A self-consistent estimate of O+ + N2 ? rate coefficient and total EUV solar flux with ? < 1050 Å using EISCAT observations

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Introduction
The ionizing solar EUV¯ux with k < 1050 A Ê and the main O + + N 2 ± reaction rate coecient in the F2 region together with the neutral composition are known to be crucial parameters for adequate modeling of the ionospheric F2 region. Unfortunately, there are serious discrepancies both between existing models of solar EUV and between dierent laboratory measurements of the O + + N 2 ± rate coecient. Therefore, aeronomic estimates of these parameters may be useful for qualifying the existing EUV models and laboratory measured O + + N 2 ± rate coecients.
The problems with modeling the solar EUV are discussed by Richards et al. (1994a and references therein). Due to the lack of measurements the¯uxes in some spectral intervals of the EUVAC model (Richards et al., 1994a) were chosen arbitrarily to a great extent. For instance, the¯uxes in the reference spectrum F74113 were increased by a factor of 2 between 150 and 250 A Ê and even were tripled below 150 A Ê to improve the agreement between the calculated and measured photoelectron¯uxes. Titheridge (1997) has proposed a factor of 4 in the EUVAC model for the radiation with k < 150 A Ê to reconcile his model calculations with the E-region observations. Total EUV uxes are also dierent in various models. For instance, a two-component EUV model by  widely used in our calculations gives a 50% larger total¯ux for radiations with k < 1050 A Ê than in the EUVAC model (Richards et al., 1994a).
The ion molecular O + + N 2 reaction is the main sink for O + ions and it controls the recombination rate in the ionospheric F2 region. This rate coecient depends both on the temperature of the reacting species and the amount of vibrationally excited N 2 which varies with geophysical conditions (e.g., Pavlov, 1986;Ennis et al., 1995;Pavlov et al., 1999 and references therein). Such conditions are hard to reproduce in laboratory measurements, therefore, dierent temperature dependencies for this reaction rate coecient can be found in literature. Measurements of this reaction rate by Albritton et al. (1977) in¯ow-drift tubes with helium buer gas for T n T v 300 K and varying ion temperatures 300 K T i 6000 K were used by St.-Maurice and Torr (1978) to derive the reaction rate depending on the eective temperature T eff m i T n m n T i =m n m i . The amount of vibrationally excited N 2 molecules is negligible at T v 300 K, therefore this rate coecient does not take into account the eects of vibrational excitation. Similar¯ow-drift tube measurements but with argon buer gas were provided by McFarland et al. (1973). Dierent rate constants obtained by these two groups are attributed to dierent buer gases used in their measurements. The rate coecient proposed by St.-Maurice and Torr (1978) widely used in aeronomic calculations provides the lowest values for this rate coecient in the ionospheric temperature range. On the other hand the temperature dependence given by McFarland et al. (1973) is very steep. It was eciently used in our calculations to mimic the increase of this reaction rate due to vibrationally excited N 2 at high solar activity (Mikhailov and FoÈ rster, 1997;Mikhailov and Foster, 1997;Mikhailov and Schlegel, 1998). Recent owing afterglow measurements by Hierl et al. (1997) carried out at T n T i T v 300±1600 K con®rmed the results by Schmeltekopf et al. (1968) and showed that the translation energy dependencies for the vibrationally excited species are the same as those for the species being at the ground level (v 0). Eects of rotational excitations were shown to be unimportant in enhancing this reaction rate. Their rate coecient is somewhat higher compared to St.-Maurice and Torr (1978) and McFarland et al. (1973) values for temperatures <1000 K, but it steeply increases for T > 1300 K due to N Ã 2 (v 2). Nevertheless, the temperature dependence for the reaction rate by McFarland et al. (1973) is even steeper for such temperatures. Therefore, it is interesting to have an independent estimate for this rate coecient using an aeronomic approach.
It should be kept in mind that aeronomic estimates are indirect ones and depend on many other parameters used in the model calculations. Therefore, on one hand specially selected reliable measurements should be used for such an analysis; on the other hand one should try to minimize the in¯uence of other parameters during calculations.
