Prediction and modelling of doubly-charged ions in the Earth's upper atmosphere

Abstract. We discuss the existence of the doubly-charged ions N 2 ++ , O 2 ++ and O ++ in the Earth's ionosphere through the theoretical computation of density profiles over the altitude range 100-500km calculated with the TRANSCAR model. Photoproductions and electron impact productions are described in detail. The photochemical equilibrium assumption verified in this context yields density profiles, validated by comparison to the measurements of the satellite Atmosphere Explorer. Those three dications are lost essentially by dissociative recombination and collisions with the neutrals N 2 , O 2 and O. The metastable lifetime of N 2 ++ is 3s and is estimated at 4s for O 2 ++ . A survey of the sensitivity to geophysical parameters (solar activity, latitude and solar zenith angle) is also shown. Emphasis is cast on the need for rate constants for chemical reactions involving doubly-charged ions. The fit of the model yields an estimated value of 1.06±0.40x10 -10 cm 3 s -1 for the rate constant of the O ++ +O reaction.


Introduction
The existence of stable doubly-charged molecular ions in laboratories dates back to the early years of the last century when Aston (1920) and then Thomson (1921) made the first discovery of what was then proposed to be either CO ++ or N ++ 2 .Half a century later aeronomers finally worked up an interest in the role of dications into the ionosphere: Vegard quoted by Meinel (1951) announced the identification of the OIII line at 500.8 nm in the aurora spectrum for the strong feature observed near 500.04nm.However, as the 496.0-nm line (which completes the doublet lines for O ++ ) did not appear in the spectrum, the discovery was finally rejected by most authors.
Correspondence to: C. Simon (cyril.simon@obs.ujf-grenoble.fr) Nearly at the same time as the lead for a spectral evidence was becoming more and more disputed, Hoffman (1967) reported the detection of O ++ in the topside ionosphere by the mass spectrometer on board Explorer 31.Several theoretical attempts were then led in the ionosphere (Walker, 1970), as well as in the magnetosphere (Nakada and Singer, 1968;Horwitz, 1981).Many authors such as Prasad and Furman (1975) and above all Avakyan (1978a, b) pointed out the necessity for a theoretical modelling approach to take into account doubly-charged ions, namely N ++ 2 , O ++ 2 and O ++ .Due to the very large uncertainties concerning the production, stability and reactivity of molecular dications, every effort from this time on converged on the modelling of O ++ densities in comparison to the few available measurements: in this context, Breig et al. (1977) used Atmosphere Explorer C satellite data recorded in 1974, and with simple assumptions, they produced the very first model of O ++ densities, though reaction rates still corresponded to fits with experimental data.Only one production reaction was put forward to account for observed densities, i.e. ionisation of O + .However, one year later, Avakyan (1978aAvakyan ( , 1980) ) and then Victor and Constantinides (1979) showed that another source for O ++ in the thermosphere had been previously neglected, namely double photoionisation of O, found to be the dominant mechanism below 500 km.Breig et al. (1982) then revised their hypotheses and produced another model that was compared with the same Atmosphere Explorer data for which the processing had been improved.They found good agreement and drew their conclusions on the validity of the reaction rate constants for O ++ +O, deduced to be k=6.6×10−11 cm 3 s −1 .However, other studies were conducted at the same time by Fox and Victor (1981) on Venus whose conclusions on reaction rate constants differed from those of Breig et al. (1982) by a factor of two, i.e. k=1.5×10 −10 cm 3 s −1 .Later on, a study of this reaction in Io plasma torus yielded the value 2.0×10 −10 cm 3 s −1 (Brown et al., 1983), which was close to the value of Fox and Victor.Very recently, a period which coincided with the maturity of both the experimental study of dications (see Mrázek et al., 2000 and references therein) and ionosphere models, Witasse et al. (2002Witasse et al. ( , 2003) ) predicted the existence of a CO ++ 2 layer in the atmosphere of Mars with the ionosphere model TRANSCAR which was specially adapted to the Martian conditions (Witasse, 2000;Morel et al., 2004).

Ion production rates
With the growing possibilities of high-resolution measurements made by satellites and ground-based facilities, we will have access to more and more ionospheric parameters: we present here the first complete computation of N ++ 2 , O ++ 2 and O ++ ion densities in the Earth's upper atmosphere.
We first describe the photochemical model and focus our description on the production's computation, both primary and secondary, and the loss processes taken into account for each ion.Dication densities are then deduced.
The second part deals with the inputs of the model, i.e. cross sections, extreme ultra-violet (EUV) solar flux and the neutral atmosphere.In the third section we show the results of the modelling and validate the model by a comparison with Atmosphere Explorer data.We can then detail the steps of the computation which yields interesting features while modifying geophysical parameters.
The last paragraph addresses the possibility of detection of such ions applied to the Earth's upper atmosphere and other planets or satellites in the solar system, notably Titan.

