Earth Planets Space, 59, 1035–1045, 2007

Rayleigh-Taylor instability after giant impacts: Imperfect equilibration of the
Hf-W system and its effect on the core formation age
Takanori Sasaki and Yutaka Abe
Department of Earth and Planetary Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan(Received April 4, 2006; Revised February 20, 2007; Accepted July 3, 2007; Online published September 28, 2007)

Hf-W chronometry provides constraints on the timing of planetary accretion and differentiation, as the segregation of a metal core from silicates should induce strong fractionation of Hf from W. In most previous studies,
it was assumed that a giant impact would perfectly reset the Hf-W chronometer. Here, we show the difficulty ofachieving perfect equilibration of the Hf-W system. Perfect equilibration requires iron to split into small droplets.
However, since the sedimentation velocities of small droplets are low, the Rayleigh-Taylor instability between theupper metal-containing and lower metal-free layers results in quick overturning of the layers, unless iron droplets
were uniformly distributed in the entire mantle. Therefore, the lower metal-free layers cannot be equilibrated. We
calculated the isotopic evolution of the Hf-W system, taking into account the partial resetting of this chronometer.
Our study led to three conclusions: (1) collision conditions and the number of giant impact events affect the age
estimation of core formation, (2) the Earth’s W isotope ratio indicates that more than two-tenths of the volume ofthe protoearth’s mantle must have been equilibrated at each giant impact, and (3) Mars should have experienced
a late, extensive equilibration event; it could have been a single giant impact.
Key words: Accretion, collisional physics, Earth, geochemistry, planetary formation.

1.

Introduction

Hafnium and tungsten are both highly refractory elements; hafnium is a lithophile element, whereas tungsten
is a moderately siderophile element that should partition
strongly into the metal phase during metal/silicate segregation. The decay of now extinct 182 Hf (half-life, 9 Myr)
to 182 W is an ideal chronometer for tracing core formation
events, because Hf is retained in the silicate mantle while W
is largely partitioned into the core during core segregation(Harper et al., 1991; Lee and Halliday, 1995; Harper andJacobsen, 1996).
It has been argued that the age of terrestrial core formation was limited by the accretion time and that it could haveoccurred 60 Myr later than the formation of iron meteorites(Lee and Halliday, 1995, 1996). Based on new measurements of the W isotope compositions and Hf/W ratios ofseveral meteorites, the age of terrestrial core formation andEarth’s accretion has recently been re-estimated by somegroups. These estimations were based on either a magma
ocean model or a two-stage model (Harper and Jacobsen,
1996; Jacobsen, 1988). The magma ocean model assumes
that growth of the Earth occurred at an exponentially decreasing accretion rate and does not consider the effects ofgiant impacts. The main growth stage for the Earth as defined in the magma ocean model of Yin et al. (2002), whichcorresponds to 63% growth, was estimated to occur at about
10 Myr. In this model, the end of Earth’s accretion would
have occurred at about 55 Myr (Kleine et al., 2004a). On
c The Society of Geomagnetism and Earth, Planetary and Space SciCopyright ences (SGEPSS); The Seismological Society of Japan; The Volcanological Societyof Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB.

the other hand, the two-stage model assumes that Earth’s
core formed at a well-defined point in time by a single event(e.g. a giant impact). This model age does not provide a realistic age for core formation, but it does provide the earliest
time when core formation can have ceased (Halliday et al.,
1996; Halliday, 2004; Kleine et al., 2004a). In this model,
the Earth’s core formation would have ceased at about 30
Myr (Kleine et al., 2002; Schönberg et al., 2002; Yin et al.,2002).
Some studies have considered the effect of partial resetting of the Hf-W chronometer (Jacobsen and Harper, 1996;
Sasaki and Abe, 2003; Halliday, 2004; Kleine et al., 2004a;
Jacobsen, 2005; Nimmo and Agnor, 2006). These studies
have suggested that without a quantitative assessment of
the equilibration process resulting from giant impacts, wecannot use Hf-W chronometry to determine accurately theage of a giant impact or the age of metal-silicate separation.
Meanwhile, Kleine et al. (2004a) derived constraints for the
degree of metal-silicate equilibration during Earth’s accretion by use of the 182 W excess of the Earth’s mantle relative
to chondrites conversely.
In this paper, we first discuss the mechanisms that cancause imperfect equilibration of Hf-W systems as a resultof giant impact events. We then discuss how Hf-W age
measurements are affected by imperfect resetting. Finally,
we discuss the core formation events on Earth and Mars.

2.

Mechanisms of Imperfect Equilibration of HfW Systems

A giant impact can produce a global magma ocean on a
protoearth, which, on first glance, seems to lead to complete
metal-silicate equilibration as a direct result of the giant

