Icarus 252 (2015) 454–465
Contents lists available at ScienceDirect
Icarus
journal homepage: www.elsevier.com/locate/icarus
Despinning and shape evolution of Saturn’s moon Iapetus triggered
by a giant impact
Miroslav Kuchta a,1, Gabriel Tobie b,c, Katarina Miljković d,e, Marie Běhounková a, Ondřej Souček f,
Gaël Choblet b,c, Ondřej Čadek a,⇑
a
Charles University in Prague, Faculty of Mathematics and Physics, Department of Geophysics, V Holešovičkách 2, 180 00 Praha 8, Czech Republic
Université de Nantes, Laboratoire de Planétologie et Géodynamique de Nantes, UMR 6112, F-44322 Nantes, France
CNRS, Laboratoire de Planétologie et Géodynamique de Nantes, UMR 6112, F-44322 Nantes, France
d
Institut de Physique du Globe de Paris, Université Paris Diderot, 35 rue Hélène Brion, 75205 Paris, France
e
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
f
Charles University in Prague, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovská 83, 18675 Praha 8, Czech Republic
b
c
a r t i c l e
i n f o
Article history:
Received 15 July 2014
Revised 15 January 2015
Accepted 8 February 2015
Available online 18 February 2015
Keywords:
Iapetus
Ices
Tides, solid body
Rotational dynamics
Thermal histories
a b s t r a c t
Iapetus possesses two spectacular characteristics: (i) a high equatorial ridge which is unique in the Solar
System and (ii) a large flattening (a c ¼ 34 km) inconsistent with its current spin rate. These two main
characteristics have probably been acquired in Iapetus’ early past as a consequence of coupled interiorrotation evolution. Previous models have suggested that rapid despinning may result either from
enhanced internal dissipation due to short-lived radioactive elements or from interactions with a subsatellite resulting from a giant impact. For the ridge formation, different exogenic and endogenic
hypotheses have also been proposed, but most of the proposed scenarios have not been tested numerically. In order to model simultaneously internal heat transfer, tidal despinning and shape evolution, we have
developed a two-dimensional axisymmetric thermal convection code with a deformable surface boundary, coupled with a viscoelastic code for tidal dissipation. The model includes centrifugal and buoyancy
forces, a composite non-linear viscous rheology as well as an Andrade rheology for the dissipative part. By
considering realistic rheological properties and by exploring various grain size values, we show that, in
the absence of additional external interactions, despinning of a fast rotating Iapetus is impossible even
for warm initial conditions (T > 250 K). Alternatively, the impact of a single body with a radius of
250–350 km at a velocity of 2 km/s may be sufficient to slow down the rotation from a period of 6–
10 h to more than 30 h. By combining despinning due to internal dissipation and an abrupt change of
rotation due to a giant impact, we determined the parameters leading to a complete despinning and
we computed the corresponding shape evolution. We show that stresses arising from shape change affect
the viscosity structure by enhancing dislocation creep and can lead to the formation of a large-scale ridge
at the equator as a result of rapid rotation change for initial rotation periods of 6 h.
Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction
Saturn’s moon Iapetus shows two spectacular characteristics:
an exceptionally large flattening and a narrow equatorial ridge
reaching heights up to 20 km above the surrounding terrain. The
observed oblate shape of Iapetus with a polar radius of
⇑ Corresponding author.
E-mail addresses: [email protected] (M. Kuchta), [email protected]
(G. Tobie), [email protected] (K. Miljković), [email protected]
(M. Běhounková), [email protected] (O. Souček), gael.choblet@
univ-nantes.fr (G. Choblet), [email protected] (O. Čadek).
1
Current address: Department of Mathematics, University of Oslo, Norway.
http://dx.doi.org/10.1016/j.icarus.2015.02.010
0019-1035/Ó 2015 Elsevier Inc. All rights reserved.
712:1 1:6 km and an equatorial radius of 745:7 2:9 km
(Thomas, 2010) would be consistent with a rotational period of
16 h for a homogeneous body or 15 h for a differentiated body
(Castillo-Rogez et al., 2007; Thomas, 2010). Such a large
(34 km) difference between the polar and equatorial radii implies
that Iapetus was rotating much faster in its early past, and then
slowed down to its present-day rotation period of about 79 days,
preserving its initial shape. Taking into account the large distance
between Iapetus and Saturn (semi-major axis a ¼ 3:56 106 km),
the despinning of Iapetus to synchronous rotation is problematic.
The time for a moon to reach tidal locking is proportional to a6
(Gladman et al., 1996). Iapetus could thus reach its present-day
M. Kuchta et al. / Icarus 252 (2015) 454–465
spin period only if the interior was efficiently dissipating the rotational energy, which required a hot interior and low viscosity (e.g.
Castillo-Rogez et al., 2007) for a sufficiently long (1 Gyr) time.
However, if this were the case, the equatorial bulge could hardly
be preserved until the present time because of fast viscous
relaxation.
The narrow equatorial ridge is another puzzle. It runs J 75% of
the satellite circumference, segmented in several discontinuous
portions (Giese et al., 2008; Singer and McKinnon, 2011;
Dombard et al., 2012; Lopez Garcia et al., 2014). Since its formation, it has been modified by cratering processes and landsliding
(Singer et al., 2012), and the high crater density indicates that it
is a very ancient feature (Denk et al., 2010). Its location on the
top of the equatorial bulge suggests a causal link with oblate shape,
however their possibly common origin is still unclear.
Previous models have suggested that rapid despinning may
result either from enhanced internal dissipation due to short-lived
radioactive elements (Castillo-Rogez et al., 2007, 2011; Robuchon
et al., 2010) or from interactions with a sub-satellite resulting from
a giant impact (Levison et al., 2011; Dombard et al., 2012). For the
ridge formation, proposed models can also be divided in two main
groups: endogenic and exogenic. In the endogenic hypothesis, the
ridge formation has been suggested to be related to a shape change
associated either with despinning (Porco et al., 2005; CastilloRogez et al., 2007; Robuchon et al., 2010) or contraction caused
by compaction of initially porous material due to warming
(Sandwell and Schubert, 2010). The location at the equator may
be attributed to a thinner lithosphere or weakening at the equator
(e.g. Beuthe, 2010). However, the origin of such lithosphere thinning or weakening does not seem consistent with the long term
support of the ridge and the absence of flexural signals (Giese
et al., 2008; Dombard et al., 2012). In the exogenic hypothesis,
the ridge formation is commonly attributed to the fall of a debris
ring, either of primordial origin (Ip, 2006) or resulting from the disruption of a subsatellite formed by giant impact (Levison et al.,
2011; Dombard et al., 2012). While this hypothesis naturally
explains the equatorial location of the ridge, it is still unclear if
the debris rain can explain the observed morphology of the ridge.
