Resonance locking as the source of rapid tidal migration in the Jupiter and Saturn moon systems

0.76 MB·

by  Jim Fuller

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed13September2016 (MNLATEXstylefilev2.2)
Resonance locking as the source of rapid tidal migration in
the Jupiter and Saturn moon systems
Jim Fuller1,2(cid:63), Jing Luan3, and Eliot Quataert3
1TAPIR, Walter Burke Institute for Theoretical Physics, Mailcode 350-17, Caltech, Pasadena, CA 91125, USA
2Kavli Institute for Theoretical Physics, Kohn Hall, University of California, Santa Barbara, CA 93106, USA
3Astronomy Department and Theoretical Astrophysics Center, University of California at Berkeley, Berkeley, CA 94720-3411, USA
6
1 13September2016
0
2
ABSTRACT
p
e
S The inner moons of Jupiter and Saturn migrate outwards due to tidal energy
dissipation within the planets, the details of which remain poorly understood. We
0
demonstrate that resonance locking between moons and internal oscillation modes of
1
the planet can produce rapid tidal migration. Resonance locking arises due to the in-

ternal structural evolution of the planet and typically produces an outward migration
]
P rate comparable to the age of the solar system. Resonance locking predicts a similar
E migrationtimescalebutadifferenteffectivetidalqualityfactorQgoverningthemigra-
. tion of each moon. It also predicts nearly constant migration timescales a function of
h
semi-major axis, such that effective Q values were larger in the past. Recent measure-
p
ments of Jupiter and Saturn’s moon systems find effective Q values that are smaller
-
o than expected (and are different between moons), and which correspond to migration
r timescales of ∼10 Gyr. If confirmed, the measurements are broadly consistent with
t
s resonance locking as the dominant source of tidal dissipation in Jupiter and Saturn.
a
Resonance locking also provides solutions to several problems posed by current mea-
[
surements:itnaturallyexplainstheexceptionallysmallQgoverningRhea’smigration,

2 it allows the large heating rate of Enceladus to be achieved in an equilibrium eccen-
v tricity configuration, and it resolves evolutionary problems arising from present-day
4 migration/heating rates.
0
8 Key words:
5
0
.
1
0
1 INTRODUCTION outward motion of inner moons produces convergent mi-
6
gration that allows moons to be captured into resonances
1 Satellite systems around the outer planets exhibit rich dy-
(see discussion in Dermott et al. 1988; Murray & Dermott
: namicsthatprovidecluesabouttheformationandevolution
v 1999). For instance, the 4:2:1 resonance of the orbits of
of our solar system. The orbits of the moons evolve due to
i Io:Europa:Ganymede indicates that Io has tidally migrated
X tidal interactions with their host planets, and every moon
outwards by a significant fraction of its current semi-major
r system within our solar system shows strong evidence for
axis,catchingbothEuropaandGanymedeintoMMRsdur-
a significant tidal evolution within the lifetime of the solar
ing the process.
system. Yet, in many cases, the origin of tidal energy dissi-
pationnecessarytoproducetheobservedorinferredorbital Thetidalenergydissipationresponsibleforoutwardmi-
migration remains poorly understood. gration is often parameterized by a tidal quality factor Q
The Jupiter and Saturn moon systems are particularly (Goldreich & Soter 1966) that is difficult to calculate from
intriguing. In both systems, the planet spins faster than firstprinciples.SmallervaluesofQcorrespondtolargeren-
the moons orbit, such that tidal dissipation imparts angu- ergydissipationratesandshortermigrationtimescales.The
lar momentum to the moons and they migrate outwards. actualvalueofQcanbesomewhatconstrainedbyobserved
The rich set of mean motion resonances (MMRs) between orbitalarchitectures:verysmallvaluesofQareimplausible
moonsprovideevidenceforongoingtidalmigrationbecause because they would imply the satellites formed inside their
Rocheradii,whereasverylargevaluesofQareunlikelybe-
cause they would not allow for capture into MMRs within
(cid:63) Email:[email protected] the lifetime of the solar system. In other words, the cur-
(cid:13)c 0000RAS
2 Fuller et al.
rent orbital architectures mandate that outward migration cies to gradually change, allowing for resonance locking to
timescales are within an order of magnitude of the age of occur.Duringaresonancelock,aplanetaryoscillationmode
the solar system. staysnearlyresonantwiththeforcingproducedbyamoon,
Inclassicaltidaltheory,tidalenergydissipationoccurs greatlyenhancingtidaldissipationandnaturallyproducing
via the frictional damping of the equilibrium tide, which outward migration on a timescale comparable to the age of
is defined as the tidal distortion that would be created by thesolarsystem.Inthispaper,weexaminethedynamicsof
a stationary perturbing body. In this case, damping can resonancelockingingiantplanetmoonsystems,findingthat
arise from turbulent viscosity due to convective motions resonancelockingislikelytooccurandcanresolvemanyof
in a gaseous envelope (Goldreich & Nicholson 1977) or by the problems discussed above. Although there are substan-
viscoelasticity in a solid core (Dermott 1979; Remus et al. tial uncertainties in the planetary structure, evolution, and
2012a,b; Storch & Lai 2014; Remus et al. 2015). However, oscillation mode spectra of giant planets, resonance locking
tidaldissipationcanalsooccurthroughtheactionofdynam- isinsensitivetomanyofthesedetailsandyieldsrobustpre-
icaltides,whichariseduetotheexcitationofwavesand/or dictionsforoutwardmoonmigrationonplanetaryevolution
oscillationmodesbythetime-dependentgravitationalforce timescales.
of the perturbing body. The dynamical tide may be com- The paper is organized as follows. In Section 2, we dis-
posed of traveling gravity waves (e.g., Zahn 1975; Ioannou cuss tidal dissipation via resonances with oscillation modes
&Lindzen1993a,b),tidallyexcitedgravitymodes(e.g.,Lai and the process of resonance locking. Section 3 investigates
1997; Fuller & Lai 2012; Burkart et al. 2012, or tidally ex- orbitalmigrationandMMRsresultingfromresonancelock-
citedinertialmodes/waves(e.g.,Wu2005a,b;Ogilvie&Lin ing.WecompareourtheorywithobservationsoftheJupiter
2004; Ogilvie 2013; Guenel et al. 2014; Braviner & Ogilvie and Saturn moon systems in Section 4. Section 5 provides
2015; Auclair Desrotour et al. 2015; Mathis 2015). discussion of our results, including implications for tidal
Thenotablefeatureofmanypreviousworksisthatthey heatingandtheorbitalevolutionofthemoons.Wesumma-
struggle to produce tidal dissipation rates large enough to rize our findings in Section 6. Details related to planetary
match those inferred for Jupiter and Saturn, given plausi- oscillation modes, their ability to sustain resonance locks,
ble internal structures of the planets. Models involving vis- and tidal heating can be found in the appendices.
