Solar Chromospheric Temperature Diagnostics: a joint ALMA-H$\alpha$ analysis

We present the first high-resolution, simultaneous observations of the solar chromosphere in the optical and millimeter wavelength ranges, obtained with ALMA and the IBIS instrument at the Dunn Solar Telescope. In this paper we concentrate on the comparison between the brightness temperature observed in ALMA Band 3 (3 mm; 100 GHz) and the core width of the H$\alpha$ 656.3 nm line, previously identified as a possible diagnostic of the chromospheric temperature. We find that in the area of plage, network and fibrils covered by our FOV the two diagnostics are well correlated, with similar spatial structures observed in both. The strength of the correlation is remarkable, given that the source function of the mm-radiation obeys local thermodynamic equilibrium, while the H$\alpha$ line has a source function that deviates significantly from the local Planck function. The observed range of ALMA brightness temperatures is sensibly smaller than the temperature range that was previously invoked to explain the observed width variations in H$\alpha$. We employ analysis from forward modeling with the RH code to argue that the strong correlation between H$\alpha$ width and ALMA brightness temperature is caused by their shared dependence on the population number $n_2$ of the first excited level of hydrogen. This population number drives millimeter opacity through hydrogen ionization via the Balmer continuum, and H$\alpha$ width through a curve-of-growth-like opacity effect. Ultimately, the $n_2$ population is regulated by the enhancement or lack of downward Ly$\alpha$ flux, which coherently shifts the formation height of both diagnostics to regions with different temperature, respectively.