Invalidity of local thermodynamic equilibrium for electrons in the solar transition region. Part 2: Analysis of a linear BGK model

In an earlier paper numerical solutions of the Landau equation were obtained which show that the tail of the electron velocity distribution functions differs substantially from a local Maxwellian distribution in the solar transition region and upper chromosphere. In this paper, It is shown that a linearized version of the BGK model kinetic equation, with collision frequency proportional to 1/v cubed, can be solved analytically for the tail of the distribution function in an atmosphere with prescribed temperature and density profiles. Results for the angle-averaged distribution so obtained are shown to be in reasonably good agreement with earlier numerical results. Accurate, easily evaluated approximations for the tail of the distribution function are derived from the exact formulas. These show that both the zeroth and first angular moments of the distribution function are nearly power laws over a wide velocity range in the low transition region. It is also shown that the heat flux into lower temperature region is carried by suprathermal electrons with velocities well above local thermal velocities. The formulas given here should be useful in the calculation of electron-ion inelastic collision rates under conditions in which the local Maxwellian approximation is invalid.