Inter-network regions of the Sun at millimetre wavelengths

Context. The continuum intensity at wavelengths around 1 mm provides an excellent way to probe the solar chromosphere and thus valuable input for the ongoing controversy on the thermal structure and the dynamics of this layer. Aims. The synthetic continuum intensity maps for near-millimetre wavelengths presented here demonstrate the potential of future observations of the small-scale structure and dynamics of internetwork regions on the Sun. Methods. The synthetic intensity / brightness temperature maps are calculated on basis of three-dimensional radiation (magneto-)hydrodynamic (MHD) simulations. The assumption of local thermodynamic equilibrium (LTE) is valid for the source function. The electron densities are also treated in LTE for most maps but also in non-LTE for a representative model snapshot. Quantities like intensity contrast, intensity contribution functions, spatial and temporal scales are analysed in dependence on wavelength and heliocentric angle. Results. While the millimetre continuum at 0.3 mm originates mainly from the upper photosphere, the longer wavelengths considered here map the low and middle chromosphere. The effective formation height increases generally with wavelength and also from disk-centre towards the solar limb. The average intensity contribution functions are usually rather broad and in some cases they are even double-peaked as there are contributions from hot shock waves and cool post-shock regions in the model chromosphere. The resulting shock-induced thermal structure translates to filamentary brightenings and fainter regions in between. Taking into account the deviations from ionisation equilibrium for hydrogen gives a less strong variation of the electron density and with it of the optical depth. The result is a narrower formation height range although the intensity maps still are characterised by a highly complex pattern. The average brightness temperature increases with wavelength and towards the limb although the wavelength-dependence is reversed for the MHD model and the NLTE brightness temperature maps. The relative contrast depends on wavelength in the same way as the average intensity but decreases towards the limb. The dependence of the brightness temperature distribution on wavelength and disk-position can be explained with the differences in formation height and the variation of temperature fluctuations with height in the model atmospheres. The related spatial and temporal scales of the chromospheric pattern should be accessible by future instruments. Conclusions. Future high-resolution millimetre arrays, such as the Atacama Large Millimeter Array (ALMA), will be capable of directly mapping the thermal structure of the solar chromosphere. Simultaneous observations at different wavelengths could be exploited for a tomography of the chromosphere, mapping its three-dimensional structure, and also for tracking shock waves. The new generation of millimetre arrays will be thus of great value for understanding the dynamics and structure of the solar atmosphere.