Introduction Historical Overview

The idea of neutron stars can be traced back to the early 1930s, when Subrahmanyan Chandrasekhar discovered that there is no way for a collapsed stellar core with a mass more than 1.4 times the solar mass, M0, to hold itself up against gravity once its nuclear fuel is exhausted. This implies that a star left with M > 1.4 M0 (the Chandrasekhar limit) would keep collapsing and eventually disappear from view.

After the discovery of the neutron by James Chadwick in 1932, scientists speculated on the possible existence of a star composed entirely of neutrons, which would have a radius of the order of R ~ (h/mnc)(h c/Gm2)l/2 ~ 3 x 105 cm. In view of the peculiar stellar parameters, Lev Landau called these objects "unheimliche Sterne" (weird stars), expecting that they would never be observed because of their small size and expected low optical luminosity.

Walter Baade and Fritz Zwicky were the first who proposed the idea that neutron stars could be formed in supernovae. First models for the structure of neutron stars were worked out in 1939 by Oppenheimer and Volkoff (Oppenheimer-Volkoff limit). Unfortunately, their pioneering work did not predict anything astronomers could actually observe, and the idea of neutron stars was not taken serious by the astronomical community. Neutron stars, therefore, had remained in the realm of imagination for nearly a quarter of a century, until the discovery of pulsars and of accreting binary neutron stars with Uhuru [16,50,94].

The discovery of the first radio pulsar was very soon followed by the discovery of the two most famous pulsars, the fast 33 ms pulsar in the Crab Nebula and the 89 ms pulsar in the Vela supernova remnant. The fact that these pulsars are located within supernova remnants provided striking confirmation that neutron stars are born in core collapse supernovae from massive main sequence stars. These exciting radio discoveries triggered subsequent pulsar searches at nearly all wavelengths. Since those early days of pulsar astronomy more than 1750 radio pulsars have been discovered by now (see, e.g., the ATNF pulsar database [66]).

Many radio pulsars had been observed by the mid-seventies, and two of them, the Crab and Vela pulsars, had been detected at high photon energies. Although the interpretation of both isolated and accreting pulsars as neutron stars with enormous magnetic fields, ~ 1012 G, had been generally accepted, the first direct measurement of a neutron star magnetic field came from an X-ray observation: an electron cyclotron line feature discovered in the accreting binary pulsar Her X-1 indicated a polar magnetic field of ^4 x 1012 G [99] (cf. chapter 15).

Particularly, important results on isolated neutron stars, among many other X-ray sources, were obtained with HEAO-2, widely known as the Einstein X-ray observatory. Einstein investigated the soft X-ray radiation from the previously known Crab and Vela pulsars and resolved the compact nebula around the Crab pulsar [2]. It discovered pulsed X-ray emission from two more very young pulsars, PSR B0540—69 in the Large Magellanic Cloud and PSR B1509-58, having periods of 50 and 150 ms, respectively. Interestingly, these pulsars were the first ones to be discovered in the X-ray band and only subsequently at radio frequencies. Einstein also detected X-rays from three middle-aged radio pulsars, PSR B0656+14, B1055—52, B1951+32, and the X-ray counterparts of two nearby old radio pulsars, PSR B0950+08 and B1929+10. In addition, many supernova remnants were mapped - 47 in our Galaxy and 10 in the Magellanic Clouds and several neutron star candidates were detected as faint, soft point sources close to the center of the supernova remnants RCW 103, PKS 1209—51/52, Puppis-A and Kes 73.

Some additional information on isolated neutron stars was obtained by EXOSAT (European X-ray Observatory Satellite). In particular, it measured the soft X-ray spectra of the middle-aged pulsar PSR B1055—52 and of a few neutron star candidates in supernova remnants (e.g., PKS 1209—51/52).

The situation improved drastically in the 1990s because of the results from ROSAT, ASCA, EUVE, BeppoSAX, and RXTE, as well as Chandra and XMM-Newton launched close to the millennium.

