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Spectral modelling of the high energy emission of the magnetar 4U 0142+614

We present an empirical spectral modelling of the high energy emission of the anomalous X-ray pulsar 4U 0142+614, based on simultaneous Swift and INTEGRAL observations from X to gamma-ray energies. We adopted models contained in the XSPEC analysis package, as well as models based on recent theoretical studies, and restricted ourselves to those combinations of up to three components which produce a good fit while requiring the lowest number of free parameters. Only three models were found to fit satisfactorily the 0.5-250keV spectrum of 4U 0142+614: i) a ~0.4keV blackbody and two power-laws, ii) a resonant cyclotron scattering model plus a power-law and iii) two log-parabolic functions. We found that only the latter two models do not over-predict the infrared/optical emission observed simultaneously from this AXP, and only the log-parabolic functions can naturally account for the upper limits set by COMPTEL in the gamma-ray range. A possible interpretation of the two log-parabolae in terms of inverse Compton scattering of soft X-ray photons by very energetic particles is discussed.


INTRODUCTION
Soft γ-ray Repeaters (SGRs) and Anomalous X-ray Pulsars (AXPs) are peculiar classes of high energy sources which share a number of properties (among others, the emission of short bursts, slow rotation periods in a narrow range, secular spin down and a X-ray luminosity of ≈ 10 34 -10 36 erg s −1 , see e.g. Woods & Thompson 2006 for a recent review). SGRs and AXPs are believed to be powered by ultra-magnetic neutron stars ("magnetars"; Duncan & Thompson 1992;Thompson & Duncan 1995).
Spectral analysis is an important tool in magnetar astrophysics since it can provide key information on the emission mechanism(s). The first attempts at modelling the spectra of AXPs were based on soft X-ray data (< 10 keV) only. It was found that a blackbody (kT ∼0.3-0.6 keV) plus a power-law (photon index Γ ∼2-4) or, limited to a few cases, two blackbodies (kT 1 ∼0.1 and kT 2 ∼0.8 keV) could successfully model the 0.5-10 keV emission. Even though the blackbody plus powerlaw spectral model has been frequently applied to the Xray spectra of magnetar candidates, a convincing physical interpretation is still missing (for the power-law component in particular).
Recent studies have attempted to overcome this limitation. Thompson, Lyutikov & Kulkarni (2002) proposed that magnetars have twisted magnetospheres threaded by currents, which can efficiently boost via resonant cyclotron scattering the soft thermal photons emitted by the star surface. Recent investigations (Lyutikov & Gavriil 2006;Fernandez & Thompson 2006) showed that such a process may lead to the formation of the power-law tail observed in the soft X-rays. Rea et al. (2006Rea et al. ( , 2007 presented an application of the Lyutikov & Gavriil (2006) resonant cyclotron scattering model (RCS model in the following) to SGRs and AXPs, showing that in a few sources this component alone can explain the X-ray spectrum below ∼ 10 keV.
AXPs and SGRs have been discovered to emit also at optical and/or NIR wavelengths (e.g. Hulleman et al. 2000;Israel et al. 2004;Durant & van Kerkwijk 2006b). Their optical/NIR flux represents a small fraction of the bolometric flux, but still can place important constraints on models.
Several AXPs, have exhibited long term variability both in their optical/infrared emission and in the X-rays (Oosterbroek et al. 1998;Israel et al. 2002;Hulleman et al. 2004;Rea et al. 2005). Unavoidably this introduces additional uncertainties in the modelling of broad band spectra, based on observations at different wavelengths taken at different times.
The AXP 4U 0142+61 has been recently observed quasi-simultaneously in the hard X-rays (with INTE-GRAL, see den Hartog et al. 2006  -4U 0142+614 soft and hard X-ray spectrum fitted with (from left to right): a blackbody plus two power-laws ( § 3.1), the RCS model plus a power-law ( § 3.2), and two log-parabolic functions ( § 3.3). Upper rows show the unfolded spectrum and the residuals of the fits: in blue the data, the residuals and the overall model and in red the single components. Lower row present the relative unabsorbed fitted models: in red the single components, in blue the overall model and the grey arrows report the COMPTEL upper limits from Kuiper et al. 2006. Data from this and other campaigns show that most of the bolometric emission (> 95%) revealed so far from this object lies in the X/γ-ray energy range. Motivated by this we present here an empirical modelling of the 0.5-250 keV spectrum of 4U 0142+614 obtained during this quasi-simultaneous campaign (see §3).
In §4 we discuss how the extrapolation of different models match the flux measured at other wavelengths. In particular, we note that only the fit with two absorbed log-parabolic functions is consistent with the 4U 0142+614 γ-ray upper limits and does not overestimate the observed optical/infrared emission. Some physical implications of our study are then briefly discussed.

