Eddington Winds: Radiation-Driven Outflows

• Eddington winds are radiation-driven outflows initiated when the radiative force approaches or exceeds gravitational pull, observed in systems from accretion disks to massive stars.
• They operate in distinct momentum-limited and energy-limited regimes, with properties governed by opacity sources, geometry, and photon-trapping effects.
• Research on these winds informs our understanding of phenomena in ULXs, AGN, and starbursts, linking radiative mechanisms to observable mass loss, velocity, and feedback impacts.

Searching arXiv for recent and foundational papers on Eddington winds to support the article. Eddington winds are outflows launched when radiation pressure becomes strong enough to overcome gravity. In its classical optically thin form, the threshold is set by the Eddington luminosity,

$L_{\rm Edd} = \frac{4\pi G M c}{\kappa},$

with radiative acceleration

$g_{\rm rad} = \frac{\kappa F}{c}.$

A wind can be launched where $g_{\rm rad} > GM/r^2$. Across the literature, the term encompasses several related but physically distinct regimes: continuum-driven winds from accretion disks near or above the Eddington limit, line-driven winds in UV-bright disks, optically thick stellar winds in radiation-pressure-dominated envelopes, dusty infrared-trapped winds in starbursts, and short-lived super-Eddington outflows from thermonuclear flashes on neutron stars. What unifies them is that the ratio of radiative to gravitational force approaches or exceeds unity, while the observable velocity, ionization, optical depth, mass loading, and feedback efficiency depend on geometry, opacity source, and the degree of photon trapping (Pinto et al., 2019, Zubovas et al., 2013, Quataert et al., 2015, Zhang et al., 2016).

1. Classical limit and general driving principles

The canonical starting point is the balance between gravity and radiative force. For fully ionized hydrogen, the electron-scattering opacity is $$\kappa \approx \kappa_{\rm es} \simeq 0.34\,\mathrm{cm^2\,g^{-1}}$$, and the Eddington luminosity is

$L_{\rm Edd} = \frac{4\pi G M c}{\kappa}.$

In accretion problems, a standard dimensionless form is

$$\dot m \equiv \frac{\dot M}{\dot M_{\rm Edd}}, \qquad \dot M_{\rm Edd} = \frac{L_{\rm Edd}}{\epsilon c^2},$$

with $\epsilon \approx 0.08$ in one ULX wind analysis and $\eta \simeq 0.1$ in several AGN and stellar contexts (Pinto et al., 2019, Zubovas et al., 2013).

A central distinction is between momentum-limited and energy-limited regimes. In mildly super-Eddington electron-scattering winds, the thrust obeys

$\dot M_w v \simeq \tau \frac{L}{c},$

with $\tau \simeq 1$ giving the familiar Eddington-wind scaling $g_{\rm rad} = \frac{\kappa F}{c}.$0. In heavily mass-loaded or optically thick flows, multiple scattering and photon trapping modify this simple limit, and the wind kinetic luminosity can approach the available radiative or deposited power (Zubovas et al., 2013, Owocki et al., 2017, Quataert et al., 2015).

A second distinction is the opacity source. In AGN and hot-star winds, line opacity can amplify the continuum force through a force multiplier, whereas in highly ionized gas line driving becomes ineffective and continuum driving dominates. In dusty starbursts, infrared opacity on grains replaces electron scattering as the relevant coupling agent, and the Eddington ratio becomes

$g_{\rm rad} = \frac{\kappa F}{c}.$1

In optically thick infrared atmospheres, the net momentum transfer can exceed $g_{\rm rad} = \frac{\kappa F}{c}.$2 by a factor $g_{\rm rad} = \frac{\kappa F}{c}.$3, with $g_{\rm rad} = \frac{\kappa F}{c}.$4 (Zhu et al., 2022, Zhang et al., 2016).

Geometry is not secondary. Uniformly bright self-gravitating disks differ qualitatively from spherical sources because the ratio of radiation force to gravity increases with height and can reach twice its surface value when the luminous and gravitating radii are comparable. This makes an Eddington-limited disk intrinsically unstable to launching a wind, whereas a spherical source requires $g_{\rm rad} = \frac{\kappa F}{c}.$5 for sustained acceleration (Zhang et al., 2010).