1035

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T. SASAKI AND Y. ABE: IMPERFECT EQUILIBRATION OF HF-W SYSTEM

impact, unless the impact leads to direct core merger. Rubie
et al. (2003) concluded that small metal droplets are formed
and that their settling through the magma ocean can achieve
chemical equilibration between metal droplets and silicate
liquid. In this section, however, we point to a difficultyin achieving perfect equilibration in Hf-W systems, evenif small droplets are formed, as suggested by Rubie et al.
(2003).
2.1 Achievable resetting ratio by the Stokes sedimentation of metal grains
To achieve perfect equilibration of the Hf-W system in a
mantle with added metal on the Earth’s surface, this metalmust split into small metal droplets and sink through themantle in order to equilibrate with the surrounding mantle.
There is isotopic and elemental exchange between the sinking metal droplets and silicate melts to maintain the distribution of W between metal and silicate, as defined by themetal-silicate partition coefficient. The degree of equilibration of the Hf-W system in the mantle depends on the
total surface area of metal droplets. The smaller the metal
droplets, the easier the mantle can be equilibrated. First, weestimated the required size of these droplets for complete
metal-silicate equilibration. In the following calculations,
we used the same physical properties (diffusivity and viscosity in the magma ocean) as Rubie et al. (2003).
During the stage of planetary accretion, newly accreted
W in the impactor’s metal component equilibrates withsome portion of the W in the silicate melt of the target andleads to a Hf/W ratio for this portion corresponding to thevalue to meet the metal-silicate partition coefficient for a
mixture between impactor metal and target silicate. Thisequilibration continues during the sedimentation of metalin the silicate mantle. Suppose that metal in the impactor
splits into lots of metal spheres with radius r sinking at the
Stokes velocity:vs =2ρs gr 2
9η(1)

where ρs is the difference in density between the silicate
melt and metal, and g is the gravitational acceleration. Typical values are ρs = 8 · 103 kg/m3 and g = 10 m/s2 . The
viscosity of the silicate melt η is 10−2 m·s/kg (Rubie et al.,2003), assuming a completely molten global magma ocean
for the Earth’s mantle. We can estimate the equilibrated
volume from the metal spheres:
veq = 2πr

√

Dτ H,

τ=

2r
vs

(2)

where D is the diffusivity of W in the silicate melt, D =10−8 (Rubie et al., 2003), and H is the depth of the magma
ocean in the target body. We adopt the present-day depth ofEarth’s mantle, H = 2.9 × 106 m. The volume of mantle inwhich the equilibration of the Hf-W system takes places ata single giant impact, Veq , is given by the sum of veq over allof the iron particles. We define the resetting ratio of the HfW chronometer to be the ratio of Veq and the total volumeof the target mantle and impactor mantle.
First, assuming that the metal in the impactor breaks into
small metal spheres of equal size, we can easily estimate theFig. 1. The resetting ratio achievable as a function of the radius offragmented metals. We consider those cases in which the metals fromthe impactor break into small fragments of equal size (upper), with asize distribution of metal spheres given by a power law (lower).

resetting ratio of the Hf-W chronometer. As shown in Fig. 1(upper), the radius of metal spheres must be less than about
20 cm to achieve perfect equilibration of the Hf-W system.
In this estimation, the target body was assumed to be thesize of the Earth, and the impactor body was assumed to be
the size of Mars, with the appropriate core sizes for eachobject.
Next, suppose that the size distribution of metal spheresis given by a power law. We can estimate Veq as a functionof the radius of the largest metal sphere:
 M
√
Veq =
2πr Dτ H f (m)dm,m(3)
 m −b2−bf (m) =
M
0
M2
M
where M is the maximum mass of metal spheres and M0 is
the mass of the impactor’s core. The cumulative power law
index b is known to be close to 1 for the impact fragments(Oddershede et al., 1993). When b = 1 in Eq. (3), theradius of the largest metal sphere should be less than about
40 cm to achieve perfect equilibration of the Hf-W system(Fig. 1, lower).
In these estimations, we assumed that the metal and silicate portions of impactors were well mixed at the impact
and placed upon the top of Earth’s mantle with iron split
into spheres of given size. Any part of the silicate composition through which the metal spheres passed was equilibrated. We also assumed all metal accreted to the Earthwas involved in the metal-silicate equilibration. This gives
an extreme case of equilibration which is independent of
the Hf/W history and the initial W composition of the impactor. For these reasons, Fig. 1 shows the upper limitsof the achievable resetting ratio for a given size of metal
droplets.

T. SASAKI AND Y. ABE: IMPERFECT EQUILIBRATION OF HF-W SYSTEM

1037

Fig. 2. Sketch of the metal droplets’ behavior in the silicate melt layer, which is assumed to have a low viscosity in the upper layer and high viscosity
in the lower layer. Infalling ejecta and impact fragments descend in the low-viscosity upper layer at the Stokes velocity and are deposited over the
lower layer. The Rayleigh-Taylor instability then results in big blobs of metallic material sinking into the core.

2.2

The Rayleigh-Taylor instability and the Earth’s
core formation
As discussed in the previous section, the metal dropletsin the mantle must be centimeter-scale in size to achieve
perfect equilibration of the Hf-W system. In addition, thedroplets must descend in the mantle from the surface tothe core without producing large blobs or metal layers toachieve perfect equilibration.
Many recent works have discussed the modeling of giant
impacts using a method known as smooth particle hydrodynamics, or SPH (e.g., Canup and Asphaug, 2001; Canup,2004). These results show that if an impactor and most partsof the target break into pieces, then the debris mixes during
the re-accretion onto the surface of the protoearth. Jacobsen (2005) observed that these SPH simulations suggest accretion with re-equilibration with the impactor and the target mantle. However, these simulations with N = 104 –105 particles, implying particle sizes of about 102 km, do
not have enough resolution to consider sizes smaller thanmeter-scale. Moreover, to date, no simulations that demonstrate how the target’s mantle breaks and mixes with theimpactors in giant impacts exist. Therefore, it is not possible to discuss the metal-silicate equilibration based on these
SPH simulations.
Although a protoearth could be undifferentiated before
the first giant impact (i.e. iron droplets are distributed entire
mantle uniformly), it would become differentiated after the
giant impact (i.e. most metal would fall into the core). Giant impacts are considered to have occurred more than oncefrom the physically (Kokubo and Ida, 1998) and geochemically (Kleine et al., 2004a) point of view. Therefore, we do
not consider the case that the iron droplets are distributedin the protoearth’s entire mantle in an undifferentiated state
before all giant impacts. Since giant impacts are expected
to occur at long intervals, the target body’s metal component after an impact is assumed to fall into the core before
the next giant impact occurs, and so the target’s mantle is assumed to hold no metal at the time of each giant impact. On
the other hand, the impactor’s metal component and mantle
are assumed to break into pieces as a result of the giant impact, so that infalling debris from the impactor is assumed
to be a mixture of metal and silicate. SPH simulations for ahead-on impact case, that is, a low-angular-momentum collision case, suggest the possibility that the impactor’s core

