Comparison of the measured and modelled electron densities and temperatures in the ionosphere and plasmasphere during 20/30 January, 1993

We present a comparison of the electron density and temperature behaviour in the ionosphere and plasmasphere measured by the Millstone Hill incoherent-scatter radar and the instruments on board of the EXOS-D satellite with numerical model calculations from a time-dependent mathematical model of the Earth’s ionosphere and plasmasphere during the geomagnetically quiet and storm period on 20/30 January, 1993. We have evaluated the value of the additional heating rate that should be added to the normal photoelectron heating in the electron energy equation in the daytime plasmasphere region above 5000 km along the magnetic field line to explain the high electron temperature measured by the instruments on board of the EXOS-D satellite within the Millstone Hill magnetic field flux tube in the Northern Hemisphere. The additional heating brings the measured and modelled electron temperatures into agreement in the plasmasphere and into very large disagreement in the ionosphere if the classical electron heat flux along magnetic field line is used in the model. A new approach, based on a new effective electron thermal conductivity coefficient along the magnetic field line, is presented to model the electron temperature in the ionosphere and plasmasphere. This new approach leads to a heat flux which is less than that given by the classical Spitzer-Harm theory. The evaluated additional heating of electrons in the plasmasphere and the decrease of the thermal conductivity in the topside ionosphere and the greater part of the plasmasphere found for the first time here allow the model to accurately reproduce the electron temperatures observed by the instruments on board the EXOS-D satellite in the plasmasphere and the Millstone Hill incoherent-scatter radar in the ionosphere. The effects of the daytime additional plasmaspheric heating of electrons on the electron temperature and density are small at the F-region altitudes if the modified electron heat flux is used. The deviations from the Boltzmann distribution for the first five vibrational levels of N2(v) and O2(v) were calculated. The present study suggests that these deviations are not significant at the first vibrational levels of N2 and O2 and the second level of O2, and the calculated distributions of N2(v) and O2(v) are highly non-Boltzmann at vibrational levels v > 2. The resulting effect of N2(v > 0) and O2(v > 0) on NmF2 is the decrease of the calculated daytime NmF2 up to a factor of 1.5. The modelled electron temperature is very sensitive to the electron density, and this decrease in electron density results in the increase of the calculated daytime electron temperature up to about 580 K at the F2 peak altitude giving closer agreement between the measured and modelled electron temperatures. Both the daytime and night-time densities are not reproduced by the model without N2(v > 0) and O2(v > 0), and inclusion of vibrationally excited N2 and O2 brings the model and data into better agreement.

Abstract. We present a comparison of the electron density and temperature behaviour in the ionosphere and plasmasphere measured by the Millstone Hill incoherent-scatter radar and the instruments on board of the EXOS-D satellite with numerical model calculations from a time-dependent mathematical model of the Earth's ionosphere and plasmasphere during the geomagnetically quiet and storm period on 20±30 January, 1993. We have evaluated the value of the additional heating rate that should be added to the normal photoelectron heating in the electron energy equation in the daytime plasmasphere region above 5000 km along the magnetic ®eld line to explain the high electron temperature measured by the instruments on board of the EXOS-D satellite within the Millstone Hill magnetic ®eld¯ux tube in the Northern Hemisphere. The additional heating brings the measured and modelled electron temperatures into agreement in the plasmasphere and into very large disagreement in the ionosphere if the classical electron heat¯ux along magnetic ®eld line is used in the model. A new approach, based on a new eective electron thermal conductivity coecient along the magnetic ®eld line, is presented to model the electron temperature in the ionosphere and plasmasphere. This new approach leads to a heat¯ux which is less than that given by the classical Spitzer-Harm theory. The evaluated additional heating of electrons in the plasmasphere and the decrease of the thermal conductivity in the topside ionosphere and the greater part of the plasmasphere found for the ®rst time here allow the model to accurately reproduce the electron temperatures observed by the instruments on board the EXOS-D satellite in the plasmasphere and the Millstone Hill incoherent-scatter radar in the ionosphere. The eects of the daytime additional plasmaspheric heating of electrons on the electron temperature and density are small at the F-region altitudes if the modi®ed electron heat¯ux is used. The deviations from the Boltzmann distribution for the ®rst ®ve vibrational levels of N 2 (v) and O 2 (v) were calculated. The present study suggests that these deviations are not signi®cant at the ®rst vibrational levels of N 2 and O 2 and the second level of O 2 , and the calculated distributions of N 2 (v) and O 2 (v) are highly non-Boltzmann at vibrational levels v > 2. The resulting eect of N 2 (v > 0) and O 2 (v > 0) on NmF2 is the decrease of the calculated daytime NmF2 up to a factor of 1.5. The modelled electron temperature is very sensitive to the electron density, and this decrease in electron density results in the increase of the calculated daytime electron temperature up to about 580 K at the F2 peak altitude giving closer agreement between the measured and modelled electron temperatures. Both the daytime and night-time densities are not reproduced by the model without N 2 (v > 0) and O 2 (v > 0), and inclusion of vibrationally excited N 2 and O 2 brings the model and data into better agreement.

Introduction
A particular solstice period between 20±30 January, 1993, represents a global``10 day campaign'' period wherein many ionospheric, and thermospheric instruments collected data, and collaborative studies of the structure, dynamics and electron energy balance of the ionosphere were performed using the Millstone Hill and Arecibo incoherent scatter radar data and other groundbased data Fesen et al., 1997;Forbes et al., 1997;Richards and Khazanov, 1997;Scali et al., 1997). During the 20±30 January, 1993, period the plasmaspheric electron temperature and density has also been measured routinely by the instruments on board the EXOS-D satellite. This satellite was launched in early 1989 and studies of the plasmasphere and its coupling with the ionosphere were performed by using EXOS-D data (Abe et al., 1993;Balan et al., 1996a, b;Denton et al., 1999). On the primary purpose is to report the results obtained from a study of the electron density and temperature variations in the ionosphere and plasmasphere carried out using the incoherent scatter radar data at Millstone Hill, the EXOS-D satellite observations and the ionosphere-plasmasphere model. We study for the ®rst time the electron density and temperature altitude pro®les obtained from the measurements by the incoherent scatter radar in the ionosphere and by the EXOS-D satellite in the plasmasphere.
