- Email: [email protected]
A fast computation of the diurnal secondary ion production in the ionosphere of Titan
Icarus 174 (2005) 285–288 www.elsevier.com/locate/icarus
Note
A fast computation of the diurnal secondary ion production in the ionosphere of Titan Jean Lilensten a,∗ , Cyril Simon a , Olivier Witasse b , Odile Dutuit c , Roland Thissen c , Christian Alcaraz d a Laboratoire de Planetologie de Grenoble, Batiment D de physique, BP 53, 38041 Grenoble Cedex 9, France b ESTEC, Solar System Division, Keplerlaan 1, NL-2201 AZ, Noordwijk, The Netherlands c Laboratoire de Chimie Physique, Bât. 350 Centre Universitaire Paris-Sud, 91405 Orsay Cedex, France d LURE-Bat. 209D, Centre Universitaire Paris-Sud, BP 34, 91898 Orsay Cedex, France
Received 22 June 2004; revised 13 December 2004
Abstract We propose an analytic model that allows rapid computation of the secondary ion production due to electron impact from the primary photo-production + + ++ + in the ionosphere of Titan. The model parameters are given for each of the 5 major ion productions (N+ 2 , CH4 , N , CH3 , N2 ) as well as for the electron production. 2004 Elsevier Inc. All rights reserved. Keywords: Titan; Ionospheres; Solar radiation
1. The problem to solve The study of planetary ionospheres is amongst the major scientific objectives of all planetary missions. In the Cassini–Huygens mission, several instruments will give us new hints on Saturn and its satellite’s ionospheres. In the scope of this work, we should emphasize 2 instruments in particular: The Ion and Neutral Mass Spectrometer (INMS http://saturn.jpl.nasa.gov/ spacecraft/inst-cassini-inms-details.cfm) analyses ion and neutral particles near Titan and Saturn. The Radio and Plasma Wave Science instrument (RPWS http://www-pw.physics.uiowa.edu/plasma-wave/cassini/home.html) aims at measuring the radio signals coming from Saturn, including the radio waves given off by the interaction of the solar wind with Saturn and Titan. RPWS will provide information on the electric and magnetic wave fields and determine the electron density and temperature near Titan and in some regions of Saturn’s magnetosphere. The modeling of ionospheric data requires the use of a very wide physical approach based on MHD equations including fluid and kinetic mechanics, as well as Maxwell laws. The ion and electron densities are the basic parameters that describe the ionosphere. These densities depend on productions, losses and dynamics. Productions may be of chemical or of physical origin. In the latter case, there are two sources: through photo-absorption (called primary photo-production) and through electron impact (called sec-
ondary production). The production through photo-ionization is quite easy to solve, because in the satellites or in the telluric planets, atmospheres are optically thin for the solar radiation, so that a Beer–Lambert law may be used. The production through electron impact is a more serious problem. It requires solving a kinetic transport equation, which is cumbersome and difficult to include in the global ion–neutral chemical codes. The use of production efficiency may overcome this difficulty. The efficiency is defined as the ratio between the secondary and the primary productions (Richard and Torr, 1988; Lilensten et al., 1989). In a first case study of the ionosphere of Titan (Galand et al., 1999), the authors proposed such a ratio for the electron production only. However, this ratio was not parametrized and, moreover, it is only useable to compute the electron production and not the different ion productions. In the very near future, it will become necessary to access the separate ion productions. This is why we analytically model the production efficiency for the electrons as well as for the + + ++ + following ions: N+ 2 , CH4 , N , CH3 , N2 . These ions are those directly produced by photo-ionization and electron impact. Of course, much more ions are created in the ionosphere through chemical reactions. These results may be used by the scientists who built Titan’s atmosphere and ionosphere models.