The model TRANSCAR applied to dications
The model TRANSCAR and its outputs are described in detail in Lilensten and Blelly (2002).The model describes the upper atmosphere between 90 and 3000 km, assuming it is composed of six ions, namely N + 2 , O + 2 , O + , N + , NO + and H + .To summarize, TRANSCAR is a coupled kinetic/fluid model which solves sequentially the Boltzmann kinetic equation for suprathermal electrons and the Boltzmann momentum equations for the 6 ions listed above.Each part is linked with the other and updated through ionisation and heating rates (produced by the kinetic transport code), on the one hand, and through electronic density and temperature yielded by the 1-D, time-dependent fluid transport model, on the other hand (Fig. 1).
For the ion productions which are the outputs of interest in the present study, two mechanisms are taken into account.First, the primary production is the ionisation of the thermosphere due to the solar EUV flux and is basically related to a Beer-Lambert law.Following this first photoproduction, sufficiently high-energy primary electrons may be released that can ionize, in turn, the neutral gas: secondary electrons and ions are then generated, called the secondary ion production.In the latter case, we have to use a physical description in terms of a kinetic transport of suprathermal electrons, given by the kinetic part of TRANSCAR.The Auger effect, originally proposed as a low-altitude source of O ++ ions by Avakyan (1978aAvakyan ( , 1980)), is not taken into account here.
The ionisation thresholds of N 2 , O 2 and O, for single and double ionisation processes are listed in Table 1.The kinetic formalism described below is relevant to doubly-charged ions, and follows closely the singly-charged ion description of Lilensten and Blelly (2002).The photochemical description proposed in Sect.2.4 is, however, specific to doublycharged ions while singly-charged ion parameters are given by the Boltzmann fluid formalism.

Primary productions
The expression of the primary production profile P j prim (z) of an ion j is given in cm −3 s −1 by a simple Beer-Lambert law for the energy E and at the altitude z: where I ∞ (E) is the solar flux on the top of the ionosphere, σ s→j i (E) the photoionisation cross section of species s producing the doubly-charged ion j , and n s (z) the density of the mother species s which can be either a singly-charged ion or a neutral.
dz represents the optical depth, a function of the absorption cross section of the species s noted σ a s (E) and chapχ (z), the Chapman function, a function of the altitude and the solar zenith angle χ .

