Issue 
A&A
Volume 520, SeptemberOctober 2010



Article Number  A76  
Number of page(s)  14  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/200912773  
Published online  06 October 2010 
Constraining compactness and magnetic field geometry of Xray pulsars from the statistics of their pulse profiles
M. Annala  J. Poutanen
1  Astronomy Division, Department of Physics, PO Box 3000, 90014 University of Oulu, Finland
Received 26 June 2009 / Accepted 5 August 2001
Abstract
Context. The light curves observed from Xray pulsars and
magnetars reflect the radiation emission pattern, the geometry of the
magnetic field, and the neutron star compactness.
Aims. We study the statistics of Xray pulse profiles in order
to constrain the neutron star compactness and the magnetic field
geometry.
Methods. We collect the data for 124 Xray pulsars, which are
mainly in highmass Xray binary systems, and classify their pulse
profiles according to the number of observed peaks seen during one spin
period, dividing them into two classes, single and doublepeaked. We
find that the pulsars are distributed about equally between both
groups. We also compute the probabilities predicted by the theoretical
models of two antipodal pointlike spots that emit radiation according
to the pencillike emission patterns. These are then compared to the
observed fraction of pulsars in the two classes.
Results. Assuming a blackbody emission pattern, it is possible
to constrain the neutron star compactness if the magnetic dipole has
arbitrary inclinations to the pulsar rotational axis. More realistic
pencilbeam patterns predict that 79% of the pulsars are doublepeaked
independently of their compactness. The theoretical predictions can be
made consistent with the data if the magnetic dipole inclination to the
rotational axis has an upper limit of 40 4.
We also discuss the effect of limited sensitivity of the Xray
instruments to detect weak pulses, which lowers the number of detected
doublepeaked profiles and makes the theoretical predictions to be
consistent with the data even if the magnetic dipole does have random
inclinations. This shows that the statistics of pulse profiles does not
allow us to constrain the neutron star compactness. In contrast to the
previous claims by Bulik et al. (2003, A&A, 404, 1023), the data
also do not require the magnetic inclination to be confined in a narrow
interval.
Key words: methods: statistical  pulsars: general  stars: neutron  Xrays: binaries
1 Introduction
Xray pulsars have been discovered in the 1970s (Giacconi et al. 1971) and have served as laboratories to study the neutron star (NS) physics since then. Most of them are members of binary systems and accrete matter through wind or via disk from a highmass companion. Because of a large magnetic field strength (typically 10^{12} G) the material is channeled onto small spots at the magnetic poles. Here the relativistically moving plasma is decelerated in a radiative shock near the surface and this subsonically settling plasma radiates in the Xray band (see e.g. ). Pulsations are observed if the magnetic field is inclined relative to the rotation axis. Even stronger field pulsars (magnetars) have been discovered recently, which operate by dissipating magnetic energy (see e.g. review by Mereghetti 2008). Studies of the pulse profiles of individual pulsars allow one to constrain the emission pattern of the hotspots (or accretion columns) at the NS surface as well as the geometry of the magnetic field (see e.g. ).
The number of known pulsars in the Milky Way and the nearby Small and Large Magellanic clouds is already above a hundred. The quality of the data is also improving because of the sensitive Xray/gammaray observatories such as RXTE and INTEGRAL and the long observing times. The first pulsars have already been discovered in M 31 (Trudolyubov et al. 2005) and in even more distant galaxies (Trudolyubov et al. 2007; Trudolyubov 2008). A large number of pulsars allows us to use a statistical approach to constrain the NS parameters. Because of the gravitational light bending, the more compact the star, the larger the fraction of the NS surface that is visible to an observer at all times. This increases the probability to observe two radiative poles in one rotational period and affects the relative fraction of the single and doublepeaked pulse profiles. The second important parameter that affects the pulse profile is the positions of the hotspots relative to the rotational axis (which are defined in the simplest case by the inclination of the magnetic dipole).
For any reasonable NS parameters one expects that both poles are visible in a large majority of pulsars. However, the number of observed single and doublepeaked profiles is not so different. This was already noticed by Wang & Welter (1981) and later by Bulik et al. (2003, B03 hereafter). This discrepancy can be explained if the inclination between the magnetic dipole and rotational axis is not randomly distributed, but if there is a strong bias towards alignment. In the present study, we consider a sample of 124 pulsars with better quality data than were available before. We study in detail the statistics of doublepeak profiles for various pencillike emission pattern. We also discuss the effect of the detection threshold that can significantly affect the observed fraction. We then derive useful analytical formulae that describe the probabilities of observing certain types of pulsars. And finally we compare our theoretical model to the data.
2 Data selection and classification of light curves
The light curves of Xray pulsars can be classified according to the number of pulses per period. The observed pulse profiles tend to simplify with increasing energy, and the multiplepeaked profiles change into double or singlepeaked (see e.g. Nagase 1989; Bildsten et al. 1997). In order to reduce possible effects of the photoelectric absorption and the cyclotron lines, the pulse classification is done at the highest possible energies (typically above 10 keV). We use the published light curves of Xray pulsars from several sources. Therefore the data are inhomogeneous. A significant number of light curves are produced in an nonappropriate energy range, which makes it difficult to conclude anything about the number of poles visible to the observer and details of the beam shape.
Table 1: Light curve classification of 60 Xray pulsars observed atenergies above 10 keV.
Table 2: Light curve classification of 64 Xray pulsars observed atenergies below 10 keV.
Bulik et al. (2003) have studied the profiles of 88 pulsars (not 89 because one of the sources, 1WGA J1958.2+3232, turned out to be an intermediate polar, Negueruela et al. 2000), which have been divided into three different groups. The first group consisted of 46 pulsars, which were easy to classify. The second group consisted of 31 pulsars, which were difficult to classify and the third group had 11 pulsars for which there were no good quality light curves available in 2003. We have scanned the latest literature for the light curves from the same sources and also added to our sample all newly discovered Xray pulsars. All together the sample now consists of 124 Xray pulsars. The pulsars are placed in two different categories depending on the kind of the data: those which have the profiles observed above 10 keV and those that do not. The first category contains 60 pulsars and the second one 64 pulsars. Their data are given in Tables 1 and 2, respectively, together with their classification (i.e. the number of pulses observed). A similar classification from B03 (who found 38 doublepeaked light curves out of 88 sources) is also shown for comparison. Only the pulsars residing in highmass Xray binaries and Betransients as well as magnetars are included, while all diskaccreting systems residing in lowmass Xray binaries are excluded. This is because the disk can seriously affect the light propagation from the secondary pole to the observer (e.g. Ibragimov & Poutanen 2009).
A couple of the classified sources (e.g. 4U 153852) show profile changes in the energy ranges above 10 keV, and the secondary pulse was not observed in every energy band. The most probable reason for this is the cyclotron absorption. These sources were classified as doublepeaked. In two sources, GX 3041 and AX J1841.00536, pulsations become weak as the energy increased. These sources were classified as singlepeaked.
In general, the probability of observing M doublepeaked pulsars out of N
sources, is given by the binomial distribution. Because our data set is
very large and the observed number of pulsars of both types is similar,
we can use the normal distribution instead. The estimation of the
probability is p=M/N and its error is
.
