Download 0 Gravity, Rotation, Shape of the Earth 0.1 Inertia. Conservation laws

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Gravity, Rotation, Shape of the Earth
0.1 Inertia. Conservation laws. Gravity
Inertia.
Energy. Angular momentum
Moment of inertia of Earth
Gravity field and potential
Inertia. Pseudo-gravity potential
0.2 Figure of equilibrium of the Earth.
Condition for equilibrium in a fluid.
Gravity field and potential
Inertia. Pseudo-gravity potential
Reference ellipsoid. Geoid.
0.3 Earth’s gravity field
Variations in sea level gravity.
Reduction of gravity measurements.
Free air and Bouguer corrections.
Anomalies.
Isostasy.
Post glacial rebound. Mantle viscosity.
Post glacial adjustments and earth rotation.
0.4 Tides. Ocean and solid Earth
Secular slowing down of Earth rotation.
Periodic variations in Earth rotation.
Length of day (LOD) variations .
Chandler Wobble (T ≈ 435 j )
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0.5 Milankovicic cycles
Earth’s orbit. Variations of orbital parameters: eccentricity (T ≈100,000 yr),
inclination of rotation axis (T ≈ 41, 000 yr), perihelion and precession of
equinoxes (T ≈ 25, 000 yr).
Cartwright, D.E. (1999). Tides: A scientific history. Cambridge. pp. 292.
Hays, J.D., Imbrie, J., & Shackelton, N.J. (1976). Variations in Earth orbit:
pacemaker of the ice ages. Science, 194. 1121-1132.
Lambeck, K. (1988). Geophysical geodesy. Oxford Science Publication. pp.718.
Smylie, D.E., & Mansinha, L. (1971). The rotation of the Earth. Sci. Am.,
225. 80-88.
Watts, A.B. (2001). Isostasy and flexure of the lithosphere. Cambridge University Press. pp. 458.
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Gravity potential. Gravity field.
1.1 Inertia
Newton’s first law of motion: A body at rest or in uniform motion will not
change its velocity unless forces are applied. (Galileo)
Newton’s second law of motion: The relationship between an object’s mass m,
its acceleration ~a, and the applied force f~ is
m~a = f~
Newton’s third law: action = reaction
1.2 Gravity
The gravity force F~ between two point masses m and m0 is obtained from
Newton’s gravity law:
~
Gmm0~u
Gmm0 R
F~ =
=
R2
R3
where R is the distance between the two masses, G = 6.67×10−11 N m2 kg−2 is
the gravity constant. The force is attractive and it is parallel to the line joining
~ is the vector joining the two masses and ~u = R/R
~
the two masses (R
is the
~
unit vector in the direction of R). This an example of action and reaction: the
attractive force of m to m0 is equal to that of m0 on m (examples Earth-Sun,
Earth-Moon systems.)
Although there is no obvious reason that it should be so, the property ”mass”
that enters in the gravity law, and that in the inertia law have always been
found to be proportional to each other: the proportionality constant was chosen to be 1. (General relativity is based on the equivalence principle that states
that inertial and gravitational masses are identical).
A particle of mass m under the gravity forces of other particles will experience
an acceleration ~g that does not depend on its mass. This acceleration which
characterizes any point in space is the gravity field, or the acceleration of
gravity.
Z
~ P 0 P dV (P 0 )
~k
X mk R
ρ(P 0 )R
=
G
~g (P ) = G
Rk3
RP3 0 P
k
3
~ k is the vector from P to mass mk (R
~ P P 0 = {(x0 −x), (y 0 −y), (z 0 −z)})
where R
and ρ is a mass density distribution. The gravity field is measured in m s−2 .
It is a conservative field, i.e. because the work done by gravity to go from one
point to another is independent of the path, it can be derived from a potential
function.
The gravity potential U in point P represents work done by gravity when a
unit mass is brought from P to ∞:
W (P ) =
It gives:
Z ∞
P
Z
˙r
~g (~r)d~
ρ(P 0 )dV (P 0 )
RP 0 P
U (P ) = G
with
~g = ∇P U
The acceleration of gravity is thus normal to the equipotential surfaces.
