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Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts, USA
Department of Physics, University of Connecticut, Storrs, Connecticut, USA
Corresponding author: V. Kharchenko, Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics, 60 Garden St., MS 14, Cambridge, MA 02134, USA. (email@example.com)
 We present a detailed theoretical analysis of non-thermal escape of molecular hydrogen from Mars induced by collisions with hot atomic oxygen from the Martian corona. To accurately describe the energy transfer in O + H2(v, j) collisions, we performed extensive quantum-mechanical calculations of state-to-state elastic, inelastic, and reactive cross sections. The escape flux of H2molecules was evaluated using a simplified 1D column model of the Martian atmosphere with realistic densities of atmospheric gases and hot oxygen production rates for low solar activity conditions. An average intensity of the non-thermal escape flux of H2 of 1.9 × 105 cm−2s−1 was obtained considering energetic O atoms produced in dissociative recombinations of O2+ions. Predicted ro-vibrational distribution of the escaping H2was found to contain a significant fraction of higher rotational states. While the non-thermal escape rate was found to be lower than Jeans rate for H2molecules, the non-thermal escape rates of HD and D2 are significantly higher than their respective Jeans rates. The accurate evaluation of the collisional escape flux of H2and its isotopes is important for understanding non-thermal escape of molecules from Mars, as well as for the formation of hot H2 Martian corona. The described molecular ejection mechanism is general and expected to contribute to atmospheric escape of H2 and other light molecules from planets, satellites, and exoplanetary bodies.
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 The interaction of the Martian atmosphere with the solar radiation and interplanetary plasma results in its evaporation due to thermal (Jeans) escape and a number of non-thermal mechanisms. The absence of an intrinsic magnetic field on Mars and its low gravitational potential make the Martian atmosphere particularly susceptible to erosion [Acuña et al., 1998]. Moreover, the efficient escape of the Martian atmosphere to space is thought to be partly responsible for the current low atmospheric pressure [Chassefière and Leblanc, 2004]. The present day degradation of the Martian atmosphere occurs mainly via non-thermal escape processes induced by ion charge-exchange, sputtering and ionospheric outflows driven by solar wind [Chassefière and Leblanc, 2004; Johnson et al., 2008]. The dissociative recombination (DR) of O2+ is a major source of hot O atoms in the upper atmosphere of Mars, responsible for the escape of oxygen and formation of a hot Martian corona [Ip, 1988; Fox, 1993; Krest'yanikova and Shematovich, 2005].
 Nascent hot O atoms collide with thermal constituents of the Martian atmosphere and eject them from the planetary gravitational field, if a sufficient kinetic energy transfer occurs. Suprathermal neutral oxygen was shown to be important for analyses of Mars' corona and the non-thermal escape of neutral atoms [Krest'yanikova and Shematovich, 2005, 2006; Fox and Hać, 2009]. Recent calculations of the non-thermal He escape from Mars carried out with accurate energy transfer parameters predicted a significant He escape flux induced by hot O atoms [Bovino et al., 2011].
 In this Letter we explore collisional ejection of molecules from the Martian atmosphere. Specifically, we report the results of a quantum-mechanical study of the energy transfer from the hot O to H2 molecules and their subsequent escape. Significant computational difficulties arise from the fact that molecular internal rotational and vibrational degrees of freedom can be excited in collisions. In addition, the reactive pathway leading to the production of OH molecules is energetically permitted. To account for the increased complexity, we have computed the cross sections for O(3P) + H2reactive collision using fully quantum-mechanical approach. Kinetic theory was used to calculate the rate of energy transfer, as well as distributions of excited rotational and vibrational (RV) states of the recoiled H2 molecules. The total escape flux of H2from Mars and RV distributions of escaping molecules have been evaluated for the low solar activity conditions. In addition, we have estimated the non-thermal escape fluxes of HD and D2 and compared them to the corresponding Jeans escape rates. Finally, the dependence of the escape flux on gravitational threshold and molecular mass is illustrated on a case of two hypothetical planets. Future investigations could also include direct collisional ejection of molecules from planetary satellites.
2. Cross Sections and Energy Transfer
 The DR of O2+ with electrons proceeds via five possible dissociation pathways, producing O(3P), O(1D), and O(1S) [Guberman, 1988; Fox and Hać, 2009]. Energetic metastable O(1D) atoms decay via spontaneous emission and quenching in collisions with atmospheric gases into O(3P) atoms [Kharchenko et al., 2005]. The cross sections for O(3P) and O(1D) colliding with He were found to be very similar [Bovino et al., 2011]. For simplicity, we assumed a similar behavior for O(3P) + H2 and O(1D) + H2 elastic collisions.
