Orbital classification in rotating bar potentials using an empirical proxy of the second integral of motion

TL;DR

This paper introduces Calibrated Angular Momentum (CAM) as an empirical proxy for the non-classical second integral of motion in rotating bar potentials and demonstrates that a CAM–${R_ ext{rms}}$ plane yields a robust, Hamiltonian-independent framework for 2D orbit classification. Through analytical insight in the Freeman bar and systematic tests on Freeman, logarithmic, and Ferrers bar models, the authors show that periodic orbits generate distinct branches in CAM–${R_ ext{rms}}$, partitioning the space into regions associated with orbital families such as $x_1$, $x_4$, and $x_2$, with clear separations even for 3D extensions and $N$-body snapshots. CAM is computationally efficient and less prone to degeneracies than traditional frequency analyses, and it naturally extends to 3D by incorporating a proxy for the third integral $I_3$. The framework provides a practical tool for orbit classification in barred galaxies, complementing Poincaré surface of section and frequency methods and offering direct applicability to realistic simulations.

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 ($σ_{L_z}$) in the corotating frame. We show that CAM is determined by the ratio of the azimuthal to radial actions (${J_φ}^\prime / {J_r}^\prime$) 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.

Figures (7)

• Figure 1: Contours of integrals of motion in the $\overline{L_z}-\sigma_{L_z}$ plane for orbits in the Freeman bar model. The left panel displays the $E_\mathrm{J}$ contours in black curves and the $\mathrm{CAM}$ contours in blue dashed curves. The right panel shows the ${J_r}^\prime$ contours in black curves and the ${J_\phi}^\prime$ contours in blue dashed curves.
• Figure 2: Orbital distributions of the logarithmic bar potential, shown in the ${\overline{L}_z}-\sigma_{L_z}$ plane (left) and the $\mathrm{CAM}-{R_\mathrm{rms}}$ plane (right), color-coded by $E_{\mathrm{J}}$. Representative periodic orbits, displayed with their morphological shapes, are superimposed on the distribution, while quasi-periodic orbits appear as colored curves composed of discrete points. Distinct branches corresponding to periodic $x_1$, $x_4$ and 3:1 resonant orbits are labeled for reference.
• Figure 3: Same as Figure \ref{['fig:log_lzsigmalz_peri']}, but for the Ferrers bar potential. Chaotic are shown as scattered points, while quasi-periodic orbits appear as colored curves composed of discrete point. Distinct branches corresponding to periodic $x_1$, $x_2$, $x_4$ and 3:1 resonant orbits are labeled for reference.
• Figure 4: Orbital distribution in the Freeman bar model, shown in the $\mathrm{CAM}-E_\mathrm{J}$ plane (left) and the $\mathrm{CAM}-{R_\mathrm{rms}}$ plane (right). Scaled orbits are plotted at their corresponding locations in both parameter spaces. The left panel is color-coded by ${R_\mathrm{rms}}$, while the right panel is color-coded by $E_\mathrm{J}$. The approximate regions of the $x_1$, $x_2$ and $x_4$ orbital families are outlined by black dashed lines, and the colored dashed curves in the right panel trace the general trends of orbits at fixed $E_\mathrm{J}$.
• Figure 5: Same as Figure \ref{['fig:freeman_cam_rms']}, but for the logarithmic bar potential. In the right panel, the approximate regions of the $x_1$, $x_4$ orbital families are outlined by black dashed lines, and the solid black horizontal line marks the corotation radius. The colored dashed curves trace the general trends of orbits at fixed $E_\mathrm{J}$.