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ABSTRACT

Recently, it has been discovered that the transition of X-ray pulsars to the low luminosity state (⁠|$L\lesssim 10^{35}\, {\rm erg\ \rm s^{-1}}$|⁠) is accompanied by a dramatic spectral change. That is, the typical power-law-like spectrum with high-energy cut-off transforms into a two-component structure with a possible cyclotron absorption feature on top of it. It was proposed that these spectral characteristics can be explained qualitatively by the emission of cyclotron photons in the atmosphere of the neutron star caused by collisional excitation of electrons to upper Landau levels and further Comptonization of the photons by electron gas. The electron gas are expected to be overheated in a thin top layer of the atmosphere. In this paper, we perform Monte Carlo simulations of the radiative transfer in the atmosphere of an accreting neutron star while accounting for a resonant scattering of polarized X-ray photons by thermally distributed electrons. The spectral shape is shown to be strongly polarization-dependent in soft X-rays (⁠|$\lesssim 10\, {\rm keV}$|⁠) and near the cyclotron scattering feature. The results of our numerical simulations are tested against the observational data of the X-ray pulsar A 0535+262 in the low-luminosity state. We show that the spectral shape of the pulsar can be reproduced by the proposed theoretical model. We discuss applications of the discovery to the observational studies of accreting neutron stars.

polarization, radiative transfer, scattering, stars: magnetic field, stars: neutron, X-rays: binaries

1 INTRODUCTION

Classical X-ray pulsars (XRPs) are accreting strongly magnetized neutron stars (NSs) orbiting around optical stellar companions (for a review, see Walter et al. 2015). One of the richest families of XRPs is a system with a Be star as a companion (BeXRP; Reig 2011). BeXRPs exhibit strong transient activity that allows us to study the radiation’s interaction with matter under conditions of extremely strong magnetic fields over a wide range of mass accretion rates covering up to six orders of magnitude (Doroshenko 2020).

From an observational point of view, the spectra of XRPs at high luminosities (>1036 erg s−1) have similar shapes, which can be well fitted by a power law with an exponential cut-off at high energies (e.g. Nagase 1989; Filippova et al. 2005). However, it was recently discovered that the decrease of the observed luminosity below this value is accompanied by dramatic changes of the energy spectra in several XRPs (Tsygankov et al. 2019a,b; Doroshenko et al. 2021; Lutovinov et al. 2021), pointing to varied physical and geometrical properties of the emission region.

In particular, originally it was shown for the transient XRPs, GX 304−1 and A 0535+26, that once their luminosity drops down to  1034–1035 erg s−1, the ‘canonical’ spectral shape of their emission undergoes a dramatic transition into a two-component structure, consisting of two humps peaking at ∼5–7 keV and ∼30–50 keV (Tsygankov et al. 2019a,b; Lutovinov et al. 2021). The only other source with a similar spectral structure observed earlier is the persistent low-luminosity XRP X Persei (Doroshenko et al. 2012), although this never showed spectral transitions as a function of luminosity. In the case of A 0535+26, a cyclotron absorption feature was also detected on top of the high-energy component of the spectrum. The appearance of the high-energy spectral component was interpreted by the recombination of electrons collisionally excited to the upper Landau levels in the heated layers of the NS atmosphere (see Fig. 1; Tsygankov et al. 2019b). However, no quantitative description had been done.

Schematic diagram of the theoretical model. The X-ray energy spectrum originates from the atmosphere of a NS with the upper layer overheated by low-level accretion. The accretion flow is stopped in the atmosphere due to collisions. Collisions result in excitation of electrons to upper Landau levels. The following radiative de-excitation of electrons produces cyclotron photons, which are partly reprocessed by magnetic Compton scattering and partially absorbed in the atmosphere. The reprocessed photons form a high-energy component of the spectrum, while the absorbed energy is released in thermal emission and forms a low-energy part of the spectrum.
Figure 1.

Schematic diagram of the theoretical model. The X-ray energy spectrum originates from the atmosphere of a NS with the upper layer overheated by low-level accretion. The accretion flow is stopped in the atmosphere due to collisions. Collisions result in excitation of electrons to upper Landau levels. The following radiative de-excitation of electrons produces cyclotron photons, which are partly reprocessed by magnetic Compton scattering and partially absorbed in the atmosphere. The reprocessed photons form a high-energy component of the spectrum, while the absorbed energy is released in thermal emission and forms a low-energy part of the spectrum.

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Dramatic spectral changes in XRPs at very low mass accretion rates can shed light on different physical processes responsible for the spectra formation at different luminosities. Depending on the luminosity, it is possible to distinguish different regimes of accretion (i.e. the interaction of radiation with accreting plasma), as follows.

  • At high mass accretion rates, the luminosity and local X-ray energy flux are high enough to stop accretion flow above the NS surface in radiation-dominated shock (Lyubarsky & Syunyaev 1982). Below the shock region, the matter slowly settles down to the stellar surface and forms the so-called accretion column confined by a strong magnetic field (Inoue 1975; Basko & Sunyaev 1976; Wang & Frank 1981; Mushtukov et al. 2015b; Gornostaev 2021). Because of magnetic field confinement and opacity reduced in a strong magnetic field (Herold 1979), the accretion column can survive under the extreme conditions of high radiation pressure. The minimal accretion luminosity, which is sufficient to stop the matter in radiation pressure dominated shock, is called the critical luminosity Lcrit (Basko & Sunyaev 1976). The critical luminosity is of the order of |$10^{37}\, {\rm erg\ \rm s^{-1}}$|⁠, and depends on the geometry of accretion flow and magnetic field strength at the NS surface (Becker et al. 2012; Mushtukov et al. 2015a).

