Crust cooling curves of transiently accreting neutron stars

Transcription

1 Crust cooling curves of transiently accreting neutron stars Probing the neutron star interior by Nathalie D. Degenaar Master thesis Astronomy and Astrophysics supervisor: dr. Rudy Wijnands Astronomical Institute Anton Pannekoek Faculty of Science (FNWI) University of Amsterdam

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3 Table of contents Preface Abstract v vii 1 Introduction Neutron stars X-ray binaries Mass transfer Neutron star or black hole? Low-mass X-ray transients Disk instability model Quiescent X-ray properties Quasi-persistent X-ray transients This master thesis Neutron star interior Key concepts Gravitational mass Degeneracy Equation of state Neutronization and the neutron drip density Superfluidity General structure Atmosphere Crust Core Mass-radius relation Accreting neutron stars Crust composition Crustal heating i

4 Table of contents 3 Neutron star cooling General cooling scenarios Neutrino emission from the core Neutrino emission from the crust Photon emission Effects of Superfluidity Cooling models Cooling of accretion heated neutron stars Heating Observed cooling curves Theoretical cooling curves Analytical results Crustal neutrino emission Sources of heat loss and heat gain Heat capacity Results Crust thickness Analytical model Connection with observable quantities Comparison with numerical calculations Implications for the observable luminosity Discussion Neutrino emission from Cooper pairs in the crust Analytical crust model Theoretical cooling curves Observational outlook Conclusion A Neutron star masses 75 A.1 Maximum mass A.2 Minimum mass B Particle terminology and data 77 C Observations of KS and MXB D Numerical Integration 83 D.1 Runge-Kutta method D.2 Adaptive stepsize ii

5 Table of contents E Exponential decay of the luminosity and interior temperature 87 Acknowledgements 91 Samenvatting 93 iii

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7 Preface This thesis is the condensation of approximately a year of theoretical work I did for my master project in the X-ray/High-energy astrophysics group of the Astronomical Institute Anton Pannekoek in Amsterdam. During this project I investigated how theoretical predictions for cooling curves of neutron stars in quasi-persistent X-ray binaries can be employed to gain a better understanding of the composition of neutron stars. I did a few calculations that fit into this framework. Chapter 1 of this thesis introduces neutrons stars and X-ray binaries and the various manifestations. This puts the neutron star systems I investigated into perspective, and fades into the formulation of my research goals. Chapter 2 gives background information on neutron star interiors, dealing with concepts that are necessary to understand the cooling of neutron stars, which will be extensively discussed in Chapter 3. In Chapter 4, I present analytical calculations that explore to what extent two specific effects are important for cooling of neutron star crusts. Chapter 5 gives a discussion of my findings as well as an outlook for future research. Various appendices are added in order to serve as a reference for the more technically interested reader. There are a lot of open questions concerning cooling neutron stars and much work to be done. I hope to continue working on cooling of accretion heated neutron stars during a part of my upcoming phd in Amsterdam. I have tried to write my thesis in such a way that it may serve as an introduction for students who want to be involved in research involving cooling neutron stars. Nathalie Degenaar Amsterdam, August 29, 2006 v

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9 Abstract There are many reasons to study neutron stars, one of which is to probe the behavior of matter under extreme densities. This thesis is dedicated to the subject of cooling neutron stars, a promising tool for deducing neutron star interior parameters. Not only isolated neutron stars serve for this purpose, but also a special group of transiently accreting neutron stars, denoted as quasi-persistent low-mass X-ray binaries (LMXBs). Currently, only two quasi-persistent LMXBs have been fully monitored in quiescence following an outburst, but there are several candidates to go into quiescence in the near future. If so, these sources can be monitored in order to expand the data on quasi-persistent X-ray transients and confront these observations with theory. Cooling models for this class of sources have already been developed, but these need fine-tuning before one can draw firm conclusions on the interior physics of neutron stars, by comparing these models with observed cooling curves. In this thesis I present two analytical calculations that are concerned with improving current theories. The importance of crustal neutrino emission resulting from the formation of neutron Cooper pairs, which is disregarded in existing cooling studies, is considered using an analytical model of the neutron star crust. The present calculations show that this crustal emission mechanism is not important for the thermal evolution of a neutron star crust. Furthermore, it is shown that the characteristic cooling timescale of a neutron star crust can be directly related to neutron star properties, using a simple analytical model for the thermal evolution. In particular, the results show that the neutron star compactness has a significant effect on the crustal cooling curves of transiently accreting neutron stars. This allows for a different interpretation of observed cooling curves and calls for new, more detailed numerical simulations for a further exploration of this effect. vii

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11 CHAPTER 1 Introduction 1.1 Neutron stars Neutron stars origin from the collapse of massive stars ( 8 M ) and represent one of the possible final stages of stellar evolution. Unlike regular stars, neutron stars do not burn nuclear fuel in their interior and as a consequence do not generate thermal pressure to support themselves against gravitational collapse. Instead, degenerate neutrons (see Section 2.2) supply the pressure to withstand the inward pull of gravity. Another profound feature of neutron stars is their exceedingly small size relative to normal stars of comparable mass. Neutron stars are thought to have a mass of roughly 1.4 M, while their corresponding radius is assumed to be about 10 km. Neutron stars are therefore denoted as compact objects, a group of stellar bodies that is thought to be supplemented with white dwarfs, black holes and possibly also quark-/strange stars. The compactness implies that matter in neutron stars is compressed to very large densities, exceeding the nuclear saturation density of ρ g cm 3. On Earth, we do not encounter matter at densities ρ > ρ 0 in nature and, neither have we succeeded to create matter at supra-nuclear densities in accelerator experiments. Nevertheless, theoretical models describing the behavior of matter at extreme densities can be tested through neutron stars. The compactness of neutron stars also entails that these objects inhabit immense gravitational fields. Neutron stars therefore provide a test for Einstein s Theory of General Relativity at large gravitational fields, much higher than the fields encountered in our Solar System. Another prime reason to study neutron stars is that the exceptional circumstances around these objects allow for highly energetic processes. The energies involved are far beyond the experimental abilities on Earth, but neutron stars provide a natural laboratory to survey these extremes. As sketched above, the study on neutron stars can increase our knowledge in a broad range of fields e.g., general relativity, stellar evolution, matter at extreme densities and high energy 1

12 Chapter 1: Introduction physics. During my master project I explored a method that should allow us to gain insight in the exact structure and composition of neutron stars and thus on the behavior of matter at supra-nuclear densities. This first introductory chapter describes the scientific context of my studies. 1.2 X-ray binaries Neutron stars come in different varieties; the majority of known neutron stars are observed as radio pulsars, but dim isolated neutron stars, magnetars (consisting of anomalous X-ray pulsars and soft gamma repeaters), central compact objects in supernova remnants and socalled rotating radio transients form other possible types. Neutron stars also occur in binary systems, manifesting themselves as double neutron star systems, in which one or both act as a radio pulsar, or as X-ray binaries. X-ray binaries are binary star systems in which a neutron star or a black hole accretes matter from a companion star. This mass transfer gives rise to the emission of X-rays (see Section 1.2.1). X-ray binaries are generally subdivided into two main groups, depending on the mass of the companion star. In low-mass X-ray binaries (LMXBs), the companion is less massive than 1M, and usually a star of late spectral type (K/M). However, LMXBs with more exotic companions such as white dwarfs, brown dwarfs or maybe planets also occur. X-ray binaries that contain a low-mass companion are characterized by circular orbits with short periods (hours to days), which indicates that these systems are old. This hypothesis is supported by the fact that LMXB systems in our Milky Way galaxy have the same distribution as the old stellar populations (see Figure 1.1). The opposing high-mass X-ray binaries (HMXBs) contain a massive star, usually exceeding 10M, of early spectral type (O/B) next to the primary neutron star or black hole. These systems typically show eccentric orbits with long periods (weeks/years) and are distributed along the Milky Way similar to star-forming regions (see Figure 1.1). This suggests that HMXBs are young star systems. Figure 1.1: Distribution of X-ray binaries in the Milky Way. In total 86 LMXBs (open circles) and 52 HMXBs (filled circles) are shown. HMXBs are concentrated towards the Galactic Plane while LMXBs extend into the Galactic Bulge. This figure is taken from Grimm et al. (2002). 2

13 1.2 X-ray binaries Mass transfer In X-ray binary systems, mass transfer from the donor star to the compact object can occur through Roche-lobe overflow (most profound in LMXBs), by way of wind accretion (in HMXBs) or due to capture of matter from a circumstellar decretion disk in HMXBs with a Be spectral type companion. Roche-lobe overflow In order to understand mass transfer through Roche-lobe overflow, consider a matter particle that co-rotates with the orbital motion of a binary system that contains stars with masses M 1 and M 2. The test particle will experience an effective potential that can be described by φ = GM 1 r r 1 GM 2 r r ( Ω B r) 2 (1.1) where G is Newton s gravitational constant, r is the position vector of the test particle, r 1 and r 2 are the position vectors of the two binary stars (see Figure 1.2) and Ω B is the angular velocity vector of the binary system. The first two terms in Equation 1.1 represent the individual gravitational potentials of the two binary stars and the last term describes the centripetal potential that arises from the orbital motion. Test particle r-r1 r r-r 2 M 1 r1 Center of mass r2 M 2 Figure 1.2: Schematic overview of the position vectors relevant for a test particle in a binary star system. Both a two- and a three-dimensional representation of the potential described by Equation 1.1 are given in Figure 1.3. The effective potential can be described by equipotential surfaces, which is shown in Figure 1.3c. Along these surfaces the effective potential experienced by a test particle is constant, i.e. φ = 0. The volumes within the equipotential surfaces that are connected by the point L 1 (see Figure 1.3c) are called the Roche-lobes of the stars. Within each Roche-lobe, matter is solely bound to the central star, while outside the lobes the effective potential is more complex. Figure 1.3d gives a three-dimensional representation of the Roche lobes of two binary stars. The orbital plane of a binary system contains 3

