Photoionization heating from UV radiation incident on the atmospheres of hot Jupiters may drive planetary mass loss. We construct a model of escape that includes realistic heating and cooling, ionization balance, tidal gravity, and pressure confinement by the host star wind. We show that mass loss takes the form of a hydrodynamic (Parker) wind, emitted from the planets dayside during lulls in the stellar wind. When dayside winds are suppressed by the confining action of the stellar wind, nightside winds might pick up if there is sufficient horizontal transport of heat. A hot Jupiter loses mass at maximum rates of ~2 x 10^12 g/s during its host stars pre-main-sequence phase and ~2 x10^10 g/s during the stars main sequence lifetime, for total maximum losses of ~0.06% and ~0.6% of the planets mass, respectively. For UV fluxes F_UV < 10^4 erg/cm^2/s, the mass loss rate is approximately energy-limited and is proportional to F_UV^0.9. For larger UV fluxes, such as those typical of T Tauri stars, radiative losses and plasma recombination force the mass loss rate to increase more slowly as F_UV^0.6. Dayside winds are quenched during the T Tauri phase because of confinement by overwhelming stellar wind pressure. We conclude that while UV radiation can indeed drive winds from hot Jupiters, such winds cannot significantly alter planetary masses during any evolutionary stage. They can, however, produce observable signatures. Candidates for explaining why the Lyman-alpha photons of HD 209458 are absorbed at Doppler-shifted velocities of +/- 100 km/s include charge-exchange in the shock between the planetary and stellar winds.
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