Table Of ContentTHE GRAVITY FIELD OF THE EARTH
from Classical and Modern Methods
MICHELE CAPUTO
Department of Geophysics and
Department of Mathematics
University of British Columbia
Vancouver, B. C., Canada
ACADEMIC PRESS New York London 1967
COPYRIGH0T 1967, BY ACADEMICP RESSIN C.
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To the memory of my father
PREFACE
The gravity field of the earth is one of the fields of forces which has
been observed and studied in its broad features for many centuries and
in detail for almost a century. One of the purposes of these studies has
been the determination of the shape of the earth or, more precisely, the
determination of the shape of the geoid, that is, the equipotential surface
of the gravity field which is identified by the surface of the oceans and
open seas. The shape of this surface has also been studied in the geometric
branch of geodesy by means of triangulations covering large areas of the
world.
For the first approximation order computations geodesists need a
reference surface. For this purpose geometric geodesy has long used the
ellipsoid because of the great convenience of the differential geometry of
this surface. In 1924 the International Ellipsoid was adopted by
the International Union of Geodesy and Geophysics (IUGG) as the
reference surface for the geometric work. In 1930, with the adoption
of the International Gravity Formula, IUGG adopted the same
ellipsoid as the basic surface also for the researches in gravimetric
geodesy.
With the adoption of the International Ellipsoid and of the Inter-
national Gravity Formula, unity of the geometric and gravimetric
geodesy was finally achieved, and the results of the two branches were
directly compared and integrated.
At the aforesaid times and later, the fit of this ellipsoid to the geoid
could be checked only for limited separate portions of the ellipsoid. The
observations of artificial satellites now give a description of the shape
and gravity field of the earth as a whole, which is more homogeneous
than the descriptions obtained previously. The results thus obtained
confirmed the validity of the assumptions made by IUGG, although they
showed that for a better fit the parameters of the International Ellipsoid
should be slightly modified.
vii
...
Vlll PREFACE
The necessity for unity is now more stringent because of the new
techniques introduced, which include direct measurement of distances
along geodetic lines; the use of targets at great altitudes; the facility
with which gravity can be measured on land, at sea, and in the space
surrounding the earth: the determination of the gravity field of the earth
by means of observations of artificial satellites; and the new important
developments of three-dimensional geodesy.
This unity is obtained with the model of the earth’s gravity field, in
which an ellipsoid of revolution is in an equipotential surface and in
which the gravity formula and the other theorems of the field are
determined in closed form.
To satisfy such needs of unity and according to the above-mentioned
method, this text will give the theory of the gravity field of the earth as it
can be treated according to the classical method which uses observations
of gravity taken over the earth’s surface and also according to the
modern method which uses observations of variation of orbital elements
of artificial satellites caused by the gravity field of the earth.
The determination of the earth’s figure and mass, done by means of
the classic method as well as by means of the modern method, is made
naturally in two steps. This is due to the fact that the amplitudes of the
ondulations of the geoid, the earth’s polar flattening, and the earth‘s
radius are in the ratio of 1 to 3 x lo2 to lo5; this implies that an
approximation of the earth’s figure and gravity field to an accuracy of
can be made as first step, neglecting the ondulations of the geoid
which will be taken into account in the second step.
The theory associated with the model of the earth’s gravity field in
which an ellipsoid is an equipotential surface is given according to the
method introduced by Pizzetti and later developed by Somigliana.
According to this method all the formulas of the field are obtained in
closed form. The part of the book dealing with the determination of the
form of the geoid from ground gravity measurements by means of the
Stokes formula is developed partly according to the method introduced
by Dini. The method of obtaining the Stokes formula by means of an
integral equation is also outlined.
The most recent method of obtaining the undulations of the geoid
and the variations of the gravity field by using unreduced gravity
measurements and integral equations is also given.
The part of the book dealing with the study of the earth’s gravity field
by means of observations of artificial satellites is developed according
PREFACE ix
to the method used by Kaula and Kozai and according to the method
used by King Hele.
Part I of the book contains good examples of the solution of physical
problems which are treated as Dirichlet problems or solved by means of
integral equations. Part 11 contains two modern mathematical
techniques developed to utilize the observations of artificial satellites for
geodetic purposes. Because of the mathematical techniques used and the
results given, this book could be used as a textbook for students in the
fields of geodesy, geophysics, or astronomy.
I wish to express my gratitude to Professor Marussi, Director of the
Istituto di Geodesia e Geofisica of Universith di Trieste, who gave two
important series of lectures on the gravity field of the earth (Marussi,
1956 and 1958), and developed my interest in this field. I express my
gratitude also to Professor L. B. Slichter, past director of the Institute of
Geophysics and Planetary Physics of the University of California,
for allowing me to prepare parts of the manuscript at his institution,
and to Professor D. Derry of the Department of Mathematics of the
University of British Columbia for helping me in the editing of the
English text.
