AIAA 2009-4257
41st AIAA Thermophysics Conference
22 - 25 June 2009, San Antonio, Texas
Transport Properties of High-Temperature
Jupiter-Atmosphere Components
A. Laricchiuta, D. Bruno, C. Catalfamo, M. Capitelli, G. Colonna, O. De Pascale,
P. Diomede, C. Gorse, S. Longo, F. Pirani and D. Giordano
CNR-IMIP Bari, via Amendola 122/D 70125 Bari, Italy
Dept Chemistry, University of Bari, via Orabona 4 70125 Bari, Italy
Dept Chemistry, University of Perugia, via Elce di Sotto 8 06123 Perugia, Italy
ESA-ESTEC, Aerothermodynamics Section, Noordwijk, the Netherlands
The designing phase of components for vehicles used in the planetary exploration is
assisted by the numerical simulation of critical entry conditions. In this framework reliable
and consistent data sets are needed for transport coefficients of the atmospheric components and also of the species formed in the dissociation/ionization regime, during the
vehicle impact. Jupiter atmosphere has been considered in this paper, moving to higher
approximations of the Chapman-Enskog theory for the calculation of single-component and
gas-mixture transport coefficients, considering equilibrium and non-equilibrium composition. To this aim existing transport cross section database for relevant interactions has been
updated and extended to exotic species, proposing a phenomenological approach for the
derivation of the corresponding elastic collision integrals in neutral-neutral and neutral-ion
interactions. Inelastic collision integrals terms, due to resonant charge-exchange channels,
have been also considered.
I.
Introduction
uture missions for the exploration of the planet Jupiter will certainly presuppose hypersonic entries into
F
the atmosphere of that planet. The design of thermal protection systems (TPS) is assisted by the accurate
knowledge of the high-temperature transport properties of the various components that constitute the gas
mixture. A vehicle flying hypersonically through the atmosphere has a drastic impact on the composition,
in fact main component (He/H2 ) undergo a number of processes (dissociation, ionization) leading to the
appearance of new chemical species. The derivation of transport coefficients for such a complex gas-mixture
+
(He, He+ , He++ , H, H+ , H− , H2 , H+
2 , H3 and electrons), in the frame of Chapman-Enskog theory, relies
on the accurate description of the microscopic dynamics of binary collisions, in an appropriately extended
temperature range.
Different models of the Jovian atmosphere have been proposed1, 2 and attempt can be found in literature of creating a reliable database of transport cross sections.3, 4 The completion of an high-order (up to
(`, s)=(4,4)) collision-integral database for Jupiter-components’ relevant interactions in an wide temperature
range [50 to 50 000 K] is still a challenging issue. The study of the dynamics on a multi-potential surfaces, for
many interactions, is hindered by the lack of electronic-structure information for the corresponding molecular
states. In the present work a phenomenological approach, already validated in the case of Earth and Mars
atmospheres,5 has been adopted. Effective odd-order collision integrals, resulting from both elastic and inelastic processes, have been estimated. In particular the resonant charge transfer process in atom-parent ion
collision could be experimentally measured or derived in the framework of the asymptotic theory, considering
also multiply charged ions.
Transport coefficients for equilibrium and non-equilibrium compositions have been derived in the ChapmanEnskog third-order approximation (2nd-order for viscosity). The following values for the parameters that
control the state of the plasma have been considered:
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Institute
Aeronautics
Copyright © 2009 by the American Institute of Aeronautics and
Astronautics,
Inc. All of
rights
reserved. and Astronautics
• fixed 1 atm pressure
• temperature varying in the range 50-50 000 K
• parametric study with composition (the equilibrium plasma composition is fixed while the temperature
is varied, allowing the analysis of strong non-equilibrium conditions).
II.
Collision integrals
The collision integrals for the interaction between two colliding species (i,j) can be defined as follows6
(`,s)
p
(`,s)
Ω̂i,j
2πν/KB T Ωi,j
i
h
`
1/2(s + 1)! 1 − 12 1+(−1)
1+`
=
(`,s)?
πσRS Ωi,j
=
4(` + 1)
(s + 1)![2` + 1 − (−1)` ]
=
Z
∞
2
(`)
e−γ γ 2s+3 Qi,j dγ
(1)
0
where γ 2 = µg 2 /2KB T , σRS is rigid sphere cross-section, Q(`) is the transport cross-section
Z
(`)
∞
Qi,j = 2π
(1 − cos` (χ))b db = 2π
0
Z
π
(1 − cos` (χ))σ(g, χ) sin χ dχ
(2)
0
σ is the elastic collision differential cross-section and χ is the deflection angle.
II.A.
Heavy particle interactions
Collision integrals for neutral-neutral and ion-neutral interactions involving atomic and polyatomic species
and multiply charged ions have been calculated by integration of the classical equation of motion,6 using
a novel algorithm, recently implemented, that can handle any potential function regardless the number of
extrema.7
The interaction potentials have been fitted with different analytical forms:
• the Hulburt-Hirschfelder potential8
ϕ(r)
=

»
„
«–
»
„
«–
r
r
ϕ0 exp −2αHH
−1
− 2 exp −αHH
−1
re
re
„
«3 »
„
«–
»
„
«–ff
r
r
r
+βHH
−1
1 + γHH
− 1 exp −2αHH
−1
re
re
re
(3)
• the modified Morse potential9
ϕ(r) = ϕ0 [1 + exp (−βMM (r − re ))]2 − ϕ0
(4)
βMM = β0 [1 + β1 (r − re ) + β2 (r − re )2 ]
(5)
ϕ(r) = ϕ0 exp (−ar − br2 − cr3 )
(6)
• the modified repulsive potential10
• the inverse power potential11
ϕ(r) =
d
rδ
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American Institute of Aeronautics and Astronautics
(7)
or modeled with the phenomenological potential12, 13
"
ϕ = ϕ0
m
n(x) − m
„ «n(x)
„ «m #
n(x)
1
1
−
x
n(x) − m x
(8)
where x = r/re and n(x) = β + 4x2 .
