The tides of Mercury and possible implications for its interior structure

17 Pages·2014·2.63 MB·English

JournalofGeophysicalResearch:Planets
RESEARCHARTICLE The tides of Mercury and possible implications
10.1002/2013JE004459 for its interior structure
SebastianoPadovan1,Jean-LucMargot1,2,StevenA.Hauck,II3,
KeyPoints:
•Wepresentmodelsofthetidal WilliamB.Moore4,5,andSeanC.Solomon6,7
deformationofMercurybasedon
MESSENGERresults 1DepartmentofEarth,Planetary,andSpaceSciences,UniversityofCalifornia,LosAngeles,California,USA,2Department
•Tidesaresensitivetosizeandden- ofPhysicsandAstronomy,UniversityofCalifornia,LosAngeles,California,USA,3DepartmentofEarth,Environmental,and
sityofthecoreandrheologyof PlanetarySciences,CaseWesternReserveUniversity,Cleveland,Ohio,USA,4DepartmentofAtmosphericandPlanetary
outershell
Sciences,HamptonUniversity,Hampton,Virginia,USA,5NationalInstituteofAerospace,Hampton,Virginia,USA,
•ThepresenceofaFeSlayerwould
increasethetidalresponse 6Lamont-DohertyEarthObservatory,ColumbiaUniversity,Palisades,NewYork,USA,7DepartmentofTerrestrial
Magnetism,CarnegieInstitutionofWashington,Washington,DistrictofColumbia,USA
Correspondenceto:
S.Padovan, AbstractThecombinationoftheradiotrackingoftheMErcurySurface,SpaceENvironment,
[email protected]
GEochemistry,andRangingspacecraftandEarth-basedradarmeasurementsoftheplanet’sspinstategives
threefundamentalquantitiesforthedeterminationoftheinteriorstructureofMercury:meandensity𝜌,
Citation: momentofinertiaC,andmomentofinertiaoftheoutersolidshellC .Thisworkfocusesontheadditional
Padovan,S.,J.-L.Margot,S.A.HauckII, m
W.B.Moore,andS.C.Solomon(2014), informationthatcanbegainedbyadeterminationofthechangeingravitationalpotentialdueto
ThetidesofMercuryandpossible planetarytides,asparameterizedbythetidalpotentialLovenumberk .Weinvestigatethetidalresponse
2
implicationsforitsinteriorstructure, forsetsofinteriormodelsthatarecompatiblewiththeavailableconstraints(𝜌,C,andC ).Weshowthatthe
J.Geophys.Res.Planets,119,850–866, m
doi:10.1002/2013JE004459. tidalresponsecorrelateswiththesizeoftheliquidcoreandthemeandensityofmaterialbelowtheouter
solidshellandthatitisaffectedbytherheologyoftheoutersolidshelloftheplanet,whichdependson
itstemperatureandmineralogy.Foramantlegrainsizeof1cm,wecalculatethatthetidalk ofMercuryis
Received12JUN2013 2
Accepted12MAR2014 intherange0.45to0.52.SomeofthecurrentmodelsfortheinteriorstructureofMercuryarecompatible
Acceptedarticleonline18MAR2014 withtheexistenceofasolidFeSlayeratthetopofthecore.Suchalayer,ifpresent,wouldincreasethetidal
Publishedonline21APR2014 responseoftheplanet.
1.Introduction
Intheabsenceofaninsitugeophysicalnetwork,whatweknowoftheinteriorofMercuryisbasedona
combinationofEarth-basedobservations,spacecraftexploration,andtheoreticalinsight.Earth-basedradar
observationsprovidemeasurementsoftheobliquityofMercuryandtheamplitudeofitsforcedlibration
[Margotetal.,2007,2012].ThroughradiotrackingoftheMErcurySurface,SpaceENvironment,GEochem-
istry,andRanging(MESSENGER)spacecraft,thegravitationalfieldoftheplanethasbeendetermined
[Smithetal.,2012].GiventhatMercuryisinaCassinistate[Colombo,1966;Peale,1969],thespinparameters
(obliquity𝜃andangleofforcedlibrationinlongitude𝛾),whencombinedwiththesecondharmonicdegree
componentsofthegravityfield(J andC ),providetwoimportantintegralconstraintsfortheinteriorof
2 22
Mercury:themomentofinertiaC[Peale,1969]and,inthepresenceofagloballiquidlayer,themomentof
inertiaoftheoutersolidshellC [Peale,1976].Thesetwomoments,alongwiththemeandensity𝜌,arethree
m
constraintsthatanymodeloftheinteriorofMercurymustsatisfy[Haucketal.,2013].
Themeasurementofthedeformationofaplanetduetoperiodictidalforcingcanbeusedtoplaceaddi-
tionalboundsontheinteriorstructure,becausethetidalresponseisafunctionofthedensity,rigidity,and
viscosityofthesubsurfacematerials.Thispropertyhasbeenappliedinthepasttosupportthehypothesis
ofaliquidcoreinVenus[KonoplivandYoder,1996]andagloballiquidoceaninTitan[Iessetal.,2012].Yoder
etal.[2003]usedthemeasurementofthetidestorevealtheliquidstateoftheMartiancoreandtoestimate
itsradius.ItisinterestingtonotethattheinteriorstructuresofVenusandMarsarecurrentlylesswellcharac-
terizedthanthatofMercury,sinceforMarsthemomentofinertiaoftheoutersolidshellisnotknown,and
forVenusonlythemeandensityandk tidaldeformationareknown,butnomomentofinertiainformation
2
isavailable.
ThemotivationforthispaperistoexploretheinformationthatcanbegainedabouttheinteriorofMercury
bythecombinationofthedeterminationsof𝜌,C,andC withthemeasurementofk ,whichwillindicate
m 2
the88dayannualtidalk .
