Analyzing X-Ray Pulsar Profiles: Geometry and Beam Pattern of Her X-1 S. Blum and U. Kraus Institut fu¨r Astronomie und Astrophysik, Abteilung Theoretische Astrophysik, Universita¨t Tu¨bingen, Auf der Morgenstelle 10, 72076 Tu¨bingen, Germany 0 0 0 ABSTRACT 2 n We report on our analysis of a large sample of energy dependent pulse profiles of a J the X-ray binary pulsar Hercules X-1. We find that all data are compatible with the 1 assumption of a slightly distorted magnetic dipole field as sole cause of the asymmetry 2 of the observed pulse profiles. Further the analysis provides evidence that the emission v3 from both poles is equal. We determine an angle Θm < 20◦ between the rotation axis 9 and the local magnetic axis. One pole has an offset δ < 5◦ from the antipodal position 4 of the other pole. Thebeam pattern shows structures that can be interpreted as pencil- 4 9 and fan-beam configurations. Since no assumptions on the polar emission are made, 0 the results can be compared with various emission models. A comparison of results 9 9 obtained from pulse profiles of different phases of the 35-day cycle indicates different / h attenuation of the radiation from the poles being responsible for the change of the p pulse shape during the main-on state. These results also suggest the resolution of an - o ambiguity within a previous analysis of pulse profiles of Cen X-3, leading to a unique r t result for the beam pattern of this pulsar as well. The analysis of pulse profiles of the s a short-on state indicates that a large fraction of the radiation cannot be attributed : v to the direct emission from the poles. We give a consistent explanation of both the i X evolution of the pulse profile and the spectral changes with the 35-day cycle in terms r of a warped precessing accretion disk. a 1. Introduction Since its discovery in 1972 by the UHURU satellite (Tananbaum et al. 1972), the X-ray binary system Hercules X-1/HZ Herculis has become the best studied of its class of about 44 known today (Bildsten et al. 1997). They are understood to be fast spinning neutron stars that are accreting matter from a massive companion star either via Roche lobe overflow or from the stellar wind of the companion. Since the neutron stars have strong magnetic fields, the accreted matter is funnelled along the field lines onto the magnetic poles, where most of the energy is released in form of X-radiation. Generally the magnetic axis and the rotation axis are not aligned. Therefore a large fraction of the detected flux from these sources is pulsed as during the course of each revolution of the neutron star the beams from the poles sweep through our line of sight. 2 Blum & Kraus Her X-1/HZ Her combines most of the properties that can be found in X-ray binaries, this made it one of the favourite sources of X-ray astronomers. From the observation of eclipses and from pulse timing analyses the orbital parameters are well determined. The masses of the neutron star and its optical companion are 1.3 M⊙ and 2.2 M⊙ respectively, the orbital period of Her X-1 is 1.7 d, and the inclination of the orbital plane is i > 80◦ (Deeter, Boynton, & Pravdo 1981). In addition to the pulse period of 1.24 s, i.e. the rotation period of the neutron star, Her X-1 also displays X-ray intensity variations on a period of about 35 days. Such a long-term variability is only known for two other pulsars: LMC X-4 and SMC X-1. The 35-day cycle of Her X-1 is nowadays ascribed to the precession of a warped accretion disk which periodically obscures the neutron star from our view (Petterson, Rothschild, & Gruber 1991, Schandl & Meyer 1994). During its high intensity or main-on state, Her X-1 has a luminosity Lx ≈ 2.5·1037 ergs s−1 (2-60 keV) (McCray et al. 1982). The maximum flux of the short-on state is typically only 30% of that of the main-on. Balloon observations in 1977 allowed for the first time the indirect measurement of the magnetic field strength of some 1012 G by the revelation of a spectral feature in the hard X-ray spectrum (Tru¨mper et al. 1978), interpreted as a cyclotron absorption line at about 40 keV. The pulse shapes of Her X-1 are highly asymmetric and depend on energy and on the phase of the 35-day cycle. In several studies phenomenological emission patterns have been used to reproduce the asymmetric pulse profiles of Her X-1. Wang & Welter (1981) fitted the geometry of two antipodal polar caps with asymmetric fan-beam patterns. In this approach the asymmetry of the emission pattern was attributed to asymmetric accretion due to the plasma becoming attached to the magnetic field lines away from the corotation radius. However it is not clear whether an asymmetric accretion stream must produce an asymmetric beam pattern (Basko & Sunyaev 1975). Another way of introducing asymmetry into the pulse shapes is via non antipodal emission regions. Leahy (1991) used two offset rings on the surface of the neutron star with symmetric pencil-beams and Panchenko & Postnov (1994) modelled two antipodal polar caps and one ringlike area which was attributed to a non-coaxial quadrupole configuration of the magnetic field. Further studies have shown that relativistic light deflection near the neutron star plays an important role when emission