A new vision on giant planet interiors: the impact of double diffusive convection

1.3 MB·English

by  Jérémy Leconte

Astronomy&Astrophysicsmanuscriptno.Jupiter11_aa_v10 (cid:13)c ESO2012
January24,2012
A new vision on giant planet interiors:
the impact of double diffusive convection
JérémyLeconte1 andGillesChabrier1,2
1 ÉcolenormalesupérieuredeLyon,CRAL(CNRS),46alléed’Italie,69007Lyon,
UniversitédeLyon,France(jeremy.leconte,[email protected])
2 SchoolofPhysics,UniversityofExeter,Exeter
Received20February2009
Abstract
2
1 While conventional interior models for Jupiter and Saturn are based on the simplistic assumption of a solid core surrounded by
0 a homogeneous gaseous envelope, we derive new models with an inhomogeneous distribution of heavy elements, i.e. a gradient
2 ofcomposition, withintheseplanets. Suchacompositionalstratificationhamperslargescaleconvectionwhichturnsintodouble-
diffusiveconvection,yieldinganinnerthermalprofilewhichdepartsfromthetraditionallyassumedadiabaticinterior,affectingthese
n
planetheatcontentandcoolinghistory.
a Toaddressthisproblem,wedevelopananalyticalapproachoflayereddouble-diffusiveconvectionandapplythisformalismtoSolar
J
Systemgaseousgiantplanetinteriors.Thesemodelssatisfyallobservationalconstraintsandyieldametalenrichmentforourgaseous
1 giantsupto30to60%largerthanpreviouslythought. Themodelsalsoconstrainthesizeoftheconvectivelayerswithintheplanets.
2 Astheheavyelementstendtoberedistributedwithinthegaseousenvelope,themodelspredictsmallerthanusualcentralcoresinside
SaturnandJupiter,withpossiblynocoreforthislatter.
] Thesemodelsopenanewwindowandraisenewchallengesonourunderstandingoftheinternalstructureofgiant(solarandextrasolar)
P
planets,inparticularonthedeterminationoftheirheavymaterialcontent,akeydiagnosticforplanetformationtheories.
E
. Keywords.Doublediffusiveconvection;PlanetinternalStructure;Jupiter;Saturn
h
p
-
o 1. Introduction flyby missions entering their atmosphere and provide stringent
r constraintsontheirinnerelementdistribution.
t
s Morethan500planetshavenowbeendiscoveredorbitingstars
a Animportantquestion,forinstance,istodeterminewhether
outside our Solar System, spanning a range from a few Earth
[ theheavyelementspresentingiantplanetinteriorsarelocatedin
massestoseveralJupitermasses. Planetsthusseemtobeubiq-
acentralcoreoraremixedinalargefractionintothehydrogen-
1 uitousinnature. Thesediscoveriesraisefundamentalquestions helium (H/He) fluid envelope. In this latter case, a major issue
v abouttheinnercomposition,evolutionandoriginofthesebod-
is to determine whether convective mixing is sufficiently effi-
3 ies, andaboutthefundamentalpropertiesofplanetsingeneral,
cienttoyieldahomogeneouslymixedenvelopeor,alternatively,
8 includingtheonesofourownSolarSystem.Characterizingtheir
4 if giant planet interiors can exhibit a continuous compositional
internalstructureandcomposition,andfromthere,betterunder-
4 gradient. Exploringsuchapossibilityiscrucialtodetermine(i)
standingplanetformationisoneofthemajorchallengesofmod-
. themaximumamountofheavyelementscompatiblewithobser-
1 ernastronomy. Thedeterminationoftheheavyelementcontent,
vationalconstraints, (ii)theefficiencyofheattransportingiant
0 for instance, provides keyconstraints toplanet formation mod-
planets. Thesetwoissuesdirectlyimpacttheplanetmechanical
2 els,inparticularontheefficiencyofsolidplanetesimalaccretion
1 (density),chemical(composition)andthermal(luminosity,tem-
intheprotoplanetarydisktobuildaplanetembryo. Whileonly
: perature)structuresatagivenage,withmajorconsequenceson
v themassofmostoftheseplanetscanbederivedfromobserva-
ourunderstandingofplanetformationandevolution.
i tions, the mean density can be inferred for about 20% of these
X
objects,astheytransittheirparentstar,constrainingtheplanet’s Traditionally, giant planet models have always been based
r grosscomposition(seeBaraffeetal.2010forarecentreview). on two major specific assumptions concerning their inner pro-
a
Although providing an important diagnostic, however, this in- file, essentially for reasons of simplicity in the planet’s model-
formation is too limited to determine the element distribution ing (see e.g. Stevenson 1985). It is conventional to assume (i)
within the planet and thus its precise compositional and ther- that the inner structure of our giants consists of a few - gener-
malstructure. Assumingplanetformationisauniversalprocess, ally 2 to 3 - superposed, well separated, homogeneous regions,
onethusmustturntoourSolarSystemplanets,inparticularthe namely, going from the planet’s center to the surface, a central
twogasgiants,JupiterandSaturn,whichencompass92%ofthe solidrocky/icycore,andasurroundinglargelydominantlyH/He
planetarymassoftheSolarSystem, toderivemoredetailedin- gaseous envelope, often split into an inner metallic region and
teriormodels. Indeed,forourowngiants,thegravitationalmo- anouteratomic/molecularone;themoredensecomponentsare
mentshave beendetermined withhighaccuracy bythe various always supposed to have been accreted first or to have quickly
settled into the centre under the action of gravity (for planetes-
imals accreted after the runaway gas accretion); (ii) that plane-
Sendoffprintrequeststo:J.Leconte taryinteriorsareadiabatic,basedonthefactthatthegiantplanet
1
Leconte&Chabrier:Anewvisionongiantplanetinteriors
heat flow must be transported by convection (Hubbard 1968). newvaluesoftheheavyelementcontentandofthecoremasses
All the present determinations of the internal - chemical, me- forourgasgiants.
chanicalandthermal-structuresoftheSolarsystemplanets,in-
cludingtheirheavymaterialcontent,arederivedassumingsuch
2. Physicalmechanismsleadingtoinhomogeneous
homogeneouslystratified,adiabaticinteriors(Saumon&Guillot
2004;Fortney&Nettelmann2010). densitystratification
Giantplanetinteriors,however,mightdepartfromthiscon-
2.1. Compositionalgradient
ventional,simplifieddescription,becauseofcomplexprocesses
for which we lack an accurate description but which may very In the present calculations, in contrast to all previous planetary
well be at play in real situations (Stevenson 1985). In this pa- models,weconsideramassfractionofheavymaterialZ(m)ata
per,wederiveinteriormodelsforJupiterandSaturnwhichrelax depthr(m)withintheplanet(i.e. atthedepthoftheiso-density
theaforementionedpreconceptions.Insteadofthehomogeneous surfaceenclosingamassmoftheplanet)thatiscontinuouslyde-
layerassumption,weexplorethepossibilityofamixed,inhomo- creasingfromthecoretothesurface,producingacompositional
geneoussolid-gasinteriorcomposition,leadingtoaheavymate- gradientwithinthegaseousenvelopeoftheplanet,
rialgradientthroughouttheplanet.Thisinturntendstosuppress
large scale convection which, due to the double diffusive insta- ∇ ≡ dlnZ = dr 1dZ ≡−HP∇Z, (1)
bility(seeSect.2.2),canturnintoeitherturbulentenhanceddif- Z dlnP dlnPZ dr Z
fusionorlayeredconvection.Asboththeseheattransportmech-
anisms are fairly inefficient compared to usual convection, this whereHP ≡−ddlnrP isthepressurescaleheight.
