Late Origin of the Saturn System

Erik Asphaug1,2 and Andreas Reufer3

1School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287, easphaug@asu.edu

2Department of Earth and Planetary Sciences, University of California, Santa Cruz, CA, 95064

3Physikalisches Institut, University of Bern, Switzerland, andreas.reufer@space.unibe.ch

Abstract

Saturn is orbited by a half dozen ice rich middle-sized moons (MSMs) of diverse geology and compo-

sition (e.g. Smith et al. , 1981; Thomas, 2010; Schenk et al. , 2011) that comprise ∼4.4% of Saturn’s

satellite mass. The rest is Titan, more massive per planet than Jupiter’s satellites combined. Jupiter

has no MSMs. Disk-based models to explain these differences exist (e.g. Sasaki et al. , 2010; Canup,

2010; Mosqueira et al. , 2010a; Charnoz et al. , 2011) but have various challenges and assumptions.

We introduce the hypothesis that Saturn originally had a ‘galilean’ system of moons, comparable to

Jupiter’s, that collided and merged, ultimately forming Titan. Mergers liberate ice-rich spiral arms,

in simulations, that self-gravitate into escaping clumps resembling Saturn’s MSMs in size and compo-

sitional diversity. We reason that MSMs were spawned in a few such collisions around Saturn, while

Jupiter’s original satellites stayed locked in resonance. The dynamical validity of our scenario depends

on whether some MSMs can be scattered or otherwise migrated to stable orbits following each colli-

sion, before they are accreted. If satellite formation concludes with a ‘late stage’ of giant impacts (e.g.

Ogihara & Ida, 2012) then MSMs could have formed originally by this mechanism. More speculatively,

solar-system-wide dynamical upheaval (e.g. Tsiganis et al. , 2005; Morbidelli et al. , 2009)might have

triggered final mergers, leaving behind young MSMs and a dynamically excited Titan.

1. Introduction

Since the invention of the telescope there has been a quest to explain Saturn’s rings (Cuzzi et al. ,

2010). Equally mysterious, but largely unknown until the space age, are Saturn’s middle-sized moons

(Figure 1). These icy bodies ∼ 300 − 1500 km diameter (Thomas, 2010) are found on a wide range

of orbits, from 3.2 to 62 Saturn radii (RY). The largest four, Iapetus, Tethys, Rhea and Dione, were

discovered by Giovanni Cassini in the late 1600s. Celebrated for some time as the ‘Sidera Lodicea’

honoring King Louis XIV, they were largely ignored for three centuries until the first detailed images

were transmitted by the Voyager spacecraft (e.g. Smith et al. , 1981; Thomas et al. , 1983; Moore &

Preprint submitted to Elsevier December 9, 2012

Ahern, 1983). Since then, the fantastic and perplexing moons of Saturn have been observed in great

detail by the ongoing Cassini mission (e.g. Porco et al. , 2005; Thomas, 2010). Middle-sized moons are

today recognized as one of the principal oddities of the outer solar system (e.g. Nimmo et al. , 2011;

Schenk et al. , 2011), massive enough yet diverse enough to motivate and constrain larger theories of

planet formation and evolution, tails that wag the dog.

1.1. Geophysical motivations

Saturn’s MSMs are altogether ∼ 1/20 as massive as Titan (Figure 2), and are perhaps 20 times

as massive as the rings. They are distinguished from Saturn’s smaller ‘moonlets’, which are icy

bodies tens of km diameter (Janus and others) that orbit close to the Roche limit. MSMs are many

thousands of times more massive than moonlets and are distributed to much greater distances from

Saturn. The inner moonlets appear to be well explained as accreted piles of outward-spreading ring

material (Charnoz et al. , 2010), whereas the origin of MSMs remains a fundamental mystery (e.g.

Canup, 2010; Mosqueira et al. , 2010a; Charnoz et al. , 2011; Sekine & Genda, 2012), not only in the

Saturn system but around other giant planets. The five major satellites of Uranus are ‘middle-sized’

(∼ 500 − 1600 km diameter), and Neptune has relics of a population of MSMs. None are found at

Jupiter.

Enceladus is one of the smallest of Saturn’s MSMs, and its major geological activity (Spencer et al.

, 2006) is extraordinary and unexplained. Other moons, notably Rhea, show past or recent signs of

deformation and activity (Schenk et al. , 2011), and possibly past rings or moons of their own. Some

show relatively uneventful surface histories, appearing as cratered frozen-down ice spheres, while at

others it is difficult to disentangle crater production from erasure (Lissauer et al. , 1988). Saturn’s

MSMs rotate synchronously in their orbits, the result of past or ongoing tidal dissipation in their

interiors. In the case of distant Iapetus, whose tides raised on Saturn are weak, de-spinning by an

accreted subsatellite has been proposed (Levison et al. , 2011). Determinations of shape and bulk

density (Thomas et al. , 2007) suggest the existence of interior mass concentrations or inhomogeneities,

while limb profiles reveal non-hydrostatic shapes even if one removes the signatures of the major craters

(Nimmo et al. , 2011). These lines of evidence suggest interior thermal evolution, and/or fossilized

rotational and tidal deformation.

FIGURE 1 NEAR HERE

Saturn’s MSMs have H2O-dominated surfaces (Cruikshank et al. , 2005), further supporting the

possibility of thermal evolution and differentiation. Their interiors are predominately ice (c.f. Durham

et al. , 2005). As a group they measure ∼ 3/4 H2O by mass, indicated by their global bulk density

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ρs (Anderson & Schubert, 2007), but around this average the bulk compositions of individual MSMs

exhibit unexplained and wide-ranging diversity. The radar properties of MSMs also indicate complex

and diverse physical and thermal histories, at least in their outer layers (Ostro et al. , 2006), and

these differences have yet to be explained. There is no obvious trend in bulk composition either

with semimajor axis as or with size. One dynamical connection with composition is notable but

perhaps coincidental, that the iciest of the inner MSMs, Tethys and Mimas (ρTethys = 0.98 g cm−3;

ρMimas = 1.15 g cm−3), are in a 2:1 mean motion orbital resonance, and so are the rockiest, Dione

and Enceladus (ρEnceladus = 1.61 g cm−3; ρDione = 1.43 g cm−3).

Much has been written about the geology of the giant satellite Titan, the subject of intensive

interest for its hydrocarbon seas and ice continents, its wet ‘tropical’ climate and massive atmosphere,

and its possible subsurface H2O ocean (Baland et al. , 2011) and astrobiological potential. If gas

giant planets are common in the universe, then so are Titans. We refer the reader to overviews

and interpretations by Lopes et al. (2007); Stofan et al. (2007); Radebaugh et al. (2007); Lorenz

et al. (2008); Moore & Pappalardo (2011) and references therein. Titan is on an eccentric orbit

(e = 0.0288, a = 20.3RY) but otherwise comparable in mass and semimajor axis to Ganymede and

Callisto. The orbit periapsis 19.7RY is closer to Saturn than the apoapsis 20.9RY by >1 planetary

radius, causing a strong non-equilibrium tide. In the absence of forcing, Titan’s orbit is expected to

circularize over a few billion years (Tobie et al. , 2005) due to the dissipation of tidal energy through

internal heating. Although Titan’s internal structure could, for all we know, be non-dissipative, its

massive atmosphere and active geology, and possible internal ocean, would seem to indicate deep-

seated activity and potentially strong dissipation. If so then either Titan started off with a huge

eccentricity, or else the orbital eccentricity is recently acquired or forced.

Rhea is the largest of Saturn’s MSMs (Drhea = 1530 km; see Figure 1). It is controversial

(Tiscareno et al. , 2010) for exhibiting evidence of past (Schenk et al. , 2011) and arguably present

(Jones et al. , 2008) rings of its own. Rhea has approximately the average bulk composition of

MSM-forming material, ρRhea = 1.24 g cm−3, and this fact turns out to be useful in discriminating

among hypotheses, below. The smallest three MSMs of Saturn – Mimas, Enceladus and Hyperion,

D ∼ 400, 500, 300 km respectively – exhibit almost inexplicable variations in their fundamental

characteristics, as if they formed by completely different mechanisms, and out of different materials.

This has led to the idea that some MSMs (for example, 210 km diameter Phoebe; Johnson & Lunine,

2005) are captured, while others have weird histories endogenic to their planet and its satellite system,

and specific to their planet’s interaction with the dynamically evolving young Solar System.

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1.2. Dynamical motivations

In addition to these studies of the shape, mass, geologic history and composition of MSMs, dynam-

ical studies provide powerful physical constraints (e.g. Peale et al. , 1980; Meyer & Wisdom, 2008) on

their orbital, rotational, tidal and collisional evolution. If orbital evolution is driven by tidal dissipa-

tion, then the geology and dynamics of icy satellites are strongly coupled (e.g. Zhang & Nimmo, 2009).

While fundamental aspects of Saturnian satellite evolution have been explored and identified, specific

scenarios are elusive due to computational and analytical limitations (integrating orbits for billions

of Kepler times, including nonlinear interactions) and the large uncertainties in starting conditions

and tidal Q. And while the present epoch of Saturnian satellite dynamical evolution is anchored in

the astrometric record, as discussed below there is presently a debate whether the record shows that

Mimas is migrating inwards (Lainey et al. , 2012) or outwards.

A broader consensus appears to be emerging if one looks back much earlier, to the waning of

the protoplanetary nebula, as to how major satellites might form and evolve dynamically around gas

giants. Although the framework is certainly debated, it is believed that major satellites accrete from

ices and silicates in the sub-nebulae of giant planets, analogous to miniature solar systems (Canup &

Ward, 2002; Mosqueira & Estrada, 2003). According to the model of Canup & Ward (2006), Galilean

satellites form sequentially in distant orbits and are dragged in by mass-dependent Type I migration

(Ida & Lin, 2008). Satellite formation and migration ends with the clearing of the nebula, and in

Jupiter’s case the last four standing are Io, Europa, Ganymede and Callisto. According to Ogihara

& Ida (2012) this epoch of differential migration led to ongoing collisions and giant impacts, and this

possibility motivates our research and that of Sekine & Genda (2012).

Peale & Lee (2002) show that the Canup & Ward (2002) formation scenario leads to the migration

of Jupiter’s major satellites into a Laplace resonance, lending credit to the theory. Io orbits in

∼ 2 : 1 resonance with Europa, which orbits in ∼ 2 : 1 resonance with Ganymede. Librations about

these mean motions are regulated by the low order Laplace relation φ = λio − 3λeuropa + 2λganymede

where φ librates about 180° and λ is the mean anomaly (position in its orbit) of each satellite. This

coordination has more recently been found to be an almost universal end state in general simulations

of satellite accretion around gas giants Ogihara & Ida (2012). The Laplace resonance of Jupiter is

stable to substantial forcing (Yoder & Peale, 1981), and forced perturbations are damped by tidal

heating, evidenced by the volcanic activity of Io. This explains the dynamical longevity of the Galilean

satellites.

Saturn ended up with something fundamentally different, and to understand this we might question

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whether the planets evolved in isolation. The architecture of the early solar system emerged chaotically

according to the Nice model (Gomes et al. , 2005; Morbidelli et al. , 2005a; Tsiganis et al. , 2005), due

to close encounters and resonances between migrating giant planets. Accordingly, the original satellite

systems of the giant planets might have weathered upheavals associated with powerful gravitational

interactions. In the classical Nice model, the jovian planets migrate on a timescale ∼ 10 Ma (Fernandez

& Ip, 1984; Hahn & Malhotra, 1999) until Saturn and Jupiter fall into a 2:1 mean motion resonance.

According to Tsiganis et al. (2005) Uranus and Neptune are thrown into the distant reaches as part

of an overall expansion of the architecture of the solar system.

Planetesimal scattering erupts according to the Nice model (Gomes et al. , 2005) when giant

planet resonances sweep through dense planet forming regions. In the vicinity of the Earth and

Moon, scattered planetesimals cause the ‘late lunar cataclysm’ (Tera et al. , 1974), the spike in the

production of lunar basins some ∼ 3.8− 4 Ga ago (reviews by e.g. Hartmann, 2003; Chapman et al. ,

2007). Evidence for a late cataclysm is also found in H-chondrite meteorites that show the resetting

of argon chronometers at around that time (Swindle et al. , 2009), and possibly also in eucrites and

other meteorites (see Scott & Bottke, 2011), supporting the idea of an onslaught of planetesimals and

a high rate of catastrophic disruption.

The Nice model argues that these chronometers record a late solar-system-wide upheaval that

led to the arrangement of the giant planets (Tsiganis et al. , 2005), the existence of the Kuiper Belt

(Levison et al. , 2008), and to the populations of irregular moons and Trojan asteroids (Morbidelli

et al. , 2005b). Alternatively, this chronometric record of meteorites and lunar basin ejecta might

only date an inner solar system catastrophe (e.g. Chambers, 2007) and have little bearing on Saturn.

Contrary to the original Nice model, Venus and Earth have nearly circular orbits: scanning migrations

from the giant planets would have greatly excited their eccentricities (Brasser et al. , 2009; Agnor &

Lin, 2012; Nesvorny & Morbidelli, 2012) . Additionally, the significant inclinations of the orbits of

the giant planets are not well explained by the original Nice model, because the massive planetesimal

disk leads to a high degree of dynamical damping.

Rather than discard the model, these two major contradictions can be resolved if giant planet

migration is impulsive and episodic instead of secular and sweeping. Morbidelli et al. (2009); Brasser

et al. (2009) develop a ‘jumping jupiter’ scenario, where the giant planets are jolted into new orbits,

reacting to close encounters by rogue ice giants including Uranus, Neptune, and perhaps one or two lost

companions. This scenario ends with the giant planets on appropriate orbits, and inclinations, while

leaving the terrestrial planets alone (Brasser et al. , 2009). We are unaware of published or presented

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studies of the dynamical effects of ‘jumping jupiter’ impulses on the original satellite systems of the

giant planets, but it is conceivable that a system-wide upheaval is recorded in the surviving systems.

Neptune’s original system appears to have been destroyed when it captured a Pluto-sized object

(Agnor & Hamilton, 2006); this would have required a substantial flux of plutoids crossing Neptune

(or vice versa), consistent with a wide-wandering planet. The original Uranus system partly sur-

vived whatever event toppled the planet’s obliquity (Morbidelli et al. , 2012). In general terms, the

magnitude of the consequences to a satellite system are inversely proportional to the planet’s mass,

so Jupiter’s moons would be the most stable by a substantial factor, during an epoch of mutual

encounters, and Saturn’s moons less so, and Uranus and Neptune’s the least stable.