The complement to ROSAT, covering the harder X-ray band 1-10 keV, was ASCA launched in 1993. The EUVE (Extreme Ultraviolet Explorer) was launched in 1992 and was sensitive in the range 70-760 A. It was able to observe several neutron stars at very soft X-rays, 0.07-0.2 keV.

The contributions to neutron star research, provided by the instruments aboard BeppoSAX, sensitive in the range of 0.1-200 keV, and RXTE (Rossi X-ray Timing Explorer), both launched in the mid-90s, were particularly useful for studying X-ray binaries, including accretion-powered pulsars.

At present Chandra, with its outstanding subarcsecond imaging capability, and XMM-Newton, with its unprecedently high spectral sensitivity and collecting power, provide excellent new data.

In the following, we will summarize the current knowledge of X-ray emission properties of neutron stars based on these missions, browsing through the various categories from young Crab-like pulsars to very old radio pulsars, including recycled millisecond pulsars as well as neutron stars showing pure thermal emission. Before doing so, however, we will briefly review the various emission processes discussed to be the source for their observed X-ray emission.

14.2 Physics and Astrophysics of Isolated Neutron Stars

Neutron stars represent unique astrophysical laboratories, which allow us to explore the properties of matter under the most extreme conditions observable in nature.1 Studying neutron stars is, therefore, an interdisciplinary field, where astronomers and astrophysicists work together with a broad community of physicists. Particle, nuclear and solid-state physicists are strongly interested in the internal structure of neutron stars, which is determined by the behavior of matter at densities above the nuclear density pnuc = 2.8 x 1014g cm-3. Plasma physicists are modeling the pulsar emission mechanisms using electrodynamics and general relativity. It is beyond the scope of this article to describe in detail the current status of the theory of neutron star structure or the magnetospheric emission models. We rather refer the reader to the literature [17,20,36,68,106] and provide only the basic theoretical background relevant to Sect. 14.3, which summarizes the observed high-energy emission properties of rotation-powered pulsars and radio-quiet neutron stars.

14.2.1 Rotation-Powered Pulsars: The Magnetic Braking Model

Following the ideas of Pacini [81,82] and Gold [37,38], the more than 1750 radio pulsars detected so far can be interpreted as rapidly spinning, strongly magnetized neutron stars radiating at the expense of their rotational energy. This very useful concept allows one to obtain a wealth of information on basic neutron star/pulsar parameters just from measuring the pulsar's period and period derivative. Using the Crab pulsar as an example will make this more clear. A neutron star with a canonical radius of R = 10 km and a mass of M = 1.4 M0 has a moment of inertia I« (2/5) MR2 « 1045 g cm2. The Crab pulsar spins with a period of P = 33.403 ms. The rotational energy of such a star is Erot = 2 n2 I P-2 « 2 x 1049 erg. This is comparable with the energy released in thermonuclear burning by a usual star over its entire live. Very soon after the discovery of the first radio pulsars, it was noticed that their spin periods increase with time. For the Crab pulsar, the period derivative is P = 4.2 x 10-13 s s-1, implying a decrease in the star's rotation energy of dErot/dt = Erot = -4n2IPP-3 « 4.5 x 1038erg s-1. Ostriker & Gunn [78] suggested that the pulsar slow-down is due to the braking torque exerted on the neutron star by its magneto-dipole radiation, which yields Ebrake = -(32n4/3c3) B\ R6 P-4 for the energy loss of a rotating magnetic dipole, where is the component of the equatorial magnetic field perpendicular to the rotation axis. Equating Ebrake with Erot, we find = 3.2 x 1019(PP)1/2 Gauss. For the Crab pulsar, this yields = 3.8 x 1012 G. From Erot = Ebrake one further finds that P « P-1, for a given B±. This relation can be generalized as P = kP2-n, where k is a constant, and n is the so-called braking index (n = 3 for the magneto-dipole braking). Assuming that

