Tucker 2. (Jeans) φ =π r exo 2 n exo <v exo >exp[-λ exo. ](1+λ exo )

DIRECT MONTE CARLO SIMULATIONS OF THERMALLY DRIVEN ATMOSPHERIC ESCAPE: APPLICATIONS TO TITAN AND PLUTO Orenthal J. Tucker, R.E. Johnson Engineering Physics, University of Virginia, Charlottesville, Va., 22904 Recent models of the atmospheres of both Pluto and Titan estimate large thermal escape rates of the principal atmospheric species in comparison to their respective Jeans theoretical rates. However, these continuum models were applied region of the atmosphere that transitions from being collisional to collisionless. In particular, the slow hydrodynamic escape model requires assumptions about the thermal conduction and temperature in the exosphere which favors large escape rates. The difficulties with the slow hydrodynamic approach are related to the model assumptions concerning the atmospheric structure at infinity. Here a kinetic model is used to account for nonequilibrium collisions in the exosphere, and it is found that thermal escape of N 2, CH 4 and H 2 from Titan and N 2 from Pluto should occur similar to Jeans rate. In addition a hybrid fluid/dsmc approach has been developed to obtain consistent results for the escape rate and macroscopic properties of Pluto s atmosphere between a continuum and kinetic approach. A summary on the DSMC model is provided with current results for escape from Titan and Pluto. 1. Introduction The evolution of a planetary body is directly coupled to the atmosphere. Whether or not a planet can support a significant atmosphere over time depends on its gravity and the atmospheric temperature. For example if the thermal energy of the atmosphere is greater than the planetary binding energy the atmosphere can escape to space and, over time, will deplete a planet s inventory of volatile elements. Therefore, many studies are aimed at understanding the mechanisms that control atmosphere loss in order to infer about the original inventory and evolution of planetary bodies. Here we consider loss to space by thermal escape of the atmospheres of Titan and Pluto. Both the Cassini and the New Horizons mission in the outer solar system are allowing scientists to critique previous ideas concerning atmospheric evolution and escape. While orbiting the Saturn system the Cassini satellite has been collecting data from Titan s atmosphere for over 7 years, and the New Horizons spacecraft will collect data on Pluto s atmosphere in 2015 as it travels to the Kupier Belt. Titan, Saturn s largest moon, has an N 2 rich atmosphere with a few percent CH 4, > 1 / 10 of a percent of H 2 and a sparse population of organic molecules. Pluto also has a N 2 rich atmosphere with a few percent CH 4 but there is also a sparse population of CO and its surface pressure is nearly (20-200) millionth of the surface pressure at Titan (Olkin et. al. 2003). Terrestrial atmospheres are dominantly heated by adsorbing solar radiation and transporting that heat by convection and conduction. Thermal escape from an atmosphere occurs when conduction distributes the kinetic energy, associated with temperature, to a molecule with a favorable velocity such that it gains energy greater than the planetary binding energy. Typically thermal escape is considered between two extremes one being an evaporative type of loss, so-called Jeans escape, and the other being a bulk type of loss referred to as hydrodynamic escape. Several decades worth of continuum models for the upper atmosphere of Pluto have concluded that the atmosphere is escaping hydrodynamically due to its low gravity (e.g.: Hunten and Watson 1982; McNutt 1989; Krasnolpolsky 1999; Strobel 2008a). However recently, Titan s atmosphere has also been suggested to be undergoing hydrodynamic escape (Strobel 2008b) even though it has 5 times the Tucker 1

gravitational binding energy of Pluto at the surface. The hydrodynamic escape from Titan and Pluto is suggested to be slow. It is assumed that conduction will efficiently convert thermal energy of molecules into bulk energy for escape. This results in escape rates that would be orders of magnitude larger than the corresponding Jeans rates. However, this research has shown that applying continuum models to the upper atmosphere where collisions are non equilibrium events leads to incorrect escape estimates and macroscopic properties for the atmosphere (e.g. temperature and density) (Tucker and Johnson 2009; Tucker et al. 2011; Volkov et al. 2011a,b). A Monte Carlo approach, as opposed to a fluid (continuum) approach, is necessary to model thermal escape from a rarefied atmosphere. In the following sections the application of a Direct Simulation Monte Carlo model applied to thermal escape is explained, with summaries presented on the current results for the atmosphere of Titan and Pluto. 