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. Building on established velocity-filtration models, we extend the stochastic multi-temperature framework developed in previous works by including magnetic-field expansion and magnetic-moment conservation. In this framework, stochastic heating generates a non-Maxwellian boundary distribution through the superposition of particle populations associated with heating events characterized by different temperatures. We consider a stationary two-component plasma confined within an expanding magnetic flux tube and subject to gravity, self-consistent 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. In the stochastic multi-temperature case, gravitational filtering enhances the contribution of the hottest particle populations at coronal heights, while magnetic-moment conservation further amplifies the resulting velocity-space anisotropy. Our results provide a fully analytical kinetic description of the combined effects of stochastic heating, gravitational filtering, and magnetic-moment conservation in an expanding coronal magnetic flux tube, and establish a collisionless benchmark for future kinetic models incorporating more realistic magnetic-field geometries, such as dipolar configurations, Coulomb collisions, and turbulent particle scattering.