 To describe the collision ejection of H2molecules, we have calculated elastic and inelastic cross sections for the center-of-mass (CM) collision energies from 0.01 to 4.5 eV. The quantum scattering code ABC [Skouteris et al., 2000], that can treat elastic, inelastic, and open reactive channels of the OH production, O(3P) + H2(v,j) → OH(v″,j″) + H, where (v, j) and (v″, j″) indicate initial and final RV levels of H2and OH, was used to solve the time-independent coupled-channel Schrödinger equation in Delves hyperspherical coordinates. In addition, for high collision energies, the elastic and inelastic cross sections were calculated using the MOLSCAT [Hutson and Green, 1994] code to ensure the convergence of the nonreactive channels. (A detailed description of the scattering calculations and resulting cross sections for the two lowest potential energy surfaces will be published elsewhere.) Extensive numerical convergence tests were carried out for the both codes.
 The O(3P) + H2(v, j) interaction was described using two lowest potential energy surfaces, Rogers' LEPS 3A″ [Rogers et al., 2000] and Brandão's BMS1 3A′ [Brandaõ et al., 2004]. Partial cross sections for initial and final rotational levels j and j′ were constructed as a statistically weighted sum of the independently calculated cross sections for the two potential surfaces, where both 3A″ and 3A′ contribute a weight factor of 1/3 [Balakrishnan, 2004]. Elastic, inelastic, and momentum transfer partial cross sections for oxygen colliding with the hydrogen molecule in three energetically lowest rotational states are given in Figure 1. We compared our reactive cross sections for OH production to the previously published results [Balakrishnan, 2003, 2004; Braunstein et al., 2004; Wu, 2010] and found them to be in close agreement within the available energy range.
 To determine the energy transfer rate from the suprathermal oxygen to atmospheric H2 and find its escape rate, we used kinetic theory with a quantum description of internal molecular structure and realistic anisotropic cross sections. Since the reactive cross sections are an order of magnitude smaller than the elastic cross sections (Figure 1), and the more massive OH molecule has a considerably higher escape threshold than H2, we neglected it in this study. However, note that a small fraction of produced OH molecules may be sufficiently energetic to escape. The transferred energy from the energetic projectile O to the frozen target H2 in the laboratory frame (LF) can be expressed as [Johnson, 1982]
where mO and are masses of O and H2, respectively, E is the collision energy in the LF, γv′,j′ = ϵv′j′/ϵ, is the ratio of ϵv′j′ and ϵ, the CM translational kinetic energies after and before the collision, respectively, and θ is the scattering angle in the CM frame. The energies ϵv′j′ were calculated quantum mechanically for the two triplet potential surfaces. Equation (1) takes into account that the energy transferred to H2 molecules is spent on increasing their translational kinetic energy and exciting their internal RV degrees of freedom.
 The fraction of energized H2 molecules capable of escaping can be calculated as
where Qvj,v′j′(θ) is the differential cross section for scattering of H2 in the initial (v, j) into the final (v′, j′) state. The critical angle θmin was determined from the condition that the translational part of the transferred energy Tv′,j′ is equal to the minimum energy required for H2 to escape from Mars, Eesc = 0.26 eV. An alternative description of the escape process could be constructed by performing Monte Carlo simulations with accurate quantum cross sections for angular distributions of the recoiled H2 molecules.
 Momentum transfer cross sections σvj, v′j′mt for inelastic collisions were calculated using [Parker and Pack, 1978]
3. Flux and Distribution of Escaping H2
 Jeans escape and collisions with hot oxygen are two major mechanisms that contribute to the escape of neutral H2 molecules and their isotopes from the Martian atmosphere. Both processes are dependent on the temperature and density of upper layers of the Martian atmosphere, although Jeans escape rate exhibits significantly stronger temperature dependence. The temperature of the exosphere (above the altitude of about 180 km), Texo, is approximately constant and estimated to be between 240 and 280 K, depending on the solar activity and gas density profiles [Krasnopolsky, 2010; Fox and Hać, 2009; Fox, 2003]. To obtain a conservative estimate of the non-thermal flux of escaping H2, we considered the Texo = 240 K, corresponding to the low solar activity. Furthermore, we assumed a thermal distribution of the initial rotational states of H2, where more than 95% of the total population is distributed between its first three rotational states, j = 0, 1, 2, with corresponding population fractions equal to 0.31, 0.46, and 0.19, respectively. Using equations (2) and (3) we have calculated the values of the thermally averaged momentum transfer cross sections and the fractions Γvjv′j′(E) of ro-vibrationally excited H2, sufficiently energetic to escape from Mars (Figure 2). The fraction Γvjv′j′(E) becomes significant at collision energies greater than 0.7 eV for H2(j′ = 0, 2). Although higher rotational states require increasingly larger projectile energies, e.g., H2(j′ = 16) can escape for E > 1.65 eV, their fraction in the RV distribution of the escaping molecules also becomes larger. Note that the initial population of higher vibrational levels of H2 at the exobase is negligible.
