The rate of thermal atmospheric escape. Andrei Gruzinov CCPP, Physics Department, New York University, 4 Washington Place, New York, NY 10003 1 1 0 2 ABSTRACT n a A formula is derived for the rate of thermal atmospheric escape, valid, and asymptotically J exact, at low Knudsen number. 5 1. The thermal escape problem ] P Consider a planetary atmosphere without stir- E ring and radiative heating or cooling above the . h nominalsurface. Theatmospherecannotbestatic p since a static atmosphere reaches thermodynamic - o equilibrium. The Boltzmann distribution would r then give a finite density at infinite distances, t s which is unphysical. a 0 [ The atmosphere must be in a state of perma- nent escape. We want to calculate the rate of es- 1 v cape and the resulting temperature and density 3 profiles. 0 The thermal escape problemhas a long history -5 1 (Jeans 1904, Parker 1958, Hunten 1982). But 1 in recent applications to planetary bodies, there . 1 hasbeendisagreementonwhichmodel–hydrody- 0 namic or free molecular flow – should be applied 1 1 (Johnson 2010). We believe that all issues have -10 : been finally settled by a direct molecular dynam- v ics simulations of Volkov et. al. (2010). Here we i X just show how the Volkov et. al. (2010) results r can be obtained analytically. a -15 It would seem that the escape rate calculation 0 10 20 30 40 is only possible via molecular dynamics or Boltz- mann numerical simulation, because the exobase Fig. 1.— Escape rate for Kn = 10−8, c = 5/2, region,wherethetransitionfromhydrodynamicto p α = 1. Je < 2.5 is the Parker regime, 2.5 < Je < free molecular flow occurs, cannot be treated an- 24 is the Fourier regime, Je > 24 is the Jeans alytically. But we show, for low Knudsen number regime. at the surface, that the exobase boundary conditions are either irrelevant or can be deduced due to large overlap between hydrodynamic and free molecular flow regions. This allows one to calculate the atmospheric escape using hydrodynamics and to derive a formula for the escape rate. For clarity, we approximate the molecular in- 1 teraction potential by a power law (this includes Formula (5) limits the Je range over which the hard-spheregasasalimitingcase). Then,byscal- Parker rate applies. The Parker regime is de- ing arguments, the escape rate must be given by scribed in 3. § M is the Jeans (1904) escape rate. It de- Jeans Ma=M (Je,Kn). (1) escape pends on Je only: Here we measurethe escape rate by the radialve- (Je+1)exp( Je) locity at the surface v, and the velocity is repre- MJeans = √2π − (7) sented by the isothermal Mach number Formula (5) extends the Kn range over which the v Jeans rate applies. The Jeans regime is described Ma . (2) ≡ T/m in 4. § p M is proportional to Kn, with the Je- Fourier The Jeans number is dependent coefficient of proportionality: GMm Je , (3) MFourier(Je,Kn)=F(Je)Kn. (8) ≡ TR Here F(Je) depends on Je, c , and p where G is the gravitational constant, M is the dlnκ planetmass,m isthe moleculemass, T isthe sur- α . (9) face temperature, R is the surface radius. The ≡ dlnT Knudsen number is Kn λ, where λ is the mean For example, for α=1, ∼ R free path. We define 1 F(Je)= (10) mκ Je c Kn , (4) − p ≡ Rρ T/m For uniformity of notation, we call it the Fourier p regime. The Fourier regime is described in 5. where κ is the thermal conductivity, ρ is the den- § To illustrate the formula, Fig. 1 shows the es- sity. cape rate as a function of Je, for Kn = 10−8, The escape rate formula is given in 2. The § α = 1, and c = 5/2. The different regimes were formula includes three branches. In 3-5, we give p § extended to the intersection points. Fig.1 is in the recipes for calculating the branches. In 6,we § qualitativeagreementwithmoleculardynamicsre- explain the origin of the recipes, describe numeri- sults of Volkov et. al. (2010). Quantitative com- calhydrodynamicswhichconfirmstheescaperate parison is done in 6, where we also show that formula,andcompareourresultstothemolecular § a simplified dissipative hydrodynamics model re- dynamics simulations of Volkov et. al. (2010). producesthemoleculardynamicsresultstoagood accuracy. 