Late Origin of the Saturn System
Erik Asphaug1,2 and Andreas Reufer3
1 School
of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287, [email protected]
2 Department
of Earth and Planetary Sciences, University of California, Santa Cruz, CA, 95064
3 Physikalisches
Institut, University of Bern, Switzerland, [email protected]
Abstract
Saturn is orbited by a half dozen ice rich middle-sized moons (MSMs) of diverse geology and composition (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 compositional 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 collision, 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 H2 O-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 H2 O by mass, indicated by their global bulk density
2
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 (ρT ethys = 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 H2 O 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 = 20.3 RY ) but otherwise comparable in mass and semimajor axis to Ganymede and
Callisto. The orbit periapsis 19.7 RY is closer to Saturn than the apoapsis 20.9 RY 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 deepseated 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.
3
1.2. Dynamical motivations
In addition to these studies of the shape, mass, geologic history and composition of MSMs, dynamical 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 dissipation, 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
4
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; Nesvornỳ & 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
5
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 survived 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 equilibrium 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
6
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.
7
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 ∼ 5 Rp − 30 Rp where Rp is the radius of
the planet. These final distances are a direct consequence of the choice of ∼ 30 Rp 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/M R2 = 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
8
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
2
specific energy ∝ vimp
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. Hitand-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
9
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% H2 O 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
10
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 combination 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 Titanlike satellite orbiting Saturn at around 5 RY . 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 H2 O, 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 icerich 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
p
2 + v2
km/s. Because vimp = v∞
esc 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 ∼ 5 RY (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 icerich 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 approximation 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 rotational 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 =
p
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)
dm is the work required to pull the matter apart
r
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 = −3/5GM 2 /R = −kM 5/3 for constant k, so
5/3
the total initial energy at infinity (for cold, non-rotating bodies) is −k(M1
5/3
+ M2 ).
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
h
i
5/3
5/3
5/3
5/3
5/3
∆E = −kMF − [−k(M1 + M2 )] = k (M1 + M2 )5/3 − (M1 + M2 ) . The energy ∆E that
becomes available during a theoretical perfect merger is then
f = γ 5/3 + (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
(4)
γ = M2 /(M1 + M2 )
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
dm
(5)
v>vesc
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 m v 2 dm. 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 Saturnsystem 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 =
p
2 + v2
v∞
esc 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, corresponding 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 analytical 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 destructive 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 ∼
the satellite radius, e.g. RH ∼ 20 Rtitan where atitan
as
R
Rs where Rs is
Y
∼ 20 RY . In our simulations all major clumping
finishes inside of ∼ 10 Rs in Figure 5d, meaning that collisions occurring inside of about as ∼ 10 RY
will begin to feel Saturn’s presence.
Additionally there is the pseudo-force in the non-inertial frame of reference orbiting Saturn (Keplerian 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 outerforming clumps in our simulations we estimate a relative shear velocity of order & 100 m/s over the
course of the collision, assuming as = 20 RY , 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 ∼ 10 RY . While the two-body approach taken
here provides a sound general representation of the phenomenon, a more informed understanding of
collisions around ∼ 3 − 10 RY 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,
2
but the specific kinetic energy is only ∼1/20 as intense, proportional to vesc
. 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
2
debris. As for shock heating, in the planar approximation the specific kinetic energy ui = 21 ρvimp
is
comparable to the energy of vaporization of H2 O. 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 simulations, 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 H2 O 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 (CastilloRogez & 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 hydrocodes, 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 Chandrasekhar (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 application 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 + P dV , 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% H2 O 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 − 2.6 RY . This is just outside the Roche limit
Rroche ≃ 2.44 Rp (ρp/ρs )1/3
(6)
where ρp , ρs are the bulk density of the planet and satellite, respectively. For strengthless compact
ice spheres, Rroche ≈ 2.1 RY . 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 & 3 RY
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 (6.6−9.2 RY ) 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 silicaterich 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 9.2 RY 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
