Draft version August 6, 2018Preprint typeset using LATEX style emulateapj v. 5/2/11

RESONANT REMOVAL OF EXOMOONS DURING PLANETARY MIGRATION

Christopher Spalding1, Konstantin Batygin1, and Fred C. Adams2,3

1Division of Geological and Planetary SciencesCalifornia Institute of Technology, Pasadena, CA 91125

2Physics Department, University of Michigan, Ann Arbor, MI 48109 and3Astronomy Department, University of Michigan, Ann Arbor, MI 48109

Draft version August 6, 2018

ABSTRACT

Jupiter and Saturn play host to an impressive array of satellites, making it reasonable to suspectthat similar systems of moons might exist around giant extrasolar planets. Furthermore, a significantpopulation of such planets is known to reside at distances of several Astronomical Units (AU), leadingto speculation that some moons thereof might support liquid water on their surfaces. However, giantplanets are thought to undergo inward migration within their natal protoplanetary disks, suggestingthat gas giants currently occupying their host star’s habitable zone formed further out. Here we showthat when a moon-hosting planet undergoes inward migration, dynamical interactions may naturallydestroy the moon through capture into a so-called “evection resonance.” Within this resonance, thelunar orbit’s eccentricity grows until the moon eventually collides with the planet. Our work suggeststhat moons orbiting within about ∼ 10 planetary radii are susceptible to this mechanism, with theexact number dependent upon the planetary mass, oblateness and physical size. Whether moonssurvive or not is critically related to where the planet began its inward migration as well as thecharacter of inter-lunar perturbations. For example, a Jupiter-like planet currently residing at 1 AUcould lose moons if it formed beyond ∼ 5 AU. Cumulatively, we suggest that an observational censusof exomoons could potentially inform us on the extent of inward planetary migration, for which noreliable observational proxy currently exists.

1. INTRODUCTION

The past two decades have brought thousands of ex-trasolar planetary candidates to light. These systemshave repeatedly challenged the notion that our SolarSystem is somehow typical (Winn & Fabrycky 2015).Notable examples include the existence of hot Jupiters(Mayor & Queloz 1995), spin-orbit misalignments (Winnet al. 2010), and the prevalence of highly compact, multi-planet systems (Lissauer et al. 2011; Rowe et al. 2015).However, as of yet, we have not been able to place themany known solar-system moons into their appropriateGalactic context. Observational surveys are now under-way with this specific goal (Kipping 2009; Kipping et al.2009, 2012, 2015). Motivated by the potential for up-coming exo-lunar detections, this work explores how thepresent-day configurations of exomoons might have beensculpted by dynamical interactions playing out duringthe epoch of planet formation.

Not long after the first detections of giant extraso-lar planets (Mayor & Queloz 1995), speculations aroseregarding what types of moons these bodies may host(Williams et al. 1997). Much of the interest has beenastrobiological in nature - if giant planets reside in thehabitable zones of their host stars, perhaps the moonsthereof capable of sustaining liquid water on their sur-faces (Heller et al. 2014). In contrast, any putative liquidwater within the moons of Jupiter and Saturn must bemaintained by way of tidal heating. Within the cur-rent observational dataset, however, our Solar System’sgiant planet configuration is by no means universal. Asignificant population of giant planets is found to residebetween ∼ 1−5 AU (Dawson & Murray-Clay 2013), just

inside the orbit of Jupiter1 (5.2 AU).Moons are expected to arise as an intrinsic outcome of

giant-planet formation (Canup & Ward 2002; Mosqueiraet al. 2010). In particular, the core accretion model dic-tates that cores comprising multiple Earth masses of ma-terial, form outside their natal disks’ ice line, before ini-tiating a period of runaway gas accretion (Pollack et al.1996; Lambrechts & Johansen 2012). Restricting atten-tion to planets with masses greater than Saturn, theirgravitational influence upon the protoplanetary disk willeventually clear a “gap” in the gas within their vicin-ity (Crida et al. 2006). Material is capable of flowingthrough the gap and entering the planet’s Hill sphere (rH;the region around the planet where its potential domi-nates the motion of test particles). The residual angularmomentum of the material is then distributed into a cir-cumplanetary disk, extending out to ∼ 0.4 rH (Martin &Lubow 2011), where moons are thought to form.

The angular momentum exchange associated with gap-clearing, in concert with viscous accretion within the pro-toplanetary disk, is expected to drive Type II migrationof young planets, taking them to shorter-period orbits.Traditional theoretical treatments have suggested thatthe Type II migration rate is similar to the accretionalvelocity of disk gas (Armitage 2010; Kley & Nelson 2012),though reality is likely more complicated (Duffell et al.2014). Regardless, it is widely suspected that migrationrates can be sufficient to reduce planetary semi-majoraxes by well over an order of magnitude within a typical

1 The apparent scarcity of planets at Jupiter’s distance is subjectto observational biases, not least owing to the associated long or-bital periods. Recent searches are uncovering more distant bodies(e.g., Knutson et al. 2014).

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disk lifetime (1-10 Myr; Haisch et al. 2001, see below).Consequently, the ‘Jupiters’ currently residing at severalAU probably formed at more distant radii. Crucially,however, there currently exists no reliable, observationalproxy that constrains the extent of migration.

In this paper, we demonstrate that if the migratingplanet hosts a moon, inward migration can lead to themoon’s destruction by way of the “evection resonance”(Yoder & Kaula 1976; Touma & Wisdom 1998; Cuk &Stewart 2012). To illustrate the problem, consider theapsidal precession of a lunar orbit around an oblate giantplanet. At large heliocentric distances, this precession ismore rapid than the planetary mean motion about thecentral star. As the planet migrates inwards, its orbitalfrequency increases before becoming approximately com-mensurate with the lunar precession frequency. Assum-ing resonant capture (see section [35]), further migrationwill pump the moon’s eccentricity upwards until its peri-center approaches the planet’s surface and the moon islost.

Our treatment here remains largely outside of therealm of hot Jupiters (giant planets with orbital periodsof several days), whose reduced Hill spheres permit satel-lites only within a few planetary radii (Domingos et al.2006; Kipping 2009). Planetary migration may thereforeremove moons around these objects without the resonantmechanism proposed here. Additionally, tidal planet-moon interactions further reduce the stability region oflunar orbits by expelling (or destroying) larger moonsover Gyr timescales (Barnes & O’Brien 2002). These is-sues make it difficult to relate current exolunar architec-tures of closer-in planets to their formation conditions.Consequently, we restrict our attention to bodies outsideof ∼ 0.5 AU from their stars.

The mechanism described herein requires both that theplanet-moon system both begins outside of resonanceand that migration proceeds until the moon is lost bycollision. These conditions are quantified in section (3).The dynamics are critically dependent on the planetaryradius planet and second gravitational moment J2 (thesedetermine the lunar precession frequency). Accordingly,we must begin with a brief discussion of reasonable pa-rameters associated with young giant planets.

1.1. Properties of young giant planets

Early models of giant planets naturally focussed onolder planets, such as Jupiter and Saturn. The advan-tage here was that interior models lost their sensitivityto initial conditions over the relatively short (∼ 20 Myr)Kelvin-Helmholtz timescale (Stevenson 1982; Marley etal. 2007). However, during the epoch of disk-driven plan-etary migration, the initial condition is crucial. Modelsextracting initial conditions from core accretion theoryinfer much smaller planetary radii Rp than so-called ”hotstart” models, such as gravitational instability (Marleyet al. 2007). For illustration, we focus on planets aris-ing from core accretion, where radii sit close to 1.2− 1.4times Jupiter’s current radius RJ, but all further argu-ments could easily be applied to larger, hot-start planets.For the sake of definiteness, we choose Rp = 1.4RJ forthe moon-hosting planet throughout this work.

Eccentricity growth will remove moons either throughphysical collision with the planet or through tidal dis-

ruption, whichever happens earlier. Tidal disruption willoccur close to the Roche radius (e.g., Canup 2010) which,expressed in terms of satellite mass ms, satellite radiusRs and planetary mass Mp may be written

RL

Rp≈ 2.5

(Mp

ms

) 13 Rs

Rp. (1)

For parameters typical of Io-like bodies around youngJupiters (Rp = 1.4RJ), RL/Rp < 1 and so moons areonly lost by way of direct collision with the planet.Therefore, we consider a moon as lost when its pericen-ter approaches Rp with the caveat that the Roche radiiof more massive, compact planets may indeed lie outsidethe planetary surface.