coelastic dissipation in the core or inertial waves in the en-
velope require the presence of a large core (solid in the for-
mer case, and either solid or more dense in the latter case) 2 TIDAL MIGRATION VIA RESONANCES
inordertoproducesignificantenergydissipation.Although
2.1 Basic Idea
substantial cores likely do exist (Guillot & Gautier 2014),
water ice in giant planet cores is likely to be liquid (Wilson Tidal dissipation and outward orbital migration of moons
& Militzer 2012b), and rocky materials (silicon and magne- canbegreatlyenhancedbyresonancesbetweentidalforcing
sium oxides) could be liquid or solid (Mazevet et al. 2015). frequenciesanddiscrete“modefrequencies”associatedwith
Inbothcases,theice/rockislikelytobesolubleinthesur- enhanced tidal dissipation. The mode frequencies may
rounding hydrogen/helium Wilson & Militzer (2012a), and occur at gravity/inertial/Rossby mode frequencies, or they
the cores may have substantially eroded (thereby erasing could correspond to frequencies at which inertial waves are
sharpdensityjumps)overthelifeofthesolarsystem.More- focused ontoattractorsto createenhancedtidaldissipation
over, viscoelastic models generally contain two free param- (Rieutord et al. 2001; Ogilvie & Lin 2004; Papaloizou &
eters (a core shear modulus and viscosity) which must be Ivanov 2005; Ogilvie & Lin 2007; Ivanov & Papaloizou
appropriately tuned in order to yield a tidal Q compatible 2007; Goodman & Lackner 2009; Rieutord & Valdettaro
with constraints. 2010;Papaloizou&Ivanov2010;Ivanov&Papaloizou2010;
Using remarkable astrometric observations spanning Ogilvie & Lesur 2012; Auclair Desrotour et al. 2015). For
manydecades,severalrecentworks(Laineyetal.2009,2012, the discussion that follows, the important characteristics of
2015) have provided the first (albeit uncertain) measure- the mode frequencies are
mentsoftheoutwardmigrationratesofafewmoonsinthe 1.Thetidalenergydissipationrateatthemodefrequencies
Jupiter and Saturn systems. These measurements indicate is much larger than the surrounding “continuum” dissipa-
that current outward migration rates are much faster than tion rate.
predicted. In fact, for a constant tidal Q, the current mi- 2.Themodefrequenciesoccupynarrowrangesinfrequency
gration rates are incompatible with the contemporaneous space, i.e, adjacent resonances do not overlap.
formationofthemoonsandtheplanets,becausethemoons 3. The mode frequencies are determined by the internal
wouldhavealreadymigratedbeyondtheircurrentpositions structure of the tidally forced body and may evolve with
overthelifetimeofthesolarsystem.Themeasurementsalso time.
indicatethattheeffectivetidalQisdifferentforeachmoon,
whichcannotbeexplainedbyequilibriumtidalmodelsthat TheouterlayersofJupiterandSaturnarecomposedof
predict a nearly constant Q. thick convective envelopes. These envelopes may allow for
In this work, we examine a mechanism known as reso- a dense spectrum of mode frequencies at which tidal dissi-
nancelocking(Witte&Savonije1999),originallydeveloped pation is greatly enhanced by the action of inertial waves
in stellar contexts, which produces accelerated orbital mi- (Ogilvie & Lin 2004). For Saturn, recent observations of
grationviadynamicaltides.Previousplanetarystudieshave its rings (Hedman & Nicholson 2013, 2014) have provided
neglectedthefactthattheinternalstructuresofplanetsmay evidence that stable stratification exists deep within Sat-
evolve on timescales comparable to their age. Such struc- urn’s interior Fuller (2014). Stable stratification (or semi-
tural evolution causes planetary oscillation mode frequen- convective layers) has also been advocated to exist based
(cid:13)c 0000RAS,MNRAS000,000–000
Giant Planet Tides 3
on Saturn’s thermal evolution (Leconte & Chabrier 2012, 108
2013).Bothstablystratifiedlayersandsemi-convectivelay- 107 Mi En Te Di Rh
erssupportgravitymodes(gmodes,seeBelyaevetal.2015)
106
that can enhance tidal dissipation.
105
To demonstrate the importance of resonances with Q Q
mfuondcteiofnresqoufetnhceiegs,mwoedecsalacnudlatfuentdhaemferneqtaulemncoiedsesa(nfdmeoigdeens)- Tidal110034 Eq.Tide Qteivdoel
oftheSaturnmodelpresentedinFuller(2014).Wedescribe 102
this process in more detail in Appendix A. We also calcu- 101
Dy.Tide
late an approximate damping rate of each mode due to the 100
turbulent viscosity acting within the convective envelope. 3 4 5 6 7 8 9 10 11
Next, we calculate the energy dissipation rate due to the Semi-majoraxis(RS)
tidal excitation and turbulent damping of the modes. The
efficiency of tidal dissipation can be expressed through the 1013 ttide
tidal migration timescale 1012 tevol
ttide =−EE˙toirdbe = a˙ma,mtide , (1) ()yrde11001101
whichdescribesthetimescaleonwhichagiantplanetmoon tti 109
migratesoutwardduetotidalenergydissipationwithinthe 108
planet. Here, E˙tide is the orbital energy transferred to the 107
moon by tides, and E =−GM M /(2a ) is the orbital
orb p m m
3 4 5 6 7 8 9 10 11
energy,whereM andM aretheplanetaryandmoonmass,
p m Semi-majoraxis(RS)
respectively,anda isthesemi-majoraxisofthemoon.The
m
efficiencyoftidaldissipationcanalsobeexpressedinterms
of the effective tidal quality factor Q, here defined as
1010
M (cid:18)R (cid:19)5
Q≡3k2Mmp amp Ωmttide. (2) ()yr
de StableFixedPoint
Here,R isradius oftheplanet,k isits Love number,and tti
p 2 109
Ωm is the moon’s angular orbital frequency. ttide
Figure 1 shows the values of Q and ttide as a function tevol tevol
of semi-major axis due to tidal dissipation via oscillation
modes of our Saturn model. The upper envelope of t is 0.0000 0.0005 0.0010 0.0015 0.0020
tide
set primarily by damping of the non-resonant tidally ex- Semi-majoraxis(RS) +8.465
cited f modes of our model, and is essentially the effect of
the equilibrium tide. The sharp dips in ttide correspond to Figure 1. Top:EffectivetidalqualityfactorQ(equation2)for
resonanceswiththegmodesofourmodel.Figure1alsoindi- SaturntidallyinteractingwithamoonwiththemassofTethys,
cates the orbital distances of Saturn’s five innermost major asafunctionofsemi-majoraxisfortheSaturnmodeldescribedin
moons(Mimas,Enceladus,Tethys,Dione,andRhea),which thetext.Eachsharpdipcorrespondstoaresonancewithoneof
lie amongst resonances with Saturn’s g modes. Saturn’sgmodes.Theresonancesarenarrowandwell-separated
even at small semi-major axes, they merely appear to overlap
At these resonances, the outward migration timescale
because of the resolution of the plot. The smooth green curve
may be reduced by several orders of magnitude. However,
Q correspondstotheQrequiredforaconstanttidalmigration
becausethewidthsoftheresonancesarenarrow,theaverage evol
time scale tevol = T(cid:12). Middle: Corresponding tidal migration
migration time scale is still quite long. A moon placed at
time scale t (equation 1), along with the constant time scale
tide
a random semi-major axis would likely have a long tidal t .Onelocationneararesonancewheret =t hasbeen
evol tide evol
migration timescale, i.e., ttide (cid:29)T(cid:12) =4.5Gyr, with T(cid:12) the marked with a red circle. Bottom: Zoom-in on a resonance. A
age of the solar system. moon in a resonance lock will remain at the stable fixed point,
We emphasize that the existence of resonance-like fea- whichmovesoutwardonatimescale∼tevol.