2. Production computation * Corresponding author.
E-mail address: [email protected] (J. Lilensten). 0019-1035/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2004.12.002
The production computation is fully described in Galand et al. (1999) and will not be repeated here. It consists of the computation of the primary
286
Note / Icarus 174 (2005) 285–288
production through photo-absorption and the computation of the secondary production through primary electron impacts. The main inputs are the neutral atmosphere, the cross sections and the solar EUV flux. Another source of ion production exists through the precipitation of energetic particles along the Saturn magnetic field. This source is not taken into account here. The present study may help in the future to discriminate this electron precipitation induced production from the production due to the photo-absorption. Much effort has been recently devoted to improving the neutral atmosphere models of Titan. Müller-Wodarg et al. (2000) developed for that purpose a 3D time-dependent thermospheric model which is used here for the description of the neutral atmosphere (Yelle model in Yelle, 1991). The exospheric temperature is 175 K. We performed our computations from 600 to 1600 km. More recently, Vervack et al. (2004) reanalyzed the Voyager 1 UVS solar occultation data. They find Nitrogen densities that are 25– 60% higher than previous analysis, methane densities that are smaller by a factor of 3 to 7 and C2 H2 densities that are roughly two orders of magnitude smaller. Their value of the thermospheric temperature is approximately 20 K colder. The photon-absorption cross sections come from Torr and Torr (1985) and Fennelly and Torr (1992) for N2 , and from Samson et al. (1989) for CH4 . The set of electron impact cross sections for N2 is detailed in Lummerzheim and Lilensten (1994). It comes from Davies et al. (1989) for CH4 (with 6 excitation states) and Märk (1975) for the double ionization of nitrogen (here σe (N++ 2 )). The computation is performed for a diurnal atmosphere and only deals with the total production due to EUV (Extreme Ultra Violet) inputs. Most of the current EUV models rely on few experiments taken onboard the Atmosphere Explorer missions (Hinteregger et al., 1973). A first representation of Solar EUV fluxes for aeronomical applications was given by Hinteregger (1981) and Hinteregger and Katsura (1981). A first reference flux SC#21REF was assembled from measurements performed in July 1976 (F10.7 = 70), and given in 1659 wavelengths. An extrapolation model (SERF 1) allows estimating the flux during other periods of solar activity. Torr and Torr (1979, 1985) proposed two reference fluxes for aeronomy called F79050N (F10.7 = 243) and SC#REFW (F10.7 = 68). A simple interpolation is one way to estimate a flux at other activity levels. Since then, several authors developed their codes in order to take better advantage of the Atmosphere Explorer measurements. Amongst them, two must be emphasized. Tobiska (Tobiska, 1991; Tobiska and Eparvier, 1998) developed a model which takes data from other sources into account (SME, OSO; AEROS; rockets and ground-based facilities) as well as the solar emission zone of each line, through a specific parameter. He proposes a formula to retrieve a solar flux from the gift of the decimetric index and its average. The second improved model is EUVAC (Richards et al., 1994). Its main difference with previous models is the reference flux chosen and the interpolation formula. The coronal flux is also constrained to be at most 80% of the total. In our calculations, we tested all the solar flux models. They give very little differences on the productions, and no significant variations on the efficiency itself. This is due to the fact that the efficiency is a production ratio. In the following, we use the Torr and Torr model where the solar EUV flux is interpolated in terms of decimetric index into 37 energy values from 248 eV down to 12.02 eV. Following Tobiska (1993), two values have been added at 2.327 nm (545 eV) and 3.750 nm (250 EV) to take into account ionization due to high-energy photons. The F10.7 value is 150, corresponding to a mean solar activity. As a mean value, it will serve as an anchor point for our simple model. In order to compute the height primary production profile, it is necessary to make a projection on the vertical axis through a “Chapman function” (Smith and Smith, 1972). At low solar angle (typically below 70◦ ), this projection behaves as a simple cosine law. The computed productions are shown in Fig. 1 for a solar zenith angle of 45◦ . This angle has been chosen to represent a planetary average (Lebonnois et al., 2001). The behavior is somewhat classical, with a secondary production becoming larger than the primary one below typically + 850 to 950 km. The cases of CH+ 4 and CH3 are special: the primary electron production is very small below about 800 km compared to the sec-
−3 s−1 at ondary production. The production of CH+ 4 reaches 0.05 ion cm + 800 km. The same value is reached at 900 km for CH3 . These ions are negligible at these low altitudes. These values are comparable to those in Cravens et al. (2004), and the electron production profile is as in Galand et al. (1999).
3. Production efficiency The production efficiency is defined as the ratio between the secondary production (in the numerator) and the primary production (in the denominator) as shown in Fig. 2. The same features seen in Fig. 1 are found. The efficiency is of the order of 4 to 5 at maximum for the three nitrogen ions, + but increases to several hundreds for CH+ 4 and CH3 . We have explored the effect of solar activity on the efficiency. On Earth, there are two effects. The first one is a modification of the input photon flux. The second one is a modification of the thermosphere composition. The two effects create a variability of the production efficiency of the order of about 15% (Lilensten et al., 1989). In Titan’s thermosphere, we only consider the effect of the amplification of the solar flux because there is still a great deal
Fig. 1. Total production (full lines), primary photo-production (dashed lines) and secondary production (dot-dashed lines). Top, electrons, (left), + ++ + + N+ 2 , (middle), and CH4 , (right). Bottom, N , (left), N2 , (middle), CH3 , (right).
Fig. 2. Production efficiencies (full lines) and polynomial fits (dashed lines). + + Top, electrons (left), N+ 2 , (middle), and CH4 , (right). Bottom, N , (left), ++ + N2 , (middle), CH3 , (right).
The ai coefficients stand for the altitudes above the transition altitudes. The bi coefficients are below the transition altitude. The transition altitude is given in the middle of the table.
