The magnetic virial theorem and the nature of flares on the Sun and other G stars

Modern observations of magnetic fields confirm basic ideas obtained earlier by A.B. Severny about field changes that occur over the course of a flare. For interpretation of new data of the magnetic field vector, the methods of the non-linear force-free field (NLFFF) extrapolation were developed. As a test of these methods, we carried out the NLFFF extrapolation for the active region AR 11158 (February 2011) using the algorithm by Rudenko and Myshyakov (2009). This NLFFF approximation gives information on the magnetic field structure in the corona of active regions (AR). When considering the only total energy of non-stationary processes, it is possible to shift from the equations describing the structural features to considering the magnetic virial theorem. We obtain new analytic expression for the free energy of the solar corona as a whole and discuss circumstances, which prevent using this analytic expression as a measure of the free energy of a size-limited AR, when magnetic fields of the photosphere of AR are not force-free. Observations and NLFFF extrapolation give the upper estimate of the energy of flares, which are able to occur in a given large active region. For the Sun, this value does not exceed 3×1032 ergs. Because the average longitudinal magnetic fields of G stars (at an age of 1-2 Gyr) are ten-times stronger than the maximum magnetic field of the Sun as a star, the energy of solar-type stellar flares can only slightly exceed 1034 ergs. These flares are associated with the energy deposit above the chromosphere and its subsequent release. Phenomena with the total energy up to 1036 ergs which were registered also on the younger G stars have the nature which differs substantially from those non-stationary processes occurring on the Sun.