Issue 
A&A
Volume 562, February 2014



Article Number  A62  
Number of page(s)  8  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201322303  
Published online  05 February 2014 
Spectral formation in a radiative shock: application to anomalous Xray pulsars and soft gammaray repeaters
^{1} University of Crete, Physics Department & Institute of Theoretical & Computational Physics, 71003 Heraklion, Crete, Greece
email: kylafis@physics.uoc.gr
^{2} Foundation for Research and TechnologyHellas, 71110 Heraklion, Crete, Greece
^{3} MaxPlanckInstitut für extraterrestrische Physik, Postfach 1312, 85741 Garching, Germany
^{4} Faculty of Engineering and Natural Sciences, Sabancı University, 34956 Orhanlı, Tuzla, İstanbul, Turkey
Received: 17 July 2013
Accepted: 21 December 2013
Context. In the fallback disk model for the persistent emission of anomalous Xray pulsars (AXPs) and soft gammaray repeaters (SGRs), the hard Xray emission arises from bulk and thermal Comptonization of bremsstrahlung photons, which are generated in the accretion column. The relatively low Xray luminosity of these sources implies a moderate transverse optical depth to electron scattering, with photons executing a small number of shock crossings before escaping sideways.
Aims. We explore the range of spectral shapes that can be obtained with this model and characterize the most important parameter dependencies.
Methods. We use a Monte Carlo code to study the crisscrossing of photons in a radiative shock in an accretion column and compute the resulting spectrum.
Results. As expected, highenergy powerlaw Xray spectra are produced in radiative shocks with photonnumber spectral index Γ ≳ 0.5. We find that the required transverse optical depth is 1 ≲ τ_{⊥} ≲ 7. Such spectra are observed in lowluminosity Xray pulsars.
Conclusions. We demonstrate here with a simple model that Compton upscattering in the radiative shock in the accretion column can produce hard Xray spectra similar to those seen in the persistent and transient emission of AXPs and SGRs. In particular, one can obtain a highenergy powerlaw spectrum, with photonnumber spectralindex Γ ~ 1 and a cutoff at 100 − 200 keV, with a transverse Thomson optical depth of ~5, which is shown to be typical in AXPs/SGRs.
Key words: accretion, accretion disks / stars: magnetars / Xrays: stars / pulsars: general
© ESO, 2014
1. Introduction
Anomalous Xray Pulsars (AXPs) and soft gammaray repeaters (SGRs) are young neutron stars that have been observed to emit Xrays both quiescently and in the form of bursts. They are commonly called magnetars, because they are thought to have superstrong magnetic fields (10^{14}−10^{15} G). For recent reviews see Woods & Thompson (2006) and Mereghetti (2008, 2013). Two main models have been proposed to explain the observational data; the classical magnetar model (e.g., Thompson & Duncan 1993, 1996) and the fallback disk one (Chatterjee et al. 2000; Alpar 2001). Both models have in common that the giant and the highly superEddington bursts are produced by magnetic field decay. The fallback disk model puts the superstrong magnetic field in a multipole component, while the classical magnetar model puts it in the dipole component. Yet, a recent interpretation (Tiengo et al. 2013) of a variable absorption feature in the spectrum of the magnetar SGR 0418+5729 as a proton cyclotron feature invokes a multipole magnetic field. Thus, multipole components could in principle operate in both models.
For the persistent emission, the explanation is different in the two models. The classical magnetar model assumes that the persistent emission is powered by magnetic field decay and that the observed period derivative Ṗ is due to magnetic braking of an isolated neutron star with a superstrong dipole magnetic field. On the other hand, in the fallback disk model the persistent Xray emission is powered by accretion onto the neutron star, the optical and the infrared radiation is produced by the disk, and the observed rotational properties are explained by the longterm disk – magnetosphere interaction. The evolution of AXPs and SGRs with fallback disks has been studied in a number of papers (e.g., Ertan et al. 2007, 2009) and a fallback disk has been observed in two magnetars: AXP 4U 0142+61 (Wang et al. 2006) and AXP 1E 2259+586 (Kaplan et al. 2009).
A detailed model for the formation of the quiescent spectra of AXPs/SGRs has been proposed by Trümper et al. (2010, 2013) using the fallback disk model. These spectra consist of two components (e.g., den Hartog et al. 2008a,b; Enoto et al. 2010); a quasithermal component at low energies, which is generally attributed to the photospheric emission from the neutron star polar cap, and a hard powerlaw component. In the model of Trümper et al. (2010, 2013), this powerlaw component is attributed to the emission from the radiative shock, located near the bottom of the accretion column. This component is emitted as a fan beam, which partly hits the surrounding polar cap, where the radiation is absorbed or scattered, leading to the quasithermal component. The geometry of the model ensures that both soft and hard components have approximately the same luminosity (as observed), while the observed luminosity ratio depends on the height of the radiative shock above the neutronstar surface and the angles between the line of sight on one hand and the magneticdipole and the spin axis on the other.
In this paper we concentrate specifically on the formation of the fan beam in the radiative shock and explore the conditions required to produce a hard powerlaw spectrum. Our picture is similar to that of Xray pulsars.
It is well established that in Xray pulsars matter from a companion star falls onto the magnetic poles of a neutron star. The accreting matter follows dipole magneticfield lines and falls supersonically. Eventually, it has to stop on the neutronstar surface. The supersonic to subsonic transition of the accretion flow occurs in a radiative shock above the neutronstar surface (Basko & Sunyaev 1976). In such a shock, which for simplicity we consider here as a mathematical discontinuity, soft Xray photons coming mainly from below the shock suffer a nearly headon collision with the infalling electrons. Because of the large momentum of the infalling eletrons with respect to that of the soft photons, the photons are scattered backward, gain energy, and return to the dense, hot, subsonic flow. We will refer to this as bulkmotion Comptonization (BMC). In the subsonic flow, the photons are scattered again, more or less isotropically. If their energy is significantly less than the thermal energy of the electrons, the photons gain energy on average (thermal Comptonization, hereafter TC). After the scattering, some photons find themselves again in the supersonic flow and the above processes are repeated, albeit with progressively decreasing probability, because the photons can escape sideways, i.e., perpendicular to the accretion column. We want to remark here that in a radiative shock, like the one we envision above, it is impossible to separate BMC from TC, unlike what happens in a spherical flow, where one can have pure BMC (Blandford & Payne 1981a,b; Payne & Blandford 1981; Mastichiadis & Kylafis 1992; Titarchuk et al. 1996, 1997).
