2023ApJ...955...38B

Many barred galaxies exhibit upturns (shoulders) in their bar-major-axis density profile. Simulation studies have suggested that shoulders are supported by looped x₁ orbits, occur in growing bars, and can appear after bar buckling. We investigate the orbital support and evolution of shoulders via frequency analyses of orbits in simulations. We confirm that looped orbits are shoulder-supporting, and can remain so, to a lesser extent, after being vertically thickened. We show that looped orbits appear at the resonance ( Ω_φ - Ω_P)/Ω_R = 1/2 (analogous to the classical inner Lindblad resonance, and here called ILR) with vertical-to-radial frequency ratios 1 ≲ Ω_z/Ω_R ≲ 3/2 (vertically warm orbits). Cool orbits at the ILR (those with Ω_z/Ω_R > 3/2) are vertically thin and have no loops, contributing negligibly to shoulders. As bars slow and thicken, either secularly or by buckling, they populate warm orbits at the ILR. Further thickening carries these orbits toward crossing the vertical ILR [vILR, ( Ω_φ - Ω_P)/Ω_z = 1/2], where they convert in-plane motion to vertical motion, become chaotic, kinematically hotter, and less shoulder-supporting. Hence, persistent shoulders require bars to trap new stars, consistent with the need for a growing bar. Since buckling speeds up trapping on warm orbits at the ILR, it can be followed by shoulder formation, as seen in simulations. This sequence supports the recent observational finding that shoulders likely precede the emergence of BP-bulges. The python module for the frequency analysis, naif, is made available.