Observation of the hydrogen corona with SPICAM on Mars Express

Icarus 195 (2008) 598–613 www.elsevier.com/locate/icarus

Observation of the hydrogen corona with SPICAM on Mars Express J.Y. Chaufray ∗ , J.L. Bertaux, F. Leblanc 1 , E. Quémerais 2 Service d’aéronomie du CNRS/IPSL, BP3, 91371 Verrières-Le-Buisson, France Received 11 May 2007; revised 3 January 2008 Available online 20 February 2008

Abstract A series of seven dedicated Lyman-α observations of exospheric atomic hydrogen in the martian corona were performed in March 2005 with the ultraviolet spectrometer SPICAM on board Mars Express. Two types of observations are analyzed, observations at high illumination (for a solar zenith angle SZA equal to 30◦ ) and observations at low illumination (for a solar zenith angle equal to 90◦ (terminator), and near the south pole). The measured Lyman-α emission is interpreted as purely resonant scattering of solar photons. Because the Lyman-α emission is optically thick, we use a forward model to analyze this data. Below the exobase, the hydrogen density is described by a diffusive model and above the exobase, it follows Chamberlain’s approach without satellite particles. For different hydrogen density profiles between 80 and 50,000 km, the volume emission rates are computed by solving the radiative transfer equation. Such an approach has been used to analyze the Mariner 6, 7 exospheric Lyman-α data during the late 1960s. A reasonable fit of the set of observations is obtained assuming an exobase temperature between 200 and 250 K and an exobase density of ∼1–4 × 105 cm−3 in good agreement with photochemical models. However, when considering the average exospheric temperature of 200 K measured by other methods [Leblanc, F., Chaufray, J.Y., Witasse, O., Lilensten, J., Bertaux, J.-L., 2006a. J. Geophys. Res. 111 (E9), doi:10.1029/2005JE002664. E09S11; Leblanc, F., Chaufray, J.-Y., Bertaux, J.-L., 2007. Geophys. Res. Lett. 34, doi:10.1029/2006GL028437. L02206; Bougher, S.W., Engel, S., Roble, R.G., Foster, B., 2000. J. Geophys. Res. 105, 17669–17692; Mazarico, E., Zuber, M.T., Lemoine, F.G., Smith, D.E., 2007. J. Geophys. Res. 112, doi:10.1029/2006JE002734. E05014] a supplementary hot population is needed above the exobase to reconcile Lyman-α measurements with these previous measurements, particularly for the observations at low SZA. In this case, the densities and temperatures at the exobase for the two populations are 1.0 ± 0.2 × 105 cm−3 and T = 200 K and 1.9 ± 0.5 × 104 cm−3 and T > 500 K for the cold and hot populations respectively at low SZA. At high SZA, the densities and temperatures are equal to 2 ± 0.4 × 105 cm−3 and T = 200 K and n = 1.2 ± 0.5 × 104 cm−3 and T > 500 K. These high values of the hot hydrogen component are not presently supported by the theory. Moreover, it is important to underline that the two population model remains relatively poorly constrained by the limited range of altitude covered by the present set of SPICAM measurements and cannot be unambiguously identified because of the global uncertainty of our model and of SPICAM measurements. For a one population solution, an average water escape rate of 7.5 × 10−4 precipitable µm/yr is estimated, yielding a lifetime of 13,000 years for the average present water vapor content of 10 precipitable microns. © 2008 Elsevier Inc. All rights reserved. Keywords: Mars, atmosphere; Ultraviolet observations

1. Introduction

* Corresponding author. Now at Southwest Research Institute, San Antonio,

TX, USA. E-mail address: [email protected] (J.Y. Chaufray). 1 Temporarily at Osservatorio Astronomico di Trieste, Trieste, Italy. 2 Now at Institut d’Astrophysique Spatiale, Orsay, France. 0019-1035/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2008.01.009

Venus, Mars and the Earth atmospheres extend into a tenuous corona of atomic hydrogen. Illuminated by the strong solar Lyman-α line at 121.6 nm, the corona re-emits by resonance scattering the same conspicuous Lyman-α emission line in UV (Shklovsky, 1959; Johnson and Fish, 1960; Chamberlain, 1963). Atomic hydrogen is produced by water vapor dissociation in the martian mesosphere and the resulting H-chemistry

The martian hydrogen corona

plays an important role in the martian thermosphere (McElroy and Donahue, 1972; Parkinson and Hunten, 1972; Fox and Dalgarno, 1979; Krasnopolsky, 1993, 2002; Fox et al., 1995). Indeed the water vapor dissociation is responsible for an “oddhydrogen” chemistry which stabilizes the CO2 abundance in the martian atmosphere by catalyzing the recombination of CO and O produced by photodissociation of CO2 (McElroy, 1972) and controls the amounts of ozone. The existence of the “odd-hydrogen” chemistry has recently been confirmed by the direct detection of H2 UV lines from FUSE observations (Krasnopolsky and Feldman, 2001; Krasnopolsky, 2002) and of the hydrogen peroxide H2 O2 using the Texas Echelon Cross Echelle Spectrograph (TEXES) mounted at the 3-m NASA/Infrared Telescope Facility (IRTF) (Encrenaz et al., 2004). The H-chemistry also plays an important role in + ionospheric processes. Reactions between H2 and N+ 2, O , + CO2 produce H and partly control the neutral non-thermal escape of N and C due to dissociative recombination of N+ 2 and CO+ c, 1997; Fox and Bakalian, 2001; Fox, 2 (Fox and Ha´ 2003). The martian hydrogen corona plays a key role in the interaction with the solar wind. This interaction, characterized by the charge exchange between solar wind ions and the neutral corona, leads to the creation of Energetic Neutral Atoms (H ENA) (Kallio et al., 1997) observed by ASPERA-3 on Mars Express (Barabash and Lundin, 2006). It has also been observed through the martian X-ray emission produced by charge exchange between solar wind heavy ions (almost fully ionized) and neutral H of the corona, as observed by XMM–Newton (Dennerl et al., 2006). Solar wind protons and H ENA produced by charge exchange in the exosphere can precipitate into the martian atmosphere and increase by a few percent, the ionization rate (Kallio and Barabash, 2001; Kallio and Janhunen, 2001; Leblanc et al., 2002). The observation of the hydrogen corona can also lead to a better understanding of the present H escape which is related to the history of the martian H2 O content. Geological and mineralogical observations suggest that water has flowed on Mars during the Noachian (Baker, 2001; Poulet et al., 2005; Bibring et al., 2006) and could have flowed in the last million years (Berman and Hartmann, 2002; Burr et al., 2002) and even more recently (Malin et al., 2006). Current estimates (Carr and Head, 2003) indicate that out of the initial H2 O reservoir equivalent to a 150 m ocean spread all over the planet, 50 m could have escaped in the interplanetary medium, 20–30 m could be trapped in the south and north polar caps and 80 m could correspond to a subsurface reservoir. One obvious process leading to H escape to space is the thermal escape or Jeans escape (Chassefière and Leblanc, 2004) which has been estimated as being equal to 3.0 × 1026 s−1 using Mariner’s estimation of density and exospheric temperature. It is worthwhile to remember that an escape flux of 3.0 × 1026 s−1 corresponds to a flux of 1.8 × 108 atoms cm−2 s−1 , easily converted into a quantity of H2 O of 1.0 × 10−3 pr µm (precipitable microns), while there is an average of 10 pr µm at present in the atmosphere. Therefore, at this escape rate, the atmosphere would be emptied in 1 × 104 years, without the H2 O supply of permanent polar

