Geochronometry-Isotope tracing-Age of the Earth
y Geochronometry (methods)
y Nuclear synthesis
y Meteorites
y Age of the Earth accretion
y Pb
y Formation
F
ti off the
th core
y Formation of the core (energy considerations)
y Formation of crust
y Plate tectonics starts
Geochronometry
y Radiogenic isotopes
y Decay mechanisms (α decay, β decay, electron capture)
y Main isotopic
p systems
y
for dating
g
y Rb-Sr
y K-Ar
y U-Pb
U b
y Th-Pb
y Other isotopes used mainly for “tracing” (Sm-Nd, Re-Os, …)
y
Geochronometry (hypotheses)
y Parent -> daughter decay probability λ
y Mineral closes at temperature (depends on type: zircons 800 deg, feldspars
350,, …))
y No daughter present at closure (or it can be accounted for)
y No loss or gain of parent or daughter after mineral closes
y Counting P/D gives the time that elapsed since the system closed
Geochronometry (particulars)
y K->Ar is a branching decay K40 -> Ar 40 or Ca 40
y U -> Pb two different isotopes of same element give two independent age
estimates ((must be concordant))
y Rb/Sr requires different minerals with variable Rb/Sr ratios (same for SmNd). Methods yield initial isotopic ratio of Sr87/Sr86 (important for tracing)
Same equations and method for other systems (U-Pb
(U-Pb, Sm-Nd)
4
Méthode Rb-Sr
Si des minéraux différents proviennent d’une même source, ils ont initialement le même rapport
isotopique 87 Sr/86 Sr.
Sr Ce rapport évolue
evolue au cours du temps par l’apport
l apport de 87 Sr radiogenique:
radiogénique:
87
87
Sr(t)
Sr(t = 0) 87 Rb(t)
=
+ 86
exp(λt) − 1
86 Sr
86 Sr
Sr
(8)
Pour des minéraux de composition différente, le rapport 87 Sr/86 Sr varie linéairement en fonction du
rapport 87 Rb/86 Sr. La pente de la droite donne l’age de fermeture de ces minéraux; l’intercept donne la
composition isotopique initiale de la source.
3
Méthode K-Ar
Methode
40
K est un isotope peu abondant de lélément K qui se désintègre en 40 Ca ou en 40 Ar. Ar étant un
gaz inerte, il n’est en général pas présent dans les minéraux. 40 Ar présent est donc le produit de la
désintégration de 40 K:
λAr 40
40
Ar(t) =
K(t) × (exp(λtot t) − 1)
(7)
λtot
où λtot = λAr + λCa .
K-Ar
y No Ar initially
y But problem of atmospheric contamination
y Correction based on Ar36
y Also Ar is easily lost
y Retrace loss by step heating of samples and Ar-Ar ages
5
Méthode U-Pb
Les méthodes U-Pb permettent d’obtenir deux déterminations indépendantes de l’age de fermeture d’un
minéral.
207
235
P b(t)
b( ) =
U (t)
( ) exp(λ
(λ235 t)) − 1
(9)
206
238
P b(t) =
U (t) exp(λ238 t) − 1
(10)
Si les deux determination
détermination donnent la meme
même valeur
valeur, l’age
l age est dit concordant
concordant. La composition isotopique
207
P b/235 U en fonction de 206 P b/238 U évolue au cours du temps le long d’une courbe dite Concordia. Il
est parfois possible d’interpréter des ages discordants et de déterminer l’age de fermeture et l’age d’un
évenement (p. ex. métamorphisme) pendant lequel il y a eu perte d’U et/ou de Pb.
Dating the synthesis of elements
Meteorite samples
chondrite
Iron
achondrite
Xe129
y Xe129 product of short half life I129
y Meteorites formed shortly after nucleosynthesis.
y Xe129 in earth atmosphere (I129 in primitive earth) comes from degasing of
mantle
y Earth and meteorites have ~ same age
Meteorites
y All meteorites have about the same age 4.55 Ga
y Some meteorites that have younger ages come from the moon. They were
ejected
j
after impact.
p
y A few are much younger (1.1 Ga). They are assumed to have been ejected
by Mars after a large impact
Martian meteorites (?)
