Terrestrial core formation – Constraints and Models

Gregor Golabek

GFD seminar

The problem

4,57 Ga ago:Protoearth largely undifferenciated silicates + 10 – 20% iron

today:differentiated planet withsilicate mantle & iron core

???

Formation of Earth

The start of the solar system

• Probable formation place of the solar system 4,57 Ga ago: Interstellar gas- & dust cloud with ~ 106 MSun

© NASA/Hubble Space Telescope

The start of the solar system

• Probable formation place of the solar system 4,57 Ga ago: Interstellar Gas- & dust cloud with ~ 106 MSun

• Infusion of heavy elements & radioaktive isotopes due to near supernova explosion

• Paradoxon: Star death in star forming region? Mass-rich stars live shorter!

• Example: M=100 MSonne, τnuclear=3 Ma

© D. Malin, Anglo-Australian Observatory

The start of the solar system

• Probable formation place of the solar system 4,57 Ga ago: Interstellar gas- & dust cloud with ~ 106 MSun

• Infusion of heavy elements & radioactive isotopes due to near supernova explosion

• Paradoxon: Star death in star forming region? Mass-rich stars live shorter!

• Example: M=100 MSun, τnuclear=3 Ma• If self gravitation >> gas pressure => collapse and star formation

• Transport of material in the beginning in radial direction• With rising density the number of collisions between hydrodynamic

components of the gas increases → transport of momentum away from the mass centre, flattening of the cloud

© NASA/Hubble Space Telescope

• Cloud starts to cool down: formation of grains (~1 mm) (~103 a) due to cohesive forces

• Meter-sized bodies grow due to attachment of grains of smaller bodies/grains from the protosolar nebula due to fior different orbital velocities (~104 a)

• Problem: ´Meter-sized barrier´ Velocity difference between small grains and meter-sized blocks

becomes to large => erosion/disruption [Brauer et al., 2008] very fast radial drift to the centre

Accretion

• Cloud starts to cool down: formation of grains (~1 mm) (~103 a) due to cohesive forces

• Meter-sized bodies grow due to attachment of grains of smaller bodies/grains from the protosolar nebula due to different orbital speed (~104 a)

• Problem: ´Meter-sized barrier´ Velocity difference between small grains and meter-sized blocks

becomes to large => erosion/disruption [Brauer et al., 2008] very fast radial drift to the centre• When the body reaches ~ 0,1 - 10 km gravitative effects become

important → deflection and impacts of particels from growing surrounding

(´feeding zone´)• Starting of the runaway growth:

Within ~ 105 a some planetesimals reach sizes ~ 3000 km

most of the mass concentrated in a few larger bodies

Accretion

N body results on runaway growth

[Kokubo and Ida, 2000]

mean mass

maximummass

[Melosh, 1990]

• Cloud starts to cool down: formation of grains (~1 mm) (~103 a) due to cohesive forces

• Meter-sized bodies grow due to attachment of grains of smaller bodies/grains from the protosolar nebula due to different orbital speed (~104 a)

• Problem: ´Meter-sized barrier´ Velocity difference between small grains and meter-sized blocks becomes

to large => erosion/disruption [Brauer et al., 2008] very fast radial drift to the centre• When the body reaches ~ 0,1 - 10 km gravitative effects become important → deflection and impacts of particels from growing surrounding (´feeding

zone´)• Starting of the ‘runaway growth’: Within ~ 0.1 -10 Ma some planetesimals reach sizes ~ 3000 km

much of the mass concentrated in a few larger bodies

→ Collisions (´Giant impacts´) between planetary embryos with radii up to Mars-size (‘Oligarchic growth’)

Accretion

Runaway and Oligarchic growth are stochastic

[Chambers, 2001]

Results for oligarchic growth

[data by Agnor from Rubie et al., 2007]

Giant impacts

[Canup and Asphaug, 2001]

0.3 h

23 h

Internal Energy[erg/g]

• Cloud starts to cool down: formation of grains (~1 mm) (~103 a) due to cohesive forces

• Meter-sized bodies grow due to attachment of grains of smaller bodies/grains from the protosolar nebula due to unterschiedliche Bahngeschwindigkeiten (~104 a)

• Problem: ´Meter-sized barrier´ Velocity difference between small grains and meter-sized blocks becomes to

large => erosion/disruption [Brauer et al., 2008] very fast radial drift to the centre• When the body reaches ~ 0,1 - 10 km gravitative effects become important → deflection and impacts of particels from growing surrounding (´feeding zone

´)• Starting of the ‘runaway growth’: Within ~ 0.1 -10 Ma some planetesimals reach sizes ~ 3000 km

much of the mass concentrated in a few larger bodies

→ Collisions (´Giant impacts´) between planetary embryos with radii up to Mars-size (‘Oligarchic growth’)

