Distribution and escape of molecular hydrogen in Titan's thermosphere and exosphere

This is a commentary on DOI:10.1029/2007JE00299710.1029/2007JE00303110.1029/2007JE00303310.1029/2008JE003135
Abstract
[1] We present an indepth study of the distribution and escape of molecular hydrogen (H_{2}) on Titan, based on the global average H_{2} distribution at altitudes between 1000 and 6000 km, extracted from a large sample of Cassini/Ion and Neutral Mass Spectrometer (INMS) measurements. Below Titan's exobase, the observed H_{2} distribution can be described by an isothermal diffusion model, with a most probable flux of (1.37 ± 0.01) × 10^{10} cm^{−2} s^{−1}, referred to the surface. This is a factor of ∼3 higher than the Jeans flux of 4.5 × 10^{9} cm^{−2} s^{−1}, corresponding to a temperature of 152.5 ± 1.7 K, derived from the background N_{2} distribution. The H_{2} distribution in Titan's exosphere is modeled with a collisionless approach, with a most probable exobase temperature of 151.2 ± 2.2 K. Kinetic model calculations in the 13moment approximation indicate a modest temperature decrement of several kelvin for H_{2}, as a consequence of the local energy balance between heating/cooling through thermal conduction, viscosity, neutral collision, and adiabatic outflow. The variation of the total energy flux defines an exobase level of ∼1600 km, where the perturbation of the Maxwellian velocity distribution function, driven primarily by the heat flow, becomes strong enough to raise the H_{2} escape flux considerably higher than the Jeans value. Nonthermal processes may not be required to interpret the H_{2} escape on Titan. In a more general context, we suggest that the widely used Jeans formula may significantly underestimate the actual thermal escape flux and that a gas kinetic model in the 13moment approximation provides a better description of thermal escape in planetary atmospheres.
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1. Introduction
[2] Titan has a thick and extended atmosphere, which consists of over 95% molecular nitrogen (N_{2}), with methane (CH_{4}), molecular hydrogen (H_{2}) and other minor species making up the rest. The first in situ measurements of the densities of various species in Titan's upper atmosphere have been made by the Ion and Neutral Mass Spectrometer (INMS) on the Cassini orbiter during its close Titan flybys [Waite et al., 2004]. In this paper we present our analysis of the vertical distribution of H_{2} throughout Titan's thermosphere and exosphere, in a global average sense.
[3] In Titan's collisiondominated thermosphere, the production of hydrogen neutrals is significant below an altitude of ∼1000 km, mainly through the photodissociation of CH_{4} and C_{2}H_{2} [Lebonnois et al., 2003]. Between this level and Titan's exobase at an altitude of ∼1500 km (defined traditionally as the level where the scale height of the atmospheric gas is equal to the mean free path of neutral collisions), the distribution of H_{2} is usually described by a diffusion model [Bertaux and Kockarts, 1983; Yelle et al., 2006]. In Titan's exosphere, the collisions between constituents are so rare that the problem becomes essentially one within the domain of the kinetic theory of freestreaming particles under the influence of Titan's gravity [Fahr and Shizgal, 1983]. The traditional exospheric model is based on a simple collisionless approach first proposed by Öpik and Singer [1961] and Chamberlain [1963], in an attempt to investigate the structure of the terrestrial exosphere. In such a model, the velocity distribution above the exobase is assumed to be a truncated Maxwellian, and particle densities can be directly calculated by integrating over the appropriate regions of the momentum space. Other choices of the velocity distribution at the exobase have also been investigated, such as the analytic power law and the κ distribution [e.g., De La Haye et al., 2007].
[4] The escape of H_{2} on Titan has been suggested to be mostly thermal and limited by diffusion [Hunten, 1973; Bertaux and Kockarts, 1983]. Conventionally, the thermal escape flux in planetary atmospheres is given by the Jeans formula. However, Yelle et al. [2006] observed an escape flux significantly larger than the Jeans value, through an analysis of the H_{2} distribution in Titan's thermosphere. The Yelle et al. analysis is based on the data acquired during the first close encounter of Cassini with Titan (known in project parlance as TA). An updated investigation of the H_{2} escape is necessary with the current more extensive Cassini/INMS data set.
[5] In this paper, we present our analysis of the H_{2} distribution in Titan's thermosphere and exosphere. The global average H_{2} density profile is obtained from a sample of INMS in situ measurements based on 14 lowaltitude encounters of Cassini with Titan. The horizontal/diurnal variations of the H_{2} distribution based on the same sample will be investigated in followup studies.
[6] The structure of this paper is as follows. We introduce the basic observations and describe data reductions in section 2. Simple onedimensional models are presented in section 3, to describe the H_{2} distribution at altitudes between 1000 and 6000 km. Regions below and above Titan's exobase are treated separately with different approaches. In section 4, we describe possible modifications of the simple exospheric model, considering in detail loss processes for H_{2} and the energy balance in the transition region between the thermosphere and exosphere. Section 5 is devoted to understanding the escape of H_{2} on Titan, in which we adopt a nonMaxwellian velocity distribution in the 13moment approximation to calculate the thermal escape flux. Finally, discussion and conclusions are given in section 6.
2. Observations and Data Reductions
2.1. Observations
[7] Our investigation of Titan's thermosphere and exosphere relies exclusively on the observations made in the closed source neutral (CSN) mode, which is specifically designed to optimize interpretation of neutral species detected in the atmosphere of Titan or other INMS targets [Waite et al., 2004]. In this mode, the inflowing gas particles enter the orifice of a spherical antechamber and thermally accommodate to the wall temperature before reaching the ionization region, the switching lens and the quadrupole mass analyzer. An enhancement in the signaltonoise ratio of the sampled neutral species is accomplished by limiting the conductance from the antechamber to the ionization region, while maintaining high conductance through the entrance aperture [Waite et al., 2004]. The geometric field of view of the CSN mode is as wide as 2π sr, and the angular response varies as the cosine of the angle between the INMS axis and the spacecraft velocity [Waite et al., 2004].
[8] The INMS data consist of a sequence of counts in mass channels 1–8 and 12–99 amu. The H_{2} densities in the ambient atmosphere are determined from counts in channel 2, which are typically sampled with a time resolution of 0.9 s, corresponding to a spatial resolution of 5.4 km along the spacecraft trajectory. In the 4 year length of the prime Cassini mission, there will be 44 close Titan flybys. Our work is based on 14 of them, known in project parlance as T5, T16, T17, T18, T19, T21, T23, T25, T26, T27, T28, T29, T30, and T32. The details of the encounter geometry for these flybys at closest approach (C/A) are summarized in Table 1, which shows that our sample preferentially selects measurements made over Titan's northern hemisphere, during nighttime, and at solar minimum conditions [see also MüllerWodarg et al., 2008].