2
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10.1002/2013JE004459
WemodelthetidalresponseofMercuryforarangeofinteriorstructuresthatarecompatiblewiththe
meandensity𝜌andthemomentsofinertiaCandC [Haucketal.,2013].Theformalismthatweemployis
m
describedinsection2,andsection3describestheinteriormodelsthatweuseandtheassumptionsthatwe
makeintheevaluationofthetidalresponse.Therheologyoftheoutersolidshellisdiscussedinsection4.
Theresultsofoursimulationsarepresentedinsection5(theminoreffectsofthepropertiesoftheinnercore
onthetidalresponseareexploredinAppendixA).Wediscusstheimplicationsofthedetectionofthetidal
responseforthephysicalcharacterizationoftheinteriorofMercuryinsection6.
2.PlanetaryTidalDeformation
Mercury’ssolartidesarecausedbythedifferenceinthegravitationalattractionoftheSunacrosstheplanet.
DenotingthemassoftheSunbyM ,theexpressionforthesolartide-generatingpotentialΦatapointP
S
insidetheplanetis
[ ( ) ]
Φ= GMS = GMS ∑∞ r′ nP (cos𝜓 ) = ∑∞ Φ , (1)
d r r n P n
S n=2 S n=2
wherethesummationfollowsfromtheexpansionfor(1∕d),anddisthedistancebetweenPandtheSun
[e.g.,ArfkenandWeber,2005].Theangle𝜓 istheanglebetweenr′andr ,thedistancesfromthecenterof
P S
massofthebodytoPandtotheSun,respectively.P indicatestheLegendrepolynomialofdegreen.Gisthe
n
gravitationalconstant.WeintroduceΦ tohighlightthedependenceofΦonthenthpowerofthefactor
n
(r′∕r ) ≪ 1.Forapointonthesurface,wesetr′ = R ,theradiusofMercury,andr equaltoa ,thesemi-
S M S M
majoraxisofMercury’sorbit,andwecanexpressthelargestcomponentofthepotentialasg𝜁P (cos𝜓 ),
2 P
whereg=(GM ∕R2)isthegravitationalaccelerationatthesurface,M isthemassofMercury,𝜁P (cos𝜓 )
M M M 2 P
istheheightoftheequilibriumtide[MurrayandDermott,1999],and
( )
M R 3
𝜁 = S M R . (2)
M a M
M M
Amongtheterrestrialplanets,𝜁isthelargestforMercury,withavalueof∼1.10 m(forcomparison𝜁 ∼
Venus
0.43m,𝜁 ∼0.16m,and𝜁 ∼0.03m).
Earth Mars
Theharmonicexpansionofthetide-generatingpotentialinequation(1)canbeusedtoidentifyallthe
differenttidalcomponents(inperiodandamplitude)generatedbytheSunatMercury[VanHoolstand
Jacobs,2003].ThelargestcomponenthasatimescaleequaltotheorbitalperiodofMercuryaroundtheSun
(∼ 88days).ThisannualtidalperturbationperiodicallymodifiestheshapeofMercuryandthusthedis-
tributionofmatterinsidetheplanet,withanaccompanyingmodificationofitsgravitationalfield.This
modificationisparameterizedwiththepotentialLovenumberk ,whichrelatestheadditionalpotential𝜙
2 2t
duetothedeformationoftheplanettothetide-generatingpotentialΦ duetotheSun:
2
𝜙 =k (𝜔)Φ . (3)
2t 2 2
Thesubscript2indicatesthatthemaindeformationisgeneratedbythelargesttermoftheexpansion,
whichcorrespondston = 2.Thefrequency𝜔indicatesthattheresponseofthebody,describedbyk ,
2
dependsontheperiod(i.e.,frequency)oftheappliedforcing,whichforthecaseconsideredhereisthe88
dayperiodsolartide.
Thestudyofthedeformationofaplanetundertheperturbationofanexternalpotentialrequiresthesolu-
tionoftheequationsofmotioninsidethebody.Usingasphericalharmonicdecompositioninlatitudeand
longitude,wetransformthesethreesecond-orderordinarydifferentialequationsintosixfirst-orderlinear
differentialequationsinradius[Altermanetal.,1959].Themotioniscontrolledbothbymaterialstresses
(elasticorviscoelastic)andgravitationalforces,thelatteroriginatingfromagravitationalpotentialthatisthe
sumoftheself-gravitationoftheplanetandtheexternaltidalpotential.Theframeworkforthesolutionis
formallythesamebothforelasticrheologiesandforviscoelasticrheologies,thankstothecorrespondence
principle[Biot,1954].TheresultspresentedinthefollowingsectionsareobtainedbymodelingMercuryas
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aseriesofhomogeneousincompressiblelayers.Eachlayerischaracterizedbythickness,density,rigidity,
andviscosity[Wolf,1994].Inevaluatingthetidalresponse,weusetheformalismdevelopedbyMooreand
Schubert[2000].
Thepossiblevaluesforthek ofaplanetrangebetween0foraperfectlyrigidbodythatdoesnotdeform,
2
and1.5,thevalueforahomogeneousfluidbody(fortheseidealizedbodiesthelimitsareindependentof
theforcingfrequency).Valuesfork havebeendeterminedforVenus[KonoplivandYoder,1996],theMoon
2
[Konoplivetal.,2013;Lemoineetal.,2013],Mars[Konoplivetal.,2011],andTitan[Iessetal.,2012].Thek of
2
theMoonisuncertainatthe∼0.5%level,aresultofthehigh-qualitydataobtainedwiththeGravityRecov-
eryandInteriorLaboratorymission[Zuberetal.,2013].ForMarstheestimateisuncertainatthe∼5%level,
aresultobtainedbycombiningdatafromalargenumberofspacecraftmissions,includingalanderandtwo
yearsoftrackingdatafromthelow-altitude,nearlycircularorbitalphaseoftheMarsReconnaissanceOrbiter.