• Core formation starts with the formation of bodies > 30 km radius

Accretion

Constraints: Isotopic Chronometry

• 182Hf → 182W + β- (T1/2=9 Ma)• chondritic starting material:

Hf/W=1• Hf isotopes are lithophile

(´rock-loving´), whereas W isotopes are siderophile (´metal-loving´)

→ Hafnium stays in the mantle, tungsten segregates mainly into the core

• Therefore: Hf/W ratio high in the Earth´s

mantle• Compare W abundance in

today´s Earth mantle and chondrites allows to get a constaint of the core formation time

• Core formation on Earth 14 - 62 Ma after the the start of the solar system [Kleine et al., 2004; Touboul et al., 2007]

[Halliday et al., 2000]

Chondrites

• Meteorite consisiting of the most primitive material in the solar system

• Same abundance of elements as sun except for volatiles

© Univ. Washington

Constraints: Isotopic Chronometry

• 182Hf → 182W + β- (T1/2=9 Ma)• chondritic starting material:

Hf/W=1• Hf isotopes are lithophile

(´rock-loving´), whereas W isotopes are siderophile (´metal-loving´)

→ Hafnium stays in the mantle, tungsten segregates mainly into the core

• Therefore: Hf/W ratio high in the Earth´s

mantle• Compare W abundance in

today´s Earth mantle and chondrites allows to get a constaint of the core formation time

• Core formation on Earth 14 - 62 Ma after the the start of the solar system [Kleine et al., 2004; Touboul et al., 2007]

[Halliday et al., 2000]

Main uncertainties

• Was there complete equilibrium between iron and silicates during differentiation?

• Is the Earth´s bulk composition chondritic?• Is the Hf/W system too short-lived to be applicable to the

problem?• Long-lived isotope systems U-U/ U-Pb give longer (~ 100 Ma)

timescales [Allègre et al., 2008]

[Nature 20.03.2008]

[Allègre et al., 2008]

Temperature constraints: Impact heating

• The temperature profile in the early Earth:

a

p

TrHM

rv

rc

rhGMrT

21

2

[Melosh, 1990]

Mars-sized bodies have a molten surface!

[Schubert et al., 1986]

Temperature constraints

But: • neglects short-lived

radioactive isotopes 26Al (t1/2= 0.7 Ma) and 60Fe (t1/2=1.5 Ma)

• T extremly dependent on accretion time and radial distance from sun

• The temperature profile in the early Earth is highly unknown! [Breuer and Moore, 2007]

?

Temperature constraints: Ceres-Vesta dichotomy

Radius: 487.5 km Radii: 280 km, 272 km, 227 km Mass: 9.35 1020 kg Mass: 2.71×1020 kg

© NASA/ Dawn Mission

Temperature constraints

[Grimm and McSween, 1993] [Bizzarro et al., 2005]

Ceres

Vesta

Conditions in the protocore

• Base of the magma ocean is defined by the peridotite liquidus [Rubie et al., 2003]

• Expect partial melting in regions beneath

• But > 60°:no inter-connection in upper part of protocore under static conditions

[Takafuji et al., 2004]

[Stevenson, 1990]

`sponge` `meatballs`

Laboratory constraints

• Base of the magma ocean is defined by the peridotite liquidus [Rubie et al., 2003]

• Expect partial iron melt available in regions beneath

• But > 60°:no inter-connection in upper part of protocore under static conditions

• pinch-off value for FeS melt interconnectivity in a solid peridotite matrix around 5 vol.% in static experiments[Yoshino et al., 2004]

[Takafuji et al., 2004]

Laboratory constraints• But: Segregation very slow without silicate melting => considerable amount of silicates must have been

molten or need large diapirs sinking down

[Bagdassarov et al., in review]

modified after [Wood et al., 2008]

`Vesta` `Mars` `Earth`

Destabilization ofglobal iron layer[Stevenson, 1981]

Sinking of iron-rich Diapirs[Ziethe and Spohn, 2007; Samuel and Tackley, 2008]

Possible core formation scenarios

Sinking of impactor cores [Dahl, 2005]

[Stevenson, 2008]

Global scenario

• Applied viscosity law:

• Use viscoplastic rheology• Use `sticky air`( = 1019 Pa s) layer surrounding

the planet to simulate free surface• Peierls strength limit: =5·1027 Pa (basically no

plasticity)• Plastic strength: y=107 Pa• Power law coefficient: n = 3.5

nRT

PVEA aan

Dnn

II exp/1/1

Initial temperature[°C]

Va=0.6 J/kg; H=12.2·10-6 W/kg; N=6

Va=0.8 J/kg; H=12.2·10-6 W/kg; N=12

Va=0.8 J/kg; H=12.2·10-6 W/kg;

N=12

Temperature [°C] log10(Viscosity)

Va=0.8 J/kg; H=12.2·10-6 W/kg; N=12

log10(Shear heating) density [kg/m3]