The O + ( 4 S) ions that predominate in the ionospheric F2-region are lost in the reactions with the loss rate L KN 2 bO 2 ; 3 where v = 0,1,. . . is the number of the vibrational level of N 2 or O 2 , the eective rate coecients of reactions (1), and (2) are determined as K v is the recombination rate coecient of O + ( 4 S) ions with N 2 (v), b v is the recombination rate coecient of O + ( 4 S) ions with O 2 (v), [N 2 [N 2 (v)] and [O 2 (v)] are the number densities of N 2 and O 2 at the v-th vibrational level. Schmeltekopf et al. (1968) measured KT N 2 v over the vibrational temperature range 300±6000 K, and found the K v /K 0 ratios from the measured KT N 2 v for T n = T i = 300 K where T N 2 v is the vibrational temperature of N 2 , T n is the neutral temperature, and T i is the ion temperature. The fundamental results of Schmeltekopf et al. (1968) were con®rmed by Ferguson et al. (1984). The measurements of K were given by Hierl et al. (1997) over the temperature range 300±1600 K for T n T i T N 2 v . These results con®rm the observations of Schmeltekopf et al. (1968), and show for the ®rst time that the translation temperature dependencies of K v is similar to K 0 . We examine the eects of N 2 (v) on the electron density and temperature during 20±30 January, 1993, by using the K v /K 0 ratios given by Hierl et al. (1997), and the value of K 0 measured by Albritton et al. (1977). Hierl et al. (1997) found a big dierence between the high-temperature¯owing afterglow and drift tube measurements (McFarland et al., 1973;Albritton et al. 1977) of b as a result of the input of the reactions between the vibrationally excited O 2 and O + ( 4 S), and determined the dependence of b on the O 2 vibrational temperature, T O 2 v , over the temperature range 300±1800 for T O 2 v T n T i . The¯owing afterglow measurements of b given by Hierl et al. (1997) were used by Pavlov (1998b) to invert the data to give the rate coecients b v for the various vibrational levels of O 2 (v > 0) for the model of the Boltzmann distribution of vibrationally excited molecular oxygen. The dierence between the measurements of b given by Hierl et al. (1997) and the scaled drift tube data is decreased with the decrease in T n . As a result, like N 2 (v), the eects of the vibrational excitation of O 2 are expected to be more important during solar maximum than at solar minimum. First studies of the O 2 (v > 0) eects on the F2 peak density, NmF2, for the 6±12 April 1990 storm (Pavlov, 1998b) and the 5±11 June, 1991, storm  found that enhanced vibrational excitation of O 2 leads up to the 40% decrease in the calculated NmF2 at solar maximum. In this paper we study the eects of O 2 (v > 0) on the electron density and temperature for the period of 20±30 January, 1993, which was at moderate solar-activity conditions.
High electron temperatures in the night-time ionosphere over Millstone Hill were observed during the periods 20±30 January 1993 Richards and Khazanov, 1997). Such anomalous nighttime electron temperature events over Millstone Hill that were ®rst observed in 1964 by Evans (1967) are most frequent in the winter months (Garner et al., 1994). The physical origin of these temperature events is still unclear. The Millstone Hill's conjugate point is sunlit the whole night for the most of the winter period, but the night-time electron temperature enhancements only occur for a few hours. The existence of anomalous night-time temperature events in the fall and spring months also does not support the idea of Evans (1967) that the heating is only caused by a simple relationship of these anomalous temperature enhancements to conjugate photoelectrons (Garner et al., 1994). Richards and Khazanov (1997) found that the plasmaspheric heating from photoelectrons must be doubled to reproduce the Millstone Hill observed topside electron temperature by the FLIP model of the ionosphere and plasmasphere during the periods 25±29 January 1993. Following Richards and Khazanov (1997), we believe that there is an additional heating rate of the electrons in the plasmasphere, and evaluate the value of this additional heating rate so that an agreement between the measured and modelled electron temperature in the ionosphere and plasmasphere is obtained during the 20±30 January, 1993, period.
The thermal electron impact excitation of the ®ne structure levels of the 3 P ground state of atomic oxygen is presently believed to be one of the dominant electron cooling processes in the F region of the ionosphere (Richards et al. 1986;Richards and Khazanov, 1997). Pavlov (1998a, c) and Pavlov and Berrington (1999) have revised and evaluated the electron cooling rates by vibrational and rotational excitation of N 2 and O 2 , and the electron cooling rate by electron impact excitation of ®ne-structure levels of atomic oxygen. Pavlov and Berrington (1999) found that the role of the cooling rate of thermal electrons by electron impact excitation of ®ne structure levels of atomic oxygen is not signi®cant at the F2-peak altitudes of the ionosphere for the geomagnetically quiet and disturbed period on 6±12 April, 1990, above Millstone Hill, and the energy exchange between the electron and ion gases and the electron cooling rates by vibrational excitation of O 2 and N 2 are the largest cooling rates above 160 km. The new analytical expressions for cooling rates given by Pavlov (1998a, c) and Pavlov and Berrington (1999) are applied to study the thermal balance of the ionosphere and plasmasphere and to perform an examination of the role of these electron cooling rates in the thermal balance of the ionosphere during 20±30 January, 1993.

