Salpeter time ($t_{\mathrm{Sal}}$)

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.