We investigate the physical features of the accretion process in magnetic X-ray binaries. The characteristics of the accretion column and the geometrical modulation effects due to the neutron star spin determine the shape of the pulse profile. In order to constrain the possible scenarios, we produce numerical simulations of the X-ray pulse profile, in different energy bands. As an example we reproduce the pulse profile and hardness ratio curve of X Persei. Fan emission from a hollow section of cone, along with a non-thermal component can reproduce the observed data of this source

Burderi, L.

Robba, N. R.

Wynn, G. A.

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IL NUOVO CIMENTO VOL. 20 C, N. 5 Settembre-Ottobre 1997Numerical simulations of X-ray binary pulse profiles()L. BURDERI(1), G. A. WYNN(2) and N. R. ROBBA(3)(1) Astronomy Group, University of Leicester - Leicester LE1 7RH, UKIstituto di Fisica, Universita` di Palermo - Palermo, Italy(2) Astronomy Group, University of Leicester - Leicester LE1 7RH, UK(3) Istituto di Fisica, Universita` di Palermo - Palermo, Italy(ricevuto il 4 Aprile 1996; approvato il 24 Maggio 1996)Summary. — We investigate the physical features of the accretion process in magneticX-ray binaries. The characteristics of the accretion column and the geometrical mod-ulation effects due to the neutron star spin determine the shape of the pulse profile.In order to constrain the possible scenarios, we produce numerical simulations of theX-ray pulse profile, in different energy bands. As an example we reproduce the pulseprofile and hardness ratio curve of X Persei. Fan emission from a hollow section of cone,along with a non-thermal component can reproduce the observed data of this source.PACS 96.40 – Cosmic rays.PACS 01.30.Cc – Conference proceedings.1. – IntroductionPulse profiles, averaged over many periods, are available for all the presently knownX-ray pulsars. These profiles are produced over a range of X-ray energies. It is commonlybelieved that the pulse shapes are determined primarily by emission and absorption pro-cesses, as well as geometrical shadowing, within the magnetosphere of a rotating neutronstar (NS), that is accreting material from its binary companion. This implies that it is pos-sible to use the X-ray pulse profiles to probe the structure of the magnetosphere, and theplasma conditions during the last stages of the accretion process. The magnetic field ofthe NS plays a crucial role in the dynamics of the accreting matter and in the productionof X-rays. Under a large number of different physical conditions the plasma is threadedby the magnetic field, at least in the last phase of the accretion process. In such casesa stable field-aligned flow is possible only if the kinetic energy density of the plasma isless than the energy density of the magnetic field, i.e. v2k< B2=4, where vkis the flowvelocity parallel to the magnetic field B. The Alfve´n radius is defined as the radius at() Paper presented at the VII Cosmic Physics National Conference, Rimini, October 26-28, 1994.G Societa` Italiana di Fisica 753754 L. BURDERI, G. A. WYNN and N. R. ROBBAmθSPINAXISFig. 1. – Schematic representation of the emission region subdivided into bricks.which the two energies are equal. Inside this radius the plasma is forced to flow along themagnetic-field lines towards the polar caps, where it is accreted by the NS. In this situ-ation the accreting material forms a column-shaped structure above each magnetic pole.The detailed structure and size of this column, and the radiative processes operating inthe plasma determine the (energy dependent) shape of the pulse profile, provided thatall the geometrical effects due to the inclination of the magnetic axis and the line of sightwith respect to the spin axis are taken into account. In order to investigate this, we havedeveloped a simulation program to produce pulse profiles for different geometrical shapesof the accretion column. This is achieved by dividing the emission region into a number ofbricks, each being characterised by its emission function and absorption coefficient. Onlythe principal emission processes are taken in account. The