A self-consistent method for daytime F2-region modeling developed by Mikhailov and Schlegel (1997) was applied to incoherent scatter (IS) observations with the EISCAT facility to ®nd the set of main aeronomic parameters. Summer and equinoctial daytime observations for magnetically quiet days with small electric ®elds (E<5 mV/m) were analyzed. The EISCAT observations during such conditions correspond to a typical midlatitude F2 region (Farmer et al., 1984;Lathuillere and Brekke, 1985) controlled by local processes (photoionization, recombination, diusion and vertical drift related to thermospheric winds). To minimize the in¯uence of other parameters the most important ones were found in a self-consistent way. So, neutral compo-sition (O, O 2 , N 2 ), vertical plasma drift W, T 120 and the shape parameter S for the neutral temperature height pro®le, as well as a factor for the  EUV model, and a factor for the O + + N 2 ± reaction rate coecient were found using N e h, T e h, T i h, V l h EISCAT observations. The aim of the study is to derive self-consistent estimates for the total solar EUV¯ux with k < 1050 A Ê at various levels of solar activity as well as for the O + + N 2 ± reaction rate coecient and to compare them with existing EUV models and laboratory measured rate coecients.

Method
The self-consistent method of Mikhailov and Schlegel (1997) is still under development and therefore several versions exist. One of them which seems most suitable and ecient, was used by Mikhailov and FoÈ rster (1999) in their analysis of the January 06±11, 1997 CEDAR storm period and was applied with some modi®cations to the present study. The dierence of this approach from the initial one by Mikhailov and Schlegel (1997) is that T ex is not found from the ion energy conservation in the F region, but from ®tting the calculated h m F2 to the observed one. This approach turned out to be more general as it uses the most reliable parameter, N e (h) observed with the IS method, while T e (h) and T i (h) pro®les depend on the ion composition model applied during the IS data analysis (e.g., LathuilleÁ re et al., 1983;AlcaydeÂ et al., 1996). There is also a problem with the speci®cation of the frictional term in the ion energy conservation equation when electric ®elds are strong enough. Moreover, for strong convection electric ®elds the ion velocity distribution is no longer Maxwellian (St.-Maurice and Schunk, 1979;Hubert and Kinzelin, 1992) and this basic assumption in the data analysis is not valid in such cases.
The main idea of the method is to ®t a theoretical N e (h) to the observed one varying the key aeronomic parameters, and to obtain a self-consistent set of the main aeronomic parameters responsible for the observed N e h distribution. The basic method provides: speci®ed by T 120 , T ex and the shape parameter S; and vertical plasma drift W. As the goal of the present analysis is to ®nd the O + + N 2 ± rate coecient and the solar EUV¯ux, these two parameters were included in the list of unknowns. The latter two parameters should be considered jointly as their ratio in fact mostly determines the electron concentration in the daytime F2 region.
The parameter h m F2 is strongly controlled both by T ex and the linear loss coecient b c 1 N 2 c 2 O 2 (Ivanov-Kholodny and Mikhailov, 1986). Our analysis has shown that no stable solution with proper h m F2 can be obtained if T ex and c 1 are searched simultaneously. Therefore T ex was not ®tted in the present study. Instead we used T ex values from the MSIS-83 model (Hedin, 1983). It was shown in our previous analyses (Mikhailov and Schlegel, 1997;FoÈ rster, 1997, 1999) that the calculated T ex for quiet time conditions coincide with the MSIS-83 model values within an accuracy of about 10% which is the accuracy of the MSIS model with respect to this parameter. The theoretical model of midlatitude F region used in this method was described by FoÈ rster et al. (1995). It takes into account transport process for O + ( 4 S) and photochemical processes only for O + ( 2 D), O + ( 2 P), O 2 X 2 P, N + , N 2 and NO + ions in the 120±520 km height range. Three O + + N 2 ± reaction rate temperature dependencies (McFarland et al., 1973;St.-Maurice and Torr, 1978;Hierl et al., 1997) overlapping the range of laboratory measurements for this rate coecient were used in our analysis. A two-component model of the solar EUV from  was used to calculate the photoionization rates in 35 wavelength intervals (100±1050 A Ê ). The photoionization and photoabsorption cross sections were obtained from Torr et al. (1979) and Richards and Torr (1988). It should be stressed that we only search for a multiplication factor for the total EUV¯ux keeping the dependence on solar activity as it is given by the Nusinov model. The same is applied to the O + + N 2 ± rate coecient where the temperature dependence is speci®ed and we search for a multiplication factor shifting the curve as a whole.
