Mercury's capture into the 3/2 spin-orbit resonance including the effect of core-mantle friction

0.62 MB·English

by  Alexandre C. M. Correia

Mercury’s capture into the 3/2 spin-orbit resonance
including the effect of core-mantle friction
AlexandreC.M.Correiaa,b, ,JacquesLaskarb
∗
aDepartamentodeF´ısica,UniversidadedeAveiro,CampusdeSantiago,3810-193Aveiro,Portugal
bAstronomieetSyste`mesDynamiques,IMCCE-CNRSUMR8028,ObservatoiredeParis,UPMC,77Av.Denfert-Rochereau,75014Paris,France
Abstract
9
0 TherotationofMercuryispresentlycapturedina3/2spin-orbitresonancewiththeorbitalmeanmotion.Thecapturemechanism
0
iswellunderstoodastheresultoftidalinteractionswiththeSuncombinedwithplanetaryperturbations(GoldreichandPeale,1966;
2
CorreiaandLaskar, 2004). However, it is now almost certain that Mercuryhas a liquid core (Margotetal., 2007) which should

ninduce a contributionof viscous frictionat the core-mantleboundaryto the spin evolution. Accordingto PealeandBoss (1977)
a
this last effect greatly increases the chances of capture in all spin-orbit resonances, being 100% for the 2/1 resonance, and thus
J
preventing the planet from evolving to the presently observed configuration. Here we show that for a given resonance, as the
3
chaoticevolutionofMercury’sorbitcandriveitseccentricitytoverylowvaluesduringtheplanet’shistory,anypreviouscapture
1
can be destabilized whenever the eccentricity becomes lower than a critical value. In our numerical integrations of 1000 orbits

]of Mercuryover 4Gyr, the spin ends99.8%of the time capturedin a spin-orbitresonance, in particularin one of the following
P
threeconfigurations: 5/2(22%),2/1(32%)and3/2(26%). Althoughthepresent3/2spin-orbitresonanceisnotthemostprobable
E
outcome, we also show that the capture probability in this resonance can be increased up to 55% or 73%, if the eccentricity of
.
hMercuryinthepasthasdescendedbelowthecriticalvalues0.025or0.005,respectively.
p
-Keywords: Mercury,spindynamics,tides,core-mantlefriction,resonance
o
r
t
s
a1. Introduction into the present3/2spin-orbitresonanceis onthe low side, at
[ mostabout7%,whichremainedsomewhatunsatisfactory.
Mercury’spresent rotation is locked in a 3/2 spin-orbitres- GoldreichandPeale(1967)neverthelesspointedoutthatthe
1
onance, with its spin axis nearly perpendicular to the orbital
v probabilityofcapturecouldbegreatlyenhancedifaplanethas
plane (PettengillandDyce, 1965). The stability of this ro-
3 amoltencore. In1974,thediscoveryofanintrinsicmagnetic
4tation results from the solar torque on Mercury’s quadrupo- field by the Mariner 10 spacecraft(Nessetal., 1974), seemed
8lar moment of inertia, combined with an eccentric orbit: the
to imply the existence of a conducting liquid core and conse-
1axis of minimum moment of inertia is always aligned with
quentlyanincrementinthecaptureprobabilityinthe3/2reso-
.
1the direction to the Sun when Mercury is at the perihe-
nance. However,accordingtoGoldreichandPeale(1967),this
0lion of its orbit (Colombo, 1965; GoldreichandPeale, 1966;
also increases the capture probabilityin all the previousreso-
9
CounselmanandShapiro, 1970). The initial rotation of Mer-
0 nances. PealeandBoss(1977)indeedremarkedthatonlyvery
cury was presumably faster than today, but tidal dissipation
: specificvaluesofthecoreviscosityallowtoavoidthe2/1reso-
valong with core-mantle friction brought the planet rotation to
nanceandpermitthecaptureinthe3/2configuration.
i
Xthe present configuration, where capture can occur. However, Morerecently,CorreiaandLaskar(2004)(hereafterdenoted
theexactmechanismonhowthisstateinitiallyaroseisnotcom-
r byPaperI)haveshownthatastheorbitaleccentricityofMer-
apletelyunderstood.
cury is varyingchaotically, from near zero to more than 0.45,
In their seminal work, GoldreichandPeale (1966) have
thecaptureprobabilityissubstantiallyincreased.Indeed,when
shown thatsince thetidalstrengthdependsonthe planet’sro-
thelargeeccentricityvariationisfactoredintothecapture,the
tation rate, it creates an asymmetry in the tidal potential that
rotationrateoftheplanetcanbeaccelerated,andthe3/2reso-
allowscaptureintospin-orbitresonances. Theyalsocomputed
nancecouldhavebeencrossedmanytimesinthepast.Perform-
thecaptureprobabilityintotheseresonancesforasinglecross-
ingastatisticalstudyofthepastevolutionsofMercury’sorbit,
ing, and found that for the present eccentricity value of Mer-
over1000cases, it was demonstratedthatcaptureinto the 3/2
cury(e=0.206),andunlessoneusesanunrealistictidalmodel
spin-orbitresonantstateisinfact,andwithouttheneedofspe-
with constant torques(which cannotaccount for the observed
cificcore-mantleeffect,themostprobablefinaloutcomeofthe
damping of the planet’s libration), the probability of capture
planet’sevolution,occurringabout55.4%ofthetime. Incon-
trast,becausetheeccentricitycandecreasetonearzero,allres-
onancesexceptthe1/1becomeunstable,allowingtheplanetto
Correspondingauthor,
∗
Emailaddresses:[email protected](AlexandreC.M.Correia) escapefromresonance. Thismechanismsuggeststhatinpres-
PreprintsubmittedtoIcarus January24,2009
enceofcore-mantlefrictiontheplanetcanescapetoaprevious trueanomalyvinseriesoftheeccentricityeandmeananomaly.
