New approach to modeling of outflowing envelopes at arbitrary optical depths

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We have considered a stationary outflowing envelope from the star accelerated by the radiative force in arbitrary optical depth case. Introduced approximations provide satisfactory description of the behavior of the matter flux with partially separated radiation at arbitrary optical depths. The obtained system of differential equations provides a continuous transition of the solution between optically thin and optically thick regions. We analytically derived approximate representation of the solution at the vicinity of the sonic point. Using this representation we numerically integrate the system of equations from the critical point to the infinity. Matching the boundary conditions we obtain the solution of the describing the problem system of differential equations. We have discuss in details the problem of the outer boundary condition. The situation of the photosphere was clarified for the obtained expanding solution. Theoretical approach submitted in this work could be useful for self - consistent simulations of the massive stars evolution with mass loss.