2. Exobase, Jeans Escape & Hydrodynamic Escape Escape occurs most efficiently in the exosphere, a tenuous region in the upper atmosphere, where molecules can travel planetary scale distances and not collide with other molecules (Johnson et. al. 2008). The exobase is referred to as the lower boundary to this region, and it is defined as the altitude where the mean free path for collisions, l = (2 1/2 nσ) -1, is comparable to the atmospheric scale height H = kt/mg (n density, σ collision cross-section, k Boltzmann constant, T temperature, m molecule mass, g gravity). In rarefied gas dynamics the ratio l/h is referred to as the Knudsen number Kn, and it provides the criterion for when a gas flows similarly to a fluid and can be modeled using solutions to the hydrodynamic equations (Kn << 1), or when it becomes increasingly collisionless and must be modeled stochastically (Kn > 0.1). At the exobase Kn ~ 1, and it is the altitude in the atmosphere where the column density is N = σ -1. Thermally driven atmospheric escape is often characterized by the Jeans parameter; it is the ratio of the gravitational potential energy to the thermal energy at the exobase λ exo = GM p m/(r exo kt exo ) where M p is the mass of the planet. Using inferred temperatures obtained from Cassini density measurements for Titan and occultation data for Pluto the Jeans parameter are λ exo ~ 50, and ~10 respectively. For large Jeans parameter values, the gravitational energy of the planet dominates the thermal energy of the atmosphere. Therefore, only a small fraction of particles which have speeds near the tail of distribution can escape. In this case the escape rate obtained from kinetic theory is: φ =π r exo 2 n exo <v exo >exp[-λ exo. ](1+λ exo ) (Jeans) where <v exo > is the mean thermal speed (Jeans 1916). In the above we assume that molecules have speeds according to the Maxwell Boltzmann speed distribution. If the gravitational energy of the planet is comparable to the thermal energy of the atmosphere at the exobase, λ exo = 2, then the atmosphere can escape the planet in bulk, like a fluid, and requires consideration with a hydrodynamic approach. Slow hydrodynamic escape is suggested as an intermediate case between hydrodynamic and Jeans escape (Chamberlain 1961; Parker 1963a, b). It is defined as an organized flow produced by thermal conduction which produces an expansion in the upper atmosphere in which the bulk flow speed can exceed the escape speed (Johnson et. al. 2008). In this formulation conduction is assumed to occur efficiently beyond the exobase and all of the heat is used to lift the Tucker 2

atmosphere out of the planetary gravitational well. The thermal escape problem at first glance appears intuitive, nevertheless it remains a controversial topic (Jeans 1916; Chamberlain 1960, 1961; Watson 1981; Parker 1963a, b; Johnson 2010; Grusinov 2011; Volkov et al. 2011a,b). The problem is formulated by considering a spherically symmetric atmosphere in which the dominate heating of the atmosphere by solar radiation occurs below some lower boundary r o. At r o the temperature T o and density n o are maintained at constant values. The lower boundary is chosen at an altitude considered to be in approximate radiative equilibrium, and escape as the result of non-thermal external processes is neglected. The goal is to describe how heat transferred from r o by thermal conduction and convection drive escape. To this end, two methods have been developed to obtain the escape rate and macroscopic properties (i.e. n(r) and T(r)) of the atmosphere. In the first approach solutions are obtained using the hydrodynamic equations by placing restrictions on density and temperature at infinity (Chamberlain 1961; Parker 1963a, b). The second method consists of explicitly modeling the atmospheric flow with a representative sample of modeling particles which in effect solves the Boltzmann equation. These methods are discussed further below. 