captureinthe2/1orhigherorderspin-orbitresonances. However,whentherotationrateωandthemeanmotionn= M˙
The present paper continues the work started in Paper I. In are close to resonance (ω pn, for a semi-integer1 value p),
≃
additionto theeffectsoftidesand planetaryperturbations,we wemustretainthetermswithargument(2θ 2pM)intheex-
−
will consider here also the core-mantle friction effect as de- pansion
scribed by GoldreichandPeale (1967), since the presence of
+
a liquidcoreinsideMercuryisnowconfirmedbyradarobser- cos(2θ 2v) 1 ∞
− = H(p,e)cos(2θ 2pM), (2)
vations (Margotetal., 2007). In the next section we give the r3 a3 −
pX=
averaged equations of motion in a suitable form for simula- −∞
tions of the long-term variations of Mercury’s spin, including whereaisthesemi-majoraxisoftheplanet’sorbitandthefunc-
theresonantmotion,viscoustidaleffects,core-mantlecoupling tionH(p,e)isapowerseriesine(Tab.1). Theexactaveraged
andplanetaryperturbations. Insection3wediscusstheconse- contributions to the spin are given in a suitable form for our
quencesofeacheffectintothe spinevolutionandevaluatethe studybyexpression(15)inCorreia(2006)2. Forzeroobliquity
capture probabilities in resonance. Finally, in last section we wehave:
perform some numerical simulations to illustrate the different
dω β
effectsdescribedinsection3. = H(p,e)sin2(θ+ϕ pM ̟), (3)
dt −cm − −
2. Equationsofmotion where̟isthelongitudeoftheperihelion,c = C /C = 0.55
m m
(Margotetal.,2007),and
We willadoptherea modelforMercurywhichisanexten-
sion of the model from Poincare´ (1910) of a perfect incom- 3Gm B A 3 B A
pressibleandhomogeneousliquidcorewithmomentsofinertia β= ⊙ − n2 − . (4)
2a3 C ≃ 2 C
A = B <C insideanhomogeneousrigidbodywithmoments
c c c
of inertia Am Bm < Cm, supported by the reference frame The Mariner 10 flyby of Mercury provided information on
≤
(~i,~j,~k), fixed with respect to the planet’s figure. Tidal dissi- the internal structure of the planet, though subject to some
pationandcore-mantlefrictiondrivetheobliquityclosetozero uncertainty. For the gravity field it has been measured J =
2
(Yoder,1997;Correiaetal.,2003).Sinceweareonlyinterested (6.0 2.0) 10 5andC =(1.0 0.5) 10 5(Andersonetal.,
− 22 −
± × ± ×
hereinthestudyofthefinalstagesofevolution(wherecapture 1987). ModelingtheinteriorstructureofMercury,ithasbeen
in spin-orbit resonance may occur), we will neglect the effect estimated for the structure constant ξ = C/(mR2) 0.3359
≃
of the small obliquity variations on the equations of motion. (Spohnetal.,2001). Thus,forthemomentsofinertiawecom-
Moreover, since for a long-term study we are not interested pute(B A)/C =4C /ξ 1.2 10 4.
22 −
− ≃ ×
in diurnal nutations, we can average over fast rotation angles
andmergetheaxisofprincipalinertiaandtheaxisofrotation 2.2. Tidaltorques
(Boue´andLaskar,2006). Therefore,themantlerotationrateis
Tidal effects arise from differential and inelastic deforma-
simplygivenbyω= θ˙+ϕ˙,whereθistherotationangleandϕ
tionsofMercuryduetothegravitationaleffectoftheSun.Their
theprecessionangle.
contributions to the spin variations are based on a very gen-
eral formulation of the tidal potential, initiated by George H.
2.1. Precessiontorque
Darwin(1880). Thedistortionoftheplanetgivesrisetoatidal
Thegravitationalpotential generatedatagenericpointof
V potential,
thespace~risgivenby(e.g.Tisserand,1891;Smart,1953):
Gm R 3 R 3
V(~r)=−Grm + G(Br3−A)P2(~ur·~j) Vg(~r,~r′)=−k2 R⊙ r! (cid:18)r′(cid:19) P2(cid:0)~ur·~ur′(cid:1) . (5)
where R is the average radius of the planet and r~ the radial
G(C A) ′
+ − P (~u ~k), (1) distancefromtheplanet’scentertotheSun. Ingeneral,imper-
r3 2 r·
fect elasticity will cause the phase angle of g to lag behind
V
where terms in (R/r)3 were neglected. G is the gravitational theperturbation(Kaula,1964),becausethereisisatimedelay
∆tbetweentheperturbationoftheSunandthemaximaldefor-
constant, m the mass of Mercury, ~u = ~r/r and P (x) =
r 2
(3x2 1)/2is the Legendrepolynomialof degreetwo. When mation of Mercury. Duringthat time, the planetrotatesby an
− angleω∆t,whiletheSunalsochangesitsposition.Assuminga
interacting with the Sun’s mass, m , the spin of Mercurywill
⊙ constanttimedelayallowsustolinearizethetidalpotentialand
undergoimportantchanges. Themiddletermintheabovepo-
simplifythetidalequations(Mignard,1979,1980;Hut,1981).
tentialwillberesponsiblefora librationinthespin, whilethe
last term causes the spin axis~k to precess around K~, the nor-
mal to the orbit. Since we are only interested in the study of
1Wehaveretainedtheuseofsemi-integersforbettercomparisonwithpre-
the long term motion, we will average the potential over the
viousresults.
rotationangleθandthemeananomaly M, afterexpandingthe 2ThereisamisprintinthesignofφinCorreia(2006).
2
Table1:CoefficientsofH(p,e)toe7.TheexactexpressionofthesecoefficientsisgivenbyH(p,e)= 1 π a 3exp(i2ν)exp(i2pM)dM.
π 0 r
R (cid:16) (cid:17)
p H(p,e)
5 13 35
1/1 1 e2 + e4 e6
− 2 16 − 288
7 123 489 1763
3/2 e e3 + e5 e7
2 − 16 128 − 2048
17 115 601
2/1 e2 e4 + e6
2 − 6 48
845 32525 208225
5/2 e3 e5 + e7
48 − 768 6144
533 13827
3/1 e4 e6
16 − 160
228347 3071075
7/2 e5 e7
3840 − 18432
73369
4/1 e6
720
12144273
9/2 e7
71680
For zero obliquity, the averaged contributions to the spin are If there is slippage between the liquid core and the man-
thengivenby: tle, a second source of dissipation of rotational energyresults
from friction occurring at the core-mantle boundary. Indeed,
dω K ω because of their different shapes and densities, the core and
= Ω(e) N(e) , (6)
dt −cm  n −  the mantle do nothave the same dynamicalellipticity and the
  two parts tend to precess at different rates (Poincare´, 1910).