could sink into the target’s core without breaking (Cameron,1997). However, because the mantle clearly cannot be equilibrated in this case, the assumption of the break-up of the
impactor’s core and mixing of metal and silicate gives us
the higher limit of equilibration degree. Moreover, SPH
simulations have generally shown that while some of the
impactor’s core may diffuse into the target’s mantle, muchof the impactor’s core would be ejected and fall back to
the Earth later (Cameron, 2000), so that not all of the impactor’s metal could be distributed in the entire mantle initially. Therefore, we assume that the iron droplets are distributed not in the entire mantle but only in the upper layerof the mantle after the giant impact. These assumptions are
the initial conditions for our consideration.
In this section, we consider two possible scenarios for the
structure resulting from giant impacts: (1) two layers areproduced, a fully molten layer (upper layer) and a partiallymolten layer (lower layer); (2) the entire mantle becomesfully molten.
2.2.1 A fully molten layer (upper layer) and a partially molten layer (lower layer) are produced In theupper molten layer, the infalling debris of the impactor
and the broken target would be easily separated into metaland silicate. Metal droplets would then descend in the upper molten layer and deposit over the lower viscous layer.
When the thickness of the accumulated metal layer exceeds
a critical value, the Rayleigh-Taylor instability becomes
important. Metallic materials would then fall throughthe lower layer into the central part of the planet (Elsasser, 1963; Stevenson, 1981; Sasaki and Nakazawa, 1986)(Fig. 2) in the form of kilometer-sized blobs. Since the sizeof such metal blobs would be too large to achieve perfectequilibration of the Hf-W system (Fig. 1), the lower layerof the mantle, at least, could not be equilibrated. Therefore, in this scenario, irrespective of the initial size of the
impactor’s falling metal droplets, the lower mantle cannot
be equilibrated.
2.2.2 The entire mantle is fully molten If an impactor’s core breaks into droplets, they will be larger thanmeter-scale at the impact, and the sinking of these metal
droplets, therefore, cannot equilibrate the whole mantle, as
shown in the Fig. 1. However, if these droplets can break up
into smaller droplets in the molten mantle, there is a chanceof perfect equilibration. Rubie et al. (2003) concluded that1038

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↓ ↓↓ ↓ ↓↓ ↓ ↓Fig. 3. Sketch of the metal droplets’ behavior in the silicate melt layer, which is assumed to be fully molten. After a giant impact, the ejecta and broken
impactor are mixed with each other and accrete to the protoearth as a mixture. The accreted mixture is separated into silicate and centimeter-scale
metal droplets, and then these droplets sink into a well-mixed metal-silicate layer (first sketch). If these droplets are small enough to achieve perfectequilibration, the Rayleigh-Taylor instability between the silicate melt layer with metal droplets and the metal-free layer occurs immediately because
the growth of the Rayleigh-Taylor instability is much faster than the Stokes sedimentation of metal droplets (second sketch). Once the Rayleigh-Taylor
instability has grown, the layer with metal droplets sink into the core as a cluster, and so the mixture layer and metal-free layer overturn immediately(third sketch). In this situation, the mixture layer that holds small metal droplets—the upper layer in the first sketch—has an equilibrated Hf-W
system. However, the metal droplets in the mixture layer cannot interact with silicate outside the cluster. Thus, there exist two regions in the mantle:
an equilibrated region (last sketch, lower layer) and a non-equilibrated region (last sketch, upper layer).

the metal could equilibrate with silicate melt in a magma
ocean because the molten metal dispersed as small droplets.
They considered the idea that the small metal droplets could
sink like “rainfall droplets” without forming big blobs andthat the silicate layer could achieve equilibration of the HfW system by the Stokes sedimentation of these small metal
droplets.
However, even in such a situation, we point out that thereis the possibility of another type of Rayleigh-Taylor instability between the silicate melt layer with metal droplets
and the silicate melt layer that is metal-free (Kobayashi etal., 1993) (Fig. 3). The Rayleigh-Taylor instability can occur between the upper mixture layer and the lower molten
silicate layer because the upper layer mixed with metal
droplets has a higher density than the lower metal-free layer.
If the growth of the instability is much faster than the sedimentation of small iron droplets in the molten mantle, thenthe overturn of metal-rich and metal-free layers should occur. In this case, the metal-free layers cannot be equilibrated. This type of Rayleigh-Taylor instability is actuallyobserved in laboratory experiments (Iga and Kimura, 1993).
In the case where an impactor’s core initially breaks into
droplets small enough to achieve perfect equilibration, suchmetal droplets would tend to stay at the top of the molten
mantle because the sedimentation velocity of small dropletsis very small, as we show in Subsection 2.3. Therefore, inthis scenario it is also likely that the upper silicate layer contains many small metal droplets, and the Rayleigh-Taylor
instability between the upper and the lower layer should occur as depicted in Fig. 3.
2.3 Timescale comparison between Stokes sedimentation and the Rayleigh-Taylor instability
When any mass of dense liquid (molten metal) falls
through a less dense liquid (molten mantle), instabilities
tend to form at the interface that may eventually disperse
the dense mass into small droplets (Rubie et al., 2003). In
this section, we consider the consequences of the assumption that the metals break up into small grains in the magma
ocean. We assumed a mixture layer of metal and silicate
on a metal-free layer. Since metal droplets are small, we
assume equilibrium is achieved in the mixture layer. However, whether or not equilibration of the whole mantle occurs can be assessed by comparing the timescales between
the Stokes sedimentation and growth of the Rayleigh-Taylor
instability at the interface of the metal-silicate mixture layer
and the metal-free silicate melt layer. In this paper, we consider two extreme cases: (1) the thick limit and (2) the thinlimit of the mixture layer (see Appendix).
Figure 4 shows that timescale ratios (growth of theRayleigh-Taylor/Stokes sedimentation) as a function of thesinking depth of metal grains by Stokes sedimentation. In
both cases, the timescale of the growth of the Rayleigh-T. SASAKI AND Y. ABE: IMPERFECT EQUILIBRATION OF HF-W SYSTEM

