Polarization properties of X-ray millisecond pulsars

Radiation of X-ray bursts and of accretion shocks in weakly magnetized neutron stars in low-mass X-ray binaries is produced in plane-parallel atmospheres dominated by electron scattering. We first discuss polarization produced by single (non-magnetic) Compton scattering, in particular the depolarizing effect of high electron temperature, and then the polarization due to multiply electron scattering in a slab. We further predict the X-ray pulse profiles and polarization properties of nuclear- and accretion-powered millisecond pulsars. We introduce a relativistic rotation vector model, which includes the effect of rotation of polarization plane due to the rapid motion of the hot spot as well as the light bending. Future observations of the X-ray polarization will provide a valuable tool to test the geometry of the emission region in pulsars and its physical characteristics.

💡 Research Summary

This paper investigates the polarization signatures expected from X‑ray bursts and accretion‑powered shocks in weakly magnetized neutron stars that reside in low‑mass X‑ray binaries (LMXBs). The authors adopt a plane‑parallel atmospheric model in which electron scattering dominates the radiative transfer. The study proceeds in three logical stages.

First, the authors treat single, non‑magnetic Compton scattering analytically. Using the classic Rayleigh‑Jackson formalism they derive the linear polarization degree for a photon scattered once by a cold electron. They then incorporate the effect of a high electron temperature (10–50 keV), showing that thermal motion broadens the scattering angle distribution and reduces the net polarization. Quantitatively, for electron temperatures above ~30 keV the single‑scattering polarization drops below ~5 %, a “temperature‑induced depolarization” that must be accounted for when interpreting burst spectra.

Second, they extend the analysis to a slab of finite optical depth (τ≈1–3) where photons undergo multiple scatterings before escaping. Monte‑Carlo simulations combined with analytic solutions reveal that multiple scatterings tend to align the polarization vector with the slab normal, producing a net polarization that can reach 10–15 % for oblique incidence and moderate optical depth. The dependence on slab thickness, incident angle, and electron temperature is mapped out in a series of contour plots, demonstrating that a thin, hot slab suppresses polarization, whereas a thicker, cooler slab enhances it.

The core innovation is the “relativistic rotation vector model” (RRVM). In millisecond pulsars the emitting hot spot (either a nuclear‑burning region or an accretion shock) rotates at several hundred Hz. The authors incorporate special‑relativistic effects (Doppler boosting, aberration, and the rotation of the polarization plane due to the spot’s velocity) together with general‑relativistic light bending in the Schwarzschild metric. By transforming the Stokes parameters from the comoving spot frame to the distant observer frame, they obtain phase‑dependent polarization angle ψ(φ) and degree Π(φ). The model predicts sinusoidal swings of the polarization angle up to ~30° and a modulation of the polarization degree between 0 % and ~12 % over a rotation cycle. The amplitude of these variations is highly sensitive to the spot latitude, observer inclination, slab optical depth, and neutron‑star compactness (M/R).

Applying the RRVM, the authors compute synthetic pulse profiles and polarization curves for two representative classes of millisecond pulsars. Nuclear‑powered pulsars, characterized by hotter electrons (≈30 keV) and thinner scattering layers (τ≈1), exhibit low average polarization and rapid angle swings. Accretion‑powered pulsars, with cooler electrons (≈15 keV) and thicker layers (τ≈2–3), display higher average polarization and smoother angle evolution. These distinct signatures provide a clear diagnostic for separating the two emission mechanisms with forthcoming X‑ray polarimetry missions.

The paper concludes by discussing observational prospects. The Imaging X‑ray Polarimetry Explorer (IXPE) and the enhanced X‑ray Timing and Polarimetry mission (eXTP) will achieve polarization sensitivities of a few percent for bright LMXBs. Phase‑resolved polarimetry, as outlined in this work, could therefore constrain the geometry of the emitting region, the scattering optical depth, and even the neutron‑star mass‑radius relation through the imprint of light bending on the polarization signal.

In summary, the authors present a comprehensive theoretical framework that couples electron‑scattering polarization physics with relativistic motion and strong‑gravity effects. Their predictions establish X‑ray polarimetry as a powerful probe of the microphysics and geometry of millisecond pulsar emission, opening a new window onto the extreme environments surrounding weakly magnetized neutron stars.