Exomoon Climate Models with the Carbonate-Silicate Cycle and Viscoelastic Tidal Heating

1 MB·

by  Duncan Forgan

Mon.Not.R.Astron.Soc.000,1–10() Printed6January2016 (MNLATEXstylefilev2.2)
Exomoon Climate Models with the Carbonate-Silicate Cycle and
Viscoelastic Tidal Heating
Duncan Forgan 1(cid:63) and Vera Dobos2
1ScottishUniversitiesPhysicsAlliance(SUPA),SchoolofPhysicsandAstronomy,UniversityofStAndrews,NorthHaugh,KY169SS,UK
2KonkolyThegeMiklosAstronomicalInstitute,ResearchCentreofAstronomyandEarthSciences,HungarianAcademyofSciences,
6 H1121KonkolyThegeMiklo´su´t15-17,Budapest,Hungary
1
0
2

n Accepted
a
J

5 ABSTRACT

] The habitable zone for exomoons with Earth-like properties is a non-trivial manifold, com-
P
paredtothatofEarth-likeexoplanets.Thepresenceoftidalheating,eclipsesandplanetaryil-
E
luminationintheexomoonenergybudgetcombinetoproducebothcircumstellarandcircum-
.
h planetaryhabitableregions.Analyticalcalculationssuggestthatthecircumplanetaryhabitable
p regionisdefinedonlybyaninneredge(withitsouterlimitsdeterminedbyorbitalstability).
- Subsequentcalculationsusing1Dlatitudinalclimatemodelsindicatedthatthecombinedef-
o
fect of eclipses and ice-albedo feedback can produce an outer edge to the circumplanetary
r
t habitablezone.Butisthisouteredgereal,oranartefactoftheclimatemodel’srelativesim-
s
plicity?
a
[ We present an upgraded 1D climate model of Earth-like exomoon climates, containing

the carbonate-silicate cycle and viscoelastic tidal heating. We conduct parameter surveys of
1
both the circumstellar and circumplanetary habitable zones, and we find that the outer cir-
v
cumplanetaryhabitableedgeremainsprovidedthemoon’sorbitisnotinclinedrelativetothat
9
oftheplanet.Addingthecarbonate-silicatecyclepushesthecircumplanetaryhabitablezone
8
outward,byallowingincreasesinatmosphericpartialpressureofcarbondioxidetoboostthe
7
0 greenhouseeffect.Viscoelastictidalheatingwidensthehabitablezonecomparedtostandard,
0 fixed-Qmodels.Weakeningthetidalheatingeffectduetomeltingallowsmoonstobehabit-
. ableathighereccentricity,andpushestheinnercircumstellarandcircumplanetaryhabitable
1
zoneboundaryinward.
0
6
Keywords:
1
astrobiology,methods:numerical,planetsandsatellites:general
:
v
i
X
r
a 1 INTRODUCTION goustoearlyexoplanetdetectionefforts,whicheventuallyyielded
thepulsarplanetsaroundPSR1257+12(Wolszczan&Frail1992)
At the time of writing, extrasolar moons (exomoons) remain un-
and the hot Jupiter 51 Pegasi b (Mayor & Queloz 1995). Moons
detected. A variety of detection methods for exomoons exist, in-
arerelativelycommoninourownSolarSystem,andtheyarepre-
cludingtransittiminganddurationvariations(Simon,Szatma´ry&
dicted to be a common outcome of the planet formation process,
Szabo´2007;Kipping2009;Heller&Armstrong2014),microlens-
theirformationincircumplanetarydiscsbeingsomethingofami-
ing (Liebig & Wambsganss 2010), and direct imaging (Peters &
crocosmofplanetformationincircumstellardiscs(e.g.Mosqueira
Turner2013;Agoletal.2015).Effortsfromseveralteams(Kipping
&Estrada2003b,a;Ward&Canup2010;Canup&Ward2006).
etal.2015;Hippke2015)haveproducedonlyupperlimitsonthe
occurrenceofrelativelymassivemoons(seee.g.Weidner&Horne Therefore,inanticipationofthosetantalisingfirstdetections
2010).Atentativedetectionofanexomoonorbitingafreefloating of exomoons, we can consider the possibility that these objects
planetviamicrolensingcannotbeconfirmedduetouncertaintiesin couldbehabitable.ThepotentialforliquidwaterinSolarsystem
thesystem’sdistancefromEarth(Bennettetal.2014). moonssuchasEuropa(Meloshetal.2004),Enceladus(Iessetal.
However, these detection methods are continuing to sweep 2014;Thomasetal.2015)andGanymede(Sauretal.2015)arethe
further into exomoon parameter space, in a manner quite analo- consequenceoftidalheatinginducedbytheirparentplanets,and
suchsubsurfacehabitatsmaybeextremelycommonintheUniverse
(Scharf2006).Relativelymassiveexomoonscouldhostasubstan-
(cid:63) E-mail:[email protected] tialatmosphereliketheEarth,andcouldhostasimilarbiosphere.