  • At low mass accretion rates (subcritical regime, when L < Lcrit), the radiation pressure is insufficient to stop accretion flow above the NS surface, and the final braking of the flow occurs in the atmosphere of a NS due to collisions between particles. Subcritical accretion results in hotspots at the stellar surface located close to the NS’s magnetic poles. Even in the subcritical regime, the accretion flow dynamics can be affected by radiation pressure, and X-ray spectra can be shaped by the interaction of photons with the accreting material (e.g. Becker & Wolff 2007; Mushtukov et al. 2015c). Only in the case of very low mass accretion rates (⁠|$\dot{M}\lesssim 10^{15}\, {\rm g\, s^{-1}}$|⁠) is the influence of interaction between accreting matter and radiation above the NS surface negligible. In this regime, almost all energy of accretion flow is released in the geometrically thin NS atmosphere, where the accreted protons transfer their kinetic energy to the plasma heating. In this case, the geometry of the emitting region is elementary, and the plane-parallel approximation for the particle heated atmosphere is a reasonable approximation for a complete description of spectra formation.

In this work, based on the Monte Carlo simulations of radiative transfer in a strong magnetic field, we propose the first simple two-slab model describing the spectral formation in XRPs at very low mass accretion rates. In Section 2, we introduce our model set-up and the basic assumptions of radiative transfer. The basic details of our numerical code are given in Section 3. Results of numerical simulations and their comparison with the observational data are given in Section 4. Summarizing our results in Section 5, we propose applications to observational studies and we argue that the proposed concept of spectra formation at extremely low states of accretion luminosity can be used in detailed diagnostics of the ‘propeller’ state in accretion and can be a base of alternative estimations of NS magnetic field strength.

2 MODEL SET-UP

2.1 Geometry of emitting region

Subcritical mass accretion rates (e.g. Basko & Sunyaev 1976; Mushtukov et al. 2015a) result in the simplest geometry of emitting regions: the X-ray photons are produced by hotspots at the NS surface. If the mass accretion rate is well below the critical luminosity value L ≪ Lcrit, the influence of X-ray energy flux on the dynamics of the accretion process is insignificant (Mushtukov et al. 2015c).

In the case of a magnetic field dominated by a dipole component, the area of the spots at the NS surface can be roughly estimated as
(1)
Here, Λ is a constant that depends on accretion flow geometry, Λ < 1 for the case of accretion through the disc with Λ = 0.5 being a commonly used value (e.g. Ghosh & Lamb 1978; Lai 2014), m is the NS mass measured in units of solar masses M⊙, R6 is the NS radius measured in |$10^6\, {\rm cm}$|⁠, B12 is the magnetic field strength at the pole of a NS measured in units of |$10^{12}\, {\rm G}$|⁠, and L37 is the accretion luminosity of an XRP measured in units of |$10^{37}\, {\rm erg\ \rm s^{-1}}$|⁠. The corresponding effective temperature of a hotspot can be estimated as
(2)
where σSB is the Stefan–-Boltzmann constant.
In the case of accretion on weakly magnetized NSs, the protons of accretion flow brake in a stellar atmosphere due to Coulomb interactions with atmospheric electrons (see Chapter 3 in Frank, King & Raine 2002). In the case of highly magnetized atmospheres, the physical picture is similar, but a significant fraction of proton kinetic energy turns into excitation of electrons to upper Landau levels (Nelson et al. 1995). Further de-excitation of electrons results in production of cyclotron photons at energy |$E_{\rm cyc}\approx 11.6\, B_{12}\, {\rm keV}$| (this is the case if |$B\lesssim 10^{13}\, {\rm G}$|⁠; see Harding & Preece 1987). It is expected that the sources of cyclotron photons decay exponentially with the optical depth |$\propto e^{-\tau /\tau _{\rm br}}$|⁠, where τbr is the typical optical depth due to Thomson scattering, where the accretion flow is braked due to collisions. The total number of cyclotron photons emitted in the atmosphere per cm2 s−1 due to accretion flow braking can be estimated as
(3)
where S10 is the area of a hotspot at the NS surface in units of |$10^{10}\, {\rm cm}$|⁠.

The temperature structure of the accretion-heated weakly magnetized atmosphere can be divided into two qualitatively different layers. Most of the energy is released in deep optically thick layers (if τbr > 1) where the plasma mass density is high enough, and cooling due to free–free emission can compensate the heating due to braking of the accretion flow. The deep heated layer is close to being isothermal, with the temperature close to the effective temperature (equation 2). However, the upper rarefied layers of the atmosphere have too low density to be able to cool with free–free emission. As a result, the upper layers are cooled by Compton scattering and overheated up to tens and even a hundred keV (see Suleimanov, Poutanen & Werner 2018). We assume that the highly magnetized accretion-heated atmospheres have a similarly qualitative structure. Thus, we consider a plane-parallel magnetized semi-infinite atmosphere with a thin overheated upper layer and an isothermal deeper layer (see Fig. 1). The temperatures of both parts are free parameters of our model, as well as the optical thickness of the overheated slab. The magnetic field is taken to be perpendicular to the NS surface, which is a good approximation for the regions located close to the magnetic poles of a NS. The resulting spectrum is computed by solving the radiation transfer equation using the Monte Carlo method.

