J/ApJ/774/148 Ab initio EOS for hydrogen-helium mixtures (Militzer+, 2013)
Ab initio equation of state for hydrogen-helium mixtures with recalibration of
the giant-planet mass-radius relation.
Militzer B., Hubbard W.B.
<Astrophys. J., 774, 148 (2013)>
=2013ApJ...774..148M 2013ApJ...774..148M (SIMBAD/NED BibCode)
ADC_Keywords: Models ; Planets
Keywords: methods: numerical; planets and satellites: interiors;
planets and satellites: individual (Jupiter, Saturn)
Abstract:
Using density functional molecular dynamics simulations, we determine
the equation of state (EOS) for hydrogen-helium mixtures spanning
density-temperature conditions typical of giant-planet interiors,
∼0.2-9g/cm3 and 1000-80000K for a typical helium mass fraction of
0.245. In addition to computing internal energy and pressure, we
determine the entropy using an ab initio thermodynamic integration
technique. A comprehensive EOS table with 391 density-temperature
points is constructed and the results are presented in the form of a
two-dimensional free energy fit for interpolation. Deviations between
our ab initio EOS and the semi-analytical EOS model by Saumon and
Chabrier (1992PhRvA..46.2084S 1992PhRvA..46.2084S, 1995ApJS...99..713S 1995ApJS...99..713S) are analyzed in
detail, and we use the results for initial revision of the inferred
thermal state of giant planets with known values for mass and radius.
Changes are most pronounced for planets in the Jupiter mass range and
below. We present a revision to the mass-radius relationship that
makes the hottest exoplanets increase in radius by ∼0.2 Jupiter radii
at fixed entropy and for masses greater than ∼0.5 Jupiter mass. This
change is large enough to have possible implications for some
discrepant "inflated giant exoplanets."
File Summary:
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FileName Lrecl Records Explanations
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ReadMe 80 . This file
table1.dat 104 391 *Equation of state derived from DFT-MD simulations
table2.dat 45 293 *Coefficients of free energy fit
for the equation of state
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Note on table1.dat: This table contains the pressure, internal energy, entropy,
and free energy from our ab initio simulations a hydrogen-helium mixture of a
solar-type composition with 18 helium in 220 hydrogen atoms (helium mass
fraction Y=0.245).
The zero of energy was set to state of isolated, neutral atoms at rest, which
is consistent with the convention in the VASP simulation code. The
groundstate energy is the isolate hydrogen molecule using PBE including
zero-point energy then becomes -0.2386100 Hartree per molecule.
Note on table2.dat: This table contains the coefficient of the Helmholtz free
energy fit that we constructed to reproduce data of table1.
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See also:
J/ApJS/215/21 : Hydrogen and helium EOS (Becker+, 2014)
J/ApJ/750/52 : Jupiter model with improved EOS (Nettelmann+, 2012)
Byte-by-byte Description of file: table1.dat
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Bytes Format Units Label Explanations
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1- 4 F4.2 a0 rs [0.7/2.4] Density parameter rs (G1)
6- 13 F8.6 g/cm3 rho [0.2/9] Mass density ρ
15- 19 I5 K T [500/80000] Temperature
21- 29 F9.3 GPa P [3.9/24357] Pressure
31- 35 F5.3 GPa e_P [0.01/6] Error in P
37- 47 F11.8 2Ry E [-0.1/1.2] Internal energy per electron;
in Hartree/electron (1Ha=2Ry=27.21eV)
49- 58 F10.8 2Ry e_E Error in E
60- 70 F11.8 2Ry F ? Helmholtz free energy per electron, in Hartree
72- 81 F10.8 2Ry e_F ? Error in F
83- 93 F11.8 k S [3.2/12.7]? Entropy in kB per electron;
atomic units (1)
95-104 F10.8 k e_S ? Error in S (1)
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Note (1): S is given in kB/electron where kB is Boltzmann's constant.
(1.38x10-23^SI)
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Byte-by-byte Description of file: table2.dat
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Bytes Format Units Label Explanations
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1 I1 --- n_F [1/4] Function tabulated: 1=F, 2=dF/dT,
3=dF/dn, 4=d2F/dn/dT) (1)
3- 14 F12.10 a0 rs [0.5/3.6] Density parameter rs (G1)
16- 27 F12.10 2Ry/k T [0.001/0.4] Temperature given in Hartree/kB
(1Ha/kB=3.15774660873e+05K)
29- 45 E17.10 --- F [-23.3/346.5] Free energy in Ha=2Ry (n_F=1),
or derivatives (n_F>1) (1)
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Note (1): Because this is a two dimensional spline fit in density n (Ne/V)
and temperature T, we need specify the certain derivatives at boundaries
of the fit domain. Column F following the code in n_F as follows:
1 = F is the free energy at this rs and T point.
2 = F contains temperature derivative of the free energy, dF/dT.
3 = F contains density derivative of the free energy, dF/dn.
4 = F contains cross derivative of the free energy, d2F/dn/dT.
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Global note:
Note (G1): The density parameter rs is given by 4/3*π*rs3=V/Ne
where Ne is total number of electrons in volume V
(rs can be viewed as the radius of a sphere containing 1 electron).
The number density of electrons is n=Ne/V.
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History:
From electronic version of the journal
(End) Greg Schwarz [AAS], Emmanuelle Perret [CDS] 06-Mar-2015