599

caps. It represents a modest 4 m of H2 O over 4 Gyr, in the present conditions. However, the Sun might have been brighter in the EUV earlier in the history of Mars, implying a hotter thermosphere and greater thermal escape. In addition, nonthermal H escape may also exist, increasing the present escape rate significantly. The importance of H escape is also readily attested by the D/H enrichment (Owen et al., 1988; Bertaux et al., 1993; Krasnopolsky and Feldman, 2001) of a factor of 5 in the atmosphere (w.r.t. the terrestrial value). But the ratio in the polar caps is unknown. Martian H Lyman-α emission was observed for the first time by the Mariner 6 and 7 flyby missions (Barth et al., 1969) and Mariner 9 orbiter mission during a period of high solar activity. Only Mariner 6 and 7 missions have observed this emission in the exosphere from 200 to 24,000 km. It has been estimated that the only mechanism responsible for this emission was the resonant scattering of solar Lyman-α photons by the martian hydrogen atoms (Barth et al., 1971). The analysis of this airglow with a radiative transfer approach coupled to a spherical Chamberlain exosphere model yielded an exobase density equal to 3.0 ± 0.5 × 104 cm−3 and an exobase temperature equal to 350 ± 100 K (Anderson and Hord, 1971, 1972). Mariner 9 Lyman-α data on the disc of Mars and at the limb was used to determine an optical thickness of H (above z = 80 km) of τ = 5 in 1971 (Anderson, 1974). More recently ASPERA-3 onboard Mars Express observed Lyman-α emission profiles at high solar zenith angle (Galli et al., 2006). This profile was shown to be in disagreement with a single cold hydrogen population. These authors deduced values of temperature and density at the exobase equal to 180 K and 1.2 × 104 cm−3 for the cold population and ∼1000 K and 6 × 103 cm−3 for the hot population, respectively. However, there are potentially serious photometric calibration problems with this observation, as we will see later. In order to determine the distribution of atomic H in the exosphere of Mars, we conducted a series of dedicated observations of the exospheric Lyman-α airglow with the SPICAM instrument on board the ESA Mars Express mission during March 2005. At that time, the solar longitude of Mars was Ls ∼ 180◦ and the Sun–Mars distance was 1.46 AU. All the observations at low solar illumination are above the south pole, while the observations at high solar illumination are near the equatorial plane in the morning. SPICAM (Spectroscopy for the Investigation of the Characteristics of the Atmosphere of Mars) is composed of one UV and one near-infrared spectrometers dedicated primarily to the study of the atmosphere of Mars (Bertaux et al., 2000, 2006). The SPICAM UV spectrometer is dedicated to stellar occultations for vertical profiling of CO2 , O3 and aerosols, to nadir viewing for O3 and aerosols and to limb grazing for airglow measurements, which means dayglow (Leblanc et al., 2006a, 2007; and this paper), nightglow observations (Bertaux et al., 2005a) and auroral emissions (Bertaux et al., 2005b; Leblanc et al., 2006b, 2008). The next section describes the geometry of observations and the data processing used to obtain the Lyman-α emission intensity profiles in physical units. The rest of the analysis assumes that the emission is entirely due to resonant scattering of solar

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photons on H atoms. In a first approach, it is assumed that the H distribution can be described by one single exospheric component: a Chamberlain’s distribution characterized by a temperature and hydrogen density at the exobase level (200 km). These two parameters are determined by fitting the data through a forward modeling of the radiative transfer problem. Because the results that we deduce are sensitive to the absolute calibration of the instrument, an iterative approach is performed to retrieve the temperature and the density at the exobase when varying such a calibration. Then a second model with two populations (one cold at 200 K, one hot) is also used to fit the data which results are discussed in the last section. 2. Geometry of observations and data processing The SPICAM light UV spectrometer has been described in detail by Bertaux et al. (2000, 2006). It is composed of an off axis parabolic mirror which reflects the light on an holographic concave toroidal UV grating. This UV grating diffracts the light onto an intensifier coupled to a 2D CCD. A retractable slit may be placed at the focal plane of the first mirror. This slit is divided in two parts: • A 50 µm × 4.6 mm narrow part with a field of view 0.02◦ wide by 1.9◦ long achieving a spectral resolution equal to 1.5 nm. • A 500 µm × 2.2 mm wide part with a field of view 0.2◦ wide by 0.98◦ long achieving a spectral resolution equal to 11 nm. For extended sources, the larger width increase the count rate by a factor of 8. The observations presented in this paper were obtained with a line of sight at least 400 km above the martian limb. At such an altitude, the only airglow emission observed by SPICAM UVS is the Lyman-α emission (Leblanc et al., 2006a). In the following, we only consider the large part of the slit, yielding more signal. During each orbit more than 1000 individual consecutive integration periods are achieved. Each measurement is the sum of 32 pixel lines of the CCD. Each period lasts one second which is composed of 640 ms of integration and of 360 ms for the digital processing. To achieve a good estimate of the Dark Current Non-Uniformity (DCNU), technical dedicated observations were used (Leblanc et al., 2006a). The estimate of the integrated intensity over the wavelength is done by direct integration between 2 wavelengths (Fig. 1), the thin line in Fig. 1 represents the estimated background level and the crosses show the spectral range between which the signal is integrated. Two typical geometries of the observation are displayed in Fig. 2. In both cases, the line of sight (LOS) is fixed in inertial space, and the Mars Nearest Point (MNP) of the LOS changes in altitude. The solar zenith angle at the tangent point of the line of sight, shown by red crosses, is equal to 33◦ in the first case and equal to 89◦ (close to the terminator) in the second case. Note that we are probing the south polar region in this case during autumn equinox northern season.

Fig. 1. Lyman-α line measured by SPICAM in the martian exosphere. The rectangular shape is induced by the width of the large part of the slit. Spectra obtained simultaneously with the narrow part of the slit allowed us to check the negligible character of the contribution of 130.4 OI emission at high altitudes.

