  We derive semi-analytic solutions for optically thick, super-Eddington
stellar winds, induced by an assumed steady energy addition $\Delta {\dot E}$
concentrated around a near-surface heating radius $R$ in a massive star of
central luminosity $L_\ast$. We show that obtaining steady wind solutions
requires both that the resulting total luminosity $L_o = L_\ast + \Delta {\dot
E}$ exceed the Eddington luminosity, $\Gamma_o \equiv L_o/L_{Edd} > 1$, and
that the induced mass loss rate be such that the "photon-tiring" parameter $m
\equiv {\dot M} GM/R L_o \le 1-1/\Gamma_o$, ensuring the luminosity is
sufficient to overcome the gravitational potential $GM/R$. Our analysis unifies
previous super-Eddington wind models that either: (1) assumed a direct
radiative flux-driving without accounting for the advection of radiative
enthalpy that can become important in such an optically thick flow; or (2)
assumed that such super-Eddington outflows are adiabatic, neglecting the
effects of the diffusive radiative flux. We show that these distinct models
become applicable in the asymptotic limits of small vs. large values of $m
\Gamma_o $, respectively. By solving the coupled differential equations for
radiative diffusion and wind momentum, we obtain general solutions that
effectively bridge the behaviours of these limiting models. Two key scaling
results are for the terminal wind speed to escape speed, which is found to vary
as $v_\infty^2/v_{esc}^2 = \Gamma_o/(1+m \Gamma_o) -1$, and for the final
observed luminosity $L_{ obs}$, which for all allowed steady-solutions with $m
< 1 - 1/\Gamma_o$, exceeds the Eddington luminosity, $L_{obs} > L_{Edd}$. Our
super-Eddington wind solutions have potential applicability for modeling phases
of eruptive mass loss from massive stars, classical novae, and the remnants of
stellar mergers.
