A Review of Pulsar Glitch Mechanisms

Gregory Brett McDonald

May 2007 ABSTRACT

In this thesis, a review of the most prominent pulsar glitch mechanisms is presented. This includes a discussion of the internal structure of neutron stars and the way in which glitches may be used to describe that structure. Particular at- tention is paid to the two-component model. This thesis also includes numerous simulations of the two-component model, which are ﬁtted to observational data in order to determine how suitable this mechanism is as a description of pulsar glitches. The results show that this mechanism is in fact still relevant, in spite of its age. Abstract 3

Acknowledgements

I would like to take this opportunity to thank a few people, without whom this thesis would not have been possible. First, I would like to thank my family for all of their love and support throughout the course of this project. Mom, Ruth and Josh, I love you guys. To Chris Engelbrecht and Fabio Frescura, thanks for all the input you gave me over the past few years, and for sparking my interest in the wonderful world of astrophysics. Thanks for the many useful conversations and discussions, and for convincing me that it would all turn out alright in the end, no matter how bleak it appeared at the time! To Marten, my co-conspirator and trusty side-kick, I will always remember the long lunchtime walks to the Student Center and the many games of squash we played when our frustrations threatened to overcome us. I hope you find every happiness in your future career. To the South African Square Kilometer Array Project Office (SASPO), thank you for your unfailing support of my studies, for providing such a wonderfully generous bursary, and the opportunity to study such a fascinating subject. I am saddened not to be continuing my work with you, but I hope that we will get the opportunity to work together in the future. And finally, I want to thank my Father for enabling me the opportunity to meet such wonderful people and study such an interesting subject.“You placed the stars in the sky, and you know them by name.” Without you I am nothing. CONTENTS

1. Overview of Neutron Stars ...... 9 1.1 A Brief Overview of Neutron Star History and Discovery . . . . . 9 1.2 Pulsar Discovery and Interpretation ...... 10 1.3 The Structure of Neutron Stars ...... 11 1.4 Superfluidity ...... 14 1.4.1 Basic Superfluid Theory ...... 15 1.4.2 Basic Vortex Theory ...... 17 1.4.3 Quantised Vortices ...... 17 1.5 Vortex Pinning and Vortex Creep in Neutron Stars ...... 19 1.5.1 Factors influencing vortex creep rate ...... 19 1.5.2 Determining the Activation Energy ...... 21

2. Glitches and Glitch Mechanisms ...... 27 2.1 Glitches ...... 27 2.2 Brief Summary of Glitch Mechanisms ...... 27 2.3 Crust-driven Glitch Mechanisms ...... 31 2.3.1 Crust Fracture Model ...... 31 2.3.2 Thermally Driven Glitches ...... 36 2.4 Core-driven Glitches ...... 42 2.4.1 Flux-tube Model ...... 42 2.4.2 Centrifugal Buoyancy Mechanism ...... 46

3. Two-component Model: Theory and History ...... 49

4. Simulations ...... 57 4.1 Overview of Simulations ...... 57 4.2 Fitting Simulations to Observational Data ...... 64 4.2.1 Glitch Occurring in the Crab Pulsar in 1996 ...... 64 4.2.2 Glitch Occurring in the Crab Pulsar in 1975 ...... 69 4.2.3 Simulation Summary ...... 71

5. Concluding Remarks ...... 76

6. References ...... 78 LIST OF FIGURES

1.1 Figure showing neutron star mass results for fourteen pulsar binary systems. The dotted lines contain the region which has values agreeing with all measurements, i.e. M = 1.35±0.4M . (Thorsett, Chakrabarty 1999) ...... 11 1.2 Neutron star gravitational mass in solar units vs. neutron star radius, as given by various equations of state. The Friedman- Pandharipande EOS is given by the dashed line in the middle. (Heiselberg, Pandharipande 2000) ...... 12 1.3 Diagram showing the various regions in a neutron star...... 14 1.4 Diagram showing the dynamics which lead to the creation of the Magnus force FM ...... 20 1.5 An exaggerated diagram showing a pinned vortex bending under the inﬂuence of the Magnus force. The solid circles show the pinning sites...... 22 1.6 Conﬁguration used for the calculation of the energy for a single-site breakaway (Link, Epstein, 1991) ...... 23 1.7 Energy of a vortex line as a function of separation for a vortex line in the single-site breakaway regime (Link, Epstein 1991)...... 24 1.8 Energy of a vortex line as a function of separation for a vortex line in the single-site breakaway regime (Link, Epstein 1991)...... 25

2.1 Figure published by Radhakrishnan and Manchester (1969) showing the ﬁrst observed glitch, which occurred in the Vela pulsar . . 28 2.2 Diagram used to illustrate crust-cracking parameters (Ruderman, 1991)...... 33 2.3 Diagram showing the structure of the cylindrical regime considered in the Thermal Glitch Mechanism, as given by Link & Epstein (1996) 38 2.4 The results obtained by Link and Epstein (1996) for a comparison of the thermal glitch mechanism simulation of magnitude −8 35 ∆Ωc/Ωc ' 7 × 10 and an energy deposition of 2.1 × 10 J (line) to data (dots) for the Crab pulsar ...... 40 2.5 The results obtained by Link and Epstein (1996) for a comparison of the thermal glitch mechanism simulation of magnitude −6 35 ∆Ωc/Ωc ' 10 and an energy deposition of 1.51 × 10 J (line) to data (dots) for the Vela pulsar ...... 41 List of Figures 6

2.6 Properties of various crust layers, as given by Negele and Vautherin (1973) ...... 45

3.1 Simple picture showing a possible conﬁguration for the two-component model...... 49 3.2 Response of pulsar to glitch, as predicted by the simple two-component model...... 52

4.1 Results of a Crab-like pulsar simulation. The results are for the post-glitch reaction of the theoretical pulsar. The coupling constant for this set of results is 5 days. The superfluid percentages are given by the legend...... 59 4.2 Results of a Crab-like pulsar simulation. The results are for the post-glitch reaction of the theoretical pulsar. The coupling constant for this set of results is 10 days. The superfluid percentages are given by the legend...... 60 4.3 Results of a Crab-like pulsar simulation. The results are for the post-glitch reaction of the theoretical pulsar. The coupling constant for this set of results is 20 days. The superfluid percentages are given by the legend...... 61 4.4 Results of a Crab-like pulsar simulation. The results are for the post-glitch reaction of the theoretical pulsar. The coupling constant for this set of results is 30 days. The superfluid percentages are given by the legend...... 62 4.5 Results of a Crab-like pulsar simulation. The results are for the post-glitch reaction of the theoretical pulsar. The coupling constant for this set of results is 50 days. The superfluid percentages are given by the legend...... 63 4.6 Fit of simulation containing one coupling constant (τ = 8.6 days) to the data for the glitch occurring in the Crab pulsar in 1996. The simulation is given by the line, while the data are given by the dots...... 65 4.7 Fit of simulation containing two coupling constants (τ1 = 10.3 days and τ2 = 0.3 days) to the data for the glitch occurring in the Crab pulsar in 1996. The simulation is given by the line, while the data are given by the dots...... 66 4.8 Table showing the values obtained by my simulation as well as those obtained by Wong et al. for the 1996 Crab glitch ...... 67 4.9 Fit of simulation containing two coupling constants (τ1 = 10.9 days and τ2 = 0.3 days) to the data for the glitch occurring in the Crab pulsar in 1996, with corrected start point. The simulation is given by the line, with data given by dots...... 68 List of Figures 7

4.10 Fit of simulation containing one coupling constant (τ = 9.9 days) to the data for the glitch occurring in the Crab pulsar in 1975. The simulation is given by the line, while the data are given by the dots...... 69 4.11 Table showing the values obtained by my simulation as well as those obtained by Alpar et al. for the 1975 Crab glitch ...... 70 4.12 Diagram showing the ﬁt obtained by Alpar et al. for the 1975 Crab Pulsar Glitch ...... 70 4.13 Table showing the coupling constant values obtained from the simulations, as well as published results ...... 72 4.14 Graph showing the distribution of coupling constants as a function of characteristic age ...... 73 4.15 Graph showing the distribution of coupling constants as a function of characteristic age magniﬁed to centre on the low characteristic age region ...... 74 4.16 Graph showing the distribution of coupling constants as a function of glitch magnitude ...... 75 List of Figures 8

Introduction

Neutron stars, through their observation as pulsars, provide fascinating laboratories for studying physical processes in extreme environments. The occurrence of glitches, their observation and subsequent analysis, offer physicists an extraor- dinary look into this world of incredibly high densities and ultra strong magnetic fields. It is here that some of the most exotic physics yet encountered is found. Yet, after nearly forty years of pulsar glitch study, there is still no conclusive model which can describe the exact mechanism by which these processes occur. In the first chapter of this work, I give a brief overview of neutron stars, including discussions around their discovery and internal structure. In Chapter 2, I introduce the concept of pulsar glitches and present a historical review of the evolution of glitch models. This will include the theoretical framework for some of the most promising mechanisms. In Chapter 3, one of the first models proposed, the two-component model, is discussed in detail. Chapter 4 contains simulations of the two-component model. In the final chapter, I summarise the findings of this thesis, including a description on how well the two-component model holds up in today’s arena of sophisticated glitch mechanisms. 1. OVERVIEW OF NEUTRON STARS

1.1 A Brief Overview of Neutron Star History and Discovery

Neutron stars present astrophysicists with one of the most extreme laboratories in the universe. They form one of three end products of stellar evolution, with the generally less massive white dwarfs and more massive black holes comprising the other two. They are composed predominantly of neutrons, with a mass of approximately 1.4 M and a radius of between ten and twelve kilometers. Their densities lie in the range 5-10 ρo, where ρo is the density of nuclear matter, i.e. 3 × 1014g cm−3. Their surface temperatures range from 1 − 3 × 106 K, and they have a surface gravity of around 1011g, as well as huge magnetic field strengths, in the range 108 − 1012G. In 1934, Walter Baade and Fritz Zwicky proposed the existence of what was, at the time, an entirely new type of star, one they viewed as an end point of stellar evolution. They wrote,“with all reserve, we advance the view that a supernova represents the transition of an ordinary star into a neutron star, consisting mainly of neutrons. Such a star may possess a very small radius and an extremely high density.”1 At the time these remarks were made, it was believed that the observation of these proposed objects would prove impossible, due to the small size, and hence low luminosities, proposed by the model. Two years earlier, in 1932, Lev Landau had suggested that at high densities, neutron matter would be energetically favoured over a mixture of protons and neutrons.2 This was proposed only a few months after Chadwick had discovered the neutron, and Landau suggested that very massive objects with high densities, would consist almost entirely of neutrons in chemical equilibrium.3 In 1939, Oppenheimer and Volkoff4 developed an equation of state for non- interacting, densely packed neutrons, and obtained a maximum neutron star mass of ∼ 0.7 M . This result is noteworthy, since we know that the average mass of a neutron star is 1.4 M . This implies that there must be interactions between the neutrons and that these play a very significant role in the makeup of neutron stars. In 1966, Wheeler published an article in which he stated that, due to the

1 Baade, W., Zwicky, F., 1934, Phys. Rev. 45, 138 2 Landau L.D., 1932, Phys. Z. Sowietunion, 1, 285 3 Xu, R. X., 2003, ACTA Astronomica Sinica, 44, 245 4 Oppenheimer, J.R., Volkoﬀ, G., 1939, Phys. Rev., 55, 374 1. Overview of Neutron Stars 10 tiny radii, and hence low luminosities, of the proposed neutron stars, it would be extremely diﬃcult, if at all possible, to detect them.5 The next year, a discovery was made that changed all of that.

1.2 Pulsar Discovery and Interpretation

In 1967, Jocelyn Bell, together with her supervisor, Antony Hewish, detected periodic signals while performing a galactic survey.6,7 At first, the sources were called LGM’s, an abbreviation for little green men, as they were thought to possibly be a result of extra-terrestrial communication. In fact, Hewish continued timing measurements for a few weeks to see whether there was any detectable Doppler shift due to extra-terrestrial planetary orbital motion. However, these analyses revealed no shift other than that due to the earth’s motion. Also, soon afterwards, other sources were discovered which exhibited similarly short periods and were detected at the same radio wavelength, and the idea that these radio signals were extra-terrestrial communication was abandoned.8 There were several proposals regarding the nature of these pulsing sources, or ‘pulsars’, but these were ruled out one-by-one until only one remained; rotating neutron stars. Simple vibrations were thought to be a possible source, but were ruled out because vibrations always contain some kind of damping, whereas the pulsar period remained remarkably constant, with any deceleration being too small to be accounted for by vibrational damping. The signal was also thought to originate from a binary stellar system, but this was discarded as a source because these systems emit gravitational radiation and speed up, while pulsars were observed to slow down, albeit on very long timescales. Theorists were thus left with rotation as the most likely source. White dwarf rotations were quickly ruled out for two main reasons. To obtain a well-defined signal, as was observed, the source region of the signal would have to be very small, too small to be a white dwarf. The other reason is that, due to the high rotation frequencies, of the order of seconds to milliseconds, a white dwarf would be ripped apart by the centrifugal force generated at the equator. Thus the only possible candidate for explaining the pulsar phenomenon was the rotating neutron star. This result was first proposed by Gold in 1968.9 To date there have been more than 2500 neutron stars detected as radio- regime pulsars, with this number increasing continually.

5 Wheeler, J. A., 1966, Ann. Rev. Astron. and Astrophys. 4, 428 6 Hewish, A., Bell, S.J., Pilkington, J.D.H., Scott, P.F., Collins, R.A., 1968, Nature, 217, . 709 7 Pilkington, J.D.H., Hewish, A., Bell, S.J., Cole, T.W., 1968, Nature, 218, 126 8 Hewish, A., Speech delivered upon receiving the Nobel Prize in 1974 9 Gold,T., 1968, Nature, 218, 731 1. Overview of Neutron Stars 11

1.3 The Structure of Neutron Stars

The aim of this chapter is to provide a brief overview of the current knowledge of neutron star interiors.

Mass and Radius Present day instruments do not allow direct measurement of physical parameters, such as radius and mass, of isolated neutron stars. Limits of 0.2M ≤ M ≤ 3.5M were placed on neutron star masses by Rhodes and Ruﬃni in 1974, using theoretical considerations alone.10 However, in certain binary systems containing pulsars, the measurement of pulsar masses is possible. In 1999, Thorsett and Chakrabarty used measurements from 14 pulsar binary systems to constrain this 11 range to very near 1.4M . These results are shown in Figure 1.1.

Fig. 1.1: Figure showing neutron star mass results for fourteen pulsar binary systems. The dotted lines contain the region which has values agreeing with all measurements, i.e. M = 1.35 ± 0.4M . (Thorsett, Chakrabarty 1999)

Lattimer and Prakash later ﬁtted black-body spectra to neutron stars with

10 Rhodes, C.E., Ruffini, R., 1974, Phys. Rev. Lett. 32, 324 11 Thorsett S.E., Chakrabarty D., 1999, ApJ, 512, 288 1. Overview of Neutron Stars 12 sufficient X-ray flux, and constrained their radii to somewhere in the range 8- 15km.12 Another method for determining the radii and masses of neutron stars is to make use of the relevant equation of state. Different equations of state (EOS) for the matter inside neutron stars give different relationships between their radii and masses. The mass versus radius relationships for different equations of state are shown in Figure 1.2.13

Fig. 1.2: Neutron star gravitational mass in solar units vs. neutron star radius, as given by various equations of state. The Friedman-Pandharipande EOS is given by the dashed line in the middle. (Heiselberg, Pandharipande 2000)

The more accepted models (called “medium stiﬀness” models), such as the Friedman-Pandharipande EOS,14 show that for the “observed” mass range, the corresponding radius range is expected to be 10-12 km.