2.2 Radiative transfer

Both the opacity and refractive index are dependent on photon polarization in a strongly magnetized plasma and vacuum. As a result, the solution of the radiative transfer problem has to account for the polarization of photons. In the general case, the polarization of X-ray photons can be described in terms of four Stokes parameters. At the same time, the radiative transfer equation turns into a set of four equations – one for each Stokes parameter. Because the plasma in a strong magnetic field is anisotropic and birefringent (Ginzburg 1970), the polarization of X-ray photon can vary along its trajectory. It makes the description of polarized radiative transfer by the Stokes parameters even more complicated. However, any X-ray photons can be represented as a linear composition of two orthogonal normal modes, which conserve their polarization state along their trajectories (see Zheleznyakov 1977). The linear composition of the normal modes changes its polarization because of the difference in the phase velocity of the modes. If the difference between the phase velocities of normal modes is large enough, the radiative transfer problem reduces and can be effectively solved for two normal modes. That is, the radiation can be described by specific intensities in two normal polarization modes |$I_E^{(s)}$|⁠, where |$s=1\, \, (s=2)$| corresponds to the X-mode (O-mode).

The ellipticity of normal modes is determined by plasma conditions (temperature, mass density and chemical composition; see Ginzburg 1970; Kirk 1980) and vacuum polarization (for a review, see Harding & Lai 2006). In this paper, we neglect the effects of vacuum polarization (see Appendix  B), and assume that the dielectric tensor in the atmosphere coincides with the dielectric tensor of cold plasma. The normal modes are transverse and use the approximated ellipticity of the modes affected by plasma effects only:
(4)
Here, Ex (Ey) is the photon’s electric field component along (perpendicular to) the |$\boldsymbol{k}\!-\!\boldsymbol{B}$| plane, where |$\boldsymbol{k}$| is the vector of photon momentum and |$\boldsymbol{B}$| is the vector of external magnetic field, and θ is the angle between the photon momentum and B-field direction (Gnedin & Syunyaev 1973).

The main processes of interaction between radiation and matter in our model are:

  • magnetic Compton scattering;

  • cyclotron absorption;

  • bremsstrahlung affected by a strong magnetic field.

The radiative transfer equation can be written as
(5)
where τT is the optical depth due to non-magnetic Thomson scattering, |$\alpha _{\rm abs}^{(s)}$| is the coefficient of true absorption (due to free–free and cyclotron mechanisms), αT is the absorption coefficient due to non-magnetic Thomson scattering and BE(T) is the Planck function. The redistribution function |$R_{s\, s^{\prime }}(E,\Omega | E^{\prime },\Omega ^{\prime })$| describes the redistribution of X-ray photons over energy, momentum and polarization state, and this is related to the probability of photon transition from energy E ′, the direction given by solid angle Ω′ and polarization state s′ to energy E, direction Ω and polarization state s due to magnetic Compton scattering. The radiative transfer equation (5) neglects non-linear effects such as induced scattering because these are expected to be negligible in low-luminosity states.

The first term on the right-hand side of equation (5) describes absorption due to bremsstrahlung and cyclotron mechanism. With the second term, we account for the photon redistribution due to Compton scattering. The third term introduces the thermal emission of photons, which is calculated with the assumption of local thermodynamic equilibrium (LTE). The last term gives the primary sources of cyclotron emission, which are distributed exponentially in our particular case |$S_{\rm ini}\propto {\rm e}^{-\tau /\tau _{\rm br}}$|⁠.

2.2.1 Compton scattering

The redistribution function due to Compton scattering in equation (5) is given by
(6)
where the double differential cross-section of scattering by the electron gas with the electron distribution over the momentum fe(pb) is
(7)
The solid angles are related to the spherical coordinates describing the direction of photon motion (θi, f, φi, f as dΩi, f = sin θi, f dθi, f dφi, f), and|$p^*_{b}$| is the electron momentum required for the photon transition |$\lbrace E_{\rm i},\Omega _{\rm i}\rbrace \longrightarrow \lbrace E_{\rm f},\Omega _{\rm f}\rbrace$|⁠.
(8)
is the ordinary scattering cross-section by the electron of momentum pb along magnetic field lines, and
(9)
(10)
(11)
account for the Doppler effect and aberration due to transition between different reference frames.
Compton scattering is affected by a strong magnetic field in the atmosphere of a NS (Daugherty & Harding 1986). The scattering cross-section strongly depends on photon energy E, momentum and polarization state. The scattering of photons whose energy is close to the cyclotron energy |$E_{\rm cyc}\approx 11.6\, (B/10^{12}\, {\rm G})\, {\rm keV}$| becomes resonant. In this paper, Compton scattering is considered in a non-relativistic manner (see Herold 1979). The differential cross-sections of Compton scattering by electrons at rest are expressed by the complex amplitudes of scattering |$a^{\rm (p)}_{s_{\rm f}s_{\rm i}}$|⁠,
(12)
where Ei is the initial photon energy, and si and sf denote the photon polarization state before and after the scattering, respectively. The complex amplitudes |$a^{\rm (p)}_{s_{\rm f}s_{\rm i}}$| depend on the exact expression for the polarization modes. The amplitudes can be combined in a matrix
(13)
which is given by
(14)
where the unitary matrix
(15)
and the matrix |$\widehat{a}^{\rm (v)}$| is composed of the scattering amplitudes calculated for the linearly polarized vacuum modes (Herold 1979):
(16)

The thermal motion of electrons significantly affects the scattering both below the cyclotron resonance and at the cyclotron resonance.1 In our Monte Carlo code, we use pre-calculated tables, which describe the redistribution of photons over the energy, momentum and polarization states.