ForVenusandTitantheestimateshaveanuncertainty≳ 10%.Anumericalsimulationofthedetermina-
tionofMercury’sk withBepiColombo,thefuturedualorbitermissiontoMercurybytheEuropeanSpace
2
AgencyandtheJapanAerospaceExplorationAgency,indicatesanexpectedaccuracyof∼1%[Milanietal.,
2001].ThisfigurerepresentsalowerboundforMESSENGER,becauseitseccentricorbitmakesthedetec-
tionofk morechallenging.Theuncertaintyonthedeterminationofthek ofMercuryasobtainedfrom
2 2
MESSENGERisexpectedtobe∼10%[Mazaricoetal.,2014].
ThemantleofEarthrespondselasticallyontheshorttimescalesassociatedwiththewavesgeneratedby
earthquakesbutflowslikeafluidonthegeologicallylongtimescalesofmantleconvection.TheMaxwell
rheologicalmodelisthesimplestmodelthatcapturesthisshort-andlong-timescalebehavior.Itiscom-
pletelydefinedbytwoparameters:theunrelaxed(infinite-frequency)rigidity𝜇 andthedynamicviscosity
U
𝜈.TheMaxwelltime,definedas
𝜈
𝜏 = , (4)
M 𝜇
U
isatimescalethatseparatestheelasticregime(forcingperiod≪ 𝜏 )fromthefluidregime(forcingperiod
M
≫𝜏 ).Thissimplerheologicalmodelissufficientlyaccurateforthecrust,whichiscoldandrespondselasti-
M
cally,andtheliquidcore,whichhaszerorigidityandthereforeafluidresponse.Theinnercore,ifpresent,has
anegligibleeffectonthetidalresponse(AppendixA),soforsimplicityweuseaMaxwellmodeltodescribe
itsrheology.Nevertheless,theMaxwellmodeldoesnotprovideagoodfittolaboratoryandfielddatainthe
low-frequencyseismologicalrange,andthusitshouldnotbeusedtomodeltheresponseofthemantleat
tidalfrequencies[e.g.,EfroimskyandLainey,2007;Nimmoetal.,2012].
Jacksonetal.[2010]exploredthreedifferentparameterizations(Burgers,extendedBurgers,andAndrade
pseudoperiod)tofittorsionaloscillationdatafromasetofmelt-freeolivinesamples.BoththeBurgersmod-
elsandtheAndrademodelprovideagoodfitforthelow-frequencydata.Thesmallnumberofparameters
requiredfortheAndrademodelmakesitmoreattractivetomodeltherheologyofMercury,forwhichwe
lackanyground-truthdata.Note,however,thatboththeBurgersmodelsandtheAndrademodelhave
notbeentestedatperiodslongerthan103s,sowhenappliedtothestudyofplanetarytidaldeformation
(period> 106s),theybothneedtobeextrapolated(foranapplicationoftheextendedBurgersmodelto
themantleoftheMoonandMars,seeNimmoetal.[2012]andNimmoandFaul[2013],respectively).
WereportheretheexpressionsfortherealandimaginarypartofthedynamiccomplianceJ(𝜔)forthe
Andrade-pseudoperiodmodel,asdescribedbyJacksonetal.[2010]:
{ ( )}
1 nπ
J (𝜔)= 1+𝛽∗Γ(1+n)𝜔−ncos , (5)
R 𝜇 2
U
{ ( ) }
1 nπ 1
J(𝜔)= 𝛽∗Γ(1+n)𝜔−nsin + (6)
I 𝜇 2 𝜔𝜏
U M
Theunrelaxedrigidityis𝜇 and𝛽∗=𝛽𝜇 .Thecoefficient𝛽,alongwithn,appearsintheexpressionof
U U
theAndradecreepJ(t)=1∕𝜇 +𝛽tn+t∕𝜈,whereΓisthegammafunctionand𝜏 theMaxwelltime.The
U M
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Table1.RheologicalModelsfortheInteriorofMercurya
Layer Model Parameter Definition Value Notes
Crust Maxwell section4.3
𝜇 Unrelaxedrigidity 55GPa
U
𝜈 Dynamicviscosity 1023Pas
Mantle Andradeb section2
𝜇 Unrelaxedrigidity 59–71GPa section4.2
U
T Mantlebasaltemperaturec 1600–1850K section3.2
b
n Andradecreepcoefficient 0.3
𝛽∗ Andradecreepparameter 0.02
P Referencepressure 0.2GPa
R
T Referencetemperature 1173K
R
d Referencegrain-size 3.1μm
R
d Grainsize 1mm–1cm
m Grainsizeexponent 1.31
V Activationvolume 10−5m3mol−1
E Activationenergy 303×103kJmol−1
B
FeS Andraded section4.4
Outercore Maxwell section2
𝜇 Unrelaxedrigidity 0Gpa
U
𝜈 Dynamicviscosity 0Pas
Innercore Maxwell AppendixA
𝜇 Unrelaxedrigidity 1011Pa
U
𝜈 Dynamicviscosity 1020Pas
aThemodelsareintroducedinsection2.
bThefixedparametersoftheAndrademodelarebasedontheresultsofJacksonetal.[2010].
cHere we report T because the temperature T in equation (7) depends on the temperature profile,
b
whichiscontrolledbyT .
b
dTheFeSlayerisassumedtohavethesamerheologyasthatofthebaseofthemantle.