In our classification of all pulsars in Tables 1 and 2 we found 55 doublepeaked light curves out of 124 sources, which gives the probability of observing doublepeaked profiles
For those pulsars which have light curves above 10 keV (Table 1), we have 33 doublepeaked pulsars out of 60 sources, which gives the corresponding probability of
This is still consistent with p_{0} within . Excluding the magnetars from the list of pulsars would change the probabilities very little to and .
3 Modeling light curves of Xray pulsars
3.1 Model setup
In order to obtain some constraints on the (distribution of) NS parameters such as compactness, magnetic field inclination, and the emissivity pattern from the statistical data (such as the fraction of the doublepeaked profiles), we need to make a set of simplifying assumptions regarding the NS and the emission. In most of the following discussion, we assume that the NSs have a dipole magnetic field and two antipodal, pointlike radiating hotspots at the magnetic poles at the NS surface. We assume also that all NS have the same compactness and that the emission from pulsars is described by the same pencilbeam pattern. As we will see below, the assumption of the same emission pattern for all pulsars will not have much effect on the results and therefore could be relaxed. Thus the pulsars differ from each other by the observer inclination, magnetic field inclination, and possibly by the emission pattern. We now discuss our assumptions one by one.
 1.
 Compactness. The gravitational lightbending effect depends only on the compactness, i.e. masstoradius ratio M/R, which we assume to be the same for all NS. This is reasonable, because the observed distribution of NS masses in radio pulsars is very narrow (Haensel et al. 2007; Thorsett & Chakrabarty 1999) and the accretion in highmass systems could not provide a significant mass increase during the life time of the system. Below we will also show that the statistics of pulse profiles depends very little on the compactness for realistic emission patterns.
 2.
 Pointlike emission regions. The lower limit on the spot size
in accreting Xray pulsars can be obtained by assuming that the
accreting matter is bound by the magnetic field lines intersecting the
Alfvén radius (Wang & Welter 1981). For a pulsar with massaccretion rate
/year, magnetic field strength of
B= 10^{12} G, and typical NS mass
and radius R=10 km, the typical size of the hotspots is about 4 degrees.
A study of the interchange instability and diffusion of plasma through the magnetic field gives a
higher estimate of about 20 degrees (Arons & Lea 1980).
Thus in any case, the radiating spot size is much smaller than the stellar radius
and the pulse profiles will not be dramatically affected because of the strong gravitational bending.
Increasing the spot size increases the probability to see the secondary spot, but unless the emission pattern
is fanlike, the number of pulses will not be affected.
For magnetars, the area of the thermally emitting region is only a few km^{2} (Mereghetti et al. 2002). Although the nature of the nonthermal persistent emission of anomalous Xray pulsars and softgamma ray repeaters above 10 keV (Kuiper et al. 2004,2006; Molkov et al. 2005; Mereghetti et al. 2005) is not known, it also can be produced in very localized regions close to magnetic poles (Beloborodov & Thompson 2007; Thompson & Beloborodov 2005). Thus the assumption of the spotlike regions seems reasonable and the corrections arising from a finite spot size are negligible.
 3.
 Emission patterns. The exact geometry and the structure of the
emission region in accreting Xray pulsars is modeldependent and
varies from planeparallel slabs (see e.g. ) to columns/mounds (see e.g. , and references therein).
At high enough energies, where the pulse profiles are rather simple (see e.g. Nagase 1989; Bildsten et al. 1997), the effects of the cyclotron and photoelectric absorption are minimized.
We parameterize the (possibly complicated) emission pattern with simple mathematical functions.
For example, we take the pencillike emission pattern with the surface flux given by
,
where
is the inclination of the spot normal to the light of sight. Detailed modeling of the pulse profiles of seven pulsars by Leahy & Li (1995) showed that such a pattern with n=24 gives a good description of the data.
These patterns also describe well theoretical dependences expected from a magnetized slab (Meszaros & Nagel 1985b) as was shown by
Leahy (1990).
The case n=1
corresponds to the blackbodylike emission. This emission pattern is
probably not physical, but it is a useful starting point for discussing
the pulsar classification scheme (Beloborodov 2002; Poutanen & Beloborodov 2006).
We also consider an alternative beaming pattern
,
where h>1 is a parameter.
The case with h<0
would correspond to the scattering in an optically thin electron
atmosphere associated with the accretion shock or heated NS surface
layer (see e.g. Viironen & Poutanen 2004), while h>0 resembles pencilbeam and
is more appropriate for optically thick emission.
Although the physics of the persistent emission from magnetars is very different (Beloborodov & Thompson 2007; Thompson & Beloborodov 2005), the simplicity of their pulse profiles and their broad peaks argues in favor of broad emission beams, which can be represented by the assumed patterns.
We first assume that all pulsars can be described by the same emission pattern and then discuss the consequences of relaxing this assumption. We will show below that the statistics of pulse profiles depends very little on the actual emission pattern.
 4.
 Antipodality. We assume that the hotspots are antipodal, even though many pulsars show asymmetric profiles. The detailed modeling of the pulse profiles (Kraus et al. 1996; Leahy & Li 1995) shows, however, that the displacement of the spots relative to the antipodal position vary from a few to about 10 degrees. This displacement, while causing the profile asymmetry, does not change the number of pulses, which is important for our analysis.
 5.
 Emission from the NS surface. Pulsars in Betransient systems show time evolution in their pulse profiles during the outbursts related to the changing massaccretion rate. This results either from variations in the emission pattern and/or changes in the accretion shock height. According to the recent cyclotron line measurements, the radiative region in accretion column does not extend higher than about 7% of the neutron star radius (Tsygankov et al. 2006). In magnetars the persistent emission most probably also originates from the stellar surface (Beloborodov & Thompson 2007; Thompson & Beloborodov 2005). Therefore a simple model assuming the radiation is produced in the vicinity of the neutron star surface is justified.
 6.
 Slow rotation. In principle, the stellar rotation could affect the profiles because of the effects of relativistic aberration and time delays (Poutanen & Gierlinski 2003; Poutanen & Beloborodov 2006), but typical pulsars in Betransient systems and highmass Xray binaries as well as magnetars rotate too slowly for these effects to be important.
 7.
 Accretion disk. The accretion disk in strong magnetic field pulsars is normally disrupted at a distance that is large compared to the NS radius, and therefore it does not affect the visibility of the radiative spots on the neutron star surface or the pulse profile. These effects become important in NS in lowmass Xray binaries such as accreting millisecond Xray pulsars (see Ibragimov & Poutanen 2009; Poutanen 2008), but we do not include these objects in our study.
3.2 Pulsar classes and observed fluxes
Let
be the angle between rotational and magnetic axes and i the inclination of the rotational axis to the lineofsight. Then the unit vector in the observer's direction is
and the unit vector in the direction of the primary spot from the NS center is
with
being the pulsar phase. As pulsar rotates the position of the spots
relative to the observer changes. For the primary spot (closest to the
observer)
At the primary spot is closest to the observer and , while at the spot is farthest away and .
Because of gravitational light bending, photons emitted at an angle
relative to the local radial direction reach the observer at angle
(see Fig. 1). The deflection angle
reaches the maximum
when photons are emitted at grazing angles
.
This defines
the visible part of the star:
where is a function of stellar compactness. For a typical neutron star mass of 1.4 and radii between 10 and 14 km, the maximum bending angle is between 45 and 25.
Figure 1: Light emitted from a neutron star at an angle to the normal is observed at impact parameter b, with the direction making angle to the spot position vector. The picture is in the plane of photon trajectory. 