2.2.2 Cyclotron absorption

Cyclotron absorption at the fundamental depends on the photon polarization state and is calculated according to Zheleznyakov (1977). In the case of kBTe ≪ mec2, where kB is the Boltzmann constant, Te is the electron temperature and me is the electron mass, the adopted absorption cross-section for extraordinary photons is given by
(17)
The cross-section of cyclotron absorption for O-mode photons is smaller: |$\sigma _{\rm s}^{(2)}\approx \sigma _{\rm s}^{(1)} (k_{\rm B}T_{\rm e}/m_{\rm e}c^2)$|⁠. In the case of kBTe ∼ mec2, the absorption cross-section is obtained numerically under the assumption that the distribution of electrons over the momentum pb on the ground Landau level is given by
(18)
where γ = [1 + (pb/(mec))2]1/2 is the Lorentz factor, y = mec2/(kTe) is the inverse dimensionless temperature and K1 is the modified Bessel function of the second kind (Mushtukov et al., in preparation). Cyclotron absorption at the fundamental results in excitation of an electron from the ground Landau level to the first excited level. Because the de-excitation rate of electrons due to emission of cyclotron photons is much larger than the de-excitation rate due to collisions between particles rcoll (Bonazzola, Heyvaerts & Puget 1979), the majority of cyclotron absorption events are followed by almost immediate photon emission and can be considered as an event of Compton scattering (see Harding & Lai 2006). The probability Pabs, true that the cyclotron absorption event will end up with a true absorption can be estimated from cyclotron rcyc and collision rcoll de-excitation rates:
(19)
Here, |$n_{\rm e,21}=n_{\rm e}/10^{21}\, {\rm cm^{-3}}$| is the local number density of electrons, which is taken in our calculations to be dependent on optical depth in the atmosphere.

2.2.3 Free–free absorption

Free–free (bremsstrahlung) opacity is dependent on polarization, direction and density. We calculate the opacity according to Lai & Ho (2003) (see Appendix  A) and under the assumption that the ellipticity of normal modes is given by equation (4). The mass density ρ in the atmosphere is calculated under the assumption of hydrostatic equilibrium: dP/dξ = −ρg, where P is the gas pressure, ξ is the vertical coordinate in the atmosphere and g is the gravitational acceleration. In the case of atmosphere composed of a few layers of fixed temperature, the mass density at optical depth τ is given by2
(20)
where τi is the optical depth at the top border of the i layer and |$T^{(i)}_{\rm keV}$| is the temperature in the layer.

The initial cyclotron photons are emitted at ∼Ecyc with thermal broadening, which depends on the direction with respect to magnetic field lines. The angular distribution of the cyclotron photons is taken to be isotropic.

3 MONTE CARLO CODE

We perform Monte Carlo simulations, tracing X-ray photons in the plane-parallel atmosphere of a magnetized NS. The magnetic field strength and its direction are fixed and assumed to be constant in the atmosphere, which is a reasonable assumption because of the small geometrical thickness of the atmosphere and the small size of hotspots at the NS surface. Non-linear effects of radiative transfer are not important at low-luminosity states and are neglected in our simulations. Tracing X-ray photons in the atmosphere, we obtain angular-dependent energy spectra of polarized radiation leaving the atmosphere (see Fig. 2).

Schematic diagram of Monte Carlo simulations performed in the paper.
Figure 2.

Schematic diagram of Monte Carlo simulations performed in the paper.

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There are two sources of seed photons in the model: the cyclotron photons, due to radiative de-excitation of electrons excited to upper Landau levels by collisions of accretion flow with the atmosphere, and thermal photons. The sources of cyclotron photons are distributed exponentially in the atmosphere. The cyclotron photons are emitted close to the cyclotron energy within the thermally broadened line. The distribution of initial thermal photons is determined by the absorption coefficients in the atmosphere and local temperature (see the third term in radiative transfer equation 5). The transfer of X-ray photons originating from different initial sources is considered separately. For each simulation, we trace the history of |$N_{\rm cyc}^{({\rm i})}$| cyclotron photons and |$N_{\rm th}^{({\rm i})}$| thermal photons. Because we account for free–free and cyclotron absorption in the atmosphere, the number of cyclotron and thermal photons leaving the atmosphere, |$N_{\rm cyc}^{({\rm f})}$| and |$N_{\rm th}^{({\rm f})}$|⁠, is smaller than the number of seed photons, |$N_{\rm cyc}^{({\rm f})}\le N_{\rm cyc}^{({\rm i})}$| and |$N_{\rm th}^{({\rm f})}\le N_{\rm th}^{({\rm i})}$|⁠. Simulating radiative transfer, we aim to reach a certain number, |$N_{\rm th}^{({\rm f})},N_{\rm cyc}^{({\rm f})}\sim 2\times 10^6$|⁠, of photons that leave the atmosphere. The results of the separately calculated radiative transfer problems are combined in the final angular- and polarization-dependent spectra:
(21)
In order to construct the final spectra accounting for both sources of seed photons, we normalize the contribution of both sources,
(22)
where A1 and A2 are constants. The normalization is performed on the basis of the energy conservation law in the atmosphere. We start with the simulation of radiative transfer of the cyclotron photons. The accretion luminosity is proportional to the total energy of seed cyclotron photons in the simulation:
(23)
However, the part of the luminosity due to thermal emission of the atmosphere is determined by the difference between the total energy of seed cyclotron photons and the total energy of reprocessed cyclotron photons leaving the atmosphere:
(24)
Using conditions (23) and (24), we obtain constants A1 and A2 in equation (22).

In order to perform Monte Carlo simulations and to track the photons, we use a set of pre-calculated tables describing photon redistribution due to magnetic Compton scatterings.

Tables A give the total scattering cross-section, where the cross-section is given as a function of photon energy, the polarization state before and after the scattering event, the angle between the B-field direction and photon momentum, the temperature and the bulk velocity of the electron gas. For each combination of the initial and final polarization states of a photon, the tables are pre-calculated in a fixed grid in photon energy and angle θi, and for a fixed temperature and bulk velocity of the gas. To obtain a scattering cross-section for a given photon energy and momentum, we use quadratic interpolation in the photon energy grid and further quadratic interpolation in the angle grid.