Five parameters are needed to fully determine the line of sight for the simulation of Lyman-α emission in the forward modeling: the position of Mars Express (3 parameters) and the direction of the line of sight (2 parameters). The H distribution is assumed to be spherically symmetric, and therefore only 4 parameters are necessary for a full description of the geometry (there is an axial symmetry of the Lyman-α radiation field around the Mars–Sun axis). Table 1 summarizes the main geometrical parameters for the set of observations used in this paper. Two kind of observations are analyzed, observations at low SZA (orbits 1512 and 1541) and observations at high SZA (orbits 1507, 1514, 1532, 1575 and 1582). Fig. 2 displayed the observations at low and high SZA for orbits 1541 and 1514. Each Lyman-α measurement is the sum of two components, an exospheric signal and a Lyman-α sky background, due to resonance scattering of solar Lyman-α photons on H atoms in the Solar System. To determine the sky background we used a model which describes the Lyman-α emission resulting from resonant scattering of solar photons on H interplanetary atoms (Lallement et al., 1985) fitted to the SWAN Lyman-α measurements on board SOHO. This method has been used for the interpretation of the PROGNOZ 5 and 6 measurements of the interplanetary Lyman-α emission (Bertaux et al., 1985). With this model, it is possible to determine the Lyman-α emission sky background from Mars position at the time of observation. For example, the values of the Lyman-α sky background are equal to 413 Rayleigh (R) and 596 R (one Rayleigh = 106 /4π photons/cm2 /ster/s) for the observations of the orbit 1541 and 1514, respectively. This background represents around 10% of the total measured intensity. To calculate the martian Lyman-α intensity, we also need the solar Lyman-α flux for the date of each SPICAM observation. For the observations presented here, we used the Solar Spectral Irradiance (SSI) provided by the SORCE (SOlar Radiation and Climate Experiment) data base (Rottman et al., 2006, and references therein). The SORCE data base provides the intensity measured between 121 and 122 nm. We assume that all the

The martian hydrogen corona

601

Fig. 2. Geometry of the observation at high solar illumination (SZA = 30◦ , panel a) and at low solar illumination (SZA = 90◦ , panel b). The ellipse represents the trajectory of Mars Express (MEX). The cross on the line represents the tangent point of the line of sight. The terminator is represented by the boundary between the illuminated face and dark face, the projection of MEX on Mars is also represented (green line). The direction of the line of sight is constant during each observation in EMEJ 2000 (Earth Mean Equator J2000) inertial frame of reference. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 Evolution of the geometrical parameters of the line of sight for the selected set of observations. Different parameters for Mars Express are given columns 1, 2, 3, 4 and for the tangent point of the line of sight columns 5, 6, 7, 8. The right ascension α (column 9) and declination δ (column 10) of the line of sight, given in EMEJ 2000 (Earth Mean Equator J2000) frame of reference are constant during both observations (Fig. 2). The distance Sun–Mars for these observations is between 1.44 and 1.47 AU No. orbit date

F10.7 (at Mars) (×1022 W m2 Hz−1 )

Mex altitude (km)

Mex latitude

Mex SZA

Tangent point altitude (km)

Tangent point latitude

Tangent point SZA

LOS (α)

LOS (δ)

1507 20/03.05 1512 22/03/05 1514 22/03/05 1532 27/03/05 1541 30/03/05 1575 08/04/05 1582 10/04/05

51.7

4242; 6459

−28.5; −49.7

36.9; 53.1

223; 4091

−87.8; −87.1

89.5; 89.2

242

−10.1

48.7

757; 2234

40.8; −0.9

48.3; 26.1

219; 2103

11.4; 11.4

29.2; 28.7

122.9

−45.9

48.7

4244; 6459

−28.6; −49.7

36.9; 55.1

226; 4093

−88.4; −87.4

89.5; 89.2

242

−10.1

42.0

4245; 6459

−25.4; −46.5

36.0; 50.4

228; 4095

−86.5; −86.0

85.3; 85.0

239.6

−12.3

41.0

826; 2236

42.8; 4.2

54.0; 31.7

100; 2217

8.8; 8.8

33.4; 33

126

−47

38.8

5055; 7071

−26.0; −44.0

39.9; 49.4

124; 3695

−87.5; −86.9

86.0; 85.6

241.7

−8.4

42.4

5052; 7069

−24.8; −42.8

40.1; 48.8

118; 3690

−88.0; −87.3

84.5; 84.0

240.7

−9.3

measured intensity is coming from the solar Lyman-α emission and estimate the intensity at the center of the emission line from the relation given by Emerich et al. (2005) linking the center line emission to the total emission. After correction of the solar longitude difference between the Earth and Mars, the solar rotation rate and the Sun–Mars distance, the solar flux deduced at Mars for this set of observation varies between 1.6 × 1011 and 1.8 × 1011 photons cm−2 s−1 A−1 , corresponding to a medium solar activity. Two intensity altitude profiles, after subtraction of the sky background Lyman-α are displayed in Fig. 3, for one observation at low SZA (a) (orbit 1541) and one observation at high SZA (b) (orbit 1514). The error bars represent the statistical noise; the shapes of the two profiles are clearly different and the inflection of the intensity profile between 800 and 1000 km for the observation at high solar illumination (a) is a sign of an optically thick emission.

3. Forward modeling: H distribution and radiative transfer Because the H Lyman-α emission is optically thick with a vertical optical thickness around unity (Anderson, 1974), it is not possible to retrieve directly the density column from the measured intensity. The only possible approach is to start from an arbitrary given H density profile that is used as input for a radiative transfer model to derive a Lyman-α emission profile that can then be compared to the data. Our model of the hydrogen density is divided in three altitude ranges. Between 80 and 120 km we assume an infinite scale height in order to describe the photoproduction in a simple way. This assumption has a small influence on the intensity above 400 km (lower than 10% at 400 km and lower than 1% at 2000 km compared to an extended pure diffusive model between 80 to 120 km). Between 120 and 200 km, the hydrogen density profiles is

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Fig. 3. Intensity profiles for the observations at low SZA (orbit 1541, panel a) and at high SZA (orbit 1514, panel b) after subtraction of a constant sky background component equal to 413 and 596 R, respectively.

estimated by using a diffusive model in a CO2 background (see Appendix A). Above 200 km, the exosphere is computed by using the Chamberlain’s description without satellite particles (Chamberlain, 1963). Details of this model are given in Appendix A. To estimate the intensity integrated along a line of sight, we use Thomas’ assumptions (Thomas, 1963) which are: • • • • • •

A uniform temperature; Maxwellian velocity distributions function; A flat solar line; Gaussian Doppler absorption profiles; Complete frequency redistribution; Density equal to zero above 50,000 km and complete absorption (CO2 absorption) below 80 km.

