Eddington Accretion Ratio

• Eddington Accretion Ratio is a dimensionless parameter that measures an accreting black hole’s radiative output relative to its Eddington limit, defining accretion efficiency.
• It distinguishes regimes of sub-, near-, and super-Eddington accretion, directly affecting disk structure, spectral emissions, and variability in active galactic nuclei.
• Empirical methods, using spectral lines and continuum luminosities, enable the study of black hole growth and the cosmic evolution of AGNs.

The Eddington accretion ratio quantifies the proximity of an accreting black hole’s radiative luminosity to its theoretical Eddington limit, the luminosity at which radiation pressure on ionized gas counteracts gravity. This ratio, denoted typically as λEdd, is central to black hole accretion physics and underpins the classification of accretion regimes, the evolution of active galactic nuclei (AGN), the interpretation of quasar demographics, the growth of supermassive black holes (SMBHs), and the physical structure of accretion disks across mass and redshift. Empirically, λEdd regulates emission line strengths, variability, X-ray/optical spectral slopes, and is key to understanding transitions between radiatively efficient and inefficient accretion flows.

1. Definition, Formulae, and Physical Meaning

The Eddington ratio is defined as the ratio of the bolometric luminosity to the Eddington luminosity: $$\lambda_{\rm Edd} \equiv \frac{L_{\rm bol}}{L_{\rm Edd}}$$ where $L_{\rm bol}$ is the total radiative output. The Eddington luminosity for a black hole of mass $M_{\rm BH}$ is

$$L_{\rm Edd} = 1.26 \times 10^{38} \left( \frac{M_{\rm BH}}{M_\odot} \right) \, {\rm erg\,s^{-1}}$$

assuming pure hydrogen and Thomson opacity.

For dusty gas, the effective opacity increases, producing a reduced “dust-limited” Eddington luminosity: $$L_{\rm Edd,dust} = \frac{4\pi G M c}{\kappa_{\rm dg}} = \frac{\kappa_{\rm es}}{\kappa_{\rm dg}} L_{\rm Edd}$$ where $\kappa_{\rm dg}$ incorporates both electron scattering and dust absorption (Erika et al., 2021).

The dimensionless (mass) accretion rate is

$$\dot{m} \equiv \frac{\dot{M}}{\dot{M}_{\rm Edd}} \,; \qquad \dot{M}_{\rm Edd} \equiv \frac{L_{\rm Edd}}{\eta c^2}$$

with radiative efficiency $\eta$ (typically ~0.1 for thin disks). In radiatively efficient flows, $\lambda_{\rm Edd} \simeq \eta \, \dot{m}$, but for super-Eddington disks, photon trapping lowers efficiency so that λEdd saturates (Marziani et al., 20 Feb 2025).

A value λEdd ≪ 1 denotes sub-Eddington, radiatively efficient accretion; λEdd ≈ 1 is the classical Eddington-limited regime; λEdd > 1 implies super-Eddington, radiatively inefficient accretion with strong advection and/or outflow (Inayoshi et al., 2015, Dotan et al., 2010).

2. Measurement Techniques and Accretion Regimes

Measurement of λEdd

• In unobscured AGN, $L_{\rm bol}$ is derived from continuum luminosities at rest-frame optical (e.g. $L_{5100\,\mathrm{\AA}}$) or UV, with empirically calibrated bolometric corrections (Gupta et al., 16 Jul 2025).
• Black-hole mass estimates typically employ virial scaling relations using broad-line widths (Hβ, Mg II, Pβ, CIV) and line or continuum luminosities (Kim et al., 2015).
• In obscured or “red” quasars, optical measurements are strongly influenced by dust attenuation. The Pβ (1.28 μm) near-infrared line is much less extinguished (by a factor ~4 versus 860 for Hβ at E[B–V]=2 mag) and is thus a robust probe for λEdd determination in dusty systems (Kim et al., 2015).
• For low accretion rate sources (e.g., Sgr A*, M87), hot, radiatively inefficient flows correspond to λEdd far below 1, measurable via Bondi accretion estimates and observed radiative output (Inayoshi et al., 2017).