Theoretical model
The model used is the IZMIRAN model of the ionosphere and plasmasphere that has been steadily developed over the years (Pavlov, 1997(Pavlov, , 1998aPavlov and Berrington, 1999). It is a one-dimensional model that uses a tilted dipole approximation to the Earth's magnetic ®eld and takes into account the oset between the geographic and geomagnetic axes. In the model, coupled time-dependent equations of continuity and energy balance, and diusion equations for electrons, and O + ( 4 S), H + , and He + ions are solved along a centred-dipole magnetic ®eld line for the concentrations, temperatures, and ®eld-aligned diffusion velocities of ions and electrons from a base altitude (160 km) in the Northern Hemisphere through the plasmasphere to the same base altitude in the Southern Hemisphere. Electron heating due to photoelectrons is provided by a solution of the Boltzmann equation for photoelectron¯ux. In the altitude range 120±700 km in the Northern and Southern Hemispheres the model solves time-dependent continuity equations for O + ( 2 D), O + ( 2 P), NO + , O 2 , N 2 , N 2 (v=1,. . .5), and O 2 (v=1,. . .5), and vibrationally excited nitrogen and oxygen quanta. An additional production of O + ( 4 S), O + ( 2 D), and O + ( 2 P) ions is that described by Pavlov (1998b), and obtained in the model by inclusion of O + ( 4 P) and O + ( 2 P Ã ) ions. The IZMIRAN model calculates [O( 1 D)] from a time dependent continuity equation in the region between 120 and 1500 km in altitude in both hemispheres. The diusion of ions and excited species are taken into account in continuity equations for NO + , O 2 , O 2 (v), N 2 (v), and O( 1 D), while densities of O + ( 2 D), O + ( 2 P), and N 2 are obtained from local chemical equilibrium. The updated IZMIRAN model uses the dissociative recombination rate coecient for N 2 ions measured by Peterson et al. (1998). The revised electron cooling rates by vibrational and rotational excitation of O 2 and N 2 , and by electron impact excitation of ®ne structure levels of atomic oxygen given by Pavlov (1998a, c) and Pavlov and Berrington (1999) are included in the model. To calculate the density of NO the algorithm given by Titheridge (1997) is used.
The heating rate of the electron gas by photoelectrons is calculated along a centred ± dipole magnetic ®eld line using the numerical method of Krinberg and Tachilin (1984) for the determination of the photoelec-tron¯uxes within a plasmaspheric ®eld tube on the same ®eld line grid that is used in solving for the temperatures. The updated IZMIRAN model solves the Boltz-mann equation for photoelectron¯ux using the updated elastic and inelastic cross sections of the neutral components of the atmosphere. For O, the elastic cross section employed in the electron transport code was drawn from the work of Williams and Allen (1989) for energies below 8.7 eV, and, above 8.7 eV, we have adopted the elastic cross section of Joshipura and Patel (1993). The N 2 elastic cross section of Itikawa (1994) for electron energies is used in our model. The O and N 2 inelastic cross sections are given by Majeed and Strickland (1997), and we employ these cross sections with some modi®cation for N 2 . The N 2 vibrational excitation cross sections used by Majeed and Strickland (1997) in calculations of the N 2 inelastic cross section were replaced by the N 2 vibrational excitation cross sections of Robertson et al. (1997) for vibrational levels v=1 and 2, and those of Schulz (1976) for v=3±10 with the normalization factor of 0.7 (see details in Pavlov, 1998 a). For O 2 , the elastic and inelastic cross sections are taken from Kanic et al. (1993).
The IZMIRAN model uses the recombination rate coecient of O + ( 4 S) ions with unexcited N 2 (0) and O 2 (0) (Albritton et al., 1977;St.-Maurice and Torr, 1978) and vibrationally excited N 2 (v) and O 2 (v) (Schmeltekopf et al., 1968;Hierl et al., 1997;Pavlov, 1998b) as described in detail by Pavlov (1998b) and . The energy balance equations for ions of the IZMIRAN model take into account the perpendicular component, E c , of the electric ®eld with respect to the geomagnetic ®eld and the rate coecients of such important ionospheric processes as the reactions of O + ( 4 S) with N 2 and O 2 , and N 2 with O 2 which depend on eective temperatures which are functions of the ion temperature, the neutral temperature and E c (Pavlov, 1997(Pavlov, , 1998b. The measured value of E c can be used as an input parameter for our theoretical model. To simulate magnetic storm eects on the neutral atmosphere the MSIS-86 model of Hedin (1987) was run using 3 h Ap indices. The IZMIRAN model uses the solar EUV¯uxes from the EUV97 model (Tobiska and Eparvier, 1998). At night our model includes the neutral ionization by scattered solar 121.6, 102.6 and 58.4 nm uxes (Pavlov, 1997).
In the Northern Hemisphere instead of calculating thermospheric wind components by solving the momentum equations, the IZMIRAN model calculates an equivalent neutral wind from the F2 peak altitude, hmF2, measurements using the modi®ed method of Richards (1991) described by Pavlov and Buonsanto (1997). For the Southern Hemisphere where we do not have observed hmF2 momentum equations for the horizontal components of the thermospheric wind are calculated in the altitude range 120±700 km to derive an equivalent plasma drift velocity, as described by Pavlov (1997).

Solar-geophysical conditions and data
The period of 20±30 January, 1993, was a period of moderate solar-activity conditions when the F10.7 solar activity index ranged from 100 to 110, while the 81-day averaged F10.7 solar activity index was about 130. The day of 20 January 1993 was geomagnetically disturbed when 3-h geomagnetic index Kp was between 1 and 4. Most of the 21±24 and 27±28 January periods were geomagnetically quiet with Kp varying between 0 and 2 except for short time periods when Kp reached 3. The maximum value of Kp was 5+ during the minor geomagnetic storm of 25±26 January. Kp varied between 1 and 4 on 29 January and between 2 and 5 on 30 January.
The Millstone Hill incoherent scatter radar experiment ran from 12:32 UT on 20 January, 1993, to 00:04 UT on 31 January, 1993. The electron densities, electron and ion temperatures, and plasma line-of-sight velocities were measured using both the ®xed zenithpointing 67-m antenna and the fully steerable 46-m antenna (for more details see Buonsanto et al., 1997). The ionospheric variations of the Millstone Hill incoherent scatter radar electron density, N e , and temperature, T e , used here were discussed in details by Buonsanto et al. (1997), Fesen et al. (1997), and Richards and Khazanov (1997).