models are used to explain the observed pulse shapes (e.g. Riffert et al. 1993, Leahy & Li 1995). In this analysis we take up the idea of a non antipodal location of the emission regions caused by a slightly distorted magnetic dipole field. We assume that the emission originating from the regions near the magnetic poles only depends on the viewing angle between the magnetic axis and the direction of observation which means that the emission is symmetric with respect to the local magnetic axis. In contrast to previous studies where specific emission models have been used to fit the pulse profiles, the method used here does not involve any assumptions on the polar emission. Instead it tests in a general way whether the pulse profiles are compatible with the assumption that they are the sum of two independent symmetric components. The method we use to analyze pulse profiles is briefly summarized in the following §2.1. In Geometry and Beam Pattern of Her X-1 3 §2.2 we list the analyzed data. The results of the analysis are presented in §2.3. We show that the data of Her X-1 are indeed compatible with the idea of a slightly distorted magnetic dipole field. Further we find indications in the contributions to the pulse profiles that the emission from both poles is identical. We determine the location of the magnetic poles and reconstruct the beam pattern, which is discussed in §3.1. In the following §3.2 we examine the dependence of the pulse shape on the phase of the 35-day cycle. We argue that the contributions to the pulse profile undergo different attenuation resulting in the observed evolution of the pulse shapes during the main-on state of the 35-day cycle. 2. Analysis 2.1. The Method This section is a short summary of the method we use to analyze the energy dependent pulse profiles of Her X-1. We will focus on the main ideas and assumptions omitting both formal derivations and technical details. A comprehensive presentation of the material including a test case has been given in Kraus et al. (1995). Consider the emission region near one of the magnetic poles of the neutron star. Radiation escapes from the accretion stream and from the star’s surface and, while close to the star, is deflected in the gravitational field of the neutron star. A distant observer who cannot spatially resolve the emission region measures the integrated flux coming from the entire visible part of the emission region. The observed integrated flux depends on the direction of observation because the direction of observation determines which part of the emission region is visible and also because the radiation emitted by the accretion stream and the neutron star is presumably beamed. This function, namely the flux of a single emission region measured by a distant observer as a function of the direction of observation, is the link between the properties of the emission region and the contribution of that emission region to the pulse profile. In the following we will call this function the beam pattern of the emission region. The contribution of the emission region to the pulse profile, which we will refer to as a single-pole pulse profile, depends both on the beam pattern and on the pulsar geometry, i.e., on the orientation of the rotation axis with respect to the direction of observation and on the location of the magnetic pole on the neutron star. In short: local emission pattern plus relativistic light deflection determine the beam pattern, beam pattern plus geometry result in a certain single-pole pulse profile and the superposition of the single-pole pulse profiles of the both emission regions is the total pulse profile. 4 Blum & Kraus a. decomposition into single-pole pulse profiles In the following we are going to assume that the beam pattern is axisymmetric with respect to the magnetic axis (i.e., to the axis that passes through the center of the neutron star and through the magnetic pole). The axisymmetric beam pattern is a function of only one variable, the angle θ between the direction of observation and the magnetic axis. Consider now the single-pole pulse profile f(φ), where φ is the angle of rotation of the neutron star. It can easily be shown that the single-pole pulse profile produced by an axisymmetric beam pattern is symmetric in the following sense: there is a rotation angle Φ, so that f(Φ−φ) = f(Φ+φ) for all values of φ. The fact that f is periodic in φ implies that the same symmetry must hold with respect to the rotation angle Φ+π. Now turn to the total pulseprofile producedas the sum of the two symmetric single-pole pulse profiles. If the emission regions are antipodal, i.e., the two magnetic axes are aligned, it turns out that the symmetry points Φ1 and Φ1+π of the first single-pole pulse profile fall on the same rotation angles as the symmetry points Φ2 and Φ2+π of the second single-pole pulse profile. Their sum, the total pulse profile, is therefore symmetric with respect to the same symmetry points. If the emission regions are not antipodal, however, the symmetry points of the two single-pole pulse profiles do not coincide (except for certain special displacements from the antipodal positions) and the total pulse profile is asymmetric. Given an observed asymmetric pulse profile, we