The upper boundary condition was speci®ed at 520 km for all geophysical conditions where the observed electron concentration was used to solve the continuity equation for O + ( 4 S). At lower boundary O + ( 4 S) was supposed to be in photochemical equilibrium. We apply the stationary form of the continuity equation for daytime hours, so only periods of relative stability in N m F2 and h m F2 daily variations around noon were chosen for the analysis. Observed T e (h) and T i (h) pro®les were used in the calculations.
The EISCAT CP-1 programme provides height pro®les of N e , T e , T i and V l every 5 min with the antenna beam directed along the local geomagnetic ®eld line. They were used to calculate median pro®les over 1.5±2 h of observations (18±25 pro®les) for the chosen period around noon when the F2 region is most stable. The relative stability of F2-region parameters is important as we use the stationary form of the continuity equation for electron concentration in our analysis. These median vertical pro®les were then smoothed by a polynomial ®tting (up to the 5th degree) before being used in calculations. Vertical plasma drift W was obtained from the observed parameters with the help of the expression (19.59) from Banks and Kockarts (1973) where V z V l sin I, m ij are diusion collision frequencies for O + , related to momentum transfer collision frequencies m Ã by the expression m ij m j =m i m j m Ã ij (see Eq. 19.13 in Banks and Kockarts, 1973), where i applies to O + ions and j applies to other neutral or ionized gas species, all other symbols are standard. Collisions of O + ions with neutral O, O 2 , N 2 and NO + , O 2 , N 2 , N + ions were taken into account. All O + ion collision frequencies were taken from Banks and Kockarts (1973). The scatter of the measured V l around the median V l h pro®le increases with height (as the observations show), so the reliability of the calculated median V l decreases at high altitudes. Vertical plasma drift W, on the other hand, is mainly controlled by thermospheric winds and electric ®elds ± both are height independent in the topside F2 region (below 520 km). So we assumed W to be constant above the heights: 350±450 km for solar minimum and 450±500 km for solar maximum conditions.

Data selection and calculations
The list of 32 selected periods is given in Table 1. Only quiet days with Ap < 12 and small observed electric ®elds (E < 5 mV/m) were selected for the analysis. F 10.7 varied in a wide range from 73 to 258 to cover various levels of solar activity. This was necessary for checking the EUV¯ux dependence on solar activity given by the EUV models, as well as to cover a range for the temperature dependence of the O + + N 2 ± rate coef-®cient as wide as possible. The condition of small electric ®elds is important to get a pure temperature dependence for the O + + N 2 ± rate coecient without additional eects related to the applied electric ®elds (Schunk et al., 1975). Only summer and equinoctial, noon time sunlit periods were selected, since the auroral F2 region may be considered during such conditions as a typical midlatitude one, as mentioned before. The method provides an excellent ®t of the calculated N e h to the observed ones especially for quiet days; examples may be found in Schlegel (1997, 1998). As the calculated ion composition for quiet days turns out to be very close to the model composition used in ®tting the theoretical to the measured autocorrelation functions during the IS data analysis, no correction of the routine EISCAT observations was required (Mikhailov and Schlegel, 1997).