The possible origin of such a compositional gradient is an
compositional gradient thus leads to significant departure from
open question (Stevenson 1985; Chabrier & Baraffe 2007). In
globaladiabaticityintheinterior. Asshownbelow, thesemod-
theconventionalscenario,alltheaccretedplanetesimalsareas-
elsdofulfilltheplanetobservationalconstraintswhileleadingto
sumed to directly sink to the core and not to evaporate in the
(i)asignificantlylargermetalcontentand(ii)significantlylarger
envelope,forsakeofsimplicity. Inreality,however,incomplete
internaltemperaturesthantheoneinferredfromhomogeneously
mixingoflargeplanetesimalsordissolutionofasubstantialfrac-
stratifiedadiabaticmodels.
tion of volatiles and rocks from small solid bodies could occur
This opens a new vision on planet structure, evolution and
formation efficiency. Such inhomogeneous interior profiles for in the envelope during the phase of planetesimal accretion on
thenascentplanet;asubstantialamountoficecouldaswellre-
Solar System giant planets had briefly been suggested several
mainintheenvelope(Iaroslavitz&Podolak2007;Hori&Ikoma
decades ago by Stevenson (Stevenson 1985) but no attempt
2011).
has ever been made to derive consistent models and to verify
The gradient might also stem from an only partial redistri-
whether such models would be consistent with the planet var-
butionbysmallscaleconvectivemotionsofstablylayered(sol-
ious observational constraints. This scenario has been revived
uble) constituents released by core erosion in the gas-rich en-
recentlyinthecontextofextrasolarplanetsandhasbeenshown
velope during the planet’s evolution (Stevenson 1982; Guillot
toprovideapossibleoratleastcomplementaryexplanationfor
et al. 2004), as seems to be supported for water by recent nu-
the anomalously large observed radii of many of these bodies
(Chabrier & Baraffe 2007). Indeed, as mentioned above, not merical simulations (Wilson & Militzer 2010). This could be
onlyaninhomogeneouslystratifiedinterioryieldsadifferentin- enhancedbytheimmiscibility(phaseseparation)ofanabundant
enoughmaterial(e.g. helium,water)inthedominantlymetallic-
terior structure and global metal content, but it decreases heat
transportefficiencythroughouttheplanet’sinteriorandthusaf- hydrogen envelope (Stevenson & Salpeter 1977a). This would
change the dynamical properties of (double diffusive) convec-
fectsitscooling,thusitsmass-radiusrelationshipatagivenage,
tionneartheregionsofimmiscibilityandaddcomplexitytothe
a crucial diagnostic to understand (transiting) extrasolar planet
problem. Therefore, to avoid extra complication, we only con-
structureandevolution. Since,asmentionedearlier,onlyloose
sider the occurence of this process for soluble material in the
constraints on the object’s internal composition are accessible
presentpaper.
for gaseous exoplanets, it is crucial to verify whether such un-
At last, rapid rotation and/or strong magnetic fields, neces-
conventional internal structures are a viable possibility for our
sarilypresentinJupiterandSaturninteriors,areknowntoham-
own giants. Furthermore, determining the maximum possible
perlarge-scaleconvection(Chabrieretal.2007),possiblylead-
amountofheavyelementsinJupiterandSaturnandtheirdistri-
ing to imperfect mixing of heavy elements in part of the enve-
butionwithintheplanetareimportantdiagnosticstounderstand
lope.
howourownSolarSystemgiantsformed.
Inordertoaddressthisissue,wefirstbrieflyreviewourcur-
rentunderstandingofthedoublediffusiveinstability,andofthe 2.2. Doublediffusiveconvection
various regimes under which it can occur in Sect.2. Then, in
Convectivesystemsinwhich(rapidlydiffusive)heatand(slowly
Sect.3, we derive an analytical formalism, based on a standard
diffusive)compositionhaveoppositedestabilizingand/orstabi-
parametrizationofconvection,similartothemixinglengthfor-
lizing effects tend to develop inhomogeneous density profiles.
malism, which describes the global transport properties of an
inhomogeneousconvective/diffusivemedium.InSect.4,wedis- The(de)stabilizingeffectofheatcanbequantifiedbythethermal
gradient,∇ ≡ dlnT,ormorepreciselybythesuperadiabaticity
cussthepossibleexistenceofanequilibriumsizefortheconvec- T dlnP
tive/diffusivelayerspresentinasemi-convectiveplanetandde- (∇T −∇ad, wherethederivativein∇ad istakenatconstantspe-
cificentropy),andtheoneoftheheavyelementsbythegradient
riveanalyticalconstraintsforthisequilibriumvalue. Finally,in
Sect.5,wederivesemi-convectivemodelsofJupiterandSaturn ofmeanmolecularweight(µ),∇µ ≡ ddllnnPµ. Theactualdynamical
thatareconsistentwiththegravitationalmomentsand withthe stateofthemediumdependsonthevalueofthedensityratio
surface abundances measured by the Galileo and Cassini mis-
α ∇ −∇
sions. Thisenablesustofurtherconstrainthenumberofpossi- R ≡ T T ad, (2)
bleconvective/diffusivecellswithintheseplanets,andtoderive ρ αµ ∇µ
2
Leconte&Chabrier:Anewvisionongiantplanetinteriors
Overturning Layered Turbulent Stable
convection convection diffusion system
Figure1. Stability diagram for a
medium presenting a destabilizing
Α (cid:209) temperature gradient and a stabilizing
RΡ(cid:45)1(cid:186) Μ Μ solute gradient, as a function of in-
Pr(cid:43)1 ΑT (cid:209)T(cid:45)(cid:209)ad creasingR−ρ1(decreasingRρ). Theusual
R (cid:45)1(cid:61)1 R (cid:45)1(cid:61)R (cid:45)1 R (cid:45)1(cid:61) instabilityLedouxcriterioncorresponds
Ρ Ρ min Ρ Pr(cid:43)Τ toR−1(cid:54)1.
ρ
(cid:12) (cid:12)
whereα ≡− ∂lnρ(cid:12)(cid:12) ,α ≡ ∂lnρ(cid:12)(cid:12) 1(Stern1960). R−1 corresponds to the point where the solute to heat buoy-
T ∂lnT(cid:12)P,µ¯ µ ∂lnµ¯(cid:12)P,T anmcinyfluxratio(≡γ−1)stopsdecreasingwhenR−1increases(see
In the case both the mean molecular weight (due e.g. to a ρ
Radko 2003 for details). Its exact value, however, depends on
higherconcentrationofsaltinsaltywater)andthetemperature
increasewithheight(∇ and(∇ −∇ )<0),thecompositional thecharacteristicsofthemediuminanontrivialwayandisdif-
µ T ad
ficult to estimate (Rosenblum et al. 2011; Mirouh et al. 2011).
gradientisdestabilizingwhilethetemperaturegradientisstabi-
For smaller ∇ gradients, the medium is unstable according to
lizing. Thiscaseisreferredtoasthefingeringcase. Inthatcase, µ
the Ledoux criterion, and the thermal forcing is strong enough
convective instability develops when R < 1, which is equiva-
ρ
toforcelargescaleoverturningconvection.
lenttotheLedouxinstabilitycriterion.Butevenif1<R <1/τ,
ρ
whereτ= D/κ istheratioofsolute(D)tothermal(κ )diffusiv- Variousargumentsseemtosupport,oratleastnottoexclude,
T T
ities,theslowerdiffusivityofelementscomparedtoheatyields the existence of layered convection under planetary conditions
theso-calleddouble-diffusiveinstabilitywhichleadstothefor- (Chabrier&Baraffe2007). Conducting3Dhydrodynamicscal-
culationsoverawidedomainofparameterspace(Prandltnum-
mation of salt fingers, and sometimes of thermo-compositional
berandatomictothermaldiffusivityratio)includingtheregime
staircases,asobservedinsomepartsoftheoceansandinlabo-
relevantforplanetaryinteriors,Mirouhetal.(2011)alwaysfind
ratoryexperiments(Turner1967).
