  Photoevaporation driven by hydrogen-ionizing radiation, also known as
extreme-ultraviolet (EUV), profoundly shapes the lives of diverse astrophysical
objects. Focusing here mainly on the dispersal of protoplanetary disks, we
construct an analytical model accounting for the finite timescales of
photoheating and photoionization. The model offers improved estimates for the
ionization, temperature, and velocity structures versus distance from the
central source, for a given EUV emission rate and spectral hardness. Compared
to the classical picture of fully-ionized and isothermal winds with
temperatures $\approx 10^4{\rm \,K}$ and speeds $\approx 10{\rm \,km\,s^{-1}}$,
our model unveils broader hydrodynamical and thermochemical states of
photoevaporative winds. In contrast to the classical picture, T~Tauri stars
with EUV luminosities around $10^{30}{\rm \,erg\,s^{-1}}$ have non-isothermal
ionized winds at lower temperatures than the classical value if the spectrum is
soft, with an average deposited energy per photoionization less than about
3.7\,eV. Conversely, if the spectrum is hard, the winds tend to be atomic and
isothermal at most radii in the disk. For lower EUV intensities, even with soft
spectra, atomic winds can emerge beyond $\sim 10{\, \rm au}$ through advection.
We demonstrate that the analytical model's predictions are in general agreement
with detailed radiation-hydrodynamics calculations. The model furthermore
illustrates how the energy efficiency of photoevaporation varies with the
intensity and spectral hardness of the EUV illumination, as well as addressing
discrepancies in the literature around the effectiveness of X-ray
photoevaporation. These findings highlight the importance of considering the
finite timescales of photoheating and photoionization, both in modeling and in
interpreting observational data.
