Collisionless stationary states of a stratified plasma in an expanding magnetic tube with stochastic heating

We investigate the collisionless kinetic structure of the upper solar atmosphere in the presence of an expanding magnetic field. We consider a stationary two-component plasma confined within an expanding magnetic flux tube and subject to gravity, self-electrostatic interactions, the Pannekoek-Rosseland electric field, and magnetic moment conservation. Starting from the Vlasov equation, we derive fully analytical expressions for the particle distribution functions, density profiles, and the parallel, perpendicular, and total temperature profiles. We show that the combined conservation of energy and magnetic moment generates a loss-cone distribution, reducing the density with respect to the corresponding unmagnetized atmosphere and producing a pronounced temperature anisotropy. For a single-temperature boundary condition, the competition between magnetic moment conservation and gravity causes the parallel temperature to develop a maximum. We derive analytical scaling laws for its location and amplitude and validate them against numerical calculations. We further show that the anisotropy persists independently of the temperature distribution at the lower boundary. In the regime of rare but intense heating events, gravitational filtering enhances the contribution of the hottest particle populations at coronal heights, while magnetic moment conservation further amplifies the resulting velocity-space anisotropy. This work provides a fully analytical kinetic description of the combined effects of gravitational filtering and magnetic moment conservation in an expanding coronal magnetic flux tube undergoing stochastic heating at its base. These results establish a theoretical framework for investigating the role of magnetic-field expansion in shaping the density and temperature structure of weakly collisional stellar coronae.