The opposite case, referred to as the diffusive case, corre- a domain where γ−1 decreases with R−ρ1, a necessary and suffi-
spondstoafluidexhibitingapositivemolecularweightgradient cientconditionforthelayeringinstability, thuslayerformation
(∇ > 0). Inthatcase,thefluidwillbeconvectivelyunstableif to occur. A central question is then the size of the layers; this
µ
this gradient is insufficient to stabilize the system against con- will be examined in Sect.4. In any event, both homogeneous
vectiveinstability,i.e. iftheLedouxinstabilitycriterion double-diffusive convection or layered convection - generically
denominated as "semi-convection" in the following - are found
α
∇ −∇ > µ ∇ ⇔ R >1, i.e. R−1 <1, (3) to yield thermal and compositional fluxes that are significantly
T ad α µ ρ ρ smallerthanthatexpectedfromstandardconvection.Indeed,the
T
presenceofdiffusiveinterfacesstronglydecreasestheefficiency
ismet2. Whathappens,however,intheregionswhicharestable
of heat transport compared with large-scale, adiabatic convec-
according to the Ledoux criterion, i.e. R−1 > 1, but unstable
ρ tion, leading in planet interiors to a significant departure from
accordingtotheSchwarzschildcriterion,
theusualadiabaticprofile,asquantifiedbelow.
∇ >∇ , (4)
T ad
is less clear, especially in the astrophysical context, where the 3. Ananalyticaltheoryforlayeredconvection
very low values of the Prandlt number (Pr ≡ ν/κ , where ν is
thekinematicviscosity)renderdifficultdirectnumTericalhydro- Inordertoinvestigatetheimpactofsuchstronglyhamperedcon-
vectionongiantplanetinternalstructure,wedevelopedasimple
dynamicalsimulations. Inparticular,theexactnatureofdouble
diffusiveconvectionifitoccurs-homogeneousoscillatorycon- sub-grid model of layered convection. As illustrated on Fig.2
andfoundinsimulations(Rosenblumetal.2011;Mirouhetal.
vectionorlayeredconvection(i.e. uniformlymixedconvective
layers separated by thin diffusive interfaces characterized by a 2011), we consider that a semi convective zone consists of a
largenumber, N,ofwellmixedconvectivelyunstablelayersof
steep jump in the mean molecular weight) - remains uncertain. l
size l, separated by thin diffusive interfaces of thermal thick-
Analyticalarguments(Radko2003)andrecent3Dhydrodynam-
nessδ ,withinwhichthelargestabilizingcompositionalgradi-
ical simulations (Rosenblum et al. 2011; Mirouh et al. 2011), T
entcompletelyinhibitsconvectivemotions.
however,seemtosuggestthepicturepresentedinFig.1. When
themeanmolecularweightgradient(∝ R−1)decreasesinasta-
ρ
ble medium, homogeneous oscillatory convection, also called 3.1. Convectivelayers
turbulentdiffusion,firstappearsfor
Withineachconvectivelayer,thefluidisexpectedtofollowthe
R−1 (cid:54)R−1 (cid:54) Pr+1 (oscillatoryconvection), (5) dynamicsfoundinturbulentRayleigh-Bénardconvectionwithin
min ρ Pr+τ acelloftypicalheightequaltothesizeofthelayer,l.Byanalogy
whilewelldefinedthermo/compositionallayersstarttodevelop with the mixing length formalism (see details in AppendixA),
wedefineadimensionlessmixinglengthparameterbydividing
when
lbythepressurescaleheight,α≡l/H .
P
1(cid:54)R−1 (cid:54)R−1 (layeredconvection). (6) In a laboratory or a numerical experiment, the efficiency of
ρ min
theconvectionischaracterizedbythethermalNusseltnumber,
1 Foraperfectgas,α =1andα =1,aswillbeusedhereafter.
µ T
2 AsdiscussedbyRosenblumetal.(2011),theanalogybetweenthe F −Fad
fingeringandthediffusivecaseismoreapparentwhentheinverseden- NuT ≡ Ftot−Fadd , (7)
sityratio,R−1,isused,aswillbedonethoughouttherestofthepaper. d d
ρ
3
Leconte&Chabrier:Anewvisionongiantplanetinteriors
165K 135K
2-4 Z 2-8 Z
� �
Jupiter Saturn
8-12 Z 11-25Z
� �
19000-30000K 11500-42000K
0-0.5M 10-21M
Rocks+Ices
Figure2. SchematicrepresentationoftheinteriorsofJupiterandSaturn,accordingtothepresentstudy,andoflayeredconvection,
withtheresultingcompositionalandthermalradialprofiles. Theabundanceofmetalsisconstantwithinthewellmixedconvective
cellsofsizel,andundergoesasteepvariationwithinthediffusiveinterfacesofthermalsizeδ (greyregions). Thankstothissteep
T
gradient,theseinterfacesarestableagainstconvectionandenergyandmatteraretransportedthereinbydiffusiveprocesses.Because
thesizeoftheselayersisverysmallcomparedwiththesizeoftheplanet,themeanthermalandcompositionalgradients((cid:104)∇ (cid:105)and
T
(cid:104)∇ (cid:105))canbeusedingoodapproximationtoinfertheglobalplanetstructure.
µ
wherebydefinitionthetotalintrinsicflux(F ), thefluxtrans- as a = 0.2. On the other hand, for homogeneous Rayleigh-
tot
ported by diffusive processes (F ), and the diffusive flux that Bénard convection (without boundaries), Garaud et al. (2010)
d
wouldbepresentinacompletelyadiabaticzone(Fad)aregiven showed that the regime predicted by the mixing length theory,
d
by(Cox&Giuli1968;Hansen&Kawaler1994) i.e. Nu = Ra1/2 (C = 1,a = 1/2;seeAppendixA),isrecov-
T (cid:63) L
FFFtaoddt≡κTρHcPPT ∇∇∇Td , (8) CerLed=E.q1I.n.(9th)eisfoslulffiowciinengt,wtoecwaliclulltahtuestchoenflsuidxertr0an.2sp(cid:54)orate(cid:54)db0y.5caonnd-
d ad vection once the super adiabaticity is known. To compute this
latter, however, we must first define a quantity which can be
where c is the heat capacity at constant pressure. It is found
P computedapriorifromthelocalthermodynamicalpropertiesof
that, forlargeRayleighnumbers, theNusseltnumberfollowsa
themediumandthetotalinternalenergyfluxtobetransported.
lawofthetype
FollowingHansen&Kawaler(1994),thisconvectiveforcingcan
Nu =C Raa, (9) bedefinedby
T L (cid:63)
where Ra is a modified Rayleigh number3, which is the ratio Φ≡ NuT ×Ra(cid:63). (11)
(cid:63)
ofthestrengthofthethermalforcingtotheoneoftheradiative
Introducingε ≡ ∇ −∇ ,werewriteEq.(11)asΦ ≡ Φ α4ε
losses d d ad 0 d
where
Ra(cid:63) ≡ αTκgT2HP3 α4(∇T −∇ad)=(cid:18)NT2l4/κT2(cid:19). (10) Φ0 ≡αTκg2HP3. (12)
T
Here, g is the local gravity acceleration, and N the Brunt-
T ItisclearfromEq.(11)thatΦ isalocalconstantofthemedium,
Väisäläfrequency. 0
whichcharacterizesitsabilitytotransportenergybyconvection,
As convection at very high Rayleigh numbers is difficult to
independently of the mixing length or of the flux to be trans-
study either experimentally or computationally, it is difficult to
ported(∝ε ).
give precise values for a and C . For the bounded Rayleigh- d
L Then,fromEqs.(9)and(11),oneseesthatinaregionwhere
Bénardproblem,theoreticalmodelssuggestthattheexponentof
convectionremainsefficientenough,
the convective flux law, a, could be equal to 1/3 (Garaud et al.
