Gravitational steady states of solar coronal loops

Abstract

Coronal loops on the surface of the sun appear to consist of curved, plasma-confining magnetic flux tubes or “ropes,” anchored at both ends in the photosphere. Toroidal loops carrying current are inherently unstable to expansion in the major radius due to toroidal-curvature-induced imbalances in the magnetic and plasma pressures. An ideal MHD analysis of a simple isolated loop with density and pressure higher than the surrounding corona, based on the theory of magnetically confined toroidal plasmas, shows that the radial force balance depends on the loop internal structure and varies over parameter space. It provides a unified picture of simple loop steady states in terms of the plasma beta βo, the inverse aspect ratio ϵ=a/Ro, and the MHD gravitational parameter Ĝ≡ga/vA2, all at the top of the loop, where g is the acceleration due to gravity, a the average minor radius, and vA the shear Alfvén velocity. In the high and low beta tokamak orderings, βo=2noT/(Bo2/2μo)∼ϵ1 and ϵ2, that fit many loops, the solar gravity can sustain nonaxisymmetric steady states at Ĝ∼ϵβo that represent the maximum stable height. At smaller Ĝ≤ϵ2βo, the loop is axisymmetric to leading order and stabilized primarily by the two fixed loop ends. Very low beta, nearly force-free, steady states with βo∼ϵ3 may also exist, with or without gravity, depending on higher order effects. The thin coronal loops commonly observed in solar active regions have ϵ≃0.02 and fit the high beta steady states. Ĝ increases with loop height. Fatter loops in active regions that form along magnetic neutral lines and may lead to solar flares and Coronal Mass Ejections have ϵ≃0.1–0.2 and may fit the low beta ordering. Larger loops tend to have Ĝ>ϵβo and be unstable to radial expansion because the exponential hydrostatic reduction in the density at the loop-top reduces the gravitational force −ρĜR̂ below the level that balances expansion, in agreement with the observation that most sufficiently large loops grow.