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The relation between
and
for slowly rotating stars is given by an elliptical integral (Pechenick et al. 1983). For stars with radii R larger than about two Schwarzschild radii
,
a very accurate linear relation between the respective cosines can be used instead (Beloborodov 2002; Poutanen & Beloborodov 2006):
where . In this approximation, the visibility condition (4) gets a simple form
where now
For the assumed dipole magnetic field with two antipodal spots, a light curve can belong to any of the four visibility classes introduced by Beloborodov (2002), according to the values of i, , and u. For a Class I pulsar, the secondary pole is invisible and the primary pole is always visible. Class II pulsars have their primary pole always visible, but the secondary appears and disappears during the rotation period. In Class III pulsars both poles appear and disappear during the rotational period. Class IV pulsars have both their poles visible at all times. See Fig. 2 for an example of visibility classes for a moderate light bending.
For random inclination i and magnetic inclination , the probability densities and are constants. Therefore, the area covered by a certain Class on the plane directly gives the probability of a random pulsar belonging to that class. Depending on the stellar compactness, the area covered by each Class changes, and for example at high u, bending is strong and the area occupied by the Class IV pulsars grows.
For the blackbody emitting spot, the visibility Class directly defines the shape of the light curve (Beloborodov 2002), while for an arbitrary emissivity pattern the number of light curve classes can differ from the number of visibility classes.
Figure 2: Beloborodov's classes of pulsars on the plane ( ). The visibility classes IIV are shown for a moderate light bending . If the magnetic inclination is constrained by , the area below the dashed line is forbidden. 