2010)4. However, experiments done by Krishnamurti (1995) (cid:32) Φ (cid:33)1/(1+a)
tend to show that this exponent could be smaller, and as low Φ= Nu ×Ra =C Ra1+a ⇒ Ra = , (13)
T (cid:63) L (cid:63) (cid:63) C
L
3 Indeed,intheastrophysicalcontext,itisconvenienttouseRa =
(cid:63) whichyieldsthesuperadiabaticity,
Pr×Ra,whereRaistheusualRayleighnumber.
se4emAtlothosuupgphotrhteexspimonuelnattiovnalsuepsressmenatleledrbthyanRo1s/3en,bsluugmgesettinagl.t(h2a0t1in1-) ε ≡∇ −∇ =(cid:32) εd (cid:33)=(cid:32) εd (cid:33)1/(1+a). (14)
terfaces act as impermeable boundaries, it should be noticed that the T T ad NuT CLΦa0α4a
heightofthelayerspresentintheirsimulationsissmallcomparedtoa
pressurescaleheight.Theirconclusionmaythusnotbevalidforlarger Therangeofsuperadiabaticityintheconvectivelayersimplied
layers. bythisequationforthevariouspossibleexponentsaisshownin
4
Leconte&Chabrier:Anewvisionongiantplanetinteriors
Fig.3 (pale red area). As seen, the uncertainty on a leads to a willusethegenericdenomination∇ forthediffusivetempera-
d
largedispersiononthissuperadiabaticity. Inthishighconvec- turegradient,keepinginmindthatdiffusionisnowduetoelec-
tiveefficiencyregime,wecanfurthercomputethemeanconvec- tronicoratomicmotions,withacharacteristicthermaldiffusiv-
tivefluxwhich,bydefinition,isgivenby ityκ .Inthepresentcalculations,weusetheconductivethermal
T
diffusivitiescalculatedbyPotekhin(1999).
ρc T
F =κ P (∇ −∇ )×Nu . (15)
conv T H T ad T
P 3.3. Meanthermalgradient
As mentioned above, this scaling law only applies to the Oncewehavecalculatedthethermalgradientintheconvective
vigorous convection regime, i.e Φ or Ra(cid:63) (cid:29) 1. Thus, in zonesofsizel,andinthediffusiveinterfacesofsizeδT,weneed
terms of the layer height, convection remains efficient as long todeterminethemeanpropertiesofawholestackofconvective-
asα(cid:29)αcrit,whereαcrit isdefinedsuchthatRa(cid:63)(αcrit)=1,and, diffusivecells. AsdiscussedinChabrier&Baraffe(2007),since
fromEq.(10),thecriticallayersizeis the convective plumes must be fed by the diffusive interfaces,
the thermal convective (N−1) and diffusive (δ2/κ ) time scales
lcrit =(cid:18)κT2/N2(cid:19)1/4 =dPr−1/4, (16) shouldbesimilarineachrTespectivelayer. TheTrefoTre
T
where d is the lengthscale of the fastest growing mode of the (cid:16)δT/l(cid:17)=(cid:16)κT/l2NT(cid:17)1/2 =Ra−(cid:63)1/4 =(cid:16)Φ/CL(cid:17)−4(11+a) . (18)
linear instability, d ≡ (νκT/NT2)1/4 (e.g. Baines & Gill 1969). Not surprisingly, the size of the interface, δT, is related to the
Analytical and numerical arguments show that the size of the lengthscaleofthemostunstablemodeofthelinearinstabilityby
fRaostseesntbgluromweintgall.a2y0e1rs1)i.sTeqhueaelffitoci1e0n-t1c0o0ndve(cid:38)ctilocrnitr(eRgaimdkeois20th0u3s; δoTfa=nudpw/Perll1i/n4g. Ienddayddwitiitohnt,hceopmotpeanrtiinagletnheermgyeabnarkriienrectirceaetneedrbgyy
appropriate in the planetary domain, where Pr ∼ 10−2 −10−1 thenegativebuoyancyinthediffusiveinterface,weseethatthe
(Chabrier&Baraffe2007). aboveconditionalsoentailsthatconvectiveovershootingcanbe
ForconditionsprevailingintheinterioroftheactualJupiter neglected(seeChabrier&Baraffe2007fordetails).
(Saturn), the mean thermal diffusivity is κT ∼ 5×10−5m2.s−1 KnowingδT/lenablesustocomputethemeanthermalgra-
(Potekhin 1999), Φ0 is equal to Φ0 ≈ 3 × 1033 (9×1032) and dienttobeusedtocomputetheplanet’sstructure(seeSect.5.1)
ε ≈10(10;seealsoFig.5),sothatα =2×10−9(3×10−9). In
d crit
δ l
thefollowing,alltheorderofmagnitudeestimatesdonethrough- (cid:104)∇ (cid:105)≡ T ∇ + ∇ . (19)
outthetextwillusethesevalues. T l+δT d l+δT T
Substituting ∇ and δ /l by their expressions in Eqs.(14) and
T T
3.2. Interfaces (18), and developing the mean gradient to first non vanishing
order,wefindthat
In the interfaces of thermal size δ , overturning convection is
T    
it(nhRhe−ρsi1bei>treedg(iPboryns+thd1eo)/sn(troPotrnng+eejτud)m),tpobuibntecmianonlteahclesuolfaurbllewyiendigiffthhute.siovHsecoirwlelagetivomerrye, (cid:104)∇T(cid:105)≈∇ad+(∇d−∇ad) Ra11(cid:63)/4 + N1uT+ORa11(cid:63)/2, NuT1R(a21(cid:63)0/4),
convection regime characterized by Eq.(5). Indeed, as shown
byRadko(2005)theconditionR−1 > R−1 ,whereR−1 isthein- orequivalently
I min I
vneortsoendlyentositeynsruatrieothweitshtianbitlhiteyionftetrhfeacien,teirsfaacesuiffitseclife,nbtuctraitlesroioonf (cid:104)∇ (cid:105)−∇ ≈(∇ −∇ ) (cid:20)(cid:16)Φ/ (cid:17)−4(11+a) +(cid:16)ΦC1/a(cid:17)−(1+aa)(cid:21). (21)
T ad d ad CL L
thewholestackoflayerswhichwouldotherwisemergeintoone
homogeneouslayer(seeSect.4.1fordetails). As expected, two terms appear in the mean super adiabaticity,
Therefore, the interface is very likely either in a stable dif- whichisshowninFig.3(darkgreyregion). Thefirstoneisdue
fusion state, or in a state of weakly turbulence enhanced diffu- to the temperature jumps at each interface and is controlled by
sion. Rosenblumetal.(2011)showthatinthisregimeNu (cid:46)2, theconvectiveoverturningtimescalewhichdeterminesthesize
T
meaningthatmostoftheenergyistransportedbypurediffusion. oftheseinterfaces. Thesecondoneisduetothesuperadiabatic-
Wewillthusassumethatthethermalgradienttobeusedinthese ity in the convective layers themselves, which is enhanced by
regions is given by the gradient needed to transport the whole their small size. The relative contributions of these two terms
outgoingenergyfluxbydiffusion: only depend on the properties of the convection, characterized
bytheexponenta.
∇ ≡ 1 HP F . (17) Interestingly, if the impact of a on the super adiabaticity in
d κ ρc T tot the convective layers (pale red area in Fig.3) is significant, the
T P
impactonthemeansuperadiabaticityofthewholestackoflay-
The thermal diffusivity κ encompasses the contribution of all ers, (cid:104)∇ (cid:105)−∇ (darkgrayregion), israthersmall. Thiscanbe
T T ad
diffusiveprocesses. Ifdiffusionisensuredbyphotons,asgener- understood as follows: for the smallest value of the exponent
allyinmostastrophysicalobjects,thediffusivethermalgradient consideredhere,a=0.2,convectionisveryinefficientandmost
reduces to the so-called radiative gradient, ∇ (e.g. Hansen ofthemeansuperadiabaticityiscontainedintheconvectivelay-
rad
& Kawaler 1994). However, in the deep interior of giant plan- ers (upper contour of the grey region). In that case, Eq.(19) is
ets and in degenerate bodies, density can be high enough for dominatedbythesecondterm,(cid:104)∇ (cid:105)≈∇ .