2. Accretion-disk Eddington winds in black-hole and neutron-star systems

In super-Eddington accretion flows, radiation pressure inflates the inner disk, produces a funnel-like geometry, and launches a wind from near and inside the spherization radius. In ULXs, the emergent spectral energy distribution is described as a broadened multi-component continuum composed of a soft blackbody-like component with $g_{\rm rad} = \frac{\kappa F}{c}.$6, a hotter broadened multicolor blackbody with $g_{\rm rad} = \frac{\kappa F}{c}.$7, and a power law with $g_{\rm rad} = \frac{\kappa F}{c}.$8 (Pinto et al., 2019). The same supercritical disk framework underlies the interpretation of optically thick photospheric winds in ultraluminous supersoft sources and SS 433–like systems, where the photospheric luminosity is regulated near the Eddington scale while the photosphere expands and the color temperature drops (Zhou et al., 2018, King et al., 2015).

Photoionized ULX winds are generally thermally stable under ULX spectral energy distributions. Stability is diagnosed on $g_{\rm rad} = \frac{\kappa F}{c}.$9-curves in the $g_{\rm rad} > GM/r^2$0 plane, where

$g_{\rm rad} > GM/r^2$1

The observed ULX wind phases lie on stable, positive-slope branches; softer and intermediate ULXs occupy $g_{\rm rad} > GM/r^2$2 with temperatures from a few $g_{\rm rad} > GM/r^2$3 to $g_{\rm rad} > GM/r^2$4 K, while harder ULXs reach $g_{\rm rad} > GM/r^2$5 and temperatures on the Compton arm near $g_{\rm rad} > GM/r^2$6 K (Pinto et al., 2019).

A related but not identical framework is the AGN Eddington-wind model that unifies ultrafast outflows and BAL QSO winds through mass loading. For a supercritical disk with

$g_{\rm rad} > GM/r^2$7

the wind velocity becomes

$g_{\rm rad} > GM/r^2$8

This directly encodes the statement that higher wind mass-loss rates lower both the terminal speed and the ionization parameter. Typical UFOs then correspond to modest mass loading, high $g_{\rm rad} > GM/r^2$9, and velocities $$\kappa \approx \kappa_{\rm es} \simeq 0.34\,\mathrm{cm^2\,g^{-1}}$$0, while BAL QSO winds correspond to $$\kappa \approx \kappa_{\rm es} \simeq 0.34\,\mathrm{cm^2\,g^{-1}}$$1 and velocities commonly $$\kappa \approx \kappa_{\rm es} \simeq 0.34\,\mathrm{cm^2\,g^{-1}}$$2 (Zubovas et al., 2013).

Large-scale 2D simulations of line-driven thin-disk winds show that escape depends on both black-hole mass and accretion-disk luminosity. At $$\kappa \approx \kappa_{\rm es} \simeq 0.34\,\mathrm{cm^2\,g^{-1}}$$3, winds have kinetic energy flux exceeding $$\kappa \approx \kappa_{\rm es} \simeq 0.34\,\mathrm{cm^2\,g^{-1}}$$4 and escape for $$\kappa \approx \kappa_{\rm es} \simeq 0.34\,\mathrm{cm^2\,g^{-1}}$$5; at $$\kappa \approx \kappa_{\rm es} \simeq 0.34\,\mathrm{cm^2\,g^{-1}}$$6, the wind power declines strongly toward lower mass, reaching $$\kappa \approx \kappa_{\rm es} \simeq 0.34\,\mathrm{cm^2\,g^{-1}}$$7 at $$\kappa \approx \kappa_{\rm es} \simeq 0.34\,\mathrm{cm^2\,g^{-1}}$$8, where the flow fails to escape (Zhu et al., 2022). This suggests that sub-Eddington line driving can reproduce some UFO phenomenology, but only when the UV field and ionization state are favorable.

At the phenomenological level, wind power, momentum, and outflow mode correlate with Eddington fraction across the black-hole mass scale. A uniform Chandra grating analysis found

$$\kappa \approx \kappa_{\rm es} \simeq 0.34\,\mathrm{cm^2\,g^{-1}}$$9

with a transition near $L_{\rm Edd} = \frac{4\pi G M c}{\kappa}.$0 such that jets dominate at low Eddington fraction and winds dominate at high Eddington fraction (King et al., 2012).

3. Thermal state, launch radii, and observational signatures in compact accretors

In ULXs, the geometry of the thick disk and wind produces a specific observational synthesis. Harder spectra correspond to more highly ionized and faster winds, consistent with low-inclination sightlines sampling the inner funnel. A straight-line fit to the ULX sample gave

$L_{\rm Edd} = \frac{4\pi G M c}{\kappa}.$1

and

$L_{\rm Edd} = \frac{4\pi G M c}{\kappa}.$2

with Spearman/Pearson coefficients 0.6–0.8 (Pinto et al., 2019). The same work connects the spectral residuals around 1 keV in soft/intermediate ULXs to Fe L-shell blends and Ne IX–X formed at $L_{\rm Edd} = \frac{4\pi G M c}{\kappa}.$3.