14 Chapter 1: Introduction five special points like L 1, which are called Lagrange points. In these special points, the three forces acting on a test particle cancel out. This implies that matter located at the inner Lagrange point, L 1, can freely transfer from one star to another, due the net cancellation of the acting forces. In this framework, if either star fills its Roche-lobe completely, matter will transfer from the Roche-lobe filling star through the inner Lagrange point L 1 to the other star in a process known as Roche-lobe overflow (Figure 1.4). This process can occur when a star expands (e.g., becomes a giant star) and fills its Roche-lobe or when the Roche-lobe of a star shrinks past the star s surface due to contraction of the orbital period. Roche-lobe overflow is the most important mass transfer mechanism for LMXBs and may also occur in some HMXBs (Savonije, 1979). Due to conservation of angular momentum, matter transferring from the rotating companion star does not directly fall onto the primary star. The infalling gas forms an accretion disk (see Figure 1.4) where it diffuses towards the central body by losing its angular momentum through viscous interactions. The release of large amounts of gravitational energy (up to erg s 1 in neutron star LMXBs and up to erg s 1 in black hole LMXBs) effectively heats the accretion disk up to temperatures of 10 7 K at the inner end. The thermal emission from the hot inner accretion disk can be observed in soft X-rays. In LMXBs that contain a neutron star, X-rays are also emitted when gas from the disk eventually reaches the neutron star surface, which results in the conversion of gravitational energy into heat and radiation. Wind accretion The majority of HMXBs contain a companion star that lies within its Roche-lobe, making mass transfer through Roche-lobe overflow impossible. However, massive stars ( 10 M ) naturally produce strong stellar winds (average mass loss rates of M yr 1 ). In HMXBs, the fraction of the stellar wind that passes the compact object within a cylinder of radius r acc will be gravitationally captured. This is illustrated by Figure 1.5. The size of the accretion radius r acc is determined by the wind velocity, v w, of the companion and by the surface gravity and the orbital velocity, v cs, of the compact object. Near the compact object, a test particle with mass m (moving with the stellar wind), has a kinetic energy E kin = m(v w + v cs ) 2 /2, while its potential energy in the gravitational field of the compact object will be E grav = GMm/r acc, where M is the mass of the compact object. Only when its potential energy is larger than its kinetic energy, will the test particle be gravitationally captured by the compact object. Thus, an approximation for the accretion radius can be found by equating this two energy term, which results in r acc 2GM (v w + v cs ) 2 (1.2) 4

15 1.2 X-ray binaries Potential y x a) c) x b) d) Figure 1.3: Roche potential in a binary system, containing stars of mass ratio 2:1, as experienced by a co-rotating test particle. a) Two-dimensional plot. b) Representation in three dimensions. c) Equipotential plot. Net forces cancel out in the Lagrange points, denoted by capital L s. d) A threedimensional visualization of equipotential surfaces through L1, the Roche-lobes of the binary stars, and L2. The specific angular momentum captured from a stellar wind is thought to have a much lower magnitude than for matter accreted through Roche-lobe overflow (Davies and Pringle, 1980). This makes it marginal whether or not an accretion disk will form in HMXBs. A few sources are inferred to have an accretion disk, but it is unclear whether a disk is present in all wind-accreting X-ray binaries. Wind accretion is not very efficient in the sense that most of the stellar wind escapes from the system. However, because of the large mass-loss rates of early type stars, even a small capture fraction produces a significant X-ray luminosity. 5

16 Chapter 1: Introduction L1 Accretion disk X-rays Roche-lobes Compact object Low-mass companion fills Roche lobe Figure 1.4: Schematic drawing of mass transfer due to Roche-lobe overflow in a LMXB. Due to conservation of angular momentum, an accretion disk is formed. This disk is heated to very high temperatures, which results in the emission of thermal X-rays. Accretion of circumstellar matter Binary systems in which the companion is a Be spectral type star represent the largest subclass of HMXBs. In this type of system, accretion is rather different from the previous described mechanisms. Be stars exhibit spectral emission lines, which are attributed to a circumstellar gaseous disk in the equatorial plane of the star (see e.g., Porter and Rivinius, 2003). Most currently known Be/X-ray binaries contain a neutron star in a wide and very eccentric orbit around its companion. These systems are generally faint in X-rays ( erg s 1 ), but show periodical X-ray brightening (L x erg s 1 ) close to the time of periastron passage of the neutron star. The rise in X-ray luminosity presumably results from accretion when the neutron star penetrates the expelled matter ring of its companion near periastron. This scenario is schematically pictured in Figure Neutron star or black hole? The accretion properties (e.g., the accretion efficiency or the size of the accretion disk) concerning neutron stars and black holes are very similar. This makes it difficult to deduce from observations whether an X-ray binary contains a neutron star or a black hole. There are however two observable X-ray phenomena, X-ray pulsations and X-ray bursts, that are characteristic for neutron stars and do not occur for black holes. If one of these two phenomena is observed, the X-ray binary indisputably contains a neutron star, because both phenomena require a surface area in order to occur. 6

17 1.2 X-ray binaries Stellar wind X-rays racc Wind capture Roche-lobes Compact object Massive companion Figure 1.5: Schematic drawing of mass transfer due to wind accretion in a HMXB. The fraction of the wind that passes the compact object within a cylinder of radius r acc will be accreted. In some cases this accretion mechanism may result in the formation of an accretion disk, but it is unclear whether this is common in HMXBs. Neutron star Direction of motion X-rays Stellar wind Periastron Eccentric orbit Be companion Circumstellar disk Figure 1.6: Schematic drawing of a HMXB with a Be companion star. Accretion occurs only when the neutron star penetrates the circumstellar disk of the companion star near periastron. X-ray pulsations If an accreting neutron star has a magnetic field larger than G, ionized accreted matter will be directed along the magnetic field lines to the neutron star s magnetic poles (Davidson and Ostriker, 1973). This is illustrated in Figure 1.7. The magnetic poles will be substantially heated due to the continuous infall of matter and become X-ray emitting hot spots. If the magnetic axis does not coincide with the rotation axis of the neutron star, these hot spots can periodically enter and leave our line of sight (depending on our orientation), so that X-ray pulsations can be observed (like a lighthouse beacon). This phenomenon will be most profound in HMXB systems. The older LMXBs tend to contain neutron stars with lower magnetic fields due to magnetic field decay. Strong magnetic field reduction in neutron stars is thought to be accretion induced, although the exact mechanism is not yet understood (see e.g., Bhattacharya and Srinivasan, 1995). To date, over a 100 accreting X-ray pulsars are 7

18 Chapter 1: Introduction known, with pulse periods ranging from a few milliseconds (in LMXBs) to tens of minutes (mostly in HMXBs). X-ray pulsations are apparent in the time domain (X-ray lightcurves) as well as in the frequency domain (power-density spectrum), as is illustrated in Figure 1.8. Rotation axis Magnetic axis Magnetic field lines Infalling gas Accretion disk Hot spot (out of sight) Hot spot emits X-rays Figure 1.7: Schematic picture of the mechanism that causes X-ray pulsations in accreting neutron stars. Figure 1.8: Upper plot: Chandra X-ray lightcurve ( kev) of IGR J , taken from Patel et al. (2004). This source shows X-ray pulses with a clear 5880 second modulation. Bottom figure: Power-density spectrum of millisecond pulsar XTE J , showing a prominent pulsar spike at approximately 300 Hz (Wijnands, 2004a). X-ray bursts When a neutron star accretes material from its stellar companion, nuclear fuel consisting mainly of hydrogen and helium falls onto the neutron star surface. As the accreted layer builds up, the temperature and pressure in the layer can become high enough for spontaneous nuclear 8

19 1.2 X-ray binaries burning to occur (Lewin et al., 1993). A schematic overview of this process is represented in Figure 1.9. The thermonuclear explosion will result in an observable short (a few seconds to a few minutes) brightening of the X-ray luminosity, denoted as a type-i X-ray burst. About a 100 known X-ray binaries are showing these X-ray bursts. An example of a lightcurve featuring X-ray bursts is represented in Figure Some neutron stars show so-called superbursts. These are rare, extremely energetic and long-duration type-i X-ray bursts, which are though to origin from the ignition of carbon well below the neutron star surface (Cumming and Bildsten, 2001). Accreted layer Accretion disk Neutron star Temperature and pressure high enough Thermonuclear burst Accretion disk Neutron star X-rays Figure 1.9: Schematic diagram of the process that results in an X-ray burst. Mass determination A lack of observed X-ray pulsations or X-ray bursts does not automatically imply that a binary contains a black hole. Another test to discover the identity a compact object in an X-ray binary is to determine the mass of the compact object. The theoretical mass range of neutron stars is restricted and depends fully on the microphysics of the interior. Because of our poor understanding of the behavior of matter at supra-nuclear densities, the maximum neutron star mass is unconstrained (as will be discussed in Chapter 2). However, there does exist a strict upper limit for neutron star masses of 3 M above which causality is violated, regardless of details of the composition in the high density-regime (Rhoades and Ruffini, 9

20 Chapter 1: Introduction Figure 1.10: X-ray lightcurve of MXB in August 1985 from Haberl et al. (1987). The regular pattern of steep rises in X-ray luminosity represents X-ray bursts. The right panel shows an enlargement of two of the seven bursts (Lewin et al., 1993). 1974). In general, compact objects in X-ray binaries with determined masses above 3 M are identified as black holes candidates rather than neutron stars. See Appendix A for a derivation of this upper mass limit for neutron stars, as well as an informative note on their lower mass limit. 1.3 Low-mass X-ray transients Now let s focus on the X-ray binaries with a low-mass companion. The group of LMXBs can be further divided into subclasses, based on differences in their X-ray luminosity properties. The persistent LMXBs typically show an X-rays luminosity of about erg s 1 and in addition the neutron star systems can show occasional bright X-ray bursts. The X- ray luminosity of these sources is due to a continuous accretion from the companion star onto the compact object through Roche-lobe overflow. Apart from the common persistent sources, there exists another class of LMXBs called X-ray transients. These sources are characterized by a relatively low luminosity of erg s 1, but sporadically show a substantial X-ray brightening, lasting weeks to months. Figure 1.11 displays typical examples of lightcurves of X-ray transients. During outbursts, the X-ray properties of the transients cannot be distinguished from that of the persistent sources (see e.g., Campana et al., 1998). The outbursts in transient systems are thus very likely due to accretion of matter onto the compact object. The time between subsequent outbursts is of the order of years to decades and this period of low luminosity is denoted as the quiescent state of the X-ray transient. An artist impression of the two distinct phases of a low-mass X-ray transient is given in Figure The transient behavior of these X-ray sources is thought to be attributed to a non-continuous accretion caused by thermal-viscous instabilities in the accretion disk. 10