tures in Figure 1 is not highly dependent on the plan-
etary model, mode damping rates, or the existence of g
modes.Calculationsthatincludetheeffectsofinertialwaves
(Ogilvie&Lin2004)producesimilarfeaturesint :sharp tation, and core dissolution which likely proceed on a time
tide
dipsatcertainfrequencies,withalongerfrequency-averaged scalecomparabletotheageoftheplanets(seeSection5.2for
tidal migration time scale. The basic picture of enhanced morediscussiononthisissue).Wethusexpectt ∼T ,to
evol (cid:12)
tidal dissipation at resonances but weak dissipation away orderofmagnitude.Themodefrequenciesevolveonasimi-
from resonances holds for many tidal models. lartimescale,andthusthelocationsoftheresonanttroughs
However, it is essential to realize that the mode fre- inFigure1willsweeppastthelocationsofthemoons.This
quencies are determined by the internal structure of the hastwoconsequences.First,moonswillpassthroughmode
planetandthereforechangeonsometimescalet .Jupiter resonances even if their initial condition placed them far
evol
and Saturn have dynamic interiors which are continuing to from resonance. Second, we show in the following section
evolve due to processes such as cooling, helium sedimen- thatmoonsmaylockintoresonancewithamodeasitsweeps
(cid:13)c 0000RAS,MNRAS000,000–000
4 Fuller et al.
past, allowing it to “surf” the resonance and migrate out- and may therefore lie in the sub-inertial range where mode
ward at a greatly reduced timescale of order ∼T . properties becomemore complex, which wediscuss more in
(cid:12)
Appendix A. For the moons considered in this work, only
Mimashasω <2Ω form(cid:62)3,andsomostresonantm(cid:62)3
f p
2.2 Resonance Locking modes do not lie in the sub-inertial regime. Typical mode
periods are P =2π/ω ∼3×104s.
The structures of planets evolve with time, as do the fre- α α
quenciesoftheiroscillationmodes.Wedefinethetimescale
on which the angular frequency ω of an oscillation mode
α
changes as 2.3 Resonance Locking with Inertial Waves
ω
ω˙ = α . (3) Resonance locking with inertial waves may be difficult to
α t
α achieve. Consider a “mode” frequency at which there is en-
In general, the time scale t is comparable to the stel- hanced tidal dissipation due to inertial waves. We assume
α
lar/planetary evolution time scale. For Jupiter and Saturn, this mode frequency (measured in Saturn’s rotating frame)
we leave t as a free parameter, and make informed esti- scales with Saturn’s rotation rate such that
α
mates of it in Section 5.2. The effect of changing mode fre-
ω =cΩ (5)
quencies can drastically alter tidal evolution timescales by α p
allowingforaprocesscalledresonancelocking,originallyex-
where c is a constant and |c|<2 (see discussion in Ogilvie
amined by Witte & Savonije 1999, 2001 (see also Fuller &
& Lin 2004). For a retrograde mode that can resonate with
Lai2012;Burkartetal.2012,2014).Duringresonancelock-
the moons, the frequency in the inertial frame is σ =(c−
α
ing, the coupled evolution of the mode frequency and the
m)Ω , where m>0 is the azimuthal number of the mode.
p
orbitalfrequencyoftheperturbingbodyproceedssuchthat
Resonance occurs when
the perturber remains near resonance with the oscillation
mode. −mΩ =(c−m)Ω , (6)
m p
Thedynamicsofresonancelockingcanbequalitatively
understood from the bottom panel of Figure 1. Consider a and a resonance lock requires
moon located at the stable fixed point, where its outward
mΩ˙ =(m−c)Ω˙ (7)
migration timescale is equal to that at which the resonant m p
locationmovesoutward.Ifitsorbitisperturbedinward(to- or equivalently
wardresonance)tidaldissipationwillincrease,andthemoon
will be pushed back outward toward the fixed point. If the Ω˙m = Ω˙p . (8)
moon’s orbit is perturbed outward (away from resonance) Ω Ω
m p
the moon’s outward migration rate will decrease, and the
resonant location will move outward and catch up with the Since Ω˙m <0 as a moon migrates outward, resonance lock-
moon. Thus, the moon can “ride the tide” and stably mi- ing requires Ω˙p < 0, i.e., it requires the planet to be spin-
grateoutwardwiththelocationofaresonance.Whilelocked ning down. Therefore, unless the value of c changes due to
in resonance, the tidalmigration timescale is drastically re- internalstructuralevolution,resonancelockingwithinertial
ducedcomparedtoitsvalueawayfromresonances,andthe waves cannot occur due to planetary contraction and spin-
moon’s orbit evolves on a time scale comparable to t . up. However, we note that c is a function of the rotation
α
Inwhatfollows,wemakesomesimplifyingassumptions. frequency(see,e.g.,Figure19ofPapaloizou&Ivanov2010)
Because the moon masses are very small (M (cid:28) M ), we and is not expected to be exactly constant. Moreover, in
m p
may safely neglect the backreaction on (i.e., the spin-down the realistic case of an evolving density profile, the value of
of) the planet. Moreover, for our purposes we can approxi- c will change because the frequencies of inertial modes de-
matethemoons’orbitsascircular(e=0)andalignedwith pend on the density profile (see, e.g., Ivanov & Papaloizou
the spin axis of the planet (i = 0). We shall account for 2007, 2010).
MMRs between moons in Section 3.1. In this paper, we use No studies of inertial waves in evolving planets have
the convention that the mode displacement in the rotating been performed. We encourage such studies to determine
frame of the planet is proportional to ei(ωαt+mφ), where m howthevalueofcwillchangeinanevolvingplanet.Ifc˙>0,
istheazimuthalnumberofthemode,suchthatmodeswith resonance locking with inertial waves may be possible and
positive frequency and positive m are retrograde modes in would provide an avenue for enhanced tidal dissipation in
the planet’s frame. generic models of giant planets.
