Simulations of the Photospheric Magnetic Activity and Outer Atmospheric Radiative Losses of Cool Stars Based on Characteristics of the Solar Magnetic Field

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Abstract
The observed disk-integrated radiative losses from the outer atmospheres of stars with convective envelopes are determined by the distribution of magnetic field over their surfaces. Earlier modeling of the random walk transport of the solar photospheric magnetic field with the classical Leighton model has given us insight into how field patterns form and evolve on large scales. This paper presents the first comprehensive simulations of the dynamic photospheric magnetic field of the Sun down to the scale of the mixed polarity network, thus incorporating all flux involved in outer atmospheric heating. The algorithm incorporates the classical diffusion model but includes ephemeral regions (which populate the network that contributes significantly to the disk-integrated chromospheric emission) and the early phase of decay of active regions (which is important for the field patterns in very active stars). Moreover, individual flux concentrations are tracked and subjected to collisions and fragmentation, and the flux dispersal is made dependent on the flux contained in the concentrations, as observed on the Sun. The latter modification causes the model to be nonlinear. Tests demonstrate that the new model successfully describes the solar magnetic field. The model is then used to simulate the field on other cool stars covering several orders of magnitude in activity and to estimate the surface-averaged radiative losses associated with that field. The stellar extrapolations are based on the statistical properties of solar bipolar regions throughout the cycle. Simulations in which only the frequency of flux emergence is changed to simulate stars of different activity are shown to be consistent with the observed nonlinear relationships between disk-averaged radiative losses from chromospheres and coronae of cool stars. Consequently, the properties of the solar magnetic field from small ephemeral regions up to large active regions are compatible with stellar observations. Stellar observations suggest that those field properties are not the only ones that can explain the flux-flux relationships, however, because also stars with polar spots or persistent active longitudes obey these same flux-flux relationships. The model is also used to understand how rapidly flux is processed in stellar photospheres in stars with activity patterns like the Sun: the average total absolute magnetic flux Φ* (Mx) at the stellar surface is found to be proportional to the mean rate of flux emergence and cancellation E* (Mx s-1) within the range from E☉ up to 10E☉, where E☉ is the flux injection rate for the active Sun. This linearity is primarily a consequence of an activity-dependent change in the shape of the flux histogram for emerging bipoles. This change reflects that active regions and ephemeral regions have a different dependence on dynamo strength. The implications of the results of the simulations for the dynamo and for the relationship between activity and stellar rotation are discussed.