EDDINGTON LUMINOSITY

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'Eddington luminosity' (sometimes also called the 'Eddington limit') is the largest luminosity that can pass through a layer of gas in hydrostatic equilibrium, supposing spherical symmetry. Using the mass-luminosity relation, it can be used to set limits on the maximum mass of a star. If the luminosity of a star exceeds the Eddington luminosity of a layer on the stellar surface, the gas layer is ejected from the star. The phenomenon is named in honor of Sir Arthur Stanley Eddington.

Contents

Derivation

Examples

References

Derivation

The 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 different throughout the star.
The pressure support of a star is given by the equation of hydrostatic equilibrium:
:$$
rac{dP}{dr} = -
ho g = -G rac{M
ho}{r^2}

The outward force of radiation pressure is given by:
:$$
rac{dP}{dr} = -rac{kappa
ho}{c}F_{rad} =-rac{sigma_T
ho}{m_p c} rac{L}{4pi r^2}

where $sigma\_T$ is the Thomson scattering cross-section for the electron and the gas is assumed to be purely made of ionized hydrogen.
Equating these two pressures and solving for the luminosity gives the Eddington Luminosity:
:
m Edd}&=4pi G M m_{
m p} c/sigma_{
m T}\
&cong 1.3 imes10^{38}left(rac{M}{M_igodot}
ight){
m erg s}^{-1}
= 3.3 imes10^4left(rac{M}{M_igodot}
ight) L_igodot
end{align}

where $M$ is the mass of the central object, the mass and the luminosity of the sun, $m\_\{$
m p} the mass of a proton and $sigma\_\{$
m T} the Thomson cross-section for the electron.
The exact value of 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.

Examples

Gamma ray bursts, novae and supernovae are examples of systems exceeding their Eddington luminosity by a large factor for very short times. In those cases, the result is a radical change in physical structure (namely, the ejection of a fraction of the star's mass).
Some X-ray binaries and active galaxies are able to maintain luminosities close to the Eddington limit for very long times.
The Eddington limit is not a true limit, and it is believed that photon-bubble instabilities (which remove the strict spherical symmetry) allow nature to have radiating flows with much higher luminosities. Super-Eddington accretion onto stellar-mass black holes is one possible model for ultraluminous X-ray sources (ULXs).

References

★ Accretion Power in Astrophysics, Juhan Frank, Andrew King, Derek Raine, , , Cambridge University Press, 2002, ISBN 0-521-62957-8