Pulsed, Polarized X-ray Emission from Neutron Star Surfaces: the Effects of Vacuum Birefringence in the Magnetosphere

TL;DR

This work extends Monte Carlo radiative transfer with MAGTHOMSCATT to include vacuum birefringence in neutron-star magnetospheres, enabling a self-consistent treatment of polarization from surface emission through magnetospheric propagation to infinity within general relativity. The authors contrast two polarization-evolution formalisms and derive the recoupling (polarization-limiting) radius, adopting the Heyl–Shaviv prescription for magnetars, to quantify how birefringence preserves or alters surface polarization signatures. Applying the model to magnetar 1RXS J1708-4009 and a low-field CCO RX J0822.0-4300, they constrain emission-region sizes and geometry while showing that VB can significantly boost linear polarization in magnetars and modestly affect lower-field sources; they also discuss the need to incorporate atmospheric vacuum resonance in future work. Overall, the formalism provides a robust framework to interpret phase-resolved X-ray polarization data and to constrain neutron-star geometry and surface emission properties.

Abstract

Intense magnetic fields in the atmospheres of neutron stars render non-trivial angular dependence of intensity and polarization of soft X-ray emission originating from their surfaces. By tracking the complex electric field vector for each photon during its atmospheric transport and propagation in general relativistic and birefringent magnetospheres, our Monte Carlo simulation, named MAGTHOMSCATT, allows for capturing the complete polarization properties, including the intricate interplay between linearity and circularity. The new inclusion in MAGTHOMSCATT of quantum electrodynamical influences on polarization in the magnetosphere is presented. We simulate the pulsed and polarized X-ray emission from the outer layers of optically thick, fully ionized atmospheres of neutron stars, with a focus on the radiation emitted from extended polar caps of magnetars, which are the most highly magnetized neutron stars. Using the recent intensity pulse profile data for the magnetar 1RXS J11708-4009, we constrain the geometric parameters, namely the angles between the magnetic axis and the observer's viewing direction relative to the spin axis, as well as the sizes of emission regions. The distributions of these parameters and the best-fit configuration are provided. In addition, we discuss the important impacts of vacuum birefringence in the magnetosphere on increasing the linear polarization degree. A comparison with the case of a weakly magnetized neutron star, RX J0822.0-4300, is also discussed. Our simulation still needs further development, particularly to incorporate the vacuum resonance effect. Nevertheless, the formalism presented here can be employed to constrain geometric parameters for various types of neutron stars.

Figures (10)

• Figure 1: Recoupling surfaces for photons with an energy of $E_{\gamma} = 2$ keV emitted from magnetar 1RXS J1708-4009 with a polar magnetic field strength of $B_p = 9.3 \times 10^{14}$ G (left panel) and CCO RX J0822.0-4300 with $B_p = 5.7 \times 10^{10}$ G (right panel). In both panels, the NS is positioned at (0, 0, 0). The colorbar on the right of each panel represents the recoupling radius in units of the NS radius, $r_{\rm rec} / R_{\rm NS}$. The z-axis in both panels align with the magnetic dipole moment. We employed the prescription by Heyl-2000-MNRAS as in Equation (\ref{['eq:r_rc2']}).
• Figure 2: Intensity sky maps obtained from the modeling of two uniform antipodal polar caps extending from the respective magnetic poles to colatitudes $\theta_{\rm cap} = 15^{\circ}$ (top row), $\theta_{\rm cap} = 30^{\circ}$ (middle row), and $45^{\circ}$ (bottom row), as functions of viewing angle $\zeta$ and rotational phase $\Phi = \Omega t$. The five columns correspond to five different values of the inclination angle, $\alpha =10^{\circ}, 30^{\circ}, 50^{\circ}, 70^{\circ}, 90^{\circ}$, as labelled. To approximate the magnetar domain, the simulations were performed at $\omega /\omega_{\hbox{B}} = 0.01$ in the LIF at the magnetic poles (see text). The number of recorded photons is $\mathcal{N}_{\rm rec} = 10^8$.
• Figure 3: Panel (a): Simulated pulse profiles for intensity $I$ for two antipodal polar caps extending from the respective magnetic poles to colatitudes of $\theta_{\rm cap} = 15^{\circ}, 30^{\circ}, \textcolor{black}{40^{\circ}}$ as a function of rotational phase in units of cycles $\Phi / (2\pi)$. The black scattered points represent the data extracted from XMM in the energy band $0.5-3$ keV by Stewart et al. (in prep.) for magnetar 1RXS J1708-4009. The red curve represents the parameter set ($\theta_{\rm cap}$, $\alpha$, $\zeta$) = ($40^{\circ}$, $80^{\circ}$, $10^{\circ}$) that provides the best fit to the data. The other three curves correspond to parameter sets that also fit the data well. Panel (b): Two-dimensional distribution of the geometric parameters ($\alpha, \zeta$), constrained with the observed intensity data on the left. The colorbar indicates the normalized probability density $P(\alpha, \zeta) = \mathcal{N}\sum_{i}P(\theta^i_{\rm cap},\alpha, \zeta)$, with $\mathcal{N}$ being a normalization constant such that the volume below the distribution equals one, i.e., $\sum P(\alpha, \zeta)\Delta\alpha \Delta\zeta = 1$. The vertical and horizontal dashed lines indicates the most probable value of $\alpha$ and $\zeta$, respectively. Panels (c) and (d): a counterpart to panels (a) and (b), but obtained with the excised dataset, that is, omitting the intensity data points between phases $0.20-0.65$. Note the different best fit parameters. See text for details.
• Figure 4: Probability density distributions of the capsize $\theta_{\rm cap}$ (panel (a)), $\alpha$ (panel (b)), $\zeta$ (panel (c), and $\alpha + \zeta$ (panel (d)); all distributions are normalized such that the area below each distribution equals one. These distributions are obtained using observed intensity data for magnetar 1RXS J1708-4009 from Stewart et al. (in prep.), with the data points between phases 0.20-0.65 being excised; see text. The vertical dashed lines indicate the best-fit values.
• Figure 5: Intensity and polarization ($Q/I$, $U/I$, $V/I$) sky maps in $\, \zeta\,$ versus $\, \Phi = \Omega t\,$ space, obtained from the modeling of two uniform antipodal polar caps extending from the respective magnetic poles to colatitudes $\theta_{\rm cap} = 30^{\circ}$ for magnetar 1RXS J1708-4009. Results obtained without including vacuum birefringence in the magnetosphere are shown on the left column (labeled VB off). The middle and right columns respectively correspond to the intensity and polarization information obtained including vacuum birefringence in the magnetosphere with the phase shift $\Delta \phi$ set to zero (labeled VB on, $\Delta \phi = 0$) and $\Delta \phi$ randomized in the range $[0, 2 \pi]$ (labeled VB on, $\Delta \phi =$ rnd). The number of recorded photons is $\mathcal{N}_{\rm rec} = 10^8$. The inclination angle is $\alpha = 60^{\circ}$.