Classifying solar wind observations by HELIOS in several populations sorted by bulk speed, has revealed slight accelerations of the wind as it expands away from the Sun in the 0.3 – 1 au radial range (Maksimovic et al. 2020). The faster the wind is, the smaller is this acceleration. Recent measurements from Parker Solar Probe, which have been added closer to the Sun, show that the HELIOS populations can be extended smoothly back to the Sun. Moreover, the well established bulk speed/proton temperature (u, Tp) correlation, together with the acceleration of the slowest winds, are clearly visible in the PSP data (Dakeyo et al. 2022, Halekas et al. 2022), and in the Solar Orbiter data at larger distance.
Based on the previous classifications, we present results of semi-empirical Parker-like models (Iso-poly) for which the solar wind is modeled as isothermal in the corona, then polytropic well after the sonic point, with polytropic indices corresponding to the observed temperature gradients. Such models allow to establish a differentiated energy balance for the heating of the wind and for the acceleration separately. We also present an estimate of the heating rates radial evolution required to coherently reproduce the solar wind observations.