Accretion Disks: How Matter Loses Its Orbit

Introduction: why matter does not fall straight in

Black holes are often portrayed as cosmic vacuum cleaners. That picture is misleading. Far away, a black hole attracts like any other object of the same mass. Matter with transverse velocity can orbit. To fall inward, it must lose enough angular momentum.

Angular momentum keeps matter rotating. In an isolated system it is conserved. A gas parcel cannot simply stop. An accretion disk is the collective machine that makes inward motion possible.

Gas exchanges energy, angular momentum, heat, magnetic field and radiation. Angular momentum moves outward. Mass moves inward. Gravitational binding energy becomes heat, light, turbulence, winds and sometimes jets.

Matter orbits rapidly and drifts inward slowly. Its mean path is a tightly wound spiral. The spiral is not a permanent groove. It is the statistical result of turbulent magnetized flow.

Orbit, disk and spiral

A circular orbit keeps constant radius. A conservative ellipse closes. A disk is matter distributed across radii. A spiral requires radius to change as angle advances.

In a thin disk, |v_φ| >> |v_r|. A parcel completes many orbits before moving far inward. Any drawing showing two or three open loops necessarily exaggerates the drift for visibility. The real spiral is much tighter. Turbulence adds deviations. The spiral is statistical and dynamic.

Disk formation

Accreting gas usually carries angular momentum. It may come from a companion, stellar wind, cloud, disrupted star, galaxy or debris. Collisions and shocks damp radial and vertical motions. The net angular momentum remains.

The flow flattens around its angular-momentum axis. Vertical balance gives approximately:

where H is the scale height, c_s the sound speed and Ω_K the Keplerian frequency. A thin disk has H/R << 1. Hotter flows are thicker.

Keplerian rotation

At distance R from mass M, the angular frequency and orbital velocity are:

Inner rings rotate faster. This is differential rotation. Specific angular momentum l = sqrt(GMR) increases with radius. To move inward, gas must transfer l elsewhere.

The angular-momentum lock

Without torque, matter remains ballistic. Gravity provides binding energy but does not erase angular momentum. Accretion requires stress.

Possible mechanisms include turbulence, magnetic fields, waves, shocks, gravitational torques and magnetized winds. In ionized disks around black holes, MRI is a central mechanism.

Mass inward, angular momentum outward

A coupling transfers angular momentum from faster inner rings to slower outer rings. The inner ring loses angular momentum and falls inward. The outer ring gains it and spreads outward. A small outer tail carries angular momentum. A wind can also carry it away.

Viscous evolution

For a thin Keplerian disk, the surface density Σ obeys the Lynden-Bell & Pringle diffusion equation:

An initially narrow ring spreads. Most mass may move inward. A small outer tail carries angular momentum. The viscous time is:

Since H/R is small, t_visc is far longer than the orbital time. This is why the spiral is so tightly wound.

Shakura–Sunyaev alpha model

Shakura and Sunyaev parameterized the turbulent viscosity:

The turbulent stress is often parameterized as T_Rφ = α P. Alpha summarizes transport efficiency. It is not a universal constant. Its value depends on magnetic field, geometry, pressure and regime. The model remains extremely useful for computing density, temperature, flux, spectrum and evolution time.

Magnetorotational instability

MRI destabilizes differentially rotating magnetized fluid. Two neighboring elements linked by magnetic tension exchange angular momentum. The outer element is accelerated outward. The inner element is braked inward. The displacement amplifies the coupling.

A key condition is:

Keplerian disks satisfy it. MRI produces magnetic turbulence and outward angular-momentum transport. The Maxwell stress, proportional to −B_R B_φ, often carries a large fraction of the angular momentum.

Turbulence and heating

Differential rotation contains free energy. The stress extracts it. It cascades and dissipates. For a steady thin disk, the radiative flux per face is:

The effective temperature satisfies σ_SB T_eff⁴ = F. Far from the inner edge: T_eff ∝ R^(−3/4). The total spectrum is a sum of annuli at different temperatures.

Energy efficiency

L = η Ṁ c². For a non-rotating black hole with thin disk:

Prograde Kerr disks can be more efficient because the ISCO is deeper. The ideal extremal limit approaches 0.42, while classic radiative spin equilibrium yields a lower practical maximum near 0.3. Hydrogen fusion converts about 0.7% of mass to energy. Efficient accretion can release far more per unit mass.

The radiation comes from matter outside the horizon. The black hole does not emit from its interior.

Temperature and spectrum

At comparable Eddington fraction, the thermal maximum of a thin disk scales approximately as:

A hot corona can produce X-rays by Comptonization. A radiatively inefficient flow can contain very hot electrons and ions. Disk, corona, jet and hot flow must be distinguished.

Eddington limit

The Eddington luminosity is:

It represents a simplified balance between gravity and radiation pressure in a spherical ionized flow. It is not an absolute ceiling. Super-Eddington flows can exist through non-spherical geometry, advection, photon trapping, winds, anisotropy and porosity.

Thin disk, slim, ADAF, SANE and MAD

The main accretion regimes are:

• Thin disk (SS73): low H/R, optically thick, radiatively efficient, thermal spectrum.
• Slim disk: high accretion rate, photon advection, moderate thickness, radiation less proportional to Ṁ.
• ADAF/RIAF: low density, radiatively inefficient, thick geometry, energy advected inward, possible winds.
• SANE: moderate disordered magnetic flux.
• MAD: strong accumulated magnetic flux, dynamically important field, intermittent inflow, powerful jets possible.

