The disk precession in a Be star-magnetar binary and its application to the rotation measure of FRB 20201124A

1. Problem Statement

Fast Radio Bursts (FRBs) are millisecond-duration radio transients with poorly understood origins. While most are one-off events, repeaters like FRB 20201124A provide crucial insights into their environments. FRB 20201124A is notable for its extreme activity and significant, complex temporal variations in its Rotation Measure (RM), observed by the Five-hundred-meter Aperture Spherical radio Telescope (FAST).

• The Phenomenon: Between April and June 2021, the RM of FRB 20201124A exhibited a rapid increase, a sharp drop, and a subsequent re-ascent over 40 days, with a total variation amplitude of ~500 rad m$^{-2}$. This was followed by a stable phase and later epochs with lower amplitude variations (100 rad m$^{-2}$).
• The Gap: Previous models attributed these variations primarily to the orbital motion of a binary system (the "cosmic-comb" model or simple binary orbital motion). However, simple orbital motion cannot fully explain the long-term evolution of the variation amplitude (the decrease in amplitude over time) or the specific oscillatory patterns observed across different epochs. The paper seeks to explain the long-term behavior and the modulation of the RM amplitude.

2. Methodology

The authors propose a Be star-magnetar binary model incorporating a precessing circumstellar disk.

Physical Model

• System Configuration: A magnetar orbits a Be star (mass $\sim 8 M_\odot$) in an elliptical orbit. The Be star possesses a dense equatorial decretion disk.
• Disk Geometry: The disk density follows a power-law profile $\rho(r, z) \propto r^{-\beta} \exp[-(z/H)^2]$. The disk is assumed to be axisymmetric but undergoes rigid precession around an axis misaligned with the orbital plane.
• Magnetic Field: The model assumes a simplified toroidal magnetic field ($B \propto r^{-1}$) in the disk, which dominates the RM contribution at the distances relevant to the FRB line of sight (LoS). The dipolar field of the star is shown to be negligible at these distances.
• RM Calculation: The RM is calculated by integrating the product of electron density ($n_e$) and the magnetic field component along the LoS ($B_\parallel$) through the disk. The integration accounts for the changing geometry of the disk relative to the magnetar's position (orbital phase) and the observer (due to precession).

Data Analysis & Fitting

• Data Sources: The authors utilized FAST observation data from three active epochs (E1: April–June 2021; E2: Sept–Oct 2021; E3: Feb–May 2022).
• Preprocessing: RM data were binned by Modified Julian Date (MJD), and mean values were subtracted to isolate variations ($\Delta$RM).
• Fitting Parameters: The model was fitted to the $\Delta$RM data using free parameters including:
  • Orbital eccentricity ($\epsilon$) and period ($P_{orb}$).
  • Disk density profile ($\rho_0, \beta$).
  • Precession parameters: Precession axis angle ($\phi$), initial disk tilt ($\theta$), and precession angular velocity ($\omega$).
  • Line-of-sight angles ($\lambda_1, \lambda_2$).
• Validation: The authors performed a dynamical analysis to verify if the derived precession period is physically consistent with gravitational torques exerted by the companion star on the disk. They also calculated the free-free absorption optical depth to ensure FRBs are not absorbed by the disk.

3. Key Contributions

1. Introduction of Disk Precession: The paper introduces the concept of Be star disk precession as a mechanism to modulate the RM variations of FRBs in binary systems, moving beyond simple orbital motion models.
2. Unified Explanation of Amplitude Evolution: The model successfully explains not just the short-term oscillations (orbital motion) but also the long-term decrease in the amplitude of RM variations observed between epochs E1 and E3.
3. Dynamical Consistency: The authors provide a theoretical derivation showing that the fitted precession period is consistent with the gravitational torque exerted by the magnetar on the Be star's decretion disk.
4. Application to Other Sources: The model is extended to other FRBs with RM variations (FRB 20180301A, FRB 20190520B, FRB 20191106C), offering predictions for their missing data.

4. Key Results

• Best-Fit Parameters:
  • Orbital Period ($P_{orb}$): $\sim 73$ days.
  • Eccentricity ($\epsilon$): $0.69$.
  • Precession Period ($P_{pre}$): $\sim 785$ days (derived from $\omega \approx 8.01 \times 10^{-3}$ rad day$^{-1}$).
  • Disk Density: Base density $\rho_0 \approx 3.49 \times 10^{-15}$ g cm$^{-3}$ with slope $\beta \approx 4.09$.
• Model Performance:
  • The model reproduces the rapid rise, sharp drop, and re-ascent of the RM in Epoch 1.
  • It accurately predicts the reduction in variation amplitude in Epochs 2 and 3 as a natural consequence of the disk precession changing the viewing angle and the density profile along the LoS over the $\sim 785$-day cycle.
  • The model predicts that the amplitude of RM variations should oscillate with the precession period.
• Physical Constraints:
  • Absorption: Calculations show the free-free absorption optical depth ($\tau_{ff}$) is $\ll 1$, confirming that the disk does not absorb the radio signals.
  • Magnetic Field: The FRB LoS passes well beyond the Alfvén radius ($R_A \sim 10 R_\odot$), confirming that the toroidal magnetic field approximation is valid and the stellar dipole field is negligible for RM generation.
• Other FRBs: The model provides plausible fits for FRB 20180301A, FRB 20190520B, and FRB 20191106C, suggesting a common origin mechanism (Be star-magnetar binaries with precessing disks) for FRBs with dramatic RM variations.

5. Significance

• Origin of FRBs: This work strengthens the hypothesis that a significant subset of repeating FRBs originates from Be star-magnetar binaries.
• Environmental Dynamics: It demonstrates that the local magneto-ionic environment of FRB sources is highly dynamic, governed by both orbital motion and super-orbital precession of the circumstellar disk.
• Predictive Power: The identification of a $\sim 785$-day precession period offers a specific, testable prediction for future observations of FRB 20201124A. If the RM amplitude oscillates with this period, it would confirm the precession model.
• Broader Context: The study bridges the gap between FRB phenomenology and the well-studied super-orbital brightness oscillations in Be/X-ray binaries, suggesting that similar physical mechanisms (disk precession) drive variability in both X-ray and radio bands.

In conclusion, the paper provides a robust physical framework linking the complex, long-term RM variations of FRB 20201124A to the precession of a Be star's decretion disk, offering a compelling explanation for the source's behavior and a template for understanding other active FRB repeaters.