A critical aspect that has not yet been modeled in detail in previous studies is the coupled evolution of the shape and viscosity
structure of the despinning body. In Robuchon et al. (2010), the
change of shape was computed from the viscosity profile controlled by thermal diffusion and convection. However, the possible
feedback of shape evolution on the viscosity structure was not considered. In both hypotheses, endogenic or exogenic, the change of
shape during the satellite slowdown (whatever the despinning
process) is expected to have a strong effect on the litospheric stress
and the rheological structure of the interior. In order to re-evaluate
the despinning hypothesis and its consequence for the shape
evolution and the ridge formation and preservation, we have
developed a new numerical tool that allows the simulation of
thermo-mechanical processes in a rotating self-gravitating body
with free surface and realistic composite rheology of ice.
Numerical models of thermal convection in planets and moons
(e.g. Tackley, 2010; Choblet et al., 2007; Šrámek and Zhong, 2010;
Běhounková et al., 2010) usually neglect the centrifugal force due
to the body spin and assume that the body is spherical. For the
majority of the solid bodies in the Solar System, dynamical flattening and topography induced by thermal convection are small in
comparison with the body radius. The surface of the body can thus
be approximated by a fixed spherical boundary and formally
described by a free-slip or no-slip boundary condition. The present-day shape of Iapetus indicates that such assumptions are
not valid for this moon, at least during the early stage of its evolution when it was rotating much faster than today. The non-spherical shape of the moon could have a pronounced impact on its
455
thermal evolution. The latitudinally varying centrifugal force due
to fast rotation as well as the highly flattened shape may have
affected the onset of convection and its style. For a fast rotating
object, the gravity force (gravitational + centrifugal force) cannot
be represented by a radial vector, and the gravitational effect of a
generally non-spherical body must be evaluated to obtain a correct
distribution of body forces. Another effect that should be taken into
account for determining correctly the viscosity structure is the
strain-rate weakening due to non-linear (dislocation) viscous
creep. This effect could potentially have a strong impact on the
rheological structure of Iapetus’ lithosphere during the despinning
stage and therefore may have influenced the final shape of the
moon and the formation of the equatorial ridge. All these effects
are included in our new numerical code, which makes it a suitable
tool for investigating the early Iapetus’ evolution.
We consider two possible evolution scenarios. The first one
implies only tidal deceleration due to internal friction similar as
in Castillo-Rogez et al. (2007) and Robuchon et al. (2010).
Following Castillo-Rogez et al. (2011), we use an Andrade rheology
to describe the viscoelastic response and the associated dissipation. This rheology is much better adapted to describe the viscoelastic response of water ice on a wide range of temperature
and frequency than the Maxwell or Burgers rheology, which were
used previously in Castillo-Rogez et al. (2007) and Robuchon et al.
(2010). However, for realistic viscoelastic rheology, thought to be
representative of Iapetus’ materials, as we will show by exploring
systematically a wide range of initial conditions and rheological
parameters, despinning of a fast rotating Iapetus is impossible in
the absence of additional external interactions. In a second scenario, we consider the effect of a giant impact that partially despins
Iapetus and compute the shape evolution due to the abrupt change
of spin as well as the long-term evolution. By using analytical estimates and 3D impact simulations, we quantify the size of the
impactor needed to trigger the despinning process and we determine what values of initial rotation can explain both the shape
and the formation of an equatorial ridge.
The structure of the paper is as follows. In Section 2, we
describe the method used to simulate the thermal, shape (2.1)
and spin (2.2) history of the moon and we present results illustrating despinning of Iapetus due to tidal friction (2.3). We demonstrate that for physically admissible initial temperature and
realistic constitutive laws, Iapetus could be tidally locked only if
its initial spin period were longer than 30 h which contradicts
the observed flattening. Motivated by this finding, we investigate
in Section 3 the possibility of despinning due to a giant impact
(3.1) and the consequences of such an event for Iapetus’ shape evolution (3.2). The results of our numerical simulations are confronted with available data and summarized in Section 4.
2. Simulation of heat transfer and internal dynamics
2.1. Numerical model of Iapetus’ thermal evolution
To simulate the thermal–mechanical evolution of an icy moon
with a freely deformable irregular outer boundary we use the modeling strategy described e.g. by Harlow and Welch (1965), Gerya
and Yuen (2007) and others, where instead of considering a
Lagrangian mesh deforming together with the free surface, the free
surface is included as an internal interface within an Eulerian
mesh. This interface corresponds to the real surface of the body
and separates the domain with realistic material properties from
external ‘‘sticky air’’ (Fig. 1), an artificial low density and low viscosity material that does not restrain the real body from deforming
(for recent applications of the sticky-air method, see, e.g., Tkalcec
et al., 2013; Crameri and Tackley, 2014; Golabek et al., 2014).
456
M. Kuchta et al. / Icarus 252 (2015) 454–465
where p is the pressure, r is the deviatoric stress tensor, q is the
density, g represents the gravitational and centrifugal acceleration,
v is the velocity, g is the viscosity, s denotes transposition of a tensor, T is the temperature, t is the time, q0 is the density at temperature T 0 , cp is the specific heat at constant pressure, k is the
thermal conductivity, Q v represents the volumetric heat production, and a is the thermal expansivity. The parameters cp and k
are temperature dependent and include the effect of the silicate
phase (Castillo-Rogez et al., 2007). Thermal expansivity a is
assumed to be constant and equal to 1:5 104 K1. The reference
density is chosen to be equal to the mean density of Iapetus,
q0 ¼ 1083 kg m3, and the reference temperature is T 0 ¼ 90 K.
The radiogenic heating is spatially homogeneous and involves only
the long-lived radiogenic isotopes (Robuchon et al., 2010).
The deformation of ice occurs via four different creep
mechanisms (Goldsby and Kohlstedt, 2001), namely diffusion
creep, dislocation creep, basal slip-accommodated GBS and GBSaccommodated
basal
slip, characterized by
viscosities
gdiff ; gdisl ; ggbs and gbasal , respectively. If we neglect the dependence
on pressure, viscosity gi of each creep mechanism can be expressed
as a function of temperature T, grain size d and the second invariant of deviatoric stress rII ,
gi ¼
Fig. 1. Sketch of the computational domain. Deformation and thermal evolution of
the moon (domain Xr , plotted in light blue) is solved in axisymmetric geometry in a
computational domain (Xc ) with a fixed spherical boundary. The space between the
surface of the moon and the outer boundary of the computational domain is filled
with an artificial low-viscosity and low-density material (‘‘sticky air’’, domain Xa ,
plotted in dark blue) having a constant temperature. The Stokes problem and the
heat equation are solved on a fixed Eulerian mesh while material surface Eq. (13) is
used to compute the moon’s surface deformation. For more details, see Section 2.1.