Under these assumptions, the intensity integrated over the line of sight is solution of the following equation:  g S(r)T (τ )e−τCO2 ds, I (r, Ω) = (1) 4π where S(r) is the source function which is related to the volume emission rate ε(r) by S(r) =

ε(r) , g

(2)

where g is the excitation frequency in s−1 (frequency of scattering of solar photons by H atoms) defined by √ g = F0 × σ0 × υD × π. (3) τ and τCO2 are the optical thickness of H self absorption and pure absorption by CO2 , given by  τ = σ0 (T ) × nH (r) dr, (4)  τCO2 = σCO2 × nCO2 (r) dr. (5) nCO2 (r) and nH (r) are the densities of carbon dioxide and hydrogen, respectively, σ0 is the cross section at the center of the Lyman-α line and σCO2 is the cross section of absorption of

Lyman-α photon by CO2 equal to 6.3 × 10−20 cm2 . In all the results documented hereafter (above 400 km) the absorption by CO2 is negligible. σ0 (expressed in cm−2 ) is varying with the exospheric temperature according to σ0 (T ) =

5.96 × 10−12 . √ T

(6)

F0 is the solar flux for Lyman-α wavelength at Mars and νD the Doppler width of the profile. T (τ ) is the transmission function which represents the probability for a photon to cross an optical thickness τ without being absorbed (Holstein, 1947). Under the previous assumptions, the source function is solution of the equation: S(r) = n(r)T (τsol )e−τCO2 ∞  dΩ + σ0 n(r) S(r )G(τ )e−τCO2 ds. 4π

(7)

r

In this equation the first term represents the primary source function or single scattering function and the second term the multiple scattering function. τsol is the optical thickness between the Sun and the point r. G(τ ) is the second Holstein function (Holstein, 1947). The assumption of complete frequency redistribution and isothermal profile has been studied by Bush and Chakrabarti (1995). Following their work, an isothermal approximation tends to overestimate the amounts of scatterings below the exobase. This simplifying assumption is convenient because we only consider the part well above the exobase. The method used to solve Eq. (7) is described in Appendix B. 4. Analysis of SPICAM Lyman-α observations Different pairs of hydrogen density nexo and temperature Texo at the exobase were defined to build the H atmosphere. Thirty values of hydrogen density nexo varying from 1 × 104 to 7 × 105 cm−3 which correspond to the range of hydrogen density at the exobase expected from the photochemical models

The martian hydrogen corona

603

(∼40 at low SZA and ∼80 at high SZA). If a model represents perfectly the data, a value of χ 2 = 1 should be found. It is difficult to estimate the uncertainty of the model, particularly the uncertainties due the following approximate: • Maxwellian distribution functions in the exosphere; • Spherical symmetry of the density profile; • Uniform temperature.

Fig. 4. Variation of the model intensity along one line of sight at ∼2930 km above the martian surface (orbit 1514) as a function of the hydrogen exobase density. The points represent model for which the intensity has been computed. The straight line represents the values of the optically thin assumption. At high density values, a small variation in intensity corresponds to a large variation of the exobase density, a saturation effect when the H medium is no longer optically thin.

(Krasnopolsky, 2002; Fox, 2003) and 13 exospheric temperatures, varying from 160 to 400 K with a uniform step equal to 20 K have been used. For each model, the radiative transfer was solved, and the predicted integrated intensities along the line of sight were computed for each observation. These integrated intensities have been then bilinearly interpolated on a more refined grid in nexo , Texo . The dependence of the intensity along one line of sight for the observation during the orbit 1514 as a function of the exobase density (for a fixed exobase temperature equal to 200 K) is displayed in Fig. 4. At low exobase density, the intensity varies linearly with the exobase density, in agreement with the values of the optically thin atmosphere, since the whole exospheric distribution is proportional to nexo , as well as the H column density in the LOS. At larger exobase density, the intensity departs from linear behavior showing saturation. For each point of the refined grid defined by the two parameters nexo , Texo , a quality of fit to the data is quantified by a value of χ 2 defined as follows: 1  (Io,i − Im,i (Texo , nexo ))2 . 2 n−2 σo,i n

χ 2 (Texo , nexo ) =

(8)

i=1

The value of χ 2 is therefore normalized to the number of fitted data points (−2 because of the two parameters). Index i refers to the ith line of sight, Io is the observed intensity, Im the computed intensity, σo the standard deviation of the observed intensity and n the number of lines of sight fitted for each orbit

Therefore, some models with a χ 2 value higher than 1, can be considered as acceptable. If we assume an overall uncertainty equal to 10% associated to the models, fits with χ 2 values lower than 100 can be considered as acceptable. Fig. C1 (orbit with SZA = 30◦ ) and Fig. C2 (orbit with SZA = 90◦ ) in Appendix C display examples of modeled profiles at different exobase temperature and hydrogen density. Figs. 5a and 5b show for high and low solar illuminations the normalized χ 2 map. The color saturation represents a χ 2 value higher than four times the minimum value. In this figure, we can find models leading to a χ 2 < 100 for any exospheric temperatures which implies that the exospheric temperature is not well constrained by the observations presented here essentially because of the relatively limited altitude range covered by SPICAM. In that sense, the observations done during Mariner’s 6 and 7 fly-bies remain probably the best set of measurements to constrain the exospheric temperature. Table 2 summarizes the (nexo , Texo ) pairs estimated for the full set of data (Table 1). The observations at solar zenith angle equal to 90◦ are best fitted with an exobase temperature higher than 300 K, which is in disagreement with temperatures estimated from Cameron bands, CO+ 2 lines by Leblanc et al. (2006a) and N2 Vegard–Kaplan bands (Leblanc et al., 2007) as well as aerobraking measurements (Bougher et al., 2000). Hydrogen densities deduce from this analysis are between 0.3 and 1.2 × 105 cm−3 , lower by a substantial factor (4 to 11) than the density derived from the photochemistry model of Krasnopolsky (2002) (4 × 105 cm−3 ) for low solar activity and a solar zenith angle equal to 60◦ . At a given hydrogen density at the exobase, the intensity increases when the exobase temperature increases because of a higher H scale height. There is some coupling between nexo and Texo and the best fit hydrogen density at the exobase is decreasing when the exospheric temperature is increasing (Fig. 5b). The best fit values for these observations at high SZA are in fair agreement with the values estimated from Mariner 6 and 7 missions during their fly-by which were nexo = 3 ± 0.5 × 104 cm−3 and Texo = 350 ± 100 K (Anderson and Hord, 1971, 1972). However we know that the exospheric temperature of Mars is closer to 200 K, from 3 different sources: airglow measurements (taken during the same conditions), as quoted before, aerobraking drag measurements (Bougher et al., 2000), and the extrapolation of SPICAM stellar occultations densities and temperature measurements up to 150 km. Because for optically thick emission, the information extracted from the observations are very sensitive to absolute intensity, we therefore investigate a possible change in the absolute calibration.

