Moment of inertia
Gravity field and potential Energy
Shape of the Earth
Reference ellipsoid and geoid
y
y
y
y
y
y
y
y
y
y
y
y
Fundamentals: Gravity and inertia.
Conservation laws.
Example: Core formation
Earth rotation
Figure of equilibrium
Reference ellipsoid.
p
Geoid
Gravity anomalies
Isostasy
Tides,, secular changes
g in rotation rate
Post glacial rebound
Milankovicic cycles
Chandler wobble
Basics
y Newton’s law F = m a
y Gravity force Fg = G M1 M2 /R2 ( Newton)
y M(inertia) = M(gravity)
y Gravity field g = GM/R2 (m s-2)
y Field conservative (potential)
y g = grad U
y Coriolis force
y Conservation of angular momentum y If r (distance to axis) changes, acceleration relative to spinning body
y If velocity changes, accelaration or deceleration to keep angular momentum
y In northern hemisphere to the right
Mars has no
symmetry about axis
of rotation (Tharsis
bulge) and has two
different principal
axes off iinertia
i in
i
equatorial plane.
R d d momentt off inertia
i ti (I/Ma
(I/M 2) off
Reduced
planets
y Earth 0.33
y Moon 0.391
y Mars 0.365
y Jupiter 0.26
y Saturn 0.2
M
t off iinertia
ti “linear”,
“li
” i.e.
i
Moment
I (sphere + shell) = I(sphere) + I(shell)
G it field
fi ld andd potential
t ti l
Gravity
y Note potential 1/R
y Field 1/R2
y Field is not constant on equipotential surfaces
y Present moment of inertia for Earth 8 x 1037 kg m2
y Energy of rotation y ~ 1029 J
y Gravitational potential energy for spherical Earth y ~ 1032 J
y Moment of inertia (0.33
(0 33 Mr2)
y Present energy of rotation
y Change in rotation energy due to core formation
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
Angular momentum is a vector
Direction and length must be conserved
(6)
(7)
(8)
Core formation
Core formation ((conservation laws))
•
•
•
•
•
•
•
•
Gravitational potential energy decreases when core forms
Moment of inertia decreases
Angular velocity of rotation increases
Rotational energy increases
Increase in energy of rotation < Decrease in gravitational potential energy
Total energy must be conserved
Difference
iff
goes into
i
heat
h
Estimates: Core formation -> 1000-2000K temperature increase