Issue 
A&A
Volume 527, March 2011



Article Number  A2  
Number of page(s)  5  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201016005  
Published online  18 January 2011 
The plasma emission model of RBS1774
Center for theoretical Astrophysics, ITP, Ilia State
University,
0162
Tbilisi,
Georgia
email: nino.chkheidze@iliauni.edu.ge
Received:
26
October
2010
Accepted:
15
December
2010
In the present paper we construct a selfconsistent theory, interpreting the observational properties of RBS1774. It is well known that the distribution function of relativistic particles is onedimensional at the pulsar surface. However, cyclotron instability causes an appearance of transverse momenta of relativistic electrons, which as a result, start to radiate in the synchrotron regime. We study the process of the quasilinear diffusion developed by means of the cyclotron instability on the basis of the Vlasov’s kinetic equation. This mechanism provides generation of measured optical and Xray emission on the light cylinder lengthscales. A different approach of the synchrotron theory is considered, giving the spectral energy distribution that is in a good agreement with the XMMNewton observational data. We also provide the possible explanation of the spectral feature at 0.7 keV, in the framework of the model.
Key words: Xrays / stars: pulsars: individual: RBS1774 / radiation mechanisms: nonthermal
© ESO, 2011
1. Introduction
RBS1774 (1RXS J214303.7+065419) has been the most recent XDIN (Xray dim isolated neutron star) to be found (Zampieri et al. 2001). Its Xray spectrum is well reproduced by an absorbed blackbody with a temperature kT ~ 100 eV and with a total column density of n_{H} ~ 3 × 10^{20} cm^{2}. Application of more sophisticated, and physically motivated models for the surface emission (atmospheric models) result in worse agreement with the data (Zane et al. 2005). According to Schwope et al. (2009), a fit to the Xray spectra extracted from RGS spectrographs onboard XMMNewton yields that the best result is obtained when the twotemperature blackbody model is used. But the same model applied to the Xray spectra extracted from three EPIC detectors does not improve the fit compared to the simple blackbody model. However, the formation of a nonuniform distribution of the surface temperature is more likely artificial and needs to be examined by convincing theory.
Alternatively, the observational properties of RBS1774 can be explained in the framework of the plasma emission model first developed by Machabeli & Usov (1979) and Lominadze et al. (1983). According to these works, in the electronpositron plasma of a pulsar magnetosphere the waves excited by the cyclotron resonance interact with particles, leading to the appearance of pitch angles, which obviously causes synchrotron radiation. We suppose that the Xray emission from this object is generated by the synchrotron mechanism. According to the standard theory of the synchrotron emission (Bekefi & Barrett 1977; Ginzburg 1981) the typical synchrotron spectrum is a powerlaw, when the present model suggests different spectral distribution. The main reason for this is that we take into account the mechanism of creation of the pitch angles, consequently restricting their values. Contrary to this, in the standard theory of the synchrotron radiation, it is assumed that along the line of sight the magnetic field is chaotic, leading to the broad interval (from 0 to π) of the pitch angles. The present model gives successful fit for the observed Xray spectrum, when the originally excited cyclotron modes enter the same domain as the measured optical emission of RBS1774. We suppose that the observed spectral feature at 0.7 keV in the Xray spectrum of RBS1774 is caused by wave damping process developed near the light cylinder due to the cyclotron instability.
In this paper, we describe the emission model (Sect. 2), derive theoretical Xray spectrum of RBS1774 and fit with XMMNewton observations (Sect. 3), explain the possible nature of the observed spectral feature at ~0.7 keV (Sect. 4), and discuss our results (Sect. 5).
2. Emission model
The distribution function of relativistic particles is one dimensional at the pulsar surface, because any transverse momenta (p_{ ⊥ }) of relativistic electrons are lost in a very short time ( ≤ 10^{20} s) via synchrotron emission in very strong magnetic fields. For typical pulsars the plasma consists of the following components: the bulk of plasma with an average Lorentzfactor γ_{p} ≃ 10^{2}, a tail on the distribution function with γ_{t} ≃ 10^{5}, and the primary beam with γ_{b} ≃ 10^{7} (see Fig. 1). However, plasma with an anisotropic distribution function becomes unstable, which can lead to a wave excitation in the pulsar magnetosphere. The generation of waves is possible during the further motion of the relativistic particles along the dipolar magnetic field lines if the condition of cyclotron resonance is fulfilled (Kazbegi et al. 1991): (1)where u_{x} = cV_{ϕ}γ_{r}/ρω_{B} is the drift velocity of the particles due to curvature of the field lines, ρ is the radius of curvature of the field lines and ω_{B} = eB/mc is the cyclotron frequency. During the generation of waves by resonant particles, one also has a simultaneous feedback of these waves on the electrons (Vedenov et al. 1961). This mechanism is described by quasilinear diffusion, leading to the diffusion of particles as along as across the magnetic field lines. Therefore, resonant electrons acquire transverse momenta (pitch angles) and, as a result, start to radiate through the synchrotron mechanism.
Fig. 1 Distribution function of a onedimensional plasma in the pulsar magnetosphere. Left corresponds to secondary particles, right to the primary beam. 