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Fig. 5. Variations of the χ 2 function as a function of the temperature and of the hydrogen density at the exobase for the observation 1541 (SZA ∼ 30◦ , panel a) and for the observation 1514 (SZA ∼ 90◦ , panel b). The χ 2 value is color coded. A perfect fit would imply χ 2 = 1, while the minimum χ 2 value is 59 for SZA ∼ 30◦ , meaning that no combination of exospheric temperature and density give a reasonable fit to the data. A reasonable fit is obtained for SZA ∼ 90◦ implying however a temperature of ∼300 K. Table 2 Best parameters (nexo and Texo ) estimated for the set of data. The data at SZA = 30◦ are not well reproduced by the models, while the temperature estimated from the data at SZA = 90◦ is higher than those deduced from the CO+ 2 line by Leblanc et al. (2006a). The temperatures in parenthesis are the smaller and higher temperatures for which it is possible to have a model fitting the data with a χ 2 value lower than 100 No. orbit

SZA

Minimal value of the χ 2 function

nexo (cm−3 )

Texo (K)

Orbit 1507 Orbit 1512 Orbit 1514 Orbit 1532a Orbit 1541 Orbit 1575 Orbit 1582

SZA = 90◦ SZA = 30◦ SZA = 90◦ SZA = 90◦ SZA = 30◦ SZA = 90◦ SZA = 90◦

2.2 49.8 1.5 2.4 59.5 2.7 1.9

1.2 ± 0.3 × 105 1.5 ± 0.3 × 105 5.7 ± 1.3 × 104 4.0 ± 1.0 × 104 8.6 ± 1.0 × 104 3.7 ± 1.0 × 104 3.9 ± 1.0 × 104

270 (160–400) 240 (170–335) 365 (160–400) 390 (160–400) 255 (190–385) 350 (160–400) 340 (160–400)

a A part of the data (corresponding to altitudes between 2400 and 3600 km) are missing for this observation.

5. Sensitivity to the absolute calibration Because the results are sensitive to the absolute calibration, we tested this sensitivity by adding another free parameter A which accounts for a change in the calibration defined by Io Ireal = . (9) A In other words, for each observation we fit the shape of the data profiles, but we let free a scaling factor for the data. For each model defined by nexo and Texo , we determine this scaling factor A by minimizing the function 2 given by  2 log(Io ) − log(Im ) − log(A) . 2 = (10) The minimum of the 2 is found when its derivative ∂2 /∂A = 0 yielding A as  1  log(A) = (11) log(Io ) − log(Im ) . n

Note that this parameter A also includes uncertainties in the solar flux at Lyman-α. We then compare data and models studying the new χ 2 function given by 1  (Io /A − Im (Texo , nexo ))2 . n−2 (σo /A)2 n

χ 2 (Texo , nexo ) =

(12)

i=1

This new function allows us to determine which models best describe the shape of the profile of the Lyman-α intensity. The normalized χ 2 function variations with the hydrogen density and temperature at the exobase are displayed in Fig. 6. At low SZA (Fig. 6a), the observed profiles are now very well fitted and the best fit is obtained for a temperature equal to Texo = 305 K and an exobase density equal to 9.8 × 104 cm−3 . The value of the factor A is equal to 0.80. At high SZA (Fig. 6b), the best fit is obtained for a temperature equal to Texo = 400 K and an exobase density equal to 4.6 × 104 cm−3 ,

The martian hydrogen corona

605

Fig. 6. Same figure as Fig. 5, but in the case of an optimized scaling factor A (Eq. (11)) applied to the nominal SPICAM calibration. In this case, the profile at low SZA is well fitted with an atmospheric model (Texo = 305 ± 20 K; nexo = 9.8 ± 1 × 104 cm−3 ). The profile at high SZA is well fitted with an atmospheric model (Texo = 400 ± 20 K; nexo = 4.6 ± 1 × 104 cm−3 ). However, the “best” scaling factor is systematically different for observations at low and high SZA. Table 3 Best parameters (nexo , Texo and A) estimated for the data. A is a scale factor of the intensity. This scale factor is systematically different between observations at low SZA (∼0.8) and observations at high SZA (∼1.0). In all cases, the exospheric temperatures estimated from the data are significantly higher than those deduced from other UV bands by Leblanc et al. (2006a) No. orbit

SZA

Minimal value of the χ 2 function

nexo (cm−3 )

Texo (K)

A

Orbit 1507 Orbit 1512 Orbit 1514 Orbit 1532 Orbit 1541 Orbit 1575 Orbit 1582

SZA = 90◦ SZA = 30◦ SZA = 90◦ SZA = 90◦ SZA = 30◦ SZA = 90◦ SZA = 90◦

1.5 1.5 1.3 2.2 1.1 1.5 1.6

4.5 ± 1.0 × 104 1.30 ± 0.5 × 105 4.6 ± 1.0 × 104 3.6 ± 1.0 × 104 9.8 ± 1.0 × 104 2.6 ± 1.0 × 104 2.7 ± 1.0 × 104

400 335 400 400 305 400 400

1.11 0.81 1.03 1.04 0.80 1.10 1.08

Table 4 Same parameters as Table 2, but for values of A (scale factor of calibration) equal to 0.9 No. orbit

SZA

Minimal value of the χ 2 function

nexo (cm−3 )

Texo (K)

Orbit 1507 Orbit 1512 Orbit 1514 Orbit 1532a Orbit 1541 Orbit 1575 Orbit 1582

SZA = 90◦ SZA = 30◦ SZA = 90◦ SZA = 90◦ SZA = 30◦ SZA = 90◦ SZA = 90◦

2.4 5.4 2.0 4.4 13.3 4.6 3.5

3.9 ± 1.5 × 105 1.8 ± 0.5 × 105 2.3 ± 0.7 × 105 1.3 ± 0.4 × 105 1.2 ± 0.3 × 105 1.2 ± 0.3 × 105 1.1 ± 0.3 × 105

200 (170–400) 255 (160–400) 225 (160–400) 245 (160–400) 245 (160–400) 225 (160–400) 230 (160–400)

a A part of the data (corresponding to altitudes between 2400 and 3600 km) are missing for this observation.

the value of the factor A is equal to 1.03. Table 3 summarizes the best fit values of the three parameters (A, nexo and Texo ) obtained for this set of observations. In the following we choose the average value A = 0.90 as the best corrective factor. Table 4 summarizes the results for the seven observations using this value. According to Krasnopolsky (2002), the exobase temperature is expected to be ∼230–240 K for a F 10.7 index equal to 40

corresponding to the average value between low and medium solar activity. From the results given in Table 4, our results are in good agreement with this solution assuming temperatures between 200 and 255 K for the full set of observations as expected for this solar activity with an exobase density varying between 1 and 4 × 105 cm−3 in good agreement with the density estimated by the photochemical model from Krasnopolsky (2002). However because it is difficult to fit both SZA profiles with a