Kinetic transport and secondary productions
The primary electrons can be of a high enough energy to ionize as well the neutral gas, even for doubly-charged ions, which have a high ionisation threshold.The steady-state kinetic transport equation describes the angular and energy redistributions of electrons and connects the spatial evolution of the electron flux from the top of the ionosphere to the low E-region.Mathematically speaking, when reviewing losses and sources, we can write (Lilensten and Blelly, 2002): where I (τ, E, µ) is the electron stationary flux (cm 2 s −1 eV −1 sr −1 ), E, E , µ and µ the energies in eV of scattered and incident electrons and their respective pitch-angles cosine.σ T k (E) stands for the total elastic and inelastic collisions' cross section for the neutral species s, while n k (z) and n e (z) represent the density of the neutral species k and the electron density.τ (z,E) is the electron scattering depth, defined as dτ  , µ), which corresponds to the photoelectron production rate, varies with the precipitated flux or the primary photoelectron flux conditions.The continuous energy loss function L(E), describing the energy transfer from the suprathermal "hot" electrons to the ambient electrons, is computed via a scheme given by Swartz and Nisbet (1972).
The secondary production of the ion j is then: (E) is here the electron-impact ionisation cross section for the production of a doubly-charged ion j through the mother species s, either an ion or a neutral.
The final overall production for the doubly-charged species j is simply the sum of the primary and the secondary productions.
2.3 Loss mechanisms for dications: a chemical model The chemical loss profiles are defined as: where k i is the reaction rate constant (cm 3 s −1 ) corresponding to the reaction of N ++ 2 , O ++ 2 or O ++ with a neutral of density n i , and τ d j is the dissociation lifetime of the molecular doubly-charged ion j .
The electronic recombination rate with N ++ 2 is taken from Seiersen et al. (2003).The rate constants for the reaction of O ++ 2 with N 2 and O 2 are taken from Chatterjee and Johnsen (1989).N ++ 2 reaction rate constants are given by new laboratory measurements from the LCP/LURE team with the CERISES apparatus (acronym for Collisions Et Réactions Avakyan (1978a) Avakyan (1978a) d'Ions Sélectionnés par des Electrons de Seuil).The experimental procedure is described in Nicolas et al. (2002) and Franceschi et al. (2003).
In the case of N ++ 2 , the lifetime of the ion ground state is 3 s (Mathur et al., 1995).For O ++ 2 , as no data is currently available, we assume that the lifetime is limited by the tunnelling through a barrier towards dissociation and we estimate it equal to the lifetime of CO ++ 2 , i.e. 4 s (Mathur et al., 1995).When produced in excited states (electronic or vibrational), the lifetime of the dication is known to decrease rapidly (Cox et al., 2003).Therefore, another unknown is the relative abundance of the dication ground state to the excited states in the "ionisation cross section measurements" reported in Table 3.This is unknown, but as the lifetime is rapidly decreasing with internal energy, and as the spectrometers used for absolute measurement are generally sensitive to species stable in the 10-microseconds-or-more time range, we estimated that the abundance of the ground state might well be 50-80% of the population that was measured.However, as we don't know more precisely the ratio, we chose to use the data from Table 3 as it is, and therefore produce an upper estimate of the expected density of the molecular dications O ++ 2 .  2 and O ++ 2 are made here on the following assumptions proposed by Krishnakumar and Srivastava (1990) for N ++ 2 and by Märk (1975) for O ++ 2 .According to these authors, a maximum of 10% of the overall N + + N ++ 2 production reported may be of N ++ 2 origin while Märk (1975) reports than O ++ 2 cross sections are less than 1% of those for the production of O + 2 , yielding the estimates shown in Table 3. † Cross sections measured at threshold.
Owing to the existence of measurements of the rate constants for the reaction of O ++ with N 2 , O 2 (discussed in Sect.4), and O, we can roughly estimate the effective lifetime of O ++ in multiplying by the corresponding neutral densities 1 k i n i : we obtain an O ++ lifetime of less than 100 s at all altitudes between 100 and 500 km.
Other loss reactions, namely that of O ++ with He and H, can be of importance above 600 km, where He and H concentrations become high and rate constants reach values of 1.1×10 −10 cm 3 s −1 for 1000 K.Because all densities are calculated below 500 km, as showen in Sect.2.4, these loss reactions are assumed to be negligible.Using the continuity equation ∂t +div(n j (z) v j (z)) = P j (z)−n j (z)L j (z), we are now able to determine the density profile of a doubly-charged ion.
The density n j (z) of a doubly-charged ion j is simply given by the photochemical equilibrium: This expression is valid at low altitudes, i.e. at altitudes less than 500 km.At higher altitudes, the velocity may become too large to insure the validity of the steady-state assumption.

Inputs of the model
To solve Eqs. ( 1) and (3), we need to know the parameters n s (z), σ s→j i (E), σ s→j ie (E), and I ∞ (E).The neutral compositions and temperature are provided by the semi-empirical model MSIS-90 (Hedin, 1987(Hedin, , 1991) )  while we use the solar flux model EUV91 with its 39-energy box scheme (Tobiska, 1991) on which the photoionisation cross sections are sampled.
The different ionisation cross sections of interest in this study are summarized in Table 3. Generally speaking, no data have been available up to now for the double photoionisation of the N 2 and O 2 molecules, while for the electron impact, a large documentation exists.The only complete set of data is for O ++ , for both photoionisation and electron impact ionisation.We use Fennelly and Torr (1992) for double photoionisation of oxygen, and Zipf (1985) for the electronimpact cross section because of their remarkably precise tables.
We take N 2 and O 2 electron-impact double-ionisation cross sections from Märk (1975): it is the only measurement available so far where the direct production of N ++ 2 is given without any spectrometric overlapping with N + .The double photoionisation of N 2 and O 2 is a more complex source of concern as no results have been published yet.We use the approximation: to yield the double photoionisation cross sections σ N 2 hν (N ++ 2 ) and σ O 2 hν (O ++ 2 ), where σ hν (i) stands for the photoabsorption cross section of the neutral species i, σ e (j ) or the total electron impact diffusion cross section of the species j and σ i e (i ++ ) for the double electron impact ionisation cross section of the neutral i.
Another source of molecular dications might be the single electron-impact ionisation of N + 2 and O + 2 : though the ionisation cross sections (of the order of 10 −17 cm 2 for N + 2 according to Bahati et al., 2001) is not negligible, the N + 2 and O + 2 densities are much smaller than the neutral ones by a factor of at least 10 6 .The result is that this contribution to N ++ 2 and O ++ 2 productions is negligible in our context, especially in the low-altitude regions considered here.

