Super-Eddington stellar winds: Unifying radiative-enthalpy versus flux-driven models

We derive semi-analytic solutions for optically thick, super-Eddington stellar winds, induced by an assumed steady energy addition ΔE˙ concentrated around a near-surface heating radius R in a massive star of central luminosity L*. We show that obtaining steady wind solutions requires both that the resulting total luminosity L_o = L* + ΔE˙ exceed the Eddington luminosity, Γ_o ≡ L_o/L_Edd > 1, and that the induced mass-loss rate be such that the 'photon-tiring' parameter, m ≡ M˙ GM/RL_o ≤ 1 - 1/Γ_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 versus large values of mΓ_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_∞² /v_esc² = Γ_o/(1 + mΓ_o) - 1, and for the final observed luminosity L_obs, which for all allowed steady-solutions with m < 1 - 1/Γ_o exceeds the Eddington luminosity, L_obs > L_Edd. Our super-Eddington wind solutions have potential applicability for modelling phases of eruptive mass-loss from massive stars, classical novae, and the remnants of stellar mergers.