The EXOS-D satellite experiments have provided new opportunities in recent years for investigating the plasmaspheric electron temperature and density and have increased our knowledge of the plasmasphere and its coupling with the ionosphere (Abe et al., 1990(Abe et al., , 1993Balan et al., 1996a, b;Denton et al., 1999). The EXOS-D data obtained during 20±30 January, 1993, were analyzed to investigate the local time and altitude variations of electron density and temperature in the plasmasphere. The plasmaspheric measured electron temperature and density from the EXOS D satellite at the Millstone Hill latitude and longitude were selected within a ®eld tube such that the magnetic latitude and longitude of the central magnetic ®eld line of this ®eld tube coincided with the Millstone Hill magnetic latitude and longitude. The distance, R B , (in the direction perpendicular to magnetic ®eld lines) between the surface ®eld lines and the central ®eld line of this ®eld tube is taken so that this distance was minimized, and the number of measurements was large enough to compare the data with the model results. As a result, the EXOS-D satellite data between 15:24:15 UT and 15:25:59 UT on 22 January with R B =3.5°, and from 14:32:14 UT to 14:36:06 UT on 24 January with R B =2°f ell within the Millstone Hill magnetic ®eld¯ux tube in the Northern Hemisphere during the 20±30 January period and are used in our work. It is well known that the electron temperature within the plasmasphere and topside ionosphere is anisotropic with respect to the geomagnetic ®eld (Clark et al., 1973;Oyama and Schlegel, 1988;Denmark and Schunk, 1987;Khazanov et al., 1996). An instrument on board the EXOS-D satellite for measuring the temperature distribution of thermal electrons with respect to the geomagnetic ®eld was installed on the tip of the satellite's solar cell paddles perpendicular to the satellite spin axis (more detailed information about the electron temperature measurements is given by Abe et al., 1990). It follows from our analysis of the EXOS-D data sets at middle latitudes that the ratio of the parallel temperature (along the geomagnetic ®eld) of thermal electrons to the electron temperature in the direction perpendicular to the geomagnetic ®eld is ranging from 1 to 1.2±1.3 (see also Abe et al., 1990). During the time periods studied between 15:24:15 UT and 15:25:59 UT on 22 January, and from 14:32:14 UT to 14:36:06 UT on 24 January, the instrument observed the electron temperature within the Millstone Hill magnetic ®eld¯ux tube in the direction of 60±90°between the normal direction of the sensor plane and the direction of the geomagnetic ®eld measuring a mixture of the parallel and perpendicular temperatures. The electron temperature estimation made for the events described includes the error of 15± 20%, which is almost comparable to the largest case of the ratio of the parallel temperature to the perpendicular temperature. As a result, we do not discriminate between the parallel and perpendicular temperatures of thermal electrons in this study.

Analysis and discussion
4.1 Altitude pro®les of the measured and modelled electron density and temperature in the ionosphere and plasmasphere Figures 1 and 2 show altitude pro®les of the measured (crosses) and modelled (lines) electron densities (left panels), and electron (middle panels) and ion (right panels) temperatures above Millstone Hill at 10:05 LT (15:04 UT) on 22 January, 1993, and 9:24 LT (14:23 UT) on 24 January, 1993. Dotted lines show the modelled results without an additional electron heating in the plasmasphere using the electron thermal conductivity coecient, K 0 e , along the magnetic ®eld line in the electron energy balance equation of the IZMI-RAN model given by Pavlov (1996). The model results shown by solid and dashed lines in Figs. 1 and 2 will be discussed later. It should be noted that the agreement between the measured N e , T e and T i (crosses), and the modelled N e , T e , and T i shown by solid lines is good except for high altitudes (above about 350 km) at left panel of Fig. 1 where the measured electron density is higher than the calculated electron density.
Left and middle panels of Figs Dotted lines in Figs. 3 and 4 show the modelled results without an additional electron heating in the plasmasphere using the electron thermal conductivity coecient K 0 e along the magnetic ®eld line in the multicomponent mixture of ionised gases given by Pavlov (1996) which is used in the IZMIRAN model (Pavlov, 1997(Pavlov, , 1998aPavlov and Berrington, 1999). In a completely ionised gas, the value of K 0 e corresponds to the exact value calculated by Spitzer and Harm (1953) when an in®nite number of terms were retained in Sonine expansion. As Figs. 3 and 4 show, the modelled electron temperatures shown by dotted lines are much less then the electron temperatures measured by the EXOS-D satellite in the plasmasphere.
The IZMIRAN model solves the Boltzmann equation for photoelectron¯ux along a centred-dipole magnetic ®eld line to calculate the heating rate of the electron gas by photoelectrons using the numerical method of Krin- (solid lines). Dotted and dashed lines show the results from the model using the electron thermal conductivity coecient K 0 e along the magnetic ®eld line in the multicomponent mixture of ionised gases given by Pavlov (1996). Solid lines 1, 2, and 3 show the results from the model using the modi®ed electron thermal conductivity coecient along the magnetic ®eld line given by Eq. (10) with the value of C=10, 20 and 30, correspondingly berg and Tachilin (1984). The energy lost by photoelectrons in heating the plasma in the plasmasphere is calculated using the analytical equation for the plasmaspheric transparency, P(E), (Krinberg and Matafonov, 1978;Krinberg and Tachilin, 1984) that determines the probability of the magnetically trapped photoelectrons with an energy, E, for entering the magnetically conjugated ionosphere. The transparency depends mainly on a single parameter proportional to the Coulomb cross section and the total content of electrons in the plasmasphere magnetic¯ux tube [the transparency approaches unity as photoelectrons pass through the plasmasphere without signi®cant absorption, and P(E)=0 if photoelectrons are absorbed by the plasmasphere].
The disagreement between the measured (crosses) and modelled (dotted lines) electron temperatures shown in Figs. 1 and 2 could be due to uncertainties of the IZMIRAN model in the amount of the energy   ary, 1993. Right panel shows the calculated ratio of the thermal electron mean free path to the characteristic length of the electron temperature variation along the magnetic ®eld line in the plasmasphere and ionosphere at 9:24 LT on 24 January 1993. The curves are the same as in Fig. 1 deposited in the plasmasphere by ionospheric photoelectrons. However, changing the value of P(E) we have found that the heating provided by trapped photoelectrons cannot account for the observed high electron temperatures in the plasmasphere in Figs. 3 and 4.
