low inclination ...... 77 4.6.2 High inclination ...... 79 4.6.3 Variation of the optical depth with QPO phase ...... 80 4.6.4 Optical depth and geometry of the corona ...... 82 4.6.5 Caveats ...... 84 4.7 Conclusions ...... 85 5. Summary and Outlook ...... 87 5.1 Summary ...... 87 5.2 Future Work ...... 89 5.2.1 Black Holes ...... 89 5.2.2 Neutron stars ...... 91 5.2.3 Cataclysmic variables ...... 93

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5.2.4 Active Galactic Nuclei ...... 95 5.2.5 Three wave interactions in the solar wind ...... 96 5.2.6 Stochastic Gravitational Wave Background ...... 97 5.3 Current and Future Missions ...... 97 5.3.1 NICER ...... 97 5.3.2 STROBE-X ...... 98 5.3.3 eXTP ...... 99 5.4 Concluding remarks ...... 100 Appendix A. Details of Observations Analysed in Chapter 3 ...... 135 Appendix B. Details of Observations Analysed in Chapter 4 ...... 138 Appendix C. Dufﬁng Oscillator ...... 157

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ABSTRACT

One of the fundamental problems in the field of high energy astrophysics is constraining the formation and evolution of compact objects such as black holes and neutron stars in binary systems. The physics of accretion onto compact objects has profound impacts on our understanding of processes such as mergers of binary compact objects, production of type Ia supernovae and the formation of millisecond pulsars. On larger scales, the structure and star formation properties of galaxies can be influenced by accretion onto super-massive black holes. Progress in this field requires both a global study of the X-ray binaries as a population, as well as detailed characterization of individual systems. Population synthesis models predict thousands of black hole binaries in our Galaxy. However, only ∼20 such objects have been confirmed. One potential explanation for this discrepancy is the presence of observational biases that prevent a large fraction of these objects from being detected. The work presented in this dissertation shows that these selection effects can account for a signification portion of black hole binaries, especially those with short orbital periods, being undetected. Despite these promising results, large uncertainties still exist in our knowledge of the Galactic binary population as details of the accretion physics that govern binary evolution are still not well understood. Studies of rapid X-ray variability provide an excellent method of probing the geometry and the physical processes that dominate in the accretion disks, especially close to the central black hole. In order to properly interpret the observed variability, sophisticated timing analysis techniques are needed. In this dissertation, I present the first systematic analysis of a novel higher order timing analysis technique known as the bispectrum, to understand rapid X-ray variability from black hole binaries. Applied to the well studied and characterised source GX 339-4, my analysis revealed for the first time that the phase coupling between the Quasi- Periodic Oscillation (QPO) and the broadband noise varies as the back hole transitions from a power law dominated hard state to a black body dominated soft state, indicating the presence of a gradual change in the accretion disk that cannot be seen in the power spectrum. These results are interpreted in the context of the

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Lense-Thirring precession model as a moderate increase in the optical depth of a precessing corona. I have also demonstrated that information from the bispectrum can be used to reconstruct the QPO waveform, including phase information from multiple higher harmonics. Following the successful application of this technique to GX 339-4, this technique was used to analyse QPOs from a much larger dataset of 13 additional black hole binaries. This revealed a likely inclination dependence in the bispectral properties seen during the state transition, with low inclination (face on) sources exhibiting behaviour opposite to that seen from high inclination (edge on) sources. This indicates that the QPO has a geometric origin. The results of this study were consistent with the interpretation from GX 339-4, and was extended to include scattering timescales along different lines of sight to explain the inclination dependence. However, since detailed numerical modelling is required to conﬁrm this interpretation, other models (such as those invoking accretion ejection instabilities) cannot be ruled out. This work has highlighted the need for progress in modelling efforts in this direction.

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LIST OF FIGURES

1.1 Depiction of an X-ray binary...... 2 1.2 Figure showing the different pathways of binary stellar evolution. . .4 1.3 Figure of the shape of the spectra in different states...... 6 1.4 Figure of the various spectral components needed to for the energy spectra in various states...... 7 1.5 Hardness intensity diagram of GX 339-4...... 9 1.6 Example power spectra of Type A, B and C QPOs...... 13 1.7 Artist impression of a precessing inner flow illuminating the accretion disk...... 14 1.8 Figure showing the biphase for two different light curves...... 17 1.9 Figure illustrating the relation between the biphase and the shape of the lightcurve...... 18 1.10 Figure illustrating the relation between the biphase and the bicoherence...... 19 2.1 Observed orbital period distribution of black hole X-ray binaries. . . 23 2.2 Example plot showing detection of the optical counterpart...... 28 2.3 Implied period distribution after correcting for selection effects. . . . 33 2.4 Power law fit to predicted number of binaries...... 34 2.5 Plot showing decay of orbital period of binaries...... 35 2.6 Best power law fits for orbital periods drawn from two different distributions...... 37 3.1 Example of hypotenuse pattern in bicoherence...... 48 3.2 Example of web pattern in bicoherence...... 49 3.3 Example of pattern in bicoherence for Type B QPO...... 50 3.4 Example of pattern in bicoherence for Type A QPO...... 51 3.5 Figure showing change in the non-linearity during state transition. . 52 3.6 Ratio of bicoherence between cross and hypotenuse...... 54 3.7 QPO waveforms reconstructed using the biphase...... 55 4.1 Hypotenuse pattern from GX 339-4...... 68 4.2 Cross pattern from XTE J1550-564...... 69

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4.3 Cross pattern from H1743-322...... 70 4.4 Evolution of bicoherence in low inclination source GX 339-4...... 72 4.5 Evolution of bicoherence in high inclination source XTE J1550-564. . 73 4.6 Bicoherence during the hardening phase of outburst from GRO J1655- 40...... 74 4.7 Figure showing inclination dependence in bicoherence as a function of QPO frequency...... 76 4.8 Bicoherence of type B QPO from XTE J1817-330...... 78 4.9 Simulation of variation in optical depth with QPO phase for high inclination sources...... 80 4.10 Simulation of variation in optical depth with QPO phase for low inclination sources...... 81 5.1 Schematic of different emissivity profiles for a rotating BH, non- rotating BH and neutron star ...... 90 5.2 Plot of HF QPOs vs LF QPOs ...... 95 5.3 STROBE-X configuration ...... 98 C1 Evolution of bicoherence of Duffing oscillator...... 158

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CHAPTER 1 INTRODUCTION

1.1 Black Hole X-ray Binaries Black holes are one of the most exotic classes of objects in the Universe. Formed from the collapse of the most massive stars, they are the some of the densest types of object known. By their very nature, isolated black holes are extremely hard to detect1, let alone examine in detail. Due to their extreme gravitational pull, the escape velocity close to the black hole is the speed of light, making them invisible via electromagnetic radiation. Thus, black holes that are part of binary systems (i.e in orbit with a companion star) present the best way to study these systems.

Recently, the supermassive black hole at the centre of the galaxy M87 was imaged with radio interferometry using the Event Horizon Telescope [1], which can reach angular resolutions ∼25 micro-arcseconds. However, this will only be possible with M87 and the Milky Way (Sgr A*) without going into space. Additionally,

stellar mass black holes (i.e black holes with masses ∼ 20M ), which are the focus of this dissertation, cannot be imaged directly as their angular separation on the sky (∼ nano-arcseconds) is orders of magnitude smaller compared to the resolution that can be achieved with X-ray imaging (∼ arcseconds). Even with tech- nology such as X-ray interferometry, it would only be able to get to a resolution of micro-arcseconds [2], which is much lower than the required levels. For this reason, black hole X-ray binaries (BHXBs) are best studied using timing and spectroscopic methods.

These binaries are broadly divided into two categories. 1. Low Mass X-ray Bi- naries (LMXBs) where the companion star is a main sequence star of mass ≤8M and 2. High Mass X-ray Binaries (HMXBs) where the companion star has a higher mass >8M . This dissertation focuses on the category of LMXBs. In almost all of these objects, the accretion occurs via to Roche-Lobe overﬂow, where material from the outer layers of the companion star is gravitationally attracted to the black

1This can only be done through gravitational microlensing events that are not long-lived.

1 Texas Tech University, Kavitha Arur, May 2020 hole. As this material has angular momentum, it forms an accretion disk around the black hole. Magnetic forces (which are modelled as viscous forces) in this material cause it to lose angular momentum, plunging it beyond the event horizon. Signiﬁcant amounts of angular momentum can also be transported away from the disk through disk winds. Half of the loss in gravitational potential energy as the material falls in is radiated away, while the other half goes into rotational energy (that is probably advected into the black hole at the inner edge of the disk). While different regions of this system are visible in different wavelengths (see Figure 1.1), the X-ray emission originates from the regions close to the black hole. This makes X-ray studies of black hole binaries an excellent way to understand the physical processes that dominate in these regions.

Figure 1.1. An artist impression of an X-ray binary depicting the different components of the system and the wavelengths at which their emission dominates. Image from [3]

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As material is transferred into the black hole’s disk, it builds up in the accretion disk. When the temperature of this disk reaches ∼10,000K, the hydrogen ionization temperature, magnetically driven viscous instabilities in the disk cause a rapid increase in the mass accretion rate onto the black hole [4][5]. This results in an X-ray outburst, where the source brightens by many orders of magnitude. These outbursts typically last for weeks to months2 before gradually fading back down to its pre-outburst (quiescent) levels. In most cases, the outbursts recur on timescales of months to years.

1.2 Population of BHXBs The study of binary systems plays a vital role in astrophysics, as these systems are progenitors of sources such as millisecond pulsars (the best atomic clocks in the Universe) and accreting black holes. The process of binary evolution also has profound implications for the for placing constraints on the rates of events such as binary compact mergers (such as BH-BH or NS-NS) that could be detected by their gravitational wave signatures and for the production of heavy elements through neutron star mergers and type Ia supernovae.

One way to understand these binary systems is through binary population synthesis models [6], where the evolution of an ensemble of binary stellar systems is modeled given a set of assumptions. This provides a way to analyse the statistical properties of the binary population. Figure 1.2 shows an illustration of the different evolutionary stages of a binary system consisting of two massive hydrogen burning main sequence stars, and the different types of objects that could be produced as a result. However, large uncertainties are present in the model parameters. Factors such as the efﬁciency of the energy transfer during the common envelope evolution phase, or the role placed by kick velocities imparted to the compact objects (due to asymmetries in the supernova) are not well constrained. Due to the complex nature of these systems, the details of binary evolution, and even the initial conditions and the efﬁciency of magnetic braking are not well un-

2one notable exception is GRS 1915+105, which has been in outburst for approximately 40 years

3 Texas Tech University, Kavitha Arur, May 2020 derstood.

Figure 1.2. Figure showing the different pathways of binary stellar evolution. T is the typical time scale of an evolutionary stage, N is the estimated number of objects in the given evolutionary stage. From [7]

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Thus, the observation and characterization of the population of such systems is crucial for strengthening our understanding of the physical process that dominate binary evolution. Black hole X-ray binaries are an ideal class of system to study binary evolution as they are detectable from great distances and show variability on a variety of timescales. However, the presence of various selection effects could prevent an accurate estimate of the population of these sources in our Galaxy.

One area of particular importance is the dearth of BHXBs with orbital periods of less than a few hours. Due to the loss of angular momentum through magnetic braking (where charged particles in the rotating magnetic ﬁeld carry away angular momentum) or through gravitational radiation, binaries are expected to evolve to shorter periods on timescales of millions of years. But despite decades of monitoring the X-ray sky, less than 100 candidates have been identiﬁed, with only a handful being short period BHXBs. In Chapter 2, we explore details of how different selection effects could affect observations of these short period BHXBs.

1.3 Properties of BHXBs In addition to studying the population of BHXBs as a whole, much can be learned about binary evolution and accretion through the detailed characterization of individual BHXBs. In the following sections, an overview of the spectral and timing properties of BHXBs is presented.

1.3.1 Spectral Properties The X-ray energy spectrum of BHXBs shows variations in its shape, and is often classiﬁed into different states based on the components that can be used to ﬁt it (See e.g [8] for an in depth review). In the Low/Hard state (LHS), the spectrum is dominated by a cutoff power law distribution, thought to be a result of (inverse) Comptonization of disk blackbody photons by thermal electrons with temperatures of kTe ∼ 50-100keV and a Thomson scattering optical depth of τ ∼1 [9] (see also [10] and references within). The spectral index3 is around Γ ≤ 2, and a high energy cut off at energies >100-200keV is also seen in this state [11]. Addition-

3 dN −Γ deﬁned as dE ∝ E

5 Texas Tech University, Kavitha Arur, May 2020 ally, a reprocessing component due to Compton reﬂection accompanied by an iron emission line is also present [12] [13].

In the High/Soft state (HSS), the spectrum is well described by a (multi-temperature) thermal black body plus a weak power law spectral index of Γ > 2. No high energy cut off has been observed for the non-thermal electron tail (which is the origin of the power law) seen in these states [14][15]. The reflection component in these states is relativistically broadened [14]. In the intermediate states (hard intermediate and soft intermediate), both the thermal blackbody and Comptonization components are seen [16]. A comparison of the shape of the spectrum in different states along with the configuration of the accretion disk is shown in Figure 1.3. Figure 1.4 shows the components needed for the fitting of the spectrum in the soft, hard and intermediate states.

Figure 1.3. Left: Figure of the shape of the spectra in different states. VHS: Very High state, TDS: Thermal Dominant State, USS: Ultra Soft State, LHS: Very Hard State. Right: A schematic of the conﬁguration of the thermal disk (red) and the Comptonising corona (blue) in the different states. The soft state is sometimes divided into the ’thermal dominant state’, where some non-thermal component is present and the ’ultrasoft state’ which has minimal contribution from the non- thermal component. From [8]

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Figure 1.4. Components of fits to typical (a) hard and (b) soft state spectra of Cyg X- 1, and (c) an intermediate spectrum of XTE J1550–564. All the spectra are corrected for absorption, and solid curves give the total spectra. The (green) long dashes, and (red) dots correspond to the unscattered blackbody and Compton reflection/Fe fluorescence, respectively. The (blue) short dashes give the main Comptonisation component, which is due to scattering by (a) hot thermal electrons with kT ∼ 75 keV and ∼ 1, (b) nonthermal electrons of a hybrid distribution, and (c) the total hybrid distribution. The (cyan) dot-dashes show (a) the soft excess, seen in the hard state of Cyg X-1 and some other sources and (b) scattering by thermal electrons of the hybrid distribution. From [10]

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While it is accepted that this Comptonizing region (often referred to as the corona) is located close to the compact object, there is no general consensus on the geometry of the corona. Possible geometries for this region include quasi-spherical region inside a truncated, geometrically thin accretion, a relativistic jet launched from the inner accretion disk or an extended corona located above and below a thin disk. Additionally, it is also uncertain how and if the geometry of the corona varies with spectral state.

Over the course of the X-ray outburst, the spectral distribution of the the black hole changes with a change in the accretion rate. However, since a hysteresis effect is also present (see below), there is no one-to-one correlation between the X-ray luminosity and spectral shape. The evolution of these spectral states can be seen on a hardness-intensity diagram (Fig 1.5), where the hardness is the ratio of soft (low energy) X-rays to hard (high energy) X-rays. At the beginning of the outburst, the source is in the LHS (lower right of the plot) and rapidly increases in luminosity. As the source begins to harden, it moves towards the upper left corner of the plot as the thermal disc contribution increases relative to that of the Comptonizing region. During this transition, the luminosity stays roughly constant. After the source reaches the soft state, the luminosity of the source drops and the spectrum hardens gradually. The sources then moves back through the intermediate states to the hard state and back into quiescence.

The transition from the hard to soft state occurs at luminosities that are a factor of 3-5 higher than the transition from the soft to hard state, resulting in a hysteresis effect. The origin of this effect is unclear, however, it has been proposed that different amounts of Compton heating or cooling causes a shift in the accretion mode (i.e between a disk dominated soft state and a hard state with a strong Comptoniz- ing corona) at different luminosities [17].

Spectroscopy has long been the favoured method for understanding astronomical systems. For the study of stars, where the sources are spherical, thermal, optically thick and are not strongly variable spectroscopy is an ideal tool. Additionally,

8 Texas Tech University, Kavitha Arur, May 2020 while stars have many well resolved lines to study, lines from X-ray binaries are smeared. Thus, for non-spherical, non-thermal and highly variable systems, as is the case in the spectral states of BHBs as described above, other methods are also required to understand the underlying physics.

Figure 1.5. Top panel:Hardness intensity diagram (also referred to as the ’q- diagram’ or the ’turtle head diagram’) of GX 339-4. Bottom panel: Plot showing relation between the total variability of the light curve and the hardness of the spectrum. Blue stars show where type C QPOs are seen, and the red squares and green triangles represent type B and type A QPOs respectively. From [18]

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1.3.2 Timing Properties BHXBs vary on a large range of timescales from the long (∼ years) duration between X-ray outbursts to ﬂickering observed on sub-second timescales. Studies of variability are more easily done in the Fourier domain than in the traditional time domain. One simple, yet powerful method to determine the timescales of variations that dominate the light curve is through the use of the squared modulus of the Fourier transform, which is in astronomy is commonly referred to as the power spectrum (or power density spectrum).

The discrete Fourier transform Xj (j = −N/2, ...N/2 − 1) of an evenly spaced time series x(t) (consisting of N bins such that xn where n = 0, 1, 2...N − 1) is given by:

N−1 X (2πijn/N) Xj = xne (1.1) j=0 for each Fourier frequency j, and the power spectrum is given by:

2 Pj = |Xj| (1.2)

While different normalizations exist for the power spectrum, in this dissertation the Leahy normalization [19], where the Poisson noise level is set to 2 is used. The power spectra of BHXBs are dominated by the Poisson noise at high (≥ 10Hz) frequencies.

In the hard and intermediate states, broadband (or band-limited) noise is also observed. This broadband noise is thought to arise from the fluctuations in the mass accretion rate propagating in towards the black hole on the local viscous timescale [20] [21]. At low (>1 Hz) frequencies, a break is seen in the power spectrum. Below this break frequency, the power spectrum is flat. This low frequency break is thought to mark the viscous timescale at the outer edge of the inner accretion flow / corona, inside a truncated accretion disk. Another feature seen in the power spectrum are quasi-periodic oscillations, which are discussed in the follow-

10 Texas Tech University, Kavitha Arur, May 2020 ing section.

The spectral and timing properties of black hole binaries are strongly correlated (see Fig 1.5), with the harder spectral states showing higher levels of variability than the soft states. Recently, there has been a rise in the development and use of spectral-timing techniques that simultaneously use spectral and temporal information to understand the relationship between the various components that contribute to the X-ray emission. For a detailed overview of these techniques, we refer the reader to [22].

1.4 Quasi-Periodic Oscillations Quasi-Periodic Oscillations (QPOs), ﬁrst detected in 1985 [23], are seen as modest width peaks in the power spectrum of the X-ray ﬂux of black holes and neutron stars. QPOs have since been observed from a number of these sources, and fall broadly into the two categories of high frequency (>100Hz) and low frequency (<10Hz) QPOs based on the frequency of the QPO feature. Chapters 3 and 4 of this dissertation focus on low frequency QPOs which are further divided into 3 sub categories based on their observed phenomenology [24].

Type A QPOs, seen in the soft intermediate state, have low (<3% rms variability and occur at frequencies around 6-8Hz. They are typically broad (Q∼ 1-3), making them hard to detect in noisy data. These QPOs are the least common, and thereby the least well described among the 3 types of QPOs. Type B QPOs, also seen in the soft intermediate state, occur at frequencies between 0.5 and 6.5 Hz. They are more visibly peaked, with Q>6. These QPOs are often seen with a second harmonic, with a weak subharmonic also being present. In both cases, the QPO is accompanied by weak broadband noise levels.

Type C QPOs, seen in the hard intermediate state and some hard states, are the most commonly observed type. Occurring at frequencies between ∼0.1 and ∼10 Hz, these QPOs are narrow (Q ∼ 7-12), and have high rms variability. Type C QPOs are usually accompanied by high broadband noise levels as well as higher

11 Texas Tech University, Kavitha Arur, May 2020 harmonics. Example power spectra of type A, B and C QPOs are shown in Fig- ure 1.6. The frequency of the QPO also shows signiﬁcant evolution, with the QPO moving to higher frequencies from the end of the hard state through the intermediate states. Additionally, the frequency of type C QPOs has also been shown to correlate with that of the low frequency break [25].

While they have been extensively studied in the X-rays, type C QPOs have also been observed in the optical, ultraviolet and infra-red wavelengths. In GX 339-4, simultaneous optical and X-ray observations showed QPOs in both bands, with the optical QPO occurring at half the frequency of the X-ray QPO [26]. In the same source, an IR QPO was seen at half the frequency of its X-ray counterpart [27]. However, in XTE J118+480, multi-wavelength data showed QPOs at the same frequency in the optical, UV and X-ray bands [28]. These ﬁndings highlight the need for multi-wavelength timing studies of a larger number of BHXBs.

While type A and B QPOs are brieﬂy discussed in Chapters 3 and 4, due to the extensive amounts of data and their higher levels of rms variability, this dissertation mainly focuses on type-C QPOs.

1.4.1 QPO Models Despite being studied for decades, the physical origin of QPOs is widely de- bated. This has resulted in many different models being proposed to explain the QPO phenomenon. These models broadly fall into the two categories of intrinsic - where an intrinsic quasi-periodic variation of the luminosity causes QPOs - and geometric, where they are produced by variations in the geometry of the system and the actual power produced is not modulated.

Some intrinsic models that have been proposed include the Accretion Ejection Instability (AEI) model where the QPO is the eigen-frequency corresponding to spiral density structures in the disc caused by perturbations in a poloidal magnetic ﬁeld [29] [30], transverse standing waves trapped in the accretion disk [31], oscillating shock waves in the accretion disk [32], and oscillating pressure waves in the

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Figure 1.6. Example power spectra of Type A (top), B (middle) and C (bottom) QPOs seen from XTE J1859+226. From [24] corona [33].

On the other hand, inclination dependence on observed QPO properties such as the rms variability of the QPOs [34], phase lag behaviour [35] and non-linear variability (see Chapter 4) strongly suggest that QPOs are of geometric origin. A promising geometric model proposed to explain low frequency (type-C QPOs) is one that attributes the QPOs to Lense-Thirring precession [36] [37], a frame drag- ging effect that causes nodal precession when the equatorial plane of the black hole spin is misaligned with the accretion disk. The recent observation of the modulation of the iron line centroid energy [38], thought to be the result of a precessing corona illuminating the approaching and receding parts of a rotating accretion disk

13 Texas Tech University, Kavitha Arur, May 2020

(see Fig 1.7), strongly supports this model. For a more detailed review on the proposed models and the observational properties of QPOs, we refer the reader to [39].

Figure 1.7. Artist impression of a precessing inner ﬂow illuminating the accretion disk. Image credit: A.Ingram/ESA/ATG Medialab

Each of these above models can reproduce the observed periodicities in the power spectrum. Thus, in order to break the degeneracies between these models, it is necessary to go beyond the power spectrum and utilise more sophisticated analysis techniques.

1.5 Non-linear Variability In linear variability, the phases associated with different frequencies are uncorrelated, while in the case of non-linear variability correlations are present between the phases [40]. Most models that model variability assume linear and stationary processes. However, these simple scenarios are not often seen in nature.

Evidence of non-linearity has been seen from BHXBs in the form of a linear rms- ﬂux relation, as well as log normal ﬂux distribution which rules out additive shot noise models [41]. Work on Cygnus X-1 established that the light curves from this source were not time-reversible, and that coupling exists between variability components on a range of different time scales [42]. In the following section higher order statistics, such as the bispectrum, that are sensitive to such non-linearity of the lightcurves are introduced.