In the outer planet moon systems, an oscillation mode
nearresonancewithamoonhasanangularfrequency(mea-
sured in the planet’s rotating frame)
3 TIDAL MIGRATION OF RESONANTLY
ω (cid:39)ω =m(Ω −Ω ) (4)
α f p m LOCKED MOONS
where Ω is the angular spin frequency of the planet, and
p While a moon is caught in a resonance lock, the resulting
ω is the forcing frequency of the moon measured in the
f tidal dynamics and outward migration rate are simple to
planet’s rotating frame. Because the planet rotates faster
calculate.Differentiatingtheresonancecriterionofequation
than the moons orbit, the resonant modes are retrograde
4 with respect to time leads to the locking criterion
in the rotating frame of the planet, but prograde in the
inertialframe.Resonantmodesalsohaveω <2Ω form=2, m(Ω˙ −Ω˙ )(cid:39)ω˙ . (9)
α p p m α
(cid:13)c 0000RAS,MNRAS000,000–000
Giant Planet Tides 5
We recall the definition of the mode evolution timescale The last equality results from the condition 15.
t = ω /ω˙ , and similarly define the planetary spin evo- The rate at which the orbital energy is increasing due
α α α
lution timescale t =Ω /Ω˙ . Equation 9 becomes to tides raised in the planet is
p p p
Ω˙ = Ωp − ωα , (10) E˙tide =E˙1,tide+E˙2,tide
m t mt
p α =Ω J˙ +Ω J˙ . (17)
1 1,tide 2 2,tide
and hence the moon’s semi-major axis evolves as
Thisisnotequaltotherateatwhichorbitalenergychanges
(cid:20) (cid:21)
a˙ 2 ω Ω
m = α − p . (11) because orbital energy may be tidally dissipated as heat
a 3 mΩ t Ω t
m m α m p withinthemoonsiftheirorbitsbecomeeccentric.Addition-
This can be rewritten ally, the relation E˙ /E =a˙ /a no longer holds.
1,tide 1 1 1
1 2(cid:20)Ω (cid:18) 1 1(cid:19) 1 (cid:21) Let us consider the limiting case in which all the tidal
= p − − . (12)
t 3 Ω t t t dissipation within the planet is caused by the inner moon.
tide m α p α Then we have E˙ =E˙ =Ω J˙ =Ω J˙. Using the
Therefore, during resonance locking, the orbital migration tide 1,tide 1 1,tide 1
relationabove,thedefinitionofQfromequation2,andthe
rate is determined by the evolutionary timescale of the
last line of equation 16, we have
planet. Note that outward migration requires 0<t <t ,
whichislessrestrictivethantheconditionforinertialαwaveps E˙1,tide =−9 k2 M1(cid:18)Rp(cid:19)5Ω = 1a˙1Ω1(cid:0)J1+J2(cid:1). (18)
discussed in Section 2.3. E 2Q M a 1 2a E
1 1 p 1 1 1
Usingequations1and2,theeffectivevalueofQduring
Since the moons’ orbits are nearly circular, then E (cid:39)
a resonance lock is 1
−Ω J /2.Moreover,iftheinnermoonremainscaughtinthe
1 1
Q = 9k2Mm(cid:18)R(cid:19)5(cid:20) ωα − Ωp (cid:21)−1. (13) resonance lock, equation 11 still holds. Substituting these
ResLock 2 M a mΩ2t Ω2t above, we find
p m α m p
Inadditiontothephysicalparametersofthesystem(mass, 9k M (cid:18)R (cid:19)5(cid:20) ω Ω (cid:21)−1(cid:20) J (cid:21)−1
radius, etc.), the primary factor controlling QResLock of a Q1,min = 22M1 ap mΩα2t − Ω2pt 1+ J2 .
p 1 1 α 1 p 1
moon are the evolution time scales tα and tp. According to (19)
the resonance locking hypothesis, Q is not a fundamental This corresponds to a minimum value of Q because it as-
1
property of the planet. Instead, a resonantly locked moon sumes no tides from the outer moon. Equation 13, in turn,
migrates outward on a timescale comparable to tα and tp, is a maximum tidal Q for the inner moon caught in MMR.
whicharefundamentalpropertiesoftheplanetinthesense The two are related by a factor (1+J /J ) which accounts
2 1
that they are determined by the planetary evolution. for the extra dissipation needed to drive the exterior moon
outward.
Additionally, there is a minimum possible Q for the
3.1 Accounting for Mean-Motion Resonances 2
outermoontoremaininMMR,becauseitwillescapefrom
As moons migrate outward, they may become caught in resonanceifitmigratesoutwardfasterthantheinnermoon.
MMRs with outer moons. If trapped into MMR with an This minimum Q is found by setting a˙ /a = a˙ /a , and
2 1 1 2 2
outermoon,aninnermoonmaystillmoveoutwardviares- letting each moon migrate outward via its own tides raised
onance locking on a time scale tα, such that both moons in the planet. In this case, we have
migrate outward on this time scale. The resonance with
M (cid:18)a (cid:19)5j−1
the outer moon effectively increases the inertia of the inner Q =Q 2 1 , (20)
2,min 1M a j
moon, such that a smaller Q is required to push it outward 1 2
onthetimescaletα.Toaccommodatetheaddedinertia,the with Q1 evaluated from equation 13. The MMR requires
innermoonmustmovedeeperintoresonancewiththeplan- a =(cid:2)(j−1)/j(cid:3)2/3a . Then
1 2
etary oscillation mode to remain resonantly locked.
ConsidertwomoonsmigratinginMMR,withtheinner Q =Q M2(cid:18)j−1(cid:19)13/3. (21)
moon denoted by subscript 1 and the outer moon denoted 2,min 1M j
1
bysubscript2.InafirstorderMMR(ignoringforsimplicity
Since the outer moon may be pushed out solely by the res-
thesplittingofresonancesduetoprecession/regression),the
onance with the inner moon, there is no maximum possible
moons’ orbital frequencies maintain the relation
value of Q to remain in MMR.
2
jΩ =(j−1)Ω . (14)
2 1
It follows that in an MMR,
4 COMPARISON WITH JUPITER AND
a˙ a˙
1 = 2 . (15) SATURN SYSTEMS
a a
1 2
Now, the total rate of change of angular momentum of the We now apply our theories to the Jovian and Saturnian
orbits of both moons is moon systems, and compare with the recent measurements
of Q from Lainey et al. (2009, 2012, 2015).
J˙=J˙ +J˙
1 2
= 21M1(cid:112)GMpa1aa˙1 + 21M2(cid:112)GMpa2aa˙2 4.1 Saturn
1 2
= 1a˙1(cid:0)J +J (cid:1). (16) Figure2showstheeffectivevaluesofQmeasuredbyLainey
2a1 1 2 etal.(2015)forthetidalinteractionbetweenSaturnandits
(cid:13)c 0000RAS,MNRAS000,000–000
6 Fuller et al.
measured values of Q using equation 2. This equation does
QResLock not take MMRs into account, which can change the actual
105 QLJ15 migration timescale. For Enceladus, the measured value of
t shouldberegardedasalowerlimit,sinceitsmigration
tide
may be slowed by outer moons. For Tethys and Dione, the
104 valueofttideisanupperlimit,sincetheymaybepushedout-
ward by inner moons. Unfortunately, all of these timescales
Q aredependentononeanother.Theyalsodependoninward
migration rates due to eccentricity damping, which in turn
103
dependonthevaluesofk andQfortidaleffectswithineach
2
moon, which are not constrained by Lainey et al. (2015).
