Hydrodynamic Escape from H2 Rich Atmospheres

Christopher. D. Parkinson Division of Geological and Planetary Sciences, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA, 91125, USA (e-mail: [email protected])

Mark I. Richardson Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA, USA; 1200 E. California Blvd., Pasadena, CA, 91125, USA

David J. Hill Graduate Aeronautics Laboratory, California Institute of Technology, Pasadena, CA, USA; 1200 E. California Blvd., Pasadena, CA, 91125, USA

ABSTRACT Atmospheric loss processes have played a major role in the evolution and habitability of the terrestrial planet atmospheres in our solar system. The hydrogen escape rate to space is the key parameter that controls the composition of the primitive terrestrial atmosphere, including its possible methane concentration. Most models of the early atmosphere assume that hydrogen escapes at the diffusion-limited rate (Walker, 1977), but this need not necessarily have been true. A CO2- or CH4-rich primitive atmosphere may have been relatively cool in its upper regions, and the escape may therefore have been limited by energy considerations rather than by diffusion. Resolving questions of hydrogen escape for these cases requires solving a set of hydrodynamic equations for conservation of mass, momentum, and energy. Hydrodynamic escape may also be important for Close-in Extrasolar Gas Giants (CEGPs) such as HD209458b, which has recently been observed to be losing hydrogen, carbon and oxygen by the Hubble Space Telescope (Vidal-Madjar et al., 2003; 2004). Although planetary hydrodynamic escape models have been created in the past (Watson et al., 1981; Kasting and Pollack, 1983; Chassefiere, 1996), the problems were solved by integrating the coupled, time independent mass, momentum, and energy equations for the escaping gas from the homopause out to infinity. Solving the one- dimensional, steady state approximation becomes problematic at the distance where the outflow becomes supersonic. A new technique has been developed for the treatment of hydrodynamic loss processes from planetary atmospheres. This method overcomes the instabilities inherent in modelling transonic conditions by solving the coupled, time dependent mass, momentum, and energy equations, instead of integrating time independent equations. We validate a preliminary model of hydrodynamic escape against simple, idealized cases (viz., steady state and isothermal conditions) showing that a robust solution obtains and then compare to existing cases in the literature as cited above. The general tools developed here can be applied to the problems of hydrodynamic escape on the early Earth and close-in extrasolar gas giant planets and results from these analyses are the subject of two companion papers.

