Photoevaporation of protoplanetary discs I: hydrodynamic models
  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). 
R.D.Alexander, C.J.Clarke, J.E.Pringle