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The observed flux from a small homogeneous spot can be expressed as
(8) 
where I is the observed intensity and is the solid angle covered by the spot on the observer's sky. The spot of area seen at impact parameter occupies a solid angle , where D is the distance to the observer. Owing to the gravitational redshift, the bolometric intensity is reduced from the emitted value I_{0} to
where the emitted intensity can be a function of the emission angle . The impact parameter b and the emission angle are related by
Combining the expressions above we get (Beloborodov 2002; Poutanen & Beloborodov 2006):
Using approximation (5), the flux takes the form
3.3 Light curves and probabilities for blackbody spots
Let us first assume that the hotspots emit blackbody radiation, i.e.
const. The flux from a spot is then simply
(13) 
Below we will use the flux normalized to , i.e. the flux from the primary spot is
where
(15) 
and for the secondary spot the flux is
Obviously the flux and the variability amplitude depend on the stellar compactness and the position of the spot on the NS surface. As discussed by Beloborodov (2002), the light curves from Class I pulsars are singlepeaked. In Class II the light curves are also singlepeaked with a plateau between the pulses. Class III is the only one contributing the doublepeaked light curves. Class IV pulsars produce flat light curves, because
In the blackbody case, the probability to observe doublepeaked light curves depends only on the maximal gravitational lightbending angle . Because only Class III pulsars produce doublepeaked profiles, the probability to observe it is given by the area occupied by Class III on  plane (see Fig. 2):
which is plotted as a solid curve in Fig. 3 (the case ). If bending is weak, , the probability is , i.e. 79%, and steadily decreases with increasing . This result differs dramatically from that obtained by B03, who assumed that class II, III, and IV are all producing doublepeaked profiles.
Figure 3: Probability to obtain doublepeaked light curves for blackbody spots as a function of the maximal gravitational light deflection angle . Solid curves represent the probability given by Eq. (23) when class IV pulsars are included. The dashed curves represent the case where Class IV pulsars are excluded, given by Eq. (24). The curves correspond to three different cases of the maximum magnetic inclination (i.e. unconstrained), and . The upper curves are given by analytical relations (20) and (18). 

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Another way to interpret the model predictions is to exclude the
Class IV light curves from the light curve analysis since they are
flat, i.e. these are not pulsars. It would mean that the total
parameter space for the pulsars would be reduced and the fraction of
single and doublepeaked light curves is changed (see Fig. 2). The area corresponding to the classes I, II, and III is
=  
=  (19) 
and the probability to observe a doublepeaked light curve is thus
which is plotted as a function of the maximal gravitational light deflection angle in Fig. 3 (dashed curve, case ). This probability is only slightly higher than that given by Eq. (18) where Class IV is included (solid curve).
Figure 4: Light curve classes for beam pattern on the i plane ( left) and the  plane ( right). The shaded areas correspond to the doublepeaked light curves. The number of light curve classes for this pattern differs from the blackbody case. The small filled circles at the left panel correspond to the parameter pairs () of the light curves plotted in Fig. 5. 

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We need to notice here that Class IV pulsars produce flat light
curves only if the emission is blackbodylike and if the radiating
spots are exactly antipodal. Small deviations from the isotropy, e.g.
if we use an Eddington approximation for the intensity,
(where h is a parameter), cause oscillations even in Class IV pulsars. For small h, the peaktopeak amplitude is (Poutanen & Beloborodov 2006):
(21) 
In addition, even a slight displacement from the exactly antipodal positions leads to pulsation of the light curve. Let us assume the secondary spot is shifted by and in latitude and azimuth: and . It is easy to show that for the blackbody emission
(22) 
Thus we see that the Class IV pulsars are actually expected to pulsate. They predominantly have singlepeaked profiles if the radiation emission pattern is close to black body (or more peaked, see Sects. 3.4.1 and 3.4.2), and spots are close to antipodal positions. Equation (18) should thus give a more realistic probability of observing doublepeaked light curves.
3.3.1 Limiting magnetic inclination
In the previous discussion, we assumed that magnetic inclination can take any value . However, as proposed by Wang & Welter (1981) and B03, there can be an upper limit to that angle . Then the fraction of pulsars belonging to different visibility classes is changed (see Fig. 2). The constrained case alters the probabilities of observing the doublepeaked profiles.
Let us consider a blackbodylike emission pattern and include all the
pulsar classes IIV. The probability to observe doublepeaked
light curves is proportional to the area of class III above the
curve
(see Fig. 2) and is the function of the bending angle
and
:
where is the crossing point between curves and . If , the probability is zero.
If we exclude Class IV from consideration, the corresponding probability becomes
where
(25) 
and
(26) 
The probabilities to observe doublepeaked profiles for two cases, including and excluding the Class IV pulsars, given by Eqs. (23) and (24), are shown in Fig. 3 by solid and dashed curves, respectively. The significant difference in these probabilities appears only for a large bending angle .
3.4 Modified pencil beam pattern
3.4.1 Beam pattern cos
Let us now consider two pointlike antipodal spots with the emitted intensity given by
.
The light curves produced by the two spots with this more radially concentrated beampattern
can be divided into six different classes based on the visibility of
the spots and whether the light curves are single or doublepeaked
(see Figs. 4 and 5).
The normalized flux from the primary spot in Beloborodov's approximation (5) is
(27) 
The flux from the secondary pole is
(28) 
In Class I, only the primary spot is visible and the light curve is singlepeaked with the maximum at pulsar phase and the minimum at . In Class IV, both spots are visible all the time and the flux has local extrema when
(29) 
i.e. at , or
The latter extrema exist only when
and then the light curve is doublepeaked. Similarly for Class II pulsars, for which the primary spot is always visible and the secondary spot is seen at phases around , we get that the light curve is singlepeaked when and doublepeaked otherwise. Condition (31) divides both classes II and IV into two subclasses according to the number of peaks: in IIa and IVa profiles are singlepeaked, while in classes IIb and IVb, they are doublepeaked. In Class III, both spots appear and disappear from the view at some pulsar phases, and it is easy to show that both and are maxima, and the light curve is always doublepeaked.
Figure 5: Light curve classes for beam pattern in the moderate case of light bending . The light curves are plotted with following parameters: and (class I), and (class III), and (class IVa), and (Class IVb), and (class IIa), and (class IIb). 