The phenomenological potential simulating the average interaction, allows the direct evaluation of collision integrals for unknown collisional systems. Potential features, such as the well depth, ϕ0 , and well position, re , can be
derived, in the framework of a phenomenological approach, by correlation formulas given in terms of fundamental
physical properties of involved interacting partners (polarizability α, charge, number of electrons effective in polarization).14–18 The validity of this approach was demonstrated in a paper19 by comparing, for some benchmark systems
relevant to Earth atmosphere, results obtained using the model potential with those calculated with more accurate
methods20–23 and applied to the Mars atmosphere.5
The m parameter depends on the interaction type (4 for ion-neutral and 6 for neutral-neutral interactions,
respectively), while a simple empirical formula, based on polarizability of colliders, has been proposed19 for the
estimation of β parameter, whose values range from 6 to 10 depending on the hardness of interacting electronic
distribution densities.
Reduced collision integrals Ω(`,s)? , calculated over a wide range of reduced temperatures, have been fitted as a
function of β 24 through the expression
ln Ω(`,s)?
=
[a1 (β) + a2 (β)x]
+a5 (β)
exp [(x − a3 (β))/a4 (β)]
exp [(x − a3 (β))/a4 (β)] + exp [(a3 (β) − x)/a4 (β)]
exp [(x − a6 (β))/a7 (β)]
exp [(x − a6 (β))/a7 (β)] + exp [(a6 (β) − x)/a7 (β)]
(9)
where x = ln T ? = ln kT /ϕ0 and parameters ai are polynomial functions of β
ai (β) =
2
X
cj β j
(10)
j=0
Parameters entering eq. (10) are presented in Refs.5, 24 for neutral-ion (m=4) and neutral-neutral (m=6) interaction.
H-H, H-H2 , H2 -H2 For these systems accurate collision integral calculations have been performed by Stallcop et
al.25, 26 based on ab-initio potential energy surfaces, considering an angular averaged potential for anisotropic atomdiatom and diatom-diatom collisions. For H-H2 and H2 -H2 data have been tabulated up to T =20 000 K, therefore
in the high-temperature region, dominated by the short-range interaction, collision integrals has been calculated by
integration of a repulsive potential used in Ref.27 and based on experimental data.
A full-range fitting expression (Eq. (11)), merging the low- and high-temperature data sets, has been used. The
sigmoidal form is the same already used for the phenomenological case, though in this case the dimensional σ 2 Ω(`,s)?
is obtained directly as a function of x = ln (T ).
σ 2 Ω(`,s)?
=
[a1 (β) + a2 (β)x]
+a5 (β)
exp [(x − a3 (β))/a4 (β)]
exp [(x − a3 (β))/a4 (β)] + exp [(a3 (β) − x)/a4 (β)]
exp [(x − a6 (β))/a7 (β)]
exp [(x − a6 (β))/a7 (β)] + exp [(a6 (β) − x)/a7 (β)]
(11)
Fitting coefficients ai are reported in Table 1.
H-He, H- -He, H-He+ , He-He+ The existence of weakly bound states of H- -He molecular ion where the H-He
neutral molecule itself is either unbound or barely bound has been theoretically investigated.28, 29 The interaction
potentials, merging short-range repulsive and long-range weakly attractive ab-initio results by different authors,28–30
have been fitted with Hulburt-Hirschfleder functions. Optimized potential parameters are (ϕ0 = 6.14 10−4 eV, re =
3.524 Å, αHH = 6.2613, βHH = 3.6046, γHH = 0.8732) for the H-He system and (ϕ0 = 6.22 10−4 eV, re = 6.15 Å,
αHH = 3.8, βHH = 7.9166, γHH = 3.2832) for H- -He.
For the H-He+ interaction the Hulburt-Hirschfelder fitting of the accurate ab-initio potential by Aubreton4 has
been considered (ϕ0 = 2.040 eV, re = 0.7743 Å, αHH = 2.1243, βHH = -0.3528, γHH = -1.7676) and collision integrals
derived by integration.
The accurate ab-initio potential energy curves for the gerade and ungerade electronic terms, arising in the He-He+
interaction, have been fitted by Hulburt-Hirschfelder potential in Eq. (3) (ϕ0 = 2.4730 eV, re = 1.081 Å, αHH =
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American Institute of Aeronautics and Astronautics
2.23, βHH = 0.2205, γHH = 4.3890) and modified repulsive potential in Eq. (6) (ϕ0 = 359.0 eV, a = 4.184 Å−1 , b =
-0.649 Å−2 , c = 0.08528 Å−3 ) in Ref.10
The effective collision integrals results from the usual averaging procedure
σ 2 Ω(`,s)? =
P
n
(`,s)?
wn σ 2 Ωn
P
n wn
(12)
being wn the statistical weight of each electronic term.
In Table 1 the fitting coefficients, ai , entering Eq. (11) are reported.
Table 1. Fitting coefficients, entering eq. (11), for σ 2 Ω(`,s)? in heavy-particle interactions.