With these complications in mind, we also plot in Fig-
102 ure2thevaluesQandt expectedfortheresonancelock-
tide
Mi En Te Di Rh ing scenario. We have set t = 50Gyr in equation 13 such
α
thatthevalueofQfromresonancelockingroughlymatches
ttide,ResLock the observed value for Rhea, which offers the best chance
102 ttide,LJ15 for comparison due to its lack of MMRs. For simplicity, we
adopt the limit of no planetary spin-up, t → ∞. The re-
p
markable result of this exercise is that resonance locking
naturally produces a very low effective Q for Rhea even for
)
r
y a realistic (and perhaps somewhat slow, t ∼ 10T ) mode
G α (cid:12)
( 101 evolution timescale.
e
d
tti ThepredictedvaluesofQandttidefromresonancelock-
ing for the inner moons (Enceladus, Tethys, and Dione)
are similar to but slightly larger than the measurements
of Lainey et al. (2015). The predicted values for Tethys
100
are most discrepant, and are incompatible with the value
Mi En Te Di Rh
t = 50Gyr that fits for Rhea. This may indicate that the
α
3 4 5 6 7 8 9 10 11 innermoonsmigrateoutwardduetoviscoelasticdissipation
Semi-majoraxis(RS) in the core as advocated in Lainey et al. (2015). However,
thefactthatt ∼10Gyrforeachofthesemoonsindicates
tide
thatdynamicaltidesand/orresonancelockingcouldstillbe
Figure 2. Top: Effective tidal quality factors Q for Saturn in-
teracting with its inner moons. The green points are the mea- occurring. Since tα is not expected to be the same for each
surementsofLaineyetal.2015.Theblueboxesarethepredicted oscillation mode, it is possible that the effective tα driving
valuesofQfortα=50Gyrandtp=∞intheresonancelocking resonance locking of the innermost moons is smaller by a
theory. For Mimas, the lower bound on the predicted QResLock factor of a few compared to the best fit value for Rhea.
is that required to maintain the resonance lock and push out We caution that the MMRs between these moons may
Tethys, assuming all the tidal dissipation within Saturn is pro- complicateboththemeasurementsandtheirinterpretation.
ducedbyaresonancelockwithMimas(equation19).Theupper
Forinstance,theeccentricityofEnceladusisexcitedbythe
bound on Q for Mimas is that required to maintain the
ResLock MMR with Dione, allowing inward migration due to tidal
resonancelock,butwithTethysmigratingoutwardduetoitsown
dissipation within Enceladus. The current outward migra-
tides (equation 13). For Tethys, the lower bound in Q is
ResLock
tion of both Enceladus and Dione may not represent an
the minimum value of Q such that it remains in mean motion
resonance with Mimas (equation 21). There is no upper bound equilibriumortime-averagedmigrationrate(seee.g.,Meyer
to Q for Tethys because it may be pushed outwards solely by &Wisdom2007),dependingonthedynamicsoftidaldissi-
Mimas. The Enceladus-Dione system behaves similarly. Rhea is pation within Enceladus. Moreover, the measured values of
notinamean-motionresonance,allowingforapreciseprediction. Q in Lainey et al. (2015) for Enceladus, Tethys, and Dione
Bottom:Correspondingoutwardmigrationtimescalettide.The werealldependentonthedissipationwithinEnceladus,and
observedvaluesofttide arecalculatedfromequation2usingthe sowebelievethesemeasurementsshouldbeinterpretedwith
measured values of Q and do not take mean-motion resonances
caution.Similarly,themigrationofTethysisaffectedbyits
intoaccount(seetext).Thedataisconsistentwitht ≈10Gyr
tide MMRwithMimas,forwhichtheoutwardmigrationwasnot
forallmoons.
constrainedbyLaineyetal.(2015),andwhichmayinfluence
the measured value of Q for the other moons.
inner moons. Although the measured value of Q is similar
forSaturn’sinteractionwithEnceladus,Tethys,andDione,
4.2 Jupiter
it is roughly one order of magnitude smaller for Rhea. The
much smaller Q for Rhea’s migration cannot be explained Figure 3 shows the predicted and measured (Lainey et al.
by any model of equilibrium tidal energy dissipation, and 2009)valuesofQandt forthemoonsofJupiter.Wehave
tide
can only be accounted for by models including dynamical again used t = 50Gyr, although the appropriate value of
α
tides.Interestingly,thebottompanelofFigure2showsthat t could be different for Jupiter. The measured t for Io
α tide
thecorrespondingoutwardmigrationtimescaleissimilarfor wasnegative,withtheinterpretationthattheinstantaneous
each moon, with t ∼10Gyr. migration of Io is inward due to eccentricity damping. This
tide
Wehavecomputedthemeasuredvaluesoft fromthe pointdoesnotappearontheplot,andlikelydoesnotrepre-
tide
(cid:13)c 0000RAS,MNRAS000,000–000
Giant Planet Tides 7
tidal dissipation in Jupiter caused by Europa to be quite
106 QResLock large, Q(cid:38)104, although we expect Europa to be migrating
QL09 outward at t ∼20Gyr due to its MMR with Io.
tide
105
5 DISCUSSION
Qtide104
5.1 Relation with Previous Tidal Theories
103 ThemeasurementsofLaineyetal.(2015)clearlyruleoutthe
commonassumptionofaconstantvalueofQgoverningthe
outward tidal migration of Saturn’s moons. The majority
102 of previous literature investigating the subject assumed a
Io Eu Ga
constant Q, and must now be interpreted with caution.
Several works (e.g., Remus et al. 2012a; Guenel et al.
ttide,ResLock
2014; Lainey et al. 2015) have sought to explain the tidal
ttide,L09(Observed) Q∼2000measuredforSaturnduetoforcingbyEnceladus,
102 ttide,L09(Calculated) Tethys and Rhea via viscoelastic dissipation within a solid
core. However, this conclusion generates problems for the
) orbitalevolutionofthemoons(seeSection5.3),forcingone
r
Gy to accept that the moons formed billions of years after the
(e rest of the solar system, or that Q was much larger (for
ttid101 no obvious reason) in the past. Additionally, these theories
cannotexplainthesmallQofSaturnduetoforcingbyRhea.
