Constraining the emission geometry and mass of the white Dwarf Pulsar AR Sco

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Constraining the Emission Geometry and Mass of

the White Dwarf Pulsar AR Sco

L du Plessis

orcid.org 0000-0002-5158-4152

Dissertation accepted in partial fulfilment of the requirements for

the degree

Master of Science in Astrophysical Sciences

at the

North-West University

Supervisor:

Prof C Venter

Co-Supervisor:

Dr AK Harding

Co-Supervisor:

Dr Z Wadiasingh

Graduation December 2020

25936328

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Abstract

Marsh et al. (2016) detected radio and optical pulsations from the binary system AR Scorpii

(AR Sco). This system, with an orbital period of 3.55 h, is composed of a cool, low-mass M-dwarf and a white dwarf with a spin period of 1.95 min. Buckley et al. (2017) found that the emission from the white dwarf is strongly linearly polarised (up to 40%) with periodically changing intensities. This periodic non-thermal emission is thought to be synchrotron emission

(Buckley et al.,2017;Takata et al.,2017) that is powered by the highly magnetised (5 × 108 G)

white dwarf that is spinning down, analogous to rotation-powered pulsars. The morphology of the polarisation signal, namely the position angle plotted against the phase angle, is similar to that seen in many radio pulsars. I demonstrate that we can fit the traditional pulsar rotating vector model (Radhakrishnan and Cooke,1969) to the optical position angle data of the white dwarf. I used a Markov-chain-Monte-Carlo technique to find the best fit for the model yielding a magnetic inclination angle of α = (86.6+3.0_−2.8)◦ and an observer angle of ζ = (60.5+5.3_−6.1)◦. Takata

et al. (2017) also detected non-thermal X-ray emission pulsed at the spin period of the white

dwarf. The optical and non-thermal X-ray light curve peaks are aligned in phase, suggesting that these are also due to synchrotron radiation. Recent analysis of Fermi LAT data in the 100 MeV to 500 GeV range by Kaplan et al. (2019) produced a 3σ upper limit at the position of AR Sco. Using these observations, I was able to do first-order spectral calculations to constrain some of the parameters for the system. Lastly, using phase-resolved optical observations, I was able to obtain α and ζ at the different orbital phases. I found that α varies slightly by ∼ 10◦ and ζ with ∼ 30◦ over the orbital period. The modelling done in this thesis supports the scenario that the synchrotron emission is magnetospheric and originates from above the polar caps of the white dwarf pulsar, with the white dwarf being an orthogonal rotator. Since the morphology of the polarisation is so well fit by the model, the non-thermal emission is believed to be tangential to the local magnetic field of the white dwarf. Future emission modelling should help elucidate the particle acceleration mechanism and further clarify the particle dynamics and emission.

Keywords: AR Sco, RVM, White Dwarf, Rotation-Powered Pulsar, Optical Polarisation, Polar-isation Swing, Numerical Modelling

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1 Introduction 1

1.1 Observational and Historical Context. . . 1

1.2 Research Aims and Motivation . . . 3

1.3 Thesis Outline . . . 4

1.4 Publications . . . 5

2 Background 6 2.1 Degenerate Stars and Their Formation . . . 6

2.1.1 Type II Supernovae and Neutron Star Formation . . . 6

2.1.2 Formation and Properties of White Dwarf Stars . . . 9

2.1.3 Type Ia Supernovae . . . 10

2.1.4 Binary Systems . . . 11

2.2 Pulsars. . . 13

2.2.1 Standard Braking Model. . . 13

2.2.2 GJ Magnetosphere . . . 14

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2.3 Radiation Mechanisms . . . 17

2.3.1 Synchrotron Radiation . . . 17

2.3.2 Curvature Radiation . . . 20

2.3.3 Inverse Compton Scattering . . . 21

2.3.4 Pair Production . . . 23

2.3.5 Pulsar Emission Models . . . 24

2.4 Polarisation . . . 26

2.4.1 Polarisation Measurements . . . 28

2.5 The Rotating Vector Model (RVM) . . . 29

3 The AR Sco System 31 3.1 Observational Introduction to AR Sco . . . 31

3.2 Model Interpretations . . . 33

3.2.1 General Considerations . . . 33

3.2.2 Geng et al. Model . . . 36

3.2.3 Takata et al. Model . . . 37

3.2.4 Potter and Buckley Model . . . 39

3.2.5 Lyutikov et al. Model . . . 41

3.2.6 Singh et al. SED Fits . . . 43

3.3 Summary . . . 44

4 Method 45 4.1 Folding . . . 45

4.2 Fixing Discontinuities . . . 47

4.3 Code Verification and Obtaining the Best Fit . . . 47

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4.5 Phase-Resolved Fitting. . . 49 4.6 SED Fitting . . . 51 5 Results 53 5.1 WD Geometric Constraints . . . 53 5.2 WD Mass Constraints . . . 54 5.3 Precession . . . 56

5.4 Orbital Phase-Resolved Fitting of the RVM . . . 58

5.5 SED Fits . . . 61

6 Discussion 65 6.1 Constraints on the Emission Geometry . . . 65

6.2 Constraining the Emission Mechanism and Certain System Parameters . . . 68

6.3 Assumptions Regarding the Particle Pitch Angle . . . 74

7 Conclusions 77 7.1 Suggestions for Future Work . . . 79

7.2 Acknowledgements . . . 79

8 MeerKAT Open-Time Proposal 80 8.1 Background . . . 80

8.2 Open-Time Proposal . . . 81

8.3 Method . . . 82

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2.1 The NS masses for diffrent types of NSs (Ozel and Freire¨ ,2016) . . . 8

2.2 Supernova Type Ia light curves . . . 11

2.3 Roche Lobes illustration . . . 12

2.4 P ˙P diagram . . . 15

2.5 Geometric pulsar magnetosphere illustration. . . 17

2.6 Polar cap model illustration (Daugherty and Harding,1982) . . . 24

2.7 Slot gap model illustration (Harding and Muslimov,2005) . . . 25

2.8 Outer gap model illustration (Cheng et al.,1986) . . . 26

2.9 Rotated polarisation ellipse . . . 27

2.10 Geometric radio emission beam of a pulsar. . . 30

2.11 Polarisation signature from pulsar radio beam . . . 30

3.1 AR Sco light curves . . . 33

3.2 Optical polarimetric observations of AR Sco . . . 34

3.3 Frequency-filtered linear flux of AR Sco . . . 34

3.4 PPA swing of AR Sco . . . 35 v

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3.5 Fourier analysis on AR Sco observations . . . 35

3.6 Illustration showing diffrent radio frequency peaks of AR Sco . . . 36

3.7 SED produced byTakata et al. (2018) model . . . 39

3.8 Evolution of particle Lorentz factor from Takata et al.(2017) . . . 39

3.9 PPA for diffrent particle pitch angles fromTakata and Cheng (2019) . . . 40

3.10 Geometric model illustration fromPotter and Buckley (2018b) . . . 41

3.11 Plama-loading model illustration Lyutikov et al.(2020) . . . 42

3.12 SED fits for AR Sco with two emission locales. . . 43

4.1 PPA data without a convention fix . . . 46

4.2 KDE smoothed data without a convention fix . . . 46

4.3 PPA data with a convention fix . . . 46

4.4 KDE smoothed data with a convention fix . . . 46

5.1 Orbitally averaged best fit . . . 54

5.2 Parameter space plots for best-fit results . . . 54

5.3 Mass constraints plot of AR Sco . . . 56

5.4 Selected best-fit results within the orbital phase results. . . 58

5.5 Orbital phase-resolved best fits . . . 59

5.6 Orbital phase-resolved results using different epoch binning . . . 60

5.7 The minimised chi-square fit for the RVM at the different orbital phases.. . . 61

5.8 SR best fit heat map . . . 63

5.9 CR best fit heat map. . . 63

5.10 SED for AR Sco using our parameter values . . . 64

6.1 Illustration of AR Sco at different orbital phases . . . 66

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6.3 The parts of the magnetosphere the observer is sampling . . . 67

