A&A 404, 1023-1032 (2003)
DOI: 10.1051/0004-6361:20030555
T. Bulik 1 - D. Gondek-Rosinska 2,1 - A. Santangelo 3 - T. Mihara 4 - M. Finger5 - M. Cemeljic6
1 - Nicolaus Copernicus Astronomical Center, Bartycka 18, 00716 Warsaw, Poland
2 - LUTH, Observatoire de Paris, Place Jules Janssen, 92195 Meudon Cedex, France
3 - IFCAI-CNR, via Ugo La Malfa 153, 90146 Palermo, Italy
4 - RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
5 - NSSTC, 320 Sparkman Dr., Hunstville, AL 35805, USA
6 - AIP, an der Sternwarte 16, 14482 Potsdam, Germany
Received 19 November 2002 / Accepted 25 March 2003
Abstract
Accretion powered pulsars exhibit a variety of lightcurves.
In this paper we propose to classify the observed lightcurves as
single or double pulsed. We analyze the lightcurves of 89 accretion powered
pulsars and assign them to these classes. We present three datasets:
first in which the classification can be easily done, second for which
the classification is more difficult and not certain, and
third for which we were unable to classify the pulsar because of a lack
of published data. We analyze a simple model in which the angles between the magnetic
and rotation
axis ,
and between the rotation axis and the line of sight
are random,
and show that it is inconsistent with the data. We also present a model in which
the angle between the magnetic axis and the rotation axis is restricted
and compare it with the data. This leads to an upper
limit on
the angle
.
We conclude that there must be
a mechanism that leads to alignment of the magnetic and spin axis
in X-ray pulsars.
Key words: X-rays: star - stars: neutron
Accreting neutron stars were discovered more than 30 years ago (Shklovsky 1967), with Cen X-3 beeing the first one discovered that showed pulsations (Giacconi et al. 1971). Currently we know nearly one hundred accreting neutron stars, and in more than eighty of them pulsations were identified (Liu et al. 2001,2000). Accreting neutron stars in binaries exhibit a wide range of X-ray light curves. They vary as a function of the photon energy, and moreover in the transient sources the pulse shapes change with the variation of the luminosity.
The pulse period in accreting sources is identified with the rotation of a magnetized star. As the matter from the companion star falls onto the neutron star it is channeled onto the magnetic poles by the strong magnetic field of the neutron star. Thus the polar caps and/or accretion columns are the places where most of the emission takes place. Several theoretical models of radiation of magnetized accretion powered neutron stars have been proposed and the beam shape is usually described in terms of pencil beams, when most of the radiation is emitted along the magnetic field, or fan beams when most of the radiation is emitted perpendicularly to the magnetic field.
The magnetized radiative transfer is solved using a difference scheme (Meszaros & Nagel 1985; Bulik et al. 1992), or using Monte Carlo scheme (Lamb et al. 1990; Isenberg et al. 1998), for a review see Meszaros (1992). In the pencil beam model radiation from each polar cap produces one pulse in the light curve. Depending on the emission cap physics and the strength of the magnetic field each pulse may have some additional structure. When during the rotation of an accreting pulsar we see two polar caps the lightcurve should exhibit two distinct pulses (peaks), and if only one cap is seen then the lightcurve is single peaked. In the case of dominant radiation from the accretion column (fan beam model) the shape of the lightcurve will depend on the geometry and on the shape of the beam. Even with simple beam shapes there can be triple or quadruple peaked lightcurves.
Each pulsar can be classified according to the number of peaks in the lightcurve depending on the location of the observer and the inclination between the rotation and the magnetic axes. We note that the observed pulsar lightcurves can be classified into two groups: the single pulsed light curves and the double pulsed light curves. At low X-ray energies this distinction may not be clear, however in the high energy band we clearly distinguish single or double pulses, and there are no triple (or multiple) pulsed lightcurves, see e.g. Bildsten et al. (1997); Nagase (1989). By high energy band we mean at least above 10 keV, or in the pulsars where cyclotron lines were found, above the cyclotron lines.
The paper is organized as follows: in Sect. 2 we summarize the observations, in Sect. 3 we analyze the expected shapes and expected fraction of single and double peaked lightcurves, in Sect. 4 we discuss the results. We summarize the conclusions in Sect. 5.
In this section we review the existing observations of X-ray
pulsars and attempt to classify their lightcurves according to
the number of pulses in their lightcurves. We summarize our
classification in Tables 1, 2, and 3. For each pulsar we have
searched the literature for X-ray observations, and verified if
the pulsar was observed in the band above 10 keV, or above the
cyclotron line energy. The pulsars with such observations are
listed in Table 1. There are 46 entries in the Table
out of which 20 have been classified as double and 26 as single.
