Tomography of cool giant and supergiant star atmospheres III. Validation of the method on VLTI/AMBER observations of the Mira star S Ori

The knowledge of the structure and dynamics of AGB-star atmospheres is crucial to better understand the mass loss. The tomographic method, that relies on the design of spectral masks containing lines forming in given ranges of optical depths in the stellar atmosphere, is an ideal technique for this purpose. It is applied to high-resolution spectro-interferometric VLTI/AMBER observations of the Mira-type AGB star S Ori. First, the interferometric visibilities are extracted at wavelengths contributing to the tomographic masks and fitted to those computed from a uniform disk model. This allows the measurement of the geometrical extent of the atmospheric layer probed by the corresponding mask. Then, we compare the observed atmospheric extension with those measured from available 1D pulsation CODEX models and 3D radiative-hydrodynamics CO5BOLD simulations. We found that while the average optical depths probed by the tomographic masks in S Ori decrease (with $<\log \tau_0> = -0.45$, $-1.45$, and $-2.45$ from the innermost to the central and outermost layers), the angular diameters of these layers increase, from 10.59 $\pm$ 0.09 mas through 11.84 $\pm$ 0.17 mas, up to 14.08 $\pm$ 0.15 mas. A similar behavior is observed when the tomographic method is applied to 1D and 3D dynamical models. Thus, this study derives, for the first time, a quantitative relation between optical- and geometrical-depth scales when applied to the Mira star S Ori, or to 1D and 3D dynamical models. In the context of Mira-type stars, the knowledge of the link between the optical and geometrical depths opens the way to derive the shock-wave propagation velocity, which can not be directly observed in these stars.