The presence in the ionosphere and plasmasphere of photoelectrons is a potential source for the onset of plasma instabilities that can change the photoelectron distribution function and heating rate of electrons used by the IZMIRAN model. The possible additional sources of the electron gas heating in the plasmasphere, such as wave-particle interactions, which can cause increased photoelectron scattering, and Coulomb collisions between ring current ions and plasmaspheric electrons and ions could be the possible mechanisms to explain the observed electron temperature enhancements in the plasmasphere. It is also possible that the physics of ionosphere-plasmasphere transport of photoelectrons is not well understood at this time. The heating could be also caused by heated¯ux tubes drifting past Millstone Hill. It is worth noting that such required additional heat of electrons, over that for the usual photoelectron heating from Coulomb collisions, was used in many papers to make the modelled and measured electron temperatures agree (Horwitz et al., 1990;Pavlov, 1994Pavlov, , 1996Pavlov, , 1997Graven et al., 1995;Balan et al., 1996b;Richards and Khazanov, 1997).
As a result, to explain the measured electron temperature we assume that an additional heating rate, Q ad , should be added to the normal photoelectron heating in the electron energy equation in the plasmasphere region above 5000 km along the magnetic ®eld line as Q ad Q day for v 90 ; 5 where v is the solar zenith angle at the Earth's surface at Millstone Hill. The value of Q day is found from the comparison between the electron temperatures observed by the instruments on board of the EXOS-D satellite in the plasmasphere and the calculated electron temperatures to allow the IZMIRAN model to accurately reproduce the measured electron temperatures. There is no nighttime EXOS-D electron temperature measurements into the Millstone Hill magnetic ®eld¯ux tube during the 20± 30 January period, and we cannot calculate the value of Q night from the comparison of the IZMIRAN model results and EXOS-D data. An algorithm presented in Sect. 4.2 for using the measured electron temperature at the F2 peak altitude to determine the amount of the IZMIRAN model night-time plasmaspheric additional heating is used in our calculations. We hope that the real mechanism of this additional heating will be found in future studies, and this study could only be considered as an adjusted approach to evaluate the value of this additional heating rate.
We found that good agreement between the measured and modelled electron temperatures in the plasmasphere is obtained if Q day =1.4 eV cm )3 s )1 (dashed lines in Figs. 1±4). Comparison between crosses and dashed lines of Figs. 1 and 2 (middle panels) and Figs. 3 and 4 (middle panels) show that this additional heating brings the measured and modelled electron temperatures into agreement in the plasmasphere and into very large disagreement in the ionosphere. The following investigation seeks to resolve this striking discrepancy.
It is generally accepted that the Spitzer-Harm theory (Spitzer and Harm, 1953) of the electron thermal conductivity along the magnetic ®eld line can be only applied for the highly collisional plasmas when the ratio of the thermal electron mean free path, k, to the characteristic length, L p , of the plasma parameter variation is less than about 10 )2 (Bell et al., 1981;Khan and Rognlien, 1981;Mason, 1981;Matte and Virmont, 1982). The conventional thermal conductivity model of Spitzer and Harm (1953) may result in electron energȳ ux which is large and physically unreal (the classical electron conduction becomes in®nite for collisionless plasma) and inconsistent with the results of weakly collisional plasma experiments (Choi and Wilhelm, 1976;Gray and Kilkenny, 1980;Gary et al., 1981;Clause and Balescu, 1982;Fechner and Mayer, 1984;Lee and More, 1984;Albritton et al., 1986). This is due to the fact that in these experiments the main contribution to the energy transport through the thermal conductivity comes from the suprathermal electrons whose mean free path is much larger than k.
The question of how the electron heat¯ux, q e , varies as k/L p > 10 )2 is still, as yet, an unanswered question. One common, ad hoc approach is the¯ux-limited model given as (Malone et al., 1975;Cowie and Mckee, 1977;Fechner and Mayer, 1984) q e q À1 clas q À1 sat À Á À1 ; 7 where q clas ÀK 0 e o=oST e is the classical electron heat ux along a magnetic ®eld line, S is the distance along the magnetic ®eld line, S is positive in the direction north to south.
The``saturated'' electron heat¯ux in Eq. (7) is given by where k is Boltzmann's coecient, V T 2kT e m À1 e À Á 0:5 is the electron thermal velocity, m e denotes the mass of an electron, and f is a coecient, known as the``¯ux limit'' parameter (Malone et al., 1975;Cowie and Mckee, 1977;Fechner and Mayer, 1984). It is explicitly indicated in Eq. (8) that the saturated heat¯ow is opposite to the temperature gradient. In the highly collisional limit, Eq. (7) reduces to the classical heat conduction equation, while in the weakly collisional limit, Eq. (7) describes the saturated heat¯ux of electrons.
The magnitude of the¯ux limit parameter is the subject of considerable discussions, and values of f in two orders of the magnitude range from about 0.01 to about 1 in the literature. These values of f are required to reproduce some laser fusion results with simulation code, and they have also been deduced in numerical Fokker-Planck calculations (Malone et al., 1975;Cowie and Mckee, 1977;Bell et al., 1981;Clause and Balescu, 1982;Matte and Virmont, 1982;Fechner and Mayer, 1984;Zawaideh and Kim, 1988). Equation (7) can be modi®ed as follows: where the eective electron thermal conductivity coef-®cient is determined as Right panels of Figs. 3 and 4 show the calculated ratio altitude pro®les of the thermal electron mean free path to the characteristic length of the electron temperature variation along the magnetic ®eld line in the plasmasphere and ionosphere at 10:05 LT on 22 January, 1993, and 9:24 LT on 24 January, 1993. Dotted (Q ad =0) and dashed (Q day =1.4 eV cm )3 s )1 ) lines represent the IZ-MIRAN model results when the classical electron heat ux along magnetic ®eld line is used. Figures 3 and 4 show that the value of k/L p is larger than 10 )2 in the altitude region between 600±650 km and 5000±5200 km in the ionosphere and plasmasphere without an additional heating of electrons, and from 300±340 km to 10 000±10 500 km in the ionosphere and plasmasphere with the additional heating. It means that the additional heating rate of electrons increases the size of the region in the daytime ionosphere and plasmasphere where k/L p is larger than 10 )2 . Application of the classical heat conduction equation to plasma in this region, that is not highly collisional, casts doubt on the accuracy of the results and creates the problem in agreement of the measured and modelled electron temperatures simultaneously in the daytime ionosphere and plasmasphere.
