DSMC simulations of thermal escape

Transcription

1 DSMC simulations of thermal escape Alexey N. Volkov, R.E. Johnson, O.J. Tucker, J.T. Erwin Materials Science & Engineering University of Virginia, USA Financial support is provided by NASA through Planetary Atmospheres Program (Grant NNX9AB68G) and by NASA s Cassini Mission at the Jet Propulsion Laboratory via a subgrant through SwRI 1

2 Outline Motivation Kinetic and continuum models of thermal escape Hydrodynamic and Jeans regimes of thermal escape Comparison of predictions based on continuum and kinetic models Conclusion

3 Motivation: Validation of continuum models of thermal escape Exosphere r Nearly freemolecular flow, l / H >> 1 Thermal escape is the escape of molecules with thermal energies above the gravitational binding energy in the collision-free part of the atmosphere Following to Parker (1964), thermal escape has been described by continuum models l / H = 1 Exobase Transitional flow Continuum models based on the Fourier law for thermal conduction can fail in the rarefied part of the atmosphere Purposes of the kinetic study: To reveal the effects of rarefaction and nonequilibrium F G = GMm r ρ, T R Continuum flow, l / H << 1 To find domains of different escape regimes (hydrodynamic vs. Jeans) To verify continuum models vs. kinetic one To develop a criterion for the transition from Jeans to hydrodynamic escape under heating by stellar radiation 3

4 Continuum model of thermal escape [Parker, Astrophys. J. 139, 1964] 4π r ρu = mφ du 1 d GM u = ( ρrt ) dr ρ dr r u GM 4πr q + mφ C T + p r Fourier law: Boundary conditions: = Ψ ω ( T T ) dt dr q = κ / / ρ( R = T ) = ρ, T ( R ) Single-component atmosphere No heating The number, Φ, and energy, Ψ, escape rates can be determined only with additional conditions usually applied in the rarefied part of the atmosphere, e.g., Critical (sonic) point exists [Parker, Astrophys. J. 139, 1964]. Flux-limited approximation, Ψ = [Watson et al., Icarus 48, 1981]. 4

5 Kinetic model of thermal escape [Volkov et al., Phys. Fluids 3, 11] Exit boundary Source surface O Results are scaled by the source Knudsen number Kn and Jeans parameter λ : l Kn = R R R 1 λ R = H F G Returning molecule v Orbiting molecule v Escaping molecule r The Boltzmann equation for onedimensional steady-state flow. Monatomic gas, γ = 5/3 HS, ω = 1/ PMM, ω = 1 Diatomic gas, γ = 1.4 PMM + Larsen-Borgnakke The source surface: Evaporation of outgoing molecules with Maxwell-Boltzmann distribution. Complete absorption of incident molecules. The exit boundary: Free escape of molecules with kinetic energies above the gravitational binding energies. Simulations are performed with the Direct Simulation Monte Carlo (DSMC) method. 5

6 Hydrodynamic and Jeans regimes 6

7 Transition from hydrodynamic to Jeans escape regime with increasing Jeans parameter Mach number distributions in a monatomic gas λ = -, Hydrodynamic regime λ = -3, Transitional regime λ = 3-, Jeans regime Narrow transitional range In the Jeans regime, the sonic point, if exists, is positioned far above the exobase, where the flow is nearly free molecular [Volkov et al., Astrophys. J. Lett. 79, 11; Phys. Fluids 3, 11] 7

8 Transition from hydrodynamic to Jeans regime: Number and energy escape rates Monatomic gas (PMM) No qualitative difference between mon- and diatomic gases, but limited quantitative difference (escape rates differ within 5%) Transitional region shifts towards larger λ with increasing number of internal degrees of freedom In the Jeans regime at λ > ~ 6, the escape rate is within factor of of the Jeans escape rate at the exobase Φ = 4πR [Volkov et al., Astrophys. J. Lett. 79, 11; Phys. Fluids 3, 11] Ψ n ( k / m) T π = 4π R n kt kt / m 8

9 Summary: Hydrodynamic vs. Jeans regime at Kn << 1 Hydrodynamic regime Jeans regime λ < 3 λ > 3 4 Sonic point is in the dense part of the flow Sonic point, if exists, is far above the exobase Bulk gas velocity corresponds to the most probable velocity of molecules Minor contribution of thermal conductivity to the energy transfer Escape rate is independent of Jeans parameter Energy escape rate is on the order of energy carried by molecules outgoing from the source surface Bulk gas velocity appears due to non-equilibrium velocity distribution Determinant role of thermal conductivity in the energy transfer At λ > ~6, the escape rate is within a factor of from the Jeans escape at the exobase Energy escape rate is small, but has non-negligible effect on the flow structure 9