900.00 −13005.8216953 90.1832943991 −2.48713723980 × 10−1 3.41028733095 × 10−4 −2.32422753454 × 10−7 6.29695597757 × 10−11 860.00 −67475569.5286 671334.608603 −2856.69707614 6.73929187783 −9.51912486391 × 10−3 8.04995662471 × 10−6 −3.77361892766 × 10−9 7.56416836991 × 10−13 840.00 −1319.31760093 7.89655564668 −1.75882509126 × 10−2 1.73351896006 × 10−5 −6.37754058807 × 10−9 840.00 −1024.73252212 6.20790147888 −1.39773444265 × 10−2 1.39192096995 × 10−5 −5.17158915328 × 10−9
900.00 −10065.1825906 69.7627403031 −1.92300338076 × 10−1 2.63526249734 × 10−4 −1.79485832046 × 10−7 4.85916422414 × 10−11
3434.37009107 −10.7435732399 1.31659471128 × 10−2 −7.04329769333 × 10−6 1.38023716317 × 10−9 25215.8704447 −91.6837202831 1.33751213406 × 10−1 −9.71140431498 × 10−5 3.50707166407 × 10−8 −5.04329711546 × 10−12 5181.38810037 −15.9912879965 1.89359763393 × 10−2 −9.85761076057 × 10−6 1.89374636145 × 10−9
2395515.98979 −16607.7380648 49.8972536715 −8.48554353612 × 10−2 8.93576754626 × 10−5 −5.96849017361 × 10−8 2.47016898554 × 10−11 −5.79371466726 × 10−15 5.89834031317 × 10−19 770.00 −72459379651.5 857549135.475 −4436093.94463 13100.8732986 −24.1586283157 2.84848233532 × 10−2 −2.09710876716 × 10−5 8.81400979538 × 10−9 −1.61914069594 × 10−12 8311.24113065 −24.8570921610 2.81818939864 × 10−2 −1.41414970383 × 10−5 2.63743456666 × 10−9
a0 a1 a2 a3 a4 a5 a6 a7 a8 Transition (km) b0 b1 b2 b3 b4 b5 b6 b7 b8
7168.39596798 −21.3232033733 2.41445310804 × 10−2 −1.20997119140 × 10−5 2.25353985295 × 10−9
CH+ 3 N+ CH+ 4 N+ 2 Electrons
Table 1 Polynomial coefficients in Eqs. (1) and (2)
N++ 2
Note / Icarus 174 (2005) 285–288
287
of uncertainty on the atmospheric composition. The relative variation of the efficiency compared to the mean solar activity is less than 8% whatever the ion or the altitude. The other important parameter is the solar zenith angle χ . When the Sun sets, the effect is to move the whole efficiency upward. There are two components in the solar angle influence. The first one follows a close cos(χ) law because the primary production is directly a function of cos(χ) through the Chapman function. Above about 70◦ , the Chapman function departs from a cosine, but not enough to change the efficiency behavior. The sec√ ond effect is closer to a cos(χ) behavior because of the transport effects (see, for example, Brekke and Hall (1988) for a discussion of the influence of the solar zenith angle on the ionosphere). On Earth, the efficiency is much smaller than in Titan’s ionosphere so that the cos(χ) term is dominant. On Titan, the atmosphere is weaker so√that the transport is greatly enhanced and the efficiencies are large. The cos(χ) value may become important. In order to build a useful model for the efficiency, it is not necessary at first to look for a law with a physical meaning. Physics is actually included in Boltzmann’s transport equation. This is why we fitted the efficiency profiles ε˜ (z, χ) with a simple polynomial law. We start with the estimated efficiency at a zenith solar angle of χ = 45◦ . In order to avoid the oscillations due to high order polynomials, we have divided the altitude range in two parts. Above a given transition altitude, we have used a logarithmic fit: N log10 ε˜ (z, 45◦ ) = ai zi − 2.
(1)
i=0
While below this transition altitude, we use a direct polynomial fit: ε˜ (z, 45◦ ) =
M
bi zi ,
(2)
i=0
where z is the altitude in kilometers. The transition altitudes as well as the ai and bi coefficients are given in Table 1. For any other solar angle, we compute a decay (in kilometer): dz = z + 60 cos(χ) − 205 cos(χ) + 130.
(3)
This results in a null decay at a zenith angle of 45◦ , −15 km at χ = 0◦ , 130 km at χ = 90◦ in total agreement with the Boltzmann modeling. The estimated efficiency then becomes: ε˜ (z, χ) = ε˜ (z − dz, 45◦ ).
(4)
In Fig. 2, we plot the resulting estimated efficiencies using dashed lines at a zenith angle of 45◦ . The error with respect to the full Boltzmann transport solution is less than 4% whatever the altitude. Since this fit is performed at mean solar activity, the variation of F10.7 index introduces an error smaller than 4% at extreme conditions. The additional error induced by the solar angle variation is slightly smaller than 5% at largest angle for CH+ 3 and smaller for the other species. Finally, we have studied the effect of the thermosphere. Following Vervack et al. (2004), we enhanced the nitrogen densities by 60%. Since the efficiency is a ratio with a dependence on the thermosphere both in the numerator and denominator, the effect on the amplitude is quite small, less than 1.5% on all nitrogen ions. As an example, the maximum of the electron efficiency goes from 4.78 in the reference atmosphere to 4.71 in the N2 enhanced atmosphere, and N2 goes from 5.11 to 5.18. The main effect is to move the whole layer upward by 20 km, keeping the shape of the efficiency. Therefore, the same parametrized law may be used with an altitude shift. We also decreased the methane density by a factor of 5. This has a very drastic effect on the altitude of production and therefore on the altitude of the efficiency, which moves downward by 90 km. However, we arrive here at the limit of our model since we are at the bottom of the altitude grid. This number should then not be considered as certain, and the decrease could well be still larger. The maximum amplitude of the methane
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ion production decreases from 0.78 to 0.17 cm−3 s−1 and the methane becomes in this condition totally negligible in the computation of the electron production.