The results of our calculations are given in Figs. 1±3. Everywhere error bars re¯ect the uncertainty of the numerical solution, as determined from 10±15 dierent estimates. Three dierent laboratory-measured tempera-ture dependencies for the O + + N 2 ± reaction rate coecient (McFarland et al., 1973;St.-Maurice and Torr, 1978;Hierl et al., 1997) were used in the calculations. Regarding the dependence on temperature we imply everywhere the eective temperature T eff (see Sect. 1). Figure 1  An interesting result is that regardless the temperature dependence used, the calculated values group around the Hierl et al. (1997) curve. This may be considered as a clear indication for the Hierl et al. (1997) temperature dependence to be the most preferable among the three ones. Our calculations reproduced even small features of this temperature dependence such as the relative maximum around 900 K and the minimum around 1200 K. The obtained result is an independent and strong support for this rate coecient to be recommended for aeronomic calculations.
The self-consistently calculated total EUV¯ux with three O + + N 2 ± rate coecients is shown in Fig. 2 versus the Nusinov (1992) EUV model. The EUVAC model by Richards et al. (1994a) is given for a comparison. Again independent of the O + + N 2 reaction rate coecient used, the calculated EUV¯uxes demonstrate the same variation with respect to the Nusinov (1992)  (1978) bottom panel. Error bars at the points express the uncertainty of the numerical solution, while the error bar on the left hand side is the total experimental error of the laboratory coecient measurements (Hierl et al., 1997) EUV model. Dashed lines characterize the 25% uncertainty band around the Nusinov (1992) model. The calculated points are within this band for high and middle solar activity, but are 30±35% below the model prediction in solar minimum conditions. On the contrary, the EUVAC model underestimates the EUV  Figure 3 gives a comparison of our calculations with MSIS-83 neutral composition at the height of the F2layer maximum. The results of calculations with the Hierl et al. (1997) temperature dependence are shown as an example. Dashes give the 25% uncertainty band around the MSIS-83 model. All three neutral species are within this band at all levels of solar activity. This is a typical result for quiet time periods provided by our method (Mikhailov and Schlegel, 1997;Mikhailov and FoÈ rster, 1999).

Discussion
The results of the temperature dependence of the O + + N 2 ± reaction rate ( Fig. 1) are interesting. On one hand they demonstrate the capabilities of the method used to extract important aeronomic parameters from the IS observations. On the other hand the close agreement of calculated values to the Hierl et al. (1997) experimental curve indicates that this laboratory-measured dependence seems to be reliable and can be recommended for practical use. Unfortunately, this dependence cannot be checked for temperatures below 850 K using daytime EISCAT observations. But our analysis of the night-time N m F2 increase eect observed at Millstone Hill (Mikhailov and FoÈ rster, 1999) when thermospheric temperature was as low as 690 K, has shown that model calculations may be reconciled with the observations if a low ( 5 Â 10 À13 cm )3 s )1 ) O + + N 2 ± rate coecient is used. Such low rate coecient values are provided by the McFarland et al. (1973) temperature dependence around 750 K (Fig. 1,  top). Although the total estimated experimental error is stated as 25% (Hierl et al., 1997) and thus allows such low rate coecient values (see error bar in the left-hand side of Fig. 1) one may think that the Hierl et al. (1997) values are too high around 700±750 K.
The other problem connected with the O + + N 2 ± rate coecient is its dependence on vibrationally excited N Ã 2 (Pavlov, 1986;Ennis et al., 1995;Pavlov et al., 1999 and references therein). A theoretical calculation of this eect is hampered by some uncertainties resulting in a worsening of the agreement between F2-region modeling and observations (Richards et al., 1994b, c). For this reason eects of vibrational excitation are not included in theoretical models, such as TDIM by Schunk (1988), CTIM by Fuller-Rowell et al. (1996, GTIM by Decker et al. (1994) and some others. On the other hand, it can be found in the literature (e.g., Pavlov and Buonsanto, 1997;Pavlov et al., 1999 and references therein) that an inclusion of eects of N Ã 2 vibrational excitation improves the agreement with observations. The eect of N Ã 2 should be most pronounced in summer at high solar activity (e.g., Ennis et al., 1995) when thermospheric temperatures are high. Our analysis includes such periods with temperatures as high as 1400 K (Fig. 1). According to Pavlov and Buonsanto (1997) the distribution of vibrationally excited nitrogen is mostly Boltz-mann-like for v 2. The amount of vibrationally excited N 2 (v 2) which could contribute to the total rate constant as K 2 (T) 38K 0 (T) (Pavlov et al., 1999) is only about 1% at T 1500 K. Therefore, no appreciable eects related to vibrationally excited N 2 can be expected at usual thermospheric temperatures; they will be seen at higher temperatures only. So, the Hierl et al. (1997) temperature dependence may be used in practice together with T eff , at least for T eff up to 1400 K.