contribution of the bricks dur-ing a rotation of the NS around its spin axis is calculated in order to build up the pulseprofile.2. – The X-ray emission regionThe accretion column has been investigated by several authors (e.g., Lamb et al. 1973;Basko and Sunyaev 1975; Arons and Lea 1980; Wang and Frank 1981; Wang and Welter1982; Me´sza´ros et al. 1983; White et al. 1983), under different assumptions. Some generalresults, utilised in the simulation code, are summarized below: The general shape of the X-ray emission region is that of a section of cone with theapex at the NS centre. The cone geometry is an approximation of the dipole fieldNUMERICAL SIMULATIONS OF X-RAY BINARY PULSE PROFILES 755Fig. 2. – Upper panel: time-averaged background-subtracted pulse profile in the 1.5–11.5 keV en-ergy band. Lower panel: hardness ratio curve in the energy band (4–11.5 keV)/(1.5–4 keV).structure close to the NS surface, which funnels the accretion flow down to the polarcaps. The central region of this cone can be free of matter: in the case of accretionvia a disc the magnetic-field lines thread the disc material over a restricted region,the result being that the innermost ring of the disc flows along the field lines pro-duces a hollow-cone structure above each polar cap. Even in the case of sphericallysymmetric accretion from a stellar wind the accretion column can be empty inside,since the first stages of accretion through the magnetosphere proceed via the freefall of diamagnetic blobs generated by Rayleigh-Taylor instabilities, and the magne-tospheric regions close to the poles are the most stable to these instabilities. The gravitational potential energy of the in-falling plasma is radiated away mainlyby means of thermal, and synchrotron emission. The free fall (or spiral-in for discaccretion) of the accreting matter is halted at the magnetosphere. The matter pen-etrates the magnetosphere via instabilities, and is threaded by the field at the Alfe´nradius, from which point the matter flows towards the polar caps. The structure ofthe emission region is dependent on the luminosity of the source. If the luminos-ity is  LEdd, the Eddington luminosity, say < 1037 erg s 1 the column height isdependent on the stopping mechanism of the plasma.If it is stopped by Coulomb collisions, the column height z is  RNS, the radiusof the NS. Here we have blackbody-like emission at a temperature  107 K, andsynchrotron radiation. In this case the synchrotron radiation, due to the presenceof the strong magnetic field ( 1012 G) near the NS crust, is important all along thecolumn structure. The synchrotron self-absorption is so high that this emission is756 L. BURDERI, G. A. WYNN and N. R. ROBBAFig. 3. – Upper panel: ratio of average spectra recorded during phase intervals H and M. Lowerpanel: same as upper panel using phase intervals S and M. (Phase intervals H, S, and M are definedin fig. 2.)blackbody-like at the synchrotron frequency (h  12 keV B12, where B12is thefield in units of 1012 G).If the plasma is stopped via a collisionless shock, the height of the accretion columncould be & RNS. In this case the main emission processes are bremsstrahlung andsynchrotron. The bremsstrahlung is emitted throughout the cone at a temperature 107K. The synchrotron emission dominates in a ring at the base of the accretionfunnel, since the intensity of the synchrotron emission varies as B2 / r 6.If the luminosity is& LEdd, the radiation pressure can stop the in-falling matter andwe can have z  RNS. In this case the emission is due to blackbody (T  107 K),and synchrotron radiation from a mound of slowly settling high-density material atthe base of the column, inflated by the radiation pressure. The Compton scattering is strongly dependent on the geometrical shape of the emis-sion region c cd=mp, where cis the Compton optical depth, cis the magneti-cally reduced Compton cross-section,  is the density of the plasma, d is the geomet-ical path inside the emission region, andmpis the proton mass. In the case of a coneof height z and diameter D, we can define kand ?, adopting an average density, as the optical depths parallel and perpendicular to the cone walls. Due to magneticeffects, kis strongly reduced for photon energies below the synchrotron frequency,as k T(=c)2, where Tis the Thompson cross-section and c/ r 3 is theNUMERICAL SIMULATIONS OF X-RAY BINARY PULSE PROFILES 757TopMiddleBottomCone emissivityBRBR emissivitySoft componentHard componentunitunitFig. 