It is well known that this firstorder Fermi energization of the photons results in a powerlaw, highenergy spectrum. Lyubarski & Sunyaev (1982, hereafter LS82) were the first to compute analytically the emergent spectrum from such a shock. They studied spectral formation due to BMC and TC in a radiative shock that is of the RankineHugoniot form. This is a reasonable approximation, given the fact that, due to the large optical depth along the accretion column, the upscattered photons escape sideways.
Accretion onto magnetic neutron stars has been studied extensively by Becker & Wolff. In a series of papers (Becker & Wolff 2005, 2007; Becker et al. 2012), they have studied spectral formation in Xray pulsars and have highlighted the role of BMC and TC for spectral formation in accretion columns. In their classic paper (Becker & Wolff 2007, hereafter BW07), they consider a smooth flow in a cylindrical column, with the flow being slowed down by radiation. The shock is treated properly, not as a mathematical discontinuity, and it is the place where BMC and TC operate with seed soft photons produced by bremsstrahlung, cyclotron, and blackbody processes inside the column. As most of the photons escape sideways, the model predicts a fan beam.
In Xray pulsars, the accretion column has a large optical depth in all directions. This means that the radiation in the column thermalizes to a significant degree and any highenergy photons present in the column are downscattered. Thus, the emergent spectrum has a cutoff at ~20 keV (BW07). In lowluminosity pulsars, on the other hand, the sideways optical depth is expected to only be a few. Thus, a highenergy powerlaw spectrum with a much larger cutoff energy can in principle be produced. This is what we investigate here.
In Sect. 2 we describe the model that we have used, giving a qualitative description of the processes that occur in the radiative shock, in Sect. 3 we present our results, in Sect. 4 we discuss them, and in Sect. 5 we draw our conclusions and state the predictions of our work.
2. The model
In order to demonstrate our main ideas, we use a relatively simple model. We consider a cylindrical accretion column with two regions: the preshock region, where matter falls with constant velocity, and the postshock one, which has density seven times the preshock one, electron temperature T_{e}, and negligible flow velocity. To fit real data, significantly more elaborate models are needed.
Mainly bremsstrahlung photons from below the shock, but also soft Xray photons from the neutronstar photoshpere surrounding the accretion column, enter the shock and are followed by a Monte Carlo code until they escape. Since the Thomson optical depth along the accretion column in the preshock region is infinite by construction in our model, the crucial parameter in our calculation is the Thomson optical depth in the transverse direction in this region.
2.1. Qualitative presentation
2.1.1. The radiative shock
If for simplicity we consider that the accreting plasma consists only of protons and electrons, the accretional energy resides in the protons. Charge neutrality guarantees that the electrons are tied to the protons. Thus, the electrons have the same freefall velocity as the protons, but insignificant accretional energy. This energy for the infalling plasma per protonelectron pair is (1)where G is the gravitational constant, M is the mass of the neutron star, R is its radius, and m_{p} is the proton mass. For a neutron star of mass M = 1.4 M_{⊙} and radius R = 12.5 km, E_{p} ≈ 150 MeV.
If a significant amount of this energy is to be transfered to the photons via BMC, it looks, at first sight, impossible to happen in the radiative shock. This is because the energy E_{γ} of a soft photon becomes after a headon collision with a relativistic electron (2)for an electron speed of, say, v = 0.6c. This energy gain by the photon is orders of magnitude smaller than E_{p}. As we demonstrate below, though, it is possible for the photons to “steal” a significant fraction of the accretional energy, because many photons scatter off the same electron.
For simplicity, we think of the shock as a mathematical discontinuity, but in reality it has a finite thickness with characteristic scale equal to the photon mean free path for electron scattering (3)where n_{e} is the electron number density in the shock and σ_{T} is the Thomson cross section. For simplicity, in this discussion, we use the Thomson cross section, but in our Monte Carlo calculation we use the angle and energydependent one, due to the magnetic field (see Sect. 2.2).
In this characteristic thickness , the “optical depth” that an infalling electron sees in the “bath” of outgoing photons is (4)where n_{γ} is the number density of photons that mediate the shock and is the electron mean free path in the “bath” of photons. For the electron number density we write (5)where Ṁ is the mass accretion rate, a_{0} is the radius of the accretion column at the shock, and is the freefall velocity of the plasma near the neutron star surface, where the shock is formed. Here we have taken M = 1.4 M_{⊙} and R = 12.5 km.
Let u_{γ} be the energy density of the photons in the shock and E_{γ} their characteristic energy. Then, the accretional luminosity can be written as (6)from which we get (7)Thus, the “optical depth” τ_{γ} becomes (8)which implies that the “optical depth” that an infalling electron sees in the “bath” of outgoing photons in the shock is of order the ratio of the accretional energy of a protonelectron pair to the typical energy of the outgoing photons. This means that, in the thickness of the shock, many photons scatter off the same electron and this is how the kinetic energy of the accreting plasma is reduced in the shock. As an analogy, one could think of the stopping of a diver by the water molecules in a pool. Thus, we envision the transfer of energy from the protons to the electrons and from them to the photons as follows:
At the “top” of the shock, consider a headon collision between an outgoing photon and an infalling electron. The energy gain of the photon is described by Eq. (2) and this energy is taken from the electron. Thus, the electron would slow down, and charge separation would occur, if it were not for the protons, which transfer a tiny fraction ~E_{γ}/E_{p} of their energy to replenish the lost energy of the electron. This process is repeated many times as the protonelectron pairs traverse the shock thickness. Thus, at the “bottom” of the shock, the protons have given up a significant fraction of their accretional energy to the photons that mediated the shock, with the electrons acting as intermediaries.
2.1.2. The postshock temperature
According to Eq. (2), the energy of a soft photon quadruples in a headon collision with an infalling electron. Therefore, after several crisscrosses of the shock, the energy of a photon increases significantly, but, at the same time, the photon looses some of its energy due to electron recoil in the postshock region. This energy is deposited in the postshock region and heats the electrons (LS82).
An estimate of the electron (and proton) temperature in the postshock region can be obtained as follows:
The energy lost to the photons via BMC, is replenished by the accretional energy of the protons. This channeling of energy stops when the kinetic energy of the protons becomes comparable to the accretional energy of the electrons E_{e} = GMm_{e}/R ≈ 75 keV, where m_{e} is the electron mass. Thus, a rough estimate of the electron temperature T_{e} in the postshock region is given by (9)where, due to the strong magnetic field, we have assumed a onedimensional Maxwellian distribution. This gives kT_{e} ~ 150 keV. In our calculations, and for completeness, we have considered a range of temperatures.
2.1.3. The size of the accretion column
In what follows, we estimate the radius of the accretion column at the radiative shock, close to the neutronstar surface.
For a neutron star with surface magneticdipole field B at the equator and magnetic moment μ = BR^{3}, the Alfven radius, to within a factor of order unity, is (Lamb et al. 1973; Frank et al. 2002) (10)The fallback disk, which is assumed to be of the Shakura & Sunyaev (1973) type, has a pressure scale height (Frank et al. 2002) (11)where α = 0.03 is the viscosity parameter, Ṁ_{15} is the accretion rate in units of 10^{15} g s^{1}, r_{10} is the radial distance in units of 10^{10} cm and Assuming that the inner disk couples to the magneticdipole field lines in an annular region of radius r_{A} and radial thickness Δr, the crosssectional area () of the accretion column near the surface of the neutron star can be estimated from magneticflux conservation, namely (12)where B(r_{A}) = (1/2)B_{p}(R/r_{A})^{3} and B_{p} = 2B is the strength of the magneticdipole field at the pole of the star.
Assuming Δr ≈ ηh, where η is a scaling factor, we find (13)Alternatively, if the innerdisk matter couples to the magneticdipole field lines between r_{A} and the corotation radius r_{co} = (GM/Ω^{2})^{1/3}, where Ω is the angular frequency of the neutron star, then for Δr ≈ r_{A} − r_{co} ≈ 5 × 10^{8} cm we find (14)which is not significantly larger than what we have found in Eq. (13).
The Thomson optical depth τ_{⊥} in the direction perpendicular to the magnetic field, just above the radiative shock, is (15)For Ṁ = 2 × 10^{15} g s^{1}, which corresponds to an Xray luminosity of 3.7 × 10^{35} erg s^{1}, and a_{0} ≈ 50 m, we find τ_{⊥} ≈ 5.8.
2.2. The Monte Carlo code
Our Monte Carlo code is similar to previously used codes in our work and it is based on Cashwell & Everett (1959) and Pozdnyakov et al. (1983). For simplicity, we assume cylindrical symmetry and a twozone model: the preshock zone (z > 0) and the postshock one (z < 0). The shock is of the RakineHugoniot form at z = 0. The cross section of the cylinder has radius a_{0}, which corresponds to a transverse Thomson optical depth τ_{⊥} in the preshock zone. In the preshock zone, matter falls freely with constant velocity v_{ff} = 0.58c, appropriate for a 1.4 M_{⊙} neutron star with a radius R = 12.5 km.
In the postshock zone, the matter is compressed by a factor of seven (appropriate for an adiabatic hydrodynamic shock with γ = 4/3), hot, and with negligible flow velocity. In Sect. 2.1.2 we estimated the electron temperature behind the shock to be kT_{e} ~ 150 keV. As this value is probably an overestimate, because we used a onedimensional Maxwellian distribution, we use three values of kT_{e}: 30, 50, and 100 keV.
Photons, mainly bremsstrahlung from below the shock, and to a lesser extend photospheric photons from the polar cap, enter the shock. Some of them escape unscattered, while the rest are scattered before escape. Of the last ones, some are scattered once, others a few times, and a relatively small number are scattered many times. It is this last category of photons, that have crisscrossed the shock many times, have increased their energy significantly, and have produced the highenergy powerlaw spectrum, as it is expected from such a firstorder Fermi energization.
The input bremsstrahlung photons to the radiative shock have temperatute T_{e} and are emitted isotropically upwards. In other words, we do not perform radiative transfer on the isotropically emitted bremsstrahlung photons in the postshock region, but instead assume an upward flux of such photons.
The input quasithermal photospheric photons to the radiative shock are approximated as blackbody ones with temperature of kT_{BB} = 0.5 keV. The polar cap that emits quasithermal photons has a radius r_{0} ~ 8 km in 4U 0142+61 (White et al. 1996; Israel et al. 1999; Juett et al. 2002). Therefore, only a relatively small fraction of the photospheric photons enter the shock (see Sect. 3.2 below).
Following BW07, we take the approximation of Arons et al. (1987) for the magnetic cross section for continuum scattering by electrons, namely (16)where E is the photon energy, σ(E) is the KleinNishina cross section, θ is the angle between the directions of the photon and the magnetic field, and (17)where (18)is the cyclotron energy. In our calculations, we consider two indicative values of the magneticfield strength, 10^{12} G and 10^{13} G.
The mean free path to electron scattering from every point in the accretion column and in any direction, as well as the energy change and the new direction of a photon after scattering, are computed using the corresponding relativistic expressions (Pozdnyakov et al. 1983). For the thermal distribution of electron velocities we use a onedimensional Maxwellian. When a photon escapes, both its energy and direction are recorded. Thus, we compute directiondependent spectra as well as angleaveraged spectra.
3. Results
In what follows, we present the results of our Monte Carlo calculations in the form of E^{2}dN/dE in arbitrary units, as a function of E in keV. The main parameter in our calculations is the transverse Thomson optical depth τ_{⊥} (see Eq. (15)), which is almost directly proportional to Ṁ, since a_{0} is of order 100 m (see Eqs. (13) and (14)). For completeness we explore also three values of the postshock temperature kT_{e} = 30, 50 and 100 keV, and two values of the magnetic field strength B = 10^{12} and 10^{13} G. For the bremsstrahlung photons the temperature is the assumed postshock value of T_{e}, while for the blackbody, photospheric temperature we take kT_{BB} = 0.5 keV.
For clarity of presentation, we treat the bremsstrahlung and the blackbody input photons separately, while in reality their upscattering in the shock occurs simultaneously (see, however, Sect. 3.3). We start with bremsstrahlung input photons.
3.1. Bremsstrahlung input
Below the shock, the main emission mechanism is bremsstrahlung (BW07). Thus, the spectrum of the emitted photons is given in the Born approximation (Greene 1959) by (19)where (20)is the Gaunt factor and K_{0} is the modified Bessel function of the second kind, zeroth order.
For our first calculation we use the following values of the parameters: τ_{⊥} = 5, kT_{e} = 50 keV, and B = 10^{12} G, and refer to them as the reference values.
In Fig. 1, we show the input bremsstrahlung spectrum (crosses), with the integral of dN/dE normalized to unity, and the resulting angleaveraged output spectrum of the Monte Carlo (stars) for the reference values of the parameters.
Fig. 1 Angleaveraged emergent spectrum (stars) in E^{2}dN/dE form, with arbitrary units, as a function of energy E in keV. The spectrum of the input photons (normalized to unity) is due to bremsstrahlung emission (crosses). The parameters are at their reference values. 