4. Conclusion Models of Titan’s atmosphere require an ion–neutral chemical scheme. The primary photo-production is quite a simple task to solve through the expression of a Beer–Lambert law. The secondary production is more difficult to compute because it necessitates solving a transport equation. However, it cannot be neglected since it may be 4 times as big as the primary one for the electrons and nitrogen ions, and up to 500 for the methane. With the proposed work, it is possible to deduce in a very fast computation five secondary ion productions and the secondary electron production from the primary one so as to eventually include them in global models of Titan’s ionosphere.
References Brekke, A., Hall, C., 1988. Auroral ionospheric quiet summer time conductances. Ann. Geophys. 6, 361–376. Cravens, T.E., Vann, J., Clark, J., Yu, J., Keller, C.N., Brull, C., 2004. The ionosphere of Titan: an updated theoretical model. Adv. Space Res. 33, 212–215. Davies, D.K., Kline, L.E., Bies, W.E., 1989. Measurements of a swarm parameters and derived electron collision cross sections in methane. J. Appl. Phys. 65, 3311–3323. Fennelly, J.A., Torr, D.G., 1992. Photoionization and photoabsorption cross sections of O, N2 , O2 and N for aeronomic calculations. At. Data Nucl. Data Tables 51, 321–363. Galand, M., Lilensten, J., Toublanc, D., Sylvestre, M., 1999. The ionosphere of Titan: ideal diurnal and nocturnal cases. Icarus 140, 92–105. Hinteregger, H.E., 1981. Representation of solar EUV fluxes for aeronomical applications. Adv. Space Res. 1, 39–52. Hinteregger, H.E., Katsura, F., 1981. Observationnal, reference and model data on solar EUV, from measurements on AE–E. Geophys. Res. Lett. 8, 1147–1150. Hinteregger, H.E., Bedo, D.E., Manson, J.E., 1973. The EUV spectrophotometer on Atmosphere Explorer. Radio Sci. 8, 349–354.
Lebonnois, S., Toublanc, D., Hourdin, F., Rannou, P., 2001. Seasonal variations of Titan’s atmospheric composition. Icarus 152, 384–406. Lilensten, J., Kofman, W., Wisemberg, J., Oran, E.S., DeVore, C.R., 1989. Ionization efficiency due to primary and secondary photoelectrons: a numerical model. Ann. Geophys. 7, 83–90. Lummerzheim, D., Lilensten, J., 1994. Electron transport and energy degradation in the ionosphere: evaluation of the numerical solution, comparison with laboratory experiments and auroral observations. Ann. Geophys. 12, 1039–1051. Märk, T.D., 1975. Cross sections for single and double ionization of N2 and O2 molecules by electron impact from threshold up to 170 eV. J. Chem. Phys. 63 (9), 3731–3736. Müller-Wodarg, I.C.F., Yelle, R.V., Mendillo, M., Young, L.A., Aylward, A.D., 2000. The thermosphere of Titan simulated by a global threedimensional time-dependent model. J. Geophys. Res. 105, 20833– 20856. Richard, P.G., Torr, D.G., 1988. Ratio of photoelectron to EUV ionization rate for aeronomic studies. J. Geophys. Res. 93, 4060–4066. Richard, P.G., Fennelly, J.A., Torr, D.J., 1994. EUVAC: a solar EUV flux model for aeronomic calculation. J. Geophys. Res. 99, 8981–8992. Samson, J.A.R., Haddad, G.N., Masuoka, T., Pareek, P.N., Kilcoyne, D.A.L., 1989. Ionization yields, total absorption and dissociative photoionization cross sections of CH4 from 110-950A. J. Chem. Phys. 90, 6925. Smith, F.L., Smith, C., 1972. Numerical evaluation of Chapman’s grazing incidence integral ch(x, χ). J. Geophys. Res. 77, 3592–3597. Tobiska, W.K., 1991. Revised solar extreme ultraviolet flux model. J. Atmos. Terr. Phys. 53, 1005–1018. Tobiska, W.K., 1993. Recent solar extreme ultraviolet irradiance observations and modeling: a review. J. Geophys. Res. 98, 18879–18893. Tobiska, J.R., Eparvier, J., 1998. EUV 97: improvements to EUV irradiance modeling in the soft X-ray and FUV. Sol. Phys. 177, 147–159. Torr, M.R., Torr, D.J., 1979. Ionization frequencies for major thermospheric constituents as a function of solar cycle 21. Geophys. Res. Lett. 6, 771– 774. Torr, M.R., Torr, D.J., 1985. Ionization frequencies for solar cycle 21: revised. J. Geophys. Res. 90, 6675–6678. Vervack, R.J., Sandel, B.R., Strobel, D.F., 2004. New perspectives on Titan’s upper atmosphere from a reanalysis of the Voyager 1 UVS solar occulations. Icarus 170, 91–112. Yelle, R.V., 1991. Non-LTE models of Titan’s upper atmosphere. Astrophys. J. 383, 380–400.