1/2
3k2p Ms G1/2 Rp5
51e2s
2as Γs
21k2s ns Mp Rs5 2
das
1
+
+
=
es
−
1/2 11/2
dt
4
Qs Ms a4s
Ms (GMp )1/2
Q p M p as
(7)
57k2p ns Ms Rp5
des
21k2s ns Mp Rs5
=
e
−
es + F
s
dt
Qp Mp a5s
2Qs Ms a5s
(8)
Here Rp is the radius of the planet, Rs is the radius of the satellite, and ns =
p
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 ∼ 3.2 RY 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 6.6 RY and 9.2 RY ,
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 as
−11/2
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 ∼ −1 RY 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 postformative 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 Titanforming 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 /2πa2s e, where as e is the radial extent
of the annulus of bodies, of total mass Mm at semimajor axis as , and where e is their characteristic
q
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 M˙m from the torus equals the
2
2
accretion rate M˙F ∼ ns Σd πRF2 (1 + vesc/vm
), accounting for gravitational focusing. The sweep up time
2
2
is then τacc ∼ Mm /M˙ F ∼ 2a2s vm /[ns RF2 (1 + vesc/vm
)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 ∼ 5 RY ,
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 ∼ 20 RY 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 sequentially, 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 ∼ 1 Rp per 1,000 years at 20 Rp . 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 postnebula 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/a˙s ), 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 = 62 RY ). 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 ∼ 60 RY , 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 − 30 RY 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 multiplesatellite 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
q
√
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 temper2
ature increase of & 500 K according to the simulation. The specific impact energy 1/2ρvimp
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 projectile 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 62 RY 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 similarsized 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/M R2 ∼ 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 Program 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 to
scale. Clockwise from Titan (upper right) are Iapetus, Hyperion, Enceladus, Tethys, Dione and Mimas, with Rhea in
the 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 the
four most massive wedges, in order, and range from pure ice to half rock; they orbit at as = 9.2, 62.1, 6.6 and 5.1 RY
respectively. 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 propose
that they are the residues of an inefficient final accretion that occurred at Saturn, but not somehow at Jupiter.
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Delta E (%)
30
20
10
0
0
0.1
0.2
0.3
0.4
Impactor Mass / Total Mass
Figure 3: Binding energy available after a perfect merger, expressed as a percentage of the final binding energy (Equation
3), is a function of the mass ratio γ = M1/M1 +M2 of the collision (Equation 3).
60
0.5
Figure 4: Equilibrium structures of spherical differentiated bodies of total mass (a) 0.33, (b) 0.50, (c) 0.75, and (d)
1.00 Mtitan . 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 H2 O mantle (50 wt%). Pressure equilibration is established
using a 1D Lagrangian hydrocode, further relaxed in 3D SPH. We then choose M1 and M2 from these four objects to
start our collision simulations. There is no consensus on the internal structure of Titan, let alone its precursors, so we
assume a relatively high entropy interior corresponding to liquid phase in the M-ANEOS equation of state. Plotted are
the 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 the
mass and bulk density of Titan in this simulation. The impactor M2 comes in from the upper left and first contacts at
θimp = 75◦ . Gravitationally bound objects (clumps) are indicated by solid white circles whose diameter corresponds
to a sphere of the object’s mean density and center of mass. The dashed green circle shows the Roche limit for icy
material around the largest remnant, M1 → MF . (a) M2 grazes M1 at t = 0. (b) M2 captured in orbit about M1 , now
back in an almost hydrostatic configuration. (c) A second collision begins at t = 23h where M2 is tidally sheared into
a spiral arm structure. The outermost parts of this arm are accelerated by the inner parts of the arm and by the core
of M2 , and escape. (d) At t = 31 hr the impactor hits the target again and forms a second spiral arm. The clumps
coming from the inner part of the second spiral arm become temporary satellites of MF . The fast clumps, to the left
of 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 and
assumed 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 are
restricted to low velocity collisions vimp = vesc with impact angle 60◦ ≤ θ ≤ 85◦ , and for varying γ. The small numbers
are the ejection velocity vej of each of these escaping clumps in km/s above the escape velocity of MF – that is, the
final orbital velocity relative to MF . The initial colliding masses, from Figure 4, are (a) γ = 0.4, M2 = 0.011 M⊕ ,
M1 = 0.017 M⊕ ; (b) γ = 0.33, M2 = 0.008 M⊕ , M1 = 0.017 M⊕ ; (c) γ = 0.25, M2 = 0.008 M⊕ , M1 = 0.023 M⊕ .
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Figure 7: Locations inside of M1 and M2 from where each of the identified escaping clumps derive, in the γ = 0.25
collision of Figure 5 which is the 75° case in Figure 6(c). Circles outline the silicate cores. The candidate MSMs are
color-coded, so that their originating particle locations are identified inside of M1 and M2 . Histograms for the change
in hydrostatic pressure ∆P are plotted in the top row, characterizing the radial depth inside M2 from which material is
exhumed. Almost all MSM-forming material originates from the mantle of M2 , for angular momentum reasons explained
in the text. Pressure release of ∼ 0.5 − 1 GPa corresponds to an originating depth ∼ 300 km inside of M2 . The second
most 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 spiral
arms. 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 deposition
of 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, requiring
a hand-off to other methods.
65