As pointed out by Agnor & Lin (2012), the Nice model only requires a late upheaval (∼ 4 Ga)

if it needs to explain the LHB. Otherwise the upheaval could have happened directly after planet

formation, when there is also a strong chronometric record of catastrophic disruption (e.g. Scott &

Bottke, 2011). These considerations leave us with two possible scenarios placing our hypothesis in

time. In the first scenario, the proposed collisional mergers happened primordially, either as the

first-formed satellites spiraled in and colliding during Type I migration or in response to an early

solar-system-wide dynamical upheaval. The MSMs would have formed as ancient relics, which raises

the question of how they survived to the present.

The second scenario begins with a late epoch of final collisions, caused when the the original Saturn

system was knocked out of kilter by resonant interactions with Jupiter, or by close encounters with

ice giants, some ∼ 4 Ga ago. In this case the same dynamical upheaval that triggers satellite collisions

also causes the late lunar cataclysm, removing the requirement of having to explain the survival of

the MSMs through the LHB. Although speculative, this latter scenario has the potential to address

the youthful geological and dynamical aspects of the system, including Titan’s eccentricity.

FIGURE 2 NEAR HERE

1.3. Concept synopsis

We show that MSMs may be the unaccreted remnants of a sequence of collisional mergers that

formed Titan. The physical basis for this proposition is the considerable mass ejection experienced in

collisions between similar-sized bodies. Excess angular momentum is accreted relative to an equilib-

rium spheroid, for typical incidence angles, even in zero energy events (vimp ≈ vesc), while gravitational

binding energy is liberated when merging bodies attain a lower gravitational potential. Thus there is

no such thing as a perfect merger, so

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MF < M1 +M2 (1)

where MF is the mass of the final merged body and M1 and M2 are the masses of two initially well

separated colliding bodies.

This inequality is fundamental to scenarios such as the formation of the Moon by a giant impact

into Earth (e.g. Benz et al. , 1989; Stevenson, 1987; Canup & Asphaug, 2001; Canup, 2004; ?),

the formation of satellites around dwarf planets (Canup, 2005; Leinhardt et al. , 2010), the origin of

chondrite meteorites (Asphaug et al. , 2011), and the astrophysics of stellar mergers (Rasio & Shapiro,

1994). It is therefore not surprising that in the detailed hydrodynamical simulations presented below,

we observe that giant impacts between Galilean-like (‘galilean’) satellites commonly lead to families of

middle sized moons forming as self-gravitating condensates in ejected spiral arms. These characteristic

results are seen in other studies of giant impacts, although not in the specific context considered here.

If Titan formed in a sequence of giant mergers, as we propose, then a few discrete populations

of MSMs might have formed, each sub-population representing one binary merger around Saturn,

with some fraction of the MSMs surviving each collision. Some of our simulations result in families

of escaping ice-rich bodies that directly resemble Saturn’s MSMs in size and diverse composition. In

other cases ∼ 30 escaping bodies are formed, each ∼ 300−500 km diameter, also diverse in composition

and predominately water ice, but nothing immediately the size of Dione or Rhea. These would have to

be subsequently accreted, and one can envision dozens of Enceladus- sized moons growing into larger

MSMs. While our simulations provide a plausible beginning to MSM formation, they require detailed

dynamical modeling to understand further.

It is not clear what fraction of MSMs produced will survive a given collision. They are born

on eccentric orbits around Saturn that cross the orbit of MF , and consequently will be accreted by

MF in a few thousand orbits, according to our simple estimation. Some fraction of them must be

scattered (or otherwise transported) rapidly outside of MF ’s sphere of influence in order to survive.

In lieu of constructing novel dynamical models of satellite scattering, we consider analogous studies

of planet-planet scattering (e.g. Chatterjee et al. , 2008) that suggest that satellite-satellite scattering

may be efficient. Modeling the post-formative phase of MSM evolution is well beyond our scope,

it being a complex system involving families of newly formed MSMs, their mutual collisional and

gravitational interactions, the remaining original galilean satellites, the debris disk from the collision,

and the secular influence of tidal dissipation. If occurring primordially, then the waning presence of

the nebula gas must also be included.

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1.4. Jupiter, Saturn, Titan

Using N -body simulations of satellite dynamical accretion and migration, starting from populations

of satellite ’seeds’ and including tides and the presence of a gravitating gas disk, Ogihara & Ida (2012)

obtain the result that Laplace resonances may be an almost universal end state for large satellites

forming around giant planets. This remarkable result depends upon whether or not the planet clears

a gap in the gas disk. If it clears a gap, then the first migrating satellite stops at the inner disk

edge, and the second migrating satellite gets caught in a 2:1 mean motion resonance with the first

satellite, and so on. Satellites stack up into resonant orbits over time and migrate thereafter in lock

step. As with Peale & Lee (2002), the models of Ogihara & Ida (2012) end up with ∼ 3 − 4 bodies

connected by resonances, ranging in semimajor axis from as ∼ 5Rp − 30Rp where Rp is the radius of

the planet. These final distances are a direct consequence of the choice of ∼ 30Rp initial orbits of the

protosatellite seeds in the simulations, to best match the final semimajor axis distribution for Jupiter.

Is the Jupiter system the norm and Saturn the anomaly? Formation of Saturn in a more distant

and depleted region of the solar nebula could possibly explain the difference in terms of e.g. disk

viscosity α and the ratio of gas to solids. So could the lower predicted field intensity of Saturn’s early

dipole (Takata & Stevenson, 1996) which would perhaps have been too weak at Saturn to clear an

inner disk cavity and put a stop to Type I migration. Sasaki et al. (2010) end up with one or two

major satellites in several scenarios for Saturn, and Canup (2010) obtains a last-standing Titan after

its sibling spirals in and is tidally disrupted. Ogihara & Ida (2012), on the other hand, comment on

the challenges of obtaining adequate matches to the Saturn system, as they seem to obtain either a

Laplace chain of satellites, or none, depending on the presence or absence of a cavity.

Pending a resolution to this debate, which does not seem to be immediately forthcoming, it is worth

contemplating that Titan is about the same mass per planetary mass (1.14 times) as all the satellites

of Jupiter combined. Moreover Titan has about the same bulk composition as Jupiter’s satellite

average. This suggests that the original physics of satellite formation may have been similar about

both planets, with Saturn obtaining a more singular final end state. Although not an explanation,

the mass and compositional equivalence helps justify our postulate that Titan resulted from the final

accretion of a comparable system of galilean satellites.

However Titan formed, it ended up with a large moment of inertia C/MR2 = 0.34 (Iess et al. ,

2010) compared to what is expected of differentiated bodies. This value is close to the hydrostatic

prediction for a completely undifferentiated but compressed body of ice-rock bulk composition, and

is similar to that determined for Jupiter’s satellite Callisto. The moment of inertia of Titan has

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been interpreted as implying an only partially differentiated interior (e.g. Barr et al. , 2010). Partial

differentiation may be difficult to achieve, however, because differentiation is exothermic (converting

gravitational energy into heat) and may run away (Friedson & Stevenson, 1983; Monteux et al. , 2007).

Differentiation may be unavoidable if giant impacts were a dominant aspect of Titan’s formation,

given the association of giant impacts with global melting. The proposed events occur at a much lower

specific energy ∝ v2imp than the Moon-forming giant impact, and involve much less shock heating;

nevertheless on the basis of our models described below we believe that a largely unmelted interior

would be inconsistent with our hypothesis. An alternative interpretation that allows for a thermally

evolved interior, is that the bulk of Titan is composed of mantle hydrates (e.g. Scott et al. , 2002;

Castillo-Rogez & Lunine, 2010; Fortes, 2012), something that could be an expected end state of

repeated collisional mixing during pairwise accretion.

1.5. Kinds of giant impacts

A late stage of satellite collisions around Jupiter or Saturn (e.g. Ogihara & Ida, 2012) is analogous

to the late stage of giant impacts central to today’s standard model of terrestrial planet formation (e.g.

Chambers & Wetherill, 1998; Agnor et al. , 1999). There have been numerous studies of the general

and specific outcomes of giant impacts, and apart from the thermal and Moon-forming consequences

(e.g. Rubie et al. , 2007; Canup, 2004) three primary outcomes have been identified (Agnor & Asphaug,

2004; Asphaug, 2009; Leinhardt & Stewart, 2012).

High accretion efficiency ξ / 1 is associated with the standard models of Moon formation. Hit-

and-run ‘bounces’ result in ξ ≈ 0 (c.f. Reufer et al. , 2012). At higher velocity there is erosion and

disruption, where ξ < 0. Accretion efficiency ξ in a giant impact is defined as

ξ = (MF −M1)/M2 (2)

where M1 ≥ M2 are the target and the projectile. For colliding bodies larger than about ∼ 1000 km

diameter (well into the gravity regime) ξ is primarily a function of the impact angle θ, whose mean

and most probable value is 45° (Shoemaker, 1962), and of φ = v∞/vesc, the relative velocity at great

distance normalized to the two-body escape velocity, and of the mass ratio γ of the colliding bodies.

Relatively well-defined relationships ξ(φ, γ, θ) emerge in these modeling studies, and the collisional

outcomes are found to be approximately scale invariant when the colliding bodies are dominated by

self gravity. For example, a 10 km melted body colliding with a 100 km melted body at 300 m/s, is

to first order scale similar to a 1000 km body colliding with a 10,000 km body at 30 km/s, all of the

same material. Departures from scale similarity are due to changes in material and to effects related

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to compressibility and rheology Asphaug (2010). In other words, giant impacts are not necessarily

“giant”; they include any collision between self-gravitating bodies at velocities comparable to their

mutual escape velocity vesc.

Each of the various kinds of collisional outcomes can in principle provide a formation scenario for

middle-sized moons. All are important to forming and evolving satellite systems around giant planets,

so we consider them now in turn.

1.5.1. Collisional disruption

Collisional disruption leaves no more than half the original largest body M1 intact, ξdisrupt ≤

−M1/2M2. This leads to one scenario for MSM formation, in which a & 2000 km diameter precursor

satellite is struck by an interloping Centaur or other massive body from outside the system, producing

a family of fragments. The disrupted target must add up to the mass of the MSMs or more, and must

be composed of about 75 wt% H2O and 25 wt% rock unless some selective fraction disappears e.g.

into Titan or Saturn.

The scenario does not appear capable of explaining why Saturn ended up with a family of MSMs,

whereas Jupiter, where catastrophic disruption is several times more likely (due to the greater focusing

of planetesimals; c.f. Charnoz et al. 2009) has none. At either planet, destroying a & 2000 km

diameter moon into pieces no larger than ∼ 1/3 its mass is thought to be almost impossible according

to catastrophic disruption scaling relations (Benz & Asphaug, 1999; Stewart & Leinhardt, 2009).

Perhaps fatally problematic to this hypothesis is that impact disruption produces a power-law size

distribution of fragments (e.g. Melosh et al. , 1992), in addition to the largest surviving remnant,

relevant examples being the asteroid collisional families (Williams & Wetherill, 1994; Davis et al. ,

2002). It does not produce monomodal sized fragments such as seen in Figures 1 and 2.

Furthermore, the catastrophic impact disruption of a differentiated gravity-controlled parent body

seldom if ever is energetic enough to exhume core material, even in extreme collisions (e.g. Scott

et al. , 2001). The binding energy of the core material is great, and it is buried under the mantle.

Catastrophic disruption of a differentiated planet or satellite therefore leaves behind its core, plus

whatever mantle remains bound to it, as the final largest remnant. Rhea, being the largest MSM,

should therefore be a rocky body compared to the other MSMs. These objections taken together

appear insurmountable.

1.5.2. Erosion by planetesimal impacts

Erosion (ξ < 0), short of catastrophic disruption, does not provide an origins scenario per se, but

must be understood in order to assess whether MSMs can survive the mass loss expected of icy bodies

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orbiting deep in the gravity field of a giant planet for billions of years. Moons of Saturn and Jupiter

experience an accelerated and focused bombardment, so their mass loss by cratering (Housen et al.

, 1983), especially during the Late Heavy Bombardment, could have been significant, removing most

or all of their ices, contrary to the geologic record.

According to the Nice model, giant planet migration required a widespread flux of planetesimals

that is consistent with the late lunar cataclysm. Barr & Canup (2010) argue that this bombardment

in the outer solar system left its mark on the geologic evolution and states of differentiation of the

Galilean satellites, and also Titan (Barr et al. , 2010). Considering the middle-sized satellites, this

flux might have eroded, by repeated cratering, the innermost MSMs of Saturn, and some might have

been catastrophically disrupted. Charnoz et al. (2009) calculate that Mimas, deep in the gravity well

of Saturn, would have had < 50% probability of survival, based on Nice-model fluxes applied to the

Benz & Asphaug (1999) catastrophic disruption criteria for collisions in ice.

These disruption criteria are derived for ‘cold’ impacts, at velocities ten times slower than for

comets accelerated by Saturn’s gravity (& 30 km/s typically at Mimas). Their probability of survival

appears to be considerably lower if mass loss is vapor driven rather than fragmentation driven. Nimmo

& Korycansky (2012) apply vapor production scaling for ice-rich hypersonic collisions (Kraus et al. ,

2011) to obtain estimates of mass loss, and reason that reaccumulation of the lost vapor is negligible.

They conclude that Mimas and Enceladus would have lost almost all their ice, and that Tethys would

have lost more than half its ice, by impact vaporization during a Nice-model-driven LHB.

To resolve this quandary while explaining the diversity of Saturn’s moons, Nimmo & Korycansky

(2012) suggest that the LHB was no more than ∼ 1/10 the intensity predicted by Gomes et al.

(2005), and highly stochastic, so that Tethys and Mimas lucked out while Enceladus and Dione did

not. The observed trend of increasing compositional diversity with decreasing size is consistent with

their hypothesis. However, limiting the flux of planetesimals to 1/10 of what is required is not a mere

adjustment to the Nice model. The survival of the MSMs could undermine the Nice model’s premise

of solar system wide upheaval, if the associated flux is as disruptive as predicted.

Another proposed resolution to their survival – and perhaps the only choice consistent with the

Nice model – is to have the MSMs form after the peak of the LHB. This is the approach taken by

Charnoz et al. (2011), who propose that MSMs accreted at the Roche limit of a massive, slowly

spreading, ice-rich particle disk. If the disk spreading timescale exceeds ∼ 0.6 Ga and spans the LHB,

as they argue is possible, then the issue of MSM disruption is put to rest. If the satellite mergers

proposed in our model were caused by the same forces of upheaval that caused the LHB, then we also

11

obtain a solution of MSM survival, by virtue of their more recent origin.