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The kinetic equation for the distribution function of the resonant particles can be written as (Machabeli & Usov 1979; Machabeli et al. 2002): (2)where G is the force responsible for conserving the adiabatic invariant , F is the radiation deceleration force produced by synchrotron emission, and Q_{ ∥ } is the reaction force of the curvature radiation. They can be written in the form: (3)(4)(5)where , α_{c} = 2e^{2}/3ρ^{2} and ψ ≈ p_{ ⊥ }/p_{ ∥ } ≪ 1 is the pitch angle.
Now let us compare the transverse components of the forces G and F. If we consider the case γψ ≫ 1 we will have: (6)where B_{s} is the magnetic field at the star surface, R_{s} is the star radius and r is the distance from the pulsar. For the typical parameter values of pulsars G_{ ⊥ } ≪ F_{ ⊥ } . Then taking into account that ∂/∂ψ ≫ ∂/∂γ the equation for the diffusion across the magnetic field can be written in the form (Chkheidze et al. 2010) (7)where (8)is the diffusion coefficient and E_{k} ^{2} is the density of electric energy in the waves.
The transversal quasilinear diffusion increases the pitchangle, whereas force F resists this process, leading to the stationary state (∂f/∂t = 0). Then the solution of Eq. (7) is (9)To evaluate p_{ ⊥ 0}, we use the quantity (10)where ω is the frequency of original waves, excited during the cyclotron resonance and can be estimated from Eq. (1) as follows ω ≈ ω_{B}/δγ_{r}. Consequently, we will get (11)The mean value of the pitchangle ψ_{0} ≈ p_{ ⊥ 0}/p_{ ∥ } ≃ 10^{3} (i.e. the assumption γ_{r}ψ_{0} ≫ 1 done at the beginning of our computations proves to be true). As a result of the appearance of the pitch angles, the synchrotron emission is generated.
3. Xray spectrum
Let us consider the synchrotron emission of the set of electrons. The number of emitting particles in the elementary dV volume is p_{ ⊥ }fdp_{ ⊥ }dp_{ ∥ }dVdΩ_{τ}, with momenta from the intervals [ p_{ ⊥ },p_{ ⊥ } + dp_{ ⊥ } ] and [ p_{ ∥ },p_{ ∥ } + dp_{ ∥ } ] , and with the velocities that lie inside the solid angle dΩ_{τ} near the direction of τ. If we write the parallel distribution function of the emitting particles as , then the emission flux of the set of electrons will be (Ginzburg 1981) (12)where I_{e} is the Stokes parameter, which is additive in this case, as the observed synchrotron radiation wavelength λ is much less than the value of n^{ − 1/3} – the average distance between particles, where n is the density of plasma component electrons. The integral (12) is easily reduced to (see Ginzburg 1981) (13)Here ϵ_{m} ≈ 5 × 10^{12}Bψγ^{2} keV is the photon energy of the maximum of synchrotron spectrum of a single electron and K_{5/3}(z) is a Macdonald function. After substituting the mean value of the pitchangle in the above expression for ϵ_{m}, we get (14)For the primary beam electrons with the Lorentz factor γ_{b} ~ 10^{7} the emitted photon energy ϵ_{m} ~ 0.1 keV comes in the energy domain of the observed Xray emission of RBS1774. Thus we suppose that the measured Xray spectrum is the result of the synchrotron emission of primary beam electrons (the resonance occurs on the right slope of the distribution function of beam electrons (see Fig. 1)), switched on as the result of acquirement of pitch angles by particles during the quasilinear stage of the cyclotron instability.