Atmosphere and Outer Crust Each neutron star is believed to be surrounded by an atmosphere of around 1cm thick, although very little is known about this region.15,16 Below this atmosphere, there exists a solid crust. Nuclear theory shows that under normal circumstances the most stable form of nuclear matter is the 56Fe 12 Lattimer, J.M., Prakash, M., 2000, ApJ, 550, 426 13 Heiselberg, H., Pandharipande, V., 2000, Ann. Rev. Nucl. Part. Phys., 50, 481 14 Friedman, P., Pandharipande, V.R., 1981, Nucl. Phys. A, 361, 502 15 Miller, M.C., 1992, MNRAS, 255,129 16 Rajagopal, M., Romani, R.W., Miller, M.C., 1997, ApJ, 479, 347 1. Overview of Neutron Stars 13 nucleus. Hence the outer layer of the solid crust is made of an 56Fe lattice. The structure changes rapidly with decreasing radius, as the matter density increases. With the increasing pressure and density, the electrons found in this region become both degenerate and relativistic. Their high energies allow for the process of inverse beta decay to occur, resulting in the conversion of protons and electrons to neutrons. As a result, regions in the inner crust contain lattices of heavier, neutron-rich nuclei, such as 78Ni, 76Fe or even 118Kr.17,18,19 Up to a density of 4 × 1011g cm−3, known as the neutron drip density for reasons which will soon become apparent, almost no neutrons exist outside of the lattice nuclei. The region from the star’s surface to neutron drip density is called the outer crust.

Inner Crust and Superfluidity Below the outer crust, at densities greater than the neutron drip density, the inner crust region begins. In this region, due to the high pressure found here, the neutrons begin to “leak” out of the nuclei. These neutrons form a neutron fluid which interpenetrates the crust’s lattice. At nuclear density (ρ0 = 2.7 × 1014g cm−3), the crust lattice dissolves altogether, leaving only fluid, the bulk of which consists of neutrons, mixed with about 5% protons and electrons by number, respectively. This region of fluidity, from nuclear density to central density (≈ 1015g cm−3) is called the core. The properties of the inner core are not known at all. There are many hypotheses regarding the matter present in the inner core, some of which include exotic matter such as free quarks, pions and kaons, as well as a solid core.20 A Diagrammatic representation of the neutron star interior is given in Figure 1.3.21

Rotation Rate The rotation period of pulsars increases slowly. It was pointed out by Gold that the rotational energy of a rotating neutron star would decrease due to magnetic dipole radiation, resulting in the pulsar slowing down.22 This spin down is gradual and largely predictable. However, there exist two types of timing irregularities. The ﬁrst is a generally noisy and fairly continuous, erratic behaviour. This is stochastic in nature and seems to be intrinsic to the nature of the neutron star. Its origin can be traced

17 Pethick, C.J., Ravenhall, D.G., 1991, NYASA, 647, 503 18 Lorenz, C.P., Ravenhall, D.G., Pethick, C.J., 1993, Phys.Rev.Lett., 70, 379 19 Pethick, C.J., Ravenhall, D.G., Lorenz, C.P., 1995, Nucl. Phys. A, 584, 675 20 Heiselberg, H., Pandharipande, V., 2000, Ann. Rev. Nucl. Part. Phys., 50, 481 21 Larson, M.B., 2001, DPhil. Thesis, Superﬂuid Eﬀects on Thermal Evolution and Rotational . Dynamics of Neutron Stars, Montana State University 22 Gold, T., 1968, Nature, 218, 731 1. Overview of Neutron Stars 14

Fig. 1.3: Diagram showing the various regions in a neutron star. to the neutron star’s internal constitution and/or processes occurring in its magnetosphere. The terminology used to describe this noise is not standardised. We shall refer to it as timing noise. The second, also stochastic in nature, is a sudden and more spectacular increase in rotation velocity, and is commonly called a ‘glitch’. This second type of irregularity forms the focus of this thesis.

1.4 Superﬂuidity

The bulk of the neutron star is superfluid. The superfluid occurs in both the outer core and the inner crust, and strongly affects the behaviour of the neutron star. In this section, we review relevant aspects of superfluid theory, as required. Unlike classical fluids, a superfluid is a coherent quantum system. According to the London model, this allows it to be described by a single-particle wave function Ψ, called the condensate wave function. A superfluid is one of the realizations of a Bose Einstein Condensate and has both zero viscosity and zero entropy. The first observed superfluid was helium II.23 Most superfluid theory has been based and tested on experimental work performed on helium II. Experimental evidence24 has shown that only a small percentage of fluid helium II exists in the condensed state near absolute zero. The rest of the fluid exists as thermal excitations, comprising the normal fluid component. The proportional size of this

23 Kapitza, P.L., 1938, Dk.Akad.Nauk SSSR, 18, 21 24 Harling, O.K., 1970, Phys. Rev. Lett., 24, 1046 1. Overview of Neutron Stars 15 normal component increases with temperature. For helium, at a temperature of 2.17K, these excitations cause the entire fluid to be normal, and the superfluid state disappears. So, for helium II experiencing temperatures lying between 0K and 2.17K, there exist two types of fluid simultaneously, one superfluid and the other a normal fluid. These two components interpenetrate one another and move at different velocities. This ‘two-fluid’ model was first proposed by Tisza in 1938.25

1.4.1 Basic Superfluid Theory Apart from the neighbourhood of a solid wall or in a vortex core (more on this later), the superfluid density is a slowly varying function of position. As a result, macroscopic volume elements may be regarded as being of uniform density. This enables some superfluid motions to be investigated by thermodynamic methods. A difference approach was adopted by London, in which he describes the macroscopic behaviour of the superfluid by means of a wave function. In the superfluid neutrons are paired together in a manner analogous to the formation of Cooper pairs in a superconductor. The paired neutrons then form a boson.This results in the fluid obtaining bosonic properties, enabling the formation of a Bose- Einstein Condensate. London postulated that the condensate can be described by a wave function of the following form:26

iS(~r,t) Ψ(~r, t) = Ψ0(~r, t)e , (1.1) which he assumed to be a solution of the Schrodinger equation. To determine the superfluid velocity, the usual momentum operator is applied to this wave function, giving: pôpΨ = −i~∇Ψ = ~pΨ, (1.2) which gives ~p = mnp~vs, (1.3) where mnp is the mass of the neutron pair. Combining these equations gives the velocity of the superfluid to be

~vs = (~/mnp)∇S, (1.4) where S is the phase, and is a function of position. Also, the population density is deﬁned as 2 ρs Ψ0(~r, t) = . (1.5) mnp The following continuity equation for the conservation of mass exists for a superﬂuid, ∂ρ s = −∇ · ~j , (1.6) ∂t s 25 Tisza, L., 1938, Nature, 141, 913 26 Landau, L.D., Lifshitz, E.M., 1987, Fluid Mechanics, 2nd Edition, (Pergamon Press) 1. Overview of Neutron Stars 16

where ~js and ρs are the mass current density and the density of the superfluid respectively. This equation is called the continuity equation for superfluids, and describes the conservation of mass in such a fluid. Troup27 showed that for large particle number N, there exists the following uncertainty relationship: δNδS ≈ 1. (1.7) As a result, N and ~S can be treated as conjugate variables, in the same way as the position and momentum of a particle are. Using Hamiltonian mechanics, the equations of motion for these quantities are ∂S ∂H = − (1.8) ~ ∂t ∂N and ∂N ∂H = . (1.9) ~ ∂t ∂S The Hamiltonian H can be given by the total energy U, consisting of the sum of the total kinetic and rest energy of the fluid respectively:

U = Us,k + U0. (1.10) Using mean values, the equation of motion for S becomes ∂S ∂U 1 = − = −(µ + m v2), (1.11) ~ ∂t ∂N 2 np s since the chemical potential µ is defined as ∂U µ = 0 (1.12) ∂N entropy, volume Taking the gradient of equation (1.11), together with (1.4), the superfluid equation of motion is found to be ∂∇S ∂~v 1 = m s = −∇(µ + m v2). (1.13) ~ ∂t np ∂t 2 np s Next, the convective derivative is used. The convective derivative is defined as D~a ∂~a = + (~v · ∇)~a, (1.14) Dt ∂t for an arbitrary vector ~a in a constant volume element traveling in a velocity field ~v. The Euler equation for an ideal fluid (defined as a fluid with zero viscosity, and hence applicable to a superfluid) is

D~vs ∂~vs ∇µ = + (~vs · ∇)~vs = − , (1.15) Dt ∂t mnp

27 Troup, G.J., 1967, Optical Coherence Theory - Recent Developments, (London: Methuen) 1. Overview of Neutron Stars 17 which, upon inspection is seen to be a form of NewtonII. Using the properties of the ∇ operator gives

D~vs ∂~vs 1 2 ∇µ = + ∇( vs ) − ~vs × (∇ × ~vs) = − . (1.16) Dt ∂t 2 mnp

By comparing the second equation in Equation (1.13) with Equation (1.16), the Landau criterion for superﬂuidity is reached:

∇ × ~vs = 0, if vs 6= 0. (1.17)

1.4.2 Basic Vortex Theory Landau’s criterion has interesting implications for the rotation of the superfluid. The circulation of the superfluid is defined to be I ~ κ = ~vs · dl (1.18) L for an integration contour L which is entirely in the superfluid. Applying Stokes’ theorem to the circulation equation, and considering the Landau criterion, gives the following interesting result: I Z ~ ~ κ = ~vs · dl = (∇ × ~vs) · dA = 0. (1.19) L A This implies that there can be no circulation in a pure superfluid, due to the con- straints of the Landau criterion. Hence, there is no rotation. This was famously demonstrated by the Andronikashvili experiment28, in which an oscillating pile of disks entrained the normal fluid component, while leaving the superfluid component at rest. However, in 1950 Osborne rotated a cylindrical bucket containing HeII, and the results indicated that both the normal and superfluid components were moving with the same angular velocity.29

1.4.3 Quantised Vortices The Landau criterion seems to indicate that there can be no macroscopic rotation of a superfluid, and yet, experimentally, there is evidence of such rotation. However, a closer look at the theory shows that it is possible for the circulation of the superfluid, and hence its rotation, to be non-zero. This can be accomplished by allowing the region inside the integration contour to be multiply connected. There are two ways in which this can be accomplished. The first is to ensure that this region inside the integration contour contains ‘holes’ in the superfluid,

28 Andronikashvili, E. L., 1946, Zh. eksp. theor. Fiz., 16, 780 29 Osborne, D.V., 1950, Proc. Phys. Soc. (London), A63, 909 1. Overview of Neutron Stars 18

for which ∇ × ~vs 6= 0, i.e. ‘holes’ of normal fluid. The second is to ensure that there are ‘holes’ inside the integration contour which are, in fact, empty of fluid. This gives rise to the concept of a vortex core; a hole in the superfluid, having cylindrical geometry, either empty of fluid or containing normal fluid, surrounded by superfluid matter experiencing irrotational flow, once again, with cylindrical symmetry. Combining (1.4) and (1.18) yields the following expression for the circulation in terms of phase I ~ ~ κ = ∇S · dl = (∆S)L. (1.20) mnp L mnp The wavefunction given in (1.1) is single-valued. Thus, a complete trip around the contour should yield an unchanged value. So, due to the nature of the wavefunction, the only possible values for a change in S are multiples of 2π and zero. Obviously, a value of zero corresponds to the Landau criterion. The multiples of 2π give non-zero circulation, and hence apply to the case of vortices. in this case, the circulation is given by h κ = n (1.21) mnp where n is an integer. Thus, the circulation is quantised. It is from the above conceptual framework that the definition of a vortex core is obtained; a vortex core is the non-superfluid region found inside a vortex, i.e. the region in which ∇ × ~vs 6= 0. Thus, although superfluid rotation cannot occur in a simply-connected region, it can occur macroscopically in the presence of the many multiply-connected regions formed by quantised vortex lines. Considering a streamline located at radius r from the centre of an isolated vortex line, the following equation applies I ~ κ = ~vs · dl = 2πrvs(r). (1.22) L Hence, the velocity of the fluid at a given distance from the vortex core is κ n~ vs = = , (1.23) 2πr mnpr and the angular momentum at that point is

L = mnpvsr = n~. (1.24) Thus, in addition to the circulation, both the velocity and angular momentum of the ﬂuid around a vortex line are quantised. The ﬁrst experimental evidence of quantised circulation was recorded by Vi- nen30 while the existence of quantised vortices in helium II was shown by Hall and Vinen.31 30 Vinen, W.F., 1958, Nature, 181, 1524 31 Hall, H.E., Vinen, W.F., 1956, Proc. Roy. Soc. A, 238, 204, 215 1. Overview of Neutron Stars 19

This theory was ﬁrst related to neutron stars by Ginzburg and Kirzhnits in 196432 when they stated that, should rapidly rotating neutron stars exist, then quantised vortices similar to those observed in helium II should exist in the neutron superﬂuid of these compact bodies. These predictions were made before the discovery of pulsars and their interpretation as rapidly rotating neutron stars.

1.5 Vortex Pinning and Vortex Creep in Neutron Stars

The concepts of vortex pinning and vortex creep are essential in many glitch mechanisms (these will be discussed later). One of the ﬁrst mechanisms proposed which involved vortex dynamics was that proposed by Anderson and Itoh.33 Their mechanism was supported by the work of Negele and Vautherin,34 who predicted that conditions in the inner crust of a neutron star would be conducive to vortex pinning. Many other mechanisms depend heavily on vortex pinning and vortex creep processes. Vortex creep is the process whereby pinned vortices unpin from one pinning site, migrate radially outwards, and repin to another suitable site. The following sections give a description of the relevant theory, the bulk of which comes from the work of Alpar et. al.35, Link and Epstein36 and Link, Epstein and Baym37.

1.5.1 Factors influencing vortex creep rate As discussed previously, the superfluid flow velocity is irrotational, i.e. ∇ × ~vs = 0. Hence, in bulk superfluid, rotation cannot take place except in the presence of quantised vortex lines. These lines contain cores in which superfluidity is destroyed, either by the presence of a normal fluid, or by the absence of fluid, allowing rotation. The macroscopic superfluid rotation rate is determined by the spatial arrangement of these vortex lines. As a result, fixing the positions of these lines results in the angular velocity of the superfluid being kept constant. The method by which it is envisaged that this fixing takes place is by the pinning of these lines to the crust. This pinning can occur in two possible guises. At stellar densities lower than 1013g cm−3, the vortex lines follow paths of lowest energy which thread between the nuclei. At higher densities, the paths of lowest energy pass through the nuclei. The former case is referred to as interstitial pinning while the latter case is called nuclear pinning.38

32 Ginzburg, V.L., Kirzhnits, D.A., 1964, Zh. Eksp. Teor. Fiz., 47, 2006 33 Anderson, P.W., Itoh, N., 1975, Nature, 256, 25 34 Negele, J.W., Vautherin, D., 1973, Nucl. Phys. A, 207, 298 35 Alpar, A., Anderson, P.W., Pines, D., Shaham, J., 1984, ApJ, 276, 325 36 Link, B.K., Epstein, R.I., 1991, ApJ, 373, 592 37 Link, B.K., Epstein, R.I., Baym, G., 1993, ApJ, 403, 285 38 Donati, P., Pizzochero, P.M., 2004, Nucl. Phys. A, 742, 363 1. Overview of Neutron Stars 20

As a result of the crust’s spin-down, there develops a rotation lag (~vδ ≡ ~vc−~vs) between the crust, and hence the vortices pinned to it (crust velocity = ~vc), and the bulk superfluid (superfluid velocity = ~vs). As the superfluid streams past a vortex line, a Magnus force is created, which pulls the vortex line radially outwards, acting against the pinning force. This force is a result of Bernoulli’s law. On the side of the vortex line where the streaming velocity and the vortex circulation add constructively, the fluid pressure is lower than on the opposite side, where the two velocities interact destructively. The resulting pressure gradient produces the Magnus force (See Figure 1.4).

Fig. 1.4: Diagram showing the dynamics which lead to the creation of the Magnus force FM .

