Tilted disks around black holes: radiative verification and boundary mapping of tilt-induced shock and heating phenomenology in the spin–inclination plane

A 2019 general-relativistic magnetohydrodynamic (GRMHD) parameter survey of tilted accretion flows (spins $a=0$–0.9, inclinations $0$–$24^\circ$, nonradiative, weakly magnetized, with deliberately reduced polar resolution) established a distinctive phenomenology: standing shocks appear at sufficiently high spin and inclinations above roughly $8^\circ$, they dominate angular-momentum transport in the inner few to ten gravitational radii largely through Reynolds rather than Maxwell stresses, at high spin they lower the heating efficiency $Q/\langle\dot M\rangle$ as tilt increases, and polar outflows align with the disk rather than the black-hole spin axis. The authors explicitly stated that 'tilted disk simulations with in-situ radiation will be required to verify these trends'. Later work has resolved parts of the surrounding landscape — tilted magnetically arrested disks realign and cease to precess, and highly tilted thin disks tear into independently precessing sub-disks — but no published simulation has tested the thick-disk trends with in-situ radiation, mapped where shocks, alignment, and disk breaking begin across spin, inclination, and disk thickness, or established convergence of both MRI turbulence and shock structure with adequately resolved polar regions. The problem is the verification the authors explicitly called for: determine which parts of the tilted thick-disk phenomenology are robust physics and which are artifacts of the omitted physics, above all the absence of in-situ radiation.

Background

A Kerr black hole is described by its mass and its dimensionless spin $a$ (with $|a|\le 1$). Gas accreting onto it need not share the hole's equatorial plane: when the disk's angular momentum is misaligned with the spin by an inclination angle $i_0$, the dragging of spacetime by the rotating hole (the Lense–Thirring effect) makes inclined orbits precess at a rate depending on radius, twisting the disk into a warp. In thin, viscously coupled disks this warp can spread outward and the inner disk can align with the hole's equatorial plane — the Bardeen–Petterson effect (Bardeen & Petterson 1975). The coupling between adjacent rings is provided by magnetohydrodynamic turbulence driven by the magnetorotational instability (MRI), the instability of any weakly magnetized, differentially rotating disk in which angular velocity decreases outward. Realistic models therefore require solving the equations of general-relativistic magnetohydrodynamics (GRMHD) numerically.

Tilted GRMHD simulations revealed a feature absent from untilted flows: Fragile & Blaes (2008) found a pair of standing shocks near the line of nodes (the azimuth where the tilted disk plane crosses the hole's equatorial plane), caused by the eccentric epicyclic motion that tilt induces in the inner disk orbits. Liska et al. (2018) showed that tilted thick disks and their jets can precess rigidly. White, Quataert & Blaes (2019) then performed the first systematic survey, evolving one magnetized torus around ten combinations of spin ($a=0$–0.9) and inclination ($i_0=0$–$24^\circ$) with the Athena++ code. These runs were explicitly nonradiative, non-self-gravitating, ideal-MHD simulations of geometrically thick, inefficiently cooled flows, initialized with little net vertical magnetic flux (the 'standard and normal evolution', or SANE, regime — as opposed to the magnetically arrested disk, MAD, regime in which accumulated vertical magnetic flux becomes dynamically dominant and chokes accretion), and they used static mesh refinement to hold the polar regions at low resolution so timesteps stayed tractable.

The survey found that the tilted disks settle into warped, twisted shapes that precess rigidly; that magnetized polar outflows align with the disk rotation axis rather than the spin axis (and that even modest tilt can suppress the moderate outflows seen at $a=0.5$); that at sufficiently high spin and inclinations above about $8^\circ$ standing shocks dominate dissipation and angular-momentum transport in the inner $5$–$10$ gravitational radii, with Reynolds (hydrodynamic) stresses exceeding Maxwell (magnetic) stresses; and that at high spin the shocks redirect gas more effectively than they heat it, so the heating efficiency — the volumetric heating rate $Q$ normalized by the accretion rate $\langle\dot M\rangle$, a proxy for radiative efficiency — decreases as tilt increases. This matters observationally: if a given heating efficiency can correspond to a range of spins once tilt is allowed, untilted models cannot be used to infer spin from luminous output. Because the survey omitted radiation, resolved the poles poorly, and stopped at $a=0.9$ and $24^\circ$, whether these trends are robust properties of tilted relativistic flows — and where their boundaries lie — was left open by construction.

Problem Statement

The objective is to establish, by direct GRMHD simulation, the robustness and boundaries of the tilt-induced phenomenology found in the nonradiative spin-inclination survey of White, Quataert & Blaes (2019). The central question is the one the survey's authors explicitly posed: their runs were nonradiative, and they stated that 'tilted disk simulations with in-situ radiation will be required to verify these trends'. Facet 1 below states this central question; facets 2 and 3 state the further required directions that the source's discussion suggested and later work has pursued, each targeting physics or coverage the original survey omitted. All three facets are required to resolve the objective, and each item listed within a facet must be answered separately.

1. Radiation: do the shock-formation thresholds in spin $a$ and initial inclination $i_0$, the reduction of heating efficiency $Q/\langle\dot M\rangle$ with increasing tilt at high spin, and the dominance of Reynolds over Maxwell stresses in the inner shocked regions persist when radiative cooling and radiation-matter coupling are treated in situ, rather than being inferred from nonradiative proxy heating?

2. Boundaries: where in the $(a,i_0)$ plane — including extreme spins beyond $a=0.9$ and inclinations beyond $24^\circ$, and disks thinner than the geometrically thick ones surveyed, as set by disk thickness or cooling efficiency — do standing shocks first appear, where does partial or complete Bardeen–Petterson alignment set in, and where do sufficiently thin tilted disks break or tear (split into independently precessing sub-disks)?

3. Resolution: are these trends converged with respect to numerical resolution of both the MRI turbulence and the hydrodynamic shock structure, with polar regions resolved well enough that the quantitative dependence of outflow strength, magnetic-field alignment, and jet-like structure on tilt and spin is firmly established (the original survey deliberately degraded polar resolution to limit timesteps)?

References

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