(For interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
The equations governing the deformation and heat flow are then
solved using a standard numerical algorithm. Here we use a staggered-grid finite-difference conservative scheme and we apply a
surface tracking method (e.g. Hyman, 1984) instead of markers
to evaluate the boundary deformation.
Since the problem is numerically challenging due to large viscosity contrasts and a quickly evolving surface we solve it in
two-dimensional (axisymmetric) spherical geometry. This choice
can be partly justified by the axisymmetric nature of the centrifugal force but its main reason is the high spatial and temporal
resolution that is needed if the spin rate of the moon changes
abruptly and which cannot be effectively reached in the fully
three-dimensional geometry. As we will see later, the shape evolution of Iapetus is mostly controlled by the change of stress, induced
by time variations in centrifugal force, which a posteriori justifies
the use of an axisymmetric model.
We assume that the material forming the bulk of the moon is
incompressible and behaves like a viscous fluid with an infinite
Prandtl number. The equations governing the thermal evolution
of such a body can be expressed in the Boussinesq approximation
as follows:
rp þ r r ¼ qg;
ð1Þ
r v ¼ 0;
r g rv þ ðrv Þs ¼ 0;
ð3Þ
@T
q0 cp
þ v rT r ðkrTÞ ¼ Q v ;
@t
q ¼ q0 ½1 aðT T 0 Þ;
ð2Þ
ð4Þ
ð5Þ
m
T ld
E
;
exp
RT
2Arn1
II
ð6Þ
where R is the universal gas constant and l; m; n; A and E are parameters characterizing a given creep mechanism, see Table 1. For
simplicity, we assume that the grain size is the same everywhere
and does not evolve in time. The effective viscosity g representing
the deformation properties of ice in Eq. (3) is then given by
1
g
¼
1
gdiff
þ
1
gdisl
þ
1
ggbs þ gbasal
þ
1
gcut
ð7Þ
;
where gcut denotes the cut-off viscosity. The use of the cut-off viscosity has both numerical and physical reasons. The numerical reason is that the accuracy of the code deteriorates if the viscosity
contrast is too large. This behavior is common to all thermal convection codes and the cut-off viscosity is a standard tool to tackle
this problem. The physical reason is related to a limited validity
of viscous rheology at very low temperatures and stresses for which
Eq. (6) gives extremely high (> 1030 Pa s) values of viscosity. The
consequence is that the body becomes effectively non-deformable
(v ! 0), but the stresses can remain unrealistically large. As will
be shown in Section 3.2, the present-day shape of Iapetus can be
well predicted only for gcut J 1026 Pa s. For smaller values of gcut ,
the viscous relaxation is too fast and the moon does not preserve
its initial flattening.
The acceleration g includes both gravitational and centrifugal
contributions,
g ¼ rV g x ðx rÞ;
ð8Þ
Table 1
Creep parameters of ice (after Goldsby and Kohlstedt, 2001).
gdiff
gdisl
gbasal
ggbs
E (kJ mol1)
T (K)
A (Pan Kl mm s1 )
l
n
m
–
9:0 108
1
1.0
2.0
6 258
4:0 1019
0
4.0
0.0
60
> 258
6:0 104
0
4.0
0.0
180
–
2:2 107
0
2.4
0.0
60
6 255
6:2 1014
0
1.8
1.4
49
> 255
4:8 1015
0
1.8
1.4
192
59
457
M. Kuchta et al. / Icarus 252 (2015) 454–465
where x is the angular velocity, r is the radius vector and V g is the
gravitational potential which can be expressed using the Newton
integral,
V g ðrÞ ¼ G
Z
X
qðr0 Þ
jr r0 j
dX0 ;
ð9Þ
with the integration being performed over the whole volume X of
the body.
The surface of the body is free to move and its temperature
remains constant during the whole evolution and equal to T 0 :
ðpI þ rÞ n ¼ 0;
T ¼ T0;
ð10Þ
where n denotes the outer unit vector normal to the surface. The
shape of the surface can be implicitly described by a function H
which satisfies the equation of material surface,
@H
þ v rH ¼ 0:
@t
ð11Þ
the presence of a very cold material in the top few kilometers does
not significantly influence the shape of the moon.
The Stokes problem, Eqs. (1)–(3), is solved in the whole computational domain Xc (Fig. 1) with no slip imposed on the external
boundary,
v ¼ 0;
for r ¼ r c :
ð14Þ
The heat Eq. (4) is solved only in domain Xr while a constant temperature T 0 is prescribed in Xa . The physical state of the material in
domain Xa can thus be formally defined by the following
conditions:
qa q; ga g; T a ¼ T 0 ;
ð15Þ
where subscript a refers to the quantities in domain Xa , and g and q
are the viscosity and the density of the real body, respectively. The
appropriate choice of the value of sticky-air viscosity ga has been
discussed by Crameri et al. (2012). They demonstrated that in the
case of a Cartesian domain the sticky air approach works well as
We assume that the radius r s of the surface can be expressed as an
explicit function of co-latitude h and time t, r s ¼ r s ðh; tÞ, i.e. that for
chosen values of h and t the position of the material surface is
uniquely defined. Function H can then be expressed as
long as the term ðga =gc Þ=ðha =LÞ is sufficiently small, where ha is
the thickness of the sticky air layer and gc and L are the characteristic viscosity and length scale of the model, respectively. In our
case, the computational domain has a radius of 1100 km. The den-
H ¼ r r s ðh; tÞ;
sity of the sticky-air is 1010 kg m3 and the viscosity of the sticky
air is eight orders of magnitude smaller than the maximum viscosity in the real body at a given time. We also tested a larger
(1200 km) computational domain and smaller values of the
ð12Þ
and Eq. (11) takes the form
@r s
v h @rs
¼ vr ;
@t
r @h
ð13Þ
where v r and v h denote the components of v .
Our computational method is based on a one-fluid approach
(Tryggvason et al., 2009) where the Stokes/Navier–Stokes equations are solved in the entire computational domain and an indicator function is used to make a distinction between the fluids
involved. In the method of Gerya and Yuen (2007), markers take
this role and the method can be viewed as a variant of the MAC
method (Harlow and Welch, 1965). In our case, the level set function (e.g. Osher and Fedkiw, 2003) is used and our method is then
similar to the level set approach described in Sussman et al. (1994)
with a few modifications: (i) the implicit level set function is
turned into an explicit representation of the interface whose evolution is governed by Eq. (13), (ii) due to a structured grid, the
material properties are assigned by a simple procedure where each
material point on a line corresponding to constant angle h checks
its distance from the interface, and (iii) the re-distancing procedure
is constructed such that the volume is conserved.
