  Sediment transport along the surface drives geophysical phenomena as diverse
as wind erosion and dune formation. The main length-scale controlling the
dynamics of sediment erosion and deposition is the saturation length
$L_\mathrm{s}$, which characterizes the flux response to a change in transport
conditions. Here we derive, for the first time, an expression predicting
$L_\mathrm{s}$ as a function of the average sediment velocity under different
physical environments. Our expression accounts for both the characteristics of
sediment entrainment and the saturation of particle and fluid velocities, and
has only two physical parameters which can be estimated directly from
independent experiments. We show that our expression is consistent with
measurements of $L_\mathrm{s}$ in both aeolian and subaqueous transport regimes
over at least five orders of magnitude in the ratio of fluid and particle
density, including on Mars.