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Above about 10 keV, the spectrum is a power law, with photonnumber spectral index Γ ≈ 1 and a peak in the E^{2}dN/dE distribution at E_{p} ≈ 80 keV. This is a natural consequence of the first order Fermi energization. As it is evident from Fig. 1, most of the energy in the emergent spectrum comes out at E_{p}.
Below about 10 keV, the output photonnumber spectrum dN/dE is essentially the input bremsstrahlung spectrum “shifted” to higher energy by a small amount. Its qualitative difference from the spectrum above 10 keV is due to the change in the cross section at the cyclotron energy E_{c} = 11.6 keV (see Eq. (18)). Below the cyclotron energy, the cross section is angle dependent and smaller than the Thomson value (see Eqs. (16) and (17)), while above the cyclotron energy it is isotropic and equal to the Thomson value for lowenergy photons.
3.1.1. Effects of transverse optical depth
As discussed above, the larger the transverse Thomson optical depth τ_{⊥}, the larger the number that a typical photon criscrosses the shock. This has a significant effect on the emergent spectrum.
In Fig. 2 we show the input bremsstrahlung spectrum (crosses), normalized to unity, and the emergent spectra for four values of τ_{⊥}:1 (triangles), 3 (squares), 5 (stars), and 7 (circles). The other parameters are at their reference values.
Fig. 2 Angleaveraged emergent spectrum in E^{2}dN/dE form, with arbitrary units, as a function of energy E in keV. The spectrum of the input photons (normalized to unity) is bremsstrahlung (crosses). The various curves are the results of different values of the transverse Thomson optical depth τ_{⊥}:1 (triagles), 3 (squares), 5 (stars), and 7 (circles). The rest of the parameters are at their reference values. 