T T
theelectronstobecomedegenerateenoughtoefficientlyconduct Whentheexponentisincreased,thesuperadiabaticityinthe
thermalenergy(seeStevenson&Salpeter(1977b)andChabrier convective layers (∝ Φ−a/(1+a)) drops dramatically, but this en-
&Baraffe(2007)forthecharacteristicradiativeandconductive tailsagrowthoftheconvectivetimescale,andthusofthethick-
opacitiesunderjovianplanetconditions). Alongthispaper, we ness of the interfaces (see Eq.(18)). For efficient convection,
5
Leconte&Chabrier:Anewvisionongiantplanetinteriors
timescalesforheatandsoluteleadsto
(cid:114)
10 δ D
(cid:182) (cid:61)(cid:209) (cid:45)(cid:209) Z ≈ . (24)
d d ad δ κ
0.1 T T
(cid:88)(cid:209)T(cid:92)(cid:45)(cid:209)ad Ontheotherhand,consideringthatthecompositionaljumpmust
d
a 0.001 coverthewholeinterfacetoensureitsmechanicalstability,one
(cid:209)
(cid:45) wouldexpectthat
(cid:209) 10(cid:45)5 (cid:182)T(cid:61)(cid:209)T(cid:45)(cid:209)ad δZ ≈δT. (25)
10(cid:45)7 Αcrit Thepresentpaperfocusesontheimpactofdouble-difffusive
convectionontheplanetmechanicalandthermalstructure. The
101(cid:45)90(cid:45)10 10(cid:45)8 10(cid:45)6 10(cid:45)4 0.01 1 impactontheevolutionwillbeaddressedinaforthcomingpaper
(see Sect.6). Therefore, this uncertainty on the heavy element
l(cid:144)H flux, discussed in the next section, does not have a significant
P
impactontheresultspresentedinthepresentstudy.Determining
Figure3. Range of mean super adiabaticity ((cid:104)∇T(cid:105)−∇ad) of a more precisely the solute transport properties in the regime of
semi-convectivemediumasafunctionofthemixinglengthpa- layered convection, however, will be of central importance to
rameter α = l/HP, for 0.2 (cid:54) a (cid:54) 0.5 (gray area between the evolutionary calculations, and 3D hydrodynamical simulations
red dashed curves; Φ0 = 3 × 1033; εd = 10; CL = 1). For inarealisticparameterrangearestronglyneeded(Mirouhetal.
comparison,thesuperadiabaticityinaconvective(paleredarea; 2011).
same parameters) and in a diffusive cell (dotted curve) are also Theterm∇ ,whichistherelevantquantitywhencomput-
µ,I
shown. Asexpectedasmoothtransitionbetweentheconvective ingtheequilibriumconditionoftheinterface,isgivenby
and diffusive regimes occurs near α (see AppendixA; here
αcrit ≈2×10−9). crit ∇µ,I =−l+δδZHP ∂∂lnZµ(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104)∇Z(cid:105). (26)
Z P,T
(cid:12)
a (cid:62) 1/4, this effect eventually dominates, and the mean super The precise value of ∂∂lnZµ(cid:12)(cid:12)(cid:12)P,T depends on the precise chemical
adiabaticity increases again with a. This yields a rather nar- composition of the considered heavy element, but is typically
rowregionforthemeansuper-adiabaticitywithintheconsidered aroundunity;thisisthevaluewewilluseinnumericalestimates
rangeofexponentvariation. below. WewillalsouseD≈10−7−10−8m2.s−1(i.e. τ≈10−2−
Interestingly, to first order, the a = 1/2 and a = 0.2 cases, 10−1),appropriateforgiantplanetinteriorconditionsStevenson
i.e.thetwoextremevaluesconsideredhere,yieldthesamemean &Salpeter1977b).
superadiabaticity. Therefore,themodelpresentedhereforsemi-
convectiondependsonlyweaklyonthepreciseprescriptioncho-
4. Layersize: analyticalpointofview
sen for convection. In the following, we will show results ob-
tainedwiththemixinglengththeory, i.e. CL = 1anda = 1/2, 4.1. Existenceofanequilibriumheight
whilekeepinginmindthatforα (cid:29) α theseresultswouldbe
crit
thesameastheoneobtainedinthea=0.2case. The determination of the size of the layers, when layered con-
vectionisoccurring,isacomplextask. Theproblemisrendered
evenmoredifficultbythefactthatthesmalllayersthatinitially
3.4. Meansolutegradientandflux form tend to merge into larger layers. The question is then to
knowif andwhenlayermergingends.
Disregardingconvectiveovershoot,thesoluteflux(F )isdeter-
Z In the fingering case, i.e. when the solute is the destabiliz-
mined by the transport at each interface. As for heat, the tur-
ingactor,Radko(2005)derivedacriterionforthelinearstability
bulenttransportofsoluteisexpectedtobeveryinefficient, and
ofthermo-compositionalstaircasesagainstmerging. Inthispic-
we will assume purely diffusive processes (see Sect.3.2). The
ture, an initially inhomogeneous medium starts forming layers
impactofamoreefficienttransportwillbediscussedinthenext
if its density ratio (R ) is smaller than the density ratio, R ,
section. Inthisregime,thesolutefluxisthusgivenby ρ min
forwhichtheheattosolutebuoyancyfluxratio(≡ γ)stopsde-
creasingwhenR increases.Then,layersareunstableandmerge
F =−ρD∇Z , (22) ρ
Z I as long as the density ratio within the interface between them
(R ) is smaller than R . Then, as R increases with the layer
wherethesubscriptI describesquantitiesdeterminedwithinthe I min I
height (under some conditions), it eventually reaches R and
interface,indicatingthatthesolutefluxF isdeterminedbythe min
Z themergingprocessstops. Atthisstage, anequilibriumheight
elementfractionvariationrateateachinterface. Neglectingthe
isreachedbytheconvectivelayersandthestaircaseisfullyequi-
smallinhomogeneitiesintheconvectivelayer,theinterfacialand
librated.
meangradientsarerelatedby
ThelinearstabilityanalysisofRadko(2005)canberedone
l+δ l+δ in the diffusive case, by simply accounting for the fact that the
∇ = Z(cid:104)∇ (cid:105) or ∇Z = Z(cid:104)∇Z(cid:105), (23) signs of the various density and temperature differences and
µ,I δ µ I δ
Z Z of the fluxes must be changed. Then, after some lengthy but
straightforward algebra, one can show that layers are unstable
where(cid:104)∇Z(cid:105)describesthemeangradientofheavyelementmass
andmergewhen
fraction, i.e. the value of dZ/dr averaged over several convec-
ttiivoen/adliffjuumsipveocceclulsrs,.anTdoδcZonissttrhaeinlethnegthvaoluveerowf hδiZc,htwcoomlpimositi-- ∂∂Rγ−−11(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) <0 ⇔ R−I1 <R−m1in, (27)
ingargumentscanbeused. Ononeside, equatingthediffusive ρ R−ρ1=R−I1
6
Leconte&Chabrier:Anewvisionongiantplanetinteriors
whichdefinesR−1 .Asforthefingeringcase,thislinearstability Therefore,inordertoavoidacompletehomogenizationofgiant
min
analysisshowsthat,iflayersform,theirmergingwillstopwhen planet interiors in less than 5Gyr, their present age, α must be
theirheightreachesafiniteequilibriumvalue. smaller than α ≈ 1×10−4 for a = 0.2 and 1.4×10−3 for
max
a = 0.5. Note however that, as mentioned in Rosenblum et al.
