Misalignment of the Lense-Thirring precession by an accretion torque

TL;DR

This work analyzes how accretion torques from an outer cold disk modify the Lense-Thirring precession of an inner hot torus in a truncated-disk geometry. Using angular-momentum conservation, the authors derive a general solution to the precession dynamics under an accretion torque and explore steady, growing, oscillating, and rotating torque prescriptions, showing that such torques can tilt and even halt precession, and can move the precession axis away from the BH spin. The key finding is that the precession axis becomes a moving target set by the accretion torque, with potential resonance effects when the torque varies periodically. These results imply that jet directions and observational signatures (QPOs, reflection spectra, polarization) may not reliably trace the BH spin or outer-disk orientation, highlighting the need to account for outer-disk torques in modeling accreting black-hole systems.

Abstract

Orbiting matter misaligned with a spinning black hole undergoes Lense-Thirring precession, due to the frame-dragging effect. This phenomenon is particularly relevant for type-C QPOs observed in the hard states of low-mass X-ray binaries. However, the accretion flow in these hard states is complex, consisting of a geometrically thick, hot corona surrounded by a geometrically thin, cold disk. Recent simulations have demonstrated that, in such a truncated disk scenario, the precession of the inner hot corona slows due to its interaction with the outer cold disk. This paper aims to provide an analytical description of the precession of an inner (hot) torus in the presence of accretion torques exerted by the outer (cold) disk. Using the angular momentum conservation equation, we investigate the evolution of the torus angular momentum vector for various models of accretion torque. We find that, in general, an accretion torque tilts the axis of precession away from the black hole spin axis. In all models, if the accretion torque is sufficiently strong, it can halt the precession; any perturbation from this stalled state will cause the torus to precess around an axis that is misaligned with the black hole spin axis. The accretion torque exerted by the outer thin disk can cause precession around an axis that is neither aligned with the black hole spin axis nor perpendicular to the plane of the disk. This finding may have significant observational implications, as the jet direction, if aligned with the angular momentum axis of the torus, may no longer reliably indicate the black hole spin axis or the orientation of the outer accretion disk

Figures (6)

• Figure 1: Geometry of our model. The direction of the black-hole angular momentum $\boldsymbol{L}_\mathrm{BH}$ coincides with the $z$-direction of the Cartesian system used in the calculations. Due to misalignment with the black-hole spin, the angular momenta of the outer thin accretion disk ($\boldsymbol{L}_\mathrm{disk}$) and inner hot torus ($\boldsymbol{L}$) have nonzero projections into the $x$--$y$ plane.
• Figure 2: Precession of the angular momentum vector $L$ of the inner torus when a steady accretion torque $\tau_\mathrm{f}=\tau_\mathrm{A}$ is applied. This figure is projected into the plane perpendicular to the black-hole spin. In the absence of an accretion torque, the angular momentum vector executes free Lense-Thirring precession $L_\mathrm{LT}(t)$ around the direction of the black hole spin. When the torque is applied, the general solution consists of free Lense-Thirring precession about a new direction (shifted by a constant, $\hat{L}$).
• Figure 3: Trajectory of the torus angular-momentum vector in the $x$--$y$ plane when the accretion torque evolves exponentially from zero to $\tau_\mathrm{f} = -1$ for various cases. A torus that is initially precessing with an angular momentum $L(0) = -0.3{\rm i}$ approaches the final state of precession about the new equilibrium point, $\mathrm{i}\tau_\mathrm{f}/\omega_{\rm p}$, slowly or rapidly for a low ($\omega_\mathrm{h} = 0.2\omega_\mathrm{p}$) or high ($\omega_\mathrm{h} = 2\omega_\mathrm{p}$) healing frequency, as shown in the left and middle panels, respectively. The right panel presents the same case as in the left (i.e., $\omega_\mathrm{h} = 0.2\omega_\mathrm{p}$), except that initially the torus has a zero angular momentum projection onto the $x$--$y$ plane, i.e., $L(0) = 0$, highlighting our finding that the change in accretion torque can induce precession in a torus. In all three examples, we took $\omega_\mathrm{p} = 1$ and the phase evolution is represented by both the colour and the decreasing thickness of the curve.
• Figure 4: Trajectory of the torus angular momentum vector in the $x$-$y$ plane for an accretion torque rotating with frequency $\omega_1 = 0.9 \omega_\mathrm{p}$ ( left panel) and $\omega_1 = 0.65\omega_\mathrm{p}$ ( right panel). In both cases, we use $L(0) = 7{\rm i}$, $\tau_{\rm A} = 1$, and $\omega_\mathrm{p} = 1$. The phase of the trajectory is represented by both the color and the decreasing thickness of the curve, with the starting point at $t=0$ and a point at some later time $t$ marked by a circle and triangle, respectively (for visualization purposes). The torus precesses around the black hole axis with a frequency of $(\omega_{\rm p} + \omega_1)/2$. Additionally, the tilt angle between the black hole and torus oscillates with frequency $(\omega_{\rm p} - \omega_1)$.
• Figure 5: Comparison of our model with the simulation data of Bollimpalli+2024. In the left panel, we show the magnitude of the accretion (dashed, fluctuating curve) and Lense-Thirring (smooth, solid curve) torques integrated over the torus region, i.e. $5-15\, GM/c^2$. The difference between the two magnitudes (black, dotted curve) remains close to zero in the initial phase of the simulation, up to $\sim\,25000\, GM/c^3$. The middle panel shows the $x$ and $y$ components of the net accretion torque on the torus (solid curves), and their respective fits using Eq. \ref{['eq:osilltorq']} (dashed curves). Finally, the right panel shows the the $x$ and $y$ components of the total angular momentum of the torus (solid curves), and their respective fits using Eq. \ref{['eq:oscillsol']} (dashed curves). The vertical, black, dashed lines in all the panels indicate the stop time of the original simulation.