The gravity field of a spherically symmetric body with mass M is identical to
the field of a point mass at the center of the sphere. Outside the sphere, the
gravity field does not depend on the radial density distribution. The potential
and the field at distance R from the center of mass:
M
R
M
g = −G 2
R
U =G
the negative sign indicates an acceleration toward the center of the sphere.
These equations could have been derived from Gauss theorem that states that
the flux of the gravity field across a closed surface is proportional to the total
mass enclosed by the surface
I
S
Z
~g · ~ndS = −4πG
V
ρdV = −4πGM
1.3 Units
Acceleration is measured in m s−2 . The practical unit in geophysics is the
mGal (1 mGal = 10−5 m s−2 . For the potential, units m2 s−2 or J kg−1 are
equivalent. Note that the potential represents an energy per unit mass. Geoid
anomalies are measured in m.
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Moment of inertia. Angular momentum and rotational energy
The moment of inertia I of a body relative to an axis:
I=
X
Z
mk rk2 =
ρr2 dV
k
where rk is the distance between the axis and mass mk . Moment of inertia is
given in kg m2 .
In general, for a non spherically symmetric body, the value of moment of
inertia depends on the direction of the axis. One should thus use the tensor
of inertia. In a coordinate system with origin at the center of mass, there are
three perpendicular axes, the principal axes of d’inertia, respective to which
the moment of inertia is maximum, minimum, and intermediary. Note that for
a spherically symmetric body, any axis through the center is principal. For an
ellipsoid of revolution, the axis of symmetry is principal axis. Any axis in the
equatorial plane is also principal axis. There are only two distinct principal
values for the tensor of inertia.
For a uniform sphere, with radius a , I = 0.4M a2 . For Earth, the polar
moment of inertia (i.e. relative to the rotation axis) C = 0.33M a2 . This value
implies an increase in density toward the center of Earth, i.e the existence of
a dense core.
2.1 Conservation laws
The rotation of a body around an axis is described by the rotation vector ω
~
which is parallel to the rotation axis and such that the velocity of a point in
the body:
~v = ω
~ × ~r
where ~r is a vector from the axis of rotation to the point. The magnitude of
the rotation vector is ω = 2π/T (where T is the period of rotation)
The rotation is stable only if the rotation axis is minimum or maximum principal axis of inertia (This is indeed the case for a gyroscope, a spinning top,
and the Earth).
The kinetic energy of the body:
Ecin =
1X
1X
1
mk vk2 =
mk ω 2 rk2 = Iω 2
2 k
2 k
2
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The net momentum is zero. The angular momentum is the sum of the moments of the momenta relative to a point on the axis of rotation. The angular
momentum J~
X
X
J~ =
mk~rk~vk =
mk rk2 ω
~ = I~ω
k
k
Without a torque exerted by external forces the angular momentum must be
conserved.
A mass attached to a rotating body experiences inertia. The effect of inertia
is sometimes referred to as the centrifugal acceleration.
~a = −ω 2~r
with ~r is the distance from the axis. And the sign indicates an acceleration
away from the axis. This acceleration corresponds to a pseudo-potential V :
V =
−ω 2 r2
2
r is the distance to the axis of rotation. On Earth, it depends on latitude
λ r = R cos(λ)
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Figure of equilibrium of a fluid.
The condition for a fluid to be in equilibrium is that all the forces acting on a
parcel balance each other. If there is no motion, the only forces are differences
in pressure and gravity. The condition is thus that
~g + ∇P = 0
This implies that all the isobaric surfaces are equipotentials, and in particular
that the outer surface of the fluid is an equipotential. (For example, mean sea
level corresponds to an equipotential of the gravity.
For Earth, it can be shown that the equilibrium figure is an ellipsoid with the
major inertia axis is the rotational axis.
