Accretion Flows and Disks

Black-hole accretion is not a sequence of named spectral shapes but a set of conservation-law regimes. Optical depth, thickness, cooling, radial advection, magnetic flux, and mass loss decide which approximation applies. This chapter starts from Eddington units and efficient thin disks, then changes one physical ordering at a time until it reaches supercritical, hot, and magnetically arrested flows. It ends by asking why real disks move between those regimes.

Helpful background. Black-Hole Geometry supplies Kerr circular orbits, horizons, and the ISCO; Relativistic Matter and Compact Objects supplies opacity, plasma, and radiation closure; Fluid and Kinetic Theory supplies conservation laws, stresses, and kinetic timescales.

Route through the flow regimes

The six pages are ordered so that every new regime exposes which earlier assumption failed.

1. Eddington Scales and Accretion Efficiency derives the radiative force-balance scale, distinguishes competing definitions of dimensionless accretion rate, and closes radiation, advection, wind, and jet power.
2. Relativistic Thin Disks derives the radial flux from mass, angular-momentum, and energy conservation and makes the inner-torque assumption explicit.
3. Slim Disks and Supercritical Accretion introduces radial pressure, transonic inflow, photon trapping, and mass-losing winds when local cooling fails at high supply.
4. Radiatively Inefficient Hot Flows treats the low-density, two-temperature branch on which Coulomb coupling and radiation lose the race against inflow.
5. SANE and Magnetically Arrested Disks compares weak-net-flux evolution with magnetic-flux saturation using reproducible horizon and energy diagnostics.
6. Disk Instabilities and State Transitions tests thermal, ionization, geometric, and magnetic explanations against their timescales and hysteresis predictions.

These are not mutually exclusive adjectives. A flow can be optically thin and magnetically arrested, or optically thick, supercritical, and wind dominated. “Thin/slim/hot” primarily describes thermodynamics and geometry; “SANE/MAD” describes large-scale magnetic flux; “stable/unstable” describes response to perturbations.

The regime map

Begin with four ratios:

$$\frac{H}{R},\qquad \tau,\qquad \frac{t_{\rm cool}}{t_{\rm in}},\qquad \phi_{\rm BH} =\frac{\Phi_{\rm BH}} {\sqrt{\langle\dot M\rangle r_g^2c}}, \qquad r_g=\frac{GM}{c^2}.$$

A reference thin disk has $H/R\ll1$, $\tau\gg1$, and $t_{\rm cool}\ll t_{\rm in}$. A slim disk remains optically thick but allows $H/R$ and $t_{\rm cool}/t_{\rm in}$ to approach unity, so pressure and advection enter. A hot flow has $H/R\sim1$ and commonly $\tau\lesssim1$, with separate ion and electron energy equations. The magnetic state cuts across these branches: sustained $\phi_{\rm BH}$ of order $40$–$60$ in one common convention, together with arrested dynamics and interchange, identifies the canonical MAD regime.

No boundary is a universal sharp number. Opacity depends on composition and temperature; optical depth depends on direction; cooling includes the electron distribution and radiation field; the numerical value of $\phi_{\rm BH}$ depends on flux conventions. State the definition and use limiting checks instead of treating a label as an observable.

The Eddington luminosity provides a useful dimensional anchor,

$$L_{\rm Edd}=\frac{4\pi GMc}{\kappa},$$

but it is not a regime classifier by itself. Anisotropy, beaming, opacity, photon trapping, winds, and spin extraction can all separate an observed isotropic-equivalent luminosity from the local radiative force. Yuan and Narayan 2014, § 1 shows why even $\dot M_{\rm Edd}$ has multiple common definitions.

Conservation before interpretation

Every page uses the same bookkeeping principle. Across a stationary control volume, the supplied rest mass must be divided among horizon accretion and outflow,

$$\dot M_{\rm supply}=\dot M_H+\dot M_{\rm wind}+\dot M_{\rm jet},$$

with wind and jet counted as disjoint outward rest-mass channels. The energy budget includes radiation, matter, electromagnetic flux, and black-hole spin; evolving control volumes also require stored mass and energy. A spectral luminosity divided by a fixed efficiency is not automatically the supplied or horizon mass rate.