14 Texas Tech University, Kavitha Arur, May 2020

1.6 Higher Order Statistics The bispectrum is the ﬁrst in the order of polyspectra, and the Fourier domain equivalent of the three point correlation function. Mathematically, the bispectrum (and other polyspectra) can be represented using cumulants. Cumulants are a set quantities that describe the shape of a probability distribution. The ﬁrst cumulant of a process is simply its mean (µ), and the second cumulant is the variance (σ2). A Gaussian process is completely described by these two values, which provide the location of the peak and the spread of the probability distribution respectively. Thus the higher order cumulants (such as the skewness) of a Gaussian process are zero, and can be used to pick out non-Gaussian processes.

Signal processing is often performed in the Fourier domain, where the periodicities in the signal can be more easily identiﬁed than in the time domain. One can go from the time domain to the frequency domain using the Fourier Transform (FT). Polyspectra are calculated using the Fourier transform of the cumulants, i.e, the FT of the n-th order cumulant gives the n-th order cumulant spectrum or the polyspectrum.

The traditional power spectrum can be written as the DFT of the auto correlation 4 function c2(τ1), where τ1 is a discrete time lag .

∗ P (f1) = DFT [c2(τ1)] = E[X(f1)X (f1)] (1.3)

This can be extended to higher orders in a similar manner:

2 ∗ B(k, l) = DFT [c3(τ1, τ2)] = E[X(k)X(l)X (k + l)] (1.4)

3 ∗ T (k, l, m) = DFT [c4(τ1, τ2, τ3)] = E[X(k)X(l)X(m)X (k + l + m)] (1.5)

where DFT2 indicates double DFT and and DFT3 indicates triple DFT. B and T are referred to as the bispectrum and trispectrum respectively. The later chapters of this dissertation will focus mainly on the bispectrum. Current astronomical data sets probably do not allow for good computation of the trispectrum, and the reader

4 2 the zero lag auto correlation is the variance i.e c2(0) = σ

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is referred to [43] for more information on the trispectrum.

1.6.1 Bispectrum The bispectrum, originally developed for the study of ocean waves [44], has since also successfully been applied in a variety of different ﬁelds such as the study of speech patterns, brain waves and identifying defects in machinery. In astronomy, the bispectrum has been used to study the non-Gaussianities in the cosmic microwave background. However, the bispectrum is now beginning to emerge as a powerful technique to analyse X-ray time series, and has recently begun to provide new and exciting insights from existing data.

The bispectrum is a complex number, where the real component of the bispectrum probes the skewness if the underlying ﬂux distribution and the imaginary component probes the asymmetry of the waveform. The bispectrum B(k, l) thus contains information about the amplitude and the phase of three frequencies k, l and k + l, with the values being symmetric about the k, l axis.

1.6.2 Biphase As a complex quantity, the bispectrum can also be represented as an amplitude and a phase, known as the biphase (β). The biphase must be deﬁned over the full 2π interval, and contains important information of the shape of the underlying light curve. The power of the biphase really starts to be evident when two time series that would produce identical power spectra, but have different underlying shapes.

As an illustrative example, consider a pulsar-like time series, where a series of δ-functions appear periodically, with zero flux present at other times. This would have a strong positive flux skewness with no asymmetry. On the other hand, an eclipser-like time series, where periodic dips in the flux are present with zero flux present at other times, have a strong negative flux skewness with no time asymmetry. While the two cases produce identical power spectra, the pulsar-like time

16 Texas Tech University, Kavitha Arur, May 2020 series will have a biphase of 0 where power is present, while the eclipser-like time series will have a biphase of π.

Figure 1.8. Figure showing the biphase for two different light curves. Left: A pulsar like light curve and right: an eclipser like light curve. Both would produce identical power spectra, but have different biphase values. From [45]

As mentioned above, time asymmetry (or how reversible a time series is in a statistical sense) can also be probed. Light curves that have a sharp rise and slow decay will have a negative imaginary component, while light curves that have a slow rise and sharp decay will have a positive imaginary component. Fig. 1.9 il- lustrates the different values of the biphase for various types of light curves.

The variance of the biphase is given by:

1 1 Var β = ( − 1) (1.6) 2K b2(k, l) There will always exist a value for the biphase of a triplet of frequencies. How- ever, for the signals considered in this dissertation, the biphase is only meaningful in the regions of sufﬁciently high bicoherence.

1.6.3 Bicoherence A closely related statistic is the bicoherence, which is a measure of how consistent the biphase is between measurements. Multiple normalization schemes for the

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Figure 1.9. Figure illustrating the relation between the biphase and the shape of the lightcurve. Image credit: S.Scaringi bicoherence have been proposed, but the most widely used normalization follows the convention of Kim & Powers [46]:

|P X (k)X (l)X∗(k + l)|2 b2 = i i i P 2 P 2 (1.7) |Xi(k)Xi(l)| |Xi(k + l)| where the squared bicoherence b2 is bounded between 0 and 1. While this the most common, it must be noted that other normalizations [47] are more sensitive to non-linearity under certain conditions.

The bias and variance of the bicoherence are given by:

1 Bias b2 ≈ (1 − b2(k, l))2 (1.8) K 2b2(k, l) Var b2 ≈ (1 − b2(k, l))3 (1.9) K

18 Texas Tech University, Kavitha Arur, May 2020

These expressions give an indication of the trade-offs that must be considered when choosing the size of the segments for the Fourier transforms. While choosing larger segments give greater frequency resolution (which scales as the inverse of the segment length), a large number of segments is required to obtain a reliable estimate of the bicoherence.

Figure 1.10. Figure illustrating the relation between the biphase and the bicoherence. The top panel shows the case where the biphase is randomly distributed. This indicates a lack of phase coupling and the bicoherence will tend towards 0. The bottom panel shows the case where the biphase is identical between segments. This indicates the presence of phase coupling and the bicoherence will tend towards 1. From [48]

In order to use these higher order statistics, it must be emphasised that a key assumption here is that the underlying stochastic process is stationary (i.e the mean and variance are not changed by a time shift).

1.6.4 Calculating the bicoherence The algorithm to calculate the bicoherence of the processed observations in this dissertation is as follows:

1. Segment the light curve into K segments.

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2. Compute the DFT of each segment

3. Calculate the power spectrum and bispectrum/bicoherence of each segment

4. Average over the segments

5. Subtract the bias

In signal processing, window functions are often used to isolate periodic signals. Since this work focuses on quasi-periodic signals, window functions have not been applied to the signal. However, in the cases where a window function is required, a Hamming window has been shown to be most sensitive to peaks in the bicoherence [48].

1.7 RXTE The majority of the data utilised in this dissertation were taken using the Rossi X- ray Timing Explorer (RXTE). Launched by NASA in 1995, RXTE revolutionised the ﬁeld of X-ray timing over the course of its 16 year mission lifetime. This mission led to the discovery of pulsations from highly magnetised neutron stars known as magnetars [49], accreting millisecond pulsars [50] and black holes with a range of masses, as well as the discovery of phenomena such as kHz QPOs from neutron stars [51][52]. RXTE has not only been used to probe variability from stellar mass black holes (the focus of this work), but also from accreting supermassive black holes at the centre of galaxies.

One instrument of particular importance to timing studies is the Proportional Counter Array (PCA). The PCA, consisting of 5 Proportional Counter Units (PCUs), had a sensitivity of 0.1 mCrab5 and a timing resolution of up to 1 µsec. The data utilised in Chapters 3 and 4 of this dissertation were taken by the RXTE/PCA. The All Sky Monitor (ASM) on board RXTE fulﬁlled a role that was very different but complementary to the PCA. With a sensitivity of 30 mCrab and a large ﬁeld of view, the ASM scanned 80% of the sky every 90 minutes, which made it ideal for

51 Crab = 2.4 × 10−8 erg cm−2 s−1

20 Texas Tech University, Kavitha Arur, May 2020 the detection of X-ray outbursts from BHXBs. The simulations performed in Chap- ter 2 assume RXTE/ASM sensitivity levels as the threshold for discovery of X-ray outbursts.

Though RXTE was decommissioned in 2012, the extensive archive of high quality data enables new research, such as the following chapters of this dissertation, to this day.

1.8 Outline of this dissertation In Chapter 2, various selection effects that could impact the detection and subsequent conﬁrmation of black hole binaries, and their effect on the orbital period distribution of these systems are examined. By performing simulations to account for some of these biases, this research shows that these selection effects are likely to be responsible for the observed dearth of short period binaries in the Galaxy.

In Chapter 3, the results of the ﬁrst systematic analysis of QPOs from a black hole binary using the bispectrum are presented. This analysis was performed on the well studied and characterised black hole source GX 339-4. This research revealed a gradual change in the nature of the non-linear phase coupling with change in the mass accretion rate.

In Chapter 4, this bispectral analysis is extended to a larger sample of black hole binaries. This showed a likely inclination dependence in the observed change in the phase coupling, with face-on sources exhibiting the opposite behaviour to those viewed edge-on.

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CHAPTER 2 SELECTION EFFECTS ON THE ORBITAL PERIOD DISTRIBUTION OF LOW MASS BLACK HOLE X-RAY BINARIES

K. Arur and T. J. Maccarone Monthly Notices of the Royal Astronomical Society, February 2018, Volume 474, Issue 1, p.69-76

2.1 Abstract We investigate the lack of observed low mass black hole binary systems at short periods (<4 hours) by comparing the observed orbital period distribution of 17 conﬁrmed low mass Black hole X-ray Binaries (BHXBs) with their implied period distribution after correcting for the effects of extinction of the optical counterpart, absorption of the X-ray outburst and the probability of detecting a source in outburst. We also draw samples from two simple orbital period distributions and compare the simulated and observed orbital period distributions. We predict that there are >200 and <3000 binaries in the Galaxy with periods between 2 and 3 hours, with an additional ≈600 objects between 3 and 10 hours.

2.2 Introduction Low Mass X-ray Binaries (LMXBs) are binary systems consisting of a black hole or a neutron star primary and a low mass main sequence or an evolved companion star that is <1M (see e.g [53] ). Mass transfer occurs from the secondary to the primary via Roche lobe overflow. All of the dynamically confirmed low mass Black Hole X-ray Binaries (BHXBs) that have been observed are transient sources, spend- ing most of the time in quiescence and occasionally going into outburst, when the source increases in X-ray luminosity by several orders of magnitude. The origin of the outbursts of transients is explained, at least in broad brush strokes, by the disc instability model (see reviews by [4] and [5]). The lack of persistent low mass BHXBs may be, in part, due to selection effects. For example, the persistent source 4U 1957+11 is likely to be a black hole ([54], [55]), but dynamical confirmation has not yet been obtained because the accretion disc outshines the donor star in optical

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Figure 2.1. Histogram of the observed orbital period distribution of BHXBs. For a list of the properties of the binaries, see Table 2.1.

wavelengths; still there are few strong candidate black holes in low mass X-ray binaries that are persistently bright sources.

The orbital periods of these observed BHXBs vary from a few hours to several weeks (See Fig. 2.1). With the exception of GRS1915+105, all BH systems have an orbital period shorter than a week. However, there appears to be a lack of BHXBs at the shorter periods (< 4 hours) relative to theoretical predictions ([56], [57], [58], [59]).

The peak X-ray luminosity of LMXBs during an outburst is directly proportional to the orbital period of the system ([60], [61], [62]). This is a result of the binaries

23 Texas Tech University, Kavitha Arur, May 2020 with a longer orbital period having a larger disc radius, and a larger radius to which the accretion disc is irradiated by the central compact object.

When the mass accretion rates onto the primary are low and the local cooling timescale becomes longer than the accretion timescale, the accretion flow is radiatively inefficient. In this case an advection dominated accretion flow (ADAF) can occur ([63] , [64]) and much of the kinetic energy in the accretion flow is advected onto the compact object. In the case of a black hole primary, this energy is advected beyond the event horizon. In the case of a neutron star primary, however, the energy is radiated when the accretion flow impacts the stellar surface. This makes short period BHXBs intrinsically fainter than their neutron star counterparts [65].

It was shown by [66] that the lack of short period BH systems could be explained by a drop to a radiatively inefﬁcient state at short periods due to the lower disc mass in these systems. In this case, the BHXBs have a lower peak luminosity in outburst, shorter outburst durations and lower X-ray duty cycles that result in the system being much harder to detect. They also calculated the detection probabili- ties of BHXBs in the Galaxy as a function of their orbital periods to show that this switch to a radiatively inefﬁcient state renders these short period systems undetectable at large distances.

In this chapter, we investigate the impact of selection effects on the orbital period distribution of BHXBs by randomly distributing BHXBs across the Galaxy and calculating the fraction of sources that can be detected. We incorporate the switch to a radiatively inefficient state by assuming a gradual reduction in the radiative efficiency below a few percent of the Eddington luminosity as stated in [66]. In addition to this switch, we also take into account the effects of absorption of the X-ray flux to model the likelihood of the detection of the X-ray outburst.

LMXBs tend to be concentrated towards the plane of the Galaxy, where the extinction levels are the highest. Radial velocity measurements obtained when the system is in quiescence are required to get dynamical mass estimates of the black

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hole candidate. In the case of short period binaries which have less luminous companion stars than do longer period binaries, these high levels of extinction can result in the optical counterpart being too dim for these measurements. Thus we have also included the probability of detecting the optical counterpart during quiescence in our calculations.

It is important to study and understand these effects because an accurate estimate of the population of BHXBs in the Galaxy can lead to more robust constraints on binary evolution models since the same physical process dominates the evolution of binaries irrespective of whether the primary is a white dwarf, neutron star or a black hole. In this chapter,we study the implied period distribution of BHXBs and estimate the total number of BHXBs in the Galaxy. In Section 2 we present our sample of BHXBs and the implied period distribution after correcting for various selection effects. In Section 3 we compare the implied period distribution to the period distribution drawn from 2 sample initial orbital period distributions and accounting for the effects of magnetic braking. This is followed by a discussion of our results and our conclusions.

2.3 Data set and correction of selection effects 2.3.1 Galaxy model and data set The data set consists of the 15 confirmed BH LMXBs and 2 strong candidate sources that have since been dynamically confirmed listed in [53]. The sources and their properties are listed in Table 2.1. In order to determine what fraction of BHXBs are detected and dynamical mass measurements are obtained, 25000 locations were randomly selected in the Galaxy. The random selection was weighted by stellar density, making it more likely for a binary to be located in the denser parts of the Galaxy. Each of the 17 BHXBs listed in Table 2.1 was then placed at each of the 25000 locations to test for detectability. SWIFT J1753.5-0127 has not been included in our data set due to controversy over whether it really has been dynamically confirmed [67].

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Table 2.1. Table of properties for the 17 BH LMXBs.

1 Source Name Alternate Name Porb(hr) mV AV Reference GRO J0422+32 V518 Per 5.1 22.4 0.74 [68], [69] A0620-00 V616 Mon 7.8 18.2 1.21 [70], [71], [72] GRS 1009-45 MM Vel 6.8 21.7 0.62 [73], [74] XTE J1118+480 KV UMa 4.1 19.0 0.06 [75], [76] GRS 1124-684 GU Mus 10.4 20.5 0.9 [77], [78] 4U 1543-475 IL Lupi 26.8 16.6 1.5 [79] XTE J1550-564 V381 Nor 37.0 22.0 4.75 [80] GRO J1655-40 V1033 Sco 62.9 17.3 4.03 [81], [82] GX 339-4 V821 Ara 42.1 19.2 3.4 [83], [84] H 1705-250 V22107 Oph 12.5 21.5 1.2 [85], [86] SAX J1819.3-2525 V4641 Sgr 67.6 13.7 0.9 [87] XTE J1859+226 V406 Vul 9.2 23.29 1.8 [88] GRS 1915+105 V1487 Aql 804.0 - 2 19.6 [89], [90] GS 2000+251 QZ Vul 8.3 21.7 4.0 [91], [92] GS 2023+338 V404 Cyg 155.3 18.5 3.3 [93], [94] XTE J1650-500 - 7.6 24.0 4.65 [95], [96], [97] GS 1354-64 BW Cir 61.1 21.5 3.1 [98], [99]

The modelling of the implied period distribution of the sources takes into account the probability of detecting the companion in quiescence, the probability of detecting an outburst with an all-sky monitor based on its location in the Galaxy, as well as the probability of the source having a detectable outburst based on the recurrence time and duration of the outburst. It is also worth mentioning that few of the BHXBs have repeated outbursts, and their recurrence times remain a key unknown. For more on the outburst recurrence times, see Section 2.3.4.

2.3.2 Extinction of Optical Counterpart To determine the fraction of binaries for which the optical counterparts are too dim to obtain radial velocity mass measurements due to the effects of extinction from the interstellar medium, the three-dimensional extinction model of [100] was used. The catalogue contains extinction proﬁles along over 64000 lines of sight in the regions of |l| ≤ 100◦ and |b| ≤ 10◦ using 2MASS data. Linear interpolation

26 Texas Tech University, Kavitha Arur, May 2020

of the data from the catalogue was used to determine the amount of extinction to each of the 25000 locations determined as described in Section 2.3.1.

For the optical counterpart, a cut off magnitude of V=22 was selected. After accounting for extinction, sources along each line of sight brighter than this magnitude are considered to be bright enough for radial velocity measurements. The quiescent magnitudes used for the sources are listed in Table 2.1. Fig. 2.2 shows an example of the process of calculating the fraction of optical counterparts that are visible. The fraction of companion stars that are visible in the optical for each source is listed in Table 2.2.

2.3.3 Detection of the X-ray outburst To determine if the outburst would be visible, the recorded values of the peak fluxes seen from the sources was used to calculate the peak luminosities of the X-ray outburst. The outburst was considered to be detectable if the flux from the source exceeded 2.3 counts per second (corresponding to a 30 mCrab flux for RXTE ASM) after accounting for the absorption column along that line of sight.

The conversion from X-ray ﬂux counts per second was done using the NASA HEASARC tool PIMMS. The values of hydrogen column density were obtained 21 −2 −1 from [101], using NH = 1.79 × 10 AV cm mag [102]. A standard photon index of Γ = 1.7 was assumed for all sources. The fraction of X-ray outbursts that can be detected for each source is listed in Table 2.2.

2.3.4 Probability of outburst detection The probability of a source having an outburst that is detectable by an all-sky monitor depends on both the recurrence time for outbursts and the duration of the outburst. The probability of a source having at least one outburst that could have been detected is given by:

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Figure 2.2. An example plot showing the detection of the optical counterpart for the source XTE J1150-564. The locations indicated by the blue ﬁlled circles are the locations where the optical counterpart can be observed. The yellow ﬁlled circles indicate locations where the optical counterpart cannot be detected due to high levels of extinction along that line of sight.

 tsurvey/trec if trec > tsurvey P(≥ 1outburst) = (2.1) 1 otherwise

where trec is the outburst recurrence time and tsurvey is the survey length. The survey length was taken to be 15 years, the lifetime of the Rossi X-Ray Timing

Explorer All Sky Monitor (RXTE ASM). The outburst recurrence time trec was as-

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sumed to be:  ˙ MD/|M2| if L < 0.02LEdd trec = (2.2) ˙ N × MD/|M2| otherwise ˙ where MD is the disc mass, M2 is the rate at which mass is transferred from the companion and N is a factor that accounts for an increase in the recurrence rate due to mass lost in the accretion disc due to wind. Here, we ran the simulations using N=2, N=3 and N=10, which corresponds to a mass loss of 50%, 66% and 90% respectively. If a source has an outburst during a survey, for it to be observed the outburst must be visible for at least 1 day [66]. For outbursts lasting less than 1 day, the probability of being observed decreases. The probability of observing an

outburst Pobs is given as:  tdet/1day if tdet < 1 day Pobs = (2.3) 1 otherwise

where tdet is the observable outburst duration. For a limiting flux Flim, this is 2 determined to be the time at which the luminosity falls below Llim = 4πd Flim. We have defined Flim to be 10mCrab, corresponding to the daily exposure sensitivity of RXTE. The luminosity is taken to be L = ηMc˙ 2 where η is the radiative efficiency and M˙ is the accretion rate onto the central compact object. The efficiency at high luminosities can be approximated to be η = 0.1. However, at luminosities below a few percent of the Eddington luminosity, the accretion becomes radiatively inefficient. Thus we assume a transition to lower efficiencies used in [66] as:

!n M˙ η = 0.1 ˙ (2.4) fMEdd

for L ≤ fLEdd where f = 0.02 and n = 1. This transition to lower efﬁciencies can reduce the duration for which the outburst is visible. In the cases of binaries with short orbital periods, this can render the outburst undetectable in all but the most favourable of locations in the Galaxy.

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It has been shown by [103] that the central accretion rate can be described by an initial exponential fall off followed by a linear decay:

1/2 " 1/2 # ˙ 3ν 1/2 3ν M = Mh (T ) − (t − T ) (2.5) Bn Bn

where Mh is the irradiated mass at a given time t and the constant Bn was set to 105 [66], 3 ρRd −3νt Mh(t) = exp 2 (2.6) 3 Rd and T is the time taken for the the irradiated radius to drop below the disc radius (T=0 for long period systems)

2 R Bnνρ T = d log (2.7) 3ν Rd

The value of density ρ = 10−8gcm−3 was chosen following [103], where the density was shown to be independent of the radius, making it suitable for the wide range of orbital periods used.

The viscosity is taken to be ν = αhcsH = αhcs(H/R)Rd [101, 104] where cs is the speed of sound and H is the scale height of the disc. We have chosen a value of ν = 2.2 × 1014cm2s−1, as this produces a reduction in the radiative efﬁciency of BHXBs with an orbital period less than 5 hours.

In addition to the above, it must be noted that all sky monitors cannot observe 100% of the sky at any given time due to the location of the Sun. It is thus possible for binaries that are close to the ecliptic and exhibit short period outbursts to be obscured by the Sun. The probability of detecting a binary having an outburst that is partially obscured is given by:  tobsc−tdet+1 1.0 − 365.25 if tdet < tobsc PSun = (2.8) 1.0 otherwise

where tobsc is the number of days per year that the binary is behind the Sun.

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Table 2.2. Detection fractions of BHXBs.

Source Detection Fraction Detection Fraction Outburst Name (X-Rays) (Optical) Detection Probability V518 Per 0.0025 0.99 1.0 V616 Mon 0.025 1.0 0.38 MM Vel 0.0051 0.99 1.0 KV UMa 0.0085 0.0034 1.0 GU Mus 0.0078 1.0 0.38 IL Lupi 0.38 1.0 0.27 V381 Nor 0.22 1.0 0.21 V1033 Sco 0.38 1.0 0.27 V821 Ara 0.33 1.0 0.37 V2107 Oph 0.051 1.0 0.69 V4641 Sgr 0.44 1.0 0.13 V406 Vul 0.011 1.0 0.56 V1487 Aql 0.40 1.0 0.053 QZ Vul 0.072 1.0 0.43 V404 Cyg 0.34 1.0 0.11 J1650-500 0.10 0.75 0.55 BW Cir 0.35 1.0 0.25

Thus the total probability of having an outburst and observing it is given by:

Pdet = Pobs × P(≥ 1outburst) × PSun (2.9)

The probability of detecting an outburst from each of the 17 BHXBs is listed in Table 2.2.

2.3.5 Implied Orbital Period Distribution using Observational Data Using the detection fractions shown in Table 2.2, the implied orbital period distribution was plotted. The results are shown in Fig. 2.3.

To compare the results with the distribution expected from theoretical models of magnetic braking, we fit a power law slope to the results obtained after correcting for the selection effects detailed above. While the fitting of a constant cannot be excluded in a statistically significant manner, it is expected that the distribution

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Table 2.3. Power law ﬁts to number of detected sources vs. Porb

Distribution No. of sources Power Law Index Data 7 2.45 ± 1.3 Spikea 500 1.60 ± 0.09 Flatb 500 2.38 ± 0.10 Spike 275 1.62 ± 0.12 Flat 275 2.38 ± 0.13 Spike 50 1.61 ± 0.31 Flat 50 2.36 ± 0.32 a - Drawn from an initial orbital period distribution where all the binaries start at a 20 hour period. b - Drawn from an initial orbital period distribution that was logarithmically ﬂat between 2 and 20 hours follows a power law due to the physical processes dominating the evolution of the system (see Equation 2.10). For this ﬁtting, any sources with an orbital period greater than 10 hours were excluded, since the evolution of these sources is likely to be dominated by expansion of the donor star, rather than by magnetic braking. The bin sizes used are unequal to ensure that none of the bins had a count of zero, and the error bars shown are Poissonian.