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Fig. 7. Best fit obtained for the observation at low SZA (a) and high SZA (b) with two populations, the first is a “cold” population with exospheric temperature equal to 200 K and the second is a “hot” population assumed to be present only above the exobase (at 200 km). The single cold population intensity contribution is shown as dashed line. Adding some hydrogen from the hot population increases the intensity everywhere, except at altitudes <800 km and SZA ∼ 30◦ .

range of calibration factor (here 23% of variation is needed following Table 3) that may be explained by the range of variation of the solar flux (estimated as being smaller than 10%), the possibility that a hot hydrogen component may be the origin of our present discrepancy between our fit at different SZA, cannot be ruled out. Indeed, the high temperature obtained in Section 3 could also be explained by the presence of a hot hydrogen population (Lichtenegger et al., 2006). In the following section we try a first investigation of this assumption. 6. Model with two populations of H atoms In the following, we will focus on two typical observations discussed before (orbits 1514 and 1541 corresponding to Fig. 2). We introduce an atmospheric model above the exobase: one which is referred as “cold,” corresponding to the thermal exospheric component that must exist at the temperature of the atmosphere at 200 km (exobase level), and a second component, characterized by a larger temperature: a “hot” component. Whatever could be the origin of this second population (discussed later), we describe this hot component as a classical Chamberlain’s distribution, defined by nh,exo and Th,exo . The exospheric temperature of the cold population (Tc,exo ) is assumed to be equal to 200 K as inferred by Leblanc et al. (2006a) from the study of the CO+ 2 airglow altitude distribution for orbits close to the ones studied here. The density of the cold population is fixed at a given value ncold , and a best fit is searched to find the couple nh,exo and Th,exo which minimizes the χ 2 . The values of ncold are selected in a range that is allowed by the intensity at low altitudes, where this population should dominate.

For the observation at high SZA, adding a second population increases the intensity at all altitudes (Fig. 7b). The density of the cold population ncold is then estimated by choosing a single-component model which gives slightly smaller intensity than the emission observed at low altitudes. At high SZA, the density of the cold population is estimated to be in the range [1.6 × 105 cm−3 ; 2.2 × 105 cm−3 ]. The effect of adding the hot population on the intensity profiles is illustrated in Fig. 7b. The increase of the intensity compared to the single cold population is higher at high altitudes than at low altitudes. The minimum χ 2 value on all the hot models is equal to 3.5, 1.8, 1.6, 3.4 for a density of the cold population equal to 1.6, 1.8, 2.0 and 2.2 × 105 cm−3 , respectively, and the hot population density at the exobase fitting the observations equal to 2.3, 1.9, 1.2, 0.8 × 104 cm−3 , respectively. The considered range of ncold is sufficient, since the χ 2 value increases at both ends of this range. At low SZA, adding a second population above the exobase increases the intensity at high altitudes, and decreases the intensity at low altitudes (radiative transfer effect) (Fig. 7a). Therefore, we considered densities of the cold population providing a modeled Lyman-α intensity slightly more intense than the emission observed at low altitude and less intense at high altitude. The density of the cold component is chosen to vary between [6 × 104 ; 1.2×105 ] cm−3 . The minimum of the χ 2 value on all the hot models is equal to 27.3, 3.2, 1.8, 6.4 for a density of the cold population equal to 0.6, 0.8, 1.0, 1.2 × 105 cm−3 , respectively, and the hot population density at the exobase fitting the observations is equal to 3.6, 3.0, 1.9, 1.3 × 104 cm−3 , respectively. The best values of the cold population are equal to

The martian hydrogen corona

1 ± 0.4 × 105 cm−3 at low SZA and to 2 ± 0.4 × 105 cm−3 at high SZA, therefore smaller by a factor of 4 and 2 compared to the values given in the model of Krasnopolsky (2002) for an exospheric temperature equal to 200 K, but much higher than the Mariner values (3 × 104 cm−3 ). Our values are also higher by a factor 8 and 16 compared to the value estimated by Galli et al. (2006) from the NPD sensor of ASPERA-3 on Mars Express, which happens to be sensitive to UV stray light (in addition to its main purpose which is to detect neutral energetic particles). Actually the intensities derived by Galli et al. (2006) are much smaller than our values at high altitude. These authors found 0.5 kR at 4000 km of altitude (their Fig. 6), while we found 3 kR at the same altitude (Fig. 7b). Therefore, there is clearly a calibration factor problem with NPD which, as we have seen in Section 5, should have a strong impact on the derived exobase density and temperature. Moreover, the optically thin exosphere assumption used by these authors is not in agreement with the observations done by SPICAM. For the observation at low SZA, the best fit (Fig. 7a) is obtained for a temperature Th,exo > 500 K and a density nh,exo equal to 1.9 × 104 ± 5 × 103 cm−3 . The ratio between the hot and the cold densities at the exobase vary between 0.11 and 0.36 with a nominal value equal to 0.20. For the observation at low solar illumination (Fig. 7b), the best fit is obtained for a temperature Th,exo > 500 K and a density nh,exo equal to 1.2 × 104 ± 5 × 103 cm−3 . At high SZA, the ratio between the hot and the cold densities at the exobase varies between 0.03 to 0.14 with a nominal value equal to 0.06, three times smaller than the ratio obtained at low SZA. Therefore for this approach, the density of the hot population decreases by a factor 1.6 with increasing SZA, while the density of the cold population increases by a factor 2. 7. Discussion 7.1. Exospheric temperatures and solar cycle Some estimates (i.e., Krasnopolsky, 2002) of the martian exospheric temperature at low and high solar activities imply a higher dependence of exospheric temperature with solar cycle than in the case of Venus. Krasnopolsky (2002) considered an exosphere temperature equal to 200 K for a period of minimum solar activity, based on Viking 1 density profile, and equal to 350 K for a period of maximum of solar activity, based on the Lyman-α observations of Mariner 6 and 7 fitted with a single component model of H distribution (Anderson and Hord, 1971). Actually the large variation with solar cycle (from 200 to 350 K) suggested by Viking 1 and Mariner 6 and 7 is inconsistent with solar cycle variation of ionospheric peak plasma densities (Lichtenegger et al., 2004). In addition, exospheric temperatures deduced from aerobraking maneuvers of Mars Global Surveyor for low solar activity yielded exospheric temperatures around 220–230 K (Keating et al., 1998; Bougher et al., 2000, see their Table 1 for a summary of different exospheric temperatures estimates for different solar F10.7 index and heliocentric distances) whereas exospheric temper-

607

atures deduced from aerobraking maneuvers of Mars Odyssey during high solar activity yielded values smaller than 240 K (Bougher, personal communication cited in Lichtenegger et al., 2006). More recently, a study of the martian exosphere using radio tracking data show suggested that the exospheric temperature could be between 100 and 200 K at 400 km (Mazarico et al., 2007). Therefore, the variation of exospheric temperature with solar activity could be much smaller than the range 200–350 K previously suggested by Mariner exospheric observations at high solar activity. Because the Mariner’s observations has been done up to 24,000 km, it is certainly more sensitive to a hot population than SPICAM’s observations up to 4000 km. The high temperature deduced from the Mariner’s data may be then due to a systematic departure between data and a single component model. 7.2. The origin of the possible hot component According to a detailed model of the martian upper atmosphere and ionosphere chemistry (Krasnopolsky, 2002), the main sources of H atoms between 80 and 300 km are the following 2 reactions, with their corresponding column rate (taken from Krasnopolsky, 2002, Table 2): + CO+ 2 + H2 → HCO2 + H

HCO+ + e → CO + H

(1.05 × 108 cm−2 s−1 ),

(1.17 × 108 cm−2 s−1 ).