Solid lines in Figs. 1±4 represent the IZMIRAN model results when the additional heating Q day = 0.45 eV cm )3 s )1 and the modi®ed electron heat¯ux along magnetic ®eld line given by Eqs. (9), (10) with C=10 (solid lines 1), C=20 (solid lines 2), and C=30 (solid lines 3) are used. The right panels of Figs. 3 and 4 show that the value of k/L p is larger than 10 )2 in the altitude region between 400±450 km and 8500±8600 km for C=10, from 400±450 km to 7700±8000 km for C=20, and from 380±440 km to 7200±7400 km for C=30 in the ionosphere and plasmasphere. The maximum value of the k/L p ratio is located in the plasmaspheric region between 1800 km and 2300 km if C=10±30. As a result of the reduced electron thermal conductivity, the modelled electron temperature is higher in the plasmasphere if the higher value of C is used. The dierence between electron temperatures given by solid lines 1, 2, and 3 is negligible below 500 km. Our calculations show that this dierence can be increased if the value of Q day is increased. It should be also noted that the eects of the additional plasmaspheric heating of electrons on the electron temperature and density are small at the F-region altitudes if the eective electron thermal conductivity coecient given by Eq. (10) is used.
The Millstone Hill radar measurements of the topside electron temperature are dicult because of the low electron densities and a lot of scatter in the data. As a result, some dierence between the measured and modelled topside electron temperatures may be due to the errors of the topside electron temperature measurements. The errors of the electron temperature measured by the instruments on board of the EXOS-D satellite can be evaluated as 15±20% while the errors in the EXOS-D electron density measurements are about factors of 2±3 at the considered altitudes. In general, we conclude that the modelled electron temperatures in the ionosphere and plasmasphere are in good agreement with the measurements if the new approach of Eqs. (9) and (10) for the electron heat¯ux along magnetic ®eld line is used. Taking into account the errors of the electron temperature measurements we believe that the value of C in Eq. (10) is between 10 and 30. The good agreement between the modelled and measured electron temperatures is achieved by using the middle value of C=20. The ®tting approach of Eqs. (9) and (10) can be considered as the ®rst step in the study of physical processes that are responsible for the high daytime plasmaspheric electron temperatures.
It should be noted that some models and simulations of anomalous transport processes have been presented for auroral plasma where the calculated T e based on classical transport coecients disagrees strongly with the observed storm time T e in the auroral ionosphere below 500 km (Fontheim et al., 1978;Jonson et al., 1979). As a result of particle precipitation or electron current¯ow, plasma instabilities and turbulence are excited in the auroral ionosphere. The plasma turbulence leads to a decrease in electron thermal conductivity along the geomagnetic ®eld and changes the thermal structure of the auroral ionosphere below 500 km giving an agreement between the measured and modelled electron temperatures (Fontheim et al., 1978;Jonson et al., 1979). Figures 3 and 4 show that the decrease in electron thermal conductivity is created in the¯uxlimited model approach at middle latitudes above 500± 700 km in the topside ionosphere and the plasmasphere to agree the measured and modelled electron temperature. However, as far as the authors know, the question about the plasma turbulence existence at middle latitudes above 500±700 km in the topside ionosphere and the plasmasphere has not been studied, and therefore this approach is not considered in our work.
4.2 Additional heating of electrons in the plasmasphere from the comparison of the measured and modelled night-time electron temperatures at the F2 peak altitude The dependence of the electron temperature on the neutral densities and temperature, the electron density, and the heating rate of electron gas is determined by the energy balance equation for electrons (Pavlov, 1997(Pavlov, , 1998aPavlov and Berrington, 1999) with the electron heat¯ux along magnetic ®eld line given by Eqs. (9), (10). Taking into account that an eect of cooling processes on the temperature of plasmasphere electrons is weak, we can integrate the steady state energy balance equation for electrons, with respect to S from magnetic equator, S=0, to the distance, X, in the Northern Hemisphere along the ®eld line, and obtain the following relation: where B is the magnitude of the geomagnetic ®eld, Q fot (S) is the heating rate of the electron gas by photoelectrons during night-time periods due to photo-electron¯ux from the sunlit conjugate hemisphere. The value of the k/L p ratio is proportional to T e jo=oS T e j. The mean night-time EXOS-D electron temperatures are less than the mean daytime EXOS-D electron temperatures in the plasmasphere (Abe et al., 1993;Balan et al., 1996a, b;Denton et al., 1999), and the night-time dierence between K e and K 0 e is less than the daytime dierence. As a result, we can assume for an estimate that the dependence of K e on T e is close to the dependence of K 0 e on T e at night. It means that at the upper boundary of the ionosphere the value of T 7=2 e is approximately proportional to Q ad . As there is no analytical solution of the steady state energy equation for electrons in the region of the ionosphere which takes into account the revised electron cooling rates of thermal electrons by electron impact excitation of ®ne structure levels of atomic oxygen and by vibrational and rotational excitation of N 2 and O 2 given by Pavlov (1998a, c) and Pavlov and Berrington (1999), we use Eq. (11) in the ionosphere to adjust the night-time model electron temperature. If the model electron temperature, T mod (t), at the time step, t, is less than the measured temperature, T exp (t), the required heating rate at the next time step, t + Dt, is approximately given by where A is a constant. Figures 5 and 6 show the measured (crosses) and calculated (lines) NmF2 (bottom panel), and hmF2 (middle panel) at the F2 peak altitude above Millstone Hill for the magnetically quiet and disturbed period of 20±30 January, 1993, for two models of additional heating of electrons in the plasmasphere (top panel). The modi®ed electron thermal conductivity coecient along the magnetic ®eld line given by Eq. (10) with the value of C=20 used in the IZMIRAN model calculations. Solid lines show the modelled results obtained when the additional heating rate of electrons was speci®ed using the algorithms given by Eqs. (5), (6), and (12). The approach of Eq. (12) uses the measured and modelled electron temperatures at the F2 peak altitude. Dashed lines represent the IZMIRAN model results when Q ad =0.45 eV cm )3 s )1 for v £ 90°and Q ad =0 for v > 90°are used. The ®gures are divided into two parts to avoid a compressed time scale.