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In all cases, a highenergy powerlaw spectrum with a cutoff is seen. The photonnumber spectral index Γ is approximately 1.5 for τ_{⊥} = 1, while Γ ≈ 0.7 for τ_{⊥} = 7.
3.1.2. Effects of magneticfield strength
As discussed in Sect. 2.2, the larger the cyclotron energy (see Eq. (12)), the smaller the cross section of the lowenergy input photons of a given energy, and the smaller the effective transverse optical depth for the same Thomson transverse optical depth τ_{⊥}. Thus, as the magneticfield strength increases, the average number that a photon crisscrosses the shock decreases and the resulting index Γ increases (i.e. steeper spectrum). This is clearly seen in Fig. 3, where the magneticfield strength is B = 10^{13} G and four values of the Thomson transverse optical depth τ_{⊥}: 1 (triangles), 3 (squares), 5 (stars), and 7 (circles). The rest of the parameters are at their reference values.
Comparing Fig. 3 with Fig. 2, we see that the spectra of Fig. 3 have no break at ~10 keV and the power laws extend from low energies all the way to E_{p} ≈ 70 keV. This is expected, because the cyclotron energy is E_{c} = 116 keV, which is above E_{p}. On the other hand, to get the same value of Γ, one needs a larger transverse Thomson optical depth in Fig. 3 than in Fig. 2. For example, to get Γ ≈ 1, one needs τ_{⊥} = 7 in Fig. 3, but only τ_{⊥} = 5 in Fig. 2. A slight deviation in the value of the peak energy E_{p} between Figs. 2 and 3 can be accommodated with a slight change in the temperature of the postshock electrons (see Sect. 3.1.3).
Fig. 3 Same as in Fig. 2, but for B = 10^{13} G. 