The Magnus force per unit length is given by ~ fMag = ρs~κ × ~vδ, (1.25) where ρs is the superﬂuid density and ~κ is a vector having a magnitude equal to the circulation of the vortex line κ and a direction aligned with the vortex line.39 In the remainder of the section, the analysis is performed in the instantaneous rest frame of the crust, in which the pinning sites and the pinned vortex line are stationary. At present, the details of the pinning force are not well understood. Hence, the pinning force is described by a maximum value Fmax, as well as the vortex- nucleus separation, r0, at which this occurs. Consider a simple parabolic model:

39 Link, B.K., Epstein, R.I., 1991, ApJ, 373, 592 1. Overview of Neutron Stars 21

" # r 2 (−Fmax 1 − − 1 0 ≤ r ≤ 2r0 r0 Fp(r) = (1.26) 0 2ro < r. This results in a pinning potential of

2 3 r ( 3 r 1 r Z − 0 ≤ r ≤ 2r0 0 0 4 r0 4 r0 Ep(r) = − Fp(r )dr = U0 × (1.27) 0 1 2ro < r.

We now have representations of the Magnus and pinning forces. There is one more force that needs to be taken into account, namely, the tension force T of the vortex line. For an isolated vortex with sinusoidal perturbation of wavenumber k, the eﬀective tension is given by Fetter40 and Link & Epstein41 as

ρ κ2 T = s (a − ln(kξ)), (1.28) 4π where a ≈ 0.116 for a vortex line with hollow core, ξ is the vortex core radius, and the logarithmic term is treated as constant. Combining these forces results in a total energy for a vortex of length L, of Z T 0 2 E = |~y (z)| + ρp(z)Ep(y) − ρs(~κ × ~vδ) · ~y(z) dz, (1.29) L 2 where ~y(z) is the line’s displacement from its pinned, equilibrium position, ~y0 ≡ d~y/dz 1, z is the distance along the line and ρp is the density of pinning sites along the line.42 The vortex-vortex interactions responsible for the vortex lattice rigidity are neglected, since the vortex-vortex spacing is much greater then the distance between pinning sites.43 Using the variation equation δE = 0, the static conﬁguration condition can be given by

00 ~ T~y + ρp(z)Fp + ρs~κ × ~vδ = 0. (1.30)

1.5.2 Determining the Activation Energy The creeping, or movement, of vortices involves three stages. These are: the unpinning of the vortex from one site, the outward radial motion of the vortex line, and ﬁnally, the repinning of the line to a new pinning site. This creeping can occur when the vortex is either thermally excited over an energy barrier, or

40 Fetter, A.L., 1967, Phys. Rev., 162, 143 41 Link, B., Epstein, R., 1991, ApJ, 373, 592 42 Link, B., Epstein, R., 1991, ApJ, 373, 592 43 Baym, G., Chandler, E., 1983, J. Low Temp. Physics, 50, 57 1. Overview of Neutron Stars 22 quantum tunnels through the barrier. We now proceed to determine the activation energy required for unpinning. In the absence of a Magnus force, the vortex line is assumed to be straight. The presence of a Magnus force results in the bowing of the line (Figure 1.5).44 In addition, the spacing between pinning sites is much greater than the vortex core radius. There are two types of vortex line breakaway, and these are highly temperature dependent. These are single-site and continuous breakaway respectively, which will be detailed separately.

Fig. 1.5: An exaggerated diagram showing a pinned vortex bending under the inﬂuence of the Magnus force. The solid circles show the pinning sites.

Single-site breakaway Consider a line segment which threads equally spaced pinning sites, located at z = 0, ±l, and ±2l. In this case, consider the superfluid velocity to be in the x- direction and the Magnus force in the negative y-direction. Define a dimensionless force ~ fMagl vδ Fss ≡ = , (1.31) Fmax vB where the subscript ss represents the single-site breakaway regime, and vB = Fmax/ρsκl is the breakaway velocity lag, at which fMag = Fmax, obtained from (1.25). Note that for Fss > 1, the vortex cannot stay pinned. For 0 < Fss < 1, there exist equilibrium configurations such that y(nl) = r1 (n = 0, ±1, ±2, ...) is the equilibrium spacing between the nucleus and the vortex line, and hence a constant. See Figure (1.6) for clarification.

44 Link. B.K., Epstein. R.I., 1991, ApJ, 373, 592 1. Overview of Neutron Stars 23

Fig. 1.6: Conﬁguration used for the calculation of the energy for a single-site breakaway (Link, Epstein, 1991)

For the calculation of the activation energy, consider the energy change involved in moving a pinning point located at z = l from y = r1 to y = r, while keeping the points at z = 0 and z = 2l ﬁxed at r1. The energy of this conﬁgura- tion is given by45

" 2 # 3 r r1 r E = U0 τss − − Fss + EP (r), (1.32) 4 r0 r0 r0 where the stiffness τss ≡ T r0/Fmaxl measures the relative importance of tension versus the pinning force. Figure 1.7 shows that this energy could have either a minimum at r1 alone, or a maximum at r2 straddled by two minima at r1 and r3, for three different configurations, namely 1,2 and 3 in the figure. The activation energy for vortex line breakaway is determined by the height of the energy barrier between r1 and r2, hence

" 3# 3/2 3 τss τss Ass = E(r1) − E(r2) = U0∆ss 1 + 1/2 + 1/2 , (1.33) 4 ∆ss ∆ss where ∆ss = 1 − Fss, as shown by Link, Epstein (1991). An interesting note can be made here. As the Magnus force approaches the maximum pinning force, Fss → 1 and ∆ss → 0. However, Ass 6→ 0. This is a result of the tension barrier resulting from the fact that adjacent pinning sites are ﬁxed. This energy can be minimised, however, if we introduce either longer loops, or multiple breakaways. In the single-site breakaway regime, only vortices with low tension are considered. As a result, (1.33) is only valid for the single-site breakaway regime if 2 ∆ss > τ . (1.34)

45 Link. B.K., Epstein. R.I., 1991, ApJ, 373, 592 1. Overview of Neutron Stars 24

Fig. 1.7: Energy of a vortex line as a function of separation for a vortex line in the single-site breakaway regime (Link, Epstein 1991).

Using this as the limit for this regime, the activation energy is given by

3/2 Ass ' U0∆ss . (1.35)

Figure 1.8 shows how, once unpinned from one energy minimum (by over- coming a small energy maximum), a vortex will migrate radially outwards and repin at the next minimum, for vδ/vB = 1. This becomes less likely for values of vδ/vB < 1 as the activation energy increases. It also shows that the vortex will not migrate inwards, towards lower r-values, since the energies increase in that direction. For densities greater than 1013g cm−3, nuclear pinning dominates. Here the pinning force is much greater than the vortex tension, resulting in single-site breakaway occurrences. 1. Overview of Neutron Stars 25

Fig. 1.8: Energy of a vortex line as a function of separation for a vortex line in the single-site breakaway regime (Link, Epstein 1991). 1. Overview of Neutron Stars 26

Continuous Breakaway Next, the case of simultaneous, multiple-site breakaway is considered. Here, the distance between pinning sites is much smaller than the scale of the perturbations, and as a result, discrete pinning is insigniﬁcant. As a result, in this regime, pinning is treated as continuous, with a pinning density of ρp(z) = 1/l. Solving (1.30) yields the equilibrium condition

F~ (y) p + f~ = 0. (1.36) l Mag

The definitions of Fc and ∆c (to be used later) are similar to those found in the single-site regime. As with the single-site case, there can be either one or three equilibrium configurations for y(z). Here, there are two minimum energy configurations. One is the stable, straight pinned configuration. The other occurs when the vortex straddles the region of maximum pinning force, located at y = r0. This configuration has an energy minimum for vortex displacement at large y(z), and contains j∗ broken pinning bonds. For continuous breakaway, the activation energy is defined as the energy difference between the first and second configurations described above. The activation energy is given by46 √ 18 2τ A (j ) = U c ∆5/4 ' 5, 09 U τ 1/2∆5/4, (1.37) c ∗ 0 5 c 0 c c where τc = τ(l), assuming that ∆c < 1/4, (i.e. ω/ωc > 3/4). The limit in which the continuous breakaway approximation is applicable is estimated by insisting that j∗ ≥ 3, giving the limit as 2 ∆c ≤ τc . (1.38) For densities less than 1013g cm−3, interstitial pinning dominates the region. The pinning here is much weaker than the vortex tension, and as a result, continuous breakaway dominates.

46 Link. B.K., Epstein. R.I., 1991, ApJ, 373, 592 2. GLITCHES AND GLITCH MECHANISMS

2.1 Glitches

For the most part, the spin-down of pulsars is exceedingly gradual and predictable. However, there are two types of timing irregularities exhibited in pulsars. One is called timing noise and manifests itself through a quasi-random walk in one or more of the rotational parameters on timescales of months to years. However, the second type of timing irregularity forms the major theme of this work, and is called a pulsar glitch. Glitches are sudden, positive jumps in spin rate. These increases occur superposed on the gradual spin-down due to electromagnetic torque. They usually occur as fractional increases in angular velocity of the order of ∆Ω/Ω = 10−9 to 10−6. The timescales of these glitches (i.e. the time taken for the fractional increase in angular velocity to occur) range from a few minutes to a few hours. The ﬁrst observed glitch was seen in the Vela pulsar (PSR 0833 -45) in 19691 and is shown in Figure 2.1. After the occurrence of a glitch, the spin down rate once again reaches an equilibrium value in a process called glitch recovery that can take months or even years. The post-glitch spin- down value could be the same as the pre-glitch value or, as is the case with some glitches occurring in the Crab pulsar, it could be some higher value. However, more often than not, a value near the original pre-glitch spin-down rate value is obtained. The amount of rotational energy gained by the pulsar during the spin up is huge; as much as 1036J. Pulsar glitches are the result of processes occurring in the interior of a neutron star, rather than in its magnetosphere. This is shown by the fact that the structure of the pulsar signal remains unchanged during the glitch event, with only the time series being altered. A satisfactory description of the mechanism of these glitches has remained elusive, in spite of the proposal of numerous mechanisms. The section that follows gives a brief summary of the mechanisms that have been proposed, followed by an in-depth theoretical look at these mechanisms in the subsequent sections.

2.2 Brief Summary of Glitch Mechanisms

As an increasing number of glitches were discovered, it was noted that the structure of the pulse is not aﬀected by a glitch, hence the processes responsible for

1 Radhakrishnan, V. and Manchester, R., 1969, Nature 222, 228 2. Glitches and Glitch Mechanisms 28

Fig. 2.1: Figure published by Radhakrishnan and Manchester (1969) showing the first observed glitch, which occurred in the Vela pulsar glitching must occur internally within the star. The first mechanism was proposed by Ruderman in 19692 and is called the spheroidality mechanism, or crustquake model. When the neutron star is born, it has a significant ellipticity due to its high rotation rate. The basic assumption of this mechanism is that the forces in the crust are insufficient to resist the changes in fluid equilibrium state. As the star slows down, the fluid equilibrium state moves from the form of an oblate spheroid towards a sphere. However, the rigidity of the crust resists this change, resulting in the build-up of mechanical stresses in the crust. Eventually some critical point is reached, upon which the crust cracks and rearranges itself to accommodate the fluid equilibrium configuration, and the entire process is repeated. During this rearrangement, which decreases the moment of inertia of the star, the conservation of angular momentum causes an increase in the crust rotation frequency. Since the magnetic field of the neutron star, which is responsible for the observed photon beam, is anchored to the crust, the spin-up of the crust will result in an observed glitch. A vertical surface motion of more than 1 cm would result in a glitch of magnitude ∆Ω/Ω ≥ 10−6. However, as was pointed out by Baym and Pines in 1971,3 ∆E ≥ 1036J of mechanical energy needs to be transferred per glitch for this mechanism. They calculated that this amount of energy would take far too long to accumulate in the crust, to account for the frequency of glitches that is observed, i.e. up to 5 glitches per decade.

2 Ruderman, M., 1969, Nature, 223, 579 3 Baym G., Pines D., 1971, Ann. Phys., 66, 816 2. Glitches and Glitch Mechanisms 29

This model has thus been abandoned. Soon after the spheroidality mechanism was discarded, it was recognised by Packard4 and later by Anderson and Itoh5 that the superfluid layers in the star’s interior could be responsible for glitches. It had been theoretically determined, from the neutron density and interior temperature of the star, that the neutron star contained a superfluid neutron interior. This was confirmed observationally by the fact that the relaxation timescales after glitch events are too long to be described by either a solid or by a normal fluid. The superfluid was viewed as an angular momentum reservoir, since the spin-down rate of the superfluid is slower than that of the crust. As a result, a differential rotation develops between these two layers. At some point, there is a transfer of angular momentum from the superfluid to the crust, spinning it up and causing a glitch. Although this theory is generally accepted, it still has some unanswered questions. Where in the star is the angular momentum reservoir found, and what percentage of the star does it constitute? What is responsible for the angular momentum transfer, and where in the star does this coupling occur? These questions are later examined more closely. At present, there are several mechanisms which have been considered in an attempt to answer these questions. Some have been tested, with varying amounts of success. As shown in Chapter 1, superfluid theory predicts that since the superfluid is rotating, it will be threaded with vortices. These vortices are believed to be pinned, or attached, either to, or in between the crust nuclei. It is these vortices and the theory related to them that forms the basis of almost all of these proposed mechanisms. Each mechanism falls into one of two categories; crust-driven mechanisms, and core-driven mechanisms. The crust-driven mechanisms include the thermal glitch model, crust-fracture model and the superfluid two-stream instability model. The core-driven mechanisms include the flux-tube pinning model, the centrifugal buoyancy model and the flux annihilation model. These mechanisms are briefly discussed here, followed by a detailed discussion in the next section. In 1976, Ruderman proposed an alternative mechanism to his mechanism of 1969. This latter mechanism has been called the crust fracture model.6 This time, he considered that the pinning of vortices in the inner crust is strong. He proposed that these vortices would exert large stresses on the crust, as they attempt to migrate outwards, and would eventually stress the crust to the point of fracture. This would result in the mass migration of vortices outwards, once again transferring angular momentum to the crust, causing a glitch. According to the catastrophic unpinning model developed by Cheng et. al. in 1988, an unknown mechanism causes hydrodynamic instability, which produces

4 Packard, R., E., 1972, Phys. Rev. Lett., 28, 16 5 Anderson, P.,W., Itoh, N., 1975, Nature, 256, 25 6 Ruderman. M., 1976, ApJ, 203, 213 2. Glitches and Glitch Mechanisms 30 catastrophic unpinning of vortices in the crust, resulting in a glitch.7,8,9 The thermal glitch model was first presented by Link and Epstein in 1996.10 They proposed that, similar to Ruderman’s initial model, as the crust slows down, mechanical stresses build up. Again, these stresses build up until a crustquake occurs, causing the release of heat into the star. Since the vortex creep rate, the rate at which vortex lines move radially outwards, is highly temperature dependent, the sudden deposition of heat causes a sudden increase in creep rate. This causes the transfer of angular momentum to the crust, causing a glitch. The flux tube model was proposed in 1998 by Ruderman et.al.11 In this model, the interaction between superfluid vortices and magnetic flux tubes results in an expanding core vortex array. This expanding array forces the magnetic flux into the crust, stressing it. This eventually causes the crust to crack, allowing the rapid expansion of the array outwards, spinning down some of the superfluid, and spinning up the crust and causing a glitch. In 1999, the flux annihilation model was proposed by Sedrakian and Cordes.12 In this model, proton flux tubes are annihilated at the crust-core boundary resulting in the outward migration of vortices, again causing the rapid expansion of the core vortex array. This causes a spin-up of the crust, culminating in a glitch. In 2000, Carter et.al. proposed the centrifugal buoyancy mechanism.13 In this model, they suggest that centrifugal buoyancy forces result in pressure gradients in the superfluid, which become sufficiently strong to crack the crust, resulting in a reconfiguration of the crust, and once again, a glitch. In 2002, Andersson, Comer and Prix14 developed a theory that is analogous to the Kelvin-Helmholtz instability15 used in plasma physics and discussed in relation to astrophysical topics such as merging galaxies16 and pulsar magneto- spheres.17 Their model differed from the previous incarnations of the Kelvin- Helmholtz instability in that it involved two interpenetrating fluids, as opposed to fluids interacting across an interface.