The fitting of the power law was done using XSPEC [105]. For the fitting, data from the observed BHXBs were converted to a spectrum file format using the flx2xsp routine in FTOOLS. Combining the optical detection fractions, the X-ray detection fractions and the probability of detecting the initial outburst gives the final fraction of the total BHXB population that can be observed. This final fraction was used as the corresponding response matrix for use with XSPEC. The results of the fit are shown in Fig. 2.4. We obtained a best fit value of 2.45 ± 1.3 for the plot assuming a 50% mass loss due to disc winds (i.e. N=2 in Equation 2.1). It was found that changing the amount of mass lost did not significantly alter the best fit value of the slope. It does, however, affect the normalisation of the power law. These results are further discussed in Section 2.5. The large error bars on the fit are due to the small number of binaries with an orbital period less than 10 hours.

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Figure 2.3. Top: Histogram of the observed period distribution of BHXBs. Middle: Distribution after accounting for extinction of optical counterpart. Bottom: Distri- bution after accounting for extinction of optical counterpart, absorption of X-ray ﬂux and the probability of detecting the X-ray outburst.

2.4 Monte Carlo Simulation 2.4.1 Generation of simulated sources We generated a population of 250,000 simulated sources by drawing from two simplified initial period distributions. The first is the spike distribution is where all the sources start at an orbital period of 20 hrs and the second is the flat distribution where the orbital periods are logarithmically distributed between 2 hrs and 20 hrs. The orbital period of these objects decay as a result of mass transfer driven magnetic braking and gravitational radiation that can be described using:

 −2/3 10−10 Porb M yr−1 < ˙ 2 if P(hr) 2 −M2 = 5/3 (2.10) −10 Porb −1 6 × 10 3 M yr if P(hr) > 3

33 Texas Tech University, Kavitha Arur, May 2020

Figure 2.4. A power law model ﬁt to the predicted number of black hole binaries in the Galaxy with an orbital period of less than 10 hours.

as shown by [106]. In the case of the orbital period being greater than 3 hours, the mass transfer is dominated by magnetic braking. For periods less than 2 hours, this transfer is driven by gravitational radiation. For periods between 2 & 3 hours, there is likely to be little to no mass transfer as the star’s structure changes as it becomes fully convective, and it brieﬂy stops ﬁlling its Roche lobe.

The age of each binary was selected randomly from a uniform distribution between 0 and 10 Gyrs, and its current orbital period was determined using Equa- tion 2.10. A plot of the current orbital period of the binary as a function of its age is shown in Fig. 2.5. In these simulations, all the binaries were assumed to have a primary of mass equal to 8M . The companion mass was taken to be

34 Texas Tech University, Kavitha Arur, May 2020

Figure 2.5. Plot of orbital period decay via magnetic braking and gravitational radiation for a system starting at an orbital period of 20 hours. The green horizontal line indicates a 2 hour orbital period, below which it is harder to observe the sources unless they are at a close distance.

m2 = 0.1Porb(hrs) for Porb < 10 hours, and m2 = 0.7 for Porb > 10 hours. This way, the current orbital period distributions of the two samples were generated.

2.4.2 Optical counterpart detection The luminosity of the companion star was calculated assuming that the star is a lower main sequence star that just ﬁlls its Roche lobe, with a temperature of 3500K. This corresponds to the temperature of typical KV type star. The Roche lobe radius of the secondary star is taken to be:

2/3 RL2 0.49q = (2.11) a 0.6q2/3 + ln(1 + q1/3) [107], where the binary separation a is given by

10 1/3 1/3 2/3 a = 3.5 × 10 m1 (1 + q) Porb cm (2.12)

35 Texas Tech University, Kavitha Arur, May 2020

A 50% contribution of light from the accretion disc in quiescence was assumed and the locations of the binaries were selected randomly (weighted by stellar density). The effect of extinction was calculated as described in Section 2.3.2.

2.4.3 Probability of an X-ray outburst and its detection The probability of detecting the X-ray outburst for these generated sources was calculated using the method detailed in Section 2.3.3. Once the detection fractions were calculated, a sample of 500 generated sources were randomly selected from each of the two distributions.

2 The peak luminosity of the X-ray outburst was taken to be Lpeak = ηc ρνRd,

where Rd is the disc radius and η is the radiative efﬁciency given by:

!n M˙ η = 0.1 ˙ (2.13) fMEdd

for L ≤ fLEdd where f = 0.02 and n = 1 The values of ν and ρ used are the same as described in Section 2.3.4 .

The disc radius Rd was approximated as Rd ≈ 0.7RL1 where RL1 is the Roche radius of the primary as given by:

−2/3 RL1 0.49q = (2.14) a 0.6q−2/3 + ln(1 + q−1/3)

2.4.4 Implied Orbital Period Distribution using Simulated Data The best ﬁts for the power law index for the plots of number of detected sources vs. the orbital period using the magnetic braking rate as shown in Equation(2.10) are shown in Table 2.3. The plots of the best ﬁts using a sample of 500 detected sources from each distribution can be seen in Fig.2.6.

2.4.5 Effect of mass transfer on the companion The ﬁt to the slope of the orbital period distribution plotted against the number of sources depends on the shape of the distribution that the sample is drawn from, as well as the relationship between the mass transfer rate and the orbital period of

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Figure 2.6. The best power law fit obtained using the X-ray spectral fitting software XSPEC for the orbital period distribution of 500 randomly generated black hole binaries with periods between 2 and 10 hours. The binaries in the left panel were drawn from an initial orbital period distribution that was logarithmically flat between 2 and 20 hours and decay via magnetic braking. The binaries in the right panel were drawn from an initial orbital period distribution where all the binaries start at a 20 hour period and decay via magnetic braking.

the system. In addition, due to the mass transfer that occurs during the lifetime of the donor star, the star can be expected to have a radius greater than the radius of an isolated main sequence star of the same mass [108]. This bloating factor can also affect the slope of the plot, making it steeper.

The model slope dN/dP is determined by the amount of time a binary spends at a particular orbital period P˙ −1 (i.e. the inverse of the rate of change of the orbital period). From the mass-period relationship, it can be assumed that:

˙ ˙ M2 P ∝ (2.15) M2 P

Since P ∝ M2 [106], this gives us:

dN ˙ −1 ∝ M2 (2.16) dP

Since Equation(2.10) does not take into account of the effect of mass transfer on the donor star, using Equation 9 from [108] results in:

37 Texas Tech University, Kavitha Arur, May 2020

−5/3 dN Porb ∝ mˆ −7/3 (2.17) dP 3 2

where mˆ 2 = M2/M2(MS). M2 is the mass of the companion star, and M2(MS) is the mass of a main-sequence star that would just ﬁll the Roche lobe at the current period.

2.5 Discussion By fitting the results of the implied period distribution from the observed binaries, a best fit value of 2.45 ± 1.3 was obtained for the plot assuming a 50% mass loss due to disc winds (i.e. N=2 in Equation 2.1). Changing the amount of mass lost to 66% and 90% did not significantly alter the best fit value of the slope.

From the results in Table 2.3, it would appear that the sample drawn from the flat orbital period distribution matches the data. The addition of the effect from the bloating of the companion star steepens the value of the slope by approximately 0.9, making the best fit value 3.28. However, due to the large error on the best fit to the slope of the observed data, the fit of the sample drawn from the spiked orbital period distribution cannot be excluded.

By integrating the implied period distribution to estimate the number of short period black hole binaries, we predict that approximately 600 black hole binaries are present in the Galaxy with an orbital period between 3 and 10 hours. Altering the amount of mass lost in winds to 66% yields an estimate of ∼1100 binaries, and a loss of 90% of the mass yields an estimate of ∼3700 binaries. It was estimated by [109] that 90% of the mass was lost due to disc wind and/or jets. This ﬁnd- ing is based on estimating the mass transfer rate of the binary from the ultraviolet luminosity of the system under the assumption that the UV comes from the accretion stream impact spot on the outer disc in A0620-00, and then comparing with the duty cycle of the outburst, and is also supported by measurements of strong disc winds in the outbursts of X-ray binaries (e.g. [110]). It has been alternatively suggested that the UV emission may come from closer to the black hole in this sys-

38 Texas Tech University, Kavitha Arur, May 2020 tem, in which case the inferred mass transfer rate would be much lower, and such strong disc winds would not be necessary [111], although such a result would not explain the actual observations of strong disc winds in outburst. Thus it is clear that a deeper understanding of the outburst duty cycle is needed to accurately estimate the total number of black hole binaries in the Galaxy.

By extrapolating the ﬁt down to 2 hours, we predict that there are ∼ 200 − 3000 binaries with periods between 2 and 3 hours. The range of the values are determined using the upper and lower limits of the powerlaw index estimate as shown in Table 2.3. Our estimate for the total number of BHXBs in the Galaxy is consistent with theoretical estimates of numbers between 102 and 104 obtained using population synthesis codes ([56], [57], [58], [59]). However, [112] have detected a radio source in the direction of M15 that they have suggested is a BHXB in quiescence. If correct, it would imply a much larger population of quiescent BHXBs (2.6 × 104 - 1.7 × 108). While this could suggest a new channel of black hole formation, it does not ﬁt well with our predictions. It must also be emphasized that X-rays have not been detected from this source, and thus the estimate of a larger population must be treated with caution. While this discovery could be anomalous, it has been suggested that binaries lack outbursts at accretion rates lower than those predicted by magnetic braking. This could be the case for short period binaries that exist in the period gap which, as discussed by [113] may have low but non zero accretion rates, as well as for the very long period systems suggested by [114]. Thus it is possible that there exists a hidden population of persistently quiescent BHXBs.

To estimate the number of binaries that would have to be detected and have dynamical mass estimates in order to determine which of the two distributions is closest to the true orbital period distribution, we ran the Monte Carlo simulation detailed in the previous section, drawing a different number of detectable sources each time. With a sample of 50 detected binaries, it is possible to distinguish between the two sample orbital period distributions at the 1-sigma level. A sample of approximately 275 binaries will be required to distinguish the two at the 3-sigma level. The next generation of X-ray missions such as LOFT [115] and STROBE-X

39 Texas Tech University, Kavitha Arur, May 2020

[116] are thus extremely important for identifying more of these short period systems and understanding their underlying distribution.

The model used to predict the peak luminosities of the X-ray outburst produces a decrease in radiative efficiency at short orbital periods (Porb 6 5 hours). How- ever, the predicted luminosity for XTE J1118+480 is higher than the luminosity derived from the observed peak flux from the system. This was also noticed by [62] who noted that this could be due to advection. If this is true in the case of all BHXBs with periods less than 5 hours, then the detection of the X-ray outburst becomes the limiting factor, as opposed to the detection of the optical counterpart. In this case, the availability of sensitive all-sky monitors is the most effective tool in searching for this population of very faint, short period BHXBs. This can already be seen by the large number of short period black hole candidates being detected by new, more sensitive Wide Field Monitors. However, a number of these candidates cannot be confirmed due to the faintness of the companion in quiescence and an inability to obtain detailed spectroscopy for these objects. A possible solution to the problem in obtaining dynamical confirmation for short period objects could be the relation proposed by [117] that allows the determination of the mass of the binary using Hα emission lines, rather than absorption lines. Using this method could extend the search for short period BHXBs to fainter limits, allowing us to confirm a larger fraction of candidate systems.

It is also worth noting that the addition of natal kicks to the simulation will result in an increase in the scale height of the BHXBs. See e.g. [118], [119] for a discussion of the scale height distribution of BHXBs and corresponding evidence that most form with large natal kicks. This is likely to result in a larger fraction of binaries being detected due to the lower levels of extinction away from the Galactic plane. This can be seen by the detection of short period systems at high galactic latitudes by MAXI [120], [121] such as MAXI J1659-152 [122], [121], MAXI J1836-194 [123], [124], MAXI J1305-704 [125], [126] and MAXI J1910-057 [127], [128]. The presence of natal kicks can also have an impact on the formation of BHXBs resulting from a wide binary interaction with a ﬂy by star [129].

40 Texas Tech University, Kavitha Arur, May 2020

2.6 Conclusions We have derived the implied period distribution of low mass BHXBs after correcting for the effects of extinction of the optical counterpart, absorption of the X-ray outburst and the probability of detecting a source in outburst. We have compared the power law ﬁt to the implied orbital period distribution to the distributions produced by drawing a sample from two simple orbital period distribution. Based on the results from the simulation, it is likely that the observed data arise from a ﬂat orbital period distribution. However, due to the small number of observed sources, the possibility of an orbital period distribution arising from the decay of the orbit of a large orbital period (20 hours) cannot be ruled out. With a sample of ∼275 detected sources, it would be possible to differentiate between the two distributions. Our simulation predicts ∼ 200 − 3000 binaries with periods between 2 and 3 hours, and an additional ∼600 binaries between 3 and 10 hours. This is consistent with numbers predicted using population synthesis models.

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CHAPTER 3 NONLINEAR VARIABILITY OF QUASI-PERIODIC OSCILLATIONS IN GX 339-4

K. Arur and T. J. Maccarone Monthly Notices of the Royal Astronomical Society, July 2019, Volume 486, Issue 3, p.3451-3458

3.1 Abstract We examine the nonlinear variability of quasi-periodic oscillations (QPOs) using the bicoherence, a measure of phase coupling at different Fourier frequencies. We analyse several observations of RXTE/PCA archival data of the black hole binary GX 339-4 which show QPOs. In the type C QPOs, we conﬁrm the presence of the ‘hypotenuse’ pattern where there is nonlinear coupling between low frequencies that sum to the QPO frequency as well as the ‘web’ pattern where in addition to the hypotenuse, nonlinear coupling between the QPO frequency and the broadband noise is present. We ﬁnd that type B QPOs show a previously unknown pattern. We also show that the bicoherence pattern changes gradually from ‘web’ to ‘hypotenuse’ as the source moves from a hard intermediate state to a soft intermediate state. Additionally, we reconstruct the QPO waveforms from six observations using the biphase. Finally, we present a scenario by which a moderate increase in the optical depth of the hard X-ray emission region can explain the changes in the non-linearity seen during this transition.

3.2 Introduction Quasi-periodic oscillations (QPOs) are features that appear as modest-width peaks (i.e. with quality factors 1 of ∼few to ∼10) in the power spectra of X-ray binaries, and are considered to be excellent probes of the regions closest to the compact objects. Low frequency QPOs (LFQPOs), are seen at frequencies between ∼0.1Hz and ∼10Hz [see eg. 130, for an overview]. Based on their amplitudes, quality factors, and other aspects of the power spectra, these LFQPOs, originally

1Ratio between QPO frequency and the FWHM of the QPO peak

42 Texas Tech University, Kavitha Arur, May 2020 classiﬁed into 3 types: A, B and C in XTE J1550-564 [25, 131], have since been observed in several black hole binaries (BHB).

Black hole binaries also show variability on longer time scales, leading to with spectral state transitions, where the spectra change from a power-law dominated hard state, to a soft state that is dominated by a strong disk blackbody component [132]. These changes can be clearly illustrated using the Hardness Intensity Diagram (HID) [133, 134, 135, 136, 137], where different portions of the q-shaped track correspond to the different states of the BHB. The four main states are the Low/Hard state (LHS), the Hard Intermediate State (HIMS), the Soft Intermediate State (SIMS) and the High/Soft state (HSS) [137]. In addition to the changes seen in the HID, detailed timing and spectroscopic analyses [e.g 138] are essential for a complete understanding of the underlying processes. The association of LFQPOs with the different spectral states suggests that understanding the origin of QPOs is fundamental in understanding these different states.

Several different models have been proposed to explain the origin of these QPOs, such as the Lense-Thirring precession of the accretion disk [36, 139], spiral structures in the accretion disk [140] and instabilities in the accretion disk [141]. While many of the models can reproduce the frequencies of the peaks in the power spectra, there is still no consensus about the mechanism by which these QPOs are produced. Thus it is important to go beyond the power spectrum, and study the non-linear variability present in the different states.

Recently, attempts have been made to understand the non-linear variability in X-ray binaries. Nonlinearity has been shown by presence of a linear rms-ﬂux relation [41] and has also been seen in the hard state of Cygnus X-1 using the bispectrum and bicoherence, which are relative measures of phase couplings among three different Fourier components [42]. Much of the phenomenology of X-ray binary variability is seen in active galactic nuclei, cataclysmic variables and young stellar objects, suggesting that most of the basic physics is generic to the process of accretion [142]. In GRS 1915+105, [143] found that there was a peak in the bico-

43 Texas Tech University, Kavitha Arur, May 2020 herence wherever harmonics of the QPO were present. It has been shown that the bispectrum can be used to distinguish between different models that produce very similar power spectra [40].

In this chapter, we use the bicoherence to examine the non-linear variability of multiple QPOs observed in GX 339-4 using the Rossi X-ray Timing Explorer (RXTE). In Section 3.3 we give an introduction to the bispectrum, followed by a brief overview of the data in Section 3.4. In Section 3.5 we present the results of our analysis and show that the type of variability changes as the source moves from the LHS/HIMS to the SIMS as well as reconstructed waveforms for a subset of the observations. We then propose a scenario to explain the observed results in Section 3.6 before presenting our conclusions.

3.3 Bicoherence The bispectrum is a higher order Fourier spectral analysis technique that can be used to study the non linear properties of the time series [144]. The bispectrum of a time series is given by:

K−1 1 X ∗ B(k, l) = Xi(k)Xi(l)X (k + l) (3.1) K i i=0

where K is the number of segments of the time series, Xi(f) is the Fourier trans- ∗ form of the ith segment of the time series at frequency f, and Xi (f) is the complex conjugate of Xi(f). The real component of the bispectrum describes the skewness of the ﬂux distribution of the light curve, while the imaginary component describes the reversibility of the time series in a statistical sense. The bispectrum is a complex number that can be represented using an amplitude and a phase. This phase, which is called the biphase, thus holds information about the shape of the underlying lightcurve. For a more detailed discussion of the biphase, see [45].

A related term is the bicoherence, which is analogous to the cross-coherence function that is more familiar to most X-ray astronomers [145]. The bicoherence

44 Texas Tech University, Kavitha Arur, May 2020 has a value between 0 and 1, where 0 indicates that there is no non-linear coupling between the phases of the different Fourier frequencies, and 1 indicates total coupling. The bicoherence is given by

|P X (k)X (l)X∗(k + l)|2 b2 = i i i P 2 P 2 (3.2) |Xi(k)Xi(l)| |Xi(k + l)| using the normalization proposed by [46]. Here, b2 measures the fraction of power at the frequency k + l due to the coupling of the three frequencies.

As the bicoherence is forced to lie between 0 and 1, it will have a non-zero mean arising from errors even when there is no phase coupling is present [42]. Thus a bias of 1/K is subtracted from any bicoherence measurements.

3.4 Data Overview GX 339-4 is a Low Mass X-ray Binary (LMXB) that consists of a black hole of mass 2.3M < MBH < 9.5M [146], and has undergone multiple X-ray outbursts since its discovery [147]. In this chapter we analyze RXTE archival observations of GX 339-4 during its outbursts in 2002, 2004, 2007 and 2010 where QPOs were detected by [18].

The data were obtained in several modes, and for our analysis we used SIN- GLE BIT,EVENT and GOOD XENON mode data. The data were ﬁltered to exclude periods of high offset, Earth occultation and passage through the South Atlantic Anomaly (SAA). Data above 30 keV are excluded, since for the softer states the hard X-rays are mostly contributed by the background. The data were then binned with a time resolution of 1/128s, and Fourier transformed using 16s long segments of the data. The Fourier transformed data were averaged and normalized according to [19] to produce the power spectra. The patterns are seen most clearly in the bicoherence plots of observations that are both long (>1ks) and have a high count rate (>750 counts/s).

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3.4.1 QPO Classiﬁcation QPOs can be characterized by three main properties that can be measured from the power spectra: the frequency of the QPO, the rms variability and the quality factor Q (Q=frequency/FWHM). LFQPOs from black holes are classiﬁed into one of three types: A, B or C [24].2 In this chapter, we will also look at how the phase coupling properties of the QPOs can be used to help understand them.

Type A QPOs generally occur in the frequency range of ∼6-8 Hz. These QPOs are typically broad (Q<3), show low rms variability and do not have any observed harmonic structures. Type B QPOs occur in the frequency range of ∼0.5-6.5 Hz, with Q>6. These sometimes show weak subharmonic or harmonic structures. Both Type A and B QPOs occur when the source is in the SIMS.

Type C QPOs appear at frequencies between ∼0.1 and ∼10 Hz, and are usually narrow (Q ∼ 7-12) with high rms variability. This class of QPOs are often observed along with the presence of higher harmonics, with a weak sub-harmonic feature sometimes being present. These QPOs are seen when the source is in LHS or HIMS. For a more detailed overview of the properties of the type A,B and C QPOs seen in GX 339-4, we refer the reader to [18].

3.5 Results 3.5.1 Bicoherence Patterns For classifying the different patterns seen in the bicoherence, we follow the convention used in [143]. Below, we include a brief description of the different patterns observed in GX 339-4. The ‘cross’ pattern has not been observed from this source. We also analyse and describe the bicoherence of type B QPOs.

2These phenomenological classiﬁcations are used because of the lack of a clear understanding of the physical origins of these QPOs.

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Table 3.1. Details of the RXTE observations with QPOs. The asterisk indicates that QPOs with f <2 Hz were classed as web if pattern is unclear. The full table is listed in Appendix A. MJD: Modiﬁed Julian Date

3.5.1.1 The ‘hypotenuse’ pattern When the ‘hypotenuse’ pattern is observed (see Fig 3.1) , a high bicoherence is

seen when the two noise frequencies f1 and f2 add up to the frequency of the QPO, making a diagonal line. This pattern also shows a region of high bicoherence when f1 = f2 = fQP O. This demonstrates that there is power at the frequency of the QPO and the ﬁrst harmonic and that there is coupling between these two frequency

components. Essentially, if there is power in the f1 = f2 part of the diagram, this indicates that signiﬁcant power comes in the form of a non-sinusoidal waveform (see Section 3.5.5 for details on the QPO waveform). If harmonics appeared in the

power spectrum for frequencies f1, f2 andf1 +f2, but stringent upper limits existed on the power of the bispectrum, that would argue that the harmonics represented different normal modes of a system which were being excited independently (consider, e.g. the case of a guitar string being plucked exactly at several of its nodes with totally uncorrelated driving forces).

3.5.1.2 The ‘web’ pattern The ‘web’ pattern (see Fig 3.2) shows features seen in both the hypotenuse and the cross pattern. The diagonal line of high bicoherence when f1 + f2 = fQP O is present. Also present is a vertical (and horizontal) streak where f1 (or f2) corresponds to fQP O and the other frequency is noise. The strength of bicoherence falls

47 Texas Tech University, Kavitha Arur, May 2020

Figure 3.1. The (a) Power spectrum and (b) the bicoherence plot showing the ’hypotenuse’ pattern from the observation 70109-04-01-00. The colour scheme of logb2 is as follows: dark blue:-2.0, light blue:-1.75, yellow:-1.50, red:-1.25

off at frequencies above 2fQP O. Regions of high bicoherence can also be seen when the fundamental frequency interacts with the second harmonic to produce power at higher harmonics.

3.5.2 Type B QPOs When a type B QPO is present, (see Fig 3.3) a high bicoherence is seen where both f1 and f2 are equal to fQP O, indicating the presence of a second harmonic (often visible in the power spectrum). However, it does not show any coupling with the broad band noise frequencies. When a sub-harmonic is present in the power

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Figure 3.2. The (a) Power spectrum and (b) the bicoherence plot showing the ’web’ pattern from the observation 92035-01-03-03. The colour scheme of logb2 is as follows: dark blue:-2.0, light blue:-1.75, yellow:-1.50, red:-1.25 spectrum, its presence is also indicated by a feature in the bicoherence plot where f1 and f2 are equal to half of fQP O. The diagonal elongation of the region from the bottom left to the top right is most likely a result of the frequency drift of the QPO over the length of the observation. This appears to be the only pattern (other than pure harmonic correlation) seen with a type B QPO.