The thin-disk result illustrates the method. With negligible wind and a zero inner torque, its Newtonian surface flux is

$$F(R)=\frac{3GM\dot M}{8\pi R^3} \left[1-\left(\frac{R_{\rm in}}{R}\right)^{1/2}\right].$$

Its relativistic Page–Thorne counterpart replaces Newtonian binding energy and angular momentum with the circular-geodesic quantities and yields $\eta=1-E_{\rm ISCO}$ only under the same radiatively efficient, zero-torque assumptions. Page and Thorne 1974, §§ II–IV, pp. 499–506 derives that reference solution.

When a premise fails, the missing term must reappear: advection in a slim or hot flow, radius-dependent $\dot M$ in a wind, inner work for a stressed edge, or electromagnetic energy in a high-flux spinning system. This accounting is more durable than a catalog of model names.

Evidence and computation

Accretion is inferred through a forward chain:

$$\text{flow dynamics} \longrightarrow \text{electron and radiation closure} \longrightarrow \text{emitted field} \longrightarrow \text{propagation and instrument response} \longrightarrow \text{data}.$$

The chapter uses observations only where they test the physical regime. The supercritical page fixes a named XMM–Newton/NuSTAR data release; the hot-flow page uses a dated Sgr A* multiwavelength campaign. In each case calibration, selection, likelihood, and systematic alternatives remain visible. A good spectral fit cannot uniquely identify geometry when multiple electron, wind, and inclination models project to similar counts.

Computational claims require a second chain: exact initial and boundary data, code and dependency versions, algorithm, convergence, inflow equilibrium, conserved-flux residuals, and rerun instructions. The SANE/MAD page separates a small executable diagnostic from the production GRMHD calculation. Cross-code agreement improves with refinement on controlled torus benchmarks Porth et al. 2019, §§ 4–6, but a verified solver does not remove uncertainty from field topology, floors, radiative closure, or finite duration.

Connections to other topics

This chapter develops flow structure and budgets. For detailed spectra and instrument likelihoods, continue to Radiation, Spectra, and Variability. Large-scale magnetic-flux supply, force-free magnetospheres, jets, and wind launching continue in Magnetospheres, Jets, and Outflows. Conditional mappings from disk radius or reflection to spin continue in Electromagnetic Mass and Spin Inference.

The evidence boundary on dated observational and frontier pages is 2026-08-18. Conservation laws and limiting solutions are more durable; source states, calibrations, simulation convergence, and proposed transition mechanisms require reassessment as new data or calculations appear.

Classify two flows

Flow A has $\tau=200$, $H/R=0.5$, $t_{\rm diff}/t_{\rm in}=3$, and $\phi_{\rm BH}=12$. Flow B has $\tau=0.2$, $H/R=0.8$, $t_{\rm cool}/t_{\rm in}=8$, and sustained $\phi_{\rm BH}=50$ with interchange. Classify the thermodynamic and magnetic regime of each. What additional fact is needed before either observed spectrum is predicted?

Solution

Flow A is optically thick and moderately thick, while diffusion is slower than inflow. Photon trapping and advection therefore make it slim or supercritical rather than a locally cooling thin disk. Its low dimensionless horizon flux is SANE-like in the stated convention, provided the time history confirms that it never reached arrested saturation.

Flow B is optically thin, geometrically thick, and unable to cool within an inflow time, so it is a radiatively inefficient hot flow. Its sustained $\phi_{\rm BH}=50$ plus interchange supports a MAD classification.

Neither classification predicts a unique spectrum. One must additionally specify electron heating and distribution, magnetic structure, radiative transfer, inclination, mass scaling, and any wind or jet contribution before forwarding the flow into observables.

References

• Page, Don N., and Kip S. Thorne. “Disk-Accretion onto a Black Hole. Time-Averaged Structure of Accretion Disk.” Astrophysical Journal 191 (1974): 499–506. DOI.
• Porth, Oliver, et al. “The Event Horizon General Relativistic Magnetohydrodynamic Code Comparison Project.” Astrophysical Journal Supplement Series 243 (2019): 26. DOI.
• Yuan, Feng, and Ramesh Narayan. “Hot Accretion Flows around Black Holes.” Annual Review of Astronomy and Astrophysics 52 (2014): 529–588. DOI.