Vancouver, British Columbia MICHELEC APUTO*
March 1967
* Present address: Istituto di Fisica, Universiti degli Studi, Bologna.
CHAPTER I
General Theory
1. Introductory considerations: the coordinates. The Pizzetti-Somig-
liana theory requires the use of some functions that were introduced by
Morera (1894) and that were later named after him. In this chapter we
shall introduce these functions to the reader and study the pertinent
properties. For this purpose, we have to consider a system of ellipsoidal
coordinates defined as follows. Let xl, x2,x 3 be Cartesian orthogonal
coordinates; the equation
+-
4 > >
H(A) = -x: x; - 1 = 0 , a, a2 a3 (1.1)
ai+A u ; + A + K
for all the A which are not roots of the equation
+ + +
R(A) = (a: A)(u; A)(u; A) = 0 (1.2)
represents a set Q of second-order algebraic surlaces that have the same
focuses. Fixing a point P(xl, x2, x,), the solutions of the third-order
algebraic equation
give the three values of the parameter A that determine the three quadrics
of the set Q that have in common the point P. These three solutions of
(1.3) are real and distinct, in fact the function K(A) is continuous and
also limA+mK (A) = -00, K(-ai) > 0, K(--ai) < 0, and K(--at) > 0.
We shall call these three solutions A,, A2, A, assuming -a; < 1, < co,
-a; < Az < -a:, -a,” < A, < -a:. The quadrics corresponding to
A, are ellipsoids; when A1 approaches -a;+ the flattenings of this
ellipsoid
+ + +
[(a: A)l/z - (a; A)””(a: ;
[(a; + Ay2 - (a; + A)’/”(a: + A y
3
4 I. GENERAL THEORY
become infinite and the ellipsoid flattens into the region of the plane
+ <
x,, x2 defined by x;/(a," a,") xi/(a,2 - a,") 1. The quadrics corre-
,-
sponding to A2 are hyperboloids of one sheet; when A, approaches
-a:+ then the hyperboloid flattens into the region defined by
x,2/(ai - a:) - x;/(a: - a;) Q 1 ;
when A2 approaches -a,"- the hyperboloid flattens into the region
defined by
+ >
xf/(a,2- a;) x&l; - a;) 1
The quadrics corresponding to 1, are hyperboloids of two sheets ; when
1, approaches -a;+ the hyperboloid flattens into the plane x2,x 3, and
when 1, approaches -a:- the hyperboloid flattens into the region
.;/(a," - a;) - .;/(a," - a,") > 1.
By means of the foregoing procedure, with each point P(x,, x2, x3)
we can therefore associate a triplet of numbers I,, A,, 1,. To each triplet
of numbers A,, A,, A,, satisfying the condition
> a, > > I, > > A, >
00 -0:
there corresponds, respectively, an elliposid, a hyperboloid of one sheet,
and a hyperboloid of two sheets that have in common a point P(x,, x,, x,)
and the points symmetric to P with respect to the coordinate planes.
The coordinates A,, A,, A, are called elliptic; it can be seen that they are
orthogonal and that the first differential form associated with them is
2. Morera's functions. To introduce the Morera functions we con-
sider the family of ellipsoids defined by the parameter 1, and study the
functions
We shall prove that $n(l,) is harmonic in the space outside the ellipsoid
defined by 1, = 0.
In fact
aH aH 2xi
-a=n- -p(A); axi - a: + A
2. MORERA'S FUNCTIONS 5
and from
(2.3)
follows
an
_ -- 2xi
axi (us + (2.4)
A)P(A)
and also we have
Therefore,
and since H(3,) = 0
>
Computing the second derivative accordingly, we have, for n 1,
m m
A 2.
and also
m
6 I. GENERAL THEORY
For n = 1 the same result can be proved directly. In order to prove the
4,
regularity of at infinity we note first that on assuming
+ +
r2= x; xi x32 ,
> + < < +
for A 0 it follows that a; A r2 a: A, and therefore
limC+mr/ A1l2= 1. Thus
- n !
- (---I), 1.3.5... + (2.10)
(2n 1)
More generally, other harmonic functions are given by
m
(2.11)
where f is a function continuous with its first and second derivatives.
Other harmonic functions can also be obtained by simple differentiation
of the harmonic functions (2.1) with respect to xi. Especially interesting
are the second-order derivatives of +,(A). From (2.8) they can be
written, for n = 2,
m
(2.12)
It is obvious that the function
(2.13)
considered here is constant on the ellipsoids of the family associated with
the parameter Al. Assuming that a mass A4 generates a gravitational
field in which the ellipsoid A1 = 0 is an equipotential surface and since
lim,.+,,, [&A4+0]= M, the function gives the gravitational potential of
