Why Cold BGK Modes Are So Cool: Dispersion Relations from Orbit-Constrained Distribution Functions

Abstract: We derive analytic dispersion relations for cold, orbitally constrained systems governed by the Vlasov equation. For magnetized plasmas, we obtain the first explicit relation for two-dimensional anisotropic BGK modes with finite magnetic field, showing that only a finite number of angular modes can become unstable and identifying a magnetic-field threshold for stabilization. In the gravitational case, we establish a bound on the growth rate of core perturbations, set by the potential's curvature. These results clarify how orbital constraints shape the spectrum and growth of kinetic instabilities in cold, collisionless media.