The Eddington Flux Limit

The Eddington Limit

Consider a fully ionised plasma so opacity is dominated by Thompson scattering (cross section σ_e) of free electrons.

An electron receives a momentum kick Δp=hν/c from a photon and the total force

F_rad=σ_eN_νΔp
where the number of photons N_ν=L/(4πhνR²), hence,

F_rad=σ_eL/(4πR²c)

Note that the electron is tied to the ion by the electrostatic force so where the electron goes so goes the ion. Also note that there is Thompson scattering by the ion but the cross section goes as (m_e/m_ion)² so is ignored.

The gravitational force is

F_g=GM(m_ion+m_e)/R²~GMm_ion/R².

Eddington's limit is reached when radiation pressure equals gravitational attraction. Setting F_g=F_rad we get the Eddington Luminosity

L_E=4πGMm_Hc/σ_e for a hydrogen plasma.

This is about L_E=1.3E+31(M/Mo)=32500Lo(M/Mo) Watt.

Note that the R has cancelled. And for M/Mo=100 we have from the mass-luminosity relation L/Lo=

Some workers extend this to a general plasma by replacing m_H my μ_em_H where μ_e=2/(1+X) and get

L_E=2.47E+31/(1+x) Watt, about 4.6E+06 L_Sun.