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 es- cape. 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 es- cape 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 scal- ing parameters x =xx ˆ o in order to avoid insta- bilities 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 scal- ing 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).