Detailed AGN case studies sharpen the Eddington-wind picture. In PG 1448+273, a broad iron K absorption profile at $L_{\rm Edd} = \frac{4\pi G M c}{\kappa}.$4 keV and $L_{\rm Edd} = \frac{4\pi G M c}{\kappa}.$5 keV is modeled as a fast, geometrically thick disk wind with $L_{\rm Edd} = \frac{4\pi G M c}{\kappa}.$6, a mass outflow rate $L_{\rm Edd} = \frac{4\pi G M c}{\kappa}.$7, momentum rate $L_{\rm Edd} = \frac{4\pi G M c}{\kappa}.$8, and kinetic power $L_{\rm Edd} = \frac{4\pi G M c}{\kappa}.$9 (Reeves et al., 2024). In PG1211+143, a time-resolved XMM-Newton campaign captured an ultra-fast inflow and, weeks later, the launch of a new outflow at $$\dot m \equiv \frac{\dot M}{\dot M_{\rm Edd}}, \qquad \dot M_{\rm Edd} = \frac{L_{\rm Edd}}{\epsilon c^2},$$0, consistent with continuum driving from radii of order tens of $$\dot m \equiv \frac{\dot M}{\dot M_{\rm Edd}}, \qquad \dot M_{\rm Edd} = \frac{L_{\rm Edd}}{\epsilon c^2},$$1 when the local accretion rate becomes super-Eddington (Pounds et al., 2023).

The relation between observed high-Eddington winds and launch mechanism is not always straightforward. XRISM Resolve spectroscopy of GX 13+1 shows a Compton-thick wind with slow and fast components at $$\dot m \equiv \frac{\dot M}{\dot M_{\rm Edd}}, \qquad \dot M_{\rm Edd} = \frac{L_{\rm Edd}}{\epsilon c^2},$$2 and $$\dot m \equiv \frac{\dot M}{\dot M_{\rm Edd}}, \qquad \dot M_{\rm Edd} = \frac{L_{\rm Edd}}{\epsilon c^2},$$3, respectively, with $$\dot m \equiv \frac{\dot M}{\dot M_{\rm Edd}}, \qquad \dot M_{\rm Edd} = \frac{L_{\rm Edd}}{\epsilon c^2},$$4 and intrinsic luminosity at or above Eddington. Yet the measured velocities and $$\dot m \equiv \frac{\dot M}{\dot M_{\rm Edd}}, \qquad \dot M_{\rm Edd} = \frac{L_{\rm Edd}}{\epsilon c^2},$$5 are much more consistent with a thermal-radiative wind launched from the outer disk than with an inner-disc radiation-pressure wind at $$\dot m \equiv \frac{\dot M}{\dot M_{\rm Edd}}, \qquad \dot M_{\rm Edd} = \frac{L_{\rm Edd}}{\epsilon c^2},$$6 (Collaboration et al., 18 Sep 2025). This directly constrains the use of “Eddington wind” as a purely luminosity-based label.

Type I X-ray bursts provide another compact-object realization. In photospheric radius-expansion bursts, unstable helium burning temporarily drives $$\dot m \equiv \frac{\dot M}{\dot M_{\rm Edd}}, \qquad \dot M_{\rm Edd} = \frac{L_{\rm Edd}}{\epsilon c^2},$$7 and launches an optically thick wind from the neutron-star surface. Time-dependent MESA simulations give a nearly constant ejected mass fraction $$\dot m \equiv \frac{\dot M}{\dot M_{\rm Edd}}, \qquad \dot M_{\rm Edd} = \frac{L_{\rm Edd}}{\epsilon c^2},$$8, implying that about $$\dot m \equiv \frac{\dot M}{\dot M_{\rm Edd}}, \qquad \dot M_{\rm Edd} = \frac{L_{\rm Edd}}{\epsilon c^2},$$9 of the nuclear energy release unbinds matter. Typical mass-loss rates are $\epsilon \approx 0.08$0, the photosphere expands to $\epsilon \approx 0.08$1 km, and after about $\epsilon \approx 0.08$2 s the wind composition transitions from He/C to heavy ashes with $\epsilon \approx 0.08$3 (Yu et al., 2018).