Additionally we have listed 31 pulsars in Table 2
for which the classification was difficult. For some of the
pulsars no lightcurve above 10 keV was available, we have
also included here three pulsars with accretion flow eclipses
(Cemeljic & Bulik 1998) that is pulsar with the light
curves exhibiting sudden dips. Such dips can be interpreted as
accretion flow eclipses, i.e. be caused by the absorption of the
X-ray flux when the accretion column passes the line of sight.
Such phenomena have been noted for at least three objects:
A0535-262 (Cemeljic & Bulik 1998), GX1+4
(Giles et al. 2000), and RX J0812.4-3114 (LS992)
(Reig & Roche 1999b). We classified them conservatively
as double, although we rather lean toward the interpretation
of single peaks with an accretion flow eclipse. There are 7
pulsars with likely double peaked light curves and 24 with
single in Table 2. We see that in each table there
are more single pulsed than double pulsed objects. For
completion we also list 13 pulsars in Table 3 for
which no information on the type of the lightcurve was
available. We define three sets of data corresponding to the
three tables: dataset
with the data of
Table 1, dataset
with the combined data
of Tables 1 and 2, and dataset
with all the data, and the entries of Table 3
are assumed to be double peaked.
Table 1: X-ray pulsars in group one, i.e. these that are easily classifiable.
Table 2: X-ray pulsars in group two, i.e. with no pulse shape above 10 keV or were difficult to classify.
Table 3: X-ray pulsar in group three, for which no reliable pulse shapes were found.
Let us assume that an accreting neutron star has a pure centered
dipole field. In such a case the geometry of accretion powered
pulsars is characterized by two angles:
- the angle
between the rotation axis and direction to the observer, and
the angle between the magnetic and rotation axes.
Within the pencil beam model the
parameter space when only one cap is seen is determined by the
condition:
![]() |
(1) |
The angles
and
also determine the number
of pulses in the lightcurve within the fan beam model. However here the number
of pulses seen depends also on the shape of the beam.
In a simple case when there are two columns
radiating with constant specific intensity the beam will
be a function only of the angle
between the magnetic axis
and the direction to the observer:
.
Interestingly, the parameter space for seeing two peaked
lightcurves is identical to the case of the pencil
beam model discussed above, the only difference being that
for the given angles
and
the maxima of the
lightcurve fall at different phases. Introducing
a more general beam pattern e.g.
leads to dominance of multiple peaked lightcurves:
in this case we expect 4.5% single peaked, 27.8%
double peaked, 40.1% triple peaked, and 27.6% quadruple
peaked lightcurves.
![]() |
Figure 1:
Parameter space for viewing accreting X-ray pulsars.
The region below the solid line corresponds to
visibility of two polar caps with gravitational light bending
neglected. The region below the dashed line
corresponds to visibility of two caps with the gravitational
light deflection
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For a set of data, like pulsars, with a well defined
property, i.e. either single or double pulsed, and a
given probability p that each pulsar will belong to one or other class,
the probability of observing n single peaked lightcurves and
m double peaked in an observation of randomly chosen
n+m objects is given by the binomial distribution:
![]() |
(2) |
![]() |
Figure 2: The probability of classifying a pulsar as single peaked as a function of the threshold below which the second peak is not detectable. |
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The real detectability of the second peak depends on the
exact shape of the beam and on the strength of the signal.
The second peak may be a small bump in the pulse shape
and when the signal to noise is low it might be undetected.
General modeling of second peak detectability is difficult
because it must include the unknown beam shape and the distribution
of signal to noise ratios. Here we present a simple model in which
we assume that we can detect the second peak if its amplitude
is at least a certain fraction of the main peak. Furthermore
we assume that the beam flux is described either
by
,
or
,
where
is the cosine
of the angle between the normal and the line of sight, and that the pulsar
is detectable from every direction. For these two
beam shapes we calculate the pulse shapes as a function of the
angles
and
.
For each pulse shape we calculate R2 the ratio of the magnitude of the
second peak in relation to the main one.
We then define a detection threshold T for the second peak:
if R2> T a pulse shape is classified as a double and otherwise
as a single.
We calculate the probability of classifying
a pulsar as single pulsed as a function
of the detection threshold T for the two assumed beam shapes
and plot the results in
Fig. 2. The probability of classifying a pulsar as single peaked
increases with increasing the threshold T, the case
T=0 corresponds to the ideal case when even the smallest
second peaks are detectable.
When T=0.1 the value of
reaches
for the
beam and
for the
beam.