can ask if it could possibly have been built up out of two symmetric contributions with symmetry points that do not coincide. If so, it must be possible to find two symmetric (and periodic) functions f1 and f2 with the pulse profile f as their sum. By writing the observed pulse profile, defined by a certain number N of discrete data points f(φk), as a Fourier sum and with an ansatz for f1 and f2 in the form of Fourier sums also, the following can easily be shown: For an arbitrary choice of symmetry points Φ1 and Φ2, there are two periodic functions f1 and f2, f1 symmetric with respect to Φ1 and f2 symmetric with respect to Φ2, such that f = f1+f2, and the two symmetric functions are uniquely determined. Exceptions to this rule occur only if (Φ1−Φ2)/π is a rational number. In this case the symmetric functions may not exist or, if they exist, may not be uniquely determined. It must also be noted that the symmetric functions obviously can only be determined up to a constant C, since f1+C and f2−C are also a solution if f1 and f2 are. Thus, in principle every choice of a pair of symmetry points corresponds to a unique decomposition of any pulse profile into two symmetric contributions. For such a decomposition to be an acceptable solution, however, f1 and f2 also have to meet the following physical criteria in order to be interpreted as single-pole pulse profiles: 1. They must not have negative values, since they represent photon fluxes. 2. They must be reasonably simple and smooth. We do not expect the polar contributons to have a shape that is more complex than the pulse profile. Especially modulations of the Geometry and Beam Pattern of Her X-1 5 single-pole pulse profiles that cancel out in the sum are not compatible with the assumption of two independent and therefore uncorrelated emission regions. 3. They must conform to the energy dependence of the pulse profile. The decomposition can be done independently for pulse profiles in different energy ranges. Since the symmetry points are determined by the pulsar geometry, the same symmetry points must give acceptable decompositions according to the criteria 1 and 2 in all energy ranges. Finally the single-pole pulse profiles should show the same gradual energy dependence as the pulse profile. Given the existence of formal decompositions for all pairs of symmetry points and the criteria mentioned above, we are left with the two-dimensional parameter space of all possible values of Φ1 and Φ2, which we search for points with acceptable decompositions. For practical purposes, the parameters we use are the quantities Φ1 and ∆ := π−(Φ1 −Φ2). The parameter space that contains every possible unique decomposition then is 0≤ Φ1 ≤π and 0 ≤ ∆ ≤ π/2. In the analysis of just one pulse profile there will in general be a number of different acceptable decompositions. This number may be significantly reduced by the energy dependence of the pulse profile. In general, the existence of symmetry points with acceptable decompositions in all energy channels is by no means guaranteed. If such a pair of symmetry points is found, then it is indeed possible to build up the observed pulse profile out of two symmetric contributions, and we can conclude that the analyzed data are compatible with the assumption that the asymmetry of the observed pulse profile is caused by the non-antipodal locations of the magnetic poles. The symmetric functions can be interpreted as the single-pole pulse profiles due to the two emission regions. A successful decomposition provides information both on the geometry and on the beam pattern. As to the geometry, we obtain a value for the parameter ∆. This parameter is related to the locations of the emission regions on the neutron star (see Figure 1). The beam pattern is related to the single-pole pulse profile via the geometric parameters, i.e., the location of the emission region on the neutron star and the direction of observation. Since these parameters are not known, one cannot directly deduce the beam pattern from the single-pole pulse profile. It can be shown, however, that an appropriate transformation of the single-pole pulse profile and the beam pattern turns the transformed single-pole pulse profile into a scaled, but undistorted copy of a section of the transformed beam pattern. Although the scaling factor is a geometric quantity and therefore not known, this still provides an intuitive understanding of what a section of the beam pattern must look like. Since in the case of Her X-1 it is possible to eventually reconstruct the beam pattern, the information obtained at this stage mainly serves as a starting point for the next step of the analysis and we will not go into details about the transformation mentioned above. 