In addition to the planetary radius, an approximationfor J2 is required. For purely rotational deformation, therelationship between J2, the Love number k2 (twice theapsidal motion constant) and the planetary spin rate Ωmay be expressed as (Sterne 1939):

J2 =1

3

(Ω

Ωb

)2

k2, (2)

where Ω2b ≡ GMp/R

3p is the break-up spin rate. Unfor-

tunately, the above expression merely expresses one un-known quantity J2 as a function of two other unknownquantities. However, k2 can be estimated by modellingthe planet as a polytrope with index χ = 3/2, (Chan-drasekhar 1957; Batygin & Adams 2013) yielding a Lovenumber k2 ≈ 0.28.

It is more difficult to speculate upon Ω/Ωb. The younggiant planet in β Pictoris b has had its spin period esti-mated at ∼ 8 hours (Snellen et al. 2014), close to whatone would expect by extrapolating the equatorial veloc-ities of the Solar-System’s planets to the mass of β Pic-toris b (about 8MJ). This result tentatively suggests thatspin rates of young giant planets are little altered be-tween 10 Myr and 4.5 Gyr after their formation, but thespin rate within the first 1 Myr remains purely specula-tive. For the sake of definiteness, we take J2 = 0.02 asa nominal value for young giant planets, slightly largerthan Jupiter’s current J2 ≈ 0.015 (Murray & Dermott1999). We note, however, that J2 may reasonable liewithin the range 0.01 > J2 > 0.1, with the upper bounddeduced from equation (2), and so further research isrequired to better constrain this quantity.

2. EVECTION RESONANCE

In this section, we quantitatively describe the dynami-cal influences upon a lunar orbit hosted by a young, giantplanet. Consider the moon’s orbit to have eccentricity e,inclination i and semi-major axis am. The effect of plan-etary oblateness, J2 is to force a precession of the lon-gitude of pericenter $ with frequency (for a derivation,see e.g. Danby 1992)

$ = νJ2≈ 3

2J2

(Rp

am

)2

nm1

(1− e2)2

(2− 5

2sin2(i)

),

(3)

where nm is the mean motion of the lunar orbit andRp is the planetary radius. Note that the magnitudeof νJ2

increases monotonically with eccentricity, but itssign changes at a critical inclination of (icrit ≈ 63.4).

3

For simplicity, in all further analyses we will assumethat the lunar orbit is coplanar with the planet’s equator(that is, we set i = 0) and, furthermore, that the planetitself has zero obliquity. These assumptions are moti-vated by the expectation that young giant planets in-herit sufficient angular momentum from their natal disksto align both their spin axes and circumplanetary diskswith their heliocentric orbits. It should be noted how-ever that spin-orbit resonances have been proposed asan explanation for Saturn’s obliquity (Ward & Hamilton2004) and so similar dynamical processes may generateobliquities in moon-hosting planets. For the purposesof this work, we simply note that mild non-coplanarityslows the lunar precession rate which, as discussed be-low, leads to a more distant encounter with the evectionresonance.

Provided the planet forms sufficiently far out, the pre-cession frequency of the exomoon orbit will exceed theplanetary mean motion np. During inward migration,np increases until, at some point, the two frequencies νJ2

and np are approximately equal (Figure 1), known as theevection resonance. This condition may be written in theform

3

2J2

(Rp

am

)2(GMp

a3m

)1/21

(1− e2)2=

(GM?

a3p

)1/2

, (4)

where Mp is the mass of the planet, ap is the planetaryorbital semi-major axis and M? is the mass of the cen-tral star. Therefore, supposing the moon to originate atlow eccentricity (e ≈ 0), resonance-crossing occurs at theheliocentric distance,

ares = Rp

[(2

3J2

)2(amRp

)7(M?

Mp

)]1/3. (5)

If the moon is caught into resonance, subsequent plane-tary migration drives the moon’s orbital eccentricity toever higher values. The physical source of eccentricitymodulation is the torque supplied by the central star(Touma & Wisdom 1998; Cuk & Stewart 2012).

In order to demonstrate the relevance of resonant cap-ture under typical parameters, consider the planetary pe-riod Tp corresponding the resonant condition above,

Tp

∣∣∣∣res

≈ 2700 days

(J2

0.015

)−1(am/Rp

aIo/RJ

) 72

. (6)

We have scaled the parameters appropriately for the cur-rent Jupiter-Io configuration. Jupiter’s orbital periodis 4,332 days, meaning that, were Jupiter to be slowlyforced in toward the Sun (and we ignore the influence ofthe other Jovian satellites), Io would encounter the evec-tion resonance at roughly 3.8 AU. Abundant extrasolargiant planets have thus far been detected with similar he-liocentric distances (∼ 1−5 AU; Dawson & Murray-Clay2013), suggesting that the conditions for evection reso-nance might frequently be encountered in young giantplanet-moon systems.

2.1. The Evection Hamiltonian

Criterion (4), describing an encounter with resonancetakes a simple form, however, there is in general no guar-antee that the moon will become captured into the reso-

0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

inward migration

resonant encounter

eccentricity growth

moon loss through collision

circular lunar orbit

0 0.5 1.0 1.5 2.00

0.2

0.4

0.6

0.8

1.0

ecce

ntric

ity 5

mT2

& 120

J2

0.01

2a1

Rp

4

. (29)

Accordingly, the adiabatic criterion is more likely tobe violated for moons that are more distant from theirhost planet, in units of planetary radii.

4. EVOLUTION WITHIN RESONANCE

If we now suppose that system satisfies all capture cri-teria, the lunar orbit’s eccentricity shall continue to growas the planet migrates further inwards. Eventually, themoon’s perigee rp = a1(1 e) will coincide with theRoche radius of its host planet. It has been demonstratedby ? that a moon reaching the Roche limit on a circularorbit is likely to have its outer layers tidally stripped,supposed to lead to the ice-rich nature of Saturn’s rings.It is not clear whether such results for circular lunar or-bits may be naively applied to eccentric moons withinthe evection resonance. In other words, we can’t say forsure that a potentially habitable moon will be destroyedand/or rendered uninhabitable once its perigee coincideswith the Roche Limit. Accordingly, in our work we dis-cuss the likelihood both of a moon reaching the RocheLimit in addition to the more extreme case of the moonphysically colliding with its host planet.

As mentioned above, when the system crosses = 1from below (by way of inward migration), the single equi-librium of e = 0 becomes unstable whilst two stable equi-libria appear at non-zero eccentricities. We may supposethat dissipative within the disk reduce the phase-spacearea of the lunar trajectory such that it remains approx-imately at a stable equilibrium throughout its evolution.Accordingly, we may determine the evolution of the lu-nar orbit’s eccentricity as its host planet migrates by solv-ing for the stable fixed points of the governing Hamilto-nian. In the previous section, we worked in the small-eccentricity limit, wherein an analytical expression maybe obtained for the fixed points. Whilst such a limit suf-fices for investigating adiabatic capture, we must workwith unrestricted eccentricities in order to accurately de-scribe evolution of the moon within resonance. Thereexists no analytic expression for the equilibrium fixedpoints under unrestricted eccentricities and so we solvethe appropriate equations numerically.

4.1. Equilibrium fixed points

Our first step is to adopt the Cartesian coordinatesx ,0 , y, as defined above (equation 17), and write thefull Hamiltonian 8 in terms of these variables. The rele-vant Hamiltonian takes the form

H =1

16

15 n2

2

2n1(x y)(x + y)(x2 + y2 4)

+ 64J2Rp

a1

2

n1(x2 + y2 2)3 + 8 n2(x

2 + y2)

.

(30)

Upon taking the derivative with respect to y and set-ting x = 0, we arrive at an 11th order polynomial inthe equilibrium value of y = yeq. From this, the equilib-rium eccentricity eeq may be found through Equation 17,setting = /2,

eeq = (1 (1 1

2y2eq))

12 . (31)

We present eeq for as a function of heliocentric distancenormalised by

a a1

M?a

41

MpR4p

13

, (32)

which reflects the importance of the ratio of lunar pre-cession frequency to planetary mean motion. The exo-moon’s trajectory is not perfectly specified by a0, withminor di↵erences arises from di↵erences in the relativelysmall quantity r = J2(Rp/a1)

2 1. However, for allreasonable choices of r, the trajectories all lie very closeto each other in eeq , a0 space.