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Thus the light curve in classes I, IIa, and IVa is singlepeaked and
the profile in III, IIb, and IVb is doublepeaked. The probability to
observe doublepeaked light curves is now independent of the neutron
star compactness (see the hatched area in Fig. 4):
This coincides with expression (18) for the Newtonian case, , and the blackbody radiation pattern.
If the magnetic inclination
is constrained to lie in the interval
,
the probability to observe a doublepeaked light curve is (shown by the solid curve in Fig. 6)
Decreasing the value of reduces the probability to obtain doublepeaked light curves and for , .
Figure 6: Probability to observe doublepeaked light curves as a function of the maximum magnetic inclination . The solid curve corresponds to any modified pencil beam patterns discussed in Sect. 3.4, and (Eq. (33)) in the case of full detectability of the secondary pulse, T=0. The dashed curve shows an example of the beam pattern with nonzero threshold T=0.2 and a moderate bending angle . The dotted lines correspond to the observed probability of having doublepeaked profiles as given by the classification of data p_{0}=0.44 and p_{1}=0.55 (see Sect. 2). 

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3.4.2 Beam pattern cos
Let us assume that the emission pattern can be described as
,
where .
The observed flux is then
.
The normalized flux from the primary pole is then
(34) 
and the secondary pole has the flux
(35) 
Once the power law index n exceeds 1, the light curves become similar to the case of n=2, i.e. the beam pattern. Doublepeaked profiles are obtained in the region . Therefore the probability to observe doublepeaked light curves in the case of n>1 and randomly distributed angles i and is . For the constrained magnetic dipole inclination, this probability is given by Eq. (33).
3.4.3 Eddington approximation cos(1 + h cos)
So far we have considered modified pencil beam patterns with varying power index n. Let us now assume the radiation intensity deviates from blackbody according to the Eddington approximation
and the observed flux
.
The normalized primary pole flux is
(36) 
and the secondary pole gives
(37) 
The shape of the light curves is determined by the anisotropy parameter h. The observations show that h should be positive rather than negative, because the negative values can produce double, triple and quadruplepeaked light curves, which are not consistent with the observations. Even if there are only single and doublepeaked light curves (for example when h=1), these doublepeaked light curves are doublehorned with equal primary and secondary peak amplitudes. This is not what we generally observe.
Therefore, we consider positive values of h. The light curves then resemble the classes obtained for modified beam pattern . Doublepeaked light curves are only obtained when and singlepeaked curves when . Again, this result does not depend on the compactness. Therefore, if angles and i can vary randomly between zero and , we would observe about doublepeaked light curves as calculated earlier (see Eq. (32)). If the angle between the magnetic field and rotational axis is constrained, then the probability of having doublepeaked light curves is just a function of as found earlier (see Eq. (33)). As can be seen, the probabilities to observe doublepeaked light curves do not depend on the constant h when it is positive.
Figure 7: Effect of the detection threshold on the fraction of the doublepeaked profiles. a) The parameter space at the  plane for the blackbody emission pattern. Upper curves are for the Newtonian case and the lower curves are for the moderate light bending . The solid curves correspond to the zero threshold and the dashed curves are for T=0.2. The areas below respective curves correspond to the probability of observing doublepeaked profiles. b) Same as a), but for the pencilbeam . For the zero threshold the curve is the same for any neutron star compactness. The curves for thresholds T= 0.1 and 0.2 are shown by dotted and dashed curves, respectively. The dotted curves are for , while the dashed ones are for . Even then the dependence on compactness is very weak. 

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3.5 Effect of the detection threshold
Limited photon statistics and (rednoise) flux variability can affect the visual assignment of pulsars into single and doublepeaked classes. If the secondary pulse is hardly visible above the constant emission level, the pulsar is likely to be classified as singlepeaked. It is difficult to quantify these effects. Following B03, we can assume that the secondary pulse is visible if its strength above the minimum flux is at least fraction T of the strength of the primary maximum. The probability to observe doublepeaked profiles decreases with increasing detection threshold T.
3.5.1 Detection threshold in the blackbody case
In the blackbody case, for the zero threshold T=0 the region, where profiles are doublepeaked, coincides with Class III. There the primary maximum is reached at phase ,
and its amplitude above the constant flux level is
(see Eq. (17)). For the secondary maximum to be detected, its amplitude,
,
should exceed the threshold
.
Thus for the nonzero T, the region of doublepeaked profiles is reduced as the following condition must be satisfied
Generally this inequality must be solved numerically. The resulting constraints are shown in Fig. 7a.
In the Newtonian limit
(i.e. ), this inequality is reduced to
Then the probability to observe doublepeaked light curves for randomly distributed i and is given by the integral
where , F(k) and E(k) are the complete elliptic integrals.
Figure 7a demonstrates how the parameter space is reduced where profiles are doublepeaked when the detection threshold is increased from zero to T=0.2 in a Newtonian ( ) and in a moderately relativistic case, . The solid curves correspond to the total detectability T=0 and the areas below them correspond to the probabilities to observe doublepeaked light curves. The dashed curves correspond to the threshold T=0.2.
3.5.2 Detection threshold for cos beam pattern
Let us consider the beam pattern , where the index n is greater than 1. We showed in Sect. 3.4.2 that the resulting profile becomes doublepeaked when . The effect of the detection threshold on the visibility of the secondary pulse has to be considered in three different light curve classes IIb, III, and IVb (see Fig. 4).
For a blackbody pattern when both poles are visible the flux is constant and equals 2u. For n>1, the minimum flux of 2u^{n} is reached at the phase given by Eq. (30) in all three considered classes (because the secondary pole is visible at this phase).
Figure 8: a) Probability to observe doublepeaked profiles for the blackbody emission pattern (n=1) as well as for beam patterns with indices n=1.1, 2, 4 and 6 for two different threshold cases T=0 and 0.2. The dashed curves correspond to T=0, while the solid curves are for T=0.2. The dotted lines correspond to the observed probability of having doublepeaked profiles as given by the classification of data p_{0}=0.44 and p_{1}=0.55 (see Sect. 2). The dashdotted line correspond to the probability to have multiplepeaked light curves for two randomly positioned spots radiating as black bodies. b) Same as panel a), but for beam patterns in the Eddington approximation. Cases with anisotropy parameter h=0, 0.5, 2 and 4 for two different thresholds T=0 and 0.2 are shown. 

