Stellar X-rays and magnetic activity in 3D MHD coronal models

Observations suggest a power-law relation between the coronal emission in X-rays, $L_{\rm{X}}$, and the total (unsigned) magnetic flux at the stellar surface, $\Phi$. The physics basis for this relation is poorly understood. We use three-dimensional (3D) magnetohydrodynamics (MHD) numerical models of the coronae above active regions, that is, strong concentrations of magnetic field, to investigate the $L_{\rm{X}}$ versus $\Phi$ relation and illustrate this relation with an analytical model based on simple well-established scaling relations. In the 3D MHD model horizontal (convective) motions near the surface induce currents in the coronal magnetic field that are dissipated and heat the plasma. This self-consistently creates a corona with a temperature of 1 MK. We run a series of models that differ in terms of the (unsigned) magnetic flux at the surface by changing the (peak) magnetic field strength while keeping all other parameters fixed. In the 3D MHD models we find that the energy input into the corona, characterized by either the Poynting flux or the total volumetric heating, scales roughly quadratically with the unsigned surface flux $\Phi$. This is expected from heating through field-line braiding. Our central result is the nonlinear scaling of the X-ray emission as $L_{\rm{X}}\propto \Phi^{3.44}$. This scaling is slightly steeper than found in recent observations that give power-law indices of up to only 2 or 3. Assuming that on a real star, not only the peak magnetic field strength in the active regions changes but also their number (or surface filling factor), our results are consistent with observations. Our model provides indications of what causes the steep increase in X-ray luminosity by four orders of magnitude from solar-type activity to fast rotating active stars.