21 1.3 Low-mass X-ray transients XTE J MXB MXB Figure 1.11: X-ray lightcurves obtained with the all sky monitor aboard the Rossi X-ray Timing Explorer of the X-ray transients XTE J (top), MXB (middle) and MXB (bottom). It is clear that a large variety in outburst durations and frequencies is observed for transient systems. MXB belongs to a special group a X-ray transients, which spend an unusually long time in outburst and are denoted as quasi-persistent systems (see Section 1.3.3). This Figure is taken from Wijnands (2004b) Disk instability model According to the disk instability model, an accretion disk is stable both when the disk is hot, and hydrogen is fully ionized, as well as when the disk is cool and hydrogen is in a neutral state. In persistent LMXBs, the mass transfer rate from the donor star to the neutron star is fast enough to ensure that the accretion disk is stable in a hot and fully ionized state. However, when the mass transfer occurs at a low rate within a certain critical range, the accretion disk fills only slowly and the relocated gas resides in the cool outer region of the disk, far away from the neutron star. As matter accumulates in the disk, both the surface density and the temperature increase. When the surface density reaches an upper critical value, a thermal instability sets in. This instability arises from the steep temperature dependence of the opacity in a partially ionized accretion disk. The disk quickly jumps to a hot state, which causes rapid infall of matter onto the central star and hence an X-ray outburst. While in outburst, matter is transferred onto the neutron star at a rate faster than can be supplied by the donor star, causing the disk to empty quickly. During quiescent episodes, the accretion disk slowly fills again. Thus for some critical mass transfer rates, the accretion disk undergoes a thermal limit cycle, oscillating between a hot, ionized state (outburst) and a cold, neutral state (quiescence). The paragraph above sketches the disk instability model in its basic form. It was originally developed to explain the transient behavior of dwarf novae (binary star systems that consist 11

22 Chapter 1: Introduction Companion star Jet Disk wind Residual accretion or thermal emission? Accretion stream Accretion disk Figure 1.12: Artist impression of the two different states of an X-ray transient. Left the outburst phase in which transients are indistinguishable from persistent X-ray binaries with respect to their X-ray properties. Right the quiescent state in which transients have a relatively low X-ray luminosity. The left hand side of this artist impression shows relativistic jets emerging from the compact object. Some X-ray binaries, most notably those hosting a black hole candidate, show this jet phenomenon which is observable at different wavelengths. These possibly have the same underlying physics as jets emerging from Active Galactic Nuclei. Jets are however not important for the scope of this master thesis and are therefore not discussed (for a review on jets in X-ray binaries see e.g., Migliari and Fender, 2005). Figure credits: Rob Hynes, rih. of a white dwarf and a low-mass, M < 8M, companion star). LMXBs and dwarf novae show many similar characteristics and the analogy has led to the interpretation of the outbursts of X-ray transients as being due to a thermal-viscous instability in the accretion disk. Complemented by additional physical mechanisms like, for example, mass-transfer variations and disk irradiation, the disk instability model is successful in explaining many of the characteristics of eruptions from dwarf novae and low-mass X-ray transients. However, unanswered questions and detailed problems currently remain, see e.g., Lasota (2001) Quiescent X-ray properties Despite the low X-ray luminosity in quiescence, transients can still be detected with sensitive imaging instruments like the X-ray satellites Chandra and XMM-Newton. According to observations, the quiescent X-ray spectra of neutron star X-ray transients can consists of two components; a soft, thermal component that dominates below a few kev, and a hard powerlaw tail that dominates above a few kev. The hard spectral component is not apparent for all sources, while in some systems it dominates completely and our understanding of its nature is very limited. Several emission mechanisms have been proposed to explain the observed thermal emission from neutron-star X-ray transients (see e.g., Campana, 2003, and references therein). For example, it has been suggested that the emission results from residual accretion during which the accreted matter interacts with the magnetic field of the neutron star. The currently most adopted explanation for the soft component, is thermal emission from the hot neutron star (Brown et al., 1998). Accretion induces nuclear reactions that release large 12

23 1.3 Low-mass X-ray transients amounts of heat deep within the crust (see Section 2.3). According to this model, the neutron star interior is heated by the nuclear reactions in the crust. During quiescence, the heat is released as thermal emission from the surface as the neutron star cools down again. The deep-crustal heating model can account for the thermal emission component of the quiescent luminosity of X-ray transients, but gives no explanation for the hard spectral component that is observed for some sources. However, it is conceivable, that the power-law tail is described by an alternative model in addition to the thermal emission model describing the soft component of the emission spectrum. If the quiescent emission is indeed dominated by thermal emission from the cooling neutron star, than the quiescent luminosity depends on the time-averaged accretion rate (> 10 4 yr) of the binary, as well as on the microphysics of the neutron star, such as the dominant cooling mechanism in the core and the conductivity of the crust (see Chapter 3). Studying neutron star X-ray transients can thus probe important questions on the neutron star interior, which is exactly the aim of my master research Quasi-persistent X-ray transients After all this background information on X-ray binaries, I can now treat the systems that are the focus of this master thesis; quasi-persistent X-ray binaries. This is a sub-group among the X-ray transients, that distinguishes itself both in observable quiescent properties and by spending an unusually long time in outburst, lasting years to decades rather than weeks to months (see Figure 1.11). The group of quasi-persistent X-ray binaries consists of a number of neutron star systems (see Table 1.1 for examples) and several that contain a black hole. The prolonged outburst period of quasi-persistent sources, as compared to regular (i.e., short-duration) transients, is thought to have a significant effect on the neutron stars in such systems (Wijnands et al., 2001). In the regular transient systems, the neutron star crust is slightly heated but quickly returns to equilibrium with the core, at the end of an outburst. However, in quasi-persistent systems the neutron star crust can be heated to considerably higher temperatures than the core, causing the crust and core to get significantly out of thermal equilibrium. Once back in quiescence, the crust thermally radiates X-rays, cooling it down until it reaches thermal equilibrium again. Because the crust and core are out of equilibrium, the initial (i.e., before crust and core are again in equilibrium) quiescent properties are dominated by crustal emission rather than by the state of the core as is the case in regular transients. Crust cooling curves reflect the properties of neutron star (see Chapter 3) and can be used to probe the neutron star interior by way of comparing observations with theoretical predictions. 13

24 Chapter 1: Introduction Source name Status EXO detected in outburst since February 1985 GS detected in outburst since September 1988 XTE J detected in outburst since February 2001 MXB KS U X turned off in September 2001 after an outburst of 2.5 year turned off in February 2001 after an outburst of 12.5 year quiescent since 1983 after at least 11 year in outburst quiescent since 1999 after at least 12 year in outburst Table 1.1: A few examples of quasi-persistent neutron-star X-ray transients along with their current state. References: EXO (Parmar et al., 1986), GS (Cocchi et al., 2001), XTE J (Markwardt and Swank, 2003), MXB (Wijnands et al., 2003), KS (Wijnands et al., 2001), 4U (Pietsch et al., 1986) and X (Guainazzi et al., 1999). 1.4 This master thesis Both the quasi-persistent neutron star LMXBs KS and MXB have been monitored with Chandra and XMM-Newton several times since they went in quiescence in 2001 (see Figure 1.13). These X-ray observations have tracked the cooling of the neutron stars in these systems and the cooling curves are available for comparison with theoretical models (Wijnands et al., 2004). Rutledge et al. (2002) calculated crust cooling curves for four different scenarios of the physics of the crust and core for one of these sources; KS As becomes apparent from Figure 3.9, one of the calculated cooling curves approaches the observed lightcurve reasonably well. However, fine-tuning of the model is necessary in order to be able to draw qualitative conclusions on the interior properties of this neutron star. Furthermore, as the cooling depends on the accretion history of the system, we cannot directly compare the observations of MXB with the model-curves calculated by Rutledge. The model should be adjusted in order to be tested on MXB This is especially worthwhile since MXB has also been observed in outburst between 1976 and 1978 (Lewin et al., 1978). As a result, the accretion history, which is set by the time-averaged accretion rate during outbursts and the average duration of both the quiescent and outburst episode, is considerably better constrained than for KS This allows for the calculation of more detailed cooling curves than for KS and consequently allows for a better survey of the neutron star properties in MXB In my masters project I aimed at understanding the theory of cooling neutron stars in quasi-persistent LMXBs and the numerical implementation needed to produce theoretical cooling curves. The major goal of my research is to improve the current available cooling curves. 14

25 1.4 This master thesis Figure 1.13: Lightcurves for KS (top) and MXB (bottom) averaged over 7 days from Rossi X-ray Timing Explorer/All Sky Monitor surveys. This figure also shows the times of observations of Chandra (solid lines) and XXM-Newton (dotted lines) of these sources during quiescence. This figure is taken from Cackett et al. (2006) L [10 erg s ] q t [yr] 100 Figure 1.14: Comparison of the observed lightcurve of KS , indicated in red (data points are taken from Cackett et al., 2006), with the theoretical cooling curves for four different scenarios of source composition (Rutledge et al., 2002). Both the observations and the theoretical models that are displayed in this figure will be further discussed in Section

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27 CHAPTER 2 Neutron star interior As was explained in Chapter 1, the thermal evolution of a neutron star is very sensitive to its microscopic properties, which will be the subject of this chapter. The first section shortly introduces a few concepts that are important when describing the interior of neutron stars, while the general structure and composition are considered in Section 2.2. This chapter ends with a discussion on the effects of accretion on a neutron star s interior. Appendix B can be consulted for particle physics terminology. 2.1 Key concepts Gravitational mass Throughout this thesis, any references to the mass of a neutron star concern the gravitational mass. This is the mass that is used in Kepler s laws of motion. The gravitational mass differs from the baryonic mass (i.e., the combined mass of the baryonic constituents of the star, if these would be measured separately) due to gravitational effects. The gravitational binding energy is large for neutron stars, and the gravitational mass is about 20% lower than the baryonic mass Degeneracy Neutron stars do not burn nuclear fuel in their interior, but are supported against gravitational collapse by the pressure from a degenerate neutron gas. Degeneracy is a quantum mechanical phenomenon that arises from the Pauli exclusion principle and the uncertainty principle of Heisenberg. According to the uncertainty principle, we can describe the one dimensional momentum phase space of a fermion gas as being constructed out of cells with size x by p (see Figure 2.1). The Pauli exclusion principle now prescribes that only two fermions with opposite spin can occupy the same cell in phase space. In normal matter, the momentum space is relatively empty (see Figure 2.1) and the fermions can freely move and gain or loose energy 17