1.5.3. Hit and run collisions

About half of the collisions in a random population are oblique (θ & 45°), and if coming in faster

than ∼ vesc the projectile can ‘skip’ off the target so that ξ ≈ 0. These are hit and run collisions, or

sometimes ‘graze and escape’. The relative intensity of the effects of a collision varies with the relative

masses M1,M2 of the colliding bodies (Asphaug et al. , 2006), so in cases of projectile survival (hit

and run, and graze and merge below) it is useful to shift the reference frame of the analysis from M1

to M2. Instead of looking at the target body that is dented and spun up and somewhat heated, we

look at the projectile M2 that is ripped to pieces, of interest especially if it escapes to become a new

planetary body, or bodies.

Because of the strongly accelerated frame of reference as seen by M2, the physics and geochemistry

of grazing planetary collisions are nonintuitive. They can strip the mantle off of M2, like the yolk from

an egg (e.g. Yang et al. , 2007; Reufer et al. , 2012), or they can tear M2 into a family of escaping

clumps, or into fans or sheets, depending rather sensitively on the collision angle and the impact

energy. Here it is the target that destroys the projectile, rather than vice versa, and the consequences

are petrologically interesting for a number of reasons (Asphaug, 2010; Asphaug et al. , 2011).

Whether M2 gets captured by M1 (accretion) or keeps going (hit and run) depends on a combi-

nation of inertial forces (mechanical stopping) and gravitational forces including tides. The combined

effect can be understood by inspecting detailed 3D hydrocode simulations. From these studies it has

been determined (Agnor & Asphaug, 2004; Asphaug, 2009) that hit and run is the most common

outcome, when averaged over impact angle, for similar-sized differentiated terrestrial planets colliding

at ∼ 1.2vesc . vimp . 2.7vesc (see also Stewart & Leinhardt, 2012). Approximately the same trend

appears to hold true for ice-rock bodies (galilean satellites and plutoids) suffering gravity-regime

collisions in this same velocity regime.

This is a common velocity regime for gravitationally stirred accretionary swarms, v∞ ≈ vesc,

because the characteristic random velocity is the escape velocity of the largest bodies in the swarm

(Safronov, 1972). Here v∞ is the initial velocity at a great distance relative to a Keplerian circular

orbit, prior to the collision, also called the random velocity vrand when applied to a swarm of bodies.

The characteristic impact velocity vimp between the largest bodies is then√2 times this. The overall

implication is that hit and run is common among oligarchic and post-oligarchic growth of planetesimals

and planetary embryos (e.g. Kokubo & Ida, 1998), whenever most of the mass is in a population of

similar sized bodies rather than a single dominant body.

12

If the late stage of satellite growth around Saturn involved an interacting population of galilean

satellites, then hit and run is worthy scenario for the origin of middle sized moons. It can produce

chains of like-sized bodies, and can do so in a velocity regime consistent with ongoing accretion.

Sekine & Genda (2012) develop this scenario by modeling a hit and run collision involving a Titan-

like satellite orbiting Saturn at around 5RY. This galilean satellite, subsequently vanished, served as

the target body M1. The vanishing of this Titan-sized body requires a massive subnebula to still be

present, an aspect of their model discussed more critically in §4.

Using SPH in 3D (described below) and the Tillotson equation of state (see Melosh, 1989) for

ices and silicates, Sekine & Genda (2012) model collisions between a differentiated circumplanetary

satellite ∼ 3 times the mass of Rhea (γ ≈ 0.05), composed ∼ 75% of H2O, and a Titan-like satellite,

at v∞ ' vesc and θ ∼ 45◦. The projectile continues downrange and is severely deformed by shocks,

mechanical shears and tides. It becomes an arm of material that self-gravitates into a population of ice-

rich clumps, reminiscent of comet Shoemaker-Levy 9 following its tidal breakup near Jupiter (Weaver

et al. , 1995; Asphaug & Benz, 1996). They support their model with low resolution simulations, in

which clumps (that is, MSMs) are discriminated by only 10 SPH particles. As discussed below, and

known from early studies of Moon formation, clumping in low resolution SPH simulations is prone to

strong artifacts. But according to Sekine & Genda’s interpretation, the rockiest MSMs derive from

the core of the disrupted projectile, while the ice-dominated bodies derive from its ice mantle.

The slowest hit and run collisions in the study by Sekine & Genda (2012) have impact velocity

vimp = 1.4 vesc, where vesc is the escape velocity of the vanished Titan-like target, around vesc ∼ 2.6

km/s. Because vimp =√

v2∞

+ v2esc their hit and run scenarios are fairly energetic in terms of a

gravitationally self-stirred population, v∞ & vesc; nevertheless this random velocity is much slower

than the orbital velocity at ∼ 5RY (their preferred collision location) and represents bound orbits

with a level of orbital excitation that is common in the N -body accretions simulated by Ogihara &

Ida (2012). Hit and run collisions are expected to occur during satellite formation, at least in the

context of pairwise accretion of similar-sized bodies, so it is a dynamically valid concept.

As noted by Sekine & Genda (2012), and confirmed by our own studies, hit and run must occur

within a specific range of parameters in order to obtain the required linear filament (the disrupted

projectile M2) that clumps into a chain of middle-sized bodies. Collisions that occur at a head-on

angle (direct hits & 45°) end up producing fans and sheets of escaping material, akin to crater ejecta,

and end with few if any major self-gravitating bodies. More grazing incidence hit and run collisions

(. 30°, depending on impact velocity and size ratio) result in one large body (in addition to M1) that

13

has been stripped of its mantle (Asphaug et al. , 2006) rather than a chain of massive clumps.

In the limited range of parameters for which hit and run does produce a chain of bodies, an ice-

rich Rhea (ρbulk = 1.24 g cm−3) is difficult to reconcile with the formation of Saturn’s MSMs by

this mechanism. As determined from higher resolution simulations, the most massive fragment in a

hit and run collision involving a differentiated progenitor is dominated by core material (Asphaug,

2010); Rhea is not. A strong density gradient in a differentiated impactor M2 causes its mantle to

be stripped off by tides before there is appreciable mass removal or disruption of the core. As for the

intense proximal collisional interactions (shocks), these preferentially accelerate the crust and outer

mantle in the contact zone; core material is sheltered from shocks. Rhea, containing more than half

the mass of the MSMs interior to Titan, would have to be the central mass concentration if it was

formed in a hit and run collision, and would be dominated by core silicates instead of being ∼ 3/4 ice

by mass.

Another problem with hit and run formation, is that it remains to be understood how the MSMs

would survive to the present day. If formed primordially, while Titan and its massive companions

are forming and migrating (Sekine & Genda, 2012), then they must survive dynamically even as the

Titan-like target M1 slips out of sight due to Type I migration. (An alternative, having Titan itself

be the target of the proposed hit and run collision, has not been considered, presumably because it

would form the MSMs too far from Saturn.) The MSMs must also survive even though their orbits

cross that of M1, and their sweep up time is short, ∼ 103 orbits, as discussed further below. And

lastly, they must survive the LHB, if such occurred, as just discussed.

We do not doubt that small bodies were formed repeatedly by hit and run collisions if there was

a formative epoch of giant impacts. The process is important in late stage accretion environments

and merits further study with regard to the Saturn system and other satellite systems. But even if

the scenario could explain Rhea in some way, it has yet to be demonstrated how these bodies would

survive the final dissipation of the nebula, their encounters with M1 whose orbit they cross, the rapid

migration and eventual tidal disruption of M1 by Saturn (following Canup, 2010), and any subsequent

or ‘late’ impact bombardments.

1.5.4. Partial accretion

Giant impacts are frequently represented as perfect mergers. This is an assumption (often implicit)

in nearly every every published N -body dynamical simulation of planetary accumulation, up to and

including the recent study of satellite accretion by Ogihara & Ida (2012). In an N -body code ξ is

almost always ≡ 1, whereas when two bodies actually collide, ξ < 0 for vimp several times vesc, and

14

ξ ≈ 0 in the hit and run regime vimp ∼ 1.2−2.7 vesc. Although perfect merger remains a computational

necessity when evolving hundreds or thousands of colliding bodies for millions of orbits, it is debated

how strongly this assumption affects the dynamical evolution of an accreting population (c.f. Kokubo

& Genda, 2010) .

Apart from any dynamical concerns, it is problematic how a convenient and necessary approxi-

mation has crept into becoming a gross misunderstanding of geophysics. Agnor et al. (1999) noted

that most terrestrial planets would acquire spin periods Prot . 1.5 hr at some point during their

accumulation, under the assumption of perfect merger. Instead of merger, the implication is rota-

tional fission if planets ever spun this fast. Indeed, Ćuk & Stewart (2012) have recently studied how

near-threshold rotation influences the Moon’s formation by giant impact. More generally, Asphaug

(2010) argues that many of the problems concerning the origins of minor planets can be reconciled

by paying attention to the inefficiencies of accretion. It is abundantly clear that by ignoring the hit

and run events just described, and more generally by ignoring the inefficiencies of accretion during

giant impacts, we neglect the closely associated production of populations of mantle-rich, middle sized

bodies – the MSMs that are the subject of our study.

Even in the lowest energy collisions, v∞ ≈ 0, comparable-sized bodies do not hold on to all of

their combined matter. Consider the case where two uniform-density incompressible spheres fall

onto each other, as perfect as a merger can be. The mutual escape velocity is vimp = vesc =√

2G(M1 +M2)/(R1 +R2), where R1 and R2 are their radii and G is the gravitational constant.

The gravitational binding energy EG =´

−GM(r)r dm is the work required to pull the matter apart

to infinity. Before the collision, the binding energy associated with two particles at infinity is zero,

so the initial binding energy is the sum of two spheres. The binding energy of a uniform density

incompressible sphere of mass M and radius R is EG = −3/5GM2/R = −kM5/3 for constant k, so

the total initial energy at infinity (for cold, non-rotating bodies) is −k(M5/31 +M

5/32 ).

The final gravitational binding energy is that of a sphere of mass MF = M1 +M2 −m, where m

represents the combined mass of any MSMs and any satellites of MF , and other escaping or bound

debris. In the assumption of a perfect merger m = 0, so the change in gravitational potential is

∆E = −kM5/3F − [−k(M

5/31 + M

5/32 )] = k

[

(M1 +M2)5/3 − (M

5/31 +M

5/32 )

]

. The energy ∆E that

becomes available during a theoretical perfect merger is then

f = γ5/3 + (1− γ)5/3 − 1 (3)

expressed as the fraction of the final gravitational binding energy EGF, which is plotted in Figure 3.

15

Here

γ = M2/(M1 +M2) (4)

is the ratio between the projectile mass (always the smaller, so that 0 < γ ≤ 1/2) and the total colliding

mass. (A note of caution, γ is often defined in the literature as M2/M1.) In the limit of equal mass,

γ = 50% and over a third of the final binding energy (37%) in a perfect merger is ‘available’, and more

than this for compressible bodies. In the canonical Moon-forming giant impact, assuming γ ≈ 0.1,

∼16% of the binding energy is available compared to the fully accreted Earth.

FIGURE 3 NEAR HERE

The energy budget for planet and satellite accretion, for large M1 and M2, includes shock heating

(for hypervelocity events, occurring between targets thousands of kilometers across), viscous and

frictional heating during shear; adiabatic heating of compression; radiative losses to space and among

particle; latent heats of melting and vaporization; and kinetic ‘losses’ going into the rotations of the

final bodies, and the final velocities of orbiting material (satellites and disks) and escaping material

m ≡ M2(1− ξ) =

ˆ

v>vesc

dm (5)

In small mass ratio collisions γ ∼ 0, which according to (3) means that f ∼ 0, so that ξ ∼ 1, i.e., M2

is effectively caught by M1. The result is an impact basin, global ejecta, and a small change in the

spin of M1. For γ ∼ 0 the binding energy effectively becomes heat (e.g. Tonks & Melosh, 1993; Rubie

et al. , 2007). But in grazing collisions, much of the energy remains that of the uncaptured parcels

m with escaping kinetic energy EKesc= 1/2

´

mv2dm. Even for incoming velocity vimp ≈ vesc some of

the parcels dm are ejected at v > vesc due to gravitational interactions (slingshots) and this causes

angular momentum exchange between parcels. The result is that some of the spiral arm material

escapes, clumping into MSMs according to our simulations.

In giant impacts, the total amount of escaping material m increases with v∞ and with impact

parameter b. The unaccreted mass can be substantial. Consider the limit of a non-impacting collision

(a flyby) with impact parameter b > 1. Nearly all of the final energy is kinetic, mostly belonging to

M2 as seen in the reference frame of MF = M1, and the self-gravitational binding energy remains

almost the same (although even in non-impacting collisions some mass is lost; Asphaug et al. 2006).

The deformation-induced (tidal) torque of the encounter changes the spin of M1 and causes heating,

along with an exchange of momentum in the center of mass frame of the collision. By and large the

fate of M2 does not matter to M1 in the two-body case. However, if M1 and M2 both orbit a central

16

body prior to the collision, they will experience subsequent close approaches and probable collisions

when their orbits come back into phase.

1.5.5. Graze and merge collisions

Graze and merge events (Leinhardt et al. , 2010) are partial accretion collisions (0.8 . ξ < 1)

occurring off axis (θ & 45°) at close to the mutual escape velocity. Basically, more angular momentum

is accreted than can be sustained in a finite fluid body (Chandrasekhar, 1969), so that spiral arms

are shed, forming disks, moons and escaping bodies. Graze and merge events constitute about half of

all collisions between comparable sized planetary bodies (depending on φ and γ; Leinhardt & Stewart

2012), in which case a few such events could be responsible for the origin of Saturn’s MSMs from an

original galilean-like population.

Even in the incompressible limit, these complex planetary interactions must be studied numerically

(e.g. Rasio & Shapiro, 1994). The computational timestep dt must be limited to resolve the high energy

material interactions (the rise time of a shock, or a particle crossing time) to a few seconds, assuming

a ∼ 100 km spatial resolution characteristic of our 200,000 particle simulations. The simulation must

also be run to late enough time for gravitational clumping and N -body interactions to play out among

the myriad accreting bodies; this requires several gravitation timescales τgrav ∼ 3π/√Gρ, a day or

two. Because intense hydrodynamics are important to late time, a hand-off to an incompressible

code is not feasible. Millions of timesteps are therefore required per simulation, each taking tens of

cpu-seconds for our calculations. This is fundamentally different from asteroid-scale collisions, where

the material interaction is much faster than τgrav, so the restrictive timestep only applies to the initial

stage of the computation (e.g. Asphaug & Melosh, 1993; Michel et al. , 2003; Durda et al. , 2004).

Graze and merge events unfold as a series of ∼ 2 − 3 increasingly inelastic collisions, leading to

outcomes (how many spiral arms, how massive a disk, how hot, how much mixing) that are sensitive

to the impact parameters. The outcome of the first collision determines the angle of incidence of the

second collision, for example, and this determines the third, leading to divergent specific outcomes over

a small change of initial parameters. Yet all graze and merge collisions have common characteristics.