3.5.3 Type A QPOs The bicoherence plots of type A QPOs do not show any discernible patterns such as in the case of type B or C QPOs. To illustrate this with a quantitative example,

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Figure 3.3. The (a) Power spectrum and (b) the bicoherence plot for a type B QPO from the observation 95335-01-01-01. The colour scheme of logb2 is as follows: dark blue:-2.0, light blue:-1.75, yellow:-1.50, red:-1.25

we use the observation 70110-01-45-00 which has a QPO at 7.2Hz (see Fig. 3.4), and examine the mean value of the bicoherence in different regions of the plot using a

frequency resolution of 1/16Hz. Along the diagonal region where f1+f2 are in the range 113 to 117 times the frequency resolution, the bicoherence has a mean value

of 0.029±0.129. In the region where both f1 and f2 are in the range 113 to 117 times the frequency resolution, the bicoherence has a mean value of 0.012±0.011. The value of the bias in the bicoherence for this observation is 0.011. Type A QPOs are not accompanied by either subharmonic or higher harmonic features.

50 Texas Tech University, Kavitha Arur, May 2020

Figure 3.4. The (a) Power spectrum and (b) the bicoherence plot for a type A QPO from the observation 70110-01-45-00. The colour scheme of logb2 is as follows: dark blue:-2.0, light blue:-1.75, yellow:-1.50, red:-1.25. No evidence for phase coupling between the QPO and a harmonic is seen.

3.5.4 Evolution during state transition During the course of an outburst,the source often moves from a hard state, which is often accompanied by a type C QPO to a soft state, which often shows the presence of a type A or type B QPO. In this chapter, we present plots of observations during the 2007 outburst. The details of all the observations analysed, can be found in B. As the source softens and the QPO evolves to higher frequencies, the bicoherence pattern changes, transitioning smoothly from the ‘web’ pattern, to the ‘hypotenuse’. As the source enters the SIMS and a type B QPO is seen, the bicoherence abruptly changes and no longer shows coupling between the QPO and the broad-

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band noise.

Figure 3.5. The power spectrum and the bicoherence plot for multiple observations of GX 339-4 during the 2007 outburst. The colour scheme of logb2 is as follows: dark blue:-2.0, light blue:-1.75, yellow:-1.50, red:-1.25. As the source evolves from HIMS to SIMS, the bicoherence pattern gradually evolves from a ‘web’ to a ‘hypotenuse’ state.

The strength of the bicoherence in the vertical/horizontal streaks steadily falls off as the power in the diagonal structure increases until the ‘hypotenuse’ pattern

52 Texas Tech University, Kavitha Arur, May 2020

is seen. This transition occurs 3-6 days before the source enters the SIMS. This change in the bicoherence pattern is observed most clearly during the 2007 outburst (see Fig. 3.5), but is also seen during the 2002 and 2010 outbursts. Due to a combination of the observations having short exposure and/or low count rate, it was not possible to see if this change occurs during the 2004 outburst.

Figure 3.6 shows the relative strength of the bicoherence along the horizontal and vertical ‘cross’ pattern and the diagonal ‘hypotenuse’ pattern as a function of QPO frequency. The ratio was calculated using the mean value of the bicoherence along the cross divided by the mean value of the bicoherence along the hypotenuse, and this was done for each of the 6 observations shown in Figure 3.5.

While estimating the bicoherence along the ‘cross’, the regions where f1 or f2 were equal to fQP O or 2fQP O were excluded, as these regions would be dominated by the coupling between the QPO and the harmonic. Additionally, for estimates of the bicoherence along the ‘hypotenuse’, values from the lowest frequency bins were excluded. Due to the length of the observations, the source was not observed for many cycles at these low frequencies, and thus the bicoherence values in these bins are not reliable. A clear downward trend is seen in Figure 3.6, indicating that the strength of the bicoherence along hypotenuse increases relative to that along the cross.

The possibility of quasi periodic oscillations being produced by a damped, forced oscillator has been previously suggested [143]. It is also well known that a damped, forced, non-linear oscillator, also known as a Duffing Oscillator [148], is able to produce quasi-periodic oscillations. [See e.g [149] for an application of the Duffing oscillator to longer timescale variability in an X-ray binary.] We find that an increase in the driving frequency of the oscillator, results in the bicoherence pattern of the resulting lightcurves evolving in a manner similar to that shown in Fig. 3.5. However, the connection between the parameters of the oscillator and physical quantities (such as viscosity, optical depth and mass transfer rate) is presently unclear.

53 Texas Tech University, Kavitha Arur, May 2020

Figure 3.6. The ratio between the mean value of the bicoherence along the ’cross’ and the ‘hypotenuse’. It can be seen that the strength of the bicoherence along the ‘cross’ with respect to that along the ‘hypotenuse’ decreases during the transition. The solid line shows a linear ﬁt with a reduced χ2 of 1.9, and a null hypothesis of 0.11 indicating that a simple power law is an adequate description of the data.

It is also worth noting that the amplitude type B and C QPOs show dependence on the inclination angle of the source [34], with the type C QPOs being stronger in the high inclination (edge on) systems, and type B QPOs being stronger in the low inclination (face on) systems. In future work, we will present a comprehensive study of the effects of inclination on the bicoherence.

3.5.5 Reconstructing the QPO waveforms It has previously been shown by [150] that the phase difference between the ﬁrst two QPO harmonics varies around a well deﬁned average, and that this can be used to reconstruct the QPO waveform.

The biphase holds information about the shape of the underlying waveform such as the skewness of the ﬂux distribution and the asymmetry of the time series [45]. It must be noted that a value for the biphase always exists, even when no

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phase coupling is present. Thus the biphase only contains useful information in regions of statistically signiﬁcant bicoherence. The bispectrum provides a formal approach with well understood statistical properties to addressing the question of phase difference between harmonics. The value of the biphase in the region

f1 = f2 = fQP O gives the phase difference between the fundamental and second harmonics. This can be used in a similar fashion to reconstruct the QPO waveform. Since the biphase contains information about the phase difference between any set of three frequencies, a waveform can also be reconstructed with any phase information from any higher harmonics that are present.

Fig 3.7 shows the waveforms of QPOs from the 6 observations shown in Fig 3.5. The relative amplitude of the harmonics was obtained from the height of the peaks in the power spectrum, while the relative phase difference between the harmonics was obtained from the biphase. It can be seen that the phase difference between the harmonics evolves between different observations, leading to a change in the QPO waveform, with the waveform changing from two maxima per cycle to one maxima per cycle.

Figure 3.7. The waveforms of QPOs from the observations in Fig 3.5, reconstructed using the values of the biphase.

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3.6 Discussion 3.6.1 Physical Interpretation In regions where there is a large bicoherence, phase coupling is present among the 3 frequencies involved. The gradual change in the pattern of the bicoherence over the state transition indicates a gradual change in the accretion disk.

Here, we suggest that this behaviour could be explained by an increase in the optical depth of the corona as the sources transitions from the HIMS to the SIMS. In this scenario, during the HIMS the disk component is low and with an optically thin (τ . 1) corona. The QPO, caused by the precession of the inner accretion ﬂow ([36], [139]) , modulates the high frequency variability originating in the inner regions of the accretion disk. This modulation produces the vertical and horizontal lines in the bicoherence. Additionally, the inverse Compton scattering of the soft photons from the thin disk by the corona, causes the low frequency variability of the disk to modulate the QPO. This amplitude modulation causes the weak diagonal line that is seen in the bicoherence. Overall, this produces the web pattern that is seen in the HIMS.

As the source moves towards the SIMS, the optical depth of the corona increases. In this state, the disk blackbody component dominates, with a larger number of soft seed photons being upscattered. This leads to a stronger modulation of the QPO from the low frequency variability, giving rise to the prominent hypotenuse pattern. If the diffusion time of photons through the optically thick corona is larger than the time scales of the high frequency variability, the HF variability is smeared out. This weakens the previously observed vertical and horizontal lines in the bicoherence. Overall this leads to the hypotenuse pattern that is observed in the SIMS.

It has been shown by [151] that there is signiﬁcant smearing of oscillations of frequency f by electron scattering when in a spherical cloud of radius R with an optical depth of τ if 2πfτR/c 1. Assuming a 10M black hole with a Comp-

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3 tonizing region of R=50Rs , an optical depth of τ 3.2 is sufﬁcient to smear out any variations on timescales faster than f∼10Hz.

A similar thermal Comptonization model that invokes an increase in the optical depth of the Comptonizing region was proposed by [152]. This model was used to reproduce the inversion of the sign of the phase lags as the spectrum softens and the QPO frequency increases above 2Hz as seen in GRS 1915+105 [153]. Sign re- versal of the phase lag has also been seen in XTE J1550-564 [131]. Detailed studies of the phase lags of QPOs from XTE J1550-564 have also revealed more complex behaviour such as a difference in the sign of the phase lags between the QPO fundamental and the harmonic [154]. Additionally, the phase lag behaviour of Type C QPOs also show an inclination dependence [35]. Thus a comprehensive study of the evolution of the phase lags in this model would require the computation of a detailed ray tracing model.

In principle, one can assess whether the changes in the optical depth involved here are consistent with model ﬁts of the spectral energy distribution. The changes in the energy spectrum depend on the interplay between the temperature of the electrons in the corona and the optical depth. A recent analysis of the spectra of GX 339-4 in the hard state over multiple outbursts [13] found that the temperature of the corona decreases with increasing luminosity, while the optical depth increases for a nominal value of the photon index. Similarly, a study of the 2007 outburst of GX 339-4 by [155] showed a monotonic decrease of the high energy cut-off during the brightening of the hard state, while the photon index stayed roughly constant. However, during the intermediate state, the cut off energy and the photon index increased, implying a decrease in the optical depth if a purely thermal electron distribution is assumed. But given the clear trends in the hard state, it is reasonably likely that this is the emergence of a non-thermal electron distribution in this state imitating an increase in the cut off energy. Indications of the presence of such a non-thermal component have been seen in Cygnus X-1 in its soft state [15]. Because often spectral cutoffs are inferred without high signal to

3 2 Schwarzschild Radius Rs = 2GM/c

57 Texas Tech University, Kavitha Arur, May 2020 noise data extending well beyond the cutoff energy, it is often hard to estimate the effects of non-thermal components to the electron distribution on the spectrum. In fact, it may be the case that after a full understanding is developed, timing-based approaches like what we start to develop in this chapter may become more robust estimators for the geometrically thick hot region’s optical depth than spectra.

With this in mind, we used the RXTE standard products data to fit Observa- tion 23 from [155], which is the intermediate state observation that is most secure in having a higher cutoff energy than the last few hard state observations before the intermediate state began. We used the same baseline model as [155], but then added a power law component to account for our putative non-thermal Comp- tonization component. When doing this, we find a spectral index of Γ = 2.46 for this extra power law component, with the non-thermal component dominating the total flux across the bandpass. If we freeze the index to 2.2, then the non-thermal component dominates only above 90 keV. In the former case, the cutoff energy for the component with a cutoff drops to 17 keV, while in the latter case it drops to 50 keV. These results are heavily dependent on the quality of the HEXTE background subtraction at the highest energies, but provide good support for our supposition that a non-thermal Comptonization component starts to become important in the intermediate state and affects the inferred location of the cutoff for the thermal electron distribution.

The relative strength of the harmonic to the QPO is dependent on the optical depth, with lower optical depths producing stronger harmonics (see [156] and references within). Such an effect can be seen in the reproduced QPO waveforms, with two maxima seen per cycle (see top panel of Fig. 3.7). As the source softens, and the ﬂow becomes optically thick, the harmonic becomes weaker, leading to a waveform with a single maximum. The maximum of this waveform occurs when the disk at the precession phase that is closest to face-on for the observer (see bottom panel of Fig. 3.7).

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3.6.1.1 Is this a reasonable value for τ? The presence of the hard X-ray lags [157] can indicate a temperature gradient in the corona in the context of the propagating ﬂuctuations model [20, 158] In this case, the viscous propagation of the material as it moves inwards through regions of increasing temperature, leads to the variability at lower energies pre- ceding those at higher energies.

Spectral ﬁtting to obtain optical depths can often yield poor ﬁts for medium signal-to-noise ratio with high τ, especially for a multi- temperature black body. Here, we make an estimate of the optical depth of the corona for a simple model. Consider a torus with a major radius R and minor radius R/2 (i.e. an aspect ratio of 2). The density of the torus can be approximated to be

˙ Mtvisc ρ = (3.3) π2R3 ˙ 2 where M is the mass accretion rate and the viscous timescale tvisc ∼ R /αcsH. H is the scale height with H/R ∼ 1 for a torus, and α is the viscosity parameter [159]. p The sound speed cs is given by kT/mp where mp is the proton mass. Integrating the density along the radius yields the surface density

Z Rout MR˙ M˙ 1 1 Σ = 2 3 dR ∼ 2 [ − ] (3.4) Rin π αcsR π αcs Rin Rout The optical depth of this torus is then given by

τ = σN = σΣ/mp (3.5)

The high energy cut off during the intermediate states has been measured to be kT ∼ 130 keV [155]. Using a value of α = 0.1 , R = 107m and a value of M˙ between 1018 and 1019 g/s gives a value of τ between 1.5 and 15. This agrees with the estimated τ required to explain the scenario described in Section 3.6.1, where at low M˙ values τ < 3.2 and at τ increases at higher accretion rates causing the smearing out of the high frequency variations.

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3.7 Conclusions We have shown that in GX 339-4, in both type B and C QPOs the fundamental frequency is coupled to higher harmonics. Additionally, type C QPOs show coupling between QPO frequency and the broadband noise, while type B QPOs do not. We have also found that the bicoherence gradually changes over the duration of the X-ray outburst from a ‘web’ pattern to a ‘hypotenuse’ pattern.

We also reconstruct the QPO waveforms for 6 observations from the values of the phase difference between harmonics. This was obtained using the value of the biphase, which contains information about the underlying lightcurve. We ﬁnd that the phase difference evolves, leading to a change in the QPO waveform.

Finally, we present the scenario of a moderate increase in the optical depth of the hard X-ray emitting corona to explain the gradual change in the bicoherence pattern that is seen as the source moves from a hard intermediate to a soft intermediate state.

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CHAPTER 4 A LIKELY INCLINATION DEPENDENCE IN THE NON-LINEAR VARIABILITY OF QUASI PERIODIC OSCILLATIONS FROM BLACK HOLE BINARIES

K. Arur and T. J. Maccarone Monthly Notices of the Royal Astronomical Society, January 2020, Volume 491, Issue 1, p.313-323

4.1 Abstract We present a systematic analysis of the effects of orbital inclination angle on the non-linear variability properties of type-B and type-C QPOs from black hole binaries. We use the bicoherence, a measure of phase coupling at different Fourier frequencies for our analysis. We ﬁnd that there is a likely inclination dependent change in the non-linear properties of type-C QPOs as the source transitions from a hard intermediate state to a soft intermediate state. High inclination (edge-on) sources show a change from a ‘web’ to a ‘cross’ pattern, while the low inclination (face-on) sources show a change from a ‘web’ to a ‘hypotenuse’ pattern. We present a scenario of a moderate increase in the optical depth of the Comptonizing region as a possible explanation of these effects. The bicoherence of type-B QPOs do not exhibit any measurable inclination dependence.

4.2 Introduction Quasi-Periodic Oscillations (QPOs) are moderately peaked features seen in the power spectra of Black Hole X-ray Binaries (BHXBs). These QPOs are often divided into two broad categories: high frequency (f > 100Hz) and low frequency (f <100Hz). Due to the short timescales of the variability, QPOs are powerful probes of the geometry of the inner regions of the accretion disk around the black hole. Here, we focus on the Low Frequency QPOs (LFQPOs).

LFQPOs were phenomenologically classiﬁed into types A, B and C, as originally identiﬁed in XTE J1550-564 ([160], [161], [24]). These QPOs have since been

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observed in several BHXBs. Of these three types, type-C QPOs are the most commonly observed, and are seen in the frequency range of 0.1-30Hz. Type-C QPOs are typically narrow (Q1 ≥ 6), and are observed in the hard intermediate state with large rms variability, often along with higher harmonics. Type-B QPOs, seen in the soft intermediate state, are generally broader (Q ∼ 6) and are seen in the 4-7 Hz range. Type-A QPOs are the least commonly observed type. These are seen in the soft state, with their weak and broad (Q ∼ 1-3) peaks making them hard to detect.

Various models have been proposed to explain the origin of these LFQPOs. The suggested QPO production mechanisms include models broadly based on two mechanisms: instabilities and geometrical effects. Instability models invoke different processes such as the presence of a transition layer [162], propagation of magneto-acoustic waves in the corona [33] or accretion ejection instabiltity [29], such as in [30], where the QPO is produced as a result of spiral density waves produced in the accretion disk due to magnetic stresses. The geometric models usually consider Lense-Thirring precession as proposed by [36], where the QPO originates from the relativistic precession of the accretion ﬂow as a solid body [163].

Because these models can all reproduce the observed range of frequencies and amplitudes, sophisticated timing techniques that go beyond the simple power spectrum are required to distinguish between models. One such technique is to characterize and understand the nature of non-linear variability. Non-linearity of the broadband noise in power spectra of accreting black holes has been shown by the presence of a linear rms-ﬂux relation [41] and a log-normal ﬂux distribution [164]. The bispectrum, a measure of phase coupling among triplets of frequencies, is a higher order time series analysis technique that can be used to break degeneracies between models that reproduce very similar power spectra [40]. The bispectrum has been used to detect the presence of non-linearity in the hard state of Cyg X-1 and GX 339-4 [42], detect the presence of coupling between broadband noise and QPO frequencies in GRS 1915+105 [143] and to study the evolution of

1 Q = fcentroid/FWHM

62 Texas Tech University, Kavitha Arur, May 2020 the type-C QPOs across state transitions in GX 339-4 [165].

Inclination dependence of various properties of QPOs have been reported previously. [34] found that the QPO amplitude is dependent on the orbital inclination angle, where the type C QPOs are stronger in high inclination (edge-on) systems, and type B QPOs are stronger in low inclination (face-on) systems.

[166] showed that the effect of QPOs on the power-colour2 properties is inclination dependent. Additionally, it was found by [35] that both the evolution and sign of the energy dependent phase lags of type C QPOs show a strong dependence on inclination. Hints of an inclination dependence was also seen by [167] on the evolution of the phase difference between the QPO and its harmonic, with high inclination sources showing a more consistent pattern than the low inclination ones.

In this chapter, we report on the inclination dependence of the non-linear variability seen from BHXBs using archival data from the X-ray Timing Explorer (RXTE)/ Proportional Counter Array (PCA). In Section 4.3 we give a brief overview of the statistical methods used in this chapter, followed by a description of the data sample and analysis in Section 4.4. In Section 4.5, we present the results on the evolution of the bicoherence patterns of QPOs from low and high inclination sources, and then discuss these results in Section 4.6. Finally, we present our conclusions in Section 4.7.

4.3 Statistical Methods 4.3.1 Bicoherence The bispectrum is a higher order time series analysis technique that can be used to study nonlinear interactions via the coupling of phases of Fourier components. In the same way that the power spectrum is the Fourier domain equivalent of the 2-point correlation function, the bispectrum is the Fourier domain equivalent of

2ratios of integrated power in different Fourier frequency ranges

63 Texas Tech University, Kavitha Arur, May 2020 the 3-point correlation function. The bispectrum of a time series divided into K segments is given by:

K−1 1 X ∗ B(k, l) = Xi(k)Xi(l)X (k + l) (4.1) K i i=0

where Xi(f) is the Fourier transform of the ith segment of the time series at ∗ frequency f, and Xi (f) is the complex conjugate of Xi(f). A related term is the bicoherence, which is given by:

|P X (k)X (l)X∗(k + l)|2 b2 = i i i P 2 P 2 (4.2) |Xi(k)Xi(l)| |Xi(k + l)| This normalization, proposed by [46], measures the fraction of power at frequency k + 1 due to the coupling of the three frequencies. The value of the bicoherence will have a value of 1 if the phase of the bispectrum (biphase) remains constant over time, and approaches zero for large number of measurements if the phases are random. Since the bicoherence is normalised to lie between 0 and 1, a bias of 1/K is subtracted from the bicoherence measurement.

The expectation value of the bispectrum is unaffected by Gaussian noise, but the variance is higher in noisy signals. However, due to the nonlinear nature of Pois- son noise, it can strongly effect the bicoherence at frequencies where the Poisson noise level is comparable to that of the intrinsic variability [164]. In this chapter, we study the bicoherence of QPOs which have low frequencies and have high rms variability, and thus are not strongly affected by the effects of Poisson noise.

4.3.2 Biphase The bispectrum, being a complex number, can be represented as a combination of a magnitude and a phase. This phase is called the biphase. The biphase is de- ﬁned over the full 2π interval, and contains valuable information about the shape of the underlying waveform. Since there always exists a biphase, it must be noted that the value of the biphase is only meaningful in the regions of statistically sig- niﬁcant bicoherence.

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The biphase of f1, f2 and f1+2 gives the average phase difference between these

three components. Thus the biphase where both f1 and f2 are equal to fQP O gives the phase difference between the fundamental and ﬁrst harmonic of the QPO. Us- ing this method, the phase difference between any two harmonics is easily calculated.

The imaginary component of the bispectrum contains information on the reversibility of the lightcurve, while the real component probes the asymmetry of the underlying ﬂux distribution. Thus certain aspects of non-linearity (such as the kurtosis of the ﬂux distribution) cannot be revealed using the bispectrum, and require the use of further higher order spectra. For a more detailed overview of the biphase, see [45].

4.4 Observations and Data Reduction 4.4.1 Data Sample For our analysis, we use the sample of archival RXTE observations of 14 Galac- tic black hole binaries analysed in [34]. A complete list of the observations that were analysed can be found in Appendix B. For 9 of these sources, either a direct estimation of the inclination angle or an upper or lower limit was obtained. The details of the inclination angles of the sources are listed in Table 4.1. Apart from these estimates, the presence or absence of absorption dips was used as the main discriminator between high and low inclination sources respectively.

The sources with inclination angles greater than ≈ 65-70 degrees are classified as high inclination, and low inclination below this value. XTE J1859+226 and MAXI J1543-564 were classified as intermediate inclination as they were not unambiguously a part of the previous two categories. (For more details on the classification of the inclination angle of these sources, see Section 2 and Appendix B of [34]).

For a reliable estimation of the bicoherence, it is necessary that the observation has a high count rate, as well as is long enough to observe multiple cycles at the frequency of interest. For this reason, only observations that have a count rate of

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Table 4.1. List of sources and outbursts used for analysis in this work.

Source Comment Outburst Inclination Reference Angle Swift J1735.5-01 Low 2005-2010 ∼40-55◦ [168] 4U 1543-47 Low 2002 20.7 ± 1.5◦ [2] XTE J1650-500 Low 2001 >47◦ [169] GX 339-4 Low 2002,04,07,2010 ≥40◦ [170] XTE J1752-223 Low 2009 ≤ 49◦ [171] XTE J1817-330 Low 2006 XTE J1859+226 Int 1999 ≥60◦ [172] MAXI J1543-564 Int 2011 XTE J1550-564 High 1998, 2000 74.7 ± 3.8◦ [173] 4U 1630-40 High 2002-03 GRO J1655-40 High 2005 70.2 ± 1◦ [174] H1743-322 High 2003,04,08,09,2010 75 ± 3◦ [175] MAXI J1659-152 High 2010 XTE J1748-288 High 1998

>750 counts/s/PCU and an observation length of >1ks are included in our sample. Additionally, observations showing signiﬁcant (>25%) variations in the mean value of the ﬂux between segments were excluded, as they are considered to be non-stationary.

4.4.2 Data Reduction and Analysis We examine the timing data of the observations using Single Bit, Event or GoodXenon mode data. The data were ﬁltered to exclude periods of high offset, Earth occultation and passage through the South Atlantic Anomaly (SAA). Data from the 2- 30keV energy band was extracted for analysis.