Tables B give the photon probabilities to be scattered into a certain segment of the solid angle (θf + Δθf, φf + Δφf). The tables are pre-calculated on a grid of photon initial parameters (energy, polarization state, momentum) and for both possible final polarization states.

The steps for tracing the photon history are as follows.

  • We make a choice about the origin of the seed photon: either thermal emission of the atmosphere or cyclotron emission. If the photon is a result of cyclotron emission, we obtain the optical depth where the photon is emitted:
    $$\begin{equation*} \tau _0=-\tau _{\rm br}\ln X_1. \end{equation*}$$
    Here, X1 ∈ (0; 1) is a random number, and for the photon energy, we assume that the photon is emitted within a thermally broadened cyclotron line. If the photon is a result of thermal emission, we obtain the optical depth of its emission in the atmosphere from pre-calculated cumulative distribution functions of photon emission, accounting for the assumed temperature structure in the atmosphere and free–free absorption coefficient.

  • We obtain the free path of the photon, accounting for scattering and absorption opacity.

  • Using the initial coordinate of the photon and free path-length, we obtain a new coordinate for the photon, where it is scattered or absorbed. If the new coordinate is located out of the atmosphere, the photon contributes to the spectra of X-ray radiation leaving the atmosphere, and we return to step (i). If the photon is still in the atmosphere, we move on to step (iv).

  • Comparing the cross-section of Compton scattering, free–free and cyclotron absorption, we specify the elementary process at the new photon coordinate. If the photon is absorbed, we stop the trace history of the photon and start tracing a new photon, that is, we return to step (i). If the photon is scattered by electrons, we obtain its new energy, momentum direction and polarization state on the basis of pre-calculated tables, and return to step (ii).

4 RESULTS OF NUMERICAL SIMULATIONS

4.1 Influence of NS atmosphere conditions on the X-ray spectra

For the case of fixed local magnetic field strength, there are five main parameters of the performed numerical simulations and resulting spectra of X-ray photons. These are: (i) the optical thickness of the overheated upper layer due to Thomson scattering τup; (ii) the typical length of accretion flow braking in the atmosphere measured in the optical depth due to Thomson scattering τbr; (iii) the temperature of the atmosphere under the overheated upper layer Tbot; (iv) the temperature of the overheated upper layer Tup; and (v) the chemical composition of the atmosphere given by the atomic number Z. Here we investigate how these parameters affect the spectrum. To separate effects caused by different reasons, we compare the results of numerical simulations with the results based on the following set of fiducial parameters: |$T_{\rm bot}=3\, {\rm keV}$|⁠, |$T_{\rm up}=80\, {\rm keV}$|⁠, τup = 0.1, τbr = 1, Z = 1, θB = 0 (red line in Figs 3 a–f).

$E\, F_E$ spectrum (in arbitrary units) of X-ray radiation leaving the atmosphere of a NS at very low mass accretion rates. The fiducial case is given by the red solid line and corresponds to the case of the atmosphere described by the following set of parameters: $E_{\rm cyc}=50\, {\rm keV}$, $T_{\rm bot}=3\, {\rm keV}$, $T_{\rm up}=80\, {\rm keV}$, τup = 0.1, τbr = 1, Z = 1. Different panels show the influence of different parameters on the final X-ray energy spectrum: (a) effect of optical thickness τup on the overheated upper layer; (b) influence of effective depth τbr, where the accretion flow is stopped by collisions; (c) influence of the temperature Tup of the overheated upper layer; (d) influence of the temperature Tbot of the atmosphere below the overheated upper layer; (e) influence of the chemical composition of the atmosphere; (f) influence of the angle θB between the magnetic field lines and normal to the NS surface. The run results from 2 × 106 photons leaving the atmosphere.
Figure 3.

|$E\, F_E$| spectrum (in arbitrary units) of X-ray radiation leaving the atmosphere of a NS at very low mass accretion rates. The fiducial case is given by the red solid line and corresponds to the case of the atmosphere described by the following set of parameters: |$E_{\rm cyc}=50\, {\rm keV}$|⁠, |$T_{\rm bot}=3\, {\rm keV}$|⁠, |$T_{\rm up}=80\, {\rm keV}$|⁠, τup = 0.1, τbr = 1, Z = 1. Different panels show the influence of different parameters on the final X-ray energy spectrum: (a) effect of optical thickness τup on the overheated upper layer; (b) influence of effective depth τbr, where the accretion flow is stopped by collisions; (c) influence of the temperature Tup of the overheated upper layer; (d) influence of the temperature Tbot of the atmosphere below the overheated upper layer; (e) influence of the chemical composition of the atmosphere; (f) influence of the angle θB between the magnetic field lines and normal to the NS surface. The run results from 2 × 106 photons leaving the atmosphere.

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In most cases, we see the X-ray energy spectrum consisting of two components. The low-energy component is a result of black-body radiation Comptonized by electrons in the atmosphere. The high-energy component is a result of the initial emission of cyclotron photons and their further Comptonization by electrons. The multiple scatterings of cyclotron photons are strongly affected by the resonance at the cyclotron energy broadened by the thermal motion. X-ray photons hardly escape the atmosphere at the energies close to the cyclotron energy, and because of that, the photons tend to escape in the red and blue wings of a cyclotron line. Note that thermal emission also contributes to the initial photons at cyclotron energy because free–free absorption is resonant at Ecyc in X-mode.

Comparing the results of different numerical simulations, we can make some conclusions, as follows.