The first captures the bulk of M2 into a non-escaping orbit about M1, and begins the transfer of

angular momentum into body rotation. The slowed down and spun up M2, unable to escape from

M1, collides again. The third collision erupts in a cosmic pinwheel. Coalescence of matter into MF

occurs to the extent that angular momentum permits, which is a complex analytical problem. Spiral

arms accelerate outer materials onto escaping trajectories, the clumps studied in §3.2. The interior of

M2 is eventually accreted by M1, its layers undergoing enormous shearing during their descent, and

17

initiating strong differential rotation within the final merged body.

Graze and merge collisions have been studied in various planet forming contexts. Canup (2005)

studied grazing scenarios for Pluto-Charon forming in a slow giant impact, and identified specific

outcomes including a collision at θ = 73° where a pair of icy planets are captured into mutual orbit

after some mass exchange. A more direct hit at θ = 58° (also vimp = vesc) results in a merged final

body MF with massive spiral arms, from which Charon forms. Direct capture leads to a rockier

Charon, due to the exchange of mantle ice, while the spiral arm scenario leads to an icy Charon. In

another suite of graze and merge simulations, Leinhardt et al. (2010) developed a scenario in which

the plutoid Haumea and its satellites, and its proposed collisional family (Brown et al. , 2007; Lykawka

et al. , 2012), formed in a low velocity accretion event, spinning off pieces of escaping and orbiting

ice-rich material. Apart from their other merits, what is appealing about these scenarios is that they

are so testable.

Models for the Moon-forming giant impact (Benz et al. , 1989; Cameron & Benz, 1991; Canup &

Asphaug, 2001; Canup, 2004) also include graze-and-merge collision. In the scenario favored by Canup

& Asphaug (2001), a Mars-sized projectile accretes onto the proto-Earth during a second collision,

the first time ‘bouncing off’ into a captured orbit. Most of the Moon is formed from the spun-out

fragments of ‘Theia’ in the standard model. Graze and merge is like unequal figure skaters pulled into

a spin: the big skater defines the center of mass, and the legs of the smaller get flung to the outside.

The dominance of Theia in the final Moon is in serious conflict with isotope geochemistry that shows

that the Moon is very Earth-like (Pahlevan & Stevenson, 2007), giving cause for new collisional and

post-collisional scenarios of lunar origins Reufer et al. (e.g. 2012); Ćuk & Stewart (e.g. 2012); Salmon

& Canup (e.g. 2012).

For the same physical reasons, most of the MSMs come from the mantle of M2 in our Saturn-

system hypothesis. If so, the MSMs are complimentary to the handful of satellites that accreted to

form Titan, much as the Moon is complementary to Earth.

2. Simulations of Galilean Satellite Collisions

We simulate the collisional merger of icy galilean-sized satellites using the well tested 3D smooth

particle hydrodynamics code of Reufer (2011). This standard SPH formulation uses a Barnes & Hut

tree-based self gravity calculation and the ANEOS equation of state as modified by Melosh (2007).

It gives similar outcomes as other SPH codes when applied to canonical scenarios such as Moon

formation (as described in Reufer et al. , 2012). Because gravitational stresses overwhelm the geologic

18

strength in collisions of this scale, especially for ice, we treat the merging satellites rheologically as

fluids. Negative pressures obtained in the equation of state are set to zero, representing weak material

or an easily cavitating fluid; this is not unrealistic at smoothing lengths of ∼ 100 m and mitigates

artificial clumping.

Just as important as the choice and implementation of EOS is the choice of realistic equilibrium

structures and initial thermal conditions. Here we are limited by not knowing whether the colliding

bodies have already differentiated, or even what state they are in (solid or liquid) at the time of the

collision. We must make an assumption, yet the interior structure and composition of Titan is itself

much debated. Early thermal conditions being unknown, we have simply modeled all colliding bodies

as 50 wt% ice, over 35 wt% rock, over 15 wt% iron. When analyzing our collisional outcomes it is

important to note that these calculations are for completely differentiated end members. They place

lower limits on the rock fraction escaping with the outer mantle, as addressed below.

The masses of the colliding bodies are chosen so that their mergers produce final bodies MF

comparable in size and bulk density to Titan. But achieving a close match to the final mass of

Titan is not important, because within the range of size we are considering, to first order these

gravity-dominated, relatively incompressible collisions scale with the sizes of the colliding bodies

(Asphaug, 2010). Collisions a few times larger or smaller in mass can be reliably scaled from our

present simulations. According to our hypothesis, Titan formed at the end of a series of mergers.

We need thus to consider a range of masses of the colliding bodies. The first merger, for instance,

produced MF less than half the mass of Titan, forming clumps that were proportionately smaller.

Because of the scale invariance, the most important variable in these models is not size but initial

composition, and the presence or absence of completely differentiated layered structures as we have

presumed.

FIGURE 4 NEAR HERE

Pressures, densities and temperatures are initialized within M1 and M2 using a 1D Lagrangian

code (Figure 4). We apply an isentropic profile to each material layer and allow the structures to

equilibrate under self-gravity with a small damping term, using the same material EOS. The entropy

assigned corresponds to a completely melted phase in each layer. As noted this is arbitrary, but tests

starting with lower-entropy targets obtain very similar results in terms of spiral arm formation and

clumping; see also the discussion of rheology below. These radial 1D equilibrium profiles are then

imprinted upon self-gravitating 3D spheres of SPH particles. These bodies M1 and M2 are further

relaxed using the 3D SPH hydrocode until all random velocities fall below 1% of the local escape

19

velocity, a threshold that has been found to be suitable for these calculations.

To begin a collision, the two relaxed spheres M1 and M2 are placed with their centers of mass

separated by 5(R1+R2) and given an initial velocity to achieve to the desired impact velocity vimp =√

v2∞

+ v2esc and impact angle θ. Initial rotations are set to zero, the assumption being that colliding

satellites would be locked in their rotations to Saturn. The possibility of rapid initial rotation has yet

to be explored, c.f Ćuk & Stewart (2012). Because the fate of M2 is a key concern, this initial infall

of M2 onto M1 needs to be explicitly modeled because it deforms and spins up the bodies, especially

M2, prior to the collision. It is part of the collision, there being a continuum from tidal collisions to

grazing collisions to direct hits.

We evolve mergers forward in time for 24-48 hr until collisional processes are terminated and the

remnant masses are unchanged, and until final velocities of ejected clumps relative to MF (escaping

or otherwise) are reliably determined. Each simulation uses a minimum of 200,000 particles, corre-

sponding to an initial smoothing length h ≃ 70 km. Calculations are done in the two body frame,

ignoring the effect of Saturn. Although we can include Saturn as an external gravity, for now this

requires an arbitrary choice and would introduce a large new parameter space that seems unjustified

at present. Saturn position in time, relative to the colliding satellites, would depend on the approach

vector of the encounter, which would be randomly varying, and the orbital distance of the collision.

To evaluate our hypothesis of gravitational clumping, we must address the well known concern

regarding tensile instabilities in SPH. We find, below, that our simulated clump masses agree with an-

alytical estimates of self-gravitational instability in infinite incompressible cylinders (Chandrasekhar,

1961). We also observe that clumping begins outside the Roche limit, in our simulations, while de-

structive tidal shearing occurs inside of Rroche, further evidence that gravity rather than artificial

forces are at work. The clumps are reasonably well resolved in the simulations, typically consisting

of 100-500 SPH particles, giving us some confidence that the masses of the largest final bodies are

physical. For instance, a ‘best case Enceladus’ (Figure 6) is resolved with 160 particles, or 7 particles

diameter, sufficient (barely) to resolve shocks and material contrasts. Clumps smaller than ∼ 100 m

are not physically resolved in these simulations, as they consist of only a few particles.

As for rheology, it has been found in recent studies of the tidal disruption of comets (Movshovitz

et al. , 2012) that the inclusion of solid, granular behavior (e.g. a catastrophically fragmented carapace

as modeled by Asphaug et al. 2011) leads to fluid-like behavior in self-gravitating chains of rubble at

geologic scale, and to similar but somewhat more heterogeneous size distributions of clumps. The total

mass m available for MSM formation in a given simulation (the accretion efficiency ξ = 1−m/M2) is not

20

sensitive to further increases in resolution (c.f. Agnor & Asphaug, 2004). Nevertheless, recognizing

that higher resolution is desirable for a reliable analysis of clumping, and perhaps a more explicit

treatment of rheology, we restrict our interpretations below to the general aspects of our results.

Saturn’s gravity does not influence any of the prompt collisional physics, which takes place deep

inside the gravity field of M1. But it may affect the process of clump formation in the spiral arms.

If clumping takes place outside of the Hill radius of the satellite, RH = as(Ms/3MY)1/3 where Ms is

the mass of the satellite, then it will occur in the dynamical field of Saturn and the physics might

be different. For final merged satellites MF with Titan-like bulk densities, RH ∼ as

RYRs where Rs is

the satellite radius, e.g. RH ∼ 20Rtitan where atitan ∼ 20RY. In our simulations all major clumping

finishes inside of ∼ 10Rs in Figure 5d, meaning that collisions occurring inside of about as ∼ 10RY

will begin to feel Saturn’s presence.

Additionally there is the pseudo-force in the non-inertial frame of reference orbiting Saturn (Kep-

lerian shear) that accelerates the opening of the spiral arms and causes more material to escape from

MF in a given merger than the two-body prediction. For example, between the inner- and outer-

forming clumps in our simulations we estimate a relative shear velocity of order & 100 m/s over the

course of the collision, assuming as = 20RY, and stronger for collisions closer to Saturn. In summary,

the presence of Saturn would serve to increase the yield of escaping material, but might alter the

physics and efficiency of clumping for mergers inside of ∼ 10RY. While the two-body approach taken

here provides a sound general representation of the phenomenon, a more informed understanding of

collisions around ∼ 3− 10RY requires giant impact simulations including the central planet.

FIGURE 5 NEAR HERE

3. Results of Simulations

A time sequence from a typical partial accretion event is plotted in Figure 5, in this case a slow,

grazing merger (vimp = vesc, θ = 75°, ξ ∼ 0.9) with mass ratio γ = M2/M1+M2 = 1/4. This is a

few times the mass ratio of typical Moon-forming giant impact simulations (Canup & Asphaug, 2001;

Canup, 2004). The scaled velocity vimp/vesc is also comparable to what is assumed in Moon formation,

but the specific kinetic energy is only ∼1/20 as intense, proportional to v2esc. The first collision slows

down M2 and captures it, and spins up both M2 and M1. The second collision launches an escaping

arm of material drawn from the outer layers of M2. The result is reminiscent of models by Moulton

(1905); Chamberlin (1916); Jeans (1919); Jeffreys (1924) in which stellar filaments are ripped from

the Sun into planets by a passing star. One problem with that classic, once-popular theory is that

21

stellar matter is expansive and will not clump. Relatively incompressible materials such as ices and

rocks, on the other hand, do bead up self-gravitationally when extruded into arms.

The final capturing collision, forming MF in Figure 5d, applies defining torques to the evolving

system and launches another spiral arm, this one mostly attached to the final body. MF super-rotates,

and gets rid of angular momentum by spinning out material mostly from the outer layers of the accreted

body M2. Centrifugal and gravitational forces segregate the shredded pieces of M2 further, according

to density. Thus the first spiral arm ends up being very ice rich, and the last is rockier in composition,

deriving from deeper layers of M2. This characteristic explains the compositional diversity of MSMs

in our model.

As mentioned above, in graze and merge collisions most of the impact kinetic energy goes into

rotation of MF , orbital energy of any subsatellites, and kinetic energy of any escaping MSMs and small

debris. As for shock heating, in the planar approximation the specific kinetic energy ui =12ρv

2imp is

comparable to the energy of vaporization of H2O. Some vaporization should occur, and it does in the

simulations. The final body is surrounding by a steam atmosphere that is present but not resolved.

Although of minor mass fraction, steam could play an important initial role in forcing the climate and

establishing the primary geology of the finished satellite.

Pierazzo & Melosh (1999) found in half-space studies of oblique impacts that shocks are greatly

diminished inside the projectile for shallow impacts (θ & 60°), and that regions within a projectile can

be sheltered. In 3D, and when the bodies are similar sized, the effects of obliquity are significantly

more pronounced, since the target body is not much larger in radius than the projectile. Grazing has

been defined for similar sized collisions (Asphaug, 2010) as being collisions where cores equal to half

the radius do not intersect along the approach vector, in which case θgraz & 30°. Most of the colliding

matter ‘misses’ the other body even for this relatively steep impact angle.

The deposition of heating is very complex in a graze and merge collision, given that a single event is

drawn out into two or three collisions. According to the simulations, the most intensive heating occurs

in the directly-contacting crustal materials, and then internally where the impactor’s core and mantle

shear against the target materials where they attain neutral buoyancy. They flatten, in our models,

but maintain tremendous lateral momentum. The final structural and thermal state of MF , including

the potential for differentiation, mixing, and hydration, are addressed in some detail in Section 6, but

more suitable numerical methods (e.g. Gerya & Yuen, 2007) are required to meaningfully interpret

these further aspects of the problem, perhaps using these or similar hydrocode outcomes as initial

conditions.

22

FIGURE 6 NEAR HERE

Figure 5 is representative of the simulations we have studied in detail so far. Twelve other sim-

ulations, all of them graze and merge events with impact angles ≥ 45°, are summarized in Figure

6. For the MSMs that are formed in each simulation, their sizes and silicate fractions are plotted,

where an MSM is defined as a body not bound to MF that is a discrete self-gravitating entity. We

find that a wide range of binary accretion parameters result in clump-forming spiral arms, and to

various possible scenarios for MSM formation. Collisions like Figure 5 result in chains of bodies that

resemble, in size and compositional variation, the present population of MSMs, so in principle a single

collision could form the MSMs directly, with the sizes and compositions they have today. But as we

examine in §4, some fraction of the escaping clumps are likely to be accreted by MF , whose orbit they

initially cross. This could require a larger production population of bodies, as is indicated in some

of the simulations in Figure 6. Also, seeing that the MSMs and Titan are in quite different orbits, it

is unnecessary to form them all in one event. A sequence of events is proposed, forming MSMs with

various characteristics, and leaving some of them behind in the aftermaths of two or three mergers.

These simulations are an initial exploration of a large parameter space. We focus on the relatively

high angular momentum mergers (θ & 45°), occurring about half the time, that spin out spiral arms

and MSM-like clumps on escaping trajectories. We find from these studies and previous work (e.g.

Asphaug et al. , 2011) that more head-on collisions result in fan-shaped rather than arm-shaped

ejecta, and tend to produce less escaping material for a given velocity, and less effective clumping.