7 Cheng H.F., Pines D., Alpar M.A., Shaham J., 1988, ApJ, 330, 835 8 Alpar M.A., Pines D., in Van Riper K.A., Epstein R.I., Ho C., eds, 1993, Isolated Pulsars, Cambridge Univ. Press, Cambridge, p.18 9 Mochizuki Y., Izuyama T., 1995, ApJ, 440, 263 10 Link, B., K., Epstein, R., I., 1996, ApJ, 457, 844 11 Ruderman M., Zhu T., Chen K., 1998, ApJ, 492, 267 12 Sedrakian A., Cordes J.M., 1999, MNRAS, 307, 365 13 Carter B., Langlois D., Sedrakian D.M., 2000, A&A, 361, 795 14 Andersson, N., Comer, G.L., Prix, R., 2002, MNRAS, 354, 101 15 Drazin, P.G., Reid, W.H., 1981, Hydrodynamic Instability (Cambridge: Cambridge Uni- versity Press) 16 Lovelace, R.V.E., Jore, K.P., Haynes, M.P., 1997, ApJ, 475, 83 17 Cheng, A.F., Ruderman, M.A., 1977 ApJ, 212, 800 2. Glitches and Glitch Mechanisms 31

2.3 Crust-driven Glitch Mechanisms

2.3.1 Crust Fracture Model This theoretical model was first formulated by Ruderman in 197618. The basic principles had, however, been mentioned previously by Anderson and Itoh in 19755, when they remarked that the pinning of neutron superfluid vortices to the crust lattice may cause that lattice to break before the vortices could unpin from it. The task was then to see whether or not a relevant theory could be proposed which agreed with observational data. Data from the Vela pulsar were tested against the initial 1976 theory, to determine whether or not the glitches occurring in this pulsar were a result of vortex pinning-induced crust cracking. However, no satisfactory support for the occurrence of this process in the Vela pulsar could be obtained. In 199119, Ruderman developed this theory further, and noted that for millisecond pulsars and LMXBs, the large-scale crust breaking stress would be larger than the unpinning force at individual pinning sites. There are two major forces which act on the crystal lattice of the crust as it spins down or spins up. These are: the force due to pinned neutron superfluid vortex lines on the nuclei to which they are pinned,20,21,22 and the force on the base of the crust due to magnetic flux tubes pinned to it. Obviously, the faster the neutron star rotates, and the lower the magnetic field of the star, the more dominant the former effect will be, and vice versa. In the crust cracking model, it is shown that for rapidly rotating neutron stars with low magnetic field, the crustal stresses are sufficient to cause plastic flow in the crust, by a continuous breaking process. The theory behind this result is developed in what follows and is based largely on the 1976 and 1991 works published by Ruderman. The model used here consists of a neutron rich lattice with heavy nuclei (Z ∼ 40), interpenetrated by superfluid. In this model most of the superfluid is contained in a spherical shell of thickness l ∼ 105 cm, with the lattice extending a 4 13 14 −3 further 10 cm. The total baryon density is between ρB = 10 and 10 g cm , and the total stellar radius is ∼ 106 cm. The shear modulus of the lower crust is given as µ ≤ 10 23 J cm−2. (2.1) ˆ The maximum strain angle (used to find the maximum shear stress µθm) was estimated by Smolukowski and Welch23 to lie between

ˆ −5 −3 θm ∼ 10 and 10 . (2.2)

18 Ruderman. M., 1976, ApJ, 203, 213 19 Ruderman, M., 1991, ApJ, 366, 261 20 Alpar M.A., Pines D., in Van Riper K.A., Epstein R.I., Ho C., eds, 1993, Isolated Pulsars (Cambridge: Cambridge University Press) 21 Ruderman, M., 1969, Nature, 223, 579 22 Alpar. M., Anderson. P., Pines. D., Shaham. J., 1984a, ApJ, 276, 325 23 Smoluokowski, R., Welch, D., 1970, Phys. Rev. Lett., 24, 1191 2. Glitches and Glitch Mechanisms 32

For the current model, the maximum strain angle physically attained before the lattice breaks is 10−3. Next, the two types of stress experienced by the crust are examined separately. 1) At the base of the crust, the magnetic flux tubes produced by the superfluid 15 proton sea in the core terminate. Each tube contains a magnetic field Bc ∼ 10 G. As the star spins down or spins up, the arrangement of vortex lines changes. As this occurs, the magnetic flux tubes respond by migrating accordingly.24. The shear stress acting on the base of the crust due to the motion of the flux tubes attached to it could reach BB B S(B) ∼ c ∼ 1019 J cm−2, (2.3) 8π 3 × 1012 G where, under the assumption that the core protons form a Type I superconductor, Bc is the critical magnetic field at which superconductivity is quenched. The critical stress at which crust breaking occurs is l S ∼ µθ ≤ 1019 J cm−2. (2.4) max R max

12 Thus, for neutron stars with B > 3 × 10 G, S(B) > Smax, and crust breaking occurs due to spin down (or up). 2) The other shearing mechanism, which dominates in weakly magnetised stars (those where S(B) < Smax), occurs as a result of the presence of superfluid vortex lines. In these stars, the magnetic field is frozen into the crust, irrespective of the underlying core magnetic field and how it develops. The rotational motion of the neutron superfluid is governed by the spatial arrangement of the array of vortex lines. In accordance with Link and Epstein25, for the densities considered 13 14 −3 here (ρB ∼ 10 − 10 g cm ), the vortex lines will be pinned to the nuclei in the crust. When the crust (and hence the pinned vortices) spins at the same rate ~ ~ (Ω) as the bulk superfluid (Ωn), there is no shear stress exerted on the crust. However, the presence of some velocity lag between the crust and the superfluid (ω ≡ Ωn − Ω) causes a force to be exerted on the lattice. This force experienced by the crust lattice is given here by ~ ~ Fv = 2~ω × (Ωn × ~r)ρnf, (2.5) where ρn is the superfluid neutron density. The dimensionless factor f is effec- tively a weighting factor, ensuring that almost the entire contribution to this force is from the pinned region. As an illustration, consider a cylinder containing a superfluid of constant density. Let the vortex lines of this superfluid be pinned in a region of length a, but not in the region above it of length a0 or in the region

24 Srinivasan,G., Battacharya, D., Muslimov, A., Tsygan, A., 1990, Current Sci., 59, 31 25 Link. B.K., Epstein. R.I., 1991, ApJ, 373, 592 2. Glitches and Glitch Mechanisms 33 below it of length a00. Thus, the weighting factor is given by f ∼ 1 + a0/a + a00/a in the pinning layer, and f = 0 above and below it. If vortex lines don’t extend through the core, which it is assumed they don’t, then f ∼ 1 26.

Fig. 2.2: Diagram used to illustrate crust-cracking parameters (Ruderman, 1991)

Consider Figure 2.2, showing a spherical crust shell of thickness l R. In this case, only the tangential component of the force density given by (2.5) is considered. The tangential displacement of the crust as a result of crust-cracking, ~s shown in the ﬁgure is given by

ρ ωΩ R3f s(θ) = n n sin(2θ). (2.6) 24µ This gives a crustal strain of

1 ds ρ ωΩ R2f η = = n n cos(2θ) (2.7) R dθ 12µ The result is the presence of a tearing strain at the poles, and a buckling strain at the equator. The next step is to determine the critical value for ω before crust breaking occurs. To do this, consider the dimensionless η which corresponds to a shear angle θ, with a maximum value before breaking of θmax. With this in mind, and

26 Ruderman, M., Nature, 223, 579 (1969) 2. Glitches and Glitch Mechanisms 34

rearranging (2.7), the critical lag (ωB) is found to be 2 −1 12θmaxµ|1 − 2 sin θ| ωB = 2 (2.8) ρnΩnfR 4 −1 13 −3 −2 µ3 0 θmax 10 s 3×10 g cm −1 ∼ 2 × 10 2 −3 s . (2.9) fR6 10 ω ρn Next, the regime in which the crust breaking will pre-empt unpinning can be determined. Consider the vortex core radius, to be27 2E ζ = f , (2.10) πkf ∆ where ∆ is the energy gap, which is constant in the case of a homogeneous neutron superfluid, and Ef and kf are the fermi energy and wave number respectively. The pinning energy, given by the difference between the energy of the nucleus well outside of the vortex and that of the vortex overlapping the nucleus, is28 2 3 2 3 3 ∆ kf ∆ kf ζ 4π 3 Ep ∼ 2 − 2 rN , (2.11) 8 Ef 3π Ef 3π N RN 3 where ( )N denotes evaluation for superfluid neutrons inside the nucleus. The value for the pinning force obtained by Alpar et. al29 is 2 3 2 Ep ∆ kf RN Fp = ∼ . (2.12) ζ 6πE(f) The magnitude of the force exerted on the pinning site due to superfluid flow past the vortex is F ≡ ωR sin θfπρnb~/mn (2.13) Using the above information, a critical angular velocity lag of 2 ~RN ωc = 3 (2.14) πR sin θf mn bz can be obtained, where bz is the internuclear distance and mn is the neutron mass. If the shear modulus is rewritten as (Ze)2 µ = 0.3 4 , (2.15) bz then from (2.8), (2.14) and (2.15)the following ratio is found 2 ωB 10πθmax(Ze) sin θ = 2 2 (2.16) ωc mnΩrbz~Rn 1 − 2 sin θ 4 −1 −13 2 −12 −2 θmax 10 s Z 2 7×10 cm 5×10 cm < 10 −3 , (2.17) 10 Ω 50 RN bz

27 Alpar. M., Anderson. P., Pines. D., Shaham. J., 1984a, ApJ, 276, 325 28 Baym G., Pines D., 1971, Ann. Phys., 66, 816 29 Alpar. M., Anderson. P., Pines. D., Shaham. J., 1984a, ApJ, 276, 325 2. Glitches and Glitch Mechanisms 35

where RN is the nuclear radius and Z is the atomic number. This equation gives a critical period of −3 2 10 PB ∼ 10 ms (2.18) θmax below which, crust breaking resulting from neutron superﬂuid vortex pinning should occur before vortex unpinning does. This value would seem to explain the lack of substantial literature support for crust breaking in the Vela pulsar, since this pulsar has a spin period of around 89 ms. It also gives compelling evidence for the presence of crust breaking in millisecond pulsars. In this model, the crust is divided into platelets, each of which has a constant number density and arrangement of pinned vortices, irrespective of crust spin-down. If for a moment, a cylindrical star that is undergoing spin-down is considered, then the shear strain occurs only at the disks (or the “poles”), is directed outwards, and is given by30

ωΩ ρ R2f r2 η = n n 1 − 3 , (2.19) 16µ R2 where r is the radial distance from the spin axis. Note that the ωB for this case is roughly 2/3 to 4/3 that of the spherical case. Using the fact that the shear 2 strain is proportional to the size of the platelet squared (Rp), the platelet size is found to be Rp ωB ∼ , (2.20) R Ω(t) − Ω(t0) where Ω(t) is the current crust rotation rate and Ω(t0) was the rotation rate when the present pinned vortex configuration was formed. On large time and spatial scales, the crust flows like a very viscous fluid, from the poles towards the equator for stars that are spinning down. As a result, a “subduction zone” is formed at the equator, where crust matter is pushed into the core. To compensate for the matter “lost” by the crust, matter from the core convects upwards, joining the platelets as they move across the non-equatorial regions of the star. In general, this matter consists mainly of neutrons, which are converted by weak interactions to p+ and e− just below the crust. Since the magnetic field is anchored to the crust, it migrates with the crust as the latter flows from the poles to the equator. As a result, once a low magnetic field millisecond pulsar has spun down sufficiently, its magnetic dipole will be perpendicular to the rotation axis. This configuration has been found in at least three such “orthogonal rotators”: PSR 1855 + 09,31 PSR 1937 + 2132 and PSR

30 Alpar. M., Anderson. P., Pines. D., Shaham. J., 1984a, ApJ, 276, 325 31 Segelstein,D., Rawley, L., Stinebring, D., Further, A., and Taylor, J., 1986, Nature, 323, 714 32 Cordes, J., Stinebring, D., 1984, ApJ(Letters), 277, 53 2. Glitches and Glitch Mechanisms 36

1957 + 2033. In these pulsars there exist double pulses of similar intensity, with a phase separation of close to 180o. For stars that spin up, the direction of the stress given above is inwards rather than outwards, as was the case in spinning down stars, resulting in crust motion from the equator to the poles. In this case, the magnetic field migrates towards the poles, resulting in spin-aligned magnetic fields, making them almost impossible to detect, hence the lack of experimental observations. The case for a crustquake-type mechanism (i.e. one in which the change in moment of inertia of the star is responsible for the glitch event) has gained renewed support in specific cases, most notably, for the Crab pulsar. Upon study of the behaviour of Crab glitches, Alpar et. al.34,35 found that crustquakes were at least partially responsible for these glitches. In addition, the permanent post-glitch rotation frequency derivative offsets observed in the Crab pulsar offer further support for the crustquake model.36,37 More support for this mechanism was offered by Crawford and Demiansky,38 when they showed that the values of the post-glitch healing parameter for the Crab pulsar agreed with those predicted for a crustquake-induced glitch occurring in a neutron star of mass ≈ 1.5M (a value expected for the Crab pulsar to possess).

2.3.2 Thermally Driven Glitches This model was developed by Link and Epstein in a paper published in 199639.In short, their simulations tracked the reaction of a neutron star to the sudden deposition of a large amount of heat in the crust, and the conclusion they reached was that certain such depositions would result in glitch events. During the evolution of the star, heating of the interior will occur via mechanisms such as internal friction, accretion, plastic or sudden relaxation of stellar structure as well as nuclear reactions. One additional source of heat would be a starquake, similar to those mentioned earlier, which causes a sudden heat deposition. Baym and Pines40 give an estimated, average accumulation rate for strain energy of dE strain ' Bθ t−1 , (2.21) dt c age 41 where B ∼ 10 J is the crust strain energy, θc is the critical strain angle at which the lattice breaks and tage is the neutron star age. On average, the Crab and Vela

33 Fruchter,A., Stinebring, D., and Taylor, J., 1988, Nature, 88, 237 34 Alpar, M.A., Chau, H.F., Cheng, K.S., Pines, D., 1994, ApJ, 427, L27 35 Alpar, M.A., Chau, H.F., Cheng, K.S., Pines, D., 1996, ApJ, 459, 706 36 Link, B., Franco, L.M., Epstein, R.I., 1998, ApJ, 508, 838 37 Franco, L.M., Link,B., Epstein. R.I., 2000, ApJ, 543, 987 38 Crawford, F., Demiansky, M., 2003, ApJ, 595, 1052 39 Link, B., K., Epstein, R., I., 1996, ApJ, 457, 844 40 Baym, G., Pines, D., 1971, Ann. Phys. N.Y.,2, 816 2. Glitches and Glitch Mechanisms 37 pulsars experience glitches every few years. If these were a result of the release of accumulated strain energy, then the resulting energy depositions could exceed 35 −2 10 (θc/10 )J. After the deposition, the heat propagates through the crust in the form of a thermal wave, resulting in crustal spin-up due to angular momentum transferred from the heated superfluid regions. The glitch “ends” either once the thermal energy has been diffused or the superfluid and crust velocities are equal. Obviously, the energy deposition results in a temperature increase in the given region, and it is this temperature increase which determines the magnitude and timescale of the glitch. Something that is interesting to note is that a given energy deposition results in a larger, faster glitch in an older pulsar, compared to a younger one, due to the lower specific heat.41 Larson and Link (1999) used the equation of state (EOS) given by Friedman and Pandharipande.42 According to this EOS, 90% of the the mass of the inner crust lies in the radius range 9.8 - 10.0 km for a 1.4M star. Accordingly, it is in this region that the contribution towards the coupling between superfluid and crust is the greatest. The average density in this region is ~ρ = 1.5 × 1014g cm−3, which is taken as the characteristic density for the simulation to follow in a later chapter. Due to the conservation of angular momentum, the star’s angular momentum is determined by

R 3 2 J(t) = Jc(t) + Js(t) = IcΩc(t) + d rr ρsΩs(r, t) (2.22) ˙ = J0 − I|Ω∞|t, (2.23) where J0, Jc and Js are the angular momenta of the entire star, the crust and ˙ the superfluid respectively and Ω∞ is the steady state spin down of the star. The superfluid angular velocity (Ωs(r, t)) can be obtained using the following relation, easily derived from the superfluid circulation equation: ∂Ω (r, t) 2 ∂ s = −v + Ω (r, t), (2.24) ∂t cr r ∂r s

43 The radial vortex creep rate vcr, for the regime where ω ωc, is given by −β/T ω vcr(ω, T ) = v0e , (2.25) where ω = Ωs − Ωc is the velocity lag between the superﬂuid and crust, and 6 −1 44 v0 ∼ 1 × 10 cm s is the vortex creep rate in the absence of pinning. β is the coupling constant for interactions between the superﬂuid and crust, which is given by Link and Epstein (1996) as

β = 0.54ΩcEp, (2.26) 41 Larson, M.B., Link, B., 1999, ApJ, 521, 271 42 Friedman, P., Pandharipande, V.R., 1981, Nucl. Phys. A, 361, 502 43 Link, B., Epstein, R.I., Baym, G., 1993, ApJ, 403, 285 44 Link, B., Epstein, R.I., 1996, ApJ, 457, 844 2. Glitches and Glitch Mechanisms 38

where Ep is the effective pinning energy. The thermal evolution of the crust is determined by the heat diffusion equation given below ∂T C = ∇ · (κ ∇T ), (2.27) v ∂t T with Cv and κT the specific heat and thermal conductivity respectively. Note that in this model, the heat generated due to friction between the two components is neglected. In their simulation used to test this model, Link and Epstein considered the portion of the crust responsible for driving the spin-up to be a cylindrical shell. The parameters of this shell are: constant superfluid density ~ρ, thickness ∆, height h, with inner and outer radii of Rc ± ∆/2, and a cylindrical co-ordinate x having x = 0 at radius Rc, as seen in Fig 2.3.

