  In this paper we consider the effect of the direct ionizing stellar radiation
field on the evolution of protoplanetary discs subject to photoevaporative
winds. We suggest that models which combine viscous evolution with
photoevaporation of the disc (e.g. Clarke, Gendrin & Sotomayor 2001)
incorrectly neglect the direct field after the inner disc has drained, at late
times in the evolution. We construct models of the photoevaporative wind
produced by the direct field, first using simple analytic arguments and later
using detailed numerical hydrodynamics. We find that the wind produced by the
direct field at late times is much larger than has previously been assumed, and
we show that the mass-loss rate scales as $R_{in}^{1/2}$ (where $R_{in}$ is the
radius of the instantaneous inner disc edge). We suggest that this result has
important consequences for theories of disc evolution, and go on to consider
the effects of this result on disc evolution in detail in a companion paper
(Alexander, Clarke & Pringle 2006b).
