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 ($\overline{L_z}$) to its temporal dispersion ($\sigma_{L_z}$) in the corotating frame. We show that CAM is determined by the ratio of the azimuthal to radial actions (${J_\phi}^\prime / {J_r}^\prime$) in the analytical Freeman bar model. We then construct a new parameter space defined by CAM versus the root-mean-square 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 the $x_2$ orbital family. We further validate the method using orbits from test-particle simulations, which show a well-ordered and non-overlapping 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.