  • The larger optical thickness of the overheated upper layer τup does not affect the low-energy part of X-ray spectra much, but it does influence the high-energy component (see Fig. 3a), affecting photon redistribution around the cyclotron line.

  • The smaller optical depth of accretion flow braking τbr leads to a stronger high-energy component of the X-ray spectrum (see Fig. 3b). This is natural because at smaller τbr it becomes easier for cyclotron photons to leave the atmosphere, starting their diffusion from a smaller optical depth. Additionally, the smaller optical depth of the accretion flow braking results in a smaller fraction of absorbed cyclotron photons, contributing to the thermal low-energy part of the X-ray spectra.

  • The overheated upper layer of the atmosphere affects the high-energy end of the X-ray spectra. The lower temperature of the upper layer Tup makes the blue wing of a cyclotron line weaker (see Fig. 3c). If the temperature of the upper layer is much smaller than the cyclotron energy Ecyc, then most of the cyclotron photons are scattered into the red wing of the line and leave the atmosphere at E ≲ Ecyc.

  • The temperature Tbot of the atmosphere below the overheated upper layer shapes the thermal radiation of the atmosphere and affects the low-energy part of the X-ray spectra. An increase of the temperature of the bottom atmosphere results in a corresponding shift of the low-energy component (see Fig. 3d).

  • The chemical composition in the atmosphere affects the cross-section of free–free absorption: the larger the atomic number, the larger the cross-section of free–free absorption. Because the low-energy component is dominated by the thermal emission of the atmosphere, this component is affected more strongly by the variations in the free–free absorption coefficient. Specifically, we see that the energy spectra tend to be slightly suppressed at low energies for larger effective atomic numbers Z (see Fig. 3e).

  • The direction of the magnetic field with respect to the NS surface does not greatly affect the final spectra integrated over the solid angle (see Fig. 3f). In our simulations, we see only a slight decrease of the width of the absorption feature at E ∼ Ecyc. This decrease is likely to be because the thermal broadening of the cyclotron resonances in the Compton scattering cross-section is weaker for photons propagating across the field direction.

The specific intensity of X-ray radiation leaving the atmosphere is angular-dependent (see Fig. 4). In particular, a strong angular dependence of the specific intensity at the red and blue wings of a cyclotron line is expected. The intensity integrated over the energies composes a pencil-beamed diagram, which is slightly suppressed in the direction orthogonal to the stellar surface. This phenomenon is natural for the case of the atmosphere with the overheated upper layer: the contribution of the overheated upper layer into the intensity is smaller in the direction perpendicular to the stellar surface.

Specific intensity IE at the stellar surface at different directions given by an angle θ between local B-field direction and photon momentum. Different lines denote different angles: θ = 0 (solid), 0.125π (dashed), 0.25π (dotted) and 0.375π (dash-dotted). Note that the intensities in the red and blue wings of a cyclotron line are strongly variable. Parameters for simulated spectrum are $E_{\rm cyc}=50\, {\rm keV}$, $T_{\rm bot}=3\, {\rm keV}$, $T_{\rm up}=80\, {\rm keV}$, τup = 0.1, τbr = 1, Z = 1.
Figure 4.

Specific intensity IE at the stellar surface at different directions given by an angle θ between local B-field direction and photon momentum. Different lines denote different angles: θ = 0 (solid), 0.125π (dashed), 0.25π (dotted) and 0.375π (dash-dotted). Note that the intensities in the red and blue wings of a cyclotron line are strongly variable. Parameters for simulated spectrum are |$E_{\rm cyc}=50\, {\rm keV}$|⁠, |$T_{\rm bot}=3\, {\rm keV}$|⁠, |$T_{\rm up}=80\, {\rm keV}$|⁠, τup = 0.1, τbr = 1, Z = 1.

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The X-ray energy flux leaving the atmosphere is polarization-dependent (see Fig. 5). The polarization dependence is particularly strong near the cyclotron energy, which is natural because the strength of cyclotron resonance, and even its existence, depends on the polarization state of a photon. At low energies E ≪ Ecyc, the flux is dominated by X-mode photons because the scattering cross-section is smaller for this polarization state (Herold 1979; Daugherty & Harding 1986; Mushtukov, Nagirner & Poutanen 2016). However, the difference in X-ray energy flux at different polarization states is not dramatic because of the inverse temperature profile in the atmosphere: the upper layer is assumed to be much hotter than the underling atmosphere. Note, however, that the exact predictions for polarization require detailed analyses of effects arising from vacuum polarization and complicated behaviour of plasma dielectric tensor under conditions of high temperatures. The detailed analysis of these effects is beyond the scope of this paper and will be discussed in a separate publication.

X-ray energy spectra at the NS surface in X-mode (blue dashed line) and O-mode (red dash-dotted line) and complete spectra (black solid line). Parameters for simulated spectrum are $E_{\rm cyc}=50\, {\rm keV}$, $T_{\rm bot}=3\, {\rm keV}$, $T_{\rm up}=80\, {\rm keV}$, τup = 0.1, τbr = 1, Z = 1.
Figure 5.

X-ray energy spectra at the NS surface in X-mode (blue dashed line) and O-mode (red dash-dotted line) and complete spectra (black solid line). Parameters for simulated spectrum are |$E_{\rm cyc}=50\, {\rm keV}$|⁠, |$T_{\rm bot}=3\, {\rm keV}$|⁠, |$T_{\rm up}=80\, {\rm keV}$|⁠, τup = 0.1, τbr = 1, Z = 1.