6.4 Different calculated emission timescales for AR Sco . . . 70

8.1 Image of FRB171019 . . . 83

B.2 Best-fit parameter space for pulsar B0301+19 . . . 89

B.3 MCMC best fit for pulsar B0301+19 . . . 89

B.8 Reduced chi-squared map for PSR J0659+1414 . . . 92

B.9 Reduced chi-squared map for PSR J0729−1448 . . . 92

C.10 Extra best-fit results at different orbital phases . . . 93

C.11 Different epoch binning for α over the orbital phase . . . 94

D.12 Heat map using p = 3.8 and γ = 12 . . . 95

D.13 SED using p = 3.8 and γ = 12 . . . 95

D.15 SED using p = 3.9 and γ = 12 . . . 96

D.16 Heat map using p = 4.0 and γ = 1.2 . . . 96

D.17 SED using p = 4.0 and γ = 1.2 . . . 96

D.19 SED using p = 4.0 and γ = 12 . . . 97

E.20 RVM atlas 1. . . 98

E.21 RVM atlas 2. . . 99

E.22 RVM atlas 3. . . 100

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Introduction

1.1 Observational and Historical Context

WDs form in the cores of intermediate-mass main sequence red giant stars (Tremblay et al.,

2016). The population of white dwarfs (WDs) found in isolation or in various binary systems is diverse. These systems include cataclysmic variables, polars, intermediary polars, and magnetic WDs. Cataclysmic variables are binary systems containing a WD and a main-sequence star, with the companion accreting material onto the WD. This process continues until the companion is depleted of material or the WD becomes unstable, causing a thermonuclear explosion. Polars also exhibit accretion from the companion onto the WD, but at a slower rate compared to cataclysmic variables. If a polar has a magnetic field larger than 107_{G, the accretion will be confined to the}

magnetic field of the WD. The rotation rates of the WD and companion are also found to be synchronised, meaning an accretion disc does not form in these systems. Intermediary polars have lower magnetic fields of 103 − 106_{G and the WD has a shorter spin period than orbital}

period, meaning this de-synchronisation leads to the formation of an accretion disc (Lubow and Shu,1975). Finally, magnetic WDs are found in isolation or in binary systems where the WD has a large magnetic field up to 109G, with its peak blackbody emission in the UV/optical regime

(Ferrario et al.,2015). Ferrario et al.(2015) also state that the range of determined masses for

magnetic WDs are around 0.8 M. All of the WD binary systems mentioned are found to emit

X-rays that originate due to the companion accreting material onto the compact WD.

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From recent observations, it is apparent that AR Sco is quite novel. Marsh et al.(2016) detected pulsations from AR Sco in the radio/optical/UV range, mostly at the 118 s beat period between the 117 s WD spin period and the 3.6 h binary period, with the non-thermal spectral component peaking in the optical/UV band. They also inferred a non-zero time derivative of the spin period, leading Buckley et al. (2017) to clasify AR Sco as the first detected rotation-powered WD pulsar. The system has a large duty cycle compared to normal radio pulsars, meaning we can obtain much stronger constraints on the system parameters such as the magnetic inclination angle and the observer angle than may be the case for typical radio pulsars. Buckley et al.

(2017) found strong pulsed linear polarisation (up to 40%) in the optical, highly modulated on the spin and beat periods. Takata et al.(2018) also detected non-thermal X-ray emission pulsed at the spin period of the WD. The optical and non-thermal X-ray light curve peaks are aligned in phase, suggesting that these are due to synchrotron radiation (SR) originating in the same locale. The multitude of multi-wavelength observations, especially optical, means there is a high data cadence enabling observers and modellers to infer the system’s parameters over the orbital period (i.e., phase-resolved) instead of averaging over large sections of the orbital phase of the system.

Since the M-dwarf was found to be inside the WD’s magnetosphere, distinguishing AR Sco as a novel system, different models have been proposed to explain the observed emission and polarisation from this source. These include standard SR calculations, particle injection from the companion leading to magnetic mirroring occurring at the poles of the WD, and magnetic wind interaction causing magnetic reconnection and plasma loading of the WD’s magnetic field lines

(Geng et al.,2016;Takata et al.,2017;Lyutikov et al.,2020). Most of the proposed models have

come to a consensus in recent years that the pulsed non-thermal emission is magnetospheric but the injection scenario, radiation mechanism, and most recently the magnetic field strength are still being debated. Thus better constraints for the system are essential to reaffirm the spatial location of the emission as well as the injection scenario and radiation mechanism governing the emission of AR Sco.

The first radio pulsar was observed in 1967 as a pulsating radio source by a graduate student Jocelyn Bell (Hewish et al., 1968). This detection was only published later, due to skepticism surrounding the observation (Hewish,1975). Pulsars are theorised to be rapidly-rotating neutron stars (NSs), where NSs are formed due to a Type II supernova, creating an extremely compact object. A seminal paper by Goldreich and Julian (1969) addressed the electrodynamics of a pulsar and around the same time a basic geometric model for radio pulsars was developed by

Radhakrishnan and Cooke (1969), predicting the polarisation signature of a pulsar, and forming

a backbone for understanding the geometry of the emission cones of pulsars. This allowed the angle between the rotation and magnetic axis and the observer’s line of sight to be constrained. Following an agglomeration of emission mechanisms published byBlumenthal and Gould (1970)

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more sophisticated models started to take form, being able to predict light curves and spectral energy distributions (SEDs). Examples include those ofSturrock (1971);Harding et al. (1978);

Arons (1983);Cheng et al.(1986);Daugherty and Harding (1996);Romani (1996). Thus, similar

to the spin-down of an NS powering the pulsar emission, the WD’s spin-down in AR Sco is believed to power the non-thermal, pulsar-like emission observed. The first suggestion of a WD operating with similar acceleration mechanisms to that of an NS was made byUsov (1988). Using Keplerian mechanics it is possible to constrain the mass of a compact object in a binary system. Interestingly, there is believed to be no observed isolated WD pulsars, due to WDs having low magnetic fields and low densities (the denity of the star affects the contribution of the general relativitic effects) compared to NSs (the highly periodic non-thermal emissions observed in pulsars are unfeasable or undetectable due to the inefficiencies of these pulsar-like acceleration mechanisms in WD pulsars). As will become evident in the case of AR Sco, if there is a companion present fulfilling certain requirements, similar mechanisms to those operating in pulsars may perhaps drive the emission for this source. The recent reclassification of the system AR Sco by Marsh et al. (2016) from a δ-Scuti star in the 1970s (Satyvaldiev, 1971) to a WD pulsar or new type of polar has caused a novel addition to the family of WD systems, since AR Sco is not believed to be a standard polar. This is due to the fact that optical/UV/X-ray spectroscopic observations of AR Sco find no presence of accretion or an accretion disc in the system (Marsh et al.,2016). The inclusion of a non-zero inferred spin-down rate and highly linearly-polarised emission1 of the source warranted the definition of AR Sco as the first observed WD pulsar, even though the emission mechanisms and particle acceleration scenarios are not exactly the same as those in standard neutron star pulsar.

1.2 Research Aims and Motivation

In this thesis I will apply the rotating vector model (RVM) to the observed polarisation position angle (PPA) data fromBuckley et al.(2017), determining if the model can predict the polarisa-tion signature observed in AR Sco. We will also calculate and propose a broad emission scenario for the source. The main objective of this thesis is thus to investigate if a basic geometric model can be used to describe AR Sco. This will allow us to determine if a WD pulsar is governed by similar emission scenarios as a standard NS pulsar.

Using the RVM, I will be able to constrain the magnetic inclination angle α, the observer angle ζ, and the WD mass. The fact that the RVM reasonably models the PPA will also

1_{The mentioned observations may not be enough evidence for classifying AR Sco as a WD pulsar, since the}

system could be an accreting WD in a propeller regime, but X-ray spectroscopic observations by Takata et al.