where
This tendency is more or less counteracted by different inter-
1+3e2+3e4/8 actionsproducedattheirinterface: thetorqueofnon-radialin-
Ω(e)= , (7) ertial pressure forcesof the mantle overthe core provokedby
(1 e2)9/2
− the non-sphericalshape of the interface; the torqueof the vis-
cous friction (or turbulent) between the core and the mantle;
1+15e2/2+45e4/8+5e6/16
N(e)= , (8) andthetorqueoftheelectromagneticfriction,causedbythein-
(1 e2)6 teractionbetweenelectricalcurrentsofthecoreandthebottom
−
of the magnetized mantle. The two types of friction torques
K =n2 3k2 m⊙ R 3 , (9) (viscous and electromagnetic)depend on the differential rota-
ξQ (cid:18) m (cid:19)(cid:18)a(cid:19) tionbetweenthecoreandthemantleandcanbeexpressedby
a single effective friction torque, Υ (MathewsandGuo, 2005;
and Q 1 = n∆t. Thistidalmodelisparticularlyadaptedtode-
− DeleplaceandCardin,2006):
scribe the planets behavior in slow rotating regimes (ω n),
which is the case of Mercury during the spin-orbit reson∼ance Υ=Ccκδ; δ=ω ωc , (10)
−
crossing. In the present work we will adopt k = 0.4 and
2 where κ is an effective coupling parameter, ω the core’s ro-
c
Q = 50, which yields K = 2.2 10 5yr 2. This choice is
− − tation rate and c = C /C = 1 c = 0.45 (Margotetal.,
× c c m
somewhat arbitrary, but based on the parameter values of the −
2007). According to MathewsandGuo (2005) we may write
otherterrestrialplanets(GoldreichandSoter,1966).
κ 2.62√νω/R , where R is the core radius and ν is the
c c
≃
effective kinematic viscosity of the core. Adopting R /R =
c
2.3. Core-mantlefrictioneffect 0.77 0.04(Spohnetal., 2001) and ν = 10 6m2s 1, we com-
− −
±
TheMariner10flybyofMercuryrevealedthepresenceofan puteforthe3/2resonance: κ = 5 10 5yr 1. Theuncertainty
− −
×
intrinsicmagneticfield,whichismostlikelyduetomotionsina overνisverylarge,accordingtoLumbandAldridge(1991)it
conductingfluidcore(forareviewseeNess,1978).Subsequent can coverabout13 ordersof magnitude. As informerstudies
observationsmadewithEarth-basedradarprovidedstrongevi- onplanetaryevolution(Yoder,1997;CorreiaandLaskar,2001,
dencethatthemantleofMercuryisdecoupledfromacorethat 2003), weusedthesamevalueasthebestestimatedso farfor
isatleastpartiallymolten(Margotetal.,2007). theEarth(Gans,1972;Poirier,1988).
3
Anotherimportantconsequenceofcore-mantlefrictionisto Over several millions years, the eccentricity of Mercury
tiltthe equatoroftheplanettotheorbitalplane,whichresults presentssignificantvariations(Fig.1)thatneedtobetakeninto
inaseculardecreaseoftheobliquity(Rochester,1976;Correia, accountintheresonancecaptureprocess(PaperI).
2006), reinforcing the previous assumption that the obliquity
remainsclosetozeroduringthelastevolutionarystages.
Sincethecoreandthemantlearedecoupled,weneedadif- 3. Spinevolutionandcaptureprobabilities
ferentialequationforeachrotationrate, thecouplingequation
beinggivenbythefrictiontorque(Peale,2005;Correia,2006): The evolution of the spin for zero obliquity is completely
describedwhenweputtogetherthecontributionsfromthedif-
dω = ccκδ and dωc =κδ. (11) ferenteffectspresentedinsection2:
dt −cm dt
ω˙ = P+T Υ/C ,
m
− (13)

2.4. Planetaryperturbations  ω˙c =Υ/Cc,
Whenconsideringtheperturbationsoftheotherplanets,the whereP is the precession torques (Eq.3), T the tidal torque
eccentricity of Mercury is no longer constant, but undergoes (Eq.6)andΥthefrictiontorque(Eq.10).Inordertobetterstudy
strong chaotic variations in time (Laskar, 1990, 1994, 2008). thecaptureprobabilitiesinspin-orbitresonances,wewilladopt
These variations can be modeled using the averaging of the achangeofvariablesthatisvalidaroundeachresonance p:
equations for the motion of the Solar System, that have been
compared to recent numerical integrations, with very good γ=θ+ϕ pM ̟, (14)
− −
agreement (Laskaretal., 2004a,b). These equations are ob-
tainedbyaveragingtheequationsofmotionoverthefastangles andthus,
thatarethemeanlongitudesoftheplanets.Theaveragingofthe
equationofmotionisobtainedbyexpandingtheperturbations γ˙ =ω pn ̟˙ and γ¨ =ω˙ ̟¨ . (15)
− − −
oftheKeplerianorbitsinFourierseriesoftheangles,wherethe
In absence of planetary perturbations, for a Keplerian orbit,
coefficientsthemselvesareexpandedinseriesoftheeccentric-
̟˙ = 0 and thus γ¨ = ω˙. Instead of the core rotation ω we
ities and inclinations. This averaging process was conducted c
willalsoadoptthedifferentialrotationδ = ω ω asvariable.
inaveryextensiveway,uptosecondorderwithrespecttothe − c
Theabovesystemofequations(13)becomesthen:
masses,andthroughdegreefiveineccentricityandinclination,
leading to truncated secular equations of the Solar System of
γ¨ = P(γ)+T(γ˙) c κ δ ̟¨ ,
c m
theform − − (16)
dw  δ˙ = P(γ)+T(γ˙)−κmδ,
dt =i[Γw+Φ3(w,w¯)+Φ5(w,w¯)] , (12) whereκ =κ/c . Ingeneral,weliketoexpressγ¨ asthesumof
m m
theprecessiontorqueP(γ)andaglobaldissipativetorqueD(γ˙)
where w is the column vector (z ,...,z ,ζ ,...,ζ ), with z =
1 8 1 8 j
suchthat
e exp(i̟ ),ζ = sin(I /2)exp(iΩ )and̟andΩ,respectively
j j j j j
the longitudeof the perihelionand node. The 16 16 matrix
Γ is the linear Lagrange-Laplace system, while Φ×3(w,w¯) and γ¨ =−βmsin2γ+D(γ˙), (17)
Φ (w,w¯)gathertermsofdegree3and5respectively.