Fig. 4. Timescale ratios between growth of the Rayleigh-Taylor instability
and Stokes sedimentation as a function of the sinking depth by Stokes
sedimentation. The upper panel (1) shows an example of a thick mixture
layer, and the lower panel (2) the case of a thin mixture layer. The radiusof metal grains is 1 cm in both cases.

Taylor instability is much shorter than that of the Stokes
sedimentation. Therefore, the Rayleigh-Taylor instability
should grow quickly, before metal grains can descend significantly through the mantle. Once the Rayleigh-Taylor instability has grown, the mixing layer should fall or overturninto the core as a cluster.
So far, we have estimated the timescale using a twodimensional rectangle box. It is also important to checkthe validity of applying the results to spherical shells. In
fact, the linear stability analysis of the problem yields thesame results in two or three dimensions, and growth rates
obtained through a spherical harmonic expansion are independent of azimuthal wavenumber (Goodman, 1990). It has
also been confirmed that the dominant wavelength of the instability will remain the same in three dimensions (Müller
et al., 1991).
2.4 Summary
After a giant impact, if two layers (low-viscosity upper
layer and high-viscosity lower layer) are formed, then themetal grains will descend, deposit over the lower layer, andthen fall through the lower layer as a large blob produced by
the Rayleigh-Taylor instability between the metal layer andthe high-viscosity lower silicate layer (Fig. 2). In this case,therefore, perfect equilibration of the Hf-W system cannot
be achieved.
If all of the mantle is rendered fully molten by a giant
impact, and the falling impactor’s metal debris has split
into small droplets (Rubie et al., 2003), these droplets candescend through the molten mantle. Then, if the mantle
and metal droplets in the metal-containing layer are mixed1039

well, this layer can be equilibrated completely. However,even in this situation, the Rayleigh-Taylor instability acts
between the metal-containing layer and the metal-free layer
to cause the overturn of these two layers (Fig. 3), so thatthese droplets cannot continue to descend to the bottom like“rainfall droplets” and cannot equilibrate the entire mantle. Thus, although the metal-containing layer can be equilibrated, the metal-free layer cannot be equilibrated for the
Hf-W system. Therefore, the complete metal-silicate equilibration by means of a giant impact cannot be expected inthis case either.
Jacobsen (2005) suggested that giant impacts would
completely reset the Hf-W chronometer in many cases. Although it is based on the observation of SPH simulations,the resolution of the models is still too coarse to be conclusive, as noted above. The consideration of the RayleighTaylor instability, to the contrary, leads to the conclusionthat perfect resetting occurs only in very limited cases; perfect equilibration is possible only when the iron droplets
are smaller than meter-scale and are distributed throughout the entire mantle in the first place, without a metal freelayer being produced. However, such a distribution of metal
droplets is unlikely, in particular for repeated giant impacts,
as noted above. Thus, we conclude that it is difficult to obtain complete equilibration of the Hf-W system during giant
impacts.
The occurrence of the Rayleigh-Taylor instability is not
limited to the core formation after giant impacts consideredhere. If a metal-free layer exists below a newly accretediron-silicate mixture, the Rayleigh-Taylor instability likely
occurs, and silicates in the metal free layer would not be
equilibrated with iron. Thus, complete metal-silicate equilibration also seems to be difficult in most case of core formation other than that after giant impacts.

3.

Isotopic Evolution of the Hf-W System in theCase of Imperfect Equilibration

In this section, we calculate the isotopic evolution of the
Hf-W chronometer in the case of imperfect equilibration of
the Hf-W system. We focus attention on deriving requiredconditions (not sufficient conditions) to meet the observational data for each equilibrating event. As such, we consider cases for which the equilibration between metal andsilicate occurs most easily. We use assumptions and simplifications which give us a higher limit of the resetting ratio of each equilibrating event. Although such assumptions
may not be realistic, we can provide a number of requiredconditions for each equilibrating event so as to obtain theobserved isotopic ratio.
The simplest model of fractionation is a two-stage modelwith a single episode of core formation occurring at t =tcf , where t is defined with time running forward from theorigin of the solar system, T0 years ago. The isotope time
evolution of reservoir j is given by 182 Hf
(tcf , tobs ) = Q · f j · 180
Hf 0 −λ182 tcf(4)· e− e−λ182 tobs
where tobs = T0 yields the present value (Harper and Jacobsen, 1996). Isotopic evolution relative to the CHondritic