1 1. Introduction the steady state hydrodynamic escape problem for Venus Use an iterative method in which the Hydrodynamic escape is an important process momentum and energy equations are simultane- in atmospheric evolution of the terrestrial plan- ously solved Not able to get an exact solution at ets and CEGPs and can irreversibly change the the critical point obtaining the supersonic solution composition of planetary atmospheres from pri- Instead, they obtained subsonic solutions and ar- mordial values. Hydrogen escape is of particular gued that the escape flux can be close to the crit- importance as it affects the oxidation state of the ical escape flux Method included infrared cooling atmosphere and because it results in the loss of by H2O and CO2 while only EUV absorption con- water vapour sidered by Watson et al. (1981) In Jeans escape, particles at the exobase mov- Chassefiere (1996) solves steady state hydro- ing in the outward direction with sufficient veloc- dynamic escape problem from lower boundary to ity (i.e. high enough kinetic energy) can escape exobase level Position of exobase level is deter- from the planettypically the vertical flow from the mined when the mean free path becomes greater atmosphere is small. Hydrodynamic escape arises than the scale height Outgoing flow at exobase is when the flow speed becomes large set to be equivalent to a modified Jeans escape Hydrodynamic escape also differs from gas- (ionization and interaction between escaping par- kinetic evaporation in that in some circumstances ticles and solar wind considered) Application to a substantial fraction of the entire thermospheric water-rich early Cytherian atmosphere energy budget is used to power escape of gas from the atmosphere; it is possible that heavier species 2. Model Description can be dragged along during hydrodynamic escape. Under this circumstance, it is expected that 2.1. Governing Equations atmospheric expansion due to hydrodynamic es- The generalised 3-dimensional form of the hy- cape will be the dominant loss process. drodynamic equations as given by Chamberlain For Instance(outstanding problems) Did early and Hunten (1987) are Venus initially have an ocean? Hydrodynamic escape modelling using a water-rich atmosphere on Conservation of mass: Venus can help assess this problem (Kasting and ∂ Pollack, 1983) Isotopic ratios (i.e. fractionation: ρ + ∇ · (ρv) = 0 (1) D/H, N, and noble gases) are very different on ∂t terrestrial planets even though they are believed Conservation of momentum: to be formed from similar material (Hunten et al., 1987; Pepin, 1991) and Greenhouse warming by ∂ (ρv) + ρ(v · ∇)v + ∇P = ∇ · (ν∇(ρv)) + ρg + T methane in the atmosphere of the early Earth? ∂t CH4 density on early Earth dependent on HDE, (2) strongly influencing its atmospheric climate and Conservation of energy: composition, i.e. (Pavlov et al., 2000; 2001) blow- ∂ off on HD209458b (Osiris) (Vidal-Madjar et al., ( E +v ·∇(E +P ) = ∇·(κ∇E)+ρv ·g +Q (3) 2003; 2004) HD Escape Equations Some Previous ∂t Models Watson et al. (1981): shooting method or where ρ = mass density, trial-and-error method to solve steady state HDE v = velocity, equation for early Earth and Venus Set of solu- p = pressure = nkT = ρ k T , mv tions at the critical point (exobase) selected which n = number density (concentration), can match the zero temperature at infinity and m = mass of molecule, set temperature at the lower boundary. Calcu- k = Boltzmann constant, lated temperature and density at the boundary g = acceleration due to gravity, very sensitive to initial settings and I couldnt re- E = kinetic energy ≡ ρcvT , produce cases using that method T = temperature, Kasting and Pollack (1983) numerically solve cv = specific heat capacity for constant volume,

2 1 2 P = potential energy ≡ 2 v , κ = thermal conductivity, ν = viscosity, T = tidal force, Q = heating term.

In 1-dimensional, spherical coordinates these equations respectively become (vz ≡ u) (Le Veque 2002) ∂ρ ∂(ρu) 2 + = − ρu ∂t ∂r r

∂(ρu) ∂(ρu2 + P ) 2ρu2 1 ∂ ∂(ρu) GM + = − + (r2ν )−ρ +T ∂t ∂r r r2 ∂r ∂r r2

∂E ∂((E + P )u) 2 1 ∂ ∂T GM + = − ((E+P )u)+ρc (r2κ )−ρu +Q ∂t ∂r r v r2 ∂r ∂r r2

In order to apply the numerical method to (ex- tra) solar system objects, we need to utilise scaling parameters x =xx ˆ o in order to avoid instabilities introduced by having large numbers en- tering into the calculation of the solution (Ped- losky, 1987). Here x is our original parameter,x ˆ is the parameter in new units and xo is the scaling factor. Introducing the scaling parameters r = Ro 2 rrˆ o, ρ =ρρ ˆ o, t = ttˆ o, u =uu ˆ o =u ˆ ,P = Pˆ ρou , to o ˆ 2 and E = Eρouo and neglecting diﬀusion terms, we may recast the previous equations to be ∂ρˆ ∂(ˆρuˆ) 2 + = − ρuˆ ∂tˆ ∂rˆ rˆ

2 ˆ 2 2 2 ∂(ˆρuˆ) ∂(ˆρuˆ + P ) 2ˆρuˆ GMto ρˆ to + = − − 3 2 + T ∂tˆ ∂rˆ rˆ ro rˆ ρoro ˆ ˆ ˆ 2 3 ∂E ∂((E + P )ˆu) 2 ˆ ˆ GMto ρˆ to + = − ((E+P )ˆu)− 3 2 + 2 Q ∂tˆ ∂rˆ rˆ ro rˆ ρoro 2 GMto where 3 ∼ 1 ρoro

The hydrodynamic ﬂow problem described by these equations can be solved using ﬁnite volume methods (viz., the Godunov technique) since these are linear advection equations (hyperbolic) (cf. Le Veque, 2002) and this is discussed in more detail in the appendix.