To find the synchrotron flux in our case, we need to know the onedimensional distribution function of the emitting particles f_{ ∥ }. Let us multiply both sides of Eq. (2) on p_{ ⊥ } and integrate it over p_{ ⊥ }. Using Eqs. (3) − (5) and the following expressions for the diffusion coefficients (Chkheidze et al. 2010) (15)And also taking into account that the distribution function vanishes at the boundaries of integration, Eq. (2) reduces to (16)Let us estimate the contribution of different terms on the righthand side of Eq. (16). The estimations show that the first term is much bigger than two other terms. Consequently, for the primarybeam electrons instead of Eq. (16), one gets (17)Considering the quasistationary case we find (18)For γψ ≪ 10^{10}, a magnetic field inhomogeneity does not affect the process of wave excitation. The equation that describes the cyclotron noise level, in this case, has the form (Lominadze et al. 1983) (19)where (20)is the growth rate of the instability. Here k_{ ∥ } can be found from the resonance condition (1) (21)Combining Eqs. (17) and (19) one finds (22)(23)which reduces to (24)Taking into account that for the initial moment the major contribution of the lefthand side of the Eq. (24) comes from f_{ ∥ 0}, the corresponding expression writes as (25)The distribution function f is proportional to n ~ 1/r^{3}, then one should neglect f_{ ∥ } in comparison with f_{ ∥ 0}. Consequently, the above equation reduces to (26)As we can see the function E_{k}(p_{ ∥ }) drastically depends on the form of the initial distribution of the primary beam electrons. Here we assume that the initial energy distribution in the beam has a Gaussian shape (27)where γ_{T} ≃ 10 – is the half width of the distribution function and n_{b} = B/Pce is the density of primary beam electrons, equal to the GoldreichJulian density (Goldreich & Julian 1969). Since γ_{T} ≪ γ_{b}, this distribution is very close to δfunction. Consequently, the electron distribution can be taken as monoenergetic.
In this case for the energy density of the waves we get (28)The effective value of the pitch angle depends on  E_{k}  ^{2} as follows (29)According to our emission model, the observed radiation comes from a region near the light cylinder radius, where the magnetic field lines are practically straight and parallel to each other (Osmanov et al. 2009), therefore, electrons with ψ ≈ ψ_{0} efficiently emit in the observer’s direction.
Using expression (18), (28) and (29), and replacing the integration variable p_{ ∥ } by x = ϵ/ϵ_{m}, from Eq. (13) we will get (30)The energy of the beam electrons vary in a small interval. In this case the integral (30) can be approximately expressed by the following function (31)We performed a spectral analysis by fitting the model spectrum (Eq. (31)) absorbed by cold interstellar matter, with the combined data extracted of the three EPIC Xray cameras of the XMMNewton Telescope. The resulting χ^{2} = 1.63 and the amount of interstellar matter n_{H} = (3.36 ± 0.2) × 10^{20} cm^{2}, which appears to be close to the total Galactic absorption in the source direction (n_{H} = 5 × 10^{20} cm^{2} Dickey & Lockman 1990). The spectral feature at ~0.7 keV that is mostly described as an absorption edge or line (Zane et al. 2005; Schwope et al. 2009) is also evident in our case from inspection of Fig. 1. We find that adding an absorption edge at 0.7 keV improves the fit, leading to a reduced χ^{2} = 1.50. The bestfitting energy of the edge is E_{edge} = 0.679 keV, and the optical depth is τ_{edge} = 0.20 (see Fig. 3). The fitting results are listed in Table 1. We suppose that existence of the absorption feature in spectra of RBS1774 is caused by wave damping at photon energies ~0.7 keV, which takes place near the light cylinder.
The model parameters of RBS1774 for combined fits to EPICpn and EPICMOS in the energy interval 0.2 − 1.5 keV.
Fig. 2 EPICpn and EPICMOS spectra of RBS1774, fitted with a model. 