(2011), turbulent transport due to the double-diffusive instabil-
4.2. Globalconstraints
ity can yield compositional Nusselt numbers, Nu 5, around 2-
µ
Althoughpreciselyestimatingthisequilibriumvalueisdifficult, 4,yieldingevenstrongerconstraints,assummarizedinTable1.
somestronglimitsonthelayersizecanbederivedtheoretically. These small values of α justify a posteriori the approximation
Let us consider a stack of layers extending over a zone of size ofcontinuousthermalandheavyelementprofileswhenconsid-
L, and define a global gradient (cid:104)∇Z(cid:105) ≈ −∆Z/L, where ∆Z is ering the planet’s entire internal structure. Note, however, that
the difference between the mass ratios of heavy element at the layeredinhomogeneitiescouldbedynamicallyregeneratedover
bottomandatthetopofthesemi-convectivezone. time.Inthatcase,layeredconvectionwillbeanongoingprocess
Ononehand,themeanmolecularweightgradientinallthe intheplanet’sinterior.
interfaces,∇ ,mustbehighenoughtosatisfythestabilitycri- Consideringthepossibilitythattheratioofthecompositional
µ,I
teriondiscussedinsections2.2and3.2. Thisimplies to the thermal size of the interface does not scale as the square
rootoftheratioofthediffusivities,andthereforeusingEq.(25)
α
αTµ∇µ,I >R−m1in×(∇d−∇ad). (28) ianfsatecatodro√fκEq/.D(214+a),≈yi1e0ld.sanincreaseofbothαmin andαmax by
T
Substituting ∇ by its expression in Eq.(26) and δ by using Thisanalysisshowsthat,fortheageoftheSolarSystemand
µ,I Z
Eqs.(24) and (18), the criterion (28) for a planetary scale (L ≈ for the conditions prevailing in gas giant interiors, there exists
R )semi-convectivezonereads a range of layer sizes for which ongoing layered convection is
p
a viable mechanism. According to our estimate, this range is
α1/(1+a) >α1/(1+a) ≡ αT (cid:114)D R−m1in(∇d−∇ad)1−4(11+(cid:12)a) , (29) relativelylargeandcorrespondsto:
min αµ κT (Φ0/CL)4(11+a)∆Z ∂∂lnZµ(cid:12)(cid:12)(cid:12)P,T 10−9−10−6 (cid:46)α(cid:46)10−4−10−2. (32)
Theuncertaintyonthelowerandupperboundsarerespectively
where α denotes a lower limit for the layer size. Under the
min due to our poor knowledge of the behavior of R−1 , and of the
present conditions in the interiors of our gas giants (see end of min
Sect.3.1 and Table1), this yields α ≈ 2×10−9 ×(R−1 )1+a, solute flux at low Prandlt number. Even given these uncertain-
with the less restrictive constraint bmeiinng obtained for R−m1in= 1, ties, the fact that convection is always in a regime of relatively
min highRayleighnumberappearstoberobustprediction. Tomake
as summarized in Table1. However, as the existence of layers
allowsR−1 tobeashighas(1+Pr)/(τ+Pr)≈101−102,these anattempttoovercometheselimitations,weexamineinSect.5
min how observational data can narrow this possible domain of α,
constraints could be severely tightened, as showed in Table1.
byfurtherconstrainingthedegreeofsuperadiabaticity,andthus
This simply confirms that, in order for layered convection to
bestable,convectivecellsmustremainlargerthanthediffusive thesizeofthelayers,inourSolarSystemgasgiants.
interfaces, and the medium is always in the convective regime,
α(cid:29)αcrit. 4.3. Analternativescenario
Westressthattheaforementionedconstraintsapplyonlytolay-
Table 1. Numerical constraints on the layer height in the two ered convection, and does not preclude the possibility that, un-
limiting cases of convection (a = 0.2 and 0.5; CL = 1) for the dersomeconditions, doublediffusiveconvectionmaymanifest
following conditions (representative of Jupiter interior): Φ0 = itself under the form of homogeneous double diffusive convec-
3×1033,εd = 10,κT =5×10−5m2.s−1,D=5×10−7m2.s−1. tion and act like a turbulent diffusion (Rosenblum et al. 2011).
However, forthistohappen, thecriterion(28)mustbeverified
constraint a = 0.2 a = 0.5 in the inefficient convection regime, δ (cid:29) l (see AppendixA).
T
α stability R−1 =1 2.4×10−9 2.4×10−9 Thefollowingcriterionmustthenhold
min R−m1in= 1+Pr 6.0×10−7 2.4×10−6
αmax homogeneization NNmuuiµn==13τ+Pr 12..05××1100−−45 12..46××1100−−34 −HP(cid:104)∇Z(cid:105)≈∆Z RLp (cid:38) R−m1in×(∇d−∇ad). (33)
µ
αmin observational Jupiter 3×10−5 As ∆Z (cid:54) 1 and R−1 (cid:62) 1, by definition, a zone of turbu-
constraints(cid:63) Saturn 4×10−6 lentdiffusioncannotmeixntendovertheentireplanet’sscaleunless
∇ (cid:46) ∇ ,i.e. ifthewholeobjectisnearlydiffusiveinthefirst
(cid:63)SeeSect.5 d ad
place.
Inanobjectwithaheatfluxhighenoughtobeconvectively
Ontheotherhand,thesolutegradientwithintheplanetwill unstable, the semi-convective zone must then be confined to a
behomogenizedwithinatypicaltimescale fractionoftheplanet, inparticular, butnotnecessarily, nearan
ρ∆ZR immiscibility region or a phase transition for instance. In this
tZ ≈ |FZ| p. (30) cinasJeu,ptihteer)to,ctaolnsdiizteioonfftohrewzhoincehmaluasrtgveeernifoyuLgh/Rjupm(cid:54)pεin−d1th(e∼h1e/a1v0y
Using Eq.(22), (23) and (24), and taking (cid:104)∇Z(cid:105) ≈ ∆Z/R , element mass fraction can be sufficient to stabilize the whole
Eq.(30)becomes p zone against convection and open a diffusive buffer in the inte-
rior (where ∇ ≈ ∇ ). From the global point of view of the
t ≈ R2pδZ ≈ √R2p (cid:32) Φ (cid:33)−4(11+a) ∝α−1/(1+a). (31) 5 AnalogouslTytothedthermalNusseltnumber,thefluxofheavyele-
Z D l κT D CL ments,foragivenNuµ,isgivenbyFZ ≡−Nuµ×ρD∇Z.
7
Leconte&Chabrier:Anewvisionongiantplanetinteriors
planet,thiswouldactasacomposition,temperatureandentropy wheremisthemassenclosedintheequipotentialofmeanradius
nearlydiscontinuity. Thispossibilityfortheexistenceofsucha r¯, ω is the rotation rate of the planet, ϕ (r¯) is a second order
p ω
diffusive-likebufferintheinteriorofourgasgiants,asaconse- correctionduetothecentrifugalpotential,whichdependsonthe
quenceofdouble-diffusiveinstability,mustbekeptinmind. figure functions. As discussed in Sect.3, the prescription to be
Inthefollowing,however,wewillnotconsiderthisscenario usedfor∇ isdeterminedbytheenergytransportprocesses.
T
any further, and we will only consider the effect of a planetary Thedeparturefromsphericityoftheiso-densitysurfacesre-
scalelayered-convectionzone. sults in a perturbation of the external gravity field V (r,θ) that
G
writes
5. NumericalresultsforSolarSystemgiantplanets GM(cid:110) (cid:88)∞ (cid:32)R (cid:33)2n (cid:111)
V (r,θ)=− 1− eq J P (cosθ) , (41)
In this section, we examine whether the presence of semi- G r r 2n 2n
n=1
convectioninJupiterandSaturninteriorscanbeconsistentwith (cid:90)
1
the various available observational constraints. We first derive J =− ρ(r,θ)r2nP (cosθ)d3r, (42)
homogeneous reference interior models in Sect.5.2. Then, in 2n MR2eqn V 2n
Sect.5.3, weincorporateourmodelforlayeredconvectioninto
whereristheradialdistancefromthecenteroftheplanet,Mthe
thestandardmethodusedtocomputeinteriorstructuremodelsof
massoftheplanet,R theequatorialradius,θthecolatitude,P
rotating gaseous planets (presented in Sect.5.1) and determine eq 2n
the area of the composition/layer size space parameter which areLegendrepolynomialsoforder2nandJ2ndenotethegravita-
tionalmoments,thatcanbecomputedoncethefigureequations
is consistent with observed gravitational moments and surface
havebeensolved. Themeasuredgravitymomentsprovidestrin-
abundances.
gentconstraintsonthedensityprofileandthepossiblelayering
withintheseplanets.