Resonancelockingcanresolvemanyofthesepuzzles.It
accounts for varying values of Q by positing a nearly con-
stantvalueoft whichgovernsthevalueofQthroughequa-
100 Io Eu Ga α
tion13.UnlikethevalueofQinequilibriumtidaltheorythat
4 6 8 10 12 14 16 can be very difficult to compute from first principles (and
Semi-majoraxis(RJ) frequentlyyieldspredictionsordersofmagnitudetoolarge),
the value of t can be calculated based on a thermal evolu-
α
tionmodelofaplanet.Thenaturalexpectationisthatt is
α
comparabletotheageofthesolarsystem,yieldingoutward
Figure 3. Top: Effective tidal quality factors Q for Jupiter migration timescales of similar magnitude, as observed.
interacting with its inner moons. The green point is the mea- Although other dynamical tidal theories (e.g., Ogilvie
surementofLaineyetal.2009.Theblueboxesarethepredicted & Lin 2004; Ogilvie & Lesur 2012; Auclair Desrotour et al.
valuesofQusingthesametα=50GyrasinSaturn(seeFigure 2015) can produce low and varying values of Q, they suffer
2). Bottom: Corresponding outward migration time scale t .
tide from two problems. First, the widths of the resonances at
Observed points are taken from Lainey et al. 2009, where the
which strong tidal dissipation occurs are somewhat narrow
measured inward migration of Io has t <0 and is not shown.
tide
(especially for small core radii), and it is not clear whether
We also plot the calculated outward migration timescale of the
Io:Europa:GanymedechainfromthemeasuredvalueofQforthe we expect to find any moons within these resonances. Sec-
Io-Jupiter interaction, assuming zero tidal dissipation produced ond, at an arbitrary orbital frequency there is no reason to
byEuropaandGanymede. expect ttide to be comparable to the age of the solar sys-
tem.Resonancelockingsolvesboththeseproblemsbecause
moons cangetcaughtin resonancelocksthatcouldlastfor
sentthelong-termmigrationrateofIo.Wehavealsoplotted billionsofyears,makingitlikelytoobserveamooninastate
avalueoft calculatedfromthemeasuredQforIo,byas- of rapid outward migration. Moreover, the value of t is
tide tide
suming it drives the migration of Europa and Ganymede in naturallyexpectedtobecomparabletotheageofthesolar
the current MMR. We caution against making a very thor- system.
oughcomparisonwiththedata,asitisunclearwhetherthe Our purpose here is not at all to dismiss tidal theories
measured value of Q for Io will be modified with updated basedondissipationofinertialwaves.Instead,weadvocate
measurements, as was the case in the Saturnian system. thatthesetheoriesnaturallyreproducetheobservationsonly
Nonetheless, we note that our predicted Q for Io falls if planetary evolution is included in the long-term behavior
very close to the measured value, especially if we take the of the system. The evolution of the planet causes the loca-
lower bound on the predicted Q corresponding to Io driv- tions of tidal resonances to migrate, allowing for resonance
ing the outward migration of Europa and Ganymede. This locks with the moons such that they migrate at a similar
correspondence can also be seen by the proximity of the rate. In this picture, the precise frequencies and strengths
predicted value of t with the calculated t which may of tidal resonances with inertial waves or g modes is not
tide tide
more accurately characterize the long-term migration than important, nor is their frequency-averaged dissipation rate.
the instantaneous observed t . This entails an outward All that matters is the mere existence of such resonances,
tide
migration timescale of t ∼ 20Gyr for Io, Europa, and andtherateatwhichtheseresonantfrequenciesevolve(see
tide
Ganymede. In this case, we expect the effective tidal Q for equation 12).
(cid:13)c 0000RAS,MNRAS000,000–000
8 Fuller et al.
5.2 Evolutionary Timescales 10
ConsantQ
In the resonance locking scenario, the outward migra- ResonanceLocking Rhea
tion timescale of moons is set by the planetary evolution
timescale tα in equation 3. Predicting this timescale is 8
not simple, as it depends on internal structural evolution
timescales, which are poorly constrained. However, we can
)
placesomeroughconstraints.First,weexpectt tobecom- RS Dione
α (
parable to or longer than the age of the solar system T . s 6
(cid:12) xi
Any process that occurs on a shorter timescale has already a
occurred, or has slowed down to timescales of ∼T(cid:12). ajor Tethys
m
planeSte’csotnhde,rmwael ecmanisseiostni,mwahteichanis ugpenpeerratleimditthrforuomgh tthhee mi- 4 Enceladus
e
release of gravitational energy. The intrinsic power radi- S Mimas
ated by Saturn is L (cid:39) 8.6 × 1023ergs−1 Guillot &
Sa
Gautier(2014),likelygeneratedviagravitationalenergyre- 2
leased through helium rain out. The corresponding Kelvin-
Helmholtz time of Saturn is
GM2
T = Sa ≈100Gyr. (22) 0
Sa R L 0 1 2 3 4 5
Sa Sa
Time(Gyr)
Therefore,weexpectthefrequenciesofoscillationmodesin
Saturntobechangingontimescales4.5Gyr(cid:46)t (cid:46)100Gyr.
α
Cooling timescales t = T /(dT /dt) ∼ 25Gyr found Figure4. Evolutionofmoonsemi-majoraxesintheSaturnsys-
cool ef ef
tem, calculated by integrating orbital evolution equations back-
by Fortney et al. (2011) and Leconte & Chabrier (2013)are
ward from the present era at t (cid:39) 4.5Gyr. Solid lines are calcu-
within this range, and are comparable with the best fit
latedwithaconstanttidalQtheory,usingmeasuredvaluesofQ
timescale t =50Gyr for Rhea in Figure 2.
α (Laineyetal.2015).WehaveusedQ=2000forMimas.Dashed
Importantly, we also expect that the mode frequencies
linescorrespondtotheresonancelockingtheory,withtα=50Gyr
increase with time (as measured in the rotating frame), as (approximatelyconsistentwithobservations,seetext).Theseor-
required for a stable resonance lock with Saturn’s moons. bitalevolutionsdonottakemean-motionresonancesintoaccount
Modefrequenciesdeterminedbyinternalstructuretypically and are only meant to illustrate the qualitative behavior of the
scalewiththeplanet’sdynamicalfrequency,whichincreases different tidal theories. The constant tidal Q theory is incom-
due to gravitational contraction. Moreover, ongoing helium patible with coeval formation of Saturn and its moons, whereas
sedimentation or core erosion that builds a stably stratified resonancelockingallowsforcoevalformation.
layer (found to be present via Saturn ring seismology, see
Fuller 2014) will cause g mode frequencies to increase (see
Appendix B). is a very strong function of semi-major axis (see Figure 1),
Whenaresonancelockisactive,theoutwardmigration such that ttide ∝ a13/2 (see equation 2). This has created
timescale is problems for conventional tidal theories, because it implies
that outward migration rates were much faster in the past,
3 Ω
ttide ≈ 2Ω −mΩ tα, (23) thereby requiring large values of Q (and correspondingly
p m
slowpresent-dayoutwardmigration)inorderforthemoons
inthelimittp →∞.ForMimas,ttide ∼1.5tα,butforRhea, to have migrated to their current positions if they formed
ttide ∼0.16tα. A resonance lock cannot persist indefinitely, coevallywithSaturn.However,themeasuredvaluesofQare
as it requires ttide → 0 as Ωm → 0. As a moon migrates muchsmallerthanthelowerlimitsdescribedbyPealeetal.
outward, the resonance lock will eventually break when the (1980),implyingsuchrapidpastmigration(inaconstantQ
required mode amplitude becomes too large. This can oc- scenario)thatthemoonscouldnothaveformedatthesame
cur due to non-linear effects, or because the stable fixed timeasSaturn.Studieswhichhaveassumedconstantvalues
point disappears when it reaches the center of the resonant of Q for Saturn in order to constrain the orbital evolution
trough in Figure 1 (i.e., the resonance saturates). For the g history(e.g.,Pealeetal.1980;Meyer&Wisdom2007,2008;
moderesonancesshowninFigure1,themodesarelinearin Zhang & Nimmo 2009) should be interpreted with caution.