Validation of the model
In this section, we compare the results of the model described above for O ++ with the lone experimental data from the magnetic ion mass spectrometer MIMS (Hoffman et al., 1973) on board Atmosphere Explorer (AE), which were recorded 30 years ago (Breig et al., 1982).

Calculated O ++ densities compared with AE-C data
The AE satellite sampled data during descent in late afternoon in the southern winter non-polar ionosphere, corresponding to orbits C-2735, C-2743, C-2754, C-2757, C-2758.The solar zenith angle ranged between 75 • and 85 • , while the magnetic indices A p remained around 16.
In Fig. 2, we plot O ++ densities as a function of altitude.On the left panel, we compare AE-C measurements to the model's predictions for three different geographical coordinates, corresponding to data recorded at low altitude (161 km), mean altitude (377 km) and high altitude (594 km).The model reproduces well the data profiles, however, with a little discrepancy between 140 and 200 km: as the measurement uncertainties are also growing larger at these altitudes, it results in a high spreading of experimental AE data points, which eventually enables one to validate our results at lower altitudes.
In the following subsections, we adopt for the model the average-altitude parameters of each satellite orbit, i.e. for orbit C-2758 χ=78.53 • , and the latitude/longitude set (-31.55 • , 54.64 • ), taken as mean conditions.
Let us now discuss our model inputs.As stated above, we use the semi-empirical model MSIS-90 for the neutral atmosphere.This model is in fairly good agreement up to 400 km with the recorded AE-C data for the neutrals.Above 400 km, it clearly overestimates the measurements and reaches a density value higher by a factor of two at 500 km.Nevertheless, depending on the neutral composition inputs (either MSIS or AE-C neutral composition recording), the results of our dication density model are not significantly modified at these high altitudes (differences up to 10% at most).We obtain once more good agreement with AE-C O ++ density profiles.
We can now test the reaction rate constants listed in Table 2.The best agreement with AE-C O ++ profiles is   found when using the k 6 and k 7 rate constants obtained in laboratory measurements by Howorka et al. (1979).As for the charge-exchange reaction O ++ +O, which has not been measured but has been derived from models, we are compelled to search for the best fit of our model while varying the coefficient k 8 .This is shown in Fig. 2 (right panel) for orbit C-2758: we find the best fit for the value k 8 =1.25×10 −10 cm 3 s −1 .
4.2 Determination of the O ++ +O reaction rate constant Breig et al. (1982) published AE-C data for different orbits corresponding to the two days before orbit C-2758.They occur in slightly different conditions.The satellite recordings occur during descents that spread from 10 August (C-2735) to 12 August 1974 (C-2757).We plot in Fig. 3 the O ++ density with the reaction rates defined in the previous section for each satellite orbit.We fit the reaction rate k 8 for each particular recording in order to obtain the best correlation.These different sets of orbital data lead us to the consensus value k8 =1.06±0.40×10−10 cm 3 s −1 at 1σ for the O ++ +O reaction.
The values fitted are summarised in Table 4 and averaged over the five orbits considered, and eventually yield the coefficient k8 .
This result ranges nicely in between the extreme values found by Fox and Victor (1981) determined on Venus, i.e. k 8 =1.5×10 −10 cm 3 s −1 , and Breig et al. (1982) on Earth, i.e. k 8 =6.6×10 −11 cm 3 s −1 .A limitation to these comparisons is that, together with the approach presented here, both authors fitted their model with experimental data while using very simple assumptions for the computation of productions and losses.According to Smith and Adams (1980), Breig et al. (1977) may have well underestimated their values in light of what is expected experimentally for other doubly-charged ions such as Xe ++ or Ar ++ , and should be of the order of 10 −10 cm 3 s −1 .
The value retained here needs of course further confirmation from laboratory studies.What's more, the temperature dependence of the reaction rate constant k 8 has to be cleared up as we observe large variations depending on the orbit of the satellite.These variations, though certainly not entirely, could partly be of thermal origin.