28 Chapter 2: Neutron star interior by changing their momentum. Therefore, normal matter can easily be compressed. However, when the particle density strongly increases, all the lowest momentum (or equivalently, kinetic energy) states below a certain critical value, corresponding to the Fermi energy, become occupied. This is illustrated by the right side of Figure 2.1. Matter is then said to be degenerate. The boundary that separates the unfilled levels from the fully occupied ones is called the Fermi surface. With all the lowest momentum states filled, additional fermions are forced to occupy higher momentum levels. This gives rise to a degeneracy pressure which counteracts further compression of the fermion gas. Degenerate matter behaves very peculiar. For one thing, in a degenerate Fermi gas, only those particles that reside close to the Fermi surface can interact with other particles. This is because an interaction requires a change in momentum, which cannot be contrived when all of the adjacent cells in momentum space are filled. Thus, degenerate fermions occupying the lowest momentum states (i.e. particles that are deep within the Fermi surface) cannot interact easily. Another extraordinary feature is that the pressure exerted by a degenerate gas is a permanent pressure that does not seize with decreasing temperature. This can be understood as the fermions in a degenerate gas cannot lose their momentum (energy) when all the neighboring cells in phase space are occupied. Thus, a cooling neutron star is not expected to suffer a gravitational collapse, no matter how much the temperature decreases. Note that the temperature of a neutron star reflects the average kinetic energy of the non-degenerate particles. The number of available states in the momentum phase space is different for all species of fermions, since it scales with the mass of the particles. Therefore the phase space available for electrons is smaller than for example that of nucleons. As a consequence, electrons will be the first to become degenerate, which happens already at densities of ρ 10 4 g cm 3. In white dwarfs, degenerate electrons provide an adequate pressure to counterbalance the inward pull of gravity. There is, however, a maximum pressure that can be exerted by a degenerate gas, since particle velocities can never become larger than the speed of light. Theory prescribes that a white dwarf more massive than the Chandrasekhar limit of 1.44 M cannot provide sufficient electron degeneracy pressure to withstand gravity. In neutron stars, the density is large enough for neutrons to become degenerate and gravity is balanced by the degeneracy pressure of neutrons. An upper limit similar to the Chandrasekhar limit for white dwarfs, exists for neutron stars. This limit is called the Tolman-Oppenheimer-Volkov limit and depends strongly on the equation of state (see next section) of the neutron star interior Equation of state The composition and behavior of neutron star matter is reflected in its equation of state (EOS). The EOS describes the relation between the density and the pressure of matter, which is a measure for the compressibility. The steeper the rise in pressure with increasing density, 18

29 2.1 Key concepts Fermi surface Momentum Momentum PF Position Normal matter Position Degenerate matter Figure 2.1: One dimensional representation of the concept of degenerate matter. Presented is the momentum phase space for a normal gas on the left and a degenerate gas on the right. The arrows in the boxes represent different (up and down) fermion spin states. Due to the Pauli exclusion principle, only fermions with opposed spin states can occupy the same box in phase space. the less compressible matter will be. Knowledge of the EOS is crucial for determining the macroscopic properties of a neutron star, such as its maximum mass and corresponding radius (see Section 2.2.4). Unfortunately the EOS of a neutron star is currently not well constrained, which is mainly due to our lack of understanding of the behavior of matter at supra-nuclear densities. There exist many theoretical models, which can each be characterized by their softness. The softer an EOS, the lower the pressure at a given density and thus the more compressible the matter is. As a consequence, soft EOSs predict neutron stars with smaller maximum masses than the hard EOSs (see Section 2.2 and Figure 2.5). The EOS of the interior neutron star is practically independent of temperature, since the main contribution to the pressure comes from highly energetic, strongly degenerate fermions. The crust and core of neutron stars are generally described by separate EOSs, although there are models that describe both the crust and the core in a unified manner Neutronization and the neutron drip density Due to a growth in electron Fermi energy with increasing density it becomes energetically favorable to convert electrons and protons into neutrons and neutrinos via inverse β-decay, p + e n + ν e, while the reverse reaction, n p + e + ν e, is suppressed. As a result, atomic nuclei get more and more neutron-rich, a phenomena which is dubbed neutronization. 19

30 Chapter 2: Neutron star interior The fractions of protons and electrons decrease drastically with increasing density, down to only a few percent in the neutron star core. The nuclear potential has a finite depth and width and hence a finite number of bound levels. At some critical density, the neutrons outnumber the available bound levels and thus only unbound levels can be taken up. Neutrons start to drip out of the nuclei and form a degenerate neutron liquid that surrounds the nuclei. This critical density is called the neutron drip point, ρ drip g cm Superfluidity Superfluidity is a special phase state in which the matter is characterized by a negligible viscosity and almost perfect thermal conductivity. Under the pressure and temperature conditions, characteristic for neutron stars, both protons and neutrons can become superfluid. Nucleon superfluidity was already predicted by Migdal (1959) and today pulsar glitches provide strong observational support for this hypothesis (see e.g., Pines, 1991). Nucleon superfluidity arises from the formation of Cooper pairs, mediated by the attractive component of the strong interaction. Cooper pairs are pairs of alike fermions that effectively act as a boson (which are not subject to the Pauli exclusion principle) and can therefore occupy the same quantum state. The formation of Cooper pairs can effectively lower the energy of the nucleons, when the temperature drops below a density-sensitive critical value, T c. For nucleons, the value of this superfluid transition temperature is extremely sensitive to the strong interaction models and many-body theories employed (for a review, see e.g., Lombardo and Schulze, 2001). However, most theoretical models predict a critical temperature in the order of K, which is high compared to the characteristic temperature in neutron stars ( 10 8 K, see Chapter 3). Therefore it is very plausible that nucleon superfluidity occurs in neutron stars. Below T = T c, particles are in the ground state energy configuration when they reside in Cooper pairs, while single particles then present exited states. The formation of Cooper pairs is thus associated with a release in energy, (dependant on T), which is carried away by neutrino/anti-neutrino pairs. This also implies that the onset of superfluidity (T < T c ) introduces an energy gap with magnitude, in the excitation spectrum of single nucleons. The energy gap is small near T = T c, but increases with decreasing temperature. Continuous formation and breaking of Cooper pairs takes place slightly below T = T c. However, if the temperature drops to T 0.2T c, the energy gap is so large that Cooper pairs generally cannot be broken any more (k B T ). This is called freezing in of Cooper pairs. It is important to consider nucleon superfluidity in the scope of this master thesis since it affects the heat capacity and neutrino luminosity of a neutron star and therefore its cooling (see Section 3.1.4). Superfluidity in neutron stars is not restricted to nucleons. Exotic particles like quarks (Bailin and Love, 1984) and hyperons (Balberg and Barnea, 1998) may 20

31 2.2 General structure also become superfluid when occurring in the neutron star interior. Neutrons can be superfluid both in the crust and the core. The strong interaction, which allows for Cooper pair formation, depends sensitively on the particle density. As a result, different types of neutron superfluidity exist for the crust and the core. Protons are only expected to be superfluid in the core and, in addition to being superfluid, protons are also expected to be superconducting. Superconducting materials are characterized by a zero electrical resistance and the tendency to exclude magnetic field lines from their interior. 2.2 General structure Neutron stars are thought to be born very hot, with interior temperatures of around K. At such high temperatures, nuclear processes take place very rapidly so that matter in the star can be considered to be in complete thermodynamic equilibrium. Matter in which all nuclear reactions have gone to completion is termed catalyzed matter. Within a few years after its birth, a neutron star will have cooled to temperatures of 10 8 K (see Chapter 3). Such temperatures are low compared to the characteristic excitation energies in nuclei which are 1 MeV, or K. Therefore, to a good approximation one may regard neutron star matter as always being in its lowest energy state, which is usually denoted as cold matter. It is on this basis that static models of neutron stars are constructed out of cold catalyzed matter, which corresponds to matter in full thermodynamic equilibrium and a minimum energy per nucleon at any given density (Shapiro and Teukolsky, 1986). This is a good description for non-accreting neutron stars, but as I will discuss in Section 2.3, the state of matter can be rather different when accretion takes place. The structure of a neutron star can be described as being composed out of five major regions, as shown in Figure 2.2: the atmosphere, the inner and outer crust and the inner and outer core. For some purposes, e.g., when discussing the temperature profile of a neutron star, it is more convenient to describe the outer layers as being composed of an envelope, in which per definition the temperature gradient is very steep, and a crustal shell. The envelope then extends from the bottom of the atmosphere to a density of g cm 3 (corresponding to roughly 100 m) and the shell down to the crust/core interface at ρ ρ 0. Such a treatment is for example adopted in Section Atmosphere The neutron star atmosphere is a thin ( cm) layer consisting of ionized nuclei and non-degenerate electrons. The composition is expected to be pure hydrogen, because heavier elements are thought to sink (Romani, 1987). However, the possibility of other ion species, such as helium or iron, dominating the atmosphere composition should not be completely disregarded. The atmosphere accounts for a negligible fraction of the total neutron star mass, but plays an important role in shaping the thermal photon spectrum emerging from the 21

32 Chapter 2: Neutron star interior drip atmosphere outer crust inner crust outer core e, Z e, Z, n e, p, n inner core hyperons? mesons? quarks? ~10-20 km? up to Figure 2.2: Schematic representation of the interior structure of a neutron star. The nuclear density is denoted by ρ 0 and the neutron drip density by ρ drip. The main constituents are given for each region, where Z denotes ions, and e, p and n refer to free electrons, protons and neutrons respectively. The most disputable factor of the neutron star interior is the inner core where it is highly uncertain what kind of particles are present and what the maximum central density can be. This microscopic uncertainty has consequences for, e.g., the predicted neutron star masses and radii and for the way neutron stars cool. neutron star surface. This emission carries valuable information on stellar parameters and thus on the internal structure of the neutron star. Observations of the thermal emission from neutron star surfaces can be used to estimate neutron star radii, by fitting an emission model to the data. For a correct interpretation of the observations, a proper model describing the neutron star atmosphere is required. Constructing such a model is not straightforward. The spectrum emerging from a neutron star can be quite different from a black body spectrum, depending on the chemical composition, magnetic field and gravity in the surface layers (see e.g., Zavlin et al., 1996, and references therein). The very steep temperature gradient in the neutron star atmosphere can also modify the emerging spectrum with respect to a black body spectrum. Since the neutron star atmosphere consists of ions and free electrons, the opacity will be dominated by free-free absorbtion (absorbtion of a photon by a free electron in the Coulomb field of an ion) and is therefore proportional to ν 3, where ν is the photon frequency. As a result of this strong dependence, high-energetic photons escape from deeper atmospheric layers, where the temperature is much 22

33 2.2 General structure higher, than low energetic photons do. This situation is illustrated in Figure 2.3. Despite the fact that the atmosphere is very thin, this effect is profound. For a particular temperature, the X-ray spectrum emerging from a neutron star is thus expected to be harder than a pure black body spectrum. One important implication is that fitting black body models to such modified spectra, results in an overestimate of the effective temperature and thus the size of the corresponding emission region is underestimated (Romani, 1987; Zavlin et al., 1996). This emphasizes that employing a correct model for neutron star atmospheres is crucial for a proper interpretation of the data. Although the thermal emission spectrum from a neutron star is not likely to be a Planck spectrum, it is important to note that, due to the bad quality of spectral data, black body models provide fits that are equally satisfying as those of complicated neutron star atmosphere models. Currently, we cannot distinct between these two observationally in quiescent neutron star X-ray binaries. Hard photons escaping Soft photons escaping Crust Steep temperature gradient Atmosphere Figure 2.3: Schematic overview of X-ray emission emerging from a neutron star atmosphere. Harder (i.e. more energetic) X-ray photons are less frequently absorbed than soft photons. As a result, hard X-rays reside from deeper atmospheric layers where the temperature is considerably higher due to a steep temperature gradient across the neutron star atmosphere Crust The crust typically covers about one tenth of the neutron star s radius and can be subdivided into an inner and an outer part. The properties of the neutron star crust are described by reliable microscopic theories and therefore the crust equation of state is reasonably well understood. The outer crust extends from the bottom of the atmosphere to the neutron drip density, ρ drip g cm 3. In this region, matter consists of electrons, which are degenerate at densities above ρ 10 4 g cm 3 and fully relativistic for ρ 10 7 g cm 3, and ions. As was already mentioned, neutron star matter can be described as being in its lowest energy state. The ground state configuration of nuclei is density dependent. At densities 10 6 g cm 3 iron will be the most dominant nucleus, but due to the rise in electron Fermi energy with increasing density, the nuclei suffer inverse β-decay and are enriched by neutrons. The composition of 23