As an illustration, in Figure 1, we present the evolu-tion of eeq as a function of a0 appropriate for a moon witha similar semi-major axis to Io orbiting a Jupiter-massplanet with twice Jupiter’s radius. We have chosen, fordefiniteness, J2 = 0.015, which is Jupiter’s current value.We stress again, however, that the scaled nature of thea0-axis allows the trajectory to be accurately applied toall other reasonable parameters, for example, if one wereto seek an equivalent trajectory for a Europa-like plane-tocentric orbit. The most important aspect of Figure 1 isthat inward migration from outside of resonance, a0 & 1causes the lunar orbit to move along the a0-axis, with loweccentricity, until it encounters the evection resonance,at which point, its eccentricity begins to grow monoton-ically as migration proceeds.

a2J2/32 (AU)

a1 (RJ)

a (33)

The eccentricity cannot grow indefinitely because atsome point the moon impacts the planet’s surface, or getstidally disrupted. The eccentricity at which moon lossoccurs depends directly upon the planetocentric lunardistance. The moon impacts the planetary surface whenthe lunar perigee equals the planetary radius, which oc-curs at an eccentricity

ec = 1 Rp

a1. (34)

Accordingly, for each known giant planet, we can providea region where moons ought to have been removed, givena migration event. If moons are found here, it places con-straints on how far giant planets migrate to their severalAU present positions. If we only find moons that are veryclose to their planets, but still well with the Hill Sphere,it suggests that such planets have migrated a long waysuch as to cross the evection resonance for distant moonsFigure.

5. DISSIPATIVE EFFECTS

The above calculation describes the dynamics withina picture where the lunar orbit is only a↵ected by con-servative gravitational forces. However, there exist two

semi-major axis ( )

central star

Fig. 1.— Dimensionless illustration of resonant capture and ec-centricity growth. The red line denotes the lunar eccentricity cor-responding to the stable equilibrium of the Hamiltonian 10. Thethicker grey line denotes the analytical expression (43) describingexact resonance. The two solutions are almost indistinguishable.

nance. Furthermore, assuming capture occurs, the subse-quent evolution of eccentricity is non-trivial to compute.In order to tackle these aspects, we adopt a Hamiltonianapproach, describing the lunar dynamics in terms of thecombined gravitational potential of the central star andthe planetary quadrupole (J2). This section focuses onthe dynamics of capture into resonance. The reader mayskip to section 3 for a discussion of the dynamical loss ofmoons assuming capture occurs.

The Hamiltonian describing the dynamics of a moonin orbit around an oblate planet has been derived else-where (e.g. Touma & Wisdom 1994, 1998). Despite theirintuitive convenience, Keplerian orbital elements do notcomprise a canonical set of coordinates. Accordingly, inorder to utilize a symplectic form, we work in terms ofreduced Poincare (or, modified Delauney; Murray & Der-mott 1999; Morbidelli 2002) variables defined as follows:

Λ ≡ m√GMp a λ ≡M +$ + Ω

Γ ≡ Λ(1−√

1− e2) γ ≡ −$ − Ω, (7)

where Ω is the longitude of ascending node (not usedowing to the assumption of coplanarity), M is meananomaly and subscripts ‘m’ and ‘p’ are used below to re-fer to the moon and the planet respectively. Physically,Λm corresponds to the angular momentum the moonwould possess on a circular orbit of semi-major axis am,and Γm describes the angular momentum difference be-tween the moon’s true orbit and a circular orbit sharingits semi-major axis. We assume that the lunar orbitalfrequency is large enough to utilise a secular approach,whereby the Hamiltonian is “averaged” over a lunar or-bit. Consequently, explicit dependence upon M is re-moved, extracting Λm as an integral of the motion. Interms of the variables (7), the Hamiltonian takes the form(Touma & Wisdom 1998)

H =− 1

2nm J2

(Rp

am

)2

Λm

(Λm − Γm

Λm

)−3

− 15

8

n2pnm

ΛmΓm

Λm

(2− Γm

Λm

)cos[2(npt+ γm)

], (8)

4

where the planet-moon system orbits the host star withmean motion np, such that λp = npt.

The dynamics are best analyzed in a frame co-orbitingwith the planet. Accordingly, we perform a canonicaltransformation of the above Hamiltonian using the newangle

γ ≡ npt+ γ (9)

to obtain the autonomous Hamiltonian

H =npΓm −1

2nm J2

(Rp

am

)2

Λm

(Λm − Γm

Λm

)−3

− n2pnm

Λm15

8

Γm

Λm

(2− Γm

Λm

)cos(2γm). (10)

The first term arises as a result of transformation (9), thesecond term describes the influence of planetary oblate-ness upon the lunar orbit, and gives rise to the precessionfrequency 3. The third term is new and describes the sec-ular perturbation upon the moon’s orbit raising from thestar. Note that the Gaussian averaging process is iner-tially equivalent to considering the orbit of the moon toact as an eccentric, massive wire. Thus, the third termarises from the torques communicated between the stel-lar gravitational potential and an eccentric wire.

It is appropriate to scale the action Γm by the integralof motion Λm, thus defining a new canonical momentum

Γm ≡Γm

Λm. (11)

In order to preserve symplectic structure, we likewisescale the Hamiltonian itself by Λm, yielding

H =npΓm −1

2nm J2

(Rp

am

)2 (1− Γm

)−3

− n2pnm

15

8Γm

(2− Γm

)cos(2γm), (12)

such that the system evolves according to Hamilton’sequations in the form

dγmdt

=∂H∂ Γm

d Γm

dt= − ∂H

∂γm. (13)

As mentioned earlier, we consider inward planetary mi-gration (increasing np), but do not explicitly consider thecase where the moon itself is migrating within a circum-planetary disk (Canup & Ward 2002). Qualitatively, theeffect of inwards moon-migration would be to postponethe crossing of an evection resonance by increasing theinfluence of the planetary quadrupole. Additionally, weassume that any variations in the radius of the planet andits J2 during the nebular epoch are negligible comparedto the influence of variations in np.

2.2. Capture into resonance

In this section, we outline the conditions under whichmoons are expected to become captured into resonance.The moon’s orbital eccentricity is likely to be small dur-ing resonance passage and so we analyze the dynamics

of capture using the small-e (and, consequently, small-

Γm) approximation to Hamiltonian (12) (e.g., Touma &Wisdom 1998):

H ≈[np −

3

2nmJ2

(Rp

am

)2]Γm − 3nmJ2

(Rp

am

)2

Γ2m

− 15

4np

(npnm

)Γm cos(2γm). (14)

Borderies & Goldreich (1984) computed the probablyfor resonant capture of a system governed by the inte-grable single-parameter Hamiltonian

H′ = −(1 + 2δ)Φ + 2Φ2 − Φ cos(2φ), (15)

and so we make progress by casting Hamiltonian (14)into a similar form.

First, we scale both the Hamiltonian and the actionsΓm by a factor η such that

Φ =Γm

ηη =

5

2

1

J2

(amRp

)2(npnm

)2

. (16)

This choice of η ensures a common factor,

ν′ =15

4np

(npnm

)(17)

between the coefficients of 2Φ2 and Φ cos(φ). Dividingthe Hamiltonian by this factor reproduces the form (15),with the caveat that time must now be measured in unitsof ν′−1. That is, we have introduced a “slow” canonicaltime

τ =15

4np

(npnm

)t. (18)

By inspection, we see that the “resonance proximity pa-rameter”

δ = −1

2+

2

15

nmnp− 1

5J2

(Rp

am

)2(nmnp

)2

, (19)

which is highly negative for planetary orbits far outsideof resonance (large am), but increases upon inward mi-gration.