Fig. 2.3: Diagram showing the structure of the cylindrical regime considered in the Thermal Glitch Mechanism, as given by Link & Epstein (1996)

The actual heat deposition consists of an energy amount E being deposited with a Gaussian proﬁle about x = 0, having a half-width of σ in the x-direction, where σ << ∆. Thus the initial temperature proﬁle has the form

∆T (x, t = 0) = ∆T (0, 0)e−x2/2σ2 . (2.28)

The magnitude of the temperature change is determined by Link and Epstein (1996) to be ∆T (0, 0) √ r E = − 2 + 2 + 3/2 , (2.29) T0 π Cv0 T0 Rc h σ 2. Glitches and Glitch Mechanisms 39

where T0 is the unperturbed temperature. In this model, the dependence of both the speciﬁc heat and thermal conductivity on temperature is neglected. Thus, the thermal diﬀusion equation changes form, becoming 2 ∂T κ0 ∂ T = 2 . (2.30) ∂t Cv0 ∂x Solving the above equation yields

c1/2 2 2 ∆T (x, t) = ∆T (0, 0) e−cx /2tσ , (2.31) t

2 where c ≡ Cv0 σ /2κ0. The actual simulation process is now considered. It is assumed that the superfluid and crust are initially in rotational equilibrium, i.e. their spin-down ˙ rates are both equal to Ω∞. Using this assumption, (2.24) and (2.25) are solved, and the initial lag obtained by subtracting the observed crust angular velocity from the determined superfluid velocity throughout the crust. The glitch is then initiated by the deposition of energy E, and (2.28) and (2.30) solved to obtain the temperature at every given time. Using these temperature values, the values of Ωs and Ωc can be obtained for every time step using (2.24), (2.25) and (2.22). The lag for each time step is then also obtained. The results obtained from this simulation now follow. For a pulsar with spin and thermal parameters similar to the Crab pulsar, an energy deposition 35 −8 of 2.1 × 10 J produced a glitch of magnitude ∆Ωc/Ωc ' 7 × 10 occurring within ∼ 100s, as well as a relaxation time of the order of a few days to a week. These results agreed well with the observations for the 1989 glitch occurring in Crab, as shown in Figure 2.4. For a pulsar with parameters similar to that of the Vela pulsar, an energy deposition of 1.51 × 1035 J was simulated. The result −6 was a glitch of magnitude ∆Ωc/Ωc ' 10 , occurring in the first few seconds, and a relaxation time greater than the simulated timescale (a few months). These results agreed well with observations of the so-called “Christmas glitch” observed in Vela, as shown in Figure 2.5. 2. Glitches and Glitch Mechanisms 40

Fig. 2.4: The results obtained by Link and Epstein (1996) for a comparison of the −8 thermal glitch mechanism simulation of magnitude ∆Ωc/Ωc ' 7 × 10 and an energy deposition of 2.1 × 1035J (line) to data (dots) for the Crab pulsar 2. Glitches and Glitch Mechanisms 41

Fig. 2.5: The results obtained by Link and Epstein (1996) for a comparison of the −6 thermal glitch mechanism simulation of magnitude ∆Ωc/Ωc ' 10 and an energy deposition of 1.51 × 1035J (line) to data (dots) for the Vela pulsar 2. Glitches and Glitch Mechanisms 42

2.4 Core-driven Glitches

2.4.1 Flux-tube Model This model was introduced by Ruderman, Zhu and Chen, in their 1998 article entitled “Neutron Star Magnetic Field Evolution”.45 Thus, most of the results found in this section originate from that paper. The flux-tube model is in fact a modification of the crust cracking model introduced earlier by Ruderman.46,47 In this newer model, the interaction between the neutron superfluid vortex lines and the superconducting proton magnetic flux tubes in the crust is discussed, as is the possible importance of these interactions in pulsar glitching. Roughly five percent, by mass, of the material in the outer core of the neutron star is believed to consist of type II superconducting protons. In a type II superconductor, the total magnetic field carried in this region is organised into quantised magnetic flux tubes, each with a flux

−7 Φ0 = π~c/e = 2π × 10 G, (2.32) where c and e are the speed of light and the electron charge respectively. The density of ﬂux tubes can thus be given by

B 19 −2 nΦ = ∼ 10 B12 cm , (2.33) Φ0 where B12 is the normalised magnetic field passing through the protons, i.e. B12 = B/1012 G. These flux tubes are expected to form a very intricate array, having a complicated, twisted structure due to the ambient magnetic field Bamb. In contrast, the neutron superfluid vortex array is expected to be arranged parallel to the rotational axis, with an areal vortex density of

4 mnΩs 10 −2 nV = ∼ cm . (2.34) ~ P (s) As the star spins down, the vortices migrate steadily, radially outwards. Hence, the vortex array expands as the star slows down. As this occurs, the vortices interact with the flux-tubes, pushing against them. There are two possible results of this pushing; (1) The flux-tube array is pushed along with the vortex array, or (2) the vortices cut through the flux-tube array, if the reaction of the flux-tubes to the vortices is too slow. As they pass into the crust, the flux-tubes merge into a smooth magnetic field. The high electrical conductivity of the crust causes the flux-tube contact points to be frozen in place on the inner crust. The combination of this fact with either

45 Ruderman M., Zhu T., Chen K., 1998 ApJ, 492, 267 46 Ruderman. M., 1976, ApJ, 203, 213 47 Ruderman, M., 1991, ApJ, 366, 261 2. Glitches and Glitch Mechanisms 43

(1) or (2) above causes the generation of stresses on the crust as the star spins down. If the build up of stress is incredibly slow, these stresses may be released by the dissipation of the Eddy currents holding the magnetic field in place at the crust-core boundary. However, if the build-up is faster than the so-called “Eddy- timescale”, the stress will be released once the yield strength is exceeded, causing the crust to move in such a way as to relieve the stress. Ruderman48 estimates −4 −3 the yield strain of the crust to be of the order of θmax ∼ 10 to ∼ 10 . The specific type of crust movement is determined by the temperature of the crust. For example, plastic flow occurs at higher temperatures, where the crust is more pliable, while at lower temperatures the crust undergoes cracking due to its brittle nature. Consider the case where flux-tubes move through the proton-electron sea with a relative velocity vΦ. The radial velocity of the vortices situated a distance r⊥ from the spin axis is given by

r P˙ v = − ⊥ , (2.35) V 2P where P and P˙ are the pulsar period and its time derivative respectively. Ruder- man et al.49 find that the maximum force which can be exerted on the flux-tube by the vortex is πn Λ F ' V B B Λ ln ∗ , (2.36) max 8 Φ V ∗ ξ where BΦ is the magnetic field within a flux tube, BV is the magnetic field within the cores of the neutron vortex lines embedded in the proton superconducting −11 sea, Λ∗ is the flux tube radius (∼ 10 cm), and ξ is the BCS correlation length for Cooper pairs in the proton-electron sea. Ruderman et al. also arrived at a maximum velocity at which the vortex line can push a flux-tube before cutting through it. This critical velocity is

Ω 1012 G v = β 10−6 cm s−1, (2.37) c 100 s−1 B where the constant of proportionality β depends only on the properties of core matter, and on neither Ω nor B. If vΦ is small enough to ensure that the flux- tube density remains constant, in spite of any electrical currents induced by their motion, then β ∼ 1. It is obvious that this model is a very simplistic one, as in reality the ge- ometrical distribution of the flux-tubes, as well as their motion, would be very complicated, causing vortices to push flux-tubes along in one region and cut them in another. 48 Ruderman, M., 1991, ApJ, 366, 261 49 Ruderman M., Zhu T., Chen K., 1998 ApJ, 492, 267 2. Glitches and Glitch Mechanisms 44

The existence of vc implies the presence of a critical radius rc, above which vortices cut through the ﬂux-tubes and below which the ﬂux-tubes are merely pushed along. Ruderman et al. give this radius as

−1 Ts Ω/100 s 6 rc ' 4 10 cm, (2.38) 10 yr B12 where Ts is the pulsar spin-down age. 6 For pulsars similar in character to Vela, rc ≥ 10 cm, where the star’s radius is R ' 106cm. Hence, in these types of pulsars, the flux tubes are all pushed along by the vortices, and hence move with a velocity vV . However, in young pulsars such as the Crab, rc << R, and as a result flux-tubes at a radius r⊥ < rc move with velocity vV , while those with a radius r⊥ > rc travel at velocity vc. For young, warm pulsars, the crust is expected to migrate via plastic flow. The transition to crust cracking is expected to occur at roughly the temperature of the Crab pulsar, i.e 108 K. In general, the glitch magnitude is given by50

∆Ω Ω˙ ∼ 10−2τ , (2.39) Ω g Ω

−2 where τg is the average time elapsed between glitches and the factor 10 is the fraction of the moment of inertia of the crust and all parts coupled to it which is expected to participate in the glitch. The interpretation of various observed features of glitches is now discussed in the framework of the ﬂux-tube crust-cracking model:

In some of the glitches observed in the Crab pulsar, the large ones in particular, there exists a permanent increase in the spin down rate of the pulsar after the glitch. The magnitude of this change is roughly ∆Ω˙ = 4 × 10−4. This Ω˙ change is far too large to be explained merely by a change of crust shape. In fact, it is orders of magnitude larger than the glitches themselves, which in the case of the Crab pulsar recover quickly. One possible explanation for this behavior is that due to the glitch, the spin-down torque of the star may have increased. Consider the case where crust cracking is followed by the sudden migration of a strongly magnetised platelet towards the equator, whereby this platelet is dis- placed by ∆s (given as ∼ 2 × 10−4R by Ruderman et al.51). This displacement results in an increase in the magnetic dipole moment of the star, which in turn causes an increase inω ˙ .

This behavior, the presence of an permanently increased spin-down after a glitch, is not observed in Vela. However, this might only be due to the fact that

50 Ruderman M., Zhu T., Chen K., 1998 ApJ, 492, 267 51 Ruderman M., Zhu T., Chen K., 1998, ApJ, 492, 267 2. Glitches and Glitch Mechanisms 45

Vela doesn’t heal enough for this to happen before another glitch occurs. The braking index, n, for Vela is 1.4 (compared to the standard value of 3). However, ˙ ˙ −1 −4 this might be a result of an unhealed ∆Ω/Ω = (3 − n)/2τgTs ∼ 2 × 10 which occurs after each glitch, but does not manifest itself due to the long recovery times compared to the short inter-glitch intervals. From (2.24) it can be seen that the −1 ˙ angular velocity of the platelets on Vela’s crust is ∼ Ts (where Ts = 2P/P ). As a result, the time interval between glitches for Vela should be τg ∼ (∆s/R)Ts ∼ 2 years. This is in the region of the observed value (∼ 3 years). Also, from (2.28), the magnitude of glitches in the Vela pulsar and others like it are predicted to be ∼ 10−6, which is approximately the value observed.

The values given by the table in Figure 2.6 are given for three layers in the crust, each characterised by lattices with unique values of Z, and were obtained by 52 Negele and Vautherin. The quantity Tb is the temperature at which the lattice in that specific region of the crust becomes brittle, thus breaking under excessive stress, rather than undergoing plastic flow, and is roughly 10% of the value of the melting temperature for the lattice. Plastic flow of the crust does not contribute to glitching; only discontinuous breaking does. Thus, when looking at the Crab pulsar (which has a temperature such that only region c is susceptible to cracking), and comparing it to that of the Vela-class pulsars (which have cooled such that all three regions undergo cracking), it can be seen that the ratio of glitch- −2 inducing moments of inertia is Ic/(Ia + Ib + Ic) ∼ 3 × 10 , which corresponds with the observed differences in glitch magnitudes (∼ 10−2) between these two classes. Also, the absence of glitches in other young pulsars, such as PSR 1509 -58 and PSR 1540 -69 (Ts = 1500 and 1700 respectively) could be explained by stating that they are still too warm to undergo crust cracking. In addition, if the shear stress required to slide layers across one another is much less than the stress required to crack either layer, then these layers may crack independently. This could explain the differing glitch magnitudes experienced by any given family of pulsars, due to the different moments of inertia of these layers.

Fig. 2.6: Properties of various crust layers, as given by Negele and Vautherin (1973)

52 Negele, J., Vautherin, D, 1973, Nucl. Phys. A, 207, 298 2. Glitches and Glitch Mechanisms 46

2.4.2 Centrifugal Buoyancy Mechanism In 2000, Carter et.al.53 proposed another mechanism for glitches. This mechanism has been labeled the Centrifugal Buoyancy mechanism. It does not require the neutron fluid to be a superfluid in the strict sense, and importantly, doesn’t require the presence of vortices, but rather requires that the fluid is perfect, effec- tively having zero viscosity. This is required to create differential rotation between the two fluid components; i.e. the neutron fluid and the charged fluid (including the crust). The basic principle involved in this mechanism is that, in the absence of a slower solid crust, the fluid would readjust itself in annular portions, with each ring increasing in angular momentum per unit mass (or centrifugal buoyancy) as the radius increases. Thus, in accordance with the Taylor-Proudman theorem, the angular velocity of the fluid is a function solely of cylindrical radius, assuming a barotropic fluid. However, the presence of a slow, solid crust post- pones this process, producing anisotropic stresses in the crust which balance the induced centrifugal force. These stresses build up until, at some critical point, the centrifugal buoyancy forces produce a ”starquake”, resulting in a discontinuous readjustment and subsequent spin-up of the crust. Some of the important physical arguments related to this theory are now discussed.