5 where
Thesystemofequationsthusobtainedcontainssome150000
terms,butitsmainfrequenciesarenowtheprecessionfrequen- β
β = H(p,e) (18)
m
cies of the orbits of the planets. The full system can thus be cm
numericallyintegratedwithaverylargestep-sizeof250years.
and
ContributionsduetothesecularperturbationoftheMoonand
generalrelativityarealsoincluded(formoredetailsandrefer- γ˙
encesseeLaskar,1990,1996;Laskaretal.,2004a). D(γ˙)= D0 V +µ , (19)
−  n
This secular system is then simplified and reducedto about  
50000terms,afterneglectingtermsofverysmallvalue(Laskar, D , V and µ being “constant” quantities. Indeed, under this
0
1994). Finally, a small correctionof the terms appearingin Γ linearized form we are able to estimate the capture probabili-
(Eq.12), after diagonalization, is performed in order to adjust ties using a simple expressionderivedby GoldreichandPeale
thelinearfrequencies,inasimilarwayasitwasdonebyLaskar
(1966):
(1990). The secular solutions are very close to the direct nu-
mericalintegrationLa2004(Laskaretal., 2004a,b) overabout π n V −1
P =2 1+ , (20)
35Myr,thetimeoverwhichthedirectnumericalsolutionitself cap " 2∆ω µ#
is valid because of the imperfections of the model. It is thus
legitimate to investigate the diffusion of the orbital motion of where ∆ω is the maximalamplitudeof librationin resonance,
Mercuryoverlongtimesusingthesecularequations. i.e.,∆ω= 2β .
m
p
4
3.1. Effectoftides where χ = c κn/[KΩ(e)]. It allows us easily to compute the
c
captureprobabilitiesusingexpression(20)providedthatweare
Let us consider first the simplified case where the spin of
abletoestimatecorrectlyγ˙ justbeforetheplanetcrossesthe
the planet is only subject to the precession and tidal torques c0
resonance:
(GoldreichandPeale,1966).Thus,wemayuseexpression(17)
toexpressγ¨,whereD(γ˙)isgivenbyexpression(6): 1
p E(e) χγ˙c0 −
K γ˙  π n − − n 
awictohnDEst((aγe˙n))t==vaTNlu=(ee)−o/fΩcmt(heΩe).(eecT)chepen−tlriimEcii(ttey)soi+sluontbiotan,inoefdthwehreontaDtio(γn˙()2fo1=r) AcPcocarpd=ing2to1e+xp2r∆esωsion(111)+wχe haveω˙ =. κδ. Sinceκ(27)
c
≪
0, that is, for γ˙/n = E(e) p ω/n = E(e). If the planet ∆ω, the core is unable to follow the periodicvariationsin the
− ⇔
encountersaspin-orbitresonanceinthewaytotheequilibrium mantle’srotationrate. Thus,onlythesecularvariationsonthe
position,capturemayoccurwithaprobabilitycomputedfrom mantle will be followed by the core and we may write: ω =
c
expression(20): ω δ,whereωistheaveragedrotationofthemantleand
−
Pcap =2(cid:20)1+ π2∆nω(p−E(e))(cid:21)−1 . (22) δ≃e−κmtZ tT eκmt′dt′ ≃ κT . (28)
t0 m
Using the present value of the eccentricity of Mercury and
cm = 1 we compute for the p = 3/2 resonance a probability Justbeforecrossingthe pth spin-orbitresonancewehaveω0 =
ofcaptureofabout7%, whichisunsatisfactorygiventhatthis pn+2∆ω/π,andthus
isthepresentlyobservedresonantstate.
γ˙ = ω pn=ω δ pn
c0 c0 − 0− −
3.2. Effectofcore-mantlefriction 2 T 2 KΩ(e)
= ∆ω ∆ω+ p E(e) . (29)
We now add the effect of core-mantlefriction to the effects π − κm ≃ π κ −
(cid:2) (cid:3)
alreadyconsideredintheprevioussection.Inthiscasewemust
When using the present eccentricity of Mercury and the val-
takeintoaccounttherotationofthecore,andthecompletero-
ues of κ and K estimated in previous sections, we compute
tationrateevolutionisgivenbysystem(16)with̟¨ =0.
χ = 19.5. Substituting this into expression (27) we obtain a
The general solution for the differential rotation in system
probabilityofcaptureof100%inthe3/2spin-orbitresonance.
(16)isgivenby:
However,asnoticedbyPealeandBoss(1977), if wecompute
t theprobabilityforthe2/1resonancewealsoget100%. Thus,
δ(t)=δ0e−κmt+e−κmt P(γ)+T(γ˙) eκmt′dt′, (23) either the planet started its rotation below the 2/1 resonance,
Z
t0 (cid:2) (cid:3) whichisunlikely,ortheremustbeanothermechanismtoavoid
where δ0 is the initial value of δ(t) for t = t0. In section 2.3 captureinthe2/1andhigherorderresonances.
weestimateforthepresentrotationofMercury1/κ 104yr,
m
≃
whichcanbeseenasthetimescaleneededtodamptheinitial
3.3. Effectofplanetaryperturbations
differentialrotationδ tozero.Thus,fort 1/κ theevolution
0 m
of the differentialrotationis givenonlyb≫ythe last termis ex- TheorbitaleccentricityofMercuryundergoesimportantsec-
pression(23). Italsomeansthatfortimeintervals∆t 1/κ , ularperturbationsfromtheotherplanets(Fig.1)anditscontri-
m
wecanwritefort<t +∆t: ≪ bution needsto be taken into account. The mean value of the
0
eccentricityise¯ = 0.198,slightlylowerthanthepresentvalue
t e 0.206, but we also observe a wide range for the eccen-
δ(t)≃δ0+Z γ¨ dt′, (24) tric≃ityvariations,fromnearlyzerotomorethan0.45(PaperI,
t0
Laskar, 2008). Even if some of these episodesdo not last for
thatis,
alongtime,theywillallowadditionalcaptureintoandescape
fromspin-orbitresonances.Moreover,thecaptureprobabilities
δ(t) δ +γ˙ γ˙ =γ˙ γ˙ , (25)
≃ 0 − 0 − c0 arealsomodifiedfordifferenteccentricities:forthesamereso-
whereγ˙ =γ˙ δ =ω pn,andω isthevalueofthecore nancewecanhavezeroor100%ofcapturesdependingonthe
rotationcf0or t 0=−t .0Thisc0ap−proximatiocn0 is veryusefulbecause eccentricityvalue(Fig.2). Forallresonances,thecaptureprob-
0
wecanexpress㨠bymeansofexpression(17)with ability is 100%wheneverthe eccentricity is close to the equi-
librium valuefor the rotationrate, E(e) = N(e)/Ω(e)(Eq.21),
K γ˙ ccκ but it tends to decrease as the eccentricity moves away from
D(γ˙) = Ω(e) p E(e)+ γ˙ γ˙
−cm  − n− cm (cid:0) − c0(cid:1) tihfitsheeqsupiilnibrisiuimncvraelausein.gIfftrhoemeclcoewnetrricroittyatiisotnoorahteigs)hs(oomrteoorelsoow-
= KΩ(e) p E(e) χγ˙c0 +(1+χ)γ˙ , (26) nances cannot be reached and the probability of capture sud-
−cm  − − n n denlydropstozero(Fig.2).