34 Chapter 2: Neutron star interior cold catalyzed matter in the outer crust is calculated in the classical paper by Baym et al. (1971). These authors show that towards larger densities, nuclei become heavier (A increases due to nuclear fusion) and more neutron rich. At the base of the outer crust, neutrons start to drip out of the nuclei. The inner crust covers the region from the neutron dip density to the nuclear density, ρ g cm 3. The inner crust is composed of electrons, free superfluid neutrons and neutron rich, heavy nuclei. With increasing density, nuclei continue to grow heavier, up to A 300, while Z remains roughly constant throughout the inner crust at Z 40 (see e.g., Douchin and Haensel, 2000). Towards greater depths, more and more neutrons reside in the free neutron fluid rather than in nuclei. Nuclei begin to dissolve and merge together around the crust-core interface Core The core makes up the largest part of the neutron star, containing approximately 99% of the total mass, and may be subdivided into an outer and an inner part. The outer core occupies the density range ρ 0 ρ 2ρ 0 and can be several kilometers in depth. In this density range, matter consists mainly of degenerate neutrons and merely a few percent of protons and electrons. Due to the growing Fermi energies of the particles with increasing density, it may become energetically favorable for other particles, besides the standard composition of p, e and n, to occur. If the Fermi energy of the electrons exceeds the muon rest energy, m µ =105.7 MeV, muons will replace a fraction of the electrons to lower the energy of the system. This is likely to occur in the outer core, so that the composition of matter is enriched with muons. Furthermore, both protons and neutrons are expected to be superfluid in the density regime of the outer core, where the proton superfluidity is accompanied by superconductivity. In the inner core of the neutron star, the density rises beyond ρ 2ρ 0 and may become as high as ρ (10 15)ρ 0, depending on the EOS model 1. Laboratory data on properties of matter above the nuclear density are incomplete and the reliability of theories decrease with increasing density. The following hypotheses have been put forward for the composition of the inner core: 1. Presence of hyperons (Bethe and Johnson, 1974). At the densities that exist in the inner core of a neutron star, the neutron Fermi energy can easily exceed the rest mass energy of the hyperons Σ, Λ, Ξ and (see Appendix B for the definition and rest masses of hyperons). It is then energetically favorable if neutrons are replaced by hyperons. The conversion of neutrons into hyperons, causes the highest momentum levels in neutron phase space to depopulate. Correspondingly the Fermi surface and 1 Note that not necessarily all neutron stars possess an inner core, since this depends on the central density of the star and thus on its mass and radius. 24

35 2.2 General structure the degeneracy pressure relax. Thus, for a given density, the inclusion of hyperons is likely to induce a softening in the EOS. Current estimates predict the formation of hyperons at densities of ρ 2ρ 0. However, it should be noted that the calculations are highly uncertain. For example, the potential between hyperons is largely unknown and is generally taken to be same as between two nucleons. 2. Meson condensation (see e.g., Sawyer, 1972 and Pandharipande et al., 1995). The rising Fermi energies in the neutron star inner core might induce the reactions n p + π and n p + K. The electron Fermi energy has to exceed the rest mass of the pion, m π =139.6 MeV, or the kaon, m K =493.7 MeV, to allow the replacement of electrons by these bosonic particles (Appendix B). At sufficiently low temperatures (T < T c ), all bosons will occupy the ground state and thereby form a so-called Bose- Einstein condensate that exhibits superfluid properties. Particles that populate the zero momentum state do not contribute to the pressure exerted by the matter. Therefore the occurrence of a meson condensate will soften the EOS. It is estimated that mesons will occur at densities a few times the nuclear density, ρ (2 3)ρ Quark deconfinement (e.g., Schaab et al., 1996). In normal matter, deconfined quarks do not exist owing to the fact that the force between quarks increases with increasing distance. However, it has been proposed that the particle densities in neutron stars can become so high, that the force between quarks can be neglected and these particles may be regarded as free. Current estimates suggest that a phase transition from nucleon matter to quark matter may occur at a density somewhere between 2 and 10 times the nuclear density. Nucleons are composed of up (m u = 4.2 MeV) and down (m d = 7.5 MeV) quarks, but these are expected to transform into strange quarks (s), soon after quark confinement sets in. Because of the presence of s quarks, matter in this state is often referred to as strange quark matter. The remaining three quark flavors (charm, top and bottom, see Appendix B) are much heavier and are thought to require densities greater than g cm 3 to be created. The massive quarks are therefore not expected to occur in neutron stars. A conversion into quarks will lower the neutron Fermi momentum and may therefore be expected to induce a softening of the EOS. The presence of exotic particles will not only affect the EOS, but will also have a great influence on the thermal evolution of neutron stars (see Chapter 3). Apart from the uncertainty in particle composition, there is a difficulty in constructing the core EOS, even if only protons, neutrons and electrons are incorporated. Problems arise because the nucleon-nucleon potential has not been experimentally determined for the energies and densities that are encountered in the central region of neutron stars. Current theories are generally based on an extrapolation of laboratory data, which induces considerable uncertainties. To illustrate the large uncertainty in the behavior of matter at densities ρ > ρ 0, 25

36 Chapter 2: Neutron star interior Figure 2.4 shows a few sample core EOS models. The models presented in this figure use different approaches in calculating the particle potentials as well as different core compositions (for details, see Lattimer and Prakash, 2004, and references therein). It is clear from this figure that the differences between the models grow with increasing density. Furthermore it is apparent that the presence of exotic particles softens the EOS. The macroscopic properties of neutron stars are very sensitive to the core EOS and therefore there also exists a wide range in predicted neutron star masses and radii, as will be discussed in the following section. Figure 2.4: Different EOS models for the neutron star core, expressed in a pressure versus energy density diagram. Note that the energy- and mass density are related by ɛ = ρc 2 so that the conversion is 1 MeV fm g cm 3. For the pressure one can use the conversion 1 MeV fm erg cm 3. The notation is as follows: n = neutrons, p = protons, K = K meson condensate and quarks = up,down and strange quarks. The stiffest possible EOS, as stated in this Figure, is based on validation of causality. The relativistic neutron gas model, displayed at the bottom, is the EOS for an ideal neutron gas, i.e. the energy of the particle (nucleon-nucleon) interactions is neglected in comparison with kinetic energy terms. The ideal neutron gas EOS predicts a maximum mass of M max 0.7 M, which is well below observed neutron star masses. This in fact points out that the effects of nucleon-nucleon interactions are crucial for describing neutron stars. The figure is taken from Weber (2005) Mass-radius relation For any equation of state, neutron star models can be constructed by numerically solving the equation of hydrostatic equilibrium (accounting for the effects of General Relativity) for 26

37 2.2 General structure different central densities. An increasing central density generally requires an increase in mass and a decrease in radius, so that the neutron star becomes more compact. For every EOS, the neutron star mass reaches a maximum, for a certain central density, which corresponds to the most compact stable configuration (Tolman-Oppenheimer-Volkoff limit, analogous to the Chandrasekhar limit for white dwarfs). Models with a central density beyond this maximum density are generally not stable 2. Using a set of models of different central densities, a mass-radius relation can be constructed for each EOS. This is set by the core EOSs; the structure of matter below the nuclear density hardly affects the correlation between the mass and the radius of a neutron star. Since the central densities that can be reached are very sensitive to the adopted equation of state, so are the mass-radius relations. This is illustrated by Figure 2.5, which displays the mass-radius relation for a selection of different EOSs. In addition to the mass-radius relations of neutron stars, this figure also shows the predictions for strange stars. This is a hypothetical self-bound state of deconfined u, d and s quarks, either or not accompanied by a crust of nucleon matter (see e.g., the review by Glendenning, 2000). The general trend, clearly seen in Figure 2.5, is that the softer an EOS, the compacter the neutron star and thus the lower its maximum possible mass. Knowing the maximum possible neutron star mass is crucial for the identification of black hole candidates in binary systems. The strong dependence of the neutron star mass and radius on its microscopic properties, opens up a possibility to constrain the EOS at supra-nuclear densities. For example, finding a high-mass neutron star ( 2M ) can eliminate entire families of soft models (assuming that only one EOS gives a correct description of the neutron star interior). There are several claims of observational indications of massive neutron stars (near or even over 2M, all in X-ray binaries), although the accuracy of those determinations should be improved in order to yield a useful constrain on the neutron star EOS (Abubekerov and Cherepashchuk, 2005). The most reliable mass estimates come from observations on double neutron star systems (in which one or both act as a radio pulsar) and these lie within a fairly small interval of M (Stairs, 2004). Unfortunately, double neutron star systems are expected to result from a very specific evolutionary scenario. Consequently, one cannot draw general conclusions on the neutron star mass function based on these precisely measured neutron star masses. Although accurate masses of several neutron stars have been determined, a precise measurement of the radius does not yet exist (see e.g., Lattimer and Prakash, 2001). Estimates of neutron star radii from observations have given a wide range of results. Perhaps the most reliable estimates stem from observations of the thermal emission from neutron star surfaces, 2 The fate of a neutron star that exceeds the Tolman-Oppenheimer-Volkoff limit has not been established. In principle, a neutron star could either collapse to form a black hole, or change composition so that it becomes supported by some other form of degeneracy pressure (for example quark degeneracy if the collapsing neutron star settles as a strange star). However, the properties of these exotic forms of degenerate matter are even more poorly known than those of neutron-degenerate matter. Coupled with very little experimental evidence, this leads most astrophysicists to assume a direct transition from a neutron star to a black hole. 27