Production and loss profiles for AE-C 2758
In Fig. 4, we plot the O ++ production and loss profiles for AE orbit No. 2758.On the left panel, the production shows a broad peak that maximizes at 10 ion cm −3 s −1 near 180 km.On the right panel, we have the detailed contributions of O ++ losses due to collisions with the neutral gas and the electrons.We can see that collisions with electrons are negligible whatever the altitude.The other plots underline the growing efficiency with altitude of the O ++ +O mechanism: above 350 km, collisions with O play the major role in the final concentration shape and magnitude.We can then conclude that at lower altitudes, where collisions with N 2 and O 2 molecules are prominent, the model is almost not influenced by the O ++ +O process.As this latter process is also subject to the largest uncertainties, the model verifies AE measurements better for altitudes being below 300 km.This lowaltitude validity acts as a cornerstone to the overall validity of the model.
So far as we have access to very few satellite data and as there is a clear lack of statistics, our present photochemical model is validated over the range of altitudes 100-500 km for O ++ .We have determined by means of a fit the charge-exchange reaction rate constant of O ++ with O. Concerning N ++ 2 and O ++ 2 , ionosphere measurements have never been performed, and provided that our reaction rates are up to date and our modelling approach is similar to that of O ++ , we can assume that the model is also valid for these two molecular dications.However, two strong limitations arise.First, due to the present lack of laboratory studies, it was not possible to take into account the reaction of molecular dications with O, although it is probably the major loss process at higher altitudes as we already showed for O ++ .As a final step, we need direct data comparison to ascertain the hypotheses and look for the best geophysical conditions to create molecular ions in the ionosphere.In the next sections, a detailed sensitivity study of dication productions and densities is carried out.

Sensitivity study of productions and densities
Three main geophysical parameters are taken into account that are summarized in Table 5. Grenoble (France) is assumed to be representative of middle latitudes and Tromsø (Norway) is a typical high-latitude site.Two solar flux conditions are studied through the f 10.7 proxy variation, i.e. f 10.7 =68 (solar minimum) and f 10.7 =243 (solar active conditions).Finally, the sensitivity of each ion profile is examined throughout an entire day at Tromsø: the solar zenith angle dependence is examined in detail.All over this work the magnetic index A p is set to 3, in order to avoid any influence from the magnetic activity.We choose day 120 of the year (early May) to compute our productions, losses and densities.

General considerations
Although the densities are computed only below 500 km, the production is of course calculated and shown up to 800 km.Primary and secondary production profiles of the main ion species are presented in Fig. 5 between 100 and 800 km.The electron contribution is simply the weighted sum of all ion species: for dications, two electrons are produced for each double-ionization of a neutral species.Its production profile shows a two-peak shape particularly clear for the primary productions, one at low altitudes near 110 km (due to O + 2 ), the other near 180 km (due to N + 2 and O + ).The lower part of the figure shows the evolution of the dication productions under quiet (bottom left) and active solar flux conditions (bottom right).
The productions rates of a doubly-charged ion are smaller than those for singly-charged ions by a factor at least equal to 100.At very high altitudes, however (>600 km), the O ++ primary production rate can represent up to 10% of that of O + .
A change of dependence in O ++ profiles is clearly seen on the left-hand side of Fig. 5, particularly sensitive for the primary productions.This evolution is interpreted as the two physical mechanisms causing the production of O ++ , i.e. the double ionisation of O and the single ionisation of O + , both through photoionisation and electron impact ionisation.The single photoionisation of O + is the prominent mechanism at high altitudes (400-800 km) while the double photoionisation of O plays the most important role at lower altitudes.Among the secondary production mechanisms (electron impact ionisation), the ionisation of O + is the major reaction for producing O ++ ions from 700 km upwards, as shown in Fig. 6.
As stated in Lilensten et al. (1989) and seen in Fig. 5 (bottom), secondary dication productions contribute up to 30% of the total ion productions above 180 km and are the main contribution at lower altitudes (100-200 km for the molecular dications, 100-350 km for O ++ ), because of the photon penetration altitude into the ionosphere.
The difference between molecular dications and O ++ is due both to the rather high double ionisation threshold of O (48.74 eV) and to the electron-impact cross sections that are 25 times larger than the double photoionisation cross sections of O.

Influence of the solar activity
In Fig. 5 (bottom), productions increase together with the solar flux index.The production rate peak increases in altitude from 150 km to 180 km from low to high solar activity.Between f 10.7 =68 and f 10.7 =243, their intensity increases by a mean factor of 2.5 at low altitudes, owing to the growth of neutral densities.Above 300 km, the solar flux intensification is more noticeable and production rates increase by a factor of 50 to 10 2 for f 10.7 =243, from minimum to maximum solar activity.

Influence of the latitude
The effect of the latitude is much smaller, as neither the general shape nor the global low-altitude peak intensities of dication production rates are observed to change significantly when moving from middle latitudes (Grenoble) to high latitudes (Tromsø).The only effect that can be seen at Tromsø is an increase in the total production rate of molecular dications, and a slight decrease in the production rate of O ++ .Differences between the two sites are less than 15% below 200 km for all three doubly-charged ions but can reach 50% for N ++ 2 and O ++ 2 at 600 km.(green), O ++ (red) over a whole day at Tromsø, for minimum solar flux conditions (f 10.7 =68).Molecular dications show two sharp peaks at low altitudes, one around 115 km ascribed to the secondary production profiles, the other near 180 km attributed to the primary production profiles.O ++ presents a wide peak around 220 km, mixing together the two production contributions.