For dynamics governed by Hamiltonians of theform (15), capture in the adiabatic limit is certain forΦ < 1/2 (Borderies & Goldreich 1984). In our case, thiscondition corresponds to a lunar eccentricity ecap, abovewhich, adiabatic capture is not guaranteed. Within thesmall e approximation, Γm ≈ e2/2 and so we find thecritical eccentricity, below which resonant locking is cer-tain, to be

ecap =

√5

2J2

(amRp

)(npnm

). (20)

At resonance-crossing, np ≈ (3/2)J2(Rp/am)nm and sothe criterion above yields the condition

e . 3

2

√5

2J

122

(Rp

am

)

= 0.03

(J2

0.02

) 12(Rp/am1/10

), (21)

5

where we have chosen Rp/am = 1/10 as a reference valuebecause, as discussed later, more distant orbits are typi-cally only lost outside of the adiabatic regime.

Note that e = 0.03 is significantly larger than the ec-centricities of the Galilean Satellites, but approaches thatof Titan (e = 0.028; Iess et al. 2012). Owing to their po-sition within a circumplanetary disk, we expect that anyyoung moons will possess eccentricities at least as smallas the Galilean Satellites and ought therefore to be cap-tured in the adiabatic regime. However, the presence ofmoon-moon resonances or other sources of eccentricity-pumping may quench the evection resonance in specificcases.

2.2.1. The adiabatic criterion

It is well known in celestial mechanics that passingthrough resonances sufficiently rapidly can prevent cap-ture (Quillen 2006) by way of leaving the ‘adiabaticregime.’ Adiabatic motion occurs when the librationtimescale of the moon within resonance is shorter thanthe timescale of resonance crossing. When satisfied, adi-abatic motion allows the lunar orbit to grow in eccen-tricity, and therefore precession frequency, keeping pacewith the rising planetary mean motion.

The adiabaticity criterion is best derived by changingto the canonical Cartesian coordinates,

x =√

2Φ cos(φ) ∝ e cos(φ)

y =√

2Φ sin(φ) ∝ e sin(φ), (22)

where the proportionalities are valid in the small-e limit.Performing the transformation, Hamiltonian (15) takesthe form

H = (1 + 2δ)

(x2 + y2

2

)− 2

(x2 + y2

2

)2

−(x2 − y2

2

).

(23)

In Figure 2, we plot contours of Hamiltonian (23) fora range of values of δ, where it can be seen that thenumber of equilibria increases from one to three to fiveupon increasing from δ = −1.5 to δ = 0.5. We maycompute when resonance is encountered by quantifyingthe fixed points of Hamiltonian (23). On the y-axis (φ =π/2, 3π/2), fixed points occur at

y = 0, y = ±√

1 + δ. (24)

and so the equilibria away from y = 0 exists for δ > −1.As δ continues to grow, equilibria appear on the x-axisat

x = 0, x = ±√δ, (25)

when δ > 0 (see Figure 2). Accordingly, as the planetmigrates inwards, resonance is encountered at δ = −1and an inner, circulation region develops at δ = 0.2 Inother words, the ‘width’ of the resonance is equivalent to∆δ = 1, corresponding to the amount of migration theplanet must undergo to take its moon from outside toinside of resonance.

A non-adiabatic crossing of resonance corresponds tothe transitioning from outer to inner circulation in less

2 The exact resonant position (5) corresponds to δ = −1/2.

than one oscillation period. This crossing time is givenby

dδ

dt≈ ∆δ

∆t=

1

∆t. (26)

We now suppose the planetary migration proceeds on thecharacteristic timescale τm, such that

1

ap

dapdt

= − 1

τm. (27)

With this prescription for ap, we may take the timederivative of δ (Equation (19)) yielding the resonancecrossing time,

∆t =5a2mn

2p

nm(a2mnp − 3R2pJ2nm)

τm. (28)

For definiteness, we evaluate ∆t when δ = −1/2, whichis equivalent to condition given in equation (2). Adoptingthis midway point, the resonance crossing time is givenby

∆t = 5npnm

τm. (29)

All that remains is to estimate the libration timescale.Let us analyze the local neighborhood of H around theresonant fixed point (xeq = 0, yeq =

√1 + δ). Recalling

that, at x = 0, φ = π/2 and Φ =√

2 y, we define thevariables

Φ = Φ− y2eq2

= Φ− 1 + δ

2

φ = φ− π

2(30)

which measure the distance away from the equilibriumfixed point. We now expand the Hamiltonian (23) as aTaylor series to second order in Φ and φ, setting δ =−1/2. After one final scaling of the variables

Φ =1√2

Φ φ =1√2φ, (31)

we arrive at the local Hamiltonian

H = −1

2ω(Φ2 + φ2), (32)

where the corresponding “harmonic oscillator” frequencyaround this fixed point is ω = 2. We now convert backinto real time units, and obtain the libration period

P lib = 2π(ωτ)−1 =4π

15

(nmnp

)1

np. (33)

Equating this quantity to the resonance crossing time,we arrive at the adiabatic criterion, expressed in termsof the planetary migration timescale:

P lib

∆t=

2

75

(nmnp

)3(2π

nm

)1

τm. 1 (34)

We may immediately substitute in the resonance crite-rion (4) to determine the requirement for adiabatic mi-gration in terms of lunar semi-major axis. The planetary

6

migration timescale is likely to scale with planetary Ke-plerian orbital period Tp (Tanaka et al. 2002) and so itmakes sense to likewise scale the adiabatic criterion:

τmTp

& 30

(J2

0.02

)−2(amRp

)4

. (35)

This dependence comes about because moons at largeram/Rp are resonant at greater ap/am, such that the typ-ical libration timescales are reduced and the adiabaticcriterion is easier to break. (As found above, more dis-tant moons are also more likely to break the requiremente < ecap.)

2.3. The Adiabaticity of Planetary Migration

In the above derivation, we supposed that the semi-major axis of the planet decays over a characteristictimescale τm. The exact value of τm, i.e., the rate of TypeII migration, is still an active area of research (Kley &Nelson 2012). In this work, we adopt the reasonable, firstorder approximation that once giant planets open a gapin the protoplanetary disk, they migrate inwards withthe accretionary flow (but see Duffell et al. 2014). Utiliz-ing the Shakura-Sunyaev form for disk effective viscosity(Shakura & Sunyaev 1973), the accretionary velocity isgiven by (Armitage 2011),

vacc ≈ −3

2α

(h

ap

)2

ΩKap (36)

where h is the scale height of the disk, α is the dimen-sionless turbulence parameter and ΩK is the Keplerianvelocity at radius ap in the disk. From this equation, wederive the form for the time evolution of the planetarysemi-major axis,

1

ap

dapdt≈ −3

2α

(h

ap

)2

ΩK , (37)

which we may estimate by supposing the disk aspect ratioh/ap ∼ 10−1. The value of α (and even the validity of itsusage) is widely debated, and probably varies throughoutthe disk, depending upon which mechanisms dominateturbulent motions (Hartmann et al. 1998; King et al.2007; Armitage 2011). That said, the inferred value isusually within the range 10−4 < α < 10−2. Substitutingthese parameter values in for the migration timescale, weobtain reasonable bounds on the adiabaticity parameter

104 . τm/Tp . 106. (38)

Using the above criteria, we may now estimate the mostdistant exolunar orbit that is guaranteed to be captured.From condition (35),

(amRp

)4

=1

30

τmTp

(J2

0.02

)2

, (39)

we obtain the requirement for adiabatic capture that

amRp

. 13

(J2

0.02

) 12(τm/Tp

106

) 14

. (40)

To put this number into perspective, 13 planetary radii ofRp = 1.4RJ sits outside of the current orbit of Ganymede(for which, am/Rp ≈ 11). Stable lunar orbits may exist

out to roughly 1/3 to 1/2 of the Hill Radius (Nesvornyet al. 2003), meaning that a Jupiter-mass planet, resid-ing beyond about 0.5 AU from its host star, may possessmoons too far out for adiabatic capture. Capture canstill occur outside of the adiabatic limit, but the prob-ability drops rapidly. Consequently, in the rest of thepaper, we focus on moons situated at am/Rp . 10, butmaintain the caveat that specific cases may exist wherecapture occurs outside the regime of guaranteed capture.