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3.1.3. Effects of postshock temperature
As discussed in Sect. 2.1.2, the postshock temperature is not known. We have provided a simple estimate of kT_{e} ~ 150 keV, but this is probably an overestimate, because we have assumed a onedimensional Maxwellian distribution. Thus, we examine two additional values: 100 keV and 30 keV.
Figure 4 is similar to Fig. 2, but for kT_{e} = 100 keV. For the same transverse Thomson optical depth τ_{⊥}, the emergent spectrum is flatter in Fig. 4 than in Fig. 2. For example, a τ_{⊥} = 3 is needed in Fig. 4 to produce a highenergy powerlaw index Γ ≈ 1, while a τ_{⊥} = 5 is needed in Fig. 2. Similarly, the peak energy E_{p} is larger in Fig. 4 than in Fig. 2. In particular, for kT_{e} = 100 keV, the peak energy is E_{p} ≈ 150 keV.
Figure 5 is similar to Figs. 2 and 4, but for kT_{e} = 30 keV. The conclusions are the same. A transverse Thomson optical depth τ_{⊥} = 6 is required to obtain Γ ≈ 1, while for kT_{e} = 50 keV a τ_{⊥} = 5 is enough.
Fig. 4 Same as in Fig. 2, but for kT_{e} = 100 keV. 

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Fig. 5 Same as in Fig. 2, but for kT_{e} = 30 keV. 

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3.2. Blackbody input
As discussed extensively in Trümper et al. (2013), in the accretion model of AXPs/SGRs the ratio of the observed luminosity L_{s} of the soft, quasithermal photospheric emission to the hard, fanbeamed, powerlaw luminosity L_{h} is of order unity, but it depends on several parameters, and in particular on the height of the radiative shock. In other words, the larger the height, the larger the radius r_{0} of the polar cap. This is true not only for geometrical reasons but also due to gravitational bending.
Since r_{0} ~ 8 km for 4U 0142+61 (White et al. 1996; Israel et al. 1999; Juett et al. 2002), while the radius of the radiative shock is typically a_{0} ~ 100 m (see Eqs. (13) and (14)), we estimate that only a small fraction (~10^{4}, see Sect. 3.3) of the photospheric photons enter the shock. The majority of them come from around the footprint of the accretion column. We make the approximation that they are described by a blackbody distribution of temperature T_{BB} and that they enter the shock at an angle θ with respect to the axis of the accretion column. For simplicity we take cosθ = 0.7. Our results are insensitive to this last approximation.
For a representative calculation with blackbody input, we use the following values of the parameters: τ_{⊥} = 5, kT_{e} = 50 keV, B = 10^{12} G, and kT_{BB} = 0.5 keV.
In Fig. 6, we show the input blackbody spectrum (crosses), with the integral of dN/dE normalized to unity, and the resulting angleaveraged output spectrum of the Monte carlo (stars) for the above values of the parameters. A hint of a broad peak at low energies (E ≲ 10 keV) is due to the input photons that escape unscattered or suffer a small number of scatterings, and can be thought of as a “shift” of the blackbody spectrum to higher energies, though it should be kept in mind that what is plotted is E^{2}dN/dE and not dN/dE.
Above about 10 keV, the spectrum is a power law, with photon number spectral index Γ ≈ 1.3 and a peak in the E^{2}dN/dE distribution at E_{p} ≈ 80 keV. This is again a natural consequence of first order Fermi energization. Despite the fact that only a small number of photons suffer many crisscrosses of the shock, most of the energy in the emergent spectrum comes out at E_{p}.
Fig. 6 Angleaveraged emergent spectrum (stars) in E^{2}dN/dE form, with arbitrary units, as a function of energy E in keV. The spectrum of the input photons (normalized to unity) is blackbody (crosses). The parameters are: τ_{⊥} = 5, kT_{e} = 50 keV, B = 10^{12} G, and kT_{BB} = 0.5 keV. 

