Classical XRay Pulsars

The first two X-ray pulsars, Cen X-3 (4.8 s) and Her X-1 (1.24 s), were discovered by Uhuru shortly after its launch in 1970. The identification as binary stars with accreting NS was in both cases by the observation of regular eclipses and by the regular modulation of the pulse arrival times (or Doppler shift of the pulse frequency) with the same binary period. The sinusoidal shape of the Doppler curve allowed to conclude that the orbits were highly circular, with upper limits on eccentricity. Characteristic features of X-ray pulsars were already recognized in these two objects, such as highly structured pulse profiles (see Fig. 15.5), high variability, including the 35-day modulation in Her X-1, long term changes of the pulse periods, as well as the general shape of the X-ray spectrum (a hard power law with an exponential cut-off). Today, we know more than 160 accreting binary X-ray pulsars (most of them in HMXB): —90 in our own galaxy, —50 in the SMC, —10 in the LMC and —10 in M31 and other galaxies of the Local Group. In addition there are eight so called Anomalous X-ray Pulsars (AXP) [25] with pulse periods in the narrow range of 5-12 s and large period derivatives. They show no companions and are thought to be powered not by accretion but by the energy released during the decay of an extremely strong magnetic field.

15.5.1.1 Spin Periods

The spin periods of accretion powered pulsars are found in three groups: the "classical" pulsars, showing a broad distribution between — 1 and — 1000 s, a small group of three members with spin periods of 33, 61, and 69 ms and the (as yet 6) accreting ms pulsars with spin periods between 1.7 and 5.4ms. The latter objects are LMXB with weakly magnetized NS and will therefore be discussed later. The first two groups contain highly magnetized NS and are mostly HMXB, except for the following four LMXB: Her X-1,4U 1626-67, GX 1+4, GRO J1744-28, and 2A 1822-371.

For HMXB, the Corbet diagram [13] shows an interesting relationship between the spin period of the NS and the orbital period. Although for Be binaries this relationship is linear, there is no obvious dependence for OB supergiant binaries and for LMXB.

15.5.1.2 Spin-Up/Spin-Down

Shortly after the discovery of binary X-ray pulsars, it was recognized that their pulse periods were not constant. Figure 15.3 shows the long-term development of four prominent X-ray pulsars, demonstrating that both spin-up and spin-down of the NS occurs and this can vary on a wide range of time scales (down to a few days). The physical reason for this behavior is the interaction of the material of the accretion disk with the magnetic field of the NS in a boundary layer at the magnetospheric radius rm (found by equating the ram pressure of a spherically symmetric inflow to the magnetic pressure of the B-field). The sign of the angular momentum transfer (to or from) the NS depends on the exact conditions in the region of interaction, in particular on the ratio of the corotation radius rco, where the magnetic field frozen to the NS has the same angular velocity as the accreting plasma in its Kepler rotation, to the magnetospheric radius rm. Obviously, when rm equals rco an equilibrium is reached and no angular momentum is transfered. This equilibrium period can be expressed as Peq — (2.7s) jU^cfMx2/7R6 3/7 L-73/7, ^30 is the magnetic moment of the NS in units of 1030 Gauss cm3, Mx is the mass of the NS in M0, R6 is the radius of the NS in 106 cm and L37 is the X-ray luminosity in units of 1037 erg s-1.

If rm < rco spin-up occurs, in that case the unperturbed Kepler rotation at the magnetospheric radius is faster than that of the B-field. If the magnetosphere expands (e.g., as a result of decreasing pressure due to a decreasing accretion rate) rm can exceed rco and spin-down results. This accretion barrier, also called the propeller effect, can completely stop the accretion such that the X-ray source shuts off. The complex details of these interactions are formulated in the accretion torque theory by Ghosh and Lamb [18,19]. The observed correlation between

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Fig. 15.3 Pulse period history of four X-ray binary pulsars (after [2])

the magnitude of spin-up dP/dt and Lx is explained by this theory (Fig. 15.4). Since the observed X-ray luminosity Lx reflects the mass accretion rate, one can understand the fluctuations in pulse period as variations in the accretion rate. Most X-ray pulsars appear to operate near to their equilibrium period. Measuring Peq then allows a direct estimate of the magnetic moment of the NS (e.g., using standard values for mass and radius of the NS).