Moon samples
Nasa has collected samples for dating
Ages range between 3.0 and 4.5 Ga
(see PDF document)
y
Time series of a Moon‐forming impact simulation. Results are shown looking down onto the plane of the impact at times t = 0.3, 0.7, 1.4, 1.9, 3, 3.9, 5, 7.1, 11.6, 17 and 23 hours (from left to right); the last frame is t = 23 hours viewed on edge Colour scales with internal hours viewed on‐edge. Colour scales with internal energy (shown on the colour bar in units of 6.67 times 108 erg g‐1), so that blue and dark green represents condensed matter, and red particles signify either the expanded phase or a hot, high‐pressure condensed phase; pressures at intermediate energies are computed y
p
by an interpolation between the Tillotson15 condensed and expanded phases. We form initial impactors and d
d d h
f
l
d
targets in hydrostatic equilibrium by pre‐colliding smaller bodies together at zero incidence, resulting in realistically evolved internal energies, stratified densities (basalt mantle + iron core) and consistent pressures. Each particle's internal energy is evolved due to the effects of expansion/compression and shock dissipation, with the latter represented by artificial viscosity terms that are linear and quadratic in the velocity divergence of converging particles; effects of mechanical strength and radiative transfer are ignored. The momentum of each particle is evolved due to pressure, viscous dissipation and gravity. Gravity is computed using a binary tree algorithm reducing the N2 calculation of particle
algorithm, reducing the N2 calculation of particle–
particle attractions into an NlogN calculation25. We use a beta spine kernel to define the spatial distribution of material represented by each SPH particle. The scale of each particle, h, is automatically adjusted to cause overlap with a minimum of 40 other particles, ensuring a 'smoothed' distribution of material even in low‐density regions. The code is explicit, requiring a Courant‐limited i
Th d i li i i i C
li i d timestep Deltat < (c/h) where c is the sound speed. For a full description of the technique, see ref. 26, from whose efforts our present algorithm derives.
Rappel
pp
Geochronometry hypotheses
Nucleosynthesis (6 to 4.6 Ga)
Age of meteorites 4.55
4 55 Ga
Meteorites follow shortly end of nucleosynthesis
Earth followed shortly end of nucleosynthesis
Moon samples
l 3.2
3 2 to 4.5
4 Ga
G
Oldest rock on Earth 4 Ga
Age of Earth from Pb 4.55 Ga
Dating core formation
y Hafnium Hf and Tungsten W
y Hf182 -> W182 (half life 9 Myears)
y Hf180 reference
y Hf stays in mantle
y W goes in core
y Initial
I iti l ratio
ti Hf182/Hf180 in
i solar
l system
t different
diff
t from
f
that
th t off mantle
tl
y εw values of carbonaceous
chondrites compared with
ith those of
the Toluca iron meteorite and
terrestrial samples analysed in this
study. The values for Toluca,
Allende G1
Allende,
G1-RF
RF and IGDL
IGDL-GD
GD are
the weighted averages of four or
more independent analyses. Also
included are data from ref. 16
(indicated by a),
a) ref.
ref 30 (b),
(b) and
ref. 2 (c). For the definition of εw
see Table 1. The vertical shaded
bar refers to the uncertainty in the
W isotope composition of
chondrites. Terrestrial samples
include IGDL-GD (greywacke),
G1-RF (granite) and BB and BE-N
(basalts).