6 Blum & Kraus b. search for an overlap region and determination of the geometry In general, the two emission regions on the neutron star may or may not be equal (i.e., have the same beam pattern). If they are equal, this fact may be apparent in the single-pole pulse profiles in the following way. Since in general the rotation axis and the magnetic axis of the neutron star are not aligned, the viewing angle θ between the magnetic axis and the direction of observation of each emission region changes with rotation angle φ. The range θ can cover for each magnetic pole depends on the location of that pole on the neutron star and on the direction of observation, where 0◦ ≤ θmin ≤ θmax ≤ 180◦. Only in the special case where both the magnetic axis and the direction of observation are perpendicular to the rotation axis, θ takes all values between 0◦ and 180◦. Since the emission regions have different locations on the neutron star, their ranges of values of θ are different. Depending on the geometry, these two ranges for θ may overlap. For an ideal dipole configuration e.g., the condition under which an overlap in the ranges of values of θ of the both poles exists is ΘO +Θm > π/2, where ΘO is the angle between the rotation axis and the line of sight, and Θm is the angle between the rotation axis and the magnetic axis. Consider an angle θ˜ in the overlap region. At some instant during the course of one revolution of the neutron, at rotation angle φ, one emission region is seen under the angle θ˜. At a different instant, at rotation angle φ′, the other emission region is seen under the same angle θ˜. If the beam patterns of the two emission regions are identical, then the flux detected from the one emission region at φ is equal to the flux detected from the other emission region at φ′. Thus, if an overlap region exists, the corresponding part of the beam pattern shows up in both single-pole pulse profiles, though at different values of rotation angle. Since the single-pole pulse profiles can be transformed into undistorted (though scaled) copies of sections of the beam patterns, such a part of the beam pattern that shows up in both single-pole pulse profiles should be readily recognizable. Note that the occurence and size of the overlap region depends on the geometric parameters and must therefore be the same for pulse profiles in different energy channels. If an overlap region is found in the single-pole pulse profiles obtained in the decomposition, this is an indication that there are two emission regions with identical beam patterns. Since each single-pole pulse profile provides a section of the beam pattern and the two sections overlap, we can then combine the two sections by superposingthe overlapping parts. As a result we obtain the total visible section of the beam pattern. Superposing the overlapping parts of the two sections of the beam pattern amounts to determining the relation between the corresponding values φ and φ′ of the rotation angle. On the other hand, the relation between φ and φ′ can be expressed in terms of the unknown geometric parameters of the system. Thus, the superposition provides a constraint on the geometry. Again omitting all details we simply note the procedure for superposing the overlapping parts of the two sections of the beam pattern. The single-pole pulse profiles f1(φ) with symmetry point Φ1 and f2(φ) with symmetry point Φ2 are transformed into functions of a common variable q through cos(φ−Φ1) = q for f1 and cos(φ−Φ2) = (q −a)/b for f2. The real numbers a and b > 0 are determined by means of a fit which minimizes the quadratic deviation between f1(q) Geometry and Beam Pattern of Her X-1 7 and f2(q) in the overlap region. At this point the constant C, which determines how the unpulsed flux has to be distributed to the single-pole pulse profiles, can also be computed. Since a and b can be expressed in terms of the unknown geometric parameters of the pulsar, their best-fit values constitute constraints on the pulsar geometry. The results of this second step of the analysis are the total visible beam pattern as a function of q and two constraints on the geometric parameters. The geometric information obtained so far (i.e., the values of ∆, a, and b) is not quite sufficient in itself to completely determine the pulsar geometry. It needs to be supplemented by an independent determination of any one additional geometric parameter or by an additional constraint. We suggest that this supplement may be obtained by means of the assumption that the rotation axis of the neutron star is perpendicular to the orbital plane. In this case, the angle ΘO between the direction of observation and the rotation axis of the neutron star is given by the inclination of the orbital plane. The assumption of ΘO = i seems to be quite plausible since accreted mass also carries angular momentum from the massive companion, and this transfer is expected to align the rotation axes of the binary stars on a timescale short compared to the lifetime of the system. However, this assumption must not hold true for all binary systems. With the inclination substituted for ΘO, the analysis of the pulse profiles determines the positions of the emission regions on the neutron star. Once the pulsar geometry is known, we also obtain the equation relating the auxiliary variable q and the viewing angle θ, so that the reconstructed beam pattern can be transformed into a function of θ. However, it turns out that the relation between q and θ involves an ambiguity which cannot be resolved within this analysis. It is due to the fact that we are not able to relate a single-pole pulse profile to one of the two emission regions. Therefore, we obtain two different possible solutions for the beam pattern and a choice between them must be based on either theoretical considerations and model calculations, or on additional information on the source. 