Eddington luminosity

The Eddington luminosity (also referred to as the Eddington limit) in a star is defined as the point where the gravitational force inwards equals the continuum radiation force outwards, assuming hydrostatic equilibrium and spherical symmetry. When exceeding the Eddington luminosity, a star would initiate a very intense continuum driven stellar wind from its outer layers. Since most massive stars have luminosities far below the Eddington luminosity, however, their winds are mostly driven by the less intense line absorption. ^[1]

Originally, Sir Arthur Stanley Eddington only took the electron scattering into account when calculating this limit, something that now is called the classical Eddington limit. Nowadays, the modified Eddington limit also counts on other continuum processes such as bound-free and free-free interaction.

Derivation

The limit is obtained by setting the outward continuum radiation pressure equal to the inward gravitational force. Both forces decrease by inverse square laws, so once equality is reached, the hydrodynamic flow is different throughout the star.

The pressure support of a star is given by the equation of hydrostatic equilibrium:

The outward force of radiation pressure is given by:

where σ_T is the Thomson scattering cross-section for the electron and the gas is assumed to be purely made of ionized hydrogen. κ is the opacity of the stellar material.

Equating these two pressures and solving for the luminosity gives the Eddington Luminosity:

where M is the mass of the central object, M_☉ the mass and L_☉ the luminosity of the Sun, m_p the mass of a proton and σ_T the Thomson cross-section for the electron.

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 are typically free protons under the conditions in stellar atmospheres. When the outward electric field is sufficient to levitate the protons against gravity, both electrons and protons are expelled together.

Thus in certain circumstances the balance can be different than it is for hydrogen. For example, in an evolved star with a pure helium atmosphere, the electric field would have to lift a helium nucleus (an alpha particle), with nearly four times the mass of a proton, while the radiation pressure would act on two free electrons. Thus twice the usual Eddington luminosity would be needed to drive off an atmosphere of pure He. On the other hand, at very high temperatures, as in the environment of a black hole or neutron star, high energy photon interactions with nuclei or even with other photons, can create an electron-positron plasma. In that situation the mass of the neutralizing positive charge carriers is ~1836 times smaller (the proton: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 ~2*1836. Thus the exact value of the Eddington luminosity depends on the chemical composition of the gas layer and the spectral energy distribution of the emission. Gas with cosmological abundances of hydrogen and helium is much more transparent than gas with solar abundance ratios. Atomic line transitions can greatly increase the effects of radiation pressure, and line driven winds exist in some bright stars.

Super-Eddington luminosities

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.^[2] The regular, line driven stellar winds can only stand for a mass loss rate of around 10^-4 – 10^-3 solar masses per year, whereas we need mass loss rates of up to 0.5 solar masses per year to understand the η Carinae outbursts. This can be done with the help of the super-Eddington continuum driven winds.

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. For accretion powered sources such as accreting neutron stars or cataclysmic variables (accreting white dwarfs), the limit may act to reduce or cut off the accretion flow, imposing an Eddington limit on accretion corresponding to that on luminosity. Super-Eddington accretion onto stellar-mass black holes is one possible model for ultraluminous X-ray sources (ULXs).

For accreting black holes, all the energy released by accretion does not have to appear as outgoing luminosity, since energy can be lost through the event horizon, down the hole. Such sources effectively may not conserve energy. 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.

Other factors

The Eddington limit is not a strict limit on the luminosity of a stellar object. The limit does not consider several potentially important factors, and a couple of 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:

• Porosity. A problem with steady, continuum driven winds is that both the radiative flux and gravitational acceleration scale with r^-2. The ratio between these factors is constant, and in a super-Eddington star, the whole envelope would become gravitationally unbound at the same time. This is not observed. A possible solution is introducing an atmospheric porosity, where we imagine the stellar atmosphere to consist of denser regions surrounded by lower density gas regions. This would reduce the coupling between radiation and matter, and the full force of the radiation field would only be seen in the more homogeneous outer, lower density layers of the atmosphere.

• Turbulence. A possible destabilizing factor might be the turbulent pressure arising when energy in the convection zones builds up a field of supersonic turbulence. The importance of turbulence is being debated, however. ^[3]

• Photon bubbles. Another factor that might explain some stable super-Eddington objects is the photon bubble effect. Photon bubbles would develop spontaneously in radiation-dominated atmospheres when the magnetic pressure exceeds the gas pressure. We can imagine a region in the stellar atmosphere with a density lower than the surroundings, but with a higher radiation pressure. Such a region would rise through the atmosphere, with radiation diffusing in from the sides, leading to an even higher radiation pressure. This effect could transport radiation more efficiently than a homogeneous atmosphere, increasing the allowed total radiation rate. In accretion discs, luminosities may be as high as 10-100 times the Eddington limit without experiencing instabilities. ^[4]

See also

• Hayashi limit

References