2. Slow hydrodynamic model When the hydrodynamic equations are applied to a 1D radial atmosphere, the continuity equation leads to a constant atmospheric flow vs. radial distance from a lower boundary: 4π r 2 n(r)u(r) =, with n(r) the number density and u(r) the flow speed vs. r the radial distance from the center of the planet. The radial momentum (pressure) equation for molecules of mass m in which the viscous term is typically dropped, can be written dp/dr = n [dφ/dr] n d(mu 2 /2)/dr (1a) where p is the pressure (nkt) and Φ the gravitational potential energy (GM p m/r). The corresponding energy equation can be written: d{ (mu 2 /2 + C p T - Φ)- 4π r 2 κ(dt/dr)}/dr = 4π r 2 Q(r) (1b) where C p is the heat capacity per molecule and κ = κ(t) the thermal conductivity (e.g., Johnson et al. 2008). The standard procedure in the slow hydrodynamic model is to use 0, but set u = 0 in Eq. (1a) and (1b) below an upper boundary that is above the exobase. In the approach described in Strobel (2008a, b) Eq. 1a and 1b are solved iteratively using assumed values of and dt/dr ro with the conditions n and T 0 as r. Throughout the remainder of the paper we will refer to that slow hydrodynamic escape model as the SHE model. The main difficulties with the applying the SHE model are the constraints placed on density and temperature at infinity without prior knowledge of what they should be. Furthermore the She model is unable properly account for the non-equilibrium effects in the exosphere. As shown in Eq. 1b temperature is, therefore, assumed to be defined even in the exosphere and the expression for thermal conductivity is essentially, independent of density. This procedure gives solutions of n(r) and T(r) for a range of escape rates. Strobel (2008a) obtains a best solution by using density data as a constraint, and other models only require a solution that conserves energy with asymptotically requiring n and T 0 as r (Parker 1963b). However, Tucker and Johnson (2009) have used Monte Carlo techniques to test the SHE model result and obtained very different escape fluxes while Tucker 3

producing solutions for density profiles consistent with data from Titan s atmosphere. 3. Direct Simulation Monte Carlo Applying a continuum model it beyond the exobase where Kn > ~1 must be done with care. To describe the transition region in the atmosphere from below the exobase to beyond Kn >~ 0.1, solutions to the Boltzmann equation or Monte Carlo simulations are required. We use the Direct Simulation Monte Carlo model, DSMC, (Bird, 1994). The atmosphere is described using a set of representative molecules and its evolution is calculated by following the motion of these particles subject to gravity and collisions. Thermal conduction is explicitly included and depends on the cross sectional area for collisions as well as the local density. A 1D simulation is carried out, consistent with the 1D continuum models being tested. In the main flow direction, radially outward from Titan and Pluto, the space is divided into cells with heights less than the local mean free path for collisions. To accurately describe an atmosphere using such simulations three conditions must be satisfied (Shematovich 2004): there should be a sufficient number of representative particles to describe the nature of the flow; the molecular motion between collisions should be independent of the nature of the collisions; and on average molecules should experience less than one collision during a time step. The upper boundary is placed many scale heights above the exobase where it is safe to assume further collisions will not significantly affect the escape rates. At the upper boundary upward moving molecules with speeds greater than the escape speed are assumed to escape while the others were specularly reflected back to the simulation region, allowing a shorter time to achieve steady state. The reflected, nonescaping molecules represent ballistic particles that will eventually return to the simulation region. If chosen high enough the upper boundary should not affect the results. Typically, on the order of a few hundred thousand particles were used to represent the atmosphere in order to have at least a few hundred particles in the very top cells. The weights for the representative molecules can be adjusted in order to meet that condition. The DSMC model does not make any assumptions about density and temperature at infinity and the results depend on how realistically molecular collisions can be represented. 