Dashed lines in the middle panel of Figs. 5 and 6 show that there is a dierence at night with the model temperatures generally lower than the data. Solid lines in the middle panel of Figs. 5 and 6 show how well the IZMIRAN model electron temperature is able to follow the measured electron temperature using the algorithm of Eq. (12). We conclude that the model electron temperature is much closer to the measured electron temperature if the algorithm of Eq. (12) is used. Our results show that there are a number of peaks in additional heating time behaviour at night, and the value of Q night is less than Q day . We found that the eect of the calculated night-time additional heating (given by Eq. 6) on NmF2 can be considered negligible.
The accuracy of the calculated night-time additional heating depends on the accuracy of the modelled nighttime photoelectron¯ux from the sunlit conjugate hemisphere. Uncertainties in the theoretical solar EUV uxes and in the measured electron impact cross sections can create up to about a factor of 2 uncertainties in the theoretical photoelectron spectra (Richards and Torr, 1988). Our calculations indicate that a factor of 2 increase in the photoelectron¯ux from the sunlit conjugate hemisphere leads to a decrease up to about a factor of 2 of the night-time additional heating if Q night > 0.01 eV cm )3 s )1 . However, a factor of 2 increase in the theoretical solar EUV¯uxes that can produce a factor of 2 increase in the photoelectron¯ux leads to the disagreement between the measured and modelled electron densities in the ionosphere during the daytime periods of 20±30 January, 1993. Additional argument against this increase is that an increase up to about a factor of 2 of the photoelectron¯ux from the sunlit conjugate hemisphere cannot produce the agreement between the measured and modelled electron temperatures during all night-time periods of 20±30 January, 1993, and we believe that an additional heating of electrons exists in the plasmasphere.
, where E 1 =3353 K and E H 1 = 2239 K are the energies of the ®rst vibrational levels of N 2 and O 2 (Radzig and Smirnov, 1980) vibrational temperatures are calculated by solving the time dependent continuity equations for vibrationally excited nitrogen and oxygen quanta (Pavlov, 1997(Pavlov, , 1998b. From the diurnal variations of the calculated vibrational and neutral temperatures shown in Fig. 7 it follows that T O 2 v < T n and T N 2 v < T n are realized in the atmosphere for some parts of the night-time periods where the production frequencies of O 2 (v) and N 2 (v) are low. This means that for these periods the populations of O 2 (v) or N 2 (v) are less than the populations for a Boltzmann distribution with temperature T n . During daytime T O 2 v and T N 2 v are larger than T n due to the enhanced thermal excitation of O 2 and N 2 as a result of high thermal electron temperatures at F2-region altitudes. We found that À41 K T O 2 v À T n 373 K and À63 K T N 2 v À T n 312 K. The value of the vibrational temperature was not more than 1293 K for O 2 and 1272 K for N 2 .
The present study suggests that the deviations of [N 2 (v)  shows that the increase in the O + + N 2 rate factor due to the vibrational excited N 2 leads to the decrease of the calculated daytime NmF2 up to a factor of 1.3. The comparison between dotted and dashed lines shows that the increase in the O + + O 2 loss rate due to vibrationally excited O 2 produces the reduction in the daytime peak density up to a factor of 1.3. The resulting eect of N 2 (v > 0) and O 2 (v > 0) included in L on NmF2 (from the comparison between solid and dotted lines in Fig. 9) is the decrease of the calculated daytime NmF2 up to a factor of 1.5. The eects of vibrationally excited O 2 and N 2 on N e are most pronounced during daytime.
The top panel of Figs. 8 and 9 shows the diurnal variations of the measured and modelled electron and ion temperatures at the F2-peak altitude. Despite the great amount of scatter in the data, the modelled temperatures generally follows the overall trends. As can be seen, the eects of adding N 2 (v) and O 2 (v) on T e are largest during the day, with increases in T e accompanying the decreases in NmF2. We found that the resulting eect of N 2 (v > 0) and O 2 (v > 0) included in L on the electron temperature at the F2 peak altitude (from the comparison between solid and dotted lines at top panel of Figs. 8 and 9) is the decrease of the calculated daytime electron temperature up to about 580 K. The eects of vibrationally excited O 2 and N 2 on N e are most pronounced during daytime.
It should be noted that the modelled electron temperature is very sensitive to the electron density, and, as a result, there is a large decrease in the modelled electron temperatures without the vibrational excited nitrogen and oxygen in the ionosphere-plasmasphere model (see upper panels of Figs. 8 and 9). Inclusion of vibrationally excited N 2 and O 2 in the loss rate of O + ( 4 S) ions which brings the measured and modelled electron densities into better agreement also tends to give close agreement between measured and modelled electron temperatures. Figures 8 and 9 show that the modelled electron densities and temperatures are in a reasonable accord with the observed values if the non-Boltzmann vibrational N 2 and O 2 distribution assumptions are used. However, the model results with the vibrational states of N 2 (v) and O 2 (v) included does not always ®t the data. There is a close relationship between electron temperature and electron density, and it is not possible to determine whether the increase in density results from a lowering of the electron temperature or vise versa, since both phenomena are coupled. Figures 8 and 9 show that when the model accurately reproduces the electron  (10) with the value of C 20 for the electron heat¯ux along magnetic ®eld line density, it also reproduces the observed electron temperature with some small errors. The modelled electron temperature falls below the measured electron temperature during the time when the modelled electron density increases above the measured electron density. These discrepancies are probably due to the uncertainties in the model inputs, such as a possible inability of the MSIS-86 model to accurately predict the thermospheric response to this storm above Millstone Hill, and uncertainties in EUV¯uxes, rate coecients, and thē ow of ionisation between the ionosphere and plasmasphere, and possible horizontal divergence of the¯ux of ionization above the station.