(cid:13)c RAS
2 DuncanForganandVeraDobos
Thehabitablezone(HZ)forsuchmoonswouldhaveaqual- to the oceans, where it forms carbonates on the sea floor and is
itativelydifferentstructuretothatforplanets.Thestandardplane- subsequentlysubductedintothemantle.Thecycleiscompletedby
taryHZconceptreliesprincipallyonthereceivedstellarflux,the volcanismexpellingCO intotheatmosphere.Thiscycleadjusts
2
planet’satmosphericalbedoanditsemissionoflongwaveradiation. the CO partial pressure in response to changes in temperature,
2
1Daltitudinalcalculationsusingthelatestlineradiativetransferal- actingasa“thermostat”todampthesefluctuations.Itisimmedi-
gorithmsandatmosphericdataallowtheconstructionofhabitable atelyclearthenthatCO adjustmentcouldbeameansbywhich
2
zonesthatdependonlyonthehoststar’sluminosityandeffective snowballstatesareavoided.Indeed,weshouldconsiderthepossi-
temperature, with the other parameters assumed to be Earth-like bilitythatthecircumplanetaryouterhabitableedgediscoveredby
(Kopparapuetal.2013althoughsomerecentworkbyKopparapu Forgan&Yotov(2014)ismerelyanartefactoftheabsenceofthe
etal.2014hasextendedthistomoreandlessmassiveplanets). carbonate-silicatecycle.
Thehabitablezoneformoonsissignificantlyaffectedbythe Equallyimportanttothepresenceandappearanceoftheouter
propertiesofthehostplanet,aswellasthepropertiesofthehost habitableedgeisthenatureofthetidalheating.BothForgan&Kip-
star.Asmentionedabove,tidalheatingcancreateoasesofhabit- ping(2013)andForgan&Yotov(2014)’sclimatemodelsreliedon
abilitybeyondtheedgeoftheplanetaryhabitablezone(Reynolds, the“constant-phase-lag”tidal heatingprescription(seee.g. Peale
McKay&Kasting1987;Scharf2006),androastmoonsinsidethe & Cassen 1978; Gladman et al. 1996; Greenberg 2009 amongst
planetaryhabitablezone(e.g.Heller&Barnes2015). others).Forganetal’spreviousapproachfixedthetidaldissipation
Ifthemoon’sorbitalinclinationrelativetotheplanetaryorbit and rigidity parameters, ignoring the temperature dependence of
islow,theneclipsescanberelativelyfrequent,reducingtheorbit themoon’sstructuralproperties,inparticularitsrigidityandtidal
averagedstellarfluxbyafewpercent(Heller2012).Moonsonvery dissipation.Typically,whatweduba“fixed-Q”approachunderes-
high inclination orbits (orbiting retrograde) are typically warmer timatesthetruetidalheatingofthebody(Ross&Schubert1989),
thanlowinclination,progrademoons,asretrogrademoonsexperi- andoncemorethisfactshouldgiveuspausewheninvestigatingthe
enceshorter,morefrequenteclipses(Forgan&Kipping2013).The outerhabitableedge.Istheouteredgeanartefactofsimplistictidal
planetitselfislikelytoemitstronglyintheinfraredthroughacom- heatingcalculations?
binationofreradiatedandreflectedstarlight,pushingthehabitable Wethereforepresentrevisedexomoonclimatecalculationsus-
zonefurtherfromthestar(Heller&Barnes2013). ing 1D latitudinal energy balance models (LEBMs), which now
Combining these factors requires us to consider the circum- include the carbonate-silicate cycle and viscoelastic tidal heating
stellar habitable zone and the circumplanetary habitable zone. (Dobos & Turner 2015). We revisit the exomoon habitable zones
From analytical calculations of the total flux received from all produced previously, in particular investigating whether the cir-
sources,Heller&Barnes(2015)showedthatthecircumplanetary cumplanetary outer habitable edge remains a feature of exomoon
habitableregiononlyhasaninneredgewheretidalheatinginduces habitability.
arunawaygreenhouseeffect.Thehabitableregionextendstoplan- Insection2wedescribethisnewmodelsetup.Section3dis-
etocentricdistancesoforderhalfaHillRadius(Domingos,Winter playsthenewlyderivedexomoonhabitablezonescalculated,and
& Yokoyama 2006) where the moon can no longer execute sta- sections4and5discussandsummarisetheresultsrespectively.
ble orbits around the planet. The circumplanetary habitable zone
consists of only an inner edge, with the outer edge as distant as
dynamicallypermissible.
Forgan & Yotov (2014) showed using 1D latitudinal energy
balanceclimatemodellingthatthispictureiscorrectiftheplanet
resideswellinsidethecircumstellarhabitablezone.Iftheplanetor-
bitsneartheouteredgeofthecircumstellarhabitablezoneandthe
orbitsoftheplanetandmoonareclosetocoplanar,thecombina-
2 LATITUDINALENERGYBALANCEMODELLING
tionofeclipsesandtheclimate’sice-albedofeedbackmechanism
caninduceasnowballstateinmoonsfromwhichtheystruggleto
2.1 SimulationSetup
escape.Thisdefinesanoutercircumplanetaryedgethatexistsfora
specificrangeofplanetaryandlunarorbitalparameters,inpartic- We adopt initial conditions essentially identical to those of For-
ularthelunarinclination.Itshouldbenotedthatingeneralasin- gan&Kipping(2013)andForgan&Yotov(2014).Thestarmass
gleeclipsewasinsufficienttogenerateasnowballstate,butrather is M = 1M , the mass of the host planet M = 1M , and
∗ (cid:12) p Jup
thecumulativeeffectofmanyeclipses,witheacheclipseslightly themassofthemoonM = 1M .Thissystemhasbeendemon-
s ⊕
increasingtheamountoffrozensurfaceafterthemoonreturnsto strated to be dynamically stable on timescales comparable to the
orbitallongitudesthatpossessstellarillumination. SolarSystemlifetime(Barnes&O’Brien2002).