 Since Γvjv′j′(E) depends only on the energy transfer efficiency and the escape energy threshold, it can be easily generalized to different astronomical objects. We illustrate this for two hypothetic planets, the first corresponding in size and mass to Earth and the second to the extrasolar super-earth Kepler-10b (4.56 Earth masses,Texo = 1800 K) [Batalha et al., 2011]. Note that only very energetic H2, mostly in higher excited rotational levels, is able to escape (Figure 2).
 A simple estimate of the total escape flux of H2(v′, j′) can be obtained from the exobase approximation [Fox, 2003; Chassefière and Leblanc, 2004; Krasnopolsky, 2010], using the density of exospheric H2, and fractions Γvjv′j′(E) calculated above. However, such an approach neglects the hot O production below the exobase and loss of the upward flux in atmospheric collisions, resulting in a large uncertainty in the computed flux. We constructed a more realistic 1D model of escape, analogous to the one used to describe the escape of He atoms [Bovino et al., 2011]. In our model the explicit consideration of energy transfer collisions is combined with the altitude-dependent rate of production of hot O atoms,f(E, h) via DR channels [Guberman, 1988; Petrignani et al., 2005; Bovino et al., 2011]. In addition, we estimated the extinction of fluxes of suprathermal O and H2 due to collisions with thermal atmospheric gases. All calculations were performed for low solar activity. We used the rate of production of hot O below 400 km by Fox and Hać  and smoothly interpolated it to the rate given by Krasnopolsky and Gladstone  at higher altitudes.
 The volume production rate of escaping hot H2(v′, j′) can be expressed as
with the transparency factors and TO defined as
The transparency factor is equal to the escape probability of hot H2(v′, j′) produced in collisions with the incident hot O of energy E at the altitude h. The second transparency factor TO is defined as the probability that the hot O atoms, produced at the altitude h2, reach the altitude h without the energy loss in collisions with other atmospheric constituents. Here, h2min and hmax are practical integration limits. The quantity is the inverse mean free path for O + H2(v, j) collisions, resulting in the energy transfer greater than the H2 escape threshold. The prefactor 1/2 indicates that, in our simplified 1D model, approximately half of the nascent energetic atoms and recoiled H2 molecules are scattered towards the planet and cannot escape regardless of the energy transferred. Summations of the flux loss of H2 and O in collisions with the i-th atmospheric gas of densityni(h) and momentum transfer cross section and σO,imt(E), respectively, included major constituents of the Martian upper atmosphere: CO2, CO, N2, O2, H2, H, Ar, and He. The momentum transfer cross sections for H-H2 [Krstic and Schultz, 1999], Ar-H2 [Uudus et al., 2005], H2-H2 [Phelps, 1990] were used from the literature. Since no data were available in the required energy range, we used approximate mass-scaled cross sections for He-H2(from Ar-H2), N2-H2, O2-H2, CO-H2(from O-H2), and CO2-H2(from O-N2 [Balakrishnan et al., 1998]).
