Context. Observations of the Sun tell us that its granular and subgranular small-scale magnetism has significant consequences for global quantities such as the total solar irradiance or convective blueshift of spectral lines. Aims. In this paper, properties of the small-scale magnetism of four cool stellar atmospheres, including the Sun, are investigated, and in particular its effects on the radiative intensity and flux. Methods. We carried out three-dimensional radiation magnetohydrodynamic simulations with the CO5BOLD code in two different settings: with and without a magnetic field. These are thought to represent states of high and low small-scale magnetic activity of a stellar magnetic cycle. Results. We find that the presence of small-scale magnetism increases the bolometric intensity and flux in all investigated models. The surplus in radiative flux of the magnetic over the magnetic field-free atmosphere increases with increasing effective temperature, Teff, from 0.47% for spectral type K8V to 1.05% for the solar model, but decreases for higher effective temperatures than solar. The degree of evacuation of the magnetic flux concentrations monotonically increases with Teff as does their depression of the visible optical surface, that is the Wilson depression. Nevertheless, the strength of the field concentrations on this surface stays remarkably unchanged at ≈1560 G throughout the considered range of spectral types. With respect to the surrounding gas pressure, the field strength is close to (thermal) equipartition for the Sun and spectral type F5V but is clearly sub-equipartition for K2V and more so for K8V. The magnetic flux concentrations appear most conspicuous for model K2V owing to their high brightness contrast. Conclusions. For mean magnetic flux densities of approximately 50 G, we expect the small-scale magnetism of stars in the spectral range from F5V to K8V to produce a positive contribution to their bolometric luminosity. The modulation seems to be most effective for early G-type stars.

The magnetic field outside of sunspots is concentrated in the intergranular space, where it forms a delicate filigree of bright ribbons and dots as seen on broad band images of the Sun. We expect this small-scale magnetic field to exhibit a similar behavior in stellar atmospheres. In order to find out more about it, we perform numerical simulations of the surface layers of stellar atmospheres. Here, we report on preliminary results from simulations in the range between 4000 K and 6500 K effective temperature with an initial vertical, homogeneous magnetic field of 50 G strength. We find that the field strength of the strongest magnetic flux concentrations increases with decreasing effective temperature at the height level where the average Rosseland optical depth is one. On the other hand, at the same level, the field is less strong than the thermal equipartition value in the coolest model but assumes superequipartition in the models hotter than 5000 K. While the Wilson depression of the strongest field concentrations is about one pressure scale height in the coolest model, it is more than four times the pressure scale height in the hottest one. We also find that the relative contribution of the bright filigree to the bolometric, vertically directed radiative intensity is most significant for the Teff = 5000 K model (0.6%–0.79%) and least significant for the hottest and coolest models (0.1%–0.46% and 0.14%–0.32%, respectively). This behavior suggests that the effect of the small-scale magnetic field on the photometric variability is more significant for K dwarf stars than for F-type and also M-type stars.