frequency𝜔isobtainedfrom𝜔 = 2π∕X ,whereX isthepseudoperiodmastervariableintroducedby
B B
Jacksonetal.[2010]:
( ) [( )( )] [( )( )]
X =T d −mexp −EB 1 − 1 exp −V P − PR , (7)
B 0 d R T T R T T
R R R
whichtakesintoaccounttheeffectsofpressureP,temperatureT,andgrainsized.ThesubscriptRindi-
catesreferencevalue.T istheforcingperiod(forMercury∼ 88days).Theexponentmcharacterizesthe
0
dependenceonthegrainsize,whichinprinciplecanbedifferentforanelasticprocesses(m )andforviscous
a
relaxation(m ).WetestedthatatthefrequencyoftheMercurytide,theeffectisminor,andweassumed
v
m = m = m.TheotherquantitiesaredefinedinTable1.Thedynamiccompliancewasevaluatedbyset-
a v
tingthevalueof𝜏 inequation(6)equaltothereferencevalue(𝜏 = 105.3s)reportedby[Jacksonetal.,
M MR
2010]andincludingtheeffectsofT,P,anddthroughthepseudoperiodmastervariabledefinedinequation
(7).ThedynamiccomplianceisrelatedtotheinversequalityfactorQ−1andtherigidity𝜇by
J(𝜔)
Q−1(𝜔)= I , (8)
J (𝜔)
R
[ ]
𝜇(𝜔)= J2(𝜔)+J2(𝜔) −1∕2. (9)
R I
Toillustratetheimportanceofchoosingarealisticrheologicalmodel,inFigure1weshowhowtherigidity
ofamaterialwith𝜇 =65GPavariesasafunctionoftheforcingfrequencyfortwotemperatures,atapres-
U
sureof5.5GPa,representativeofconditionsatthebaseofthemantleofMercury[Haucketal.,2013].Both
theMaxwellrheologicalmodelandtheAndrademodelareplotted.Theybothpredictafluidresponse(i.e.,
zerorigidity)athightemperaturesand/orlongforcingfrequencies,buttheMaxwellmodelunderestimates
nonelasticeffectsatforcingperiodsthatareshorterthantheMaxwelltime.Thiseffectisparticularlyrele-
vantforMercury,forwhichthecore-mantleboundarytemperaturemaybeabove1600K[RivoldiniandVan
Hoolst,2013;Tosietal.,2013].
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3.Methods
Throughoutthiswork(exceptsection3.1),
weusemodelscompatiblewiththeavail-
ableconstraints,i.e.,meandensity𝜌,
momentofinertiaC,andmomentofiner-
tiaofthesolidoutershellC (section1).
m
Bycompatiblewemeanthatthedistribu-
tionsof𝜌,C,andC inthesetofinterior
m
modelsconsideredhereareapproxi-
matelyGaussianwithmeansandstandard
deviationsthatmatchthenominalvalues
oftheobservablesandtheir1standard
deviationerrors.Themeandensity𝜌has
aGaussiandistributionwithmeanand
standarddeviationequalto5430 kg/m3
Figure1.ComparisonoftheAndrade(solidlines)andMaxwell and10 kg/m3,respectively.ForCand
(dashedlines)rheologicalmodelsatapressureof5.5GPafortwo
C ,wechooseGaussiandistributions
differenttemperatures:T = 1400K(red)andT = 1800K(green). m
ThesolidcoloredverticallinesrepresenttheMaxwelltimes.The withmeansandstandarddeviations
dash-dottedlineindicatestheforcingfrequencyofMercury’stide. definedbytheobservedvaluesanderrors
NotethatatT = 1800KtheMaxwellmodeloverestimatestherigid- reportedbyMargotetal.[2012].Accord-
ityatthetidalfrequencybyabout35%comparedwiththeAndrade
ingly,C∕M R2 = 0.346 ± 0.014and
model.Theunrelaxedmodulususedintheseexampleis65GPa. M M
C ∕C=0.431±0.025[Margotetal.,2012].
m
ThesmallabundanceofFeandrelativelylargeabundanceofSatthesurfaceofMercuryimplystrongly
reducingconditionswithintheplanet[Nittleretal.,2011].Undertheseconditionsbothsiliconandsulfur
likelypartitionedintothecoreduringMercury’sformationanddifferentiation[Haucketal.,2013].Ofthe
fivecompositionalmodelsfortheinteriorofMercuryanalyzedbyHaucketal.[2013],wefocusontwosets
thathaveaSi-bearingcore,becausetheyareconsistentwiththeinferredreducingconditions.Themajor
differencebetweenthetwosetsisthepresenceorabsenceofasolidFeSlayeratthetopofthecore.We
labelthesetwosetstheFeSsetandNoFeSset,respectively.
ThepossiblepresenceofanFeSlayerwasinitiallypredicatedonthebasisoftheinferredhighlyreducing
conditionsandthethen-bestestimateofthehighmeandensityoftheoutersolidshell[Smithetal.,2012].
Improvedvaluesoftheobliquity𝜃[Margotetal.,2012]ledtoarevisedvalueforthemeandensityofthe
outersolidshell[Haucketal.,2013]andmadethedensityargumentforthepresenceoftheFeSlayer
lesscompelling.Nevertheless,thegeochemicalargumentsupportingthepresenceoftheFeSlayerisstill
valid[Haucketal.,2013],andaconductivelayerabovetheconvectiveliquidcoreisoneofthepossible
explanationsforMercury’sweakmagneticfield[Christensen,2006;Andersonetal.,2012].
3.1.RadialDensityProfile
Theradialdensityprofilesthatweusedasinput[Haucketal.,2013]aregivenasseriesofconstant-property
layers,goingfromthecentertothesurface.Thecrust,mantle,andFeSlayeraremodeledas
constant-densityshells.Thissimplificationisjustifiedbythesmallthicknessoftheoutersolidshellof
Mercuryandbytherelativelylowsurfacegravitationalacceleration,butdoesnotaffectthecharacterization
oftheinterioroftheplanetonthebasisofthemeasuredvaluesof𝜌,C,andC [Haucketal.,2013].Thecore
m
(inner+outer)isrepresentedwith∼1000layersinordertotakeintoaccounttheeffectsofself-compression
andtemperatureintheequationofstateforcorematerials,aswasdonebyHaucketal.[2013].However,the
Lovenumberk isaglobalparameter,summarizingtheresponseoftheplanettotidalforcing,anditisnot
2
verysensitivetothefinedensitystructure.Weverifiedthatk calculationscanbeperformedaccuratelywith
2
simplified,4-or5-layermodelsinsteadoftheoriginal∼ 1000-layermodels.Inordertoestablishthispoint,
weusedarandomsampleof100modelsdrawnfromtheFeSsetandconstrainedonlybythemeandensity
𝜌ofMercury.Forthistestwedidnotapplythemomentofinertiaconstraints(CandC ),asthisallowedus
m
toexplorealargerparameterspaceandresultedinamorerobusttest.Foreachoneofthe100modelswe
computedanaveragedversion,characterizedbyfiveconstant-densitylayers.Computedk valuesforthe
2
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∼1000layermodelsandthecorre-
0.7
sponding5-layermodelsareshown
0.6
inFigure2(top).Theirratio(Figure2,
0.5 bottom)indicatesthaterrorsintro-
ducedbyusingthesimplifiedmodels
0.4
are≲ 2%.Inviewofthisresultandof
0.3
the∼10%accuracyofMESSENGER’sk
2
0.2 determination(section2),inwhatfol-
lowsweshowresultsobtainedwith
1.01 thesimplified4-or5-layermodels.This
1.00 approachreducesthecomputational
0.99
costby∼3ordersofmagnitude.