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4.2 Comparison with observational data

In order to verify our model, we compared the results of the simulations with the data obtained during a low-luminosity state of transient XRP A 0535+262 with LX = 7 × 1034 erg s−1. The data were adopted from Tsygankov et al. (2019b) and cover a broad-energy band from 0.3 to 79 keV using the Swift/XRT and NuSTAR instruments.

In Fig. 6, we represent the model calculated for the following set of parameters (see the red line): Ecyc = 48 keV, Z = 1, θB = 0, Tbot = 3.5 keV, Tup = 100 keV, τup = 0.1 and τbr= 0.5. The theoretical model represents the spectrum integrated over the solid angle at the NS surface and is based on Monte Carlo simulation with 2 × 106 photons, leaving the atmosphere. In order to compare the theoretical model with the observed X-ray spectra, we have accounted for spectral changes resulting from the gravitational redshift. We assume that the photon energy detected by a distant observer E∞ is related to the photon energy at the NS surface E as
(25)
where u = RS/R, and the Schwarzschild radius |$R_{\rm S}\approx 3m\, {\rm km}$|⁠. The mass and radius of a NS are taken to be |$M=1.4\, \mathrm{M}_\odot$| and |$R=12\, {\rm km}$|⁠. As can be seen, our theoretical predictions are able to describe the complex spectral shape of the source, including all observed features.
The observed $E\, F_E$ spectrum of A 0535+262 at an accretion luminosity of $~7\times 10^{34}\, {\rm erg\ \rm s^{-1}}$ is given by black circles (NuSTAR FPMA and FPMB data) and squares (Swift/XRT). The simulated spectrum is represented by the red line. Parameters for the simulated spectrum are $E^{\infty }_{\rm cyc}=39\, {\rm keV}$, $T_{\rm bot}=2.8\, {\rm keV}$, $T_{\rm up}=100\, {\rm keV}$, τup = 0.1, τbr = 0.5 and Z = 1. The run results from 5 × 105 photons leaving the atmosphere.
Figure 6.

The observed |$E\, F_E$| spectrum of A 0535+262 at an accretion luminosity of |$~7\times 10^{34}\, {\rm erg\ \rm s^{-1}}$| is given by black circles (NuSTAR FPMA and FPMB data) and squares (Swift/XRT). The simulated spectrum is represented by the red line. Parameters for the simulated spectrum are |$E^{\infty }_{\rm cyc}=39\, {\rm keV}$|⁠, |$T_{\rm bot}=2.8\, {\rm keV}$|⁠, |$T_{\rm up}=100\, {\rm keV}$|⁠, τup = 0.1, τbr = 0.5 and Z = 1. The run results from 5 × 105 photons leaving the atmosphere.

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We note, however, that our theoretical model shows a lack of X-ray photons in the red wing of a cyclotron line at energy |$\sim 15\!-\!20\, {\rm keV}$|⁠. It is hard to eliminate this discrepancy in the energy spectrum integrated over the solid angle. We suppose that this problem can be solved if we account for the exact geometry of NS rotation in the observer’s reference frame and the precise process of pulse profile formation. We also note that the radiative transfer at the high-energy part of the X-ray spectra can be affected by the effect of vacuum polarization (see Appendix  B), which was not taken into account in our simulations. This analysis is beyond the scope of the paper and a matter of further investigation, which will be discussed in a separate publication.

5 SUMMARY AND DISCUSSION

5.1 Spectra formation at low mass accretion rates

We performed numerical simulations for spectra formation in XRPs at very low-luminosity states, when the interaction between the radiation and accretion flow above the NS surface does not affect the X-ray spectra and dynamics of the accretion flow. Our simulations are coherent with a physical model (see Fig. 1) where the accretion flow is braked in the upper layers of the NS atmosphere because of collisions between particles, and most of the kinetic energy is released initially in the form of cyclotron photons. The spectra leaving the atmosphere of a NS are a matter of radiative transfer, strongly affected by magnetic Compton scattering. The essential ingredient of the model is an overheated upper layer of the NS atmosphere, proposed earlier by Suleimanov et al. (2018) for the case of low-level accretion on to weakly magnetized NSs. Simulated radiative transfer in the atmosphere was performed under the assumption of LTE and accounts for Compton scattering of X-ray photons by thermally distributed electrons, cyclotron photons and free–free absorption. The two components in the spectrum correspond to Comptonized thermal radiation (low-energy hump) and Comptonized cyclotron photons (high-energy hump), which originate from collisions of accreting particles with the electrons in the NS atmosphere and further radiative transition of electrons to the ground Landau level. The absorption feature on top of the high-energy hump is a result of the resonant scattering of X-ray photons at cyclotron energy, which forces cyclotron photons to leave the atmosphere in the wings of a cyclotron line.

Using the constructed model, it was possible to reproduce the observed spectrum of X-ray pulsar A 0535+262 at a very low-luminosity state (see fig. 6 of Tsygankov et al. 2019b). Qualitative agreement between simulated and observed X-ray spectra confirms assumptions about the underlying physical model. We argue that two-component spectra should be typical for low-level accretion on to strongly magnetized NSs.

5.2 Applications to the observational studies

5.2.1 Investigation of the ‘propeller’ state

A decrease of the mass accretion rates in XRPs down to very low values results in the transition of the source either to the ‘propeller’ state, when the accretion flow cannot penetrate through the centrifugal barrier set up by the rotating magnetosphere of a NS (e.g. Illarionov & Sunyaev 1975; Lipunov 1987; Ustyugova et al. 2006), or to the regime of stable accretion from a cold disc (see Tsygankov et al. 2017). The propeller effect was recently detected in a few XRPs (e.g. Tsygankov et al. 2016; Lutovinov et al. 2017). Moreover, in some sources, transitions into the propeller state were discovered to be accompanied by dramatic changes of X-ray energy spectra (Tsygankov et al. 2016). Specifically, in the energy range below ∼10 keV, the spectra were shown to become significantly softer with a shape changed from a power law to a blackbody with a typical temperature around 0.5 keV. However, it is still unknown whether the centrifugal barrier blocks the accretion process entirely, and the detected soft X-ray spectra are observational evidence of the cooling NS surface, or whether leakage of matter through the barrier is still possible and responsible for some fraction of the observed emission (Wijnands & Degenaar 2016; Rouco Escorial et al. 2017).