4. Results Before presenting the results from the DSMC simulations a brief overview is given of the current SHE model. For a baseline case in which all the heat input into the atmosphere is deposited below the lower boundary [Eq. 1b Q(r) = 0 above r o in (Strobel 2008a, b)], solutions with a significant escape flux (φ H =1x10 27 s -1 and 5x10 26 s -1 ) were obtained for both the atmospheres of Titan and Pluto respectively. Even though Titan has a Jeans parameter 5 times larger than that for Pluto, the SHE model concluded that Titan s atmosphere should escape at a higher rate. Furthermore the exobase temperatures and densities derived from the SHE model suggest the Jeans rates would be several orders of magnitude smaller (φ J ~10 9 s -1 and ~10 19 s -1 ) for Titan and Pluto respectively. DSMC is ideally suited to test these results from SHE model since no assumptions are made about the temperature or density at infinity, and thermal conduction is inherently a function of the local density. Therefore a series of simulations were done in Tucker and Johnson (2009) to test the SHE model results for Q(r) = 0 for Titan. In this result the temperature and density at the lower boundary (3875 km) used in the DSMC simulation was normalized to the solution from the SHE model in (Strobel 2008b). A comparison n(r) and T(r) of the DSMC and SHE model results are shown in Fig. 1. The density profiles Tucker 4

Radial Distance (km) between the methods are similar, but the temperature for the DSMC result is nearly isothermal throughout the simulation region. The simulations were performed for a suitable amount of time such that if 1 test particle escaped during the simulation, the escape rate would have been 6 orders of magnitude less than the SHE model result. Since no test particles escaped during our DSMC simulation, our upper bound to the escape rate is ~10 21 s -1. The gravitational binding energy at the DSMC exobase (4000km) is ~ 0.6eV, and the mean energy of the N 2 molecules at 141K is ~0.01eV. Only a very small fraction of particles at the tail of the distribution will have sufficient energy to escape, therefore, the rate should be similar to the Jeans rate. In our simulations, achieving escape for such a large Jeans parameter is limited by the capabilities of the random number generator. However, by artificial increasing the temperature to 600K at the lower boundary for Titan we were able to force escape to occur (Tucker and Johnson 2009). We found that even at λ exo ~ 11 escape still occurred similar to the Jeans theoretical values (1.5*Jeans) as opposed to the hydrodynamic values obtained in Strobel (2008a) which are orders of magnitude larger than the Jeans rates. Therefore, we considered escape from Pluto using the DSMC approach where λ exo ~ 10. Density (cm -3 ) 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 4600 4500 4400 4300 4200 4100 4000 3900 100 105 110 115 120 125 130 135 140 145 Temperature (K) Figure (1): N 2 n (solid curves (triangles) - top axis) and T (dotted curves (inverted triangles) bottom axis) vs. radial distance in Titan s atmosphere: DSMC (open triangles) (Tucker and Johnson (2009)) and SHE model (filled triangles) using lower boundary conditions at 3875 km for Q = 0 from in Strobel (2008b) (r exo = 4000 km represented by horizontal line). 5. Combined fluid/dsmc Approach The DSMC simulations of thermal escape from Pluto s atmosphere, where λ exo ~ 10, obtained a rate ~1.6*Jeans similar to the artificial Titan result discussed in section (4) (Tucker et al. 2011). Many studies of gas flows have shown for the limit Kn 0 the continuum and kinetic approach should give the same result (Bird 1994). Therefore we developed a fluid/dsmc approach to obtain consistent solutions for n(r) and T(r) with the hydrodynamic equations (fluid) and kinetic approach (DSMC) for Pluto s atmosphere. This is achieved by solving the hydrodynamic equations (Eq. 1a and 1b) in the region of atmosphere where Kn < 0.1and applying the DSMC simulation to regions where Kn > 0.1. Upon integration of Eq. 1a and 1b with u = 0 we obtain the following expressions for pressure and heat flow. p = p o exp[- ro (GM p m/r 2 )/kt dr] (2a) with p = nkt and Tucker 5