But at high enough energies, even head-on partial accretion produces copious clump-forming ejecta

(e.g. Durda et al. , 2004). According to Ogihara & Ida (2012) many of the satellite collisions involve

dynamically excited encounters, v∞ & vesc, and while this may be exaggerated by their assumption

of perfect N -body accretion, it suggests that we consider more energetic, messier events.

Future collisional modeling at much higher resolution and with better physics, looking more broadly

at the wide range of collisional parameters, and following the outcomes to much later time, will be

required to understand what if any specific scenario fits. Conversely, if our hypothesis is correct that

the MSMs formed alongside Titan as a consequence of its accretion, then their composition and size

distribution may be able to tell us something about the dynamics of satellite formation, much as the

Moon informs us of the dynamics of terrestrial planet formation.

3.1. Formation of clumps

After the major collisional interactions have played out, we identify gravitationally bound clumps

of SPH particles using the recursive algorithm described in Benz & Asphaug (1999). The algorithm

23

identifies compact gravitating clumps, plus other gravitationally bound clusters including binary pairs

and clumps with sub-satellites and bound particles (representing mass below the resolution limit). One

such configuration, a binary pair, is seen at the very lower left of Figure 5e. We assume that most

of these sub-satellite (and sub-sub-satellite) systems will coalesce. The caveats stated above apply

regarding the veracity of clumps. Below the resolution limit (about ∼ 100 km in our simulations) we

expect there to be a substantial population of smaller bodies resulting from the collision, although we

do not speculate on its mass and extent.

Clump diameters are indicated in Figure 5 by white circles, of radius equal to a compact sphere of

the same mass and bulk composition. The circle is offset from the center of figure in the final frame,

accounting for the family of ∼ 100− 1000 km bodies that are orbiting MF . The red line in Figure 5e

shows the approximate boundary between the bound objects and the unbound MSMs. Bodies bound

to MF will ultimately collide, we expect, and mostly around its equator. The MSMs will interact

dynamically and collisionally with Saturn, and with MF , and with each other, and with any remaining

galilean satellites coordinated with MF .

The inner circle in Figure 6 is the diameter of the rock core assuming complete differentiation.

None of the inner core of M2 escapes M1 in any of these simulations. The number above each subfigure

is the escaping velocity v∞ of each clump, in km/s. It is computed from the orbital energy of each

clump, above the gravitational potential of MF . In a satellite system, it is possible for clumps with

v∞ < 0 to escape, and those with v∞ > 0 to be accreted by MF , but we consider v∞ = 0 to be the

dividing line between bound and unbound objects (the red line in Figure 5e).

Being gravitational condensations from spiral arms, the clumps form with rapid rotation Prot ∼

4−9 hr. A few binary and multiple systems form in our simulations with too much angular momentum

to sustain as a single compact body, and become co-orbital binaries (e.g. Scheeres, 2004). These

subsatellites are probably ultimately unstable, so the timescale of their accretion onto Titan (as the

final MF ) is geologically relevant, as discussed in §6. Also related to these simulation outcomes is the

general circumstance of post-merger ring systems (c.f. Levison et al. , 2011) and possible evidence for

more recent equatorial rings on MSMs (e.g. Schenk et al. , 2011).

Our simulations have assumed the ideal case of a pure H2O mantle overlying silicate rock (Figure 4).

However, complete differentiation is thought by some to be inconsistent with Titan’s moment of inertia.

Alternatively, the high moment of inertia can be explained by an extensive hydrated silicate mantle

(Scott et al. , 2002; Castillo-Rogez & Lunine, 2010; Fortes, 2012) rather than discrete rock and ice

layers. Either scenario, a hydrated silicate mantle nearer the surface, or an incompletely differentiated

24

interior, would put more rock into the spiral arm structures emerging from these collisions. While

either scenario can be studied (serpentine is available as a module for ANEOS, for example), for

the present we assume complete ice-silicate differentiation in our models, placing lower limits on the

fraction of silicates in the MSMs for a given simulation.

It is important to recognize that tidal and other disruptive forces will themselves tend to segregate

high density from low density materials in expanding tidal arms. The materials experience strong

oscillations in inertia and gravity over the course of their ∼ 24 hr of expansion followed by clumping.

Segregation by density was noted in simulations of mixed rubble piles (modeled comets) undergoing

tidal disruption (Asphaug & Benz, 1996). Moreover, in the case of a hydrated silicate interior (Castillo-

Rogez & Lunine, 2010; Fortes, 2012), it is possible that water can come out of hydration during pressure

unloading of deep materials (top row of Figure 6) which transition from kilobars of pressure inside

of M2 to hundreds of bars inside the finished MSM. Thus, a well-differentiated progenitor is not a

required initial condition, in order to produce ice-dominated MSMs by this mechanism.

Modelers are generally concerned about the veracity of clump formation in self-gravitating hy-

drocodes, or in particle codes. The sizes of the clumps in our simulations are overall consistent with

those predicted by Chandrasekhar (1961) in his analytical study of the gravitational instability of

incompressible fluid cylinders. This is an appropriate benchmark since the arm-forming material can

be reasonably approximated as incompressible condensed phases (water, ice and rock). As discussed

earlier, in all cases the vapor fraction in the collisional aftermath is minor. Clumping is observed to

occur well outside the Roche limit, as expected. In the static case of an infinite cylinder of radius

R, the fastest growing self-gravitational instability is of wavelength 2πR/1.0668, according to Chan-

drasekhar (1961), forming a ‘string of pearls’ where each pearl ultimately collapses into a sphere of

radius about twice that of the cylinder. To first order, we obtain the clump sizes that are predicted

compared to the tidal arm’s diameter during clumping.

Because the spiral arms are elongating and shearing even while they are clumping, the formal appli-

cation of Chandrasekhar (1961) is only approximate. Still, it suggests how MSM diameters are related

to the dimensions (radius, mantle thickness) of the progenitor M2. As noted, it furthermore provides

evidence that the clumping seen in our simulations is physical, and the result of self-gravitation, and

that the phenomenon of clumping is adequately resolved.

In conclusion, our series of simulations, although limited in scope, is sufficient to demonstrate that

most galilean satellite mergers at θ & 45◦ result in characteristic clump-forming spiral arms, forming

primarily from the icy outer layers of M2. The clumps in these spiral arms resemble today’s MSMs

25

in important respects in several simulations (θ ∼ 60 − 80°; vimp ≈ vesc). More grazing collisions can

produce dozens of ∼ 500 km bodies that would have to accrete pairwise to become today’s MSMs.

Also, several collisional mergers would better explain the MSMs than any single event.

3.2. Provenance of MSM-forming material

Because of our assumption of a pure-ice mantle, the simulated MSMs are mostly ice, although a few

dredge up deeper materials. Figure 7 plots the originating location within M2 for each of the MSMs

that are formed in the simulation of Figure 5. The top panel shows the drop in pressure, relative to

the original pressure, expressed as a histogram of total mass exhumed over a given ∆P . This release

in pressure corresponds to an available enthalpy dH = dU + V dP + PdV , which under isentropic,

isothermal unloading (not an unjustifiable approximation initially) is dH ≈ dP/ρ. The second row

shows the color codings for the 8 escaping MSMs produced in this simulation; these do not include the

clumps orbiting MF . For each clump, the inner circle shows the fraction of silicate material if any, as

in Figure 6. These colors correspond to the originating location in the target and projectile M1 and

M2 prior to the collision. In the context of this hypothesis, Enceladus could be a piece of the deep

interior of M2, originating from kilobars original pressure inside of M2 (see histogram), while Tethys

could be a piece of its water-rich exterior.

FIGURE 7 NEAR HERE

Rhea, the largest of Saturn’s MSMs, has over half the mass of satellites inside the orbit of Titan.

It does not immediately present itself in the outcomes of the simulations we have run so far. It is quite

possible that we have missed a category of collisions in our limited explorations so far, that might

leave behind such a large ice-rich object. For instance, higher velocity events at intermediate impact

angles produce more copious escaping debris according to previous studies (ξ ≈ 1/2) and might result

in substantial clumps. In this work we have focused on low-energy spiral arm forming collisions with

e ∼ 0, v∞ ∼ 0 and ξ & 0.8.

Another possibility is that Rhea is an accumulation of smaller MSMs. An amalgam of MSMs would

end up with average composition, consistent with the ∼ 75 wt% H2O composition of Rhea. For that to

have occurred, several times the present number of MSMs would have had to form originally, and then

accrete pairwise into larger bodies. Such a scenario is compatible with a number of our simulations,

especially considering the production of MSMs in repeated mergers. Subsequent accretions would

tend to circularize the orbits of MSMs interior to MF , and this is one possible explanation for their

rapid post-formative dynamical evolution – the topic we now address.

26

4. Post-Formation

Middle-sized moons are a typical outcome when Galilean-like satellites undergo pairwise accretion

in giant impacts. We have briefly considered what would cause these collisions to occur, a topic

discussed further in §5. Here we ask, how long-lived are these moons? As with the hit and run model

of Sekine & Genda (2012), the MSMs that form in our simulations are born on orbits that cross that

of the final merged satellite MF . How long these satellites can last – hundreds of years, or billions

of years – is now the question, along with their evolution much longer term into an orbital spacing

resembling today’s population. And however they formed, the MSMs must also survive the collisional

onslaught of the Late Heavy Bombardment, if there was such an epoch at Saturn.

We start by generally considering the issues of MSM migration in the context of the two general

classes of MSM formation models. The first class of models (Canup, 2010; Charnoz et al. , 2011) spawns

MSMs at the outer edge of a massive, viscously spreading ice disk; from here they are transported

outwards. Iapetus probably cannot be explained in the context of these models; see Mosqueira et al.

(2010a). The second class class of models has them forming stochastically in one or more giant

impacts occurring between Saturn’s original satellites, either by hit and run involving a Titan-sized

precursor (a body that has since disappeared; Sekine & Genda 2012) or as we propose, by a sequence

of collisional mergers ultimately forming Titan.

4.1. Spreading disk models

The inner moonlets of Saturn are found at as ∼ 2.3− 2.6RY. This is just outside the Roche limit

Rroche ≃ 2.44Rp(ρp/ρs)1/3 (6)

where ρp, ρs are the bulk density of the planet and satellite, respectively. For strengthless compact

ice spheres, Rroche ≈ 2.1RY. These bodies were shown by Karjalainen & Salo (2004); Charnoz et al.

(2010) to have probably originated in an ongoing or recent process following the viscous spreading of

the main rings beyond Rroche, where gravitational clumping can occur. But these are minor bodies,

only tens of km diameter, and the preponderance of considerably more massive ice-rich bodies & 3RY

from Saturn – the MSMs – requires a different explanation.

Canup (2010) proposes that MSM formation occurs by an analogous process, but forming out of

a proportionally more massive ice disk. The source of this disk is proposed to be the tidally stripped

mantle of a Titan-like precursor that spiraled into Saturn, the consequence of early Type I migration.

SPH simulations show that Saturn’s tides can rip away the icy mantle of a differentiated progenitor

27

satellite when it spirals inside the Roche limit, leaving the silicate-dominated core to continue spiraling

into the planet (assuming the nebula is still present) and leaving massive quantities of ice from which

substantial moons can form. Like the Sekine & Genda (2012) model, the Canup (2010) model invokes

a precursor Titan-like satellite that no longer exists, but in this case instead of disappearing entirely,

its icy mantle is recycled, becoming the source material for MSM production.

The ice-rich disk material beyond Saturn’s corotation radius (presently Rc = 1.84RY) spreads

viscously beyond Rroche, leading to the gravitational coagulation of ice-rich bodies hundreds of km

diameter according to Canup (2010). If the coagulation efficiency is high, then the tidally-stripped

mantle of a single Titan-sized satellite could spawn hundreds of original MSMs by the proposed

mechanism. Today’s MSMs could derive from this progenitor population, assuming they can survive

the waning of the nebula, the inward migration of the final major satellites, and the subsequent

bombardment, as discussed in §1.2 and §1.5.2. With regard to the Sekine & Genda (2012) model, it

is also worth noting that when M1 disappears into Saturn in their model, the result should according

to Canup (2010) be a massive disk containing several times the mass of these newly-formed MSMs.

The influence of such a disk has yet to be considered in the context of the hit and run scenario, and

could lead to an alternative formation scenario.

In order to explain the most massive MSMs, Dione and Rhea, the disk models have to first form

them, which can occur just outside of Rroche, then must thereafter transport them from Rroche to

their present orbits (6.6−9.2RY) over the course of solar system history. Additionally they must avoid

one another’s sphere of influence, lest they merge through mutual collisions into one dominant body,

as is thought to have occurred in Earth’s protolunar disk (Ida et al. , 1997; Canup et al. , 1999). This

requires they start off with very rapid migration, which Canup (2010) proposes is recoil from the disk

in response to torques set up by inner Lindblad resonances. Once the moons have migrated beyond

the dominant inner Lindblad resonance, however, this recoil is no longer effective and the MSMs need

to migrate in some other way to attain their presently circularized and well-separated orbits.

Charnoz et al. (2011) develop a comprehensive model for the long term evolution of a massive

ice-rich disk around Saturn: first, its spreading beyond Rroche, then, the formation of the MSMs, and

lastly, their long term migration to the present orbits of Dione and Rhea. In addition to their rapid

initial migration from Rroche, three significant problems are addressed: the survival of the MSMs to

the present day, their semimajor axis distribution around Saturn, and the compositional diversity of

the MSMs. Concerning the latter, they develop a heterogeneous accretion scenario where silicate-

rich aggregates in the disk coagulate more rapidly than the ice-dominated aggregates, being more

28

dense, and thus become rock-rich cores. But it might prove challenging, in more detailed models, to

balance heterogeneous accretion against runaway coagulation about one most massive and dense core,

considering the rapid dynamical timescale.

The disk models of Canup (2010); Charnoz et al. (2011) have implications for geologic structure

that are ultimately helpful for discriminating among hypotheses. If Enceladus, Tethys and other

bodies coagulated around rocky nuclei and mantles of ice, perhaps in the manner one expects for

primordial cometesimals (though out of warmer crystalline materials), this needs to be shown to be

consistent with geological observations, bulk densities, and other geophysical constraints, to the same

degree that more geophysically motivated models need to be consistent with dynamical constraints.

The inner moonlets of Saturn (e.g. Pan and Atlas) do have the appearance of disk-accretionary

forms, consistent with their formation as gentle gravitational coagulates (Charnoz et al. , 2007; Hirata

& Miyamoto, 2012). The MSMs, on the other hand, include some of the most geologically complex

and diverse objects in the solar system, some of them active and apparently differentiated, implying

either primordial heating and formation in the first few million years (e.g. Castillo-Rogez et al. ,

2009), or intensive episodes of tidal heating (e.g. Stevenson, 2006; Zhang & Nimmo, 2009), or some

combination.