We obtained background corrected STANDARD 2 data in the bands B = 3.6-6.1 keV (corresponding to Std2 channels 4-10 in epoch 5) and C= 6.5-10.2 keV (corresponding to Std2 channels 11-20). The hardness ratio is then deﬁned to be HR = C/B [176, 34].

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In order to extract the power spectra, the data from each observation were binned with a time resolution of 1/128s (with the exception of XTE J1748-288, where a binning of 1/256s was used due to a higher QPO frequency). The data are Fourier transformed using 8s or 16s segments and averaged. The power spectra were then normalised according to [19]. The Poisson noise is not subtracted in the calcula- tion of the bicoherence, so for the regions of the power spectrum at the highest frequencies, the bicoherence may be suppressed. In practice, this is not important for the purposes of this chapter because of the aforementioned focus on high count rate observations, and on the lower frequency parts of the power spectrum. The bicoherence and biphase of the observations were calculated using the equations shown in Section 4.3.

4.5 Results 4.5.1 Bicoherence Patterns For the classiﬁcation of the different phenomenological patterns seen in the bicoherence, we follow the conventions used in [143] and [165].

4.5.1.1 Hypotenuse The ‘hypotenuse’ pattern is seen where a high bicoherence is seen in the diagonal region where the two frequency components add up to the frequency of the QPO

(see Fig. 4.1). This pattern also shows a region of high bicoherence where both f1 and f2 are equal to fQP O. This indicates coupling between the fundamental and harmonic frequency components. This pattern is seen in predominantly the low inclination sources in our sample.

4.5.1.2 Cross In the ‘cross’ pattern, high bicoherence is seen for frequency pairs where one frequency is that of the QPO, and the other frequency can be of any value. This leads to the prominent vertical and horizontal streaks as seen in Fig. 4.2. This pattern is seen in predominantly the high inclination sources in our sample.

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Figure 4.1. The power spectrum (Top panel) and the bicoherence plot (Bottom panel) showing the ’hypotenuse’ pattern from the low inclination source GX 339-4 (observation 70109-04-01-01). The colour scheme of logb2 is as follows: dark blue:- 2.0, light blue:-1.75, yellow:-1.50, red:-1.25

4.5.1.3 Web The ‘web’ pattern is a hybrid class, where both the diagonal feature of the ‘hypotenuse’ and the vertical and horizontal streaks from the ‘cross’ pattern are seen. An example of this pattern can be seen in Fig. 4.3. This pattern is seen in both the high and low inclination sources in our sample, mostly in observations where the QPO frequency is low (< 2 Hz) and the source is in LHS/HIMS.

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Figure 4.2. The power spectrum (Top panel) and the bicoherence plot (Bottom panel) showing the ’cross’ pattern from the high inclination source XTE J1550-564 (observation 30188-06-11-00). The colour scheme of logb2 is as follows: dark blue:- 2.0, light blue:-1.75, yellow:-1.50, red:-1.25

4.5.2 Evolution of type C QPOs 4.5.2.1 Low inclination sources It has previously been shown in [165] that during the softening phase of the outburst, as GX 339-4 moves from HIMS to SIMS, a gradual change is seen in the bicoherence. The pattern moves from being a ‘web’ to a ‘hypotenuse’, as the strength of the bicoherence along the diagonal steadily increases, and that along the vertical and horizontal streaks falls off. This is shown in Fig 4.4 for the 2007 outburst where this change is seen most clearly. This transition is also observed during the outbursts that occurred in 2002 and 2010. Due to fact that only a few observations

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Figure 4.3. The power spectrum (Top panel) and bicoherence plot (Bottom panel) showing the ’web’ pattern from the high inclination source H1743-322 (observation 80146-01-36-00). The colour scheme of logb2 is as follows: dark blue:-2.0, light blue:-1.75, yellow:-1.50, red:-1.25 are available during the 2004 outburst, we do not observe any change in the bicoherence in this period.

Three of the six low inclination sources in our sample3 only have a few observations of QPOs during state transition. From XTE J1752-223, two out of the three observed type C QPOs show the ‘hypotenuse’ before a type B QPO is observed. There are only 2 observations of type C QPOs from XTE J1817-330, both with QPO rms < 3%. Similarly, the three observations of Type C QPOs from 4U 1543-47 be-

3As we expand our study of the non-linear properties of QPOs to multiple sources, it must be noted that our sample of type C QPOs from low inclination sources is dominated by GX 339-4.

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fore the source transitions into SIMS also have very low (< 3%) QPO rms.

Swift J1735.5-01 entered the HIMS, but instead of transitioning into the soft states, returned to the hard state after a few months. Thus the data from this source are dominated by type C QPOs with frequencies f <1Hz.

XTE J1650-500 is the only other low inclination source in our sample that has signiﬁcant coverage of during the 2001 outburst. During this outburst, a transition from ‘web’ to ‘hypotenuse’ is observed, but is not as clearly visible as GX 339-4.

4.5.2.2 High inclination sources In the sample of high inclination sources, we observe a consistent pattern of change through the state transition in the bicoherence from XTE J1550-564, GRO J1655- 40, H 1743-322 and MAXI J1659-152. In these objects, as the source transitions from the HIMS towards the SIMS, the strength of the bicoherence along the ‘hypotenuse’ decreases, while the strength of the bicoherence along the vertical and horizontal streaks increases. This leads to a gradual transition from a ‘web’ pattern to a ‘cross’ pattern. Fig 4.5 shows an example of such a transition as seen in XTE J1550-564.

No patterns were detected in the bicoherence of 4U 1630-47, but this may be due to poor sensitivity. For this source, almost all the observations had a QPO rms<3%. The 2 observations that have a high QPO rms both have count rates < 100 counts/sec.

4.5.2.3 Hardening phase of the outburst When the source is in the HIMS during the hardening phase of the outburst, 3 of the high inclination sources [GRO J1655-40, H1743-322 and XTE J1748-288] brieﬂy show the ‘hypotenuse’ pattern in their bicoherence (see Fig 4.6). However, due to either a lack of further observations (in the case of GRO J1655-40) or a re-softening of the source (H1743-322), the evolution of this pattern could not be analysed. In the case of XTE J1748-288, only one observation showed the ‘hypotenuse’ pattern, with all subsequent observations exhibiting no detectable pattern in their bicoher-

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Figure 4.4. The power spectrum and the bicoherence plot for multiple observations of GX 339-4 during the 2007 outburst. The colour scheme of logb2 is as follows: dark blue:-2.0, light blue:-1.75, yellow:-1.50, red:-1.25. As the source evolves from HIMS to SIMS, the bicoherence pattern gradually evolves from a ‘web’ to a ‘hypotenuse’ pattern. ence.

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Figure 4.5. The power spectrum and the bicoherence plot for multiple observations of XTE J1550-564 during the outburst in 1998. The colour scheme of logb2 is as follows: dark blue:-2.0, light blue:-1.75, yellow:-1.50, red:-1.25. As the source evolves from HIMS to SIMS, the bicoherence pattern gradually evolves from a ‘web’ to a ‘cross’ pattern.

4.5.2.4 Intermediate inclination sources XTE J1859+226 and MAXI J1543-564 could not be unambiguously placed into either the low inclination or high inclination categories. All ﬁve observations of type C QPOs from MAXI J1543-564 have low count rates (<100 counts/sec), and thus re-

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Figure 4.6. The power spectrum (Top panel) and bicoherence plot (Bottom panel) showing the ’hypotenuse’ pattern during the hardening phase of an outburst from the high inclination source GRO J1655-40 (observation 91702-01-79-00). The colour scheme of logb2 is as follows: dark blue:-2.0, light blue:-1.75, yellow:-1.50, red:-1.25 liable bicoherence measurements could not be obtained. However, extensive coverage of the 1999 outburst from XTE 1859+226 was available for our analysis. We ﬁnd that this source shows a transition from ‘web’ to ‘cross’ pattern, consistent with the behaviour of high inclination sources. This is also consistent with previous ﬁndings of [34] where the QPO and noise rms properties of XTE J1859+226 and of [35] where the relative difference between the QPO centroid frequencies in different energy bands and the phase lag behaviour indicate that XTE J1859+226 is a high inclination source.

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4.5.2.5 Quantifying the inclination dependence The inclination dependence of this change is illustrated in Fig. 4.7, where we plot the ratio of mean values of the bicoherence along the hypotenuse and cross as a function of QPO frequency. In this plot we include the sources that have a reliable bicoherence estimates during the softening phase of the transition over a range of QPO frequencies. This includes GX 339-4 and XTE J1650-500 for the low inclination sample (blue points) and H1743-322, XTE J1550-564 and MAXI J1659- 152 for the high inclination sample (red points).

For the estimation of the bicoherence along the ‘cross’, regions where 1.3fQP O > f > 1.8fQP O and where 1.3f2QP O > f > 1.8f2QP O were excluded, as these regions would be dominated by the coupling between the QPO and the harmonic. Simi- larly, when estimating the bicoherence along the ‘hypotenuse’ the region around f1=f2=0.5fQP O was excluded for observations where the QPO frequency is greater than 2Hz to avoid regions dominated by coupling between the QPO and the subharmonic. It can be seen in Fig. 4.7 that this ratio decreases in low inclination sources above 2Hz, indicating a strengthening ‘hypotenuse’ and a weakening ‘cross’. The opposite behaviour is seen in the high inclination sources, indicating a strengthening ‘cross’ and a weakening ‘hypotenuse’. XTE J1859+226, which is of unknown inclination (green points) shows a high ratio of cross to hypotenuse, similar to that of the high inclination sources.

4.5.3 Statistical Signiﬁcance While a distinct difference can be seen in the behaviour between high and low inclination sources, the sample size of the sources is small. In this section, we investigate the possibility that the behaviour of the bicoherence is speciﬁc to the source and the inclination dependence has arisen from random chance. Thus our null hypothesis is that the behaviour of the bicoherence does not depend on the inclination angle of the source.

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Figure 4.7. The ratio between the mean value of the bicoherence along the ‘hypotenuse’ and ‘cross’ as a function of QPO frequency. The green squares indicate the source XTE J1859+226 which is of unknown inclination, but shows a behaviour consistent with high inclination sources.

First, we consider the sample of 3 high inclination (H1743-322, XTE J1550-564 and MAXI J1659-152) and 2 low inclination (GX 339-4 and XTE J1650-500) sources. Assuming that each source has a 50% chance of falling into either behaviour category, using a binomial distribution gives a p-value of p=0.0625 that the behaviour coincides with source inclination.

Based on the QPO and noise rms properties and the phase lag behaviour, if XTE J1859+226 is included in our sample as a high inclination source the p-value now becomes p=0.03125. These values are strongly suggestive of an underlying inclination dependence, especially when considering the inclination dependence of other variability properties. However, it is also possible that any individual source

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is more likely to exhibit one bicoherence behaviour over the other. Thus the addition of more sources is required to make a more deﬁnite claim on the statistical signiﬁcance of the inclination dependence.

4.5.4 Type B QPOs Almost all of the observations of type B QPOs show a high bicoherence in the

region where both f1 and f2 are equal to fQP O, indicating the presence of a second harmonic (often visible in the power spectrum). The presence of a sub harmonic

is also occasionally indicated by a feature in the bicoherence plot where f1 and f2 are equal to half of fQP O. However, no coupling is seen between the QPO and the broadband noise frequencies. Whether this is due to the weakness of the broadband noise in the SIMS, or a true lack of coupling as a result of type-B QPOs arising from a physically different mechanism is presently unclear. An example of the bicoherence from a Type B QPO is shown in Fig. 4.8. Additionally, unlike type C QPOs, the type B QPOs do not show any measurable inclination dependence in their bicoherence patterns.

4.6 Discussion In this section, we expand on the model of a moderate increase in the optical depth of the corona as outlined in [165] to explain the inclination dependence in the gradual change of the bicoherence. While the results are model independent, we ﬁrst consider an interpretation in the context of the Lense-Thirring precession model with the optical depth varying over the precession timescales, as this model is well developed and easy to apply simple prescriptions to.

4.6.1 Low inclination As described in [165], an increase in the optical depth of the corona (from τ ∼ 1 to τ ∼ 4) could lead to the bicoherence pattern gradually changing from a ‘web’ to a ‘hypotenuse’ pattern. In this interpretation, the initial ‘web’ pattern seen in the HIMS is a combination of two effects: 1. The low frequency variability originating

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Figure 4.8. The power spectrum (Top panel) and bicoherence plot (Bottom panel) showing the typical pattern seen from a Type B QPO. The plot is from the low inclination source XTE J1817-330 (observation 92082-01-02-04). The colour scheme of logb2 is as follows: dark blue:-2.0, light blue:-1.75, yellow:-1.50, red:-1.25 from the outer regions of the accretion disk modulate the QPO as the soft photons from the thin disk are upscattered by the corona. This produces the diagonal region of high bicoherence, which is produced by amplitude modulation. 2. The precession of the inner accretion ﬂow modulating the high frequency variability that originates from the inner region of the accretion disk. This produces the vertical and horizontal regions of high bicoherence.

As the source moves from the HIMS to SIMS, the QPO frequency increases and the spectrum of the source is dominated by the disc black body component. As

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there is a larger number of seed photons available for inverse Compton scattering by the corona, the ‘hypotenuse’ becomes stronger in this state. On the other hand, an increase in the optical depth of the corona results in the high frequency variations being smeared out as the time scales for the diffusion of the photons through the corona is larger than the time scale of the variation. This results in the horizontal and vertical stripes having lower bicoherence as the phases of the HF variability and the QPO are no longer coupled. Overall, this results in the bicoherence moving from a ‘web’ to a ‘hypotenuse’ pattern as seen in Fig. 4.4.

4.6.2 High inclination In the case of high inclination sources, the initial detection of a ‘web’ pattern is likely due to the combination of the same two effects described in Section 4.6.1.

However, as the source transitions to the HIMS, the bicoherence pattern gradually changes to a ‘cross’ pattern. Such a change is possible if the shape of the optically thick corona is radially extended with a low scale height (e.g as in a toroidal region). In this scenario, the optically thick (τ ∼ 4) corona is viewed at high inclination (i.e edge on). As this region precesses, the optical depth along the line of sight to the observer is either low or high depending on the phase of the precession (see Fig 4.9). When the torus is viewed edge-on, the high frequency variability is scattered out due to the higher τ along the line of sight. However, when the torus is viewed more face-on, the τ along the line of sight is lower, and the high frequency variability is not scattered out. This causes the high frequency variability to be strongly coupled to the QPO, leading to the prominent ‘cross’ pattern that is observed.

While the spectrum is dominated by the disc black body component, the diagonal ‘hypotenuse’ component is not observed in the high inclination sources during the softening phase. This can be explained in the above scenario if the optical thick corona is radially extended. In this case, due to the high optical depth, the increase in the X-ray luminosity in response to an increase in the mass accretion rate is delayed due to multiple scattering events. If the timescale of the scatter-

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Figure 4.9. Left Panel: An illustration of the variation in the optical depth along the line of sight for different QPO phases for high inclination sources. Right Panel: A plot of the variation of the optical depth with QPO phase. The dotted line indicates the phases where the optical depth is high, and the solid line indicates the phases where it is low for a torus of H/R=0.3. A misalignment angle of 10◦, azimuthal angle of 5◦ and an inclination angle of 80◦ is assumed. In this case, the optical depth varies with the QPO phase. ings is longer than the timescale of the accretion rate ﬂuctuations, this results in a diluted coupling between the two, leading to a loss of coherence and the lack of a ‘hypotenuse’ feature. This scenario also explains the brief emergence of the ‘hypotenuse’ pattern in the high inclination sources, as the optical depth of the Comptonizing corona drops in the hardening phase of the outburst.

4.6.3 Variation of the optical depth with QPO phase To provide a more quantitative illustration of the scenario described in the previous section, we assume the geometry and coordinate system described in [177]. Here, the BH spin is misaligned with the spin of the binary by an angle β. The

‘binary’ coordinate system has z-axis zˆb that aligns with the orbital motions axis of the binary. In these binary coordinates the vector pointing from the BH to the observer is given by

oˆ = (sin i cos Φ, sin i sin Φ, cos i) (4.3)

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Figure 4.10. Left Panel: An illustration of the variation in the optical depth along the line of sight for different QPO phases for low inclination sources. Right Panel: A plot of the variation of the optical depth with QPO phase. The dotted line indicates the phases where the optical depth is high, and the solid line indicates the phases where it is low for a torus of H/R=0.3. A misalignment angle of 10◦, azimuthal angle of 5◦ and an inclination angle of 40◦ is assumed. In this case, the optical depth remains low throughout the QPO cycle.

where i is the inclination angle and Φ is the azimuth of the observer. Assuming that the z-axis of the accretion ﬂow zˆf precesses around the BH spin axis, main- taining a misalignment angle of β as the precession angle ω increases. Thus in the ‘BH’ coordinate system, the observers line of sight can be written as

oˆ = (sin θ0, 0 , cos θ0) (4.4)

where cos θ0, the angle between the observer and the BH spin axis, is given by:

cos θ0 = sin i cos Φ sin β + cos i cos β (4.5)

Thus the z-axis of the ﬂow zˆb can be written in this coordinate system as:

zˆf = ( sin β cos (ω − ω0), sin β sin (ω − ω0), cosβ) (4.6)

where ω0 is the precession angle at which the projection of zˆf onto the BH equa-

torial plane aligns with the BH x-axis (see Fig.1 of [177]). ω0 is given by

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sin i sin Φ tan ω0 = (4.7) sin i cos Φ cos β − cos i sin β

The projection of the ﬂow z-axis zˆf onto the observer line of sight as it precesses is plotted in Fig.4.9 for high inclination and in Fig.4.10 for low inclination sources.

We also assume a simple model for an accretion ﬂow such that if the angle between the observer and the line perpendicular to zˆf (the ‘viewing angle’) is greater than a value ψ, the optical depth along the line of sight is low. If the viewing angle is lower than ψ, the optical depth along the line of sight is high. ψ is given by the inverse tangent of the H/R. It can be seen in Fig. 4.9 that for high inclination example, the optical depth along the line of sight varies with QPO phase. However, for the low inclination example, the optical depth along the line of sight remains low throughout the QPO cycle.

4.6.4 Optical depth and geometry of the corona Oscillations of frequency f are shown to be smeared by electron scattering when in a spherical cloud of radius R with an optical depth of τ if 2πfτR/c 1 [151]. Thus, it is possible to estimate a lower limit for the optical depth along the different lines of sight, based on the frequencies involved in the scenario outlined above.

For the low inclination sources, the high frequency (∼ 10Hz) variability of the

‘cross’ pattern becomes obscured. Assuming a 10M black hole with a comptonising region of 100 gravitational radii, an optical depth of τ 3.2 is required when the corona is viewed face on.

In the case of high inclination sources, the low frequency (∼ 1Hz) variability of the ‘hypotenuse’ pattern is obscured. This requires an optical depth of τ 32 when the corona is viewed edge on. This combination suggests that the corona is radially extended, with H/R ∼ 0.3 (since the optical depth increases exponentially with radius).

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To investigate the value of τ for the proposed scenario, here we estimate the optical depth of a radially extended torus assuming an alpha disk model [178]. The density of the torus with major radius R can be estimated to be:

˙ Mtvisc ρ(R) = (4.8) π2R3 2 where the viscous timescale tvisc ∼ R /αcsH, with H being the scale height. The p sound speed cs ∼ kT/mp where mp is the proton mass.

The optical depth τ = σT S/mp, where the radially integrated surface density S is given by:

Z Rout MR˙ 2 M˙ 1 1 S(R) = 2 3 dR ≈ 2 − (4.9) Rin π αcsHR π α(H/R)cs Rin Rout

Here we assume a value of 0.1 for α and H/R = 0.3 as estimated above. A value of kT ∼ 130 keV was assumed as this was the high energy cut off measured during the intermediate state of GX 339-4 [155].

19 7 For a mass accretion rate of 10 g/s and a value of Rin = 10 m (with Rin <<

Rout), this yields a τ = 38.6 that is consistent with the scattering time scales required. However, it must be noted that this value for the inner radius of the accretion ﬂow is much larger than those assumed in the Lense-Thirring models. For 5 5 smaller inner radii such as Rin = 3 × 10 m (∼ 5Rg), τ = 1287. At Rin =10 m (∼

1.5Rg), the optical depth becomes τ = 3860.

It should also be emphasized that the optical depth depends on the details of geometry of the torus, and thus can easily vary by a factor of ∼3 by changing the extent to which the torus is radially extended.

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4.6.5 Caveats We emphasise that the calculations in the interpretation above have many sim- plifying assumptions, and more complete numerical simulations are needed in order to make detailed comparisons between the model and observations.

While the calculations above are based on the the model of [151], the geometry assumed for the scattering timescales is based on a source of the seed photos that is embedded in a spherical Comptonizing cloud. In the case of a radially extended toroidal structure illuminated by seed photons from a disk, the scattering time scales are likely to be much shorter.

The optical depth estimates are made under the assumption of a constant sound speed and a constant H/R throughout the region. However, the scale height of the disk H is set the balance between vertical pressure and gravity, and thus de-

pends on the sound speed such that H = cs/Ωk where Ωk is the Keplerian angular velocity. In this case s H cs cs R = q = (4.10) R GM c Rg R R3 For the sound speed at kT = 130keV, the value of H/R equals 0.3 at R ≈ 107m, and becomes thinner at small radii. Additionally, it is likely that thicker accretion ﬂows require a 2 temperature plasma, where the protons are at a much higher temperature than the inferred electron temperature.

Changes in the optical depth will not have one-to-one correspondences with changes in the fitted source spectra, unless the corona is single temperature, has a purely thermal electron distribution and maintains a specific radial density profile. In the brightest hard states and in intermediate states, where these quasi-periodic oscillations are most often seen, it is likely that the corona has a range of temperatures and it is not clear how its geometry may be changing with time. Further- more, many observations in these states which extend into the soft gamma-rays show non-thermal tails [15]. However, the observed spectra provide useful con-

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straints on the optical depth. In the cases where extremely high optical depths are predicted, the spectra would tend towards that of a blackbody, in contrast with the observations. If a broad range of temperatures is present and/or the electron energy distribution includes a substantial non-thermal component, higher optical depths can be accommodated while still allowing the spectrum not to look like a blackbody, but as τ approaches hundreds, as is needed to have scattering-induced light travel time delays produce the relationships seen here if the coronal size is small, then the spectrum must tend toward looking like a blackbody corresponding to the outer region’s temperature; the model proposed is likely viable only if the coronal size is at the upper end of the possible range and the range of temperatures is quite large.

We have focused our discussion of the bispectral analysis on its meaning in the context of the Lense-Thirring precession model, but the observational results are model-independent. We emphasize that interpretation is presented as an outline to motivate future detailed modelling efforts to explain the non-linear behaviour of QPOs. It is possible that other models could also produce such non- linearity. For example, in the accretion ejection instability (AEI) model, the harmonics are produced are a result of general relativistic effects and show an inclination dependence [179]. Also in this model, the QPO is linked to the band limited noise, making it possible to reproduce the hypotenuse pattern (P. Varniere,` pri- vate communication). However, due to computational expenses involved in the detailed numerical simulations of this model, analytic approximations are additionally not straightforward, and current numerical simulations do not have data with the combination of sufﬁcient length and time resolution to probe the non- linearity quantitatively, so future numerical work will be required to compare the AEI model to the bispectral data.

4.7 Conclusions We have presented the ﬁrst systematic analysis of the inclination dependence of the non-linear properties of QPOs from BHXBs. We ﬁnd that:

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1. Type C QPOs from high inclination sources show a gradual change from ‘web’ to ‘cross’ bicoherence patterns.

2. Type C QPOs from low inclination sources show a gradual change from ‘web’ to ‘hypotenuse’ bicoherence patterns.

3. The evolution of Type C QPOs from the intermediate inclination source XTE J1859+226 is consistent with our sample of high inclination sources.

4. Type-B QPOs do not show any inclination dependence in their bicoherence patterns.

We also propose a scenario of an increase in the optical depth of a radially extended corona to explain the change in the bicoherence in both the high and low inclination sources.