4. Stellar Eddington winds and the role of the Eddington parameter

In massive and very massive stars, Eddington winds are tied to the approach to the electron-scattering Eddington parameter

$\epsilon \approx 0.08$4

A central result of the WNh-star literature is that Wolf-Rayet–type mass loss is governed more directly by $\epsilon \approx 0.08$5 than by surface helium enrichment (Gräfener et al., 2011). Near the Eddington limit, the effective gravity is reduced according to

$\epsilon \approx 0.08$6

and the escape speed scales as

$\epsilon \approx 0.08$7

This favors optically thick winds initiated deeper in the atmosphere, where the iron-opacity bump becomes dynamically important (Gräfener et al., 2011).

For the Arches WNh sample, an empirical fit of the form

$\epsilon \approx 0.08$8

was found to describe the WR-regime mass-loss trend (Gräfener et al., 2011). A later CAK-based extension to optically thick main-sequence winds recast the mass-loss dependence explicitly in terms of $\epsilon \approx 0.08$9, giving the LMC calibration

$\eta \simeq 0.1$0

with transition values around $\eta \simeq 0.1$1 (Bestenlehner, 2022).

A different stellar regime arises when super-Eddington energy deposition occurs near the surface, as in luminous blue variables, classical novae, and some pre-supernova envelopes. In this optically thick limit photons are trapped and the wind behaves as a radiation-dominated fluid rather than a flux-driven CAK outflow. The characteristic control parameter is

$\eta \simeq 0.1$2

and the main regimes are set by $\eta \simeq 0.1$3 relative to the local escape speed $\eta \simeq 0.1$4 (Quataert et al., 2015). When $\eta \simeq 0.1$5, most of the deposited power becomes wind kinetic luminosity; when $\eta \simeq 0.1$6, most of the power goes into unbinding mass. This framework was subsequently unified with flux-driven descriptions through the photon-tiring parameter

$\eta \simeq 0.1$7

yielding the bridge relation

$\eta \simeq 0.1$8

for optically thick super-Eddington stellar winds (Owocki et al., 2017).

Rotation alters continuum-driven stellar winds in a systematic way. With gravity darkening, the surface flux becomes latitude dependent and the net continuum driving is concentrated toward the poles. The associated weighting function is

$\eta \simeq 0.1$9

and the local mass flux scales as $\dot M_w v \simeq \tau \frac{L}{c},$0 (Shacham et al., 2012). This produces bipolar outflows and, just above the Eddington limit, can even spin up the star because the wind removes relatively little angular momentum per unit mass.

5. Supermassive stars, dusty media, and galactic-scale Eddington winds

In supermassive stars, continuum-driven Eddington winds are modified by porosity. Instabilities in radiation-pressure-dominated atmospheres reduce the effective opacity according to

$\dot M_w v \simeq \tau \frac{L}{c},$1

with $\dot M_w v \simeq \tau \frac{L}{c},$2 as a theoretically suggested threshold in one treatment (Dotan et al., 2012). The resulting mass-loss rate is written

$\dot M_w v \simeq \tau \frac{L}{c},$3

with $\dot M_w v \simeq \tau \frac{L}{c},$4 set by clump geometry. These winds shorten the evolution of non-rotating supermassive stars and suppress ionizing-radiation output by pushing the photosphere to recombination temperatures, but they are not sufficient to evaporate the star before collapse to a supermassive black hole unless rotation stabilizes the configuration (Dotan et al., 2012).

Accretion disks can also enter a porous super-Eddington regime. In super-Eddington slim-disk models with a reduced effective opacity, the disk remains slim rather than becoming globally thick, while launching a significant wind with thick-disk geometry. The total luminosity exceeds Eddington above about $\dot M_w v \simeq \tau \frac{L}{c},$5 times the standard critical mass accretion rate, the central object accretes above the Eddington accretion rate above about $\dot M_w v \simeq \tau \frac{L}{c},$6 times the critical rate, and the wind becomes spherical above about $\dot M_w v \simeq \tau \frac{L}{c},$7, where the model breaks down (Dotan et al., 2010).

On galactic scales, radiation pressure on dust defines a separate class of Eddington wind. In rapidly star-forming, infrared-thick environments, the relevant Eddington flux is

$\dot M_w v \simeq \tau \frac{L}{c},$8

and the infrared optical depth is

$\dot M_w v \simeq \tau \frac{L}{c},$9

Variable-Eddington-tensor simulations found efficient momentum coupling with

$\tau \simeq 1$0

where $\tau \simeq 1$1 decreases with increasing $\tau \simeq 1$2 (Zhang et al., 2016). This differs from flux-limited-diffusion results that produced a declining trapping factor and weak coupling.