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indicate that the emission is magnetopheric in origin and close to tangetial to the local magnetic field. Using the extensive polarimetric data from Potter and Buckley (2018a), I will obtain the orbitally phase-resolved α and ζ values to investigate potential precession present in the system or variation in these parameters. By calculating emission timescales, I will constrain the dominating emission source, confirming if it is indeed SR. Additionally, using the observed multi-wavelength data I can fit the corresponding SR and curvature-radiation (CR) spectrum to constrain the typical magnetic field strength, curvature radius, minimum particle Lorentz factor, and maximum particle Lorentz factor.

1.3 Thesis Outline

The structure of this thesis is presented in the following order:

In Chapter 2, I introduce the basic physics and formation of WDs and NSs. Next, the fun-damentals of pulsar electrodynamics and emission mechanisms are explained. I conclude this Chapter with the introduction of polarisation and polarisation measurements.

I discuss AR Sco’s historical observations, emphasising the novelty of the system in Chapter 3. Additionally, I will introduce some of the models for AR Sco proposed by various authors, with minor critique on their assumptions.

In Chapter 4, I will discuss the implementation steps and other assumptions of our model to obtain the geometric and emission constraints. This includes the model calibration and problems encountered while implementing the model, as well as the goodness-of-fit tests used in the analysis.

For Chapter 5, I will indicate and discuss the constraints obtained by fitting the RVM to the different data sets. I will emphasise the geometric constraints like the magnetic inclination angle and observer angle due to the model fitting the data so well. Any significant resulting features are also highlighted. Additionally, I outline future work to be pursued by presenting preliminary phase-resolved data fitting, obtaining α and ζ vs. orbital phase.

I do additional calculations and interpret the results in Chapter 6 to propose a broad emis-sion scenario, obtain constraints on the emisemis-sion mechanism, and constrain parameters like the magnetic field, and the maximum particle Lorentz factor.

In Chapter 7, I present the conclusion of the results and calculations of this thesis. Additionally, I will mention future research in the context of a more sophisticated emission model to better understand the source.

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I discuss the MeerKAT Open-Time proposal and imaging results in Chapter 8, linking the pulsed radio emission from AR Sco to other potential FRB candidates or similar discoveries. In Appendix A, I include a detailed derivation for the RMV as well as present additional calculations.

I show additional plots and a parameter atlas for the RVM in Appendix B to E.

1.4 Publications

The following publications resulted from this work:

Du Plessis, L., Z.Wadiasingh, C. Venter, and A. K. Harding, Constraining the Emission Geometry and Mass of the White Dwarf Pulsar AR Sco Using the Rotating Vector Model, Astrophysical Journal, 887 (1), 44, doi:10.3847/1538-4357/ab4e19, 2019a.

Du Plessis, L., Z. Wadiasingh, C. Venter, A. K. Harding, S. Chandra, and P. J. Mein-tjies, 2019b, In proceedings of 6th annual High Energy Astrophysics in Southern Africa (HEASA2018) conference, editor D.A Buckley, Modelling the Polarisation Signatures De-tected from the First White Dwarf Pulsar AR Sco, arXiv e-prints, arXiv:1907.01311. Du Plessis et al., Probing the Non-thermal Emission Geometry of AR Sco via

Phase-Resolved Polarimetric Data, (Astrophysical Journal paper, in prep)

Du Plessis et al., Modelling the non-thermal emission from the white dwarf pulsar AR Sco, (HEASA 2020 Proceedings of Science, in press)

An article containing the results discussed in Chapter 8 from the First MeerKAT Open-Time proposal is also being written.

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2

Background

In this Section I will give an introduction of two degenerate stars: neutron stars NSs and their formation, and then white dwarfs WDs and their formation. The Section then proceeds to discuss binary systems, since many contain an NS or a WD as a compact object for the binary system. Next, pulsars are introduced with a focus on standard pulsar electrodynamics and the concept of spin-down. The different radiation mechanisms believed to operate in pulsars are then be introduced, as well as the early emission models for pulsars. We will then look at the basic theory underpinning polarisation and how polarisation is actually measured. Lastly, the RVM will be discussed that is a geometric model used to describe the polarisation signatures of radio pulsars.

2.1 Degenerate Stars and Their Formation

2.1.1 Type II Supernovae and Neutron Star Formation

NSs are created from Type II supernovae that are defined by the presence of Hydrogen and Helium emission lines in the supernova remnant’s spectrum. A type II supernova occurs due to a core collapse of a massive star, Type Ib and Type Ic supernova follow a similar formation channel as a Type II supernova but lead to the formation of WDs instead of NSs. Type Ib and Ic supernovae will not be discussed in detail in this thesis. A star’s main source of energy

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comes from nuclear fusion and, indirectly, its gravitational potential providing the necessary pressure for nuclear fusion. In nuclear fusion, hydrogen atoms are fused to form helium and eventually heavier elements. Heavier stars with mass ≥ 8MJ undergo fusion and eventually

form iron, beyond which more energy is required for stability than generated by the fusion, while for lower-mass stars the fusion chain stops at carbon burning (Longair,1983).

A star stops its fusion process when it reaches the iron burning phase, since iron is the most stable element with the highest binding energy per nucleon, leaving a star with an iron core. Once the iron core reaches a critical temperature and density, the photodisintegration phase starts where the iron ions absorb the generated radiation and starts to decay. In this phase, electron capture occurs to form neutrons and neutrinos, where the neutrinos carry away vast amounts of energy from the star, e.g., 3.1 × 1045erg s−1 for a 20MJ star (Carroll and Ostlie, 2017). Usually, the radiation pressure from the core is in equilibrium with the gravitational force but in a very short time-scale the core pressure decreases rapidly. This causes the star to fall in on itself, compressing the core to extremely high densities until a new force creates a new equilibrium. This force is known as the electron degeneracy pressure force in the case of an WD or neutron degeneracy pressure in the case of a NS. When the collapsing core suddenly encounters the degeneracy pressure, a shock wave forms as the infalling matter rebounds off the dense core, dispersing the hydrogen-helium envelope that surrounded the core.

NSs consist of densely-packed neutrons forming a superfluid in the stellar interior. In the case of an NS their resistive force to gravity comes from neutron degeneracy pressure due to the fact that as the density increases in a star, the relativistic gas becomes degenerate (Longair,

1983). Calculating a simple equation of state of degenerate stars for both the relativistic and non-relativistic gas yields the relations p ∝ ρ4/3and p ∝ ρ5/3respectively, where p is the pressure and ρ is the density. A 1.44MJNS is estimated to have a radius R

NSof about 10 km. Calculating

the average density of an NS with these parameters yields ρ ≈ 6.65 × 1022g cm−3, which is almost three times larger than the density of an atomic nucleus.

Electron capture occurs more frequently at high densities, turning a proton into a neutron, since the lower energy levels are filled. This means that there is no space for an electron and the neutron does not revert back to a proton via β-decay. Neutrons experience no repulsive Coulomb forces, therefore higher densities can be reached in an NS compared to those in a WD. As a side note, the internal structure of an NS is not well understood because of the uncertainties in the equation of state of degenerate nuclear matter (Longair,1983). There are many proposed theories of the internal composition and structure of an NS.

The basic structure of an NS is described by Lorimer and Kramer (2005) as an outer layer consisting of solid heavy nuclei embedded in a degenerate electron gas that is super-conductive.

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The inner crust consists of a mixture of neutron-rich nuclei, degenerate neutron gas and degen-erate electron gas. As the density increases, a neutron fluid is formed with very few protons or electrons present. Finally, at the core, the density is so great that the state of matter at that density can only be speculated. Hence the assumption is made that an NS is a super-conductive sphere with large internal magnetic fields in the order of 1010− 1016_G.

It is well known that WDs have a Chandrasekhar mass limit that states there is a mass limit for a WD before the star experiences gravitational instabilities causing the star to collapses. This mass is calculated to be MCH= 1.44MJ(Tayler,1994;Longair,1983). Similarly, there exists a

mass limit for an NS after which it will become unstable due to gravitational instabilities. Carroll

and Ostlie (2017) state this limit as 2.2MJ for a stationary NS and 2.9MJ for a rotating NS.

This compares well to inferred NS masses as shown in Figure2.1.

