Heat Transport in Steep Temperature Gradients. I. Small Flaring Solar Loops

Results on nonlocal heat transport which properly takes into account the presence of fast electrons with mean free paths much longer than the temperature scale height L are reviewed. In terms of the mean free path for the slow bulk electrons, lambda(s), the nonlocal effects are important whenever lambda(s)/L greater than 0.001, with the following consequences. The heat flux in the hot part of the gradient is reduced relative to the Spitzer-Haerm value q(SH) which does not take into account the heat carried away by the fast electrons. The heat flux in the cold part of the gradient is enhanced relative to the value q(SH) which does not take into account the heat deposited by the fast electrons. These quite general results, which should have several applications in astrophysics, are applied to the problem of thermal hard X-ray burst models. It is shown that heat is not bottled up as effectively as in some past models, and temperatures achieved for realistic energy input rates are consequently not as high. As a result such sources can be effective only in the soft part (10-30 keV) of the hard X-ray range for energy input rates up to 6,400 ergs/cu cm s. The analysis is based on a fluid model and does not consider the X-ray signature of fast electrons which escape to distances far beyond the conduction fronts formed. It is shown that such electrons could at most be effective in the soft part of the hard X-ray range.