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CHAPTER 5 SUMMARY AND OUTLOOK

5.1 Summary Studies of the population of compact binaries in the Galaxy are important for our understanding of the evolution of binary systems. These systems are progenitors for type Ia supernovae, millisecond pulsars and gravitational wave events as a result of compact binary coalescence. However, there are large gaps in our knowledge of binary evolution, and certain processes such as the phase of common envelope evolution are not well understood. One way to test our knowledge of binary evolution is through the study of population of compact binaries in our Galaxy and compare their statistics to those predicted by binary population synthesis models.

One major discrepancy between observations and models is the dearth of black hole binaries at short (<4 hour) orbital periods. A pile up of sources is expected at these periods as the binary orbit decays due to loss of angular momentum via magnetic braking and gravitational radiation. One possible explanation for the lack of short period binaries is that black hole binaries present in the Galaxy are not being detected due to various selection effects.

In order to better understand how different selection effects affected the observed orbital period distribution of black hole X-ray binaries, a simulation that accounted for the probability of detection of the initial X-ray outburst (based on the maximum luminosity, duration and frequency of the outburst), as well as the probability of follow up of the optical counterpart was conducted. One key phenomenon that was taken into account was that at short periods black holes have intrinsically fainter X-ray outbursts compared to their neutron star counterparts. This is due to a radiatively inefﬁcient accretion ﬂow, where the energy is advected beyond the event horizon. After correcting for these effects, the results of the simulation show that approximately 600 binaries are expected with orbital periods between 3 and 10 hours, with an additional 200-3000 objects expected to lie be-

87 Texas Tech University, Kavitha Arur, May 2020 tween 2 and 3 hour orbital periods. The large uncertainty in these numbers is due to the small number of short period sources currently known.

Rapid X-ray variability from black hole binaries also provides one of the best tests of general relativity in the strong ﬁeld limit as it probes the regions closest to the black hole. One feature that can commonly be seen in the X-ray power spectra of these systems are quasi-periodic oscillations. Decades of research into the origin of these oscillations have resulted in multiple classes of models being proposed to explain this phenomenon. However, as many of these models can successfully reproduce the observed periodicities, there has been no general consensus as to how QPOs are produced.

One powerful way to break the degeneracies between different QPO models is through the use of a higher order time series analysis technique known as the bispectrum. The bispectrum is the Fourier domain equivalent of the three point correlation function, and probes the time reversibility (in a statistical sense) and the ﬂux asymmetry of the underlying light curve. It also provides a method of measuring the strength of non-linearity of the time series through the phase coupling between different frequency components.

The work in this dissertation detailed the results of the first systematic application of the bispectrum to all observations of the black hole source GX 339-4 that showed the presence of a QPO. This analysis showed for the first time that the nature of the non-linear coupling between the QPO and the broadband noise gradually changed with a change in the mass accretion rate during the transition from the hard intermediate to the soft intermediate state. These results are interpreted in the context of the precessing accretion flow model as a moderate increase in the optical depth of the corona. Using the information contained in the biphase, it was also possible to reproduce the QPO waveform. The waveform of the QPO also showed a change, going from having two maxima to a single maxima during the state transition. This finding also supports the idea of increasing optical depth of

88 Texas Tech University, Kavitha Arur, May 2020 the corona.

Following the promising results from GX 339-4, this analysis technique was applied to a much larger sample consisting of all black hole binaries that had sufﬁ- cient coverage by RXTE during an outburst. The results of this analysis showed that there was a likely inclination dependence in the nature of the variation in the bicoherence during the course of the outburst. Namely, high inclination (edge on) sources varied from a web pattern to a cross pattern in their bicoherence, while low inclination (face on) sources varied from a web pattern to a hypotenuse pattern.

Extending the interpretation of an increasing optical depth of a precessing corona, the results from this larger sample can be interpreted as a torus shaped corona where the presence or absence of phase coupling between the QPO and noise components depends on the optical depth along the line of sight. If the timescale of the variation is larger than the Compton scattering timescale, the phase coupling is preserved. On the other hand, if the Compton scattering time scale is larger than the timescale of the variations, the phase coupling is likely to be smeared out and not seen in the bicoherence. Additionally, the presence of a non-thermal electron distribution would prevent a one to one correspondence between the optical depth of the corona and the spectra of the black hole. This highlights the need for detailed numerical simulations to examine the bicoherence of QPOs and the associated broadband noise for different QPO models.

This inclination dependent behaviour indicates that QPOs have a geometric origin. However, other models such as those invoking accretion ejection instabilities also exhibit inclination dependence and cannot be ruled out without further inves- tigation.

5.2 Future Work 5.2.1 Black Holes In addition to applications to quasi-periodic variability, the bispectrum (specifically the biphase) can also be used to study the aperiodic variability from black

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Figure 5.1. Schematic of different emissivity profiles for rotating (left), non-rotating black holes (middle) and neutron stars (right). The difference in the symmetry will result in different biphases for each case. hole binaries. A model that describes this variability is the propagating fluctuations model [20][21], where the observed variability is due to mass transfer rate fluctuations in the accretion disk. Each annulus produces variability on the local viscous timescale that propagates inwards, coupling multiplicatively with those generated further in. The observed signal is the product of fluctuations at all radii, modulated by an emissivity profile that determines how the accretion rate is converted into radiation. The variability thus carries an imprint of the emissivity profile (the luminosity as a function of radius). The emissivity increases towards small disk radii as the material encounters progressively deeper gravitational potential wells, followed by a sudden drop as it encounters the region beyond the innermost stable circular orbit (ISCO) where the material plunges in. For a non-rotating black hole, this turn down in X-ray luminosity is expected to be steep. However, it has been suggested that gravitational light bending effects can play an important role for rotating black holes [180]. In such cases, a smoother turn down of the X-ray should be expected, leading to a profile with longer decay times compared to the non-rotating case, at the highest frequencies. Figure 5.1 shows a schematic of the emissivity profiles for rotating and non rotating black holes, as well as neutron stars (also see Section 5.2.2).

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As the biphase describes the shape of a light curve across a broad range of frequencies, this can be used to examine the emissivity profile of the accretion disk. Specifically, asymmetries in the shape of the emissivity profile can be probed using the biphase. Some work has been done on this front by Scaringi et al. [181] who found that in three cataclysmic variables (see also Section 5.2.3) and in a black hole binary, the biphase at high frequencies is positive. This indicates that the underlying flux distribution is positively skewed, consistent with the log normal flux distribution expected from the propagating fluctuations model. However, below the high frequency break in the power spectra, the biphase shows a sudden decrease. There is presently no explanation for this decrease in the biphase, and these results could indicate that the propagating fluctuations model does not suf- ficiently explain the behaviour of the disk at frequencies below the high frequency break. Application of this method to a larger sample, as well are more detailed modelling efforts are required to better understand this effect.

5.2.2 Neutron stars Neutron star X-ray binaries are further classiﬁed into ’Z’ and ’Atoll’ sources based on the paths they trace out on a color-color diagram [182]. Z sources consist of low mass neutron stars accreting at rates greater than ∼50% of the Eddington rate, while in Atoll sources the accretion rate is much lower (∼ 0.1 LEdd).

Neutron star X-ray binaries also exhibit QPOs in their X-ray flux. In Z sources, three types of QPOs have been identified. These are the normal branch oscillations (NBOs), flaring branch oscillations (FBOs) and horizontal branch oscillations (HBOs) and are named after the region of the colour-colour diagram where they appear [183]. NBOs, FBOs and HBOs are thought to be analogous to the type A, B and C QPOs from black holes [24]. From Atoll sources, FBO-like and HBO-like QPOs have been observed, based on the similarities of their properties with the Z-source QPOs. Since weakly magnetised neutron stars exhibit phenomenology in their variability similar to black holes [184], it is expected that the same physical process is responsible for the oscillations in both cases. Thus a careful comparison between the two is systematic differences between, for example type C QPOs and

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HBOs, could help distinguish between the two kinds of systems where the nature of the compact object is unknown.

High frequency QPOs in the kilohertz range (known as kHz QPOs), and often in pairs, are also seen from neutron stars [51] [52] and these are some of the fastest phenomena observed from X-ray binaries. These oscillations occur at frequencies close to the orbital frequency at the innermost region of the accretion disk, and the highest frequencies of these QPOs observed from these sources help provide constraints on the mass and radii of the NS. This is crucial in determining the correct model that describes the neutron star equation of state (EoS). The NS EoS is also of signiﬁcant interest in the broader physics community, as these constraints can help inform the interpretation of nuclear physics experiments and constrain the nuclear matter symmetry energy.

As in the case of low frequency QPOs, multiple models have been proposed to explain the presence of these kHz QPOs. One signiﬁcant aspect of the kHz QPOs (individually referred to as the upper or lower kHz QPO based on its frequency) is that the frequency separation between the peaks is often close to either the spin or half the spin frequency of the neutron star. One prominent model that has been put forth to explain this is the beat frequency model. In this scenario, the twin kHz peaks are produced by a beating between the Keplerian orbital frequency close to the ISCO and the spin frequency [185]. However, this model is not directly compatible with observations of cases where the frequency separation corresponds to half the NS spin frequency [186]. This led to the development of a spin-disk resonance model [187] that also involves a beat interaction, this time between a resonant wave travelling around at the spin frequency and a radiation pattern rotating at the orbital frequency. Alternate resonance models have also been proposed invoking resonances between the orbital frequency and the vertical epicyclic frequency or between the orbital and radial epicyclic frequency to explain the occurrence of these kHz QPOs [188].

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The waveforms of these high frequency QPOs can be reconstructed using the biphase in a similar manner to that of the low frequency QPOs. Examination of the proﬁle of these oscillations could provide a way to potentially distinguish between these mechanisms, especially as a function of accretion rate and inclination angle. Additionally, the study of any interactions between the low and high frequency QPOs could also provide valuable insight into the correct model for the origin of these QPOs. However, as this requires the computation of the bicoherence with high frequency resolution over a large range of frequencies, such studies are computationally very intensive. It is also possible that the intrinsic width of the kHz QPO will be large enough to obscure any interaction with the lower frequency QPO.

Aperiodic variability from neutron stars can also be examined using the biphase in a manner similar to black holes, where the emissivity profile of the accretion disk can be probed. In the case of neutron stars, a significant amount of energy is expected to be released as the material from the disk impacts the boundary layer. This will result in a different shape of the emissivity profile. This asymmetry in the profile can be probed using the biphase, and could provide a potential way to distinguish between neutron star and black hole sources in cases where this would otherwise not be possible (e.g lack of type I X-ray bursts which are a signature of thermonuclear burning on the surface of neutron stars).

5.2.3 Cataclysmic variables Cataclysmic variables (CVs) are binary systems where the accretor is a white dwarf. Due to the larger size of the white dwarf (compared to neutron stars or the event horizon of stellar mass black holes), the characteristic timescales are similarly longer. The larger physical size of the accretion disk also leads to cooler temperatures, with these disks mainly emitting in the optical and ultra-violet wavelengths. Some work has already been done on aperiodic variability from CVs, as described in Section 5.2.1.

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Dwarf novae oscillations (DNOs) are periodic low amplitude modulations seen from CVs during outbursts [189], with correlations that indicate a relationship between the mass accretion rate and the period of the DNO. While DNOs are mostly studied in the optical, they are also observed in the X-ray ﬂux of some CVs. The optical and X-ray DNOs appear to occur at similar periods, but simultaneous detections have not been achieved yet. Models based on beats have been proposed here too, with beating occuring between the rotation period of the magnetosphere and the Keplerian rotation period of the inner edge of the disc [190]. Detection of evidence of such beating using the bispectrum would provide a way to measure the rotation rate of non magnetic white dwarfs, which is currently an important but poorly addressed question in astronomy. However, the predicted periodicities are on the order of tens of seconds, and thus require data with good timing resolution. While such fast optical timing data of CVs are not widely available at present, the capabilities of the Transiting Exoplanet Survey Satellite (TESS), which offers cadences as fast as 20 seconds, will make such observations possible in the near future.

QPOs with very low Q factors have also been seen from CVs, often appearing with harmonics. These QPOs can appear simultaneously with DNOs [191], and in many systems the period of the QPO is the beat period of ”double DNOs” where two DNOs are seen in the power spectrum. If these QPOs and DNOs are considered to be analogous to the low and high frequency QPOs from X-ray binaries respectively, these CV frequencies extend the relation between the two sets of frequencies from X-ray binaries, as shown in Figure 5.2 (see [192] for an overview of QPOs from CVs). The study of the bicoherence of these features could provide insight into if these oscillations are produced by the same physical mechanism as those from X-ray binaries, or if the similarities are purely phenomenological. Conﬁrmation that these are produced by the same mechanism would indicate that QPOs are produced as a result of the accretion process rather than due to general relativistic effects.

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Figure 5.2. Frequencies of HF and LF QPOs for X-ray binaries plotted against each other (ﬁlled squares: BHB; open squares: NS) and 26 CVs (ﬁlled circles). Each CV is only shown once in this diagram. The dashed line marks PQPO/PDNO = 15. From [192]

5.2.4 Active Galactic Nuclei Accreting supermassive black holes at the centre of galaxies (also called Active Galactic Nuclei or AGN) also show phenomenology similar to those seen from black hole binaries (see for e.g [193]). The variability seen from AGN occur on much longer timescales, with the timescales scaling linearly with the mass of the black hole. While there have been some claims of detection of QPOs from AGN, the long timescales of variability mean that these periodicities have not been sam- pled over many cycles. In order to apply higher order statistics to QPOs from these objects, long duration observations with good sampling such as those provided by all sky monitors are required [194]. It is currently not possible to get sufﬁciently

95 Texas Tech University, Kavitha Arur, May 2020 high signal to noise ratio on AGNs with the current generation of all sky monitors. However, with upcoming X-ray missions it might be possible to compare the bispectral properties of QPOs to provide valuable insight into the similarities between the variability properties of AGN and BHBs, as well as a better understanding of how accretion scales with the mass of the black hole.

Application of the biphase to the aperiodic variability to probe the emissivity proﬁle is much more feasible with data that is currently available. Comparison with model predictions (e.g [21]) will similarly be possible with minor modiﬁca- tions due to mass scaling.

5.2.5 Three wave interactions in the solar wind The bispectrum also has the potential to be applied to other areas of astronomy. One such example is the study of three wave interactions in upstream solar wind.

Various types of radio bursts are seen from the Sun, named type I through V (additional sub-types of bursts also exist). Out of these, solar type III bursts are the most proliﬁc (See [195] for an overview of Solar type III bursts). From these bursts, electromagnetic emission is seen at the second harmonic of the local electron plasma frequency. The generally accepted model for this type of bursts is through the ”plasma emission mechanism” where a two beam instability gener- ates plasma oscillations (also known as Langmuir waves) at the local plasma frequency. These Langmuir waves are then converted to electromagnetic radiation through scattering by plasma ions [196].

The second harmonic is thought to arise from the coalescence of two Langmuir waves which produce a transverse wave at twice the local plasma frequency [197]. This three wave interaction requires phase coherence between the two Langmuir waves and the resultant EM wave, which could be probed using the bicoherence. Indeed, the bicoherence was used to look at phase coupling using electric ﬁeld data from the WIND satellite during a period of intense Langmuir wave activity [198], where the data showed some evidence of phase coupling in the Langmuir waves.

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However, as these waves are inherently ”bursty”, the data are non-stationary and result in a large variance in the bicoherence.

5.2.6 Stochastic Gravitational Wave Background Gravitational waves resulting from the merger of binary black holes and binary neutron stars have been detected by the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo interferometers [199]. Such detections indicates the existence of a stochastic gravitational wave background made up of a large number of such merger events beyond current detection thresholds.

The inspiral and merger of black hole binaries causes an increase in the frequency of the gravitational wave emission due to decreasing orbital distance and an increase in their orbital speeds. This produces a gravitational wave “chirp”, where the frequency and amplitude of the signal increases rapidly with time, and abruptly turns off after reaching a peak. The sensitivity of the bispectrum to the time-asymmetry of the underlying signal could potentially provide a method to constrain the rate of such events that occur at distances that are too far away to be individually detected, as while the signal-to-noise for each event would be low, a large number of such signals will be present in the data. However, it must be noted that more work is needed in this area to determine its feasibility.

5.3 Current and Future Missions 5.3.1 NICER The Neutron star Interior Composition ExploreR (NICER) is a high throughput X-ray instrument which was installed on the International Space Station in June 2017. The X-ray Timing Instrument (XTI) aboard NICER has good spectral resolution (few percent) and excellent timing resolution (up to 100 nanoseconds) of X-rays in soft energy bands (0.2-12 keV). The soft energy response combined with the large effective area makes NICER an ideal instrument for studying the bispectrum of low frequency QPOs from neutron star and black hole binaries. The good spectral resolution also enables calculations of the bispectrum in different energy

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Figure 5.3. Artist impression of STROBE-X when deployed. From [200]

bands, which allows us to probe the contribution of different parts of the accretion disk using a single observation in cases where the source is bright enough.

5.3.2 STROBE-X The Spectroscopic Time-Resolving Observatory for Broadband Energy X-rays (STROBE-X) is an probe class mission that has been selected for concept study for NASAs 2020 Decadal survey [116] [200]. STROBE-X will carry 3 instruments: the X-ray Concentrator Array (XRCA), the Large Area Detector (LAD) and the Wide- Field Monitor (WFM) (see Fig 5.3).

The WFM offers a sensitivity that is 15 times larger than that of the RXTE / ASM, which will provide unprecedented coverage of the X-ray sky, ideal for the detection of faint X-ray outbursts such as those discussed in Chapter 2. Additionally, the XRCA will have coverage of the soft X-ray energy range (0.12-12 keV) while the

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LAD provides coverage of the hard X-rays (2-30 keV). Both instruments will have excellent timing capabilities, and provide an order of magnitude improvement in effective area over the previous best instrument in their respective energy ranges (NICER and RXTE/PCA respectively). The timing capabilities of STROBE-X over such a large energy range make STROBE-X an ideal mission for the application of higher order timing analysis such as those discussed in Chapters 3 and 4 of this dissertation.

5.3.3 eXTP Another way to make progress on constraining the X-ray emission geometry is through measurement of polarization from black hole X-ray binaries. General relativistic ray tracing models of the corona with different geometries such as a ’sand- wich’ like corona, a clumpy corona consisting of many small spherical clouds or a spherical corona show some commonalities such as low levels (∼ a few percent) of horizontal polarization at low energies (mainly from the thermal disk) and vertical polarization (up to 10%) at higher energies above the thermal peak. However, differences in the polarization amplitude at different energies and details of the transition between the two orientations can distinguish between these geometries [201]. Polarization measurements can also be used to test QPO models. For example, the Lense-Thirring model predicts the modulation of the degree and angle of the polarization on the QPO frequency [150].

One instrument that has been proposed which would be ideal for these kind of studies is the Chinese enhanced X-ray Timing and Polarimetry mission (eXTP) [202]. eXTP combines polarimetric capability with high spectral and timing resolution to enable such studies of the coronal geometry while disentangling polarization from different components such as the disk, corona and the jet. Additionally, the Spectroscopic Focusing Array (SFA) and the Large Area Detector (LAD) are both silicon drift detectors that will provide a large total effective area as well as excellent time resolution with minimal deadtime even at high count rates, enabling the use of statistics such as the bispectrum for a large number of sources. Coverage

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of the high energy range by the LAD (2-30 keV) will also allow enable such studies to be performed on high frequency QPOs [203], which are spectrally harder.

5.4 Concluding remarks Currently, some models are not well developed enough to be able to make predictions on what the bispectra of the QPOs are expected to be. In the cases where the models are well developed, such as in the case of Lense-Thirring precession or accretion ejection instability, detailed numerical simulations are required in compute the bispectra of the resultant light curves. Currently, lightcurves with sufﬁ- cient resolution and length for bispectral analysis are not available due to computational limitations. However, such efforts to enable comparison with observational results using these higher order methods are currently underway, and further analysis will help motivate these modelling efforts. As outlined in the sections above, higher order analysis techniques such as the bispectrum are a powerful way to better understand the accretion process in X-ray binaries and beyond. I believe that especially with more sensitive detectors on the horizon, such techniques hold great potential in the ﬁeld of X-ray timing.

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129 Texas Tech University, Kavitha Arur, May 2020 era, R. Prix, G. A. Prodi, L. Prokhorov, O. Puncken, M. Punturo, P. Puppo, M. Purrer,¨ H. Qi, J. Qin, V. Quetschke, E. A. Quintero, R. Quitzow-James, F. J. Raab, D. S. Rabeling, H. Radkins, P. Raffai, S. Raja, M. Rakhmanov, C. R. Ramet, P. Rapagnani, V. Raymond, M. Razzano, V. Re, J. Read, C. M. Reed, T. Regimbau, L. Rei, S. Reid, D. H. Reitze, H. Rew, S. D. Reyes, F. Ricci, K. Riles, N. A. Robertson, R. Robie, F. Robinet, A. Rocchi, L. Rolland, J. G. Rollins, V. J. Roma, J. D. Romano, R. Romano, G. Romanov, J. H. Romie, D. Rosinska,´ S. Rowan, A. Rudiger,¨ P. Ruggi, K. Ryan, S. Sachdev, T. Sadecki, L. Sadeghian, L. Salconi, M. Saleem, F. Salemi, A. Samajdar, L. Sammut, L. M. Sampson, E. J. Sanchez, V. Sandberg, B. Sandeen, G. H. Sanders, J. R. Sanders, B. Sassolas, B. S. Sathyaprakash, P. R. Saulson, O. Sauter, R. L. Sav- age, A. Sawadsky, P. Schale, R. Schilling, J. Schmidt, P. Schmidt, R. Schn- abel, R. M.S. Schoﬁeld, A. Schonbeck,¨ E. Schreiber, D. Schuette, B. F. Schutz, J. Scott, S. M. Scott, D. Sellers, A. S. Sengupta, D. Sentenac, V. Sequino, A. Sergeev, G. Serna, Y. Setyawati, A. Sevigny, D. A. Shaddock, T. Shaffer, S. Shah, M. S. Shahriar, M. Shaltev, Z. Shao, B. Shapiro, P. Shawhan, A. Shep- erd, D. H. Shoemaker, D. M. Shoemaker, K. Siellez, X. Siemens, D. Sigg, A. D. Silva, D. Simakov, A. Singer, L. P. Singer, A. Singh, R. Singh, A. Singhal, A. M. Sintes, B. J.J. Slagmolen, J. R. Smith, M. R. Smith, N. D. Smith, R. J.E. Smith, E. J. Son, B. Sorazu, F. Sorrentino, T. Souradeep, A. K. Srivastava, A. Staley, M. Steinke, J. Steinlechner, S. Steinlechner, D. Steinmeyer, B. C. Stephens, S. P. Stevenson, R. Stone, K. A. Strain, N. Straniero, G. Stratta, N. A. Strauss, S. St- rigin, R. Sturani, A. L. Stuver, T. Z. Summerscales, L. Sun, P. J. Sutton, B. L. Swinkels, M. J. Szczepanczyk,´ M. Tacca, D. Talukder, D. B. Tanner, M. Tapai,´ S. P. Tarabrin, A. Taracchini, R. Taylor, T. Theeg, M. P. Thirugnanasamban- dam, E. G. Thomas, M. Thomas, P. Thomas, K. A. Thorne, K. S. Thorne, E. Thrane, S. Tiwari, V. Tiwari, K. V. Tokmakov, C. Tomlinson, M. Tonelli, C. V. Torres, C. I. Torrie, D. Toyr¨ a,¨ F. Travasso, G. Traylor, D. Triﬁro,` M. C. Tringali, L. Trozzo, M. Tse, M. Turconi, D. Tuyenbayev, D. Ugolini, C. S. Un- nikrishnan, A. L. Urban, S. A. Usman, H. Vahlbruch, G. Vajente, G. Valdes, M. Vallisneri, N. Van Bakel, M. Van Beuzekom, J. F.J. Van Den Brand, C. Van Den Broeck, D. C. Vander-Hyde, L. Van Der Schaaf, J. V. Van Heijningen,