5.1. Hydrostaticequilibriumandfiguresoftheplanet As, in practice, Legendre polynomial expansions are trun-
cated at a given order n, a closure equation is provided by the
Solar System giant planets are rapidly rotating bodies (the pe-
equationofstate(EOS)ofthemixturealongtheplanet’sinterior
riodofrotationisabout10hours),withthecentrifugalpotential
profile. Such an EOS is generally given by the so-called ideal
representingabout10%ofthegravitationalpotential.Thismod-
volumelawforthemixture:
ifiesthehydrostaticequilibriumconditionbetweenthepressure
gradient and the gravitational force in the interior, which now 1 X Y Z
= + + , (43)
writes ρ ρ ρ ρ
X Y Z
∇P=−ρ∇(V +V ), (34)
G rot whereX,Y andZ denotethemassfractionsofH,Heandheavy
where elements, respectively. For the H/He fluid, the most widely
(cid:90) ρ(r(cid:48)) usedEOSistheSaumon-Chabrier-vanHornEOS(Saumonetal.
V (r)=−G d3r(cid:48) (35)
G |r−r(cid:48)| 1995;SCvH).Fortheheavymaterial,wehaveusedthe"Rock"
EOS of Hubbard & Marley (1989) for silicates and the "Ice"
and
ANEOS equation of state (Thompson & Lauson 1972) for
V (r,θ)=−(cid:90) ξ ω2(ξ(cid:48))ξ(cid:48)dξ(cid:48) (36) volatiles (CH4, NH3, H2O). The impact of the differences be-
rot p tween various EOS’s on exoplanet structure and evolution has
0
beenexploredinBaraffeetal.(2008).
denote respectively the gravitational and centrifugal potentials,
withdifferentialrotation ω (ξ),whereξisthedistancefromthe Once such EOS’s, P[ρ(Xi)], are specified, structure models
p
positionrtotherotationaxis, andG thegravitationalconstant. with various compositions are calculated by solving iteratively
In the present study, ω is assumed to be constant and given the aforementioned hydrostatic equilibrium condition for a ro-
p
tating body and the third-order level-surface theory (Zharkov
by the magnetospheric rotation rate. Because of the symmetry
&Trubitsyn1978)toobtainamodelwhichreproducestheob-
ofthecentrifugalpotentialwithrespecttoboththerotationaxis
served values of the radius, R , and gravitational moments J
and the equatorial plane, surfaces of equal densities for these eq 2
andJ measuredbythePioneerandVoyagermissions(seeTable
objects are supposed to be generalized ellipsoids of revolution 4
2).
whoseexactshapeisgivenby
 
r(r¯,θ)=r¯ 1+(cid:88)s2n(r¯)P2n(cosθ), (37) 5.2. Referencehomogeneousmodels
n
In conventional giant planet models, the abundances of heavy
wherer¯isthemeanradiusoftheequipotential,P aretheusual
2n elements are chosen to be constant in the gaseous H/He enve-
Legendre polynomials, θ is the colatitude and the s are a set
2n lope,withapossiblediscontinuityatthetransitionbetweenthe
offigurefunctions. Theselattercanbederivedusingthetheory
molecular and metallic regions (Chabrier et al. 1992). Under
of figures for rotating bodies (Zharkov & Trubitsyn 1978), and
the actual conditions found in Jupiter and Saturn, the thermal
must be solved iteratively with the set of perturbed 1D hydro-
gradientthatwouldbeneededtotransportthewholefluxbydif-
staticequilibriumequations
fusive processes, ∇ , is always larger than ∇ and the whole
d ad
∂P 1 Gm ω2 interiorisconvectiveinthehomogeneouscaseaccordingtothe
∂m =−4π r¯4 + 6πpr¯ +ϕω(r¯), (38) Schwarzschild criterion. As convection is very efficient (see
Sect.3), thesuperadiabaticityneededtotransporttheoutgoing
∂r¯ 1
= , (39) energyisontheorderof10−8−10−9,sothatthestructurecanbe
∂m 4πr¯2ρ solvedbysetting∇ =∇ inEq.(40).
T ad
∂T = T ∂P∇ , (40) Inordertohaveareferencecase, weusetheformalismde-
∂m P∂m T scribed in Sect.5.1 to derive homogeneous, adiabatic interior
8
Leconte&Chabrier:Anewvisionongiantplanetinteriors
Table 2. Observed characteristics of Solar System gaseous gi- 108
ants(Guillot2005andreferencestherein;thenumbersinparen- (cid:72)a(cid:76) Saturn
thesesaretheuncertaintyinthelastdigitsofthevalue). 107
P
106
Jupiter Saturn
(cid:76)
Mp[1026kg] 18.986112(15) 5.684640(30) 3m 105 T
R [107m] 7.1492(4) 6.0268(4)
eq (cid:144)
R [107m] 6.6854(10) 5.4364(10) g 104
pol k Ρ
Prot[104s] 3.57297(41) 3.83577(47) Ρ(cid:72) 103
T [K] 165.(5) 135.(5)
F1tobtar[W.m−2] 5.44(43) 2.01(14) (cid:76)K, 102
J ×102 1.4697(1) 1.6332(10) (cid:72) (cid:72)b(cid:76) Jupiter
J2×104 -5.84(5) -9.19(40) T 107 P
4
,
Z(Yat/m(/XZ+(cid:12) Y))atm 0.2328-(450) 0.2125-(835) (cid:76)Bar 106
(cid:72) 105 T
P
104 Ρ
models representative of the usual 2-layer composition. As we 103 Semi(cid:45)Convective
use the interpolated SCvH EOS, we do not consider the effect Adiabatic
102
of a Plasma Phase Transition, and we are thus left with only 0.0 0.2 0.4 0.6 0.8 1.0
two free parameters, namely the core mass (M ) and the metal
c m(cid:144)M
massfractioninthegaseousenvelope(Z )6. Thetemperature, p
env
densityandpressureprofilesofourbestrepresentativehomoge-
neous models of Jupiter and Saturn are shown in Fig.4 (solid Figure4. Pressure(black),temperature(red)anddensity(blue)
curves). Thesearecomposedofasolidcoreofmass M = 3.9 profilesasafunctionofdepth(expressedbytheLagrangianco-
c
and25.6M surroundedbyaH/Hegaseousenvelopewithacon- ordinate, i.e. the mass m), for the reference adiabatic (solid
⊕
stantmetalfractionZ = 0.11andZ = 0.05forJupiterand curves)andsemi-convective(dashedcurves)casesforSaturn(a)
env env
Saturn, respectively (these results are summarized in Table 3). andJupiter(b). Theincreasedthermalgradientduetotheinef-
Thesereferencemodelsyieldinteriorenrichmentthatareconsis- ficientheattransportinthesemi-convectivecase(withNl =104
tentwithpreviousdeterminations(Chabrieretal.1992;Saumon for Jupiter and 104.5 for Saturn) strongly increases the internal
&Guillot2004;Guillot2005). temperature. Thiscausesapartialredistributionofthecorema-
terialwithinthegaseousenvelope.
5.3. Inhomogeneousmodels
wederivedseveralsetsofmodelsforwhichthecompositionof
We now derive semi-convective, inhomogeneous interior mod-
thecorevariesfrompureicetopurerock.
els for Jupiter and Saturn. We stress that all these models are
consistent,withintheobservationaluncertainties,withthemea-
suredgravitationalmomentsofJupiterandSaturn (seeTable2; 5.3.1. Numberoflayers
Campbell&Synnott1985;Campbell&Anderson1989).
