Steady, weakly relativistic jets are observed from black-hole X-ray binaries in the low/hard spectral state and from SS 433, but how the accretion flow supplies the ejected gas with the positive specific energy it needs to escape remains unsettled. Inoue (2022) proposed a purely hydrodynamic scenario in which the inner accretion flow separates into two layers and one layer receives energy from the other — through turbulent mixing in a two-layer advection-dominated accretion flow (ADAF) for the low/hard state, and through vertical photon diffusion in a super-Eddington slim disk for SS 433 — with a two-step shock sequence near the black hole converting the energized upper layer into an escaping, collimated outflow. The paper itself describes the study as primitive and leaving many questions to solve: the cross-layer energy-transfer efficiency, the strengths and locations of the two shocks and the fraction of mass diverted into the jet, the validity of the assumed inviscid inner supersonic flow, and the radiative vertical-structure estimates are all unquantified, magnetic fields are deliberately absent from the treatment, and no self-consistent multidimensional calculation has tested the picture. The problem is to determine, by theory and/or multidimensional radiation(-magneto)hydrodynamic simulation together with observational confrontation, whether this two-layer energy-transfer mechanism can quantitatively reproduce the observed steady jets — terminal speeds of order 0.1 c in the low/hard state and the observed 0.26 c for SS 433, opening angles of a few degrees, and the jet-power–accretion-rate relation — or to refute it.

Background

Relativistic outflows (jets) from accreting black holes appear and disappear in lockstep with the accretion state. In the low/hard spectral state of a black-hole X-ray binary — a dim, hard X-ray state whose inner flow is standardly modeled as an advection-dominated accretion flow (ADAF), i.e. a geometrically thick, optically thin, radiatively inefficient flow in which most dissipated energy is carried inward rather than radiated — a steady, weakly relativistic compact jet with terminal speed of order 0.1 c is always present. In the high/soft state, dominated by a cool, optically thick, geometrically thin disk, no jet is seen. The Galactic microquasar SS 433 — a stellar-mass black hole system whose accretion and jets mimic a small-scale quasar — persistently ejects steady jets at 0.26 c. A long-standing puzzle is the energy bookkeeping: the specific energy of the ejected gas (the sum of kinetic, thermal, and gravitational energy per unit mass) must be positive for the gas to reach infinity, yet the accreted gas at the outer boundary of the flow carries slightly negative specific energy, so something inside the flow must turn the sign around. The mainstream view attributes jet launching to magnetic processes (extraction of black-hole spin energy or magnetically driven disk winds); a deliberately hydrodynamic alternative was proposed by Inoue (2022), building on a companion picture of two-layer (accretion plus excretion) flows emerging from the accretion ring feeding the binary (Inoue 2021) and on a study of the mass flows in the SS 433–W 50 system (Inoue 2022, PASJ 74, 991). For the low/hard state, a geometrically thick ADAF is supposed to sandwich a geometrically thin disk that itself becomes another ADAF further in; turbulent mixing at the layer interface would transfer a fraction of the main flow's outward-transported viscous energy into a low-mass upper layer, giving it positive specific energy of order 0.01 in units of GM/RsGM/R_{s} (M the black-hole mass, RsR_s the Schwarzschild radius) and hence a terminal speed near 0.1 c. For SS 433, vertical photon diffusion through a super-Eddington slim disk — a geometrically thick, optically thick disk regime, set at accretion rates near or above the Eddington rate where radiation pressure balances gravity, in which photons diffuse out more slowly than the gas flows in — would similarly energize an upper layer to normalized specific energy of order 0.05, terminal speed near 0.22 c. Near the black hole the flow is treated as transonic: the outer subsonic, viscous ADAF passes a saddle-type sonic point (where the flow speed equals the local sound speed) into an inner supersonic flow that is taken to be inviscid with constant specific angular momentum, because the usual diffusive, turbulence-driven angular-momentum transport is acausal in a supersonic region. The jet itself is launched by a two-step shock sequence: the supersonic upper flow hits an oblique shock where the main flow thins toward the equatorial plane, and the part entering heights above roughly RsR_s, where the centrifugal force exceeds radial gravity (the centrifugal barrier), is decelerated by a second shock; the over-pressured gas, having positive specific energy, escapes and becomes supersonic again within a few Schwarzschild radii, collimating under the centrifugal-versus-gravitational force balance to an opening angle of order 2 degrees. All of these elements are semi-analytic and schematic: the mixing efficiency is asserted rather than computed, the shocks are drawn rather than solved, magnetic fields are absent, and the observational comparison is at the order-of-magnitude level.