We assume that at time t ¼ 0, the body is in a perfect hydrostatic equilibrium (r ¼ 0) and its shape is fully determined by spin
period Pini . The interior of the body has a homogeneous temperature T ini , except the top 20 km where the temperature profile
is approximated by the Gauss error function. The initial state of
the moon in our model is thus fully determined by only three parameters: initial temperature T ini , initial spin period P ini and grain
size d.
For simplicity, we assume an isothermal interior as an initial
state, with only a thin conductive boundary layer on top. Such a
temperature distribution is a reasonable representation of the
thermal state of Iapetus about 5–10 Myr after accretion provided
that the accretion was completed shortly after the formation of calcium–aluminum-rich inclusions (see Fig. 3a in Robuchon et al.,
2010, and the discussion therein). The assumption of the hydrostatic shape is well justified provided that the relaxation time
gint =l K 106 years, where l is the shear modulus of ice and gint
denotes the viscosity in the interior of the moon, determined by
temperature T ini and grain size d. It can be shown from Eq. (6) that
this condition is satisfied for all initial temperatures and grain sizes
considered in this study. As we will show in the following section,
3
sticky-air viscosity (gmax =ga 106 ) and did not find significant differences between the solutions.
The method described above has been implemented in C programming language and tested. Besides the standard time-step
and mesh refinement testing, we also checked whether the integral
heat balance was satisfied and the total volume of domain Xr preserved when the shape of the body was evolving. For small surface
deformations, we compared the topography predicted by our code
with that computed directly from the normal stress. Finally, we
checked whether a correct hydrostatic limit was obtained if the
body was deformed only by rotation. The calculations were performed with the resolution of about 1.5 km in radius and 0.3° in
latitude. The choice of the time step was controlled by the
Courant criterion, with C between 0.1 and 0.5.
2.2. Tidal dissipation and despinning model
To explain its present-day flattened shape, the initial spin period of Iapetus should have been smaller than 16 h, but in any case
not smaller than 4 h, corresponding to the Roche limit (CastilloRogez et al., 2007; Robuchon et al., 2010). Even though Iapetus
orbits at about 60 Saturn’s radii, it is subjected to non-zero tidal
forces induced by Saturn’s gravitational field. Iapetus could have
reached its present-day spin period of about 80 days only if the
interior efficiently dissipated the rotational energy. Before the
satellite reaches synchronous rotation, its tidally-distorted shape
rotates relative to the satellite body frame. Owing to the anelastic
properties of the interior, the fast-rotating satellite does not perfectly respond to the tidal forcing. As a consequence, the misalignement of the tidal bulge with respect to Saturn acts as a torque that
tends to slow down the satellite spin.
The evolution of Iapetus’ spin rate as a function of time t is then
related to the amplitude of the tidal bulge and its angular phase lag
(MacDonald, 1964):
2
dx 3k2 ðtÞGMp a5
¼
dt
2Q ðtÞCD6
ð16Þ
where k2 the tidal Love number, Q 1 the specific dissipation function, M p Saturn’s mass, a Iapetus’ equatorial radius, C the polar
458
M. Kuchta et al. / Icarus 252 (2015) 454–465
moment of inertia and D the semi-major axis of the orbit. The tidal
Love number k2 characterizes the time variation of the degree-2
gravitational potential due to the tidally-induced mass redistribution. The specific dissipation function Q 1 corresponds to the ratio
between the energy dissipated over one cycle and the peak energy
stored during the tidal cycle. It is related to the angular phase lag in
a periodic oscillation and can be expressed in the phase domain as
the ratio between the imaginary part and the modulus of the comc
plex Love number k2 . These two quantities can be computed from
the averaged rheological structure of the satellite. While the equations governing the thermal and shape evolution are solved in twodimensional geometry, a spherically symmetric model is used when
computing tidal deceleration. This simplification is necessary
because of the absence of a numerical tool to evaluate dissipation
in a general non-spherical body. The available codes either include
laterally dependent viscosity but assume that the body has a spherical shape (e.g. Běhounková et al., 2010), or they can be applied to
non-spherical bodies but with constant material parameters (e.g.
Breiter et al., 2012). Including the static oblateness might be important to better compute the dynamical equatorial bulge and therefore the tidal torque, but only when the body was rotating very
fast (
10 h). As it will be shown later (see Section 2.3), tidal dissipation is not sufficient to significantly slowdown a fastly-rotating
proto-Iapetus. Even if the static oblateness is included, it should
not change our conclusions.
To compute the viscoelastic response of the interior, we assume
an incompressible Andrade model, which is more appropriate than
Maxwell and Burgers rheology to describe the dissipative process
at tidal frequencies (Castillo-Rogez et al., 2011). The complex compliance for an Andrade model is given by:
Jv ¼
1
i
a
l g v
þ bðivÞ Cð1 þ aÞ;
ð17Þ
is the laterally averaged viscosity, v
where l is the shear modulus, g
is the tidal frequency, v ¼ 2ðx nÞ, with n being the mean orbital
motion, C is the Gamma function, and a and b are parameters
describing the frequency dependence and the amplitude of the
transient response, respectively. Comparison with available
experimental data for ice and rock indicates that the Andrade model
is a good approximation to describe the anelastic attenuation at
tidal frequencies (cf. Castillo-Rogez et al., 2011). Even though
experimental data on the anelasticity of ice are available (e.g.
Tatibouet et al., 1987; Cole, 1995, 1998), the uncertainty in the
rheological description and parameters at tidal frequencies is still
large. Moreover, the anelastic properties of Iapetus’ icy interior
may be affected by the presence of impurities, the nature and concentration of which are unconstrained. For simplicity, we adopt the
Andrade model assuming a ¼ 0:3 and b ¼ la1 ga . The shear mod
ulus is assumed uniform and constant through time. Viscosity g
varies only with radius and is computed by averaging the viscosity
field over colatitude:
g ðrÞ ¼ exp
Z p
r 1
s
ln g r ; h sin hdh ;
2 0
a
ð18Þ
where r ranges from 0 to a, and r s ¼ r s ðhÞ is the radius of the surface
at co-latitude h. Using Eq. (17), the complex shear modulus is
derived from the averaged viscosity profile and is used to compute
k2 ðtÞ and Q ðtÞ at each time step using the method of Tobie et al.
(2005), see Appendix A in the same paper for further details. The
evolution of the spin rate is then determined from Eq. (16) by a simple integration forward in time.
2.3. Initial conditions and their impact on the spin evolution
To understand the relative importance of individual initial parameters in the spin evolution, we have performed several hundreds
of thermal convection runs started from different combinations of
parameters T ini ; P ini and d and covering the time period of 4.5 Gyr.
We considered three values of initial temperature (T ini = 190, 230
and 270 K), five different values of grain size (d = 0.01, 0.1, 1, 1.0
and 10 mm), and the initial period P ini ranging from 16 to 300 h.
