Aims. We use advanced 3D NLTE radiative magnetohydrodynamic simulations of the solar atmosphere to carry out detailed tests of chromospheric diagnostics at millimeter and submillimeter wavelengths. Methods. We focused on the diagnostics of the thermal structure of the chromosphere in the wavelength bands from 0.4 mm up to 9.6 mm that can be accessed with the Atacama Large Millimeter/Submillimeter Array (ALMA) and investigated how these diagnostics are affected by the instrumental resolution. Results. We find that the formation height range of the millimeter radiation depends on the location in the simulation domain and is related to the underlying magnetic structure. Nonetheless, the brightness temperature is a reasonable measure of the gas temperature at the effective formation height at a given location on the solar surface. There is considerable scatter in this relationship, but this is significantly reduced when very weak magnetic fields are avoided. Our results indicate that although instrumental smearing reduces the correlation between brightness and temperature, millimeter brightness can still be used to reliably diagnose electron temperature up to a resolution of 1. If the resolution is more degraded, then the value of the diagnostic diminishes rapidly. Conclusions. We conclude that millimeter brightness can image the chromospheric thermal structure at the height at which the radiation is formed. Thus multiwavelength observations with ALMA with a narrow step in wavelength should provide sufficient information for a tomographic imaging of the chromosphere.
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