However,giventhatthisedgeexistsprimarilyduetoastrong Theplanet’sorbitisgivenbyitssemi-majoraxisa andec-
p
positivefeedbackmechanismintheclimatemodel,wemustbecau- centricitye ,andthemoon’sorbitbya ande respectively.We
p m m
tiousofthemodel’sveracity.Inparticular,wemusttakecaretoin- assumethattheplanetresidesatthebarycentreofthemoon-planet
cludeallfeedbackmechanisms,bothpositiveandnegative.Theice- system, which is satisfactory given the relatively large planet-to-
albedofeedbackmechanismisastrongpositivemechanismwhich moonmassratio.Theinclinationoftheplanetrelativetothestellar
encourages rapid freezing of a planet’s surface as increasing ice equator, i = 0 (i.e. the planet orbits in the x−y plane). The
p
coverincreasesthelocalalbedo. inclinationofthemoonrelativetotheplanet’sequator,i.e.thein-
Thecarbonate-silicatecycleisanegativefeedbackmechanism clinationofthemoonrelativetothex−yplane,i ,iszerounless
m
that regulates the atmospheric partial pressure of carbon dioxide statedotherwise.Theorbitallongitudesoftheplanetandmoonare
(CO ). CO is removed from the atmosphere through precipita- definedsuchthatφ =φ =0correspondstothex-axis.Wealso
2 2 p m
tionandsilicateweathering,whichoccursatincreasingrateswith assume that the moon’s obliquity has been efficiently damped by
increasingtemperature.Thisweatheringprocessreturnsthecarbon tidalevolution,andwethereforesetittozero.
(cid:13)c RAS,MNRAS000,1–10
AdvancedClimateModelsofEarth-likeExomoons 3
2.2 LatitudinalEnergyBalanceModelswithViscoelastic
TidalHeating,PlanetaryIlluminationandCarbonate
SilicateCycles I =9.468980−7.714727×10−5β−2.794778T
−3.244753×10−3βT −3.4547406×10−4β2
TheLatitudinalEnergyBalanceModel(LEBM)employedinthis
worksolvesthefollowingdiffusionequation: +2.212108×10−2T2+2.229142×10−3β2T
+3.088497×10−5βT2−2.789815×10−5β2T2
−3.442973×10−3β3−3.361939×10−5T3
+9.173169×10−3β3T −7.775195×10−5β3T2
(cid:18) (cid:19)
∂T ∂ ∂T
C ∂t −∂x D(1−x2)∂x =(S+Sp)[1−A(T)]+ζ−I(T), −1.679112×10−7βT3+6.590999×10−8β2T3
(1) +1.528125×10−7β3T3−3.367567×10−2β4
whereT =T(x,t)isthetemperatureattimet,x≡sinλ,andλis
−1.631909×10−4β4T +3.663871×10−6β4T2
thelatitude(between−90◦and90◦).Thisequationisevolvedwith
theboundarycondition dT =0atthepoles.The(1−x2)termis −9.255646×10−9β4T3 (4)
dx
a geometric factor, arising from solving the diffusion equation in
wherewehavedefined
sphericalgeometry.
Cistheatmosphericheatcapacity,thediffusioncoefficientD
(cid:18) (cid:19)
P
controlslatitudinalheatredistribution,SandSparethestellarand β =log CO2 . (5)
P
planetaryinsolationrespectively,ζisthesurfaceheatinggenerated CO2,⊕
bytidesinthemoon’sinterior,Iistheatmosphericinfraredcooling The diffusion equation is solved using a simple explicit forward
andAisthealbedo. time, centre space finite difference algorithm. A global timestep
Judicious selection of D allows us to reproduce a fiducial wasadopted,withconstraint
Earth-Sunclimatesystemwiththecorrectlatitudinaltemperature
variations measured on Earth (see e.g. North, Cahalan & Coak- (∆x)2C
ley 1981; Spiegel, Menou & Scharf 2008). Planets that rotate δt< . (6)
2D(1−x2)
rapidlyexperienceinhibitedlatitudinalheattransport,duetoCori-
olis forces truncating the effects of Hadley circulation (cf Farrell Thistimestepconstraintensuresthatthefirsttermonthelefthand
1990;Williams&Kasting1997).ThepartialpressureofCO also sideofequation(1)isalwayslargerthanthesecondterm,prevent-
2
playsarole.WefollowWilliams&Kasting(1997)byscalingD ingthediffusiontermfromsettingupunphysicaltemperaturegra-
accordingto: dients.Theparametersarediurnallyaveraged,i.e.akeyassumption
ofthemodelisthatthemoonsrotatesufficientlyquicklyrelativeto
their orbital period around the primary insolation source. This is
broadlytrue,asthestaristheprincipalinsolationsource,andthe
moonrotatesrelativetothestarontimescalesofafewdays.
(cid:18) ω (cid:19)−2(cid:18) P (cid:19) The atmospheric heat capacity depends on what fraction of
D=5.394×102 d CO2 , (2)
ωd,⊕ PCO2,⊕ themoon’ssurfaceisocean,focean,whatfractionislandfland =
1.0−f ,andwhatfractionoftheoceanisfrozenf :
ocean ice
C =f C +f [(1−f )C +f C ]. (7)
whereω istherotationalangularvelocityoftheplanet,andω land land ocean ice ocean ice ice
d d,⊕
istherotationalangularvelocityoftheEarth,andP =3.3× Theheatcapacitiesofland,oceanandicecoveredareasare
CO2,⊕
10−4bar.
