Eddington Scales and Accretion Efficiency

The Eddington luminosity is a reference force-balance scale, not a universal luminosity ceiling. Accretion efficiency is an energy-accounting ratio, not a fixed property of a black hole. Used together, they organize observations and simulations only after opacity, geometry, mass-flow surface, and energy channels have been declared. This page derives the scale, separates the competing definitions of dimensionless accretion rate, and shows where binding-energy efficiency enters.

Required background. Fluid and Kinetic Theory: continuum kinematics and balance laws supplies material, momentum, and energy fluxes.

Helpful background. Black-Hole Geometry supplies horizon, circular-orbit, and ISCO quantities; Relativistic Matter and Compact Objects supplies opacity, radiation–matter coupling, and fluid closure.

Radiative force balance

Restore $c$ temporarily to expose dimensions. An isotropic luminosity $L$ produces flux

$$F(r)=\frac{L}{4\pi r^2}.$$

If matter has flux-mean opacity $\kappa$, the outward radiative acceleration is

$$a_{\rm rad}=\frac{\kappa F}{c} =\frac{\kappa L}{4\pi r^2c}.$$

Balancing this against the Newtonian inward acceleration $GM/r^2$ gives

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

In the site units $c=1$, this is $4\pi GM/\kappa$. The cancellation of $r^2$ is specific to spherical flux and inverse-square gravity. It does not say that the local force balance is radius independent in a disk, a funnel, an optically thick atmosphere, or a relativistic spacetime.

For fully ionized gas dominated by electron scattering,

$$\kappa_{\rm es}=\frac{\sigma_T}{\mu_e m_p},$$

where $\mu_e$ is the mean mass per free electron in proton-mass units. Hydrogen-rich material has a smaller opacity per unit mass than pure ionized hydrogen because helium adds mass without the same number of electrons per baryon. Bound–free, free–free, line, pair, dust, and Klein–Nishina effects can change the flux-mean opacity in other temperature and frequency regimes.

The force-balance derivation passes three immediate checks. Its dimensions are energy per time; it scales linearly with $M$; and increasing $\kappa$ lowers the luminosity required to accelerate matter. If any imported formula reverses the opacity dependence, a convention or algebra error has occurred.

Which mass accretion rate?

At least four rates can differ:

$$\dot M_{\rm supply}, \qquad \dot M(r), \qquad \dot M_{\rm H}, \qquad \dot M_{\rm grow}.$$

They denote external supply, inward mass flux at radius $r$, horizon-crossing rest mass, and retained black-hole growth. Winds can make $\dot M(r)$ decrease inward. Radiation and mechanical extraction make $\dot M_{\rm grow}$ smaller than the rest-mass flow through the horizon in the global energy accounting.

For a radiative luminosity $L_{\rm rad}$, define

$$\eta_{\rm rad}=\frac{L_{\rm rad}}{\dot M_{\rm H}c^2}.$$

Two common “Eddington accretion rates” are

$$\dot M_{\rm Edd}^{(A)}=\frac{L_{\rm Edd}}{c^2}, \qquad \dot M_{\rm Edd}^{(B)}=\frac{L_{\rm Edd}}{\eta_0c^2},$$

where $\eta_0$ is a chosen reference efficiency, often $0.1$. The corresponding dimensionless rates obey $\dot m_B=\eta_0\dot m_A$. A statement such as “$\dot m=10$” is incomplete until the definition and the radius at which $\dot M$ is measured are given. Yuan and Narayan 2014, § 1 including footnote 1, and §§ 2.4–2.5 makes this convention difference explicit in the context of hot flows.

The Eddington ratio

$$\lambda=\frac{L_{\rm bol}}{L_{\rm Edd}}$$

is an observed or inferred luminosity ratio. If $L_{\rm bol}$ is the total escaping radiative power and the same horizon mass rate is used, then $\lambda=\eta_{\rm rad}\dot m_A=(\eta_{\rm rad}/\eta_0)\dot m_B$. It is not generally equal to either definition of $\dot m$: radiative efficiency, bolometric correction, anisotropy, photon trapping, and winds intervene.

Binding energy and efficiency

For a stationary, radiatively efficient, geometrically thin disk with negligible torque at its inner edge, the specific orbital energy at the ISCO determines the ideal efficiency,

$$\eta_{\rm NT}=1-E_{\rm ISCO}(a_*).$$

For a Schwarzschild black hole, $E_{\rm ISCO}=\sqrt{8/9}$ and

$$\eta_{\rm NT}=1-\sqrt{\frac89}\simeq0.0572.$$

Spin changes the ISCO and the binding energy, with the sign defined relative to the disk orbit. Page and Thorne 1974, §§ II–IV, pp. 499–506 derives the relativistic conservation law and flux under these assumptions.