The gravity potential of an ellipsoid can be written as:
U (R) =
G
1
−GM
3
+ 3 (C − A)
cos2 θ −
R
R
2
2
with C moment of inertia relative to the rotation axis and A equatorial moment of inertia (in the direction perpendicular to the axis of rotation), θ
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colatitude. Including the pseudo-potentiel of rotation, we get:
−GM
GM a2 J2
U (R) =
+
R
R3
3
1
1
cos2 θ −
− ω 2 R2 sin2 θ
2
2
2
with J2 = (C − A)/M a2 ) (it is a dimensionless number).
If the equatorial et polar radii are a et c. The flattening is defined f = (a−c)/a.
The flattening is determined by the condition that the potential is the same
at the pole and the equator.
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ω 2 a2
f = J2 +
2
2ge
with ge gravity at the equator. As the Earth polar radius is c is about 20km
shorter than the equatorial radius a, the flattening is about 1/300. It has been
measured and calculated very precisely. The observed and predicted values of
the flattening ≈ 1/300 are very close, but there is a small difference due to
recent changes in Earth’s rotation rate.
The theoretical equipotential surface defining mean sea level is the reference ellipsoid.
The observed mean sea level equipotential is the geoid. Geoid anomalies are
due to lateral variations in density within the Earth. They are measured in
m. The fact that they are small is another indication that the Earth is close
to hydrostatic equilibrium.
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Gravity anomalies
The gravity acceleration is the direction perpendicular to the equipotential. It
is not constant on an equipotential. (The acceleration is inversely proportional
to the distance between equipotentials.) As the polar radius is shorter than
the equatorial, it is thus expected that g will be larger at the pole that at the
equator.
For a homogeneous (rotating) Earth, sea level gravity depends only on latitude
φ and is given by:
gref = ge (1 + α sin2 φ + β sin2 (2φ))
with ge = 9.780327m s−2 , α = 0.0053024, β = 0.0000058. This is the International Gravity Formula 1967.
The difference between observed and calculated density defines the gravity
anomalies. As gravity measurements are made at an elevation different from
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sea level, the observed field must be reduced to sea level with different corrections.
• The free air correction accounts for the changing distance to Earth center
of mass. This is a positive correction (0.3086mgal m−1 ) when gravity is
measured above sea level.
• The Bouguer correction accounts for the masses between the observation
surface and sea level. It is calculated for an infinite slab of density ρ and
thickness H. It is equal to 2πGρH. It depends on the density of the rocks and
is −0.118mgal m−1 for rock density = 2670kg m−3 . Note that the Bouguer
correction is exactly half the correction that we would make for a spherical
shell of thickness H. The correction is 4πρH.
There are other corrections to account for variations in topography. Very precise measurements even require tidal loading corrections.
Thus, free air (or Faye) anomaly is the difference:
∆gF = gmes + 0.3086H − gref
with H is the elevation and g is measured in mGal. The Bouguer anomaly is
∆gB = gmes + 0.3086H − 2πGρH − gref
Bouguer anomalies are negatively correlated with elevation, while free air are
not. This observation is the basis for the concept of isostasy, i.e. topography
of the Earth surface must be compensated at depth.
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Tides
Tides are caused by the difference between the gravitational pull of Moon and
Sun, and the inertial forces due to Earth’s orbital rotation around the Sun and
Moon (or more precisely the center of mass of the Earth-Moon system). At
the center of mass of the Earth, gravity forces balance the inertia. But gravity
varies with distance, so the gravity pull is stronger on the day side (toward
the Sun) and weaker on the night side. So on both sides there will be a small
force away from the center of Earth. Likewise for the Moon. The solid Earth,
itself is deformed by these tidal forces and the the effect of this deformation
can be observed in the time variations of the gravity field.
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5.1 Lois de Kepler
Les observations astronomiques de Kepler lui avaient permis de formuler certaines lois du mouvement des planétes autour du soleil. La troisième loi relie
la période T de l’orbite d’une planète à sa distance moyenne au soleil R. Cette
loi peut-être dérivée de façon très simpliste en exprimant que l’accélération
due à la force gravitationnelle doit être égale à l’accélération inertielle. Pour
une orbite circulaire de rayon R autour d’un corps de masse M, il faut donc:
ω2R =
GM
R2
En remplacant ω = 2π/T , on obtient:
T2 =
4π 2 R3
∝ R3
GM
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