GEOPHYSICAL RESEARCH LETTERS, VOL. 19, NO. 19, PAGES 1915-1918, OCTOBER 2, 1992 ANALYTIC DESCRIPTION OF TIRE ELECFRON TEMPERATURE BEHAVIOR IN THE UPPER IONOSPHERE AND PLASMASPHERE. G. V. Khazanov, A. F. Nagy, T. I. Gombosi Space Physics Research Laboratory, Department ofAtmospheric, Oceanic and Space Science, University of Michigan M. A. Koen Irkutsk Polytechnical Institute S. J. Cariglia MIT Haystack Observatory Approximate analytic solutions to thewell-known and commonly used time-dependent electron energy balance equation for the upper ionosphere and plasmasphere have been obtained andarediscussed. The various potential heating sources for the terrestrial plasmasphere are summarized and the corresponding electron temperatures and related characteristic heating and cooling times are calculated. A comparison between the analytic expressions for the temperature variations and relevant measurements shows excellent agreement. 1. Introduction The terrestrial ionosphere/plasmasphere system is a very complex one, but significant progress has been made during the last decade in developing comprehensive theoretical models to describe its behavior [e.g. Schunk etal.,1986; Roble etal., 1987; Guiter and Gombosi, 1990]. These major models are all based onextensive numerical approaches. However, simple analytic solutions are still very useful, because they usually allow a much better understanding of how the system responds to awide spectrum of relevant parameters. Analytic solutions are also useful in testing numericalmodel calculations. Inthis paper we solve the well known and commonly used energy balance equation for thermal electrons in the upper ionosphere and plasmasphere. We obtain simple analytic expressions for the electron temperature and heat flows into the ionosphere, as well as characteristic heating and cooling times. These questions have been considered before, using numerical models, but our analytic expressions allow a simple and to some degree more helpful insight into the processes controlling the electron temperatures inthese regions. 2. Electron EnergyEquation The theoretical basis necessary to calculate the electron temperatures inionosphere and plasmasphere iswell-known [e.g., Banks and Kockarts, 1973; Schunk and Nagy, 1978]. The electron energy equation, for altitudes above 200 km, in the absence offield aligned currents, simplifies down tothe following: n '- 2 as,,, j Copyright 1992 by the American Geophysical Union. Paper number 92GL01940 0094-8534/92/92GL-01940503.00 where Teisthe electron temperature, B isthe magnitude ofthe magnetic field, s is the distance along the magnetic field line, k the Boltzmann constant, ne the electron density, *reis the appropriate constant associated with the thermal conductivity, Lei and Len are the electron-ion and electron-neutral cooling rates respectively and Qe iselectron heating rate. 3. Sources of Electron Heating The main sources of ionospheric electron heating are collisions between superthermal and thermal electrons. Photoelectron production, due to EUV andX-ray solar radiation, peaks between about 100 and 200 km, which isalso the altitude range where precipitating electrons deposit most of their energy. These superthermal electrons move along the geomagnetic field lines, changing direction and losing energy via collisions with neutral andcharged particles [e.g. Nagy and Banks, 1970; Banks et ai., 1974]. Superthermal electrons escaping from the ionosphere experience small-angle scattering, when they move through the plasmasphere, due toCoulomb interactions with the ambient thermal plasma. As aresult of this scattering process, some of the electrons are scattered out of the loss cone and become trapped. Sanatani and Hanson [1970] and Nagy and Banks [1970] gave some qualitative discussion ofelectron trapping and the resulting increase in plasmaspheric heating, but the first attempts of quantitative calculations were those of Gastman [1973], Takahashi [1973] and Lejeune and Worsmer [1976]. Khazanov and Gefan [1982], using theappropriate transport equations for superthermal electrons, obtained the following simple analytic expression for the thermal electron heating rate inthe ionosphere and plasmasphere: Qe (s) = Esq(s) + asn• (s) (2) where Es isthe mean energy gained locally bythe thermal electrons from the superthermal ones, q(s) is the total ionization rate and as isthe strength ofnonlocal heating. There are a number of other, potentially important, processes besides photoelectron heating of the thermal plasma inthe plasmasphere. There are numerous indirect indications of such heating processes such as high ion and electron temperatures (at times exceeding 10,000 OK), enhanced heavy ion (O +and O ++)densities at high altitudes, and electron temperature enhancements with associated SAR arc emissions in the ionosphere [Chandler et al., 1988]. Cole [1965] proposed that Coulomb collisions of the thermal electrons with energetic ring current protons willresult in energy deposition inthe outer plasmasphere and that this energy will be transported down into the ionosphere via 'heat conduction, resulting inenhanced temperatures and 6300 A oxygen airglow emissions, known as SAR arcs. Kozyra et al. 1915