X-ray Polarimetry in the Low Statistics Regime using the Bayesian Approach Reveals Polarization Angle Variations

TL;DR

X-ray PA variations in accreting compact objects can be robustly constrained even in the low-count regime by a Bayesian framework that treats the measured azimuthal distribution with a sinusoidal likelihood and non-informative priors. By reparameterizing with $\mathcal{P}=p/\mathrm{MDP}$, the PA posterior $\rho(\Psi_0|p_m)$ remains informative even when $p_m/\mathrm{MDP}$ is small, enabling time-resolved inferences without Gaussian assumptions. Applying this method to IXPE observations of GX 13+1, XTE J1701-462, and Sco X-1 reveals PA often swinging between two discrete angles rather than following a simple linear rotation, with some cases showing PA correlated with flux and hence changes in the Comptonization geometry or optical depth. These results demonstrate the practical potential of low-count Bayesian polarimetry to probe fast accretion-driven geometry changes and motivate joint spectral-polarimetric analyses and higher-time-resolution follow-ups.

Abstract

X-ray polarimetry of accreting compact objects has revealed fast time variations in the polarization angle (PA), suggesting that the geometry and/or optical depth of the Comptonization region is changing rapidly. This prompts investigations into how fast such variability can be. Conventionally, the data are often binned to examine the time variability such that the measurement in each bin is above the minimum detectable polarization (MDP). Here we demonstrate that this is unnecessary, and even below the MDP, one can infer the posterior distribution of PA reliably using the Bayesian approach and still be able to place useful constraints on the physics in many cases, due to small relative uncertainties on PA (e.g., $Δ$PA $\approx$ 10--30$^\circ$ compared with a dynamical range of 180$^\circ$). With this approach, we discovered that the PA variation in one of the Imaging X-ray Polarimetry Explorer (IXPE) observations of GX 13+1 is not following a linear rotation mode as suggested previously. Instead, the PA swings between two discrete angles, suggesting that there are two emitting components, e.g., the boundary layer and the spreading layer, competing with each other. In XTE J1701-462, we confirmed previous results for a variable PA in the normal branch, and furthermore, revealed that the variation timescale could be as short as 1.5 hours. During the IXPE observation of Sco X-1, a hint is found for the PA in the highest flux level to be different from the average but consistent with previous measurement results with PolarLight and OSO-8.

Figures (7)

• Figure 1: Examples of bivariate posterior distributions of $(p_0, \Psi_0 - \Psi_\text{m})$ at different $p_\text{m}$ and MDPs. The contours represent the 68%, 90%, and 99% confidence levels. A modulation factor $\mu = 0.3$ is assumed.
• Figure 2: Credibility intervals of $p_\text{0}$ and $\Psi_\text{0}$ at different confidence levels maier2014. The dashed lines in the bottom panel represent the corresponding PA uncertainties assuming Gaussianization in the high statistics regime kislat2015.
• Figure 3: Posterior distributions of $p_0/\text{MDP}$maier2014 and $\Psi_0$ at different low-statistical levels, with $p_\text{m} / \text{MDP} =$ 0.2, 0.4, 0.6, and 0.8, respectively.
• Figure 4: Temporal variation of polarization properties and source count rate in the three IXPE observations of GX 13+1. Error bars indicate the 68% credible intervals. Filled points indicate $p_{\rm m} / \mathrm{MDP} \geq 1$. In Obs1, the green data points are calculated using the binning scheme in bobrikova2024a. The blue lines represent the three best-fit models: constant (solid), linear (dotted), and bimodal (dashed).
• Figure 5: Energy spectra of GX 13+1 in the two states with different PAs, where $\text{PA}1 \approx -43\arcdeg$ and $\text{PA}2 \approx 26\arcdeg$. Only the IXPE DU1 data are plotted for clarity.