(`, s)
a1
a2
H-H26
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
15.09506044
14.14566908
13.39722075
12.97073246
12.69248000
22.08948804
17.94703897
18.78590499
13.82986524
-1.25710008
-1.17057105
-1.09886403
-1.06479185
-1.04857945
-1.85066626
-1.42488999
-1.59291967
-1.01454290
9.57839369
9.02830724
8.50097335
8.18885522
7.97861283
8.50932055
7.66669340
7.97734302
7.48970759
a3
-3.80371463
-3.00779776
-2.86025395
-2.78105132
-2.73621289
-7.66943974
-4.76239721
-5.66814860
-3.27628187
a4
0.98646613
0.74653903
0.85345727
0.89401865
0.90816787
0.77454531
1.26783524
1.01816360
2.08225623
a5
9.25705877
9.10299040
8.90666490
8.73403138
8.57840253
9.69545318
9.53716768
9.32328437
9.21388055
-0.93611707
-0.68184353
-0.67571329
-0.65658782
-0.63732002
-0.62104466
-0.73914215
-0.60882006
-1.32086596
H-H2 25
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
12.49063970
12.02124035
11.69204285
11.45792771
11.00483923
7.45048892
10.84507417
11.55088396
-15.25288758
-1.14704753
-1.19514025
-1.24240232
-1.29677120
-1.27212994
-1.43326160
-1.42859529
-1.41480945
-1.39293852
8.76515503
8.76515503
8.76515503
8.76515503
8.76515503
9.59201391
9.20889644
8.98739895
9.59147724
-3.52921496
-3.45192920
-3.49608019
-3.64478512
-3.51537463
-1.35066206
-1.29890434
-1.39880703
-1.62599901
0.32874932
0.45922882
0.63354264
0.85298582
0.85298582
7.15859874
3.37747184
2.32276221
28.71128123
12.77040465
12.77040465
12.77040465
12.77040465
12.77040465
9.88881724
9.83307970
9.89142509
9.68396961
-3.04802967
-2.29080329
-2.29080329
-2.29080329
-2.29080329
-1.39484886
-1.30321649
-1.26804718
-1.63186985
H2 -H2 26
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
24.00841090
23.02146328
21.17218602
20.05416161
19.06639058
27.54387526
26.22527642
24.59185702
24.57128293
-1.61027553
-1.70509850
-1.57714612
-1.51326919
-1.45577823
-1.98253166
-1.94538819
-1.83729737
-1.80855250
3.88885724
3.88885724
3.88885724
3.88885724
3.88885724
3.88885724
3.88885724
3.88885724
3.88885724
-8.89043396
-10.46929121
-9.72209606
-9.38278743
-9.14716131
-12.91940775
-13.40557645
-12.78050876
-11.86035430
0.44260972
0.36330166
0.59112956
0.70004430
0.81250655
0.34707960
0.40398208
0.62739891
0.36590658
8.88408687
8.26405726
8.15580488
8.00952510
7.85268967
8.72131306
8.42662474
8.27557505
8.38682707
-1.05402226
-1.02331842
-1.46063978
-1.57063623
-1.66995743
-0.88296275
-0.96878644
-1.33071440
-1.00746362
He-H28, 29
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
10.06955597
7.19869772
6.55743323
6.35904452
6.25347577
11.79267184
9.11713279
8.55265897
8.45675669
-0.78208832
-0.68092029
-0.71505725
-0.73482728
-0.74253418
-0.91854409
-0.82598144
-0.88142795
-0.74217526
8.90551185
9.54561159
9.75769768
9.66498209
9.51511012
9.46049069
10.00526403
10.32649048
9.61034137
-4.17119119
-3.73700195
-3.90437553
-4.00054192
-4.02417882
-4.15691291
-3.68708893
-3.83898513
-3.82155303
76.18192207
76.18192207
76.18192207
76.18192207
76.18192207
76.18192207
76.18192207
76.18192207
76.18192207
-2.53113293
-8.84268703
-11.7792635
-13.1043859
-13.6296726
-2.52045886
-10.9038647
-15.7314905
-8.95019085
-2.89309888
-7.48001507
-9.63359232
-10.4621344
-10.6948126
-2.67556855
-8.42960742
-11.8299368
-7.14064928
He-H- 28, 29
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
16.83383128
11.26280636
8.49517155
6.92882973
5.99889967
18.96022395
14.24553114
11.68946064
13.37723617
-1.40086033
-1.02780542
-0.88101058
-0.78943626
-0.73327723
-1.59117471
-1.27240767
-1.13472411
-1.18163311
9.03792344
9.48063008
9.91582728
10.13073161
10.17888281
9.72963354
9.96040497
10.21270634
9.67449296
-3.76712840
-2.93672204
-2.56415853
-2.45943130
-2.44506151
-3.60520521
-2.85245325
-2.43973469
-3.11159621
76.18192207
50.16628602
45.66368787
43.96165902
42.80471500
72.97489590
56.52153525
52.00801821
53.82204847
0.0128648
0.0128648
0.0128648
0.0128648
0.0128648
0.0128648
0.0128648
0.0128648
0.0128648
-2.87368310
-4.15375249
-4.89901464
-5.28289363
-5.47961017
-3.19508067
-4.18966597
-4.81326828
-4.03136340
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American Institute of Aeronautics and Astronautics
a5
a7
(`, s)
a1
a2
a3
a4
a5
a5
a7
H-He+ 4
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
7.26019648
5.31621068
4.17730650
3.17455995
1.68374817
4.47457332
3.75792422
2.98401159
11.80660156
-0.45830355
-0.25198722
-0.13011997
-0.00742372
0.19151490
-0.22703750
-0.14961184
-0.06111433
-1.03306109
8.75289388
8.39297559
8.12931880
7.90685901
7.70176676
9.11487604
8.85486132
8.63912765
8.70928231
-1.05196080
-0.93572327
-0.85589802
-0.80603726
-0.76610507
-0.84002241
-0.77896426
-0.74591089
-1.32152742
7.88022406
8.72583156
9.22623392
9.70796087
10.49889283
13.68381322
13.27800873
13.14184555
7.24503625
5.84488923
5.65613996
5.50181035
5.34685366
5.20973048
4.46462878
4.46462878
4.46462878
4.46462878
-2.83633436
-2.63420758
-2.54217036
-2.47598237
-2.40196402
-3.36880863
-3.22466335
-3.11327610
-3.57325027
He-He+ 10
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
23.80276836
22.57798208
21.77686866
21.11384220
20.64796097
30.01038539
29.62345576
29.31400621
25.03330307
-1.53417198
-1.37458298
-1.29515584
-1.22362410
-1.18639584
-1.71614435
-1.80193239
-1.84347108
-1.29500767
7.14997977
6.71823986
6.44908263
6.23671845
6.07978682
5.80735212
5.65162107
5.48758712
6.24002334
-3.74339421
-3.53660052
-3.42650635
-3.33679994
-3.28123742
-4.49537445
-4.53517495
-4.55520888
-3.72851310
2.24179600
2.18797641
2.18797641
2.18797641
2.18797641
2.18797641
2.18797641
2.18797641
2.18797641
8.83793442
8.56245919
8.32964088
8.14201577
7.98030797
9.31326960
9.07520846
8.87814997
8.65029782
-0.99003900
-0.88202165
-0.88202165
-0.88202165
-0.88202165
-0.88202165
-0.88202165
-0.88202165
-0.88202165
He2+ -He, He2+ -H, He2+ -H2 Collision integrals for interactions involving the He2+ dication has been derived in
the framework of a polarization model. The potential function
ze2 αpol
(13)
8πε0 r4
the polarizability of neutral collider, can be considered as a special case for the general
ϕ(r) = −
being z the ion charge and αpol
σ 2 Ω(`,s)?