Figure 2.1: The NS masses for diffrent types of NSs (Ozel and Freire¨ ,2016)

If the progenitor star rotates even slightly, the conservation of angular momentum would cause the collapsed core to rotate much faster. The relation between the initial angular velocity Ωi

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and the final angular velocity Ωf is given by: Ωf = Ωi Ri Rf 2 , (2.1)

where Ri is the initial radius and Rf is the radius of the collapsed core (Lorimer and Kramer, 2005). Thus an important property of an NS is that it rotates rapidly around its rotational axis. If the magnetic flux, φ =H B.da through a sphere, namely the core, is conserved, meaning the magnetic field before the core collapse Bi and the magnetic field after the collapse Bf can be

related by the following equation:

Bf = Bi

Ri

Rf

2

. (2.2)

This leads to a large increase, ∼ 108, in the magnetic field. This gives rise to the theory that pulsars are fast rotating neutron stars where these large rotating magnetic fields induce electric fields.

2.1.2 Formation and Properties of White Dwarf Stars

A WD is believed to form and reside at the core of low- or intermediate-mass stars (< 8MJ),

until the aging star expels its outer layer to reveal the WD residing on the inside. WDs are found in the lower left corner of the Hertzsprung–Russell diagram because of their high surface temperatures and their small radii (Tayler,1994). The following luminosity relation holds:

LWD= πacR2WDTe4, (2.3)

with a = 4σ/c being the radiation density constant, c the speed of light in vacuum, σ the Stefan-Boltzmann constant, RWDthe surface radius of the WD, and Te the effective temperature. The

core pressure Pc of a WD can be calculated using(Carroll and Ostlie,2017):

Pc≈

2 3πGρ

2_R2

WD, (2.4)

where G is the gravitational constant, ρ is the density and RWD is the radius of the WD.

Taking the values for Sirius B, Pc ∼ 106 times larger than the core pressure of the sun. Using

the differential equation for the temperature gradient dT /dr = −3κρ Lr/8πacT3r2 (Carroll and

Ostlie, 2017) and Equation (2.3) for the surface temperature of the WD, an equation can be

obtained for the core temperature Tc of a WD:

Tc≈ 3κρ 4ac LWD 4πRWD 1 4_, _(2.5)

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where κ is the opacity and LWD is the luminosity of the WD (Longair,1983).

Since a WD has a high core temperature (Tc∼ 8 × 107for Sirius B), it is believed to have surface

layers of hydrogen and helium and an ionised oxygen and carbon core. The hydrodynamic equi-librium of a WD is balanced by the electron degeneracy pressure caused by the Pauli exclusion principal, allowing only one electron with specific quantum numbers in a particular energy state. The electron degeneracy pressure due to non-relativistic electron gas is calculated inCarroll and

Ostlie (2017) to be: P = (3π 2₎2₃ 5 ~2 me Z A ρ mH 5₃ , (2.6)

where ~ = h/2π and h is Plank’s constant, me is the mass of an electron, mH is the mass of

a hydrogen atom, Z is the elemental number, and A is the mass number. Using the values for Sirius B this equates to P ≈ 1.9 × 1022N m−2, which is comparable to the central pressure of the WD calculated with Equation (2.4), Pc∼ 1022N m−2. This means that it is possible for the

electron degeneracy pressure to maintain the hydrostatic equilibrium of the WD.

2.1.3 Type Ia Supernovae

Type Ia supernovae are classified by the absence of hydrogen and helium lines in their spectrum, meaning there was no hydrogen-helium envelope present before the thermonuclear explosion

(Leibundgut,2000). These supernovae also have a very characteristic light curve shown in Figure

2.2used for secondary classification to distinguish them from Type Ib/Ic supernovae. Supernovae Type Ia are believed to occur in binary systems involving a WD and its companion. The supernova occurs when the WD reaches a maximum mass, known as the Chandrasekhar mass limit as discussed previously.

When this limit is exceeded, the star becomes unstable and ends in a thermonuclear explosion known as a supernova. Two situations are known to cause a type Ia supernova as described

by Leibundgut (2000). The first case is when two orbiting WDs lose energy by emission of

gravitational waves and eventually merge, causing the newly formed star to become unstable and form a black hole or NS. The second case is known as a cataclysmic variable, where a WD has a giant or a main sequence star as a companion. The companion star accretes material onto the WD’s surface, where the WD then burns this material, increasing its mass until the WD becomes unstable, resulting in a supernova Type Ia. Since supernovae Type Ia have very characteristic light curves, they can be calibrated with known variable stars and used as standard candles to determine cosmic distances.

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Figure 2.2: a) Light curves of multiple Type Ia supernovae. b)Looking at the luminosity-width relation, the light curves of a) overlap once dispersion is taken into account (Longair, 1983). Here dispersion refers to the fact that different electromagnetic wavelength propagate at different velocities through a medium.

2.1.4 Binary Systems

Binary star systems are a common occurrence in the Universe, with different types of binary systems being observed, namely visual, astronomic, eclipsing, spectrum, and spectroscopic bina-ries. Only two of these binary systems will be briefly discussed in this thesis. Eclipsing binaries are two orbiting stars that periodically eclipse one another, causing visible signatures in the observed luminosity. By observing these systems, one can calculate the orbital period, radii, and effective temperature of the stars. Spectroscopic binaries are two stars with discernible spectra, where Doppler shifting causes the spectral lines to be shifted from their wavelengths in their rest frames. This makes it possible to calculate the radial velocities of the stars, constraining their masses (Carroll and Ostlie,2017). Using the observed properties we can apply Kepler’s law for two orbiting bodies in a circular orbit given by

P2 a3 =

4π2 G(M1+ M2)

, (2.7)

where P is the orbital period, a is the orbital separation, G is the gravitational constant and M1

and M2 are the masses of the stellar bodies. It is common practice to use M1 as the compact

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angle i, the angle between the plane of the orbit and the plane of the sky, a mass function can be derived to constrain the mass of the compact object (Hartle,2003)

M3 2 (M1+ M2)2 sin3i = P V 3 1 2πG, (2.8)

where V1 is the orbital velocity of the first body. This equation can be written in terms of the

mass ratio M1/M2 for further simplification. Thus, by estimating i and M2 from the spectrum,

it is possible to constrain M1.

Figure 2.3: Equipotential surfaces of a binary system where the stellar bodies are shown as black dots and centre of mass as an x (Carroll and Ostlie,2017). The inner Lagrangian point is located at the intersecting point numbered 1.

A close binary system can now be defined where the two stars in the system share the same envelope. If we assume the binary system has a circular orbit, then using Newton’s gravitational attraction force, equipotential surfaces can be calculated for each star. Equipotential surfaces are surfaces constructed by connecting points with equal gravitational potential shown in Figure

2.3. Close to each star, these equipotential surfaces are spheres and become deformed as the distance from the star increases. The point where the two stars’ equipotential surfaces make contact is called the inner Lagrangian point and these two equipotential surfaces are called the critical Roche lobes. The inner Lagrangian point is also the point where mass passes through from the donor star to the compact object. The size of the Roche lobe is measured by its average radius r, such that the volume inside the Roche lobe is equal to the volume of a sphere with radius r (Paczy´nski,1971). The average radius in terms of the mass ratio M1/M2 = q is given

by

r1

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r1 a = 0.46224 M1 M1+ M2 1₃ for 0 < q < 0.8, (2.10) where a is the orbital separation. Using r1 with Equation (2.9) yields a larger value than using

Equation (2.10) for q > 0.523 (Paczy´nski, 1971). Paczynski postulates that for a close binary system there is a critical surface with average radius rcr where, if the star has a radius greater

than rcr, matter will flow out from the star and possibly accrete onto the companion.

As a side note using classical GR, the precession of the system can be calculated. The Schwarzschild metric for a non-rotating black hole and the appropriate killing vectors (a vector that does not change the components of the metric namely (1, 0, 0, 0)) can be used to find the conserved quan-tities of the system. These conserved quanquan-tities, namely energy and momentum, can then be used to calculate a free particle’s orbit and the precession of a stellar object (Hartle,2003). The precession per orbit is given by

φprec=

6πGM

c2_{a (1 −}2₎, (2.11)

where a is the semi-major axis or orbital separation for a circular orbit, and is the eccentricity of the orbit. Using the Kerr metric for a rotating black hole, the same result can be derived. The advantages of the Kerr metric is that Lense-Thirring precession (the precession of a free-falling gyroscope due to the rotation of a massive object) is also derivable.