0.98
0 20 40 60 80 100 3.2.PressureandTemperature
ProfilesintheMantle
Figure2.Effectoftheradialdensityprofileonthemagnitudeofk2.(top) Tocalculatearheologicalprofile
Foreachofasetof100models,thevalueofk hasbeencalculatedboth
2 forthemantleofMercurywiththe
forthe∼1000layerversion(k )andforthefive-layerversion(k ).
2(M) 2(Av) Andraderheologicalmodeldescribed
(bottom)Theratioofthetwodeterminations,whichinmostcasesis
within2%ofunity.Themodelsusedforthisplotareconstrainedonlyby insection2,thepressureandtem-
themeandensityofMercury. peratureasafunctionofdepthmust
becalculated.
AttheradiusrinthemantlethepressureissimplyobtainedasanoverburdenloadP(r)=g[𝜌 h +𝜌 h (r)],
c c m m
wherethesubscripts“c”and“m”refertothecrustandmantle,respectively.Thecrustalthicknessish ,and
c
h (r)isthethicknessofthemantleabover(i.e.,r+h (r)+h =R ).
m m c M
Weobtainedthetemperatureprofilebysolvingthestaticheatconductionequationwithheatsourcesin
sphericalcoordinates[e.g.,TurcotteandSchubert,2002]inthemantleandcrust:
( )
1 d dT
k r2 +𝜌H=0. (10)
r2dr dr
Inequation(10)kisthethermalconductivity,𝜌isthedensity,andHistheheatproductionrate.
Weassumedahomogeneousdistributionofheatsourcesinthecrustandinthemantle.Thedistributionin
thecrustmightbeexponentialasinthecrustoftheEarth,butweverifiedthatthiswouldonlymarginally
affectthedeepmantletemperatureprofile.ThevalueofHatthesurface,H ,hasbeeninferredfromMES-
0
SENGERmeasurementsandisequaltoH = 2.2×10−11 Wkg−1[Peplowskietal.,2011].Weadoptedthe
0
surfacevalueH fortheheatproductionrateinthecrust,H =H .Forthedistributionofheatsourcesinthe
0 c 0
mantleweusedH =H ∕2.5,whichiscompatiblewiththeenrichmentfactorderivedbyTosietal.[2013].
m 0
Thevalueofkissetto3.3Wm−1K−1.
Asboundaryconditions,weappliedthesurfacetemperatureT andthetemperatureatthebaseoftheman-
S
tleT .T issetto440K,avalueobtainedwithasimpleequilibriumtemperaturecalculation.Therefore,in
b S
ourmodelsthetemperatureprofileiscontrolledbythetemperatureT .Therearecurrentlyfewconstraints
b
onT ,buttwoindependentsetsofworkers[RivoldiniandVanHoolst,2013;Tosietal.,2013]pointtothe
b
range1600–1900K.Wedefinedtwoend-memberprofiles:acoldmantlewithT = 1600Kandahotman-
b
tlewithT = 1850K.WeconsiderT = 1850Kasourhotmantlecase,since,fromtheperidotitesolidusof
b b
Hirschmann[2000],T = 1900Kwouldresultinpartialmeltingatthebaseofthemantle(Figure3).Wedid
b
notconsiderinthisworkthepresenceofpartialmelting.
Therheologicalmodelsdescribedinsection2stronglydependonthetemperature.Ourend-membertem-
peratureprofilesareobtainedundertheassumptionofaconductivemantle.Thisassumptionisconsistent
withtheresultsofTosietal.[2013],whichindicatethatthemantleofMercuryismostlikelyconductiveat
thepresenttime.Nevertheless,apresent-dayconvectivemantleisnotexcluded[Micheletal.,2013;Tosi
etal.,2013].AconvectivemantlefortheT =1850Kcasewouldresultinpartialmelting(Figure3).Acon-
b
vectiveprofilewithT =1600Kwouldbemoredissipativeanddeformablethantheconductivecase(since
b
intheconvectiveenvelopethetemperatureisapproximatelyconstantandequaltoT ),butthiseffectis
b
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Journal of Geophysical Research: Planets
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similartoaconductivecasewitha
higherT .Ourtwoend-membertem-
b
peratureprofilesthuscapturethe
possibleeffectsoftemperaturevaria-
tionsinthemantleofMercury,under
theassumptionthatthereisnopartial
meltinginthemantle.
Tomodeltherheologyoftheman-
tleasafunctionofdepth,wedivided
itintosublayers.Startingfromthe
core-mantleboundary,wedivided
themantlein40kmthicksublayers,
asillustratedinFigure3.Foreach
sublayerthepressureandthetemper-
atureatthemidpointwerecalculated.