This number is not the efficiency of every flow at that spin. A nonzero inner stress changes the energy release. Advection can carry thermal energy inward. Winds carry rest mass, energy, and angular momentum away. A jet can extract black-hole rotational energy, so the sum of outward radiative and mechanical power divided by horizon rest-mass flux can exceed unity without violating energy conservation; the black-hole mass and spin supply the difference. Conversely, a hot two-temperature flow can have very small $\eta_{\rm rad}$ even while releasing substantial mechanical energy.

A complete steady budget may be written schematically as

$$\dot M_{\rm supply}c^2 =\dot E_{\rm H}+L_{\rm rad}+P_{\rm wind}+P_{\rm jet},$$

for cold supply from a distant boundary, with signs and measurement surfaces declared. Here $P_{\rm wind}$ and $P_{\rm jet}$ must include the rest-mass energy of any escaping matter. If instead they denote only mechanical and electromagnetic power above rest mass, use $\dot M_{\rm H}c^2$ on the left after subtracting the conserved outflow rest-mass flux. The horizon energy flux $\dot E_{\rm H}$ is not automatically $\dot M_{\rm H}c^2$ because binding, enthalpy, and electromagnetic flux enter; it already includes spin extraction and may be negative.

Where the scale fails as a ceiling

The Eddington derivation assumes spherical, steady, outward radiation acting on material with opacity $\kappa$. A disk radiates anisotropically. A funnel can collimate light, making the isotropic-equivalent luminosity exceed $L_{\rm Edd}$. Optically thick inflow can trap and advect photons. Porous or magnetized atmospheres can redistribute flux. Lines or dust can enhance the force; at high photon energies the electron-scattering cross section changes. General-relativistic redshift distinguishes local and asymptotic luminosity.

These effects do not make the reference scale meaningless. They specify what must be added. The useful question is not “Did the source violate Eddington?” but “Which opacity, frame, solid angle, mass-flow surface, and radiation–momentum budget map the measured luminosity to the flow?” Abramowicz and Fragile 2013, § 1 and §§ 5.1–5.3 places the scale within the thin, thick, and advective disk regimes.

Detailed spectra and instrument likelihoods are beyond this page’s boundary. The later radiation chapter develops source-specific spectral forward models, while calibrated detector response and likelihood construction remain with the applicable observing analysis; here they enter only through a declared bolometric luminosity and uncertainty.

Common pitfalls

Omitting the opacity. The familiar numerical coefficient assumes a composition and interaction. State $\kappa$ before comparing sources.

Using $\dot M_{\rm Edd}$ without its efficiency convention. The two definitions above differ by a factor of ten when $\eta_0=0.1$.

Equating supply with growth. Winds, radiation, jets, and binding energy separate the rates. Name the radius and flux surface.

Calling $L>L_{\rm Edd}$ impossible. The reference balance is spherical and local. Anisotropy, trapping, time dependence, opacity, and outflows change the inference.

Compare the two accretion-rate conventions

Use $\kappa=0.34\ {\rm cm^2\,g^{-1}}$ and

$$L_{\rm Edd}=1.47\times10^{38} \left(\frac{M}{M_\odot}\right) {\rm erg\,s^{-1}}.$$

Compute $L_{\rm Edd}$ and both $\dot M_{\rm Edd}$ definitions for $M=10M_\odot$ and $M=10^8M_\odot$, taking $\eta_0=0.1$. Express the rates in $M_\odot\,{\rm yr^{-1}}$.

Solution

For $10M_\odot$,

$$L_{\rm Edd}=1.47\times10^{39}\ {\rm erg\,s^{-1}}.$$

Dividing by $c^2$ and converting units gives

$$\dot M_{\rm Edd}^{(A)} =2.60\times10^{-8}M_\odot\,{\rm yr^{-1}}, \qquad \dot M_{\rm Edd}^{(B)} =2.60\times10^{-7}M_\odot\,{\rm yr^{-1}}.$$

The $10^8M_\odot$ values are larger by $10^7$:

$$L_{\rm Edd}=1.47\times10^{46}\ {\rm erg\,s^{-1}},$$ $$\dot M_{\rm Edd}^{(A)}=0.260M_\odot\,{\rm yr^{-1}}, \qquad \dot M_{\rm Edd}^{(B)}=2.60M_\odot\,{\rm yr^{-1}}.$$

Both conventions are internally valid. A numerical $\dot m$ changes by a factor of ten between them, so the definition must accompany the value. If a wind removes mass between the outer disk and horizon, neither outer rate alone determines $\eta_{\rm rad}=L/(\dot M_{\rm H}c^2)$.

References

• Abramowicz, Marek A., and P. Chris Fragile. “Foundations of Black Hole Accretion Disk Theory.” Living Reviews in Relativity 16 (2013): 1. DOI.
• 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.
• Yuan, Feng, and Ramesh Narayan. “Hot Accretion Flows Around Black Holes.” Annual Review of Astronomy and Astrophysics 52 (2014): 529–588. DOI.