Log Energy (keV)

Figure 1.10. Multiwavelength spectrum of the Vela pulsar (from Pavlov et al. 2002 with the kind permission of the authors). The solid line shows the fit to the Chandra X-ray observations with the model spectrum of the thermal (neutron-star hydrogen atmosphere model) plus nonthermal (power-law) radiation components (including the effect of interstellar absorption). The dot-and-dashed line is the extrapolation of the fit to the ultraviolet and optical bands. The dotted line is the same extrapolated fit but excluding the effect of interstellar absorption. The various symbols show the spectrum detected with other observatories in the optical and gamma-ray bands.

gated object is J1210-5226 (=1E 1207.4-5209) in the center of the supernova remnant G296.5+10.0 (t & (3 - 20) x 103 years). It is the first isolated neutron star found to exhibit pronounced spectral features (X-ray absorption spectral lines) in its radiation spectrum (Sanwal et al., 2002) although the interpretation of these features seems ambiguous. Its radiation contains the thermal component which can be interpreted with the aid of a hydrogen atmosphere model as the thermal radiation from the surface of the neutron star with the effective temperature ~ (1.4 - 1.9) x 106 K. Some other objects (e.g., J2323+5848 in Cassiopeia A, t ~ 320 years) show thermal-like radiation which cannot be emitted from the entire stellar surface. The radius of the emission region, inferred from observations, is 0.5-1 km, much smaller than the expected neutron star radius. This radiation may be produced by a spot on the neutron star surface, but in this case the absence of pulsations of the observed radiation requires explanation.

There is also a class of dim isolated neutron stars not associated directly with supernova remnants. Their emission is characterized by black-body X-ray spectra with the effective temperatures ~ (0.5 - 1) x 106 K. They are probably nearby isolated neutron stars. The most famous is RX J1856.5-3754 discovered by Walter et al. (1996). Parallax measurements give the distance 140 ± 40 pc which makes this object one of the closest observed neutron star (Kaplan et al., 2002)9 the age ~ 5 x 105 years is estimated from kinematics of proper motion. Observations show no spectral lines and no pulsations. The effective surface temperature is estimated to be Ts & 4.3 x 105 K (Ho et al., 2006). Another example - RX J0720.4-3125 - is a dim object which shows periodic variations with a long period P = 8.4 s. Its characteristic age is t ~ 1.3 x 105 years, and Ts ~ 5 x 105 K. This object shows a phase-dependent absorption feature (Haberl et al., 2004) and precession (Haberl et al., 2006).

Soft gamma repeaters + anomalous X-ray pulsars = magnetars. Soft gamma repeaters and anomalous X-ray pulsars are two other types of isolated neutron stars. They seem to form a larger class of magnetars (see, e.g., Thompson 2002, Kaspi 2004, and references therein).

Soft gamma repeaters (SGRs) are sources of repeating soft gamma-ray and X-ray bursts. Typical bursts last for ~ 0.1 s and have energies ~ 1041 erg. Their bursting activity is highly irregular. Years of quiet states are interlaced with weeks of hundreds of bursts. By 2006 four soft gamma repeaters and two candidates have been discovered. The first discovered object, SGR 052566, is in the Large Magellanic Cloud, whereas other ones are in the Galactic plane. The most remarkable events were three gigantic gamma-ray bursts, much stronger than typical bursts. The first one was detected from SGR 0525-66 on March 5, 1979 (Mazets et al., 1979a), the second one was detected from SGR 1900+14 on August 27, 1998 (Hurley et al., 1999) and the third from SGR 1806-20 on December 27, 2004 (e.g., Hurley et al. 2005; Mazets et al. 2005). The energy of the third burst was especially huge and exceeded 1046 erg.

Periodic pulsations with large periods, from 5 to 8 s, have been detected in X-rays from the three sources. Two of them show pulsations in quiescent states which have enabled one to measure P. In particular, one has got P = 5.2 s and P = 6.1 x 10"11 for SGR 1900+14. Then Eq. (1.15) gives the characteristic age t ~ 1.3 x 103 years. Using Eq. (1.13) (with all the reservations about its validity!), we immediately obtain an enormous characteristic magnetic field Beff ~ 5.7 x 1014 G. There are other arguments that soft gamma repeaters are young, slowly rotating and rapidly spinning down neutron stars with superstrong magnetic fields B ~ 1014-1015 G.

