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ABSTRACT We present a new method to infer the 3D luminosity distributions of edge-on barred galaxies with boxy-peanut/X (BP/X) shaped structures from their 2D surface brightness distributions. Our method relies on forward modelling of newly introduced parametric 3D density distributions for the BP/X bar, disc and other components using an existing image fitting software package (imfit). We validate our method using an N-body simulation of a barred disc galaxy with a moderately strong BP/X shape. For fixed orientation angles, the derived 3D BP/X-shaped density distribution is shown to yield a gravitational potential that is accurate to at least 5 per cent and forces that are accurate to at least 15 per cent, with average errors being $$\sim 1.5~{{\ \rm per\ cent}}$$ for both. When additional quantities of interest, such as the orientation of the bar to the line of sight, its pattern speed, and the stellar mass-to-light ratio are unknown they can be recovered to high accuracy by providing the parametric density distribution to the Schwarzschild modelling code FORSTAND. We also explore the ability of our models to recover the mass of the central supermassive black hole. This method is the first to be able to accurately recover both the orientation of the bar to the line of sight and its pattern speed when the disc is perfectly edge-on. 

ABSTRACT We apply the barred Schwarzschild method developed by Tahmasebzadeh et al. (2022) to a barred S0 galaxy, NGC 4371, observed by IFU instruments from the TIMER and ATLAS3D projects. We construct the gravitational potential by combining a fixed black hole mass, a spherical dark matter halo, and stellar mass distribution deprojected from 3.6 μm S$^4$G image considering an axisymmetric disc and a triaxial bar. We independently modelled kinematic data from TIMER and ATLAS3D. Both models fit the data remarkably well. We find a consistent bar pattern speed from the two sets of models with $$\Omega _{\rm p} = 23.6 \pm 2.8 \, \mathrm{km \, s^{-1} \, kpc^{-1} }$$ and $$\Omega _{\rm p} = 22.4 \pm 3.5 \, \mathrm{km \, s^{-1} \, kpc^{-1} }$$, respectively. The dimensionless bar rotation parameter is determined to be $$\mathcal {R} \equiv R_{\rm cor}/R_{\rm bar}=1.88 \pm 0.37$$, indicating a likely slow bar in NGC 4371. Additionally, our model predicts a high amount of dark matter within the bar region ($$M_{\rm DM}/ M_{\rm total}$$\sim 0.51 \pm 0.06$$), which, aligned with the predictions of cosmological simulations, indicates that fast bars are generally found in baryon-dominated discs. Based on the best-fitting model, we further decompose the galaxy into multiple 3D orbital structures, including a BP/X bar, a classical bulge, a nuclear disc, and a main disc. The BP/X bar is not perfectly included in the input 3D density model, but BP/X-supporting orbits are picked through the fitting to the kinematic data. This is the first time a real barred galaxy has been modelled utilizing the Schwarzschild method including a 3D bar. 

Abstract 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.