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For Class IIb, the primary maximum is reached at ,
when only the primary pole is visible. On the other hand, both poles contribute to the flux at the secondary maximum
.
Thus the secondary pulse will be detected if
For Class III, the primary maximum is produced by the primary pole and the secondary maximum by the secondary pole only, thus the condition for the secondary pulse to be seen is
Class IV pulsars have both their primary and secondary poles contributing to the flux all the time, and therefore the threshold to the secondary pulse to be detected is
For n=2 the boundary between single and doublepeaked profiles given by relations (41)(43) are shown in Fig. 7b at the  plane. The areas below the corresponding curves give the probability to observe doublepeaked profiles. Figure 8a shows this probability as a function of compactness for various indices n and two thresholds T=0 and 0.2. When T=0, i.e. even the smallest bumps in the light curve can be detected, all beam patterns with n>1 predict . For T=0.2, the probability to observe doublepeaked profiles decreases. For typical , a slight deviation from n=1 to n=1.1 immediately increases the value of from 0.3 to 0.45. The maximum of is reached at , and at n>2 the probability starts decreasing again. This probability depends weakly on as well as on index n (see Figs. 7b and 8a). If the magnetic inclination is constrained, the fraction of doublepeaked profiles decreases as shown in Fig. 6 for the n=2 case.
3.5.3 Detection threshold in the Eddington approximation case
The emission pattern in the form given by the Eddington approximation produces light curve classes identical to the classes for the modified pencil beam . Therefore the consideration of the threshold is similar to the previous section. Doublepeaked light curves are obtained only in classes IIb, III, and IVb, and increasing the threshold will increase the probability of observing singlepeaked light curves in these classes.
The local extrema are at the same pulsar phases as for the
beam pattern, i.e. at
and
At the minima the flux is
(43) 
In Class IVb, the secondary pulse is detected if the following condition holds:
(44) 
In Class IIb, the secondary pulse is detected if
(45) 
Finally in Class III, the secondary pulse is detected if
(46) 
The influence of the increasing threshold on the probability to observe doublepeaked light curves for various anisotropy parameters h is shown in Fig. 8b. The case h=0 corresponds to the blackbody emission pattern and therefore gives the same dependences as plotted in Fig. 8a (curves marked with n=1). With full detectability of the secondary pulse T=0 the probability to observe doublepeaked profiles is constant for any h>0. This is similar to the result obtained for the modified pencil beam pattern with index n>1 discussed in Sect. 3.5.2.
When the threshold T is increased, the probability to have doublepeaked light curves falls and becomes almost independent of the bending angle and anisotropy parameter h.
4 Results
Based on our classification of pulsars into single and doublepeaked in Sect. 2, we can now compare the corresponding fractions with the predictions of various radiation models.
4.1 Blackbody emission pattern and comparison to B03
Let us consider first the blackbody radiation pattern and assume that secondary pulses are fully detectable (i.e. T=0). If the magnetic inclination is unconstrained (i.e. ), taking p=p_{0} (Eq. (1)) we get the maximum deflection angle (1 error), if we include Class IV pulsars into consideration, or if we do not, see upper curves in Fig. 3. As the effect of excluding Class IV pulsars is small, we take below that Class into consideration. If we assume that there is an upper limit of magnetic inclination , we get that (at 95% confidence) for any deflection angle, and for the Newtonian case (i.e. ) we have . In general, the probability to observe a doublepeaked profile was found to be given by Eq. (23), and the corresponding constraints on both and are shown Fig. 9. For typical neutron stars with , we have (at 95% confidence). Thus our results favor not very compact neutron stars (with radius of above 12 km for ) and nearly random magnetic dipole inclinations. Figure 10 shows various neutron star massradius relations and our best constraint corresponding to .
Figure 9: Constraints on the maximum magnetic inclination and maximum deflection angle for the blackbody pattern and full detectability of the secondary pulse. The solid line gives the most probable values and the dashed lines bound the 95% confidence region. 

Open with DEXTER 
Figure 10: Different massradius relations of neutron and strange stars equation of states (curves 17 and SS, see Suleimanov & Poutanen 2006) and the obtained constraints on the compactness assuming a blackbody emission pattern and unconstrained magnetic inclination. The most probable value for the compactness ( ) is shown by the solid red line and the 95% confidence region is bound by red dotted lines. The dotdashedline shows the most probable value for the compactness (corresponding to ) obtained for the detection threshold T=0.2. The blue dashed line corresponds to the upper limits on mass as a function of radius obtained by B03. 