(1) (2)

All the other reactions produce together ∼3 × 107 H atom cm−2 s−1 . While the exobase level is around 200 km, the ionosphere extends to higher altitudes. If a reaction occurs below the exobase, the H atom will be thermalized before reaching the exobase, providing the source of the cold H component of the exosphere, which dominates the low altitude part of the exosphere. But if the reaction occurs above the exobase (and the ionosphere extends well above 200 km), the H atom will be created with some extra energy (∼1–8 eV, according to Lichtenegger et al., 2004). This would be an important source of “hot” atoms, populating the external exosphere. In their recent study, Lichtenegger et al. (2006) include reactions (1) and (2) in order to deduce a hot hydrogen density profile. They perform a Monte-Carlo simulation of the fate of newly formed H atoms, and find that the ratio between the hot and cold hydrogen density at the exobase is about 10−3 , which is in strong disagreement with our estimate (0.06 to 0.2). Uncertainties on the reaction (1) coefficient rate estimated between [0.4, 1.4] × 10−10 cm3 s−1 (see Scott et al., 1997, and references therein) cannot explain such a discrepancy. Another potential source of hot H atoms is charge exchange of solar wind and planetary protons with neutral atoms. A quantitative estimate of the production rate and resulting neutral exosphere density is beyond the scope of this paper. The velocity V of the SW proton is a crucial factor for the fate of the newly formed H neutral, and there is a region where the solar wind is decelerated by the obstacle of Mars, as modeled for instance by Modolo et al. (2005). From an initial velocity of 400 km/s, it drops to 100 km/s at the bow shock (∼2000 km of altitude) and

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below a few tens of km/s inside the Magnetic Pile-up boundary. If the solar wind proton velocity V > Vesc ∼ 5 km/s, it will produce a so-called Energetic neutral atom (ENA) that will escape Mars, but can be seen in Lyman-α if V < 100 km/s (the linewidth of the exciting solar Lyman-α line). Pick-up planetary protons are dominant below the MPB (Modolo et al., 2005). ENA resulting from the charge exchange between pick-up planetary protons and neutral atoms that would be produced with a low velocity, on satellite orbits, could accumulate for some time in orbit around Mars (tens of days), before being ionized by solar EUV. A first approach, described by Chen and Cloutier (2003) shows the presence of a partial satellite and hot population due to the Lyman-α pressure and charge exchange between solar wind protons and the neutral corona. But these authors did not include the charge exchange between planetary protons and neutral component of the exosphere which could also contribute to a hot or satellite hydrogen population. However, even by including this contribution, the density of the hot hydrogen as derived in this study cannot be explained by the theory. Clearly this apparent discrepancy between observations and their interpretations and theoretical works suggests that significant progresses are needed in modeling a radiative transfer approach able to describe both 3D structure and non-Maxwellian distribution of the atmospheric populations as well as in looking for possible source of these putative hot exospheric hydrogen populations. 7.3. Escape flux In the case of the solution with one population (Section 5), the Jeans escape flux is found to be ∼1.4 ± 0.6 × 108 cm−2 s−1 (∼2.2 ± 1 × 1026 s−1 ). This escape flux represent upper limit to the real escape flux because, as said above, the Jeans formula is obtained by assuming a Maxwellian distribution at the exobase. Finally, for convenience for other authors dealing with the evolution of Mars, if we express the escape fluxes integrated over the whole Mars Area (at ground level), and convert it into the equivalent of liquid water we find an average escape rate of 7 ± 3 × 10−4 precipitable µm of liquid water/year corresponds to the total disappearance of water in the atmosphere in ∼13,000 years, if it were not replenished from polar caps sublimation. This escape rate is in good agreement with the escape rate deduced from Mariner’s missions (Anderson and Hord, 1971).

second component at a much higher temperature is suggested by the discrepancy between the relatively larger temperature found with one population than the previous measurements of the exospheric temperature below 400 km. In this case, the density and temperature at the exobase (Table 4) for the two populations are respectively nc,exo = 1.0 ± 0.2 × 105 cm−3 and Tc,exo = 200 K and nh,exo = 1.9 ± 0.5 × 104 cm−3 and Th,exo > 500 K at SZA ∼ 30◦ , and nc,exo between [1.6–2.2] × 105 cm−3 and Tc,exo = 200 K and nh,exo = 1.2 ± 0.5 × 104 cm−3 and Th,exo > 500 K at SZA ∼ 90◦ . The cold component is denser near the terminator than at SZA = 30◦ , while the hot component is denser at SZA = 30◦ than at the terminator. Though the reactions between CO+ 2 and H2 , and dissociative recombination of HCO+ , certainly contribute to produce a hot population, they are apparently not sufficient to account for the hot population observed in the exosphere of Mars. Possible alternatives might be found in the influence of the solar wind on the atmosphere/ionosphere/exosphere system, but the origin of the hot population is not yet clearly understood. This paper is the first paper dedicated to the study of the Lyman-α observations done by SPICAM on Mars Express. More observations are needed to clearly resolve the problem of the variations of the martian exobase temperature with the solar cycle and the possible role of a hot population. Observations with a higher range of altitude variations and at other SZA have been scheduled by SPICAM on Mars Express and HST. Another possibility comes from the observation of the Lymanβ line by UV spectrometer on ROSETTA (Stern et al., 2007). This line is optically thin and can provide a more accurate estimate of the temperature. Acknowledgments Mars Express is a space mission from European Space Agency (ESA). We wish to express our gratitude to all ESA members who participated in this successful mission. We would like to thank E. Dimarellis, A. Reberac and J.-F. Daloze for help in planning the observations and data processing, and R. Lallement and S. Ferron for the values of the Lyman-α sky background estimated with their model. We thank our collaborators at the three institutes for the design and fabrication of the instrument (Service d’Aeronomie/France, BIRA/Belgium, and IKI/Moscow), and in particular Emiel Vanraansbeck at BIRA for careful mechanical design and fabrication. We wish to thank CNRS and CNES for financing SPICAM in France.