The EUV¯ux calculations (Fig. 2) give practically identical results for the three dierent temperature dependencies of the used O + + N 2 ± rate coecient. The¯uxes are close to the  EUV model at solar maximum, about 25% smaller than the model values at medium activity and about 30±35% at solar minimum. On the contrary, the EUVAC model gives close EUV values at solar minimum, but underestimates the¯uxes by 30±35% at solar maximum. The overall agreement of the calculations to the EUV models seems surprising keeping in mind the quality and scatter (due to calibration problems) of the initial experimental material used for the EUV models derivation (Nusinov, 1984;Bruevich and Nusinov, 1984;Richards et al., 1994a). The calculated EUV¯uxes demonstrate a somewhat steeper dependence on solar activity than both EUV models predict. This is the ®rst independent (aeronomic) check of the existing EUV models. The dierence obtained in the dependence on solar activity needs additional analysis which is outside of the scope of this work. Nevertheless, the results of our calculations allow us to conclude the following: As the ionospheric F2 region is formed by the whole spectrum with k < 1050 A Ê , this deviation from the EUV models may be taken into account in F2-region practical calculations. A correction may be introduced on average by shifting the total EUV ux of EUVAC and the Nusinov (1992) model by 25%, correspondingly. The uncertainties with the k < 250 A Ê radiation (see Introduction) are not important for the F2 region calculations as this radiations contribute to the ionization rate at lower altitudes only.

Conclusions
The main results of our analysis are the following: 1. A self-consistent method developed by Mikhailov and Schlegel (1997) was extended to extract additional important aeronomic parameters (total solar EUV¯ux and O + + N 2 ± reaction rate coecient) using routine incoherent scatter F2-region observations. The method was applied to 32 daytime summer and equinoctial EISCAT observations for quiet periods (Ap < 12) and various levels of solar activity (F 10.7 73±258) to check the existing solar EUV models and laboratory measured temperature dependencies of the O + + N 2 ± reaction rate.
2. Three groups of independent calculations with temperature dependencies of the O + + N 2 ± rate coecient given by McFarland et al. (1973), St.-Maurice and Torr (1978), and Hierl et al. (1997) have shown the calculated rate-coecient values to group around the Hierl et al. (1997) dependence in the 850±1400 K temperature range regardless of the rate coecient used in the calculations. Therefore, the recently measured temperature dependence of the O + + N 2 ± rate coef-®cient can be considered as the most preferable and is recommended for aeronomic calculations.
3. The self-consistently determined total EUV solar uxes with k < 1050 A Ê show similar variations with solar activity level, independent of the O + + N 2 ± rate coecient used in the calculations. The calculated EUV uxes are close to the  EUV model at solar maximum, apart from about 25% dierences from the model values at medium activity and 30±35% dierences at solar minimum. On the contrary, the EUVAC model by Richards et al. (1994a) gives close values at solar minimum, but underestimates the EUV uxes by 30±35% at solar maximum. The calculated uxes show a steeper dependence on solar activity than both models predict. As the ionospheric F2 region is formed by the whole spectrum with k < 1050 A Ê , in practice both EUV models may be recommended for F2-region electron density model calculations if the total ux shifted by 25% for the EUVAC and Nusinov models, correspondingly.