Constraining compactness and magnetic field geometry of X-ray pulsars from the statistics of their pulse profiles

The light curves observed from X-ray pulsars and magnetars reflect the radiation emission pattern, the geometry of the magnetic field, and the neutron star compactness. We study the statistics of X-ray pulse profiles in order to constrain the neutron star compactness and the magnetic field geometry. We collect the data for 124 X-ray pulsars, which are mainly in high-mass X-ray 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 double-peaked. 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 point-like spots that emit radiation according to the pencil-like emission patterns. These are then compared to the observed fraction of pulsars in the two classes. 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 pencil-beam patterns predict that 79% of the pulsars are double-peaked 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 deg. We also discuss the effect of limited sensitivity of the X-ray instruments to detect weak pulses, which lowers the number of detected double-peaked 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), the data also do not require the magnetic inclination to be confined in a narrow interval.

💡 Research Summary

The authors set out to test whether the statistical distribution of X‑ray pulse profiles can be used to constrain two fundamental properties of neutron stars: their compactness (mass‑to‑radius ratio) and the geometry of their magnetic field. They assembled a sample of 124 X‑ray pulsars, the majority of which are high‑mass X‑ray binaries, and classified each light curve according to the number of distinct peaks observed within a single rotation. The sample splits almost evenly into single‑peaked and double‑peaked profiles, a result that is not trivially expected from simple geometric arguments.

To interpret these numbers they adopt the classic two‑antipodal‑spot model, in which the star’s surface hosts two point‑like hot spots located at the magnetic poles. The emission from each spot is assumed to follow a “pencil‑like” beaming pattern. Two specific beaming prescriptions are examined. The first is an isotropic blackbody pattern, which represents the limiting case of a wide‑angle emission. The second is a more realistic narrow pencil beam, typical of the fan‑beam or pencil‑beam components seen in many accretion‑powered pulsars.

For the isotropic case the visibility of each spot depends sensitively on the star’s compactness because gravitational light‑bending widens the visible region. By varying the compactness, the magnetic dipole inclination (α) relative to the rotation axis, and the observer’s inclination (ζ), the authors compute the probability that both spots are seen during a rotation, i.e., that the light curve is double‑peaked. In this scenario the observed 50 % double‑peak fraction can be reproduced for a range of compactness values, provided that the magnetic dipole can assume any inclination. Thus, in principle, the pulse‑profile statistics could place limits on the compactness.

When the narrow pencil‑beam pattern is used, the situation changes dramatically. Because the beam is confined to a small solid angle, the probability of seeing both beams is largely independent of compactness and is instead set by the geometry of the magnetic axis. The model predicts that about 79 % of randomly oriented pulsars should appear double‑peaked, a figure that is far higher than the observed 50 %. To reconcile theory with observation the authors find that the magnetic dipole inclination must be limited to α ≤ 40° ± 4°. In other words, the magnetic axis would need to be relatively well aligned with the spin axis.

However, the authors also consider the effect of instrumental sensitivity. Real X‑ray detectors have finite signal‑to‑noise ratios, and a weak secondary peak can be missed, causing a genuinely double‑peaked source to be classified as single‑peaked. By incorporating a realistic detection threshold into their simulations, they show that the apparent double‑peak fraction can be reduced from the theoretical 79 % to the observed 50 % even when the magnetic dipole orientations are completely random. This demonstrates that the apparent discrepancy can be explained without invoking any special alignment of the magnetic field.

The key conclusions are therefore twofold. First, the simple statistics of pulse‑profile shapes do not provide a robust constraint on neutron‑star compactness because the result is highly degenerate with assumptions about the beaming pattern and observational sensitivity. Second, contrary to earlier claims (Bulik et al. 2003), the data do not require the magnetic dipole inclination to be confined to a narrow range; the observed distribution can be reproduced either by imposing an artificial alignment (α ≤ 40°) or by acknowledging the limited detectability of weak secondary peaks.

The paper highlights the importance of accounting for instrumental selection effects and of using realistic emission models when attempting to infer fundamental neutron‑star parameters from population‑level pulse‑profile studies. Future work that combines high‑resolution timing, phase‑resolved spectroscopy, and multi‑wavelength observations will be needed to break the degeneracies identified here and to obtain meaningful constraints on the equation of state and magnetic geometry of neutron stars.