ze2 αpol
.
8πε0
In this case collision integrals assume a closed form11
r
„
«
4(` + 1)
dδ
2
(`)
=
A
(δ)
Γ
s
+
2
−
(14)
(s + 1)![2` + 1 − (−1)` ]
kT
δ
inverse power potential in Eq. (7) with δ=4 and d = −
where Γ is the Gamma function and A(`) is a temperature-free coefficient correlated to the transport cross section.
The A` (4) values have been estimated by Smith31 for `=1, 2 and 3 in the
3
attractive
case (A1 (4)=0.5523, A2 (4)=0.3846, A3 (4)=0.6377), leading to simple
Table 2. Polarizability values [Å ]
for Jupiter atmosphere species.
relations
r
αpol
2 (1,1)?
He
0.205
σ Ω
= 424.443z
T
H2
0.81
H
He+
HH+
2
H+
3
0.6668
0.03
27.0
0.424
0.4663
σ 2 Ω(1,2)?
=
0.833 σ 2 Ω(1,1)?
σ 2 Ω(2,2)? = 0.870 σ 2 Ω(1,1)?
(1,3)?
=
2
(1,1)?
0.729 σ Ω
σ 2 Ω(2,3)? = 0.761 σ 2 Ω(1,1)?
σ 2 Ω(1,4)?
=
0.656 σ 2 Ω(1,1)?
σ 2 Ω(2,4)? = 0.685 σ 2 Ω(1,1)?
=
2
2
σ Ω
2
σ Ω
(1,5)?
(1,1)?
0.602 σ Ω
σ 2 Ω(3,3)? = 0.842 σ 2 Ω(1,1)? (15)
Polarizability values are reported in Table 2.
H+ -H, H+ -H2 , H+ -He Accurate potential energy curves are available for the two electronic terms, 2 Σg and 2 Σu ,
32
of the H+
correlating with H(2 S)-H+ , allowing the rigorous multi-potential procedure. The gerade term
2 system
has been fitted with the Hulburt-Hirschfelder potential (Eq. (3) with parameters ϕ0 = 2.791 eV, re = 1.060 Å, αHH
= 1.28, βHH = -0.5, γHH = -2.0) while the modified Morse potential proposed in Ref.33 (Eq. (4) with parameters ϕ0
= 3.018 eV, re = 1.058 Å, β0 = 1.3870 Å−1 , β1 = 0.0135, β2 = 0.0117) has been adopted for the ungerade state.
The H+ -H2 and H+ -He systems have been studied by Krstic34, 35 within the frame of a quantum approach,
obtaining accurate diffusive and viscosity transport cross–sections in the energy range [0.1-20 eV] and [0.1-100 eV],
respectively. The derivation of collision integrals is straightforward by integration in Eq. (1), however the lower energy
limit does not allow the σ 2 Ω(`,s)? estimation at low-temperatures.
In order to estimate the transport cross sections, Q(`) , for E ≤0.1 eV the long-range potential for the H+ -He
+
has been described by a polarization model (Eq. (13)), while the potential surface for H+
3 in the limit of H -H2
36
dissociation has been fitted with a Hulburt-Hirschfelder potential (ϕ0 = 4.78 eV, re = 0.73291 Å, αHH = 1.3767,
βHH = -0.590, γHH = -1.653). The σ 2 Ω(3,3)? values, required by the adopted approximation in the Chapman-Enskog
method, for H+ -H2 has been calculated by the Hulburt-Hirschfelder potential and in the case of H+ -He interaction
has been set equal to σ 2 Ω(1,1)? . Collision integrals have been fitted by using the expression in Eq. (11) and ai
coefficients given in Table 3.
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American Institute of Aeronautics and Astronautics
Table 3. Fitting coefficients, entering eq. (11), for σ 2 Ω(`,s)? in H+ -neutral interactions.