  5
Table2: Criticaleccentricityec fortheresonance p. Ife < ec,theresonance Table3:Captureprobabilitiesinseveralspin-orbitresonances(inpercentage).
pbecomesunstable,andthesolutionmayescapetheresonance(PaperI).The Intheleft panel (Tonly)weconsider theeffect oftides alone, while inthe
criticaleccentricityecisobtainedwhenβH(p,e)<K[Ω(e)p N(e)]. rightpanel(T+CMF)bothtidesandcore-mantlefrictioneffectsaretakeninto
−
account. Thefirstcolumn(Pcap)referstothetheoretical estimationgivenby
p e expression(20),whilethenextcolumn(num.)referstotheestimationobtained
c
runninganumericalsimulationwith2000closeinitialconditions,differingby
5/1 0.211334
π/1000inthelibrationangle. Planetaryperturbations arenotconsideredand
9/2 0.174269 weusedaconstanteccentricitye=0.206.
4/1 0.135506
7/2 0.095959 Tonly T+CMF
3/1 0.057675 p Pcap num. Pcap num.
5/2 0.024877 (%) (%) (%) (%)
2/1 0.004602 5/1 1.6 0.3
− −
3/2 0.000026 9/2 3.1 1.3
− −
1/1 4/1 0.1 5.9 4.8
− −
7/2 0.1 0.1 11.4 10.9
3/1 0.3 0.4 22.6 22.8
5/2 0.7 1.4 46.6 46.2
Foranon-constanteccentricitye(t),thelimitsolutionofthe 2/1 1.8 1.7 100.0 100.0
rotationratewhen D(γ˙) = 0(Eq.17)isnolongerω/n = E(e), 3/2 7.7 7.2 100.0 100.0
butmoregenerally:
1 t K c κ
ω(t)= N(e(τ)) c δ(τ) g(τ)dτ, (30)
g(t)Z0 cm (cid:20) − K (cid:21) 4.1. Simulationswithoutplanetaryperturbations
Before considering the effect of planetary perturbationswe
where
can test numerically the theoretical estimates of the capture
K t probability given by expressions (22) and (27). Since cap-
g(t)=exp Ω(e(τ))dτ . (31)
cmnZ0 ! ture in resonance is a statistical process we need to perform
manyintegrationswithslightlydifferentinitialconditions. For
The dissipation torques can thus drive the rotation rate sev- that purpose we ran 2000 simulations using a fixed eccentric-
eraltimesacrossthesamespin-orbitresonance,increasingthe ity (e = 0.206), initial rotation period of 2 days, zero obliq-
chancesofcapture. uity and different initial libration phase angles with step-size
Anotherimportantconsequenceofanon-constanteccentric- of π/1000 rad. Results are listed in Table 3. We can see that
ityisthatallresonancesbutthe1/1maybecomeunstable. In- there is a good agreement between the theoretical previsions
deed, the amplitudeofthe librationtorquedependson the co- andthenumericalestimationoftheprobabilities. Asdiscussed
efficientH(p,e)(Eq.3), whichgoestozerowiththeeccentric- in sections 3.1 and 3.2 the probability of capture when con-
ity, except for the 1/1 resonance (Tab.1). Whenever the am- sidering only the effect of tides is very small ( 7% for the
∼
plitudeofthelibrationrestorationtorquebecomessmallerthan 3/2 resonance), while it becomes very important when core-
theamplitudeofthedissipationtorque,equilibriuminthespin- mantle friction is added (100% for the 3/2 resonance). They
orbit resonance can no longer be sustained and the resonance arealsoinconformitywiththoseobtainedinthepreviousstud-
is destabilized. Critical eccentricities for each resonance are ies by GoldreichandPeale (1967) and PealeandBoss (1977).
listedinTable2,obtainedwhenthetorquesbecomeequivalent Astheyallnoticed,whentheeffectfromcore-mantlefrictionis
(Eqs.3,6): considered,theprobabilitiesofcapturearegreatlyenhancedfor
all spin-orbitresonances. In particular, capture in the 2/1 res-
Ω(e)p N(e) β
− = . (32) onancealsobecomes100%,preventingasubsequentevolution
H(p,e) K tothe3/2resonance.
4.2. Inclusionofplanetaryperturbations
4. Numericalsimulations Whenplanetaryperturbationsaretakenintoaccount,theec-
centricity presents chaotic variations with many excursions to
Wewillnowusethedynamicalequationsestablishedinsec- higherandlowervaluesthantoday(Laskar,1990,1994,2008,
tion2tosimulatethefinalevolutionofMercury’sspinbyper- PaperI).Itisthenimpossibletoknowitsexactevolutionatthe
forming massive numerical integrations. The main goal is to time the planet first encounteredthe spin-orbitresonances. A
illustrate the effects described in section 3, in particular the statisticalstudywithmanydifferentorbitalsolutionsistheonly
probabilitiesofcaptureandescapefromspin-orbitresonances. possibility to get a global picture of the past evolution of the
Mercury geophysical models and parameters in use are those spinofMercury. InPaperI we performedsucha studybyin-
listed in section 2. We recall here the most uncertain values: tegrating1000orbitsover4 Gyr in the paststarting with very
k =0.4,Q=50andν=10 6m2s 1. close initial conditions. This statistical study was only made
2 − −
6
possiblebytheuseoftheaveragedequationsforthemotionof
Table4:Captureprobabilitiesinseveralspin-orbitresonances(inpercentage).
theSolarSystem(Laskar,1990,1994).
