Five methods for determining pattern speeds in galaxies

Context. After matter distribution and kinematics, the bar or spiral pattern speeds are next fundamental parameters determining dynamics of a galaxy. Aims. New refined methods for instantaneous scalar vector from restricted domain developed applications in N -body simulations galaxies such as Milky Way, which stellar coordinates become increasingly better known. Methods. The general feature used throughout follows fact that time derivative function is linearly proportional to its rotation rate particle velocities. Knowing these therefore allows retrieving speed by linear optimization. Similarly, if an invariant depends on position velocities, then vectors space can be retrieved. accelerations also velocity space. Results. first three based assumed rotational invariance functions at each point space: (1) 6D method, measuring space, (2) differential/regional 3D Tremaine-Weinberg evaluated over regions with high signal-to-noise ratio, (3) Jacobi integral yielding potential speed. Extensions derive center position, speed, acceleration introduced third methods. last two average coordinates: (4) 2D moment inertia using singular value decomposition, (5) Fourier method (3D m = 2 mode), giving mode speeds. Pattern derived fourth fifth Conclusions. Depending available data specific problems, different provide choice approaches.