2.2. The Data The analysis presented in this paper is based on pulse profiles of the main-on and short-on states of Her X-1. The analyzed sample contains a total of 148 pulse profiles from 20 different observations. References, the platform of the detectors, year of observation, the total energy range, the state of the 35-day cycle, the number of separate observations and the total number of pulse profiles of the respective observations are listed in Table 1. The data reduction including background subtraction has been done by the respective authors. In order to compare the pulse profiles from different observations, the pulse profiles of the main-on have been aligned in phase so that their common features match best. Since the pulse profiles of the short-on are markedly different, their features have been aligned with respect to the main-on as suggested by Deeter et al. (1998). At energies below 1 keV, the pulses of Her X-1 have a sinusoidal shape which is interpreted 8 Blum & Kraus 2......................n........................................................................................................................................................................d................................................................................................................................M.......(cid:14)......P..........................................................................................................................o............................a..................................................................l.....................(cid:1)............g..................e.............R................................................................n........(cid:2)......(cid:14)............................................................o.............................e............................................2..................t...(cid:15)...........t...............................................................................................................................................................................a.........................i....................................................................................c....................................t...............................................................................i...................................................o....(cid:14)(cid:14)..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................n.......................................................................................................................................................................A..................................................................................................................................................(cid:2)......x.............................................................................................................i...........................................1..........s................................................................................................................................................................................................................................................................................................................................................(cid:15).......................................................................................................................................................................................................1.................................(cid:14)....................................................................................s...............................................................................................................................................t..........................................................................................................M......................P....................................................ao........................................gl.......................e...................n...........(cid:2)......................................e................................O..........t....................................i............................c.............................................................L.................................................................................i..............................................n...................................................................e............................o.................f.......................S....................i.........g..................h...................t......................................................................... Fig. 1.