38 Chapter 2: Neutron star interior Figure 2.5: Different mass-radius relations rising from different core EOSs. The notation is as follows: n = neutrons, p = protons, K = K meson condensate and quarks = up, down and strange quarks. This figure is taken from Weber (2005). which yields a radiation radius R = 1 r s /R, where r s = 2GM/c 2 is the Schwarzschild radius. The actual neutron star radius will satisfy R < R and thus R gives an upper limit. When the mass of the neutron star is estimated, R can be used to calculate R. As already discussed, the value of the radiation radius is strongly dependent on the adopted model for the emission spectrum (see Section 2.2.1). Significant constraints on the EOS at supra-nuclear densities could be realized when both the mass and radius of a single neutron star are deduced from observations. One of my research results presented in this thesis explores a method that allows one to constrain the neutron star radius when the mass is determined from observations and vice versa (Section 4.2). This method is applicable to neutron stars with an inverse temperature gradient between the core and the crust, as is the case for example for the quasi-persistent X-ray transients (recall Section 1.3.3). 2.3 Accreting neutron stars Crust composition Matter that is accreted onto a neutron star, is likely to accumulate in the surface layers where it eventually undergoes nuclear burning into heavier elements. In the presence of hydrogen, 28

39 2.3 Accreting neutron stars the nuclear burning of accreted material proceeds via rapid proton captures (rp-process, Wallace and Woosley, 1981). The composition of the ashes is set by the endpoint of the rpprocess (Haensel and Zdunik, 2003), which is rather uncertain. Early studies predicted that the nuclear burning of accreted matter would produce mainly iron, 56 Fe, nuclei. However, calculations that extend the network of possible nuclear reactions, resulted in ashes composed of nuclei heavier than iron, with A up to 112 (Schatz et al., 2001). Under the weight of freshly accreted material, the ashes from the nuclear burning sink deeper into the neutron star crust. Due to ongoing accretion the crust is compressed, which induces a sequence of nuclear reactions that involve electron captures in the outer crust, neutron emissions beyond the neutron drip density and pyconuclear reactions (density sensitive nuclear fusion processes) when ρ > g cm 3. The growing layer of processed accreted matter replaces the original crust, that consisted out of cold catalyzed matter. The crust of an accreting neutron star is thus constructed out of non-catalyzed matter (nuclear reactions have not gone to completion yet). Furthermore, the temperature in the accreted crust is likely too low to overcome the nucleon Coulomb-barrier, which is necessarily for thermonuclear fusion (temperature sensitive fusion processes). As a result, it is not possible to form the large nuclei, characteristic for cold catalyzed matter, in an accreted crust. Until pyconuclear fusion reactions become possible (beyond the neutron drip, see Section 2.3.2), the number of nucleons, A, in a nucleus stays constant and is set by the nuclear burning on the surface. Haensel and Zdunik (1990a) calculated the composition of an accreted neutron star crust, assuming that the ashes of the nuclear burning are pure iron ( 56 Fe). Their calculations showed, that for example at a density of about g cm 3, an accreting neutron star is characterized by 56 Ca (Z = 20) nuclei. For reference, at the same density, the dominant nucleus of cold catalyzed matter will be 122 Zr (Z = 40) (Baym et al., 1971). Because the endpoint of the rp-process might be elements heavier than iron (Schatz et al., 2001), Haensel and Zdunik (2003) also calculated the crustal composition for the scenario in which the nuclear burning ashes are dominated by 106 Pb (Z = 46). To get an impression of the effect of the initial composition, for this scenario matter will mainly consist of 106 Kr (Z = 36) at a density ρ = g cm 3. The model studies of Haensel and Zdunik (2003) show that the composition of the outer crust (i.e. for densities below the neutron drip) is very sensitive to the initial conditions of the ashes, set by the rp-process. However, a chain of processes that occurs after the neutron drip (see Section 2.3.2) causes the nuclei in the inner crust to be rather similar for the different models. The difference in composition between accreted and non-accreted neutron stars will also be reflected in their EOSs. In the outer crust, the pressure is mainly supplied by degenerate electrons, with only small contributions from the nuclei. The number of free electrons in an accreted crust will not very different from that of cold catalyzed matter, and thus the EOSs will also be quite alike. Deep in the inner crust, at densities ρ g cm 3, the number of free neutrons is very large and these account for the main contribution to the pressure. 29

40 Chapter 2: Neutron star interior In this density regime, again the EOS of the an accreting neutron star will not deviate very much. However, just above the neutron drip, where degenerate neutrons start to contribute to the pressure, the number of free neutrons is not at an equilibrium value, but larger, for an accreting neutron star. As a result, an accreted crust will be characterized by a higher pressure for densities g cm 3 ρ g cm 3, when compared with the crust of a non-accreting neutron star. This is illustrated by Figure 2.6, which displays simple model EOSs for both accreting (initial composition of 56 Fe) and non-accreting neutron star crusts. Thus, the EOS of a neutron star inner crust will be harder when the star accretes matter. This also implies that the crust will be slightly thicker. Figure 2.6: Comparison between the crust EOS of an accreting neutron star (solid line) and sample EOSs for non-accreting neutron star crusts (dashed and dotted curves). See Haensel and Zdunik (1990a) and references therein for details on the different EOSs Crustal heating The chemical reactions that take place in a non-catalyzed neutron star crust represent a potential source of energy. This is best explained by following the evolution of an accreted matter element with initial composition (A, Z), as it is pushed deeper into the neutron star (Haensel and Zdunik, 1990b). With increasing depth, the matter element will cross the threshold density for electron capture, which allows the reaction (A, Z) + e (A, Z 1) + ν e (2.1) 30

41 2.3 Accreting neutron stars In dense matter, stable nuclei have even numbers of neutrons, N = A Z, and even numbers of protons, Z (even-even nuclei, see Preston and Bhaduri, 1975). The electron capture produces an odd-odd nucleus which is highly unstable in very dense matter, so that immediately a second electron is captured (A, Z 1) + e (A, Z 2) + ν e + Q d (2.2) This reaction takes place above the threshold for electron capture by an odd-odd nucleus (which is lower than the threshold for Reaction 2.1), and therefore results in the release of energy Q d, that will be deposited in the neutron star crust. Q d is typically in the order of 0.01 MeV/nucleon (Haensel and Zdunik, 1990b). The two-step electron captures stated above, occur each time the matter element passes the threshold density for an electron capture and thus lead to a systematic decrease of Z. Note that A remains constant since no fusion is taking place. When a matter element sinks beyond the neutron drip density, electron capture induces the emission of a neutron, so that the reaction sequence becomes (A, Z) + e (A, Z 1) + ν e (2.3) (A, Z 1) + e (A k, Z 2) + kn + ν e + Q c (2.4) where k (the number of emitted neutrons) is necessarily an even number. This is also a non-equilibrium reaction, resulting in the release of an amount of energy, Q c, which again will be deposited as heat in the crust. As the matter element deeper and deeper into the inner crust, each time a threshold density for electron capture is crossed, a chain of electron captures and neutron emission follows. The energy released in these reactions is typically MeV/nucleon (Haensel and Zdunik, 1990b) and results from de-excitation of the created nucleus as well as from the neutrons, which are emitted with an energy well above the Fermi energy of the degenerate neutron liquid. Due to the systematic decrease of the value of Z by electron captures, the Coulomb-barrier for nucleon-nucleon reactions will be lowered. Combined with the fact that the separation distance between the nuclei decreases with rising density, this opens up the possibility for a chain of reactions involving pyconuclear fusion. When a threshold for fusion is crossed (this happens for the first time when ρ g cm 3 ), Reactions are followed by (A, Z ) + (A, Z ) (2A, 2Z ) + Q 1 (2.5) (2A, 2Z ) (2A k, 2Z ) + k n + Q 2 (2.6) Q 3 (2.7) where Z = Z 2 and the third line corresponds to an unspecified chain of electron captures and neutron emissions. This series of reactions deposits an amount Q 1 + Q 2 + Q 3 in the 31

42 Chapter 2: Neutron star interior neutron star crust. The fusion reaction (2.5) releases the largest amount of energy, MeV/nucleon, in the form of excitation energy of the final nucleus (Haensel and Zdunik, 1990b). An accreted matter element passes through the chain of reactions as it is pushed deeper within the neutron star crust under the weight of newly accreted material. This results in the deposition of a considerable amount of energy (> 1 MeV per accreted baryon) in the neutron star crust. Most heat is released in the pyconuclear reaction chains ( ) that occur deep in the inner crust at densities ρ g cm 3. As was already mentioned in Section 2.3.1, the composition of the inner crust is practically independent of the initial composition of the nuclear burning ashes. Since most of the heat is released in the inner crust, the total energy deposited (per accreted baryon) in the crustal reactions is rather similar for different initial compositions of the burning ashes (Haensel and Zdunik, 2003). The nuclear reactions discussed in this section have a profound effect on the thermal evolution of transiently accreting neutron stars, as will be discussed in Section

43 CHAPTER 3 Neutron star cooling As was discussed in Chapter 2, the equation of state and composition of neutron star matter are largely unknown. One potential powerful method to probe the neutron star interior, is to track and model their thermal evolution, which is very sensitive to the neutron star parameters. This chapter is dedicated to what we can learn from cooling of neutron stars. Section 3.1 will deal with the dominant cooling mechanisms and thermal evolution of neutron stars in general, while Section 3.2 takes a closer look at the special case of cooling neutron stars in quasi-persistent X-ray binaries. Both observations and the general scheme of calculating theoretical cooling curves for transiently accreting neutron stars will be discussed. 3.1 General cooling scenarios Neutron stars are born extremely hot in supernova explosions, with interior temperatures around T K. During their life, neutron stars lose energy, both by the emission of neutrinos in various particle interactions, and by thermal photon emission from the surface. Since neutron stars do not burn nuclear fuel in their interior, they cannot compensate for these energy losses, and consequently will cool in time (except when accretion takes place, which is the subject of Section 3.2). One can distinguish two main cooling stages for neutron stars, as is illustrated by Figure 3.1. The neutrino cooling era starts with a thermal relaxation phase right after the neutron star formation, in which the star cools particularly fast due to very efficient neutrino emission from the stellar core. Already within a day, the temperature in the central region of the neutron star will have dropped down to K (Burrows and Lattimer, 1986). However, as we will see in the following sections, the cooling mechanisms that operate in the neutron star crust are much less efficient and the crust remains at a higher temperature. The crust and core will be strongly out of thermal equilibrium at this early stage and cool independently of each other. During thermal relaxation, the neutron star surface luminosity carries no information on the thermal state of the core, but will only reflect the state of the thermally decoupled crust. The size of the cooler interior grows, as energy from the 33