9By the time of this writing, the most recent parallax measurements give the distance ~ 160 pc (D.L. Kaplan et al., in preparation).

Anomalous X-ray pulsars (AXPs) are sources of pulsed X-ray emission. The pulsation periods range from 6 to 12 s, and the X-ray luminosities range from ~ 1033 to ~ 1035 erg s"1. These pulsars differ from the classical X-ray pulsars in X-ray binaries (§ 1.4.6) by the absence of any evidence that they enter binary systems. By 2005 five AXPs were discovered, together with several candidates. Some of them have been detected in optical. In most of the cases pulsar timing has been performed and the values of P. have been measured. The estimated characteristic ages are slightly higher than for soft gamma repeaters, but the characteristic magnetic fields are of the same order of magnitude. For instance, for 1E 1048.1-5937 one has P = 6.4 s, P = 3.3 x 10-11, t ~ 3.1 x 103 years, and Beff ~ 4.7 x 1014 G.

Therefore, AXPs have much in common with soft gamma repeaters. A solid piece of evidence that these sources are related was provided by the discovery of bursting activity of AXPs (in particular, two bursts, separated by 16 days, from 1E 1048.1-5937, Gavriil et al. 2002; and over 80 bursts detected in June 2002 from 1E 2259+586, Kaspi et al. 2003). It is currently assumed that soft gamma repeaters and AXPs belong to the same class of neutron stars, which are called magnetically powered pulsars or magnetars - neutron stars with superstrong magnetic fields. The magnetar hypothesis was put forward, on theoretical grounds, by Duncan & Thompson (1992) and Paczynski (1992). Soft gamma repeaters are thought to be younger and transform into a AXPs in the course of their evolution. The sources of both types can be powered by huge magnetic fields located in neutron star interiors. Bursts are thought to be associated with episodic releases of stresses caused by the evolution of magnetic fields in neutron star crusts. The superstrong magnetic field is estimated to decay in ~ 104 years hampering the activity of these sources when they become older. Further observations are required to confirm these ideas.

1.4.6 Neutron stars in binary systems - X-ray binaries

Neutron stars have been observed in binaries with other neutron stars, white dwarfs, and nondegenerate stars. These systems are useful for measuring neutron star masses, testing theories of stellar evolution, and solving many other problems (see, e.g., § 9.1). We are still waiting for a discovery of a neutron star in binary with a black hole. Observed binaries can be divided into wide systems (without mass exchange) and more compact systems (with mass transfer, which often results in accretion onto a neutron star). If the mass transfer is absent, a neutron star behaves usually as an isolated object (§ 1.4.5). A mass transfer in a compact binary makes this binary an X-ray source. Such systems are called X-ray binaries, and they are outlined below. Many of them are observed not only in X-rays but also in other spectral bands. Some X-ray binaries contain black holes rather than neutron stars.

There is a rich phenomenology of X-ray binaries containing neutron stars (see, e.g., Lipunov 1992 and Lewin et al. 1997; also see Table 1.1). Their X-ray emission is generated either at (near) neutron star surfaces and/or in accretion disks. Generally, these binaries are divided into high-mass X-ray binaries (HMXBs), M2 > (2 — 3) Mq, and low-mass X-ray binaries (LMXBs), M2 < Mq, with respect to companion masses M2. By 2006, about 100 low-mass X-ray binaries and about 40 high-mass X-ray binaries containing neutron stars have been discovered. Neutron star companions in high-mass binaries are usually massive O-B stars, while in low-mass binaries they are dwarf stars (particularly, red dwarfs). Massive O-B stars produce strong stellar wind; its accretion on a neutron star may be nearly spherical. Life times of massive main-sequence stars, and hence life times of high-mass X-ray binaries are sufficiently short. A strong accretion from dwarf companions in compact low-mass X-ray binaries occurs if a dwarf star fills its Roche lobe and the plasma outflows through the first Lagrange point. This accretion regime is favorable for the formation of an accretion disk. X-ray emission produced by a population of (unresolved) high-mass X-ray binaries in distant galaxies serves as the indicator of the star formation rate in these galaxies (see, e.g., Grimm et al. 2003).