We would have classified the pulse shapes as double
peaked with the second peak as small as a few percent of the
main peak amplitude. We take the
case T=0.1, together with the
beam
i.e.
and find that
the probabilities of obtaining the observed datasets
are
for the
,
for
,
and
for
.
We note that we have assumed that a pulsar is detectable from
all directions, while for strongly beamed cases there is a
fraction of the parameter space where a
pulsar is undetectable because of small flux.
In the calculations below we consider the model
with T=0.
Including the effects of the gravitational light deflection
only reduces the probability of seeing one pulsed objects and makes
the values of the above probabilities smaller.
We present the probability of seeing one polar cap as a function of
in the left panel of Fig. 3.
Another possibility to consider is that the pulsar beam is not wide
but strongly collimated, and the beam has an opening angle
of
.
In this case for a given
there
is only a small range of angles
for which the pulsar is
visible at all. The restriction on the parameter space for small
is shown in Fig. 1 as the shaded region.
The probability of seeing only one accretion cap as a function of the
angle
is shown in the right panel of Fig. 3.
Another possibility to consider is to allow for non
aligned dipole, i.e. to assume that the two polar caps
are not antipodal but can be placed anywhere on
the stellar surface. A straightforward calculation neglecting the
gravitational light bending effects shows
that the probability of seeing just one polar cap drops
in this case to 17%.
Thus the probabilities computed above
are conservative
and inclusion of the gravitational light bending and beaming
make the result even more significant.
![]() |
Figure 3:
Left panel: the probability of seeing only one polar cap
as a function of the maximal gravitational light deflection. The non-relativistic
case corresponds to
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Figure 4:
The probability of seeing only one polar cap
as a function of the maximal gravitational light deflection when the beam is described
by the from F2. The non-relativistic
case corresponds to
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The effects of gravitational light deflection in the case
of fan beams do not alter the results.
We calculate the beam shape modified by the gravitational light bending using
the approximate formulae of Beloborodov (2002).
For the simple
fan beam F1 the probability of seeing the single or double
peaked lightcurves does not depend on
.
This is due to the fact
that gravitational light bending does not change qualitatively
the shape of the beam: it has a minimum along the magnetic axis and
a single maximum in the direction perpendicular to the magnetic axis.
We present the probability of seeing a single pulsed lightcurve
for the fan beam of the form F2 in Fig. 4.
Here the shape of the beam depends strongly on the value of
.
For small values of
the probability
of seeing single pulsed lightcurves slightly decreases, and then it rises to
0.21 at
.
At this point the gravitational
light bending causes the two beams from the two
columns to merge and the beam shape is qualitatively the same
as for the case F1.
![]() |
Figure 5:
The probability of observing the data
in models pencil beam parameterized by the the maximum angle
between the magnetic and rotation axis
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Figure 6:
The probability of observing the data
in fan beam models parameterized by the the maximum angle
between the magnetic and rotation axis
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Figure 7: Each panel presents the region containing the models that are consistent with the data at the level of 60%, 90%, and 99%. The three panels correspond to the three datasets. |
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A more general model taking into
account the general relativistic effects includes two parameters: the
maximal angle between the magnetic and rotation axis
and the maximum gravitational light deflection angle
.
In the framework of the pencil beam model we compute the probability
of seeing a pulsar with only one polar cap as a function of these two angles.
In Fig. 7 we present the
regions containing the models that are consistent with
the data with the probability higher than 40%, 10%, and 1%.
The non relativistic results correspond to
the line
in Fig. 7.
Since the probability of seeing only one polar cap decreases
with increasing
,
we also set a limit
on its maximum value. The region corresponding to the
1% agreement limit for the dataset
can
be expressed as
.
Thus if there were no gravitational light deflection
effects we find that the inclination angle of the magnetic
axis to the rotation axis should be smaller than
.
For the case with a moderate gravitational light bending
the magnetic axis inclination must
be smaller than
.
We also
see that the amount of gravitational light bending
cannot exceed
at the 99% confidence level.
The limits are even stronger when we take into account
dataset
,
however for the set
the constraints are similar to these obtained
for the dataset
.
![]() |
Figure 8:
The value of the gravitational light bending as a
function of the emission angle, measured from the radial direction
for photons emitted at different initial radii. The
maximum gravitational light deflection is the value of deflection
for the initial angle of ![]() |
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![]() |
Figure 9: The ratio of the stellar radius to the gravitational radius as a function of mass for several neutron star equations of state. The labels are BPAL12 - Prakash et al. (1997), SBD - Sahu et al. (1993), BB1 and BB2 - Baldo et al. (1997), while SS1 and SS2 correspond to the MIT Bag model of strange quark matter (Witten 1984) with two different densities at zero pressure. |
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To discuss the limit on
obtained above
we will consider two cases: (i) emission from a polar
cap, and (ii) emission from the accretion column. In the first
case the upper limit on the
obtained above
can be interpreted as a limit on the compactness
of the star. We plot the amount of gravitational
light deflection as a function of the emission angle
for a few values of the ratio of the neutron star radius to the
gravitational radius
in Fig. 8.