44 Chapter 3: Neutron star cooling hot crust is conducted towards the core. After approximately years, depending on the neutron star structure, the expanding cooling front will reach the top of the neutron star crust and the surface temperature will drop considerably. The crust and core are now in thermal equilibrium and the stellar interior has a uniform temperature. Only in the outermost layer of the neutron star, where the thermal conductivity is much lower, there will continue to exist a temperature gradient. This envelope extends to a density of approximately ρ g cm 3, which corresponds to a depth of 100 m. After thermal relaxation, the neutron star continues to cool predominantly by neutrino emission, however the surface temperature now tracks that of the isothermal interior, and the core cooling processes will control the observable luminosity. As we will see further on, the neutrino emissivities are much more sensitive to a temperature decrease than the photon luminosity and consequently, at a certain stage, the thermal surface radiation will govern the interior temperature evolution. This photon cooling phase sets in when the interior temperature falls below 10 8 K, which happens roughly after the neutron star formation. 9 T neutrino stage photon stage 38 Log T [K] 8 7 L T s L tot Log L [ergs -1 ] 6 relaxation 30 L Log t [yr] Figure 3.1: General evolution of the surface temperature, T s, and the core temperature, T, as would be experienced by an observer in infinity. The neutrino-, photon- and total luminosity in infinity are also displayed, indicated by the subscripts ν, γ and tot respectively. Note that only the photon luminosity (and effective temperature) can actually be observed, since neutrino emission can not easily be detected. 34

45 3.1 General cooling scenarios The above sketched cooling stages are general for all neutron stars, but the duration of the different episodes, as well as the magnitude of the surface temperature can differ strongly, depending on the neutron star interior parameters. The following sections outline the most important heat loss mechanisms in both the neutron star crust and core and discuss how the thermal evolution is affected by the neutron star structure and composition Neutrino emission from the core Neutrinos are generated in numerous reactions in the neutron star interior and can freely escape the star, thereby providing an efficient source of cooling. The most powerful neutrino emission processes are produced in the core, where matter consists of free neutrons, protons, electrons and possibly different forms of exotic matter (see Section 2.2.3). Table 3.1 at the end of this chapter summarizes the dominant neutrino emitting processes, together with their estimated emissivities (i.e. the energy carried away by the neutrinos per second per unit volume). The various processes will be briefly discussed below. I will first consider the core neutrino emission processes that occur in standard nuclear composition (neutrons, protons, electrons and possibly muons) and will discuss how the cooling of a neutron star is affected by the presence of exotic particles, at the end of this section. Urca reactions The processes of β-decay and electron capture among nucleons produce a neutrino and an anti-neutrino according to n p + e + ν e (3.1) e + p n + ν e (3.2) A pair of these subsequent reactions is referred to as direct Urca (durca) and serves as the most efficient neutrino emission mechanism in the interior of neutron stars (see Table 3.1). However, in degenerate matter, only particles with energies within k B T of the Fermi surface can participate in reactions (see Section 2.1.2). If the proton and electron Fermi momenta are too small compared with that of neutrons, the reaction given by Equation 3.1 cannot satisfy conservation of momentum and is therefore prohibited. Note that the Fermi momentum of a fermion species j is given by p F (j) = (3π 2 n j ) 1/3, where n j is the number density of the fermion species and j = n, p, e. According to Lattimer et al. (1991), the proton fraction x p = n p /n b (n b is the total number density of baryons) must exceed the critical value of x c 0.11, in order to open up the durca reactions (for standard nuclear composition). Which density corresponds to this critical proton fraction depends on the employed EOS, but will generally be a few times the nuclear density (Lattimer et al., 1991). Thus, only neutron stars with high central densities are expected to suffer durca neutrino cooling in the core. 35

46 Chapter 3: Neutron star cooling If the proton fraction does not exceed the critical value, β-decay can only proceed if there is a momentum sink available for the excess neutron Fermi momentum, which can be provided by a bystander particle. In this case, the reactions are denoted as modified Urca (murca) and have the general form n + N N + p + e + ν e (3.3) e + p + N N + n + ν e (3.4) where N denotes the bystander nucleon. If muons are present, additional durca and murca reactions will occur with electrons replaced by muons (and electron-neutrinos replaced by muon-neutrinos). The emissivities of these reactions will be the same as for the electron processes, but the critical proton fraction, necessary to open up the durca reactions, will be slightly higher (Lattimer et al., 1991). Nucleon bremsstrahlung In absence of the strong durca reactions, the neutrino luminosity from standard nucleon matter is not only determined by the murca reactions, but also by the process of neutrino bremsstrahlung from collisions of free nucleons: n + n n + n + ν + ν n + p n + p + ν + ν (3.5) p + p p + p + ν + ν where ν + ν denotes a neutrino/anti-neutrino pair of arbitrary flavor, either e, µ or τ. These bremsstrahlung processes are mediated by the strong interaction between nucleons and the estimated emissivities depend on the strong interaction model employed. Emission from Cooper pair formation As discussed in Section 2.1.5, nucleons are likely to become superfluid in the neutron star interior. Superfluid neutrons are thought to occur in both the core and the crust, while protons will only undergo a transition into the superfluid state in the core. The onset of nucleon superfluidity gives rise to a new neutrino generation mechanism, concerned with the creation of Cooper pairs. This process has been proposed and calculated in the pioneering article of Flowers et al. (1976). The schematic reactions for neutron and proton superfluidity are n + n [nn] + ν + ν (3.6) p + p [pp] + ν + ν (3.7) 36

47 3.1 General cooling scenarios Cooper pairs will be continuously formed and broken for temperatures just below a critical temperature T c (recall Section 2.1.5). However, if the temperature approaches T 0.2T c, the energy gap for excitation into the single particle state will become so large that the Cooper pairs are frozen in. If Cooper pairs are no longer broken, neither can there be new formation and consequently the neutrino emission associated with this process vanishes for temperatures below T 0.2T c. This emission behavior is enclosed in the control function F (T/T c ), which has a Gaussian shape. Thus, neutrinos are emitted due do Cooper pair formation for temperatures 0.2T c T T c. Within this temperature range, the emissivity of this process is comparable or even larger than that of the murca reactions (see Table 3.1) and can therefore make an important contribution to the neutron star cooling. In addition to opening up a new channel for neutrino emission from the neutron star interior, superfluidity of nucleons affects other neutrino emission processes. The effects of superfluidity on the thermal evolution of neutron stars, will be discussed separately in Section Effects of exotica in the core As was discussed in Section 2.2.3, exotic particles may occur in the inner cores of neutron stars, where the density rises far beyond the nuclear density. Both hyperons and deconfined quarks can undergo various neutrino emission processes. Similar to that in standard nucleon matter, the most important reactions will be durca, murca and neutrino bremsstrahlung radiation. Prakash et al. (1992) showed that the threshold density for hyperon durca processes to occur, almost coincides with the density at which hyperons are expected to appear. The neutrino luminosities from hyperon durca processes are about times less than the typical luminosity from the nucleon durca reactions (if these are not forbidden), but significantly larger than any of the slow cooling mechanisms (Prakash et al., 1992). Table 3.1 shows just an example of a hyperon durca reaction, a few similar processes involving other hyperons or combined reactions of nucleons and hyperons are also possible (see e.g., Prakash et al., 1992). According to Iwamoto (1980), the occurrence of deconfined quarks immediately opens up quark durca reactions, with neutrino luminosities only slightly below that of the nucleon durca processes. Thus the appearance of hyperons or deconfined quarks in the neutron star core will increase the total neutrino luminosity of the star when compared to standard cooling. Unlike nucleons, hyperons and quarks, π and K mesons are bosons and are not subject to the Pauli exclusion principle. As a consequence, there is no difficulty in conserving momentum in a meson condensate and durca processes involving these particles can occur unrestricted. The presence of meson condensates induces a strong mixing between neutron and proton states so that a description of quasi-particles (denoted by q) becomes appropriate. It is beyond the scope of this thesis to discuss this quantummechanical phenomenon, but it is important to note that the occurrence of meson condensates can account for a neutrino luminosity much larger than that originating from the slow cooling processes (see Table 3.1). 37

48 Chapter 3: Neutron star cooling Furthermore, mesons in the neutron star interior are formed in the reactions n p+π and n p+k (recall Section 2.2.3) and will thus increase the fraction of protons over the total number of baryons. It has been claimed by Thorsson et al. (1994), that the presence of kaon or pion condensates in the neutron star core will increase the fraction of protons in excess of the nucleon durca threshold. Summarizing, any exotic phases of matter in the inner core of a neutron star will lead to efficient neutrino emission by durca-like processes and consequently to a fast cooling rate. The core neutrino emission processes discussed in this section basically fall into two categories. The family of durca processes, occurring for different particle species, account for high neutrino emissivities and will therefore cause the neutron star to cool quickly in time (Q ν T9 6 ). The fast durca reactions are expected to occur at densities several times the nuclear density, where the proton fraction can exceed the durca threshold, or where the occurrence of exotic particles opens up new durca processes. On the other hand, when the central density in a neutron star does not exceed ρ 2ρ 0, the dominant cooling mechanism will be the much slower murca process (Q ν T9 8 ), which can operate everywhere in the core, in every neutron star. These two branches of neutron star cooling are denoted as standard (slow) and enhanced (fast) cooling. To illustrate the difference between the scenarios of fast and slow cooling, Figure 3.2 displays cooling curves for neutron stars with masses ranging from ( ) M, all with the same radius. The particular core EOS used for these calculations allows for durca reactions in the core if the mass of the neutron star exceeds 1.35 M (note that this is just an example and that the threshold mass for durca to occur depends fully on the employed core EOS). Clearly, there is an enormous difference in thermal evolution for the enhanced and standard cooling models. The more massive the neutron star, the larger the central region where durca processes can operate. As a consequence, the more massive neutron stars cool at a much higher rate and have lower core temperatures. Furthermore, the neutron stars with high masses have large surface redshifts and consequently thin crusts. Since diffusion of heat will proceed more rapidly if the crust is thin, the thermal relaxation time is shorter for more massive stars (see also Section 4.2). This effect is also apparent in Figure 3.2. The standard cooling curves are less sensitive to the neutron star mass Neutrino emission from the crust In the crust of a neutron star there also exists a broad scala of neutrino emission mechanisms, which determine the evolution of the surface temperature during the relaxation stage. I will only consider neutrino emission processes from crustal layers with densities ρ g cm 3, because the outermost layer is very thin and will not contribute significantly to the total energy loss from the crust. The dominant neutrino emission from the deep crustal layers of a cooling neutron star comes from plasmon decay, neutrino bremsstrahlung due to electron- 38