Since direct measurements of grain size on icy moons are not available, our choice of values of d is based on the data from Earth and
the models of grain size evolution. Analysis of ice samples from
Antarctic glaciers (Durand et al., 2006), corresponding to pressures
and temperatures similar to those in early Iapetus, gives grain size
between 1 and 4 mm. Self-consistent models of grain size evolution (Barr and McKinnon, 2007; Barr and Milkovich, 2008) suggest
that the grain size evolves in time to an equilibrium value which
depends on stress and the amount of impurities. In absence of silicate microparticles, the grain size of ice is controlled by dynamic
recrystallization which is balanced by grain size reduction due to
motion of dislocations during flow. Barr and McKinnon (2007) predict that for large icy satellites the grain can grow to a size between
30 and 80 mm. However, the presence of silicate microparticles or
other impurities, such as salt crystals, clathrate or gas bubbles, can
effectively inhibit grain growth and keep grains small – of the
order of 1 mm, or even smaller, depending on the amount of
impurities.
The values of initial temperature T ini are based on the estimates
by Robuchon et al. (2010) who demonstrated that the temperature
in Iapetus’ interior shortly after accretion mainly depended on the
concentration of the short-lived radiogenic isotopes (SLRI) in the
silicate fraction. The maximum temperature shortly after accretion
could range between 125 K for a negligible amount of [26 Al] and
250 K if [26 Al] = 72 ppb. The initial temperatures considered in
our study are from the interval predicted by Robuchon et al.
(2010), except for T ini = 270 K, which is considered as an extreme
case. This model is only used to illustrate the case with the largest
possible dissipation.
Fig. 2 illustrates the evolution of the spin period for initial temperature 230 K, grain size 1 mm and selected values of the initial
period. The results plotted in Fig. 2 can be divided into two groups:
the models with P ini P 45 h (red curves) which reach the synchronous rotation before 4.5 Gyr, and the others (Pini 6 44 h, blue
curves) for which the despinning is not completed. The initial period of 16 h, corresponding to the present-day flattening of the
moon, gives an almost flat despinning curve with a spin period of
about 20 h after 4.5 Gyr of evolution. For completeness, we note
that all models considered in Fig. 2 have been found to cool by conduction only during the first tens of Myr and then the heat transfer
is enhanced by convective processes.
The example given in Fig. 2 suggests that for each initial temperature T ini and grain size d one can find a minimum value of initial spin period P ini for which the moon reaches the synchronous
rotation. For a given pair of parameters T ini and d this minimum
period must be determined numerically by testing various initial
spin periods Pini . To minimize the number of computer runs, we
used the bisection method. The results corresponding to initial
temperatures 190, 230 and 270 K are depicted in Fig. 3 by blue,
red and black circles, respectively. The dashed line in the middle
of the figure divides the models into those characterized by purely
conductive cooling, mostly with a large grain size and/or a low initial temperature, and those that cool by convective heat transfer.
The convective cooling is found to be especially important for the
thermal evolution of models with initial temperature 230 and
270 K and grain size smaller than or equal to 1 mm.
M. Kuchta et al. / Icarus 252 (2015) 454–465
Spin period [hours]
1500
50
47
1000
45
44
500
0
0
43
16
1000
2000
3000
4000
Time [Myr]
Fig. 2. Evolution of the spin period for models with purely tidal despinning,
computed for initial temperature 230 K, grain size 1 mm and initial periods ranging
from 16 to 50 h. Note that only the models with initial period of 45 h or longer (red
curves) reach synchronous rotation before 4.5 Gyr. (For interpretation of the
references to color in this figure legend, the reader is referred to the web version of
this article.)
Initial spin period [hours]
300
250
459
et al. (2010). The authors of the former paper have demonstrated
that Iapetus can successfully reach the synchronous orbit if its
interior stays warm for a sufficiently long time and a realistic
anelastic rheology is considered. However, their thermal model
only includes conductive cooling and neglects the effect of thermal
convection which may increase the cooling rate significantly.
Robuchon et al. (2010) pointed out the role of thermal convection
but used a simplified rheological description of both viscous flow
and anelastic deformation. In the present study, we consider both
conductive and convective cooling and we use rheological parameters that are as realistic as possible. We show that for these
parameters the moon cools too quickly and cannot be despun by
tidal dissipation if the initial spin period is 16 h or shorter. We
note that the fast cooling of models with a high initial temperature
and/or a small grain size is mainly related to convective heat transfer and is only weekly influenced by the non-spherical shape of the
body. In first approximation, the rate of cooling is proportional to
the surface area of the moon. For the shortest (16 h) spin period
considered above the surface area of the flattened moon is only
about 0.4% larger than that of the sphere of the same volume.
The effect of non-sphericity is important only for models with very
short (6 h) periods where the increase of surface area is significant (10%) and where the flattened shape as well as latitudinal
changes in gravity influence also the geometry of convection
currents.
190 K
2.4. Summary
200
230 K
150
conductive cooling
100
270 K
convection
50
16 h
0
0.01
0.1
1
10
100
Grain size [mm]
Fig. 3. Minimum initial spin period needed for Iapetus to be tidally locked within
4.5 Gyr, computed for three different initial temperatures (T ini = 190, 230 and 270 K,
plotted in blue, red and black, respectively) and five values of grain size (d = 0.01,
0.1, 1, 10 and 100 mm). The dashed line in the middle of the figure marks the
approximate transition between the models where thermal convection is developed
and those cooling by pure conduction. The horizontal dotted line marks the spin
period of 16 h corresponding to the observed flattening of the moon. (For
interpretation of the references to color in this figure legend, the reader is referred
to the web version of this article.)
The horizontal dotted line in the bottom part of the figure
marks the 16-h spin period corresponding to the observed flattening of Iapetus. Note that for all the tested combinations of parameters T ini and d the minimum despinning period is always
found larger than this limit, except the model with a very small
grain size (0.01 mm) and a very high initial temperature (270 K).
The models with more realistic values of parameters (d between
0.1 and 10 mm, T ini 6 230 K) require an initial period of at least
27 h for the synchronous rotation to be reached. Such models cannot explain the present-day flattening of the moon if we assume
that its initial state was close to hydrostatic equilibrium.
However, the observed value of flattening cannot be obtained even
for the model with d ¼ 0:01 mm, T ini ¼ 270 K and the initial period
of 16 h: Since the viscosity of the moon is very low in this case, its
equatorial bulge rapidly relaxes as the spin period decreases, and
its final shape is almost perfectly spherical.