(`, s)
a1
a2
a4
a5
a5
a7
H+ -H
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
46.68783791
46.68783791
46.68783791
46.68783791
46.68783791
46.68783791
46.68783791
46.68783791
46.68783791
-0.33303803
-0.33303803
-0.33303803
-0.33303803
-0.33303803
-0.33303803
-0.33303803
-0.33303803
-0.33303803
4.25686770
3.92217635
3.65740159
3.43102576
3.23079831
4.10212490
3.89701552
3.73496748
3.95678840
a3
-2.03851201
-2.00886829
-2.01434735
-2.04002032
-2.07543755
-1.85454858
-1.76267951
-1.69596577
-2.00381603
14.98170958
14.98170958
14.98170958
14.98170958
14.98170958
14.98170958
14.98170958
14.98170958
15.72150840
8.59618369
8.24501842
7.97534885
7.76086951
7.58613195
8.86285119
8.61913831
8.41234103
8.30656354
-1.65616736
-1.65616736
-1.65616736
-1.65616736
-1.65616736
-1.65616736
-1.65616736
-1.65616736
-1.79347178
H+ -H2 35
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
21.26444082
22.04614090
22.20902097
22.03893787
21.58227738
16.29263601
15.17470368
14.08165081
11.21674909
-1.66425043
-1.82881022
-1.90013155
-1.91163206
-1.87980396
-1.32780746
-1.24428548
-1.15372544
-0.56122635
9.43954208
9.15603332
8.91515513
8.70196666
8.51589918
10.00998032
9.70964984
9.48976675
9.08694827
-1.18521519
-1.09802478
-1.04865323
-1.02144928
-1.00037698
-0.70566506
-0.61129207
-0.58297710
-1.28700668
17.72506447
16.51669691
15.76252584
15.03274560
14.35787253
17.72506447
17.72506447
17.72506447
25.96742671
5.16010621
4.84818012
4.62340367
4.48334473
4.41475379
5.16010621
4.99747949
4.89671501
4.83728681
-2.45568134
-2.34842201
-2.26445563
-2.16080985
-2.05912196
-2.45568134
-2.45568134
-2.45568134
-2.37965195
H+ -He34
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
2.71952379
0.90687201
-1.51856588
-4.74037779
-7.26104273
-3.94194326
-5.23082562
-6.28592988
2.71952379
-0.04746783
0.18953042
0.53435861
1.00045484
1.37303488
0.73466954
0.92841556
1.09364996
-0.04746783
8.74213997
8.34680329
7.99486464
7.68664217
7.46364505
8.79837656
8.50739551
8.27457168
8.74213997
-0.88498112
-0.81094456
-0.78866372
-0.76331244
-0.72571710
-0.85969721
-0.83699163
-0.81943409
-0.88498112
91.77715465
67.45506771
53.68311327
45.61529587
41.33193035
76.18192207
63.85598616
56.74233232
91.77715465
2.49590845
2.84282065
3.16719614
3.46378279
3.63435585
2.86197842
3.05252604
3.16129013
2.49590845
-2.82208634
-2.68838648
-2.55404045
-2.42943739
-2.34213326
-2.85970495
-2.75743238
-2.68295935
-2.82208634
phenomenological interactions The phenome-
Table 4.
Parameters of phenomenological potential for
nological approach has been adopted for the other interac- interactions relevant to Jupiter atmosphere.
tions relevant to Jupiter atmosphere. Polarizability values
β
ϕ0 [meV]
re [Å]
m
for jupiter species, taken from literature37–40 or estimated
He-He
8.0
0.947
2.974
6
through an empirical formula41 as for H- , have been reHe-H2
9.29
1.865
3.189
6
ported in Table 2 and potential parameters for different
He-H+
9.73
30.664
2.574
4
2
interactions, needed for the used of bi-dimensional fit, pre9.66
29.471
2.606
4
He-H+
3
sented in Table 4.
H-H7.05
15.151
4.975
4
A special note is required in the case of He-He interH-H+
8.00
73.104
2.769
4
2
action. The adopted parameter values, describing the po7.98
70.982
2.797
4
H-H+
3
sition and the depth of the potential well and reported in
H2 -H7.27
18.896
4.962
4
Table 4, have been taken from accurate theoretical results
H2 -He+
9.22
177.500
2.223
4
H2 -H+
8.97
83.283
2.809
4
in literature.42, 43 A special comment is required for the
2
H2 -H+
8.93
80.981
2.838
4
value assigned to the parameter β, in fact, in the general
3
approach, for hard interactions involving atoms of noble
gases β is usually set approximately 9 or 10, but in the
He-He case the phenomenological potential predicts a short-range interaction too repulsive and, by comparison with
different potentials,44, 45 β = 8 has been chosen, giving accurate description of both short and long-range part of the
interaction potential.
II.B.
Charge Transfer
The contributions to odd-` terms of collision integrals coming from inelastic channels, i.e. the resonant charge transfer
cross–section in atom-parent ion and molecule-molecular ion interactions has been also estimated and, following ref.,46
the actual collision integral for relevant interactions has been estimated through the relation
q
(`,s)? 2
(`,s)?
) + (Ωch-ex )2
` = odd
(16)
Ω(`,s)? = (Ωel
(`,s)?
Ωch-ex , in this paper, have been evaluated by means of a closed formula, suggested by Devoto,47 for a linear dependency
of the square root of charge transfer cross–section on collision velocity and then fitted with the following formula
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American Institute of Aeronautics and Astronautics
σ 2 Ω(`,s)? = d1 + d2 x + d3 x2
(17)
where x = ln (T ). Table 5 reports fitting parameters for different systems, together with the references in literature
for resonant charge exchange cross–sections. For atom-parent ion interactions experimental results have been considered.48, 49 The cross-section for the double charge transfer in He-He2+ collisions has been theoretically derived
in Ref.50 In the case of H–H+ accurate charge-transfer cross sections have been obtained by Krstic34 by quantum
51
approach. H2 –H+
obtained by
2 interaction has been included, using resonant cross–sections by Yevseyev et al.,
extending the asymptotic theory to diatoms. For H-H interaction the experimental cross–sections by Huels49 have
been used, that compares well with theoretical results obtained in the framework of the asymptotic theory52 and
with a perturbed-stationary-states approach.53
(`,s)?