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131 Texas Tech University, Kavitha Arur, May 2020

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134 Texas Tech University, Kavitha Arur, May 2020

CHAPTER A DETAILS OF OBSERVATIONS ANALYSED IN CHAPTER 3

Table A.1. Details of the RXTE observations with QPOs used in Chapter 3. The asterisk indicates that QPOs with f <2 Hz were classed as web if pattern is unclear. MJD: Modiﬁed Julian Date

ObsId MJD Year Freq. Type State Mean Pattern [Hz]a Count 40031-03-02-05 52388.054 2002 0.20 C LHS 1055 Web* 70109-01-05-01G 52391.318 2002 0.22 C HIMS 1055 Web* 70109-01-06-00 52400.83 2002 1.26 C HIMS 1120 Web* 70108-03-01-00 52400.853 2002 1.30 C HIMS 1111 Web 70110-01-10-00 52402.492 2002 4.20 C HIMS 1091 Web 70109-04-01-00 52405.58 2002 5.46 C HIMS 1293 Hypotenuse 70109-04-01-01 52405.713 2002 5.45 C HIMS 1289 Hypotenuse 70110-01-12-00 52410.528 2002 8.1 C HIMS 1250 None 70109-01-07-00a 52411.601 2002 7.0 B SIMS 1288 Type B 70109-01-07-00b 52411.601 2002 5.8 B SIMS 1288 Type B 70110-01-45-00 52524.948 2002 7.2 A SIMS 1355 None 70109-01-23-00 52529.580 2002 7.4 A SIMS 1228 None 70109-01-24-00 52536.358 2002 8.0 A SIMS 1112 None 60705-01-68-01 53222.24 2004 1.03 C HIMS 318 Web* 60705-01-69-00 53225.40 2004 1.3 C HIMS 334 None 90704-01-01-00 53226.43 2004 2.0 C HIMS 342 Web* 90704-01-02-00 53233.389 2004 4.4 B SIMS 431 Type B 60705-01-84-02 53333.899 2004 5.2 B SIMS 760 Type B 92035-01-02-01 54133.922 2007 0.28 C LHS 918 Web* 92035-01-02-02 54135.033 2007 0.30 C LHS 934 Web* 92035-01-02-03 54136.015 2007 0.37 C LHS 963 Web* 92035-01-02-04 54136.997 2007 0.43 C LHS 1024 Web 92035-01-02-08 54137.851 2007 0.55 C HIMS 1062 Web* 92035-01-02-07 54138.83 2007 0.90 C HIMS 1078 Web*

135 Texas Tech University, Kavitha Arur, May 2020

Table A.1 Continued. ObsId MJD Year Freq. Type State Mean Pattern [Hz]a Count 92035-01-02-06 54139.942 2007 0.99 C HIMS 1078 Web 92035-01-03-00 54140.204 2007 1.13 C HIMS 1091 Web 92035-01-03-01 54141.055 2007 1.68 C HIMS 1106 Web 92035-01-03-02 54142.036 2007 2.45 C HIMS 1156 Web 92035-01-03-03 54143.019 2007 3.52 C HIMS 1249 Web 92428-01-04-00 54143.870 2007 4.34 C HIMS 1299 Hypotenuse 92428-01-04-01 54143.951 2007 4.23 C HIMS 1304 Web 92428-01-04-02 54144.086 2007 4.13 C HIMS 1317 Hypotenuse 92428-01-04-03 54144.871 2007 4.99 C HIMS 1357 Hypotenuse 92035-01-03-05 54145.114 2007 5.80 C HIMS 1362 Hypotenuse 92035-01-04-00 54147.011 2007 6.7 B SIMS 1549 Type B 92085-01-02-06 54160.896 2007 7.8 A SIMS 1221 None 92085-01-03-00 54161.669 2007 7.1 C HIMS 1178 Hypotenuse 92085-01-03-01 54162.665 2007 6.4 B SIMS 1252 Type B 92085-01-03-02 54163.698 2007 7.3 C HIMS 1117 None 92085-01-03-03 54164.557 2007 7.0 C HIMS 1102 Hypotenuse 92085-01-03-04 54165.527 2007 7.7 A SIMS 1058 None 95409-01-13-03 55288.367 2010 0.2 C LHS 717 Web* 95409-01-13-04 55290.722 2010 0.29 C LHS 754 Web* 95409-01-13-02 55291.649 2010 0.32 C LHS 779 Web* 95409-01-14-02 55297.87 2010 1.25 C HIMS 884 Web 95409-01-14-06 55299.766 2010 2.43 C HIMS 944 Web 95409-01-14-04 55300.336 2010 2.38 C HIMS 942 Web 95409-01-14-07 55300.923 2010 2.92 C HIMS 962 Web 95409-01-15-01 55303.604 2010 5.65 C HIMS 1077 Hypotenuse 95409-01-15-02 55304.714 2010 5.6 B SIMS 1173 None 95409-01-15-06 55308.983 2010 5.9 B SIMS 1110 Type B 95409-01-16-05 55315.695 2010 6.1 B SIMS 1011 Type B 95409-01-17-02 55318.441 2010 6.67 C HIMS 881 None

136 Texas Tech University, Kavitha Arur, May 2020

Table A.1 Continued. ObsId MJD Year Freq. Type State Mean Pattern [Hz]a Count 95409-01-17-05 55321.718 2010 5.3 B SIMS 903 Type B 95409-01-17-06 55322.230 2010 5.2 B SIMS 859 Type B 95409-01-18-00 55323.210 2010 5.5 B SIMS 908 Type B 95335-01-01-07 55324.189 2010 5.3 B SIMS 888 Type B 95335-01-01-00 55324.254 2010 5.3 B SIMS 871 Type B 95335-01-01-01 55324.393 2010 5.1 B SIMS 848 Type B 95335-01-01-06 55326.280 2010 4.9 B SIMS 790 None

137 Texas Tech University, Kavitha Arur, May 2020

CHAPTER B DETAILS OF OBSERVATIONS ANALYSED IN CHAPTER 4

Table B.1. Details of the RXTE observations with QPOs used in Chapter 4. The asterisk indicates that QPOs with f <2 Hz were classed as web if pattern is unclear. MJD: Modiﬁed Julian Date

ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count Low Inclination Swift J1753.5-01 91094-01-01-00 53553.043 2005 0.718 0.639 C 344 None 91094-01-01-01 53555.186 2005 0.717 0.823 C 523 Web 91094-01-01-02 53555.582 2005 0.714 0.842 C 544 Web 91094-01-01-03 53556.169 2005 0.722 0.824 C 554 Web 91423-01-01-04 53557.218 2005 0.706 0.895 C 592 Web 91094-01-01-04 53557.480 2005 0.713 0.832 C 586 Web 91423-01-01-00 53558.201 2005 0.716 0.671 C 554 Web 91094-01-02-01 53559.715 2005 0.711 0.738 C 571 Web 91094-01-02-00 53560.495 2005 0.727 0.714 C 561 Web 91094-01-02-02 53561.478 2005 0.721 0.698 C 557 Web 91094-01-02-03 53562.593 2005 0.741 0.630 C 533 Web 91423-01-02-00 53563.051 2005 0.725 0.614 C 522 Web 91423-01-02-05 53563.510 2005 0.721 0.589 C 516 Web 91423-01-02-06 53564.887 2005 0.744 0.577 C 497 Web 91423-01-03-00 53567.050 2005 0.733 0.486 C 451 None 91423-01-03-07 53567.968 2005 0.766 0.457 C 429 Web 91423-01-03-02 53568.558 2005 0.756 0.523 C 437 Web 91423-01-03-03 53569.737 2005 0.726 0.578 C 447 Web 91423-01-03-04 53570.524 2005 0.743 0.510 C 419 Web 91423-01-03-05 53571.506 2005 0.735 0.516 C 408 Web 91423-01-03-06 53572.884 2005 0.762 0.478 C 382 Web 91423-01-04-00 53573.866 2005 0.769 0.460 C 368 Web

138 Texas Tech University, Kavitha Arur, May 2020

Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 91423-01-04-01 53574.652 2005 0.750 0.463 C 366 Web 91423-01-04-02 53575.505 2005 0.764 0.442 C 351 Web 91423-01-04-03 53576.755 2005 0.760 0.425 C 335 Web 91423-01-04-04 53577.733 2005 0.771 0.440 C 335 None 91423-01-04-05 53578.192 2005 0.759 0.468 C 337 None 91423-01-04-06 53579.437 2005 0.765 0.471 C 327 None 91423-01-05-01 53583.632 2005 0.746 0.373 C 279 None 91423-01-05-02 53585.664 2005 0.780 0.326 C 255 Web 91423-01-06-00 53587.630 2005 0.772 0.295 C 241 None 4U 1543-47 70133-01-01-00 52442.812 2002 0.137 5.973 C 3032 None 70133-01-06-00 52454.591 2002 0.155 6.092 C 3240 None 70133-01-08-00 52456.729 2002 0.125 6.646 C 2580 None 70133-01-11-00 52459.075 2002 0.224 7.726 B 2440 Type B 70133-01-15-00 52460.484 2002 0.233 8.015 B 2092 Type B 70133-01-16-00 52461.406 2002 0.232 7.509 B 1927 Type B 70133-01-30-00 52474.930 2002 0.218 11.102 C 237 None 70133-01-31-01 52477.217 2002 0.299 9.443 C 179 None 70133-01-31-00 52477.272 2002 0.295 9.716 C 176 Hypotenuse 70124-02-01-00 52478.678 2002 0.331 9.009 C 145 None 70124-02-02-00 52479.735 2002 0.340 9.219 C 116 None 70124-02-03-00 52480.655 2002 0.485 5.564 C 117 None 70128-01-01-01 52481.029 2002 0.495 5.383 C 108 None 70128-01-01-00 52481.162 2002 0.546 4.317 C 106 None

XTE J1650-500 60113-01-03-00 52160.385 2001 0.695 1.992 C 1076 None 60113-01-04-00 52161.114 2001 0.698 2.230 C 1113 None 60113-01-05-00 52162.451 2001 0.679 1.305 C 1115 None

139 Texas Tech University, Kavitha Arur, May 2020

Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 60113-01-05-01 52162.592 2001 0.680 2.745 C 1119 None 60113-01-06-00 52163.445 2001 0.668 1.540 C 1102 Web 60113-01-07-00 52165.157 2001 0.641 1.914 C 1032 None 60113-01-08-00 52166.094 2001 0.611 2.354 C 1030 Web 60113-01-08-01 52166.563 2001 0.623 2.072 C 1024 None 60113-01-09-00 52167.487 2001 0.576 2.800 C 1007 None 60113-01-09-01 52167.556 2001 0.588 2.722 C 997 Hypotenuse 60113-01-09-02 52167.625 2001 0.582 2.706 C 997 None 60113-01-10-00 52168.410 2001 0.561 3.025 C 981 Hypotenuse 60113-01-10-01 52168.480 2001 0.573 3.009 C 976 Hypotenuse 60113-01-11-00 52169.403 2001 0.548 3.266 C 958 Hypotenuse 60113-01-11-01 52169.472 2001 0.550 3.351 C 955 Hypotenuse 60113-01-11-02 52169.542 2001 0.528 3.572 C 964 Hypotenuse 60113-01-12-00 52170.253 2001 0.505 4.096 C 946 Hypotenuse 60113-01-12-01 52170.465 2001 0.487 4.359 C 952 Hypotenuse 60113-01-12-02 52170.603 2001 0.458 4.848 C 950 Hypotenuse 60113-01-12-03 52170.808 2001 0.473 4.660 C 947 None 60113-01-12-04 52170.871 2001 0.449 5.138 C 943 Hypotenuse 60113-01-13-00 52171.527 2001 0.419 5.721 C 926 Hypotenuse 60113-01-13-01 52171.664 2001 0.394 6.350 C 917 None 60113-01-13-02 52171.771 2001 0.367 6.835 C 907 None 60113-01-14-00 52171.903 2001 0.318 5.954 C 906 None 60113-01-15-00 52173.095 2001 0.247 1.548 B 855 Type B GX 339-4 40031-03-02-05 52388.054 2002 0.766 0.2 C 1055 Web* 70109-01-05-01G 52391.318 2002 0.763 0.22 C 1055 Web 70109-01-06-00 52400.830 2002 0.697 1.26 C 1120 Web* 70108-03-01-00 52400.853 2002 0.694 1.3 C 1111 Web 70110-01-10-00 52402.492 2002 0.562 4.2 C 1091 Web

140 Texas Tech University, Kavitha Arur, May 2020

Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 70109-04-01-00 52405.580 2002 0.354 5.46 C 1293 Hypotenuse 70109-04-01-01 52405.713 2002 0.356 5.45 C 1289 Hypotenuse 70110-01-12-00 52410.528 2002 0.266 8.1 C 1250 None 70110-01-45-00 52524.948 2002 0.222 7.2 A 1355 None 70109-01-23-00 52529.580 2002 0.209 7.4 A 1228 None 70109-01-24-00 52536.358 2002 0.204 8 A 1112 None

60705-01-68-01 53222.240 2004 0.728 1.03 C 318 Web* 60705-01-69-00 53225.400 2004 0.714 1.3 C 334 None 90704-01-01-00 53226.430 2004 0.705 2 C 342 Web* 90704-01-02-00 53233.389 2004 0.271 4.4 B 431 Type B 60705-01-84-02 53333.899 2004 0.254 5.2 B 760 Type B

92035-01-02-01 54133.922 2007 0.771 0.28 C 918 Web* 92035-01-02-02 54135.033 2007 0.771 0.3 C 934 Web* 92035-01-02-03 54136.015 2007 0.766 0.37 C 963 Web* 92035-01-02-04 54136.997 2007 0.759 0.43 C 1024 Web 92035-01-02-08 54137.851 2007 0.748 0.55 C 1062 Web 92035-01-02-07 54138.830 2007 0.731 0.9 C 1078 Web* 92035-01-02-06 54139.942 2007 0.686 0.99 C 1078 Web 92035-01-03-00 54140.204 2007 0.670 1.13 C 1091 Web 92035-01-03-01 54141.055 2007 0.621 1.68 C 1106 Web 92035-01-03-02 54142.036 2007 0.547 2.45 C 1156 Web 92035-01-03-03 54143.019 2007 0.461 3.52 C 1249 Web 92428-01-04-00 54143.870 2007 0.411 4.34 C 1299 Hypotenuse 92428-01-04-01 54143.951 2007 0.419 4.23 C 1304 Web 92428-01-04-02 54144.086 2007 0.424 4.13 C 1317 Hypotenuse 92428-01-04-03 54144.871 2007 0.380 4.99 C 1357 Hypotenuse 92035-01-03-05 54145.114 2007 0.343 5.8 C 1362 Hypotenuse

141 Texas Tech University, Kavitha Arur, May 2020

Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 92035-01-04-00 54147.011 2007 0.262 6.7 B 1549 Type B 92085-01-02-06 54160.896 2007 0.217 7.8 A 1221 None 92085-01-03-00 54161.669 2007 0.295 7.1 C 1178 Hypotenuse 92085-01-03-01 54162.665 2007 0.259 6.4 B 1252 Type B 92085-01-03-02 54163.698 2007 0.288 7.3 C 1117 None 92085-01-03-03 54164.557 2007 0.296 7 C 1102 Hypotenuse 92085-01-03-04 54165.527 2007 0.210 7.7 A 1058 None

95409-01-13-03 55288.367 2010 0.783 0.2 C 717 Web* 95409-01-13-04 55290.722 2010 0.781 0.29 C 754 Web* 95409-01-13-02 55291.649 2010 0.775 0.32 C 779 Web* 95409-01-14-02 55297.870 2010 0.672 1.25 C 884 Web 95409-01-14-06 55299.766 2010 0.564 2.43 C 944 Web 95409-01-14-04 55300.336 2010 0.563 2.38 C 942 Web 95409-01-14-07 55300.923 2010 0.515 2.92 C 962 Web 95409-01-15-01 55303.604 2010 0.346 5.65 C 1077 Hypotenuse 95409-01-15-02 55304.714 2010 0.256 5.6 B 1173 None 95409-01-15-06 55308.983 2010 0.236 5.9 B 1110 Type B 95409-01-16-05 55315.695 2010 0.262 6.1 B 1011 Type B 95409-01-17-02 55318.441 2010 0.308 6.67 C 881 None 95409-01-17-05 55321.718 2010 0.243 5.3 B 903 Type B 95409-01-17-06 55322.230 2010 0.238 5.2 B 859 Type B 95409-01-18-00 55323.210 2010 0.259 5.5 B 908 Type B 95335-01-01-07 55324.189 2010 0.255 5.3 B 888 Type B 95335-01-01-00 55324.254 2010 0.247 5.3 B 871 Type B 95335-01-01-01 55324.393 2010 0.240 5.1 B 848 Type 95335-01-01-06 55326.280 2010 0.223 4.9 B 790 None XTE J1752-223 94331-01-06-00 55215.912 2009 0.611 2.247 C 832 None

142 Texas Tech University, Kavitha Arur, May 2020

Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 94331-01-06-01 55216.958 2009 0.490 4.079 C 828 Hypotenuse 94331-01-06-02 55217.878 2009 0.414 5.340 C 836 Hypotenuse 95360-01-01-00 55218.800 2009 0.241 3.511 B 843 Type B 95360-01-01-02 55220.686 2009 0.228 4.118 B 786 None 95360-01-11-00 55288.771 2009 0.528 6.462 C 130 None XTE J1817-330 91110-02-05-00 53766.787 2006 0.193 5.439 B 2817 None 91110-02-07-00 53768.154 2006 0.193 5.427 B 2649 Type B 91110-02-17-00 53778.054 2006 0.149 4.858 B 1545 None 91110-02-18-00 53780.097 2006 0.151 5.060 B 1391 None 91110-02-24-00 53786.067 2006 0.152 5.328 B 1085 None 91110-02-25-00 53786.895 2006 0.188 10.584 C 1089 None 91110-02-29-00 53789.710 2006 0.287 5.682 B 1232 Type B 91110-02-30-00 53790.097 2006 0.276 5.614 B 1197 Type B 92082-01-02-03 53790.622 2006 0.252 4.847 B 1099 Type B 92082-01-02-04 53790.689 2006 0.266 4.977 B 1144 Type B 91110-02-31-00 53791.749 2006 0.177 11.423 C 848 None Int. Inclination XTE J1859+226 40124-01-04-00 51462.768 1999 0.718 1.201 C 778 Web 40124-01-05-00 51463.833 1999 0.557 3.051 C 1026 Cross 40124-01-06-00 51464.109 1999 0.514 3.625 C 1136 Cross 40124-01-07-00 51464.633 1999 0.512 3.663 C 1124 Cross 40124-01-08-00 51465.305 1999 0.477 4.364 C 1238 Cross 40124-01-09-00 51465.498 1999 0.447 4.949 C 1279 Cross 40124-01-10-00 51465.902 1999 0.422 5.772 C 1461 Cross 40124-01-11-00 51466.896 1999 0.417 5.920 C 1472 Cross 40124-01-13-00 51467.961 1999 0.344 6.102 B 2532 None 40124-01-14-00 51468.427 1999 0.356 8.561 C 2066 None

143 Texas Tech University, Kavitha Arur, May 2020

Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 40122-01-01-03 51469.093 1999 0.351 6.117 B 2397 Type B 40122-01-01-02 51469.159 1999 0.343 6.047 B 2460 Type B 40122-01-01-00 51469.292 1999 0.345 6.032 B 2396 Type B 40124-01-16-00 51469.493 1999 0.365 7.731 C 1825 None 40124-01-17-00 51469.893 1999 0.381 7.500 C 1659 None 40124-01-18-00 51471.024 1999 0.429 5.889 C 1313 Cross 40124-01-19-00 51471.224 1999 0.454 5.179 C 1213 Cross 40124-01-20-00 51471.890 1999 0.408 6.433 C 1388 None 40124-01-21-00 51472.503 1999 0.412 6.313 C 1317 Cross 40124-01-15-00 51473.245 1999 0.425 6.160 C 1238 None 40124-01-23-00 51473.822 1999 0.395 6.906 C 1346 None 40124-01-23-01 51473.890 1999 0.373 7.684 C 1528 None 40124-01-24-00 51474.429 1999 0.372 5.815 B 1941 Type B 40124-01-25-00 51474.820 1999 0.417 6.199 C 1229 None 40124-01-26-00 51475.154 1999 0.377 7.526 C 1422 Cross 40124-01-27-00 51475.428 1999 0.374 5.476 B 1811 None 40124-01-28-00 51476.428 1999 0.358 7.695 C 1479 None 40124-01-28-01 51476.501 1999 0.364 7.505 C 1367 None 40124-01-29-00 51477.152 1999 0.369 7.684 C 1429 None 40124-01-30-00 51478.017 1999 0.360 5.031 B 1751 Type B 40124-01-31-00 51478.777 1999 0.388 7.257 C 1242 None 40124-01-36-00 51483.107 1999 0.362 4.741 B 1438 Type B 40124-01-37-00 51483.945 1999 0.346 4.822 B 1434 Type B 40124-01-37-01 51484.077 1999 0.333 4.542 B 1354 Type B 40124-01-37-02 51484.276 1999 0.333 4.455 B 1359 Type B 40124-01-39-00 51485.875 1999 0.204 5.687 B 926 None 40124-01-40-00 51486.828 1999 0.188 5.292 B 879 None 40124-01-40-01 51486.873 1999 0.196 5.244 B 863 None 40124-01-41-00 51487.009 1999 0.180 4.744 B 844 None

144 Texas Tech University, Kavitha Arur, May 2020

Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 40124-01-42-00 51488.409 1999 0.228 11.632 B 836 None 40124-01-49-01 51497.255 1999 0.189 5.202 B 561 None 40124-01-61-00 51546.036 1999 0.190 4.585 B 176 None MAXI J1543-564 96371-02-01-00 55691.089 2011 0.750 1.056 C 56 None 96371-02-01-01 55692.085 2011 0.698 1.742 C 60 None 96371-02-01-02 55693.066 2011 0.597 2.971 C 72 None 96371-02-02-00 55694.095 2011 0.525 4.376 C 84 None 96371-02-02-01 55694.884 2011 0.503 5.724 C 91 None High Inclination XTE J1550-564 30188-06-03-00 51064.007 1998 0.991 0.122 C 1000 Web* 30188-06-01-00 51065.068 1998 0.967 0.288 C 1793 Web* 30188-06-01-01 51065.343 1998 0.950 0.395 C 1743 Web* 30188-06-01-02 51066.068 1998 0.911 0.809 C 2148 Web 30188-06-01-03 51066.345 1998 0.884 1.034 C 2330 Web 30188-06-04-00 51067.271 1998 0.831 1.533 C 2773 Web 30188-06-05-00 51068.346 1998 0.717 2.381 C 3098 Cross 30188-06-06-00 51069.275 1998 0.638 3.297 C 3510 Cross 30188-06-07-00 51070.132 1998 0.646 3.182 C 3539 Cross 30188-06-08-00 51070.275 1998 0.648 3.161 C 3518 Cross 30188-06-09-00 51071.201 1998 0.609 3.634 C 3887 Cross 30188-06-10-00 51071.997 1998 0.693 2.563 C 3274 Cross 30188-06-11-00 51072.345 1998 0.598 3.977 C 4024 Cross 30191-01-01-00 51074.138 1998 0.546 5.713 C 5157 Cross 30191-01-02-00 51076.000 1998 0.613 4.940 B 13200 None 30191-01-02-00 51076.000 1998 0.613 13.081 C 13200 None 30191-01-27-01 51095.609 1998 0.584 4.481 C 2883 Cross 30191-01-27-00 51096.571 1998 0.559 5.544 C 3084 Cross

