Radiative equilibrium temperature of the SPH particles (points) and 1D semi-analytical wind solutions (lines) in a single-star configuration for the low mass-loss (Ṁ_AGB = 10⁻⁸ M_⊙ yr⁻¹, upper panel) and the high (Ṁ_AGB = 3 × 10⁻⁶ M_⊙ yr⁻¹, bottom panel) mass loss-rates models. The free-wind approximation (red SPH points and 1D solid line profile), the geometrical approximation (light-green and dashed line), the Lucy approximation (orange and dash-dotted line), and the attenuation approximation (blue and dashed-double dotted line) are shown. The geometrical, Lucy, and attenuation approximation overlap in the upper plot. The vertical lines in the lower panel indicate the corresponding dust condensation radius for the geometrical, Lucy, and attenuation approximation.