In the disk origin model of Charnoz et al. (2011), tidal dissipation inside of Saturn, together with

eccentric perturbations due to mutual satellite interactions, are responsible for raising the orbits of

the MSMs from their formation, near Rroche, to as far as 9.2RY in the case of Rhea. Tides expand

satellite orbits when the satellite raises a tidal bulge on the rotating planet. For a planet rotation

that is faster than the orbital period, the satellite lags the bulge by an angle ǫ. The quality factors

Qs and Qp (satellite and planet) express the inverse of the fractional dissipation of energy per cycle,

so that the lag angle 2ǫ ∼ 1/Q. The tidal Love numbers k2p and k2s are inversely proportional to the

material rigidity, so that k2/Q is the fundamental (though notoriously unconstrained) coefficient of

tidal dissipation.

Using a model based upon e.g. Kaula (1964); Peale et al. (1980) the following differential equations

are used by Charnoz et al. (2011) to obtain good fits to the distribution of MSMs interior to Titan.

They model their evolution starting just outside the Roche limit of a massive, spreading Saturnian

ice disk. To achieve these good fits, they must invoke unexpectedly intense dissipation inside Saturn,

something that has far-reaching consequences. In their model the semimajor axis as and eccentricity

es of a satellite s evolves in time t as

29

dasdt

=3k2pMsG

1/2R5p

QpM1/2p a

11/2s

[

1 +51e2s4

]

+2a

1/2s Γs

Ms(GMp)1/2−

21k2snsMpR5s

QsMsa4se2s (7)

desdt

=57k2pnsMsR

5p

QpMpa5ses −

21k2snsMpR5s

2QsMsa5ses + F (8)

Here Rp is the radius of the planet, Rs is the radius of the satellite, and ns =√

G(M+m)/a3 = 2π/P

is the mean angular motion of the orbit, where P is the orbital period. Both Q and k2 depend on

interior structure and composition.

What physically accounts for tidal dissipation inside of giant planets is largely unknown. It is

conjectured that Q may be frequency dependent by orders of magnitude, leading to resonant epochs

of intense dissipation (e.g. Stevenson, 2006). The sloshing of oceans or liquid layers, on the surface

or below, lead to low values of Q, so that Q ≈ 10 applies for the oceanic Earth (Goldreich, 1966)

and is presumed in models of an ocean-layered Enceladus by Matsuyama (2012). As for Q of the

giant planets, astrometry of the historically evolving orbits of the satellites of Jupiter (Lainey et al.

, 2009) leads to a value k2X/QX ∼ 10−5, implying QX ≈ 3 × 104 for Jupiter. This is at the lowest

(most dissipative) end of the previously expected range of QX (e.g. Goldreich & Nicholson, 1977).

Consequently, Qp of a gas giant is thought to be orders of magnitude greater than Qs of a satellite,

so that the second expression des/dt is generally negative, leading to circular orbits.

In these expressions, eccentricity forcing by external perturbers (other satellites, the Sun, Jupiter,

rogue ice giants) is represented simply by a scalar constant F . The angular momentum contribution

from e.g. rings or moonlet swarms is represented by Γs. Not included are mutual satellite interactions,

which Charnoz et al. (2011) treat using a separate ‘toy’ model for eccentric perturbations. It is

otherwise very challenging to treat mutual interactions explicitly in such a complex system, and

requires a fundamentally different hybrid N -body approach. Mutual interactions can lead to resonant

couplings and coordinated satellite migrations (e.g. Meyer & Wisdom, 2008; Zhang & Nimmo, 2009),

and if coupled migration dominates, then a more explicit approach is required to model the long term

dynamical evolution of the system.

A lower bound QY & 1.8× 104 was derived for Saturn by Peale et al. (1980) on the basis of the

assumption that Mimas originated outside the corotation radius Rc and migrated to its present orbit

at ∼ 3.2RY over solar system history. This is considered a lower limit on QY because Mimas would

otherwise have migrated further than this, if das/dt > 0. But much greater tidal dissipation inside

Saturn is required in order for Dione and Rhea to migrate to their present orbits at 6.6RY and 9.2RY,

respectively, and Charnoz et al. (2011) require QY ∼ 1.6 × 103 for their best fits. This is so much

30

lower than the Peale et al. (1980) value because of the dependence of das/dt on a−11/2s in Equation

6 greatly diminishes the effect of tidal migtation with distance.

Lainey et al. (2012) studied the detailed astrometry of Saturn’s moons, using an historical baseline,

to obtain k2Y/QY ∼ 2×10−5, or QY ≈ 103, in good agreement with the requirement of Charnoz et al.

(2011). The result is highly controversial, not only for the unexpectedly low value of Q, but especially

because the study concludes that Mimas is migrating rapidly inwards, coming closer to Saturn at a

rate das/dt ∼ −1RY per 50 Ma, opposite the Peale et al. assumption. If so, Mimas might have

formed by some other event – it is difficult to even speculate. Geology suggests this could not be

the case, because in order to spiral in at such a rate Mimas would have to be undergoing a level of

internal tidal dissipation far greater than the energy output of erupting Enceladus, while maintaining

the appearance of a rather ancient, quiet moon. Apart from Mimas, the Lainey et al. (2012) result

seems to require a profoundly different interior structure for Saturn than for Jupiter. Of course, this

might explain why the Saturn and Jupiter satellite systems are so different.

In summary, the models that create MSMs at the Roche limit of a spreading disk propose two

stages of post-formative dynamical evolution: First, an inner Lindblad gravitational interaction with

the disk pushes them out rapidly, and beyond one another’s influence. Next, over much longer time,

tidal interactions with the planet, and eccentric perturbations with one another, force their migration

out to their present orbits. An intense, almost Earth-like tidal dissipation inside Saturn is required

to explain the orbits of the MSMs out to Rhea. A separate origins scenario is required to explain the

satellites Hyperion and Iapetus beyond Titan.

4.2. Stochastic Formation

Stochastic formation – that is to say, formation by giant impacts – involves very different post-

formative circumstances and dynamical requirements . A primary advantage to stochastic formation is

that MSMs originate wherever collisions occur, instead of forming just outside of Rroche and migrating

hundreds of thousands of km by tidal torque. For instance, Mimas, Enceladus and Tethys may have

formed in one collision, and Dione and Rhea in another, and perhaps Hyperion in the final Titan-

forming merger. Thus a principal dynamical requirement is relaxed in our scenario compared to the

disk models, because the satellites might have been born close to their present locations.

An initial phase of scattering or rapid migration is required, both in our model and in the hit

and run stochastic model of Sekine & Genda (2012), because the MSMs start off on MF -crossing

orbits. Both models require that the newly-formed MSMs move well beyond the Hill sphere of MF ,

or vice-versa that MF be dragged away, on a timescale faster than the characteristic timescale of

31

accretion. A rough estimate of this timescale is obtained by assuming the largest body MF sweeps

through and accretes a particle disk of surface density Σd ≈ Mm/2πa2se, where ase is the radial extent

of the annulus of bodies, of total mass Mm at semimajor axis as, and where e is their characteristic

eccentricity e = vm/vk, and where vk is the Keplerian velocity√

GMY/as, and vm is the characteristic

random velocity of the moons relative to MF . The sweep up rate Mm from the torus equals the

accretion rate MF ∼ nsΣdπR2F (1+ v2

esc/v2

m), accounting for gravitational focusing. The sweep up time

is then τacc ∼ Mm/MF ∼ 2a2svm/[nsR2F (1 + v2

esc/v2

m)vk].

In Saturn’s case, the mass available to be swept up may be estimated as today’s mass fraction

of MSMs to Titan, or Mm ≈ 1/20MF . In the hit and run scenario, the encounter velocity vm is of

order the pre-encounter velocity ∼ vesc. In the graze and merge scenario, vm ∼ 1/3vesc based on

our simulations. The sweep up timescale is consequently a factor of ∼ 2 longer in the hit and run

scenario compared to our graze and merge scenario, otherwise the dynamics are similar. Assuming

RF ≈ Rtitan, then the sweep-up timescale is ∼ 105(as/RY)4 seconds for hit and run (vm ∼ vesc) and

twice as fast for graze and merge (vm ∼ 1/3 vesc).

For a collision that might have created the inner MSMs of Saturn, we can approximate as ∼ 5RY,

in which case the sweep up time τacc is of order 2 years, or τacc ∼ 1,000 orbital periods P . For a

Titan-forming merger, as ∼ 20RY and τacc ∼ 500 yr ∼ 10,000 P . A population of bodies is not

a uniform torus of matter, and at the tail end of the distribution of encounters a fraction of the

escaping MSMs will not be swept up so quickly; nevertheless some further process is needed by which

a fraction of these newly formed moons traverse a few Hill radii RH away from MF in ∼ 103 − 104

orbits. Once outside its immediate influence, other kinds of evolution (e.g. tidal migration) can play

out over billions of years.

We identify and evaluate five possible scenarios that might account for the rapid initial scattering

and migration in the aftermath of giant satellite impacts. For context we focus on our scenario of

graze and merge collisions. Any or all of the following mechanisms might apply together or sequen-

tially, depending on factors such as timing (primordial or late), the orbital distances where successive

mergers take place, the presence or absence of debris disks and yet-to-be-accreted massive satellites,

and perturbations from outside the Saturn system. The variety and uncertainty of the candidate

mechanisms, and the complexity of their potential interactions, requires a detailed study beyond our

present scope.

32

4.2.1. Gas-driven migration

Collisional mergers of original major satellites might have occurred as the last phase of accretion

around Saturn, in the presence of nebular gas. If so, then post-collision satellite migration could be

dominated by the same nebula interactions that led to the collisional mergers in the first place. Type I

migration being mass dependent, MF would be dragged towards Saturn after the collision much faster

than the MSMs, leaving them behind. In the scenario proposed by Sekine & Genda (2012) MF spirals

in until it accretes with Saturn; in our model MF only needs to migrate several RH on a timescale

faster than it would accrete the remaining pieces of M2, a few thousand orbits.

This rate appears to be consistent with the differential rates of migration in published simulations.

In Ogihara & Ida (2012) migration rates are typically several planetary radii per 105 Kepler periods, of

order ∼ 1Rp per 1,000 years at 20Rp. Although this a few times slower than required by the estimates

above, the uncertainties of the initial conditions and the various model parameters (especially, the

parameterized disk viscosity α) make it plausible that Type I migration, perhaps in conjunction with

the mechanisms described below, could have led to well separated orbits in the aftermath of primordial

final mergers.

The problem with differential Type I migration is that it leaves the MSMs exterior to MF , when

in fact most of Saturn’s MSMs are interior to Titan. So while the MSMs can in principle be separated

from the immediate sphere of influence of MF on a sufficiently rapid timescale by this mechanism,

they would find themselves exterior to the major body’s orbit. This might conceivably work for

Iapetus and Hyperion (see below), forming in an early collision where MF migrates inward leaving

them behind. But a different mechanism is required to explain the inner MSMs – for instance, a ‘lost

Titan’ spiraling into Saturn, in the manner of Sekine & Genda (2012).

As noted above, if the inner MSMs formed primordially, then they might not have survived a

canonical Late Heavy Bombardment (e.g. Nimmo & Korycansky, 2012). Therefore, in addition to

Type I migration, which might be sufficient, we now turn to mechanisms that can work in the post-

nebula gas free environment.

4.2.2. Tides

The early Moon may have experienced semimajor axis evolution of order tens of km per year (c.f.

Touma & Wisdom, 1994), in the ballpark of what is required. However, the mass ratio (first term in

Equation 7) is significantly greater in the Earth-Moon case, so that around Saturn such rapid rates

of tidally-driven migration are impossible. The model of Charnoz et al. (2011) requires QY ∼ 1600;

this is already orders of magnitude smaller than the commonly accepted range of values for Saturn.

33

A migration rate several thousands of times faster, as required to achieve dynamical separation from

MF in a few thousand orbits, implies that QY ≪ 1 (since QY ∼ 1/as), so we can simply rule it out.

Inward migration by tidal dissipation within the MSMs would likewise occur at a rate far slower than

required to achieve sufficiently rapid initial dynamical separation from MF .

Tides do, of course, play a major role in semimajor axis evolution over geologic time. There is

considerable evidence of this process at work, for instance in the apparent size-ordering of MSMs with

semimajor axis, and in the orbital coupling of the satellites (e.g. Meyer & Wisdom, 2008; Zhang &

Nimmo, 2009). If MSMs form at a few discrete locations during successive mergers, then the required

magnitude of tidally-driven migration is greatly reduced, so that tidal evolution could represent a

final adjustment of Saturn’s satellite system achieved by a more traditionally accepted value of QY,

with stochastic events shaping its initial architecture. But we are still left to explain the dynamical

separation of MSMs from MF .

4.2.3. Disk collisions

Another possible mechanism for rapid initial migration, is the collision by the MSMs each periapse

with an interior particle disk, that might have been spawned by the collision or been pre-existing. This

mechanism would slow down the MSMs since the circular velocity vk is slower than their periapse

velocity at q. The interior disk would thus damp the orbital velocity in inverse proportion to the

satellite radius Rs, and in direct proportion to the disk density Σd and the mean motion ns. The

source of the interior particle disk could be residuum from the final stage of mergers and collisions, or

it could be e.g. the remains of a massive disk created by tidal disruption (Canup, 2010). If occurring

at around the time of the LHB, it could also be a collisional disk formed by the impact disruption of

satellites (Charnoz et al. , 2009).

And so, there are various reasons to expect an interior particle disk, and inward-driven drag would

act to order the MSMs appropriately in size. However the mechanism would have to be studied in

detail to understand whether sufficient rates of migration could be achieved without destroying the

MSMs, or accumulating them into one large interior satellite. Additionally, there should be a geologic

signature to this form of migration. It also remains to be studied whether a disk that is sufficiently

massive to exert sufficient drag force on the MSMs, would itself be gravitationally unstable and

accumulate into moons.

4.2.4. Mutual collisions

A third candidate mechanism is mutual collisions among dozens or perhaps hundreds of precursor

MSMs, and thousands of smaller new satellites that are likely to be produced in these energetic

34

mergers, only the largest of which are resolved in Figure 5. A more numerous but smaller sized initial

system of moons is consistent with several of our simulation outcomes (Figure 6). In this scenario,

precursors to the MSMs are formed by the merger of M1 and M2, and accumulate into larger bodies

(finished MSMs) in subsequent pairwise encounters.

