Misalignment of the Lense-Thirring precession by an accretion torque

1. Problem Statement

In low-mass X-ray binaries (LMXBs) in the hard spectral state, the accretion flow is often modeled as a geometrically thick, hot inner torus (corona) surrounded by a geometrically thin, cold outer disk. A leading hypothesis for the origin of Type-C Low-Frequency Quasi-Periodic Oscillations (LFQPOs) is the Lense-Thirring (LT) precession of this inner torus due to the frame-dragging effect of a spinning black hole.

The Gap: Previous analytical models and simulations of LT precession generally assumed "free precession," where the torus precesses uniformly around the black hole's spin axis with a constant tilt angle. However, recent General Relativistic Magnetohydrodynamic (GRMHD) simulations (Bollimpalli et al. 2023) revealed that the interaction between the inner torus and the outer thin disk introduces an accretion torque. This torque significantly alters the precession dynamics, causing the precession axis to shift and potentially stall, a phenomenon not captured by standard free-precession models.

2. Methodology

The authors developed an analytical framework based on the conservation of angular momentum to describe the evolution of a tilted, accreting torus.

• Governing Equation: They utilized a Newtonian angular momentum conservation equation modified to include relativistic effects:
  $$\frac{d\mathbf{L}}{dt} = \boldsymbol{\tau}_p + \boldsymbol{\tau}_{acc}$$
  Where:
  • $\mathbf{L}$ is the angular momentum vector of the torus.
  • $\boldsymbol{\tau}_p = \boldsymbol{\omega}_p \times \mathbf{L}$ is the Lense-Thirring torque (precession frequency $\boldsymbol{\omega}_p$ aligned with the black hole spin).
  • $\boldsymbol{\tau}_{acc}$ is the accretion torque, representing the net rate of angular momentum advection from the outer disk onto the inner torus.
• Complex Variable Approach: To solve the coupled differential equations for the $x$ and $y$ components of angular momentum, the authors introduced complex variables ($L = L_x + iL_y$), transforming the system into a linear inhomogeneous differential equation:
  $$\frac{dL}{dt} = i\omega_p L + \tau_{acc}$$
• Torque Models: The authors derived general solutions for four distinct scenarios of $\tau_{acc}$:
  1. Steady Torque: Constant accretion rate.
  2. Linearly Growing Torque: Modeling a slow "outburst" or ramp-up.
  3. Exponentially Changing Torque: Modeling state transitions or rapid adjustments.
  4. Oscillating/Rotating Torque: Modeling periodic variations in the outer disk (e.g., diskoseismology modes) or rotating accretion flows.

3. Key Contributions

• Analytical Derivation of Misaligned Precession: The paper provides the first analytical description showing that the presence of an accretion torque causes the torus to precess around an axis that is misaligned with both the black hole spin axis and the outer disk plane.
• Mechanism of Axis Shift: The authors demonstrate that the equilibrium point of precession (the center of the precession cone) is determined by the balance between the LT torque and the accretion torque. Specifically, for a steady torque, the precession center is shifted by $\pi/2$ in phase relative to the torque direction.
• Excitation of Precession: A critical finding is that a change in accretion torque can induce free Lense-Thirring precession even in a torus that was initially aligned with the black hole spin (i.e., non-precessing).
• Resonance Phenomena: The study identifies a resonance condition where the frequency of a rotating accretion torque matches the LT precession frequency ($\omega_1 \approx \omega_p$). This leads to a dramatic increase in the tilt amplitude, potentially causing the torus to approach a vertical orientation ("standing on its edge").

4. Results

• General Behavior: In all models, if the accretion torque is sufficiently strong, it can halt the precession (stall the torus). However, any perturbation from this stalled state causes the torus to resume precession around a new, misaligned axis.
• Steady Torque: The torus precesses at the standard LT frequency, but the mean angular momentum vector points in a direction determined by the vector sum of the LT and accretion torques, not the black hole spin.
• Oscillating Torque: When the accretion torque oscillates, the tip of the angular momentum vector traces an ellipse rather than a circle. This matches the behavior observed in recent GRMHD simulations.
• Validation with Simulations: The authors validated their analytical model against simulation data from Bollimpalli et al. (2023, 2024). By fitting an oscillating torque model (Eq. 20) to the simulation's accretion torque, they successfully reproduced the time evolution of the torus's angular momentum components ($L_x, L_y$). The model confirmed that the outer disk's influence explains the "stalling" and subsequent small-loop precession observed in simulations.
• Resonance Effects: In the rotating torque scenario, near-resonance conditions lead to "beats" where the tilt angle oscillates at a low frequency $(\omega_p - \omega_1)$ while the system precesses at a high frequency.

5. Significance and Observational Implications

• Jet Orientation: If relativistic jets are anchored to the inner torus and aligned with its mean angular momentum axis, the jet direction will not necessarily trace the black hole spin axis. This challenges the assumption that jet direction is a direct proxy for black hole spin orientation.
• QPO Interpretations: The findings suggest that Type-C QPOs may arise from precession around a complex, time-varying axis rather than a simple cone around the spin axis. This affects the interpretation of QPO frequencies used to constrain black hole spin and mass.
• Spectral and Polarization Variability: The misalignment and changing tilt angle imply that the illumination of the outer disk by the inner torus (and the resulting reflection spectra and X-ray polarization) will vary differently than predicted by standard models. This could explain complex correlations between QPO phase and spectral properties observed in IXPE and other X-ray missions.
• Jet Formation Mechanisms: The results favor jet formation mechanisms influenced by the accretion flow (e.g., Blandford-Payne or hybrid models) over pure Blandford-Znajek mechanisms, which strictly align jets with the black hole spin.

In conclusion, this paper establishes that accretion torques from outer disks fundamentally alter the dynamics of inner tori, leading to misaligned precession axes that have profound implications for interpreting X-ray variability, jet physics, and black hole spin measurements in LMXBs.