Influence of the solar zenith angle
Together with the solar activity, the solar zenith angle variation is the parameter which has the strongest influence on productions.The larger the solar zenith angle, the more intense the solar flux absorption, because of the larger column density of atmosphere along the path of the photons: it implies that at a given altitude, fewer productions will be seen and the peaks will be shifted towards higher altitudes (Fig. 7).At χ=90 • , for instance, and according to the model, no ions are produced at 100 km, regardless of the solar conditions.This simple reasoning is verified for N ++ 2 and O ++ 2 .As seen in Fig. 7, peaks for molecular dications come in twos and are centred at very low altitudes (115 km and 120 km for N ++ 2 and O ++ 2 , respectively) due to secondary productions and the other at 180 km due to the primary production.O ++ single production peak is wider and centred around 220 km, a difference due to the density of atomic oxygen which becomes prominent over N 2 and O 2 , from 200 km upwards.

Brief summary
The cumulative contribution of these 3 ions can be up to 5% of the total primary electron production and more than 10% for the secondary electron production, depending on the solar flux conditions and the solar zenith angle.As seen before, the total ion production rate is the sum of primary and secondary production rates.In the altitude range considered, secondary production rates contribute 50% of the total O ++ ion production rate.For N ++ 2 , it reaches an average value of 15%, and for O ++ 2 a value of less than 5%.According to the results of the model, O ++ 2 is the rarest produced species of all three dications by a factor of at least 10.  production reaction O + +hν→ O ++ +e − becomes prominent above 500 km and leads to a more slowly decreasing slope for the total production.

Densities
Figure 9 shows the major ion densities above Tromsø, for the minimum solar activity model.Near 160 km, the F 1 ionospheric layer is clearly seen, characterised by a peak in NO + density.We plot the electron density in black.Doublycharged ions are represented by dashed lines.
We note that, among dications, the highest densities are expected for O ++ which reaches 60 to 100 ions per cm 3 at 500 km, and becomes the ion with the fourth highest density, after O + , H + and N + .At 500 km, the O ++ density is 10 5 times larger than N ++ 2 and O ++ 2 .Peaks in the density of N ++ 2 and O ++ 2 (1 cm −3 and 0.01 cm −3 , respectively) are then predicted between 200 km and 250 km.

Solar flux effect
O ++ densities increase by a factor of 2.5 between 150 and 500 km, from low (f 10.7 =68) to high (f 10.7 =243) solar activity (Fig. 10,top).At the same time, N ++ 2 densities increase by a factor of 3 between 100 and 300 km, and by an increasing factor of 10 to 10 5 above 300 km.O ++ 2 follows the same evolution as N ++ 2 : a mean factor 2 separates the two solar flux results between 100 and 200 km, while above 300 km, the factors by which the densities increase reach 10 5 , showing that the differences become dramatically larger when altitudes increase.

Latitude influence
The influence of latitude is shown in Fig. 10 (bottom left).Generally speaking, the densities are more important at high latitudes for N ++ 2 and O ++ 2 whereas it is the opposite for O ++ .The differences are ranging from a constant 40% for O ++ to a highly variable factor for N ++ 2 and O ++ 2 depending on the solar conditions and the altitude (less than 10% at low altitudes or for active solar conditions, continuouslyincreasing in quiet conditions above 250 km, eventually reaching 40% at 500 km).

Solar zenith angle evolution
In Fig. 10 (bottom right), we consider two extreme cases to study the solar zenith angle effect on dication densities at mean latitude: local zenith (χ =30.47 • ), and twilight (χ =90 • ).Significant changes at low altitudes are to be seen for χ=90 • : peaks appear higher in altitude and their amplitude drops on average by a factor of 1.5.
Around 500 km, O ++ densities are slightly larger for twilight conditions, a tendency which does not recur with molecular doubly-charged ions.This original feature results from a more efficient O ++ production at high altitudes, as explained in the previous section.latitudinal relative differences between Tromsø and Grenoble are shown for each three species N ++ 2 (blue), O ++ 2 (green) or O ++ (red) in active (dashed curves) and quiet (solid curves) solar conditions.Bottom right: densities of the three dications for χ=30.94• (solid curves) and for χ=90 • (dashed curves).

Detection of doubly-charged ions in planetary ionospheres
Two methods are available, namely mass spectrometers experiments and optical measurements.