49 3.1 General cooling scenarios Figure 3.2: Thermal evolution of the effective temperature for different neutron star masses. The employed neutron star model allows for durca cooling if the neutron star mass exceeds 1.35 M. This figure is taken from Page and Applegate (1992). ion collisions and Cooper pair formation. In addition, the presence of high magnetic fields gives rise to electron synchrotron emission. The emissivities of these processes are displayed in Figure 3.3, for two different internal temperatures, T = and K respectively. The synchrotron emission is plotted for three magnetic field strengths B = 10 14, and G. The different mechanisms will only be briefly explained below, but an extensive discussion of these processes can be found in Yakovlev et al. (2001). Plasmon decay A free electron cannot emit a neutrino pair, since this is forbidden by energy-momentum conservation. However, an electron interacting with its environment, can. Plasmon decay is an example of such an interaction and can be extremely efficient at high temperatures and not too high densities (see Figure 3.3). The name plasmon is given to a collective oscillation of a free electron gas (due to interactions with photons), which can spontaneously de-excite under the emission of a neutrino pair (see e.g., Yakovlev et al., 2001). The process can be written as e e + ν + ν (3.8) 39

50 Chapter 3: Neutron star cooling 18 plasmon e-brems. 14 Cooper pairs Log Q [erg cm s ] syn(13) syn(12) syn(14) Log Q [erg cm s ] e-brems. neutron drip syn(14) syn(13) syn(12) Log g cm Log g cm Figure 3.3: Density dependence of the dominant neutrino emission processes in the neutron star crust at T = K (left) and T = K (right). Synchrotron emission is denoted as syn, where the number in parenthesis is the log of the magnetic field strength. Furthermore, e-brems is short for bremsstrahlung from electron-ion interactions and plasmon refers to plasmon decay. These calculations were done by Yakovlev et al. (2001). where e denotes a plasmon. Emission from the formation of Cooper pairs In the neutron star crust, neutrons can become superfluid and the formation of Cooper pairs is accompanied by the emission of neutrinos. Cooper pair formation in the crust is very similar to the analogue process in the core, which was discussed in Section Neutrino emission from Cooper pair formation only comes in to play when the temperature drops below the critical temperature for the onset of superfluidity. In the crust, T c is typically several times 10 8 K. Synchrotron emission In addition to the above discussed crustal emission processes, the presence of a magnetic field will enforce electrons in the crust to move in a helical motion around the magnetic field lines. The accelerated electrons produce synchrotron emission according to the reaction e e + ν + ν (3.9) 40

51 3.1 General cooling scenarios As can be seen in Figure 3.3, this process will only make a notable contribution to the total neutrino emission from the crust at low temperatures and for high magnetic fields, G. Neutrino bremsstrahlung from electrons In the neutron star crust, electrons can scatter of the Coulomb field of nuclei. This results in the emission of a neutrino/anti-neutrino pair of arbitrary flavor referred to as neutrino bremsstrahlung: e + (A, Z) e + (A, Z) + ν + ν (3.10) Note that the emissivity of neutrino bremsstrahlung is nearly independent of density, in contrast to that of Cooper pair formation and plasmon decay Photon emission Throughout the life of a neutron star, heat from the hot interior will be transported through the surface layers and radiated away in the form of photons. The photon luminosity emerging from a neutron star surface is L γ = 4πR 2 σ B Te 4 (3.11) where T e is the effective temperature and σ B is the Stefan-Boltzman constant. Since R = RΓ and T e, = T/ Γ 2 (where Γ = (1 2GM/Rc 2 ) 1/2 ), the observable photon luminosity will be L γ, = L γ /Γ 4. The relation between the surface temperature and that of the neutron star interior is determined by the thermal conductivity in the outer neutron star layer, the envelope, where heat is transported by electrons. If the envelope consists mainly of light elements, electrons undergo very few collisions and have a large mean free path. Consequently, heat is transported very quickly. Heavy elements, on the other hand, have lower electron opacities and will cause a less efficient heat transport. These differences will affect the observable cooling curves, as is illustrated by Figure 3.4. This figure displays the thermal evolution of both the surface- and core temperature of a neutron star for two extreme envelope compositions. The curve denoted by L describes the temperature evolution of a crust constructed only out of light elements, while the model denoted by H incorporates a crust consisting of iron-like elements. During the neutrino cooling stage (t 10 4 yr), the interior temperatures are similar for both models, since the envelope does not contribute to the total energy loss of the star at this stage. However, the observable surface temperature of a neutron star with a light envelope will be higher, owing to the more efficient transport of heat from the interior. Furthermore, due to the high thermal conductivity, a neutron star with a light envelope will shift to the photon cooling stage at a much earlier time. The relation between the effective temperature, T eff, and the internal temperature, T, of a neutron star is roughly T eff T 0.5 (Gudmundsson et al., 1982; Potekhin et al., 1999). Thus, 41

52 Chapter 3: Neutron star cooling Figure 3.4: Effect of the envelope composition on the thermal evolution of the internal temperature (left) and the observable surface temperature (right). Both temperatures are redshifted to infinity for reference (Page et al., 2004). the photon luminosity scales with the internal temperature according to L γ T 2. Due to the much stronger temperature dependence of L ν (recall Table 3.1) compared with L γ, neutrino emission will drive the cooling at early times, but will drop below the photon luminosity for sufficiently low temperatures ( 10 8 K). Therefore, we can distinguish two different cooling stages: the neutrino cooling stage and the photon emission phase, as was discussed in the introduction Section Effects of Superfluidity The effects of nucleon superfluidity on the thermal evolution of a neutron star are twofold. As we saw above, superfluidity provides an additional neutrino emission mechanism, and can thereby accelerate the neutron star cooling, especially when durca reactions are prohibited. On the other hand, nucleons residing in Cooper pairs cannot participate in the particle reactions involving single nucleons, such as the durca, murca and bremsstrahlung processes. The transition to the superfluid state basically makes the nucleons inert and all the neutrino emission reactions involving those particles will seize. Thus, opposed to opening up a new neutrino emission mechanism, nucleon superfluidity will suppress all neutrino emission processes in the core (neutrino emission reactions in the crust do not involve free nucleons). Figure 3.5 illustrates the different effects of nucleon superfluidity in the core on the thermal evolution of a neutron star. During the neutrino cooling stage (shallow slope), the pairing in itself reduces the cooling rate, due to the fact that all neutrino emission processes associated with the superfluid particles are suppressed. However, superfluidity also opens up an additional channel for heat loss and induces a net acceleration of the neutron star cooling. In the neutron star crust, the heat capacity is set by that of degenerate neutrons and will decrease drastically when neutrons become superfluid (this will be made qualitative in Section 4.1). A small heat capacity will allow for more rapid cooling during the photon cooling stage (steep slope), due to the fact that heat can be less efficiently stored in the neutron star. This is merely a simple example, the effects of superfluidity on the thermal evolution of neutron stars 42

53 3.1 General cooling scenarios are comprehensive (see e.g., Yakovlev et al., 1999, for an illustrative review). 6.5 pairing, no CP emission Log T e [K] no pairing pairing and CP emission Log t [yr] Figure 3.5: Comparison of the cooling curve of a 1.4 M (slow cooling) neutron star, both including (dotted line) and in absence (middle curve) of nucleon superfluidity in both the crust and the core. For reference, there is also a curve plotted in which superfluidity is included, but the accompanying neutrino emission from Cooper pair formation is omitted. This illustrates the two-fold effect of nucleon Cooper pairing (Page et al., 2004) Cooling models As the previous sections have shown, the observable thermal evolution of neutron stars is sensitive to the core EOS, the neutron star mass (since it sets the central density of the star, that in its turn determines which neutrino emission processes are allowed for), nucleon superfluidity and the composition of the crust. When all the possibilities are considered, a wide range of possible thermal histories, or cooling curves, result. If thermal emission is observed, and ages for these neutron stars can be estimated (e.g., via the age of an associated supernova remnant), theoretical cooling curves can be compared with observations and allow to explore the interior properties of neutron stars. Figure 3.6 shows an example of such an approach. It displays theoretical cooling curves for four qualitatively different core EOSs and compositions (Yakovlev and Pethick, 2004). The upper curve is for a neutron star with a low central density, which cools slowly via nucleon-nucleon bremsstrahlung. Each model is limited by this low-mass curve and by a faster cooling curve, which corresponds to a more massive 43

54 Chapter 3: Neutron star cooling neutron star. Between these two boundaries, lie a sequence of cooling curves for neutron stars with intermediate masses (hatched regions). The theoretical models are compared with observations of a selection of isolated neutron stars, from which thermal emission is observed and ages can be estimated. The observations displayed in Figure 3.6 are discussed in Yakovlev and Pethick (2004). The data seems consistent with any of the theoretical models and no EOS can be rejected. However, it is clear that should cooler neutron stars ever be observed, this would have important implications for the composition of matter at supra-nuclear densities. Figure 3.6: Theoretical cooling curves for four different core EOSs and compositions. The models are compared with observations of isolated neutron stars that show thermal surface emission and for which ages can be estimated. This figure is taken from Yakovlev and Pethick (2004). 3.2 Cooling of accretion heated neutron stars Heating The general cooling scenario, as described in Section 3.1, will be altered when a neutron star accretes matter from a companion. As was discussed in Section 2.3, accreted matter sinks deep into the neutron star crust where it undergoes chemical reactions that release significant amounts of heat. Brown et al. (1998) showed that deep crustal heating can efficiently maintain 44

55 3.2 Cooling of accretion heated neutron stars the core of an accreting neutron star at a temperature (5 10) 10 7 K. Normally, thermal surface radiation from accreting neutron stars would not be observable, being overwhelmed by X-ray emission from the hot accretion disk. However, it has been proposed that transiently accreting neutron stars will emit observable thermal emission during periods of quiescence (e.g., Brown et al., 1998). If this hypothesis is true, the cooling curves of these sources can also be used to study various neutron star parameters by confronting theory with observations. During accretion, a temperature inversion between the crust and the core may develop, due to the fact that the crust is efficiently heated by nuclear reactions. When accretion halts, the crust cools down via various sources of heat loss (see Figure 3.7) and the luminosity of the neutron star will reflect the thermal state of the crust. This scenario resembles the situation in the thermal relaxation phase of young neutron stars (see Section 3.1). The thermal state of the neutron star core and crust in quiescence is set by the timeaveraged accretion rate (over many outbursts) and the neutron star properties. Crust cooling curves allow us to probe the properties of neutron stars, such as, the kind of core cooling process at work and the thermal conductivity of the crust (which in its turn depends on properties of the ion lattice). While it is expected that the crust can cool significantly between outbursts, the core will only cool significantly on much longer timescales. Deep crustal heating zone L crust isothermal core L cond L core L Crust-core interface Figure 3.7: Schematic overview of the dissipation of heat in an accretion heated neutron star. Most of the heat is deposited near the crust/core interface in the deep crustal heating zone. This heat is dissipated through photon or neutrino emission from the crust or through conduction into the core Observed cooling curves There exists a considerable group of identified quasi-persistent neutron star LMXBs, residing either in outburst or in quiescence. A few were listed in Table 1.1. Of two quasi-persistent 45