From a macroscopic point of view, an important quantity
Anadditionalconstraintontheoutermostvalueofthecom-
describing layered convection is the number of convective-
positional gradient is provided by the surface abundance of diffusivelayers,N. Thisnumberisroughlyequaltotheratioof
l
heavy elements in the planets measured by the 1995 Galileo
thesizeofthesemi-convectivezone,comparabletotheplanet’s
EntryProbemission.Indeed,elementalabundancesoftheatmo-
radius, R , if this zone extends over the whole planet, to the
spheresofsolargiantplanetsareobservedtodiffersignificantly height ofpa typical convective/diffusive cell, l+δ . As shown
T
fromeachotherandfromthesolarcomposition,beingenriched in Sect.4, in the regime of interest, l+δ ≈ l, and N ≈ R /l.
byafactor∼ 2−4and∼ 2−8withrespecttotheSun’satmo- Because H ≈ R in the deep interior, thTe number ofllayerspin
P p
sphereforJupiterandSaturn,respectively,asshowninTable2
the planet is thus approximately equal to N ∼ α−1, and in the
(Guillot2005). Moreover,theplanet’stotalmeanabundancesof followingwewillalwaysreferindifferentlytloeitherαor
H and He (X¯ and Y¯) must recover the values of the protosolar
nebula,i.e. Y¯/(X¯ +Y¯)≈0.275. N ≡α−1 ≡ H /l. (44)
l P
In our calculations, the adjustable parameters to fulfill all
these constraints are chosen to be the mass of the core (Mc), AsshowninSect.4,Nlisconstrainedtoliewithintherange
the mean heavy element mass fraction in the gaseous envelope 102−4 ≤ Nl ≤ 106−9. Note that, given the small size of the
(Z¯ ), and the global compositional variation in the envelope diffusive-convectivelayerscomparedwiththesizeoftheplanet,
env
(∆Z ,thedifferencebetweenthemetalmassfractionjustabove the discontinuous (staircase-like) temperature and composition
env
the central core and the one in the atmosphere). To assess the profiles canbe well approximatedby continuous mean thermal
robustness of our results with respect to the equation of state andcompositionalgradients((cid:104)∇T(cid:105)and(cid:104)∇µ(cid:105),respectively)tode-
chosen to describe the thermodynamics of the heavy material, terminetheplanet’sglobalstructure,asillustratedinFig.2.
This possible range of numbers of layers is further con-
6 AsalreadyfoundbyChabrieretal.(1992)andSaumon&Guillot strainedbyournumericalcalculations,whichshowthat,inorder
(2004),withtheSCvHEOS,simplehomogeneousmodelssuchasour toreproduceourgiantplanetobservationalconstraints,nomore
referencecasecannotreproduceJ tobetterthanafewpercentserror. than∼2.5×105layers(α ≈4×10−6)caninrealitybepresent
4 min
9
Leconte&Chabrier:Anewvisionongiantplanetinteriors
0.5
100 (cid:72)a(cid:76) Saturn
(cid:72)a(cid:76)Saturn 0.45
0.4
10 (cid:209)
d
0.35
(cid:76) 0.3
Z
nt 1 ‹(cid:209)T› (cid:72)n 0.25
e
di io 0.2
alGra 100 (cid:209)ad (cid:72)b(cid:76)Jupiter fract 0.15 110055.5LLaayyeerrss
m s 0.1 104.5Layers
her 10 mas 0.05 110034.5LLaayyeerrss
T (cid:209) al 103 Layers
d et (cid:72)b(cid:76) Jupiter
M 0.2
1
‹(cid:209)T› 0.15
(cid:209) 0.1
ad
107 108 109 1010 1011 0.05
Pressure(cid:72)Pa(cid:76) 0.
0.0 0.2 0.4 0.6 0.8 1.0
Figure5. Conductive (top red curves) and adiabatic (bottom m(cid:144)M
p
black curves) thermal gradient profiles for Jupiter and Saturn.
Thedashedcurvescorrespondtothemeanthermalgradientfor Figure6. Metalabundanceprofilesasafunctionofdepth(ex-
modelswith104.5and105.4layersforJupiterandSaturn,respec- pressed by the Lagrangian mass m) for Saturn (a) and Jupiter
tively. The shaded area represents the allowed range of super (b) for different numbers of layers. The abundance increases
adiabaticityinpresenceofsemi-convection,consistentwiththe withthenumberoflayers,tokeepthedensityprofileunchanged
observationalconstraints. whenconvectionbecomesinefficient. Theextremecases(105.4
layersforSaturnand104.5 layersforJupiter)correspondtothe
semi-convectiveprofilesportrayedinFig.4.
inSaturnand∼ 3×104 (α 3×10−5)inJupiter(seeTable1).
min
Indeed,alargernumberoflayersleadstosohightemperatures
intheinteriorthattheinducedmeandensitydecreasecannotbe thantheusualadiabaticassumption,asheatandmaterialredis-
counterbalancedbyanincreaseoftheheavyelementmassfrac- tributionsarepartlyinhibitedbydiffusiveprocesses. Thepres-
tion compatible with the observed surface abundances. This is sure and density profiles, on the other hand, remain barely af-
duetothefactthat,thelargerthenumberoflayers,thesmaller fected,beingstronglyconstrainedbythegravitationalmoments.
thesizeofeachconvectivecell,reducingthemaximumheighta Hence, at basically fixed density profile, a higher temperature
convectiveeddycantraveltotransportheatbeforebeingstopped profilemustbecompensatedbyalargeramountofheavymate-
bythenegativebuoyancypresentinthediffusiveinterface. rial within the envelope. This is illustrated in Fig.6, where we
show the abundance profiles, as calculated in AppendixB, cor-
Alargenumberoflayersthusdecreasesconvectiveheat(and
composition) transport efficiency. This leads to an increase of respondingtosemi-convectivemodelswithdifferentnumbersof
layers. The bottom curve (solid) in each panel corresponds to
themeansuperadiabaticity,asportrayedonFig.5,whichinturn
modelswith1000layerswhiletheothercurvescorrespondtoa
immediatelyimpliesariseoftheinternaltemperature, asillus-
graduallyincreasingnumberoflayers.
trated on Fig.4. It is important to stress that super adiabaticity
Therefore, in order to compensate the radius increase (den-
is the physical quantity most directly constrained by the data.
sity decrease) due to the hotter interior, semi-convection yields
Thus,whereastheallowedrangeofnumberoflayers(orequiv-
asignificantlylargertotalmetalcontentcomparedwithconven-
alently of layer sizes) may slightly depend on the model used
tional homogeneous models. This can be seen in Fig.7, which
toparametrizesemi-convection(seeSect.3), theallowedrange
shows the amount of heavy elements in the core and envelope
of super adiabaticity displayed in Fig.5 should remain weakly
affected. for the various cases discussed here, as summarized in Table
3. ForSaturn, upto50M ofheavyelementscouldbepresent
⊕
intheplanetwhileforJupitertheheavymaterialcontentcould
5.3.2. Enhancedheavymaterialenrichment reach63 M . Thiscorrespondstoabout25and10timestheso-
⊕
lar abundances, respectively7. Since these values only depend
Thepressure, densityandthermalprofilesobtainedinthemost
on the allowed amount of super adiabaticity, they should not
extremesemi-convectivecasecompatiblewiththeobservational strongly depend on our modeling of diffusive/convective trans-
constraintsdiscussedaboveareshowninFig.4(dashedcurves).
As seen on the figure, and as expected from the above discus- 7 NotethattheabundancesofheavyelementsbroughttoJupiterand
sion, the non-adiabatic envelope profile obtained in the semi- Saturn, in particular water, could already be enriched compared with
convectivecaseyieldssubstantiallyhigherinternaltemperatures thesolarvalue(Gautieretal.2001)
10