The maximal gravitational light deflection of about
corresponds to the case with ratio
for the
photons emitted parallel to the surface.
The limit on
can be understood as the upper
limit on the compactness of the neutron stars.
In Fig. 9 we plot the ratio
as a function of
the stellar gravitational mass for a few representative equations
of state. The ratio
decreases with increasing mass.
The upper limit on the ratio
corresponds
for a given equation of state to the lower limit
on the mass of the accreting neutron stars. Masses
of neutron stars in some accreting objects have been
estimated from the observations of binary motion,
e.g. in the case of 4U1538-52
(Pakull et al. 1983; Reynolds et al. 1992) and they all seem
to be consistent with the fiducial
with
some uncertainty. The general consensus seems to be
that the masses of all neutron stars are consistent with
the mean value of
1.4
,
and this is also
the value found for the binary pulsars
(Thorsett & Chakrabarty 1999),
however in LMXBs this range may be much wider
and reach above
(Bulik et al. 2000).
If the neutron stars have masses around
and
the beams are wide
then our results point to the
very stiff equations of state e.g. SDB (Sahu et al. 1993).
A second possibility is that we see the emission from the
accretion shock above the surface of the neutron star with the
typical properties of the fan beam and that this emission is
responsible for the observed flux at angles above 90with respect to the normal. Our considerations above imply
that the emission is visible up to the angles exceeding the
perpendicular to the normal by
.
In the
non-relativistic case this means that such shocks can lie no
higher than
above the surface of the neutron
star. If the effects of the gravitational light bending around a
neutron star are included the maximum height allowed is even
smaller.
We have obtained
a limit on the possible inclination angles of the magnetic axis
with respect to the rotation axis
.
A hint of such a distribution was already shown by
Leahy (1991). He analyzed the lightcurves
of 20 pulsars using a simple model with the beam dependence on the
emission angle as
and
where
is the direction to the normal of the surface.
For each pulsar he found the angles
and
,
however the fits were degenerate in these angles, so that
it was not clear a priori which angle is which. The resulting
distribution was inconsistent with the assumption that both
angles are random, and Leahy (1991) concluded
that the mean angle between the magnetic and rotation axis
is as small
.
Inclusion of the gravitational
light bending effects (Leahy & Li 1995)
does not alter this conclusion.
Bulik et al. (1995)
modeled lightcurves of 4U 1538-52 and Vela X-1 observed by Ginga,
and obtained the inclination angles below
for their
fits with best significance. Kraus et al. (1996) analyzed the
pulse profiles of Cen X-3 and reconstructed the geometry and
beam pattern for this object. They found that the most likely
inclination angles of the magnetic axis to the rotation axis
are
.
An analysis of
the pulse shape of Her X-1 (Blum & Kraus 2000)
also lead to a constraint on the angle between the magnetic and
rotation axis
.
We have mentioned in Sect. 2 three special objects
A0535-262, GX1+4,and RX J0812.4-3114 for which accretion flow
eclipses have been reported (possibly this list also includes
GS1843-24, Finger et al. 1999, and XTE J1946+274, Wilson et al. 2003).
If such eclipses happen when the accretion column passes through
the line of sight then they should take place when
,
where
is the angular width of the
accretion column. The region corresponding to
is shaded in Fig. 1. We see that such
eclipses should frequently take place in double pulsed light
curves, if the magnetic axis is randomly oriented with respect
to the rotation axis. Moreover within the single pulsed light
curves a significant fraction, - approximately 20-30%
should exhibit such dips, and this fraction increases as the
amount of gravitational light bending increases.
Yet the only three cases when such dips have been found are in
single pulsed light curves. The lightcurve of A0535-262 seems
to be double peaked, however Cemeljic & Bulik (1998) pointed
out that it can be interpreted as a single pulse divided by the
accretion flow eclipse. This lack of double peaked pulsars with
accretion flow eclipses (dips) may mean that the region of
parameter space for double peaked light curves with dips is
either empty or scarcely populated! Again we are drawn to the
conclusion that the magnetic axis needs to be aligned with
the rotation axis. If the magnetic axis inclination is restricted
to the region
,
then one expects that the
accretion flow eclipses to be observed only in the
one pulsed lightcurves, as is the case. While this by itself is
not a significant result, the identification of accretion flow
eclipses is consistent with the conclusion that
the range of allowed values of
is limited.