Firstly, this analysis was performed in a Newtonian framework, by considering an idealised two-component model. The components considered here are (i) the neutron ”superfluid” and (ii) the crust (along with the electrons and protons which are bound to the crust by the strong magnetic fields, as well as the non- superfluid neutrons) . These two components are independent, and are assumed to obey the Euler-like equations

i j i i i i ρc(∂0vc + vc ∇jvc) = −∇ Pc − ρc∇ φ + fc (2.40) and i j i i i i ρn(∂0vn + vn∇jvn) = −∇ Pn − ρn∇ φ + fn (2.41)

In the above equations, ∂0 denotes partial diﬀerentiation with respect to Newto- nian time, φ is the Newtonian gravitational potential, P() and f() are the relevant pressure and force density vectors, and ρ() the respective mass densities, with the subscripts c and n denoting the crust and neutron ﬂuid components respectively. By rewriting the velocity in terms of the angular velocity and cylindrical radius, $, (i.e. v() = Ω()$) and considering the fact that in the crust, there will be an i extra force density term, fs, due to the aforementioned anisotropic stresses, the Euler equations become 1 Ω2∇i$2 − ∇i(φ + m−1µ ) = ρ−1(f i − f i) (2.42) 2 c c c n s 53 Carter, B., Langlois, D., Sedrakian, D.M., 2000, A&A, 361, 795 2. Glitches and Glitch Mechanisms 47

1 Ω2 ∇i$2 − ∇i(φ + m−1µ ) = −ρ−1f i , (2.43) 2 n n n n where µ() are the chemical potentials for the relevant components. In the pres- i ence of vortex pinning, a major component of the stress force fn would be that responsible for counteracting the Magnus-type force of the form

i i 2 fJ = ρn(Ωn − Ωc)Ωn∇ $ . (2.44)

By combining (2.42) and (2.43), Carter et al. obtained the following equation: 1 1 ∇iP + ρ(∇iφ − Ω2∇i$2) = f i + f i − ρ (Ω − Ω )2∇i$2, (2.45) 2 c J s 2 n n c where P and ρ are the total pressure and mass density, given by their sums over the respective components. The solid stress force is made up of three components: the neutron fluid stress fn, a possible neutron drip delay stress fx and a stress due to the deficit in centrifugal buoyancy of the crust, fb. This final term is given by i i 2 2 i fb = ρc(Ωn − Ωc)(∇ ($ Ωn) − $ ∇ Ωc). (2.46) i However, the crust is rigid, hence ∇ Ωc = 0, which gives

i i 2 fb ' ρc(Ωn − Ωc)Ωn∇ $ . (2.47)

From (2.44) and (2.47), the relationship between the Magnus and centrifugal buoyancy deﬁcit forces is given by

i ρn i fJ ' fb, (2.48) ρc assuming vortex pinning is present. If the assumption is made that the neutron drip delay force is negligible, and if vortex pinning is not present, then the solid stress force is simply given by i i fs ' fb. (2.49) The corresponding value for a regime where catastrophic unpinning of vortices is considered to be the cause of the glitch (and centrifugal buoyancy effects are neglected) is i i fs ' −fJ . (2.50) The negative sign is present in (2.50) since the stress inducing effects of the two mechanisms are in opposite directions, with pinning forces causing stress radially outwards, and centrifugal buoyancy forces pulling inwards. Also, the magnitudes of these forces are fairly similar, with the constant of proportionality being the ratio of the density of the crust to the density of the neutron fluid (as seen in (2.48)). In summary, this mechanism is one which is expected to operate at all times, even 2. Glitches and Glitch Mechanisms 48 in the absence of vortex pinning, or where vortex pinning in virtually ineffective. The only requirement is that there exist two independent fluid components, as discussed above. This is something that is predicted in nearly every modern model of the neutron star. Carter et al. predict that centrifugal buoyancy forces could produce glitches of magnitude I ∆Ω ≥ − n Ω˙ , (2.51) I where I and In are the moments of inertia of the entire star and the neutron fluid respectively, Ω˙ is the pulsar period spin-down, and is an efficiency coefficient, having a value between 0 and 1. This is, however, only a crude quantitative measure of the glitch magnitude. 3. TWO-COMPONENT MODEL: THEORY AND HISTORY

In 1969, shortly after the first glitch was observed in the Vela pulsar, Baym et al. formulated a model to explain the behaviour of a pulsar after a glitch.1 This is generally referred to as the two-component model, and has been revised many times since its inception. This model assumes that the neutron star has a solid crust of quasispherical form, consisting of a lattice of nuclei; see Figure (3.1). The region immediately beneath this shell is completely occupied by neutron superfluid matter, and possibly a solid core, in the case of heavier neutron stars. The superfluid contains both proton and electron impurities; 1% - 5% of each, where the protons also form a superfluid, and the electrons a normal fluid. The electrons are bound to the core and to the charged crust lattice by means of the strong magnetic field present in the neutron star.

Fig. 3.1: Simple picture showing a possible conﬁguration for the two-component model.

The superﬂuid component interacts with the rest of the star through the presence of quantised vortices present in the superﬂuid, the theory of which has been

1 Baym, G., Pethick, C., Pines, D., Ruderman, M., 1969, Nature, 224, 872 3. Two-component Model: Theory and History 50 discussed previously. The normal fluid component present in the vortex cores interacts with the crust component. These vortices are attached to the hard shell, and possibly to the solid inner core where applicable, and hence move accordingly. Baym et al. (1969) showed that only this configuration gives post-glitch relaxation times of the order of weeks to years as observed. Other configurations, such as normal fluid inside a solid crust or a superfluid containing only neutrons, gave relaxation times which were far too small compared to observations (10−7 −10−17sec compared to ≈ 107sec). The normal fluid present in the neutron vortex cores also interacts with the normal fluid present in the superfluid proton vortices. However, this coupling is relatively weak. This model is one of the longest standing and most tested mechanisms for pulsar glitches. In view of this, a summary of the major results obtained using this theory will follow, including the viewpoints expressed both for and against it. In 1980, Tsakadze and Tsakadze2 published the results of various experiments, performed over a number of years, to determine whether the glitching behaviour observed in pulsars could be reproduced using superfluid helium-4. The experiments involved various configurations of rotating cylindrical and spherical con- tainers filled with superfluid helium-4. It was expected that these experimental results should agree with observations of pulsars, if only qualitatively. The experimental results did, in fact, agree with that expectation.3 For a pulsar with the physical parameters discussed above, Baym et al. (1969) predicted a relaxation timescale of ≈ 107 sec, while extrapolations from the experiments performed by Tsakadze and Tsakadze gave values of 5 × 106 to 107 sec. Observations of the first observed glitches in the Crab and Vela pulsars, in 1969, gave relaxation timescales of 3 × 105 sec and 3.7 × 107 sec respectively. It is obvious that even the rough experimental and theoretical approximations by Tsakadze & Tsakadze and Baym et al. respectively, give results which agree fairly well with observed values. Also, experimental work done by Tsakadze & Tsakadze showed that the model of a neutron star which does not contain a solid core is preferable for the Crab and Vela pulsars.2 The two-component model serves as a method to predict the post-glitch behaviour of a pulsar, where the two components which comprise this model are the solid crust and the neutron (and proton) superfluid, having moments of inertia Ic and In and angular velocities Ωc and Ωn respectively. Possibly the most critical factor in this model is the parameter describing the interaction between the two components. This parameter is called the coupling constant τc and is sometimes referred to as the relaxation timescale. A simple look at the equations for angular momentum conservation yields the following equations of motion for the two components after a glitch event:

2 Tsakadze, J.S., Tsakadze, S.J., 1980, Journ. Low Temp. Phys., 39, 649 3 Tsakadze, J.S., Tsakadze, S.J., 1972, Phys. Lett. A, 41A, 197 3. Two-component Model: Theory and History 51

˙ Ic IcΩc = −α − (Ωc − Ωn) (3.1) τc and ˙ Ic InΩn = (Ωc − Ωn), (3.2) τc where α is the torque term arising from the electromagnetic braking of the pulsar, which in this model, is assumed constant in Ωn over the timescales con- 7 ˙ sidered (typically ∼ 10 s) and is given by IΩc. Considering this, and taking τc to also be constant for these timescales (as it is expected to be for a given pulsar), then the solutions of the above equations of motion are

α In −t/τ Ωc = − + Ae + B (3.3) Ic + In Ic + In and −t/τ ατ Ωn = Ωc − Ae + , (3.4) Ic where τ ≡ τcIn/(Ic +In), In and Ic are the superfluid and crust moments of inertia respectively and A and B are arbitrary constants, arising from integration. If these constants are given by ∆Ω0 (i.e. the absolute magnitude of the glitch) and Q (the so-called ‘healing parameter’) respectively, then the following equation can be obtained: −t/τ Ωc = Ω0(t) + ∆Ω0[Qe + 1 − Q]. (3.5) The healing parameter gives the fraction of angular velocity change which is recovered after a glitch, as shown in Figure 3.2.4 Equation (3.5) shows that the two-component model gives an exponential decay of the crustal spin rate after a glitch. As a result, common practise is to perform the purely mathematical fitting of an exponential curve to glitch data, in order to obtain a value for the coupling constant. What follows is a brief look at some of the work which has been performed based on the two-component model, since its formulation in 1969. In 1971, Boynton et al.5 fitted the model to the glitch which occurred in the Crab pulsar at the end of September 1969, obtaining a healing parameter Q = 0.9 and a coupling constant of 4 < τ < 16 days. In 1974, Pines et al.6 stated that a key test of a glitch model is whether or not the values of Q and τ are constant for all observed glitches on a given

4 Shapiro, S.L., Teukolsky, S.A., 1983, Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects, (Wiley-Interscience) 5 Boynton, P.E., Groth, E.J., Partridge, R.B., Wilkinson, D.T.,(1971), The Crab Nebula, Davies and Smith (eds), (The IAU) 6 Pines, D., Shaham, J., Ruderman, M.A., (1974), Physics of Dense Matter, Hansen, C.J., (ed.), (The IAU) 3. Two-component Model: Theory and History 52

Fig. 3.2: Response of pulsar to glitch, as predicted by the simple two-component model. pulsar. They also stated that up until that point in time, there were no observed inconsistencies with the two-component model. In 1978, Manchester et al.7 applied the two-component model to a glitch in PSR 1641-45, which occurred in 1977. They found the coupling constant to be τ ≈ 85yr. In 1979, George Greenstein8 proposed a thermal glitch mechanism. He said that the major features of the post-glitch relaxation after a glitch of this kind can be described by the two-component model. However, he stated that the coupling constant (or post-glitch relaxation timescale) τ is not necessarily constant for a given pulsar, but depends on the post-glitch temperature of the star. This mechanism suited smaller glitches, such as those occurring in the Crab pulsar, very well. In 1981, the ﬁrst real opposition to the two-component model appeared, in two separate papers. Firstly, Boynton9 obtained a power spectrum of timing ﬂuctuations from pulsar timing data. He showed that for the calculated values of Q and τ for the Crab pulsar, according to the two-component model, there should be a prominent, characteristic, two-component signature in the power spectrum. However, in his analysis, Boynton found none. Instead, he found that the signature pointed towards rigid-body behaviour. He then went on to say that the two-component model works well for macroscopic glitches, but not for minor

7 Manchester, R.N., Newton, L.M., Goss, W.M., Hamilton, H.A., 1978, MNRAS, 184, Short Communication, 35 8 Greenstein, G., 1979, ApJ, 231, 880 9 Boynton, P.E., 1981, Pulsars, Sieber, W., Wielebinski, R. (eds), The IAU, 279 3. Two-component Model: Theory and History 53 perturbations, which give results resembling rigid-body behaviour. The second objection came in the same year from Downs10. In this paper, he thoroughly analysed twelve years of observational data for the Vela pulsar. He mentioned that only 20% of each Vela period jump decayed exponentially towards the pre-glitch spindown rate, having a recovery time of around one year. He also showed that the coupling time τ isn’t the same from glitch to glitch for Vela, in contradiction with what was proposed by Pines et al. in 1974. Another important observation is that the values of Q obtained in this analysis were much lower than expected, for large glitches (∆P/P ' 10−6). These Q-values were used to calculate neutron star masses which are roughly half those obtained using more direct methods. As a result, Downs stated that the two-component model is not dominant in cases of large spinup. He concluded that the interior structure of the Vela pulsar is too complex to be explained by the simplistic two-component model. Later that year, Alpar et al.11 commented on the merit of Downs’ conclusion that the Vela pulsar’s relaxation behaviour is too complex to be described by the simplistic two-component model presented by Baym et al. They used this result as the basis for their model in which spinups are caused by a change of vorticity in the superﬂuid, and the post-glitch relaxation is caused by vortex creep mechanisms,12 discussed in chapter 2 of this thesis. This newer model relies on the processes of vortex pinning and unpinning as the main drivers of the glitch phenomenon. The main diﬀerences between this model and the two-component model are as follows:

1. The vortex unpinning model involves a much smaller fraction (∼ a few percent) of the neutron superfluid, i.e. the pinned portion of the superfluid, as compared to the entire superfluid in the two-component model.

2. The vortex unpinning model is a non-linear theory, in that the vortex creep rate (i.e. the rate at which vortices migrate radially outwards) depends exponentially on the size of the glitch, as well as on the glitch-induced change in superﬂuid angular velocity.

3. The vortex unpinning model produces relaxation timescales which are proportional to the internal temperature of the star. As a result, analyses of timing data using this model would enable the determination of the internal temperature, and hence an estimate of the surface temperature, of the star.

A few months after Alpar et al. ﬁrst proposed the vortex unpinning model, in early 1982, Downs produced a second detailed paper on pulsar timing analysis13,

10 Downs, G.S., 1981, ApJ, 249, 687 11 Alpar, M.A., Anderson, P.W., Pines, D., Shaham, J., 1981, ApJ, 249, L29 12 Alpar, M.A., Anderson, P.W., Pines, D., Shaham, J., 1984a, ApJ, 276, 325 13 Downs, G.S., 1982, ApJ, 257, L67 3. Two-component Model: Theory and History 54 this time for PSR 0525 +21. Once again, twelve years of data were analysed, and in this period, two small glitches were observed (∆P/P ∼ −10−9). Downs contended that the original two-component model was not a good fit to these glitches. The reason for this is that after the spinup, the period derivative P˙ did not return to the pre-jump value, as is required by the model. Instead, the post- glitch behaviour seemed to mimic that displayed by the Vela pulsar, as discussed in Downs’ previous investigation. The following year, in 1983, Demiańskyand Prózyński14 published their results on the 1975 Crab pulsar glitch. They too found a permanent change in P˙ after the glitch event, citing a change in the external torque on the star as the cause. This further added to the evidence supporting the claim that the two-component model is too simplistic to describe the processes occurring during a glitch. Link et al.15 proposed that this observed behaviour is a result of an increased external torque caused by the rearrangement of the stellar magnetic field. In 1985, Cordes and Downs16 aptly commented on the then recent findings regarding the two-component model. They stated that the model wasn’t to be scrapped, but that a redefinition of the two components was required. They emphasised the fact that the most successful mechanisms were those that involved a transfer of angular momentum from the more rapidly rotating superfluid to the slower crust. In 1987, the two-component model received some observational support from the analysis of the 1986 glitch in the Crab pulsar, performed by Lyne and Pritchard17. They found that the data for this glitch could be easily fitted using a simple exponential, of the form of that obtained using the two-component model. They did, however, find that the data are best fitted by two separate exponentials. However, the two exponential factors (τ1 = 2.5±0.5 days and τ2 = 5.0±0.5 days) are multiples of each other. Hence, they could not conclude whether these do in fact represent two separate exponentials, or a single more complicated one. They went on to say that the two-component model still represents an applicable model; one that can be best understood in terms of the vortex unpinning model of Alpar et al. It is interesting to note that the size of this glitch is rather small, with ∆P/P = −9.2 ± 1 × 10−9, similar in size to the 1969 Crab glitch, which was also fitted using a two-component-derived exponential. The Crab glitch of 1975 was a lot larger (∆P/P = 4 × 10−6), and its data could not be fitted using the simple two-component model. This adds weight to the comments made by Downs that the two-component model fails in the large glitch regime. In 1990, Flanagan18 produced results from the analysis of the 1988 Vela