Note
A fast computation of the diurnal secondary ion production in the ionosphere of Titan Jean Lilensten a,∗ , Cyril Simon a , Olivier Witasse b , Odile Dutuit c , Roland Thissen c , Christian Alcaraz d a Laboratoire de Planetologie de Grenoble, Batiment D de physique, BP 53, 38041 Grenoble Cedex 9, France b ESTEC, Solar System Division, Keplerlaan 1, NL-2201 AZ, Noordwijk, The Netherlands c Laboratoire de Chimie Physique, Bât. 350 Centre Universitaire Paris-Sud, 91405 Orsay Cedex, France d LURE-Bat. 209D, Centre Universitaire Paris-Sud, BP 34, 91898 Orsay Cedex, France
Received 22 June 2004; revised 13 December 2004
Abstract We propose an analytic model that allows rapid computation of the secondary ion production due to electron impact from the primary photo-production + + ++ + in the ionosphere of Titan. The model parameters are given for each of the 5 major ion productions (N+ 2 , CH4 , N , CH3 , N2 ) as well as for the electron production. 2004 Elsevier Inc. All rights reserved. Keywords: Titan; Ionospheres; Solar radiation
1. The problem to solve The study of planetary ionospheres is amongst the major scientific objectives of all planetary missions. In the Cassini–Huygens mission, several instruments will give us new hints on Saturn and its satellite’s ionospheres. In the scope of this work, we should emphasize 2 instruments in particular: The Ion and Neutral Mass Spectrometer (INMS http://saturn.jpl.nasa.gov/ spacecraft/inst-cassini-inms-details.cfm) analyses ion and neutral particles near Titan and Saturn. The Radio and Plasma Wave Science instrument (RPWS http://www-pw.physics.uiowa.edu/plasma-wave/cassini/home.html) aims at measuring the radio signals coming from Saturn, including the radio waves given off by the interaction of the solar wind with Saturn and Titan. RPWS will provide information on the electric and magnetic wave fields and determine the electron density and temperature near Titan and in some regions of Saturn’s magnetosphere. The modeling of ionospheric data requires the use of a very wide physical approach based on MHD equations including fluid and kinetic mechanics, as well as Maxwell laws. The ion and electron densities are the basic parameters that describe the ionosphere. These densities depend on productions, losses and dynamics. Productions may be of chemical or of physical origin. In the latter case, there are two sources: through photo-absorption (called primary photo-production) and through electron impact (called sec-
ondary production). The production through photo-ionization is quite easy to solve, because in the satellites or in the telluric planets, atmospheres are optically thin for the solar radiation, so that a Beer–Lambert law may be used. The production through electron impact is a more serious problem. It requires solving a kinetic transport equation, which is cumbersome and difficult to include in the global ion–neutral chemical codes. The use of production efficiency may overcome this difficulty. The efficiency is defined as the ratio between the secondary and the primary productions (Richard and Torr, 1988; Lilensten et al., 1989). In a first case study of the ionosphere of Titan (Galand et al., 1999), the authors proposed such a ratio for the electron production only. However, this ratio was not parametrized and, moreover, it is only useable to compute the electron production and not the different ion productions. In the very near future, it will become necessary to access the separate ion productions. This is why we analytically model the production efficiency for the electrons as well as for the + + ++ + following ions: N+ 2 , CH4 , N , CH3 , N2 . These ions are those directly produced by photo-ionization and electron impact. Of course, much more ions are created in the ionosphere through chemical reactions. These results may be used by the scientists who built Titan’s atmosphere and ionosphere models.
