Recent observation of some luminous transient sources with low color temperatures suggests that the emission is dominated by optically thick winds driven by super-Eddington accretion. We present a general analytical theory of the dynamics of radiation pressure-driven, optically thick winds. Unlike the classical adiabatic stellar wind solution whose dynamics are solely determined by the sonic radius, here the loss of the radiation pressure due to photon diffusion also plays an important role. We identify two high mass loss rate regimes ($dot{M} > L_{rm Edd,}/c^2$). In the large total luminosity regime the solution resembles an adiabatic wind solution. Both the radiative luminosity, $L$, and the kinetic luminosity, $L_k$, are super-Eddington with $L < L_k$ and $L propto L_k^{1/3}$. In the lower total luminosity regime most of the energy is carried out by the radiation with $L_k < L approx L_{rm Edd,}$. In a third, low mass loss regime ($dot{M} < L_{rm Edd,}/c^2$), the wind becomes optically thin early on and, unless gas pressure is important at this stage, the solution is very different from the adiabatic one. The results are independent from the energy generation mechanism at the foot of the wind, therefore they are applicable to a wide range of mass ejection systems, from black hole accretion, to planetary nebulae, and to classical novae.
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