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Comparing Figs. 1 and 6, we see that for the same transverse optical depth τ_{⊥} = 5 the photon number spectral index Γ is approximately equal to 1 in Fig. 1 and approximately 1.3 in Fig. 6. This means that bremsstrahlung is more efficient in producing the highenergy power law than the blackbody. This is, of course, natural because the bremsstrahlung contains photons with energy comparable to kT_{e}, while the typical blackbodyphoton energy is 2kT_{BB} ≪ kT_{e}. Thus, fewer scatterings are needed for some bremsstrahlung photons to reach the peak energy E_{p} ≈ 80 keV than for all the blackbody ones.
3.3. Blackbody versus bremsstrahlung input
Despite the fact that the quasithermal, photospheric photons can in principle produce highenergy powerlaw spectra similar to the ones oberved in AXPs/SGRs by Comptonization in the radiative shock (e.g., Fig. 6), we will show below that their role is rather insignificant compared to that of the bremsstrahlung ones. The reasons are the following:

1)
Most of the photospheric photons do not enter the shock.For simplicity we assume that the polar cap is flat. This doesnot introduce any significant error, because gravitationalbending applies to both the hard Xrays from the shock to thepolar cap and to the soft Xrays from the polar cap to the shock.Let the area of the shock seen by a photospheric photon be for τ_{⊥} = 5, where is the shock thickness (see Eq. (3)). The fractional solid angle subtended by the area A at radius r of the polar cap is dΩ/4π = A/4πr^{2} and for isotropic emission from the polar cap we find that the fraction of polar cap photons that reach the shock is (21)for r_{0} = 8 km and a_{0} = 100 m.

2)
The number of photospheric photons that enter the radiative shock is smaller than the corresponding bremsstrahlung ones. This is because, for accretional luminosity L, the photospheric luminosity that enters the shock is about 10^{4}(L/2), while the corresponding bremsstrahlung luminosity, i.e., the remaining luminosity in the postshock region is (see Sect. 2.1.2) ~ (m_{e}/m_{p})L. For a typical photon energy of 1 keV for the photospheric emission and less than 1 keV for bremsstrahlung, we see that bremsstrahlung dominates the soft photon input to the shock.

3)
As discussed in Sect. 3.2, for equal inputs to the radiative shock, the bremsstrahlung is more effectively upscattered to the highenergy power law than the photospheric input.
4. Discussion
It is known observationally (e.g., Enoto et al. 2010) that in AXPs/SGRs the hard Xray luminosity L_{h} is comparable to the soft Xray one L_{s}. In the accretion picture presented by Trümper et al. (2010, 2013), this is a natural consequence of the fact that only about half of the produced hard Xray luminosity in the radiative shock is observed directly (L_{h}), while the other half is intercepted by the polar cap and the major fraction of it is reemitted as soft Xrays (L_{s}).
With our reported calculations in Sect. 3.1, we have shown that the radiative shock can produce a fanbeam, hard Xray, powerlaw spectrum similar to those seen in AXPs/SGRs. Furthermore, in Sect. 3.3 we have shown that nearly all the photospheric luminosity L_{s} is observed unaltered. Thus, the total Xray spectrum, soft and hard, of an AXP/SGR with L_{h} = L_{s} can be easily produced by adding to the hard spectrum shown in Figs. 1–5 a blackbody spectrum with a peak in E^{2}dN/dE having the same height as the peak of the hard Xray spectrum.
As an example, we consider the hard Xray spectrum of Fig. 1 (stars) and add to it a blackbody spectrum of temperature T_{BB} = 0.5 keV that has the same height in E^{2}dN/dE as the hard Xray spectrum. The result is shown in Fig. 7.
In view of the above, it is natural to expect that lowluminosity Xray pulsars should exhibit soft and hard Xray spectra similar to those seen in AXPs and SGRs. Indeed, as discussed in Trümper et al. (2013), 4U 0352+309 (XPer) and 4U 2206+54 show highenergy powerlaw tails extending to ~100 keV, with photon number spectral index Γ ~ 1.
Fig. 7 Characteristic spectrum expected from lowluminosity accretion onto a magnetic neutron star. The highenergy spectrum is produced in the radiative shock near the bottom of the accretion column. The parameters are: τ_{⊥} = 5, kT_{e} = 50 keV, and B = 10^{12} G. We have added to it a blackbody spectrum of temperature kT_{BB} = 0.5 keV such that its peak has the same height as the peak at 80 keV. 