1 Although black holes are even more compact than neutron stars, they can only be observed through the interaction with their surroundings.

the initial rotation period P0 at the time t0 of the neutron star formation was much smaller than today, at t = t0 + t, we obtain t = P/[(n — 1)P], or t = P/(2P) for n = 3. This quantity is called the characteristic spin-down age. It is a measure for the time span required to lose the rotational energy Erot(t0) — Erot(t) via magnetodipole radiation. For the Crab pulsar one finds t = 1258 yrs. As the neutron star in the Crab supernova remnant is the only pulsar for which its historical age is known (the Crab supernova was observed by Chinese astronomers in 1054 AD), we see that the spin-down age exceeds the true age by about 25%. Although the spin-down age is just an estimate for the true age of the pulsar, it is the only one available for pulsars other than the Crab, and it is commonly used in evolutionary studies (e.g., neutron star cooling).

A plot of observed periods vs. period derivatives is shown in Fig. 14.1, using the pulsars from the ATNF online pulsar database [66]. Such a P-P diagram is extremely useful for classification purposes. The colored symbols represent those pulsars that were detected at X-ray energies until fall 2006. The objects in the upper right corner represent the soft-gamma-ray repeaters (SGRs) and anomalous X-ray pulsars (AXPs) , which have been suggested to be magnetars (neutron stars with ultra strong magnetic fields).

Although the magnetic braking model is generally accepted, the observed spin-modulated emission, which gave pulsars their name, is found to account only for a small fraction of E. The efficiencies, n = L/E, observed in the radio and optical bands are typically in the range ~10—7 — 10—5, whereas they are about 10—4 — 10—3 at X-ray and gamma-ray energies, respectively. It has, therefore, been a long-standing question of how rotation-powered pulsars lose the bulk of their rotational energy.

The fact that the energy loss of rotation-powered pulsars cannot be fully accounted for by the magneto-dipole radiation is known from the investigation of the pulsar braking index, n = 2 — PPP—2. Pure dipole radiation would imply a braking index n = 3, whereas the values observed so far are n = 2.515 ± 0.005 for the Crab, n = 2.8 ±0.2 for PSR B1509—58, n = 2.28 ±0.02 for PSR B0540—69, 2.91 ±0.05 for PSR J1911—6127, 2.65 ± 0.01 for PSR J1846-0258, and n = 1.4 ± 0.2 for the Vela pulsar. The deviation from n = 3 is usually taken as evidence that a significant fraction of the pulsar's rotational energy is carried away by a pulsar wind, i.e., a mixture of charged particles and electromagnetic fields, which, if the conditions are appropriate, forms a pulsar-wind nebula observable at optical, radio, and X-ray energies. Such pulsar-wind nebulae (often called plerions or synchrotron nebulae) are known so far only for a few young and powerful (high E) pulsars and for some center-filled supernova remnants, in which a young neutron star is expected, but only emission from its plerion is detected.

Thus, the popular model of magnetic braking provides plausible estimates for the neutron star magnetic field B±, its rotational energy loss E, and the characteristic age t, but it does not provide detailed information about the physical processes that operate in the pulsar magnetosphere and are responsible for the broad-band spectrum, from the radio to the X-ray and gamma-ray bands. Forty years after the discovery of pulsars, the physical details of their emission mechanisms are still barely known.