One should mention here that there
may possibly be a selection bias against seeing double pulsed objects.
In the case of nearly perpendicular rotators the two
pulses will be similar. It is therefore possible that
the lack of double pulsed objects is due to misidentification of the
pulse period which should be two times longer than reported.
The fraction of such misidentified objects depends on the
exact shape of the beam. For wide uncollimated beams
the two pulses look similar for a range of angles as large as the width of the beams. However, our limit
is
,
so the beams would have to be very
wide. Additionally, the beam shape
would have to be very fine tuned in order to compensate for the viewing
geometry effects over such a large range of angles.
Another possible selection effect against seeing double
peaked pulsars was shown by Beloborodov (2002).
Assuming that both accretion caps lie
exactly opposite one another on the surface of the neutron star
and are identical,and
that the radiation beam is well described by
(or intensity is independent of the angle of emission),
he shows that the pulse shapes are flat in the region where both polar
caps are visible due to gravitational light bending.
This decreases the parameter space for seeing double peaked
pulsars. However detailed modeling of individual objects
tends to show that the above assumptions are not satisfied,
the data usually requires non antipodal caps, complicated beam patterns,
and sometimes differences between the two polar caps
(Kraus et al. 1996; Leahy & Li 1995; Blum & Kraus 2000; Bulik et al. 1995).
The sample of accreting pulsars presented here is dominated by transient
Be/X-ray systems. One should note that for the disk-fed persistent
sources (Her X-1, SMC X-1, SMC X-2, Cen X-3, LMC X-4, RX J0648.1-4419) there
are three double peaked objects out of six sources, still
below the expected 79 percent,
but this is not a significant discrepancy.
In the case of wind-fed persistent sources
(Vela X-1, 1E 1145.1-6141, 4U1907+09, 4U1583-53, GX 301-2, 2S 0114+650)
there are four double peaked out of six objects.
However, because of the small number of these sources it is
impossible to obtain any significant conclusion just from
a statistical analysis and, as we already mentioned above,
detailed modeling of two wind-fed pulsars (4U 1538-52 and Vela X-1)
and one disk-fed (Cen X-3) resulted in low values of
.
We analyzed the shapes of accretion powered pulsar light
curves, looking for the expected fraction of single and double
peaked light curves, as well as these exhibiting accretion flow
eclipses. We find that the expected number of single peaked
lightcurves is small; certainly less than about one in five
lightcurves should be single peaked
both in the framework of the pencil beam model
and a simple fan beam model. A general fan beam model
predicts predominantly triple and quadruple peaked
pulse shapes and only a small percentage of single
peaked lightcurves.
On the other hand the
observations show that quite a large fraction of
accreting pulsars have single peaked lightcurves in hard X-rays
with the rest being double peaked.
In order to explain this discrepancy between
theory and observations
we are drawn to conclude that the
magnetic axes in accretion-powered pulsars tend to be aligned
with the rotation axis. Such alignment decreases the expected
number of double peaked lightcurves, as seen in
Fig. 1. We have compared the models
parametrized by the maximal inclination of the magnetic to the
rotation axis
and the value of the
maximum gravitational light deflection
with the
data. We obtain upper limits at the
99% confidence level
and
.
Our results are based on the classification of the neutron star lightcurves as single or double pulsed. In order to avoid possible misclassification we have attempted to use the data above 10 keV, or above the cyclotron line wherever possible. Moreover, we have attempted to classify as many pulsars as double when we were in doubt. In order to rid ourselves of the possible systematic effects we have repeated the analysis for the three datasets, and in the third dataset we included all the unclassifiable pulsars as double peaked objects to be conservative. The deficit of double pulsed objects is significant for every dataset that we analyzed. Moreover we note that the paucity of double peaked pulsars is so strong that our results still hold even if there are still a few misidentifications in our analysis. We have also shown that if the pulsar beam is strongly collimated, the deficit of double peaked objects becomes more significant.
Acknowledgements
We thank Peter Jonker, Biswajit Paul, and Silvia Zane for helpful comments on the contents of the Tables, and the anonymous referee for helpful comments on this paper. This research has been supported by the KBN grants 5P03D01721 and PBZ-KBN-054/P03/2001, and the EU Programme "Improving the Human Research Potential and the Socio-Economic Knowledge Base'' (Research Training Network Contract HPRN-CT-2000-00137). TB is grateful for the hospitality of the Observatoire de Paris in Meudon and for the support of the CNRS/PAN Jumelage programme.