2. Production computation * Corresponding author.
E-mail address: [email protected] (J. Lilensten). 0019-1035/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2004.12.002
The production computation is fully described in Galand et al. (1999) and will not be repeated here. It consists of the computation of the primary
286
Note / Icarus 174 (2005) 285–288
production through photo-absorption and the computation of the secondary production through primary electron impacts. The main inputs are the neutral atmosphere, the cross sections and the solar EUV flux. Another source of ion production exists through the precipitation of energetic particles along the Saturn magnetic field. This source is not taken into account here. The present study may help in the future to discriminate this electron precipitation induced production from the production due to the photo-absorption. Much effort has been recently devoted to improving the neutral atmosphere models of Titan. Müller-Wodarg et al. (2000) developed for that purpose a 3D time-dependent thermospheric model which is used here for the description of the neutral atmosphere (Yelle model in Yelle, 1991). The exospheric temperature is 175 K. We performed our computations from 600 to 1600 km. More recently, Vervack et al. (2004) reanalyzed the Voyager 1 UVS solar occultation data. They find Nitrogen densities that are 25– 60% higher than previous analysis, methane densities that are smaller by a factor of 3 to 7 and C2 H2 densities that are roughly two orders of magnitude smaller. Their value of the thermospheric temperature is approximately 20 K colder. The photon-absorption cross sections come from Torr and Torr (1985) and Fennelly and Torr (1992) for N2 , and from Samson et al. (1989) for CH4 . The set of electron impact cross sections for N2 is detailed in Lummerzheim and Lilensten (1994). It comes from Davies et al. (1989) for CH4 (with 6 excitation states) and Märk (1975) for the double ionization of nitrogen (here σe (N++ 2 )). The computation is performed for a diurnal atmosphere and only deals with the total production due to EUV (Extreme Ultra Violet) inputs. Most of the current EUV models rely on few experiments taken onboard the Atmosphere Explorer missions (Hinteregger et al., 1973). A first representation of Solar EUV fluxes for aeronomical applications was given by Hinteregger (1981) and Hinteregger and Katsura (1981). A first reference flux SC#21REF was assembled from measurements performed in July 1976 (F10.7 = 70), and given in 1659 wavelengths. An extrapolation model (SERF 1) allows estimating the flux during other periods of solar activity. Torr and Torr (1979, 1985) proposed two reference fluxes for aeronomy called F79050N (F10.7 = 243) and SC#REFW (F10.7 = 68). A simple interpolation is one way to estimate a flux at other activity levels. Since then, several authors developed their codes in order to take better advantage of the Atmosphere Explorer measurements. Amongst them, two must be emphasized. Tobiska (Tobiska, 1991; Tobiska and Eparvier, 1998) developed a model which takes data from other sources into account (SME, OSO; AEROS; rockets and ground-based facilities) as well as the solar emission zone of each line, through a specific parameter. He proposes a formula to retrieve a solar flux from the gift of the decimetric index and its average. The second improved model is EUVAC (Richards et al., 1994). Its main difference with previous models is the reference flux chosen and the interpolation formula. The coronal flux is also constrained to be at most 80% of the total. In our calculations, we tested all the solar flux models. They give very little differences on the productions, and no significant variations on the efficiency itself. This is due to the fact that the efficiency is a production ratio. In the following, we use the Torr and Torr model where the solar EUV flux is interpolated in terms of decimetric index into 37 energy values from 248 eV down to 12.02 eV. Following Tobiska (1993), two values have been added at 2.327 nm (545 eV) and 3.750 nm (250 EV) to take into account ionization due to high-energy photons. The F10.7 value is 150, corresponding to a mean solar activity. As a mean value, it will serve as an anchor point for our simple model. In order to compute the height primary production profile, it is necessary to make a projection on the vertical axis through a “Chapman function” (Smith and Smith, 1972). At low solar angle (typically below 70◦ ), this projection behaves as a simple cosine law. The computed productions are shown in Fig. 1 for a solar zenith angle of 45◦ . This angle has been chosen to represent a planetary average (Lebonnois et al., 2001). The behavior is somewhat classical, with a secondary production becoming larger than the primary one below typically + 850 to 950 km. The cases of CH+ 4 and CH3 are special: the primary electron production is very small below about 800 km compared to the sec-
−3 s−1 at ondary production. The production of CH+ 4 reaches 0.05 ion cm + 800 km. The same value is reached at 900 km for CH3 . These ions are negligible at these low altitudes. These values are comparable to those in Cravens et al. (2004), and the electron production profile is as in Galand et al. (1999).
3. Production efficiency The production efficiency is defined as the ratio between the secondary production (in the numerator) and the primary production (in the denominator) as shown in Fig. 2. The same features seen in Fig. 1 are found. The efficiency is of the order of 4 to 5 at maximum for the three nitrogen ions, + but increases to several hundreds for CH+ 4 and CH3 . We have explored the effect of solar activity on the efficiency. On Earth, there are two effects. The first one is a modification of the input photon flux. The second one is a modification of the thermosphere composition. The two effects create a variability of the production efficiency of the order of about 15% (Lilensten et al., 1989). In Titan’s thermosphere, we only consider the effect of the amplification of the solar flux because there is still a great deal
Fig. 1. Total production (full lines), primary photo-production (dashed lines) and secondary production (dot-dashed lines). Top, electrons, (left), + ++ + + N+ 2 , (middle), and CH4 , (right). Bottom, N , (left), N2 , (middle), CH3 , (right).
Fig. 2. Production efficiencies (full lines) and polynomial fits (dashed lines). + + Top, electrons (left), N+ 2 , (middle), and CH4 , (right). Bottom, N , (left), ++ + N2 , (middle), CH3 , (right).
The ai coefficients stand for the altitudes above the transition altitudes. The bi coefficients are below the transition altitude. The transition altitude is given in the middle of the table.