X-ray binaries can be sources of regular (periodic) and irregular emission; they can also be subdivided into persistent and transient sources. The latter sources are called X-ray transients. X-ray binaries can be observed as X-ray pulsars, X-ray bursters, sources of quasiperiodic X-ray oscillations (QPOs), etc. X-ray pulsars are powered by accretion. These accretion-powered pulsars should not be confused with rotation-powered pulsars or AXPs, §§ 1.4.4 and 1.4.5. The complicated phenomenology of X-ray binaries reflects the complex nature of these sources which is far from being clear. The same source can manifest itself in different ways. For instance, Vela X-1 is a classical (accretion powered) X-ray pulsar, a persistent source of regular X-ray pulsations. A 0538-66 is also an X-ray pulsar whose activity is transient. XTE J2123-058 (§ 9.1.1 c) is an X-ray transient which demonstrates X-ray bursts and quasiperiodic oscillations.

X-ray transients. They are X-ray sources which go from active (or 'on') to quiescent (or 'off') states and back on timescales of some hours and longer. As a rule, quiescent states last longer than active ones. The first transient, Cen X-2, was discovered by Harries et al. (1967) in April 1967 with rocket-born X-ray detectors.

X-ray transients do not form a uniform class of objects. For instance, the X-ray pulsar A 0538-66 with the spin period P = 69 ms demonstrates transient X-ray activity with the well determined period Pb = 16.66 days, which is the orbital period in a highly eccentric binary. Strong accretion, which powers the pulsar, occurs only near periastron passages and turns the system into active states for short periods of time. However, the majority of transients show irregular sequence of active states (weeks-months or even years) interspersed with longer periods of quiescence (months-years or even decades). Active states can be switched on by many mechanisms, particularly, by instabilities in accretion disks, irregular outflow of matter from a donor star, or by the changes of the accretion regime near a neutron star surface.

Some X-ray transients, for instance 4U 0115+63, have hard spectra in active states, with spectral fluxes extended to some tens keV. They are called hard X-ray transients (HXTs), and they are usually identified with high-mass X-ray binaries (with Be companions). Other transients have softer spectra (extended to < 1 — 2 keV in active states). Accordingly, they are called soft X-ray transients. They are compact low-mass X-ray binaries. Some soft X-ray transients in quiescence (for instance, Aql X-1) show thermal-like radiation component which can be fitted by neutron-star atmosphere models. As suggested by Brown et al. (1998), this radiation emerges from the interiors of warm neutron stars (with the surface temperatures ~ 106 K) being produced by deep crustal heating in the inner crust. The heating mechanism proposed by Haensel & Zdunik (1990a) consists in pycnonuclear burning of accreted matter sinking in the crust under the weight of newly accreted material. There is a close correspondence (see, e.g., Yakovlev & Pethick 2004 and references therein) between the theory of thermal states of transiently accreting neutron stars and the theory of neutron star cooling (§ 1.3.7). Some X-ray transients have short active states (hours-days).

X-ray pulsars in X-ray binaries. They are accretion-powered rotating and strongly magnetized neutron stars in compact binaries. The first source recognized as an X-ray pulsar was Cen X-3 (in observations of Schreier et al. 1972 with the Uhuru X-ray satellite). There are about 35 X-ray pulsars in our Galaxy known by 2006. They mainly enter high-mass X-ray binaries (HMXBs) with intense accretion. An example is Vela X-1, a binary with the orbital period Pb ~ 9 days. It consists of a neutron star and a companion, GP Vel, a B0.5 Ib supergiant whose mass is (23-28) Mq (§ 9.1.1 b). The pulsar spin period is P = 283 s. GP Vel nearly fills its Roche lobe and produces a powerful stellar wind. The star is bulky and creates eclipses of the X-ray source. Many X-ray pulsars are very slow rotators (P > 100 s) but not all (e.g., P = 69 ms for A 0538-66).

It is thought that the accretion is channeled by the pulsar magnetic field into a thin accretion column near the neutron star surface. Strong X-ray emission from these columns corotating with neutron stars creates regular X-ray pulsations. The accretion energy release rate can be estimated as

Eacc - GMM/R - 8.4 x 1035M-10 (M/Mq)/R6 erg s-1, (1.16)

where M_ 10 is the mass accretion rate M in units of 10"10 Mq yr-1. X-ray luminosities of X-ray pulsars range from ~ 1035 to ~ 1039 erg s-1. As arule, an intense accretion spins up a neutron star, decreasing the pulsar spin period P (in contrast to rotation-powered pulsars). However, rotation of some pulsars (e.g., Vela X-1) is nearly spun up to the equilibrium limit; their spin periods seem to undergo variations around these steady-state values.