Open with DEXTER 
Let us now look at the effect of the detection threshold which reduces the probability to observe doublepeaked profiles (see Fig. 8a). For unconstrained magnetic inclination, we evaluate the probability to see doublepeaked profiles by numerically computing the area on the  plane where inequality (38) is satisfied. Compared to the case T=0, the maximum bending angle for T=0.2 decreases by just a few degrees to , which (for fixed M) would favor even larger neutron star radii (see Fig. 10).
A similar study was made by B03, who obtained a different result for a number of reasons. Although our classification of Xray pulsar light curves differs from B03 (we have 60 doublepeaked profiles out of 124, while B03 have 38/88), the probabilities p_{0} are similar within a few percent. However, B03 assumed that singlepeaked light curves would be observed only if one magnetic pole is seen (Class I). This led to the conclusion that the probability of having singlepeaked profiles is about 21% for and this value decreases as compactness grows. Therefore in their point of view there is a conflict between the theory and observations, because the percentage of observed singlepeaked light curves is much higher. As a result, B03 needed to constrain the magnetic field geometry (by introducing maximum magnetic inclination) in order to attain consistency between the observations and the theory. In our view, Class III is the only place to obtain doublepeaked light curves. Therefore, we get (and for reasonable value of the compactness the magnetic dipole inclination is random), while B03 got (compare our Fig. 9 to Fig. 7 in B03).
4.2 Modified pencilbeams and implications for neutron star compactness and magnetic field geometry
The pulsars do not shine as blackbodies and therefore we now take a look at the effect of changing the emission pattern. For the beam pattern, the probability of having doublepeaked profiles is given by Eq. (33). If magnetic inclination is not constrained, , which significantly exceeds the observed fraction. Varying , we can get an agreement between the observations and the model. Requiring (see Fig. 6), we get , while if we account only for pulsars observed at energies above 10 keV and take , then . We note here that the probability to observe doublepeaked profiles (33) does not depend on the deflection angle and thus we cannot get any constraints on the compactness of the neutron star (for a full detectability T=0). Other pencil beam models, and , give identical results. Thus even if the parameters describing the beam patterns of various NS have a large spread, the results are not affected.
The detection threshold may play an important role in classifying the pulse profiles. The expectation value for the fraction of the doublepeaked profiles becomes lower and makes it rather close to the observed one for T=0.2, even if deviations from the blackbody pattern are not very large (see Fig. 8). Taking n=2, the predicted fraction of doublepeaked profiles is about 0.57, which is within 3 of the observed value p_{0}=0.44 and within 1 of the value p_{1}=0.55 obtained for pulsars observed above 10 keV (see Table 1). Increasing n makes the agreement even better. Again we stress that the predicted fraction depends only weakly on the neutron star compactness, which therefore cannot be constrained. Another conclusion is that the magnetic dipole inclination can be random. These results are not affected by our assumptions about the emission pattern if it is sufficiently far from the blackbody, because indices n in the interval between about 1.5 and 6 and the anisotropy parameter h>0.5 predict similar fractions of doublepeaked profiles (see Fig. 8).
The observed light curves are not always symmetric indicating the secondary pole is not at the antipodal position. To check how our assumption on antipodal positions of the spots affect the results, we have simulated the pulse profiles for two randomly positioned spots on the neutron star surface radiating as blackbodies. The probability to observe multiplepeaked light curves depends on the compactness of the neutron star reaching the maximum of 37% in the Newtonian limit (see the dasheddotted curve in Fig. 8a). This is much less than what is observed (between 44% and 55%). A significant predicted fraction of multiplepeaked profiles (with number of peaks three or larger) even for the blackbody pattern contradicts the fact that such profiles are not observed. Other, more beamed patterns will just increase the fraction of multiplepeaked profiles. Thus even though the observed light curves indicate slight asymmetry, the simulations show that the spot positions cannot be completely random. This is consistent with the detailed models of the pulse profiles in several pulsars that show less than 10 displacement of the dipole (Kraus et al. 1996; Leahy & Li 1995). These small displacements will not affect our conclusions.
5 Conclusions
 1.
 We have collected pulse profiles of 124 Xray pulsars and magnetars and classified them according to the number of pulses visible in one rotational period. At energies above 10 keV, where the effects of photoelectric absorption and cyclotron line are minimal, 55% of the pulsars have doublepeaked profiles, while for all pulsars this fraction is 44%.
 2.
 We considered a simple model with two pointlike antipodal spots emitting radiation according to different types of pencilbeam patterns (where ) and (with ). The light curves produced by these cases are either single or doublepeaked depending on the model parameters.
 3.
 We obtained some constraints on the radiation model parameters in the blackbody case, n=1 (h=0). The relative fraction of doublepeaked profiles here depends on the neutron star compactness. The most probable values for the maximal gravitational light deflection angle is (for unconstrained magnetic field, and with a full detectability of the secondary pulse). We also obtained a lower limit on the maximum magnetic inclination to be , and for reasonable NS compactnesses we have . When we include the effect of the detection threshold, the limit on the compactness is reduced to (at 95% confidence) for T=0.2.
 4.
 Any pencilbeam pattern (if not a blackbody) predicts a fixed fraction of doublepeaked profiles of 79%, which is inconsistent with the data. Restricting the maximum magnetic inclination reduces the fraction of doublepeaked profiles. Comparison to the data gives us the most probable value for the maximum magnetic inclination of . The neutron star compactness, however, cannot be constrained at all.
 5.
 A limited detection sensitivity to weak pulses also reduces the fraction of doublepeaked profiles. We found that this fraction depends weakly on the neutron star compactness and is consistent with the data for a large range of pencilbeam patterns at . In this case, we do not find good evidence that the magnetic inclination has a strict upper limit.
 6.
 The overall conclusion is that contrary to the previous claims made by B03, the statistical method based on the classification of pulsar profiles by number of peaks cannot constrain the compactness of the neutron star. We also do not find univocal evidences in favor of the alignment of the magnetic dipole. It seems that the detailed analysis of the pulse profiles of individual pulsars and their evolution is the only way to obtain any useful constraints on the neutron star compactness and the magnetic field geometry.
This research was supported by Space institute, University of Oulu, and the Väisäla foundation (AM). J.P. acknowledges support from the Academy of Finland grant 110792. We also acknowledge the support of the International Space Science Institute (Bern, Switzerland), where part of this investigation was carried out. We thank Alexander Lutovinov and the referee for the valuable comments.
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All Tables
Table 1: Light curve classification of 60 Xray pulsars observed atenergies above 10 keV.
Table 2: Light curve classification of 64 Xray pulsars observed atenergies below 10 keV.
All Figures
Figure 1: Light emitted from a neutron star at an angle to the normal is observed at impact parameter b, with the direction making angle to the spot position vector. The picture is in the plane of photon trajectory. 

