Recent observational studies of X-shaped structures revealed that values of their opening angles lie in a narrow range: from 20 • to 43 • with smaller X-shaped structures appearing to follow a characteristic opening angle ∼ 27 • -31 • . We use self-consistent numerical simulations to uncover what parameters of host galaxies govern the opening angle spread. We constructed a series of equilibrium models of galaxies with high spatial resolution, varying the dark halo and bulge contribution in the overall gravitational potential, the initial disc thickness of models and the Toomre parameter Q and followed their evolution for almost 8 Gyrs. Each model demonstrated the formation of clear X-structures with different flatness. We have obtained that opening angles lie in the range from 25 • to 42 • throughout the entire evolution. These values are roughly consistent with observational data. The greatest variation in the opening angles is obtained by varying the mass of the dark halo. The initial thickness of the disc and the Toomre parameter are responsible for smaller variations of the angle and shorter X-structures. An increase of both parameters changes the morphology of in-plane bars and X-structures. In some cases we observed even double X-structures. The main effect of the bulge is to prevent buckling at intermediate and late stages of the disc evolution. Comparison of models with different halo masses indicates that the smallest observable values of opening angles can be associated with the presence of a heavy dark halo (more than 3 masses of the disc within the optical radius).

X-structures are often observed in galaxies hosting the so-called B/PS (boxy/peanuts) bulges and are visible from the edge-on view. They are the most notable features of B/PS bulges and appear as four rays protruding from the disk of the host galaxy and distinguishable against the B/PS bulge background. In some works, their origin is thought to be connected with the so-called banana-shaped orbits with a vertical resonance 2:1. A star in such an orbit performs two oscillations in the vertical direction per one revolution in the bar frame. Several recent studies that analyzed ensembles of orbits arising in different N-body models do not confirm the dominance of the resonant 2:1 orbits in X-structures. In our work, we analyze two N-body models and show how the X-structure in our models is gradually assembled from the center to the periphery from orbits with less than 2:1 frequency ratio. The most number of such orbits is enclosed in a “farfalle”-shape (Italian pasta) form and turns out to be non-periodic. We conclude that the X-structure is akin to the envelope curve of regions of high density caused by the crossing or folding of different types of orbits at their highest points, and does not have a “backbone” similar to that of the in-plane bar. Comparing the orbital structure of two different numerical models, we show that the dominance of one or another family of orbits with a certain ratio of the vertical oscillations frequency to the in-plane frequency depends on the parameters of the underlying galaxy and ultimately determines the morphology of the X-structure and the opening angle of its rays.

Based on the spectral analysis of individual orbits of stars from different N-body models, we show that the face-on morphology of the so-called ‘face-on peanut’ bars (boxy bars) and barlenses is supported by different types of orbits. For ‘face-on peanut’ bars, the so-called boxy orbits come to the fore, and they are responsible for the unusual morphology of the bar in the central regions. In the models with compact bulges, the bars show a barlens morphology in their central parts. We found that the barlens supporting orbits come in two types, one of which gives a square-like shape and the other have a rosette-like shape in the frame co-rotating with the bar. Such a shape is typical for orbits around stable loop orbits in nearly axisymmetric potentials only slightly distorted by the bar. They were already known from some of the previous studies but their role in barlens shaping was barely investigated. Although quite simple, the rosette-like orbits are found to be the main building block of a barlens in our models. The detailed analysis of all bar orbits also allowed us to distinguish the x2 orbital family and isolate the structure supported by orbits trapped around this family. The x2 family is well-known, but, apparently, for the first time in N-body models we have revealed the structure it supports by means of spectral dynamics and highlight its contribution to the barlens. We found that the x2 family population increases with an increase in central matter concentration.