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5. Conclusions and predictions
We have demonstrated with a simple quantitative model that accretion from a fallback disk onto a neutron star with normal (10^{12} − 10^{13} G) dipole magnetic field can produce hard Xray, powerlaw spectra similar to the ones observed from steadystate or transient AXPs and SGRs. Furthermore, we have shown that the soft part of the Xray spectrum, that comes from the polar cap, does not enter in a significant way in the formation of the hard part of the spectrum and can simply be added to it.
The powerlaw index Γ of the hard Xray spectrum depends mainly on the accretion rate, or equivalently on the transverse optical depth of the accretion column at the radiative shock, and less on the strength of the magnetic field and the postshock temperature. A transverse Thomson optical depth of ~5 is enough to produce a highenergy powerlaw index Γ ≈ 1.
The cutoff at high energies (E ~ 100 keV), is due to the limited kinetic energy of the accreting electrons, and it depends on the postshock temperature. If the accretion rate is sufficient for the formation of a radiative shock, the cutoff energy cannot be larger than ~300 keV. Thus, if our picture is correct, we predict that no AXP or SGR will be found to have a cutoff energy of 400 keV or larger in their steadystate or transient spectra.
Acknowledgments
We thank an anonymous referee for useful comments, which have improved our paper. One of us (NDK) acknowledges partial support by the “RoboPol” project, which is implemented under the “ARISTEIA” Action of the “OPERATIONAL PROGRAM EDUCATION AND LIFELONG LEARNING” and is cofunded by the European Social Fund (ESF) and National Resources. NDK also acknowledges a grant from the European Astronomical Society in 2013. Ü.E. acknowledges research support from TÜBİTAK (The Scientific and Technical Research Council of Turkey) through grant 113F166.
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All Figures
Fig. 1 Angleaveraged emergent spectrum (stars) in E^{2}dN/dE form, with arbitrary units, as a function of energy E in keV. The spectrum of the input photons (normalized to unity) is due to bremsstrahlung emission (crosses). The parameters are at their reference values. 

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In the text 
Fig. 2 Angleaveraged emergent spectrum in E^{2}dN/dE form, with arbitrary units, as a function of energy E in keV. The spectrum of the input photons (normalized to unity) is bremsstrahlung (crosses). The various curves are the results of different values of the transverse Thomson optical depth τ_{⊥}:1 (triagles), 3 (squares), 5 (stars), and 7 (circles). The rest of the parameters are at their reference values. 

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In the text 
Fig. 3 Same as in Fig. 2, but for B = 10^{13} G. 

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In the text 
Fig. 4 Same as in Fig. 2, but for kT_{e} = 100 keV. 

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In the text 
Fig. 5 Same as in Fig. 2, but for kT_{e} = 30 keV. 

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In the text 
Fig. 6 Angleaveraged emergent spectrum (stars) in E^{2}dN/dE form, with arbitrary units, as a function of energy E in keV. The spectrum of the input photons (normalized to unity) is blackbody (crosses). The parameters are: τ_{⊥} = 5, kT_{e} = 50 keV, B = 10^{12} G, and kT_{BB} = 0.5 keV. 

Open with DEXTER  
In the text 
Fig. 7 Characteristic spectrum expected from lowluminosity accretion onto a magnetic neutron star. The highenergy spectrum is produced in the radiative shock near the bottom of the accretion column. The parameters are: τ_{⊥} = 5, kT_{e} = 50 keV, and B = 10^{12} G. We have added to it a blackbody spectrum of temperature kT_{BB} = 0.5 keV such that its peak has the same height as the peak at 80 keV. 

Open with DEXTER  
In the text 
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