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 x1 orbits, are present in growing bars, and can appear after bar buckling. Here we investigate the orbital support and evolution of shoulders via frequency analyses of orbits in simulations. We confirm that looped orbits are important to shoulders, and can remain so, to a lesser extent, after they have been vertically thickened. We show that looped orbits are found at the Inner Lindblad Resonance (ILR) with typical ratios of vertical to radial frequencies $1 \lesssim\Omega_z/\Omega_R \lesssim 3/2$ ($\textit{warm}$ ILR). $\textit{Cool}$ ILR orbits (those with $\Omega_z/\Omega_R > 3/2$), which are vertically thin and have no loops, contribute negligibly to shoulders. As bars slow and thicken, either secularly or by buckling, they populate warm ILR orbits. Further thickening carries these orbits towards the vertical ILR (vILR) at $\Omega_z/\Omega_R=1$, where they convert in-plane motion to vertical motion, become chaotic, kinematically hotter and less shoulder-supporting. Because of this heating by the vILR, persistent shoulders require bars to trap new orbits, consistent with the need for a growing bar. Since buckling speeds up the trapping of stars on warm ILR orbits, it can be followed by the formation of shoulders, as seen in simulations. This evolutionary sequence supports the recent observational finding that shoulder formation likely precedes the emergence of BP-bulges. The python module used for the frequency analysis, $\texttt{naif}$, is made publicly available.