[ (C p T - Φ) - 4 r 2 κ(dt/dr)] = <E > ro + 4 r o 2 (r) (2b) Here <E > ro is the energy flow across r o, and, as in Strobel (2008a), (r) = r o -2 [ ro r 2 Q(r) dr] with o as r. The conditions we use to solve Eq. 2a and 2b are a fixed n o and T o at r o and the heat flow <E > ro and escape rate φ were determined iteratively as discussed below and shown schematically in Fig. 2. We do not assume, as in SHE, that T 0 as r. In the most dense region of the atmosphere (Kn << 0.1), we numerically solve Eq. 2a and 2b to obtain n(r) and T(r) from r o to r od an altitude below the exobase. The solutions provide (n od and T od ) at r od and are then used as the lower boundary conditions to the DSMC model. The DSMC model discussed in section (3) is applied to a region from r od to r top an altitude many scale heights above the exobase. At r top we calculate both the particle escape rate φ, and the average energy carried off by escaping molecules, <E > =<E > ro. The new <E > ro and φ are then used to resolve Eq. 2a and 2b, which in turn provide new conditions at r od for the DSMC model. This procedure is iterated until the pair (, <E > ro ) is obtained that produce a convergent n(r) and T(r) in the region where the fluid equation solution and the DSMC results overlap 0.1 < Kn < 1. Again this approach differs from the SHE because no assumptions are made about the tendency of density and temperature at infinity. Furthermore in the SHE model it is customary to set <E > ro = 0, we directly calculate this quantity and from the amount and energies of escaping molecules. Figure (2): Schematic of the fluid/dsmc iterative procedure. In Fig. 3 we compare results for solar minimum conditions at Pluto obtained with the fluid/dsmc method to the SHE model The implementation of solar heating will be discussed further in Tucker et al. (2011), the following discussion is a summary of the result. The escape rate obtained from the fluid/dsmc method 1.2 x10 27 s -1 is fortuitously close the SHE model result, 1.5 x10 27 s -1, but the resulting macroscopic properties for the atmosphere are very different as shown in Fig. 3. Therefore, using the corresponding exobase values the fluid/dsmc model give a rate 1.9*Jeans while the SHE model suggests the rate is >10 3 *Jeans. Furthermore, the fluid/dsmc model determined the energy flow into the lower boundary to be <E > ro = 1.5 x10 13 ergs s -1 and not 0 as assumed by the SHE model. For solar minimum conditions this energy amounts to a small fraction of the actual heat deposited, but it still affects the temperature gradient from lower boundary. Tucker 6

Radial Distance (km) Density (cm -3 ) 10 1 10 3 10 5 10 7 10 9 10 11 10 13 10 15 10000 a-fluid/dsmc 9000 b-she 8000 7000 6000 5000 4000 3000 2000 a-(u) b-(u) a-(n) b-(n) b-(t) a-(t) 0 10 20 30 40 50 60 70 80 90 100 110 Temperature & Flow Speed (K, m/s) Figure (3): Result for solar minimum conditions: n(cm - 3 ) (top axis), T(K) & u(m/s) (bottom axis) vs. radial distance. Comparison of Fluid/DSMC n(dashed curves), T(solid curves) & u(dotted curves) to SHE model results from Strobel (2008a) n(circles), T(triangles) & u(squares): exobase 6200 km Fluid/DSMC (solid curve on right axis) and 3530 km SHE (dashed curve). 6. Summary/Future Work Pluto s atmosphere has been widely considered to be undergoing hydrodynamic escape (Hunten and Watson 1982; McNutt 1989; Krasnolpolsky 1999; Strobel 2008a), and more recently measurements made by the Cassini spacecraft of density data in Titan s upper atmosphere (Waite et al. 2005) have been used to postulate hydrodynamic escape from Titan s atmosphere (Strobel 2008b). In particular the a recent application of SHE model to the atmospheres of Pluto and Titan Strobel (2008a, b) found the mass loss rate of from Titan (λ exo = 50) and Pluto (λ exo = 10) to be several orders of magnitude larger than the corresponding Jeans rates. In this research a kinetic approach is used to consider the thermal escape problem, and contrary to the slow hydrodynamic results it is found that escape occurs on molecules by molecules basis. Specifically we use the socalled DSMC model which in effect solves the Boltzmann equation. Such simulations include thermal conduction explicitly via the molecular collisions. Therefore they can serve as a test of the results from the SHE model which is unable to properly account for the effect of collisions on temperature and escape in the exosphere region. For Titan the DSMC results have shown that in the absence of any external energy input into the atmosphere by non-thermal processes N 2 escape occurs at rates similar to Jeans. For Pluto, we used a combined fluid/dsmc iterative procedure to obtain consistent escape rates, density and temperature profiles between the DSMC model and the solutions to the equilibrium fluid equations without the assumption that n, T 0 as r. In principle the DSMC can be used to simulate the entire simulation region but as Kn becomes << 0.1 the calculation becomes computationally expensive. Therefore, we numerically solve the equilibrium equations in the region of the atmosphere where Kn<<0.1, and use the DSMC model in the tenuous region Kn>0.1 (section 5). In the case of escape from Pluto atmosphere for solar minimum conditions the fluid/dsmc method again results in an escape rate similar to Jeans, and the temperatures and density profiles are significantly different from the SHE model result. Below we list the general conclusions from our ongoing studies. 