3. EVOLUTION WITHIN RESONANCE

In this section, we calculate the evolution of the moon’seccentricity within resonance, assuming the planet-moonsystem satisfies the capture criteria given by equa-tions (20) and (40). Furthermore, we derive the con-ditions under which the lunar pericenter am(1 − e) co-incides either with the Roche radius of its host planet,or the planetary radius itself. The Roche radii of young,Jupiter-mass planets are likely to reside inside the plan-etary radius (see section [1.1] above) and so we considera planet-crossing orbital trajectory as the criterion formoon loss, which occurs at an eccentricity

ecoll = 1− Rp

am. (41)

We only consider the case whereby moons are lost atthe planetary radius, but mention that higher-mass, com-pact gas giants might lose moons through tidal stripping,potentially generating a primordial ring system (Canup2010). Furthermore, planets forming under the “hotstart” regime, as opposed to core-accretion, will have sig-nificantly larger radii, lowering the required eccentricityfor moon-loss (Marley et al. 2007).

3.1. Lunar eccentricity growth

As mentioned above, when the system crosses δ = −1from below (by way of inward migration), the singleequilibrium at e = 0 becomes unstable and undergoesa bifurcation into two stable equilibria appearing at non-zero eccentricities (Figures [1, 2]). Provided the lunareccentricity begins relatively small, the resonant orbitwill perform small-amplitude oscillations about the ec-centricity corresponding to the equilibrium fixed pointof the Hamiltonian. As long as the evolution proceedswithin the adiabatic regime, quasi-conservation of phase-space area guarantees that the oscillation amplitude willremain small (see Figure 3). Moreover, dissipative pro-cesses, stemming from tides or disk interactions, willreduce the amplitude of these oscillations, causing themoon to very closely track the fixed points.

Cumulatively, we may determine the evolution of themoon’s eccentricity by solving for the stable fixed pointsof the governing Hamiltonian. The small-e case consid-ered in the previous section was sufficient for analysis ofadiabatic capture but we must work with unrestrictedeccentricities in order to accurately describe evolution ofthe moon within resonance. The first-order approxima-tion is to assume that the equilibrium eccentricity corre-sponds to an exact balance between the lunar precessionperiod and planetary mean motion, described by equa-tion (4). It is sensible to work in terms of a dimensionless

7

0 0.5 10.51p

2p

2

0

0.5

0.5

1

p2

1

= 1.5 = 0.5 = 0.5 inward migration(inside resonance)

resonant capture

separatrix0 0.5 10.51

p2

p2

0

0.5

0.5

1

p2

1

0 0.5 10.51p

2p

2

0

0.5

0.5

1

p2

1

e-growth within resonancee cos()1/2

esi

n(

)

1/2

x =

y=

Fig. 2.— Contours of the small-eccentricity Hamiltonian derived in the text under three scenarios. On the left, the moon is outside ofresonance and all trajectories circulate about the origin. The middle panel represents the situation when the moon is inside resonance andthe stable, null equilibrium eccentricity in the left panel has bifurcated into two, non-zero stable solutions and one unstable solution. Onthe right, we illustrate the situation once the moon leaves resonance. If capture occurs, the trajectory remains near the upper or lowerequilibria. Unsuccessful capture causes the system to remain within the circulating region around the null-eccentricity equilibrium.

semi-major axis

a ≡ apares

, (42)

such that, in solving the criterion (4) we find the moon’sresonant eccentricity growth is well-described by

eeq =

√1− a 3

4 . (43)

One can show, both perturbatively and through numeri-cal solution, that the above expression corresponds veryclosely with the exact equilibrium of Hamiltonian (10).Such an equilibrium may be obtained by a similar ap-proach as was used above to calculate the small-e equi-libria, except that the resulting polynomial is non-trivialto solve.

We plot both the approximate solution (43) and the ex-act, numerical solution in Figure 1 to demonstrate theirsimilarity. Furthermore, in Figure 3, we compare theanalytic expression (43) (the black line) to a direct nu-merical solution of Hamilton’s equations (the oscillating,green line), where it is apparent that the approximate so-lution is more than adequate to describe the eccentricitygrowth of the moon.

3.2. Condition for Moon-Loss

With an analytic solution for eeq in hand, we are now ina position to calculate the semi-major axis acoll at whicha resonant moon will collide with the planet. We sup-pose the moon to be lost at eeq = ecoll which, from (43)and (44), occurs when

1− Rp

am=

√

1−(acollares

) 34

. (44)

For convenience, we define the dimensionless variable

r ≡ amRp

, (45)

such that the condition for a moon being lost (after sub-bing in for ares) becomes

(1− 1

rloss

)2

= 1−(acollRp

) 34

r− 7

4

loss

(M?

Mp

)− 14

J122

(2

3

)− 12

,

(46)

where rloss is the dimensionless semi-major axis of amoon lost at heliocentric distance ap . An analytic solu-tion exists for rloss above, but its functional form is rathercomplicated (though analytic approximations exist).

As mentioned above, in order to capture a moon intothe evection resonance, the planet must originate outsideof resonance (a0 > ares, where a0 is the location of theplanet at the time of moon formation). This conditionmay be recast in terms of the the most distant lunar orbitthat would encounter resonance (r = rMax) as the planetmigrates from ap = a0. Rearranging the expression forares, we obtain

rMax =

[(3

2

)2(a0Rp

)3(Mp

M?

)J22

]1/7(47)

In other words, any moons forming further than rMax

from their host planet will not become captured duringsubsequent inward migration. Combined with the con-dition for moon loss r = rloss above, we may define an“exclusion zone”, within which, moons may be lost viathe evection resonance as a planet migrates from a0 toacoll:

rloss < rexcl < rMax , (48)

where expressions for rloss and rMax are given by equa-tions (44) and (47).

Crucially, the excluded region’s outer edge rMax de-pends only upon where the planet began its inward mi-gration, a0. Consequently, if such a region is observedamong future exomoon detections, its outer edge may beused to directly constrain where the planet-moon systemformed, irrespective of where the planet resides presently.

8

0 4 8 12 16 20

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

luna

r ecc

entri

city

moon loss

green, blurred trajectory - numerical solution black, solid trajectory - analytic solution

planetary migration

planetary semi-m

ajor axis (AU)

2

1.6

1.2

0.8

0.4

00

planet-crossing eccentricity

collides at ~ 0.63 AU

time (104 years)

Fig. 3.— Numerical solution of the capture and subsequent lossof a moon with semi-major axis equal to that of Io’s current value.The green, jagged line follows the numerical solution where theblack line close to its center illustrates the analytic solution de-rived in the text for the equilibrium of the the Hamiltonian (equa-tion (43)). The horizontal, red line denotes the eccentricity at thewhich the lunar orbit crosses the planet’s surface and the blue line,decreasing from left to right, depicts planetary migration. We con-sider parameters typical to the Io-Jupiter system except with theplanetary radius inflated to 1.4RJ. Notice that the analytic so-lution is almost indistinguishable from the mean lunar trajectory.The addition of dissipation collapses the numerical curve on top ofthe analytic one, provided the dissipation is not too severe (e.g.,excessive tides, see Figure 5).

This is fortunate, because giant planets have long beensuspected to undergo planetesimal-driven migration fol-lowing the epoch of disk-driven migration (Tsiganis et al.2005). We note, additionally, that the extent of post-diskplanetary migration may in principle be inferred from thediscrepancy between the current planetary position andthat derived from expression (44) for rloss.

3.3. Illustrative example of exclusion zone

The condition (48) is general, but for clarity, in Fig-ure 4 we present the extent of moon-removal appropriateto a Jupiter-mass planet around a Sun-like star. Wedisplay the specific regions of moon-loss for a planet cur-rently found at 0.5 AU, 1 AU and 1.4 AU, as a functionof a0. It is clear that, provided the planet-moon sys-tem formed sufficiently far out, a significant extent ofmoon-space may be removed. For example, a hypothet-ical Jupiter, currently found at 1 AU, could have lost aEuropa-distanced moon (r ≈ 6.8) had the system origi-nated at & 5.3 AU.

It can be seen from Figure 4 that for each current plan-etary location (horizontal line), there exists a minimuminitial location (a0 = acrit), below which no moons arelost (where the horizontal lines meet the red curve). Thissituation corresponds to when the migration extent is notsufficient to take any one lunar orbit all the way from cir-cular to planet-crossing. We may approximate acrit as afunction of the final position af by solving

rloss

∣∣∣∣af

= rMax

∣∣∣∣a0=acrit

, (49)

which yields the solution,

acrit = Rp

[(2

3J2

)2(M?