We use a simple piecewise function to determine PCO2 Cland =5.25×109ergcm−2K−1, (8)
(Spiegeletal.2010):
C =40.0C , (9)
ocean land

10−2bar T (cid:54)250K 9.2Cland 263K <T <273K
PCO2 =1P0−2−(T−250)/27bar T25(cid:62)0K29<0KT <290K (3) Cice =20C.0land TT <>226733KK. . (10)
CO2,⊕
These parameters assume a wind-mixed ocean layer of 50m
(Williams&Kasting1997).Increasingtheassumeddepthofthis
layerwouldincreaseC (seee.g.North,Mengel&Short1983
ocean
OurprescriptionallowsDtovarywithlatitude,dependingonthe fordetails).Thealbedofunctionis
local temperature. This is not guaranteed to produce Hadley cir-
culation(seee.g.Vladiloetal.2013fordetailsonhowD canbe (cid:20) (cid:21)
T −268K
modifiedtoachievethis).AsweallowpartialpressureofCO2 to A(T)=0.525−0.245tanh 5K . (11)
vary, we adopt Williams & Kasting (1997)’s prescription for the
coolingfunction,I(T,P ): Thisproducesarapidshiftfromlowalbedo(∼0.3)tohighalbedo
CO2
(cid:13)c RAS,MNRAS000,1–10
4 DuncanForganandVeraDobos
(∼ 0.75) as the temperature drops below the freezing point of dT ),weassumetheplanet’sorbitisnotsynchronous,andwe
planet
water,producinghighlyreflectiveicesheets.Figure1ofSpiegel, fixdT =0.Theplanetaryalbedoisfixedat0.3.
planet
Menou&Scharf(2008)demonstrateshowthisshiftinalbedoaf-
fects the potential for global energy balance, and that for planets
in circular orbits, two stable climate solutions arise, one ice-free, 2.2.1 Fixed-QandViscoelasticTidalHeating
andoneice-covered.Spiegeletalalsoshowthatsuchafunction
issufficienttoreproducetheannualmeanlatitudinaltemperature Intheinterestofcomputationalexpediency,wemakeasimpleap-
distributionontheEarth. proximationfortidalheating,byfirstlyassumingthetidalheating
Notethatwedonotconsidercloudsinthismodel,whichcould perunitareais(Peale,Cassen&Reynolds1980;Scharf2006):
modify both the albedo and optical depth of the system signif-
icantly. Also, we assume that both stellar and planetary flux are 21ρ2 R5 e2 (cid:18)GM (cid:19)5/2
governedbythesamealbedo,whichintruthisnotlikelytobethe ζ = m m m p (19)
38 ΓQ a3
case(seeDiscussion). m
ThestellarinsolationfluxS isafunctionofbothseasonand whereΓisthemoon’selasticrigidity(whichweassumetobeuni-
latitude.Atanyinstant,thebolometricfluxreceivedatagivenlati- form throughout the body), R is the moon’s radius, ρ is the
m m
tudeatanorbitaldistanceris moon’sdensity,M istheplanetmass,a ande arethemoon’s
p m m
orbitalsemi-majoraxisandeccentricity(relativetotheplanet),and
(cid:18)1AU(cid:19)2 Qisthemoon’stidaldissipationparameter.
S =q0cosZ r , (12) We consider both fixed-Q tidal heating, where Q and Γ are
held constant, and viscoelastic models where Q and Γ become
whereq isthebolometricfluxreceivedfromthestaratadistance functionsofthemoon’sorbitalparameters.Throughout,weassume
0
of1AU,andZisthezenithangle: thattidalheatingoccursuniformlyacrossthemoon’ssurface.
Inthefixed-Qcase,weassumeterrestrialvalues:Q = 100,
(cid:18)M (cid:19)4 Γ = 1011dynecm−2 (appropriate for silicate rock). In the vis-
q =1.36×106 ∗ ergs−1cm−2 (13) coelastic case, we calculate the product QΓ using the models of
0 M
(cid:12) Dobos&Turner(2015).Inbothcases,thedensityofthemoonis
fixedatρ =5gcm−3.
m
cosZ =µ=sinλsinδ+cosλcosδcosh. (14) Intheviscoelasticmodelthetidalheatingiscalculatedby:
δ isthesolardeclination,andhisthesolarhourangle.Asstated
21 R5n5e2
previously,wesetthemoon’sobliquityδ0tozero.Thesolardecli- E˙tidal =− 2 Im(k2) mG . (20)
nationiscalculatedas:
where n is the mean motion of the moon and Im(k ) is the
2
complexLovenumber,whichdescribesstructureandrheologyin
sinδ=−sinδ cos(φ −φ −φ ), (15)
0 ∗m peri,m a thesatellite(Segatzetal.1988).Henning,O’Connell&Sasselov
where φ∗m is the current orbital longitude of the moon relative (2009)givesthevalueofIm(k2)fortheMaxwellmodel:
to the star, φ is the longitude of periastron, and φ is the
peri,m a
longitudeofwintersolstice,relativetothelongitudeofperiastron. 57ηω
Wesetφperi,m =φa =0forsimplicity. −Im(k2)= 4ρgR (cid:20)1+(cid:16)1+ 19µ (cid:17)2 η2ω2(cid:21), (21)
Wemustdiurnallyaveragethesolarflux: m 2ρgRm µ2
where η is the viscosity, ω is the orbital frequency and µ is the
S =q0µ¯. (16) shearmodulusofthesatellite.Thetemperaturedependencyofthe
viscosity and the shear modulus is described in Dobos & Turner
This means we must first integrate µ over the sunlit part of the
(2015).SinceonlyrockybodieslikeEarthareconsideredassatel-
day, i.e. h = [−H,+H], where H is the radian half-day length
litesinthiswork,thesolidusandliquidustemperaturesatwhichthe
at a given latitude. Multiplying by the factor H/π (as H = π if
materialoftherockybodystartsmeltingandbecomescompletely
alatitudeisilluminatedforafullrotation)givesthetotaldiurnal
liquidwerechosentobe1600Kand2000K,respectively.Weas-
insolationas
sumethatdisaggregationoccursat50%meltfraction,whichleads
toabreakdowntemperatureof1800K.