Flyby  Date  Alt (km)  LST (h:min)  SZA (deg)  Latitude (deg)  Longitude (deg)  F10.7 () 

T5  16 Apr 2005  1027  23:17  128  74  89  83 
T16  22 Jul 2006  950  17:21  105  85  45  72 
T17  7 Sep 2006  1000  10:30  44  23  −56  87 
T18  23 Sep 2006  962  14:25  90  71  3.0  70 
T19  9 Oct 2006  980  14:20  81  61  2.6  75 
T21  12 Dec 2006  1000  20:20  125  43  95  102 
T23  13 Jan 2007  1000  14:02  53  31  2.1  81 
T25  22 Feb 2007  1000  00:35  161  30  −16  76 
T26  10 Mar 2007  981  01:45  150  32  2.1  71 
T27  25 Mar 2007  1010  01:43  144  41  2.1  74 
T28  10 Apr 2007  991  01:40  137  50  2.0  69 
T29  26 Apr 2007  981  01:36  130  59  1.6  81 
T30  12 May 2007  960  01:32  122  69  1.2  71 
T32  13 Jun 2007  965  01:18  107  85  −1.2  71 
2.2. Extraction of the H_{2} Density Profile
[9] The counts in mass channel 2 are mainly contributed by H_{2} molecules, with minor contributions from hydrocarbon species (CH_{4}, C_{2}H_{2}, etc.) ignored. Only inbound data are included in our analysis. This is because the INMS chamber walls have a certain probability to adsorb molecules entering the instrument orifice, which may further undergo complicated wall chemistry before being released with a time delay [Vuitton et al., 2008]. Such a wall effect mainly takes place near C/A when the density in the ambient atmosphere is high, and primarily contaminates the outbound counts. In Figure 1, we compare the inbound and outbound H_{2} density profiles averaged over all flybys in our sample (see below for the details on extracting the H_{2} density profile from the raw measurements). The outbound H_{2} densities are systematically higher than the inbound densities, with the deviation increasing at high altitudes. This is an indication of the importance of wall effect since other effects such as horizontal/diurnal variations should be smoothed by averaging. A more detailed discussion of the wall effects, including both the simple adsorption/desorption processes and the more complicated heterogeneous surface chemistry on the chamber walls, will be presented elsewhere (J. Cui et al., Analysis of Titan's neutral upper atmosphere from Cassini Ion Neutral Mass Spectrometer measurements, manuscript in preparation, 2008).
[10] Corrections for thruster firings are required before the counts in channel 2 can be converted to H_{2} densities [Yelle et al., 2006]. The Cassini spacecraft thrusters operate with hydrazine (N_{2}H_{4}), with H_{2} being a significant component of the thruster effluent. This effect is usually viewed as large excursions from the expected counts in channel 2, interspersed in the altitude range over which measurements were made. In most cases, contamination by thruster firings is serious near C/A when thrusters fire frequently to offset the torque on the spacecraft due to atmospheric drag [Yelle et al., 2006]. Regions thought to be contaminated by thruster firings are identified by correlating with accumulated thruster operation time, accompanied by eyeball checking [Yelle et al., 2006]. These regions are rejected from our analysis below.
[11] The counts in channel 2 tend to a constant level at very high altitudes above ∼8000 km as a result of residual H_{2} gas present in the INMS chamber. This effect causes significant overestimates of the H_{2} densities of the ambient atmosphere at high altitudes if not properly removed. For each flyby, we use the inbound INMS data above an altitude of 10,000 km to evaluate the mean background signal in channel 2, assuming it is constant for any individual flyby. For the inbound pass of T25, the INMS measurements do not extend to altitudes above ∼6000 km, and the background count rate averaged over all the other flybys is adopted. The mean background count rate varies from flyby to flyby, ranging between 70 and 240 counts s^{−1}.
[12] With thruster firings removed and background subtracted, counts in channel 2 are converted to H_{2} number densities with a preflight laboratory calibration sensitivity of 3.526 × 10^{−4} counts (cm^{−3} s)^{−1} [Waite et al., 2004]. Since the preflight sensitivities were obtained for mixtures of reference gases with their isotopes, it is necessary to separate the contributions from H_{2} and HD. The procedure used to correct for isotopic ratios will be presented elsewhere (Cui et al., manuscript in preparation, 2008).
[13] The conversion of INMS counts to densities of the ambient gas depends on the ram enhancement factor, which is a function of molecular mass, the angle of attack, and the spacecraft velocity relative to Titan [Waite et al., 2004]. The conversion from channel 2 counts, C_{2}, to H_{2} densities, n is
where 3.526 × 10^{−4} counts (cm^{−3} s)^{−1} is the sensitivity, 0.031 s the integration time, and ℜ_{H2} the dimensionless ram enhancement factor. For the passes considered here, the INMS is pointed in the ram direction near C/A, allowing accurate density determination of the ambient gas. To improve the statistical significance of our analysis, we only include measurements with ram enhancement factors ℜ_{H2} ≥ 5, corresponding to ram angles ≤68° for a typical spacecraft velocity of 6 km s^{−1}. Response decreases significantly with increasing ram angle. For ram angles ≥90°, the spacecraft configuration prevents the recording of any useful data of the ambient gas.
[14] To obtain the average H_{2} distribution, the raw inbound measurements are binned by 50 km below an altitude of 2000 km, binned by 100 km between 2000 and 4000 km, and binned by 500 km above 4000 km. Such a profile is shown by the solid circles in Figure 1. Vertical error bars in Figure 1 represent the standard deviation of altitude for each bin. Horizontal error bars reflect uncertainties due to counting statistics, not necessarily associated with any horizontal/diurnal variations. The open circles in Figure 1 correspond to the average outbound H_{2} profile determined in the same manner, which is contaminated by the wall effect (see above).
2.3. N_{2} Density Profile and Barometric Fitting
[15] The determination of an average N_{2} density profile is necessary for deriving the physical conditions of the background component. Here the contamination by thruster firings is unimportant because the N_{2} density of the ambient atmosphere is much higher than that of the spacecraft effluent. For the detailed procedure of determining N_{2} densities from counts in channels 14 and 28, see MüllerWodarg et al. [2008]. The average N_{2} distribution below an altitude of 2000 km is obtained by combining N_{2} profiles from all flybys, binned by 50 km. Both inbound and outbound data of N_{2} are included, since wall effects are not important for this species below 2000 km where the N_{2} molecules in the ambient atmosphere are much more abundant than those formed on the chamber walls through surface chemistry.
[16] Such an average N_{2} profile is shown in Figure 2, with ±1σ uncertainties. Only the portion below 1500 km is presented. The thick solid line gives the best fit barometric relation of the observed N_{2} distribution, with a most probable temperature of 152.5 ± 1.7 K. Also shown in Figure 2 is the average H_{2} distribution in the same altitude range, as well as several model profiles calculated from the diffusion equation (see section 3.1). For comparison, we notice that the analysis of the inbound Voyager 1 UV solar occultation data by Vervack et al. [2004] estimated a temperature of 153 ± 5 K. Yelle et al. [2006] determined a similar temperature of 149 ± 3 K based on the INMS data acquired during the TA flyby. These previous determinations are in agreement with our value of ∼153 K, which is also consistent with the empirical twodimensional temperature distribution given by MüllerWodarg et al. [2008], derived on the basis of a nearly identical INMS sample but with a different approach.
