Probing the Extended Atmospheres of AGB Stars: I. Synthetic imaging of 1D hydrodynamical models at radio and (sub-)millimeter wavelengths

We investigate the observable characteristics of the extended atmospheres of AGB stars across a wide range of radio and (sub-)mm wavelengths using state-of-the-art 1D dynamical atmosphere and wind models over one pulsation period. We also study the relationships between the observable features and model properties. We further study practical distance ranges for observable sources assuming the capabilities of current and upcoming observatories. We present time-variable, frequency-dependent profiles of pulsating AGB stars' atmospheres, illustrating observable features in resolved and unresolved observations, including disc brightness temperature, photosphere radius, and resolved and unresolved spectral indices. Notably, temporal variations in disc brightness temperature closely mirror the temperature variability of the stellar atmosphere. We find that while the photospheric radius decreases due to gas dilution in the layers between consecutive shocks, the increase in the observed stellar radius reflects shock propagation through the atmosphere during the expansion phase, providing a direct measurement method for the shock velocity. Furthermore, our models indicate that enhanced gas temperatures after the passage of a strong shock might be observable in the high-frequency ALMA bands as a decrease in the brightness temperature with increasing frequency. We demonstrate that synthetic observations based on state-of-the-art dynamical atmosphere and wind models are necessary for proper interpretations of current (ALMA and VLA) and future (SKA and ngVLA) observations and that multi-wavelength observations of AGB stars are crucial for empirical studies of their extended atmospheres.