Geoscience Reference
In-Depth Information
1 Fundamentals of potential
theory
1.1
Attraction and potential
The purpose in this preparatory chapter is to present the fundamentals of
potential theory, including spherical and ellipsoidal harmonics, in sucient
detail to assure a full understanding of the later chapters. Our intent is to
explain the meaning of the theorems and formulas, avoiding long derivations
that can be found in any textbook on classical (before 1950) potential theory;
we recommend Kellogg (1929). A simple rather than completely rigorous
presentation is offered in our topic.
Nevertheless, the reader might consider this chapter perhaps more di-
cult and abstract than other parts of the topic. Since later practical appli-
cations will give concrete meaning to the topics of the present chapter, the
reader may wish to read it only cursorily at first and return to it later when
necessary.
According to Newton's law of gravitation, two points with masses
m
1
,m
2
,
separated by a distance
l
, attract each other with a force
F
=
G
m
1
m
2
l
2
.
(1-1)
This force is directed along the line connecting the two points;
G
is Newton's
gravitational constant. In SI units (Systeme International d'unites) based on
meter [m], kilogram [kg], and second [s], the gravitational constant has the
value
G
=6
.
6742
·
10
−
11
m
3
kg
−
1
s
−
2
.
(1-2)
The Newtonian gravitational constant
G
is somewhat of a scandal in
measuring physics. It is on the one hand one of the most important physi-
cal constants, and at the same time one of the least accurately determined
ones. The international authority in this field is the Committee on Data
for Science and Technology (CODATA), see under www.codata.org. In July
2002, CODATA recommended the value of
G
mentioned above, more pre-
cisely it assigned the value
G
=(6
.
6742
10
−
11
m
3
kg
−
1
s
−
2
.The
±
0
.
0010)
·
symbol
denotes the standard uncertainty, also called standard deviation
or standard error. This corresponds to a relative standard uncertainty of
1
.
5
±
10
−
4
·
or 150 ppm which is a deplorably high inaccuracy for such an