3. Preliminary Models of H_{2} Distribution
3.1. H_{2} Distribution in Titan's Thermosphere
[17] The H_{2} distribution in Titan's collisiondominated thermosphere is usually modeled as a minor species (H_{2}) diffusing through a stationary background gas (N_{2}), following the formulation of Chapman and Cowling [1970]. This is similar to the 5moment approximation to the Boltzmann momentum transport equation [Schunk and Nagy, 2000], with the additional assumption that the two interacting species have the same temperature. Adopting a constant H_{2} temperature of 152.5 K based on the barometric fitting of the N_{2} distribution (see section 2.3), the H_{2} distribution is then described by the diffusion equation as
where R is Titan's radius, k is the Boltzmann constant, m is the mass of H_{2} molecules, g is the local gravity, F_{s} is the H_{2} flux referred to the surface, n is the H_{2} number density, D is the molecular diffusion coefficient for H_{2}−N_{2} gas mixture adopted from Mason and Marrero [1970], and T_{0} = 152.5 K is the N_{2} (also H_{2}) temperature. In equation (2), we have implicitly used the condition of flux continuity, since photochemical production and/or loss of H_{2} is negligible above ∼1000 km [e.g., Lebonnois et al., 2003]. Eddy diffusion is ignored here since the altitude range under consideration is well above Titan's homopause at ∼850 km [Wilson and Atreya, 2004; Yelle et al., 2008]. More specifically, the molecular diffusion coefficient is 2 × 10^{9} cm^{2} s^{−1} at the lower boundary of 1000 km, about a factor of 70 greater than the eddy diffusion coefficient of ∼3 × 10^{7} cm^{2} s^{−1} [Yelle et al., 2008]. We also ignore the higherorder thermal diffusion process in equation (2), since the temperature gradient over the relevant altitude range is small, of order 1 K over ∼500 km (see section 4.1). From the kinetic theory, the ratio of the molecular diffusion term to the thermal diffusion term is ∼5kT_{0}F/q, where q is the heat flux, F = F_{s}(R/r)^{2} is the local H_{2} flux, and the hard sphere approximation is assumed [see Schunk and Nagy, 2000, equation (4.129b)]. Adopting the solution for heat flux presented in section 4.1, the molecular diffusion term is estimated to be a factor of 60 larger than the thermal diffusion term at 1000 km.
[18] We solve equation (2) with a fourthorder RungeKutta algorithm, with the boundary condition of n = 2.7 × 10^{7} cm^{−3} at 1000 km inferred from the INMS data. The variation of gravity with altitude has been taken into account. In equation (2), the H_{2} flux referred to the surface, F_{s}, is treated as a free parameter. On the basis of a χ^{2} goodnessoffit test between 1000 and 1500 km, the most probable value of the H_{2} flux is found to be F_{s} = (1.37 ± 0.01) × 10^{10} cm^{−2} s^{−1}. In Figure 2 we show the model H_{2} distribution calculated from equation (2) with different choices of F_{s}. Also shown in Figure 2 are the INMS data binned by 50 km. The solid line corresponds to the most probable value of F_{s} = 1.37 × 10^{10} cm^{−2} s^{−1}, whereas the dotted line corresponds to the Jeans flux of 4.46 × 10^{9} cm^{−2} s^{−1}, calculated with an exobase temperature of 152.5 K. The most probable H_{2} flux is about a factor of 3 higher than the Jeans value, implying an enhanced escape of H_{2} on Titan. The interpretation of such a large H_{2} escape flux will be discussed in section 5.
[19] Figure 2 shows that the diffusion model, assuming full thermal coupling between H_{2} and N_{2}, provides a reasonable description of the observed H_{2} distribution below ∼1500 km. On the basis of a similar analysis of the Cassini/INMS data acquired during the TA flyby, Yelle et al. [2006] obtained an H_{2} flux of (1.2 ± 0.2) × 10^{10} cm^{−2} s^{−1} (referred to the surface), which is consistent with our result.
3.2. H_{2} Distribution in Titan's Exosphere
[20] To model the H_{2} distribution above Titan's exobase, we adopt a kinetic approach based on the solution of the collisionless Boltzmann equation [Chamberlain and Hunten, 1987]. Following the idea originally conceived by Öpik and Singer [1961] and Chamberlain [1963], any particle in the exosphere naturally falls into one of four categories based on orbital characteristics, i.e., ballistic, satellite, escaping, and incoming hyperbolic particles. At any given point in the exosphere, each of the above types occupies an isolated region in the momentum space. Ballistic and escaping particles intersect the exobase, with velocities either smaller or greater than the escape velocity. These two categories represent particles which are directly injected from the thermosphere. On the other hand, satellite particles have perigees above the exobase, and therefore have a purely exospheric origin. Because in any collisionless model, there is no mechanism to establish a steady population of satellite particles, this category is excluded from our calculations. The incoming hyperbolic particles, which obviously require an external origin, are also excluded.
[21] Assuming a Maxwellian velocity distribution function (VDF) at the exobase, Liouville's theorem implies that the VDF for H_{2} molecules above this level is also Maxwellian, but truncated to include only regions in the momentum space occupied by either ballistic or escaping particles with trajectories intersecting the exobase. The H_{2} densities in the exosphere can be determined by integrating over the Maxwellian VDF within the truncated regions. Analytical results for these integrations are given by Chamberlain [1963]. The model exospheric profile only depends on the density and temperature of H_{2} at the exobase, which are treated as two free parameters in the model fitting. The most probable values of these parameters are found to be n_{exo} = (4.34 ± 0.02) × 10^{5} cm^{−3} and T_{exo} = 151.2 ± 2.2 K, where the exobase is placed at an altitude of 1500 km. Although the definition of the exobase level is itself subject to uncertainty, the results presented in this section are not sensitive to the exact choice, as a result of the large H_{2} scale height (∼1000 km near the exobase). However, in section 5.3 we show that a more realistic exobase height is ∼1600 km, which has important implication on the derived thermal escape flux.
[22] The exospheric H_{2} distribution calculated from such a collisionless model is shown in Figure 3, overplotted on the INMS data. Different profiles correspond to different choices of the exobase temperature, with the solid line representing the most probable value of 151.2 ± 2.2 K, consistent with the N_{2} temperature of 152.5 K within 1σ. The dotted line shows the exospheric H_{2} profile calculated from collisionless Monte Carlo simulations, taking into account Saturn's gravitational influence (see section 5.1 for details).
4. Thermal Effect and Loss Processes of H_{2}
[23] In section 3, we show that the simple collisionless model reasonably describes the observations of H_{2} in Titan's exosphere. Here we investigate two physical mechanisms that may potentially modify the exospheric H_{2} distribution: (1) the thermal disequilibrium between H_{2} and N_{2} caused by escape and (2) the external loss processes of H_{2} above Titan's exobase. We will show below that neither of the mechanisms has a substantial influence on the H_{2} distribution. However, taking into account the actual thermal structure may have important implications in interpreting the observed H_{2} escape, which is discussed in section 5.3.
4.1. Temperature Decrement for H_{2} Near Titan's Exobase
[24] Early observations of the terrestrial exosphere have shown a significant temperature decrement for atomic H, as large as ∼100 K near the exobase [Atreya et al., 1975]. To interpret this, Fahr [1976] has suggested that a correct description of the exospheric model must satisfy energy continuity, in addition to momentum and particle conservation. This condition requires that the energy loss due to particle escape be balanced by an appropriate energy supply through thermal conduction, which is naturally associated with a temperature gradient for the escaping component [Fahr, 1976; Fahr and Weidner, 1977]. For Titan, such a thermal effect implies a temperature difference between the background N_{2} gas at T_{0} and the diffusing H_{2} gas at T < T_{0}. However, this effect should be assessed quantitatively, such that the calculated temperature reduction for H_{2} does not contradict the INMS observations. In section 3, we have already seen that the H_{2} gas is approximately in thermal equilibrium with N_{2}, as indicated by the closeness of their temperatures near the exobase.
