A Generalized Model for the Proton Expansion in Astrophysical Winds. I. The Velocity Distribution Function Representation

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Abstract
We construct a new approach to model the velocity distribution function (VDF) for the protons in stellar atmosphere expansions or planetary polar winds. The generalized Grad method of construction is used, and comparisons with the bi-Maxwellian polynomial expansion model are made in applications to the solar wind in the context of the measurements made by the Helios probes between 0.3 and 1 AU. A fitting procedure based on a sum of two Maxwellian functions is used to check the convergence property of both polynomial expansions and to calculate the predicted polynomial expansion profiles along the magnetic field orientation for typical proton VDFs in the solar wind. The generalized model is better adapted than the bi-Maxwellian polynomial expansion function to reproduce the long-tail features of a majority of the observed proton VDFs; moreover, our model does not display negative values of the VDF, contrary to the bi-Maxwellian expansion for normalized heat flux larger than unity. A 16 moment approximation, which corresponds to a third order of development, allows us to provide an associated set of generalized transport equations better closed than the equivalent system associated with a bi-Maxwellian polynomial expansion.