Effective Eddington Limit for Dust

• Effective Eddington limit for dust is defined as the luminosity threshold where radiation pressure on dusty gas balances gravity, modulated by dust opacities and the spectral energy distribution.
• This concept differentiates regimes in star-forming regions, AGN obscurers, and dusty accretion flows, separating stable hydrostatic conditions from radiative blowout scenarios.
• Methodologies involve opacity averaging, radiative transfer simulations, and observational diagnostics to understand momentum coupling, screening effects, and anisotropic gas dynamics.

Effective Eddington limit for dust denotes the luminosity or flux at which radiation pressure acting on dusty gas balances gravity, with the relevant coupling set by dust absorption and scattering rather than Thomson electron scattering. Because dust opacities can be much larger than the electron-scattering opacity, the critical luminosity for dusty gas is often far below the classical Eddington limit; however, the operative threshold depends on the spectral energy distribution (SED), the appropriate opacity average, the gas column density, attenuation and reprocessing, and the geometry of the source and absorber. In the literature, the concept is used for star-forming galaxies, star-forming subregions, AGN obscurers and winds, and dusty accretion flows, where it separates regimes of long-lived absorption or hydrostatic support from regimes of radiative blowout or outflow (Socrates et al., 2013, Arakawa et al., 2022, Vasudevan et al., 2013).

1. Formal definition and mathematical framework

In dusty media, the radiation force density is written as

$$\frac{\rho}{c}\int d\nu\, \kappa_\nu\, \mathbf{F}_\nu ,$$

where $\rho$ is the mass density, $\kappa_\nu$ is the total opacity, and $\mathbf{F}_\nu$ is the radiative flux. Setting the integrated radiation force equal to gravity yields a dusty Eddington luminosity

$$L_{\rm Edd} = \frac{4\pi G M_{\rm enc} c}{\kappa_F},$$

with $M_{\rm enc}$ the enclosed mass and $\kappa_F$ the flux-mean opacity (Socrates et al., 2013).

The flux-mean opacity is defined by

$$\kappa_F \equiv \frac{\int d\nu\, \kappa_\nu L_\nu}{\int d\nu\, L_\nu},$$

so the Eddington limit is determined by the opacity weighted by the emergent SED rather than by a single monochromatic dust opacity. In star-forming subregions, an analogous optically thin form is

$$\frac{L_{\rm Edd}}{M_{\rm tot}} = \frac{4\pi G c}{\langle \kappa_{\rm RP}\rangle},$$

where $\langle \kappa_{\rm RP}\rangle$ is the radiation-pressure mean opacity and $\rho$0 includes stars and gas (Blackstone et al., 2023).

For AGN, the classical electron-scattering Eddington luminosity is

$\rho$1

and the Eddington fraction is

$\rho$2

When dusty gas is considered, the effective absorption cross-section per proton due to dust, $\rho$3, is $\rho$4 times the Thomson value, so the effective Eddington luminosity is lower by this factor; the balance condition is commonly expressed as $\rho$5 in the $\rho$6 plane (Arakawa et al., 2022).

A phase-specific generalization used for AGN outflows is

$\rho$7

with $\rho$8 chosen for dusty or dust-free gas and $\rho$9 the luminosity fraction relevant for the force. In this formulation, dusty gas beyond the sublimation front can have $\kappa_\nu$0 much larger than dust-free gas at the same classical $\kappa_\nu$1 (Kudoh et al., 2024).

2. Opacity averaging, screening, and momentum coupling

A central result in dusty systems is that the opacity relevant for the force is usually not the UV peak opacity. In dust-rich star-forming galaxies, “the value of the flux-mean opacity that mediates the radiation force onto matter is orders of magnitude smaller than the UV or optical dust opacity,” because much of the emergent spectrum is in the far infrared (FIR), where dust opacity is much lower. Empirical values of $\kappa_\nu$2 are given as $\kappa_\nu$3 for such galaxies (Socrates et al., 2013).