900.00 −13005.8216953 90.1832943991 −2.48713723980 × 10−1 3.41028733095 × 10−4 −2.32422753454 × 10−7 6.29695597757 × 10−11 860.00 −67475569.5286 671334.608603 −2856.69707614 6.73929187783 −9.51912486391 × 10−3 8.04995662471 × 10−6 −3.77361892766 × 10−9 7.56416836991 × 10−13 840.00 −1319.31760093 7.89655564668 −1.75882509126 × 10−2 1.73351896006 × 10−5 −6.37754058807 × 10−9 840.00 −1024.73252212 6.20790147888 −1.39773444265 × 10−2 1.39192096995 × 10−5 −5.17158915328 × 10−9
900.00 −10065.1825906 69.7627403031 −1.92300338076 × 10−1 2.63526249734 × 10−4 −1.79485832046 × 10−7 4.85916422414 × 10−11
3434.37009107 −10.7435732399 1.31659471128 × 10−2 −7.04329769333 × 10−6 1.38023716317 × 10−9 25215.8704447 −91.6837202831 1.33751213406 × 10−1 −9.71140431498 × 10−5 3.50707166407 × 10−8 −5.04329711546 × 10−12 5181.38810037 −15.9912879965 1.89359763393 × 10−2 −9.85761076057 × 10−6 1.89374636145 × 10−9
2395515.98979 −16607.7380648 49.8972536715 −8.48554353612 × 10−2 8.93576754626 × 10−5 −5.96849017361 × 10−8 2.47016898554 × 10−11 −5.79371466726 × 10−15 5.89834031317 × 10−19 770.00 −72459379651.5 857549135.475 −4436093.94463 13100.8732986 −24.1586283157 2.84848233532 × 10−2 −2.09710876716 × 10−5 8.81400979538 × 10−9 −1.61914069594 × 10−12 8311.24113065 −24.8570921610 2.81818939864 × 10−2 −1.41414970383 × 10−5 2.63743456666 × 10−9
a0 a1 a2 a3 a4 a5 a6 a7 a8 Transition (km) b0 b1 b2 b3 b4 b5 b6 b7 b8
7168.39596798 −21.3232033733 2.41445310804 × 10−2 −1.20997119140 × 10−5 2.25353985295 × 10−9
CH+ 3 N+ CH+ 4 N+ 2 Electrons
Table 1 Polynomial coefficients in Eqs. (1) and (2)
N++ 2
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of uncertainty on the atmospheric composition. The relative variation of the efficiency compared to the mean solar activity is less than 8% whatever the ion or the altitude. The other important parameter is the solar zenith angle χ . When the Sun sets, the effect is to move the whole efficiency upward. There are two components in the solar angle influence. The first one follows a close cos(χ) law because the primary production is directly a function of cos(χ) through the Chapman function. Above about 70◦ , the Chapman function departs from a cosine, but not enough to change the efficiency behavior. The sec√ ond effect is closer to a cos(χ) behavior because of the transport effects (see, for example, Brekke and Hall (1988) for a discussion of the influence of the solar zenith angle on the ionosphere). On Earth, the efficiency is much smaller than in Titan’s ionosphere so that the cos(χ) term is dominant. On Titan, the atmosphere is weaker so√that the transport is greatly enhanced and the efficiencies are large. The cos(χ) value may become important. In order to build a useful model for the efficiency, it is not necessary at first to look for a law with a physical meaning. Physics is actually included in Boltzmann’s transport equation. This is why we fitted the efficiency profiles ε˜ (z, χ) with a simple polynomial law. We start with the estimated efficiency at a zenith solar angle of χ = 45◦ . In order to avoid the oscillations due to high order polynomials, we have divided the altitude range in two parts. Above a given transition altitude, we have used a logarithmic fit: N log10 ε˜ (z, 45◦ ) = ai zi − 2.
(1)
i=0
While below this transition altitude, we use a direct polynomial fit: ε˜ (z, 45◦ ) =
M
bi zi ,
(2)
i=0
where z is the altitude in kilometers. The transition altitudes as well as the ai and bi coefficients are given in Table 1. For any other solar angle, we compute a decay (in kilometer): dz = z + 60 cos(χ) − 205 cos(χ) + 130.
(3)
This results in a null decay at a zenith angle of 45◦ , −15 km at χ = 0◦ , 130 km at χ = 90◦ in total agreement with the Boltzmann modeling. The estimated efficiency then becomes: ε˜ (z, χ) = ε˜ (z − dz, 45◦ ).
(4)
In Fig. 2, we plot the resulting estimated efficiencies using dashed lines at a zenith angle of 45◦ . The error with respect to the full Boltzmann transport solution is less than 4% whatever the altitude. Since this fit is performed at mean solar activity, the variation of F10.7 index introduces an error smaller than 4% at extreme conditions. The additional error induced by the solar angle variation is slightly smaller than 5% at largest angle for CH+ 3 and smaller for the other species. Finally, we have studied the effect of the thermosphere. Following Vervack et al. (2004), we enhanced the nitrogen densities by 60%. Since the efficiency is a ratio with a dependence on the thermosphere both in the numerator and denominator, the effect on the amplitude is quite small, less than 1.5% on all nitrogen ions. As an example, the maximum of the electron efficiency goes from 4.78 in the reference atmosphere to 4.71 in the N2 enhanced atmosphere, and N2 goes from 5.11 to 5.18. The main effect is to move the whole layer upward by 20 km, keeping the shape of the efficiency. Therefore, the same parametrized law may be used with an altitude shift. We also decreased the methane density by a factor of 5. This has a very drastic effect on the altitude of production and therefore on the altitude of the efficiency, which moves downward by 90 km. However, we arrive here at the limit of our model since we are at the bottom of the altitude grid. This number should then not be considered as certain, and the decrease could well be still larger. The maximum amplitude of the methane
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ion production decreases from 0.78 to 0.17 cm−3 s−1 and the methane becomes in this condition totally negligible in the computation of the electron production.