Fig. 14.1 The P — P diagram: distribution of rotation-powered pulsars (small black dots) over their spin parameters. The straight lines correspond to constant ages t = P/ (2P) and magnetic field strengths B± = 3.2 x 1019(PP)1/2 as deduced in the frame of the magnetic braking model. Separate from the majority of ordinary-field pulsars are the millisecond pulsars in the lower left corner and the high magnetic field pulsars - soft gamma-ray repeaters (dark blue) and anomalous X-ray pulsars (light blue) in the upper right. Although magnetars and anomalous X-ray pulsars are not rotation-powered, they are included in this plot to visualize their estimated superstrong magnetic fields. X-ray detected pulsars are indicated by colored symbols. Red filled circles indicate the Crab-like pulsars. Green stars indicate Vela-like pulsars, green diamonds the X-ray detected cooling neutron stars, and red squares million years old pulsars

Fig. 14.1 The P — P diagram: distribution of rotation-powered pulsars (small black dots) over their spin parameters. The straight lines correspond to constant ages t = P/ (2P) and magnetic field strengths B± = 3.2 x 1019(PP)1/2 as deduced in the frame of the magnetic braking model. Separate from the majority of ordinary-field pulsars are the millisecond pulsars in the lower left corner and the high magnetic field pulsars - soft gamma-ray repeaters (dark blue) and anomalous X-ray pulsars (light blue) in the upper right. Although magnetars and anomalous X-ray pulsars are not rotation-powered, they are included in this plot to visualize their estimated superstrong magnetic fields. X-ray detected pulsars are indicated by colored symbols. Red filled circles indicate the Crab-like pulsars. Green stars indicate Vela-like pulsars, green diamonds the X-ray detected cooling neutron stars, and red squares million years old pulsars

As a consequence, there exist a number of magnetospheric emission models, but no generally accepted theory.

14.2.2 High-Energy Emission Models

Although rotation-powered pulsars are most widely known for their radio emission, the mechanism of the radio emission is poorly understood. However, it is certainly different from those responsible for the high-energy (infrared through gamma-ray) radiation observed from them with space observatories. It is well known that the radio emission of pulsars is a coherent process, and the coherent curvature radiation has been proposed as the most promising mechanism (see [68] and references therein). On the other hand, the optical, X-ray, and gamma-ray emission observed in pulsars must be incoherent. Therefore, the fluxes in these energy bands are directly proportional to the densities of the radiating high-energy electrons in the acceleration regions, no matter which radiation process (synchrotron radiation, curvature radiation, or inverse Compton scattering) is at work at a given energy. High-energy observations thus provide the key for the understanding of the pulsar emission mechanisms. So far, the high-energy radiation detected from rotation-driven pulsars has been attributed to various thermal and nonthermal emission processes including the following:

- Nonthermal emission from charged relativistic particles accelerated in the pulsar magnetosphere (cf. Fig. 14.2). As the energy distribution of these particles follows a power-law, the emission is also characterized by power-law-like spectra in broad energy bands. The emitted radiation can be observed from optical to the gamma-ray band.

- Extended emission from pulsar-driven synchrotron nebulae. Depending on the local conditions (density of the ambient interstellar medium), these nebulae can be observed from radio through hard X-ray energies.

- Photospheric emission from the hot surface of a cooling neutron star. In this case, a modified black-body spectrum and smooth, low-amplitude intensity variations n b

Fig. 14.2 Geometry of the acceleration zones as they are defined in the polar cap model (left), according to Ruderman and Sutherland [90], and outer gap model (right), according to Cheng, Ho and Ruderman [28, 29]. The polar cap model predicts "pencil" beams emitted by particles accelerated along the curved magnetic field lines. According to the outer gap model, the pulsar radiation is emitted in "fan" beams. Being broader, the latter can easier explain two (and more) pulse components as observed in some X-ray and gamma-ray pulsars with the rotational period are expected, observable from the optical through the soft X-ray range.

- Thermal soft X-ray emission from the neutron star's polar caps which are heated by the bombardment of relativistic particles streaming back to the surface from the pulsar magnetosphere.

In almost all pulsars, the observed X-ray emission is due to a mixture of different thermal and nonthermal processes. Often, however, the available data do not allow to fully discriminate between the different emission scenarios. This was true for ROSAT, ASCA, and BeppoSAX observations of pulsars and is - at a certain level -still true in Chandra and XMM-Newton data.

In the following subsections, we will briefly present the basics on the magne-tospheric emission models as well as material relevant to thermal emission from the neutron star surface.

