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 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>z</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">¯</mml:mo> </mml:mrow> </mml:mover> </mml:math> ) to its temporal dispersion ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>z</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub> </mml:math> ) in the corotating frame. We show that CAM is determined by the ratio of the azimuthal to radial actions ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>J</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ϕ</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo accent="true">′</mml:mo> </mml:mrow> </mml:msup> <mml:mo>/</mml:mo> <mml:msup> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>J</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo accent="true">′</mml:mo> </mml:mrow> </mml:msup> </mml:math> ) 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.