Our theoretical model of spectra formation provides a natural way to distinguish low-level accretion from the cooling NS surface. The hard component of X-ray spectra is a direct result and a specific feature of the accretion process. As a result, low-level accretion in the case of the leakage of the centrifugal barrier should result in two-component X-ray spectra, while the propeller state without penetration of material through the barrier should result in single-hump soft spectra.

5.2.2 Measurements of magnetic field strength

Two-component X-ray energy spectra at low mass accretion rates provide a way to estimate the magnetic field strength at the NS surface. Indeed, the hard component of X-ray spectra is formed around cyclotron energy, which is directly related to the field strength: |$E_{\rm cyc}\approx 11.6\, B_{12}\,$| keV. Thus, the detection of a hard-energy hump provides a way to estimate the cyclotron energy in the case when the cyclotron absorption feature is not seen in the source spectrum.

For instance, according to our model, the spectra of X Persei (Di Salvo et al. 1998) imply that the cyclotron energy |$E_{\rm cyc}\gtrsim 100\, {\rm keV}$| and the corresponding magnetic field strength |$B\gtrsim 10^{13}\, {\rm G}$|⁠. Note that this estimation is consistent with the results based on timing analyses and torque models applied to this particular source (Doroshenko et al. 2012).

ACKNOWLEDGEMENTS

This work was supported by the Netherlands Organization for Scientific Research Veni Fellowship (AAM), the Väisälä Foundation (SST) and the Academy of Finland travel grant 324550. VFS thanks Deutsche Forschungsgemeinschaft (DFG) for financial support (grant WE 1312/51-1). The authors also thank the Russian Science Foundation (grant 19-12-00423) for financial support. We are grateful to an anonymous referee for a number of useful comments and suggestions that helped us improve the paper.

DATA AVAILABILITY

The calculations presented in this paper were performed using a private code developed and owned by the corresponding author; please contact A. Mushtukov for any request/question about the calculations. Data appearing in the figures are available upon request. The observational data used in the paper are adopted from those reported in Tsygankov et al. (2019b).

Footnotes

1

The thermal broadening of the resonance is calculated according to one-dimensional electron distribution and natural width of Landau levels approximately calculated according to Pavlov et al. (1991).

2

We use the optical depth due to non-magnetic Thomson scattering, assuming that the opacity is |$\kappa _{\rm T}=0.34\, {\rm cm^2\, g^{-1}}$|⁠.

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APPENDIX A: FREE–FREE ABSORPTION IN A STRONG MAGNETIC FIELD

The opacity due to free–free absorption is calculated in our simulations according to Lai & Ho (2003). The opacity is dependent on the magnetic field strength, the polarization of X-ray photons and their energy, and the direction of the momentum with respect to the local direction of the B-field. The opacity for mode j of ellipticity Kj can be written as
(A1)
where e0, e± are the spherical components of the photon’s unit polarization vector with
(A2)
(A3)
Here, the ellipticities Kj are given by equation (4) and Kz,j is taken to be 0. The coefficients in equation (A1) are given by
(A4)
where
(A5)
(A6)
The dimensionless quantities are
(A7)
where the energies corresponding to electron cyclotron frequency, ion cyclotron frequency and the electron plasma frequency are given by
(A8)
(A9)
(A10)
The dimensionless dumping rates due to electron–ion collisions and photon emission by electrons and ions are given by
(A11)
where |$g^{\rm ff}_{\perp ,\parallel }$| are magnetic Gaunt factors, which are calculated according to Suleimanov, Pavlov & Werner (2010) in our simulations.

APPENDIX B: ON THE INFLUENCE OF VACUUM POLARIZATION

Assuming constant temperature T in the atmosphere, we can obtain the dependence of the mass density on the vertical coordinate ξ,
(B1)
where |$g\simeq 1.3\times 10^{14}\, m R_6^{-2}\, {\rm cm\, s^{-2}}$| is the acceleration due to gravity at the NS surface. If the opacity is given by Thomson opacity |$\kappa _{\rm e}=0.34\, {\rm cm^2\, g^{-1}}$|⁠, the local mass density is related to the optical depth in the atmosphere as
(B2)
The critical mass density, when the contribution of vacuum polarization becomes comparable to the contribution of plasma to the dielectric tensor, can be estimated as (Lai & Ho 2003)
(B3)
where Ye = Z/A and f ∼ 1. At ρ ≪ ρV, the dielectric tensor is dominated by vacuum effects, while at ρ ≫ ρV, the dielectric tensor is dominated by plasma effects. Using equations (B2) and (B3), we obtain the optical depth due to Thomson scattering corresponding to ρV:
(B4)
Therefore, for the case of physical conditions expected in XRP A 0535+262, we obtain
$$\begin{equation*} \tau _{\rm T,V}\sim 3\times 10^{-5}T_{\rm keV}E_{\rm keV}^2, \end{equation*}$$
which means that the polarization of X-ray photons in the low-energy hump is described well by equation (4). However, the polarization of X-ray photons in the high-energy hump might be affected by the effects of vacuum polarization, which will be investigated in a separate publication.
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