A complementary analytic treatment of uniformly bright self-gravitating disks showed that the Eddington ratio increases with height and approaches

$\tau \simeq 1$3

For $\tau \simeq 1$4, a disk at the Eddington limit is therefore unstable to wind launching, with a characteristic terminal speed

$\tau \simeq 1$5

in the idealized disk-only case (Zhang et al., 2010). This establishes a geometric route to large-scale dusty Eddington winds that does not require a globally super-Eddington spherical source.

6. Energetics, feedback, and outstanding ambiguities

The energetics of Eddington winds are often summarized through mass-loss rate, momentum flux, and kinetic power. In photoionized compact-object winds,

$\tau \simeq 1$6

and using $\tau \simeq 1$7 one may write

$\tau \simeq 1$8

(Pinto et al., 2019). In AGN and X-ray binaries this makes the inferred mechanical power very sensitive to velocity. This is why the highest-velocity UFOs often appear mechanically closer to jets than to slow winds (King et al., 2012).

Feedback implications depend on both power and coupling. In AGN, near-Eddington winds with $\tau \simeq 1$9 underpin the canonical momentum-driven derivation of

$g_{\rm rad} = \frac{\kappa F}{c}.$00

while hyper-Eddington winds with larger thrust would predict smaller black-hole masses at fixed $g_{\rm rad} = \frac{\kappa F}{c}.$01 than are observed (Zubovas et al., 2013, King et al., 2015). In hot and super-Eddington accretion flows more broadly, recent theoretical work emphasizes that winds often dominate jets in momentum coupling even when jets dominate in energy, largely because winds are wide-angle and couple more efficiently to the ambient medium (Yang et al., 2024).

Several ambiguities remain. One concerns the relative roles of continuum driving, line driving, thermal driving, and magnetic driving. In luminous NLS1s such as PG 1448+273, the measured $g_{\rm rad} = \frac{\kappa F}{c}.$02 and high momentum rate challenge pure single-scattering continuum driving and are compatible with magnetohydrodynamic assistance or a strong UV contribution (Reeves et al., 2024). In ULX pulsars and other neutron-star systems, strong magnetic fields can truncate the inner disk and alter or suppress radiation-driven disk winds (Pinto et al., 2019). In GX 13+1, high Eddington ratio does not translate into a classical ultrafast inner-disk wind along the observed line of sight, illustrating that luminosity alone does not determine the observed wind type (Collaboration et al., 18 Sep 2025).

Another uncertainty concerns thermal stability, clumping, and code dependence. ULX wind stability at low $g_{\rm rad} = \frac{\kappa F}{c}.$03 is sensitive to UV uncertainties and self-shielding (Pinto et al., 2019). Photoionization codes such as SPEX, XSTAR, and CLOUDY differ at the 10–30% level in $g_{\rm rad} = \frac{\kappa F}{c}.$04 and ionic fractions (Pinto et al., 2019). In stellar and supermassive-star contexts, the nonlinear porosity law and the corresponding wind function remain parametrized rather than first-principles quantities (Dotan et al., 2012, Dotan et al., 2010).

Taken together, the literature supports a broad but technically specific usage of the term. Eddington winds are not a single phenomenological class but a family of radiation-regulated outflows. In accretion flows they include thermally stable, highly ionized winds in ULXs, BAL/UFO unification through mass loading in AGN, optically thick photospheric winds in ultraluminous supersoft sources, and episodic continuum-driven launches from the inner disk in luminous Seyferts (Pinto et al., 2019, Zubovas et al., 2013, Zhou et al., 2018, Pounds et al., 2023). In stars they include $g_{\rm rad} = \frac{\kappa F}{c}.$05-controlled Wolf-Rayet–type mass loss, optically thick super-Eddington eruptions with photon trapping, and rotation-shaped continuum winds (Bestenlehner, 2022, Quataert et al., 2015, Shacham et al., 2012). In dusty and galactic environments they include infrared-multiscattering winds and radiation-driven outflows from self-gravitating disks (Zhang et al., 2016, Zhang et al., 2010). A plausible implication is that the enduring significance of the concept lies less in any one velocity or spectral signature than in a common dynamical condition: radiative forcing becomes comparable to or greater than gravity, and the excess luminosity is redirected into mass loss, momentum transport, and feedback.