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Dynamic atmospheres and winds of cool luminous giants, I. Al$_2$O$_3$ and silicate dust in the close vicinity of M-type AGB stars

High spatial resolution techniques have given valuable insights into the mass loss mechanism of AGB stars, which presumably involves a combination of atmospheric levitation by pulsation-induced shock waves and radiation pressure on dust. Observations indicate that Al$_2$O$_3$ condenses at distances of about 2 stellar radii or less, prior to the formation of silicates. Al$_2$O$_3$ grains are therefore prime candidates for producing the scattered light observed in the close vicinity of several M-type AGB stars, and they may be seed particles for the condensation of silicates at lower temperatures. We have constructed a new generation of Dynamic Atmosphere & Radiation-driven Wind models based on Implicit Numerics (DARWIN), including a time-dependent treatment of grain growth & evaporation for both Al$_2$O$_3$ and Fe-free silicates (Mg$_2$SiO$_4$). The equations describing these dust species are solved in the framework of a frequency-dependent radiation-hydrodynamical model for the atmosphere & wind structure, taking pulsation-induced shock waves and periodic luminosity variations into account. Condensation of Al$_2$O$_3$ at the close distances and in the high concentrations implied by observations requires high transparency of the grains in the visual and near-IR region to avoid destruction by radiative heating. For solar abundances, radiation pressure due to Al$_2$O$_3$ is too low to drive a wind. Nevertheless, this dust species may have indirect effects on mass loss. The formation of composite grains with an Al$_2$O$_3$ core and a silicate mantle can give grain growth a head start, increasing both mass loss rates and wind velocities. Furthermore, our experimental core-mantle grain models lead to variations of visual and near-IR colors during a pulsation cycle which are in excellent agreement with observations.