Table 5. Fitting coefficients, entering eq. (17), for σ 2 Ωch−ex in neutral-ion interactions.
(`, s)
II.C.
d3
(`, s)
d1
d2
d3
He-He+ 48
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
38.6185
37.5764
36.8042
36.1948
35.6877
d1
-3.1289
-3.0869
-3.0551
-3.0302
-3.0085
d2
6.3410(-2)
6.3429(-2)
6.3424(-2)
6.3459(-2)
6.3422(-2)
He-He2+ 50
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
12.1147
11.7937
11.5550
11.3658
11.2103
-0.9650
-0.9524
-0.9426
-0.9347
-0.9283
1.9236(-2)
1.9245(-2)
1.9229(-2)
1.9223(-2)
1.9218(-2)
H-H- 49
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
363.2187
349.9484
340.1784
332.4694
326.1197
-39.8121
-39.0841
-38.5383
-38.1009
-37.7370
1.0922
1.0922
1.0922
1.0922
1.0922
51
H2 -H+
2
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
38.2368
36.9454
35.9923
35.2418
34.6202
-3.8755
-3.8101
-3.7608
-3.7220
-3.6888
9.8299(-2)
9.8307(-2)
9.8299(-2)
9.8329(-2)
9.8306(-2)
H-H+ 34
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
63.5437
61.8730
60.6364
59.6591
58.8493
-5.0093
-4.9431
-4.8936
-4.8544
-4.8213
9.8797(-2)
9.8766(-2)
9.8767(-2)
9.8785(-2)
9.8776(-2)
Electron-molecule interactions
Collision integrals for electron-neutral species interactions have been calculated by straightforward integration based
on the corresponding transport cross–sections, Q` (E) as functions of electron energy.
Electronic scattering from hydrogen atoms has been deeply investigated theoretically54, 55 finding an excellent
agreement with absolute crossed-beam measurements.56, 57 The diffusion-type collision integral has been calculated
here by integration of the momentum transfer cross section by Ref.,27 whereas the corrections Q(2) /Q(1) and Q(3) /Q(1)
have been obtained by integrating the elastic differential cross sections by Ref.54 for the low energy range [0.582-30 eV]
and by Ref.55 for energies above 50 eV.
The momentum transfer cross section Q(1) for e-H2 interaction by Ref.58 has been integrated, including low (E <
0.01 eV) and high-energy (E > 25 eV) data by Biagi.59 The corrections to higher momentum transport cross sections,
Q(2) /Q(1) and Q(3) /Q(1) , have been obtained by integrating the elastic differential cross sections by Refs.58, 60
For e-He interaction the momentum transfer and elastic cross sections were taken from Biagi59 and the Q2 /Q1
and Q3 /Q1 ratios were deduced from the known Q1 /Q0 by assuming a model angular dependence of the differential
cross–section:
1
(1 − h(E) cos θ)2
(18)
where h is here an adjustable parameter that depends on the electron energy, obtained from the Q1 /Q0 ratio.
Electron-neutral interaction collision integrals have been fitted, as a function of x = ln (T ), using the following
formula
" „
«2 #
g3 xg5 exp [(x − g1 )/g2 ]
x − g7
σ 2 Ω(`,s)? =
+ g6 exp −
+ g4
(19)
exp [(x − g1 )/g2 ] + exp [−(x − g1 )/g2 ]
g8
Fitting coefficients gj , entering Eq. (19), are presented in Table 6, for relevant interaction in the Jupiter atmosphere.
Concerning interactions between charged particles, corresponding collision integrals have been obtained assuming
a screened Coulomb potential.61
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American Institute of Aeronautics and Astronautics
Table 6. Fitting coefficients, entering eq. (19), for σ 2 Ω(`,s)? in electron-neutral interactions.
(`, s)
g1
g2
e-He59
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
10.59136372(+00)
10.38275022(+00)
10.21366567(+00)
10.06183007(+00)
9.92613479(+00)
11.02594479(+00)
10.95544093(+00)
10.68567439(+00)
10.27636518(+00)
-1.52697125(+00)
-1.47464657(+00)
-1.44621765(+00)
-1.41920632(+00)
-1.39377540(+00)
-2.75226422(+00)
-2.24065670(+00)
-2.03462360(+00)
-1.60869251(+00)
2.18892432(-02)
6.90741396(-02)
0.15768321(+00)
0.26516138(+00)
0.37258117(+00)
0.25837224(+00)
1.42500349(+00)
1.47732284(+00)
0.18373753(+00)
-17.3791485(+00)
-3.37723956(+00)
-0.40718547(+00)
-0.15710196(+00)
-8.83214413(-02)
-1.80065315(+00)
-0.65562457(+00)
-0.44873292(+00)
-0.44953771(+00)
e-H54, 55
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
10.35291134(+00)
10.09959930(+00)
9.84443341(+00)
9.65959253(+00)
9.50113220(+00)
10.33445440(+00)
10.08612484(+00)
9.89312188(+00)
9.99023294(+00)
-1.58301162(+00)
-1.50068352(+00)
-1.42568830(+00)
-1.38293428(+00)
-1.36043711(+00)
-1.44880911(+00)
-1.39070408(+00)
-1.34820033(+00)
-1.41457896(+00)
12.45844036(+00)
12.54524872(+00)
12.97194554(+00)
13.18865311(+00)
13.23885240(+00)
12.08534341(+00)
12.39823810(+00)
12.63138071(+00)
12.53875975(+00)
-0.23285190(+00)
-7.29529868(-02)
-9.24067489(-02)
-6.34310879(-02)
-4.44172748(-02)
-1.86163984(-02)
-3.26532858(-02)
-1.96030794(-02)
-2.82169003(-02)
e-H2 58, 60
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
7.61552567(+00)
7.82045948(+00)
7.91713411(+00)
7.86089850(+00)
7.76607601(+00)
8.44904482(+00)
8.28235333(+00)
8.14899029(+00)
8.09697330(+00)
-1.31152238(+00)
-1.28509145(+00)
-1.18533267(+00)
