Orthogonal vertical velocity dispersion distributions produced by bars

In barred galaxies, the contours of stellar velocity dispersions ($\sigma$) are generally expected to be oval and aligned with the orientation of bars. However, many double-barred (S2B) galaxies exhibit distinct $\sigma$ peaks on the minor axis of inner bars, which we termed "$\sigma$-humps," while two local $\sigma$ minima are present close to the ends of inner bars, i.e., "$\sigma$-hollows." Analysis of numerical simulations shows that $\sigma_z$-humps or hollows should play an important role in generating the observed $\sigma$-humps+hollows in low-inclination galaxies. In order to systematically investigate the properties of $\sigma_z$ 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 $\sigma_z$ along the bar and enhance it perpendicular to the bar, thus generating $\sigma_z$-humps+hollows. Such a result suggests that $\sigma_z$-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 $\sigma$-humps+hollows. Thus the ubiquitous presence of $\sigma$-humps+hollows in S2Bs implies that inner bars are vertically thin. The addition of a bulge makes the $\sigma_z$-humps more ambiguous and thus tends to somewhat hide the $\sigma_z$-humps+hollows. We show that $\sigma_z$ 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.