A generalized linear matrix method for normal modes in collisionless stellar disks

Abstract

We generalize the linear matrix method for computing normal modes in collisionless stellar disks to distribution functions with sharp edges at zero angular momentum ($L=0$). The generalization adds boundary-integral terms to the matrix equation without increasing its size. We validate the method by computing $m=2$ modes for two Kuzmin--Toomre disk models (Miyamoto $n_{\rm M}=3$ and Kalnajs $m_{\rm K}=6$ families) and comparing the eigenvalues with those obtained from an independent nonlinear matrix method based on logarithmic-spiral expansions. A systematic convergence study over grid resolution and harmonic truncation yields eigenvalues accurate to ${\sim}\,0.003$ in both pattern speed and growth rate. Unlike the nonlinear method, the linear method naturally incorporates gravitational softening, enabling the computation of eigenmodes for softened disk models. The implementation in Julia with GPU acceleration is openly available.