2.2 Pulsars

2.2.1 Standard Braking Model

Pulsars are generally considered to be extremely fast rotating and highly magnetised NSs as is evident from Equations (2.1) and (2.2). The large rotating magnetic fields induce a large electric field that can accelerate particles. At the magnetic poles of the NS, there are two radio emission beams with high-energy emission regions located at higher altitudes from the surface of the star. A pulsar may be described as a rotator with the source having a magnetic inclination angle with respect to the rotation axis, causing the radio emission beams to sweep out across the sky and allowing an observer to see a pulsating source if the beam crosses their line of sight. Since the pulsar is radiating energy away, a spin-down model is used to describe the spin evolution of a pulsar due to the slowing rotational period over time if the star is not spun up by accretion

(Lorimer and Kramer,2005). The rotational kinetic energy is taken as the energy source for the

star and is converted partly to particle acceleration and partly to pulsed emission. Traditionally, the rotational energy loss rateE˙rot is equated to the vacuum magnetic dipole radiation loss rate

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(Ostriker and Gunn,1969), yielding d dt 1 2ImΩ 2 = ImΩ ˙Ω = − 2µ2Ω4 3c3 sin 2_α, _(2.12)

where Im is the moment of inertia, Ps = 2π/Ω is the spin period, ˙Ω = 4π2ImP˙s/Ps3 is the rate

of change in the angular frequency, µ = BR3/2 is the magnetic moment, and α is the magnetic inclination angle. Solving Equation (2.12) for B, the magnetic field at the surface of the star can be calculated as B0≡ r 3c3 8π2 Im R6_sin2_αP ˙P . (2.13)

Using I = 1045g cm2, R = 10 km and α = 90◦, reduces Equation (2.13) to B0 = 6.4 × 1019

p

P ˙P G (2.14)

at the poles. If we characterise the spin-down as ˙Ω = −kΩn, where k is a constant and Ω is the angular frequency then the braking index n can be found by solving for n when calculating ¨Ω/ ˙Ω. To calculate the age of a pulsar we integrate ˙Ω = −kΩn _{and assume n > 1 and that the initial}

angular velocity is Ω0 Ω. This yields τ = P/ (n − 1) ˙P and by setting n = 3 (for magnetic

dipole radiation), τ = P/2 ˙P , which is the characteristic age of a pulsar (Ostriker and Gunn,

1969).

Since the magnetic field and the age of a pulsar is dependent on P and ˙P we can form P ˙P graph shown in Figure2.4. The dashed lines indicate the age and surface magnetic field strength of a pulsar. This also may aid in understanding the evolution cycle of a pulsar as it ages. This is due to the proposed idea that pulsar recycling can occur if the pulsar is in a binary system, the pulsar accretes from the companion, and the pulsar’s magnetic field is low enough that the pulsar can be spun up. The pulsar then becomes accretion-powered causing the pulsar to rotate faster and the radio emission to stop during the accretion phase. As the pulsar becomes rotation-powered when the accretion stops due to the pulsars’s larger magnetic field and fast rotation. The radio emission of the pulsar resumes as the accretion stops, thus forming a millisecond pulsar.

2.2.2 GJ Magnetosphere

After Gold (1968) proposed some of the first models for the pulsar mechanism, Goldreich and

Julian (1969) started the investigation into the electrodynamics of a pulsar. They began by

assuming that an NS is a perfectly conducting sphere with an aligned magnetic field and spin axis (µ k Ω). A dipole magnetic field structure (r = k sin2θ, in polar coordinates, where k indicates the specific field line) is assumed outside the star, with the interior magnetic field

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Figure 2.4: P ˙P diagram where the positive slope dashed lines indicate the age of the pulsar and the negative slope dashed lines indicate the magnetic field strength (Gotthelf et al.,2013).

being Bin = B0~ez k µ (Padmanabhan,2001). Since inside the conductor E · B = 0, Goldreich

and Julian (1969) showed that the standard electrodynamics Maxwell equations yield

Ein+

(Ω × r)

c × Bin = 0 Ein= −

B0Ωr sin θ

c (sin θer+ cos θeθ) ,

(2.15)

where r, θ, and φ are the base spherical coordinates and the φ dependence is removed due to the axisymmetry of a sphere. In the stellar interior the condition ∇Ein = 0 is met, meaning

that the Laplace equation can be solved to find the potential, where R is the radius of the NS and r the distance from the surface,

Φ = −B0ΩR

5

2cr3 sin

2_θ. _(2.16)

Assuming there is no charge in the surrounding magnetosphere, the Lorentz invariant value E ·B can be calculated using the potential and assuming a dipole magnetic field

E · B = − ΩR c R r 7 B₀2cos3θ 6= 0. (2.17)

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The electric field component parallel to the local magnetic field can then be calculated as E · B | B | ≈ − ΩR c B0cos3θ ∼ 2 × 108B12P−1 statvolt cm−1, (2.18)

where B12 = B0/1012 G. This is about 108 times the gravitational force binding a proton,

meaning that particles are ripped from the surface of the NS and filling the magnetosphere. This means an NS cannot be surrounded by a vacuum (Goldreich and Julian,1969). If we take into account that a particle’s speed cannot exceed the speed of light in a vacuum, a cylindrical radius can be defined using the dipolar magnetic field structure,

RLC=

c Ω =

cP

2π, (2.19)

where particles travelling along these the magnetic field lines at this radius are co-rotated at approximately the speed of light. This is known as the light cylinder radius. This also defines the last closed magnetic field line forming the magnetosphere. Using the magnetic dipole equation (r = k sin2θ), Equation (2.19), and a small-angle approximation, sin−1θ ∼ θ, the polar cap angle θpc can be calculated as (Rybicki and Lightman,2008)

θpc'

r ΩR

c . (2.20)

Using this angle, a corresponding polar cap radius RPC = R sin θ ∼ RpΩR/c can be derived.

A potential difference can hence be derived between the magnetic pole and θPC by substituting

RPC into Equation (2.16), yielding

−∆Φ = 1 2B0R ΩR c 2 . (2.21)

The particles pulled from the NS surface redistributes themselves such that there is no Lorentz force acting on them, thus creating a force-free magnetosphere. Gauss’s equation can then be solved to find the Goldreich-Julian charge density

ρGJ = ∇ · E 4π = Ω · B 2πc 1 1 − (Ωr/c)βtsin θ ' −Ω · B 2πc , (2.22)

where βt is the temporal component of the β four-vector. We then obtain an electron number

density of

ne' −

Ω · B

2πec, (2.23)

where e is the electron charge.

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along these closed magnetic field lines of the WD and stay confined to the magnetosphere. The open magnetic field lines are forced open, since particles travelling along the magnetic field lines cannot exceed the speed of light in vacuum as they are co-rotating. Particles liberated from the polar caps are thus accelerated along the open magnetic field lines away from the NS, and emit radiation.

Figure 2.5: A pulsar magnetosphere showing the light cylinder and the slant dashed line indi-cating where Ω · B = 0 (ρGJ = 0)(Goldreich and Julian,1969).

