Eddington luminosity

The Eddington luminosity, also referred to as the Eddington limit, is the maximum luminosity a body (such as a star) can achieve when there is balance between the force of radiation acting outward and the gravitational force acting inward.

The state of balance is called hydrostatic equilibrium.

When a star exceeds the Eddington luminosity, it will initiate a very intense radiation-driven stellar wind from its outer layers.

[1] The Eddington limit is invoked to explain the observed luminosities of accreting black holes such as quasars.

Originally, Sir Arthur Eddington took only the electron scattering into account when calculating this limit, something that now is called the classical Eddington limit.

The Eddington limit is obtained by setting the outward radiation pressure equal to the inward gravitational force.

Both forces decrease by inverse-square laws, so once equality is reached, the hydrodynamic flow is the same throughout the star.

is the opacity of the stellar material, defined as the fraction of radiation energy flux absorbed by the medium per unit density and unit length.

Therefore, the rate of momentum transfer from the radiation to the gaseous medium per unit density is

If the luminosity exceeds the Eddington limit, then the radiation pressure drives an outflow.

The mass of the proton appears because, in the typical environment for the outer layers of a star, the radiation pressure acts on electrons, which are driven away from the center.

Because protons are negligibly pressured by the analog of Thomson scattering, due to their larger mass, the result is to create a slight charge separation and therefore a radially directed electric field, acting to lift the positive charges, which, under the conditions in stellar atmospheres, typically are free protons.

The derivation above for the outward light pressure assumes a hydrogen plasma.

In an evolved star with a pure helium atmosphere, the electric field would have to lift a helium nucleus (an alpha particle), with nearly 4 times the mass of a proton, while the radiation pressure would act on 2 free electrons.

Thus twice the usual Eddington luminosity would be needed to drive off an atmosphere of pure helium.

At very high temperatures, as in the environment of a black hole or neutron star, high-energy photons can interact with nuclei, or even with other photons, to create an electron–positron plasma.

In that situation the combined mass of the positive–negative charge carrier pair is approximately 918 times smaller (half of the proton-to-electron mass ratio), while the radiation pressure on the positrons doubles the effective upward force per unit mass, so the limiting luminosity needed is reduced by a factor of ≈ 918×2.

The exact value of the Eddington luminosity depends on the chemical composition of the gas layer and the spectral energy distribution of the emission.

Atomic line transitions can greatly increase the effects of radiation pressure, and line-driven winds exist in some bright stars (e.g., Wolf–Rayet and O-type stars).

The role of the Eddington limit in today's research lies in explaining the very high mass loss rates seen in, for example, the series of outbursts of η Carinae in 1840–1860.

[3] The regular, line-driven stellar winds can only explain a mass loss rate of around 10−4~10−3 solar masses per year, whereas losses of up to ⁠ 1 / 2 ⁠ solar mass per year are needed to understand the η Carinae outbursts.

Gamma-ray bursts, novae and supernovae are examples of systems exceeding their Eddington luminosity by a large factor for very short times, resulting in short and highly intensive mass loss rates.

Some X-ray binaries and active galaxies are able to maintain luminosities close to the Eddington limit for very long times.

Super-Eddington accretion onto stellar-mass black holes is one possible model for ultraluminous X-ray sources (ULXSs).

[4][5] For accreting black holes, not all the energy released by accretion has to appear as outgoing luminosity, since energy can be lost through the event horizon, down the hole.

Then the accretion efficiency, or the fraction of energy actually radiated of that theoretically available from the gravitational energy release of accreting material, enters in an essential way.

The limit does not consider several potentially important factors, and super-Eddington objects have been observed that do not seem to have the predicted high mass-loss rate.

Other factors that might affect the maximum luminosity of a star include: Observations of massive stars show a clear upper limit to their luminosity, termed the Humphreys–Davidson limit after the researchers who first wrote about it.

[8] Only highly unstable objects are found, temporarily, at higher luminosities.

Efforts to reconcile this with the theoretical Eddington limit have been largely unsuccessful.