Va=0.8 J/kg; H=12.2·10-6 W/kg; N=12

density [kg/m3]

[Stevenson, 1981]

Va=0.8 J/kg; H=12.2·10-6 W/kg; N=24

Results diapirism intermediate? fragmentation

Va=0.8 J/kg; H=12.2·10-6 W/kg; N=596

Peierls strength limit: = 1·109 Pa

Early stage (t=0.007 Ma)

density [kg/m3]log10(Viscosity)

Later stage (t ~ 1 Ma)

log10(Viscosity) density [kg/m3]

Results

• Transition:• High H/ low V => diapirism (internal activation)• Low H/ high V => fragmentation (global activation)• Preliminary results as most massive diapirs control

shift of the protocore• More homogeneous diapir distribution can stabilize

protocore & increase core formation time• Formation of a transient iron layer• Shear zones may provide pathways for following

diapirs• Group behaviour (`Rayleigh-Taylor-like`) of diapirs

possible

modified after [Wood et al., 2008]

Iron can be left over in partially molten regions of the protomantleWhat happens with the iron? => need two phase flow

`Vesta` `Mars``Earth`

Experiments applying shear stress

• Shear stresses can interconnect the isolated iron pockets and create melt rich bands

• Minimum shear stress needed for interconnection of melt pockets with porosity< 5% (100 MPa)

[Hustoft and Kohlstedt, 2006]

Dark: olivineLight: Fe-S

Physics and model setup

Two phase flow equations solved numerically using FDCON by H. SchmelingLimitation: porosity < 25%Iron partially molten, silicates solid

(10 – 400) km

0

Dt

DMu

t fff

zf

sf gnPk

uu

011

Dt

DMu

t sss

0

j

ijz x

Pgn

The retention number Rt

• The retention number [Tackley and Stevenson, 1993] controls the development of channelling instability:

• Values for Rt is highly uncertain and may lie in the range between 10-10 – 10-2

=> numerical models (Rt ≥ 1) underestimate the development time of melt channelling

12

2

20

2

00

2 n

DiapirDarcy

Stokesff

r

h

v

v

aη

bhη=

kη

hη=Rt

The channelling instability• Assumptions: White noise on temperature

field and porosity depen-dent viscosity:

• Channels form in partially

molten material parallel to the maximum compressive stress

• Total stresses of about 0.1 to 0.2 are needed to induce melt channels

[Müller, 2005]

nII)a(η=η 1

10exp

Applied on the core formation scenarios

Rayleigh-Taylor instability Sinking iron core

Rt = 1.0

Evolution of the channelling instability

Rt = 1.0

Channels may reach centre of the planet in this case in t < 30 Ma

Numerical experiments indicate a 1/r scaling for thedevelopment time of melt

channels

Same timestep (for poro-sity) for inducing diapirswith: (A) 25 km(B) 50 km (C) 100 km (D) 150 km radius

Impactor cores preferred?

Where does the channelling instability form?

• The deviatoric stress must be high enough to enable channelling

=> channels form in a sphere with c·r, with

c = 2 – 3 for r > 25 km• Results indicate that

diapirs must have at least 5 km radius to interconnect melt pockets to induce melt channels

Interactive behaviour - Rt = 1.0

Dependency of the stress field on the distance between diapirs

• Proximity of neighbouring diapirs reduces the size of the draining zones; protracts channelling

• Same behaviour also with pseudoplasticity up to n=4

0

0.5

1

1.5

2

2.5

3

3.5

-1 1 3 5 7 9 11

lambda/2

r(1/e)

What‘s about the chemical equilibration?

• The theory indicates that the growth rate is constant with wave number k = 2/

• Expect existence of melt channels with very small wavelengths below numerical resolution feeding the major channels

(“root tree mechanism“)

[Müller & Schmeling, 2008]

What‘s about the chemical equilibration?

Regions between melt channels are highly depleted of iron melt

Melt channels start to drain the iron diapir itself

As long as the melt velocity in the channels is not too high chemical equilibrium is possible(further modelling needed)

Porosity [%]

Conclusions

• Iron melt channels can be induced by diapirs with r > 5 km

• Undisturbed melt channels can reach the centre of the planet in t < 30 Ma, but certainly depends on the radius

• Impactor cores may be preferred to induce channelling instability due to their larger radii

• They are able to drain iron melt from the partially molten regions of the protocore

• The “root tree mechanism“ of the channelling instability may enable chemical equilibration below the boundary of the magma ocean

What‘s next?• Apply model to 3D (almost

done) with porosities up to 100%

• Investigate more complex rheologies = f(stress,T)

But need more T constraints!• What is the switch between

diapirism mechanism and dykes (not density contrast)?

=> propagation of melt via dykes to the core?

=> creation of new diapirs?• Combine global model with

channelling instability (local resolution enhancement)

Thanks for your attention!