1040

T. SASAKI AND Y. ABE: IMPERFECT EQUILIBRATION OF HF-W SYSTEM

Uniform Reservoir (CHUR) began after tcf years from theinitial state. In the above equations λ182 is the 182 Hf decay
constant (0.0779 Myr −1 (Vockenhuber et al., 2004)), andQ = 104 ·(180 Hf/182 W)CHUR = 1.49 × 104 (Kleine et al.,2004b). The Hf/W fractionation in a reservoir j relative to
CHUR is defined by the f -value:fj =(180 Hf/183 W) j
−1
(180 Hf/183 W)CHUR(5)

We use values f j = 12 for the region (Harper et al.,
 Earth’s= 1 × 10−4 (Kleine et1991; Yin et al., 2002) and 180 Hf
Hf 0al., 2002; Schönberg et al., 2002; Yin et al., 2002; Kleine etal., 2005) from a survey of estimates of Hf/W ratios in thesilicate Earth. In fact, there is an uncertainty in the f -value,and estimates range from 10 to 40 (Newsom et al., 1996).
The effects of varying the f -value on our conclusions are
discussed at the end of this section.
Recent studies on planetary formation have shown thatseveral tens of Mars-sized protoplanets are formed throughthe successive accretion of planetesimals in the terrestrial
planet region. Then, during the final stages of planetary
formation, the protoearth suffers collisions with other protoplanetary bodies, known as giant impacts (Kokubo and
Ida, 1998; Wetherill and Stewart, 1989). We therefore consider two different stages: the protoplanet formation stage
and the giant impact stage.
In the protoplanet formation stage, through each impactof a planetesimal, a part of the target body is equilibrated,
and the value of this part is reset to the initial value.
All of the impactor’s mantle is assumed to be equilibrated
during the impact, and each impact equilibrates the target’s
mantle by k times the impactor’s volume, dV . The degreeof equilibration, k, resulting from each planetesimal impact
is a parameter in this model. Thus, each impact equilibrates
the target mantle in a volume k · dV . Therefore, the nth
impact, occurring at t = tn , equilibrates a fraction k · dVofVn
the target mantle, where Vn is the volume of the target bodyat the time of the nth impact, and updates the value to(tn , T0 ) in the equilibrated region. Meanwhile, the rest of
the target mantle, of fractional volume (1−k · dV), retains its
Vnvalue at the time of the (n − 1)th impact. In this calculation,
for simplicity, we assumed that the whole mantle is well
mixed after each impact and that the value for any part ofthe mantle has the average value of the whole mantle at thetime of each impact. Therefore, the isotopic fractionation
by nth impact is given by182

n =k ·dVdV· (tn , T0 ) + (1 − k ·) · n−1VnVn(6)where (tn , T0 ) is given by Eq. (4). We iterated this calculation for each planetesimal impact a hundred million times atregular time intervals before the formation of protoplanets.
In other words, we assumed that the planetesimal accretion
rate was constant in this calculation. Here we define the ageof protoplanet formation as the age of the end of the firststage, or the end of planetesimal accretion to protoplanet. A
plot of versus the age of protoplanet formation is shown in
Fig. 5. k is varied from 0.1 to 10; that is, each impact equilibrated the target’s mantle by 0.1 to 10 times the impactor’sFig. 5. Isotopic evolution of as a function of protoplanet formation age
in the Earth’s region. Parameter k [Eq. (6)] is 0.1, 0.5, 1, 5, 10 from top
to bottom.

volume. This results in values of 10–12 at 10 Myr (Fig. 5).
During this stage, the parameter k does not greatly affect theisotopic fractionation until the formation of a protoplanet at
10 Myr, i.e., a factor of 100 difference in the parameter k
results in an difference of only 2. Therefore, during thisstage, the Hf-W system’s isotopic evolution does not play
an important role, as long as it is assumed that protoplanet
formation was completed in less than 10 Myr (Kokubo and
Ida, 2000).
In the giant impact stage, each giant impact partiallyequilibrates the target body and, thus, partially resets its
chronometer. We define the resetting ratio as the ratio of
equilibrated volume of mantle. The resetting ratio in relation to a giant impact, p, and the number of giant impacts,
n, are parameters in this model. All of the impactor’s mantle was assumed to be equilibrated by the giant impact for
simplicity, which leads to a larger estimation of the equilibration rate of the Hf-W system in our calculation. This
assumption also means that the W isotope composition of
the impactor mantle is reset at giant impact and that the results are independent of the Hf-W history of the impactor.
This simplification gives us a lower limit of the required resetting ratio of giant impact, which is discussed at the endof this section. Each giant impact is assumed to equilibratea fraction p of the target’s mantle. We also assumed that thewhole mantle is well mixed after each giant impact. The
value of the protoplanet, which was experienced during thenth giant impact at a time tng , is given byn = p·(tng , T0 ) + (1 − p) · n−1(7)

We used the value shown in Fig. 5 for the initial value
at the first giant impact. We assumed that the first giant
impact occurred at 10 Myr, so the initial value of was
fixed at 10, according to the result of Fig. 5. Subsequent
giant impacts were assumed to occur at regular intervals.
The Earth’s mantle has a value of 0 and chondrites havea value of −1.9, which means that the Earth’s mantle has
an excess of about 2 units (Kleine et al., 2002, 2004a;
Schönberg et al., 2002; Yin et al., 2002). We calculated
the age of the last giant impact satisfying this Earth’s observational data as a function of the resetting ratio of each
giant impact. In our calculations using Eq. (6) and Eq. (7),the entire Earth’s core was assumed to be involved in the