1) Both Pluto and Titan are not undergoing hydrodynamic escape (Tucker and Johnson 2009; Tucker et al. 2011). 2) The SHE model significantly overestimates the heat flux in the exosphere region (Kn>1) (Volkov et al. 2011a, b). 3) Escape occurs on a molecule by molecules basis for atmospheres where λ ro >~3, where λ ro is the Jeans parameter defined at r o (Volkov et al. 2011a, b). Tucker 7

Radial Distance (km) Radial Distance km 4) A consistent solution for φ, <Eφ> and the macroscopic properties of an atmosphere can be obtained between the hydrodynamic equations and a kinetic model in the region Kn << 1 to Kn ~1 using the described fluid/dsmc approach (Tucker et al. 2011). 5) The energy flow <Eφ> ro is not 0 as assumed in the SHE model (Tucker et al. 2011). This assumption by the SHE affects solutions obtained by Eq. 2b. At the VSGC 2010 conference the DSMC results presented only considered hard sphere collisions and the internal energy of molecules was neglected. In additions for the fluid/dsmc method solar heating above r o was not considered. We note for Titan even with such assumptions regarding the collisional model used in the DSMC simulations were able to reproduce the Cassini density for the N 2 and without requiring significant mass loss as in the SHE model. However, the accuracy of the DSMC results depends upon how well collisions are described. Since the 2010 meeting we have performed simulations including internal energy of molecules, and using a temperature dependent cross-section (Tucker et al. 2011). While these changes do slightly change the quantitative results slightly (e.g. φ, n(r), T(r), u(r) etc.), the qualitative conclusion remains that thermal escape for bound planetary atmospheres λ exo >~ 3 occurs on a molecule by molecules basis. In moving forward the main goal is to evaluate thermal escape from multicomponent atmospheres. To this end we are using the DSMC model with Cassini density data to model thermal escape of N 2, CH 4 and H 2 from Titan s atmosphere, see Fig. 4. It is important to note the most recent SHE model results that consider multiple species have backed off the assertion that that N 2 escapes Titan at rates excessively larger than Jeans (Strobel 2009). These new results attribute the dominant portion of the escaping mass flux to CH 4 2.5x10 27 s -1 (λ exo = ~30) and H 2 9x10 27 (λ exo = ~4). However, using the corresponding exobase values for that result the above rates would are 10 10 and ~3 times the Jeans values respectively. Still CH 4 is considered to escape at large rates even though it has a mass 8 times that of H 2. The SHE model results appear to contradictorily suggest that hydrodynamic escape of an atmosphere is more favorable for larger Jeans parameters. We attribute this discrepancy to the assumptions inherent to the SHE model approach. The preliminary DSMC result shown in Fig. 4, again suggest that CH 4 will not escape thermally at rates significantly larger than Jeans similar to (Tucker and Johnson 2009), and for H 2 we obtain a rate that is 1.2*Jeans. 4200 4100 4000 3900 3800 3700 3600 r exo = 4000 km 4200 4100 4000 3900 3800 3700 3600 118 120 122 124 126 128 130 132 134 10 6 10 7 10 8 10 9 10 10 Density (cm -3 ) Temperature (K) Figure (4): Comparison of n(r) vs. radial distance: DSMC result (N 2 (solid curve,) CH 4 (dash dotted curve), H 2 (dotted curve)) to Cassini data (N 2 (triangles,) CH 4 (squares), H 2 (circles)) for the major species in Titan s atmosphere. The inserted graph shows the corresponding single species temperatures for the DSMC result. Also as shown in Fig. 4 the current DSMC results do not match the Cassini Tucker 8