(basalts)
y εw versus 180Hf/184W for different fractions of the H diff
f
i
f h H chondrites Ste Marguerite (a) and Forest Vale (b). NM‐1, NM‐2 and NM 3 refer to different and NM‐3 refer to different nonmagnetic fractions, M is the magnetic fraction. We interpret the positive correlation of εw with 180Hf/184W as an internal ith 8 Hf/ 8 W i t
l Hf–W isochron whose slope corresponds to the initial /
182Hf/180Hf ratio at the time of closure of the Hf–W system. y Time of core formation in Myr after CAI condensation for f CAI d
i f Vesta, Mars, Earth and Moon versus planet radius as deduced from Hf W systematics. For the from Hf–W systematics. For the Moon, the two data points refer to the endmember model ages. The Moon plots distinctly to the l ft f th left of the correlation line l ti li defined by Vesta, Mars and Earth, suggesting a different p
formation process. y
Timing of core formation. The Earth g
,
g
formed through accretion, absorbing planetesimals (lumps of rock and ice) through collisions. Did the Earth accrete undifferentiated material that then separated into shell and core —
in which case, did the planet reach its present mass before differentiating present mass before differentiating, or was it a more gradual process? Alternatively, core formation might have happened rapidly inside growing planetesimals, so that the Earth's core is a combination of these previously formed cores Isotopic previously formed cores. Isotopic evidence supports the latter model, and now Yoshino et al.1 demonstrate a mechanism for the physical process. Core formation (conservation laws)
y Gravitational potential energy decreases when core forms
y Moment of inertia decreases
y Angular velocity of rotation increases
y Rotational energy increases
y Increase in energy of rotation < Decrease in gravitational potential energy
y Total
T t l energy mustt be
b conservedd
y Difference goes into heat
y Estimates: Core formation -> 1000-2000K temperature increase
1
Tenseur d’inertie. Moment d’inertie. Moment cinétique et
énergie de rotation
Dans un système de coordonnées cartésiens, le tenseur d’inertie d’un corps est défini par:
ρx2 dV
I = ρyxdV
ρzxdV
ρxydV
2
ρy dV
ρzydV
ρxzdV
ρyzdV
2
ρz dV (1)
Le tenseur d’inertie est symétrique.
Dans un système de coordonnées dont l’origine est le centre de masse, il existe trois axes
perpendiculaires, axes principaux d’inertie par rapport auxquels la valeur du moment
d’inertie est maximum, intermédiaire, ou minimum. Les moments principaux d’inertie
sont donc:
2
I=
mk rk = ρr 2 dV
(2)
k
où rk est la distance entre la masse mk et l’axe. Pour un corps à symétrie sphérique, tout
axe passant par le centre de masse est axe principal d’inertie. Pour une sphère uniforme,
de rayon a, I = 0.4Ma2 . Pour la terre, le moment d’inertie (polaire) par rapport à l’axe
de rotation, C = 0.33Ma2 . La valeur du moment d’inertie de la Terre implique une
augmentation de la densité vers le centre de la Terre.
La rotation d’un corps est stable si l’axe de rotation est l’axe principal d’inertie maximum
ou minimum.
Le moment d’inertie s’exprime en kg m2 .
2. Géodésie et Gravité
3
Conservation of angular momemtum:
ΔI Δω
+
=0
I
ω
Energy
E=
E
Energy
b
budget
d
Iω 2
2
ΔI
Δω
−ΔI
ΔE
=
+2
=
E
I
ω
I
(6)
(7)
(8)
2. Géodésie et Gravité
2
Lois de conservation d’un corps en rotation
La rotation d’un corps autour d’un axe est décrite par le vecteur de rotation ω qui est
parallèle à l’axe de rotation et tel que la vitesse d’un point est:
v = ω × r
(3)
ou r est un vecteur de l’axe de rotation vers le point. La rotation d’un corps est stable si
l’axe de rotation est axe principal minimum ou maximum (c’est le cas de la toupie ou du
gyroscope). L’énergie cinétique d’un corps en rotation:
Ecin =
1
1
1
mk vk2 =
mk ω 2 rk2 = Iω 2
2 k
2 k
2
(4)
La quantité
L
tit´ de
d mouvementt totale
t t l estt nulle.
ll Le
L momentt cinétique
i ´ti
estt le
l momentt de
d la
l
quantité de mouvement par rapport à un point sur l’axe de rotation J est:
J =
k
mk rkvk =
k
mk rk2 ω = Iω
(5)
En l’
E
l’absence
b
d
de fforces extérieures
´i
quii exercent un couple,
l le
l moment cinétique
i ´i
doit
d i ˆêtre
conservé.
He
y It is assumed that volatiles were lost during accretion
y Very little He in atmosphere (too light, lost to space)
y He in mantle
y He3 is primitive, He4 primitive + decay of radioelements
y He4/He3 ratio (initial ratio same as that of universe)
y H
He4/He
/H 3 ratio
ti grows with
ith time
ti
y Some degasing
y Shows mantle is not well mixed
Tracing with isotopes
y Crust
y Mantle
y Rb/Sr high
y Rb/Sr low
y Sm/Nd low
y Sm/Nd high
y Sr87/Sr86 increases
y Nd143/Nd144 decreases
d
compared
d
with mantle