3. Results and Discussion

3 A. Method of solution Following Le Veque (2002), our equations above represent the generalised system of equations of the form

κ(x)qt + f(q)x = Ψ(q; x; t)(A.1) where q = q(x; t) ∈ Rm and in one space dimension, our model can be used to solve this system. The standard case of a homogeneous conservation law has κ ≡ 1 and Ψ ≡ 0,

qt + f(q)x = 0 (A.2) The ﬂux function f(q) can also depend explicitly on x and t as well as on q. Hyperbolic systems that are not in conservation form, e.g., qt + A(x; t)qx = 0 (A.3) can also be solved. The basic requirement on the homogeneous system is that it be hyperbolic in the sense that a Riemann solver can be speci ed that, for any two states Qi−1 and Qi, returns a set of Mw waves p p Wi−1/2 and speeds si−1/2 satisfying

Mw X p Wi−1/2 = Qi − Qi−1 ≡ Qi−1/2 p=1

− The Riemann solver must also return a left-going ﬂuctuation A ∆Qi−1/2 and a right-going ﬂuctuation + A ∆Qi−1/2. In the standard conservative case (A.2) these should satisfy

− + A ∆Qi−1/2 + A ∆Qi−1/2 = f(Qi) − f(Qi−1) (A.4) and the fluctuations then define a “flux-difference splitting” as described in Chapter 4 of Le Veque (2002). Typically − X p − p + X p + p A ∆Qi−1/2 = (si−1/2) Wi−1/2; A ∆Qi−1/2 = (si−1/2) Wi−1/2 (A.5) p p where s− = min(s; 0) and s+ = max(s; 0). In the nonconservative case (A.3), there is no “flux function” f(q), and the constraint (A.4) need not be satisfied. Only the fluctuations are used for the first-order Godunov method, which is implemented in the form ∆t Qn+1 = Qn − [A+∆Q + A−∆Q ](A.6) i i ∆x i−1/2 i+1/2 assuming κ ≡ 1. A general description of the user supplied Riemann solver is given below. Typically the + − Riemann solver first computes waves and speeds and then uses these to compute A ∆Qi−1/2 and A ∆Qi−1/2 internally in the Riemann solver. The waves and speeds must also be returned by the Riemann solver in order to use the high-resolution methods described in Chapter 6 of Le Veque (2002). These methods take the form ∆t ∆t Qn+1 = Qn − [A+∆Q + A−∆Q ] − (Fˆ − Fˆ )(A.7) i i ∆x i−1/2 i+1/2 ∆x i+1/2 i+1/2 where M 1 Xw ∆t Fˆ = |sp |(1 − |sp |W˜ p (A.8) i−1/2 2 i−1/2 ∆x i−1/2 i−1/2 p=1 ˜ p p p p Here Wi−1/2 represents a limited version of the wave Wi−1/2, obtained by comparing Wi−1/2 to Wi−3/2 if p p p s > 0 or to Wi+1/2 if s < 0. When a capacity function κ(x) is present, the Godunov method becomes

n+1 n ∆t + − Qi = Qi − [A ∆Qi−1/2 + A ∆Qi+1/2](A.9) κi∆x

4 See Chapter 6 of Le Veque (2002) for a discussion of this algorithm and its extension to the high-resolution method. If the equation has a source term, a routine must also be supplied that solves the source term equation qt = Ψ over a time step. A fractional step method is used to couple this with the homogeneous solution, as described in Chapter 17 of Le Veque (2002). Boundary conditions are imposed by setting values in ghost cells each time step, as described in Chapter 7 of Le Veque (2002). The Riemann solver is the crucial user-supplied routine that species the hyperbolic equation being solved. L The input data consists of two arrays ql and qr. The value ql(i, :) is the value Qi at the left edge of the i’th R cell, while qr(i, :) is the value Qi at the right edge of the i’th cell. Normally ql = qr and both values agree n with Qi , the cell average. More flexibility is allowed because in some applications, or in adapting clawpack to implement different algorithms, it is useful to allow different values at each edge. For example, we might want to define a piecewise linear function within the grid cell and then solve the Riemann problems between these values. This approach to high-resolution methods is discussed in Chapter 6 of Le Veque (2002). Note that the Riemann problem at the interface xi−1/2 between cells i − 1 and i has data