The relative magnitudes of the cooling rates are of particular interest to understand the main processes that determine the electron temperature. We found that the energy exchange between electrons and ions, and the electron cooling rates by vibrational excitation of N 2 and O 2 are the dominant cooling channels above 180 km during daytime. We found also that the contribution of the cooling of electrons by low-lying electronic excitation of O 2 (a 1 D g ) and O 2 (b 1 S g + ), by excitation of O to the 1 D state, and by rotational excitation of O 2 can be neglected above 160 km altitude since it represents not more than a few percent of the total cooling rate during the quiet and geomagnetic storm period 20±30 January, 1993. The atomic oxygen ®ne structure cooling rate of thermal electrons is not the dominant electron cooling process in agreement with the conclusions of Pavlov and Berrington (1999).

Conclusions
The Millstone Hill incoherent-scatter radar and the EXOS-D satellite measurements, and the IZMIRAN ionosphere-plasmasphere model simulations of electron density and temperature for the geomagnetically quiet and disturbed period of 20±30 January, 1993, were used to study the thermal electron energy budget of the ionosphere and plasmasphere, the eects of N 2 (v) and O 2 (v) on the electron density and temperature at moderate solar-activity conditions, and to evaluate the current capabilities of the IZMIRAN model. The model used is an enhanced and updated version of the IZMIRAN model that we have steadily developed over the years. The updated model uses the revised electron cooling rates by vibrational and rotational excitation of O 2 and N 2 , and by electron impact excitation of ®ne structure levels of atomic oxygen given by Pavlov (1998a, c) and Pavlov and Berrington (1999) in calculations of the electron temperature, and the updated elastic and inelastic cross sections of the neutral components of the atmosphere to solve the Boltzmann equation for photoelectron¯uxes.
We have examined the thermal electron energy budget of the mid-latitude ionosphere and plasmasphere. Our initial comparison between the measured and modelled electron temperatures show that the modelled electron temperatures are much less then the electron temperatures measured by the instruments on board of the EXOS-D satellite in the plasmasphere if the Fig. 9. Observed (crosses) and calculated (lines) NmF2 (bottom panel), hmF2 (middle panel), and the electron temperature, T em , at the F2 peak altitude (top panel) above Millstone Hill for the magnetically quiet and disturbed period 26±30 January, 1993. The curves are the same as in Fig. 8 ionosphere-plasmasphere model without an additional heating of electrons is used. The heating provided by trapped photoelectrons cannot account for the observed high electron temperatures in the plasmasphere. We have evaluated the value of the additional heating rate that should be added to the normal photoelectron heating in the electron energy equation in the daytime plasmasphere region above 5000 km along the magnetic ®eld line to explain the high electron temperature between 15:22:39 UT and 15:26:39 UT on 22 January and from 14:32:14 UT to 14:39:02 UT on 24 January measured by the instruments on the EXOS-D satellite board into the Millstone Hill magnetic ®eld¯ux tube in the Northern Hemisphere. The additional heating brings the measured and modelled electron temperatures into agreement in the plasmasphere and into very large disagreement in the ionosphere if the classical electron heat¯ux along magnetic ®eld line is used.
A new approach is presented to model the electron temperature in the ionosphere and plasmasphere. A new eective electron thermal conductivity coecient along the magnetic ®eld line is derived from the¯ux-limited model to make the measured and modelled electron temperature agree. The ratio of the eective electron thermal conductivity coecient to the classical electron thermal conductivity coecient of Spitzer and Harm (1953) is a function of T e ; jo=osT e j; N e , and the ®tting¯ux-limited theory parameter which value is estimated to be between 10 and 30 from the comparison of the measured and modelled electron temperature in the plasmasphere. The calculations show that the good agreement between the modelled and measured electron temperatures is achieved by using the middle value of C=20. This new approach leads to a heat¯ux which is less than that given by the Spitzer-Harm theory. The reason for the low thermal conductivity of electrons in the topside ionosphere and in the most part of the plasmasphere where the value of k/L p is larger than 10 )2 is the deviation of the electron distribution from the Maxwellian distribution. The heat¯ux in this region is carried not only by thermal electrons but also by a number of superthermal electrons whose mean free path is much larger than the thermal electron mean free path. The electron temperatures observed by the instruments on board of the EXOS-D satellite in the plasmasphere are explained by the additional heating of electrons in the plasmasphere and the decrease of the thermal conductivity in the topside ionosphere and the most part of the plasmasphere found for the ®rst time in this study. We found also that the eects of the daytime additional plasmaspheric heating of electrons on the electron temperature and density are small at the F-region altitudes if the electron heat¯ux modi®cation is used.
In order to reproduce the high night-time observed electron temperatures at the F2 peak altitude, an additional plasmaspheric heating rate of electrons is required, and a new algorithm is presented to determine the amount of this plasmaspheric heating. It was found that the additional plasmaspheric heating greatly improves the model results at night producing a model electron temperature that is much closer to the measured electron temperature than in the standard case without this heating. Our results show that there are a number of peaks in additional heating time behavior at night, and the value of Q night is less than Q day . We found that the eect of the calculated night-time additional heating on NmF2 can be considered negligible.
The deviations from the Boltzmann distribution for the ®rst ®ve vibrational levels of N 2 and O 2 were calculated. The present study suggests that the deviations from the Boltzmann distribution are not signi®cant at the ®rst vibrational levels of N 2 and O 2 and the second level of O 2 , and the calculated distributions of N 2 (v) and O 2 (v) are highly non-Boltzmann at vibrational levels v > 2. The calculations also showed that the O 2 and N 2 vibrational temperatures during the quiet periods are less than during the magnetic storm periods. During daytime the high vibrational temperatures stem from the enhanced thermal excitation of O 2 and N 2 as a result of high thermal electron temperatures at F2region altitudes.
We found that the resulting eect of N 2 (v > 0) and O 2 (v > 0) on the NmF2 is the decrease of the calculated daytime NmF2 up to a factor of 1.5 for the non-Boltzmann N 2 (v) and O 2 (v) vibrational distribution assumptions. The modelled electron temperature is very sensitive to the electron density, and this decrease in electron density results in the increase of the calculated daytime electron temperature up to about 580 K at the F2 peak altitude. Both the daytime and night-time densities are not reproduced by the model without N 2 (v > 0) and O 2 (v > 0), and inclusion of vibrationally excited N 2 and O 2 brings the model and data into better agreement. The eects of vibrationally excited O 2 and N 2 on the electron density and temperature are most pronounced during daytime.