(cid:18) (cid:19)
S =q H µ¯= q0 (Hsinλsinδ+cosλcosδsinH). Theviscoelastictidalheatingmodelalsodescribestheconvec-
0 π π tivecooling ofthebody. Theiterative methoddescribedby Hen-
(17) ning, O’Connell & Sasselov (2009) was used for calculating the
Theradianhalfdaylengthiscalculatedas convectiveheatloss:
cosH =−tanλtanδ. (18) q =k Tmantle−Tsurf , (22)
BL therm δ(T)
Weimplementplanetaryilluminationandeclipsesofthemoon
inthesamemannerasForgan&Yotov(2014)(seealsoHeller& where k is the thermal conductivity (∼ 2W/mK), T
therm mantle
Barnes2013).WhileHelleretalallowfortheplanettobeinsyn- andT arethetemperatureinthemantleandonthesurface,re-
surf
chronousrotationandhaveasignificanttemperaturedifferencebe- spectively,andδ(T)isthethicknessoftheconductivelayer.Weuse
tween the dayside and nightside (expressed in the free parameter δ(T)=30kmasafirstapproximation,andthenfortheiteration
(cid:13)c RAS,MNRAS000,1–10
AdvancedClimateModelsofEarth-likeExomoons 5
3 RESULTS
d (cid:18) Ra (cid:19)−1/4 3.1 TheCircumstellarExomoonHabitableZone
δ(T)= (23)
2a2 Rac 3.1.1 Theeffectofthecarbonate-silicatecycle
is used, where d is the mantle thickness (∼ 3000 km), a is the Figure 1 shows the classification of simulations as a function of
2
flowgeometryconstant(∼1),Ra isthecriticalRayleighnumber planetary orbital parameters when the carbonate silicate cycle is
c
(∼ 1100)andRaistheRayleighnumberwhichcanbeexpressed active, and fixed-Q tidal heating is used. The left column shows
by data for zero moon eccentricity, the right for moon eccentricity
e =0.05.
m
ThesedatashouldbecomparedwithFigures2and3ofFor-
Ra= αgρd4qBL . (24) gan&Yotov(2014),whichshowapronounced“C”shapeintheir
η(T)κktherm outerHZboundaries.Bycontrast,ourlatestresultspossessquite
a reducedsensitivity ofthe outer boundaryto eccentricity. While
Hereαisthethermalexpansivity(∼ 10−4)andκisthethermal more eccentric orbits do permit larger planetary semi-major axes
diffusivity: κ = ktherm/(ρCp) with Cp = 1260J/(kgK). The forhabitablemoons,theallowedincreaseinap isnotparticularly
iterationoftheconvectiveheatfluxlastsuntilthedifferenceofthe large.
lasttwovaluesishigherthan10−10W/m2. TheinnerHZboundaryhasmovedsignificantlyfurtherout-
We assume that with time, the tidal heating and the convec- ward.InourpreviousworkwithouttheCScycle, thetypicalHZ
tiveheatlossreachastableequilibriumstate.Afterfindingthesta- boundarywasinteriorof0.85au,whereasnowthisboundaryex-
bleequilibriumtemperature,thetidalheatfluxiscalculated,from istsat0.9auandbeyond.ReplacingtheSpiegel,Menou&Scharf
whichtheQΓproductcanbeobtained. (2008)coolingfunctionwiththeWilliams&Kasting(1997)cool-
ingfunctionresultsinlessefficientcooling(forafixedCO par-
2
tialpressure).Thenumberoftransientlyhabitableclassificationsis
greatlyreduced,butthisisnotdirectlyduetoclimatemodulation
2.2.2 HabitabilityIndices via the CS cycle. This is a consequence of less efficient cooling.
Systems which would before have been able to turn away briefly
We calculate habitability indices in the same manner as most from irreversible heating using the cooling function of Spiegel,
groups do (Spiegel, Menou & Scharf 2008; Vladilo et al. 2013; Menou & Scharf (2008) can no longer do so, and must therefore
Forgan2014).Thehabitabilityfunctionξis: followtrajectoriestowardsgreenhouseorhabitablestates.
With a small amount of moon eccentricity (right column of
(cid:26) Figure1),fixed-Qtidalheatingcomesintoplay.Therelativelyinef-
1 273K<T(λ,t)<373K
ξ(λ,t)= (25) ficientcoolingoftheWilliams&Kasting(1997)functionbecomes
0 otherwise.
increasinglyevident.Atlowmoonsemi-majoraxis(toprow),this
resultsinthehabitablezonebeingpushedfurtheroutwardsinplan-
Wethenaveragethisoverlatitudetocalculatethefractionofhab-
etarysemimajoraxis,wellbeyond1au.Habitablemoonswellbe-
itablesurfaceatanytimestep:
yond1auhavelongbeenpredicted,especiallyicymoonswithsub-
surface liquid water oceans (Reynolds, McKay & Kasting 1987;
1(cid:90) π/2 Scharf2006),butthisisthefirstindicationsofsuchbehaviourin
ξ(t)= ξ(λ,t)cosλdλ. (26) LEBMcalculationsofEarth-likemoonsatrelativelyloweccentric-
2
−π/2 ity.