Spectra of some X-ray pulsars show prominent electron cyclotron lines which serve to directly measure pulsar magnetic field B. The cyclotron lines are observed at photon energies ~ hwc ~ (30 — 50) keV (where wc is the electron cyclotron frequency) and give B ~ (3 — 5) x 1012 G. These lines were predicted by Gnedin & Sunyaev (1974) and discovered by Tamper et al. (1978) in the spectrum of Her X-1.

X-ray bursters. On September 28, 1975 Grindlay et al. (1976) discovered two X-ray bursts from the X-ray source 4U 1820-30 in the globular cluster NGC 6624 with the Astronomical Netherlands Satellite (ANS). Since then X-ray bursters have been observed many times (see, e.g., Strohmayer & Bildsten 2004). The total number of detected X-ray bursters is about 50. An X-ray burst lasts usually from a few to a few tens of seconds. Bursts repeat quasiperiodically with the recurrence time of several hours. X-ray bursts are thought to occur on the surfaces of neutron stars in compact low-mass X-ray binaries (LMXBs). A low-mass companion fills its Roche lobe and ejects matter, which is accreted by a neutron star. In some cases, X-ray eclipses have been detected. X-ray bursters concentrate within the Galactic bulge; many (but not all) are observed in globular clusters. The bursts are detected in soft X-rays, their spectra are much softer than the spectra of X-ray pulsars. Soft X-ray transients in active states are usually bursting sources.

All X-ray bursts are subdivided into two nonequal parts: type I and type II bursts. Type I bursts are widespread phenomena. X-ray luminosity LX in burst maxima often reaches the Eddington limit, LEdd ~ 1038 erg s-1, Eq. (1.3). An X-ray energy emitted during one burst constitutes typically ~ 10-2 of the energy emitted during a recurrence (quasi)period. Type I bursts are explained by explosive thermonuclear burning of accreted matter on the surfaces of neutron stars with low magnetic fields (B < 108 — 109 G). The nuclear energy release 5 MeV per one accreted nucleon) is just ~ 10-2 of the accretion energy 200 MeV per nucleon) responsible for the persistent X-ray emission. The instability of accreted matter with respect to nuclear burning was predicted by Hansen & Van Horn (1975); it was related to X-ray bursters by Woosley & Taam (1976). The state of the theory by 2004 is described, for instance, by Strohmayer & Bildsten (2004) and Woosley et al. (2004).

Among all X-ray bursts, one can clearly distinguish the so called superbursts. They were discovered in the system 4U 1735-444 by Cornelisse et al. (2000).

They are rare events, but very strong. They last 2-12 hours and their recurrence times are of several years. The total energy release in a superburst can be as high as ~ 1042 erg, several orders of magnitude higher than in an ordinary X-ray burst (see Strohmayer & Bildsten 2004 for review). They are usually explained by unstable carbon burning in deep layers of the outer crust of an accreting neutron star (although this explanation meets some difficulties; see, e.g., Page & Cumming 2005). Such a burning has been studied theoretically (Woosley & Taam, 1976; Taam & Picklum, 1978; Brown & Bildsten, 1998) before the discovery of superbursts.

Type II X-ray bursts are demonstrated by two sources - neutron stars in transiently accreting LMXBs. These bursts are very frequent (with variable burst intervals which can be as short as tens of seconds); the burst energy correlates with burst intervals. It is most likely that these bursts are associated with the nonstationary accretion onto a neutron star (Lamb & Lamb, 1977), and the burst energy is supplied by accretion. The first source, MXB 1730-335, is the famous rapid burster discovered in March 1976 during observations with the Small Astronomical Satellite (SAS 3) observatory (Lewin et al., 1976). It shows also type I bursts. The second source, GRO J1744-28, was discovered in December 1995 with the Burst and Transient Source Experiment (BATSE) aboard the Compton Gamma Ray Observatory (Fishman et al. 1995, Kouveliotou et al. 1996). In addition to type II X-ray bursts it demonstrates periodic X-ray pulsations (revealing neutron star spin period P = 0.467 s). It is called the bursting pulsar.

