The new observations of Parker Solar Probe and Solar Orbiter are used to constrain and compare with the exospheric model of the solar wind. The observations at low radial distances show suprathermal electrons already well present in the Strahl in the antisunward direction and a deficit in the sunward direction, confirming the exospheric feature of almost no incoming particles. The presence of suprathermal Strahl electrons at the exobase increases the velocity to higher values, as well as a low exobase, because that increases the electric potential acccelerating the wind. That causes a clear difference between the slow wind and the higher speed wind originating from low density solar corona. The kinetic exospheric model of the solar wind is also improved by considering regularized Kappa distributions that have no diverging moments through consideration of a cut-off at relativistic velocities. The model becomes valid even for kappa indices lower than 2, which is important since low values of kappa are observed in the fast solar wind.