Solar Wind protons: Gaussian and Inverse-Gaussian components

Secondary proton populations (ion beams) are often observed in the solar wind. This non-thermal feature is characterized by a velocity shift of the order of the Alfven velocity with respect to the core gaussian proton population. Using the measurements of the Proton-Alpha Sensor (PAS) onboard Solar Orbiter (SWA instrument), we show that the beams are not 'simply' shifted gaussian distributions. They are asymmetric, with a more populated high-energy side ('heavy tail') than given by the gaussian function, in the parallel-to-B direction. It appears that the 'Inverse Gaussian Distribution' (IGD), another family of statistical distributions, provides excellent fits of the beam profile in the parallel-to-B direction (the perpendicular profile remains essentially gaussian). Then, remarkably good models of the whole proton distribution are obtained by superposing a gaussian for the core and an IGD for the beam. This modelling (Gaussian + Inverse Gaussian) applies in different situations: relatively slow and fast winds, single and double-bump populations, situations of radial or inclined magnetic field, low or high level of turbulence.

A vast literature describes the statistical properties of IGD and its potential applications. It was discovered by solving the 'first time passage' problem for random walk (Schrodinger, 1915), then proved to offer excellent models in various physical, chemical, biological and financial contexts. It is also related to the 'Levy flight' theory, the superdiffusion and solutions of fractionnal Fokker-Planck equation. That so good fits are obtained with a single mathematical form (the IGD) should indicate that a stable, robust and rather ubiquitous underlying process explains the beam formation and its characteristics. We explore how the acceleration by kinetic Alfven waves can be adapted to explain the observations.