AswesawinForgan&Yotov(2014),asmoonsemimajoraxis
Each simulation is allowed to evolve until it reaches a steady or
increases,thepicturebeginstolookverysimilartothezeromoon
quasi-steadystate,andthefinaltenyearsofclimatedataareused
toproduceatime-averagedvalueofξ(t),ξ¯,andthesamplestan- eccentricitycase,astidalheatingbecomesnegligiblecomparedto
theothercontributorstothemoon’senergybudget,ascanbeseen
dard deviation, σ . We use these two parameters to classify each
ξ
intherightcolumnofFigure1.
simulationsasfollows:
(i) HabitableMoons-thesemoonspossessatime-averagedξ¯>
0.1,andσ < 0.1ξ¯,i.e.thefluctuationinhabitablesurfaceisless 3.1.2 Addingviscoelastictidalheating
ξ
than10%ofthemean. Figure2showstheeffectofusingmorerealistic,viscoelastictidal
(ii) HotMoons-thesemoonshaveaveragetemperaturesabove heating.Wearenowmodellingtheinnermeltingofthebody,which
373Kacrossallseasons,andarethereforeconventionallyuninhab- allowsthevaluesofQandΓtovarysignificantly,highlightingthe
itable,andξ¯<0.1. factthatthevaluesusedforthesequantitiesinthefixed-Qmodel
(iii) SnowballMoons-thesemoonshaveundergoneasnowball are somewhat arbitrary, and can vary greatly for different rocky
transitiontoastatewheretheentiremoonisfrozen,andarethere- satellites.
foreconventionallyuninhabitable.Aswithhotmoons,werequire Wecanimmediatelyseethatallowingtidalparameterstovary
ξ¯< 0.1forthemoontobeclassifiedasasnowball,butgiventhe withtemperatureresultsinaslightwideningofthehabitablezone.
natureofthesnowballtransitionasitismodelledhere,theseworlds Theinnerboundaryoncemoreextendsasfaras0.9au(whenthe
typicallyhaveξ¯=0. moon semimajor axis is sufficiently large), again with little evi-
(iv) Transient Moons - these moons possess a time-averaged denceofthe“C”shapeintheboundarycurvesasseenpreviously.
ξ¯> 0.1,andσξ > 0.1ξ¯,i.e.thefluctuationinhabitablesurface Again,lowamsimulationsarehabitablewellbeyond1au,andin-
isgreaterthan10%ofthemean. deedbeyondthelimitsofoursimulationparameterspace.
(cid:13)c RAS,MNRAS000,1–10
6 DuncanForganandVeraDobos
Figure1.Theexomoonhabitablezoneasafunctionofplanetaryparameters,withthecarbonate-silicatecycleactiveandfixed-Qtidalheating.Therows,
fromtoptobottom,aremoonsemimajoraxesof0.1,0.15and0.2HillRadiirespectively.Theleftcolumnshowssimulationswherethemooneccentricity
iszero,andtherightcolumnshowsmoonswitheccentricityem = 0.05.Thecoloursofthepointsindicatetheirclassificationaccordingtotheschemein
section2.2.2.Theredsquaresindicate“hotmoons”withglobalmeantemperaturesabove373K,thebluediamondsindicate“coldmoons”,withglobalmean
temperaturesbelow273K,thegreencirclesindicate“habitablemoons”withmeantemperaturesbetween273and373Kandlowfluctuationsinthemean,
andtheblackcrossesindicate“transientmoons”,wheremeantemperaturesareinthe273-373Krange,butfluctuatestrongly.
Atlargea ,thedifferencesbetweenthefixed-Qruns(bottom 3.2 TheExomoonCircumplanetaryHabitableZone
m
rightofFigure1)andtheviscoelasticruns(bottomrightofFigure
2)becomenegligible.Inthislimit,thetidalheatingisaverysmall
We now consider the circumplanetary region, and show simula-
contributiontothetotalenergybudgetoftheexomoonclimate,and
tion data as a function of moon semimajor axis and eccentricity.
indeedboththeserunsareverysimilartothezeroeccentricityrun
The planet’s orbit is fixed as circular with a = 1 au. Figure 3
p
displayedinthebottomleftofFigure1.
showstheresultingclassificationswhentheCScycleisactive,and
(cid:13)c RAS,MNRAS000,1–10
AdvancedClimateModelsofEarth-likeExomoons 7
Figure2.Theeffectofthecarbonatesilicatecycleandviscoelastictidalheating.ThisisidenticaltotherightcolumnofFigure1,i.e.therowsfromtopto
bottom,aremoonsemimajoraxesof0.1,0.15and0.2HillRadiirespectively,andthemoonshaveeccentricityem=0.05.
when fixed-Q tidal heating is applied (left column) compared to The outer habitable edge remains at lower eccentricities, but has
viscoelastictidalheating(rightcolumn). beenpushedtohighera alongwiththeinneredge.However,the
m
widthofthezonehasincreasedfromaround0.02HillRadiito0.04
Beginningwiththezeroinclinationplotforfixed-Qtidalheat- HillRadii(atleast,fore >0.04).
