Orbital Classification in Rotating Bar Potentials Using an Empirical Proxy of the Second Integral of Motion

Abstract We present a novel method for classifying two-dimensional orbits in rotating bar potentials based on an empirical proxy for the second integral of motion, calibrated angular momentum (CAM), which is defined as the ratio of the time-averaged angular momentum ( L z ¯ ) to its temporal dispersion ( σ L z ) in the corotating frame. We show that CAM is determined by the ratio of the azimuthal to radial actions ( J ϕ ′ / J r ′ ) in the analytical Freeman bar model. We then construct a new parameter space defined by CAM versus the rms radius ( R rms ) and apply this framework to orbits in several representative rotating bar potentials. In the CAM– R rms plane, periodic orbits generate well-defined branches separating distinct regions corresponding to different orbital families. Several of these branches enclose isolated areas that can be associated with specific orbital families, such as the x 2 orbital family. We further validate the method using orbits from test-particle simulations, which show a well-ordered and nonoverlapping distribution of orbital families in the CAM– R rms plane. Since CAM is fundamentally linked to intrinsic orbital properties and readily applied to three-dimensional orbits in N -body simulations, our results establish the CAM– R rms plane as a robust and efficient framework for orbit classification in rotating bars that complements conventional methods.