Boltzmann moment equation approach for the numerical study of anisotropic stellar disks

We present the Boltzmann moment equation approach for the dynamics of stars (BEADS-2D), which is a finite-difference Eulerian numerical code designed for the modelling of anisotropic and non-axisymmetric flat stellar disks. The BEADS-2D code solves the Boltzmann moment equations up to second order in the thin-disk approximation. This allows us to obtain the anisotropy of the velocity ellipsoid and the vertex deviation in the plane of the disk. We study the time-dependent evolution of exponential stellar disks in the linear regime and beyond. The disks are initially characterized by different values of the Toomre parameter Q_s and are embedded in a dark matter halo, yielding a rotation curve composed of a rigid central part and a flat outer region. Starting from a near equilibrium state, several unstable modes develop in the disk. In the early linear phase, the very centre and the large scales are characterized by growing one-armed and bisymmetric positive density perturbations, respectively. This is in agreement with expectations from the swing amplification mechanism of short-wavelength trailing disturbances, propagating through the disk centre. In the late linear phase, the overall appearance is dominated by a two-armed spiral structure localized within the outer Lindblad resonance (OLR). During the non-linear evolutionary phase, radial mass redistribution due to the gravitational torques of spiral arms produces an outflow of mass, which forms a ring at the OLR, and an inflow of mass, which forms a transient central bar. This process of mass redistribution is self-regulatory and it terminates when spiral arms diminish due to a shortage of matter. Finally, a compact central disk and a diffuse ring at the OLR are formed (the abstract is abridged).