Figure3.Temperature(solidline)asafunctionofdepthforamodelwith Thecomplexcomplianceforeach
a2020kmradiuscoreanda50kmthickcrust.Thevaluesfortheheatpro- sublayerwasobtainedwithequations
ductionrateinthecrustandmantleareindicated.Thedotsrepresentthe (5)and(6)ofsection2.Therigidity
midpointofeachmantlesublayer,forwhichthetemperatureandpressure
wascalculatedwithequation(9).The
areusedtoderivearheologyforthesublayer.Theperidotitesolidusof
viscosityisthengivenbytheexpres-
Hirschmann[2000]isalsoshown.
sion𝜈 = 1∕(J𝜔).Itistheviscosityof
I
aMaxwellmodelwiththesamecomplexcompliance,i.e.,withthesamerheology.Thevalueofrigidityand
viscositysocalculatedweretakenasrepresentativeofthefullsublayer.
TheAndraderheologicalmodelhasbeensuccessfullyappliedtothedescriptionofdissipationinrocks,
ices,andmetals[i.e.,Efroimsky,2012,andreferencestherein].Themodeldescribedinsection2currently
representsthebestavailableAndrademodelparameterizationthatincorporatestheeffectsoftemperature,
pressure,andgrainsizeontherheology.However,theparametersthatarekeptfixedinthemodel(listed
inTable1)arebasedonlaboratorydataonolivine[Jacksonetal.,2010].Inwhatfollowsweapplythe
Andrademodelofsection2todifferentmineralogicalmodelsforthemantleofMercury.Wethusassume
thatthefixedparametersofolivinecanbeappliedtootherminerals.Thisassumptionisnotstrictlycorrect,
especiallyformantlemodelsinwhicholivineisnotthedominantphase,butthebroadapplicabilityofthe
Andrademodeltodescribematerialsaschemicallyandphysicallydifferentasicesandsilicatesindicates
thatthemodelweuseshouldprovideagooddescriptionoftherheologyofsilicateminerals.
4.AssessmentoftheRheologyoftheOuterSolidShell
Theunrelaxedrigidityisaparameterrequiredtocharacterizetherheologyandthustheresponsetothe
tidalforcing.Differentmineralshavedifferentrigidityvalues,sothemineralassemblagesofthemantleand
crustdeterminetheirrigidities.Inthissectionweassesstheimpactofthecompositionontherigidityof
themantleandthecrust.Table2containsdataformineralsthatareusedbelowinmodelingtherigidityof
themantleandcrustofMercury.Inaddition,wedescribeourassumptionsinmodelingtheresponseofthe
FeSlayer.
4.1.MineralogicalModelsfortheMantle
Forthemineralogyofthemantle,weusetheworksofRivoldinietal.[2009]andMalavergneetal.[2010]
asreferences.Malavergneetal.[2010]calculatedtheexpectedmineralogyofthemantleofMercuryasa
functionofpressure,giventwodifferentassumedbulkcompositionsforthewholeplanet,anenstatite
chondrite(EH)andaBencubbin-likechondrite(CB).TheEHchondriteprovidesagoodcompositionaland
mineralogicalmatchtothedataoftheX-RaySpectrometer(XRS)onMESSENGER[Weideretal.,2012],which
arecompatiblewiththedatafromtheGamma-RaySpectrometer(GRS)[Evansetal.,2012].TheXRSandthe
GRSaresensitivetothetoptensofmicrometersandcentimetersofnear-surfacematerial,respectively,and
theconsistencybetweentheresultsofthetwoinstrumentsindicatesthatthetoptensofcentimetersof
Mercury’sregolithareverticallyhomogeneous[Evansetal.,2012].Despitetheapparentgoodagreement
betweenXRSandGRSresultsandenstatitechondritecompositions,themetalfractioninEHchondritesis
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Table2.MineralsRelevanttotheMantleandCrustofMercurya
| |
Abbr. 𝜌0 𝜇0 𝜇R′||0 𝜇T′||0
(kg/m3) (GPa) (GPa/K)
Garnet Grt 3565+760𝜒 92+7𝜒 1.4 −0.010
Fe Fe
Orthopyroxene Opx 3194+799𝜒 78+10𝜒 1.6 −0.012
Fe Fe
Clinopyroxene Cpx 3277+380𝜒 67−6𝜒 1.7 −0.010
Fe Fe
Quartz Qtz 2650 44.5 0.4 −0.001
Spinel Spl 3580+700𝜒 108−24𝜒 0.5 −0.009
Fe Fe
Plagioclase Pl 2750 40.4 2.5 −0.002
Merwinite Mw 3330 81 1.4 −0.014
Olivine Ol 3222+1182𝜒 81−31𝜒 1.4 −0.014
Fe Fe
aAbbr.denotesmineralabbreviation[SiivolaandSchmid,2007].Asubscript
“0”indicatesstandardambienttemperatureandpressure(298K,105Pa);density
is 𝜌; 𝜇, 𝜇′, and 𝜇′ are the rigidity and its pressure and temperature deriva-
P T
tives, respectively; and 𝜒 is the mole fraction of iron. Data in this table are
Fe
takenfromthecompilationsofSobolevandBabeyko[1994],Vacheretal.[1998],
Cammaranoetal.[2003],Verhoevenetal.[2005],andRivoldinietal.[2009].
lowerthanthebulkvalueforMercury.TheCBchondritesanalyzedbyMalavergneetal.[2010]haveahigher
metalliccomponentandthusmightrepresentanotherpossiblebuildingblockforMercury.Rivoldinietal.
[2009]calculatedtheexpectedmineralogyforasetoffivemodelsofthemantleofMercury.Theseincluded
thefollowing:anenstatitechondritemodel(EC),similartotheEHcaseofMalavergneetal.[2010];amodel
inwhichthebuildingblocksforMercuryarematchedcompositionallybythechondrulesoftwometal-rich
chondrites(MC)[TaylorandScott,2005];amodelbasedonfractionationprocessesinthesolarnebula(MA)
[MorganandAnders,1980];therefractory-volatilemodel(TS)ofTaylorandScott[2005];andtheevaporation
modelofFegleyandCameron[1987].Thelatterisnotconsistentwiththehighabundanceofsulfur,potas-
sium,andsodiuminMercury’ssurfacematerials[Nittleretal.,2011;Peplowskietal.,2011;Evansetal.,2012].