THE OBSERVATIONS
We used Swift X-ray Telescope (XRT) data obtained on 2005 July 11-12th for an on-source exposure time of 7400 s in Photon Counting (PC) mode (that produced a standard 2D image). The data were analysed with the FTOOL task xrtpipeline (version build-14 under HEADAS 6.0). We applied standard screening criteria to the data, and hot and flickering pixels, high background intervals and bright limb were removed. Screened event files were then used to derive light curves and spectra. We included data between 0.5-8 keV, where the PC response matrix is well calibrated (we used the latest v.8 response matrices). We extracted photons from an annular region (3 pixels inner radius, 30 pixel outer radius) in order to avoid pile-up contamination and we considered standard grades for the PC mode 0-12.
For the hard X-ray part of the spectrum, we used INTEGRAL data taken by the IBIS/ISGRI instrument (20-250 keV energy range) quasi-simultaneously with the Swift XRT ones. These consist of 868 ks divided in 265 Science Windows, collected between 2005 June 29th and July 17th, in a pointed observation (see den Hartog et al. 2006).

SPECTRAL MODELLING
We fitted (making use of XSPEC 11.3 and 12.1) the X/γ-ray spectrum of 4U 0142+614 by using any combination of the following components: a) for the soft Xray part -two blackbodies, a blackbody plus a powerlaw, a neutron star atmosphere model (nsa model, only available for a maximum magnetic field of B=10 13 G), a comptonized blackbody (compbb), a disk blackbody (diskbb), a log-parabolic function, a RCS model (as in Rea et al. 2006Rea et al. , 2007, and b) for the hard X-ray part -a power-law, a bremsstrahlung model (bremss), a logparabolic function. In all cases we included photoelectric absorption (abundances set to solar values from Lodders 2003) and a constant to account for cross-calibration uncertainties in the two instruments.
All combinations, except for the three models discussed below (see also Fig. 1, 2, and Tab. 1), gave unsatisfactory reduced χ 2 values (χ 2 ν > 1.8). All the three models that successfully reproduce the spectrum require 8 free parameters. Of course, we found that other models, involving components not listed above (e.g. a cut-off power-law instead of the hard X-ray power-law), gave satisfactory results. However, in all cases more than 8 free parameters were required.

The canonical empirical model
This is the empirical model which is usually adopted for AXP spectra and consists of a single blackbody and two power-laws. For 4U 0142+614 the best fit gave a blackbody temperature of kT ≃ 0.42 keV, a very steep power-law below ∼ 10 keV with Γ ≃ 3.77, and a flatter power law for the high-energy part of the spectrum, Γ ≃ 0.73 (see also Kuiper et al. 2006;den Hartog et al. 2006). We tried also a slightly more general model which includes a cut-off around 300 keV in the attempt to make the extrapolation of the canonical model compatible with the existing upper limits above ∼ 1 MeV  (note that we did not include the MeV upper limits in the fit). We found that the quality of the fit did not improve much, despite the addition of one more free parameter in this modelling. Note that the model consisting of two blackbodies plus a power-law did not provide a good fit, mainly because the two blackbodies alone cannot account for the soft X-ray part of the emission.

The Resonant Cyclotron Scattering model
This combination makes use of the RCS model, based on the spectra computed by Lyutikov & Gavriil (2006) and recently implemented in XSPEC by Rea et al. (2006). As shown by Rea et al. (2006Rea et al. ( , 2007, this component can explain well the soft X-ray emission of a few magnetars, whereas an additional power-law is required for sources of this class that have a strong hard X-ray emission. The model parameters, together with the ranges over which each parameter was allowed to vary, are: the velocity of the magnetospheric e − currents (in units of c) 0.1 < β th < 0.5; their (Thomson) scattering optical depth, 1 < τ 0 < 10; and the temperature of the seed surface emission (assumed to be a blackbody), 0.1 keV< kT < 3 keV.
The efficiency of RCS drops above ∼ 10 keV, but even in the 6-10 keV energy range the addition of another model component is required to fit the X-ray spectrum of 4U 0142+614. We find a best fit of the whole 0.5-250 keV spectrum with a RCS plus power-law model with a surface temperature of kT ≃ 0.33 keV, magnetospheric plasma parameters of β th ≃ 0.22 and τ 0 ≃ 1.84, and a power law index of Γ ≃ 0.67. The RCS component dominates the emission below 6 keV.

The log-parabolic model
The log-parabolic model is an empirical model commonly used to fit blazars spectra, but also successfully applied to a few radio pulsars displaying hard X-ray emission, e.g. the Crab and the Vela pulsars (Kuiper et al. 2001;Massaro et al. 2006a,b). We applied this model to our data and found that it fits well the soft and hard X-ray spectrum of 4U 0142+614 (see also Kuiper et al. 2006). Log-parabolic spectra (Landau et al. 1986) can be obtained when relativistic electrons are accelerated by a statistical acceleration mechanism in which the probability of acceleration depends on energy, and cool via synchrotron losses (Landau et al. 1986;Massaro et al. 2004) or by inverse Compton scattering of synchrotron seed photons.
The log-parabolic spectral distribution can be regarded as the simplest generalization of a power-law distribution, as it comprises a mild symmetric spectral curvature. We adopted the following expression, F (E) = KE −α−β log E , where E is the energy in keV, α is the photon index at 1 keV energy and β controls the curvature of the parabola in a log-log representation. The log-parabola peaks at E p = 10 (2−α)/2β ; its free parameters, besides the normalization constant K, are α and β or, equivalently, E p and β.
The two log-parabolae we fitted have peaks at ≃ 1.32 and 412 keV, and curvatures of β =-2.52 and -0.7. They provide a very good fit to the data, and have the same number of free parameters as the models discussed above (see Fig. 1 and Tab. 1).