It was a long-standing problem to detect periodic X-ray pulsations associated with neutron star rotation in X-ray bursters of type I, but it was solved. Wijnands & van der Klis (1998) discovered regular P = 2.5 ms pulsations from an outburst of the X-ray transient SAX J1808.4-3658 in the observations with the Rossi X-ray Timing Explorer. That was the first discovered accreting millisecond pulsar and the first observational evidence that millisecond pulsars are associated with LMXBs. Now we know other examples (e.g., XTE J1814-338, Strohmayer et al. 2003). Another example - neutron-star spin pulsations were observed during a superburst of 4U 1636-53 (Strohmayer & Markwardt, 2002). These observations indicate the presence of nonuniform regions (hot spots) on the surfaces of X-ray bursters. Comparing theoretical models of these spots with observations one will be able to obtain useful constraints on neutron star masses and radii (e.g., Strohmayer 2004, Bhattacharyya et al. 2005).

Very powerful X-ray bursts have super-Eddington luminosities, so that the radiative pressure in the neutron star atmosphere exceeds the gravity. Such a burst initiates a huge expansion of the neutron star atmosphere by the radiative pressure (up to a few hundred kilometers) followed by a contraction to to the initial state. These bursts serve as nearly standard candles (with LX slightly higher than LEdd) useful to estimate distances to the bursters.

Sources of quasiperiodic X-ray oscillations. Some X-ray binaries are the sources of quasiperiodic X-ray oscillations (QPOs). These oscillations are not exactly periodic - not pulsar clocks; also see § 9.3.2. They were discovered by van der Klis et al. (1985) in observations of the LMXB GX 5-1 (4U 1758-25) with the European X-ray Observatory Satellite (EXOSAT), operated from 1983 to 1986. Great progress in observations of quasiperiodic oscillations has been made with the Ginga (Japanese for galaxy) satellite (1987-1991) and with the Rossi X-ray Timing Explorer (launched in 1996).

Quasiperiodic X-ray oscillations have been observed from X-ray binaries containing neutron stars, black holes and white dwarfs. We focus on the binaries with neutron stars (see, e.g., van der Klis 2000). By 2006 about 20 such objects were discovered and identified as compact LMXBs; some of them are X-ray bursters. In 16 systems, the neutron star spin period has been determined. Drifts of quasiperiodic oscillation frequencies have been analyzed, as well as the tracks of the sources on the so called color-color diagram (hardness ratio in harder X-ray channels versus hardness ratio in softer channels). One distinguishes "atoll" and "Z"sources which have corresponding tracks. It is especially important to analyze the power spectrum of X-ray flux fluctuations which extends from tens of Hz to about 1 kHz (to 1.330 kHz for 4U 0614+09) with a cutoff afterwards. A power spectrum may contain several (up to three) pronounced peaks, which may drift from one observation to another.

Theoretical interpretation of a zoo of observational properties of quasiperiodic oscillations is not simple. It is likely that oscillations occur in the accretion disks around neutron stars withlow magnetic fields (B < 108 —109 G). Oscillation frequencies can be associated with the Keplerian frequency of the innermost stable orbit of matter elements in a disk, or with some resonant frequency in the disk itself, or with combination of these frequencies and neutron-star spin frequency. After understanding the real nature of quasiperiodic X-ray oscillations, their observation will be very useful to put stringent constraints on neutron star masses and radii.

1.4.7 Neutron stars in the Galaxy

X-ray binaries in the Galaxy concentrate to the galactic bulge and the galactic plane. By contrast, the distribution of radio pulsars (see, e.g., Arzoumanian et al. 2002, and references therein) is drastically different from the distribution of X-ray binaries and normal stars. Some radio pulsars are observed at high galactic latitudes, and many of them demonstrate strong proper motion, with the velocities v > 500 km s_1. Thus, neutron stars populate much wider space and move with much higher velocities than other stars. Observational constraints on radio pulsar (velocity and spatial) distribution in the Galaxy are still rather uncertain. Nevertheless, there are strong indications of the two-component velocity distribution with characteristic velocities of ~ 100 km s_1 and ~

