Shock-induced Partial Alignment in Geometrically Thick Tilted Accretion Disks Around Black Holes

Gupta & Dexter (2024), The Astrophysical Journal, 974:209

This paper asks a focused question: in thick, tilted accretion flows, can standing shocks systematically reduce the disk’s inclination before the gas plunges into the black hole?

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Introduction

Hot accretion flows are central to black hole astrophysics because they are radiatively inefficient and operate far below the Eddington limit, making them relevant for the low/hard state of X-ray binaries and low-luminosity AGN, including Sgr A* and M87 [1,2]. These systems are often assumed to be aligned with the black hole spin, but for geometrically thick, low-accretion-rate flows, classic alignment routes (Bardeen–Petterson [3], cumulative mass accretion [4], magneto-spin alignment [5] ) are not expected to be efficient.

At the same time, the physical origin and geometry of shocks in thick, tilted disks has remained unclear. Prior works argued that convergence of eccentric orbits near apocenter could trigger shocks, but their analysis was not able to explain the unique geometry of the shocks or does it depends on bending and twisting of the disk and a black hole spin [6,7,8]. In light of these uncertainties, we attempt to explain the governing mechanism behind shock structure and their potential role in disk alignment.

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Computational setup

We performed ideal, 3D, parallelized general-relativistic magnetohydrodynamic (GRMHD) simulations using the publicly available HARMPI code [9], evolving a SANE disk around a Kerr black hole with spins: \(a=0.5\), \(0.75\), and \(0.9375\).

We initialize a Fishbone–Moncrief equilibrium torus [10] with inner radius and pressure maximum at \(r_{\rm in}=12\,r_g\) and \(r_{\rm max}=25\,r_g\), where \(r_g \equiv \frac{G M_{\rm BH}}{c^2}\) The torus is seeded with a single loop of poloidal magnetic field (\(\beta=100\), \(\Gamma=5/3\)) to trigger magneto-rotational instability (MRI). To study misalignment, we rotate the torus by \(24^\circ\) so that the initial disk angular momentum lies in the X–Z plane.

We evolve the system in modified spherical-polar Kerr–Schild coordinates \((r,\theta,\phi)\) with resolution \(320\times256\times160\). The domain covers \((0.88\,r_H,\,10^5 \,r_g)\times(0,\pi)\times(0,2\pi)\), using a super-exponential radial coordinate to causally disconnect the event horizon \(r_H\) and outer radius.

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Results

3.1 Warped and twisted disk structure

The tilted disks settle into a warped and twisted steady state. The outer disk warped beyond the initial \(24^\circ\) misalignment, while closer to the black hole the inclination decreases with radius, indicating partial inner alignment. The key geometric point is that the disk is not a rigid tilt: the angular-momentum direction varies with radius (warp) and with azimuth (twist) as shown below.

3.2 Standing shocks and a toy model for their origin

By analyzing the total heating rate \(\dot{q}\) we find that the standing shocks emerge near \(\sim6\,r_g\). To connect their location and geometry to the warped disk structure, we use a simple orbit-crowding toy model: we construct centered elliptical orbits (eccentricity \(e=0.1\)) in a disk-aligned frame, defined as a frame whose vertical axis is aligned with the steady-state disk angular momentum vector at each radius. We then map those fluid orbits into the black-hole frame so the warped, twisted geometry appears. The resulting crowding loci reproduce key features of the shock geometry seen in the simulations. In the left-panel video, the orbit bundle is carried through this mapping step-by-step, and the regions where trajectories bunch up mark the expected crowding loci. The right panel shows the predicted loci overlaid on the simulated shock indicator.

3.3 Shock-induced partial alignment

By analyzing the rate of change of disk tilt in the Lagnrangian frame, i.e. by tracking the motion of fluid parcels as they travel through spcae, we find that the regions of decrease tilt is associated with the location of the standing shocks, as shown below. This suggest that the standing shocks may be responsible for partially aligning the inner disk with the black hole spin. Furthermore, we find that the alignment rate increases with black hole spin magnitude, but in all cases the gas accretes before full alignment can be achieved.

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Conclusions & discussion

We carry out idealized 3D GRMHD simulations of prograde, weakly magnetized, geometrically thick accretion flows misaligned from the black hole spin axis. The disks develop a warped and twisted steady state with the outer disk misaligning further away from the black hole equatorial plane. However, closer to the black hole, there is evidence of partial alignment, as the inclination angle decreases with radius. Standing shocks form near \(\sim6\,r_g\), and we show that these shocks could possibly contribute directly to partial alignment, with a stronger effect at higher spin. Beyond demonstrating this shock-induced pathway, the orbit-crowding toy model provides an interpretable bridge between warped disk geometry and expected shock locations. The effect is real but limited: under these conditions, shocks do not align the flow fast enough to fully align the gas before inflow.

However, Our toy model is intentionally minimal and should be read as a *locator* for where shocks are likely to form, not a full predictor of shock morphology. In particular, it cannot predict the shock geometry from the initial misalignment alone; doing so would require tracking the complete fluid propertyvariations (e.g., density, pressure, and velocity) across the shock front. These, standing shocks are natural sites for plasma acceleration and may seed nonthermal electron populations in collisionless systems (e.g., Sgr A*), and can also affect the accretion rate and hence X-ray observations, by affecting the transport of angular momentum.

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References

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Project Details

Author: Sajal Gupta

Year: 2020-2024

Contributors: Jason Dexter

Method: 3D ideal GRMHD (HARMPI), tilted SANE disks, shock diagnostics via heating-rate proxy

Keywords: tilted accretion flows, GRMHD, standing shocks, warp/twist, partial alignment