The physical reason is radiative screening and reprocessing. In the diffusion-with-absorption description,

$\kappa_\nu$4

and the screening length is

$\kappa_\nu$5

For strong screening, $\kappa_\nu$6, the escaping UV flux is suppressed roughly as

$\kappa_\nu$7

Thus UV radiation pressure is confined largely to a thin surface layer, while most luminosity escapes as reprocessed FIR with weaker coupling (Socrates et al., 2013).

On smaller scales, momentum deposition through a dusty column is also limited by finite optical depth. Monte Carlo calculations for star-forming subregions found that the transfer function

$\kappa_\nu$8

is an excellent approximation for beamed radiation and a very good approximation for isotropic sources except for subtle corrections at large or small $\kappa_\nu$9 due to boundary scattering. Here

$\mathbf{F}_\nu$0

This provides a compact prescription linking extinction, SED, and the effective momentum transfer to gas (Blackstone et al., 2023).

Idealized radiation-hydrodynamic simulations of radiatively driven dusty winds further show that high optical depth does not guarantee large momentum amplification. After wind acceleration begins, radiation Rayleigh-Taylor instability forces the gas into a configuration that reduces the rate of momentum transfer by a factor $\mathbf{F}_\nu$1 compared to estimates based on the optical depth at the base of the atmosphere. The resulting momentum transfer is roughly $\mathbf{F}_\nu$2 multiplied by $\mathbf{F}_\nu$3, where $\mathbf{F}_\nu$4 is evaluated using the photospheric temperature; this is much smaller than the optical depth inferred from interior temperatures (Krumholz et al., 2013).

3. Star-forming galaxies, starbursts, and subregions

Galaxy-integrated analyses generally find that bright star-forming galaxies radiate well below their dusty Eddington limits. On empirical grounds, high-redshift ULIRGs radiate at two orders of magnitude below their Eddington limit, while the local starbursters M82 and Arp 220 radiate at a few percent of their Eddington limit. The SEDs of these systems peak in the FIR rather than the UV, and radiation pressure on dust does not greatly affect the large-scale gas dynamics of star-forming galaxies (Socrates et al., 2013).

Observational arguments for dusty radiation-pressure regulation were developed through comparisons of $\mathbf{F}_\nu$5 and molecular gas tracers. The linear $\mathbf{F}_\nu$6-$\mathbf{F}_\nu$7 correlation was presented as evidence that galaxies may be regulated by radiation pressure feedback, and star-forming galaxies were found to approach but not dramatically exceed Eddington. At the same time, many systems lie significantly below Eddington, with intermittency of star formation, dust-to-gas ratio, and CO- and HCN-to-$\mathbf{F}_\nu$8 conversion factors identified as key uncertainties (Andrews et al., 2011).

For self-gravitating mixed gas-star discs, one-dimensional models yield a maximum Eddington-limited star formation rate per unit area $\mathbf{F}_\nu$9, corresponding to a critical flux $$L_{\rm Edd} = \frac{4\pi G M_{\rm enc} c}{\kappa_F},$$0. Above this level, mass is ejected in bulk and further star formation is halted; below it, the models imply simple vertical hydrostatic equilibrium and radiation pressure is ineffective at driving turbulence or ejecting matter. This effective limit does not explain the Kennicutt-Schmidt relation, but it does impose an upper truncation on it (Crocker et al., 2018).