T. SASAKI AND Y. ABE: IMPERFECT EQUILIBRATION OF HF-W SYSTEM

Fig. 6. The age of the last giant impact as a function of the resetting ratioof each giant impact, fitting to the observational data ( = 2) from Earthsamples. The number of giant impacts is assumed to be five. The initialstate is = 10 at t = 10. The formation age of the Earth for perfectresetting (resetting ratio = 1) is about 30 Myr, in agreement with aprevious study (Yin et al., 2002).

metal-silicate equilibration. This would not be a realistic
assumption. However, these calculations give us the higher
limit of equilibration at each giant impact event. Therefore,
from the viewpoint of obtaining the observed isotopic ratio,
we can obtain the lower limit of the resetting ratio required
for each giant impact. That is, we use assumptions that leadto the “higher limit of equilibration” on calculating isotopic
evolution to obtain “the lower limit of required resetting ratio” to meet the observed epsilon value. The number ofgiant impacts, n, was varied from 2 to 10. Figure 6 shows
the result for n = 5, for example. The estimated age ofthe last giant impact depends on the resetting ratio of each
giant impact, which must be greater than 0.3 to yield theobserved value. The effect of the number of giant impacts
is shown in Fig. 7. It shows that the resetting ratio of each
giant impact and the number of giant impacts both affectthe estimation of the age of the last giant impact. The results indicate that the average resetting ratio of each giant
impact must be greater than 0.2 to yield a good fit with theobservations, even if giant impacts occurred ten times.
Although we use the f -value of 12 in Eq. (4) to solve
Eq. (6) and Eq. (7), as mentioned earlier, the f -value in
Eq. (4) is considerably uncertain (from 10 to 40). Therefore,
we check how does this uncertainty affect our conclusions
here. Figure 8 shows the age of last giant impact assumed
to have occurred ten times to form the Earth for f = 10,
12, 20, 30, 40, which was calculated in a manner similar tothe case of f = 12. It shows that the lower limit of average
resetting ratio of each giant impact is still about 0.2 in therange from f = 10 to f = 40, which would not alter ourconclusions.
Our calculations tend to give an overestimation of theequilibration rate of the Hf-W system, as all of the impactor’s core and mantle is assumed to be equilibrated bya giant impact. In practice, because some fraction of the
impactor’s core (mantle) may be added to the target’s core(mantle) without equilibrating, the required resetting ratio
may be larger than that was estimated in this section. Therefore, our value of 0.2 should be regarded as a lower limit of
the required resetting ratio of each giant impact, for a total

1041

Fig. 7. The age of the last giant impact as a function of the resetting ratioof each giant impact, fitting to the observational data ( = 2) from Earthsamples. The number of giant impacts is 2 to 10 from left to right. The
initial state is = 10 at t = 10.

Fig. 8. The age of the tenth giant impact as a function of the resetting
ratio of each giant impact, fitting to the observational data ( = 2) from
Earth samples. The f -value is 10, 12, 20, 30, 40 from bottom to top.
Each initial -value was calculated using Eq. (6) and Eq. (7) for eachf -value.

number of giant impacts less than or equal to ten.
Nimmo and Agnor (2006) considered two extreme scenarios after giant impacts: complete metal-silicate equilibration and core merging without any significant equilibration. They claimed that the observed isotopic data require
re-equilibration of impacting bodies with the target mantle
and ruled out direct core merging even for the largest impacting bodies. Although the direct core merging is one
type of imperfect equilibration, it is not the only possible
way of imperfectness. The Rayleigh-Taylor instability discussed in Section 2 is another type of imperfect equilibration, which leaves some fraction of silicate without equilibration with iron, while direct core merging leaves somefraction of iron without equilibration with silicate. Here
we showed that the observational data does not rule out incomplete equilibration due to the Rayleigh-Taylor instability. Our results indicate that the collision conditions and thenumber of giant impacts are essential parameters to estimate
the age of the core formation event.

4.

Discussion

We have shown that complete metal-silicate equilibration
by a giant impact cannot be expected even if the impact

