Rheological Properties of
Super-Earth’s Mantle
Shun-ichiro Karato
Yale University
Department of Geology and Geophysics
LEAPS workshop, Pasadena, 2010
1
How does a super-Earth evolve?
mantle convection, thermal evolution
Plate tectonics is a key to habitable surface
environment.
Does plate tectonic operate on super-Earths?
tidal heating
orbital evolution
How much have exo-planets migrated since their
formation?
 Rheological properties
2
Tidal dissipation and evolution of super-Earths
1
Q
µ h,
1
()
1
a
h
(low viscosity  higher heating rate, faster orbital evolution)
3
Could plate tectonics operate on a super-Earth?
How does the resistance and driving force for plate tectonics change with
planetary mass?
resistance: plate thickness
 Rayleigh number
driving force: convective stress
 Rayleigh number
Ra =
r ga (T -Ts ) d 3
kh
A large Rayleigh number  high stress, thin plate  promote plate tectonics
How does the Rayleigh number change with planetary mass?
(Valencia et al., 2007)
P-effect on viscosity is often ignored. Is it justifiable?
4
T-P conditions
P to ~1 TPa (1000 GPa)
T to 5000 K
5
Viscosity of planetary materials depends strongly on T and P.
P-effect is potentially very large!
h = h0 exp
*
( HRT* ) = h0 exp ( E*+PV
RT )
(H*=300-600 kJ/mol, V*=3-10 cc/mol for typical mantle minerals)
6
T-mass relationship
Mass dependence of P: P~M 2/3
energy balance
7
Viscosity-mass relationship
8
Internal structure of a super-Earth
(B1 B2 transition)
(dissociation of
post-perovskite)
A model of a super-Earth (Earth-like composition)
A: upper mantle
B: lower mantle
C: core
9
h = h0 exp
*
( HRT* ) = h0 exp ( E*+PV
RT )
A linear approximation, H*=E*+PV* is not valid at high-P.
V* decreases with depth (pressure) (smaller P-effect), but viscosity increases with P
at a given T.
(Karato, 2010)
10
Viscosity changes when mechanisms of atomic motion change.
V*vacancy >0
V*interstitial <0
vacancy mechanism
interstitial mechanism
(from (Ito and Toriumi, 2007))
(from Karato (1978))
11
Viscosity changes also with crystal structure.
normalize viscosity
normalized temperature
B1
In most of super-Earth’s mantle, MgO
is the softest phase.
MgO changes its structure from B1 to
B2 at ~0.5 TPa.
(modified from Karato (1989))
12
Materials with B2 structure are softer than those with B1 structure.
h µ exp
10-4
( )
*
Gm
RT
10-5
stress=1 MPa
sC
C
10-7
l
10-8
aC
N
B2
l
strain-rate, s -1
10-6
10-9
B1
10-10
10-11
-12
10 1
1.2
1.4
1.6
1.8
2
Tm/T
(data from Franssen (1994) and Heard-Kirby (1981))
(data from Rowell-Sanger (1981))
Fig. 3a (Karato)
Dissociation of post-perovskite (MgSiO3=MgO+SiO2) increases the
volume fraction of a weak MgO.
13
I: mechanism change in diffusion
II: B1 B2 transition
III: dissociation of post-perovskite
(+ metallization?)
14
Conclusions
• Effects of pressure on rheological properties are large.
• If a conventional parameterization is used, viscosity increases so
much with planetary mass and plate tectonics would be difficult for a
large super-Earths.
• Viscosity of the mantle of a super-Earth likely decreases with
pressure and hence decreases with planetary mass.
 plate tectonics is possible in large planets.
• Low viscosity of the deep mantle  high tidal dissipation 
rapid orbital migration + substantial heating.
(effects of tidal dissipation is much larger for rocky planets than for
gaseous planets: influence of tidal dissipation on orbital migration of superEarths will be important to 1-2 AU)
15