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4. Possible nature of the spectral feature
During the farther motion in the pulsar magnetosphere, the Xray emission of RBS1774 that is generated on the light cylinder lengthscales, might come in the cyclotron damping range (Khechinashvili & Melikidze 1997): (32)The condition for the development of the cyclotron instability may be easily derived for the small angles of propagation with respect to the magnetic field. Representing the dispersion of the waves as (33)and neglecting the drift term, the resonance condition (32) may then be written as (34)where θ ≈ ψ is the angle between the wave vector and the magnetic field. Taking into account that one finds from Eq. (34) the frequency of damped waves (35)If we assume that the resonant particles are the primary beam electrons, then the estimation shows that on the light cylinder lengthscales ϵ_{0} = (h/2π)ω_{0} ≃ 0.7 keV.
Fig. 3 EPICpn and EPICMOS spectra of RBS1774 fitted with a model, including an absorption edge at ~0.7 keV. 

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5. Discussion
According to the generally accepted point of view, the Xray spectrum of RBS1774 is purely thermal and is best represented by a Planckian shape. A fit with a pure blackbody component absorbed by cold interstellar matter gives χ^{2} = 1.81 (Schwope et al. 2009). Including a Gaussian absorption line at ~0.7 keV (as the largest discrepansies between model and data are around 0.7 keV) improves the fit χ^{2} = 1.50 (parameters are listed in Table 1). However, the nature of this spectral feature is not fully clarified as yet. The most likely interpretation is that it is due to proton cyclotron resonance, which implies ultrastrong magnetic field of B_{cyc} ~ 10^{14} G (Zane et al. 2005; Rea et al. 2007). Although, the required strong magnetic field is inconstistent with timing measurements giving G (Kaplan & van Kerkwijk 2009).
We are not about to reject the existing thermal emission models, but in present paper we propose an alternative explanation of the observed Xray spectrum of RBS1774. It is supposed that the emission of this source is generated by the synchrotron mechanism. The distribution function of relativistic particles is one dimensional at the pulsar surface, but plasma with an anisotropic distribution function is unstable which can lead to wave excitation. The main mechanism of wave generation in plasmas of the pulsar magnetosphere is the cyclotron instability, which develops on the light cylinder lengthscales. During the quasilinear stage of the instability, a diffusion of particles arises along and across the magnetic field lines. Therefore, plasma particles acquire transverse momenta and, as a result, the synchrotron mechanism is switched on. If the resonant particles are the primary beam electrons with γ_{b} ≃ 10^{7} their synchrotron emission enter the same energy domain as the measured Xray spectrum of RBS1774.
We construct a selfconsistent theory interpreting the observations of RBS1774. Differently from the standard theory of the synchrotron emission (Ginzburg 1981), which only provides a powerlaw spectrum with the spectral index less than 1, our approach gives the possibility to obtain different spectral energy distributions. In the standard theory of the synchrotron emission, it is supposed that the observed radiation is collected from a large spacial region in various parts of which, the magnetic field is oriented randomly. Thus, it is supposed that along the line of sight the magnetic field directions are chaotic and when finding emission flux, Eq. (13) is averaged over all directions of the magnetic field (which means integration over ψ varying from 0 to π). In our case the emission comes from a region of the pulsar magnetosphere where the magnetic field lines are practically straight and parallel to each other. And differently from standard theory, we take into account the mechanism of creation of the pitch angles. Thus we obtain a certain distribution function of the emitting particles from their perpendicular momenta (see Eq. (10)), which restricts the possible values of the pitch angles. In the framework of the model we obtain the following theoretical Xray spectrum of RBS1774 F_{ϵ} ∝ ϵ^{0.3}exp( − ϵ/ϵ_{m}) and perform a spectral analysis by fitting data from the three EPIC detectors simultaneously. The fit with a model spectrum absorbed by cold interstellar matter yields χ^{2} = 1.63 (see parameters in Table 1).
During the farther motion in the pulsar magnetosphere, the Xray emission of RBS1774 comes in the cyclotron damping range (see Eq. (32)). If we assume that damping happens on the left slope of the distribution function of primary beam electrons (see Fig. 1), then the photon energy of damped waves will be ϵ_{0} = (h/2π)ω_{0} = (h/2π)2ω_{B}/γ_{b}ψ^{2} ≃ 0.7 keV. Taking into account the shape of the distribution function of beam electrons, we interpret the large residuals around ~0.7 keV (see Fig. 2) as an absorption edge. Including an absorption edge improves the fit leading to a reduced χ^{2} = 1.50. The bestfitting energy of the edge is E_{edge} = 0.679 keV, and the optical depth is τ_{edge} = 0.20 (see Table 1). However, adding an absorption edge to the model spectrum does not produce a statistically significant improvement of the fitting. According to Schwope et al. (2009) if one uses the RGS Xray spectra of RBS1774 in place of EPIC spectra, the resulting χ^{2} is changed just marginally when a Gaussian absorption line is included at ~0.7 keV. Thus, we conclude that the nature of the feature at 0.7 keV is uncertain and might be related to calibration uncertainties of the CCDs and the RGS at those very soft Xray energies. The same can be told about a feature at ~0.3 keV (the large residuals around 0.3 keV are evident from inspection of Figs. 2 and 3). A feature of possible similar nature was detected in EPICpn spectra of the much brighter
prototypical object RXJ1856.43754 and classified as remaining calibration problem by Haberl (2007). Consequently, more data are necessary to finally prove or disprove the existence of those features.
The frequency of the original waves, excited during the cyclotron resonance can be estimated from Eq. (1) as follows ν ≈ 2πω_{B}/δγ_{b} ~ 10^{14} Hz. As we can see the frequency of cyclotron modes comes in the same domain as the measured optical emission of RBS1774 (Zane et al. 2008; Schwope et al. 2009).
Acknowledgments
The author is grateful to George Machabeli for valuable discussions and Axel Schwope for providing the Xray data.
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All Tables
The model parameters of RBS1774 for combined fits to EPICpn and EPICMOS in the energy interval 0.2 − 1.5 keV.
All Figures
Fig. 1 Distribution function of a onedimensional plasma in the pulsar magnetosphere. Left corresponds to secondary particles, right to the primary beam. 

Open with DEXTER  
In the text 
Fig. 2 EPICpn and EPICMOS spectra of RBS1774, fitted with a model. 

Open with DEXTER  
In the text 
Fig. 3 EPICpn and EPICMOS spectra of RBS1774 fitted with a model, including an absorption edge at ~0.7 keV. 

Open with DEXTER  
In the text 
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