10 Continuum model vs. kinetic one in the hydrodynamic regime 1

11 Isentropic model for hydrodynamic escape [Parker, Interplanetary dynamical processes, 1963] General continuum model Isentropic model 4π d dr r C nu = Φ T V lg = ρ γ 1 P T P = 4π T ρ r nu = = Φ T γ 1 γ 1 ρ d dr C u T + GM r p = = 1 d P 4πr Q (4πr q) mφ dr P C p u GM u GM T + = C pt + r R γ = C p /C V, Poisson ratio Assumption: Physically-realized solutions go transonic with du/dr > everywhere Conclusion: The unique transonic solution goes through the critical (sonic) point Such a solution exists only at γ < 3/ γ < λ γ < γ 1 No appropriate solution for a monatomic gas Quantitative difference between mon- and diatomic gases No solutions for a full range of Jeans parameters in hydrodynamic regime 11

12 DSMC results on flow structure in the hydrodynamic regime Gas velocity distribution in monatomic gas, Solutions with the decreasing velocity (du/dr < ) exist Solutions exist for a full range of λ specific for the hydrodynamic regime No quantitative difference between mon- and diatomic gases Reconsideration of Parker s assumptions is necessary! 1

13 Critical analysis of the isentropic model (I) 4π r nu = Φ T ρ = T γ 1 γ 1 ρ C p T + u GM u GM = C pt + r r, dma dλ = Ma λ γ 1 λ γ + 1 γ + 1 λ ( γ 1) Ma γ γ + 1 ( γ 1) Ma γ 5 3γ γ + 1 Ma λ(r) = GMm / (kt(r)r), Local Jeans parameter Ma(r)=u(r)/(γkT(r)/m) 1/, Local Mach number Isentropic solution in the form λ = λ(ma) does not depend on other gas dynamic parameters and distance 13

14 Critical analysis of the isentropic model (II) Mach number Jeans parameter Mach number distance, dma dλ = Ma λ γ + 1 λ ( γ 1) Ma γ γ 1 γ γ λ ( γ 1) Ma Ma γ + 1 γ γ + 1 dma dr = Ma (1 Ma γ + 1 λ ( γ 1 Ma ) r γ ) I, Solution family with arbitrary Ma, but dma/dr > at Ma > 1 II, Solution family with arbitrary Ma, but dma/dr < at Ma > 1 III, Subsonic solution family IV, Supersonic solution family One can expect that physically-realized solutiona are close to isentropic solutions of family I 14

15 Isentropic model vs. kinetic one (λ <-3) Monatomic gas (PMM) Diatomic gas Kn =1-3 Experiment Flow zones Knudsen layer dλ/dr depends on γ Continuum ~isentropic zone dλ/dr < Non-equilibrium far field Thin, non-equilibrium Knudsen layer, dλ/dr > 15

16 Isentropic model vs. kinetic one (λ <-3) Monatomic gas (PMM) Diatomic gas Kn =1-3 Experiment Flow zones Far field dλ/dr depends on γ Continuum ~isentropic zone dλ/dr < Non-equilibrium far field Thin, non-equilibrium Knudsen layer, dλ/dr > 16

17 Summary: Isentropic model vs. kinetic one Parker s hypothesis is invalid and solutions with decreasing velocity are typical. The gas velocity never drops to zero, since in the collision-free far field the flow never follows the isentropic model. The escape rate is formed within the Knudsen layer [Volkov et al., Phys. Fluids 3, 11]. In order to predict the escape rate based on surface parameters in the hydrodynamic regime, one needs to consider the flow in the Knudsen layer. Experiment Arguments based on the existence of Parker s critical solution (e.g., Levi and Podolak, Icarus, 9) can not be used to limit somehow the unknown parameters of gas expansion. 17

18 Continuum model vs. kinetic one in the Jeans regime 18

19 Comparison of continuum and kinetic models in the Jeans regime (λ >3-4) Continuum simulation is performed with Φ and Ψ found in kinetic simulations Perfect agreement of solutions below the exobase Continuum model fails to predict the structure of non-equilibrium flow above the exobase Continuum matching solution does not have a critical point and is completely subsonic. Parker s hypothesis on the existence of the critical point seems to be incorrect if the critical point lies above the exobase, where the hydrodynamic description fails. 19

20 Heat flux in the Jeans regime (λ >3-4) Fourier heat flux is calculated as dt q F = κ( T ) dr 5 k 1 k κ( T ) = T 3 d Pr π m 1/ where T(r) is found in DSMC simulations Continuum model fails above the exobase because the Fourier law predicts invalid heat flux. In order to use a continuum model Φ and Ψ should be defined either with condition applied only to the continuum part of the flow or taken from kinetic simulations.

21 Conclusion Kinetic simulations predict abrupt changes in the flow structure and escape rates at transition from hydrodynamic to Jeans escape. There is no qualitative difference between monatomic and diatomic species. In the hydrodynamic regime, the assumptions and limitations of Parker s theory of isentropic expansion do not hold. In the Jeans regime, the escape rate is within a factor of of the Jeans escape rate at the exobase, unlike the predictions of continuum models. In the Jeans regime, continuum model can match the kinetic model only below the exobase if Φ and E are preliminary found in kinetic simulations. Continuum matching solution does not go transonic. Results of kinetic simulations obtained without heating can be used to establish a criterion for transition from Jeans to hydrodynamic escape in atmospheres heated by stellar radiation. 1