Eddington ratio ($f_{\mathrm{Edd}}$)

The Eddington ratio is the luminosity divided by the Eddington luminosity. At fixed radiative efficiency it also measures the accretion rate relative to the Eddington rate.

An Eddington accretion rate converts the Eddington luminosity into a mass supply rate using an adopted reference efficiency ${\eta }_0$: ${\dot{M}}_{\mathrm{Edd}}=L_{\mathrm{Edd}}/({\eta }_0{c}^2)$. The common choice ${\eta }_0=0.1$ gives ${\dot{M}}_{\mathrm{Edd}}=10L_{\mathrm{Edd}}/c^2$.

For radiatively efficient release of Newtonian surface binding energy, $L\sim GM\dot{M}/R_{\ast }$ and $L_E=4\pi GMc/\kappa$ imply the displayed rate. Its efficiency is $\eta \sim GM/(R_{\ast }{c}^2)$, so at fixed opacity the maximum rate scales with stellar radius. This differs from defining an Eddington supply rate with an arbitrarily fixed reference efficiency. Geometry, advection and mass loss can modify the simple limit.

The Salpeter time is the e-folding time for Eddington-limited black-hole growth. If $t_E={\sigma }_{T}c/(4\pi G{m}_p)$, then $t_{\mathrm{Sal}}=\eta t_E/[(1-\eta )f_{\mathrm{Edd}}]$.

For fixed efficiency, growth from $M_i$ to $M_f$ within $\Delta t$ at no more than the Eddington luminosity requires $[\eta /(1-\eta )]t_E\log (M_f/M_i)\le \Delta t$. Solving gives an upper bound, because greater radiative efficiency reduces retained mass growth at a fixed luminosity. A duty cycle below unity tightens the requirement; mergers and super-Eddington episodes invalidate its stated premises.

At constant radiative efficiency of black-hole accretion $\eta$ and the Eddington luminosity, a black hole grows exponentially with the Salpeter time $t_S=\eta t_E/(1-\eta )$, where $t_E={\sigma }_{T}c/(4\pi G{m}_p)$. Its doubling time is $t_S\log 2$. The factor $1-\eta$ accounts for the supplied rest energy that escapes as radiation instead of increasing the hole's mass.