WeusedthecompositionofthesesixmodelstoestimatearangefortherigidityofthemantleofMercury.
ThemineralogicalcompositionofthesemodelsislistedinTable3.
4.2.RigidityoftheMantle
MESSENGERconfirmedthatthesurfaceofMercuryhasanextremelylowironabundance[Nittleretal.,2011;
Evansetal.,2012]andshowedthatasubstantialfractionofthesurfaceisvolcanicinorigin[Denevietal.,
2013].ThelowsurfaceabundanceofFeOisanindicationthatthesourceregionsofvolcanicmaterialare
alsoFeOpoor,sinceFeOdoesnotundergomajorfractionationduringpartialmelting[TaylorandScott,
2005].However,underthehighlyreducingconditionsinferredforMercury,partoftheironinthesilicate
shellispresentassulfidesandmetal[Zolotovetal.,2013].Incalculatingtherigidityofthemantle,weassume
thatthesilicatemineralscontainnoiron.Inotherwords,weassumethat𝜒 = 0inTable2.Theeffectsof
Fe
smallamountsofiron-richmineralsaresmallcomparedwiththeuncertaintiesintroducedbytheunknown
mineralogyofthemantleofMercury.Itshouldbenoted,however,thatatleastforolivinetherheological
Table3.ModelsfortheMantleofMercurya
Model Grt Opx Cpx Qtz Spl Pl Mw Ol 𝜇 (GPa)
c
CB - 66 4 22 4 4 - - 59
EH - 78 2 8 - 12 - - 65
MA 23 32 15 - - - - 30 69
TS 25 - - - 8 - 2 65 71
MC 15 50 9 - - - - 26 68
EC 1 75 7 17 - - - - 60
aTwocapitallettersidentifythemodel(detailsinsection4.1).CBandEH:Malavergne
etal.[2010],MA:MorganandAnders[1980],TSandMC:TaylorandScott[2005],andEC:
Wasson[1988].Thecentralpartofthetablegivesthemineralogicalcontentintermsofthe
vol%ofitscomponents[afterMalavergneetal.,2010;Rivoldinietal.,2009].Mineralabbre-
viationsaredefinedinTable2.Adashindicatesthatthemineralisabsent.Thecomposite
rigidity𝜇 isevaluatedastheHillrigidityatT=1173KandP=0.2GPa.
c
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propertiesshowastrongdependenceontheiron
Table4.CompositionandRigidityoftheCrustof
Mercurya content[Zhaoetal.,2009].
Model Ol Opx Pl Spl Qz 𝜇 (GPa) ForeachmineralogicalmodelofthemantleinTable3,
c
SP 2 44 26 6 22 53−58 wecalculatethecompositerigidityatthereference
SP 8 30 57 5 - 51−53 conditionsofT =1173KandP =0.2GPa,required
Na R R
IcP-HCT 2 59 29 1 9 57−60 fortheAndrademodel(equation(7)).First,foreach
a“Model” column: SP stands for smooth plains mineraltherigidityatT andP isobtainedfromthe
R R
andIcP-HCTstandsforintercraterplainsandheavily parametersinTable2withtheexpression
cratered terrain. SPNa takes into account the differ- [ ]
encethatmightarisewithadifferentNaabundance d𝜇 d𝜇
𝜇 (T ,P )= 𝜇 +(T−T ) +(P−P ) . (11)
[see Stockstill-Cahill et al., 2012]. The central part U R R 0 R dT R dP
of the table gives the mineralogical composition
in weight percent. In the last column the compos- ThecompositerigidityisobtainedwithHill’sexpres-
ite rigidity is calculated with Hill’s expression. The
sion,whichisanaveragebetweentheReussandthe
rangein𝜇 foreachmodelisgivenbythedifferent
c Voigtrigidities[Wattetal.,1976].Table3liststhecom-
amountsofend-members(i.e.,forsteriteandfayalite
intheolivinesolid-solutionseries). positerigiditiessoderivedforthemantlemodels.The
rangeis59–71GPa.
4.3.RigidityandViscosityoftheCrust
ThesurfaceofMercurypresentsacompositionalandmorphologicaldichotomybetweentheyounger
smoothplains(SP)andtheolderintercraterplainsandheavilycrateredterrain(IcP-HCT)[Peplowskietal.,
2011;Weideretal.,2012].ThemajorityoftheSP,whichcover∼ 27%ofthesurfaceofMercury,isvolcanic
inorigin[Denevietal.,2013].FromthesurfacecompositionaldatareturnedbyMESSENGER,Stockstill-Cahill
etal.[2012]modeledtheexpectedmineralogyoftheIcP-HCTandthenorthernvolcanicplains(NVP).The
NVPisalargecontiguousareaofvolcanicsmoothplains[Headetal.,2011]thatshowssimilarcomposition
toothersmoothplains(i.e.,Calorisbasininterior)[Weideretal.,2012].Therefore,themineralogyoftheNVP
canbetakenasrepresentativeofothersmoothplainsareas.TheresultsofStockstill-Cahilletal.[2012]are
summarizedinTable4.Thetablealsoincludesthemineralogyforsmoothplainswhentheeffectofuncer-
taintiesintheNaabundancearetakenintoaccount(SP ).Thesethreemineralogiesareusedtoestimate
Na
therigidityofthecrustofMercury.
Variationsinthetemperatureandpressureofthecrustwithdepthhavenegligibleeffectsontherigiditiesof
theindividualminerals.Therefore,thecompositerigidityisobtainedwithHill’sexpressionusing𝜇 (Table2)
0
astherigidityforeachmineral.
Therangeincrustalrigidityis51–60GPaandwilllikelyencompasstheactualrigidityofthecrustifthe
IcP-HCTrepresentstheoldercrustandtheSParerepresentativeoftheyoungercrustproducedbythemost
recentwidespreadepisodesofpartialmeltingofthemantle.Weusethecentralvalueof55GPaastherigid-
ityofthecrust.Itsviscosityissetat1023Pas.Thischoiceisnotcriticalsincethecrustiscoldandresponds
elasticallyattheforcingfrequencyofthetide.