WeperformedastatisticalstudyofthepastevolutionsofMercury’sspin,with
In figure 3 we show five examples of the eccentricity evo- theintegration of1000orbits over4Gyr, ainitial rotation periodof10days
lution through the 4.0Gyr. We choose some cases illustrative andzeroobliquity.Inthe“1stcap.”columnwelisttheresonancesinwhichthe
ofthechaoticbehavior,wherewecanseethattheeccentricity spinwasfirstcaptured(beforebeingdestabilized). Inthe“final”columnwe
listtheresultsafterthefull4Gyrofsimulations. Forcomparisonwealsolist
canbeassmallaszero,butitcanalsoreachvaluesashighas
theresultsobtainedwithaconstanteccentricity e = 0.206(“const.”) andthe
0.5. Insomecasestheeccentricitycanremainwithin[0.1,0.3] finalresultsobtainedinPaperI(“C&L04”),withamodelwithoutcore-mantle
throughoutthe evolution, while in other cases it can span the friction.
wholeinterval[0,0.5](Laskar,2008).
numberofcaptures
Owing to the chaotic evolution, the density function of the
p const. 1stcap. final C&L04
1000 solutions over 4Gyr is a smooth function (Fig.4), well
(%) (%) (%) (%)
approximatedbyaRiceprobabilitydistribution(Laskar,2008).
6/1 0.1
The eccentricity excursions to higher values allow the planet − − −
11/2 0.4
to crossthe 3/2resonanceseveraltimes, andthusincreasethe − − −
5/1 0.3 1.3
probabilityofcapture.Thisbehaviorbecomesveryimportantif − −
9/2 1.3 2.7
theevolutionisdrivenbytidalfrictionalone. Eventhoughthe − −
4/1 4.7 5.3
probabilityofcaptureinasinglecrossingofthe3/2spin-orbit − −
7/2 10.3 8.7 4.7
resonanceisonlyaround7%,multiplecrossingsincreaseitup −
3/1 19.0 15.5 11.6
to55%(PaperI). −
5/2 29.8 26.5 22.1
−
2/1 34.6 31.2 31.6 3.6
4.3. Planetaryperturbationswithcore-mantlefriction
3/2 8.1 25.9 55.4
In presence of an efficientcore-mantle friction the multiple −
1/1 0.2 3.9 2.2
crossingsofthe 3/2resonanceare nolongerneeded,sincethe −
none 0.2 38.3
captureinthisresonanceafterasinglecrossingisalready100% − −
(Fig.2). Nevertheless, eccentricity excursions to lower values
candestabilizetheequilibriuminanyspin-orbitresonancedif-
ferent from the 1/1 (Tab.2). This effect was already present
32%respectivelyforthe5/2andthe 2/1resonances). The 5/2
whenthecore-mantlefrictionwasnotconsidered(PaperI),but
andthe2/1spin-orbitresonancesbenefitfromthefactthatthe
with small influence on the results, while here it becomes of
planet must cross them first. On the other hand, the 3/2 reso-
capitalimportance. Indeed,itmayallowtheevasionfrompre-
nanceismorestableandthechancesofcapturearehigherwhen
viouscapturesinhigherorderresonancesthanthe3/2andper-
crossed.
mitsubsequentevolutiontothepresentobservedspinstate.
Inordertocheckthisnewscenario,wehaveperformedasta- SincetheeccentricityofMercury4.0Gyragocanbearound
tisticalstudyofthepastevolutionsofMercury’sorbit,withthe 0.4 (e.g. Fig.3), at the momentof the first encounterwith the
integrationofthesame1000orbitsover4Gyrinthepastusedin spin-orbitresonances,captureinresonancesashighasthe6/1
PaperI.We nowadditionallyincludetheeffectofcore-mantle can occur(Fig.2). Because the probabilityof capture is small
friction as described by GoldreichandPeale (1967), i.e., we andbecausetherearenotmanyorbitalsolutionsreachingsuch
willconsiderthefulldynamicsofthespingovernedbyEq.(13). high values for the eccentricity, we only count about 10% of
Assuming an initial rotation period of Mercury of 10h, we capturesinresonancesaboveorequaltothe4/1(Tab.4). Once
estimatedthatthetimeneededtode-spintheplanettotheslow captured,theseequilibriacanbemaintainedaslongastheec-
rotationswouldbeabout300millionyears. Wewillthenstart centricityremainsabovetherespectivecriticalvalues(Tab.2).
ourintegrationsalreadyintheslow-rotationregime,witharota- Contrary to the results predicted for a constant eccentricity
tionperiodof10days(ω 8.8n),zeroobliquityandastarting (Tab.4), we also registereda few trajectoriesdirectlycaptured
≃
timeof 4Gyr,althoughthesevaluesarenotcritical.InTable4 inthe3/2resonancejustafterthefirstpassagethroughthereso-
−
weshowtheamountofcapturesforeachresonanceattheendof nancearea. Whenweusedthepresentvalueoftheeccentricity
thesimulations(column“final”).Wealsolisttheresonancesin (e = 0.206), the 3/2 resonance could not be attained because
whichthespinwasfirstcapturedbeforebeingdestabilized(col- forthatvaluecaptureprobabilityinthe2/1resonanceis100%
umn“1stcap.”)andwerecalltheresultsobtainedforaconstant (Fig.2).However,foreccentricityvalueslowerthanabout0.19,
eccentricity (e = 0.206)and in Paper I, with a modelwithout theprobabilityofcaptureinthisresonancedecreases,aswellas
core-mantlefriction. forhigherorderresonances.Forinstance,whentheeccentricity
After running 1000 trajectories we observe that the spin of is0.09,captureinthe2/1resonancedropsto50%. Thus,since
Mercury preferably chooses one of the three final configura- theeccentricityofMercuryisvarying,itmayhappenthatabout
tions:5/2,2/1or3/2(Tab.4).With26%ofcaptures,thepresent 4.0Gyrago its value was muchlowerthan todayand the spin
configurationnolongerrepresentsthemostprobablefinalout- managedtoavoidallthespin-orbitresonanceshigherthanthe
come, as it was in absence of core-mantle friction (Paper I). 3/2 and was directly captured in the present observed config-
However,itisstill amongthemostprobablescenarios, theal- uration. We estimate nevertheless that the probability for this
ternativesreceivingcomparableamountsofcaptures(22%and scenariotooccurisverylow,onlyabout8%(Tab.4).
7
4.4. Criticaleccentricities evenfor verylow valuesof the eccentricity (Fig.2), the major
Over4.0Gyrofevolutiontheeccentricityhasmanychances possibilityofevolvingintothe1/1resonanceisbydestabilizing
ofexperiencingaperiodofverysmallvalues(e.g.Fig.3).Even the3/2. Thisbecomesapossibilityiftheeccentricityisalmost
whenaperiodofloweccentricitydoesnotlastforalongtime,a zero,thatis,fore<3 10 5(Tab.2).