[25] To investigate the thermal effect, we adopt a 13moment approximation to the kinetic theory, which has been extensively used in modeling the terrestrial polar wind [Lemaire et al., 2007; Tam et al., 2007]. In such an approximation, the Boltzmann energy transport equation is given by Schunk and Nagy [2000] as
where F = F_{s}(R/r)^{2} is the local H_{2} flux with F_{s} adopted as the most probable value derived from the diffusion equation (R is Titan's radius), m and m_{0} are the molecular masses of H_{2} and N_{2}, and T and T_{0} are their temperatures of which the latter is fixed as 152.5 K. The quantity, ν in equation (3) is the H_{2}−N_{2} neutral collision frequency, which is related to the diffusion coefficient, D, in equation (2) through νD = kT/m [Schunk and Nagy, 2000]. Φ_{E} represents the local energy flux, given as
where c_{p} = (5/2)(k/m) = 1.03 × 10^{8} ergs K^{−1} g^{−1} is the specific heat of H_{2} at constant pressure, G is the gravitational constant, M is Titan's mass, u = F/n is the drift velocity of H_{2}, κ and η are the thermal conductivity and the viscosity coefficient. We adopt κ = 1.1 × 10^{4} ergs cm^{−1} K^{−1} and η = 5.5 × 10^{−5} g cm^{−1} s^{−1} for the appropriate temperature [Rowley et al., 2003]. The first term on the righthand side of equation (4) corresponds to the intrinsic energy flux of H_{2}, with contributions from the internal energy, the bulk kinetic energy and the gravitational energy added together. The other two terms represent the energy transfer through thermal conduction and viscosity. The derivation of equation (4) is provided in the appendix. Equation (3) characterizes the local energy balance of H_{2} on Titan. The equation includes the effect of energy transfer from N_{2} to H_{2} through neutral collisions given by the righthand side (the meanings of the two terms will be addressed below). The effects of thermal conduction, viscosity, as well as adiabatic cooling due to H_{2} outflow are included in the divergence term on the lefthand side of equation (3).
[26] We solve equation (3) for the H_{2} thermal structure at altitudes between 1000 and 2500 km. Boundary conditions have to be specified to complete the problem, including one for T and one for dT/dr. We assume that the H_{2} and N_{2} gases are in thermal equilibrium at the lower boundary, i.e., T = T_{0} = 152.5 K at 1000 km. The boundary condition for the temperature gradient is determined by the requirement of energy continuity at the upper boundary, which can be expressed as
where v_{esc} is the escape velocity at the upper boundary and f(v, θ) is the VDF for H_{2} molecules, which is assumed to be independent of the azimuthal angle but allows for dependence on the polar angle. The simplest scheme is to adopt the drifting Maxwellian distribution. However, the realistic VDF for H_{2} molecules at the upper boundary is not strictly Maxwellian. To correct for this, we adopt the VDF for H_{2} molecules in the 13moment approximation. The appropriate form of such a distribution function will be presented in section 5.3. Here, we emphasize that both the drifting Maxwellian and the 13moment VDF depend on the values of some unknown parameters at the upper boundary (e.g., temperature and drift velocity). This requires that equations (3), (4), and (5) be solved in an iterative manner to ensure selfconsistency.
[27] Figure 4 presents the calculated thermal structure for the diffusing H_{2} component, which predicts an H_{2} temperature of 150.1 K at 1500 km, or a temperature decrement of 2.4 K. This is consistent with the INMS observations, which give a most probable exobase H_{2} temperature of 151.2 K based on the collisionless Chamberlain approach (see section 3.2). However, the predicted thermal effect is too small to get firm supports from the data, since the uncertainties in the temperature determination are considerably larger. A similar calculation in the 13moment approximation has been carried out by Boqueho and Blelly [2005] on various neutral components in the Martian atmosphere, which shows that the thermal structure of relatively light species such as O present a modest temperature decrement of order 1 K near the exobase (see Boqueho and Blelly's Figure 8), comparable to our results.
[28] In Figure 5 we show the relative magnitudes of various terms in equation (3) which represent the energy gain/loss rates associated with heat conduction (solid), viscosity (shortdashed), adiabatic outflow (dotted), as well as H_{2}−N_{2} neutral collisions (longdashed). Heating and cooling terms are shown in the left panel and right panel, respectively. First, we notice that though the background N_{2} gas is warmer than H_{2}, neutral collisions between the two components do not necessarily mean ‘heating’. The energy transfer through collisions is represented by the righthand side of equation (3), which consists of two terms. The first term characterizes the energy transfer due to random motion of the colliding particles, which always acts to heat the H_{2} gas. The second term shows that the bulk diffusive motion of H_{2} through the stationary N_{2} gas is decelerated by their mutual interactions, acting as a cooling mechanism. Whether the net effect of neutral collisions is heating or cooling depends on the relative magnitudes of these two mechanisms. According to our model calculations, the effect of neutral collisions between H_{2} and N_{2} is heating below ∼1160 km and cooling above. The effect of thermal conduction can be either heating (above ∼1320 km) or cooling (below ∼1320 km), which is always an important energy term in the local energy budget, except near 1300 km. The effect of viscosity also switches between heating and cooling (at an altitude of ∼1350 km). Finally, adiabatic outflow is always a cooling mechanism, and is important above ∼1800 km. Figure 5 shows that well below the exobase, the energy gain through neutral collisions is primarily balanced by energy loss through thermal conduction. However, well above the exobase, the local energy budget is a balance between energy gain through thermal conduction and energy loss through both viscous dissipation and adiabatic outflow. In the transition region between the thermosphere and exosphere, the energy budget is more complicated and an individual energy term may switch between heating and cooling as mentioned above.
[29] The energy budget of H_{2} implied in the 13moment model is more complicated than that described in early works [Fahr, 1976; Fahr and Weidner, 1977], in which the thermal structure of the diffusing component was obtained by assuming equality between the escaping energy flux, Φ_{esc} and the conductive heat flux. This corresponds to a simplified case of the boundary condition given by equation (5), which ignores both the intrinsic and viscous energy fluxes. To examine the relative magnitudes of various energy fluxes, in Figure 6 we show different terms from the righthand side of equation (4) as a function of altitude. The dotted, shortdashed and longdashed lines represent the conductive heat flux, the viscous energy flux, and the intrinsic energy flux, respectively. The total energy flux is shown by the thick solid line in Figure 6, along with the condition of energy flux continuity (given by the thin solid line). Above ∼1600 km, the total energy flux tends to scale as 1/r^{2}, implying negligible effects of neutral collisions according to equation (3). Correspondingly, the exobase of Titan can be placed at ∼1600 km based on Figure 6. At this altitude, the total downward energy flux counteracts roughly 50% of the upward conductive heat flux, indicating that the neglect of intrinsic and viscous energy fluxes in the early works is not justified here. The exobase height of ∼1600 km implied by the variation of total energy flux is higher than the traditional choice of ∼1400–1500 km estimated from a comparison between the atmospheric scale height and mean free path. The implication of this result on the thermal escape flux is discussed in section 5.3.
[30] Finally, we mention that although the thermal effect for H_{2} on Titan is not significant, in terms of the absolute value of the temperature decrement, we will show in section 5.3 that the associated heat flux provides an important modification to the velocity distribution of H_{2} molecules. In fact, the consequence of the nonMaxwellian VDF for the escape flux is so large that it may completely invalidates the Jeans formula.
4.2. External Loss of H_{2} in Titan's Exosphere
[31] Titan's exosphere is subject to solar EUV radiation, and is, most of the time, within Saturn's magnetosphere. The exospheric distribution of H_{2} on Titan may therefore be affected by its interactions with either solar photons or magnetospheric particles through external loss processes. These processes include photoionization and photodissociation, electron impact ionization, as well as charge transfer reactions with energetic protons/ions in the magnetosphere.