500 km s"1, respectively. Both components contain a comparable number of sources, and ~ 10% ofradio pulsars have velocities > 1000 kms"1. Theescape velocity from the Galactic potential is estimated to range from 450 km s"1 to 650 km s"1 (Leonard & Tremaine, 1990). Thus, a sizable fraction of all radio pulsars are sufficiently fast to escape from the Galaxy. This means that the Galaxy possesses an extended halo of radio pulsars which are evaporated into the intergalactic space. The fastest is PSR B2224+65 in the Guitar Nebula whose projected (perpendicular to line of sight) velocity ~ 1600 km s"1 is nearly parallel to the Galactic plane (Cordes et al., 1993). Another pulsar, PSR J1740+1000, may be moving much (about twice) faster but its velocity measurement (McLaughlin et al., 2002) is still ambiguous.

It is believed that high pulsar velocities are gained at neutron star birth due to pulsar kicks. The origin of huge kick velocities is a subject of debates (see, e.g., Lai et al. 2001, Arzoumanian et al. 2002 and references therein).

1.5. Neutron stars as "superstars" in physics and astrophysics

The Pines theorem. The theorem was formulated by David Pines in a talk given at the conference on "Neutron Stars: Theory and Observation" (The NATO Advanced Study Institute, Crete, Greece, September 3-14, 1990). The formulation is fairly simple:

Neutron Stars are Superstars.

Proof. After reading §§ 1.1-1.4 the proof is trivial. Indeed, neutron stars are superdense objects; superfast rotators; superfluid and superconducting inside; super accelerators of high-energy particles; sources of superstrong magnetic fields; superprecise timers; superglitching objects; superrich in the range of physics involved. Neutron stars are related to many branches of contemporary physics and astrophysics, particularly to nuclear physics; particle physics; condensed matter physics; plasma physics; general theory of relativity; hydrodynamics; quantum electrodynamics in superstrong magnetic fields; quantum chromodynamics; radio-, optical-, X-ray and gamma-ray astronomy; neutrino astronomy; gravitational-wave astronomy; physics of stellar structure and evolution, etc.

Let us stress that neutron stars contain the matter of essentially supranuclear density in their interiors (§ 1.3.1). This state of matter cannot be reproduced in laboratory because the nuclear matter is known to be highly incompressible under laboratory conditions. The compression in neutron stars is produced by enormous gravitational forces. This enables one to treat neutron stars as unique natural laboratories of superdense matter under the most extreme conditions.

One can test theoretical models of dense matter by comparing observations of neutron stars with theoretical predictions.

Observational manifestations of neutron stars are really numerous (§ 1.4). The need to observe these objects has triggered the development of foremost telescopes and detectors, from the best modern radio telescopes to laser interferometers for detecting gravitational waves. One cannot imagine the modern observational astrophysics without the astrophysics of neutron stars.

In other words, neutron stars are fascinating objects to observe and to study theoretically.

1.6. About the rest of this book

We hope that, after reading this chapter, the reader has become familiar with the main ideas of neutron star physics and can decide if it is worthwhile to read further. Naturally, the subject is too wide to be discussed in one book. Moreover, the field is rapidly developing; many problems have not been solved yet; many outstanding discoveries could be expected soon after this writing.

In the next chapters we will focus on the internal structure of neutron stars. We will discuss thermodynamic properties of the matter in all neutron star layers, from the surface to the center (Chapters 2-5, 7 and 8). Our main concern will be to consider the structure, composition and equation of state of dense matter, particularly, the basic problem - the equation of state in inner neutron star cores (§ 1.3.2). We will also discuss the models of neutron star structure, masses and radii of neutron stars (Chapters 6 and 8), and observational tests for these models (Chapter 9). We will try to be pedagogical and describe not only the results of sophisticated theories but explain these theories and underlying ideas. Although the book is written by theoreticians, we will summarize the necessary data coming from the nuclear physics experiments and from some neutron star observations.

In the present book we do not discuss in detail kinetic properties of neutron star matter, neutrino emission mechanisms, thermal evolution of neutron stars, evolution of their magnetic fields, and associated observational problems. We hope to address these issues in a separate book. The present book also does not touch many other problems, for instance, neutron star birth in supernova explosions, the structure and evolution of protoneutron stars, the evolution of neutron stars in binary systems, the physics of neutron star magnetospheres. Some references to these subjects can be found in this and subsequent chapters of the present book.

Chapter 2

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