Title: Orbital support and evolution of flat profiles of bars (shoulders)

Abstract: Many barred galaxies exhibit upturns (shoulders) in their bar major-axis density profile. Simulation studies have suggested that shoulders are supported by looped $x_1$ 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 $(\Omega_\varphi - \Omega_\mathrm{P})/\Omega_R=1/2$ (analogous to the classical Inner Lindblad Resonance, and here called ILR) with vertical-to-radial frequency ratios $1 \lesssim\Omega_z/\Omega_R \lesssim 3/2$ (vertically warm orbits). Cool orbits at the ILR (those with $\Omega_z/\Omega_R > 3/2$) 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 towards crossing the vertical ILR [vILR, $(\Omega_\varphi - \Omega_\mathrm{P})/\Omega_z=1/2$], where they convert in-plane 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.