Ab Initio EOS for Planetary Matter and Implications for Giant Planets

Transcription

1 Introduction Ab initio EOS H-He phase separation Ab Initio EOS for Planetary Matter and Implications for Giant Planets Winfried Lorenzen, Bastian Holst, Nadine Nettelmann, Ronald Redmer Planet Formation and Evolution: The Solar System and Extrasolar Planets Conclusion

2 Outline 1 Introduction 2 Ab initio EOS and its implications 3 H-He phase separation and its implications 4 Conclusion

3 Outline 1 Introduction 2 Ab initio EOS and its implications 3 H-He phase separation and its implications 4 Conclusion

4 Model of Jupiter LM-REOS He H 2 1 bar 170 K Constraints: mass, radius metals H,H + 4 Mbar 8800 K surface temperature surface helium fraction (x mol 7.2%) mean helium fraction (x mean 8.6%) 39 Mbar K gravitational moments J 2, J 4 LM-REOS [1] Linear mixing of H, He and H 2 O ab initio EOS data [1] N. Nettelmann et al. Astrophys. J. 683, 1217 (2008) see also poster P6.8

5 Finite Temperature Density Functional Theory Molecular Dynamics (FT-DFT-MD) VASP - Vienna Ab-initio Simulation Package input: volume, temperature (thermostat) output: energy U, pressure P also: electrical conductivity, optical reflectivity

6 Outline 1 Introduction 2 Ab initio EOS and its implications 3 H-He phase separation and its implications 4 Conclusion

7 Example: EOS of hydrogen Also available: helium, water 10 1 Saumon, Chabrier, van Horn (1995) FVT + (Holst et al., 2007) FT-DFT-MD (Holst et al., 2008) P [Mbar] K K 5000 K Saumon et al. ApJS 99, 713 (1995) B. Holst et al. ρ [g/cm 3 ] Contrib. Plasma Phys. 47, 368 (2007) 10 0 B. Holst et al. Phys. Rev. B 77, (2008)

8 Hugoniot of hydrogen Benchmark of the method 2 Pressure [Mbar] 1 Theories: Kerley 98 LM Ross FVT PIMC FT-DFT-MD Lenosky FT-DFT-MD Desjarlais FT-DFT-MD Holst Experiments: Z-Pinch Gas-Gun Explosives ρ/ρ 0 B. Holst et al. Phys. Rev. B 77, (2008)

9 Implications for Jupiter Results continuous dissociation feasible metal distribution colder core larger P T

10 Outline 1 Introduction 2 Ab initio EOS and its implications 3 H-He phase separation and its implications 4 Conclusion

11 Excess Gibbs free energy of mixing Definition G(x) = G(x) xg(1) (1 x)g(0) Helium fraction N He x = { N He +N H 0 : H x = 1 : He Contributions to G(x) G(x) = U(x) + p V (x) T S(x) Ideal entropy of mixing S(x) = k B [x ln x + (1 x) ln (1 x)]

12 Excess Gibbs free energy of mixing Example: 4 Mbar G (ev) K 5000 K 6000 K 7000 K 8000 K 9000 K K K Helium fraction x

13 Miscibility gap Mbar 24 Mbar Temperature (K) Mbar T m T m T m Helium fraction x Lorenzen et al. Phys. Rev. Lett., accepted Pfaffenzeller et al. Phys. Rev. Lett. 74, 2599 (1995)

14 Miscibility gap Temperature (K) 24 Mbar 10 Mbar Mbar 5000 T p T p T p x c 0.8 x c x c 1 Helium fraction x plateaus T p = T melt (He) critical He fractions x c a B n 1/3 H 0.25 [Mott-criterion] N. F. Mott Metal-Nonmetal Transitions (Second Edition) Taylor and Francis, London, 1990

15 Miscibility gap T Jupiter along isentrope 10 Mbar 24 Mbar Temperature (K) Mbar 2 Mbar 4 Mbar 0 x mean Helium fraction x

16 Consequences for Jupiter and Saturn Temperature (K) Saturn Jupiter P T P core FT-DFT-MD MC CP-MD Jupiter isentrope Saturn isentrope Pressure (Mbar) [CP-MD] Pfaffenzeller et al. Phys. Rev. Lett. 74, 2599 (1995) [MC] Schouten et al. Phys. Rev. B 44, 6630 (1991)