Mass spectrometers
Originally, mass spectrometer experiments were the first device used on board satellites to yield some information about the ion composition of the atmosphere.Thus, Atmosphere Explorer and Pioneer were used to look for doublycharged ions in the atmosphere of the Earth and Venus.With the Cassini-Huygens mission to the Saturnian system, much effort has been produced most recently to obtain a proper atmosphere/ionosphere model of Titan to be made available to the community.Predictions have then been cast on the possi-ble existence of doubly-charged species in Titan's ionosphere (Lilensten et al., 2005a and2005b).
This technique allows for the quantitative detection of a given ion through its mass to charge ratio m/q.
On Earth, the determination of O ++ (m/q=8) is particularly easy because it does not overlap with other species.However, this method is not suitable for molecular dications such as N ++ 2 and O ++ 2 as N + and O + have, respectively, the same mass to charge ratio.
According to the present model, the detection of the three dications, N ++ 2 , O ++ 2 and O ++ , will be made easier in solar maximum conditions and at noon when densities are at their highest.Latitude is not a crucial parameter and should not be worried about when recording data in the ionosphere.
2 Spectroscopy of dications Among doubly-charged ions, N ++ 2 has perhaps been the most widely studied species for over two decades.The ground state of N ++ 2 , denoted by X 1 + g , is 43.00 eV above the ground neutral state (Ahmad, 2002, see Table 1).So far, 8 excited electronic states have been identified, such as the first ones a 3 u and b 3 − g , situated at 0.57 and 1.48 eV, respectively, above the N ++ 2 ground state (Taylor and Partridge, 1987).The ground state X 1 + g of O ++ 2 is located at 36.13 eV above the ground neutral state, and the first excited state A 3 + u at 4.16 eV above the O ++ 2 ground state (Fournier et al., 1992).Moreover, the O ++ 2 vibrational structure has been investigated by a threshold photoelectron coincidence technique (TPESCO) (Hall et al., 1992 and references therein).
The O ++ ground state 3 P 0 is situated 48.74 eV above its ground neutral state (He et al., 1995).Many excited states exist, among which 1 D 2 , 1 S 0 and 5 S 2 0 lie 2.513 eV, 5.354 eV and 7.479 eV, respectively, above the O ++ ground level.These states are notably involved in the formation of emission lines, discussed in the next paragraph.

Fluorescence
The fluorescence of O ++ is well-known since its doublet centred around 500 nm has been used as a tracer of electron densities and temperatures in gaseous nebulae since the 1940s (e.g.Menzel and Haller, 1941).
Due to the fast dissociation or predissociation of molecular dication excited states, fluorescence towards the ground state is generally a rare phenomenon, but was at least observed for N ++ 2 (Cossart et al., 1985;Cossart and Launay, 1985;Olsson et al., 1988;Ehresmann et al., 2000).For O ++ 2 , Avakyan (1978b) also reports the possible identification of auroral emissions around 243 nm which he ascribes to the A 3 + u -X 1 + g transition: despite the fact that no further confirmation of a fluorescence feature in O ++ 2 has yet been found, three emission wavelengths have recently been tabulated by Avakyan (1998), as shown in Table 6, where we list the fluorescence transitions and their respective wavelengths for N ++ 2 , O ++ 2 and O ++ .The O ++ transitions are situated in the visible blue spectrum and in the UV range.
The N () excited state of specific interest to us is D 1 + u , which lies 7.8 eV above the N ++ 2 ground state (Ahmad, 2002;Olsson et al., 1988).This state fluoresces in transitions to the N ++ 2 ground state, with bands centred around 159 nm (Table 6).The fluorescence lifetime of the N ++ 2 D 1 + u electronic state is 6.0±0.5 ns, according to Olsson et al. (1988).No collisional deactivation is therefore liable to occur in the ionosphere during this very short lifetime.Using the results of Ehresmann et al. (2003), a rough approximation leads us to conclude that nearly 10% of the total N ++ 2 ions created by double photoionisation at the cross-section peak near 65 eV, are produced in this state.As densities for N ++ 2 reach, at the most, 1 ion per cm 3 , we would expect, in a very optimistic hypothesis, that a mere 0.1 ion per cm 3 is produced in the fluorescent state.Depending on the solar flux intensity, we can then estimate the maximum intensity likely to be produced along the line of sight by integrating ion productions over our altitude range.The intensity ranges from 0.6 R in quiet solar conditions to 2.3 R in active solar conditions.This value is high enough to be detected by spectrophotometers whose accuracy lies typically within the sub deci-Rayleigh range.
Regarding O ++ , it is by far the most abundant doublycharged species in the Earth's ionosphere and is therefore the most likely to be detected optically in aurora or diurnal highresolution spectra.Besides the forbidden transitions N 1 and N 2 , three other lines exist for O ++ in the UV range (see Table 6; NIST database at http://physics.nist.gov/cgi-bin/AtData/main asd) but have never been identified in the diurnal ionosphere spectrum.The radiative lifetime of the 5 S 0 2 metastable level of O ++ was measured to be 1.22±0.08ms (Johnson et al., 1984).A fruitful comparison with mass spectrometer results on board satellites should be of great interest in the years to come.