1042

T. SASAKI AND Y. ABE: IMPERFECT EQUILIBRATION OF HF-W SYSTEM

causes the whole mantle to become fully molten. We also
showed that the age estimate varies dramatically, depending on the resetting ratio and the number of giant impacts,
and that the resetting ratio of each giant impact is required
to be greater than 0.2 on average. In the fully molten mantle situation, the equilibrated region (silicate-metal mixture
layer) and non-equilibrated region (metal-free silicate melt
layer) could overturn in a very short time, as shown in
Fig. 3. If metal in the mixture layers (Fig. 3) becomes finely
dispersed as small droplets, as described by Rubie et al.
(2003), then only in these layers can the equilibration between metal and silicate be complete. An initially metalfree mantle, however, could not be equilibrated, because
the overturn between the mixture layer and metal-free layer
must occur before the metal droplets are able to sink significantly through the metal-free layer. This indicates thatthe volume of the initially well mixed metal-silicate layer
resulting from each giant impact should be more than twotenths of the volume of the protoearth’s mantle.
Such a mixture layer may correspond to the impactor’score, mantle, and the target’s ejecta layer produced by agiant impact. Therefore, whether or not the mixture layer
is well mixed and perfectly equilibrated between metal andsilicate depends on how the target and impactor break, melt,
vaporize, and accrete after a giant impact. In addition,
SPH simulations of giant impacts suggest that the collisionconditions of giant impacts greatly affect how the bodies
break up. We therefore consider that the Hf-W chronometer
also reflects collision conditions, such as the initial inward
velocity and incident angle of each impactor.
Giant impacts would also have some important consequences for metal-silicate equilibration, including the partial vaporization of the target’s mantle and possible breaking not only of the impactor’s core but also of the target’score. At the very high temperatures just after a giant impact,some part of the target’s upper layer of mantle could vaporize and mix effectively with the impactor’s metal. Suchvaporizations would enhance metal-silicate equilibration in
the upper layer; however, the vaporization of the deep mantle is unlikely because of its very high pressure. A giant impact could break the target’s core and distribute irondroplets in the mantle. These droplets could then equilibrate some part of the mantle; however, unless the irondroplets are evenly distributed throughout the whole mantle,
the Rayleigh-Taylor instability will again prevent the equilibration of the whole mantle. Therefore, even in this case,perfect equilibration would only occur under very limited
conditions, such as the complete vaporization of the target’s
mantle by a giant impact or the distribution of material froma broken target core throughout the whole metal-free layerof the mantle.
It is well known that the observed mantle abundance ofmoderately siderophile elements such as Ni and Co is larger
than that predicted from low-pressure partitioning experiments (Ringwood, 1966). On the other hand, high-pressure
experimental studies have shown that the high pressure andtemperature partitioning can resolve this problem (e.g. Li
and Agee, 2001). As shown below, our description of quick
overturn of the mixture layer shown in Fig. 3 is consistent
with a high pressure and temperature equilibrium between

Fig. 9. Isotopic evolution of as a function of protoplanet formation age
in the Mars region. Parameter k [Eq. (6)] is 0.1, 0.5, 1, 5, 10 from top
to bottom.

Fig. 10. The age of the core formation event of Mars as a function of
the resetting ratio, fitting to the observational data ( = 2, lowest value
observed in SNC meteorites).

metal and silicate for moderately siderophile elements ina magma ocean scenario (e.g. Righter et al., 1997). Figure 3 shows that the segregated mixture layer including liquid metal droplets is deposited at the base of the molten
mantle. Since the growth of the Rayleigh-Taylor instability of the mixture layer is very fast, equilibration between
metal droplets and the silicate mantle occurs after the mixture layer has settled down, i.e., metal-silicate equilibration
is expected to occur under high pressure at the base of themolten mantle. Each giant impact brings about the overturn of the molten mantle, and so multiple giant impacts
could mix the molten mantle many times. Unlike tungsten,other siderophile elements are not produced by the decayof other elements. After multiple giant impacts, almost
all siderophile elements except W in the mantle are equilibrated with metal under high pressure at the base of themolten mantle. Thus, the abundance of these elements islikely controlled by high-pressure equilibration.
We also considered the core formation event of Mars
from the viewpoint of Hf-W chronometry. We calculated
the isotopic evolution of in the Mars region using Eq. (6)with f = 2 (less than the protoearth’s value: f = 12). Thisvalue was chosen because Mars is more oxidizing than the
Earth. W does not efficiently partition into the core, andso the Hf/W ratio for the bulk silicate of Mars should belower than that of the Earth (Kong et al., 1999; Kleine et

T. SASAKI AND Y. ABE: IMPERFECT EQUILIBRATION OF HF-W SYSTEM

al., 2002). The result is shown in Fig. 9. The result shows= 3–4 at 10 Myr, which is slightly larger than the value
for Mars [ = 2, the lowest value observed in SNC meteorites, (Lee and Halliday, 1997; Kleine et al., 2002, 2004a;
Foley et al., 2005)], with a timescale of growth of protoplanets from planetesimals of 105 –106 years in the terrestrial planet region (Kokubo and Ida, 2000). This indicatesthat Mars could have experienced some sort of equilibrating event after the formation of the protoplanets and result
is consistent with Nimmo and Agnor (2006), who impliedthat Mars may not simply be a stranded planetary embryo.
We calculated the isotopic evolution for Mars in the same
way as for the Earth. Figure 10 shows the age of core formation of Mars as a function of the resetting ratio of thecore formation event, fitting the lowest -value observed in
Martian meteorites. This result indicates that the resetting
ratio of the core formation event of Mars is required to begreater than 0.3. Therefore, Mars could have experienced
an event that equilibrated more than three-tenths the volumeof Mars’ mantle, such as a single giant impact.

5.

Conclusions

To achieve perfect equilibration, the iron droplets mustbe less than meter-scale, which makes the sedimentation
velocities of the droplets low. The sedimentation velocity
is therefore much slower than the growth of the RayleighTaylor instability between the upper, metal-containing silicate layer and the lower, metal-free silicate layer. In otherwords, if a metal-free layer exists in the first place, its equilibration is inhibited by the overturn of metal-containing
and metal-free layers. When the equilibration is imperfect,
the collision conditions and the number of giant impacts
must be known in order to estimate the age of core formation of the Earth. Nevertheless, measurements of the tungsten isotope of terrestrial and SNC meteorites can provide
constraints on giant impact events on Earth and Mars.
Acknowledgments. We thank Q.-Z. Yin for many helpful suggestions and comments, and M. Ozima and S. Sugita for valuable
discussions. We also thank T. Kleine and another anonymous reviewer for their constructive comments, which led us to greatly
improve this paper. This study was completed under the support
of a JSPS Research Fellowship.