17 Consequences for Jupiter and Saturn Temperature (K) P T Saturn Jupiter P core FT-DFT-MD MC CP-MD Jupiter isentrope Saturn isentrope Pressure (Mbar) [1] Fortney and Hubbard Astrophys. J. 608, 1039 (2004) Consequences three-layer models for Jupiter can be justified by miscibility gap transition pressure P T can be determined saturns whole interior is in the demixing region this is necessary to reproduce the correct age of Saturn [1] miscibility gap might also indicate four layers

18 Outline 1 Introduction 2 Ab initio EOS and its implications 3 H-He phase separation and its implications 4 Conclusion

19 Summary FT-DFT-MD gives reliable EOS data for warm dense matter simulations for mixtures go beyond linear mixing miscibility gap is deeply connected to metallization in hydrogen EOS of real mixture should be used as input for planetary models phase separation of H and He is relevant for Jovian conditions complete phase separation in Saturn yield the correct age for Saturn basis for new, advanced planetary models

20 FT-DFT-MD Explanation of miscibility gap Planetary modelling Appendix 5 FT-DFT-MD 6 Explanation of miscibility gap 7 Planetary modelling

21 FT-DFT-MD Explanation of miscibility gap Planetary modelling Simulation details DFT finite temperature DFT [1] GGA (PBE) [2] plane wave basis sets Baldereschi mean value point [3] Molecular dynamics 64 electrons in a box periodic boundary conditions 0.5 fs time steps 1 3 ps simulation time [1] N. David Mermin Phys. Rev. 137, 1441 (1965) [2] John P. Perdew, Kieron Burke, and Matthias Ernzerhof Phys. Rev. Lett. 77, 3865 (1996) [3] A. Baldereschi Phys. Rev. B 7, 5212 (1973)

22 FT-DFT-MD Explanation of miscibility gap Planetary modelling Dynamic conductivity σ(ω) Kubo-Greenwood formula [1, 2] σ(ω) = 2πe2 2 m 2 ωω W (k) i,j k F ij D ij 2 δ(ɛ j,k ɛ i,k ω) D ij = Ψ j,k α Ψ i,k F ij = F (ɛ i,k ) F (ɛ j,k ) F(E) DOS E [ev] [1] R. Kubo J. Phys. Soc. Jpn. 12, 570 (1957) [2] D. A. Greenwood Proc. Phys. Soc. London 71, 585 (1958)

23 FT-DFT-MD Explanation of miscibility gap Planetary modelling Optical properties Kramers-Kronig relation σ 2 (ω) = 2 π P σ 1 (ν)ω ν 2 ω 2 dν Dielectric function ɛ 1 (ω) = 1 1 ɛ 0 ω σ 2(w) ɛ 2 (ω) = 1 ɛ 0 ω σ 1(w) Index of refraction n(ω) = k(ω) = 1 Reflectivity 2 [ ɛ(ω) + ɛ 1(ω)] 1 2 [ ɛ(ω) ɛ 1(ω)] r(ω) = [1 n(ω)]2 +k(ω) 2 [1+n(ω)] 2 +k(ω) 2

24 FT-DFT-MD Explanation of miscibility gap Planetary modelling Explanation for the plateaus Helium phase diagram Pressure (Mbar) FCC solid liquid melting Loubeyre et al. Datchi et al. HCP 24 Mbar 10 Mbar 4 Mbar LIQUID T 10 4 M T M T M Temperature (K) Loubeyre et al. Phys. Rev. Lett. 71, 2272 (1993) Datchi et al. Phys. Rev. B 61, 6535 (2000)

25 FT-DFT-MD Explanation of miscibility gap Planetary modelling Explanation for the critical He-fractions The Mott criterion [1] Mbar 10 Mbar 4 Mbar Mott [1] a B n H 1/ a B n H 1/3 ~ Helium fraction x [1] N. F. Mott Metal-Nonmetal Transitions (Second Edition) Taylor and Francis, London, 1990 x c a B n 1/3 H 0.25

26 FT-DFT-MD Explanation of miscibility gap Planetary modelling Planetary modelling