The boundaries are not sharp. A single source can change regime.

ISCO and inner edge

The ISCO is the innermost stable circular orbit. For Schwarzschild:

In the classic thin model, stress is often assumed zero at the ISCO. MHD simulations show that fields may transmit stress inside it. The ISCO remains a major dynamical landmark, not a wall.

Spin and efficiency

The dimensionless spin is a* = Jc/(GM²). It changes the ISCO, efficiency, frame dragging, precession and possible jet power. Measurement methods include thermal continuum, X-ray reflection, polarimetry, EHT imaging and dynamics. Each method depends on assumptions.

• a* = 0 (Schwarzschild): R_ISCO = 6 r_g, η ≈ 0.057
• a* = 0.998 (fast prograde): R_ISCO ≈ 1.24 r_g, η ≈ 0.32
• a* = −1 (retrograde): R_ISCO = 9 r_g, η ≈ 0.038

Relativistic appearance

A classical picture would show a flat disk around a dark center. Relativity changes this appearance. Gravity bends light. The back side of the disk appears above and below the black hole. The approaching side is Doppler boosted. The receding side is dimmed. Gravitational redshift reduces the energy of photons emitted deep in the potential.

Photons may complete a fraction of an orbit before reaching the observer. The result depends on spin, inclination, emissivity profile, opacity, plasma velocity and wavelength.

Lensing, Doppler and redshift

Define the frequency shift factor g = ν_obs/ν_emit. Relativistic invariance of specific intensity gives:

The approaching side can be brighter and bluer. The receding side is dimmer and redder. The ideal photon ring is linked to rays approaching unstable photon orbits. A real image contains direct emission, lensed images, plasma structure and instrumental reconstruction.

What EHT measures

EHT combines Earth-scale radio interferometry. At 1.3 mm, it reaches resolution near the horizon scale. In 2019, M87* showed an asymmetric ring, central depression, size consistent with Kerr and mass of about 6.5 billion solar masses. In 2022, EHT published Sgr A*.

M87* and Sgr A*

M87*: about 6.5 billion solar masses, powerful jet, slower evolution, dynamically important organized magnetic field. Sgr A*: about four million solar masses, extremely underluminous, minute-scale variability, hot inefficient flow.

The similar normalized ring geometry comes partly from Kerr geometry. Plasma physics and accretion regimes differ. M87* and Sgr A* are not identical apart from their size.

Magnetic fields and jets

The Blandford–Znajek mechanism extracts rotational energy through magnetic fields. Schematically:

The disk supplies magnetic flux. In a MAD state, the field becomes strong enough to disrupt accretion. Simulations show turbulence, interchange, intermittent accretion, magnetic funnel and relativistic jet.

Variability and QPOs

Real disks vary. Possible causes include MRI turbulence, reconnection, hotspots, precession, Ṁ variations, thermal instabilities and shocks. Characteristic timescales are dynamical t_dyn ~ 1/Ω, thermal t_th ~ t_dyn/α and viscous t_visc ~ (1/α)(R/H)² t_dyn.

Quasi-periodic oscillations exist in some X-ray binaries. Their origin remains debated: orbital frequencies, Lense–Thirring precession, modes, resonances or corona geometry.

Spiral waves and shocks

Some disks contain visible spiral patterns. In a binary, the companion exerts a tidal torque that can produce density waves, spiral shocks or tidal arms. Gas flows through these patterns. They are not identical to the mean accretion trajectory.

Three meanings of "spiral" in a disk must remain separate:

• Trajectory spiral: mean path of a gas parcel drifting inward.
• Pattern spiral: density wave or shock propagating through the disk.
• Apparent luminous spiral: observed emission pattern, influenced by lensing and kinematics.

Open questions

• Particle heating partition between ions and electrons in hot flows.
• Formation and structure of the corona.
• Magnetic flux accumulation and transition to MAD state.
• Collimation and composition of relativistic jets.
• Mass loss in winds and its role in the energy budget.
• Interpretation of quasi-periodic oscillations.
• Degeneracies in GRMHD models fitted to EHT data.
• Horizon-scale time structure in Sgr A*.

Where is the spiral?

An accretion disk looks like a continuous set of rings, not a single spiral line. Azimuthal velocity dominates: |v_φ| >> |v_r|. In a thin disk, a parcel completes many rotations before a significant decrease in radius. Its mean trajectory forms an extremely tightly wound spiral.

The real trajectory is turbulent, magnetized, compressible and sometimes crossed by waves or shocks. The spiral lies in fast rotation plus slow radial migration. It connects the transport equations to quasars, X-ray binaries, images of M87* and Sgr A*, and jets that influence entire galaxies.

Conclusion

An accretion disk solves the angular-momentum problem. Gravity pulls inward. Rotation resists direct infall. Magnetic turbulence carries angular momentum outward. Mass drifts inward. Gravitational binding energy becomes radiation, heat, winds and jets.

The spiral lies in fast rotation plus slow radial migration. Matter does not merely lose its orbit. It transfers that orbit to the rest of the system.