Dorelli and Scudder (1999) modelled this effect, while describing the zeroth-order local electron VDF by a kappa function with exponent , and retaining only the linear terms in an expansion in terms of the pitch angle variable . This linearization badly fails in the distant solar wind, as was shown by Dum et al. (1980), and was also criticised for its application to the lower corona by Landi and Pantellini (2001), both authors arguing that higher-order polynomials must be retained to model the collisional energy exchange between the thermal core electrons and suprathermal halo electrons. Also some time ago, Anderson (1994) criticised the exospheric velocity filtration model and argued that a collisional treatment of this effect is needed.

According to Dorelli and Scudder (1999), if one considers in the electron VDF power-law suprathermal tails (kappa function) and then accounts for the associated velocity filtration effect, one gets a corrected energy balance for a steady-state transition region (for a general review of the classical approach see the book of Mariska, 1992 about the TR), reading:

where and are positive constants depending on the -exponent, and denotes the combined external force related to the gravitational and electric potential. Here is the altitude, and and are the radiative loss function, respectively local heating function. If the net force is attractive (), then the last term represents a heating source adding to . In conclusion, non-local electron heat flow coupled to the force through velocity filtration must be taken into account in the internal energy balance in the TR. This result is a kinetic consequence of heat flow in a weakly collisional and a non-uniform medium such
as the corona, and will likely remain valid in any realistic future model. Landi and Pantellini (2001) have
already demonstrated this with their more refined kinetic model of electron heat conduction. In the solar
corona the collisional mean free path for a thermal particle (electrons or protons) is small, of the order of
10^{–2} to 10^{–3} times the typical scale height, , of macroscopic fluid quantities like density or temperature.
Despite this relative smallness of , the coronal plasma cannot be described satisfactorily by
theories supposing that the local VDFs are close to Maxwellians (see our previous discussion
again).

It was shown in particular that if the electron VDFs at the base of the corona have sufficiently strong suprathermal power-law tails, the heat flux may indeed flow upwards, i.e., in the direction of increasing temperature. Using kappa functions as prototypes for non-thermal VDFs at the base, they found that heat conduction can only be adequately described by the classical law provided that . This value is much smaller than the one found by Dorelli and Scudder (1999). The results from both groups are illustrated and compared in Figure 9. Landi and Pantellini (2001) further showed that, unless extremely strong electron tails are assumed near the base of the corona (), a local heating mechanism (most likely by waves) is needed to sustain the steep temperature gradient between the base of the corona and the location of its temperature maximum.

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