Accretion Disks: How Matter Loses Its Orbit

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;
• 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:

dphi/dt > 0 dR/dt < 0

In a thin disk:

$|v_\varphi|\gg|v_r|$

A parcel completes many orbits before moving far inward.

Real trajectories are turbulent rather than perfectly smooth.

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:

$H\approx\frac{c_s}{\Omega_K}$

A thin disk has:

H/R << 1

Hotter flows are thicker.

$\Omega_K=\sqrt{\frac{GM}{R^3}}$

$v_\varphi=\sqrt{\frac{GM}{R}}$

Inner rings rotate faster.

Specific angular momentum is:

$l=\sqrt{GMR}$

To move inward, gas must transfer l elsewhere.

Without torque, matter remains ballistic.

Gravity provides binding energy but does not erase angular momentum.

Accretion requires stress from:

• turbulence;
• magnetic fields;
• waves;
• shocks;
• gravity;
• winds.

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.

This is the central accretion paradox.

For a thin Keplerian disk:

$\frac{\partial\Sigma}{\partial t}=\frac{3}{R}\frac{\partial}{\partial R}\left[R^{1/2}\frac{\partial}{\partial R}\left(\nu\Sigma R^{1/2}\right)\right]$

An initially narrow ring spreads.

Most mass may move inward.

A small outer tail carries angular momentum.

The viscous time is:

$t_{\mathrm{visc}}\sim\frac{R^2}{\nu}$

or:

$t_{\mathrm{visc}}\sim\frac1\alpha\left(\frac RH\right)^2\frac1\Omega$

It is far longer than the orbital time.

Shakura and Sunyaev parameterized: ^[1]

$\nu=\alpha c_s H$

and often:

$T_{R\varphi}=\alpha P$

Alpha summarizes transport efficiency.

It is not a universal constant.

MRI destabilizes differentially rotating magnetized fluid. ^[2]

Two neighboring elements linked by magnetic tension exchange angular momentum.

The outer element is accelerated outward.

The inner element is braked inward.

A key condition is:

dOmega^2/d ln R < 0

Keplerian disks satisfy it.

MRI produces magnetic turbulence and outward angular-momentum transport.

For a steady thin disk:

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

and:

$\sigma_{\mathrm{SB}}T_{\mathrm{eff}}^4=F$

Far from the inner edge:

$T_{\mathrm{eff}}\propto R^{-3/4}$

The total spectrum is a sum of annuli at different temperatures.

$L=\eta\dot M c^2$

For Schwarzschild:

eta ≈ 0.057

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.

The radiation comes from matter outside the horizon.

At comparable Eddington fraction:

$T_{\max}\propto M^{-1/4}$

Stellar-mass black-hole thin disks are hotter and may peak in soft X-rays.

Supermassive thin disks are cooler and often peak in ultraviolet.

Hot coronae and radiatively inefficient flows are separate components.

$L_{\mathrm{Edd}}=\frac{4\pi GMm_p c}{\sigma_T}$

approximately:

$1.26\times10^{38}\left(\frac M{M_{\mathrm{Sun}}}\right)\,\mathrm{erg/s}$

It is not an absolute ceiling because geometry, advection, winds and anisotropy matter.

Thin disk: radiatively efficient and optically thick.

Slim disk: high accretion rate and photon advection.

ADAF/RIAF: hot, thick and radiatively inefficient.

SANE: moderate disordered magnetic flux.

MAD: strong accumulated magnetic flux, intermittent inflow and powerful jets.

For Schwarzschild:

$R_{\mathrm{ISCO}}=\frac{6GM}{c^2}$

The horizon is:

$R_H=\frac{2GM}{c^2}$

Spin moves the prograde ISCO inward and the retrograde ISCO outward.

The ISCO is not a surface.

Magnetic stress may persist inside it.

$a^*=\frac{Jc}{GM^2}$

Spin changes:

• ISCO;
• efficiency;
• frame dragging;
• precession;
• possible jet power.

Measurements depend on models.

Relativistic appearance

Gravity bends light.

The back side appears above and below the black hole.

The approaching side is Doppler boosted.

The receding side is dimmed.

Gravitational redshift changes photon energy.

The appearance depends on spin, inclination, emissivity, opacity and wavelength.

Define:

$g=\frac{\nu_{\mathrm{obs}}}{\nu_{\mathrm{emit}}}$

Specific intensity transforms as:

$I_{\mathrm{obs}}=g^3 I_{\mathrm{emit}}$

The observed ring includes direct emission, lensed images, plasma structure and instrument reconstruction.

EHT combines Earth-scale radio interferometry. ^[3]

M87* showed an asymmetric ring and central depression consistent with a Kerr shadow.

Sgr A* showed a similar normalized scale but much faster variability.

EHT reconstructs radio emission.

It does not photograph an event-horizon surface.

M87*:

• about 6.5 billion solar masses;
• powerful jet;
• slower evolution;
• dynamically important magnetic field.

Sgr A*:

• about four million solar masses;
• extremely underluminous;
• minute-scale variability;
• hot inefficient flow.

Similar ring geometry does not imply identical plasma physics.

Blandford–Znajek power scales schematically as:

$P_{\mathrm{BZ}}\propto\frac{\Phi_{\mathrm{BH}}^2\Omega_H^2}{c}$

The disk supplies magnetic flux.

A MAD state can support strong jets.

The jet is launched in plasma outside the horizon.

Polarized EHT images probe magnetic structure near M87*.

Disks vary through turbulence, reconnection, hotspots, precession and accretion fluctuations.

Timescales include:

dynamical; thermal; viscous.

QPO origins remain debated.

Some disks contain visible spiral density waves or shocks.

These are distinct from the mean spiral path of matter.

Three meanings must remain separate:

• parcel trajectory;
• wave pattern;
• observed brightness pattern.

Four tabs:

• parcel trajectory;
• viscous ring;
• temperature and spectrum;
• relativistic appearance.

Every simplification must be labeled.

Important unknowns include:

• particle heating;
• corona formation;
• magnetic-flux accumulation;
• jet collimation;
• winds;
• QPO interpretation;
• EHT model degeneracies;
• horizon-scale time variability.

Accretion disks solve the angular-momentum problem.

Gravity pulls inward.

Rotation resists direct infall.

Magnetic turbulence carries angular momentum outward.

Mass drifts inward.

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.