the sense that fluid displacements are orders of magnitude Figure 4 demonstrates the qualitative nature of orbital
smaller than their wavelengths. Resonant saturation could evolution assuming the effective value of Q measured by
occurifthemodedampingratesareafewordersofmagni- Laineyetal.(2015)isconstantintime.Here,wehaveinte-
tude larger than those calculated in Appendix A3. We find grated equation 2 backward in time for each moon, finding
this unlikely, given the long mode lifetimes of the f modes that the orbital semi-major axis decreases to zero in less
resonatingwithSaturn’srings(Hedman&Nicholson2013). thantheageofthesolarsystem.Inotherwords,themoons
must have formed billions of years after Saturn to be com-
patiblewiththetheoryofaconstanttidalQ.Althoughthis
5.3 Orbital Evolution of the Moons
scenario has been proposed (Charnoz et al. 2011; Crida &
Resonance locking yields qualitatively different orbital evo- Charnoz 2012), we find the resonance locking solution de-
lution compared to any constant Q theory. For a constant scribed below to be simpler. A model in which the moons
Q,thetidalmigrationtimescaleforamoonofagivenmass formed well outside of the Roche radius, but still hundreds
(cid:13)c 0000RAS,MNRAS000,000–000
Giant Planet Tides 9
of millions of years after Saturn (Asphaug & Reufer 2013), (2015) (the calculation above corresponded to Q ∼ 1000),
remains possible in the resonance locking framework. or if we account for outward migration of Dione due to its
IncontrasttoconstanttidalQmodels,resonancelock- tidalinteractionwithSaturn.Inanycase,resonancelocking
ing predicts that t increases at small orbital distances can account for thermal emission as high as E˙ ≈ 16GW
tide
(see equation 23) where orbital frequencies are higher, for (Howett et al. 2011), even if Enceladus is currently in an
a constant value of t . The corresponding values of Q (see equilibrium configuration. We note that the heating rate of
α
equation2)aremuchlargerbecauseoftheQ∝a−13/2 scal- (Howett et al. 2011) is controversial, and an updated esti-
ing for a constant t . Hence, the effective values of Q for mateforheatemittedfromEnceladus’stigerstripesaloneis
tide
themoonswerelikelymuchlargerinthepastthantheyare 5 GW (Spencer et al. 2013, see also Porco et al. 2014). The
at present, resolving the incompatibility of the small mea- moon’stotalradiatedpowerremainsunclear,butwepredict
sured values of Q with the age of the solar system. This is it may be considerably greater than 5 GW. It is not neces-
demonstrated in Figure 4, where we integrate equation 23 sarytoinvoketheexistenceoftidalheatingcyclestoexplain
backwardintimeforeachmoon,usingt =50Gyr.Wecau- Enceladus’ observed heat flux, although it remains possible
α
tion that the value of t may also have been smaller in the that periodic or outbursting heating events do occur.
α
past,somewhatoffsettingtheorbitalfrequencydependence For Io, we calculate an equilibrium heating rate of
of equation 23, and a detailed orbital evolution should take E˙ ≈5×104GW from equation C5, using t =20Gyr
heat tide
both these effects into account. (see Figure 3). This is roughly half the observed heat flux
Note also that resonance locking entails the effective of 105GW (Veeder et al. 1994). The difference may stem
valuesofQmayvarybyordersofmagnitudeovertime,de- from the apparent inward migration of Io due to eccentric-
pending on whether a given moon is in a resonance lock. itytides(Laineyetal.2009),currentlycreatingmoderately
Each moon may have spent large amounts of time not in- enhancedheatingcomparedtothelong-termaverage.Alter-
volved in resonance locks and migrating outwards on long natively,theaveraget maybecloserto10Gyr,inwhich
tide
timescales, until eventually encountering a resonance and case we expect E˙ ≈105GW as observed.
heat
migratingoutwardsmorerapidly(orbeingpushedoutward Finally, resonance locking predicts that the tidal heat-
by a MMR). ingrate(equation24)isonlyaweakfunctionofsemi-major
A potential problem with resonance locking is that it axis. In contrast to a constant tidal Q scenario, we expect
does not always guarantee convergent orbital migration for thepastheatingratesofmoonslikeEnceladusandIotobe
twomoons.Foraconstantt ,equation23impliesashorter comparable(withinafactorofafew)tothecurrentheating
α
migrationtimescaleforoutermoons.Iftwomoonsareboth rates, as long as they were in their current MMRs. As in
caughtinaresonancelockwiththesamet ,theinnermoon Section 5.3, we stress that the large current heating rates
α
will not catch up to the outer moon to establish a MMR. do not require the moons to have formed after their host
Similarly, if an outer moon is pushed outward by a MMR planets.
such that it passes through a mode resonance, it could lock
into resonance with the mode and escape the MMR. This
couldexplainhowRheawasabletoescapeMMRswithother
5.5 Titan and Callisto
inner moons of Saturn; it simply migrated outward faster.
However,theobservedMMRsofinnermoonsrequireexpla- It is possible that Titan and Callisto have experienced sig-
nation.Itisnotexpectedthattα isexactlythesameforall nificant outward tidal migration despite their larger semi-
modes,anditispossiblethatsomeinnermoonshavelocked major axes compared to the inner moons. We posit that
into resonance with modes with low values of tα such that Titanand/orCallistocouldcurrentlybemigratingoutward
theymigrateoutfaster.Sincethemodedensityofourmodel via resonance locking. If so, we predict that t ∼2Gyr,
tide
is higher near inner moons (see Figure 1), the inner moons and Q∼20 for the Saturn-Titan tidal interaction. For the
may have had more chances to lock with low tα modes. A Jupiter-Callisto interaction, we predict ttide ∼ 2Gyr, and
more detailed planetary thermal evolution model would be Q∼1. Note that values of Q<1 are possible for migra-
required to investigate this possibility. tion driven by dynamical tides. A future measurement of
such a low Q driving Titan or Callisto’s migration would
bestrongevidenceforresonancelocking.Inthiscase,these
5.4 Tidal Heating
moonsmayhavemigratedoutwardbyasignificantfraction
InAppendixC,wecalculatethetidalheatingofmoonsim- of their current semi-major axis during the lifetime of the
plied by resonance locking. As in conventional tidal theo- solarsystem.ThismigrationmayhavecausedTitantopass
ries, the outward migration does not induce tidal heating through MMRs that excited its eccentricity to its current
in a moon unless the moon’s eccentricity is increased due level (C´uk et al. 2013).
to a MMR, leading to heat deposition by eccentricity tides. AlthoughresonancelockingcouldhaveoccurredforTi-
In this case, the equilibrium tidal heating rate of the inner tan or Callisto in the past, these moons may not currently
moon (assuming no tidal migration of the outer moon) is be in resonance locks. The required mode amplitudes may
notbeachievable(largeramplitudesarerequiredformoons
1 |E |
E˙ (cid:39) 2,orb . (24) oflargermassandsemi-majoraxis)duetonon-lineareffects
heat,1 j−1 t
tide orresonancesaturation.ItispossiblethatTitanorCallisto
Using t = 35Gyr for Enceladus (see Figure 2), we were previously in resonance locks, which eventually broke
tide
calculate its equilibrium heating rate to be E˙ ≈50GW. duetotheincreasinglyshortmigrationtimescales(andcor-
heat
The actual heating rate may be lower by a factor of a few respondingly large required mode amplitudes) as their or-
if we used the measured Q ∼ 2000 found by Lainey et al. bital frequencies decreased (see equation 23).