Problem Statement

Determine quantitatively whether the two-layer energy-transfer mechanism of Inoue (2022) can launch the observed steady jets from black-hole accretion flows, or establish that it cannot. The quantitative questions left unresolved by the semi-analytic treatment include: (1) the efficiency and physics of cross-layer energy transfer — the turbulent mixing rate between the thick ADAF and the embedded thin disk and the fraction of the main flow's viscous-transport energy that can be deposited into the low-mass upper layer in the low/hard-state case, and the corresponding radiative-transfer quantities in the super-Eddington slim-disk case; (2) the dynamics of the two-step shock sequence near the black hole — strengths and locations of the oblique shock and the centrifugal-barrier shock, post-shock thermodynamics, and the fraction of inflowing mass converted into escaping jet matter; (3) the status of the inviscid inner-supersonic-flow assumption, i.e. whether a causal treatment of angular-momentum transport in the supersonic regime changes the transonic structure on which the scenario depends; (4) whether the vertical layering and flux estimates of the slim-disk photon-diffusion picture survive self-consistent radiation-hydrodynamic calculation; and (5) the quantitative observational consequences — jet terminal speeds (of order 0.1 c in the low/hard state, and the predicted ~0.22 c for SS 433 versus the observed 0.26 c), opening angle (of order 2 degrees), jet power as a function of accretion rate, and the contrast between the low/hard and high/soft states. These questions bear on one objective — a demonstrated quantitative account of the steady jets in both regimes, or a demonstration that the mechanism cannot provide it. A complete account requires resolving all five items across both regimes — the low/hard-state ADAF case and the super-Eddington slim-disk case; these enumerated items are the necessary conditions, not an open-ended list. A refutation is established by demonstrating, for an item, that the mechanism's assumptions cannot realize it, rather than merely leaving the item unquantified. Magnetic fields are deliberately omitted from the scenario; whether including them in angular-momentum transport, energy partitioning, and collimation would modify or overturn the hydrodynamic picture is an open question bearing on that objective.

References

  1. Hajime Inoue (2022), 'Steady jet ejections from the innermost region of advection-dominated accretion flow around a black hole', Publications of the Astronomical Society of Japan 74(6), 1263–1274. DOI: 10.1093/pasj/psac081, arXiv:2210.10976. https://arxiv.org/abs/2210.10976
  2. Hajime Inoue (2021), 'Properties of the accretion ring in an X-ray binary, and accretion and excretion two-layer flows from it', Publications of the Astronomical Society of Japan 73(4), 795–808. DOI: 10.1093/pasj/psab043. https://doi.org/10.1093/pasj/psab043
  3. Hajime Inoue (2022), 'Mass flows and their behaviors in the SS 433–W 50 system', Publications of the Astronomical Society of Japan 74(4), 991–1003. DOI: 10.1093/pasj/psac050, arXiv:2206.14353. https://doi.org/10.1093/pasj/psac050
  4. Hajime Inoue (2023), 'Transient jet ejections associated with limit-cycle behaviors in the very high state of black hole binaries' (RETRACTED), Publications of the Astronomical Society of Japan 75(3), 677–690. DOI: 10.1093/pasj/psad028, arXiv:2304.10847. https://doi.org/10.1093/pasj/psad028 (retraction notice: 'Retraction of: Transient jet ejections associated with limit-cycle behaviors in the very high state of black hole binaries', PASJ 75(4), 833, 2023, DOI: 10.1093/pasj/psad045)
  5. Hajime Inoue (2024), 'Relativistic Jet Ejections from Slim Disks', The Astrophysical Journal 967(1), 3. DOI: 10.3847/1538-4357/ad3937, arXiv:2404.03684. https://arxiv.org/abs/2404.03684
  6. G. Radureau and C. Michaut (2025), 'Impact of frequency-dependent radiation on the dynamics and structure of radiative shocks', Physical Review E 111(4), 045213. DOI: 10.1103/physreve.111.045213. https://doi.org/10.1103/physreve.111.045213
  7. Hajime Inoue (2026), 'Energy transfer from jets to surrounding matter to form lateral lobes in SS 433/W 50', Publications of the Astronomical Society of Japan 78(3), 878–887. DOI: 10.1093/pasj/psag025, arXiv:2603.14399. https://arxiv.org/abs/2603.14399