Conclusion and perspectives
Throughout this work, we study the densities of stable, doubly-charged ions produced in the Earth's dayside ionosphere.For the first time a comprehensive investigation of the problem is led for N ++ 2 , O ++ 2 and O ++ .The photokinetic model TRANSCAR, based on Boltzmann's formalism, is used to yield productions (primary and secondary).In order to compute the densities of each doubly-charged ion, a photochemical part is built up in which reaction rate constants are included.The main inputs of the model are given by laboratory experiments.Emphasis is laid on the need for laboratory measurements concerning double photoionisation cross sections of N 2 and O 2 , as well as the reactions of dications with O.
We then validate our approach for O ++ by the comparison with the only measurements available, those recorded by the satellite Atmosphere Explorer in the mid-seventies.The reaction rate constant concerning the O ++ +O → O + +O + reaction is predicted to be 1.06±0.40×10−10 cm 3 s −1 and needs to be confirmed by laboratory experiments.
A sensitivity study is also proposed, showing the most favourable conditions for the productions and existence of doubly-charged ions, via the variation of the solar flux intensity (high f 10.7 ), the geographical coordinates (middle latitudes) and the solar zenith angle (local zenith).
More comparisons with satellite data should, of course, be carried out in the future, both by measurements of the diurnally varying spectra (need of high resolution spectra) and by mass spectrometers on board satellites.
To bring this study to a conclusion, the effects of solarwind precipitations could be investigated.
This study is the third of a series of predictions undertaken on Mars (Witasse et al., 2002), Titan (Lilensten et al., 2005a and2005b) and the Earth.Together with the work performed by Fox and Victor (1981) on Venus and Brown et al. (1983) on Io, we can suggest that doubly-charged ions are a common feature of planets and satellites with thick atmospheres.

Fig. 1 .
Fig. 1.Synopsis of the fluid/kinetic transport model TRANSCAR.We show only the outputs that are involved in the present study, together with the basic couplings.
are less than 1% of those for the production of O 2 + , yielding the estimates shown in Table 3. † Cross sections measured at threshold.* The original measurements concerned the products N + + N ++ 2 and O + + O ++ 2 together.Maximum estimates on cross sections for the production of N ++

Fig. 4 .
Fig. 4. O ++ production and loss profiles for orbit 2758.Left panel: total O ++ production profile.Right panel: loss mechanisms taken into account in the model.Collisions with N 2 (solid curve), with O 2 (dashed curve), with O (dash-dotted curve) and with the electrons (triangles) are represented.The sum of all contributions is in full circles.

Fig. 6 .
Fig. 6.Detail of the secondary productions of O ++ in Grenoble and for quiet solar conditions, χ=63.51 • .Productions of O ++ via double ionisation by electron impact of O are shown in open circles.The single ionisation of O + is represented by black triangles.The solid curve is the total secondary production of O ++ (in cm −3 s −1 ) through both mechanisms.

Figure 8
Figure8shows the total production above Tromsø for all eight ions at minimum solar activity conditions and for χ =63.51 • .Two peaks are visible for N ++ 2 and O ++ 2 , one around 110 km, the other at 180 km.For O ++ , the

Fig. 9 .
Fig. 9. Density profiles of the major ionic species in the ionosphere above Tromsø, f 10.7 =68 and at χ=63.51.N + 2 is plotted in solid dark blue, O + 2 in solid green, O + in solid red, N + in magenta, H + in dashed black, NO + in solid light blue, N ++ 2 in dashed blue, O ++ 2 in dashed green and O ++ in dashed red.The electron density is represented in black solid.The three doubly-charged ion densities are computed under the photochemical equilibrium assumption.

Table 1 .
Ionisation thresholds for some common species.

Table 1 .
Ionisation thresholds for some common species

Table 2 .
. A comprehensive list of reaction rate constants for N ++ 2 , O ++ 2 , and O ++ corresponding to the reactions with the neutrals N 2 , O 2 , O and the electronic recombination.

Table 2 .
A comprehensive list of reaction rate constants for N +

Table 3 .
Cross sections available for the ionisation of N 2 , O 2 and O.

Table 3 .
Cross sections available for the ionisation of N 2 , O 2 and O.

Table 4 .
Summary of the reaction rate constants obtained for the reaction O ++ +O on each AE orbit.

Table 4 .
Summary of the reaction rate constants obtained for the reaction O ++ + O on each AE orbit.