 Using equation (4) we have calculated volume production rate of the escaping H2(v′ = 0, j′) molecules induced in H2(v = 0, j = 0 − 2) + O collisions for a range of altitudes from hmin = 130 km to hmax = 800 km (Figure 3). Note that, by symmetry arguments, for a homonuclear H2 only Δj = 0, 2, 4..transitions are allowed [Cohen-Tannoudji et al., 1986]. The resulting escape rates of H2 molecules are the largest for the elastic collisions, followed by the three times smaller rates for the first two excited rotational states, j′ = 2 and j′ = 4. The rates remain significant for the final rotational levels up to j′ = 10. The altitude profile of the production rate of H2 capable of escaping is similar to the production rate profile of He [Bovino et al., 2011]. This was expected, since the escape of both species is driven by collisions with the nascent fast O atoms, produced mostly below 150 km for the considered atmospheric and solar conditions. The calculated altitude profile can be used to compute the non-thermal escape flux of H2 molecules. We calculated ϕj′, the non-thermal flux for H2(v′ = 0, j′) molecules as
Total collisional and thermal fluxes were calculated as sums over all rotational levels and found to be 1.9 × 105 cm−2 s−1 and 1.1 × 106 cm−2 s−1, respectively. A comparison of Jeans and non-thermal rates of escape of H2(v′ = 0, j′) molecules, escaping in different rotational states j′ from Martian dayside, is given in Figure 4. To simplify the calculation we assumed the average solar conditions and neglected the latitude dependence of the production rate of hot O atoms. Jeans rate is about eight times greater than the non-thermal rate of the escaping H2 for the lowest three rotational states, while for j′ > 3 the latter starts to dominate. The distinct character of the two RV distributions is a clear signature of different physical escape mechanisms. While, in case of H2, the thermal rate is almost an order of magnitude higher than the collisionally-induced rate of escape, relative importance of the two processes changes for heavier isotopologues, namely HD and D2 (Table 1).
Table 1. Total Collisionally-Induced Escape Rates of H2, HD, and D2 From the Martian Atmosphere
Jeans escape rate (s−1)
1.1 × 106
3.3 × 10−6
Non-thermal escape rate (s−1)
1.9 × 105
 We have evaluated the rate of production of hot H2molecules via collisions with energetic O atoms in the upper Martian atmosphere and computed the absolute value of collisionally-induced outflow of H2molecules and their isotopologues. Namely, the escape rate of molecular hydrogen induced by collisions with hot oxygen from the Martian atmosphere was calculated and found to be about six times smaller than the corresponding Jeans escape rate for the low solar activity. Since a detailed evaluation would require calculations of the non-thermal escape rates over one or more solar cycles, as well as taking into account related temperature and density variations of the Martian upper atmosphere, we allow a factor of two uncertainty of this ratio. For heavier molecules, the collisional escape will dominate over thermal, as we have illustrated in case of HD and D2 isotope molecules. In fact, the described process of molecular ejection induced by collisions may be one of the most important escape mechanisms of HD and D2 from Mars. The calculated escape fluxes of H2 isotopologues are small in comparison with the atomic H and D escape rates. Nevertheless, they could be important for formation of hot H2 corona and its isotopic composition.
 The described mechanism of molecular escape, where collisions provides sufficient translational energy to exceed the escape threshold and simultaneously excite internal molecular degrees of freedom, is rather general. It could be used to evaluate non-thermal escape fluxes and RV distributions of heavier molecules, such as CO, N2, or CH4, from Mars, Solar system bodies, and exoplanets. For the escape flux induced by O atoms produced in DR the upper limit on the mass of the escaping molecule is about 30 uon Mars. Similarly, we estimate that the non-thermal escape of H2from a planetary atmosphere is possible for planetary masses up to about 3.4 Earth masses. A number of solar system bodies as well as the lightest currently confirmed exoplanets belong in that mass range. These limits do not include other non-thermal sources of hot atoms.
 The escaping H2 molecules exhibit a characteristic internal energy distribution, with a significant fraction of populated higher rotational states. Since H2 molecules do not have a permanent electric dipole moment, they decay to the ground state mainly via collisions with the background gases present in an extended planetary corona. This is true for all escaping molecules without a permanent dipole moment. Molecules that have a permanent electric dipole moment, such as CO, radiate characteristic emission and can be detected directly. The presence of hot and rotationally or vibrationally excited H2 molecules in the upper atmosphere may be detected directly in specific absorption and solar radiation scattering spectra, if the column density of hot molecules is sufficiently large. In addition, it may be possible to indirectly detect the presence of H2, or other rotationally excited light molecules, in the extended Martian corona by performing a spectroscopic analysis of the emission of nascent OH(v, j) produced in reactive collisions between hot H2 and coronal O atoms.
 Finally, a significant amount of ro-vibrationally excited H2molecules remain in the Martian atmosphere after colliding with hot O atoms. The cross sections and energy transfer parameters presented in this study can be used to determine non-thermal translational and ro-vibrational distributions of hot H2 gas in the upper atmosphere of Mars.
 We are grateful to D. Wang, A. Kuppermann, and J. Brandão for providing Fortran subroutines for constructing potential energy surfaces, and to N. Lewkow for reading and suggestions. M.G. and V.K. were supported by NASA grants NNX09AF13G and NNX10AB88G.
 The Editor thanks two anonymous reviewers for assisting with the evaluation this paper.