(cid:13)c 0000RAS,MNRAS000,000–000
10 Fuller et al.
6 CONCLUSIONS likely decreased by a factor of order unity over the lifetime
of the solar system (see Figure 4), allowing them to have
Wehaveproposedthataresonancelockingprocessaccounts
migrated into mean motion resonances with one another,
fortherapidoutwardmigrationofsomeoftheinnermoons
yet still to have formed coevally with Jupiter and Saturn.
ofJupiterandSaturn.Duringaresonancelock,theoutward
We cannot disprove the hypothesis that Saturn’s medium-
migration rate is greatly enhanced due to a resonance be-
sizedinnermoonsformedafterSaturn,butthecurrentrapid
tween the moon’s tidal forcing frequency and an oscillation
migration does not require such a scenario. One possible
mode of the planet. The oscillation mode could correspond
problem with resonance locking is that it does not guaran-
to a gravity mode, or a frequency of enhanced energy dis-
teeconvergentmigrationofmoons.Detailedplanetary/tidal
sipation via inertial waves. In either case, the frequency of
evolution models are needed to determine whether it can
theseoscillationmodeschangeastheplanetscoolandtheir
generallyaccountforthemoons’observedmeanmotionres-
internal structures evolve. When a mode frequency crosses
onances.
a forcing frequency, the moon can be caught in a resonance
lock and then migrates outward at a rate comparable to
the planet’s evolutionary timescale, set by equation 11. In
some respects, the dynamics of resonance locking is similar 7 ACKNOWLEDGMENTS
to locking into mean motion resonances during convergent
We thank Burkhard Militzer, Carolyn Porco, Francis
orbital migration.
Nimmo,YanqinWu,DongLai,andPeterGoldreichforuse-
Resonancelockingcanonlyexplaintheoutwardmigra- ful discussions. JF acknowledges partial support from NSF
tion of Jupiter and Saturn’s moons if mode frequencies in- under grant no. AST-1205732 and through a Lee DuBridge
creaseintherotatingframeoftheplanet,suchthattheirres- FellowshipatCaltech.JLissupportedbyTACandCIPSat
onant locations move outward (away from the planet), and UCBerkeley.EQwassupportedinpartbyaSimonsInves-
moonscan“surf”theresonancesoutward.Thiswilllikelybe tigator award from the Simons Foundation and the David
thecaseforgmodes,butthepictureislessclearforinertial and Lucile Packard Foundation.
waves(seeSection2.3).Theoccurrenceofresonancelocking
isonlyweaklydependentonexactmodedampingratesand
their gravitational coupling with moons, which we find to
be amenable to resonance locking (see Appendix B). Once REFERENCES
aresonancelockwithamoonisestablished,themoon’smi-
Asphaug E., Reufer A., 2013, Icarus, 223, 544
grationrateisdeterminedpurelybytheoutwardmigration
Auclair Desrotour P., Mathis S., Le Poncin-Lafitte C.,
oftheresonantlocation,andisnotdependentonthedetails
2015, A&A, 581, A118
oftheplanetarystructure,modefrequencies,dampingrates,
BelyaevM.A.,QuataertE.,FullerJ.,2015,MNRAS,452,
or eigenfunctions (see equation 12).
2700
Resonance locking predicts an outward migration BildstenL.,UshomirskyG.,CutlerC.,1996,ApJ,460,827
timescale of order the age of the solar system, as suggested Braviner H. J., Ogilvie G. I., 2015, MNRAS, 447, 1141
byobservationsoftheSaturnandJupitersystems(seeFig- Burkart J., Quataert E., Arras P., Weinberg N. N., 2012,
ures2and3,Laineyetal.2009,2012,2015).Inotherwords, MNRAS, 421, 983
resonancelockingtypicallypredictsloweffectivetidalqual- BurkartJ.,QuataertE.,ArrasP.,2014,MNRAS,443,2957
ity factors Q governing the migration of moons caught in Charnoz S., et al., 2011, Icarus, 216, 535
resonance locks. Resonance locking also predicts different Crida A., Charnoz S., 2012, Science, 338, 1196
effective values of Q for different moons (via equation 13), C´uk M., Dones L., Nesvorny´ D., 2013, preprint,
and naturally accounts for the very small value effective Q (arXiv:1311.6780)
measured for the migration of Rhea. In the resonance lock- Dermott S. F., 1979, Icarus, 37, 310
ing paradigm, the outward migration rate ttide is roughly Dermott S. F., Malhotra R., Murray C. D., 1988, Icarus,
constant for each moon (with some variation, see equation 76, 295
23), rather than the value of Q being constant. Dintrans B., Rieutord M., 2000, A&A, 354, 86
Similarly, resonance locking can account for the large Dintrans B., Rieutord M., Valdettaro L., 1999, Journal of
observedheatingratesofIoandEnceladus.Thesehighheat- Fluid Mechanics, 398, 271
ing rates arise from the short outward migration timescale FortneyJ.J.,IkomaM.,NettelmannN.,GuillotT.,Marley
resulting from resonance locking, resulting in a correspond- M. S., 2011, ApJ, 729, 32
ingly large equilibrium heating rate (equation 24). Cyclic Fuller J., 2014, Icarus, 242, 283
heating events need not be invoked (except perhaps mild Fuller J., Lai D., 2012, MNRAS, 420, 3126
cyclic variation for Io) to account for the current heating, Fuller J., Lai D., 2014, MNRAS, 444, 3488
althoughofcourseitremainspossiblethatheatingcyclesdo Goldreich P., Nicholson P. D., 1977, Icarus, 30, 301
occur. Goldreich P., Soter S., 1966, Icarus, 5, 375
Finally,resonancelockingresolvestheproblemsarising Goodman J., Lackner C., 2009, ApJ, 696, 2054
fromcurrent-daymigration/heatingrates,whichimplythat Guenel M., Mathis S., Remus F., 2014, A&A, 566, L9
somemoonsformedlongaftertheirplanetsifthetidalQis GuillotT.,GautierD.,2014,preprint, (arXiv:1405.3752)
constant. Instead, resonance locking predicts that the out- Hedman M. M., Nicholson P. D., 2013, AJ, 146, 12
ward migration timescale t is nearly constant, such that HedmanM.M.,NicholsonP.D.,2014,MNRAS,444,1369
tide
theeffectivevaluesofQwerelargerinthepast.Theconse- Howett C. J. A., Spencer J. R., Pearl J., Segura M., 2011,
quence is that the orbital frequencies of inner moons have Journal of Geophysical Research (Planets), 116, 3003
(cid:13)c 0000RAS,MNRAS000,000–000