4.4.RheologyoftheFeSLayer
TheprocedureusedtocalculatetherigidityofthemantlemineralscannotbeusedfortheFeSlayerbecause
ofalackoflaboratorydata.AttherelevantpressuresandtemperaturesoftheoutercoreofMercury,theFeS
wouldbeintheFeSVphase[Feietal.,1995].ForFeSVthebulkmodulusanditspressureandtemperature
derivativeshavebeenmeasured[Urakawaetal.,2004].Thereisnorigiditydetermination,however.Even
therigidityoftroilite(orFeSI,thephaseatstandardpressureandtemperature)hasneverbeenmeasured
[HofmeisterandMao,2003].Nevertheless,anargumentillustratedbyHofmeisterandMao[2003,Figure7]
sets𝜇 =31.5GPa.FromthephasediagramofFeS[Feietal.,1995],theconditionsatthebaseofthe
FeSI
mantle(P ∼5.5GPa)areclosetothemeltingcurveforFeSV.Thecorrespondinghomologoustemperature
T ,theratioofthetemperatureofthematerialtothesolidustemperature,isT > 0.85.Itisoftenassumed
H H
thattheviscosityisproportionaltotheexponentialoftheinverseofthehomologoustemperature[e.g.,
BorchandGreen,1987].Therefore,theviscosityoftheFeSlayeratthetopofthecorewouldbeclosetothe
lowviscosityofthemelt.TheseconsiderationsindicatethattheFeSlayer,ifpresent,isweak.
WeconsidertheeffectsoftheFeSlayeronlyinthecoldmantlecase(T = 1600K),sinceforhighertem-
b
peraturestheFeSwouldbeliquid(seethephasediagraminFeietal.[1995]).WeassumethattheFeSwill
havethesamerheologicalpropertiesasthebaseofthemantle.Thisassumptionisconservativebecauseat
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Figure4.Thevalueofk asafunctionoftheradiusoftheliquidcorefortheNoFeSset.Forthesedatathetemperature
2
atthebaseofthecoreisT =1725K,andthemantleunrelaxedrigidityis𝜇 =65GPa.(left)Colorsindicatethemean
b U
densityofmaterialbelowtheoutersolidshell(OSS).Thearrowsindicatehowthedatapointswouldshiftwithachange
intherheologicalpropertiesoftheOSS.(right)Sameasforleftpanel,butherecolorsindicatethemeandensityof
theOSS.
T =1600K,theT ofFeSislargerthanT ofthesilicates,andfromthevalueof𝜇 ,theunrelaxedrigidity
H H FeSI
ofFeSVislikelytobesmallerthanthatforthesilicates.
5.Results:TidalResponseandInteriorProperties
TheresultsillustratedbelowshowthatmodelsofMercurywithaliquidoutercorehavek ≳ 0.3.Foracom-
2
pletelysolidmodelofMercury(i.e.,amodeldevoidofaliquidoutercore),thevalueofk wouldbereduced
2
byapproximatelyanorderofmagnitude.Giventhisvariationinthemagnitudeofk betweenacompletely
2
solidinteriorandonewithaliquid(outer)core,ameasurementofthetidalresponsewouldprovideacon-
firmationofthepresenceofaliquid(outer)core.ItsexistencehasalreadybeeninferredfromEarth-based
radarmeasurements[Margotetal.,2007]andalsofromtheinterpretationofthemagneticfielddetectedby
theMESSENGERMagnetometer[Andersonetal.,2012].Therefore,theresultspresentedbelowfocusonthe
modelsofMercurywithaliquid(outer)corethathavebeendescribedinsection3.
5.1.TheMainParametersControllingtheTidalDeformation
ThetidalresponseofMercuryislargelycontrolledbythestrengthandthicknessoftheoutersolidshell
(OSS),muchlikethesimilarcaseforEuropa’siceshell[MooreandSchubert,2000].Thisresultisaconse-
quenceofthepresenceofaliquid(outer)core,whichdecouplestheshellfromthedeformationofthe
deeperinterior.Duetothecombinedmassandmomentsofinertiaconstraints,thethicknessoftheOSS
dependsonthedensityofthecore.ThisoutcomeisshowninFigure4(left),wherethetrade-offbetween
coredensityandliquidcoreradius(i.e.,OSSthickness)isseeninthecolorscalethatstronglycorrelatesboth
withtheradiusoftheliquidcoreandk .Forthesemodels,𝜇 = 65GPa,andT = 1725K.Amodification
2 U b
oftherheologicalpropertiesoftheOSS,throughavariationofthetemperatureatthebaseofthemantle
T and/oroftheunrelaxedrigidity𝜇 ,wouldmodifytheresponseasindicatedbythearrowsinthefigure.
b U
Note,however,thatthereisonlyaweakdependenceonthedensityoftheOSSitself,asseeninFigure4
(right),wherethecolorsshowthedensityoftheOSSandspannearlytheentirerangeoftheresponse.The
smalleffectsofasolidinnercoreonthetidalresponsearediscussedinAppendixA.
ThesamesetofmodelsusedinFigure4areshowninFigure5intheformofaplotshowinghowthecom-
patiblemodels(section3)aredistributed.TheavailabilityforMercuryofthethreeconstraints𝜌,C,andC
m
resultsinadistributionofthedatathatisrelativelynarrow,whichmakesthedeterminationofk inprinciple
2
veryuseful.Ithasthepotentialforimprovingthedeterminationofthelocationoftheradiusoftheouter
liquidcoreandthemeandensityofthematerialbelowtheoutersolidshellandofprovidinginsightsinto
therheologicalproperties(temperatureandrigidity)oftheoutersolidshell.Thisimprovedknowledge
willdependbothontheprecisionofthek determinationandontheeffectsoftheuncertaintiesinthe
2
temperatureandrigidityoftheoutersolidshell.
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