14.2.2.1 Magnetospheric Emission Models

So far, there is no consensus as to where the pulsar high-energy radiation comes from (see for example [68]). There exist two main types of models - the polar cap models, which place the emission zone in the immediate vicinity of the neutron star's polar caps, and the outer gap models, in which this zone is assumed to be close to the pulsar's light cylinder2 to prevent materializing of the photons by the one-photon pair creation in the strong magnetic field, according to y+ B ^ e+ + e- (see Fig. 14.2). The gamma-ray emission in the polar cap models forms a hollow cone centered on the magnetic pole, producing either double-peaked or single-peaked pulse profiles, depending on the observer's line of sight. The outer gap model was originally proposed to explain the bright gamma-ray emission from the Crab and Vela pulsars ([28,29]) as the efficiency to get high-energy photons out of the high B-field regions close to the surface is rather small. Placing the gamma-ray emission zone at the light cylinder, where the magnetic field strength is reduced to BL = B (R/Rl)3, provides higher gamma-ray emissivities, which are in somewhat better agreement with the observations. In both types of models, the high-energy radiation is emitted by relativistic particles accelerated in the very strong electric field, E ~ (R/cP)B, generated by the magnetic field corotating with the neutron star. These particles are generated in cascade (avalanche) processes in charge-free gaps, located either above the magnetic poles or at the light cylinder. The main photon emission mechanisms are synchrotron/curvature radiation and inverse Compton scattering of soft thermal X-ray photons emitted from the hot neutron star surface.

In recent years the polar-cap and outer-gap models have been further developed, incorporating new results on gamma-ray emission from pulsars obtained with the Compton Gamma-Ray Observatory. At the present stage, the observational data can be interpreted with any of the two models, albeit under quite different assumptions

2 The light cylinder is a virtual cylinder whose radius, RL = cP/(2n), is defined by the condition that the azimuthal velocity of the corotating magnetic field lines is equal to the speed of light.

on pulsar parameters. The critical observations to distinguish between the two models include measuring the relative phases between the peaks of the pulse profiles at different energies. Probably, the AGILE and GLAST gamma-ray observatories, which are supposed to be launched in November 2007, will provide valuable information to further constrain both models.

14.2.2.2 Thermal Evolution of Neutron Stars

Neutron stars are formed at very high temperatures, ~1011 K, in the imploding cores of supernova explosions. Much of the initial thermal energy is radiated away from the interior of the star by various processes of neutrino emission (mainly, Urca processes and neutrino bremsstrahlung), leaving a one-day-old neutron star with an internal temperature of about 109 — 1010K. After ~100yr (typical time of thermal relaxation), the star's interior (densities p > 1010 g cm—3) becomes nearly isothermal, and the energy balance of the cooling neutron star is determined by the following equation:

CT) dT = —Lv(Ti) — Ly(Ts)+£Hk , where Ti and Ts are the internal and surface temperatures, C(Ti) is the heat capacity of the neutron star. Neutron star cooling thus means a decrease of thermal energy, which is mainly stored in the stellar core, because of energy loss by neutrinos from the interior (Lv = f Qv dV, Qv is the neutrino emissivity) plus energy loss by thermal photons from the surface (Ly = 4nR2oTs4). The relationship between Ts and T is determined by the thermal insulation of the outer envelope (p < 1010 g cm—3), where the temperature gradient is formed. The results of model calculations, assuming that the outer envelope is composed of iron, can be fitted with a simple relation

where g is the gravitational acceleration at the neutron star surface [43]. The cooling rate might be reduced by heating mechanisms Hk, like frictional heating of super-fluid neutrons in the inner neutron star crust or some exothermal nuclear reactions.