[32] Whether a particular loss process appreciably influences the exospheric H_{2} distribution relies on a comparison between the corresponding loss timescale and the dynamical time of H_{2} molecules spending above the exobase, following their own orbits. To investigate this, we draw a random sample of ∼22,000 particles from the Maxwellian distribution with a temperature of 152.5 K. The trajectories of these particles, assumed to be injected from Titan's exobase in random upward directions, are calculated and averaged. For ballistic particles, the inferred mean dynamical time increases from 1 × 10^{3} s (on ascending trajectories) and 9 × 10^{3} s (on descending trajectories) at an altitude of 2000 km, to 7 × 10^{3} s (ascending) and 5 × 10^{4} s (descending) at 6000 km. The mean dynamical time for escaping particles varies from 5 × 10^{2} s at 2000 km to 3 × 10^{3} s at 6000 km. Averaged over all particle types and weighted by their number fractions, the total mean dynamical time is found to be 5 × 10^{3} s at 2000 km and 2 × 10^{4} s at 6000 km. Clearly, any external loss process is more efficient at depleting particles on ballistic trajectories, since the loss probability scales exponentially with the dynamical timescale (see equation (7)).
[33] Assuming that a particular loss process is characterized by a constant timescale of τ_{loss}, the H_{2} density distribution can be calculated by
where λ = (GMm)/(kT_{exo}r) and λ_{exo} is the value at the exobase with r_{exo} = 4075.5 km. For simplicity, we have implicitly assumed a Maxwellian velocity distribution and adopted T = T_{exo} = 152.5 K at all altitudes above the exobase. Taking into account the nonMaxwellian VDF as well as the actual thermal structure should not alter our results significantly. In equation (6), ζ′_{bal} and ζ′_{esc} are the partition functions for ballistic and escaping particles, which take into account the loss processes. Extending the formalism of Chamberlain [1963] to include loss processes, we can express the partition functions as
where t(λ, ξ, χ) is the dynamical time required by a particle to travel from the exobase to a given point in the exosphere, following its own orbit. The integration limits in equations (7) and (8) are given by χ_{1} = λ^{2}(λ_{exo} − λ + ξ^{2})/(λ_{exo}^{2} − λ^{2}), χ_{2} = λ − ξ^{2}, and ξ_{1} = λ(1 − λ/λ_{exo}) [Chamberlain and Hunten, 1987]. The exponential factor of exp[−t(λ, ξ, χ)/τ_{loss}] represents the probability that an H_{2} molecule survives the loss process under consideration. t(λ, ξ, χ) can be calculated by
for ξ > 0 and
for ξ < 0, where v_{th} = (2kT_{exo}/m)^{1/2} is the thermal velocity of H_{2} at the exobase, and λ_{m} corresponds to the maximum radius reached by an H_{2} molecule along its orbit (only for ballistic particles). Equations (9) and (10) correspond to the situations in which the H_{2} molecule is on the ascending and descending portions of its trajectory, respectively. Escaping particles do not have descending trajectories, and should be excluded from equation (10).
[34] In Figure 7 we show the model H_{2} profiles calculated from equations (6)–(10), overplotted on the INMS measurements. Different lines represent different choices of the constant loss timescale, τ_{loss}, with the solid one giving the reference case with no external H_{2} loss. Figure 7 indicates that a loss timescale of order 10^{5} s is required to have an appreciable effect on the observed exospheric H_{2} distribution. We show below that all reasonable loss processes of H_{2} have typical timescales much longer than ∼10^{5} s, therefore the exospheric distribution of H_{2} molecules cannot be significantly modified by these loss processes.
4.2.1. Photoionization and Photodissociation
[35] H_{2} molecules are ionized by solar EUV photons with energy above 15.4 eV. Assuming an exosphere that is optically thin to the solar EUV radiation, the photoionization timescale, t_{ion} can be calculated from
where is the wavelength, _{ion}() is the photoionization cross section of H_{2} molecules, and π() is the solar spectral irradiance. We adopt the analytic formulae for H_{2} photoionization cross section from Yan et al. [1998], which combines experimental results at low energies and theoretical calculations at high energies. For the solar EUV irradiance, we adopt the sounding rocket measurements made on 3 November 1994, appropriate for solar minimum conditions during solar cycle 22 [Woods et al., 1998]. The corresponding F10.7 cm flux is 86 at 1 AU, comparable with the average value of 77 for our INMS sample. With the solar irradiances scaled to the value at Titan, equation (11) gives t_{ion} = 8.8 × 10^{8} s.
[36] H_{2} molecules are also destroyed through dissociation by solar EUV photons at energies between the Lyman continuum and Lyα. Destruction of H_{2} by photodissociation is accomplished through the Solomon process, i.e., upward transitions to electronic excited states followed by spontaneous decays to the vibrational continuum of the ground state [e.g., Abgrall et al., 1992]. We adopt the parameters for individual transitions in the Lyman and Werner bands published by Abgrall et al. [1992] and the spontaneous radiative dissociation rate from Abgrall et al. [2000]. The photodissociation timescale, t_{dis} can then be expressed as
where i and j refer to the lower and upper state of an electronic transition, _{ij} and _{ij} are the f value and central wavelength of the corresponding transition, _{j} is the probability that an H_{2} molecule at the electronic excited state j spontaneously decays to the vibrational continuum of the ground state. Here π is in units of photons s^{−1} cm^{−2} Å^{−1}, and _{ij} in units of Å. The summation is over all dipoleallowed transitions in the Lyman and Werner bands. In obtaining equation (12), we have also assumed a Doppler line profile, and the integrated results are independent of the adopted exobase temperature. From equation (12) we estimate an H_{2} photodissociation timescale of t_{dis} = 7.2 × 10^{8} s.
4.2.2. Interactions With Saturn's Magnetosphere
[37] Titan has a nonexistent or very weak intrinsic magnetic field, and its exosphere is, most of the time, directly subject to bombardment by energetic ions/protons and electrons in Saturn's corotating plasma. Here we investigate the interactions between Titan's exospheric H_{2} molecules and Saturn's magnetosphere through the following processes:
where p is proton and e* is energetic electron. Rather than integrating over the full energy distribution of the incident fluxes, we approximate the timescale, t_{i} for each of the above reactions by
where i stands for either proton, electron, or singly ionized oxygen, n_{i}, E_{i}, and m_{i} are the density, energy, and particle mass of species i, and _{i}(E_{i}) is the cross section for reaction between i and H_{2} at incident energy E_{i}.
[38] To calculate t_{i}, we adopt results from the extended plasma model for Saturn, constructed on the basis of the data from Voyager 1 and 2 plasma (PLS) experiments [Richardson and Sittler, 1990; Richardson, 1995]. We use model parameters obtained at L ≈ 20 to represent plasma conditions close to Titan's orbit around Saturn. For protons, we use n_{p} = 0.1 cm^{−3} and E_{p} = 50 eV; for O^{+}, we use n_{O+} = 0.13 cm^{−3} and E_{O+} = 280 eV [Richardson, 1995]. The electron energy distribution in Saturn's outer magnetosphere is characterized by a cold thermal component and a hot suprathermal component [Sittler et al., 1983]. For hot electrons, we use n_{e,hot} = 0.019 cm^{−3} and E_{e,hot} = 600 eV [Richardson, 1995]. The cold thermal electron component is highly time variable, and the temperature variation is anticorrelated with the density variation [Sittler et al., 1983]. Here we use values from Voyager 1 inbound measurements made at L ≈ 15 (day 317, 10:29), with n_{e,cold} = 0.4 cm^{−3} and E_{e,cold} = 21 eV [Sittler et al., 1983]. The cross sections for reactions (13)–(15) are adopted as _{p} = 2.0 × 10^{−16} cm^{−2} at an incident energy of 48 eV [McClure, 1966]; _{O+} = 8.1 × 10^{−16} cm^{−2} at 300 eV [Nutt et al., 1979]; _{e,cold} = 3.3 × 10^{−17} cm^{−2} at 21 eV and _{e,hot} = 3.6 × 10^{−17} cm^{−2} at 600 eV [Kim and Rudd, 1994]. With these values, we estimate the characteristic timescales for reactions (13)–(15) as t_{p} ≈ 5.1 × 10^{9} s, t_{O+} ≈ 1.6 × 10^{9} s, t_{e,hot} ≈ 1.0 × 10^{9} s, and t_{e,cold} ≈ 2.8 × 10^{8} s.