14 Demiańsky, M., Prózyński,M., 1983, MNRAS, 202, 437 15 Link, B., Epstein, R.I., Baym, G., 1992, ApJ, 390, L21 16 Cordes, J.M., Downs, G.S., 1985, ApJ Supplement Series, 59, 343 17 Lyne, A.G., Pritchard, R.S., 1987, MNRAS, 229, 223 18 Flanagan, C.S., 1990, Nature, 345, 416 3. Two-component Model: Theory and History 55

“Christmas” glitch. She found that there was an additional coupling timescale when the data were fitted using exponential functions. This timescale was much shorter than those previously discovered (τ = 0.4 ± 1 days), most probably due to the fact that observations hadn’t been made so soon after a glitch prior to this one. This extra coupling constant provided evidence for the presence of two superfluid components, both linearly coupled to the crust and its accompanying constituents. This finding was supported by Alpar et al.19 The analysis of the glitches occurring in PSR 1737 - 30, performed by Michel et al.20 showed that the simple two-component model fitted the data very well, although exact parameter values weren’t given. This showed that in this pulsar at least, there is no change in moment of inertia (i.e. Q = 1), and supported the case for the continued validity of the two-component model. Some time later, in 1996, Lyne et al.21 performed an analysis on a glitch in PSR B1757 -24. It was one of the largest recorded glitches, having a magnitude of ∆P/P ≈ 2 × 10−6. The decay was fitted using an exponential function, as given by the two-component model, having a coupling constant of τ = 42 ± 14 days, and a post-glitch spin-down rate equal to the pre-glitch value (i.e. Q = 1 once again), further strengthening the case of the two-component model as a contributor to pulsar glitching. Also in that year, Shemar and Lyne22 presented a collection of results on the analysis of 25 glitches occurring in ten pulsars. These results presented some interesting findings. Firstly, the older pulsars did not show substantial recovery towards pre-glitch frequencies (i.e. they displayed small Q-values), while the younger pulsars displayed substantial recoveries towards pre-glitch rotation frequencies. However, almost all the glitches showed some form of recovery, which was easily modeled using a single exponential having a time constant of the order of 100 days, followed by a long-term relaxation, of at least 1000 days (in most cases this value was greater than the inter-glitch spacing). The few glitches for which this was not found to be the case were those for which the timing data was either too sparse or too noisy for accurate analysis. These findings were later confirmed by Lyne and Shemar, with the input of Graham-Smith23 in their statistical analysis of 48 glitches occurring in 18 pulsars, including those studied in their 1996 paper. In 2005, Shabanova24 published results on three glitches in PSR B1822 -09. These glitches exhibited exponential recovery, similar to that described by the two-component model, with timescales of τ ∼ 100 days, 235 days and 80 days

19 Alpar, M.A., Pines, D., Cheng, K.S., 1990, Nature, 348, 707 20 Michel, F.C., Bland Hawthorn, J., Lyne, A.G., 1990, MNRAS, 246, 624 21 Lyne, A.G., Kaspi, V.M., Bailes M., Manchester, R.N., Taylor, H., Arzoumanian, A., 1996, MNRAS, 281, L14 22 Shemar, S.L., Lyne, A.G., 1996, MNRAS, 282, 677 23 Lyne, A.G., Shemar, S.L., Graham-Smith, F., 2000, MNRAS, 315, 534 24 Shabanova, T.V., 2005, MNRAS, 356, 1435 3. Two-component Model: Theory and History 56 respectively. In summary, the two-component model is the oldest model still in use for explaining the glitch phenomenon. It has been scrutinised and revised numerous times, taking on various guises, with a more complex model being created each time, such as some of those mentioned in Chapter 2. However, the basics of the model have remained the same; there exist two separate rotating components in a neutron star, which are coupled to one another by some coupling parameter. Theoretically, consensus on the composition and proportion of the two components hasn’t been reached yet. Neither has there been agreement on the method of coupling between the two components, although vortex pinning and unpinning is the prime candidate. Observationally, there has been debate over the agreement of the model with the observational data of some glitches. This has, however, been the exception rather than the rule. For most glitches analysed, the data could be explained using an exponential fit, based on the two-component model. In addition, the experiments mentioned above which were performed by Tsakadze and Tsakadze deserve some careful thought. These experiments consisted of ex- actly the kind of system described by the original two-component model, with a superfluid component interacting with a solid crust via some form of coupling. These experiments yielded results which agreed with observed pulsar glitch data, if scaled-up appropriately. This experimental evidence adds weight to the case for the two-component model’s validity. It is the experimental evidence, both that found in the lab as well as that obtained from glitch data analysis, combined with the simplicity of the model, which preempts the following two sections. In the first section, a series of simulations is shown, displaying the dependence of the model on various parameters. In the second section, the model is compared to actual glitch data. However, it is important to recall the comments made by Downs,25, where he showed that for some of the larger glitches, the two-component model did not provide an adequate fit, as well as those made by Flanagan26 who mentioned evidence of a three component system. These factors will be considered in the analysis presented in the following chapters.

25 Downs, G.S., 1981, ApJ, 249, 687 26 Flanagan, C.S., 1990, Nature, 345, 416 4. SIMULATIONS

4.1 Overview of Simulations

I performed the following simulations using the MATLAB programming language. They are the results of the direct, simultaneous, numerical solution of Equations (3.1) and (3.2), using a Fourth Order Runge-Kutta method.1 The results which follow in this section are those for a simulated, theoretical pulsar. They show the reaction of the two-component theory to a change in various parameters, such as superfluid percentage and coupling constants. The aim of these simulations was to test whether or not the simulation and coding methods which I used were accurate. Hence, the results of these simulations were checked to see whether they gave results consistent with that expected for such a system. In the following simulations, the parameters which were used were chosen to represent a Crab-like pulsar, with a rotation rate of Ω = 189 rad/s, a spin-down rate of Ω˙ = 1 × 10−9 rad/s2 and a glitch size of ∆Ω/Ω = 1 × 10−8. An important note is that the glitch size doesn’t affect the shape of the glitch recovery, but merely the vertical position (or amplitude) of the starting point of the recovery. For each individual simulation presented, the value of α (the external magnetic braking torque on the star) remains constant for the duration of the time span considered (i.e. 60 days).2 There is no observational evidence that the pulsar electromagnetic torque changes during a glitch.3 Figures (4.1) - (4.5) show the exponential post-glitch decay of Ω for the simulated pulsar described above, for varying superfluid percentages and coupling constants. Several observations may be made for the model described above. Firstly, for a constant decay timescale (or coupling constant), and increasing superfluid percentage, the exponential term of the solution begins to dominate, whereas for low superfluid percentage, the linear term is dominant and the exponential term virtually disappears. In fact, for instances where only a very small superfluid component exists, the exponential recovery is indistinguishable from the original spin-down rate. For instances where a large superfluid component is considered, there is a permanent change in spin-down rate after a glitch; the

1 Zill, D., Cullen, M.R., Differential Equations with Boundary-Value Problems, 2001, 5th ed., Brooks/Cole, California 2 Shapiro, S.L., Teukolsky, S.A., 1983, Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects, (Wiley-Interscience) 3 Alpar, M.A., Chau, H.F., Cheng, K.S., Pines, D., 1996, ApJ, 459, 706 4. Simulations 58 pulsar spins down at a slower rate than before the glitch. Secondly, as the value of the coupling constant increases, so does the time taken to reach a constant post-glitch spin-down rate. This is in agreement with the theoretical interpretation of the relaxation timescale (another name for the coupling constant). An interesting point is that this dependence on relaxation timescale is only observable for stars containing a higher percentage of superfluid. The results of the simulations, given by Figures (4.1) - (4.5), show that these simulations agree well with theoretical predictions, and hence, confirm the accuracy of the methods and coding used to perform them. 4. Simulations 59

Fig. 4.1: Results of a Crab-like pulsar simulation. The results are for the post-glitch reaction of the theoretical pulsar. The coupling constant for this set of results is 5 days. The superﬂuid percentages are given by the legend. 4. Simulations 60

Fig. 4.2: Results of a Crab-like pulsar simulation. The results are for the post-glitch reaction of the theoretical pulsar. The coupling constant for this set of results is 10 days. The superﬂuid percentages are given by the legend. 4. Simulations 61

Fig. 4.3: Results of a Crab-like pulsar simulation. The results are for the post-glitch reaction of the theoretical pulsar. The coupling constant for this set of results is 20 days. The superﬂuid percentages are given by the legend. 4. Simulations 62

Fig. 4.4: Results of a Crab-like pulsar simulation. The results are for the post-glitch reaction of the theoretical pulsar. The coupling constant for this set of results is 30 days. The superﬂuid percentages are given by the legend. 4. Simulations 63

Fig. 4.5: Results of a Crab-like pulsar simulation. The results are for the post-glitch reaction of the theoretical pulsar. The coupling constant for this set of results is 50 days. The superﬂuid percentages are given by the legend. 4. Simulations 64

4.2 Fitting Simulations to Observational Data

I once again used the MATLAB programming language to provide numerical solutions to Equations (3.1) and (3.2), using the Fourth-Order Runge-Kutta method. I then compared these simulations to data obtained for various observed glitches. I then fitted the simulations to the data using a least-squares fitting procedure, by varying the following parameters: coupling constant value, number of coupling constants and superfluid percentage. Note that the start of the simulation relative to the glitch epoch is fixed for each fit. The default is to allow the simulation to run from the moment the glitch occurs. However, in some instances it was necessary to change this by making the simulation run some finite amount of time after the glitch epoch in order to obtain a better fit, as will be seen later. The relevance and necessity of each of the variable parameters will be discussed in due course. For each of the figures below, the residual angular velocity (in s−1) is plotted against time (in days). The residual angular velocity is obtained by subtracting the pre-glitch spin-down model from the angular velocities produced by the glitch model. The results of the fitting of the simulations to observational data are given below.

4.2.1 Glitch Occurring in the Crab Pulsar in 1996 The data set used (given by dots in all the relevant figures) is from Wong et al.4 The data set was obtained by observing the glitch which occurred in the Crab pulsar (PSR B0531 +21) in 1996, and then subtracting the pre-glitch spin-down model. The following parameters in the simulation were kept constant during the −1 ˙ −13 −2 fitting procedure for this data set; Ω0 = 29.887774244 s , Ω = 3 × 10 s and ∆Ω/Ω = 10−6 while the time span considered is from the glitch epoch, to 23.6 days later. First, I considered a model containing only one coupling constant. The fit of this simulation to the data is given by Figure (4.6). The least squares fit resulted in a configuration with τ = 8.6 days and the superfluid percentage = 40.7%, with a χ2 value (comparing the simulation with the data) of 3.3097 × 10−14.

4 Wong, T., Backer, D.C., Lyne, A.G., 2001, ApJ, 548, 447 4. Simulations 65

Fig. 4.6: Fit of simulation containing one coupling constant (τ = 8.6 days) to the data for the glitch occurring in the Crab pulsar in 1996. The simulation is given by the line, while the data are given by the dots. 4. Simulations 66

Fig. 4.7: Fit of simulation containing two coupling constants (τ1 = 10.3 days and τ2 = 0.3 days) to the data for the glitch occurring in the Crab pulsar in 1996. The simulation is given by the line, while the data are given by the dots.

Next, I considered a model containing two coupling constants. The fit of this simulation to the data is given by Figure (4.7). The least squares fit resulted in a configuration where τ1 = 10.3 days, τ2 = 0.3 days and the superfluid percentage = 45.4%, with a χ2-value of 2.7505 × 10−14. Hence, the simulation involving two coupling constants is a better fit to the data than that containing only one. The inclusion of additional coupling constants, hence increasing the number of parameters, does not improve the fit significantly, and hence, such instances have been excluded. Something that was observed early on in the simulation process was that, even for the best fitting models, the fit at the beginning of the time series was always very poor, but improved as time evolved. It became apparent that the reason for this was that, although the simulations ran from the instant after a glitch event, the data set starts some finite time after the glitch epoch. The Epoch of the glitch was obtained from Wong et al. (2001). Corrections for this were then made, and the model containing two coupling constants was refitted. The results of this corrected model are shown in Figure (4.9). The least squares fit resulted in a configuration where τ1 = 10.9 days, τ2 = 0.3 days and 4. Simulations 67

Fig. 4.8: Table showing the values obtained by my simulation as well as those obtained by Wong et al. for the 1996 Crab glitch the superfluid percentage = 46.1%, with a χ2 value of 4.9788 × 10−15. This therefore represents an order of magnitude improvement in accuracy over the previous simulations. As a result, these values for τ1,2 and superfluid percentage are taken as the most correct, according to the simulations. The published values given by Wong et al. for the coupling constants are τ1 = 10.3±1.5 days and τ2 = 0.5 days. These values were obtained by fitting two exponentials to the curve, using a semi-empirical model, taking physical processes into account which are more complex than the two-component model, and involve processes such as vortex creep. A comparison of my results with those obtained by Wong et al. is given in the table in Figure (4.8) The results obtained by my simulation of the two-component model agree with the generally accepted values obtained by Wong et al. This builds further confidence that the fairly simplistic two-component model is still a relevant mechanism. 4. Simulations 68

Fig. 4.9: Fit of simulation containing two coupling constants (τ1 = 10.9 days and τ2 = 0.3 days) to the data for the glitch occurring in the Crab pulsar in 1996, with corrected start point. The simulation is given by the line, with data given by dots. 4. Simulations 69

Fig. 4.10: Fit of simulation containing one coupling constant (τ = 9.9 days) to the data for the glitch occurring in the Crab pulsar in 1975. The simulation is given by the line, while the data are given by the dots.