The conclusions that can be inferred from Fig. 3 differ from
those presented by Castillo-Rogez et al. (2011) and Robuchon
Our numerical simulations of thermal and spin evolution of
Iapetus, including a realistic material description of convective viscosity and anelasticity, suggest that the moon could not reach synchronous rotation solely by means of tidal friction. If we exclude
the models with extremely high initial temperature (270 K) and
very small grain size (<10 lm), then the tidal despinning is completed within 4.5 Gyr only for models with initial spin periods
J 30 h. However, such models can hardly explain the flattening
of Iapetus observed at present and consistent with a spin period
of about 16 h. Based on these findings, we can hypothesize that
the Iapetus’ initial spin period was of the order of 16 h or shorter,
and that the moon’s rotation was subsequently slowed down to a
period of at least 30 h by an additional external interaction, for
instance the collision with a large object.
3. Role of a giant impact
When a large object collides with a satellite, an exchange of
angular momentum occurs and the spin state of the satellite is
instantaneously altered. In this section, we investigate the possible
effect of a giant impact on the evolution of Iapetus’ spin. By considering the angular momentum budget, we estimate the size and
velocity of the impactor as well as the geometry of the encounter
needed to significantly slow down the spin rate of early Iapetus.
We then simulate the long-term evolution of Iapetus’ interior
and spin following the impact event.
3.1. Despinning by giant impact
Giant impacts are thought to have been common during the late
stage of the formation of Saturn’s satellites (e.g. Mosqueira et al.,
2010; Levison et al., 2011; Sekine and Genda, 2012; Zhang and
Nimmo, 2012; Asphaug and Reufer, 2013). In a way analogous to
the standard model of terrestrial planet formation (e.g. Chambers
and Wetherill, 1998; Agnor et al., 1999), the satellites likely formed
from the accretion of large ice-rock blocks up to more than 100 km
in radius (e.g Asphaug and Reufer, 2013). Several disruption events
M. Kuchta et al. / Icarus 252 (2015) 454–465
00
=5
xm
J/g
5%
0%
=7
xm
5%
=5
xm
where mi is the mass of the impactor, v i is the impact velocity, RI is
the radius of Iapetus, i is the impact angle measured with respect to
the plane tangent to the surface, and C I is the largest principal
moment of intertia of Iapetus. For a fixed value of momentum
exchange fraction xm , the maximum change in the spin rate occurs
for low impact angles. However, the fraction of momentum
exchange xm is expected to also depend on the impact angle, with
low impact angles being less efficient to transfer mass and momentum from the impactor to the target.
For low impact angles, only a small fraction of the impactor is
buried in the satellite during the collision, and most of the impactor mass does not interact with the satellite target and continues
its trajectory. For a 20° impact with a 170-km radius impactor,
the fraction of the exchanged mass is less than 10%, resulting in
a very small change of spin rate. The fraction of the exchanged
mass is even smaller for larger impactors. Using these geometrical
considerations, we find that the optimal configuration for
mass/momentum exchange is for an impact angle between 50°
and 60°, and that in this setting the exchanged fraction reaches values ranging from 55% to 70%. For higher impact angles, the
exchanged mass is larger but the impacts are less efficient to
reduce the angular momentum of the satellite.
The other critical issue concerns the complete disruption, or at
least strong loss of mass, of the satellite for too energetic impact
events. The threshold for catastrophic disruption depends on the
impact geometry (impact angle, size ratio between the impactor
and the target spheres), the impact velocity, and the target body
gravity and strength (e.g. Benz and Asphaug, 1999). Iapetus is large
enough to fall within the gravity rather than the strength dominated regime. The threshold for catastrophic disruption, Q D , which is
defined as the ratio of the projectile kinetic energy to the target
mass, can be estimated from the analysis of Benz and Asphaug
(1999). For Iapetus-size spherical body, Q D may typically range
between 40 and 200 J/g.
Fig. 4 presents the impactor mass and impact velocity required
to slow down a proto-Iapetus, spinning with an initial period of 6, 8
and 10 h to a period of 30 h, for xm ranging from 55% to 70%, an
impact angle of 55°, and a collision occurring at the equator parallel to the equatorial plane. This estimate shows that, to slow
Iapetus down from 6-h spin to more than 30-h spin without disrupting the satellite, one needs an impactor with a radius larger
than 340 km (a little bit smaller than a half of Iapetus’ radius) hitting Iapetus with a low ([2 km s1) velocity. At higher velocities,
full disruption is very likely based on the disruption limit provided
by Benz and Asphaug (1999). For initial spin periods of 8 and 10 h,
smaller impactors with a somewhat higher velocities are possible.
Such large impacts are expected to have affected the entire
=2
70%
ð19Þ
Q*
5
0%
=7
x m 5%
=5
xm
Dx ¼ xm mi v i RI cos i=C I
Initial spin period
10 h
8h
6h
6
x m=
probably occurred during this accretional phase, and the satellites
we observe today around Saturn are the ones that survived this
violent period. The irregularly shaped satellite Hyperion may be
considered as a remnant of this accretion process (e.g. Mosqueira
et al., 2010). In the case of Iapetus, a giant impact must have been
strong enough to significantly reduce its spin, but not too violent to
disrupt the satellite. In order to assess the viability of a giant
impact scenario, we need to consider these two aspects.
For the change of spin, simple angular momentum arguments
can be used. To change the spin rate of the satellite without
modifying significantly the direction of the spin axis, we need to
consider an impactor hitting the surface near the equator with a
trajectory parallel to the equatorial plane. In this case, assuming
that a fraction xm of the impactor momentum is transferred to
the satellite, the magnitude of the change in angular rotation rate
about the spin axis is (Wieczorek and Le Feuvre, 2009):
Impactor velocity [km/s]
460
Disruption limit based on
Benz and Asphaug (1999)
4
3
Q*=
2
1
150
40 J/g
i= 55°
200
250
300
350
Impactor radius [km]
Fig. 4. Impactor velocity and radius required to slow down a proto-Iapetus
spinning with an initial period of 6, 8 and 10 h to a period of 30 h for a giant impact
event with i ¼ 55 and fraction of momentum transfer xm varying between 55% and
70% (cf. Eq. (19)).
hemisphere, and possibly to have generated some dichotomy in
the crustal composition and lithospheric properties. Our preliminary simulations for a slow and large impactor using the
iSALE-3D shock pkysics hydrocode (Elbeshausen et al., 2009;
Elbeshausen and Wünnemann, 2011) indicate that the increase
in temperature within the impact site may be between 30 and
50 K for an initial temperature of 230 K.
3.2. Consequences for the shape evolution
Using the numerical method described in Section 2.1, we have
simulated the thermal and shape evolution of Iapetus after the
impact. The impact is implemented in the code in terms of an
abrupt change in centrifugal force, hence only through its effect
on despinning. Other effects, such as redistribution of mass at
the surface, and local increase of temperature, cannot be properly
included because of the two-dimensional nature of our model.