15.5.1.3 Pulse Profiles

The pulse profiles of X-ray pulsars can look very different, from simple sinusoidal type shapes to double and multiple peaked profiles. Figure 15.5 gives a few examples. In a particular source, the profiles are energy dependent, generally showing "simpler" profiles at higher photon energies. They are also not always constant in time but can show dramatic changes, both systematically repeated or seemingly irregular. In analysing and modeling these profiles, a great deal can be learned about the geometry of the hot spots on the surface of the NS, their emission characteristics and the emission mechanism itself, as well as about the geometry of the observation, e.g., the line of sight with respect to the axis of rotation and the axis of the magnetic

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Fig. 15.4 Correlation between pulse period derivative -dP/dt and P x L3/7 [19,34]

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Fig. 15.4 Correlation between pulse period derivative -dP/dt and P x L3/7 [19,34]

Fig. 15.5 Pulse profiles of selected X-ray pulsars (after [33])

field, and in fact about the structure of the magnetic field itself (dipole and multipole components). In doing the model calculations, one has to take into account the gravitational bending of the photon's path in the strong gravitational field of the NS.

15.5.1.4 Continuum Spectra and Cyclotron Lines

The X-ray continuum spectra of X-ray pulsars are rather characteristic, they are successfully described by a hard power law spectrum that is exponentially cut off. The modeling parameters are the power law photon index (typically between 0 and 1), the energy at which the cut-off sets in (<20keV) and the folding energy of the exponential cut-off (^several kilo electrovolt). Various types of mathematical fitting functions exist. Physically, the spectra are thought to arise through the emission (thermal bremsstrahlung) of the hot regions at the base of the accretion column at the

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Fig. 15.6 The high energy X-ray spectrum of Her X-1 as observed by the MPE/AIT balloon detector, leading to the discovery of the first cyclotron line (Fig. 15.6 of [37])

magnetic poles of the NS (the accretion mound) and modifications by comptoniza-tion through fast electrons (thermal or nonthermal). Interestingly, the heuristical fit functions have so far been more successful than attempts to do physical modeling.

An important modification of such a spectrum was discovered through a balloon observation of Her X-1 where a line feature was detected and interpreted as due to cyclotron radiation [37] (see Fig. 15.6). The strong magnetic field in the polar region of a NS forces discrete energy levels (Landau levels) of electrons with respect to their motion perpendicular to the direction of the field. The cyclotron energy, Ec, defined as the difference between the (to first order equidistant) energy levels is given by Ec = heB/me c — 11.6 B12 keV, where B12 is the field strength in units of 1012 Gauss. The early data did not allow to distinguish between emission and absorption. This was achieved through higher quality observations [26, 35, 42] and model calculations [41], which showed that cyclotron lines are formed through resonant scattering of photons trying to escape from the accretion mound, which has led to the term CRSF - Cyclotron Resonant Scattering Feature. The observed line energy is at E = Ec/(1 + z) with z being the gravitational redshift (—0.3). The CRSF

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Fig. 15.6 The high energy X-ray spectrum of Her X-1 as observed by the MPE/AIT balloon detector, leading to the discovery of the first cyclotron line (Fig. 15.6 of [37])

at 37keV in Her X-1, therefore, corresponds to a field strength of 4 x 1012 G. The indirect estimate using accretion torque theory (see earlier) leads for Her X-1 to a magnetic moment of 1030 Gauss cm-3, equivalent to 2 x 1012 G (for a dipolar field). This is, keeping the model dependency in mind, consistent with the cyclotron line measurement. In any case, cyclotron lines provide the means for a direct measurement of the magnetic field strength at the emission site.

Today, at least 15 cyclotron line objects are known (for recent reviews see [23, 36]). In most objects just one line is seen, in three objects two lines are observed, in one object three lines are seen, the record holder with five lines is 4U 0115+63.

It has been found that the line spectra are changing with pulse phase. This type of analysis, pulse phase spectroscopy, has revealed that the cyclotron energy, Ec, in most objects varies substantially (by up to 20%) throughout the pulse. During the rotation of the NS, the observer apparently sees emission regions with different magnetic field strength. Such analysis has still to be performed in a systematic way on all cyclotron line objects using archival data of RXTE and BeppoSAX. New high resolution observations by INTEGRAL in coordination with theoretical calculations offer the prospect for a deeper understanding.

15.5.1.5 Aperiodic Variability in Pulsars

It had taken some time to recognize that many of the classical accretion powered pulsars also show rapid X-ray variability, which can in part be described in terms of quasi-periodic oscillations (QPO) mostly seen in LMXB [38] (as discussed in Sect. 15.6). The physics is probably similar in both cases (see later), most likely involving special regions of the accretion flow, e.g., the magnetospheric boundary at the inner edge of the accretion disk. In at least 13 pulsars (of all types) QPOs have now been seen, with frequencies mainly in the 2-200 mHz range, in the case of Cen X-3 possibly reaching the kilohertz regime [14].

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