DISCUSSION AND INTERPRETATION
We have shown that among spectral models with a relatively limited number of free parameters (N par = 8), only three successfully fit the X-ray spectrum of 4U 0142+614 (see §3).
The 0.5-250 keV fluxes we derived, are consistent among all the three best models. N H in the RCS and log-parabolic models is closer to the value measured for this source using X-ray edges in grating spectra (∼ 6 × 10 21 cm −2 ; see Durant & van Kerkwijk 2006a).
Comparing the extrapolations of these models with the radio, infrared and optical fluxes measured from quasi-simultaneous observations (Durant & van Kerkwijk 2006b, den Hartog et al. 2006, as well as γ-ray upper limits obtained around 0.2 MeV by COMPTEL (Kuiper et al. 2006), we found that the canonical model ( § 3.1) overpredicts all of them. The RCS model ( § 3.2) and the two log-parabolae ( § 3.3) improve over the canonical model since they do not overpredict the low energy emission. In any case, the optical/NIR emission of 4U0142+614 (and likely of other magnetar candidates) accounts for only less than 5% of the overall source luminosity, and might be due to secondary processes as compared to those responsible for the X/γ-ray flux. The optical emission is pulsed, so it is likely to be non-thermal in origin (Kern & Martin 2000). On the contrary, the IR emission can be ascribed to a fossil disk (see e.g. Wang et al. 2006), reprocessing a fraction of the soft/hard X-ray emission of the magnetar.
The two log-parabolae ( § 3.3) are the only model that naturally accounts for the fast decline of the spectrum in the γ-rays, implied by the COMPTEL upper limits. Nevertheless, in the other two models these limits would not be violated if the hard X-ray power law were characterised by a cut-off at ∼ 300 keV (though this would require an additional free parameter in the fit). Note that in a more detailed physical model the high-energy power law required to fit the INTEGRAL data may quickly decline around ∼ 300 keV, e.g. as a result of the maximum energy up to which the spectrum of the emitting particles extends. Thompson & Belobodorov (2005) recently proposed that the hard X-ray magnetar emission may be attributed to bremsstrahlung photons emitted by a thin surface layer heated up to ∼ 100 keV by returning currents. However, we find that the 4U 0142+614 spectrum is harder at high energies than envisaged this scenario (Fig. 2). A bremsstrahlung model seems hardly compatible with the hard X-ray spectrum of this source and the COMPTEL upper limits.
Although two log-parabolic functions are often used to model a Synchrotron Self Compton (SSC) emission, in the case at hand this interpretation appears unlikely for several reasons. First, it is difficult to reconcile the soft part of the X-ray spectrum with synchrotron emission, since the low energy log-parabola has a curvature β = −2.52 which translates in a curvature of ∼ 12 for the e − spectrum. This is very close to the cut-off curvature of the high energy synchrotron emission of a single particle. Moreover, we performed detailed simulations (see Tramacere 2002;Tramacere & Tosti 2003 for details) in order to infer which local magnetic field and electron density is needed to power the 4U 0142+614 emission by a SSC mechanism. We obtained B ∼ 10 8 G and n e ∼ 10 15 cm −3 , respectively. A charge density of the same order is predicted in the simple twisted magnetosphere model by Thompson, Lyutikov & Kulkarni (2002), but close to the neutron star surface. Instead, the low value of the magnetic field requires that such e − are located at an altitude of several hundred kms in the magnetosphere. We may speculate that inverse Compton scattering by a population of relativistic particles high up in the magnetosphere may explain the hard Xray emission of magnetars, for other (than synchrotron) spectral distributions of seed photons, e.g. that produced by the RCS model. Interestingly, the presence of highly relativistic particles well above the star surface has been also suggested by Thompson & Beloborodov (2005). These authors pointed out that seed positrons injected at ∼ 100 km above the surface can undergo runaway acceleration and upscatter X-ray photons above the threshold for pair production. In their model, the e ± pairs would then be responsible for efficient synchrotron emission in the 20-200 keV band.
In this Letter we introduced and fitted spectral models, as purely empirical laws. As in many other studies where the total emission of X-ray pulsars is modelled, no attempt was made to fit separately the spectral behaviour of the pulsed and the non-pulsed components. A more extensive study based on data of other magnetar candidates and attempting a more detailed physical interpretation of these results is in progress and will be reported elsewhere.
NR acknowledges support from an NWO-Postdoctoral Fellowship and a Short Term Visiting Fellowship awarded by The University of Sydney, and SZ thanks PPARC for financial support. This work was partially supported through ASI and MIUR funds. NR thanks S.