5 Conclusion

>From self-consistent kinetic simulation of a solar type wind we find that, unless an efficient isotropization mechanism for the electron velocity distribution (e.g. wave-particle interaction), or some source of suprathermal electron distributions are invoked (e.g. shock produced), the formation of a transonic wind is only compatible with a sufficiently high collisionality in the vicinity of the sonic point $r=r_*$. In oder words, the coronal density must exceed a threshold density for the wind acceleration to be sufficiently strong to become supersonic. Given the admittedly oversimplifications in our model, combined with the fact that the parameters we use are rather unrealistic (excessively high coronal temperature and low $m_{\rm p}/m_{\rm e}$ ratio) makes it impossible for us to specify an upper limit for the thermal Knudsen number at the sonic point $r_*$. We do merely show that the number cannot be arbitrarily large, the limiting value most likely being of the order unity, or less. As already stated, non Maxwellian boundary conditions, feeding an excess of suprathermal particles into the system (e.g. kappa distributions) may help overcoming the low Knudsen number condition. However, the existence of non thermal particle distributions rises the question of their origin. We chose not to address this question and assume Maxwellian boundary conditions which have the advantage of not requiring the introduction of additional ad hoc parameters into the model. Thus, in the absence of any electron scattering mechanism (apart from collisions), and unless special boundary conditions are imposed at the base of the simulation domain, collisions appear to be the essential ingredient for the wind to become accelerated to supersonic velocities, mainly because collisions are necessary to convert the electron heat flux into bulk energy of the plasma. The enthalpy gradient does also contribute to the acceleration of the wind. However, the acceleration associated with the radially decreasing enthalpy is found to be weakly dependent on the plasma collisionality and doesn't seem to be the discriminating factor in the acceleration of the wind to supersonic velocities.

In simulations where a transonic wind forms we find that the proton potential has a maximum near (but above) the sonic point. Typical values of the electric field near the sonic point $r_*$ are found to be of the order of Dreicer's field, or larger. The presence of such strong electric field intensities may contribute in making the electron heat flux depart from the classical Spitzer-Härm formula (which requires the electric field being much weaker than Dreicer) but the main reason for the observed heat flux to depart from the Spitzer-Härm prediction is due to the presence of a strong ``non collisional" heat flux $q_{NC} \propto nvT_{\rm e}$. The latter appears to contribute significantly to the total electron heat flux, even in the region where the wind velocity is much smaller than the sound speed. We are aware of the fact that extrapolating the above results to the "real" Sun is a perilous exercise. However, we do not expect the qualitative behavior of a system with real coronal temperature and real proton to electron mass-ratio to behave in a substantially different way from the high density case discussed in this paper. In particular, increasing the proton to electron mass ratio from 400 to 1836 implies a factor two increase in the electron thermal velocity, only. Given that the electron thermal velocity at the base of the system is already one order of magnitude larger than the escape velocity for the $m_{\rm p}/m_{\rm e}=400$ case, not much difference is expected in a system with twice this thermal velocity. The skeptical reader may also argue that using a realistic mass ratio would substantially modify the transport properties of the plasma. This is certainly true, but we expect the modifications to be small, essentially because neither the classical electron heat flux $q_{\rm SH}$ nor the collisionless heat flux $q_{\rm NC}$ depend on the mass ratio, at least as long as $m_{\rm p}/m_{\rm e}\gg 1$. We expect the excessively high coronal temperature used in our simulations to be a more crucial limitation in the process of transposing the above results to the solar case just because the proton thermal velocity and the escape velocity are of the same order, i.e. $\sqrt{\gamma_{\rm p}}=O(1)$. Indeed, the coronal temperature of the real Sun is roughly 3 times smaller that the value we use here. This implies a factor $\sqrt{3}$ difference for the order unity quantity $\gamma_p$. The impact is certainly non negligible from a quantitative point of view but there aren't any reasons for us to believe that the qualitative aspects of our results do not apply to the solar case. For instance, whether or not the collisionless heat flux near the solar sonic point is really one order of magnitude stronger that the classical Spitzer-Härm flux remains an open question since this finding discords with recent results from multi-moment models.