4. – Schematic representation of the spatial distribution of the cone emission in terms of hardand soft components.synchrotron frequency. So in the case of a short cone magnetic reduction is impor-tant throughout and we have k< ?, i.e. the radiation is mainly emitted along themagnetic axis: pencil emission. For an extended column, the magnetic reduction be-comes unimportant with increasing z and we have k> ?since z > D. In thiscase the radiation is mainly emitted perpendicular to the cone axis: fan emission.Taking all of these effects into account we can schematically describe the geometryof the emission region as a section of hollow cone (sometimes filled at the centre). Theemission mechanism along the cone is thermal bremsstrahlung (/ 2T 1=2), the Comptonscattering along the cone (c=Rcd=mpdx), determines the character of the source, asbeing “fan” or “pencil” emission. An optically thick blackbody-like emitting ring or slabis present at the base of the cone, the energy of which depends on whether thermal orsynchrotron processes dominate.3. – Simulating the pulse profilesThe detailed physical processes operating in the accretion column are very complexand, as yet, not completely understood. In particular, the extreme physical conditions andthe high intensity of the magnetic field ( 1012G) near the NS crust make the problem dif-ficult to treat. Moreover, the physics of collisionless shocks is unclear. As a consequence,no self-consistent solutions for an accretion column exist. However, interesting qualitativeresults can be obtained assuming the density and temperature profiles along the emission758 L. BURDERI, G. A. WYNN and N. R. ROBBA1.2Medium Zone-180 +180Hard ZoneMedium ZoneSoft ZoneSoft Zone11.5-72 0 +72Hard Zone / Medium ZoneSoft Zone / Medium Zone0.90.80.70.611.41.7HRpulse phase (deg)Spectral ratioenergy (keV)Fig. 5. – Hardness ratio and pulse-phase spectroscopy for the toy model of sect. 4.region. The pulse profile produced by the spin of the NS can then be calculated for par-ticular values of the observer inclination angle (i), and the co-latitude of the magnetic axiswith respect to the spin axis (m). The emission region is divided into a number of “bricks”(fig. 1), each of which is small enough such that its surface presents a single angle to theline of sight. With a suitable transformation matrix it is possible to determine which bricksare visible for a given i and m, i.e. unobscured by the body of the NS, at any particularspin phase. The emission process considered is thermal bremsstrahlung, Thomson scat-tering is also considered: given the light path (d) over which a particular brick is seen,the optical depth ( ) isRd0T=mpdx, where  is the density, mpthe proton mass andTthe Thomson cross-section. The assumption of optically thin bremsstrahlung is validproviding that the solid angle over which   0 is 4. This condition is satisfied in thecase of a thin-walled, hollow cone as   1 except over the small phase interval when theobserver looks directly down the cone walls. If a blackbody-like region is present at thebase of the cone in a short ring or slab, its contribution is also computed; for this regionthe occultation effects of the NS body could be very important. It is possible to producepulse profiles in the hard and soft spectral regimes and the relative hardness ratio curve(HR), depending on the temperature profile.NUMERICAL SIMULATIONS OF X-RAY BINARY PULSE PROFILES 759Fig. 6. – Upper panel: simulated light curve. Lower panel: simulated hardness ratio.4. – Application to X PerseiAs an example we try to reproduce the pulse profile of the X-ray source X-Persei ob-served by the Ginga satellite. This source is particularly interesting because of the verycomplex phase-modulated HR. The spectrum of this source is consistent with thermalbremsstrahlung at a temperature of 7–12 keV (Becker et al. 1979, White et al. 