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Many barred galaxies exhibit upturns (shoulders) in their bar-major-axis density profile. Simulation studies have suggested that shoulders are supported by loopedx₁orbits, 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 (verticallywarmorbits).Coolorbits 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.

Many disc galaxies host galactic bars, which exert time-dependent, non-axisymmetric forces that can alter the orbits of stars. There should be both angle and radius dependences in the resulting radial rearrangement of stars (‘radial mixing’) due to a bar; we present here novel results and trends through analysis of the joint impact of these factors. We use an N-body simulation to investigate the changes in the radial locations of star particles in a disc after a bar forms by quantifying the change in orbital radii in a series of annuli at different times post bar formation. We find that the bar induces both azimuth angle- and radius-dependent trends in the median distance that stars have travelled to enter a given annulus. Angle-dependent trends are present at all radii we consider, and the radius-dependent trends roughly divide the disc into three ‘zones’. In the inner zone, stars generally originated at larger radii and their orbits evolved inwards. Stars in the outer zone likely originated at smaller radii and their orbits evolved outwards. In the intermediate zone, there is no net inwards or outwards evolution of orbits. We adopt a simple toy model of a radius-dependent initial metallicity gradient and discuss recent observational evidence for angle-dependent stellar metallicity variations in the Milky Way in the context of this model. We briefly comment on the possibility of using observed angle-dependent metallicity trends to learn about the initial metallicity gradient(s) and the radial rearrangement that occurred in the disc.