Mp

)r7loss

]. (50)

3.4. Time required for moon-loss

Having related the extent of planetary migration to arange of lost lunar orbits, we now consider how muchtime must pass in order to lose these moons and whetherit may occur adiabatically. We mentioned above thatthe Type II migration timescale is expected to scale in-versely with planetary mean motion. Accordingly, moonsat larger distances from their planets (larger r), whichare captured when their host planets cross more distantheliocentric radii, will be subject to much longer migra-tion timescales than closer-in moons. Furthermore, im-posing more rapid migration timescales would begin toimpinge upon the adiabatic criterion (34). In what fol-lows, we calculate the time taken for adiabatic moon-lossand compare it to the lifetime of a typical protoplanetarydisk.

Suppose the planetary semi-major axis evolves accord-ing to equation (27), and that the migrationary timescaleτm = ξ Tp. In this case, we may calculate the time inter-val ∆te within which a given planet may migrate fromsome outer distance a0 = ares to an inner semi-majoraxis ac. We do this through the solution of

1

ap

dapdt

= − 1

2π ξ

√GM?

a3p(51)

whereby we obtain

∆te =4π

3ξ

√a3resGM?

[1−

(2

r− 1

r2

)2], (52)

where we have made use of relationship (44), derivedabove, which states that

ac = ares

(2

r− 1

r2

) 43

. (53)

We illustrate the timescale ∆te in Figure 6 for the casesξ = 10 4, 105, 106 - all reasonable numbers given thecurrent knowledge of Type II migration (Armitage 2010;Kley & Nelson 2012). The critical planetocentric dis-tance, below which moons may be adiabatically lost maybe found by solving equation (52) for the value of r suchthat ∆te equals some nominal disk lifetime τdisk.

There are two competing effects at play. First, a planetmust migrate slowly enough to capture its moon into res-onance. Second, the planet must traverse a sufficient ex-tent in semi-major axis, for its moon to crash into theplanetary surface before the protoplanetary disk dissi-pates. What Figure 6 suggests is that moons are un-likely to be adiabatically lost around Jupiter-mass plan-ets if they lie further than ∼ 10 planetary radii away.In other words, if migration is slow enough for adiabaticcapture, the disk dissipates before moon-loss at largerradii is complete. If migration is rapid, such as ξ = 10 4,the planet can traverse the required distance in time, butonly closer orbits (r ∼ 3) satisfy the adiabatic criterion,making capture of more distant moons rare. Here, we fo-cus on Jupiter-like planets, but the range of lunar orbitsover which adiabatic loss may occur expands for moremassive planets, larger J2 and less massive stars.

4. DISSIPATIVE EFFECTS

9

initial planetary position

planetary positions after migration

1.5 AU

1 AU

0.5 AU

excl

uded

regi

on

more distant lunar orbits

lost

1 AU

0.5 AU

1.6 AU

J2 = 0.02, Mp = MJ, Rp = 1.4RJ

a0 (AU)

moon-loss region

amoon

Rplanet

0 5 10 15 20 25 30 35 40

2

4

6

8

10

12

14capture probability drops

Fig. 4.— The region of lunar orbit parameter space resulting in moon-loss. The shaded region between each horizontal line and the redcurve illustrates lunar orbits lost when the host planet migrates from a0 (horizontal axis) to the location denoted by the horizontal lines(1.6, 1.0 and 0.5 AU). Requiring migration to occur sub-adiabatically (slowly) limits capture of orbits to r . 10 (see text), correspondingto a0 ∼ 15 AU. As an illustrative example, a Jupiter-like planet migrating from 15 AU to 1 AU may lose moons between r ∼ 6.8 − 10.

0. 0.30103 0.69897 1.

-1.

-0.522879

0.

0.477121

1.

k/Q= 0.0

15ms

= MIo

k/Q= 0.0

15ms

= MEarth

planetary migration

collision with planet

enhanced dissipation

tide lib

typical dissipation

1 2 5 10

0.3

1

3

10

0.1

(amoon/Rp)

apla

net(A

U)

diminishing

capture

probabilitytides unimportant

where capture is certain

resonance crossing

k/Q= 0.1

5ms

= MEarth

Fig. 5.— An illustration of the required degree of migration forcapture and loss of moons as a function of their planetocentric lo-cation. For clarity, we refer to the semi-major axes of the moon andplanet as amoon and aplanet (where ‘m’ and ‘p’ were used as sub-scripts in the text). The dashed, grey and black lines indicate theloci where tidal dissipation of eccentricity occurs as rapidly as libra-tion. To the right of these lines, dissipation can break resonance.As can be seen, everywhere moons may be captured adiabatically,tides are unimportant, except when we artificially enhance the dis-sipation to roughly model the influence of continents and oceansin a similar configuration to the modern day Earth.

In the calculations presented above, we described thedynamics of a lunar orbit under the influence of purelygravitational forces. However, there exist two majorsources of dissipation that may complicate the picture.First, moons around gas giants are thought to origi-

nate from within a circumplanetary disk of gas and dust(Canup & Ward 2002, 2006). Analogously to planetsembedded in circumstellar disks, moons are thought tointeract with their disks such as to lead to inward migra-tion of the moons in addition to a potential modulation ofeccentricity. The second dominant source of dissipationis tidal planet-moon interactions, the strength of which isdependent upon both lunar eccentricity and semi-majoraxis (Mignard 1981; Hut 1981). We may generally sup-pose that dissipative influences will significantly alter thepicture described above if the dissipation timescale isshorter than the libration timescale of the conservativeHamiltonian (12).

4.1. Tides

In this section, we discuss the influence of tidal dis-sipation upon the evection resonance. Specifically, theeffect of the evection resonance is to increase eccentric-ity. Therefore, it is important to determine whethertidal damping of eccentricity will counteract its resonantgrowth. Recall that, in the conservative problem, weworked with the canonical variable Γ = 1 −

√1− e2,

the rate of change of which is directly obtainable fromHamiltonian (12) by Hamilton’s equation

10

˙Γ =e

(1− e2)12

e = − ∂H∂γm

= −npnpnm

15

4Γ(2− Γ) sin(2γm)

= −npnpnm

15

4(1−

√1− e2)(1 +

√1− e2) sin 2γm,

(54)

where we now suppose that sin(2γm) → 1 because thiscorresponds to the maximum restoring torque that theconservative dynamics can apply. The tidal dampingmust overcome this eccentricity forcing if it is to breakthe system out of resonance. Therefore, we may writethe conservative eccentricity growth as

de

dt

∣∣∣∣grav

∼ npnpnm

15

4e (1− e2)

12 . (55)

The degree to which tidal eccentricity damping oper-ates is somewhat uncertain, especially for general eccen-tricity. However, we obtain an approximate expressionfor the tidal damping by utilising the tidal formulae ofHut (1981). Specifically, we may approximate the tidalevolution of eccentricity by

de

dt

∣∣∣∣tides

=− 27km nm2Qm

(Mp

ms

)(Rs

am

)5

e (1− e2)−132

×[f3(e2)− 11

18(1− e2)

32 f4(e2)

Ωm

nm

]

(56)

where, for Ωm, we consider the satellite to be in the equi-librium spin state (Ωm = 0). In Appendix A, we providea brief derivation of the functional form of the equilib-rium Ωm, which evaluates to

Ωm = nmf2(e2)

(1− e2)32 f5(e2)

. (57)

In the above equations, Rs is the satellite’s physicalradius and ms is its mass. In addition, we must specifythe tidal love number k2 and the quality factor Q, whichare highly uncertain even in well-studied solar systembodies, such as the Galilean satellites, let alone hypo-thetical exomoons (Lainey et al. 2009). Accordingly, wechoose three reasonable cases. First, we consider twomoons with dissipative parameters appropriate for Io,with k2/Q ≈ 0.015, but with one having the mass ofEarth and the other the mass of Io. Owing to the de-pendence of the tidal damping upon satellite radius andmass, the Earth-mass moon will dissipate eccentricitymore rapidly, assuming similar k/Q. As a third case, wenote that a truly “Earth-like” moon will come completewith oceans and continents, from which the vast majorityof Earth’s tidal dissipation stems (Egbert & Ray 2000).Therefore, in the interest of completeness, we considera scenario where the moon has the mass and radius ofEarth, but tidal parameters ten times that of Io. Thefactor of ten is somewhat arbitrary and is taken simplyto illustrate an extreme case. However, we note that the