4.2.2 Glitch Occurring in the Crab Pulsar in 1975 The data set used for this comparison was obtained from Alpar et al.5 The data set is that obtained for the glitch which occurred in 1975 in the Crab pulsar, with the pre-glitch spin-down and persistent changes in Ω, Ω˙ and Ω¨ once again removed. Once again, I performed a least squares fit to determine the optimal model parameters to fit the data. First, I considered a model containing only one coupling constant. The fit of this simulation to the data is given by Figure (4.10). The least squares fit resulted in a configuration with τ = 9.9 days and the superfluid percentage = 90.2%, with a χ2 value (comparing the simulation with the data) of 2.9813 × 10−12. Simulations for more than one coupling constant didn’t improve the fit significantly. Hence, the model containing only one coupling constant is considered sufficient to describe the data. The fit obtained by Alpar et al. is provided in Figure (4.12), and was obtained

5 Alpar, M.A., Chau, H.F., Cheng, K.S., Pines, D., 1996, ApJ, 459, 706 4. Simulations 70

Fig. 4.11: Table showing the values obtained by my simulation as well as those obtained by Alpar et al. for the 1975 Crab glitch

Fig. 4.12: Diagram showing the fit obtained by Alpar et al. for the 1975 Crab Pulsar Glitch by fitting a semi-empirical model, which takes into account the fairly complex physical processes of vortex creep and crust cracking, to the data. The parameter values obtained by Alpar et al. were τ1 = 9.7 days as well as a second coupling constant value of τ2 = 190 days. This second coupling constant was not discernable using the simulations performed in my work, but the first one obtained by Alpar et al. agrees very well with that obtained by simulation. The results obtained from my simulation as well as those obtained by Alpar et al. are found in the table in Figure (4.11). This once again instills confidence in the two-component model’s relevance, as the results obtained in my simulation of the model agree very closely with the results obtained using Alpar’s more complex model. 4. Simulations 71

4.2.3 Simulation Summary The results obtained by my simulations for the Crab pulsar, as well as other literature values, are summarised in the table in Figure (4.13). As can be seen by the table, the values for the coupling constants in the Crab pulsar are fairly consistent at ≈ 10 and ≈ 0.5 days respectively, irrespective of the complexity of the physical processes taken into account during the fitting of the data. Still, the values of these constants show significant variation about their average values, from glitch to glitch (e.g.: the values of τ2 for the Crab pulsar range between 6.6 and 14.4 days) The short timescale coupling constant is not mentioned in the Alpar et al. analysis of the 1975 Crab glitch. However, the paper by Alpar et al. (1996) gives the post-glitch parameter fittings of three more Crab glitches, all of which show the short timescale coupling constant. Two of these other three fits also show a long timescale coupling constant similar to that obtained for the 1975 glitch. Unfortunately, time series data for these other three glitches could not be obtained in time to be included in the simulations performed in this thesis. In order to explore the possible origin of the coupling constants further, we consider a collection of glitches for which coupling constants have been fitted, found in literature.6,7,8,9,10,11,12,13,14,15,16 I created two different plots using these values. The first is given in Figure (4.14) and shows the distribution of coupling constants as a function of pulsar characteristic age. Initially, it appeared that there may be some sort of trend, whereby the coupling constant increased sharply with age, and then decreased exponentially. The lack of availability of fitted coupling constant data made it difficult to determine whether this is in fact the case or not. However, Figure (4.15) shows a magnification of the low characteristic age region. Note that there are three types of data points in this figure; glitches which occurred in the Crab pulsar, glitches which occurred in the Vela pulsar and glitches which occurred in other pulsars. This figure shows that there is in

6 Boynton, P.E., Groth, E.J., Partridge, R.B., Wilkinson, D.T.,(1971), The Crab Nebula, Davies and Smith (eds), (The IAU) 7 Lyne, A.G., Pritchard, R.S., 1987, MNRAS, 229, 223 8 Wong, T., Backer, D.C., Lyne, A.G., 2001, ApJ, 548, 447 9 Alpar, M.A., Chau, H.F., Cheng, K.S., Pines, D., 1996, ApJ, 459, 706 10 Flanagan, C.S., 1990, Nature, 345, 416 11 Lyne, A.G., Kaspi, V.M., Bailes, M., Manchester, R.N., Taylor, H., Arzoumanian, Z., 1996, MNRAS, 281, L14 12 Downs, G.S., 1982, ApJ, 257, L67 13 Kaspi, V.M., Manchester, R.N., Johnston, S., Lyne, A.G., D’Amico, N., 1992, ApJ, 399, L155 14 Johnston, S., Manchester, R.N., Lyne, A.G., Kaspi, V.M., D’Amico, N., 1995, A&A, 293, 795 15 Shemar, S.L., Lyne, A.G., 1996, MNRAS, 282, 677 16 Lyne, A.G., Shemar, S.L., Graham Smith, F., 2000, MNRAS, 315, 534 4. Simulations 72

Fig. 4.13: Table showing the coupling constant values obtained from the simulations, as well as published results fact no discernable pattern in the distribution of coupling constants with respect to characteristic age. The second plot created using the coupling constant values found in literature is given in Figure (4.16) and shows the distribution of coupling constants as a function of glitch magnitude. It is obvious from this plot that there is no discernable pattern relating coupling constants to glitch sizes. The aim of this simulations chapter was to determine whether or not the two-component model is sufficient to describe a glitch. In the case of both of the Crab glitches to which the simulation was applied, the two-component model was a good fit. Although many more tests will be required, and in spite of the fact that it has already been established that the two-component model does not fit all pulsar glitches, the good fit obtained in this work, as well as the consistency of the results with published values which were obtained using more complicated physical models, indicate that the two-component model can still provide a good description of glitches, for the Crab pulsar at least. 4. Simulations 73

Fig. 4.14: Graph showing the distribution of coupling constants as a function of characteristic age 4. Simulations 74

Fig. 4.15: Graph showing the distribution of coupling constants as a function of characteristic age magniﬁed to centre on the low characteristic age region 4. Simulations 75

Fig. 4.16: Graph showing the distribution of coupling constants as a function of glitch magnitude 5. CONCLUDING REMARKS

Neutron stars present scientists with some of the most extreme environments known to exist in the universe. The phenomenon of pulsar glitches has long been believed to hold important keys for unlocking the secrets held by these alien bodies. However, one thing that has become clear in this work is that the task of prying these secrets free is no easy one. The first problem faced in this field of study is the inability to recreate the observed environment in a terrestrial laboratory. Not only are the densities involved far too high for any earthly material, but the magnetic fields involved are many orders of magnitude stronger than the largest produced on earth. However, some useful laboratory experiments have been performed by people such as Tsakadze and Tsakadze, who performed superfluid experiments in an attempt to under- stand pulsar glitches better. They determined that the rotation of superfluid matter in the interior of a pulsar plays an integral part in the glitch process. Over the years, there have been many mechanisms proposed in an attempt to explain the glitching phenomenon. These models range in complexity from the fairly simplistic spheroidality and two component mechanisms proposed by Ru- derman and Baym et al. respectively, both in 1969, to fairly complicated models such as Ruderman’s 1998 flux tube model, and Sedrakian and Cordes’ 1999 flux tube annihilation model. However, in spite of a near plethora of seemingly viable mechanisms, there remains no resolution regarding the manner in which glitches occur. One particular problem is the increasing difficulty associated with simulating the increasingly complex physical glitch models. One model that has stood up to many tests is the two-component model which was proposed by Baym et al. in 1969. In the work presented in this thesis, the two-component model was fitted to two glitches observed in the Crab pulsar, yielding remarkably good fits. The coupling constants which were obtained from these fitting procedures were in close agreement with the decay timescales published previously. The interesting thing to note is that the literature values were obtained by purely mathematical methods, while the values obtained in this work came from the fitting of a physical model, albeit a simplistic one. This agreement indicates that for the Crab pulsar, the two-component model provides a good description of the glitch process. In addition, if the start point of the simulation relative to the glitch epoch were to be made a variable parameter, one would expect the fit to be improved even further. 5. Concluding Remarks 77

There are still some improvements which can be made. If one were to extend the two-component model to the limit where the thickness of each component is infinitely small, and there are thus an infinite number of components, by the use of a suitably derived differential equation, one would expect to obtain results which are more accurate than those given by the original two-component model. One question that still needs to be answered is, how well does the two- component model fit the glitches found in other pulsars? Unfortunately, timing data for other pulsars were not available for analysis in this thesis, so this could not be tested. The suitability of the two-component model to all glitching pulsars has been questioned in the past. In fact, the concern that has been raised has not been whether or not the two-component model applies, but whether it is sufficient in its simplistic form to describe glitches in all pulsars. This has been the root from which many of the more complicated models have sprung. This thesis was intended to provide an overview of the various glitch models which have been proposed in the past, and to determine whether or not the two- component model still has a part to play in the modern era of pulsar glitch mechanisms. The results obtained in this work have indicated that it most certainly does. However, this mechanism still does not describe pulsar glitch mechanisms in their entirety. Given the increasing complexity of today’s mechanisms and the corresponding difficulty involved in simulating them, can a testable mechanism be found that can conclusively describe pulsar glitch mechanisms, something that has remained so elusive to date? Only time and more research will tell. 6. REFERENCES

• Alpar, M.A., Anderson, P.W., Pines, D., Shaham, J., 1981, ApJ, 249, L29 • Alpar, A., Anderson, P.W., Pines, D., Shaham, J., 1984, ApJ, 276, 32 • Alpar, M.A., Pines, D., Cheng, K.S., 1990, Nature, 348, 707 • Alpar M.A., Pines D., in Van Riper K.A., Epstein R.I., Ho C., eds, 1993, Iso- lated Pulsars, Cambridge Univ. Press, Cambridge, p.18 • Alpar, M.A., Chau, H.F., Cheng, K.S., Pines, D., 1994, ApJ, 427, L27 • Alpar, M.A., Chau, H.F., Cheng, K.S., Pines, D., 1996, ApJ, 459, 706 • Andersson, P.W., Itoh, N., 1975, Nature, 256, 25 • Andersson, N., Comer, G.L., Prix, R., 2002, MNRAS, 354, 101 • Andronikashvili, E. L., 1946, Zh. eksp. theor. Fiz., 16, 780 • Baade, W., Zwicky, F., 1934, Phys. Rev. 45, 138 • Baym, G., Pethick, C., Pines, D., Ruderman, M., 1969, Nature, 224, 872 • Baym G., Pines D., 1971, Ann. Phys., 66, 816 • Baym, G., Chandler, E., 1983, J. Low Temp. Physics, 50, 57 • Boynton, P.E., Groth, E.J., Partridge, R.B., Wilkinson, D.T.,(1971), The Crab Nebula, Davies and Smith (eds), (The IAU) • Boynton, P.E., 1981, Pulsars, Sieber, W., Wielebinski, R. (eds), The IAU, 279 • Carter B., Langlois D., Sedrakian D.M., 2000, A&A, 361, 795 • Cheng, A.F., Ruderman, M.A., 1977 ApJ, 212, 800 • Cheng H.F., Pines D., Alpar M.A., Shaham J., 1988, ApJ, 330, 835 • Cordes, J., Stinebring, D., 1984, ApJ(Letters), 277, 53 • Cordes, J.M., Downs, G.S., 1985, ApJ Supplement Series, 59, 343 • Crawford, F., Demiansky, M., 2003, ApJ, 595, 1052 • Demiańsky, M., Prózyński,M., 1983, MNRAS, 202, 437 • Donati, P., Pizzochero, P.M., 2004, Nucl. Phys. A, 742, 363 • Downs, G.S., 1981, ApJ, 249, 687 • Downs, G.S., 1982, ApJ, 257, L67 • Drazin, P.G., Reid, W.H., 1981, Hydrodynamic Instability (Cambridge: Cam- bridge University Press) • Fetter, A.L., 1967, Phys. Rev., 162, 143 • Flanagan, C.S., 1990, Nature, 345, 416 • Franco, L.M., Link,B., Epstein. R.I., 2000, ApJ, 543, 987 • Friedman, P., Pandharipande, V.R., 1981, Nucl. Phys. A, 361, 502 • Fruchter,A., Stinebring, D., and Taylor, J., 1988, Nature, 88, 237 • Ginzburg, V.L., Kirzhnits, D.A., 1964, Zh. Eksp. Teor. Fiz., 47, 2006 6. References 79

Gold,T., 1968, Nature, 218, 731 • Greenstein, G., 1979, ApJ, 231, 880 • Hall, H.E., Vinen, W.F., 1956, Proc. Roy. Soc. A, 238, 204, 215 • Harling, O.K., 1970, Phys. Rev. Lett., 24, 1046 • Heiselberg, H., Pandharipande, V., 2000, Ann. Rev. Nucl. Part. Phys., 50, 481 • Hewish, A., Speech delivered upon receiving the Nobel Prize in 1974 • Hewish, A., Bell, S.J., Pilkington, J.D.H., Scott, P.F., Collins, R.A., 1968, Na- ture, 217, 709 • Johnston, S., Manchester, R.N., Lyne, A.G., Kaspi, V.M., D’Amico, N., 1995, A&A, 293, 795 • Kapitza, P.L., 1938, Dk.Akad.Nauk SSSR, 18, 21 • Kaspi, V.M., Manchester, R.N., Johnston, S., Lyne, A.G., D’Amico, N., 1992, ApJ, 399, L155 • Landau L.D., 1932, Phys. Z. Sowietunion, 1, 285 • Landau, L.D., Lifshitz, E.M., 1987, Fluid Mechanics, 2nd Edition, (Pergamon Press) • Larson, M.B., Link, B., 1999, ApJ, 521, 271 • Larson, M.B., 2001, DPhil. Thesis, Superfluid Effects on Thermal Evolution and Rotational Dynamics of Neutron Stars, Montana State University • Lattimer, J.M., Prakash, M., 2000, ApJ, 550, 426 • Link, B.K., Epstein, R.I., 1991, ApJ, 373, 592 • Link, B., Epstein, R.I., Baym, G., 1992, ApJ, 390, L21 • Link, B.K., Epstein, R.I., Baym, G., 1993, ApJ, 403, 285 • Link, B., K., Epstein, R., I., 1996, ApJ, 457, 844 • Link, B., Franco, L.M., Epstein, R.I., 1998, ApJ, 508, 838 • Lorenz, C.P., Ravenhall, D.G., Pethick, C.J., 1993, Phys.Rev.Lett., 70, 379 • Lovelace, R.V.E., Jore, K.P., Haynes, M.P., 1997, ApJ, 475, 83 • Lyne, A.G., Pritchard, R.S., 1987, MNRAS, 229, 223 • Lyne, A.G., Kaspi, V.M., Bailes M., Manchester, R.N., Taylor, H., Arzouma- nian, A., 1996, MNRAS, 281, L14 • Lyne, A.G., Shemar, S.L., Graham-Smith, F., 2000, MNRAS, 315, 534 • Manchester, R.N., Newton, L.M., Goss, W.M., Hamilton, H.A., 1978, MNRAS, 184, Short Communication, 35 • Michel, F.C., Bland Hawthorn, J., Lyne, A.G., 1990, MNRAS, 246, 624 • Miller, M.C., 1992, MNRAS, 255,129 • Mochizuki Y., Izuyama T., 1995, ApJ, 440, 263 • Negele, J.W., Vautherin, D., 1973, Nucl. Phys. A, 207, 298 • Oppenheimer, J.R., Volkoff, G., 1939, Phys. Rev., 55, 374 • Osborne, D.V., 1950, Proc. Phys. Soc. (London), A63, 909 • Packard, R., E., 1972, Phys. Rev. Lett., 28, 16 • Pethick, C.J., Ravenhall, D.G., 1991, NYASA, 647, 503 • Pethick, C.J., Ravenhall, D.G., Lorenz, C.P., 1995, Nucl. Phys. A, 584, 675 6. References 80

• Pilkington, J.D.H., Hewish, A., Bell, S.J., Cole, T.W., 1968, Nature, 218, 126 • Pines, D., Shaham, J., Ruderman, M.A., (1974), Physics of Dense Matter, Hansen, C.J., (ed.), (The IAU) • Radhakrishnan, V. and Manchester, R., 1969, Nature 222, 228 • Rajagopal, M., Romani, R.W., Miller, M.C., 1997, ApJ, 479, 347 • Rhodes, C.E., Ruﬃni, R., 1974, Phys. Rev. Lett. 32, 324 • Ruderman, M., 1969, Nature, 223, 579 • Ruderman. M., 1976, ApJ, 203, 213 • Ruderman, M., 1991, ApJ, 366, 261 • Ruderman M., Zhu T., Chen K., 1998, ApJ, 492, 267 • Sedrakian A., Cordes J.M., 1999, MNRAS, 307, 365 • Segelstein,D., Rawley, L., Stinebring, D., Further, A., and Taylor, J., 1986, Nature, 323, 714 • Shabanova, T.V., 2005, MNRAS, 356, 1435 • Shapiro, S.L., Teukolsky, S.A., 1983, Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects, (Wiley-Interscience) • Shemar, S.L., Lyne, A.G., 1996, MNRAS, 282, 677 • Smoluokowski, R., Welch, D., 1970, Phys. Rev. Lett., 24, 1191 • Srinivasan,G., Battacharya, D., Muslimov, A., Tsygan, A., 1990, Current Sci., 59, 31 • Thorsett S.E., Chakrabarty D., 1999, ApJ, 512, 288 • Tisza, L., 1938, Nature, 141, 913 • Troup, G.J., 1967, Optical Coherence Theory - Recent Developments, (London: Methuen) • Tsakadze, J.S., Tsakadze, S.J., 1972, Phys. Lett. A, 41A, 197 • Tsakadze, J.S., Tsakadze, S.J., 1980, Journ. Low Temp. Phys., 39, 649 • Vinen, W.F., 1958, Nature, 181, 1524 • Wheeler, J. A., 1966, Ann. Rev. Astron. and Astrophys. 4, 428 • Wong, T., Backer, D.C., Lyne, A.G., 2001, ApJ, 548, 447 • Xu, R. X., 2003, ACTA Astronomica Sinica, 44, 245 • Zill, D., Cullen, M.R., Diﬀerential Equations with Boundary-Value Problems, 2001, 5th ed., Brooks/Cole, California