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tokens: ), equation ( 37 ) becomes At far from the centre, the flow speed is almost constant, and this equation is solved to yield a solution and the flux roughly varies as . Similarly, at far from the centre, where B 0 is dominated and the flow speed is constant, equation ( 38 ) with equation ( 39 ) is also solved to yield an approximate solution: and the mean intensity roughly varies as at least on the main dependence. In Fig. 8 , the emergent intensities I + ( p, z out ) at the outer boundary R out normalized by I * are plotted as a function of the impact coordinate p normalized by R * . The normalized luminosity is Γ * = 5, the typical optical depth τ * = 3, the blackbody intensity B 0 / I * = 1, and the values of β out are 0.1, 0.3, 0.5 from top to bottom. In contrast to the scattering case (Fig. 4 ), the emergent intensity in the outer envelope is quite large. This is also because the source emissivity is uniformly distributed in the entire space in the isothermal wind. Similar to the relations obtained in the scattering case (Fig. 5 ), in this isothermal wind we can also the relations on the parameters, although we do not discuss them for simplicity. Finally, we should mention the opacity effect. In this isothermal wind, we assume that the absorption opacity κ 0 is constant. In the realistic case, the absorption opacity depends on the physical quantities, e.g. the free–free opacity is proportional to . Hence, it is in the isothermal case. In this case, the results are quite similar to the scattering case. The reason is because the density is concentrated in the centre and quickly decreases towards the outer envelope, and therefore, the emissivity is also concentrated in the central core. 5 CONCLUDING REMARKS In this paper, we have examined the radiatively driven relativistic spherical winds from the central object with mass M and luminosity L * under Newtonian gravity and special relativity, by numerically solving the relativistic radiative transfer equation and relativistic hydrodynamical ones for spherical geometry, simultaneously and consistently. Namely, using the initial trial velocity profile, we iteratively solved the relativistic formal solution of the transfer equation to obtain specific intensities and moment quantities in the inertial and comoving frames. Using the comoving flux obtained, we then solved the relativistic hydrodynamical equations to obtain the refined velocity profile, and the mass-loss rate as an eigenvalue. We repeat these double-iteration procedures until both the intensity and velocity converge. We found that the momentum-driven winds with scattering are quickly accelerated near the central luminous core to reach the terminal speed (Fig. 2 ), similar to the radiative outflows without gravity (Fukue 2017b ). The behaviour of moment quantities and emergent intensity is quite similar to the previous results (figs 3 and 4 in Fukue 2017b ). The results of numerical solutions are roughly fitted by a relation of , which