However, our 2D approach with an adjustable surface in response
to a change of centrifugal forces provides a reasonable estimate of
the shape evolution subsequent to an impact-induced rotation
change.
We first examine the case when the moon initially spins very
fast (P ini ¼ 6 h). As discussed in Section 2.3, the impact-induced
despinning is possible only for impactors with a radius of at least
340 km hitting the proto-Iapetus with a relative velocity of
2 km s1. Fig. 5 illustrates the evolution of the moon, initially spinning with a period of 6 h, shortly before and after the impact which
slowed its rotation down to a period of 50 h. The simulation is performed for the grain size of 1 mm and the initial temperature of
230 K. The impact event is prescribed after 10 Myr of evolution.
The panels in the very left column show the physical state of
Iapetus shortly (5 kyr) before the impact. The moon at this stage
is strongly flattened (a–c about 370 km), cools by conduction and
is still close to hydrostatic equilibrium, as indicated by the low values of strain rate. The viscosity in its interior is 1017 Pa s owing to
rather high initial temperature but significantly increases towards
the cold surface.
The increase of the spin period from 6 to 50 h due to the impact
is accompanied by a significant (about 99%) reduction of centrifugal force, which leads to a rapid collapse of the equatorial bulge.
Deformation of the moon at this stage is enabled by strain-rate
weakening of the uppermost low-temperature layer where dislocation creep is the dominant creep mechanism. During a few tens
of kyr after the impact, the strain rate gradually increases to values
M. Kuchta et al. / Icarus 252 (2015) 454–465
461
Fig. 5. Shape evolution of Iapetus computed for initial period of 6 h, initial temperature 230 K and grain size 1 mm. The impact event is mimicked by an abrupt increase of
spin period, from 6 to 50 h, imposed 10 Myr after the beginning of the simulation (on the time axis at the top of the figure, this instant corresponds to time zero). The panels in
the top, middle and bottom row illustrate the evolution of strain rate, viscosity and prevailing creep mechanism, respectively.
larger than 1013 s1 everywhere and the viscosity of the uppermost layer drops from 1027 Pa s to 1020 Pa s as the dislocation
creep strongly depends on the strain rate (or stress). The moon
then quickly adapts to the new distribution of the body force and
its flattening rapidly diminishes (to about 60 km after 250 kyr).
Note, that while the initial shape of the moon was close to a rotational ellipsoid, its shape during the deformation is more complex,
showing a pronounced fold around the equator. Once the main initial phase of the deformation is over, the strain rate starts to
decrease and the viscosity of the uppermost cold layer returns to
its initial large value. Owing to this viscosity increase, the shortwavelength topographic feature developed at the equator stops
evolving and its shape remains almost unchanged during the rest
of the simulation.
In the first 10 Myr of evolution and during the rapid shape
change subsequent to the impact, thermal conduction remains
the main mechanism through which the moon cools. The first convection currents develop at the poles and around the equator about
20 Myr after the impact (Fig. 6a) and thermal convection then contributes to the moon’s cooling for more than 500 Myr. The typical
values of strain rate in the convective part of the moon are
1013 s1 in the initial phase of convection (Fig. 6b) and they
gradually decrease to 1015 s1 as the radius of the warm
(T J 210 K) domain diminishes (Fig. 6c). After 700 Myr of evolution, convection becomes less vigorous and it gradually ceases
(Fig. 6d). After another 300 Myr (not shown in Fig. 6), the
temperature in the innermost part of the moon drops below
180 K and the body stops deforming (strain rate K 1020 s1 ).
During the whole period of convective cooling, warm parts of the
moon deform mainly by diffusion creep (not shown in Fig. 6) while
dislocation creep prevails in the uppermost cold (T K 150 K) layer.
Other two creep mechanisms (basal slip-accommodated GBS and
GBS-accommodated basal slip) are always less significant than
the diffusion and/or dislocation creep.
The simulation shown in Figs. 5 and 6 reproduces rather well
the two main characteristics of the present-day shape of Iapetus,
namely the abnormally large flattening and the narrow ridge at
the equator. The formation of the short-wavelength ridge is related
to the fact that the rotating moon initially had a larger (by almost
10%) surface area than a sphere of the same volume. When the
rotation of Iapetus was suddenly slowed down by an impact, a part
of the high-viscosity lithosphere became redundant because the
surface area of the moon started to diminish. This redundant material could either sink into the weak icy interior or buckle. Our model indicates that shortly after the impact, buckling was the
dominant mechanism. The buckling was facilitated by the non-linear stress-dependent rheology which tended to localize the deformation close to the equator. Later, after the onset of thermal
convection, some of the redundant material was flushed away by
downward flows. At this stage the equatorial ridge was already
formed and the viscosity of the surface was too high for the ridge
to be affected by internal flows. Our model indicates that, in the
process of the ridge formation, the non-linear stress-dependent
462
M. Kuchta et al. / Icarus 252 (2015) 454–465
(b) 100 Myr
(c) 300 Myr
(d) 700 Myr
210
Temperature [K]
(a) 23 Myr
180
150
120
90
Strain rate [s-1]
10-13
10-15
10-17
10-19
10-21
Fig. 6. Temperature and strain-rate evolution of Iapetus computed for the same parameters as in Fig. 5 and shown for the time interval from 23 to 700 Myr. At the beginning
of this period, the shape of the body is already stabilized and the moon starts cooling by convection, which rapidly decreases its temperature.
rhelogy played the crucial role – if only a temperature dependent
viscosity were considered in our model, no narrow ridge would
develop and the moon shape would maintain its initial degree-2
pattern.
The exact value of the flattening in our model depends on the
viscosity cut-off considered in the simulation, see Eq. (7), but it is
generally comparable to the observed value provided that
gcut P 1026 Pa s. For gcut ¼ 1027 Pa s (the case shown in Figs. 5
and 6), we obtain a c ¼ 50 km after 4.5 Gyr of evolution, with
the equatorial ridge rising 10 km above the surrounding plains.
As shown in Fig. 7, the predicted shape of the ridge strongly
depends on the initial temperature and grain size.