m
ing(toprow,leftcolumn),wecancomparedirectlytothebottom
leftpanelofFigure4inForgan&Yotov(2014),andseeimmedi- Ifviscoelastictidalheatingisused(rightcolumnofFigure3),
atelythattheCScyclepushesthecircumplanetaryhabitablezone then we can see the habitable zone becoming even wider. While
outward.Theouterboundaryinpreviousworkwascalculatedtobe theweakertidalheatingtendstopinchthehabitablezoneslightly
approximately0.1HillRadiiate =0.1-inthecurrentwork,the atlowe ,athighe thisweakertidalheatingisaboon,prevent-
m m m
outerboundarynowextendsbeyond0.14HillRadiiate = 0.1. ingcatastrophicheatingofthemoon.Thispermitsextremelywide
m
(cid:13)c RAS,MNRAS000,1–10
8 DuncanForganandVeraDobos
Figure3.Thehabitablezoneasafunctionofmoonorbitalparameters.Inallcases,thecarbonatesilicatecycleisactive.Theleftcolumnshowstheresults
forfixed-Qtidalheating,therightcolumntheresultsforviscoelastictidalheating.Eachrowshowsadifferentvalueofmooninclination:im = 0(top),
im=10◦(middle),andim=45◦(bottom).
circumplanetaryHZs,atleastaswideas0.06HillRadii,andlikely comeapparent,eclipsesmustbesufficientlyfrequentfortheiref-
widerthanwemeasureinthisparameterspace.Perhapsmostno- fectstobecumulativeandinduceasnowballstate.Increasingin-
tably,fora (cid:46)0.07HillRadii,theouteredgeisnotpresentatany clinationreducestheeclipserate(seee.g.Heller2012)untilitis
m
eccentricity. These moons experience rapid, brief eclipses which close to zero1. With both implementations of tidal heating, rela-
are now easily buffered by thermal inertia and the carbonate sili-
catecycle,regardlessofthetidalheatingprescriptionused.
If the inclination of the moon is increased (middle and bot- 1 Forpolarlunarorbits(im = 90◦),theeclipserateisnotexactlyzero-
tomrows)wecanseethattheouteredgebeginstodisappear,even eclipsescanstilloccurtwiceperyear,whenthelunarnodesalignwiththe
atrelativelylargemoonsemimajoraxis.Fortheouteredgetobe- planet’sorbitalpositionvector
(cid:13)c RAS,MNRAS000,1–10
AdvancedClimateModelsofEarth-likeExomoons 9
tivelylargeinclinations(45◦)erasetheouterhabitableedgecom- cyclesusingLEBMs(Imkeller2001;Benzi2010,Forgan&Mead,
pletely.Atintermediateinclinations(10◦),weseeaslightlymixed inprep.)whichpredicttemperatureoscillationsduetoEarth’sor-
picture.Theeclipserateshouldbeeffectivelyzero,yetthefixed-Q bitalvariationnearlyanorderofmagnitudelowerthanthoseob-
tidal heating run shows that an edge still exists (albeit much fur- served in paleoclimate data (Zachos et al. 2001). The solution to
therfromtheplanet).Thisisanindicationthatheretheice-albedo this issue is to add short period random noise to the LEBM (to
feedbackmechanismisactivatingduetothemoon’sdistancefrom mimicweatherpatternsonscaleslessthanafewdays),whichcan
thestarratherthanduetoeclipses,whichinturnmaysuggestthat allowmuchlargertemperaturevariationsthroughthephenomenon
thealbedo prescriptionwe usemay infact betoo simplefor this of stochastic resonance (e.g. Benzi et al. 1982). This clearly has
scenario. implications for the outer circumplanetary habitable edge, which
maybemovedoutwarddependingonwhethersuchresonancescan
overcomethecoolingeffectofeclipses,oncethemoonmovesback
intosunlight.
4 DISCUSSION
Wehaveseenthatsomeexomoonorbitspermitsurfaceliquid
While we have gone further than previous work in instituting a waterduetonon-zeroeccentricity.However,tidaldissipationwill
carbonate-silicatecycleinthemodel,ourimplementationisrela- eventuallyreducethemoon’seccentricitytozero.Interactionswith
tivelysimple.WeassumethatthepartialpressureofCO responds neighbouring moons can pump eccentricity to higher values. In-
2
instantlytochangesintemperature.Inpractice,CO levelschange deed,wemightexpectthatmoonsexperienceclimatecyclesanal-
2
duetoimbalancesbetweensilicateweatheringratesandvolcanic ogoustoEarth’sMilankovitchcycles,presumablyonagreatlyre-
outgassing(aswellasanthropogenicsources).Williams&Kasting duced timescale. Close encounters between planets hosting satel-
(1997)computeweatheringratesinordertofindquasi-staticequi- lites can also excite moon eccentricities (see e.g. Deienno et al.
libriumvaluesforCO ,wheretherelaxationtimescaleisassumed 2014).Aswithplanets,thehabitabilityofanexomoonovergeolog-
2
tobeoforderhalfamillionyears!Suchanapproachissomewhat icaltimescaleswilldependonitsdynamicallandscape,andreplac-
impracticalforouraims,wherewewishtorunalargesuiteofsimu- ingfixedKeplerianorbitswithanNBodycalculationwillprovide
lationswithalimitedruntimeneededtoachieveequilibrium.Also, usefulinsights(Forgan,submitted,Forgan&Mead,inprep.).