Spatially resolved studies complicate the galaxy-averaged picture. In NGC 6946 and NGC 5194, optically thin regions around young stellar populations are $$L_{\rm Edd} = \frac{4\pi G M_{\rm enc} c}{\kappa_F},$$1 times super-Eddington, but depending on the assumed height of the dusty gas and the age of the stellar population, only $$L_{\rm Edd} = \frac{4\pi G M_{\rm enc} c}{\kappa_F},$$2 of the sightlines are super-Eddington. Young clusters with ages $$L_{\rm Edd} = \frac{4\pi G M_{\rm enc} c}{\kappa_F},$$3 Myr have high $$L_{\rm Edd} = \frac{4\pi G M_{\rm enc} c}{\kappa_F},$$4 and high $$L_{\rm Edd} = \frac{4\pi G M_{\rm enc} c}{\kappa_F},$$5 ($$L_{\rm Edd} = \frac{4\pi G M_{\rm enc} c}{\kappa_F},$$6), and these regions may be accelerated to $$L_{\rm Edd} = \frac{4\pi G M_{\rm enc} c}{\kappa_F},$$7 km s$$L_{\rm Edd} = \frac{4\pi G M_{\rm enc} c}{\kappa_F},$$8 by radiation pressure alone (Blackstone et al., 2023).

M82 illustrates the scale dependence explicitly. In the superwind, the escaping UV luminosity perpendicular to the disk is $$L_{\rm Edd} = \frac{4\pi G M_{\rm enc} c}{\kappa_F},$$9 erg s$M_{\rm enc}$0, and on scales of $M_{\rm enc}$1 kpc above the plane the Eddington ratio is $M_{\rm enc}$2. On smaller scales of $M_{\rm enc}$3 kpc, $M_{\rm enc}$4 with factor-of-few uncertainties, while within the starburst itself the single-scattering Eddington ratio is of order unity. Radiation pressure is therefore weak compared to gravity on kpc scales, but it may be important in launching the outflow near the starburst (Coker et al., 2013).

4. AGN obscuration, the $M_{\rm enc}$5 plane, and dusty blowout

In AGN, the effective Eddington limit for dusty gas is commonly represented as a boundary in the plane of absorption column density $M_{\rm enc}$6 and Eddington fraction $M_{\rm enc}$7. Below the boundary, gravity dominates and long-lived absorption clouds can survive; above it, radiation pressure dominates and dusty gas is expelled as an outflow. The absence of objects in the region where the effective limit is exceeded is a central empirical motivation for the framework (Arakawa et al., 2022).

Detailed boundaries have been obtained with CLOUDY photoionization calculations for spherical, constant-density dusty shells. For each scenario, CLOUDY computes the mass-weighted mean radiative acceleration and compares it to the gravitational acceleration. Dust abundance has the largest effect on the $M_{\rm enc}$8 diagram, while the impact of the inner radius of the dusty shell, shell width, and AGN spectral shape is relatively negligible; changing shell radius and width shifts the boundary by $M_{\rm enc}$9. The presence of other central masses, such as a nuclear star cluster, raises the effective Eddington limit by increasing the gravitational force (Arakawa et al., 2022).

Observationally, there are very few AGN with larger Eddington ratio than the effective Eddington limit of the dusty gas, and the covering factor of dusty gas with $\kappa_F$0 exhibits a clear drop at the effective Eddington limit. Using extinction $\kappa_F$1 to isolate the dusty component and X-ray column density to separate dusty from dust-free gas, the dusty covering factor changes from $\kappa_F$2 at very low $\kappa_F$3 to $\kappa_F$4 or $\kappa_F$5 at higher $\kappa_F$6, supporting radiation-driven erosion of the dusty torus (Mizukoshi et al., 2024).

The same framework also predicts a forbidden region for short-lived absorption. Reanalysis of three AGN previously thought to lie close to the effective limit—NGC 454, 2MASX J03565655-4041453, and XSS J05054-2348—moved all three away from the region expected for short-lived absorption into the long-lived absorption region after improved XMM-Newton spectroscopy and refined bolometric luminosities and Eddington ratios. Evidence for absorption variability remained in NGC 454 and 2MASX J03565655-4041453, suggesting that radiation pressure may influence absorption changes even when it is insufficient to drive clearly detectable outflows (Vasudevan et al., 2013).