Open with DEXTER  
In the text 
Figure 2: Beloborodov's classes of pulsars on the plane ( ). The visibility classes IIV are shown for a moderate light bending . If the magnetic inclination is constrained by , the area below the dashed line is forbidden. 

Open with DEXTER  
In the text 
Figure 3: Probability to obtain doublepeaked light curves for blackbody spots as a function of the maximal gravitational light deflection angle . Solid curves represent the probability given by Eq. (23) when class IV pulsars are included. The dashed curves represent the case where Class IV pulsars are excluded, given by Eq. (24). The curves correspond to three different cases of the maximum magnetic inclination (i.e. unconstrained), and . The upper curves are given by analytical relations (20) and (18). 

Open with DEXTER  
In the text 
Figure 4: Light curve classes for beam pattern on the i plane ( left) and the  plane ( right). The shaded areas correspond to the doublepeaked light curves. The number of light curve classes for this pattern differs from the blackbody case. The small filled circles at the left panel correspond to the parameter pairs () of the light curves plotted in Fig. 5. 

Open with DEXTER  
In the text 
Figure 5: Light curve classes for beam pattern in the moderate case of light bending . The light curves are plotted with following parameters: and (class I), and (class III), and (class IVa), and (Class IVb), and (class IIa), and (class IIb). 

Open with DEXTER  
In the text 
Figure 6: Probability to observe doublepeaked light curves as a function of the maximum magnetic inclination . The solid curve corresponds to any modified pencil beam patterns discussed in Sect. 3.4, and (Eq. (33)) in the case of full detectability of the secondary pulse, T=0. The dashed curve shows an example of the beam pattern with nonzero threshold T=0.2 and a moderate bending angle . The dotted lines correspond to the observed probability of having doublepeaked profiles as given by the classification of data p_{0}=0.44 and p_{1}=0.55 (see Sect. 2). 

Open with DEXTER  
In the text 
Figure 7: Effect of the detection threshold on the fraction of the doublepeaked profiles. a) The parameter space at the  plane for the blackbody emission pattern. Upper curves are for the Newtonian case and the lower curves are for the moderate light bending . The solid curves correspond to the zero threshold and the dashed curves are for T=0.2. The areas below respective curves correspond to the probability of observing doublepeaked profiles. b) Same as a), but for the pencilbeam . For the zero threshold the curve is the same for any neutron star compactness. The curves for thresholds T= 0.1 and 0.2 are shown by dotted and dashed curves, respectively. The dotted curves are for , while the dashed ones are for . Even then the dependence on compactness is very weak. 

Open with DEXTER  
In the text 
Figure 8: a) Probability to observe doublepeaked profiles for the blackbody emission pattern (n=1) as well as for beam patterns with indices n=1.1, 2, 4 and 6 for two different threshold cases T=0 and 0.2. The dashed curves correspond to T=0, while the solid curves are for T=0.2. The dotted lines correspond to the observed probability of having doublepeaked profiles as given by the classification of data p_{0}=0.44 and p_{1}=0.55 (see Sect. 2). The dashdotted line correspond to the probability to have multiplepeaked light curves for two randomly positioned spots radiating as black bodies. b) Same as panel a), but for beam patterns in the Eddington approximation. Cases with anisotropy parameter h=0, 0.5, 2 and 4 for two different thresholds T=0 and 0.2 are shown. 

Open with DEXTER  
In the text 
Figure 9: Constraints on the maximum magnetic inclination and maximum deflection angle for the blackbody pattern and full detectability of the secondary pulse. The solid line gives the most probable values and the dashed lines bound the 95% confidence region. 

Open with DEXTER  
In the text 
Figure 10: Different massradius relations of neutron and strange stars equation of states (curves 17 and SS, see Suleimanov & Poutanen 2006) and the obtained constraints on the compactness assuming a blackbody emission pattern and unconstrained magnetic inclination. The most probable value for the compactness ( ) is shown by the solid red line and the 95% confidence region is bound by red dotted lines. The dotdashedline shows the most probable value for the compactness (corresponding to ) obtained for the detection threshold T=0.2. The blue dashed line corresponds to the upper limits on mass as a function of radius obtained by B03. 

Open with DEXTER  
In the text 
Copyright ESO 2010
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