Because collisions in a swarm occur preferentially at periapse (e.g. Wyatt et al. , 2010), mutual

collisions tend to circularize these satellite orbits interior to MF . With random velocities vm ∼ 1 km/s

they would collide at a few to several times their escape velocity, so that the expectation (Agnor &

Asphaug, 2004; Asphaug, 2010) is for a mix of hit and run and partial accretion events, plus some

fraction of erosive events. The inefficiencies of accretion for these random velocities would serve to

expand the torus of debris around MF and potentially accelerate the migration of the remaining bodies

according to the previous mechanism.

As noted, a final stage of MSM accumulation is consistent with Rhea, the most massive, having a

bulk composition that is about the same as the bulk composition of the system of MSMs. But this

is also one of the scenario’s potential shortcomings, for it may be difficult to preserve a nearly pure

ice body such as Tethys if it is built by the mergers of several random bodies, some of which might

be silicate-rich. The larger MSMs should all be of average composition, if randomly assembled from

progenitors. It might furthermore be difficult to preserve the relatively monomodal size distribution

of the MSMs, seeing as one largest body could dominate the mass accretion.

4.2.5. Resonant scattering

The timescale of resonant scattering can be short in the presence of several massive migrating

satellites. In our scenario the MSMs are born into a strongly coupled 1:1 interaction with MF .

If there was initially a Laplace-like resonance set up among the major satellites, then the MSMs are

born into a mean motion resonance with any yet-to-be accreted major satellites as well. The combined

interaction might be expected to scatter some fraction of them away from MF , and if so, the retention

process following each merger could be stochastic and lossy. This would be the case whether it is

primordial gas, or some other perturbation, that causes the largest bodies to migrate.

Study of this scenario requires numerical integrations of the N major bodies (the major satellites

and the family of MSMs) evolving in time, likely in the presence of a particle disk, and (if primordial)

in the presence of a waning nebula. In lieu of direct modeling of this post-formation dynamical

evolution, which is well beyond our scope, we reason by analogy with solar system formation models

that scattering could chaotically expand the architecture of the system. One immediate analogy is the

Nice model’s proposed origin of Uranus and Neptune, two ice giants tossed out of the planet forming

35

region (along with the trans-Neptunian population) during the 2:1 mean motion resonance between

Jupiter and Saturn (Tsiganis et al. , 2005). If the orbital evolution and architecture of the solar system

is the result of dynamical instabilities related to resonances among the major bodies, then the same

may be true of MSM populations evolving in the presence of massive galilean satellites in resonant

orbits about Saturn. Also relevant is the study of evolving exoplanetary systems by Chatterjee et al.

(e.g. 2008), who find that resonant scattering can eject smaller planets (analogous to the MSMs in

our scenario) and leave behind larger planets with increased eccentricity (analogous to the eccentric

Titan, perhaps).

4.3. Forming Iapetus

Iapetus is the second most massive of Saturn’s MSMs after Rhea. It is one of the most ice rich,

and by far the most distant (as = 62RY). Iapetus is one of the strangest moons geologically, with

its gigantic equatorial ridge and its bizarre black-white coloration, and it is thought to be one of the

most ancient (Castillo-Rogez et al. , 2009). Models for Saturn system formation cannot ignore the

origin of such a massive distant body, and theories are challenged by the fact that Titan would be

an impervious barrier to outward migration; this would seem to preclude its formation out of a inner

ice-rich disk.

Mosqueira et al. (2010a) argue that Iapetus is the key to understanding Saturn’s moons, and

propose that it formed from the capture of icy planetesimals by an extended gaseous subnebula about

Saturn. However, it has been noted that the large orbital inclination of Iapetus, almost 15°, is not

an expected consequence of disk accretion of captured planetesimals. Ward (1981) proposes that this

inclination arises from the warp in Saturn’s Laplace plane, in the context of a rapidly evolving disk,

while Mosqueira et al. (2010a) argue that it arises from post-formation stochastic interactions.

Forming Iapetus as a massive clump from a binary merger such as Figure 5 could in principle

account for its high inclination, as well as its very ice rich composition. In our model, the original

eccentricities and inclinations of MSMs are expected to increase with increasing distance from Saturn,

because the final v∞ of the clumps, relative to MF , do not depend on the semimajor axis as of the

collision, whereas the Keplerian velocity vk decreases with as. Consequently the MSMs forming far

from Saturn, by this process, may be expected to begin with high inclinations.

Assuming that the impact parameters of circumplanetary collisions are reasonably isotropic, in

the frame of the target satellite M1 (e.g. Alvarellos et al. , 2005; Zahnle et al. , 2008), then in typical

collisions a substantial component of the impact parameter is out of the plane. Escaping clump

velocities v∞ ∼ 1 km/s (from Figure 6) would correspond to inclinations i & 10° for MSMs originating

36

at ∼ 60RY, where the orbital velocity is only vk = 3 km/s. It may seem counter to this model that

Iapetus has no companion MSMs, given that many of our simulations end up with families of ice-rich

bodies. But not all mergers produce large populations of MSMs, and it is not at all certain that all of

the bodies that are produced would survive, especially if Iapetus formed as debris from a primordial

merger.

A binary merger formation scenario for Iapetus would seem to require that that progenitor satellites

of Saturn orbited initially at distances ranging out to much farther than the ∼ 5− 30RY adopted by

Ogihara & Ida (2012) for their starting distances for ‘satellitesimals’ being fed into their simulations.

The primary goal of Ogihara & Ida (2012) is to explain the Jupiter system, not Saturn, so it may be

that different starting conditions would apply. But also, it is possible that scattering from multiple-

satellite interactions would strand some MSMs at greater distances than the formation region – Iapetus

might in this case be a sort of “KBO” of the Saturn system. Apart from assessing the consistency of

such conjectures, they must be properly explored using explicit integrations of the complex system.

5. A Late Origin

It seems unlikely that the innermost MSMs could have survived the Late Heavy Bombardment

(Charnoz et al. , 2009; Nimmo & Korycansky, 2012). But there are other ways to explain the late

lunar cataclysm (e.g. Chambers, 2007) that do not require a cataclysm at Saturn, and as emphasized

by Agnor & Lin (2012), the planetary interplay that led to the expansion of the solar system in the

Nice model (Tsiganis et al. , 2005) could well have occurred much earlier than the ∼ 3.8 − 4.0 Ga

timeframe proposed by Gomes et al. (2005). So there might never have been a system-wide LHB,

which is certainly one way of explaining the MSMs’ survival.

Another way out, proposed by Charnoz et al. (2011), is for Saturn’s MSMs to have formed

during or after the LHB, thereby postdating the peak of the bombardment. We now consider whether

a late origin – in this case, coinciding with the LHB – can apply to our model. Specifically, we

consider whether an original system of Saturnian satellites, initially in a Laplace-type resonance,

might have stored transformative quantities of gravitational binding energy until powerful external

perturbation triggered a late dynamical collapse. Apart from explaining the survival of the MSMs

through the LHB, geophysical motivations for considering a late origin include the youthful-seeming

geology, icy composition, and dynamical excitation of Saturn’s satellites. The dynamical motivation is

that Laplace-type resonances appear to be an almost universal end state for satellite formation around

gas giant planets, in simulations (Ogihara & Ida, 2012). A resonant chain at Saturn, proportional to

Jupiter’s, would have been more loosely coupled.

37

Long range external perturbations are inefficient at modifying satellite orbits, however, because

regular satellites orbit deep within the planet’s Hill sphere. Still, one finds prospects in the recent

literature for at least three possible mechanisms. First is the gravitational scattering of satellites

by planetesimals fluxing inside the planet’s Hill sphere. If scattering by planetesimals can exert a

sufficient influence to move giant planets by tens of AU (e.g. Hahn & Malhotra, 1999) then the same

planetesimals fluxing through the satellite system might influence satellite orbits to a comparable

degree, apart from any direct collisions that may occur.

A second possible long range perturbation is Jupiter’s resonant gravitational forcing. Over many

thousands of encounters, Jupiter in its 2:1 mean motion resonance (or 3:2; Batygin & Brown, 2010)

might have driven the eccentricities of one or more of Saturn’s original satellites. During these resonant

interactions, Saturn’s original satellites would have been perturbed by Jupiter to a much greater extent

than Jupiter’s moons were perturbed by Saturn, accounting for the fact that only the least massive

system went unstable.

We also consider the ‘jumping jupiter’ variant of the Nice model (Morbidelli et al. , 2009; Brasser

et al. , 2009), introduced to avoid destabilizing the orbits of the inner planets while not overly damping

the inclinations of the giant planets. According to this model, Jupiter and Saturn migrated impulsively

by discrete interactions with roaming ice giants. In one characteristic scenario (Brasser et al. , 2009)

“Saturn’s semi-major axis evolves rapidly to ∼ 9.2 AU and Uranus’ semi-major axis to ∼ 6.5 AU.

Jupiter first exchanges orbits with Uranus: Jupiter moves out to 5.52 AU while aU reaches 3.65 AU

and the perihelion distance of Uranus, qU , decreases to ∼ 2 AU. [...] Then Jupiter scatters Uranus

outwards to aU ∼ 50 AU, itself reaching aJ ∼ 5.2 AU .” Although the perturbation of satellite

systems during such encounters has yet to be studied, analogous studies of solar systems experiencing

close stellar encounters (e.g. Holman & Wiegert, 1999) suggest that a close-passing ice giant could

potentially destabilize Saturn’s moons.

If giant planet migration led to a solar system wide Late Heavy Bombardment (the Nice model)

then the survival of ice-rich Mimas and Tethys and other bodied needs to be explained (e.g. Nimmo

& Korycansky, 2012). The issue of survival is resolved if MSMs are formed as a consequence of the

same solar system chaos. Moreover, there is a geophysical appeal to a late origin when it comes to

explaining the active geology of Enceladus (Spencer et al. , 2006), Titan’s anomalous eccentricity (e.g.

Sohl et al. , 1995), and other unique aspects of the Saturn system. Despite these merits, the scenario

of a late dynamical collapse (∼ 4 Ga) of an original Saturn system is speculative until the effects of

giant planet migration on satellite evolution are modeled dynamically. ‘Late stage’ giant impacts, at

38

the end stage of primordial accretion (c.f. Ogihara & Ida, 2012), may form the MSMs primordially,so

if their survival can be otherwise explained, there is no compelling reason for tying our hypothesis to

the Nice model.

6. Titan

Titan is the last of the final merged bodies (MF ) according to our model, the consequence of

several giant impacts. For simplicity we inspect the same simulation as in Figure 5 and assume it

is representative of a final Titan-forming merger. Here, two bodies of mass ratio γ = M2

M1+M2

∼ 1/4

merge at vimp = vesc = 2.6 km/s. The final Titan-forming merger could of course have been different,

e.g. relatively head on (not forming MSMs, perhaps only Hyperion), or involving bodies more equal

in size (γ ∼ 1/2), or at higher velocity (e.g. v∞ ∼ vesc). Certainly, the internal structures of M1 and

M2 may have been different from the perfectly differentiated bodies assumed in the calculation. Both

M1 and M2 would be the result of previous mergers, and would bear the record of those prior events

in their structure and thermal state.

As described in §3, the bulk of the silicate mantle of M2 and all of its inner core are deposited into

the deep interior of M1. These sink through the ice to form sub-layers in the core and mantle, as seen

in Figures 5 and 8. New layers might later be convectively mixed or otherwise diffused, or perhaps

hardened as strata. Predictive studies of the sheared-out core and mantle of M2 sinking through the

ice of M1, and eventually mixing, are well beyond the timeframe and the physical capabilities of SPH,

and require other numerical approaches (e.g. Gerya & Yuen, 2007). Much of the gravitational binding

energy in this merger (∼ 25% above the binding energy compared to a perfect merger; Equation 3)

goes into the spin-up of Titan, the escape of the MSMs, and the orbital energy of clumps captured as

subsatellites (the bodies inside the red line in Figure 5e), as well as the binding, thermal and rotational

energies of each clump. MF is heated by shocks and friction, and by adiabatic compression. There is

so far no general study of the energy budget for similar sized collisions.

Titan would be excited in its orbit by this collision, and the merger would give it an eccentricity

that could be several times that of the present day. While the dynamical exchanges leading up to

a satellite collision are complicated and specific to the encounter, and probably chaotic, an order of

magnitude estimate of the effect of giant impact can be made (following Sohl et al. , 1995) by assuming

that M1 (proto-Titan) remains on a circular coplanar orbit until the collision. The final eccentricity

then corresponds to the change in orbital velocity ∆v ≈√

GMY/as(√1 + e−1); consequently today’s

e ≈ 0.03 corresponds to ∆v ≈ 80 m/s. By comparison, the characteristic ∆v associated with the

39

binary merger in Figure 5 is several times larger, of order the mass ratio times the escape velocity,

γvesc ∼ 500 m/s. To first order, a Titan-forming merger can easily account for Titan’s high orbital

eccentricity, and contribute a long term source of internal heating (e.g. Tobie et al. , 2005).

FIGURE 8 NEAR HERE

Far less energy is contributed by direct impact heating, than is contributed via excited spin and

eccentricity that lead to tidal heating. But impact heating is delivered in hours to days. Figure 8

shows the same time-step as Figure 5e, plotting the rise in temperature ∆T according to the ANEOS

equation of state. Shock heating is represented by the code’s artificial viscosity model, which solves

the Hugoniot shock relation by decelerating rapidly convergent particles and causing heating du and

compression. The method is not designed to model frictional heating, although this may be dominant

at these relatively low velocities (hundreds of m/s). A further caveat is that artificial viscosity can

inhibit the mixing of condensed phases, especially at low resolution, and can generate a shear viscosity

that can dominate the diffusion physics especially in a 2D continuum such as a disk or sheet. Also,

real progenitors would be more complex and have more complicated interactions. So we exercise

caution in interpreting the thermal results, and show them primarily because they trace the shock

and interfacial dynamics of the simulation.

Heat and mass deposition are hemispheric in this giant impact, involving a characteristic temper-

ature increase of & 500 K according to the simulation. The specific impact energy 1/2ρv2imp is equal

to the energy of complete vaporization of ice for vimp ≈ 2.6 km/s, causing shock heating to occur in

the colliding surface materials. Titan ends up with a steam atmosphere for some time, present in our

simulations but not well resolved. The bulk of the internal heating occurs where the mantle and core

are decelerated; they are sheared into pancakes and have attained their depths of neutral buoyancy.

Slugs of dense materials (rock and iron) crash through thousands of kilometers of ice and rock inside

M1, at hundreds of meter per second. The mantle acquires an outer heated region composed of the

outer mantle of the impactor, while the core of the target acquires an outer region that is the projec-

tile core. Because the compositions of M1 are modeled as identical to those of M2, the mantle and

core of M2 are comparatively less dense, originally, in our simulations than those of M1 (Figure 4).