-1.10926763(+00)
-1.06076517(+00)
-0.85941165(+00)
-0.77764787(+00)
-0.71893844(+00)
-1.08367568(+00)
1.08767030(+00)
0.84661899(+00)
0.77902584(+00)
0.77235681(+00)
0.78630351(+00)
0.45608703(+00)
0.39162611(+00)
0.33943549(+00)
0.67651715(+00)
-8.58478684(-02)
7.30252753(-03)
4.56053378(-02)
5.98096935(-02)
6.51985996(-02)
-3.28370076(-02)
3.15214680(-02)
8.34297336(-02)
5.92335717(-02)
(`, s)
g6
g3
g4
g7
g8
g9
e-He59
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
1.99019765(+00)
1.54430366(+00)
1.22119154(+00)
1.01178482(+00)
0.87073597(+00)
1.11428478(+00)
0.31741278(+00)
0.25766199(+00)
1.13659747(+00)
19.13183426(+00)
36.06175647(+00)
36.06175647(+00)
36.06175647(+00)
36.06175647(+00)
36.06175647(+00)
36.06175647(+00)
36.06175647(+00)
36.06175647(+00)
-3.87345716(+00)
-115.629356(+00)
-41.6895622(+00)
-27.9895600(+00)
-21.8337945(+00)
-41.1176569(+00)
-9.11869898(+00)
-3.57516627(+00)
-42.9018487(+00)
52.39042000(+00)
82.85050640(+00)
24.95446099(+00)
16.41208651(+00)
12.72530780(+00)
27.16469738(+00)
5.60148399(+00)
2.96346809(+00)
25.63994176(+00)
e-H54, 55
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
5.36628573(-02)
4.37301268(-02)
2.32754987(-02)
1.11968653(-02)
7.88864536(-03)
3.30723152(-02)
1.75287406(-02)
4.55766915(-03)
2.70547916(-02)
-5.34372929(+00)
-5.73166847(+00)
-5.71948057(+00)
-5.77977010(+00)
-5.83593794(+00)
-6.45649277(+00)
-6.48186913(+00)
-6.50687636(+00)
-6.11376507(+00)
9.35561752(+00)
9.09798179(+00)
8.83259870(+00)
8.64525855(+00)
8.49000000(+00)
9.15932646(+00)
8.90783546(+00)
8.71405704(+00)
8.89657156(+00)
-2.15463427(+00)
-2.13265127(+00)
-2.05797013(+00)
-2.02715634(+00)
-2.01418763(+00)
-2.13494419(+00)
-2.08119318(+00)
-2.04690115(+00)
-2.09400530(+00)
e-H2 58, 60
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(2,2)
(2,3)
(2,4)
(3,3)
0.60307406(+00)
0.77338688(+00)
0.84160605(+00)
0.86441504(+00)
0.86956793(+00)
0.88418654(+00)
0.98366646(+00)
1.07846844(+00)
0.86030007(+00)
5.01125522(+00)
4.59038620(+00)
4.29393447(+00)
4.18795370(+00)
4.14851371(+00)
3.62672485(+00)
3.59756557(+00)
3.56463327(+00)
4.05861889(+00)
9.07403916(+00)
8.92912750(+00)
8.81056581(+00)
8.67601998(+00)
8.54553153(+00)
9.45483919(+00)
9.25577830(+00)
9.10019221(+00)
9.00694175(+00)
1.92416429(+00)
1.72315372(+00)
1.59949292(+00)
1.53347205(+00)
1.49396365(+00)
1.67160237(+00)
1.57423504(+00)
1.49404727(+00)
1.62336471(+00)
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American Institute of Aeronautics and Astronautics
III.
Transport properties
The collision integrals discussed in the previous section have been used to calculate the transport properties of
Jupiter atmosphere. The calculation of the relevant coefficients entails the solution of systems of integro-differential
equations. The coefficients are then expanded in a series of orthogonal Sonine polynomials and the series truncated
at the desired order of approximation. The calculation thus reduces to the solution of systems of linear algebraic
equations.62 The third Chapman-Enskog approximation has been used throughout in order to provide accurate results
also in the ionization regime. Calculated transport coefficients include multicomponent diffusion coefficients (Dij ),
thermal diffusion coefficients (DiT ), translational thermal conductivity (λ), viscosity (η) and electrical conductivity
(λe ) coefficients.63
III.A.
Single component and equilibrium H2 -He mixture transport coefficients
VISCOSITY [104 kg m-1 s-1]
10.0
1.0
pure He
0.1
pure H2
pure H
0.0
102
103
104
105
TEMPERATURE [K]
Figure 1. Viscosity for pure species (lines) compared with
results by Biolsi1 (open markers) and by Hansen2 (closed
markers).
The mixture equilibrium compositions have been taken
from.62 Pure gas transport coefficients are calculated for
each of the three neutral species considered. These data
provide reference values for those who wish to use approximate mixture rules in their CFD calculations. Singlecomponent viscosities are reported in figure 1 as a function of temperature and compared in different temperature ranges with two series of data in literature.1, 2 An
excellent agreement is found with the only exception of
the high-temperature coefficient for helium, attributable
to differences in the adopted values of collision integrals
for He-He interaction. In figures 2 and 3 the comparison
is shown of viscosity and translational thermal conductivity for equilibrium Jupiter atmosphere at 1 atm pressure
with results in refs.1, 2 Despite the good agreement in
the low-temperature region, dominated by neutral interactions, increasing the temperature (T >10 000 K) differences are found where ionized species appear. This
seems to give an indication of the relevance of the adopted
Jupiter-atmosphere model with respect to the simplified
5-species-model adopted by Biolsi1 and to the calculations
by Hansen,2 based on mixing rules.