2. The escape rate formula In the next three sections we calculate the es- In the low-Knudsen limit, cape rate postulating the properties of the flow. The assumptions made can be shown to be self- consistent. But, in fact, what makes the results MParker(Je), Je 1, such that the corresponding R R Herevistheradialvelocity,uistheabsolutevalue density ρ=v /(r2v) goes to zero at large r. This 0 ofthetangentialvelocity,f θexp( (u2+v2)/2) requirement limits the range of possible values of ∝ − is the truncated Maxwellian. θ = 0 if simultane- v . 0 ously the energy is positive, (u2+v2)/2 Je>0, For c < 3, and also for c > 3, G < − p p and the radial velocity v is negative; otherwise 2c /(c 1), the smallest possible value is v = p p 0 θ =1. − c /(c 1), in which case the sonic point is p p − pat the surface. For cp > 3 and larger G, 5. Fourier regime G > 2c /(c 1), the sonic point moves out to p p − r >1, and v0 decreases. FixtheJeansnumberJe>cpandlettheKnud- It is the smallest possible v that we call sen number decrease. The escape rate then ap- 0 M . Time-dependent ideal hydrodynamics proaches Parker simulationsandsimulationsofVolkovetal(2010) MFourier =F(Je)Kn. (16) confirm that the flow does approach the smallest The coefficient F(Je) is calculated below. v allowed by stationary ideal hydrodynamics. 0 In the Fourier regime, the negative hydrody- The above results can be written as namic energy flux carried by the flow is almost exactlycompensatedbythethermalconductivity: c p M = , (13) Parker rcp−1 r2κT′ =Φ(c T G). (17) p − r for c <3, and also for c >3, Je<2c /(c 1). p p p p For c > 3, 2c /(c 1) < Je < c , M −de- Here Φ = ρvr2 =const is the flux per steradian, p p p p Parker creases, reaching zer−o at Je = c . The precise M = m = 1 was set by scaling, prime denotes p value of M is then given implicitly by an the r-derivative, kinetic energy and viscosity are Parker negligibly small. 1 More precisely, dissipative effects must become negligible Usingscalings,weputT =1atr=1,andthen after a possible boundary layer is passed. Whether the solveeq.(17)forr >1. WeadjustΦinsuchaway boundarylayerispresent,dependsonhowtheatmosphere isfed. Iftheboundarylayerispresent,wesimplymovethe nominalsurfaceup,beyondtheboundarylayer. 2 (cid:16)M2a2 +cp−Je(cid:17)cp−52 =(cid:16)cpc−p1(cid:17)cp−1(cp− 52)cp−254JMe2a. 3 as to get the temperature going to zero at large usestationarynon-idealhydrodynamics. Theflux radii (otherwise the flow either stalls or becomes conservation,theenergyfluxconservation,andthe singularat finite radii,neverreachinganexobase, momentum equations are 6). We want to solve the problem for arbitrary § Φ=ρvr2, α≡Fidrsltnκco/ndslindTer3j.ust the case α = 1. One can Qρvv=′ =Φ(v22(ρ+Tc)p′T −GρGr+)−4 134η(rr23vη((vv′′− vrv)))−′.r2κT′, check numerically or analytically, that there is − − r2 3r3 − r (25) only one solution of eq.(17) with correct asymp- Here Φ, Q are constant fluxes, η is the viscosity4. totic behavior at large r: The boundary conditions at r =1 are 1 T = (18) ρ=1,T =1,v =Φ (26) r GivenG=Jeandκ(T =1)=Kn,wemustfind Substitution into eq.(17) then gives a pair (Φ,Q) that gives the “true” flow 5. Which κ flows are “true” is decided by the boundary con- =(G c )Φ. (19) T − p ditions at the exobase, and it would appear that onecannotproceedwithoutamoleculardynamics Or, in different notations, simulation of the exobase region. 1 But it turns out that for Kn 1 the mere F(Je)= . (20) ≪ Je c requirement that the exobase exists allows to cal- p − culate the escape flow in the Parker and Fourier For arbitrary α < 1, numerical integration of regimes. In the Jeans regime, this trick no longer eq.(17) shows that there is still a unique choice of works– there are many outflows with anexobase. Φ which gives a non-singular zero-at-infinity tem- What allows to solve the problem in the Jeans perature. One can show that for 0 0, eqs.(28, 32) give a density growing at in- (33) finity. We then integrate eq.(30) backwardsin radius, to In summary, small-velocity flows have no r < 1. Simultaneously we integrate eq.