Mixing, along with expansion and degassing, are likely to be intense processes during and following

the collision, but we are not capable of modeling these processes accurately in a 3D simulation.

The source of Titan’s geological activity may be internal (e.g. Sohl et al. , 1995) or exogenic

(Moore & Pappalardo, 2011). Formation by late accretion is both: depositing an internal heat source

exogenically. The immediately buried heat (Figure 8) could sustain geologic activity over billions of

40

years inside of a non-convecting Titan, and for tens of Ma if the structure is convective (Grasset et al.

, 2000; Monteux et al. , 2007). After this transitory heating comes a longer term source of heating

from the intense tides raised by the excited rotation of Titan (MF has a period of about 6 hours in

most simulations) and its post-collision orbital eccentricity, expected to be e & 0.1 following a merger

according to the estimates above. Dissipation of impact-induced rotational and orbital excitation

could last for billions of years (e.g. Tobie et al. , 2005).

Another source of energy is the fall back of the collection of subsatellites that are bound to Titan

following the final merger. These bodies, ∼ 100 − 1000 km diameter in the simulations, might be

solidified by the time of their reaccumulation, and the events would occur randomly in time. Because

their collision velocities into MF are slow (∼ 1 − 2 km/s), the accretions might result in a “splat”

(Belton et al. , 2007; Jutzi & Asphaug, 2011), leading to a crustal or compositional anomaly along the

equator if Titan had solidified to ∼ 100 km depth by this time. The collection of subsatellites that

is formed, is sensitive to the specifics of the collisional encounter, and a final Titan-forming collision

may have ended up quite different from Figure 8.

A sequence of similar sized collisional mergers forming Titan might have provided sufficient global

heating to result in a completely differentiated body (e.g. Monteux et al. , 2007). This is a critical

consequence of our model because it is often argued that complete differentiation can be ruled out

by Titan’s high moment of inertia. But it is also possible that the mass of Titan is dominated by

a mantle of hydrated silicates (Scott et al. , 2002; Castillo-Rogez & Lunine, 2010; Fortes, 2012), the

consequence of global shock and frictional heating, pressure loading and unloading of collisions and

close encounters (Asphaug et al. , 2006), and the shearing and folding and turbulent mixing of the

layers of M1 and M2. With most of the water bound in hydrated mantle silicates, there would less of

a low density ice shell.

Although speculative until the orbital dynamics are better understood, the scenario has advantages

in accounting for why Titan stands apart geophysically from the Galilean satellites of Jupiter. It

accreted differently from them, in a few similar-sized collisions. Jupiter’s Galilean satellites attained

primordial stability in the Laplace resonance, which is thought to have survived dynamically to the

present day. They consequently suffered few if any of these types of collisions involving similar-sized

bodies. According to dynamical models, once satellites accrete to galilean dimensions (by the accretion

of much less massive components γ ≃ 0) they migrate rapidly into resonant orbits (Peale & Lee, 2002;

Ogihara & Ida, 2012) and are sheltered from mutual collisions. If something triggered final collisions

forming Titan, it would evolve fundamentally differently according to Equation 3.

41

7. Conclusions

The following peculiarities of Saturn’s middle-sized moons (MSMs) (e.g. Porco et al. , 2005; Ostro

et al. , 2006; Thomas, 2010) are diagnostic of their origin and evolution. But the pieces of the puzzle

are hard to put together:

• Several are composed almost entirely of water ice.

• There is no apparent relationship between their size and composition.

• The icy innermost MSMs have avoided a calamitous impact bombardment.

• The bulk composition of the largest, Rhea, is the average for all the MSMs.

• The smallest MSMs are incredibly diverse (Mimas, Enceladus, Hyperion).

• The most rock rich, Enceladus, is active today.

• Several of them show past tectonic activity and nonhydrostatic shape.

• Some show evidence for sub-satellites and rings.

• Radar observations reveal considerable variation in near-surface properties.

The dynamical pairings of Saturn’s moons are also interesting: Of the five MSMs interior to Titan,

the two iciest and the two rockiest are in 2:1 mean motion resonances. Saturn also has two pairs

of Trojan moons that are leading and trailing Tethys and Dione. These dynamical couplings could

be the result of orbital evolution over billions of years (Meyer & Wisdom, 2008; Zhang & Nimmo,

2009), or could be related to MSM formation. Beyond Titan, the satellites Iapetus and Hyperion are

also thought to hold important dynamical clues to the system’s origin (Peale, 1978; Farinella et al.

, 1997; Ward, 1981; Mosqueira et al. , 2010b), together with geological clues. Icy Iapetus, orbiting

at 62RY near the 5:1 mean motion resonance with Titan, has a stunning black-and-white surface

coloration (the feature that caused it to disappear in Cassini’s telescope) and an enormous equatorial

ridge. Hyperion is another oddball, an irregular (∼ 360× 270× 210 km) porous ice (ρ = 0.57 g cm−3)

body in a 3:4 mean motion resonance with Titan, and tumbling in a chaotic rotation (Wisdom et al.

, 1984).

In search for a formation mechanism that is consistent with these observations, we propose that

Saturn originated with a system of massive satellites similar to Jupiter’s, and that this system was

rendered unstable, leading to a phase of giant impacts not experienced at Jupiter. This led to the

42

final accretion of Titan in place of a Galilean-like system, leaving behind MSMs as clumped remnants.

These collisions may have occurred when Type I migration was still active, or it may have happened

‘late’. If the cause was the same chaos that drove giant planet migration (e.g. the Nice model) then

MSM formation would postdate the LHB. However, our hypothesis is agnostic of the Nice model.

The SPH simulations presented here of icy satellite collisions are comparable in many respects to

previous simulations of scenarios explaining the Pluto-Charon system (Canup, 2005) and the dwarf

planet Haumea (Leinhardt et al. , 2010). The resemblance between MSMs and KBOs in size, icy

composition and overall diversity is remarkable, and it is not farfetched to speculate that KBOs

themselves had a similar origin, being formed as spun-off remnants from the collisional mergers of

larger planets, for instance the roaming ice giants or their progenitors.

In the case of merging galilean satellites, we find that moderately off axis collisions generally lead

to the formation to MSM-like bodies. A primary unknown is whether these bodies can attain stable

orbits before they are accreted by the merged satellite MF in a few thousand orbits. By analogy with

the strong observational and theoretical evidence for planetary scattering during planetary accretion

(e.g. Chatterjee et al. , 2008), we argue that a fraction will survive; other modes of migration and

interaction will further modify their orbits. The problem requires a detailed dynamical investigation.

Fortunately, the geological and geophysical predictions of our model can be tested against the

increasingly well resolved record of Saturn’s moons. The MSMs are formed with rapid rotations,

Prot ∼ 4 − 9 h, in our models, many times ending up with their own subsystems of satellites. The

late accretion of subsatellites could in principle lead to equatorial anomalies on Titan, Iapetus and

Rhea, possibly consistent with their non-hydrostatic figures (Thomas, 2010; Nimmo et al. , 2011) and

related oddities (Schenk et al. , 2011). Their overall ice-dominated composition is consistent with

their deriving primarily from the exterior layers of M2, and the diversity of their composition and

geology is consistent with several of them incorporating silicates dredged up from kilobar pressures, a

P − T range that might encompass fundamental ice phase transitions including domains of hydrated

silicates (Scott et al. , 2002) and clathrates. Encaladus could be feeding off this enthalpy by a pathway

of decomposition (e.g. Kieffer et al. , 2006) while large, low-density moons like Tethys could derive

from ice shells or ammonia-rich oceans to undergo quieter evolution.

Titan would start with a fast rotation and an excited orbit if it formed in the merger of similar-

sized moons. This would leave Titan with a storehouse of energy (Sohl et al. , 1995) to be released

over millions or billions of years by tidal dissipation (Tobie et al. , 2005). The phase of thermal and

compositional readjustment could last for the duration of accretion, as studied by Monteux et al.

43

(2007). Shock and dynamical friction have a more immediate effect, melting and vaporizing surface

ices and depositing heat (along with the bulk of M2) deep into the lower mantle and outer core of

M1. This intensive deep heating might have led to a differentiated body, ice being easy to melt but

collision velocities being relatively slow in the case of galilean satellites. Once partial melting begins,

then shear dissipation increases during tides, further increasing the internal temperatures. All in all,

global melting may be an inevitable consequence of giant impacts occurring to build Titan. If so,

then the rather large moment of inertia of Titan, C/MR2 ∼ 0.34 (Iess et al. , 2010) could require a

hydrated mantle (Scott et al. , 2002; Castillo-Rogez & Lunine, 2010; Fortes, 2012), perhaps consistent

with repeated global-scale collisional mixing and shock and frictional heating of ices and silicates.

Acknowledgements: E.A. was supported by NASA’s Planetary Geology and Geophysics Pro-

gram under the award “Small Bodies and Planetary Collisions”, and by the University of California,

Santa Cruz. A.R. was supported by the Swiss National Science Foundation and the University of

Bern. We are grateful to our colleagues for their invaluable advice and criticism, and the hard work

of the referees, and critical feedback from Francis Nimmo and Sébastien Charnoz among others.

44

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Figure 1: A diversity of moons. Montage of the regular satellites of Saturn at common phase angle (∼ 45°) and toscale. Clockwise from Titan (upper right) are Iapetus, Hyperion, Enceladus, Tethys, Dione and Mimas, with Rhea inthe center. NASA/JPL/SSI; montage by E. Lakdawalla.

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Figure 2: Satellites of Saturn plotted as a percentage of the total mass. Rhea, Iapetus, Dione and Tethys are thefour most massive wedges, in order, and range from pure ice to half rock; they orbit at as = 9.2, 62.1, 6.6 and 5.1RYrespectively. Enceladus, Mimas, Hyperion and Phoebe, which together are the last visible sliver, are equally diverse.Explaining the middle-sized ice rich moons has led to scenarios including accretion out of a massive ice disk (Canup,2010; Mosqueira et al. , 2010a; Charnoz et al. , 2011) and hit and run collisions (Sekine & Genda, 2012). We proposethat they are the residues of an inefficient final accretion that occurred at Saturn, but not somehow at Jupiter.

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0 0.1 0.2 0.3 0.4 0.5

Impactor Mass / Total Mass

0

10

20

30

Delta E

(%

)

Figure 3: Binding energy available after a perfect merger, expressed as a percentage of the final binding energy (Equation3), is a function of the mass ratio γ = M1/M1+M2 of the collision (Equation 3).

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Figure 4: Equilibrium structures of spherical differentiated bodies of total mass (a) 0.33, (b) 0.50, (c) 0.75, and (d)1.00Mtitan. For simplicity of the modeling and interpretation, they are composed of a completely differentiated iron core(15 wt%), an inner silicate mantle (35 wt%), and an outer H2O mantle (50 wt%). Pressure equilibration is establishedusing a 1D Lagrangian hydrocode, further relaxed in 3D SPH. We then choose M1 and M2 from these four objects tostart our collision simulations. There is no consensus on the internal structure of Titan, let alone its precursors, so weassume a relatively high entropy interior corresponding to liquid phase in the M-ANEOS equation of state. Plotted arethe pressure p, temperature T and density ρ as a function of depth r inside the initial satellite.

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Figure 5: Five consecutive snapshots (a-e, clockwise) of a 1:3 mass ratio collision (γ = 0.25) occurring between satellites(a) and (d) in Figure 3 at their mutual escape velocity vimp = vesc = 2.6 km/s. The target is approximately themass and bulk density of Titan in this simulation. The impactor M2 comes in from the upper left and first contacts atθimp = 75◦. Gravitationally bound objects (clumps) are indicated by solid white circles whose diameter correspondsto a sphere of the object’s mean density and center of mass. The dashed green circle shows the Roche limit for icymaterial around the largest remnant, M1 → MF . (a) M2 grazes M1 at t = 0. (b) M2 captured in orbit about M1, nowback in an almost hydrostatic configuration. (c) A second collision begins at t = 23h where M2 is tidally sheared intoa spiral arm structure. The outermost parts of this arm are accelerated by the inner parts of the arm and by the coreof M2, and escape. (d) At t = 31 hr the impactor hits the target again and forms a second spiral arm. The clumpscoming from the inner part of the second spiral arm become temporary satellites of MF . The fast clumps, to the leftof the red line in (e), escape at vej > vesc and are tallied in Figure 6 as potential MSMs.

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(a)

(b)

(c)

Figure 6: Diameters of the escaping clumps that are formed in collisions such as Figure 5, scaled to the sizes andassumed compositions of Saturn’s MSMs (top row, from Thomas 2010). The dark inner circle shows the rock fraction,if present, assuming total differentiation, for a density ρrock = 3 g cm−3 and ρice = 0.93 g cm−3. Simulations arerestricted to low velocity collisions vimp = vesc with impact angle 60◦ ≤ θ ≤ 85◦, and for varying γ. The small numbersare the ejection velocity vej of each of these escaping clumps in km/s above the escape velocity of MF – that is, thefinal orbital velocity relative to MF . The initial colliding masses, from Figure 4, are (a) γ = 0.4, M2 = 0.011M⊕,M1 = 0.017M⊕; (b) γ = 0.33, M2 = 0.008M⊕, M1 = 0.017M⊕; (c) γ = 0.25, M2 = 0.008M⊕, M1 = 0.023M⊕.

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Figure 7: Locations inside of M1 and M2 from where each of the identified escaping clumps derive, in the γ = 0.25collision of Figure 5 which is the 75° case in Figure 6(c). Circles outline the silicate cores. The candidate MSMs arecolor-coded, so that their originating particle locations are identified inside of M1 and M2 . Histograms for the changein hydrostatic pressure ∆P are plotted in the top row, characterizing the radial depth inside M2 from which material isexhumed. Almost all MSM-forming material originates from the mantle of M2, for angular momentum reasons explainedin the text. Pressure release of ∼ 0.5− 1 GPa corresponds to an originating depth ∼ 300 km inside of M2. The secondmost massive clump (second from left) includes silicate material extracted from the base of the mantle, ∆P ∼ 3 GPa,suggesting a potential source of enthalpy for Enceladus-like bodies.

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Figure 8: The computed change in temperature ∆T in the final merged body MF for the simulation in Figure 5.Also shown are the bound sub-satellites of MF ; these resemble MSMs in most respects and form in the same spiralarms. A steam atmosphere is present around MF but not well resolved. The deep silicate portion of MF is heated by∆T ≈ 500−700 K, caused when the impactor rock merges into the target rock and abruptly decelerates. The depositionof heat appears hemispheric and is deeply buried, but see caveats in the text regarding the fidelity of the heat budget.This represents initial conditions for post- and syn-collisional thermal, dynamical and geochemical evolution, requiringa hand-off to other methods.

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