100.0
TRANSLATIONAL THERMAL
CONDUCTIVITY [W m-1 K-1]
-1 -1
VISCOSITY [10 kg m s ]
10.0
4
1.0
0.1
0.0
101
10
2
3
10
4
10
10.0
1.0
0.1
0.0
101
5
102
10
103
104
105
TEMPERATURE [K]
TEMPERATURE [K]
Figure 2. Viscosity of equilibrium Jupiter atmosphere
at 1 atm pressure compared with results by Biolsi1
(open circles) and by Hansen2 (open diamonds).
Figure 3. Translational thermal conductivity (including electrons) of equilibrium Jupiter atmosphere at
1 atm pressure compared with results by Biolsi1 (open
circles).
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III.B.
Non-Equilibrium H2 -He mixture trans-
port coefficients
VISCOSITY [104 kg m-1 s-1]
The investigation of non-equilibrium
Table 7. Non-equilibrium compositions, cases ci
conditions has been performed considering a frozen composition system ci
species
c1
c2
c3
c4
c5
and studying the temperature depenHe
2.0000(-01)
1.6094(-01)
9.4874(-02)
2.1026(-03)
7.6259(-05)
dence of transport coefficients. The
H2
8.0000(-01)
4.4846(-01)
8.8133(-06)
1.2473(-12)
1.1946(-15)
chosen compositions, reported in taH
3.7016(-36)
3.9060(-01)
6.0956(-01)
6.3193(-03)
2.5981(-03)
ble 7, correspond to equilibrium valHe+
2.8943(-25)
5.0593(-06)
5.4128(-02)
2.4971(-02)
He2+
1.9669(-22)
4.2152(-06)
2.9125(-02)
ues at fixed temperature, miming strong
H+
4.0107(-11)
4.7042(-06)
1.8852(-09)
3.7565(-11)
non-equilibrium conditions and allowing
2
H+
8.2049(-10)
1.2914(-11)
2.2508(-19)
1.5231(-23)
an insight on the relative role of different
3
H+
1.2420(-09)
1.4943(-01)
4.4356(-01)
4.3079(-01)
species. Five different cases are considH1.0450(-11)
7.0068(-06)
2.6139(-09)
2.3248(-11)
ered exploring extreme conditions from
e2.0921(-09)
1.4944(-01)
4.9770(-01)
5.1401(-01)
the c1 mixture, containing only neutral
components H2 (80%) and He (20%) and
corresponding to equilibrium composition at T =300 K, to the c5 illustrating the behavior of the completely ionized gas (T =50 000 K), passing through c2 ,
c3 , c4 intermediate cases, representing the equilibrium composition at T =3 500 K, 13 000 K, 30 000 K, respectively.
Viscosity is controlled by the collisions between heavy particles. Therefore a decrease is expected with ionisation
101
degree because the electrons do not contribute and bec2
c1
cause the Coulomb cross-sections are larger, as shown in
figure 4. In figure 5 the translational thermal conductivity
c3
is shown as a function of temperature for different compo0
10
sitions. It is seen that the thermal conductivity increases
equilibrium
composition
with the dissociation degree, while decreasing with the ionization degree. Also, the temperature dependence is much
-1
stronger for the Coulomb interactions than for collisions
c
10
4
involving neutral species. The electrical conductivity, in
c5
figure 6, is essentially due to electron diffusion. Therefore, non negligible values are attained only if the electron
component is present. The equilibrium case is reported in
10-2
0 100
1 104
2 104
3 104
4 104
5 104
each figure to emphasize the large discrepancies arising in
TEMPERATURE [K]
comparing the results obtained in this assumption.
Figure 4. Viscosity as a function of temperature for
strongly non-equilibrium conditions of Jupiter atmosphere, at 1 atm pressure, compared with equilibrium case
(see table 7 for fixed ci compositions).
ELECTRICAL CONDUCTIVITY [S m-1]
TRANSLATIONAL THERMAL
CONDUCTIVITY [W m-1 K-1]
35.0
30.0
c3
25.0
c5
c4
20.0
equilibrium
composition
15.0
c2
10.0
5.0
0 100
c1
1 104
2 104
3 104
4 104
5 104
3.5 104
c4
3.0 104
c5
2.5 104
equilibrium
composition
2.0 104
1.5 10
c3
4
1.0 104
5.0 103
0.0 100
1 104
TEMPERATURE [K]
2 104
3 104
4 104
5 104
TEMPERATURE [K]
Figure 5. Translational thermal conductivity as a
function of temperature for strongly non-equilibrium
conditions of Jupiter atmosphere, at 1 atm pressure,
compared with equilibrium case.
Figure 6. Electrical conductivity as a function of temperature for strongly non-equilibrium conditions of
Jupiter atmosphere, at 1 atm pressure, compared with
equilibrium case.
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IV.
Conclusion
In this paper a considerable effort has been devoted to the construction of a reliable database of transport cross
sections for interactions relevant to Jupiter atmosphere components. High-order collision integrals have been derived
for neutral-neutral, neutral-ion and electron-neutral interactions in a wide temperature range [50-50,000 K]. Moreover
suitable fitting formulas have been provided that could be easily implemented in existing numerical transport codes.
Based on a new thermodynamic model, transport coefficients have been derived for pure neutral species (H2 , He
and H), equilibrium and non-equilibrium Jupiter atmosphere compositions, emphasizing the relevance of the adopted
model and extending the range of validity to very high temperatures.
Acknowledgemets
The present work has been partially supported by ESA (Project 17283/03/NL/PA) and by MIUR PRIN 2007
(2007H9S8SW 003).
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