(28) and exobase, and large-velocity flows are ideal, giv- calculate Kn(r). The integration is terminated ing the Parker flow. The lack of collisions beyond when the desired value of Kn is reached. We ad- the exobase cannot change the escape rate of the just G so that the desired value of Je is reached highly supersonic outflow. simultaneouslywiththe desiredvalueofKn. This Fourier regime – At fixed Je > c , as Kn de- gives the escape rate for these Kn and Je. One p creases, the set of pairs (Φ,Q) which give solu- also gets correct temperature and density profiles tions with an exobase shrinks to a single point for G 1 and also for r 1 at any G. We have ≫ ≪ (M ,0). If (Φ,Q)=(M ,0), the flow ei- checked, by numerical integration of the full sys- Fourier Fourier 6 ther becomes singular at a finite radius or goes to tem (25), that neglecting viscosity and inertia is infinite radii, without ever reaching local Kn 1. justified, again for G 1 and also for r 1 at ∼ ≫ ≪ So in this case too, we can claim that M is any G. Fourier an exact asymptotic even without knowing what We have used the simplified hydrodynamics happens at the exobase. (30,33) to establish the logarithmic accuracy of Theonlywayto reachanexobaseistosetQ= the escape rate formula (5) in the Fourier-Jeans 0 in eq.(30), and then tune Φ, as described in 5. transition region (Appendix). § Jeans regime – Fix Kn and let Je increase. Comparison with Volkov et. al. (2010) – Now The outflow velocity becomes so small that ther- comparethepredictionsofourescaperateformula malconductionineq.(30) dominatesandwe geta eq.(5) to the molecular dynamics simulations of 5 Volkov et. al. (2010). We put c =5/2, α=1/2. the Parker wind model ( 3). Both temperature p § WealsorememberthatourdefinitionofKnisdif- and density decrease algebraically. In the Jeans ferent by a constant factor from the conventional regime, we have a nearly isothermal slowly escap- hard-sphere Knudsen Kn =0.34Kn. ing atmosphere, with exponential density. In the hs We find that for Je = 3,10,15, Kn = Fourierregime,thermalconductivity balancesthe hs 10−2,10−3,10−4 the simplified hydrodynamics ideal energy flow ( 5). The temperature and the § (30,33) (with Kn = 3) predicts the escape rates density decrease algebraically. 0 which agree with the results of Volkov et. al. (2010) to about 10% for all points but one. I thank Robert Johnson for useful discussions. Let us discuss this discrepancy. For Je = 3, the escape rate formula (5) gives for Kn = Appendix hs 10−2,10−3,10−4 (using eqs.(16, 23, 24)) The escape rate formula (5) is asymptotically exact in the following sense: Ma =6.4, 6.4, 6.4 (34) Knhs Je=constcp,Kn 0: 1, (38) =5.6, 6.0, 6.2 (35) → MFourier → Kn hs M escape Kn=const,Je : 1, (39) Fig.4 of Volkov et al (2010) gives →∞ M → Jeans ln(M ) Ma Je>c ,Kn 0: escape 1. =5.0, 6.8, 4.1 (36) p → ln(min(M ,M )) → Kn Fourier Jeans hs (40) The simplified hydrodynamics works as expected In words: (i) As Knudsen decreases at fixed – as we go deeper into the Fourier regime, pre- Jeans, the Parker and Fourier formulas approach dicted Ma approaches MFourier. The first two val- the true escape rate everywhere but near the ues of Volkov et. al. are also reasonable – the Parker-Fourier transition. As Knudsen decreases, error of MFourier decreases as we get deeper into the Jeans number width of the Parker-Fourier theFourierregime. Butthelastpointshouldhave transition region shrinks to zero. (ii) As Jeans been evencloserto the Fouriervalue of 6.4. Some increases at fixed Knudsen, the Jeans formula ap- 50% error is unexpected. proachesthe true escaperate. (iii) Inthe Fourier- We believe that it is the molecular dynamics Jeans transition interval, both formulas are valid, results at Je=3, Kn =10−4 which have a large but only with logarithmic accuracy. hs error. Wehavechecked,byintegratingthefullsys- tem (25) numerically, that a hydrodynamic flow, REFERENCES startingatKn =10−4,Ma=4.1 10−4,Je=3, hs never goes above Kn =1.5 10−×3 for any Q. It Jeans, J. H., 1904, “The dynamical theory of hs × gases”, Chapter XVII, University Press, Cam- seems unlikely that molecular dynamics can no- bridge. ticeably deviate from hydrodynamics at such low Knudsen numbers. Parker,E. N., 1958, ApJ, 128, 664 Hunten, D. M., 1982, Planet. Space Sci., 30, 773 7. Summary Johnson, R. E., 2010,ApJ, 716, 1573 Formula (5) gives the thermal escape rate at Volkov, A. N., et al, 2010, arXiv:1009.5110 low Knudsen. The escape occurs either in Parker, in Fourier, or in the Jeans regime. In the Parker regime, we get a supersonic ideal hydrodynamical wind. This2-columnpreprintwaspreparedwiththeAASLATEX The temperatureanddensity profilesaregivenby macrosv5.2. 6

The rate of thermal atmospheric escape

Similar The rate of thermal atmospheric escape