[39] To summarize, we list all the relevant timescales in Table 2. Various external loss processes have characteristic timescales between 3 × 10^{8} s and 5 × 10^{9} s, which are much longer than the dynamical time of H_{2} molecules above the exobase. This indicates that the exospheric distribution of H_{2} on Titan is not significantly modified by these loss processes.
Process  t_{loss} (s)  Note 

 
Photoionization  9 × 10^{8}  solar minimum 
Photodissociation  7 × 10^{8}  solar minimum 
Electron impact ionization  1 × 10^{9}  hot electrons 
3 × 10^{8}  cold electrons  
Charge transfer  5 × 10^{9}  H_{2} + p → H_{2}^{+} + H 
2 × 10^{9}  H_{2} + O^{+} → H_{2}^{+} + O  
Dynamical timescale^{a}  4 × 10^{3}  upward ballistic 
3 × 10^{4}  downward ballistic  
2 × 10^{3}  escaping 
5. Escape of H_{2} on Titan
[40] We have shown in section 4.1 that the H_{2} escape flux inferred from the diffusion model is about a factor of 3 higher than the Jeans value, implying an enhanced escape of H_{2} on Titan. This flux enhancement could of course suggest that nonthermal processes may play an important role. The nonthermal escape of nitrogen neutrals from this satellite has been extensively studied in previous works. A total loss rate of nonthermal N atoms was estimated to be <10^{25} s^{−1} on the basis of Voyager/UVS observations of airglow emissions [Strobel et al., 1992], consistent with the more recent value of 8.3 × 10^{24} s^{−1} based on Cassini/INMS observations [De La Haye et al., 2007]. The production of suprathermal nitrogen neutrals might be contributed by collisional dissociation and dissociative ionization, atmospheric sputtering by magnetospheric ions and pickup ions, as well as photochemical processes [e.g., Lammer and Bauer, 1993; Cravens et al., 1997; Shematovich et al., 2001, 2003; Michael et al., 2005]. Nonthermal escape also dominates over thermal escape for most other planetary atmospheres in the solar system. However, because of the rapid thermal escape of H_{2} on Titan, it has long been proposed that nonthermal escape of H_{2} is not important for this satellite [Hunten, 1973; Bertaux and Kockarts, 1983].
5.1. Saturn's Gravitational Influence
[41] The escape of H_{2} on Titan is complicated by the potential influence of Saturn's gravity. McDonough and Brice [1973] first proposed the possibility that particles escaping from Titan may be captured by Saturn's strong gravitational field and form into a toroidal cloud near Titan's orbit [see also Smyth, 1981; Hilton and Hunten, 1988]. Here, we investigate to what extent the escape of H_{2} on Titan can be influenced by Saturn's gravity.
[42] As Saturn's gravity is taken into account, all H_{2} molecules with trajectories reaching above the Hill sphere (roughly at 20 Titan radii) are able to escape from the satellite, since these particles would be progressively perturbed by Saturn's gravity and eventually end up orbiting with either the planet or the satellite. This implies that the actual H_{2} flux at the Hill sphere should include both upward ballistic flow and escaping flow.
[43] To estimate this effect, a Monte Carlo simulation is performed to numerically integrate the trajectories of test particles in a collisionless exosphere. The test particles start the simulation at the altitude of Titan's exobase with a velocity vector randomly selected from the upward flux of a Maxwellian distribution with a temperature of 152.5 K [Brinkmann, 1970]. The trajectories of the test particles are then integrated forward with an adaptive stepsized BulirschStoer routine according to the equations of motion for the circular restricted threebody problem, with Titan and Saturn treated as the perturbing bodies. The particles are followed until they either return to the exobase or reach the outer boundary of the simulation with a kinetic energy greater than Titan's gravitational potential. The H_{2} density profile above Titan's exobase calculated from the Monte Carlo simulation is shown as the dotted line in Figure 3. Its difference with the traditional collisionless model calculated with the same exobase temperature (given by the solid line), is completely due to the inclusion of Saturn's gravitational influence. The H_{2} flux is calculated in spherical bins over Titan using the trajectories of one million test particles. We find that Saturn's gravitational influence causes the flux in the simulation to be 23% higher than the Jeans value. This demonstrates that Saturn's gravity is only responsible for a small fraction of the enhanced escape of H_{2} on Titan.
5.2. Effect of Diffusive Motion
[44] The conventional way to calculate the thermal escape flux on planetary atmospheres is to use the Jeans formula, which is based on an integration of the Maxwellian distribution for all escaping particles. A preliminary correction to the Jeans flux can be obtained by noting that the nonzero escape flux is naturally associated with the bulk diffusive motion for the escaping component [e.g., Chamberlain and Campbell, 1967]. Therefore the VDF at Titan's exobase should be taken as a drifting Maxwellian distribution, with the form
where = − is the random velocity with being the drift velocity. The subscript “5” is used to emphasize that the drifting Maxwellian is essentially the 5moment approximation to the full kinetic model, as compared with the 13moment approximation introduced in section 4.1.
[45] We integrate equation (17) over all escaping particles at the exobase, with a temperature of 152.5 K and a drift velocity of 1.3 × 10^{4} cm s^{−1} from the diffusion model. This gives an H_{2} flux of 6.2 × 10^{9} cm^{−2} s^{−1}, more than a factor of 2 smaller than the value derived from the diffusion model. Therefore taking into account the bulk motion does not help to interpret the required flux enhancement.
5.3. NonMaxwellian Feature of the Velocity Distribution Function
[46] A useful technique for obtaining approximate expressions for the VDF is to choose the drifting Maxwellian distribution as the zerothorder function and expanding the real VDF in a complete orthogonal series [Schunk and Nagy, 2000]. In the 13moment approximation, the expansion is truncated to include velocity moments up to the heat flux vector and stress tensor. Such a truncated series expansion has the form
where p is the partial pressure of H_{2}, c_{r} is the radial component of the random velocity, and other quantities have been defined in equations (3), (4) and (17). The last two terms on the righthand side of equation (18) represent contributions from viscosity and thermal conduction, respectively.
[47] Thermal escape flux can be obtained by integrating equation (18) over all particles with kinetic energy exceeding the gravitational potential. An inherent assumption in this procedure is that the region above the level for performing such an integration is completely collisionfree, therefore any particle injected from that level with v > v_{esc} is able to escape without a further collision to alter its trajectory. The lowest choice of this level can be estimated as ∼1600 km from the vertical variation of energy flux shown in Figure 6. Here we apply equation (18) to a range of altitudes between 1600 and 2500 km, with all physical parameters such as temperature and heat flux adopted from the 13moment calculations in section 4.1. The mean thermal escape flux calculated at these levels is 1.1 × 10^{10} cm^{−2} s^{−1} referred to Titan's surface, with a variation of ∼20% depending on the exact altitude where the integration over equation (18) is performed.