1976, Whiteet al. 1982), or a power law resulting from subsequent Comptonization (Sunyaev andTitarchuk 1980) as reported by Murakami et al. (1987) and Robba and Warwick (1989).The light curve, the HR and the pulse-phase spectroscopy from the Ginga data are shown760 L. BURDERI, G. A. WYNN and N. R. ROBBATABLE I. – Variation of the visible flux from different cone regions with spin phase. (degrees) Cone section BRtop middle bottom0 100% 100% 0% 100%72 100% 100% 33% 0%180 100% 100% 100% 0%in figs. 2 and 3 (Robba et al. 1996). Fan beam bremsstrahlung emission plus a blackbody-like region near the NS crust can explain the general shape of the pulse profile and themain features of the hardness ratio and the pulse-phase spectroscopy. As conceptualaid we present a toy model of the cone and the blackbody-like region (BR) in fig. 4, theemissivity and the hard/soft fraction for various zones of the cone and the BR are alsodisplayed. We assume that i ' m, to allow the line of sight to pass close by the magneticaxis, and choose the phase angle () in such a way that  = 0 corresponds to lookingalong the magnetic axis. Table I shows the visible flux from different cone regions at anumber of spin phases, as a percentage of the total flux from the region considered. Whilethe top and middle regions of the cone are always visible, the fan emission nature of thebremsstrahlung component means that the emission from the bottom region of the coneis negligible when the observer’s line of sight is aligned along the magnetic axis, but isonly partially shaded at intermediate angles. The BR is mostly visible only when the theline of sight is close to the magnetic axis, due to the combined effect of the large openingangle of the cone, the variation in the magnetic field strength over the cone walls and theabsorption due to the cone walls (Robba et al. 1996). In table II the resulting intensityof the hard and soft bands are shown, together with the HR for different spin phases,and the results of phase spectroscopy. In fig. 5 the HR and the phase spectroscopy areschematically sketched, from which it is possible to see that this toy model explains themain features of the HR and of the phase spectroscopy.The results of a full numerical simulation of the X Persei light curve and HR, using themethod outlined in sect. 3, are presented in fig. 6. Here a linear gradient is adopted for theTABLE II. – Flux (arbitrary units) and Hardness Ratio for different phases. Spectral ratios be-tween the Hard Zone (H), the Medium Zone (M), and the Soft Zone (S) as defined in fig. 5. (degrees) status soft hard HRtop middle bottom BR top middle bottom BR (hard/soft)3 4 0 0 1 4 0 70 H 7 12 1.73 4 1 0 1 4 3 072 S 8 8 1.03 4 3 0 1 4 9 0180 M 10 14 1.4phase H/M 0.7 0.9spec. S/M 0.8 0.6NUMERICAL SIMULATIONS OF X-RAY BINARY PULSE PROFILES 761density and temperature profiles along the cone walls, with Ttop= 10 keV, Tbot= 3keV,top= 0:1bot, cone opening half-angle = 30, angular wall width = 9, i = 40, m = 30,and a cone height of 5RNS. Comparison with fig. 2 shows the simulation results to be ingood agreement with the observational data.REFERENCESARONS J. and LEA S. M., Astrophys. J., 235 (1980) 1016.BASKO M. M. and SUNYAEV R. A., Astron. Astrophys., 42 (1975) 311.BECKER R. H. et al., Astrophys. J. Lett., 227 (1979) L21.LAMB F. K. et al., Astrophys. J., 184 (1973) 271.MESZAROS P., HARDING A. K., KIRK J. G. and GALLOWAY D. J., Astrophys. J. Lett., 266 (1983)L33.MURAKAMI T. et al., Publ. Astron. Soc. Jpn., 39 (1987) 253.ROBBA N. R. and WARWICK R. S., Astrophys. J., 346 (1989) 469.ROBBA N. R. et al., Astrophys. J., 472 (1996) 341.SUNYAEV R. A. and TITARCHUK L. G., Astron. Astrophys., 86 (1980) 121.WANG Y.-M. and FRANK J., Astron. Astrophys., 93 (1981) 255.WANG Y.-M. and WELTER G. L., Astron. Astrophys., 113 (1982) 113.WHITE N. E. et al., Mon. Not. R. Astron. Soc., 176 (1976) 201.WHITE N. E. et al., Astrophys. J., 263 (1982) 277.WHITE N. E. et al., Astrophys. J., 270 (1983) 711.

Numerical simulations of X-ray binary pulse profiles

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Numerical simulations of X-ray binary pulse profiles

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