weshouldbeawarethatcarbonate-silicatecyclemodelsimplicitly
assumeagreatdealabouttheparentbody.ModelsofCO conden-
2
sationonthenightsideoftidallylockedplanets(Joshi,Haberle&
Reynolds1997;Edsonetal.2012)showtheimportanceofafully
resolved surface, with appropriate land-ocean surfaces and inter-
5 CONCLUSIONS
faces.TheLEBMassumesafixedoceanfractionatalllatitudes,in
effectplacinglimitsontheextentofcontinents. Wehavereturnedtoour1Dlatitudinalenergybalancemodelsof
Higher dimension simulations will also be of importance Earth-like exomoon climates, which contain stellar and planetary
whenattemptingtocalculateanisotropictidalheating.Wehaveas- insolation, atmospheric circulation, infrared cooling, eclipses and
sumed that the energy produced in the interior by tidal forces is tidal heating as the principal contributors to the moon’s radiative
dissipateduniformlyacrossthesurface,butobservationsofSolar energy budget. We modify these models further by incorporating
systemmoonssuchasIoshowbothlongitudinalandlatitudinalde- the carbonate silicate cycle, a negative feedback mechanism to
pendenceintidalheating(Segatzetal.1988),andthatthisislikely regulate planetary temperatures (opposing the positive ice-albedo
tobeacommonfeatureinalltidallyheatedmoons(Peale,Cassen feedback system already present). We also investigate improved
&Reynolds1979;Beuthe2013).Thepreciseformofthisspatial viscoelasticmodelsoftidalheating,whichallowthemoon’srigid-
distributionwilldependona)atwhatlocationintheinteriordissi- ityandtidaldissipationparameterstovaryasafunctionoftemper-
pationoccurs,andb)thetransportmechanismbywhichthisheat ature.
reaches the surface. This becomes especially complicated for icy Wefindthatthecombinationofbothadditionsresultsinmany
moons,wheretheiceshellaroundasubsurfaceoceanrespondsto more potentially habitable configurations, even in our somewhat
tidal forcing as a membrane over a fluid layer, affecting its tidal limited parameter space. In terms of planetary orbits, the habit-
Lovenumber(Beuthe2015).LEBMscouldaccommodatea1Dra- able zone is pushed outward, and is somewhat wider if the exo-
dialmodelofamoon’sinteriorstructuretogivealatitudinaldepen- moonorbitsclosetotheplanet.Thecircumplanetaryhabitablezone
denceontidalheating,butareofcourseincapableofmodelling,for is significantly wider with viscoelastic tidal heating, and extends
example,Io’sreducedtidalheatingatthesubplanetarypoint. muchfurtherfromtheplanet(providedthemoon’seccentricityis
Weshouldalsonotethatourprescriptionforplanetaryillumi- largerthanabout0.02,whichisslightlylessthanthatofTitan).We
nationassumesthatthemoon’salbedoisthesameforbothplan- findthatthereisawell-definededgetothecircumplanetaryhab-
etary and stellar radiation. For example, given that the planet is itablezone,insidetheorbitalstabilitylimit,despitethecarbonate
likelytoemitstronglyintheIR,anicymoon’salbedoislikelyto silicatecycleandviscoelastictidalheating,duetoacombination
beatleastanorderofmagnitudelowertotheplanet’sradiationthan of eclipses and ice-albedo feedback. The outer edge requires the
thestar’s.Asimplemodificationwouldbea“dualpassband”sys- planetandmoonorbitstobequitecloselyaligned,sothateclipses
temwhereplanetaryilluminationcalculationsarecarriedoutwith aresufficientlyfrequent.Weshowthatifthemoon’sorbitissuffi-
aseparatealbedo(e.g.Heller&Barnes2015).Itmaywellbethe cientlyinclinedthateclipsesareunlikely,theouteredgecompletely
casethattheouterhabitableedgecanbepushedfurtheroutwardby disappears,evenforrelativelylargemoonsemimajoraxis.
moreefficientplanetaryillumination. Giventhenonlinearnatureofthisouteredge’sorigin,future
AddingtheCScycledoesappeartoreducefluctuationsinthe workinthisareamustincludetheuseoffull3Dgeneralcirculation
moon’stemperature,butweshouldbecognisantthattheLEBMin models(GCMs)withtheabovephysicsimplemented,todetermine
itscurrentformactuallyunderestimatestemperaturefluctuationsin thepropertiesofthisouteredge,especiallyinthecaseofslowro-
general. This is evident from studies of the Earth’s Milankovitch tatingortidallylockedmoons.
(cid:13)c RAS,MNRAS000,1–10
10 DuncanForganandVeraDobos
ACKNOWLEDGMENTS Melosh H., Ekholm A., Showman A., Lorenz R., 2004, Icarus,
168,498
DFgratefullyacknowledgessupportfromthe”ECOGAL”ERCad-
MosqueiraI.,EstradaP.R.,2003a,Icarus,163,198
vanced grant. VD thanks La´szlo´ L. Kiss for useful comments on
MosqueiraI.,EstradaP.R.,2003b,Icarus,163,232
themanuscript.VDhasbeensupportedbytheHungarianOTKA
North G., Cahalan R., Coakley J., 1981, Rev. Geophys. Space
grantK104607,theLendu¨let-2009YoungResearchersProgramof
Phys.,19,91
theHungarianAcademyofSciences,theESAPECScontractNo.
NorthG.R.,MengelJ.G.,ShortD.A.,1983,JournalofGeophys-
4000110889/14/NL/NDe,theTE´T-14FR-1-2015-0012project,and
icalResearch,88,6576
theNKFIHK-115709grantoftheHungarianNationalResearch,
PealeS.,CassenP.,1978,Icarus,36,245
DevelopmentandInnovationOffice.Thisworkreliedonthecom-
PealeS.,CassenP.,ReynoldsR.,1980,Icarus,43,65
puteresourcesoftheStAndrewsMHDcluster.Theauthorsthank
PealeS.J.,CassenP.,ReynoldsR.T.,1979,Science(NewYork,
thereviewerfortheirinsights,whichhelpedrefineandclarifythe
N.Y.),203,892
argumentsmadeinthismanuscript.
PetersM.A.,TurnerE.L.,2013,ApJ,769,98
Reynolds R. T., McKay C. P., Kasting J. F., 1987, Advances in
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RossM.,SchubertG.,1989,Icarus,78,90
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