Appendix A.
First we evaluate the characteristic growth time of theRayleigh-Taylor instability for the (1) thick mixture layerand (2) thin mixture layer.
(1) Thick mixture layer: Suppose a fluid layer with thickness b and density ρ1 (metal grains + silicate melt) lies on
top of another fluid layer with the same thickness b but withdensity ρ2 (silicate melt), with ρ1 > ρ2 . Both fluid layers
have the same viscosity η. The upper boundary of the top
layer and the lower boundary of the bottom layer are assumed to be rigid surfaces. This assumption gives slower
growth of the Rayleigh-Taylor instability than occurs in thefree surface case. We can therefore estimate the upper limit(longer limit) of the growth time. The growth time of a disturbance τr t1 depends on the wavelength λ of the interface1043

displacement (Turcotte and Schubert, 2001)4ηθ 2 + θ cosh θ sinh θ,·(ρ1 − ρ2 )gb
sinh2 θ − θ 2
2π b
θ=
λ

τr t1 =

(A.1)

When the heavy fluid lies on top, the disturbance with theshortest time constant grows and dominates the instability.
The wavelength that gives the smallest value for τr t1 is
λ = 2.568b(A.2)

The growth time of the fastest growing disturbance is obtained by substituting Eq. (A.2) into Eq. (A.1) with the result
τr t1 =

13.04η
(ρ1 − ρ2 )gb(A.3)(2) Thin mixture layer: Suppose a fluid layer with thickness b and density ρ1 lies on top of the semi-infinite fluidlayer with density ρ2 (ρ1 > ρ2 ). Both fluid layers have thesame viscosity η and both upper and lower boundaries arefree because the upper layer is assumed to be a thin layer.
We can derive the relationship between the growth time ofa disturbance τr t2 and the wavelength λ of the interface distortion with the same procedures as for situation (1).
The relationship between physical values at the surfaceof the upper layer (z = H ; subscript number 1) and theboundary between two layers (z = 0; subscript number 2)
is described below:
W2
 U2 

 S2  =F2−θ sinh θ + cosh θθ cosh θθ sinh θ−θ cosh θ + sinh θ−θ cosh θθ sinh θ + cosh θ θ cosh θ + sinh θ−θ sinh θ−θ sinh θθ cosh θ + sinh θ θ sinh θ + cosh θ−θ cosh θ−θ cosh θ + sinh θθ sinh θθ cosh θ−θ sinh θ + cosh θW1 U1 
·
(A.4)
 S1 
F1

where W, U are velocities, S is shear stress, and F is surface traction. θ is the dimensionless layer thickness. We
applied the following boundary conditions: (1) shear stressis zero and surface fraction corresponds to the surface displacement at z = H ; (2) the velocities and shear stress arecontinuous and the surface fraction corresponds to the surface displacement at z = 0. Therefore,
S1 = U1(A.5)

S2 = 0(A.6)

F2 = −τ W2(A.7)

F1 = (1 + δτ )W1

(A.8)

1044

T. SASAKI AND Y. ABE: IMPERFECT EQUILIBRATION OF HF-W SYSTEM

It is then possible to derive the following equationAδτ 2 + (B1 δ + B2 )τ + C = 0
A = cosh θ sinh θ − θ + sinh θ
B1 = cosh2 θ − θ 2 + θ + cosh θ sinh θ
2(A.9)(A.10)(A.11)
B2 = cosh θ + sinh θ + 2 cosh θ sinh θ (A.12)
C = (cosh θ + sinh θ )2
(A.13)
ρ2 − ρ1
δ =(A.14)
ρ12

2

which can be solved to deriveτr t2 =

ηθ
ρ1 gb
B1 δ + B2
·
Aδ




4AB2 δ
1−−1(A.15)(B1 δ + B2 )2On the assumption that the lower layer is semi-infinite,higher order terms of θ can be assumed to be negligible
and Eq. (A.15) is expanded:√√√2ρ1 2 − 2ρ1 ρ2 + 4( 2 + 1)ρ2 2 ητr t2 =
(A.16)
ρ1 ρ2 (ρ1 − ρ2 )gb
The density of the upper layer, ρ1 is obtained byρ1 =1· {ρsi · Vsi + ρmetal · Vmetal }Vsi + Vmetal(A.17)

where Vsi , Vmetal and ρsi , ρmetal represent volumes and densities of the silicate melt and impactor’s core, respectively.
Typical values are ρsi (= ρ2 ) = 4 × 103 kg/m3 , and ρmetal =12 × 103 kg/m3 .
Next we evaluated the characteristic sedimentation timeof metal grains by Stokes sedimentation. According to
Eq. (1), the timescale τs for an iron sphere to sink a distanceof l by Stokes sedimentation is given by
τs =

l
9ηl
=
vs
2ρs gr 2(A.18)

Thus, we can compare the timescales of Stokes sedimentation and the growth of the Rayleigh-Taylor instability using
Eq. (A.3), Eq. (A.16), and Eq. (A.18).
τr t1
26.08 · ρs · r 2=(A.19)τs
9 · (ρ1 − ρ2 ) · bl

√
√
√
2·
2ρ1 2 − 2ρ1 ρ2 + 4( 2 + 1)ρ2 2 ρs · r 2
τr t2
=
τs
9 · ρ1 ρ2 (ρ1 − ρ2 ) · bl
(A.20)
Assuming that b in Eq. (A.19) is 1.45 × 106 m, the half
depth of the present Earth’s mantle, and b in Eq. (A.20) isvery small, e.g. 1 × 104 m, we can estimate the ratio τr t1 /τsand τr t2 /τs .
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