On sub-parsec scales, two-dimensional radiation-hydrodynamics simulations with $\kappa_F$7 SMBHs show that for $\kappa_F$8 radiation force exceeds gas pressure, leading to stronger outflows and a larger dust sublimation radius. The transition from thermally driven to radiation-pressure-dominated dusty outflows occurs around $\kappa_F$9, and the analytical solutions emphasize the dust sublimation scale as a determinant of terminal velocity and column density (Kudoh et al., 2024).

5. Geometry, anisotropy, and attenuation

The dusty effective Eddington limit is strongly geometry-dependent. In high accretion rate AGN observed with VLTI/GRAVITY, VLTI/MATISSE, and VLTI/MIDI, the inner dust emission sizes are measured to be $$\kappa_F \equiv \frac{\int d\nu\, \kappa_\nu L_\nu}{\int d\nu\, L_\nu},$$0 mas, with no signs of elongation. In H0557-385, the dust sizes between $$\kappa_F \equiv \frac{\int d\nu\, \kappa_\nu L_\nu}{\int d\nu\, L_\nu},$$1 are roughly constant at $$\kappa_F \equiv \frac{\int d\nu\, \kappa_\nu L_\nu}{\int d\nu\, L_\nu},$$2 sublimation radii, which was interpreted as a direct view of the wind-launching region. Together with the absence of polar elongation, this implies that any wind would be launched in a preferentially equatorial direction or blown out by strong radiation pressure. The size-wavelength relation in both H0557-385 and the super-Eddington source I Zw 1 indicates a preferentially disky equatorial dust distribution, and the results were taken as evidence that the Eddington ratio shapes the inner dust structure (Drewes et al., 17 Jan 2025).

Attenuation can also permit accretion above the nominal dusty-gas Eddington threshold. In Hoyle-Lyttleton accretion onto black hole accretion disks with dusty gas, the radiation force is weakened by attenuation via dust absorption, so gas can accrete even if the disk luminosity exceeds the Eddington luminosity for dusty gas. For gas flowing from the rotation axis direction with $$\kappa_F \equiv \frac{\int d\nu\, \kappa_\nu L_\nu}{\int d\nu\, L_\nu},$$3, the accretion rate is about $$\kappa_F \equiv \frac{\int d\nu\, \kappa_\nu L_\nu}{\int d\nu\, L_\nu},$$4 of the Hoyle-Lyttleton value if $$\kappa_F \equiv \frac{\int d\nu\, \kappa_\nu L_\nu}{\int d\nu\, L_\nu},$$5, but zero for $$\kappa_F \equiv \frac{\int d\nu\, \kappa_\nu L_\nu}{\int d\nu\, L_\nu},$$6. Because the radiation flux in the disk plane is small, accretion from near the disk plane is less impeded: for $$\kappa_F \equiv \frac{\int d\nu\, \kappa_\nu L_\nu}{\int d\nu\, L_\nu},$$7 and $$\kappa_F \equiv \frac{\int d\nu\, \kappa_\nu L_\nu}{\int d\nu\, L_\nu},$$8, the accretion rate is $$\kappa_F \equiv \frac{\int d\nu\, \kappa_\nu L_\nu}{\int d\nu\, L_\nu},$$9 of the Hoyle-Lyttleton value for $$\frac{L_{\rm Edd}}{M_{\rm tot}} = \frac{4\pi G c}{\langle \kappa_{\rm RP}\rangle},$$0, although accretion is impossible for $$\frac{L_{\rm Edd}}{M_{\rm tot}} = \frac{4\pi G c}{\langle \kappa_{\rm RP}\rangle},$$1 (Erika et al., 2021).