2.3 Radiation Mechanisms

2.3.1 Synchrotron Radiation

From classical electrodynamics it can be shown that accelerating charges radiate, which produces the Lamour equation for the radiated power per charge. A charged particle gyrates around a magnetic field with an angular frequency ωg = qB/γmc known as the relativistic angular

gyrofrequency, which is easily derivable from the equation of motion by setting v · a = 0. We thus get cyclotron radiation when these charges gyrate at non-relativistic speeds, leading to a spectrum of harmonics that are radiated. In the relativistic case, this is known as synchrotron radiation (SR), whereby a simple geometric method can be used to show that the emitted frequencies are boosted up to the range νω ∼ γ3νg (Longair, 1983). For SR there is a

non-zero velocity component parallel to the local magnetic field and a perpendicular acceleration component, with ak = 0. Using the Lorentz formula that gives the general expression for the

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acceleration a · a yields P = 2q 2_{a · a} 3c3 = 2q 2_γ4 3c3 a2_⊥+ γ2a2_k = 2q 2_γ4 3c3 ωgv 2 ⊥ = 2q 2_γ4 3c3 qB γmc 2 (v sin η)2 (2.24)

where ak = 0, a⊥= ωgv⊥, and η is the pitch angle (Jackson,1962). Substituting some constants

yields

P = 2σTcβ2γ2UBsin2η, (2.25)

where the Thomson cross section is given by σT = 8πe4/3m2c4 and the magnetic energy density

UB= B2/8π.Taking the average over an isotropic pitch angle distribution (Padmanabhan,2001)

we obtain

P = 4 3σTcβ

2_γ2_U

B. (2.26)

Another important property is that the emission cone is highly beamed towards the observer and found to be approximately sin θ ∼ 1/γ. A critical frequency can be defined (Blumenthal and

Gould,1970) by ωc= 3 2γ 3_ω gsin η, (2.27)

which arises from the fact that the acceleration rate cannot exceed the gyrofrequency, otherwise that would mean the particle is not bound to the magnetic field, thus giving the limit tacc≥ tSR.

This then allows us to calculate the power per unit frequency (Blumenthal and Gould, 1970) given by P (ω) = √ 3q3B sin η 2πmc2 F (x), (2.28) where F (x) = xR∞

x K5/3() d, K5/3 is the modified Bessel function of the order 5/3 and is also

referred to as a power spectrum with x = ω/ωc. Longair (1983) notes that the function has the

following asymptotic limits

F (x) ∼ √ 4π 3Γ 1₃ x 2 1/3 , x 1 F (x) ∼ π 2 1/2 exp−xx1/2, x 1 (2.29)

where Γ(x) is the Gamma function and can be calculated by using Γ(n+1) = nΓ(n) and Γ(1) = 1 for a real number, since Γ is the generalisation of the factorial. If the electron energy distribution is assumed as a power law with a number density of electrons N (E)dE with the electron energy

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distribution from E to E + dE, then it can be expressed as the relation (Rybicki and Lightman,

2008)

N (E)dE = κE−pdE, (2.30)

where p is the power law index of the electrons and κ a normalisation constant. We can now use Equation (2.28), the power per unit frequency, and calculate the total power radiated as a function of frequency Ptot(ω) = Z E2 E1 P (ω) N (E) dE ∝ Z γ2 γ1 F ω ωc γ−pdγ ∝ ω−(p−1)/2, (2.31)

where the spectral index is a = (p − 1) /2. If Fν is taken to be the flux, then Fν ∝ PSR(ν) ∝ ν−a.

The result is then found to be νFν ∝ ν−(p−3)/2, which is used to plot spectral energy distribution

figures, since νFν =erg/cm2/s gives an energy flux per area. We use νFν since dFν/d(ln(ν)) =

dνFν/dν, which is the energy flux per logarithmic frequency interval. Additionally, the

SR-reaction-limited Lorentz factor can be calculated by equating the particle acceleration to the SR loss rate, ecEk= ˙ESR= 4σTuBcγ2/3. Assuming that the pitch angle distribution stays constant

and solving for γ yields

γSRR= _E k6πe σTB2p 1/2 . (2.32)

To derive the SR-reaction-limited Lorentz factor accurately, one would need to include a mech-anism to replenish the perpendicular momentum of the particles, since Ek only replenishes the

parallel momentum component. In the scenario of synchrotron self-absorption, where a relativis-tic charged parrelativis-ticle in the magnerelativis-tic field can absorb a photon or if stimulated emission occurs, the spectral index can change. If one assumes the absorbed photon energy is much smaller than the charged particle energy it interacted with, the absorption coefficient can be calculated with a dependence of ν−(p+4)/2 (B¨ottcher et al., 2012). This indicates that the opacity of a non-thermal source increases with smaller frequencies until the source becomes optically thick. This frequency is known as the synchrotron self-absorption frequency and taking this effect into account causes a = 5/2 in that regime (Rybicki and Lightman, 2008). Usually this regime is located in the radio or low-frequency optical band.

Next the function G(x) is defined similarly to how F (x) was defined (Rybicki and Lightman,

2008)

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where K_3/2is the modified Bessel function of order 3/2. The power radiated can then be written in terms of the two linear polarisation components (Longair,1983)

P⊥(ω) = √ 3q3B sin η 4πmc2 (F (x) + G(x)) Pk(ω) = √ 3q3B sin η 4πmc2 (F (x) − G(x)) . (2.34)

The ratio of power yielded for the two polarisations for one electron can be shown to be P⊥/Pk=

7 (Longair,1983). Longair also showed that the fractional polarisation is

Π = P⊥(ω) − Pk(ω) P⊥(ω) + Pk(ω)

= G(x)

F (x). (2.35)

Using an electron power law distribution the fractional polarisation reduces to Π = p + 1

p + 7/3. (2.36)

2.3.2 Curvature Radiation

In the previous segment it was noted that SR is associated with change in perpendicular momen-tum or energy. Curvature radiation (CR) is associated with longitudinal energy change, namely a charged particle following a curved path. As shown in Section 2.2.2, particles are liberated from the pulsar surface and accelerated along the magnetic field lines, making CR a more im-portant radiation process for pulsars’ primary particles than SR (Sturrock, 1971). For CR it is highly efficient if the particles can be accelarated above a Lorentz factor of ∼ 107. Assuming a sufficiently large radius of curvature to use classical electrodynamics, Jackson (1962) defines a critical frequency beyond which the radiation is exponentially suppressed as

ωc= 3 2γ 3 c ρc , (2.37)

where ρcis the curvature radius. The critical energy can be calculated by

EC = ~ωc= 3~cγ 3

2ρc

, (2.38)

which can be obtained through similar means as for SR in the previous subsection. The curvature radius close to the stellar surface is approximated inHarding et al.(1978) as

ρc≈

4Rθpc

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where θpc is the polar cap angle as defined in Equation (2.20) and θ the polar angle with regards

to the magnetic axis. At large distances, the curvature radius can be approximated as the light cylinder radius ρc≈ RLC, although this becomes even larger for force-free magnetosphere

(Kalapotharakos et al., 2014). The CR power spectrum is given in Jackson (1962) and Erber

(1966) as dPCR dE =√3αf γc 2πρc F ω ωc , (2.40)

where F is defined similarly as before F (x) = xR_x∞K5/3() d, and αf = e2/~c is the fine

structure constant. Integrating over all frequencies yields (Jackson,1962)

Ptot=

2e2c 3ρ2

c

β4γ4. (2.41)

For detailed derivations of these equations, seeJackson (1962). By substituting standard pulsar radii and magnetic fields into Equation (2.41), CR photons are found to have GeV energies and can produce electron-positron pair cascades in young pulsars. The power spectrum of CR for a single particle energy is also found to be similar to that of SR. Equation (2.31) for an electron energy distribution can also be calculated for CR by substituting PSR for PCR in Equation

(2.40). Similarly as shown for SR we can obtain a curvature radiation reaction limited Lorentz factor by equating the acceleration rate to the CR loss rate, ecEk= ˙ECR= 2e2cγ4/3ρ2c Harding

et al. (2005). Solving for γ yields

γCRR∼ (1.5Ek/e)3/4ρ1/2c ∼ 5 × 107. (2.42)

2.3.3 Inverse Compton Scattering

Compton scattering is a well-known process of an interaction between a high-energy photon and a low-energy electron. The scattered photon experiences a shift in wavelength due to energy and momentum transfer to the electron, namely

∆λ = h

mc(1 − cos φ), (2.43)

where φ is the scattering angle of the photon. Inverse Compton scattering (ICS) is the interaction between a low-energy photon and a high-energy electron, where the electron upscatters the photon to higher energies. This is an important process, as energy electrons move in high-density photons fields in astrophysical settings (Longair,1983). Relativistic electrons can boost photon energies depending on the initial energy and scattering angle of the photon. For the Thomson (elastic) limit γ mec2 the maximum energy boost is γ2, meaning Eph ≈ γ2. In