Regime Classification

λEdd regime	Accretion Physics	Disk Structure	Emission Characteristics
λEdd ≪ 0.01	Advection-dominated (ADAF/CDAF)	Geometrically thick, opt. thin	Hard X-ray-dominated, low variability
0.01 < λEdd < 1	Standard thin disk, disk-corona	Geometrically thin, opt. thick	Big blue bump, UV/optical, strong lines, variable X-ray
λEdd ≥ 1	Slim disk, photon trapping, strong winds	Geometrically thick, inflow/outflow, wind	Saturated L/L_Edd, soft-excess, wind/jet lines, reduced efficiency

Thus, transitions in λEdd mark structural and radiative changes in the accretion flow (Qiao et al., 2012, Inayoshi et al., 2015, Zhang et al., 2 Jun 2025).

3. Physical Implications and Theoretical Scaling

Transition in Disk and Corona Properties

• At λEdd < 0.01, thin-disk emission vanishes and the accretion flow is an ADAF or CDAF. The X-ray spectrum becomes harder, radiative efficiency drops, and the disk is geometrically thick and optically thin (Inayoshi et al., 2017, Qiao et al., 2012).
• For λEdd > 0.01, the disk is optically thick, and an inner thin disk survives. Increasing λEdd softens the X-ray spectrum due to enhanced Comptonization of disk photons in the corona; the optically thin corona becomes less dominant (Qiao et al., 2012, Gupta et al., 16 Jul 2025, Jiang et al., 2019).

Ionization State and Line Diagnostics

• The disk’s surface ionization parameter, ξ (ξ = 4π FX / nH), scales nearly linearly with λEdd in AGNs, i.e., log ξ = (1.008 ± 0.162) log λEdd + (3.14 ± 0.164), over a wide range. Departures from theoretical steeper slopes are attributable to changes in coronal fraction, surface density, and spin (Ballantyne et al., 2011).
• High-λEdd systems (“extreme Population A”, xA) show strong Fe II emission, elevated UV outflow velocities, and peculiar broad-line ratios (e.g., RFeII ≳ 1, FWHM(Hβ) < 4000 km/s), with counterparts in UV line diagnostics at higher redshift (Marziani et al., 20 Feb 2025, Panda et al., 1 Oct 2025).

Super-Eddington Accretion

• For much greater than unity λEdd, photon-trapping and advection in slim disks (H/R≳0.1–1) limit Lbol/Ledd to a slow logarithmic rise, while the accretion rate and mass flux can be orders of magnitude above the classical Eddington value (Inayoshi et al., 2015, Dotan et al., 2010, Zhang et al., 2 Jun 2025). Emergent luminosity is set by photon diffusion through the envelope and wind, with low radiative efficiency ($\eta \sim 1/\dot{m}$).
• Strong, optically thick winds are present; as the wind scale height approaches the radius (H/r→1), the system transitions to quasi-spherical outflows (Dotan et al., 2010).

4. Population Demographics and Cosmic Evolution

Redshift and Mass Dependence

• Large surveys show that for SMBHs, λEdd decreases both with decreasing redshift and increasing black hole mass. For example, at z ≈ 1.5, λEdd ≃ 0.4 for M_BH ≃ 10^7.25 M_⊙, falling to λEdd < 0.04 for M_BH ≃ 10^{10.25} M_⊙ (Aggarwal, 27 Apr 2024, Shirakata et al., 2019).
• The decline in λEdd with cosmic time at fixed mass is well established across both observational and semi-empirical models (Aggarwal, 27 Apr 2024, Shankar et al., 2011, Carraro et al., 2022).
• The mean Eddington ratio at fixed stellar mass, ⟨λEdd⟩, follows strong evolution, e.g., ⟨λEdd⟩(z) ≃ 0.01 [(1+z)/1.45]^{4.3}, with near-Eddington accretion at z ≳ 2 (Carraro et al., 2022).