Neutrino emission from the neutron star interior is the dominant cooling process for at least the first 105 yrs. After ~106 years, photon emission from the neutron star surface takes over as the main cooling mechanism. The thermal evolution of a neutron star after the age of ~ 10-100 yr, when the neutron star has cooled down to Ts = 1.5 — 3 x 106K, can follow two different scenarios, depending on the still poorly known properties of super-dense matter (see Fig. 14.7). According to the so-called standard cooling scenario, the temperature decreases gradually, down to ^0.3 — 1 x 106 K, by the end of the neutrino cooling era and then falls down exponentially, becoming lower than ~0.1 x 106K in ~107yr. In this scenario, the main neutrino generation processes are the modified Urca reactions, n + N — p + N + e + ve and p + N + e — n+N + ve, where N is anucleon (neutron

Table 14.1 Nuclear reactions and their neutrino emissivity as a function of neutron star temperature [84]

Neutrino emissivity used in neutron star cooling models

Neutrino emissivity used in neutron star cooling models

Table 14.1 Nuclear reactions and their neutrino emissivity as a function of neutron star temperature [84]

Process

Nuclear reaction

Emissivity (erg s-1 cm-3)

Direct URCA-Process

n ^ p + e- + ve p + e- ^ n + ve

-1027 x T96

^-condensate

n + n- ^ n + e- + ve n + e- ^ n + n- + ve

-1026 x T96

Quark-URCA-Process

d ^ u + e- + ve u + e- ^ d + ve

~1026 «ct96

Kaon condensate

n + K- ^ n + e- + ve n + e- ^ n + K- + ve

-1024 x T96

Modified URCA-Process

n + n ^ n + p + e- + ve n + p + e- ^ n + n + ve

~1020 x T98

Direct coupled Elektron-neutrino-process

Neutron-neutron and Neutron-protonbremsstrahlung Elektron-ion-neutrinoBremsstrahlung

Y+ e- ^ e- + Ve + Ve Yplasmon ^ Ve + ve e+ + e- ^ Ve + Ve n + n^n + n+V + V n + p ^ n + p + V + V

e- + (Z, A) ^ e- + (Z, A) + Ve + Ve

« t96

T9 is the temperature in units of 109 K. Each particle (n, p, e—), which takes part in a reaction contributes to the temperature dependence with a T and each neutrino with a T3. The reactions denoted as direct-Urca, ^-condensate, Quark-URCA-process, and Kaon condensate are taken into account in the so-called accelerated cooling models. They have an order of magnitude higher neutrino emissivity in comparison with the other nuclear reactions. The higher the neutrino emissivity is the more efficient is the neutron star cooling or proton) needed to conserve momentum of reacting particles (cf. Table 14.1). In the accelerated cooling scenarios, associated with higher central densities (up to 1015 g cm—3) and/or exotic interior composition (e.g., pion condensation, quark-gluon plasma), a sharp drop of temperature, down to 0.3 — 0.5 x 106 K, occurs at an age of —10 — 100 yr, followed by a more gradual decrease, down to the same —0.1 x 106 K at —107 yr. The faster cooling is caused by the direct Urca reactions, n — p + e + ve and p + e — n + ve, allowed at very high densities.

The neutron star models used in these calculations are based on a moderate equation of state that opens the direct Urca process for M > 1.35M0, the stars with lower M undergo the standard cooling. Recent studies have shown that both the standard and accelerated cooling can be substantially affected by nucleon superfluidity in the stellar interiors (see [83,100,112] for comprehensive reviews). In particular, many cooling curves exist intermediate between those of the standard and accelerated scenarios, depending on the properties of nucleon superfluidity, which are also poorly known.

Thus, the thermal evolution of neutron stars is very sensitive to the composition and structure of their interiors, in particular, to the equation of state at super-nuclear densities. Therefore, measuring surface temperatures of neutron stars is an important tool to study super-dense matter. Since typical temperatures of such neutron stars correspond to the extreme UV - soft X-ray range, the thermal radiation from cooling neutron stars can be observed with X-ray detectors sufficiently sensitive at E< 1 keV.

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