145 Texas Tech University, Kavitha Arur, May 2020

Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 30191-01-28-01 51097.569 1998 0.574 4.742 C 2758 Cross 30191-01-28-00 51097.809 1998 0.600 4.191 C 2583 Cross 30191-01-28-02 51098.276 1998 0.571 4.959 C 2793 Cross 30191-01-29-00 51099.208 1998 0.573 4.835 C 2686 Cross 30191-01-29-01 51099.608 1998 0.564 4.961 C 2638 Cross 30191-01-30-00 51100.275 1998 0.536 6.452 C 3115 Cross 30191-01-31-00 51101.607 1998 0.521 6.764 C 3082 None 30191-01-31-01 51101.941 1998 0.520 6.717 C 3026 Cross 30191-01-32-00 51106.953 1998 0.467 5.403 B 3761 Type B 30191-01-34-01 51109.737 1998 0.452 4.913 B 3277 Type B 30191-01-41-00 51126.591 1998 0.265 10.291 B 653 None 40401-01-02-00 51180.846 1998 0.311 1.535 C 4515 None 40401-01-48-00 51239.082 1998 0.303 18.061 C 3792 None 40401-01-50-00 51241.802 1998 0.362 5.870 B 4181 None 40401-01-51-00 51242.508 1998 0.395 5.649 B 3978 None 40401-01-53-00 51245.354 1998 0.482 6.333 B 4525 Type B 40401-01-51-01 51248.093 1998 0.488 5.741 B 4397 Type B 40401-01-56-00 51249.400 1998 0.490 6.276 B 4059 Type B 40401-01-56-01 51249.471 1998 0.491 6.277 B 4021 None 40401-01-57-00 51250.693 1998 0.510 6.690 C 2545 Cross 40401-01-58-00 51253.225 1998 0.482 5.673 B 3545 Type B 40401-01-58-01 51254.092 1998 0.484 3.110 B 3143 Type B 40401-01-59-01 51255.091 1998 0.341 5.757 B 2357 None 40401-01-61-01 51258.089 1998 0.293 9.142 B 1885 None 40401-01-61-00 51258.497 1998 0.307 5.486 B 1865 None

50137-02-02-00 51646.335 2000 0.821 0.257 C 482 Web* 50137-02-02-01 51646.613 2000 0.807 0.265 C 488 Web* 50137-02-03-00 51648.739 2000 0.806 0.239 C 524 Web*

146 Texas Tech University, Kavitha Arur, May 2020

Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 50137-02-03-01G 51649.751 2000 0.807 0.240 C 537 Web* 50137-02-04-00 51650.733 2000 0.807 0.233 C 555 Web* 50137-02-04-01 51651.384 2000 0.802 0.249 C 568 Web* 50137-02-05-00 51652.248 2000 0.816 0.279 C 606 Web* 50137-02-05-01 51653.531 2000 0.810 0.314 C 648 Web* 50137-02-06-00 51654.721 2000 0.796 0.320 C 676 Web* 50137-02-07-00 51655.665 2000 0.787 0.420 C 703 Web* 50134-02-01-00 51658.600 2000 0.719 1.257 C 822 Cross 50134-02-02-00 51662.169 2000 0.379 4.421 B 2026 Type B 50134-01-02-00 51674.695 2000 0.374 3.580 B 944 None 50134-01-04-00 51676.400 2000 0.447 6.915 C 613 None 50134-01-05-00 51678.452 2000 0.556 4.476 C 506 Web 50135-01-02-00 51682.306 2000 0.659 2.340 C 348 Web 50135-01-03-00 51683.771 2000 0.725 1.124 C 280 Web* 50135-01-04-00 51684.764 2000 0.741 0.981 C 249 None 50135-01-05-00 51686.296 2000 0.756 0.684 C 209 None 50135-01-06-00 51687.224 2000 0.765 0.552 C 184 None 4U 1630-47 70417-01-09-00 52636.800 2002 0.743 4.651 B 2136 None 70417-01-09-00 52636.800 2002 0.743 12.51 C 2136 None 80117-01-02-01 52793.385 2003 0.598 12.416 C 1676 None 80417-01-01-00 52794.914 2003 0.764 4.835 B 2317 None 80417-01-01-00 52794.914 2003 0.764 12.257 C 2317 None 80117-01-04-00 52798.966 2003 0.579 11.543 C 1664 None 80117-01-02-04 52800.844 2003 0.551 12.453 C 1755 None 80117-01-05-00 52801.400 2003 0.778 4.663 B 2416 None 80117-01-05-00 52801.400 2003 0.778 12.326 C 2416 None 80117-01-06-01 52802.860 2003 0.765 7.795 C 1706 None 80117-01-07-01 52806.550 2003 0.732 4.628 B 2452 None

147 Texas Tech University, Kavitha Arur, May 2020

Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 80117-01-07-01 52806.550 2003 0.732 14.80 C 2452 None 80117-01-07-02 52808.251 2003 0.727 4.532 B 2498 None 80117-01-07-02 52808.251 2003 0.727 14.484 C 2498 None 80117-01-08-00 52810.092 2003 0.616 11.467 C 1662 None 80117-01-08-01 52812.268 2003 0.751 13.799 C 2439 None 80117-01-10-00 52819.230 2003 0.728 4.547 B 2207 None 80117-01-10-00 52819.230 2003 0.728 14.048 C 2207 None 80117-01-11-02 52829.551 2003 0.774 14.039 C 2579 None 80117-01-12-01 52834.502 2003 0.568 11.661 C 1682 None 80117-01-16-00 52862.849 2003 0.553 13.335 C 1744 None 80117-01-17-00 52868.084 2003 0.557 13.14 C 1819 None 80117-01-21-01 53066.823 2003 0.731 0.794 C 65 None 80117-01-22-00 53069.765 2003 0.771 4.024 C 61 None GRO 1655-40 91702-01-01-00 53436.725 2005 0.797 0.491 C 207 Web 91702-01-01-01 53437.072 2005 0.804 0.504 C 220 None 91702-01-01-02 53437.142 2005 0.802 0.519 C 222 None 91702-01-01-03 53438.054 2005 0.759 0.889 C 334 Web 91702-01-01-04 53438.757 2005 0.750 1.333 C 436 Web 91702-01-01-05 53439.107 2005 0.732 1.526 C 476 Cross 91702-01-02-00G 53440.680 2005 0.454 6.442 B 1950 None 91702-01-58-00 53508.507 2005 0.549 6.793 B 9431 None 91702-01-71-03 53628.917 2005 0.478 9.858 C 318 None 91702-01-71-04 53628.983 2005 0.466 10.373 C 324 None 91702-01-76-00 53628.196 2005 0.340 13.178 C 340 None 91702-01-76-01 53628.590 2005 0.386 12.691 C 331 None 91702-01-79-00 53630.490 2005 0.538 8.679 C 279 Hypotenuse 91702-01-79-01 53629.376 2005 0.488 9.736 C 315 None 91702-01-80-00 53631.474 2005 0.562 7.762 C 243 Hypotenuse

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Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 91702-01-80-01 53632.456 2005 0.640 4.738 C 202 Hypotenuse H1743-322 80138-01-02-00G 52729.791 2003 0.847 0.161 C 299 Web* 80138-01-03-00G 52733.742 2003 0.611 2.216 C 610 Web 80138-01-04-00G 52735.722 2003 0.513 3.664 C 768 Web 80138-01-05-00G 52737.569 2003 0.591 2.072 C 1109 Web 80138-01-06-00 52739.655 2003 0.537 3.136 C 1241 Web 80138-01-07-00 52741.834 2003 0.422 7.143 C 1169 None 80146-01-01-00 52743.220 2003 0.413 8.517 C 1938 Web 80146-01-02-00 52744.203 2003 0.443 5.672 C 1539 Web 80146-01-03-00 52746.179 2003 0.465 4.756 C 707 Web 80146-01-03-01 52747.616 2003 0.422 7.016 C 1027 None 80146-01-11-00 52751.696 2003 0.376 5.528 B 1804 None 80146-01-12-00 52751.961 2003 0.358 5.787 B 1725 None 80146-01-15-00 52754.532 2003 0.381 5.179 B 1619 None 80146-01-16-00 52755.907 2003 0.309 4.492 B 1346 None 80146-01-26-00 52763.601 2003 0.331 5.106 B 2212 None 80146-01-27-00 52764.852 2003 0.336 5.277 B 2215 Type B 80146-01-29-00 52766.561 2003 0.436 5.527 C 1562 Web 80146-01-30-00 52767.810 2003 0.469 4.431 C 1286 Web 80146-01-31-00 52768.534 2003 0.434 5.301 C 1302 Web 80146-01-32-00 52769.717 2003 0.449 4.862 C 1230 Cross 80146-01-33-01 52770.375 2003 0.416 5.879 C 1495 None 80146-01-33-00 52770.649 2003 0.435 6.220 C 1547 None 80146-01-34-00 52771.755 2003 0.556 2.774 C 1075 Web 80146-01-35-00 52771.974 2003 0.587 2.264 C 1013 Web 80146-01-36-00 52772.678 2003 0.642 1.827 C 996 Web 80146-01-37-00 52773.662 2003 0.639 1.899 C 990 Web 80146-01-38-00 52774.518 2003 0.546 3.202 C 1031 Web

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Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 80146-01-39-00 52775.569 2003 0.613 2.134 C 950 Web 80146-01-40-00 52776.623 2003 0.670 1.701 C 708 Web 80146-01-41-00 52777.607 2003 0.587 2.498 C 976 Web 80146-01-42-00 52778.461 2003 0.538 3.200 C 1008 Web 80146-01-43-01 52779.530 2003 0.507 3.811 C 1077 Cross 80146-01-43-00 52779.580 2003 0.497 3.769 C 1063 Web 80146-01-44-00 52780.566 2003 0.504 3.778 C 1075 Web 80146-01-45-00 52781.552 2003 0.515 3.606 C 1043 Cross 80146-01-46-00 52782.672 2003 0.468 4.630 C 1087 Cross 80146-01-47-00 52783.461 2003 0.430 6.665 C 1422 Cross 80146-01-48-00 52784.510 2003 0.360 5.399 B 2075 None 80146-01-49-00 52785.431 2003 0.361 5.253 B 2149 None 80146-01-50-00 52786.291 2003 0.421 9.440 C 2494 Hypotenuse? 80144-01-01-00 52786.826 2003 0.345 5.073 B 2116 None 80144-01-01-01 52787.032 2003 0.318 5.420 B 1984 None 80135-02-02-01 52787.229 2003 0.303 4.738 B 1767 None 80135-02-02-00 52787.603 2003 0.328 5.349 B 2118 Type B 80144-01-01-02 52788.016 2003 0.356 5.455 B 2059 None 80146-01-51-00 52788.454 2003 0.299 4.557 B 1558 Type B 80146-01-51-01 52788.520 2003 0.279 4.221 B 1634 Type B 80146-01-52-00 52789.244 2003 0.326 5.137 B 1879 Type B 80144-01-02-00 52789.918 2003 0.330 5.231 B 1809 None 80144-01-02-01 52790.056 2003 0.339 5.570 B 1800 None 80144-01-03-01 52790.121 2003 0.315 4.963 B 1625 None 80146-01-53-01 52790.190 2003 0.298 5.082 B 1646 None 80146-01-53-00 52790.241 2003 0.316 4.931 B 1834 Type B 80146-01-54-00 52791.571 2003 0.321 4.853 B 1713 Type B 80146-01-55-00 52792.273 2003 0.321 5.173 B 1716 None 80146-01-52-01 52792.466 2003 0.318 5.077 B 1729 Type B

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Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 80146-01-56-00 52793.591 2003 0.288 4.861 B 1778 Type B 80146-01-58-00 52795.492 2003 0.281 3.864 B 1519 None 80146-01-59-00 52796.215 2003 0.322 4.991 B 1777 Type B 80146-01-60-00 52797.529 2003 0.332 5.373 B 1771 None 80146-01-62-00 52799.438 2003 0.292 4.549 B 1554 Type B 80146-01-65-00 52801.876 2003 0.306 4.938 B 1611 Type B 80146-01-66-00 52802.926 2003 0.312 5.160 B 1610 Type B 80146-01-67-00 52803.517 2003 0.304 5.152 B 1532 Type B 80146-01-68-00 52804.570 2003 0.334 5.323 B 1600 None 80146-01-69-00 52805.425 2003 0.318 5.363 B 1570 None 80137-01-20-00 52932.092 2003 0.408 7.846 C 148 None 80137-01-25-00 52937.020 2003 0.501 6.953 C 106 None 80137-01-26-00 52938.005 2003 0.567 5.820 C 97 None 80137-01-27-00 52939.121 2003 0.569 5.838 C 88 None 80137-01-28-00 52942.143 2003 0.683 2.973 C 62 None 80137-02-01-00 52944.114 2003 0.719 2.340 C 49 None

90058-11-03-00 53228.405 2004 1.052 0.127 C 14 None 90058-16-05-00 53427.026 2005 0.837 0.106 C 51 None 90058-16-07-00 53427.942 2005 0.824 0.119 C 58 Web* 91050-06-01-00 53595.361 2005 0.469 6.099 C 305 None 91428-01-03-00 53630.623 2005 0.747 1.845 C 65 None

93427-01-03-01 54492.179 2008 0.501 6.989 C 129 None 93427-01-04-00 54498.837 2008 0.691 3.789 C 84 None 93427-01-04-02 54500.801 2008 0.718 2.449 C 66 None 93427-01-04-03 54502.828 2008 0.719 2.307 C 57 None 93427-01-09-00 54742.981 2008 0.878 0.332 C 193 None 93427-01-09-01 54746.507 2008 0.835 0.376 C 207 Web*

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Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 93427-01-09-03 54747.487 2008 0.827 0.432 C 206 Web* 93427-01-09-02 54748.208 2008 0.857 0.469 C 212 Web* 93427-01-10-00 54750.366 2008 0.810 0.580 C 211 None 93427-01-10-01 54752.262 2008 0.865 0.676 C 213 Web 93427-01-10-02 54755.235 2008 0.808 0.745 C 202 Web 93427-01-11-00 54756.253 2008 0.790 0.858 C 200 Web 93427-01-11-01 54758.153 2008 0.773 1.027 C 193 Web 93427-01-11-03 54762.142 2008 0.538 5.582 C 239 None 93427-01-12-04 54767.841 2008 0.589 3.845 C 170 None 93427-01-12-02 54768.638 2008 0.647 3.297 C 160 None 93427-01-12-05 54769.140 2008 0.649 2.933 C 152 None 93427-01-13-00 54770.120 2008 0.672 2.595 C 142 Web 93427-01-13-05 54770.395 2008 0.687 2.275 C 138 None 93427-01-13-04 54771.762 2008 0.689 2.377 C 129 Web 93427-01-13-01 54772.151 2008 0.705 2.204 C 126 None 93427-01-13-02 54773.268 2008 0.766 2.242 C 119 None 93427-01-13-06 54774.642 2008 0.710 2.106 C 114 Web 93427-01-13-03 54775.565 2008 0.735 1.813 C 110 None 93427-01-14-00 54777.859 2008 0.721 1.880 C 107 Web 93427-01-14-01 54778.769 2008 0.764 1.790 C 106 None 93427-01-14-02 54779.040 2008 0.776 1.526 C 104 Web 93427-01-14-03 54780.018 2008 0.778 1.470 C 103 None 93427-01-14-04 54781.779 2008 0.794 1.716 C 102 None 93427-01-14-05 54782.892 2008 0.738 2.191 C 104 None 93427-01-14-06 54783.808 2008 0.712 2.081 C 102 None 93427-01-15-00 54784.454 2008 0.740 1.796 C 97 None 93427-01-15-01 54785.701 2008 0.730 1.918 C 91 None 93427-01-15-02 54786.482 2008 0.740 1.758 C 86 None 93427-01-15-03 54787.726 2008 0.753 1.533 C 78 None

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Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 93427-01-15-04 54788.642 2008 0.785 1.009 C 69 None 93427-01-15-06 54788.843 2008 0.848 1.026 C 67 None 93427-01-15-05 54789.494 2008 0.788 0.850 C 62 None

94413-01-02-00 54980.396 2009 0.818 0.909 C 260 Web 94413-01-02-02 54980.846 2009 0.825 1.004 C 260 Web 94413-01-02-01 54981.954 2009 0.774 1.193 C 270 Web 94413-01-02-05 54982.279 2009 0.838 1.277 C 266 Web 94413-01-02-04 54983.326 2009 0.759 2.017 C 275 Web 94413-01-02-03 54984.373 2009 0.560 3.579 C 321 Cross 94413-01-03-02 54990.266 2009 0.396 3.752 B 554 None 94413-01-03-03 54990.327 2009 0.399 3.728 B 565 None 94413-01-07-00 55016.317 2009 0.587 4.965 C 110 None 94413-01-07-01 55019.453 2009 0.653 3.431 C 85 None 94413-01-07-02 55021.417 2009 0.642 3.766 C 79 None

95405-01-01-01 55217.575 2010 0.385 1.608 B 276 None 95405-01-02-00 55219.400 2010 0.403 1.939 B 259 None 95405-01-02-02 55220.514 2010 0.640 3.061 C 135 None 95405-01-02-05 55221.610 2010 0.435 1.832 B 216 None 95405-01-02-06 55223.386 2010 0.612 3.823 C 151 None 95405-01-02-04 55224.674 2010 0.410 2.346 B 268 None 95405-01-03-00 55226.529 2010 0.713 2.040 C 114 None 95405-01-03-04 55227.770 2010 0.718 2.218 C 105 None 95405-01-03-01 55228.609 2010 0.718 2.201 C 102 None 95405-01-03-02 55230.587 2010 0.767 1.339 C 84 None 95405-01-04-01 55233.398 2010 0.779 0.893 C 63 None 95360-14-01-00 55418.406 2010 0.815 1.006 C 231 Web 95360-14-02-01 55419.089 2010 0.841 1.038 C 227 None

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Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 95360-14-02-00 55420.238 2010 0.859 1.173 C 235 Cross 95360-14-03-00 55421.282 2010 0.763 1.477 C 245 Web 95360-14-02-03 55422.024 2010 0.790 1.766 C 248 None 95360-14-02-02 55423.204 2010 0.618 2.956 C 270 Cross 95360-14-03-01 55424.056 2010 0.558 4.792 C 314 None 95360-14-04-00 55425.144 2010 0.390 3.563 B 515 None 95360-14-23-01 55457.119 2010 0.691 2.557 C 92 None 95360-14-24-00 55458.633 2010 0.751 1.565 C 78 None MAXI J1659-152 95108-01-02-00 55470.453 2010 0.584 2.594 C 524 Web 95108-01-03-00 55471.502 2010 0.573 3.062 C 486 Cross 95108-01-04-00 55471.772 2010 0.539 3.064 C 507 None 95108-01-05-00 55472.073 2010 0.532 3.332 C 492 Cross 95108-01-06-00 55472.479 2010 0.481 4.594 C 570 Cross 95108-01-07-00 55472.880 2010 0.455 4.408 C 578 None 95108-01-08-00 55473.119 2010 0.449 4.852 C 556 Cross 95108-01-09-00 55473.459 2010 0.462 4.745 C 583 None 95108-01-10-00 55473.731 2010 0.455 4.883 C 547 None 95108-01-11-00 55474.575 2010 0.468 4.637 C 545 None 95108-01-12-00 55474.838 2010 0.446 4.799 C 515 Cross 95108-01-13-00 55475.418 2010 0.422 6.173 C 578 None 95108-01-14-00 55475.756 2010 0.458 5.021 C 521 None 95108-01-15-00 55476.059 2010 0.455 5.097 C 524 Cross 95108-01-16-00 55476.397 2010 0.406 6.035 C 587 None 95108-01-17-00 55476.668 2010 0.390 7.023 C 652 None 95108-01-18-01 55477.002 2010 0.395 7.308 C 690 None 95108-01-20-00 55478.038 2010 0.416 6.545 C 556 None 95108-01-21-00 55478.491 2010 0.385 7.284 C 673 None 95108-01-22-00 55479.131 2010 0.408 6.326 C 531 None

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Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 95108-01-23-00 55479.672 2010 0.446 5.340 C 451 None 95108-01-24-00 55480.177 2010 0.452 6.010 C 486 None 95108-01-25-00 55480.652 2010 0.422 5.822 C 468 None 95108-01-26-00 55481.027 2010 0.415 6.672 C 497 None 95108-01-27-00 55481.693 2010 0.395 3.888 B 648 None 95108-01-28-00 55482.408 2010 0.400 4.103 B 660 None 95108-01-30-00 55483.903 2010 0.444 6.005 C 416 None 95118-01-01-00 55484.231 2010 0.415 6.859 C 450 None 95118-01-01-01 55484.687 2010 0.389 3.857 B 580 None 95118-01-02-00 55485.145 2010 0.367 3.469 B 562 None 95118-01-07-00 55490.706 2010 0.382 3.287 B 416 None 95118-01-09-00 55491.814 2010 0.373 5.080 B 372 None 95118-01-10-00 55493.253 2010 0.384 3.564 B 366 None 95118-01-12-00 55495.035 2010 0.357 1.972 B 315 None 95118-01-16-01 55501.226 2010 0.459 5.944 C 184 None 95118-01-17-00 55502.018 2010 0.529 4.775 C 171 None 95118-01-17-01 55503.056 2010 0.606 3.363 C 149 None 95118-01-18-00 55504.057 2010 0.665 2.549 C 132 None 95118-01-19-00 55505.018 2010 0.715 2.183 C 121 None 95118-01-20-00 55506.194 2010 0.676 2.047 C 113 None 95118-01-21-00 55508.088 2010 0.705 1.624 C 101 None 95358-01-02-00 55467.040 2010 0.665 1.647 C 477 Cross 95358-01-02-01 55468.086 2010 0.648 2.263 C 434 Web 95358-01-02-02 55469.088 2010 0.569 2.737 C 520 Cross 95358-01-03-00 55470.243 2010 0.562 2.792 C 517 Web 95358-01-03-01 55471.111 2010 0.541 3.197 C 454 Cross XTE J1748-288 30188-05-01-00 50968.841 1998 0.000 17.157 C 931 Hypotenuse 30171-02-01-00 50970.404 1998 0.709 31.555 C 1290 None

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Table B.1 Continued. ObsId MJD Year HRa Freq. Type Mean Pattern [Hz]a Count 30185-01-01-00 50971.331 1998 0.732 31.401 C 1283 None 30185-01-02-00 50972.266 1998 0.741 23.652 C 1071 None 30185-01-03-00 50973.533 1998 0.768 20.053 C 967 None 30185-01-04-00 50974.154 1998 0.752 19.872 C 937 None 30185-01-05-00 50975.536 1998 0.749 22.819 C 898 None

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CHAPTER C DUFFING OSCILLATOR

The Dufﬁng oscillator is a damped, non-linear oscillator driven by a periodic driving force and is described by:

3 x¨ + γx˙(t) + k1x(t) + k2x (t) = F0cos(ωt) (Ci)

where x denotes the displacement, γ is the damping factor and k1 gives the stiff- ness of the spring. k2 is a measure of the non-linearity of the oscillator. F0 and ω give the amplitude and (angular) frequency of the driving force respectively.

One promising feature of the Dufﬁng oscillator over the other oscillators is that it produces a subharmonic feature (see Figure C1) at a frequency that is half of the driving force. Subharmonics are often seen with both type B and type C QPOs, and are one of the most puzzling QPO properties, with no obvious explanation as to what causes them.

Solving for the displacement of the oscillator and exponentiating the displacement gives a simulated ”lightcurve”. This was then used to calculate the bicoherence for various values of the constants listed above. As shown in Figure C1, an increase in the driving frequency of the oscillator results in a gradual change in the bicoherence pattern, going from a ’web’ to a ’hypotenuse’ pattern similar to the change seen in GX 339-4 and other low inclination sources.

Physically, this could be due to driving by Lense-Thirring precession, with damping by the disk viscosity. However, the connection between the parameters of the oscillator and physical quantities (such as viscosity, optical depth and mass transfer rate) is presently unclear.

157 Texas Tech University, Kavitha Arur, May 2020

Figure C1. The power spectrum and the bicoherence plot for Dufﬁng oscillator. The colour scheme of logb2 . The bicoherence pattern gradually evolves from a ‘web’ to a ‘hypotenuse’ pattern with an increase in the driving frequency

158