The Gravity Field of the Earth
Linking Observations to the Earth
Readings: Chapter 6; Physics text: Chapter 15; Calculus text: Section on three-dimensional
Table of Contents
Newton's Law of Gravitation
where G is the constant of gravitation defined as (6.67259 ± 0.00085) x 10-11 m3kg-1
s-2 by the Committee on Data for Science
and Technology (CODATA), m1 and m2 are the masses of the two attracting masses, l is the distance separating the bodies,
and the negative sign indicates that F is directed toward the m1 mass.
Assigning the attracted mass, m2,
a unit mass of 1, Newton's law of gravitation is usually written as
Introducing an arbitrary coordinate system the force vector F can be decomposed as follows
- (x',y',z') is the location of m1, and
- (x,y,z) is the location of m2.
Potential of Gravitation
The potential of gravitation is a measure of the amount of work required to transport a unit mass a distance l from the attracting mass to infinity. Integrating
In vector notation, the potential of gravitation V and the gravitational force vector F are related by
In terms of all three axes using a gradient of the vector, this expression can be written as
From the above equation, we see that the gradient of the gravitational potential is only dependent on the separation distance of the objects, and is independent of
the coordinate system.
Expanding from Two Masses to Multiple Masses
Since the potential is a scalar, the effect of several masses acting on a single point mass is simply the sum of the individual potentials. That is,
For a solid body, M, the summation is replaced by a volume integral over the body. That is
denotes the density which varies throughout the earth and v denotes its mass volume.
However, the Earth is not a
stationary object. In fact due to is daily rotation, objects on the surface of the earth experience varying amounts of centrifugal
force. This force is directed away from the z rotational axis, and be expressed as
where f is the centrifugal force acting on the body,
w is the angular velocity of the earth's rotation (7.292115 x 10-5 rad/s), and p
is the distance of the object from the rotation axis.
The centrifugal potential is
F = ½ w2(x2
Note again that the potential is a function of the distance the object is from the rotational axis. The potential of gravity is the sum of the gravitational and
centrifugal potentials, or
The gravity force vector
g is then
The important point here is that the total force acting on a body is a summation of both the gravitational and centrifugal potential. Centrifugal potential is
greatest at the equator and decreases to zero as one moves toward the rotational axis. Thus the force of gravity is less at the equator than at the poles. For
example, the magnitude of g is approximately 978.049 gal at the equator. At the poles, the magnitude of g is approximately 983.221 gal.
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August 31, 2009