1 2 3 4 5 6 7 80.0001

0.001

0.01

0.1

1

10

(m/T2)

= 106

(m/T2)

= 105

(m/T2)

= 104

increasing

migration rate

adiabatic thresholds

typical disk

lifetime

adiabatic moon-loss not rapid enough

102

101

103

104

1

10

(amoon/Rplanet)

adia

bati

cm

oon

-los

sti

mes

cale

(Myr)

adia

bat

icm

oon-los

sti

mes

cale

Myr(0

.01/J

2)

(0.01/J2)

Fig. 6.— Time taken to adiabatically lose a moon by way of theevection resonance. Capture into resonance occurs with certaintyonly below a threshold migration rate. The vertical lines indicatethe most distant moon where the migration time τm/Tp = 104

(grey) and τm/Tp = 105 (blue) lead to adiabatic capture. A mi-gration time of 104 can traverse the resonant dynamics within adisk lifetime but is too rapid for adiabatic capture of moons be-

yond ∼ 2.9 planetary radii. Times scale as J−12 and we have chosen

J2 = 0.01 for illustration.

model proposed by Touma & Wisdom (1998) requiredEarth’s dissipation to be about 25 times weaker in thepast to match the moon’s current position, so an orderof magnitude amplification is at least feasible.

In Figure 5, we plot the locus of parameters where

de

dt

∣∣∣∣tides

=de

dt

∣∣∣∣grav

(58)

for the three different cases described above. In general,tides act over too long of a timescale to break the res-onance. However, where tides are artificially enhanced(the dotted line in Figure 5), moons residing beyondr ∼ 3 − 4 may be broken out of the resonance beforedestruction. Accordingly, the remote possibility existsthat some habitable, Earth-like moons have been savedfrom annihilation by the very oceans and continents thatmake them habitable3.

4.2. Influence of a Circumplanetary disk

Despite decades of work, the exact mechanisms govern-ing turbulence, migration and planet formation withincircumstellar disks remain elusive. Therefore, to claim aprecise understanding of the analogous disks encirclingyoung planets would be premature. However, the prop-erties of moons around our own gas giants, Jupiter andSaturn, have helped guide sophisticated models of cir-cumplanetary disks (Canup & Ward 2002, 2006; Martin& Lubow 2011). In particular, moons are thought to un-dergo inward migration within such disks, a process wehave thus far neglected.

A theory was put forward in Canup & Ward (2006)to explain the conspicuously uniform mass ratio (∼ 104)between the masses of the planets Saturn, Jupiter andUranus, and their respective satellite systems. The the-ory relies upon disk-driven migration (of the Type I va-riety) carrying previous generations of moons into the

3 This is somewhat of a fun speculation rather than a seriousstatement.

11

host planet, leaving only a surviving remnant whose for-mation time was similar to their migration time. If thispicture is persistent across extrasolar giant planets, thenthe possibility exists that no one particular moon willever stick around long enough for the evection resonanceto remove it. However, with data limited to our ownSolar System, it is not yet clear whether significant mi-gration of moons does indeed occur during the epoch ofplanetary migration.

The treatment of migration within a circumplanetarycontext possesses several key differences from that withincircumstellar disks. First, there are currently no obser-vational constraints upon accretion rates within disks en-circling planets. Indeed, once the planet has acquired itsgaseous envelope in a runaway fashion, there is no strictrequirement that the circumplanetary material accreteat all. In Appendix B, we provide a brief calculation ofthe steady-state disk mass arising from a maximum pos-sible accretion rate of about one Jupiter mass per millionyears. For turbulence parameters typical of circumstel-lar disks, the total disk mass could be smaller than thatof Io, but conversely, if turbulence was generated by thedisk reaching a gravitationally unstable state, then thedisk could easily be orders of magnitude more massive.Owing to such uncertainties, for the sake of this work, wesimply mention that significant inward lunar migrationwould delay resonance-crossing, with the details left forcase-by-case considerations.

If moon formation does indeed occur by way of a rapidcreation and destruction of moons as they sequentiallycascade into the host planet (Canup & Ward 2006), thenthe evection resonance will only apply to the final gen-eration of moons, when its efficacy depends upon howmuch more planetary migration occurs after this point intime. Similar arguments apply to disk-driven eccentric-ity damping. The evection resonance will only proceedonce disk-driven eccentricity damping slows down, whichought to occur within a similar epoch to when satelliteloss ceases and so we meet the same conclusion, thatonly the final generation of moons is subject to evection-induced moon loss.

5. DISCUSSION

In this paper, we have identified a mechanism by whichmoons may be dynamically lost as their host planetundergoes Type II migration to shorter-period orbits.Specifically, inward migration increases the planetary or-bital frequency until it becomes commensurate with theJ2-forced pericenter precession rate. Capture into this“evection” resonance, followed by subsequent migration,drives the lunar orbit’s eccentricity higher until the mooncollides with the planet. We have shown that this mech-anism is generally constrained to remove moons closerthan about ∼ 10 planetary radii from their host plan-ets. More distant moons may enter the resonance, butin their case, moon-loss is unlikely to occur within thetypical lifetime of a protoplanetary disk.

5.1. Determining migration extent

It has long been suspected that giant planets mustform outside of the ice lines of their natal disks (Pol-lack et al. 1996), but exactly where they form is still anopen question. An observational prediction of the mech-anism proposed here is that moons should be more abun-

dant at small and large planetocentric distances but rareat intermediate distances. The unoccupied region maybe used to constrain where migration began, via Equa-tion (48). Such an inference has important implicationsfor the formation pathway of giant planets. In partic-ular, core-accretion is not thought to operate efficientlyat larger heliocentric distances. (Pollack et al. 1996; Ar-mitage 2011). In these cooler regions (beyond about 20-40 AU), gravitational instability and disk-fragmentationhave been tentatively proposed, but are generally disfa-vored.

Disk fragmentation is occasionally invoked to explainvery distant giant planets, such as those directly imagedin the system HR 8799 (heliocentric distances & 40 AU;Marois et al. 2008, 2010). We have shown that moonscannot be lost adiabatically via the evection resonancewhen the planet originates at such distant radii. How-ever, upon migration from ∼ 10 AU, we do expect moonsto be lost, depending upon where the planet ends itsmigration. Accordingly, we broadly expect evection-induced moon-loss to be more indicative of core accretionthan of fragmentation, but caution that planets formedvia the “hot start” of disk fragmentation are likely tohave enhanced radii than those arising from core accre-tion.

An important caveat is that we assume the planet andits moon form almost simultaneously. Were the moonto form after significant planetary migration, the regionover which moons are lost could be significantly reduced.The extreme case thereof would be moons gravitation-ally caught subsequent to the disk-hosting phase, sce-nario we neglect. Proposed moon formation times arehighly uncertain and mixed. For example, Callisto hasbeen proposed to form before Jupiter’s hypothesized in-ward migration (Heller et al. 2015). On the other hand, ifmoons are continuously being formed and lost within cir-cumplanetary disks, no one single moon might be aroundlong enough for loss via the evection resonance (Canup& Ward 2006). Only once systems of exomoons are de-tected can we thoroughly test the competing hypothesesregarding moon formation and so for now we state thatformation locations inferred from exolunar systems rep-resent a lower bound on the actual location where theplanet itself formed.

In addition to determining the location at which aplanet formed, we may also be able to constrain howmuch the planet has migrated outwards since the epochof Type II migration. The three outer planets of our solarsystem are thought to have undergone significant post-nebular migration by way of planetesimal scattering (Tsi-ganis et al. 2005). Notwithstanding any influence suchscattering has upon the moons, the observed inner edgeof an excluded region may constrain where the planetresided at the end of Type II migration. A comparisonto its observed location may extract some informationregarding post-disk migration. It would be optimisticto expect particularly precise estimates via this method,but the general existence, or non-existence, of significantoutward migration would help place the so-called “Nice”model of our Solar System into its Galactic context.