[48] The 13moment approximation provides a further correction to the thermal escape flux calculated from either the widely used Jeans formula or the drifting Maxwellian distribution. The H_{2} flux calculated in such an approximation is a factor of 2.4 higher than the Jeans value. Considering the minor enhancement due to Saturn's gravity (see section 5.1), we suggest that the large H_{2} flux of 1.4 × 10^{10} cm^{−2} s^{−1} on Titan, as inferred from the diffusion model, can be interpreted by thermal escape alone, and nonthermal processes are not required.
[49] The correction to the Jeans flux based on the 13moment approximation comes primarily from the effect of thermal conduction. To investigate the contribution of thermal conduction alone, we run our 13moment model with the viscosity term ignored in both equations (3) and (18). This gives a similar thermal structure of H_{2} and an H_{2} escape flux very close to the value obtained in the full 13moment approximation.
[50] In Figure 8 we show the 13moment VDF calculated from equation (18) (normalized by the drifting Maxwellian), as a function of vertical and horizontal velocities (scaled by either the local thermal velocity or the local escape velocity). The upper panel represents the VDF at the lower boundary of 1000 km, which shows that the velocity distribution of H_{2} molecules is close to Maxwellian, representing a situation with near thermal equilibrium between H_{2} and N_{2}. With increasing altitude, the deviation from the Maxwellian VDF becomes significant, which is clearly seen in Figure 8 (bottom), calculated at our upper boundary of 2500 km. Several features can be identified from Figure 8:
[51] 1. Compared with the drifting Maxwellian, the 13moment VDF presents a depletion of particles with v < −v_{esc}, corresponding to an absence of incoming hyperbolic particles. This is expected for any exospheric model since the collision frequency at such high altitudes is too low to allow a steady population of incoming hyperbolic particles to be established.
[52] 2. The 13moment VDF shows an enhanced population of particles with v > v_{esc}, especially along the radial direction. These particles are expected to carry the conductive heat flux required by the local energy budget. Figure 9 gives the velocity distribution (scaled by the drifting Maxwellian) in the radial direction at the upper boundary of 2500 km, which shows the depletion of slow particles as well as the accumulation of fast particles more explicitly.
[53] The implications of the results presented here deserve some further concern. First, we notice that the continuity of escape flux is satisfied exactly in the traditional Jeans formalism, since the upward and downward ballistic flows are in perfect balance, with the integration over escaping particles alone giving the accurate total flux. However, this is not exactly true in the 13moment approximation. The values of the thermal escape flux derived at different altitudes (between 1600 km and 2500 km) but all referred to the surface show some variation at about 20% level, implying that the continuity of escape flux is not perfectly satisfied. More specifically, the integration over all escaping particles gives an estimate of the thermal escape flux of ∼9 × 10^{9} cm^{−2} s^{−1} at 1600 km and ∼1.3 × 10^{10} cm^{−2} s^{−1} at 2500 km, where both flux values are referred to the surface. Such a feature of imperfect continuity of escape flux might be related to the collisional nature of the 13moment model, which allows transitions between ballistic and escaping particles in the exosphere in response to rare collisions. In such a model, the perfect balance between upward and downward ballistic flows is clearly not ensured, and the integration over escaping particles gives a representation of the thermal loss rate, rather than an exact physical value.
[54] Second, the traditional exobase level is placed at 1400–1500 km for Titan, based on a comparison between the local scale height and mean free path. Here an inherent assumption is that all gas components are stationary. However, the H_{2} gas is escaping with a considerable drift velocity, which contributes to an additional collisional term in the energy equation (the second term on the righthand side of equation (3)). This term, representing the deceleration of bulk motion by molecular diffusion, helps to raise the actual exobase level by several hundreds km. We notice that with this term ignored, the total energy flux shows vanishing divergence at ∼1450 km, consistent with the traditional choice of the exobase height. The choice of the exobase level has important effects on the derived thermal escape rate. In fact, when integrating equation (18) over all escaping particles at an altitude of 1500 km, we obtain a flux value about 17% higher than the Jeans value. However, the flux calculated at this altitude does not necessarily mean any realistic physical flux, since the effect of neutral collisions is not negligible as we emphasized above. At higher altitudes where collisions can be safely ignored and the procedure of integrating the VDF over all escaping particles is justified, the perturbation of the VDF by thermal conduction becomes strong enough to raise the thermal escape flux significantly above the Jeans value.
[55] Finally, we notice that in the traditional collisionless model, calculating the thermal escape flux at altitudes above the exobase relies on the integration of the Maxwellian VDF over a truncated region of the momentum space, to include only escaping particles that intersect the exobase. In fact, it is through this procedure of truncation that the continuity of escape flux in the traditional Jeans formalism is naturally satisfied. However, such a truncation is not necessary in the 13moment model, since escaping particles reaching any level above the exobase may come from all directions as a result of rare collisions in the exosphere. The 13moment VDF that smoothly occupies the entire momentum space is a more realistic representation of the actual velocity distribution, as compared with the truncated Maxwellian in the collisionless model.
[56] It has been suggested in previous works that a realistic VDF at the exobase shows signatures of particle depletion on the highvelocity tail of the Maxwellian distribution as a result of thermal escape [Fahr and Shizgal, 1983, and references therein]. This effect tends to reduce the thermal escape rate, contrary to the result presented here. However, the earlier works neglected the effects of thermal conduction that we show to be of paramount importance. Further investigations including application of the 13moment equation to escape from other planetary atmospheres are needed to understand the full implications of our results. Despite this, we notice that earlier Monte Carlo simulations [e.g., Chamberlain and Campbell, 1967; Brinkmann, 1970] share the common procedure of picking random source particles from an assumed Maxwellian VDF at the lower boundary (either drifting or nondrifting). However, Figure 6 shows that the heat flux tends to a finite value of 1.6 × 10^{−4} ergs cm^{−2} s^{−1} near an altitude of 1000 km, which implies the presence of the nonMaxwellian VDF well below the exobase. To assess the importance of this effect, we examined the solution to equation (3) by adopting a heat flux at the lower boundary equal to half of the value satisfying energy continuity. This gives a thermal escape flux in the 13moment approximation ∼30% lower than the Jeans value. By comparison, with the boundary condition satisfying energy continuity, the 13moment flux is higher by a factor of ∼2.5. This suggests that the reduced thermal escape rate claimed in earlier works is probably associated with the (incorrect) neglect of thermal conduction well below the exobase.
6. Discussions and Conclusions
[57] We extract the average H_{2} density profile at altitudes between 1000 and 6000 km for Titan's thermosphere and exosphere by combining the INMS measurements from 14 lowaltitude encounters of Cassini with Titan. The average N_{2} distribution, obtained from the same sample, suggests a temperature of 152.5 K, consistent with previous results [e.g., Vervack et al., 2004].
[58] Below the exobase, the observed H_{2} distribution is well described by the diffusion model, with a most probable H_{2} flux of 1.37 × 10^{10} cm^{−2} s^{−1}, referred to the surface. The model assumes full thermal equilibrium between H_{2} and N_{2}. Above Titan's exobase, the H_{2} density profile can be described by a simple collisionless model, including both ballistic and escaping molecules. The collisionless model assumes a Maxwellian VDF for H_{2} at the exobase, with a most probable exobase temperature of 151.2 K. Interactions with solar EUV photons and energetic particles in Saturn's magnetosphere have negligible effects on the exospheric distribution of H_{2} on Titan.