density data for H 2. Therefore, we are performing DSMC simulations with more realistic cross-sections, based on available experimental data. Below the exobase H 2 is a trace species in Titan s atmosphere, but above the exobase it becomes the dominant species, therefore it is important to describe the collisions appropriately. Furthermore, these simulations are very time consuming so we are evaluating techniques to reduce the computational time. After a series of simulations are performed to evaluate how the choice of collision model used in the DSMC simulation affects the n(r), T(r) and φ for H 2, the effect of non-thermal processes occurring in the upper atmosphere on the density distribution will be considered. Acknowledgements This work is supported by the NSF and the VSGC. Special thanks to Justin Erwin and Alexey Volkov for discussions, and to Darrell Strobel for providing solar heating data. References Bird, G.A., 1994. DSMC procedures in a homogenous gas. In: Molecular Gas Dynamics and the Direct Simulation of Gas Flows, pp. 218 256. Clarendon Press, Oxford, England. Chamberlain, J. W. 1960. Interplanetary Gas. II. Expansion of a model of the Solar Corona. Astrophys. J 131, 47-56. Chamberlain, J. W. 1961. Interplanetary Gas. III. Hydrodynamic Model of the Corona. Astrophys. J 133, 675-687. Chamberlain, J. W., Hunten D., 1987. Theory of Planetary Atmosphere. Academic Press, New York. Cui, J., Yelle, R.V., Volk, K., 2008. Distribution and escape of molecular hydrogen in Titan s thermosphere and exosphere. J. Geophys. Res. 113. J. Cui, R.V. Yelle, V. Vuitton, J.H. Waite Jr., W.T. Kasprzak, D.A. Gell, H.B. Niemann, I.C.F. Müller-Woodarg, N. Borggren, G.C. Fletcher, E.L. Patrick, E. Raaen, B.A. Magee, Icarus, 200, 581-615 (2009). Grusinov, A., 2011. The rate of thermal atmospheric escape, arxiv:1101.1103v1 [astro-ph.ep] 5 Jan 2011. Hirschfelder, J. O., Curtiss C. F., Bird, R. B., 1964. Molecular Theory of Gases and Liquids. New York: Wiley; 2nd corrected printing. Hubbard, W.B., Yelle, R.V., Lunine, J.I., 1999. Nonisothermal Pluto Atmosphere Models. Icarus 84, 1-11. Hunten, D. M.,Watson, A.J., 1982. Stability of Pluto s Atmosphere. Icarus 51, 655 657. Jeans, J. H., M. A., F. R. S., 1916. The Dynamical Theory of Gases: The Outer Atmosphere. Cambridge University Press, 351-363. Johnson, R. E., 1990. Energetic Charged- Particle Interactions with Atmospheres and Surfaces. Springer-Verlag, Berlin. Johnson, R.E., 2009 Phil. Trans. R. Soc. A. 367,753-771. Johnson, R.E., M.R. Combi, J.L. Fox, W-H. Ip, F. Leblanc, M.A. McGrath, V.I. Shematovich, D.F. Strobel, J.H. Waite Jr, "Exospheres and Atmospheric Escape", Chapter in Comparative Aeronomy, Ed. A. Nagy, Space Sci Rev 139: 355-397, (2008). Johnson, R.E., Combi, M. R., Fox, J. L., Ip, W-H., Leblanc, F., McGrath, M. A., Kasting, J.F., Pollack, J.B., 1983. Loss of water from Tucker 9

Venus I. Hydrodynamic Escape of Hydrogen. Icarus 53 479-508. Olkin, C.B., Wasserman, L.H., Franz, O.G., The mass ratio of Charon to Pluto from the Hubble Space Telescope astrometry with fine guidance sensors, Icarus,164, 254-259, (2003). Shematovich, V. I, Strobel, D. F., Waite J.H., Jr., Exospheres and Atmospheric Escape. Comparative Aeronomy, edited by. A. Nagy et al., Space Sci. Rev, doi 10.1007/s11214-008- 9415-3 (2008) Krasnopolsky, V.A., 1999. Hydrodynamic flow of N 2 from Pluto. J. Geophys. Res. 104, 5955 5962. McNutt, R.L., 1989. Models of Pluto s upper atmosphere. Geophys. Res. Lett. 16, 1225 1228. hydrodynamic to Jeans escape. Astrophys. J., 729, L24 Volkov, A. N., Tucker, O. J., Erwin, J. T. and Johnson, R. E., 2011b. Kinetic Simulations of Thermal escape from a single component atmosphere. submitted. Waite Jr., J.H., et. al., 2005. Ion neutral mass spectrometer results from the first flyby of Titan. Science 308, 982-986. Watson, A.J., Donahue, T.M., Walker, J.C.G., 1981. The dynamics of a rapidly escaping atmosphere: Applications to the evolution of Earth and Venus. Icarus 48, 150 166. Yelle, R.V., Borggren, Cui, J., Muller- Wodarg, I.C.F., 2008. Methane escape from Titan s atmosphere. J. Geophys. Res 113. Strobel, D.F. N2 escape rates from Pluto s atmosphere Icarus 193, 612-619 (2008a). Strobel, D. F., Titan's hydrodynamically escaping atmosphere. Icarus 193, 588 594 (2008b). Strobel, D.F. Titan's hydrodynamically escaping atmosphere: Escape Rates and Exobase Structure, Icarus 202, 632 641 (2009). Tucker, O.J. and Johnson R.E., 2009. Thermally driven atmospheric escape: Monte Carlo simulations for Titan s atmosphere. PSS 57, 1889-1894. Tucker, O.J., Erwin, J.T., Volkov, A.N., Johnson, R.E., Deighan, J.I., 2011. Thermally driven escape from Pluto s atmosphere: A fluid/hybrid model. (to be submitted) Trafton, L., 1980. Does Pluto Have a Substantial Atmosphere?. Icarus 44, 53-61. Volkov, A. N., Johnson, R. E., Tucker, O. J. and Erwin, J. T. 2011a. Thermally-driven atmospheric escape: Transition from Tucker 10