An activity indicator, which can provide a robust quantitative mapping between the stellar activity and the physical properties of its atmosphere, is important in exploring the evolution of the observed active phenomena across main-sequence stars of different spectral types. Common activity indicators do provide qualitative correlations with physical properties such as T_eff and the rotation period, among others. However, due to the large variability in their values, even for a single star, defining robust quantitative mappings between activity and physical properties is difficult. Millimetre (mm) wavelengths probe the different atmospheric layers within the stellar chromosphere, providing a tomographic view of the atmospheric dynamics.

The project aims to define a robust activity indicator by characterising mm brightness temperature spectra (T_B(ν)) of the cool main-sequence stars (T_eff ∼ 5000–7000 K) compiled by Paper I in this series. The sample contains 13 stars, including the Sun.

We derived the mm T_B(ν) spectral indices (α_mm) for cool stars, including the Sun, based on observations in the 30–1000 GHz range. The derived values for α_mm are explored as a function of various physical parameters and empirical power-law functions were derived. We also compared α_mm estimates with other activity indicators.

Despite the estimation errors, α_mm values could distinguish the cool stars well, unlike common activity indicators. The low estimation errors on the derived trends of α_mm vs. physical parameters suggest that α_mm could be a robust activity indicator.

We note that α_mm, which is linked to chromospheric thermal stratification and activity in cool stars, can well distinguish and physically characterise the stars more robustly than common activity indicators. We emphasise the need for multi-frequency data across the mm band for stars, with a range of physical parameters and gathered at multiple epochs during their activity cycles. This will help to explore α_mm in a statistically robust manner and to study the emergence of chromospheric heating on the main sequence.