Fig. 8 illustrates the thermal and deformation evolution of the
moon for an initial spin period of 8 h. All other initial parameters
as well as the time of the impact and the post-impact spin period
are the same as in the simulation discussed above. Since the initial
flattening is now significantly (by more than 50%) smaller than in
the previous case, the abrupt reduction of the spin rate due to the
impact induces only a limited strain-rate weakening (compare
Figs. 5 and 8). As a consequence, the shape changes of the moon
are spread over a much longer period of time, no equatorial ridge
develops and the moon’s surface maintains its ellipsoidal shape
during the whole evolution. Due to a rapid convection cooling,
the uppermost cold and highly viscous layer of the moon quickly
thickens and its flattening thus remains large (a c w 140 km after
4.5 Gyr of evolution if gcut ¼ 1027 Pa s). A similar thermal and shape
evolution as for the 8-h spin period is also obtained for the initial
periods of 10 and 12 h (not shown here). In both cases the simulation gives a larger final flattening than observed (90 and 65 km,
respectively). For models with initial periods between 8 and 12 h,
a flattening comparable to the observed value can be obtained only
by significantly reducing the lithosphere viscosity (by decreasing
the cut-off viscosity, Eq. (3)), while in the model with the 6-h initial
period, the weakening of the lithosphere arises self-consistently
from the strong increase of stress. The 6-h model also provides a
natural way to rapidly relax the moon’s shape during a short period of time and to abruptly slow the relaxation down once the
stress decreases, thus preserving an equatorial bulge during the
rest of the evolution.
4. Discussion and conclusions
Assuming that the oblate shape of Iapetus is related to its initially fast spin, we demonstrate that the present-day figure of the
moon and its synchronous rotation can be explained by a slow
(2 km/s) collision with a large impactor of radius between 250
and 350 km and subsequent tidal despinning. The abrupt deceleration of the moon due to a sufficiently large impact leads to a collapse of the equatorial bulge and development of a narrow ridge
along the equator. The rapid change of the shape is facilitated by
strain-rate softening accompanied by a significant (several orders
of magnitude) decrease in viscosity. Once the main phase of deformation is over, the strain rate significantly diminishes and the viscosity in cold regions close to the surface again dramatically
increases, causing the shape of the moon to be ‘‘frozen’’ for the rest
of the evolution. To our knowledge, this is the first time that such
softening/hardening effect of the lithosphere is modeled for icy
satellites, thus opening new perspectives beyond Iapetus for other
icy moon studies.
In all tested cases, the ridge predicted for the 6-h initial period
differs from the real one in two respects. First, the model ridge is
always broader and less steep than the observed one. This is not
surprising because the constitutive laws used in our simulations
properly describe the deformation of ice only if it behaves as a viscous material, but they are not appropriate for granular or porous
ice forming the uppermost part of the moon and exhibiting brittle
and/or plastic behavior. The implementation of appropriate constitutive equations in a few top kilometers of our model would
463
M. Kuchta et al. / Icarus 252 (2015) 454–465
Fig. 7. Final shapes of the moon obtained after 4.5 Gyr of evolution for initial spin period of 6 h and different values of initial temperature T ini and grain size d: (a) T ini ¼ 170 K,
d ¼ 0:1 mm, (b) T ini ¼ 190 K, d ¼ 1 mm, (c) T ini ¼ 210 K, d ¼ 1 mm, and (d) T ini ¼ 230 K, d ¼ 1 mm.
Before
-5 kyr
2 Myr
30 Myr
50 Myr
100 Myr
After impact
Strain rate [s-1]
10-12
10-15
10-18
Viscosity [Pas]
10-21
1025
1023
1021
1019
Creep mechanism
1017
GBS
Disl
Diff
Fig. 8. Model with an initial spin period of 8 h. Other parameters are the same as in Figs. 5 and 6 (T ini ¼ 230 K, d ¼ 1 mm, impact imposed 10 Myr after the beginning of
simulation).
464
M. Kuchta et al. / Icarus 252 (2015) 454–465
require extremely high spatial resolution and would be highly
challenging from the numerical point of view. Second, as a consequence of the axisymmetric geometry, the predicted ridge is
always circum-equatorial, while the real ridge is observed only
along about 70% of the Iapetus’ circumference (Giese et al.,
2008). While it is possible that the original ridge on Iapetus was
also formed along the entire equator and its part was subsequently
destroyed (e.g. by later impacts), it is more probable that the difference between our prediction and the observation is caused by simplifications adopted in our model, in particular by neglecting threedimensional heterogeneities formed during accretion and due to
the giant impact. The deformation and mass redistribution processes induced by the impact are likely to have established a certain level of asymmetry in the moon’s structure which affected
the consequent shape evolution.
It is also possible that the giant impact only slowed down the
moon, which facilitated the tidal braking and allowed to preserve
a sufficiently large flattening until present, and that the ridge
was formed by the fall of a debris ring as proposed by Levison
et al. (2011) and Dombard et al. (2012). Besides the model with
the 6-h initial period, we have also investigated the shape evolution of models with lower initial spin rates, corresponding to periods between 8 and 12 h. A sufficiently large deceleration, leading
ultimately to synchronous rotation, can in this case be achieved
with a significantly smaller and/or slower impactor. Although these models successfully predict a large final flattening (often larger
than observed), they do not lead to the rise of a significant equatorial ridge, at least not when a purely viscous rheology of ice is considered. Further simulations including plastic and brittle
deformation will be needed to clarify a possible role of realistic
rheologies in formation of the ridge.
The results of our modeling suggest that the concept of impact
despinning is a feasible alternative to models relating the despinning of Iapetus to only internal processes (Castillo-Rogez et al.,
2007; Robuchon et al., 2010; Castillo-Rogez et al., 2011). Our
numerical tests without impact, including only tidal despinning
(Section 2.3), indicate that it is very difficult, if not impossible, to
achieve synchronous rotation if the initial period is 16 h or shorter.
The despinning is completed within 4.5 Gyr only if the initial temperature is very high (270 K) and/or the grain size very small
(0:01 mm). But even if the model finally achieves synchronous
rotation, the predicted flattening of the moon is negligible and
the equatorial ridge does not develop. In contrast, these features
can be well predicted using a model with impact despinning.
Preliminary simulations of giant impacts on Iapetus, which we
have performed using the iSALE-3D hydrocode (Elbeshausen
et al., 2009; Elbeshausen and Wünnemann, 2011), indicate that
the impact scenario proposed here is viable. Possible impact scenarios include icy projectiles that are 240–300 km in radius hitting
a completely icy proto-Iapetus at 2–3 km s1 at 45° to 55° impact
incidence angle. Such impacts would have a global but non-destructive effect on Iapetus. Part of the ejecta would escape and part
would fall back onto the surface under Iapetus gravity. The complexity of this impact outcome is yet to be investigated. Although
the details of the impact event and subsequent deformation processes still remain elusive, our modeling results suggest that the
impact scenario can provide a selfconsistent explanation of the
two main topographic characteristics of Iapetus and its presentday synchronous rotation.
Acknowledgements
The related research received funding from the European
Research Council under the European Community’s Seventh
Framework Program (FP7/2007–2013 Grant Agreement No.
259285) and the Charles University Grant SVV-2014-260096. O.S.
acknowledges the support by project LL1202 in the program
ERC-CZ funded by the Ministry of Education, Youth and Sports of
the Czech Republic.
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