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to the particle energy, namely Eγ≈ γmec2. In the Thompson regime, the radiation is scattered

elasticity whereas in the Klein-Nishina regime, the radiation can be polarised but with lower levels of linear polarisation than SR (Longair, 1983). The power radiated by an electron in an isotropic radiation field in the Thompson regime is given by (Blumenthal and Gould,1970)

PIC =

4 3σTcβ

2_γ2_U

rad, (2.44)

where Urad is the photon energy density. Interestingly, it is found that the ratio of the loss rates

can be expressed as PIC PSR = Urad UB , (2.45)

in the Thompson regime (Rybicki and Lightman,2008). This is useful when comparing flux peaks of SR and IC in spectral energy distribution (SED) plots, since this may lead to constraints on the magnetic field if Urad may be estimated. As the energy of the initial photon increases, the

regime changes from Thompson to Klein-Nishina, where the Thompson cross section σT needs

to be replaced by the Klein-Nishina cross section σKN (Klein and Nishina,1929)

σKN = 3 2σT 1 + x x3 2x (1 + x) 1 + 2x − ln (1 + 2x) + 1 2xln (1 + 2x) − 1 + 3x (1 + 2x)2 , (2.46)

where x = γhν/mec2. Taking the Klein-Nishina effects into account and assuming a black body

as the source of soft photons, the loss rate can be approximately written as (Schlickeiser and

Ruppel,2010) ˙ EIC = 4σTcUradγ2 3 γ_KN2 γ2 KN+ γ2 , (2.47) where γKN= 3√5mec2 8πkBT , (2.48)

with Urad = 2σSBT4R?2/cR02 the soft-photon energy density, and T the soft-photon temperature.

In these equations, me is the electron mass, σT the Thomson cross section, σSB the

Stefan-Boltzmann constant, k the Stefan-Boltzmann constant, R? the emitting body’s radius and R0 is the

distance to the photon field. If the assumption is made that the electron distribution is given by a power law, the number of electrons are Ne ∝ γ−p and if the soft-photon distribution is

approximated by a black body, the power spectrum is given by (Blumenthal and Gould,1970) dN dEγ tot ∝ E_γ−(p+1)/2, Thomson regime dN dEγ tot

∝ E_γ−(p+1), Klein − Nishina regime.

(2.49)

It can be seen that the spectrum in the Thompson regime is similar as for classical SR, but in the Klein-Nishina regime the spectrum becomes steeper. The cut-off in the spectrum occurs due

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to the electron giving all of its energy to the photon in the interaction.

2.3.4 Pair Production

The process of pair production operates in environments with efficient radiation processes and high magnetic and electric fields (Daugherty and Harding,1982). Pair production can occur in one of two regimes, namely single-photon pair production, which requires high magnetic fields and two-photon pair production, which requires high photon densities. In the case of single-photon pair production, the single-photons can interact with a strong magnetic field to produce an electron-positron pair. If these photons have large enough energies, the probability of a number of photons nph traveling a distance d in the magnetic field to create a number of pairs np can

be calculated for a uniform magnetic field asErber (1966) np = nph

1 − e−αat(χ)d_, _(2.50)

where nph and np are number densities, αat(χ) the photon attenuation coefficient and χ =

0.5 hν/mec2 (B⊥/Bcr) the Erber parameter. The critical magnetic field Bcr = m2ec3/e~ is a

useful parametrisation used in Daugherty and Harding (1983) where the gyration energy of a particle is equal to its rest mass. The attenuation coefficient, which is the photon absorption per length for a photon propagating perpendicular to a uniform magnetic field (Erber,1966), is given as αat(χ) = 1 2 αf λc B⊥ Bcr T (χ) , (2.51)

where λc= h/mc is the Compton wavelength and T (χ) a modified Bessel function. The

asymp-totic properties of T (χ) are

T (χ) = 0.46e−3χ4 _, _{χ 1}

T (χ) = 0.60χ−1/3, χ 1

(2.52)

These limits can be inserted into the attenuation equation to be written in the following form

(Luo et al.,2000). αat(χ) = 0.46 αf λc χ ph , (2.53)

where ph = Eph/mec2 is the photon energy in normalised units. From this equation we see

that if B⊥ increases the attenuation coefficient increases, meaning the pair production becomes

more efficient. An approximated formulation for the conditions necessary for single-photon pair production is given bySturrock (1971) as

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Two-photon pair production occurs when two photons with sufficiently high enough photon energies, namely Eph > mec2 collide, creating an electron-positron pair. The cross-sectional area

for this collision can be calculated in terms of the photon energy in the centre of momentum frame as shown bySvensson (1982).

2.3.5 Pulsar Emission Models

A number of emission models and hybrid emission models exist although most incorporate the ideas of the older fundamental emission model like the polar cap model ofDaugherty and Harding

(1982), the slot gap model of Arons (1983), and the outer gap model of Cheng et al. (1986). A more modern class of emission model includes current sheet models where the magnetic field beyond the light cylinder forms current sheets that produce the high-energy radiation.

Figure 2.6: An illustration of a pair photon cascade fromDaugherty and Harding(1982). The produced SR and CR photons are shown as well as the secondary e± _particles.

The polar cap model first proposed bySturrock (1971), speculates that the emission originates in the regions just above the pulsar’s magnetic poles, however the polar cap pair cascades were first calculated by Daugherty and Harding (1982). As discussed previously, particles can be ripped from the stellar surface and accelerated by the induced electric field along the magnetic fields. These accelerated particles emit CR at low altitudes and, if the photon energy is high enough, produces electron-positron pairs through interaction with the magnetic field. The transverse energy of these particles are quickly lost through SR because of the large magnetic fields. Since the particles have different charges, they are accelerated in opposite directions by the electric field. The particles reabsorbed at the the stellar surface heat the polar cap even further. Particles

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accelerated away from the surface will emit CR or SR and once again interact with the magnetic field, which leads to the creation of particle cascades. This is illustrated in Figure2.6. At some radius above the polar cap, the charge density will exceed that of the Goldreich-Julian charge density and the induced electric field will be screened out. This radial boundary is known as the pair formation front (PFF). This region below this boundary is considered to be radially uniform above the stellar surface. Primary particles are thus accelerated by a non-zero electric field and radiate gamma rays through CR above the stellar surface until they reach the PFF, where they keep radiating until their energy is depleted.

Figure 2.7: Schematic fromHarding and Muslimov(2005) showing the PFF, and thickness of the slot gap ∆ξSG. It also shows how the slot gap asymptomatically approaches the last closed

field lines.

The slot gap model proposed byArons (1983) is similar to the polar cap model. The difference is that the PFF is not uniform in the slot gap model. This is due to the boundary condition of Φ = 0 at the last closed field line, since the induced parallel electric field changes as one moves away from the centre of the magnetic pole where Ek (the electric field component parallel to

the magnetic field) drops until it becomes zero at the first closed magnetic field line. With a lower electric field, the particles can move farther from the stellar surface before creating pairs, meaning the PFF will be located at higher altitudes, as shown in Figure 2.7. The solutions for the parallel electric field and the slot gap width were calculated by Muslimov and Harding

(2003), also invoking general relativistic frame dragging that increases Ek.

The outer gap model proposed byCheng et al.(1986) stipulates that at the postulated neutral current sheet where Ω · B = 0, shown in Figure 2.5and 2.8, positive charges flow back through this neutral sheet while negative charges flow past the light cylinder. They propose an excess of negative charge where the charge density deviates from the Goldreich-Julian charge density,

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Figure 2.8: An illustration fromCheng et al.(1986) showing the different particle acceleration regions for the outer gap model.

forming an electric field in the last open field line. In this region Ek 6= 0, meaning particles can

be accelerated to emit CR, SR and IC emission. This region consists of different subsequent regions and more involved physics, which is beyond the scope of this thesis.