−
×
singlepassageoftheeccentricitybelowacriticalvalue(Tab.2)
canbeenoughtodestabilizethecorrespondingspin-orbitreso- 4.5. Differentscenariosofevolution
nance.
The critical eccentricity needed to destabilize the 2/1 spin-
Allorbitalsolutionsweregeneratedstartingfrominitialcon-
orbitresonanceise 0.0046(Tab.2). Whenevertheorbital
ditionsclosetothepresentvalues(seeLaskar,2008),andthere- 2/1 ≈
eccentricity is below this value, the spin will then evolve to-
fore converge to the same final evolution. The eccentricity
wardsthe3/2resonanceorbelow(Fig.7). However,thisisnot
behavior is thus identical for the last 50-60Myr (Fig.1), be-
the only possibility of achieving this last configuration if the
forewhichthechaoticdiffusiondominates(Fig.5). Duringthe
planet was first captured in a higher-orderresonance. Indeed,
last50Myrtheeccentricitycertainlyreachedvalueslowerthan
as discussed for the 1st capture column (Tab.4), if the eccen-
0.13,thusthe4/1andabovespin-orbitresonancescannotrep-
tricity islowerthan 0.19atthe time the planetcrossesthe 2/1
resentapossiblefinaloutcomeforMercury(e 0.136).For
4/1 ≈ resonance(Fig.2),thechancesofcapturearelowerthan100%,
the7/2andlowerorderspin-orbitresonances,captureuntilthe
opening some space for subsequent evolution to the 3/2 reso-
present day is not forbidden by the last 50Myr of Mercury’s
nance.Forinstance,forapreviouscaptureinthe5/2resonance,
evolution, but depends on the true orbital evolution of the ec-
an eccentricityof e 0.025will destabilize it and produce
centricity (Fig.5). The higher is the critical eccentricity, the 5/2 ≈
a capture probabilityof only 14.4%in the 2/1 resonance, i.e.,
loweristheprobabilityofremainingtrapped,becausemoreor-
when the 5/2 resonance is destabilized there is about 85% of
bitalsolutionswillcomebelowthisvalue.
chanceofendinginthe3/2presentconfiguration(Fig.7).
Infigure6weplotthecumulativedistributionoftheminimal
Inordertoexemplifythemultitudeofpossibleevolutionary
eccentricitiesattainedforeachoneofthe1000orbitalsolutions
scenarios, we performed another kind of experiment. Adopt-
thatweused.Wealsomarkwithstraightlinesthecriticalvalues
ing a particular orbital solution, which presents a gradual de-
oftheeccentricityforeachspin-orbitresonance(Tab.2)anduse
crease in the eccentricity (Fig.3a), we integrated close initial
dots to representthe amountof capturesobtainednumerically
conditions for the spin. Since for this orbital solution the ec-
forspin-orbitresonancesthatarestillstablebeloweachcritical
centricity is high at the time of the first encounter, there is a
value of the eccentricity. Since a large amount of the orbital
greatchanceofcapturingthespininaspin-orbitresonancewith
solutions experience at least one episode with an eccentricity
p > 5/2. Infigure 7 we plotthe behaviorof fourtrajectories,
below0.05(log e 1.3),about84%ofthefinalevolutions
10 ≈ − each one initially captured in a differentspin-orbit resonance,
will end in the 5/2 spin-orbit resonance or lower (Tab.4). By
when the eccentricity approaches a zone of very low values.
comparingtheeccentricityinstabilitythresholdsforeachspin-
As expected, the spin-orbit resonances are sequentially aban-
orbit resonancewith the amountof capturesobtained numeri-
doned as the eccentricity assumes small values. In particular,
cally below that resonance we see that there is a good agree-
we observe that the resonances are quit immediately after the
ment, suggestinga strongcorrelationbetweenthe orbitalevo-
eccentricityisbelowthecriticalvalueslistedinTable2. After
lutionoftheeccentricityandthepercentageofcapturesineach
being destabilized, the spin can evolve directly to the present
resonance.Thereasonwhythereisnotfullagreementbetween
3/2 configuration, or can be trapped in an intermediate spin-
thetwoisbecausespin-orbitresonancesbelowcriticalvaluesof
orbitresonance.Inthisexample,theeccentricitybecomelower
theeccentricitycanalsobeattainedbytrajectoriesthatescaped
than e 0.0046 around 1.79Gyr and captures in the 2/1
captureinhigherorderresonances,thatis,theycanbeattained 2/1 ≈ −
resonancebecomedestabilizedafterthatdate.
eveniftheeccentricityisneverbelowthecriticalvalueforthat
Wepurposelyplotonesituation,wherethespindoesnotend
resonance(Fig.2).
inthe3/2resonance,however. Inthiscase, atthemomentthe
When comparing the results after 4.0Gyr with those after
eccentricitybecomeslower than e , the spin is still captured
the first capture, we verify that the 5/2 resonance (and above) 2/1
inthe5/2resonance. Thisresonancelogicallybecomesdesta-
loseasignificantamountofpreviouslycapturedsolutions. The
bilizedandtherotationratedecreases.Nevertheless,atthemo-
amountoforbitscapturedinthe2/1resonanceremainsroughly
ment the spin encountersthe 2/1 resonance the eccentricity is
thesame,becausethenumberoftrajectoriesquittingthisreso-
again higherthan e , and thereforethere is a chance of cap-
nanceismoreorlesscompensatedbytheincomingtrajectories 2/1
tureinthisresonance,preventingasubsequentevolutiontoward
fromhigherorderresonances.The3/2istherealwinnerofthis
the3/2state(Fig.7).
transitionprocess,astheamountoftrajectoriesthatendinthis
lastconfigurationisabout4timeslargerthanitwasinitially.An
4.6. Constraintsontheorbitalevolution
identicalscenariowasalreadyobservedinPaperI,exceptthat
only a few captures occurred in spin-orbit resonances higher Wehaveseeninprevioussectionsthatthereisanimportant
thanthe2/1andtheywereallsubsequentlydestabilized(Tab.4). correlationbetween the minimaleccentricityattainedby Mer-
As in Paper I, we also noticed about 4% of the trajectories curythroughitsorbitalevolutionandtheprobabilityofcapture
capturedinthe1/1spin-orbitresonance(Tab.4).Sincetheprob- in agivenresonance(Fig.6). Theloweris theminimaleccen-
abilityofcaptureinthe3/2spin-orbitresonanceisalmost100% tricity, the higher is the probability of achieving a low order
8
Table5: Captureprobabilities inspin-orbitresonances (inpercentage), when Table6:ProbabilityfortheeccentricityofMercurytohavedescendedbelowa
theeccentricitydescendsbelowagivencriticalvalue(Tab.2). specificcriticallevel(Pec)giventhatitsrotationiscapturedinthe3/2spin-orbit
resonancetoday. Resultsforeachcriticaleccentricity ec (Tab.2)areobtained
p e3/2 e2/1 e5/2 e3/1 e7/2 e4/1 fromEq.(33)withPcap=25.9%(Tab.4).
