Constraining the properties of nonradiative heating of the coronae of cool stars and the Sun

The dominant mechanism that heats the coronae of the Sun and of other cool stars remains to be identified, despite numerous solar and stellar studies. We address the problem from a statistical point of view, by approximating the emission expected from the ensemble of loops in stellar coronae. We develop a prototype of a global atmospheric, empirical model that employs (1) simulations of the surface magnetic field of the Sun and active stars throughout sunspot cycles, (2) potential field computations of the corresponding coronal field, and (3) an approximation of atmospheres for 2000 coronal loops for randomly selected field lines in each flux configuration, representative of all environments from very quiet to the interior of active regions. The latter requires specification of the flux density that passes through the base of the loops to heat the corona. We parameterize P-H as a function of the base field strength B-base (in G), loop half-length l (in Mm), and footpoint velocity upsilon (in km s(-1)). We find a best fit for a heating flux density of PH approximate to 2 x 10(7)(B-base/100)(1.0+/-0.5)(l/24)(-0.7 +/-) (0.5) ergs cm(-2) s(-1) (the allowed ranges of the exponents are shown). This parameterization matches the observed soft X-ray losses from the coronae of the Sun and more active stars with rotation periods down to 5 days, throughout their activity cycles, as well as the characteristic coronal temperatures, and the relationships between disk-averaged radiative and magnetic flux densities. We compare this parameterization to models previously published in the literature and find that dissipation of current layers and turbulence are the most likely candidate heating mechanisms, for which both low-frequency driving and high-frequency driving meet the criteria comparably well. We find, moreover, that the heating scale length of similar to20 Mm inferred from solar observations matches the characteristic e-folding height of the field strength over solar active regions, which suggests that coronal heating depends on the local field strength. Our modeling suggests that there is no need for a strong selection mechanism to determine which loops are heated and which are not, but that the sensitive dependence of the heating on the base field strength causes the appearance of a corona that consists of bright loops embedded in less bright environments. We compare the differential emission measures for the simulated coronae to those of the Sun and more active cool stars, and we also discuss the apparently weak velocity dependence of the best-fit parameterization for P-H.