6.1. Thermal Effect
[59] By assuming continuity of energy flux at the upper boundary of 2500 km, we obtain a numerical solution to the H_{2} thermal structure for Titan's upper atmosphere, which presents a small temperature decrement of ∼2 K between 1000 and 2500 km. This is a result of the local energy budget in Titan's upper thermosphere, in which thermal conduction plays an essential role and naturally accounts for the temperature decrement for H_{2}. The energy budget implied in the 13moment approximation is more thorough and complicated than that suggested in early works for the terrestrial exosphere [e.g., Fahr, 1976]. The variation of the total H_{2} energy flux suggests an exobase level of ∼1600 km, which is significantly higher than the traditional choice of 1400–1500 km, as a result of the deceleration of the H_{2} bulk motion by molecular diffusion through N_{2}.
6.2. Enhanced Escape
[60] The orthogonal series expansion in the 13moment approximation defines a nonMaxwellian VDF that includes the effects of both thermal conduction and viscosity (see equation (18)). Integrating over such a VDF for all escaping particles at a range of altitudes above the exobase gives a mean H_{2} flux of 1.1 × 10^{10} cm^{−2} s^{−1} referred to the surface, with an uncertainty of ∼20% associated with the exact altitude level (between 1600 and 2500 km) where the integration is performed. Below 1600 km, the effect of collisions between H_{2} and N_{2} cannot be ignored, and the integration of the VDF over all escaping particles to estimate the thermal escape flux is not justified. The escape flux implied by the 13moment model is significantly higher than the Jeans value and roughly matches the flux inferred from the diffusion equation. The 13moment model interprets the enhanced escape as a result of the accumulation of H_{2} molecules on the highenergy portion of the VDF, primarily associated with the conductive heat flux. In this work, the enhanced escape of H_{2} on Titan is still thermal in nature. Nonthermal processes are not required to interpret the loss of H_{2} on Titan.
[61] In a recent work by Strobel [2008], the thermal escape process on Titan was investigated by solving the hydrodynamic equations for a single component N_{2} atmosphere, which gave a hydrodynamic escape rate of 4.5 × 10^{28} amu s^{−1} (the sum of H_{2} and CH_{4} escape), restricted by power limitations. Assuming that the ratio between individual loss rates is equal to the corresponding limiting flux ratio, Strobel [2008] obtained an H_{2} loss rate of 5.3 × 10^{27} s^{−1}, or an H_{2} flux of 6.3 × 10^{9} cm^{−2} s^{−1}, referred to Titan's surface. This indicates that by treating the thermal escape process as hydrodynamic rather than stationary (as implicitly assumed in the Jeans formula), the derived H_{2} escape rate exceeds the Jeans value by ∼40%, to be compared with the 13moment calculation of the flux enhancement of about a factor of 3.
[62] The approach followed by Strobel [2008] is quite different from that adopted here, in the sense that he assumed constant composition and worked entirely in the fluid, rather than the kinetic regime. Escape in Strobel's model is due entirely to bulk outflow of the atmosphere whereas in our calculations escape is driven primarily by the perturbations to the VDF due to the heat flow. Instead, the primarily effect of bulk outflow is to raise the exobase to a higher level at ∼1600 km, where the perturbation of the VDF by thermal conduction becomes strong enough to have an appreciate effect on the thermal escape rate. In terms of perturbations to the VDF, the heat flow is more important than the drift velocity. On the other hand, Strobel [2008] did carefully treat the energy balance in the upper thermosphere and argued that it is the energetics that causes the large loss rates and the breakdown of Jeans escape. In that sense, his results are consistent with those found here in terms of the importance of energy continuity to the molecular escape rate. The characters of Strobel's fluid solutions are determined by requiring energy continuity and that the energy escape flux be consistent with the mass escape flux. These requirements coupled with the NavierStokes equations imply an escape rate significantly greater than Jeans escape. The same requirement on consistency between mass and energy escape also appears in our calculations, in which the boundary condition on the energy flux forces a negative temperature gradient. The associated heat flux alters the VDF and thereby enhances the escape rate in the kinetic description. It is worth restating that one of the fundamental assumptions in Jeans escape is that the atmospheric energetics is unaffected by escape. In both Strobel's approach and ours, it is the effect of the escape process on the atmospheric energetics that causes the high escape rates and the failure of Jeans escape. What is most surprising is that Jeans escape fails for such small values of the energy flux. An energy flux corresponding to a temperature drop across the transition region of several kelvin causes more than a factor of 2 increase in the escape flux. It appears important to determine under what conditions the Jeans escape formula can reliably be used.
[63] Finally, we summarize all the relevant fluxes in Table 3, in which F_{s} is the thermal escape fluxes calculated in various ways (all referred to Titan's surface) and R is the corresponding escape rates. A complete interpretation of the enhanced escape has to rely on a proper consideration of both bulk outflow and thermal conduction. The former controls the exact level of the exobase, while the latter drives significant departures from the Maxwellian so that the actual thermal escape rate is significantly higher than the Jeans value. In a more general context, enhanced escape induced by bulk outflow and thermal conduction is expected to be a common feature for planetary atmospheres. The 13moment kinetic model presented in this paper will be applied to other planetary systems in followup studies.
F_{s}^{a} (cm^{−2} s^{−1})  R^{b} (s^{−1})  Note 

 
1.4 × 10^{10}  1.2 × 10^{28}  inferred from data 
4.5 × 10^{9}  3.8 × 10^{27}  Jeans escape 
5.4 × 10^{9}  4.5 × 10^{27}  Jeans escape including Saturn's gravity 
6.2 × 10^{9}  5.2 × 10^{27}  drifting Maxwellian 
1.1 × 10^{10}  9.2 × 10^{27}  13moment 
Appendix A:: Energy Density and Flux in the 13Moment Approximation
[64] In the 13moment approximation, the continuity, momentum and energy equations for a diffusing neutral component can be expressed as
where n, p, u_{i}, q_{i}, τ_{ij} are the density, pressure, drift velocity vector, heat flux vector and stress tensor of the neutral species, with i, j = 1, 2, 3 characterizing components along the three orthogonal spatial coordinates (x_{i}), and δM_{i}/δt and δE/δt are the momentum and energy integrals [Schunk and Nagy, 2000]. Here, for repeated indices, the Einstein summation convention is assumed. Equation (A3) can be recast as
Using equation (A2) to eliminate u_{i}∂p/∂x_{i}, we get
where u^{2} = u_{i}u_{i}. The gravity term in equation (A5) can be expressed as
where r = (x_{i}x_{i})^{1/2}, M is the planet mass, G is the gravitational constant, and we have used equation (A1) to eliminate ∂(nu_{i})/∂x_{i} in the last equality. The term, mnu_{i}u_{j}(∂u_{i}/∂x_{j}) in equation (A5) can be recast as
which gives
Finally, we write u_{i}(∂τ_{ji}/∂x_{j}) as
Using equations (A6), (A8), and (A9), equation (A5) can be expressed as
where we have used the fact that τ_{ij} is symmetric.
[65] We further recast equation (A10) as
Clearly, ε and ϕ_{j} represent the energy density and energy flux, with the definitions of
where we have replaced p by nkT with k being the Boltzmann constant and T being the gas temperature, c_{v} = (3/2)(k/m) and c_{p} = (5/2)(k/m) are the specific heat capacities at constant volume and pressure. The terms, u_{i}τ_{ji} and q_{j} in equation (A11) represent energy fluxes associated with viscous dissipation and thermal conduction, respectively. Assuming spherical symmetry, we can express the radial components of these energy fluxes as
where κ is the thermal conductivity and η is the viscosity coefficient.
Acknowledgments
[66] We are grateful to D. F. Strobel, I. C. F. MüllerWodarg, D. M. Hunten, and R. Malhotra for helpful discussions. This work is supported by NASA through grant NAG512699 to the Lunar and Planetary Laboratory, University of Arizona.