TEMPORAL GRAVITY VARIATIONSThe body of the earth and its gravity field are subject to temporal

variations of secular, periodic, and abrupt nature, which can occur globally, regionally, and locally. These variations also influence the orientation of the earth. Modern geodetic measurement and evaluation techniques are used to detect these variations to a high level of accuracy. If time-independent results are required, geodetic observations must be corrected for temporal variations. By determining temporal variations, the science of geodesy contributes to the investigation of the kinematic and dynamic properties of the terrestrial body.

Gravity changes with time may be divided into effects due to: 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 (C) Variations caused by terrestrial mass displacements.

A. A time dependent Gravitational Constant and variations of the Earth's Rotation.

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:

B. Tidal accelerations and Tidal Potential

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.

Geometry of the Earth-Moon systemThe 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 systemThe 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

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 Earth’s 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.

Here M L and M T are the mass of Moon and Earth, respectively, see Table 1. With mean values of r and R (Table 2), 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.

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.

Table 1. Mass of Earth, Moon and Sun

Table 2. The mean distance between Earth and Moon, and Earth and Sun

According to Newton's second law, this force leads to 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 (6.67384×10-11 N m2 kg-2 ) and MT(5.974x1024kg) and r(6.37 x106m ¿ 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:

b t=b−bo

If we apply the law of gravitation to (b t=b−bo), we obtain for the moon (m) the formula:

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 b t= 0 for lm = rm. Corresponding relations hold for the earth-sun and earth-planet systems.

Tidal PotentialTides are a measure of changes in gravity, caused by the attraction of the

moon and sun. Tidal potential is the gravitational potential that varies with the position of the Moon and Sun relative to the Earth .

Components of the tidal potential – Deformation of the solid earth due to gravitational potential (solid

earth tide)– Movement of ocean water due to changing potential (ocean tides) – Deformation of the solid earth due to the changing load of ocean

tides (ocean tidal loading).

• 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:

In the above equations,ξt is the (prescribed) tidal forcing and η 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.

(a)(b)(c

C. Variations caused by terrestrial mass displacements.

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.