2.4 Polarisation

If the electric field of an electromagnetic wave is oscillating in one direction (in a plane), the wave is said to be linearly polarised. This direction of oscillation and the direction of propagation define the plane of polarisation. Let us now consider an electric field vector composed of two linearly polarised components, namely

E = (E1ˆx + E2y)eˆ −iωt, (2.55)

where ˆx and ˆy are the unit vectors, ω is the angular frequency and t is the time (Rybicki and

Lightman,2008). The complex amplitudes can then be written as E1 = ξ1eiφ1 and E2= ξ2eiφ2.

This leads to the electric field x and y components

Ex= ξ1cos(ωt − φ1), Ey = ξ2cos(ωt − φ2), (2.56)

where φ1 and φ2 are the phase shifts. If the two components’ amplitudes differ, using a rotated

(primed) axis we may write in the case of elliptical polarisation:

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Figure 2.9: The principal axis of the polarisation ellipse where the electric field components are rotated by an angle χ (Rybicki and Lightman,2008).

where β is the ratio between the x0 and the y0 component (or the phase shift between these two components) and ξ0 is the amplitude. Rotating Equations (2.57) by χ yields

Ex= ξ0(cos β cos χ cos ωt + sin β sin χ sin ωt),

Ey= ξ0(cos β sin χ cos ωt − sin β cos χ sin ωt).

(2.58)

Equating these two equations with Equation (2.56) produces four equations given inRybicki and

Lightman (2008), where solving for ξ0, β, and χ leads to the Stokes parameters.

I ≡ ξ₁2+ ξ₂2= ξ2₀ Q ≡ ξ₀2cos 2β cos 2χ U ≡ ξ₀2cos 2β sin 2χ V ≡ ξ₀2cos 2β,

(2.59)

where I is the energy flux or intensity, since I2= Q2+U2+V2, and V is the circular polarisation. The equations above can also be written in terms of linear polarisation components or right- and left-handed components. The Stokes parameters given as linear polarisation components are:

I =E2 x + Ey2 Q =E2 x − Ey2 U = 2 hExEycos δi V = 2 hExEysin δi (2.60)

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where δ = φ1 − φ2 is the phase difference and the brackets represent the time-averaged value

(Trippe,2014). Using the Stokes parameters, we can calculate the elliptical polarisation

param-eters ξ0 = √ I sin 2β = V I tan 2χ = U Q, (2.61)

where the parameters Q and U are used to measure the orientation of the ellipse with respect to the x-axis. It is important to note that the degree of linear polarisation (mL=

p

Q2_{+ U}2_/I)

and χ the polarisation angle, gives the length and orientation of a vector centred in the vector space of Q and U respectively (Trippe,2014).

2.4.1 Polarisation Measurements

There is a lot of nuance with measurements of polarisation since there are different changes to the polarisation plane and amplitude to take into account as an electromagnetic wave propagates through different media. A few examples of these changes include scattering, reflection, Faraday rotation, Faraday depolarisation, birefringence, and the Chandrasekar-Fermi effect. These are discussed byTrippe (2014). There are also different formalisms used when working with polari-sation, but for the purpose of this thesis, only the basics will be discussed for some background on the subject.

In the radio-frequency regime, the recorded radiation can be analysed as waves with amplitudes and different phases. This means that two perpendicular dipole antennae along the x and y axes can be used to measure the two polarisation components. The linear polarisation components can then be used to obtain the Stokes parameters by using Equations (2.60).

In the optical-frequency regime, the light intensity is measured instead of wave amplitudes. Thus polarisers, polarisation wave splitters, and polarisation plane turners are necessary for polarimetric observations. The different linear polarisation intensities can then be used to obtain the different Stokes parameters (Trippe,2014)

Q I = I(0◦) − I(90◦) I(0◦_{) + I(90}◦₎ U I = I(45◦) − I(135◦) I(45◦_{) + I(135}◦₎. (2.62)

These values, namely I(0◦) etc., refer to the measured intensity when the polariser is rotated by the specified degrees in brackets. The Stokes parameters can also be obtained by measuring the

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intensities at a minimum of four different phases and using Equation (65) discussed in Trippe

(2014).

For X-ray and γ-ray polarimetic observations, Bragg diffraction can be used but this will not be discussed since it is beyond the scope of this thesis. Instrumental effects also play a large role in polarisation observations but is beyond the scope of this thesis.

2.5 The Rotating Vector Model (RVM)

Pulsars can be schematically thought of as rotating NSs with two radio emission cones located near the magnetic poles. Figure 2.10 defines a magnetic inclination angle α of the magnetic dipole moment µ with respect to the rotation axis Ω. The beam geometry of a pulsar shown in Figure2.10also defines an observer angle ζ, which is the angle between the rotation axis and the observer’s line of sight. The impact angle is β = ζ − α, where ψ is the polarisation position angle (PPA), φ is the sweeping or azimuthal angle, W is the pulse width and ρ is the half-opening angle of the beam. The RVM is a geometrical model normally used for radio pulsars, where the PPA is predicted by the following equation (Radhakrishnan and Cooke,1969)

tan(ψ − ψ0) =

sin α sin(φ − φ0)

sin ζ cos α − cos ζ sin α cos(φ − φ0)

. (2.63)

The parameters φ0 and ψ0 are used to define a fiducial plane. The derivation of Equation (2.63)

is done in Appendix A and a parameter atlas for α and β, showing how these two parameters affect the resulting PPA in Equation (2.63), which is shown in Appendix E.

The RVM makes the following assumptions: a zero-emission height, all the emission is tangent to the local magnetic field, the pulsar’s co-rotational speed at the emission altitude is non-relativistic, the emission beams are circular, the magnetic field is well approximated by a static vacuum dipole magnetic field, and the plane of polarisation is perpendicular to the local magnetic field. It is important to note that the RVM is a geometric model, meaning that one can not produce spectra or light curves using the RVM alone.

Figure2.11shows that at the edges of the emission beam, the PPA changes slowly with φ, but as it approaches the centre, (φ = φ0), it increases more rapidly, forming the well-known canonical

S-shape. The steepest gradient of Equation (2.63) is given by

dψ dφ max = sin(α) sin(β), (2.64)

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Figure 2.10: Schematic of the radio emis-sion beam of a pulsar, indicating all the ge-ometric angles associated with the emission beam (Lorimer and Kramer,2005).

Figure 2.11: The emission beam and the path of the line of sight across the beam. The bottom image shows the ob-served PPA as it changes along this path

(Lorimer and Kramer,2005).

which is useful to help determine the continuity of the model at specific parameter choices. An example of this is where α = ζ meaning β = 0, leading to Equation (2.64) blowing up due to the division of zero.

In the next Chapter, an observational introduction to the binary system AR Sco will be given and various proposed models will be discussed that try to explain these observations.

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The AR Sco System

In this Chapter, I will highlight the primary observations that led to the novel classification of AR Sco. Next, I will introduce the fundamental observational properties that are key observables for models to reproduce, namely the light curves, phase-resolved polarisation signatures, and SED plot. I will then discuss the different interpretations that led to the different proposed models for AR Sco. Additionally, I will highlight controversies including the spatial location of the mechanism driving the particle acceleration, the particle injection source, and the particle emission mechanism. Finally, each model is discussed individually as well as some of their shortcomings, since these drive the motivations for this thesis as well as future research.

3.1 Observational Introduction to AR Sco

AR Sco is an intriguing binary system containing a putative WD and an M-dwarf companion. The system exhibits pulsed non-thermal radio, optical, and X-ray emission as shown in Figure

3.1, likely of SR origin (Marsh et al., 2016;Buckley et al., 2017; Takata et al., 2018). AR Sco has a binary orbital period of Pb = 3.56 hours, a WD spin period of Ps = 1.95 minutes, and

a beat period of 1.97 minutes. The light cylinder radius1 and orbital separation are RLC =

c/Ω ∼ 6 × 1011 cm and a ∼ 8 × 1010 cm, respectively, with Ω the spin angular frequency. Thus, the M star is located within the WD’s magnetosphere at a ≈ 0.13RLC. A change in spin

1

This is the radius where the corotation speed equals that of light in vacuum.