Impact of Resonant Compton Scattering on Magnetar X-Ray Polarization with QED Vacuum Resonance

Abstract

Recent obeservations have revealed significant soft X-ray polarizations from several quiescent magnetars, including the intriguing $90^°$ polarization angle (PA) swing as a function of photon energy for some sources. We present a general semi-analytical framework for calculating energy-dependent soft X-ray polarization signatures from magnetars, consistently incorporating both QED vacuum resonance in the atmosphere and resonant Compton scattering (RCS) in the magnetosphere. Starting from the polarized radiative transfer equation for RCS and treating vacuum-resonance-induced mode conversion as an input, we employ a first-order approximation in RCS optical depth to evaluate the effect of different magnetospheric plasma density (which depends on magnetic twist), drift velocity and temperature, and viewing geometry on the observed radiation. Our analysis reveals that magnetic twist and plasma drift velocity are the critical parameters controlling the impact of RCS on both the absolute polarization degree and its variation across the soft X-ray spectrum. We find that sufficiently strong RCS can wash out the PA swing caused by vacuum resonance. Furthermore, in addition to the QED vacuum resonance effect, significant relativistic signatures arising from plasma drift velocity ($β_0 \gtrsim 0.5$) may introduce an extra $90^\circ$ PA swing in the spectrum. Our calculation framework, based on single-scattering approximation, bypasses the need for complex, multi-dimensional Monte Carlo simulations, providing an analytical pathway for modeling full-surface emission and rotational-phase-resolved radiation from magnetic neutron stars, in support of current and future X-ray polarization missions.

Figures (7)

• Figure 1: Schematic picture of the radiative transfer with resonant Compton scattering and the computation of the observed polarization flux. The surface element of the NS is $\text{d}\boldsymbol A_s=\widehat{\boldsymbol\Omega}_s\text{d}A_s$. The scattering surface element (located at $\boldsymbol r_{\text{sc}}=r_{\text{sc}}\widehat{\boldsymbol{\Omega}}_{\text{sc}}$) is denoted by $\text{d}\boldsymbol A$. The scattered radiation propagates along the direction $\widehat{\boldsymbol{\Omega}}\simeq \widehat{\boldsymbol{\Omega}}_{\text{obs}}$, and the incident radiation (before scattering) propagates along $\widehat{\boldsymbol\Omega}'=(r_\text{sc}\widehat{\boldsymbol{\Omega}}_0-R\widehat{\boldsymbol\Omega}_s)/|r_\text{sc}\widehat{\boldsymbol{\Omega}}_0-R\widehat{\boldsymbol\Omega}_s|$. The magnetic axis of the magnetar is along the $z$-axis, and $\widehat{\boldsymbol{\Omega}}_\text{obs}$ is in the $yz$ plane, with the angle between $\widehat{\boldsymbol{\Omega}}_\text{obs}$ and $z$-axis being $\theta$.
• Figure 2: Flux and polarization spectra for different scattering strengths compared to the unscattered case in the simplified setup with $B_{\mathrm p} = 10^{14}\ \mathrm{G}$, $kT = 0.6\ \mathrm{keV}$, $R = 10\ \mathrm{km}$, a hot spot at $(\theta_s,\phi_s) = (65^\circ,30^\circ)$, viewing angle $\theta=25^\circ$, and varying $\zeta \equiv \xi_\tau/\beta_0 = 0.8, 2.5, 7.0$ (see Eqs.\ref{['density']} and \ref{['opticaldepth']}). Left and middle: the relative O-mode and X-mode fluxes. Right: the corresponding linear polarization degree $P_L(\omega)$ as a function of energy.
• Figure 3: Heatmap of the characteristic resonant optical depth in the full model. The parameters are: $B_\mathrm p = 10^{14}~\mathrm{G}$, $R = 10~\mathrm{km}$, $kT_e = 10~\mathrm{keV}$, $kT_s = 0.6~\mathrm{keV}$, a hot spot at $(\theta_s, \phi_s) = (53^\circ, 37^\circ)$, viewing angle $\theta=35^\circ$. Contours for $\tau_\alpha = 1$ and $\tau_\alpha = 0.4$ delineate the validity of single-scattering/first-order approximation regime.
• Figure 4: Flux and polarization spectra for different observer viewing angles in the full model. The parameters are: $B_\mathrm p = 10^{14}~\mathrm{G}$, $R = 10~\mathrm{km}$, $kT_e = 10~\mathrm{keV}$, $kT_s = 0.6~\mathrm{keV}$, $\xi_\tau = 0.2$, $\beta_0 = 0.5$, and a hot spot at $(\theta_s, \phi_s) = (90^\circ, 37^\circ)$. Results are shown for the viewing angle $\theta = 37^\circ, 68^\circ, 112^\circ, 143^\circ$.
• Figure 5: Flux and polarization spectra for different magnetospheric twist parameters in the full model. The parameters are: $B_\mathrm p = 10^{14}~\mathrm{G}$, $R = 10~\mathrm{km}$, $kT_e = 10~\mathrm{keV}$, $kT_s = 0.6~\mathrm{keV}$, $\theta=35^\circ$, $\beta_0 = 0.5$, and a hot spot at $(\theta_s, \phi_s) = (53^\circ, 37^\circ)$. Results are shown for magnetic twist $\xi_\tau = 0.3, 0.8, 1.6$, and are compared with the unscattered case.