8. Summary and conclusions The first observations of the Lyman-α emission by SPICAM on Mars Express at altitudes in the range 400–4000 km are presented for two different solar zenith angles (SZA = 30◦ and 90◦ ). A reasonable solution within the uncertainties of the observations and of our modeling is found for an exobase density equal to 1–4 × 105 cm−3 and an exobase temperature between 200 and 250 K at SZA = 30◦ and 90◦ , in good agreement with Krasnopolsky (2002). A second solution corresponding to a martian exosphere composed of two populations: one cold component at the thermal exobase temperature of 200 K and a

Appendix A The CO2 atmosphere is defined by resolving the hydrostatic equation n(r) = n(r0 )e

−

r

dr  r0 Hn (r  )

,

where Hn is the density scale height given by     1 1 dT 1 dT GMm 1 + + = = . Hn (r) HP T dr kT (r)r 2 T dr

(A.1)

(A.2)

The martian hydrogen corona

r0 = 80 km and the CO2 density at 80 km is set as being equal to 2.6 × 1013 cm−3 (Krasnopolsky, 2002). The temperature profile is given by the analytic expression used by Krasnopolsky (2002) T (r) = T∞ − (T∞ − 125)e

2

− (r−90) 11.4T∞

.

(A.3)

k is the Boltzmann constant, M is the Mars’ mass, m is the CO2 mass, r is the distance from the center of the planet, T∞ a reference temperature and HP the scale height of the atmospheric pressure. The hydrogen density profiles, solution of the diffusive equation, is (e.g., Hunten, 1973) 

dnH 1 + αT dT 1 + − DH ΦH (r) = −(DH + K) dr HH T dr

 1 1 dT +K (A.4) + nH . Hn T dr DH is the diffusion coefficient of H into CO2 given by DH (r) =

AT (r)s , nCO2 (r)

(A.5)

where A = 8.4 × 1017 cm2 s−1 and s = 0.6 (Hunten, 1973). K(r) is given by the analytic expression from Krasnopolsky (2002), αT is the thermal coefficient factor equal to −0.25 for H (Krasnopolsky, 2002). In this model we neglect loss and hydrogen production. The escape flux at a given altitude r is also given by 2 re ΦH (re ), ΦH (r) = (A.6) r where re is a reference distance from the center of Mars, here we choose the exobase as a reference. In our model we assume that the escape flux ΦH (re ) is equal to the Jeans escape flux: ΦH (re ) = ΦJeans = u(Te )nH (re ),

(A.7)

(A.8)

where nH (re ) is the hydrogen density at a referential altitude (here we choose the exobase at 200 km) and a(r) and b(r) can be expressed following:  r f1 (r  ) dr  ,

a(r) = exp − 

re r

b(r) = exp − re

f1 (r  ) dr 

r ×



 r 

f2 (r ) exp re



f1 (r ) dr



 

dr ,

(A.10)

re

where f1 (r) and f2 (r) are respectively given by 

1 1 + αT dT 1 f1 (r) = + DH DH + K HH T dr

 1 1 dT +K + , Hn T dr 2 1 re . f2 (r) = DH + K r

(A.11) (A.12)

The two parameters n(re ) and T (re ) are kept free. The H density above the exobase is determined using a spherical model based on the classical evaporative theory (Chamberlain, 1963). This method has been used to calculate the density at 250 km in the frame of Mariner’s 6 and 7 Lyman-α emission analysis (Anderson and Hord, 1971). We set re = 200 km and do not consider satellite particles. The complete H density profile from 80 km to 50,000 km is therefore constrained by the knowledge of two parameters: the temperature and the hydrogen density at the exobase. Appendix B To solve Eq. (7), we use an iterative approach where the zero order solution is the first term (Bertaux, 1974; Quémerais and Bertaux, 1993). In this approach Eq. (7) is discretized and represented by the vectorial relation S = S 0 + A · S,

(B.1)

where A is the influence matrix whose coefficients are given by (Quémerais and Bertaux, 1993)  dΩij σ 0 Ni G(τij )e−τCO2 ,ij dsj . aij = (B.2) 4π Ωij

where u(Te ) is the effusion velocity and nH (re ) the hydrogen density at the exobase. The real H escape flux has been estimated as being up to 30% larger than the Jeans escape in the case of the Earth (Brinkmann, 1970). However, because such a correction is not well known for Mars, we do not introduce such a correction in our calculation. The solution of Eq. (A.7) can be written under the form: nH (r) = a(r)nH (re ) − b(r)ΦH (re ),

609

(A.9)

The dimensionless coefficient aij represents the fraction of the emissivity due to multiple scattering at point Pi which contributes to the total emissivity at point Pj . dΩij is the solid angle under which a point Pi see Pj . Because of the axisymmetry, each point is defined by a couple (r, θ ) in our 2D grid represents a torus in the 3D space. Then we should scan the whole space to determine the lines starting from Pi that intersects the torus corresponding to Pj to estimate aij coefficient (Bertaux, 1974; Quémerais and Bertaux, 1993). The S vector is calculated by an iterative procedure: S n+1 = S 0 + A · S n ,

(B.3)

while the following condition is checked,

S n+1 − S n

 ε = 1 × 10−5 ,

Sn+1

(B.4)

where the norm used for a vector is corresponding to the maximal value of a vector.

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Appendix C

Fig. C1. Comparison between SPICAM data (small squares) and theoretical hydrogen Lyman-α intensity profiles for the orbit 1541 (SZA = 30◦ ) at different exobase temperatures. There is one panel for different exobase temperature. On each panel are displayed the intensity profiles corresponding to exobase density equal to 1 × 104 , 2 × 104 , 3 × 104 , 5 × 104 , 1 × 105 , 2 × 105 , 3 × 105 cm−3 at 200 km.

Fig. C2. Same figure as Fig. C1 but for the orbit 1514 (SZA = 90◦ ).

The martian hydrogen corona

611

Fig. C3. Comparison between SPICAM data (small squares) and theoretical hydrogen intensity profiles for the orbit 1541 (SZA = 30◦ ). For a model with two components, the cold component is defined by exobase conditions: nc,exo = 1 × 105 cm−3 and Tc,exo = 200 K, while the hot temperature T varies from 500 to 1000 K. For each T , intensity profiles corresponding to hot hydrogen density at the exobase equal to 0.4 × 104 , 0.8 × 104 , 1.2 × 104 , 1.6 × 104 , 2 × 104 , 2.4 × 104 , 2.8 × 104 cm−3 are displayed.

Fig. C4. Same figure as Fig. C3 but for the orbit 1514 (SZA = 90◦ ), the cold population is defined here by nc,exo = 2 × 105 cm−3 and Tc,exo = 200 K.

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