Orthogonal vertical velocity dispersion distributions produced by bars
In barred galaxies, the contours of stellar velocity dispersions ($σ$) are generally expected to be oval and aligned with the orientation of bars. However, many double-barred (S2B) galaxies exhibit distinct $σ$ peaks on the minor axis of inner bars, which we termed "$σ$-humps," while two local $σ$ minima are present close to the ends of inner bars, i.e., "$σ$-hollows." Analysis of numerical simulations shows that $σ_z$-humps or hollows should play an important role in generating the observed $σ$-humps+hollows in low-inclination galaxies. In order to systematically investigate the properties of $σ_z$ in barred galaxies, we apply the vertical Jeans equation to a group of well-designed three-dimensional bar+disk(+bulge) models. A vertically thin bar can lower $σ_z$ along the bar and enhance it perpendicular to the bar, thus generating $σ_z$-humps+hollows. Such a result suggests that $σ_z$-humps+hollows can be generated by the purely dynamical response of stars in the presence of a, sufficiently massive, vertically thin bar, even without an outer bar. Using self-consistent $N$-body simulations, we verify the existence of vertically thin bars in the nuclear-barred and S2B models which generate prominent $σ$-humps+hollows. Thus the ubiquitous presence of $σ$-humps+hollows in S2Bs implies that inner bars are vertically thin. The addition of a bulge makes the $σ_z$-humps more ambiguous and thus tends to somewhat hide the $σ_z$-humps+hollows. We show that $σ_z$ may be used as a kinematic diagnostic of stellar components that have different thickness, providing a direct perspective on the morphology and thickness of nearly face-on bars and bulges with integral field unit spectroscopy.