We have thus far neglected that, just as planets canmove around after disk-dispersal, planetary tides canlead to significant evolution of satellite orbits. Indeed, ithas even been proposed that the larger moons of planets

12

within about 0.6 AU of their host stars may be entirelylost (e.g. Barnes & O’Brien 2002). In principle, it ispossible to utilize tidal theory to infer where the moononce was around any given planet, but uncertainties arelimited by the theory governing planet-moon tides, anactive field of research even within our own solar sys-tem’s satellites. With this in mind, we might supposethat younger planets are better targets, as they will haveexperienced less tidal evolution.

In general, backing out the journey taken by a planetand its moon within their natal disk from the presentlunar configuration is unlikely to become immensely pre-cise for any one target. However, given a large enoughsample of exomoon systems, we may begin to determinetrends, or populations of planets with significantly differ-ent migrational histories that have thus far gone unno-ticed, providing yet more impetus to continue searchingfor these objects.

5.2. Implications for habitable moons

Part of the motivation for this work was to determinewhether Type II migration of giant planets may signifi-cantly reduce the number of moons occupying habitablezones. We have shown that these worlds are indeed sub-ject to destruction through the evection resonance, butover a fairly restricted parameter space. Nevertheless,the mechanism is capable of reducing the population ofhabitable moons, except for the somewhat unlikely casethat the satellite is as dissipative as Earth currently is,which is anomalously high even relative to Earth’s geo-logical history.

As an illustration of the potential for habitable moon-loss, consider the horizontal line labelled “1 AU” in Fig-ure 4, appropriate to a Jupiter-mass planet currently sit-uated at 1 AU. Moons might be lost outside of ∼ 6.3planetary radii upon migration from & 5 AU. However,in order to lose moons beyond about 8 planetary radii,migration must take place within a disk lifetime, whichcorresponds to super-adiabatic motion for these parame-ters, making moon-loss unlikely. Accordingly, the moon-loss region is somewhat narrow in this specific case, butother cases may have significantly greater excluded re-gions.

5.3. Additional considerations

The assumptions adopted in our work inevitably leaveroom for future extensions to the framework. In par-ticular, Jupiter’s moons Io, Europa and Ganymede arelocked in a 1:2:4 mean motion resonance, leaving open

the question of how the picture changes if there are mul-tiple moons around the migrating planet. Mean motionresonances are likely to quench the evection resonance asthe apsidal recession driven through moon-moon inter-actions dominates over the evection-induced precession(Murray & Dermott 1999; Morbidelli 2002). However,the picture is less clear when the moons are not lockedin mutual resonances.

All discussion thus far has been with regard to de-stroying moons. However, suppose that the moon iscaught into resonance, but the planet subsequently mi-grates only a small distance. The lunar orbit will thus beleft eccentric but not planet-crossing. Further contrac-tion of the planet after dissipation of the disk is analogousto inward migration within the disk because it reducesthe coefficient of the J2 term in the Hamiltonian. Ac-cordingly, the moon would be pushed to yet higher ec-centricities, potentially leading to a later collision withthe planet. Alternatively, if sufficient contraction hasoccurred, the Roche Limit may lie outside the planetaryradius and, in lieu of a collision, tidal forces would ripthe moon apart to form a ring system. Indeed, tidalstripping from the icy, surface layers of a past moon hasbeen invoked to explain the rings of Saturn (Canup &Ward 2006), though it is unclear whether the evectionresonance might have had a role in their formation.

In tis work, we have introduced a novel mechanism forthe removal of moons orbiting young, giant planets. In-ward migration is expected to be almost ubiquitous inthe formation of these planets, suggesting that the cap-ture of moons into evection resonance is potentially acommon process. We highlight that the resonance maybe prevented by one of several mechanisms. Sufficientlyrapid planetary migration can prevent capture and mi-gration of the moon itself to shorter-period planetocen-tric orbits can delay or prevent resonance-crossing. Fi-nally, the presence of other moons in the system, whetheror not they exist in mean-motion resonance, can some-times overpower evection. Such complications must betreated on a case-by-case basis as future exo-lunar detec-tions emerge.Acknowledgements CS acknowledges support fromthe NESSF student fellowship in Earth and PlanetaryScience. We acknowledge enlightening discussions withDave Stevenson. We would also like to thank the ref-eree, Matija Cuk, for his thoughtful report that greatlyimproved the manuscript. This research is based in partupon work supported by NSF grant AST 1517936.

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(Vol. 8, p. 434).

A. TIDAL EQUATIONS

In this section, we describe the equations used in determining tidal dissipation rates. Following Hut (1981), we adoptthe following equations describing the tidal evolution of satellite spin-rate (Ωs) and eccentricity:

de

dt=− 27

ksTq(1 + q)

(Rs

a

)8e

(1− e2)132

×[f3(e2)− 11

18(1− e2)

32 f4(e2)

Ωs

nm

],

dΩs

dt=3

ksT

q2

I

(Rs

am

)6nm

(1− e2)6

×[f2(e2)− (1− e2)

32 f5(e2)

Ωs

nm

]

(59)

where, adopting a constant-Q framework,

T ≡ R3

GM2nmQ.

(60)

In the above equations, q = Mp/ms where Mp is the mass of the perturbing body (the planet) and ms is the massof the body upon which a tide is being raised (the satellite). Likewise, Rs refers to the satellite’s physical radius, Qs isthe tidal quality parameter of the satellite and ks its tidal Love Number. The mass of the satellite ms is much smallerthan the mass of the planet Mp, such that q(1 + q) ≈ q2. The various functions fi are defined as

14

f2(e2) = 1 +15

2e2 +

45

8e4 +

5

16e6

f3(e2) = 1 +15

4e2 +

15

8e4 +

5

64e6

f4(e2) = 1 +3

2e2 +

1

8e4

f5(e2) = 1 + 3e2 +3

8e4.

(61)

The moment of inertia of the satellite I is not important here because we suppose the satellite to be tidally locked(Ωs = 0) such that

Ωs = nf2(e2)

(1− e2)32 f5(e2)

, (62)

as claimed in the main text. When the appropriate substitutions are carried out, we arrive at equation (57).

B. CIRCUMPLANETARY DISK MODEL

In what follows, we gain insight by considering a steady-state disk model, constrained to drive an accretion rate Mlower than about 1 Jupiter mass every million years. If we adopt the Shakura & Sunyaev (1973) parameterizationfor effective viscosity, ignore any mass inflow and impose zero-torque inner boundary conditions (Armitage 2011), thesteady-state solution for surface density Σ reads

Σ =M

3πα

1√GMpa

(h

a

)−2(1−

√Rp

a

), (63)

where a is the planetocentric distance of the disk gas and h is the pressure scale height of the circumplanetary disk.We may now estimate the mass of the disk by integrating from the planetary surface to some outer radius, aout, whichwe estimate as the last non-crossing orbit inside the Hill radius (Martin & Lubow 2011), i.e.,

rout ≈ 0.4 rH =2

5ap

(Mp

3M?

)1/3

. (64)

Using the above conditions, and taking the limit aout Rp, we obtain a disk with mass given by

Mdisk ≈2M

3αΩout

(h

a

)−28

45

(Mp

3M?

)1/2

, (65)

where Ωout is the orbital angular velocity of gas at the outer edge of the disk, which is related to the planet’s meanmotion via

Ω2out = 3

(5

2

)3

n2p. (66)

Finally, we prescribe an accretion rate of 1MJ myr−1, such that M ∼ MJ/τacc with τacc = 1 myr, leading to a diskmass

Mdisk

Mp≈ 10−3

Tpτacc

1

α

(h/a

0.2

)−2(Mp/M?

10−3

)1/2

. (67)

Considering a planet at 1 AU around a Sun-like star, Tp = 1 year and so the disk mass we obtain is roughly

Mdisk

Mp≈ 10−9

1

α. (68)

If α ∼ 10−3, then the inferred disk mass around a Jupiter-mass planet is significantly smaller than Io. However, thesource of α is a mystery in these disks. Turbulence almost certainly commences once gravitational instability sets inbut this requires Mdisk ∼ (h/a)Mp, suggesting very small α ∼ 10−8.

15

Such a diminutive α is not entirely unreasonable within the gravitationally-driven turbulence regime, provided diskshave very long cooling times (Gammie 2001). However, we are veering yet further into the unknown with theseconsiderations and so we leave the details for future work.