Duty Cycle, AGN Fractions, and Specific Growth

• In continuity-equation models, widespread AGN “downsizing” is reconciled by requiring λEdd distributions to shift to lower typical values with time and with increasing M_BH; broadening P(log λEdd) at late times is also required (Shankar et al., 2011).
• The observed evolution of mean X-ray luminosity versus stellar mass, ⟨LX⟩–M*, is fundamentally set by the mean λEdd at each epoch, rather than by the AGN duty cycle, which is nearly constant (Carraro et al., 2022).

Selection Effects and Observational Biases

• Selection thresholds (luminosity or flux limits) naturally truncate the low-λEdd portion of observed distributions, misrepresenting the true population and biasing inferred growth rates, especially for low-mass and high-mass extremes (Shirakata et al., 2019).

5. λEdd in AGN/Quasar Phenomenology and Diagnostics

Variability and Empirical Relations

• Optical variability amplitude (Fvar) exhibits a nearly universal, redshift-independent anti-correlation with λEdd: log λEdd = (−0.71 ± 0.06) log Fvar − (1.52 ± 0.06) (Panda et al., 1 Oct 2025). This relation is leveraged for rapid λEdd estimation in time-domain surveys.
• The scatter in the broad-line region radius–luminosity (R–L) relation is reduced by parameterizing as a function of RFeII (proxy for λEdd), enabling AGN standardization for cosmological work (Panda et al., 1 Oct 2025).

Changing-Look AGNs and State Transitions

• Changing-Look AGNs (CLAGNs) cluster in the low-λEdd regime (log λEdd ≈ −2 to −1.7), with transitions between AGN “types” driven by rapid changes in accretion rate (Panda et al., 1 Oct 2025).
• The two-regime X-ray/optical continuum and emission-line behavior in X-ray binaries and AGN fundamentally reflects transitions in λEdd across ~0.01 (Qiao et al., 2012, Gupta et al., 16 Jul 2025).

Multiwavelength Diagnostics

• In dusty accretion environments, the mid-to-far infrared flux ratio (e.g., f_{14/140} = F_{14μm}/F_{140μm}) tightly traces λEdd, enabling estimation even where UV/optical fluxes are unobservable (Yajima et al., 2017).
• The fraction of power emitted in X-rays (the X-ray bolometric correction) is principally set by λEdd, not by luminosity alone; above λEdd ≃ 0.01, increasing λEdd produces stronger disk Compton cooling, decreasing X-ray output as a fraction of Lbol (Gupta et al., 16 Jul 2025).

6. Broader Context: Feedback, Growth, and Cosmological Impact

• High λEdd and super-Eddington accretion phases are required for the early, rapid growth of SMBHs to ≳10^9 M_⊙ at z ≳ 6, as revealed by both theoretical arguments and observations of high-z quasars (Marziani et al., 20 Feb 2025, Shirakata et al., 2019).
• Super-Eddington accretion phases are accompanied by strong winds and outflows, providing feedback mechanisms capable of influencing star formation and establishing black hole–galaxy scaling relations (Marziani et al., 20 Feb 2025).
• λEdd serves as a “second parameter” in quasar main-sequence schemes: line ratios, SED hardening, variability, and size-luminosity offsets all fundamentally depend on this parameter (Panda et al., 1 Oct 2025).
• The non-universality of radiative efficiency η, and its strong dependence on both mass and redshift inferred through λEdd distributions, compels revision of black hole growth and evolutionary models based on assumed constant η (Aggarwal, 27 Apr 2024, Shankar et al., 2011).

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In summary, the Eddington accretion ratio λEdd is a dimensionless parameter encapsulating the balance between radiative output and the fundamental limit set by black hole mass and radiative physics. It governs the mode of accretion, sets key emission properties, drives the cosmic evolution of quasars and AGNs, and serves as a critical variable in linking theoretical models with observations across environments, epochs, and mass scales.