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The Ca II H & K lines are strong chromospheric diagnostics that can be used to determine the temperature stratification and magnetic structure of the solar atmosphere. The Atacama Large Millimetre/Submillimetre Array (ALMA) offers complementary information on the thermal structure of stellar atmospheres using mm continuum radiation. The overall aim is to establish more robust solar/stellar activity indicators using ALMA observations in comparison with classical diagnostics, such as the s index and infrared triplet (IRT) index. A study was conducted using 1.5D radiative transfer codes RH1.5D and advanced radiative transfer (ART), along with an enhanced network atmosphere model generated by the state-of-the-art 3D radiation magnetohydrodynamics (rMHD) Bifrost code, to compute synthetic spectra for both Ca II lines and mm continua. To account for the limited spatial resolution of ALMA, we simulated the effect using a Gaussian point spread function (PSF). Additionally, we analysed the correlations and slopes of scatter plots between the Ca II indices and mm continuum for the original and degraded resolutions, focusing on the entire simulation box, quiet Sun regions, and enhanced network patches separately. The activity indices generated from these lines could further be used to compare the spectra of Sun-like stars with the solar spectrum. The Ca II activity indices and mm brightness temperatures are weakly correlated at the high resolution, with the highest correlation observed at a wavelength of 0.3 mm, corresponding to ALMA band 10. As the resolution decreases, the correlation consistently increases. Conversely, the slopes exhibit a decreasing trend with increasing wavelength, while the degradation of resolution does not noticeably affect the calculated slopes. Consequently, these relationships could be valuable for calibrating the mm continuum maps obtained through ALMA observations., Comment: 17 pages, 17 figures

The Ha line, one of the most studied chromospheric diagnostics, is a tracer for magnetic field structures, while its line core intensity provides an estimate of the mass density. The brightness temperatures from Atacama Large Millimetre-submm Array (ALMA) observations provide a complementary view of the activity and the thermal structure of stellar atmospheres. These two diagnostics together can provide insights into the physical properties of stellar atmospheres. In this paper, we present a comparative study between synthetic continuum brightness temperature maps for mm wavelengths (0.3 mm to 8.5 mm) and the width of the Ha 6565{\AA} line. The 3D radiative transfer codes Multi3D and Advanced Radiative Transfer (ART) are used to calculate synthetic spectra for the Ha line and the mm continua respectively, from an enhanced network atmosphere model with non-equilibrium hydrogen ionisation generated with the state-of-the-art 3D rMHD code Bifrost. We use Gaussian Point Spread Function (PSF) for simulating the effect of ALMA's limited spatial resolution and calculate the Ha vs. mm continuum correlations and slopes of scatter plots for the original and degraded resolution of the whole box, quiet sun and enhanced network patches separately. The Ha linewidth and mm brightness temperatures are highly correlated and the correlation is highest at a wavelength 0.8 mm i.e. ALMA Band 7. The correlation increases with decreased resolution. On the other hand, the slopes decrease with increasing wavelength. The degradation of resolution does not have a significant impact on the calculated slopes. With decreasing spatial resolution the standard deviations of the observables, Ha linewidth and brightness temperatures decrease and the correlations between them increase, but the slopes do not change significantly. Hence, these relations may prove useful to calibrate the mm continuum maps observed with ALMA., Comment: 20 pages, 26 figures

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 physics of activity across spectral types. But the common activity indicators show large variability in their values which makes defining a robust quantitative scale 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 mm-based activity indicator for the cool main-sequence stars ($\mathrm{T_{eff}} \sim$ 5000 - 7000 K). We derive the mm-brightness temperature ($\mathrm{T_B(\nu)}$) spectral indices ($\mathrm{\alpha_{mm}}$) for cool stars including the Sun using archival data in the 30 - 1000 GHz range. The derived values for $\mathrm{\alpha_{mm}}$ are explored as a function of various physical parameters and empirical power-law functions were derived. $\mathrm{\alpha_{mm}}$ estimates were also compared with other activity indicators. Despite the estimation errors, $\mathrm{\alpha_{mm}}$ values could well distinguish the cool stars, unlike common activity indicators. The low estimation errors on the derived trends of $\mathrm{\alpha_{mm}}$ versus physical parameters suggest that $\mathrm{\alpha_{mm}}$ could be a robust activity indicator. $\mathrm{\alpha_{mm}}$, 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 activity cycles. This will help explore $\mathrm{\alpha_{mm}}$ in a statistically robust manner and study the emergence of chromospheric heating on the main-sequence., Comment: Accepted in A&A Letters

Due to their wide wavelength coverage across the millimetre to centimetre (mm - cm) range and their increased sensitivity, modern interferometric arrays facilitate observations of the thermal and non-thermal emission from different stellar atmospheric layers. We study the spectral energy distribution ($S_{obs}(\nu)$) of main sequence stars using archival mm - cm data with the aim to study their atmospheric stratification as a function of stellar type. The main-sequence stars with significant detection in mm bands were identified in the ALMA Science Archive. These data were complemented with spectral flux data in the Ultra violet to centimetre range as compiled from various catalogues and observatory archives. We compare the resultant $S_{obs}(\nu)$ of each star with a photospheric emission model ($S_{mod}(\nu)$) calculated with the PHOENIX code. The departures of $S_{obs}(\nu)$ from the model are quantified in terms of a "spectral flux excess" parameter ($\Delta S/S_{mod} (\nu)$) for every star in the sample. The initial sample consists of 12 stars across a range of spectral type from A1 to M3.5 and the Sun-as-a-star as reference. The stars with $T_{eff} = 4000 - 7000\,K$ (F - M type) showed a systematically higher $\Delta S/S_{mod}$ in the mm - cm range, with the values rising with decreasing $\nu$. The steepness of this rise is higher for cooler stars, though the fully convective 3000 K star in the sample deviated from this trend. For the A-type stars, $\Delta S/S_{mod} \sim 0$ within errors. The high $\Delta S/S_{mod}$ in cool stars points to the presence of hotter upper atmospheric layers, i.e. a chromosphere and corona, like for the Sun. The mm - cm $\Delta S/S_{mod}$ spectrum offers a way to estimate the efficiency of the heating mechanisms across various atmospheric layers and thereby to understand their structure and activity. We emphasise the need for more mm - cm data., Comment: Accepted for publication in A&A