What determines the flatness of X-shaped structures in edge-on galaxies?

Recent observational studies of X-shaped structures revealed that the values of their opening angles lie in a narrow range: from 20 to 43 degrees with smaller X-shaped structures appearing to follow a characteristic opening angle similar to 27-31 degrees. We use self-consistent numerical simulations to uncover which 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 Gyr. Each model demonstrated the formation of clear X-structures with different flatnesses. We have found that opening angles lie in the range from 25 to 42 degrees 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 even observed 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 three masses of the disc within the optical radius).