R L leftstate : Qi−1 = qr(i − 1; :); rightstate : Qi = qr(i; :) (A.11)

This notation is rather confusing since normally we use ql to denote the left state and qr to denote the right state in specifying Riemann data. Our Reimann solver also has input parameters auxl and auxr that contain values of the auxiliary variables (see Section 1.9, Le Veque (2002)). Normally auxl = auxr = aux when the Riemann solver is called from the standard clawpack routines. Our solver must solve the Riemann p problem for each value of i, and return the following: the left and right going ﬂucuation, the vector Wi−1/2 representing the jump in q across the p’th wave in the Reimann solution at xi−1/2, for p = 1,2,3, ..., and the p wave speed si−1/2 for each wave. For Godunov’s method, only the ﬂuctuations amdq and apdq are actually used, and the update formula (A.6) is employed. The waves and speeds are only used for high-resolution correction terms (A.8) as described in Chapter 6 of Le Veque (2002). The values in any auxiliary arrays (see Section 1.9) are also passed into the Riemann solver routine since these arrays typically contain spatially- varying information that is needed in solving the Riemann problem. For consistency with the ql and qr notation, two arrays auxl and auxr are passed in, but in the standard clawpack implementation these arrays are identical and simply agree with aux in the main routine. So one can also use auxl(i,:), for example, as the value in the ith grid cell.

Acknowledgements C. D. Parkinson wishes to acknowledge that some of this work was supported by the National Aeronautics and Space Administration through the NASA Astrobiology Institute under Cooperative Agreement No. CAN-00-OSS-01 and issued through the Oﬃce of Space Science. C. D. Parkinson would like to thank Y. L. Yung, R. L. Shia, and M. C. Liang for helpful discussions implementing viscous damping and resolving boundary layer issues as well as J. F. Kasting for numerous helpful discussions regarding the steady state solution and illuminating the Earth Earth problem for me.

5 References Chasseﬁere, E., Hydrodynamic escape of hydrogen from a hot water-rich atmosphere: The case of Venus, J.G.R., 48, 26,039-26,055, 1996. Chamberlain, J. W. and D. M. Hunten, Theory of planetary atmospheres, an introduction to their physics and chemistry, Academic Press, Toronto, 1987. Kasting, J.F., Pollack, J.B., Loss of Wather from Venus, I. Hydrodynamic escape of hydrogenIcarus, 53, 479-508, 1983. Le Veque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cam- bridge, 2002. Pedlosky, J., Geophysical Fluid Dynamics, 2nd ed. New York: Springer-Verlag, 1987. Vidal-Madjar, A., Lecavelier des Etang, A., Desert, J.-M., G.E. Ballester, Ferlet, R., Hebrard, and Mayor, M., An Extended Upper Atmosphere Around Planet HD209458b, Nature, 422, 143-146, 2003. Vidal-Madjar, A., Desert, J.-M., Lecavelier des Etang, A., Hebrard, G., Ballester, G.E., Ehrenreich, D. Ferlet, R., McConnell, J.C., Mayor, M., Parkinson, C. D., Detection of Oxygen and Carbon in the Upper Atmosphere of the Extrasolar Planet HD209458b, Ap. J., 604, L69, 2004. Walker, J. C. G., Evolution of the Atmosphere. New York: MacMillian, 1977. Watson, A.J., Donahue, T.M., Walker, J.C.G. , The dynamics of a rapidly escaping atmosphere: Appli- cation to the evolution of Earth and Venus, Icarus, 48, 150-166, 1981.