Landau Modes are Eigenmodes of Stellar Systems in the Limit of Zero Collisions

We consider the spectrum of eigenmodes in a stellar system dominated by gravitational forces in the limit of zero collisions. We show analytically and numerically using the Lenard–Bernstein collision operator that the Landau modes, which are not true eigenmodes in a strictly collisionless system (except for the Jeans unstable mode), become part of the true eigenmode spectrum in the limit of zero collisions. Under these conditions, the continuous spectrum of true eigenmodes in a collisionless system, also known as the Case–van Kampen modes, is eliminated. Furthermore, because the background distribution function in a weakly collisional system can exhibit significant deviations from a Maxwellian distribution function over long times, we show that the spectrum of Landau modes can change drastically even in the presence of slight deviations from a Maxwellian, primarily through the appearance of weakly damped modes that may be otherwise heavily damped for a Maxwellian distribution. Our results provide important insights for developing statistical theories to describe thermal fluctuations in a stellar system, which are currently a subject of great interest for N-body simulations as well as observations of gravitational systems.

Amplification of fast magnetosonic waves and the cut-off spectrum

The propagation of fast magnetosonic waves in an inhomogeneous medium with planar flow in investigated. The equations describing the rays along which the waves propagate are derived, as well as the equations for the variations of the wave amplitude along the rays. These equations are solved for the case of steady flow that depends only on the radius. In addition, it is shown that a spectrum of eigenmodes may exist if the steady flow contains a shock. For that purpose, the reflection coefficient of a fast magnetosonic wave form a shock is derived, and it is shown that the waves can be localized in a region bounded by a shock and a critical surface. A criterion for stability of the spectrum is derived.