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.