— Intrinsic pulsar geometry: the locations of the magnetic poles on the neutron star surface can be described by means of their polar angles Θ1 and Θ2 with respect to the rotation axis and by the angular distance δ between the location of one magnetic pole and the point that is antipodal to the other magnetic pole. Table 1. Analyzed Data Instr. Year of Energy Range 35-Day No. of Pulse Reference Platform Obs. (keV) State Obs.a Profilesb Kuster et al. 1998 RXTE 1997 2 - 19 turn-on 1 25 Soong et al. 1990a HEAO-1 1978 12 - 55 main-on 1c 5 Kahabka 1987 EXOSAT 1984/1985 0.9 - 29 main-on 4 48 Kunz 1996d Kvant 1987/1988 16 - 30 main-on 1c 1 Scott 1993 Ginga 1988-1990 1 - 37 main-on 8 40 Stelzer 1997d RXTE 1996 2 - 26 main-on 1 9 Kahabka 1987 EXOSAT 1984 0.9 - 23 short-on 1 11 Scott 1993 Ginga 1989 1 - 14 short-on 3 9 aNumber of separate observations bTotal number of pulse profiles in different energy subranges cSeveral pointings have been integrated dprivate communication Geometry and Beam Pattern of Her X-1 9 as reprocessed hard X-radiation at the inner edge of the accretion disk (McCray et al. 1982). Since the origin of these soft X-rays is not the region near the magnetic poles, the analysis is restricted to higher energies. Above 1 keV the pulse profiles of Her X-1 are highly asymmetric and their typical energy dependence has been examined in a variety of studies (see Deeter et al. 1998, and references therein). In the analysis the pulse profiles are written as Fourier series. Since the higher Fourier coefficients are presumably affected by aliasing and also may have fairly large statistical errors, the highest coefficients are set to zero. This has a smoothing effect depending on the number of Fourier coefficients concerned. An example of the typical energy dependence of the pulse profiles and their representation in the analysis is given in the top row of Figure 2. It shows pulse profiles in three different energy ranges of an EXOSAT observation (Kahabka 1987) during the main-on state. The observed pulse profiles are plotted with crosses. The profiles plotted as solid lines are inverse Fourier-transformed using 32 out of originally 60 Fourier coefficients. 2.3. Results a. decomposition into single-pole pulse profiles In a first run, the decomposition method has been simultaneously applied to the 103 pulse profiles of the 15 observations of the main-on state. Due to the large number of distinct pulse shapes and due to the fact that they have a relatively low level of unpulsed flux, the positive flux criterion has led to an exclusion of about 90% of the whole parameter space of possible symmetry points Φ1 and Φ1+∆. Further sorting out the decompositions (i.e. the single-pole pulse profiles) that are qualitatively too complicated to match the criterion of two independent emission regions only left over one type of decomposition. The energy dependence of this type of decomposition is as smooth as that of the pulse profiles. Thus we have found acceptable decompositions in a small range of Φ1 and Φ1 +∆ which are all of the same type. This type of decomposition is unique in the sense that a small deviation from the ’best-values’ of the symmetry points results in decompositions that look similar but become more and more complicated the larger the deviation becomes until they do not match the physical criteria any more. A systematic variation of the best-values of the symmetry points, which could be caused by free precession of the neutron star, is not observed. The lower panels in Figure 2 show the decompositions of the typical pulse profiles of the respective top panels. The unpulsed flux has been distributed to the single-pole pulse profiles according to the constant C as derived in the second step of the analysis (see § 2.1). The single-pole pulse profiles show that the energy dependence of the pulse profiles is mainly due to the change of one polar contribution (dashed curve) where an additional peak appears above 10 keV, whereas the pulse shape of the other pole (solid curve) does not change much. Interestingly, the contributions of the emission regions we obtain look very similar to those of Panchenko & Postnov (1994) obtained from a model calculation mentioned in § 1. Similar components were also obtained by Kahabka (1987) in an attempt to model the observed pulse shapes by means of 3 to 5 gaussians, a sinusoidal component and a constant flux. 10 Blum & Kraus (cid:10)EXOSAT 6.0-8.3 keV main-on(cid:10) (cid:10)EXOSAT 10.0-13.0 keV main-on(cid:10) (cid:10)EXOSAT 20.0-23.0 keV main-on(cid:10) 1.0 0.5 0.0 x u fl 0.5 0.0 0.5 0.0 (cid:10)0.0(cid:10) (cid:10)0.5(cid:10) (cid:10)1.0(cid:10) (cid:10)1.5(cid:10) (cid:10)0.0(cid:10) (cid:10)0.5(cid:10) (cid:10)1.0(cid:10) (cid:10)1.5(cid:10) (cid:10)0.0(cid:10) (cid:10)0.5(cid:10) (cid:10)1.0(cid:10) (cid:10)1.5(cid:10) φ Fig. 2.— Examples of analyzed pulse profiles and their decompositions into single-pole contributions. The panels in the top row show observed pulse profiles: the crosses correspond to the bin-width (60 phase bins) and the estimated statistical error of the observations, whereas the solid curves are the inverse Fourier- transformed pulse profiles using 32 Fourier coefficients. The pulse profiles have been normalized to have a maximum flux of unity. The energy range is indicated above each column. The energy dependence is typical for pulse profiles of the main-on state. The middle and bottom panels show the decompositions of the pulse profiles into the two symmetric contributions from the poles with symmetry points at Φ1 ≈ 95◦ (φ ≈ 0.26) and Φ1+∆ with ∆≈9◦ (φ≈0.29). The single-pole pulse profiles add up exactly to the pulse profile in the respective top panel.