Post on 30-Jan-2016
description
Temporal
Gravity
Variations
Temporal Gravity
Variation Gravity changes with time may be divided into
effects due to:
A. A time dependent Gravitational Constant and
variations of the Earth's Rotation.
B. Tidal accelerations and Tidal Potential
C. Variations caused by terrestrial mass
displacements.
• Newton's law of universal gravitation states that an attractive force F is set up between any two point masses, varying proportional with the product of the masses (𝑚1and 𝑚2) and inversely proportional with the distance l between the masses:
• The gravitational constant is the proportionality constant used in Newton’s Law of Universal Gravitation, and is commonly denoted by G.
G = 6.67384×10-11 N m2 kg-2
• The earth's rotational vector ω is subject to secular, periodic, and irregular variations, leading to changes
of the centrifugal acceleration z. In a spherical
approximation, the radial component of z enters
into gravity. By multiplying with
(φ = geocentric latitude), we obtain:
• Differentiation yields the effect of changes in
latitude (polar motion) and angular velocity (length
of day) on gravity:
• Tidal acceleration is caused by the superposition of lunisolar gravitation (and to a far lesser extent planetary gravitation) and orbital accelerations due to the motion of the earth around the barycenter of the respective two-body system (earth-moon, earth-sun etc.).
• For a rigid earth, the tidal acceleration at a given point can be determined from Newton's law of gravitation and the ephemerides (coordinates) of the celestial bodies (moon, sun, planets). The computations are carried out separately for the individual two-body systems (earth-moon, earth-sun etc.), and the results are subsequently added, with the celestial bodies regarded as point masses.
Figure 1: Illustration of the Earth-Moon system with the Earth to the left and the Moon to
the right (figure greatly out of scale). O, P and L are the centre of the Earth, an arbitrary
point on Earths surface and the centers of the Moon, respectively. r is the Earth's radius
vector (from point O to P), R is the position vector from the centre of the Earth to Moon's
centre (from O to L), and q is the position vector from an arbitrary point P on Earth's
surface to L. The line between O and L is sometimes called the center line and the angle the
zenith angle or the center angle.
• Geometry of the Earth-Moon system The configuration of the Earth-Moon system used for deriving the
properties of the tidal equilibrium is displayed in Figure 1. It follows from
the figure that r + q = R
• Centre of mass of the Earth-Moon system
The center of mass of the Earth-Moon system is
located along the center line OP at a distance xR (0 < x
< 1) from point O (Fig. 1). We then get that
or
Here 𝑀𝐿 and 𝑀𝑇 are the mass of Moon and Earth, respectively, see Table 1. With mean values of r and R (Table 1), we get that
x ≈ 0.73 r
implying that the center of mass of the Earth-Moon system is
located about one quarter of Earth's radius from the surface of the
Earth.
Mass of: Symbol Value
Earth 𝑀𝑇 5.974𝑥1024 kg
Moon 𝑀𝐿 7.347𝑥1022 kg
Sun 𝑀𝑆 1.989𝑥1030kg
Table 1. Mass of Earth, Moon and Sun
• Gravitational forces and accelerations in the Earth and
Moon system
The gravitational force at the Earth's center because
of the presence of the Moon,𝐹𝑇𝐿, is
where R/R is the unit vector along the center line from Earth to
Moon.
According to Newton's second law, this force leads
to an acceleration at the center of Earth
Similarly, the gravitational acceleration at point
P caused by the Moon is
At point P, there is also a gravitational
acceleration g towards the center of the Earth caused by
Earth's mass:
By inserting the numerical values of G, MT and r
(appendix A), one obtains 𝑔 = 9.8𝑚/𝑠2 expected. Furthermore, the equation above gives the relationship
• Tidal Acceleration
We consider the geocentric coordinate system to be moving
in space with the earth but not rotating with it (revolution without
rotation). All points on the earth experience the same orbital
acceleration in the geocentric coordinate system (see Fig. 2 for the
earth-moon system). In order to obtain equilibrium, orbital
acceleration and gravitation of the celestial bodies have to cancel in
the earth's center of gravity. Tidal acceleration occurs at all other
points of the earth. The acceleration is defined as the difference
between the gravitation b, which depends on the position of the
point, and the constant part 𝑏𝑜, referring to the earth's center:
𝒃𝒕 = 𝒃 − 𝒃𝒐
The tidal acceleration deforms the earth's gravity field
symmetrically with respect to three orthogonal axes with origin
at the earth's center. This tidal acceleration field experiences
diurnal and semidiurnal variations, which are due to the rotation
of the earth about its axis.
Fig. 2 Lunar gravitation, orbital acceleration, and tidal acceleration
If we apply the law of gravitation to (𝒃𝒕 = 𝒃 − 𝒃𝒐), we obtain for the moon (m)
Here, 𝑀𝑚 =mass of the moon,
and 𝑙𝑚 and 𝑟𝑚 = distance to the moon as reckoned from
the calculation point P and the earth's center of gravity Ο respectively. We have 𝑏𝑡 = 0 for 𝑙𝑚 = 𝑟𝑚. Corresponding relations hold for the earth-sun and earth-planet systems.
• Laplace's tidal equations
When tidal forcing is introduced to the
(quasi)linearized version of the shallow water equations,
the obtained equations are known as Laplace's tidal
equations (LTE). Tidal flow is then described as the flow
of a barotropic fluid, forced by the tidal pull from the
Moon and the Sun. The phrase “shallow water
equations" reacts that the wavelength of the resulting
motion is large compared to the thickness of the fluid.
The horizontal components of the momentum equation
and the continuity equation can then be expressed as:
(a)
(b)
(c)
In the above equations, t is the (prescribed) tidal forcing and
is the resulting surface elevation, h is the ocean depth.
The horizontal momentum equations are linear, but
inclusion of a friction term will typically turn the equations non-
linear. Likewise, the divergence terms in the continuity equation are
nonlinear because of the product uh and vh. Solution of LTE
requires discretization and subsequent numerical solution.
• The terrestrial gravity field is affected by
a number of variations with time due to
mass redistributions in the atmosphere,
the hydrosphere, and the solid earth.
These processes take place at different
time scales and are of global, regional,
and local character.
• Long-term global effects include postglacial
rebound, melting of the ice caps and glaciers, as
well as sea level changes induced by atmospheric
warming; slow motions of the earth's core and
mantle convection also contribute. Subsidence in
sedimentary basins and tectonic uplift are
examples of regional effects. Groundwater
variations are primarily of seasonal character,
while volcanic and earthquake activities are
short-term processes of more local extent.
• The magnitude of the resulting gravity variations depends on the amount of mass shifts and is related to them by the law of gravitation. Research and modeling of these variations is still in the beginning stages. Large-scale variations have been found from satellite-derived gravity field models, but small-scale effects can be detected only by terrestrial gravity measurements. Simple models have been developed for the relation between atmospheric and hydrological mass shifts and gravity changes, Generally, gravity changes produced by mass redistributions do not exceed the order of 10−9 to 10−8g.