7/2 1.2 2.6 3.8 4.7
− − P e e e e e e
3/1 2.0 4.2 7.2 10.7 11.6 3/2 2/1 5/2 3/1 7/2 4/1
− P 25.0 73.2 55.1 37.2 27.7 25.9
5/2 3.2 5.2 16.0 20.7 22.1 cap/ec
− P 3.2 27.1 40.3 64.5 99.7 100.0
2/1 3.1 5.4 23.9 30.5 32.6 31.6 orb
P 3.1 76.6 85.7 92.6 98.8 100.0
3/2 25.0 73.2 55.1 37.2 27.7 25.9 ec
1/1 68.8 15.4 9.7 6.1 4.2 3.9
none 3.1 0.7 0.5 0.3 0.2 0.2
scended below e 0.0046(butonly a 3% chance of going
2/1
≈
belowe 0.00003).
3/2
≈
spin-orbit resonance. In particular, each time the eccentricity
descendsbelowagivencriticalvalueforaspin-orbitresonance 5. Conclusions
(Tab.2),thespinwillevolveintoalowerresonance.
DuetotheincreasingevidenceofamoltencoreinsideMer-
Sinceweknowthedistributionoftheminimaleccentricities
cury(Nessetal.,1974;Margotetal.,2007),viscousfrictionat
(Fig.6),wecanestimatetheprobabilityofendinginaspecific
the core-mantleboundaryis expectedand its consequencesto
spin-orbitresonancegiventhe valueofthe minimaleccentric-
thespinmustbetakenintoaccount.Animportantconsequence
ityofaconsideredorbit,P . Forthatpurposeweeliminate
cap/ec is a considerable increase in the probability of capture for all
allthetrajectoriesforwhichtheminimaleccentricityisabove
spin-orbit resonances; in particular, for the 2/1 and the 3/2 it
thecriticalvalue(Tab.2)andthencountthenumberofcaptures
can reach 100% (PealeandBoss, 1977). Since it is believed
in each resonancefor the remainingorbitalsolutions. Results
that Mercury’s initial rotation was much faster than today, a
arelistedinTable5. Whileforanarbitraryorbitalsolutionthe
destabilizationmechanismis then requiredto allowthe planet
probabilityofcaptureinthepresent3/2spin-orbitresonanceis
toescapefromthe2/1andhigherorderresonancesandsubse-
only25.9%,thisvaluerisesto55.1%ifweassumethattheec-
quentlyevolvetothepresentobserved3/2configuration.
centricityofMercurywasbelowe 0.025atsometimein
5/2 ≈ Withtheconsiderationofthechaoticevolutionoftheeccen-
thepast,orevenupto73.2%iftheeccentricitydescendsbelow
tricityofMercuryweshowthatsuchdestabilizationmechanism
e 0.0046. Resultsforthecriticaleccentricitye 0.136
2/1 ≈ 4/1 ≈ existswhenevertheeccentricitybecomessmallerthanacritical
are the same as the global results shown in Table 4, because
value for each spin-orbit resonance (Tab.2). This mechanism
the eccentricity of Mercury was below that value in the most
wasalreadydescribedinPaperI,butbecomesofcapitalimpor-
recent 50 Myr, where the chaotic behavior is not significant
tance when core-mantle friction is taken into account. There
(Fig.5). Notice also that there are always a few captures left
aretwomainpossibilitiestoevolveintothe3/2configuration:
inresonancesabovethecorrespondingcriticalvalue. Thiscan
beexplainedbythesameeffectdescribedinthelastparagraph Theeccentricitybecomeslowerthanthecriticalvaluefor
•
ofsection4.5andillustratedinfigure7. the 2/1 spin-orbit resonance (e 0.005) and evolves
2/1
≈
Inversely, since we know that the rotation of Mercury is intothe3/2.
presentlycapturedinthe3/2spin-orbitresonance,wecanesti-
Theeccentricitybecomeslowerthanthecriticalvaluefor
matetheprobabilityfortheeccentricitytohavedescendeddur-
•
a higher order resonance than the 2/1, and then crosses
ingitspastevolutionbelowaspecificcriticallevel,P . Using
ec this resonance with an eccentricity lower than e < 0.19.
conditionalprobabilities,wehavethen
Thisallowsanonzeroprobabilityofescapingthe2/1res-
P = Pcap/ec ×Porb , (33) onance,andsubsequentevolutionintothe3/2.
ec P
cap
The other mechanism of capturingin the 3/2 resonancede-
whereP istheprobabilityofendinginaspecificspin-orbit
cap/ec scribedinPaperI,consistedinareturningtothe3/2spin-orbit
resonancegiventhe value of the minimaleccentricity(Tab.5),
resonanceafteranincreaseintheeccentricity.Thiseffectisnot
P the probability for the eccentricity to reach that minimal
orb as important when we take into account core-mantle friction,
eccentricity (Fig.6) and P the global capture probability in
cap sincethemostpartofthetrajectoriesarecapturedinresonance
the specific spin-orbit resonance (Tab.4). Results for the 3/2
afterasinglepassage.
spin-orbitresonance are given in Table 6. These probabilities
After running1000 orbitalsolutions, starting from 4Gyr in
for the orbital evolution of the eccentricity are the same as if
thepastuntiltheyreachedthepresentdate,thespinoftheplanet
weselectonlytheevolutionsthatfinishedinthe3/2spin-orbit
wascapturedinaspin-orbitresonance99.8%ofthetime. The
resonance and then look at the minimal eccentricity distribu-
mainresonancestobefilledandtherespectiveprobabilitywere
tion.Fromtheaboveanalysisweconcludethatthereisastrong
(Tab.4):
probability that the eccentricity of Mercury reached very low
values; in particular there is about a 77% chance that it de- P =22.1%, P =31.6%, P =25.9%. (34)
5/2 2/1 3/2
9
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