One-dimensional radiative-hydrodynamics simulations of dusty gas accretion onto $$\frac{L_{\rm Edd}}{M_{\rm tot}} = \frac{4\pi G c}{\langle \kappa_{\rm RP}\rangle},$$2 black holes likewise show that dust modifies both the force balance and the variability pattern. At dust-to-gas mass ratios similar to the solar neighborhood, the time-averaged luminosity is smaller than for primordial gas by one order of magnitude, and the time-averaged Eddington ratio ranges from $$\frac{L_{\rm Edd}}{M_{\rm tot}} = \frac{4\pi G c}{\langle \kappa_{\rm RP}\rangle},$$3 to $$\frac{L_{\rm Edd}}{M_{\rm tot}} = \frac{4\pi G c}{\langle \kappa_{\rm RP}\rangle},$$4 for initial gas densities $$\frac{L_{\rm Edd}}{M_{\rm tot}} = \frac{4\pi G c}{\langle \kappa_{\rm RP}\rangle},$$5. The effect of dust opacity alone is secondary compared to the radiation pressure on dust in regulating black-hole growth, and the flux ratio between $$\frac{L_{\rm Edd}}{M_{\rm tot}} = \frac{4\pi G c}{\langle \kappa_{\rm RP}\rangle},$$6 and $$\frac{L_{\rm Edd}}{M_{\rm tot}} = \frac{4\pi G c}{\langle \kappa_{\rm RP}\rangle},$$7 is closely related to the Eddington ratio (Yajima et al., 2017).

A non-spherical example appears in the classical nova V339 Del. Multiwavelength SED modeling found a super-Eddington white-dwarf luminosity of $$\frac{L_{\rm Edd}}{M_{\rm tot}} = \frac{4\pi G c}{\langle \kappa_{\rm RP}\rangle},$$8 erg s$$\frac{L_{\rm Edd}}{M_{\rm tot}} = \frac{4\pi G c}{\langle \kappa_{\rm RP}\rangle},$$9, lasting from approximately day 2 to at least day 100. The dust was inferred to reside in a slow, equatorially concentrated outflow or disk-like structure, allowing radiation to escape relatively unimpeded along polar directions while preserving dust in the equatorial ejecta for a long time. This suggests that a global luminosity comparison to a spherical dusty Eddington limit can be misleading when the opacity distribution is strongly anisotropic (Skopal, 2019).

6. Common simplifications and their limitations

A recurring simplification is to equate the dusty effective Eddington limit with a single opacity, often the UV opacity. The literature instead distinguishes the UV or optical dust opacity from the flux-mean or radiation-pressure-mean opacity that actually mediates the force. In dusty starbursts, the effective opacity can be $\langle \kappa_{\rm RP}\rangle$0, far below the UV value, because the emergent spectrum is FIR-dominated (Socrates et al., 2013).

A second simplification is to infer global dynamical importance from global luminosity alone. Galaxy-integrated studies find high-redshift ULIRGs, M82, and Arp 220 well below their dusty Eddington limits, yet resolved analyses of NGC 6946 and NGC 5194 identify optically thin lines of sight around young stellar populations that are $\langle \kappa_{\rm RP}\rangle$1 times super-Eddington, with $\langle \kappa_{\rm RP}\rangle$2 of sightlines super-Eddington depending on geometry and age assumptions (Blackstone et al., 2023).

A third simplification is to treat large optical depth as guaranteeing large momentum boosts. Radiation-hydrodynamic simulations show that radiation Rayleigh-Taylor instability and the resulting flux-density anti-correlation reduce the momentum transfer substantially, so wind momentum fluxes in ULIRGs and star clusters should not significantly exceed the direct radiation field $\langle \kappa_{\rm RP}\rangle$3 by more than a modest factor (Krumholz et al., 2013).

Collectively, these results suggest that the effective Eddington limit for dust is best understood not as a universal number but as a family of thresholds that depend on the SED, dust abundance and grain properties, column density, transfer function, dust sublimation radius, additional gravitational mass, and the angular structure of both radiation and gas. That interpretation is consistent with the contrast between long-lived absorbers and forbidden regions in AGN, the coexistence of globally sub-Eddington galaxies with locally super-Eddington subregions, and the ability of attenuation or anisotropy to permit accretion or emission states that would be excluded by a spherical, optically thin estimate (Arakawa et al., 2022, Crocker et al., 2018, Drewes et al., 17 Jan 2025).