How well do we know density in the Earth?

Velocity in the Earth is well known

So far, we have seen how to extrapolate K,G and using an equation of state

K(Tf,P=0)

G(Tf,P=0)

(Tf,P=0)

K(T,P)

G(T,P)

(T,P)

What does it all mean?

Thermo-chemicalParameterization:•Temperature•Fraction of Pv•Fraction of total Fe

Inferring the Earth’s interior

• If we know density we can link laboratory measurements to models of the Earth interior (temperature and composition)

• Vp2=(K+4/3G)/• Vs2=G/• V2=K/

Total mass of the Earth

Maskelyne (18th)

4.5 g/cm3

Today

5.515 g/cm3

Radius R=6371 km (known since Newton 17th, Kepler)

Mass M=5.9739*1024 kg (Kepler)

Average density =5.515 g/cm3

Density of surface rocks 2.5 g/cm3

Density in the centre 13 g/cm3

Moment of inertia about the axis of rotation J2

Full sphere: J2=0.4MR2

Hollow sphere J2=0.66MR2

Astronomical observation (shape and rotation of the Earth) J2=0.33MR2

Density from seismology

dr

dc

cdr

d

dr

dT

Tdr

dP

Pdr

d

We can write

with T=temperature, P=pressure, phase transition and c=chemical variation

Density from seismology

In a homogeneous, self-compressed layer, far from phase transitions,

d/dr=0, dc/dr=0 and dP/dr=-g

g is the gravitational acceleration

Density from seismology

In a convecting mantle, the temperature gradient is close to adiabatic

CKP

TPs

S

TT

dP

dT

which gives

dr

dPT

dr

dT

CP

Density from seismology

We finally get

CP

gTg

dr

d

using

KVV

PK

TS

SP

S

S

P

22

3

4,,

1

Density from seismology

This Adams-Williamson’s law

g

dr

d

Where describes the deviation from adiabacity

Density from seismology

Which can be rewritten as

gdr

d

gdP

dK S

11

The Earth is abiabatic if the Bullen parameter

11

dr

d

gdP

dK S

Temperature in the Earth

Composition in the Earth

Assume that the mantle (core) is adiabatic and homogeneous, make a zero pressure extrapolation

Stacey PEPI 2004

Composition in the Earth

An approach based on high pressure and high temperature mineral physics data (Deschamps and Trampert, EPSL 2004)

Heating (1-3)

Adiabatic compression (4-7)

Method

• Pressure is known from PREM for each depth• We vary potential temperature (end temperature is

calculated along adiabat)• We vary average composition (Pv, Fe) between

certain limits• An adiabatic compression is done for each mineral• VRH average is calculated• Finally, Vp, Vs and is compared to PREM

We can’tresolve the trade-offs