4. Conclusion Models of Titan’s atmosphere require an ion–neutral chemical scheme. The primary photo-production is quite a simple task to solve through the expression of a Beer–Lambert law. The secondary production is more difficult to compute because it necessitates solving a transport equation. However, it cannot be neglected since it may be 4 times as big as the primary one for the electrons and nitrogen ions, and up to 500 for the methane. With the proposed work, it is possible to deduce in a very fast computation five secondary ion productions and the secondary electron production from the primary one so as to eventually include them in global models of Titan’s ionosphere.
References Brekke, A., Hall, C., 1988. Auroral ionospheric quiet summer time conductances. Ann. Geophys. 6, 361–376. Cravens, T.E., Vann, J., Clark, J., Yu, J., Keller, C.N., Brull, C., 2004. The ionosphere of Titan: an updated theoretical model. Adv. Space Res. 33, 212–215. Davies, D.K., Kline, L.E., Bies, W.E., 1989. Measurements of a swarm parameters and derived electron collision cross sections in methane. J. Appl. Phys. 65, 3311–3323. Fennelly, J.A., Torr, D.G., 1992. Photoionization and photoabsorption cross sections of O, N2 , O2 and N for aeronomic calculations. At. Data Nucl. Data Tables 51, 321–363. Galand, M., Lilensten, J., Toublanc, D., Sylvestre, M., 1999. The ionosphere of Titan: ideal diurnal and nocturnal cases. Icarus 140, 92–105. Hinteregger, H.E., 1981. Representation of solar EUV fluxes for aeronomical applications. Adv. Space Res. 1, 39–52. Hinteregger, H.E., Katsura, F., 1981. Observationnal, reference and model data on solar EUV, from measurements on AE–E. Geophys. Res. Lett. 8, 1147–1150. Hinteregger, H.E., Bedo, D.E., Manson, J.E., 1973. The EUV spectrophotometer on Atmosphere Explorer. Radio Sci. 8, 349–354.
Lebonnois, S., Toublanc, D., Hourdin, F., Rannou, P., 2001. Seasonal variations of Titan’s atmospheric composition. Icarus 152, 384–406. Lilensten, J., Kofman, W., Wisemberg, J., Oran, E.S., DeVore, C.R., 1989. Ionization efficiency due to primary and secondary photoelectrons: a numerical model. Ann. Geophys. 7, 83–90. Lummerzheim, D., Lilensten, J., 1994. Electron transport and energy degradation in the ionosphere: evaluation of the numerical solution, comparison with laboratory experiments and auroral observations. Ann. Geophys. 12, 1039–1051. Märk, T.D., 1975. Cross sections for single and double ionization of N2 and O2 molecules by electron impact from threshold up to 170 eV. J. Chem. Phys. 63 (9), 3731–3736. Müller-Wodarg, I.C.F., Yelle, R.V., Mendillo, M., Young, L.A., Aylward, A.D., 2000. The thermosphere of Titan simulated by a global threedimensional time-dependent model. J. Geophys. Res. 105, 20833– 20856. Richard, P.G., Torr, D.G., 1988. Ratio of photoelectron to EUV ionization rate for aeronomic studies. J. Geophys. Res. 93, 4060–4066. Richard, P.G., Fennelly, J.A., Torr, D.J., 1994. EUVAC: a solar EUV flux model for aeronomic calculation. J. Geophys. Res. 99, 8981–8992. Samson, J.A.R., Haddad, G.N., Masuoka, T., Pareek, P.N., Kilcoyne, D.A.L., 1989. Ionization yields, total absorption and dissociative photoionization cross sections of CH4 from 110-950A. J. Chem. Phys. 90, 6925. Smith, F.L., Smith, C., 1972. Numerical evaluation of Chapman’s grazing incidence integral ch(x, χ). J. Geophys. Res. 77, 3592–3597. Tobiska, W.K., 1991. Revised solar extreme ultraviolet flux model. J. Atmos. Terr. Phys. 53, 1005–1018. Tobiska, W.K., 1993. Recent solar extreme ultraviolet irradiance observations and modeling: a review. J. Geophys. Res. 98, 18879–18893. Tobiska, J.R., Eparvier, J., 1998. EUV 97: improvements to EUV irradiance modeling in the soft X-ray and FUV. Sol. Phys. 177, 147–159. Torr, M.R., Torr, D.J., 1979. Ionization frequencies for major thermospheric constituents as a function of solar cycle 21. Geophys. Res. Lett. 6, 771– 774. Torr, M.R., Torr, D.J., 1985. Ionization frequencies for solar cycle 21: revised. J. Geophys. Res. 90, 6675–6678. Vervack, R.J., Sandel, B.R., Strobel, D.F., 2004. New perspectives on Titan’s upper atmosphere from a reanalysis of the Voyager 1 UVS solar occulations. Icarus 170, 91–112. Yelle, R.V., 1991. Non-LTE models of Titan’s upper atmosphere. Astrophys. J. 383, 380–400.